From 7fbc25cd6cea49a7940512f63944efe28a631fe5 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 7 Jun 2017 05:54:09 +0200
Subject: [PATCH 01/64] =?UTF-8?q?Les=20fichiers=20pdf/md5=20des=20figures?=
 =?UTF-8?q?=20modifi=C3=A9es.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 auto/pictures_tikz/tikzAMDUooZZUOqa.md5    |   2 +-
 auto/pictures_tikz/tikzAMDUooZZUOqa.pdf    | Bin 24866 -> 24991 bytes
 auto/pictures_tikz/tikzCWKJooppMsZXjw.md5  |   2 +-
 auto/pictures_tikz/tikzCWKJooppMsZXjw.pdf  | Bin 32165 -> 32187 bytes
 auto/pictures_tikz/tikzCercleTrigono.md5   |   2 +-
 auto/pictures_tikz/tikzCercleTrigono.pdf   | Bin 31860 -> 31881 bytes
 auto/pictures_tikz/tikzTgCercleTrigono.md5 |   2 +-
 auto/pictures_tikz/tikzTgCercleTrigono.pdf | Bin 32501 -> 32518 bytes
 8 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/auto/pictures_tikz/tikzAMDUooZZUOqa.md5 b/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
index 653cc956d..a8896e52f 100644
--- a/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
+++ b/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {85A99FF3936E7506B736DF05A3966190}%
+\def \tikzexternallastkey {3345F266A94F0440D55ED2B0B5D88AD8}%
diff --git a/auto/pictures_tikz/tikzAMDUooZZUOqa.pdf b/auto/pictures_tikz/tikzAMDUooZZUOqa.pdf
index d09ff7249c287851f0bc53440a5908f20628cb95..3e92878a9b501c1949274882af9f93901690e6b5 100644
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diff --git a/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5 b/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
index a514d5041..f46935921 100644
--- a/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
+++ b/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6FAF8888BC7B4AEAEFA57F6D91268658}%
+\def \tikzexternallastkey {6D0C8FB875137B88ECCD08E37B34FF12}%
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diff --git a/auto/pictures_tikz/tikzCercleTrigono.md5 b/auto/pictures_tikz/tikzCercleTrigono.md5
index ecdf3ddd0..1dfeabd2a 100644
--- a/auto/pictures_tikz/tikzCercleTrigono.md5
+++ b/auto/pictures_tikz/tikzCercleTrigono.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6B12D470F848F82D122FDEF013DA0F81}%
+\def \tikzexternallastkey {C70CB22CF717378D3DB2778DF83DC686}%
diff --git a/auto/pictures_tikz/tikzCercleTrigono.pdf b/auto/pictures_tikz/tikzCercleTrigono.pdf
index 961d3cd86555357fd5afba454871fcece5de7edd..e7ca193c64fe58d17aabf241ae323aa61d904c6a 100644
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diff --git a/auto/pictures_tikz/tikzTgCercleTrigono.md5 b/auto/pictures_tikz/tikzTgCercleTrigono.md5
index ea87b4672..0f527aef0 100644
--- a/auto/pictures_tikz/tikzTgCercleTrigono.md5
+++ b/auto/pictures_tikz/tikzTgCercleTrigono.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {9098D410EECADE035E8EC84DD59942EF}%
+\def \tikzexternallastkey {6123A04DDC9811FB76E71C6DB97C148A}%
diff --git a/auto/pictures_tikz/tikzTgCercleTrigono.pdf b/auto/pictures_tikz/tikzTgCercleTrigono.pdf
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From a8b81d826c1ae144e209e07f159904e1593052c0 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 8 Jun 2017 01:43:52 +0200
Subject: [PATCH 02/64] =?UTF-8?q?Correction=20d'une=20r=C3=A9f=C3=A9rence?=
 =?UTF-8?q?=20dans=20l'index=20th=C3=A9matique.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/front_back_matter/6_theme.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/tex/front_back_matter/6_theme.tex b/tex/front_back_matter/6_theme.tex
index 3337bb80c..950c68444 100644
--- a/tex/front_back_matter/6_theme.tex
+++ b/tex/front_back_matter/6_theme.tex
@@ -6,6 +6,6 @@
         \item
             Le cercle circonscrit à une courbe donne un cercle de rayon minimal contenant une courbe fermée simple, proposition \ref{PROPDEFooCWESooVbDven}.
     \item Enveloppe convexe du groupe orthogonal \ref{ThoVBzqUpy}.
-    \item Enveloppe convexe d'une courbe fermée plana comme intersection des demi-plans tangents, théorème \ref{PROPooWZITooTFiWsi}.
+    \item Enveloppe convexe d'une courbe fermée plana comme intersection des demi-plans tangents, proposition \ref{PROPooOORPooCXrIQi}.
         \end{enumerate}
 

From 61948998e1fcd5a7222b8ed353d0ba41856ebe5c Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Fri, 9 Jun 2017 05:19:35 +0200
Subject: [PATCH 03/64] Bug fix. The testing script was passing a wrong
 directory to `test_recall.py`, so that nothing was actually tested.

---
 testing/TestRecall.py  | 1 -
 testing/test_recall.py | 9 +++++++--
 testing/testing.sh     | 7 +++----
 3 files changed, 10 insertions(+), 7 deletions(-)

diff --git a/testing/TestRecall.py b/testing/TestRecall.py
index 0bdac1a13..ceef7913f 100755
--- a/testing/TestRecall.py
+++ b/testing/TestRecall.py
@@ -29,7 +29,6 @@ def wrong_file_list(directory):
     auto_pictures_tex_dir=os.path.join(directory,"auto/pictures_tex")
     wfl=[]  # wrong file list
     mfl=[]  # missing file list
-    print("src_phystricks_dir : ",src_phystricks_dir)
     for filename in pstricks_files_iterator(auto_pictures_tex_dir):
         with open(filename,'r') as f:
             get_text=f.read()
diff --git a/testing/test_recall.py b/testing/test_recall.py
index 9b9a3e87c..ae0351041 100755
--- a/testing/test_recall.py
+++ b/testing/test_recall.py
@@ -11,9 +11,14 @@
 import sys
 from TestRecall import wrong_file_list
 
-directory=sys.argv[0]
+directory=sys.argv[1]
+
+try:
+    mfl,wfl=wrong_file_list(directory)
+except NotADirectoryError :
+    print("[test_recall.py] the passed directory does not exist")
+    raise
 
-mfl,wfl=wrong_file_list(directory)
 
 for f in mfl:
     print("missing recall : ",f)
diff --git a/testing/testing.sh b/testing/testing.sh
index 662c73ab4..115d1d698 100755
--- a/testing/testing.sh
+++ b/testing/testing.sh
@@ -61,9 +61,6 @@ cd $BUILD_DIR
 rm $LOG_FILE
 touch  $LOG_FILE
 
-# Frido's compilation is together with everything's verification
-# because we want to balance the two threads.
-
 compile_frido ()
 {
     cd $CLONE_DIR
@@ -93,9 +90,11 @@ test_picture ()
 
     cd $CLONE_DIR/testing
     ./test_recall.py $AUTO_PICTURES_TEX >> $LOG_FILE
+    if [ $? -eq 1 ]; then
+        echo "test_recall.py had a problem " >> $LOG_FILE
+    fi
 }
 
-
 if [[  "$@" == "--pictures"  ]] || [[  "$@" == "--full"  ]]
 then
     test_picture

From 684c96a777c6736d2f4bf27031348c34cd0b00e8 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Fri, 9 Jun 2017 08:51:41 +0200
Subject: [PATCH 04/64] Bug fix : 'testing.sh' was providing a wrong directory
 to 'test_recall.py'

---
 testing/testing.sh | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/testing/testing.sh b/testing/testing.sh
index 115d1d698..5ab0e5258 100755
--- a/testing/testing.sh
+++ b/testing/testing.sh
@@ -89,7 +89,7 @@ test_picture ()
     ./testing.sh
 
     cd $CLONE_DIR/testing
-    ./test_recall.py $AUTO_PICTURES_TEX >> $LOG_FILE
+    ./test_recall.py $BUILD_DIR >> $LOG_FILE
     if [ $? -eq 1 ]; then
         echo "test_recall.py had a problem " >> $LOG_FILE
     fi

From 6855aeda9a05d68545b976ca2e1a870a8dbc568e Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 10 Jun 2017 14:31:47 +0200
Subject: [PATCH 05/64] Some modified pictures.

---
 auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks  |  2 +-
 auto/pictures_tex/Fig_CercleTrigono.pstricks   |  2 +-
 auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks  |  6 +++---
 auto/pictures_tex/Fig_TgCercleTrigono.pstricks |  2 +-
 auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks  |  2 +-
 testing/testing.sh                             | 11 +++++------
 6 files changed, 12 insertions(+), 13 deletions(-)

diff --git a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
index 24d735a74..4c1ce07d4 100644
--- a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
+++ b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
@@ -75,7 +75,7 @@
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 \draw (-3.828301160,-1.204882387) node {\( -x+\lambda i\)};
-\draw (0.5487913378,-0.6676704275) node {\( \arg(z)\)};
+\draw (0.5312334950,-0.7091140396) node {\( \arg(z)\)};
 
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diff --git a/auto/pictures_tex/Fig_CercleTrigono.pstricks b/auto/pictures_tex/Fig_CercleTrigono.pstricks
index db08d1ee6..5097fdb00 100644
--- a/auto/pictures_tex/Fig_CercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_CercleTrigono.pstricks
@@ -95,7 +95,7 @@
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 \draw [color=brown,->,>=latex] (0,0) -- (1.732050808,1.000000000);
-\draw (1.012732295,0.2561455226) node {$\theta$};
+\draw (0.9161397128,0.2302636180) node {$\theta$};
 
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diff --git a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
index 2d0b6e358..e7f084afb 100644
--- a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
+++ b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
@@ -80,7 +80,7 @@
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 \draw []  (1.735039751,1.464349294) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246914);
+\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246913);
 \draw []  (2.657517092,2.437667613) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.247852509,3.098210238);
 \draw []  (0.9681152512,1.020091151) node [rotate=0] {$\bullet$};
@@ -96,7 +96,7 @@
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 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.671786357,3.458304371);
 \draw []  (-0.6188917196,1.057674546) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357359,1.183270382);
+\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357358,1.183270382);
 \draw [] (-3.00,0) -- (3.00,0);
 \draw [] (0,0) -- (3.83,3.21);
 \draw (0.7745541682,0.2495862383) node {\( \alpha\)};
@@ -104,7 +104,7 @@
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 \draw []  (1.915111108,1.606969024) node [rotate=0] {$\bullet$};
 \draw (2.186183463,1.329052136) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.38)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.456,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
 \draw (4.139119438,3.522532237) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
index 96ed0d514..54059faa9 100644
--- a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
@@ -89,7 +89,7 @@
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-\draw (-0.1463280799,0.5970250798) node {$\varphi$};
+\draw (0.3487948559,0.6970250798) node {$\varphi$};
 
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diff --git a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
index 1399869d5..4a92b4ea1 100644
--- a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
+++ b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
@@ -121,7 +121,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156037) -- (0,4.065868530);
+\draw [,->,>=latex] (0,-4.525156038) -- (0,4.065868530);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
diff --git a/testing/testing.sh b/testing/testing.sh
index 5ab0e5258..094cf14fb 100755
--- a/testing/testing.sh
+++ b/testing/testing.sh
@@ -89,7 +89,7 @@ test_picture ()
     ./testing.sh
 
     cd $CLONE_DIR/testing
-    ./test_recall.py $BUILD_DIR >> $LOG_FILE
+    ./test_recall.py $CLONE_DIR >> $LOG_FILE
     if [ $? -eq 1 ]; then
         echo "test_recall.py had a problem " >> $LOG_FILE
     fi
@@ -105,21 +105,20 @@ then
     test_death_links&
 fi
 
-compile_everything&
-compile_frido
+#compile_everything&
+#compile_frido
 
 
 cd $MAIN_DIR
-git status >> $LOG_FILE
 
 wait
 
 cd $CLONE_DIR
 
 
-echo "Result : -----------"
+echo "log file : $LOG_FILE "
+echo "----------------"
 
 cat  $LOG_FILE 
 
 echo "--------------------"
-echo "Beware that this is the result for the branch $1. I did not compile here."

From 633a5e12f350125347a148d58dffd99b51acb30c Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Mon, 12 Jun 2017 06:58:04 +0200
Subject: [PATCH 06/64] Quelque remplacement de fichiers 'recall' et renommage
 Exercicebis -> ExerciceGraphesbis

---
 auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks |  6 +--
 ...tricks => Fig_ExerciceGraphesbis.pstricks} | 44 +++++++++----------
 .../Fig_CWKJooppMsZXjw.pstricks.recall        |  2 +-
 .../Fig_CercleTrigono.pstricks.recall         |  2 +-
 ...=> Fig_ExerciceGraphesbis.pstricks.recall} | 44 +++++++++----------
 .../Fig_TgCercleTrigono.pstricks.recall       |  2 +-
 .../Fig_XOLBooGcrjiwoU.pstricks.recall        |  2 +-
 .../phystricksExerciceGraphesbis.py           |  2 +-
 tex/exocorr/corrDS2010bis-0002.tex            | 14 +++---
 tex/exocorr/exoDS2010bis-0002.tex             |  6 +--
 10 files changed, 62 insertions(+), 62 deletions(-)
 rename auto/pictures_tex/{Fig_Exercicebis.pstricks => Fig_ExerciceGraphesbis.pstricks} (97%)
 rename src_phystricks/{Fig_Exercicebis.pstricks.recall => Fig_ExerciceGraphesbis.pstricks.recall} (97%)

diff --git a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
index e7f084afb..2d0b6e358 100644
--- a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
+++ b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
@@ -80,7 +80,7 @@
 \draw []  (2.086833187,1.759539037) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.410240275,2.935347274);
 \draw []  (1.735039751,1.464349294) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246913);
+\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246914);
 \draw []  (2.657517092,2.437667613) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.247852509,3.098210238);
 \draw []  (0.9681152512,1.020091151) node [rotate=0] {$\bullet$};
@@ -96,7 +96,7 @@
 \draw []  (2.896085542,4.007090670) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.671786357,3.458304371);
 \draw []  (-0.6188917196,1.057674546) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357358,1.183270382);
+\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357359,1.183270382);
 \draw [] (-3.00,0) -- (3.00,0);
 \draw [] (0,0) -- (3.83,3.21);
 \draw (0.7745541682,0.2495862383) node {\( \alpha\)};
@@ -104,7 +104,7 @@
 \draw [] (0.500,0)--(0.500,0.00353)--(0.500,0.00705)--(0.500,0.0106)--(0.500,0.0141)--(0.500,0.0176)--(0.500,0.0211)--(0.499,0.0247)--(0.499,0.0282)--(0.499,0.0317)--(0.499,0.0352)--(0.499,0.0387)--(0.498,0.0423)--(0.498,0.0458)--(0.498,0.0493)--(0.497,0.0528)--(0.497,0.0563)--(0.496,0.0598)--(0.496,0.0633)--(0.496,0.0668)--(0.495,0.0703)--(0.495,0.0738)--(0.494,0.0773)--(0.493,0.0807)--(0.493,0.0842)--(0.492,0.0877)--(0.492,0.0912)--(0.491,0.0946)--(0.490,0.0981)--(0.490,0.102)--(0.489,0.105)--(0.488,0.108)--(0.487,0.112)--(0.487,0.115)--(0.486,0.119)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.132)--(0.481,0.136)--(0.480,0.139)--(0.479,0.143)--(0.478,0.146)--(0.477,0.149)--(0.476,0.153)--(0.475,0.156)--(0.474,0.159)--(0.473,0.163)--(0.472,0.166)--(0.470,0.169)--(0.469,0.173)--(0.468,0.176)--(0.467,0.179)--(0.465,0.183)--(0.464,0.186)--(0.463,0.189)--(0.462,0.192)--(0.460,0.196)--(0.459,0.199)--(0.457,0.202)--(0.456,0.205)--(0.454,0.209)--(0.453,0.212)--(0.451,0.215)--(0.450,0.218)--(0.448,0.221)--(0.447,0.224)--(0.445,0.228)--(0.444,0.231)--(0.442,0.234)--(0.440,0.237)--(0.439,0.240)--(0.437,0.243)--(0.435,0.246)--(0.433,0.249)--(0.432,0.252)--(0.430,0.255)--(0.428,0.258)--(0.426,0.261)--(0.424,0.264)--(0.423,0.267)--(0.421,0.270)--(0.419,0.273)--(0.417,0.276)--(0.415,0.279)--(0.413,0.282)--(0.411,0.285)--(0.409,0.288)--(0.407,0.291)--(0.405,0.294)--(0.403,0.296)--(0.401,0.299)--(0.398,0.302)--(0.396,0.305)--(0.394,0.308)--(0.392,0.310)--(0.390,0.313)--(0.388,0.316)--(0.385,0.319)--(0.383,0.321);
 \draw []  (1.915111108,1.606969024) node [rotate=0] {$\bullet$};
 \draw (2.186183463,1.329052136) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.38)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.456,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
 \draw (4.139119438,3.522532237) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_Exercicebis.pstricks b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
similarity index 97%
rename from auto/pictures_tex/Fig_Exercicebis.pstricks
rename to auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
index c8885319c..e519f6799 100644
--- a/auto/pictures_tex/Fig_Exercicebis.pstricks
+++ b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
@@ -22,7 +22,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosinusPICTcosinus.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosinusPICTcosinus.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -60,7 +60,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosinusPICTcosinus}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosinusPICTcosinus}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -94,7 +94,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosinus}
+\label{LabelFigExerciceGraphesbissscosinus}
 }                  % Closing subfigure 1
 %
 \subfigure[]{%
@@ -108,7 +108,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebisssvalabsoluecosinusPICTvalabsoluecosinus.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -146,7 +146,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebisssvalabsoluecosinusPICTvalabsoluecosinus}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -178,7 +178,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebisssvalabsoluecosinus}
+\label{LabelFigExerciceGraphesbisssvalabsoluecosinus}
 }                  % Closing subfigure 2
 %
 \subfigure[]{%
@@ -192,7 +192,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosxplusunPICTcosxplusun.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosxplusunPICTcosxplusun.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -4 $}}%
 \immediate\write\writeOfphystricks{totalheightof2e7da68c8ac350edb45389e799ce4e0c0e8c72dd:\the\lengthOfforphystricks-}
@@ -234,7 +234,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosxplusunPICTcosxplusun}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosxplusunPICTcosxplusun}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -272,7 +272,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosxplusun}
+\label{LabelFigExerciceGraphesbissscosxplusun}
 }                  % Closing subfigure 3
 %
 \subfigure[]{%
@@ -286,7 +286,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissssinusPICTsinus.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissssinusPICTsinus.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -324,7 +324,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissssinusPICTsinus}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissssinusPICTsinus}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -358,7 +358,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissssinus}
+\label{LabelFigExerciceGraphesbissssinus}
 }                  % Closing subfigure 4
 %
 \subfigure[]{%
@@ -372,7 +372,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosexPICTcosex.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosexPICTcosex.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -410,7 +410,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosexPICTcosex}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosexPICTcosex}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -445,7 +445,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosex}
+\label{LabelFigExerciceGraphesbissscosex}
 }                  % Closing subfigure 5
 %
 \subfigure[]{%
@@ -459,7 +459,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissssqrtcosPICTsqrtcos.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissssqrtcosPICTsqrtcos.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -2 $}}%
 \immediate\write\writeOfphystricks{totalheightoff66654a967831d75e2a133d03fb0753c049acbc1:\the\lengthOfforphystricks-}
@@ -497,7 +497,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissssqrtcosPICTsqrtcos}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissssqrtcosPICTsqrtcos}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -531,7 +531,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissssqrtcos}
+\label{LabelFigExerciceGraphesbissssqrtcos}
 }                  % Closing subfigure 6
 %
 \subfigure[]{%
@@ -545,7 +545,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosquattroxPICTcosquattrox.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -583,7 +583,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosquattroxPICTcosquattrox}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -617,7 +617,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosquattrox}
+\label{LabelFigExerciceGraphesbissscosquattrox}
 }                  % Closing subfigure 7
 %
 %AFTER SUBFIGURES
@@ -626,6 +626,6 @@
 %PSPICTURE
 %AFTER PSPICTURE
 %AFTER ALL
-\caption{\CaptionFigExercicebis}\label{LabelFigExercicebis}
+\caption{\CaptionFigExerciceGraphesbis}\label{LabelFigExerciceGraphesbis}
             \end{figure}
             
diff --git a/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall b/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
index 24d735a74..4c1ce07d4 100644
--- a/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
+++ b/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
@@ -75,7 +75,7 @@
 \draw [] (0,0) -- (-3.00,-1.00);
 \draw []  (-3.000000000,-1.000000000) node [rotate=0] {$\bullet$};
 \draw (-3.828301160,-1.204882387) node {\( -x+\lambda i\)};
-\draw (0.5487913378,-0.6676704275) node {\( \arg(z)\)};
+\draw (0.5312334950,-0.7091140396) node {\( \arg(z)\)};
 
 \draw [] (-0.474,-0.158)--(-0.470,-0.172)--(-0.465,-0.185)--(-0.459,-0.198)--(-0.453,-0.211)--(-0.447,-0.224)--(-0.441,-0.236)--(-0.434,-0.249)--(-0.426,-0.261)--(-0.419,-0.273)--(-0.411,-0.285)--(-0.403,-0.297)--(-0.394,-0.308)--(-0.385,-0.319)--(-0.376,-0.330)--(-0.366,-0.340)--(-0.356,-0.351)--(-0.346,-0.361)--(-0.336,-0.370)--(-0.325,-0.380)--(-0.314,-0.389)--(-0.303,-0.398)--(-0.292,-0.406)--(-0.280,-0.414)--(-0.268,-0.422)--(-0.256,-0.430)--(-0.243,-0.437)--(-0.231,-0.444)--(-0.218,-0.450)--(-0.205,-0.456)--(-0.192,-0.462)--(-0.179,-0.467)--(-0.166,-0.472)--(-0.152,-0.476)--(-0.138,-0.480)--(-0.125,-0.484)--(-0.111,-0.488)--(-0.0970,-0.491)--(-0.0830,-0.493)--(-0.0689,-0.495)--(-0.0547,-0.497)--(-0.0406,-0.498)--(-0.0264,-0.499)--(-0.0121,-0.500)--(0.00211,-0.500)--(0.0163,-0.500)--(0.0306,-0.499)--(0.0448,-0.498)--(0.0589,-0.497)--(0.0731,-0.495)--(0.0871,-0.492)--(0.101,-0.490)--(0.115,-0.487)--(0.129,-0.483)--(0.143,-0.479)--(0.156,-0.475)--(0.170,-0.470)--(0.183,-0.465)--(0.196,-0.460)--(0.209,-0.454)--(0.222,-0.448)--(0.235,-0.442)--(0.247,-0.435)--(0.259,-0.427)--(0.271,-0.420)--(0.283,-0.412)--(0.295,-0.404)--(0.306,-0.395)--(0.317,-0.386)--(0.328,-0.377)--(0.339,-0.368)--(0.349,-0.358)--(0.359,-0.348)--(0.369,-0.337)--(0.379,-0.327)--(0.388,-0.316)--(0.396,-0.305)--(0.405,-0.293)--(0.413,-0.282)--(0.421,-0.270)--(0.429,-0.258)--(0.436,-0.245)--(0.443,-0.233)--(0.449,-0.220)--(0.455,-0.207)--(0.461,-0.194)--(0.466,-0.181)--(0.471,-0.168)--(0.476,-0.154)--(0.480,-0.141)--(0.484,-0.127)--(0.487,-0.113)--(0.490,-0.0990)--(0.493,-0.0850)--(0.495,-0.0710)--(0.497,-0.0568)--(0.498,-0.0427)--(0.499,-0.0285)--(0.500,-0.0142)--(0.500,0);
 
diff --git a/src_phystricks/Fig_CercleTrigono.pstricks.recall b/src_phystricks/Fig_CercleTrigono.pstricks.recall
index db08d1ee6..5097fdb00 100644
--- a/src_phystricks/Fig_CercleTrigono.pstricks.recall
+++ b/src_phystricks/Fig_CercleTrigono.pstricks.recall
@@ -95,7 +95,7 @@
 \draw [,->,>=latex] (0.8660254038,-0.2000000000) -- (1.732050808,-0.2000000000);
 \draw (0.8660254038,-0.4824550000) node {$\cos(\theta)$};
 \draw [color=brown,->,>=latex] (0,0) -- (1.732050808,1.000000000);
-\draw (1.012732295,0.2561455226) node {$\theta$};
+\draw (0.9161397128,0.2302636180) node {$\theta$};
 
 \draw [] (0.400,0)--(0.400,0.00212)--(0.400,0.00423)--(0.400,0.00635)--(0.400,0.00846)--(0.400,0.0106)--(0.400,0.0127)--(0.400,0.0148)--(0.400,0.0169)--(0.400,0.0190)--(0.399,0.0211)--(0.399,0.0233)--(0.399,0.0254)--(0.399,0.0275)--(0.399,0.0296)--(0.399,0.0317)--(0.399,0.0338)--(0.398,0.0359)--(0.398,0.0380)--(0.398,0.0401)--(0.398,0.0422)--(0.398,0.0443)--(0.397,0.0464)--(0.397,0.0485)--(0.397,0.0506)--(0.397,0.0527)--(0.396,0.0548)--(0.396,0.0569)--(0.396,0.0590)--(0.395,0.0611)--(0.395,0.0632)--(0.395,0.0653)--(0.394,0.0674)--(0.394,0.0695)--(0.394,0.0715)--(0.393,0.0736)--(0.393,0.0757)--(0.392,0.0778)--(0.392,0.0798)--(0.392,0.0819)--(0.391,0.0840)--(0.391,0.0861)--(0.390,0.0881)--(0.390,0.0902)--(0.389,0.0922)--(0.389,0.0943)--(0.388,0.0964)--(0.388,0.0984)--(0.387,0.100)--(0.387,0.103)--(0.386,0.105)--(0.386,0.107)--(0.385,0.109)--(0.384,0.111)--(0.384,0.113)--(0.383,0.115)--(0.383,0.117)--(0.382,0.119)--(0.381,0.121)--(0.381,0.123)--(0.380,0.125)--(0.379,0.127)--(0.379,0.129)--(0.378,0.131)--(0.377,0.133)--(0.377,0.135)--(0.376,0.137)--(0.375,0.139)--(0.374,0.141)--(0.374,0.143)--(0.373,0.145)--(0.372,0.147)--(0.371,0.149)--(0.371,0.151)--(0.370,0.153)--(0.369,0.155)--(0.368,0.156)--(0.367,0.158)--(0.366,0.160)--(0.366,0.162)--(0.365,0.164)--(0.364,0.166)--(0.363,0.168)--(0.362,0.170)--(0.361,0.172)--(0.360,0.174)--(0.359,0.176)--(0.358,0.178)--(0.357,0.180)--(0.357,0.181)--(0.356,0.183)--(0.355,0.185)--(0.354,0.187)--(0.353,0.189)--(0.352,0.191)--(0.351,0.193)--(0.350,0.194)--(0.349,0.196)--(0.347,0.198)--(0.346,0.200);
 \draw (2.134373262,1.274708000) node {$P$};
diff --git a/src_phystricks/Fig_Exercicebis.pstricks.recall b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
similarity index 97%
rename from src_phystricks/Fig_Exercicebis.pstricks.recall
rename to src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
index c8885319c..e519f6799 100644
--- a/src_phystricks/Fig_Exercicebis.pstricks.recall
+++ b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
@@ -22,7 +22,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosinusPICTcosinus.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosinusPICTcosinus.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -60,7 +60,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosinusPICTcosinus}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosinusPICTcosinus}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -94,7 +94,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosinus}
+\label{LabelFigExerciceGraphesbissscosinus}
 }                  % Closing subfigure 1
 %
 \subfigure[]{%
@@ -108,7 +108,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebisssvalabsoluecosinusPICTvalabsoluecosinus.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -146,7 +146,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebisssvalabsoluecosinusPICTvalabsoluecosinus}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -178,7 +178,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebisssvalabsoluecosinus}
+\label{LabelFigExerciceGraphesbisssvalabsoluecosinus}
 }                  % Closing subfigure 2
 %
 \subfigure[]{%
@@ -192,7 +192,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosxplusunPICTcosxplusun.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosxplusunPICTcosxplusun.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -4 $}}%
 \immediate\write\writeOfphystricks{totalheightof2e7da68c8ac350edb45389e799ce4e0c0e8c72dd:\the\lengthOfforphystricks-}
@@ -234,7 +234,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosxplusunPICTcosxplusun}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosxplusunPICTcosxplusun}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -272,7 +272,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosxplusun}
+\label{LabelFigExerciceGraphesbissscosxplusun}
 }                  % Closing subfigure 3
 %
 \subfigure[]{%
@@ -286,7 +286,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissssinusPICTsinus.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissssinusPICTsinus.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -324,7 +324,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissssinusPICTsinus}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissssinusPICTsinus}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -358,7 +358,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissssinus}
+\label{LabelFigExerciceGraphesbissssinus}
 }                  % Closing subfigure 4
 %
 \subfigure[]{%
@@ -372,7 +372,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosexPICTcosex.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosexPICTcosex.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -410,7 +410,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosexPICTcosex}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosexPICTcosex}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -445,7 +445,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosex}
+\label{LabelFigExerciceGraphesbissscosex}
 }                  % Closing subfigure 5
 %
 \subfigure[]{%
@@ -459,7 +459,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissssqrtcosPICTsqrtcos.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissssqrtcosPICTsqrtcos.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -2 $}}%
 \immediate\write\writeOfphystricks{totalheightoff66654a967831d75e2a133d03fb0753c049acbc1:\the\lengthOfforphystricks-}
@@ -497,7 +497,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissssqrtcosPICTsqrtcos}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissssqrtcosPICTsqrtcos}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -531,7 +531,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissssqrtcos}
+\label{LabelFigExerciceGraphesbissssqrtcos}
 }                  % Closing subfigure 6
 %
 \subfigure[]{%
@@ -545,7 +545,7 @@
 %OPEN_WRITE_AND_LABEL
 \ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
 \ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
-\immediate\openout\writeOfphystricks=FIGLabelFigExercicebissscosquattroxPICTcosquattrox.phystricks.aux%
+\immediate\openout\writeOfphystricks=FIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox.phystricks.aux%
 %WRITE_AND_LABEL
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -3 $}}%
 \immediate\write\writeOfphystricks{totalheightof4d313dc1a6b4e76710b316f19deffc9362a63da7:\the\lengthOfforphystricks-}
@@ -583,7 +583,7 @@
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
 %BEGIN PSPICTURE
-\tikzsetnextfilename{tikzFIGLabelFigExercicebissscosquattroxPICTcosquattrox}
+\tikzsetnextfilename{tikzFIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox}
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %GRID
 %PSTRICKS CODE
@@ -617,7 +617,7 @@
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
-\label{LabelFigExercicebissscosquattrox}
+\label{LabelFigExerciceGraphesbissscosquattrox}
 }                  % Closing subfigure 7
 %
 %AFTER SUBFIGURES
@@ -626,6 +626,6 @@
 %PSPICTURE
 %AFTER PSPICTURE
 %AFTER ALL
-\caption{\CaptionFigExercicebis}\label{LabelFigExercicebis}
+\caption{\CaptionFigExerciceGraphesbis}\label{LabelFigExerciceGraphesbis}
             \end{figure}
             
diff --git a/src_phystricks/Fig_TgCercleTrigono.pstricks.recall b/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
index 96ed0d514..54059faa9 100644
--- a/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
+++ b/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
@@ -89,7 +89,7 @@
 \draw (0.7565851552,0.3380451309) node {$\theta$};
 
 \draw [color=red] (0.500,0)--(0.500,0.00441)--(0.500,0.00881)--(0.500,0.0132)--(0.500,0.0176)--(0.500,0.0220)--(0.499,0.0264)--(0.499,0.0308)--(0.499,0.0352)--(0.498,0.0396)--(0.498,0.0440)--(0.498,0.0484)--(0.497,0.0528)--(0.497,0.0572)--(0.496,0.0615)--(0.496,0.0659)--(0.495,0.0703)--(0.494,0.0746)--(0.494,0.0790)--(0.493,0.0834)--(0.492,0.0877)--(0.491,0.0920)--(0.491,0.0964)--(0.490,0.101)--(0.489,0.105)--(0.488,0.109)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.131)--(0.481,0.135)--(0.480,0.139)--(0.479,0.143)--(0.478,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.472,0.164)--(0.471,0.169)--(0.469,0.173)--(0.468,0.177)--(0.466,0.181)--(0.465,0.185)--(0.463,0.189)--(0.461,0.193)--(0.459,0.197)--(0.458,0.201)--(0.456,0.205)--(0.454,0.209)--(0.452,0.213)--(0.450,0.217)--(0.448,0.221)--(0.446,0.225)--(0.444,0.229)--(0.442,0.233)--(0.440,0.237)--(0.438,0.241)--(0.436,0.245)--(0.434,0.248)--(0.432,0.252)--(0.429,0.256)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.420,0.271)--(0.418,0.275)--(0.415,0.278)--(0.413,0.282)--(0.410,0.286)--(0.408,0.289)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.397,0.303)--(0.395,0.307)--(0.392,0.310)--(0.389,0.314)--(0.386,0.317)--(0.384,0.321)--(0.381,0.324)--(0.378,0.327)--(0.375,0.331)--(0.372,0.334)--(0.369,0.337)--(0.366,0.341)--(0.363,0.344)--(0.360,0.347)--(0.357,0.350)--(0.354,0.353)--(0.351,0.356)--(0.348,0.359)--(0.344,0.362)--(0.341,0.366)--(0.338,0.368)--(0.335,0.371)--(0.331,0.374)--(0.328,0.377)--(0.325,0.380)--(0.321,0.383);
-\draw (-0.1463280799,0.5970250798) node {$\varphi$};
+\draw (0.3487948559,0.6970250798) node {$\varphi$};
 
 \draw [color=cyan] (0.500,0)--(0.500,0.0132)--(0.499,0.0264)--(0.498,0.0396)--(0.497,0.0528)--(0.496,0.0659)--(0.494,0.0790)--(0.491,0.0920)--(0.489,0.105)--(0.486,0.118)--(0.483,0.131)--(0.479,0.143)--(0.475,0.156)--(0.471,0.169)--(0.466,0.181)--(0.461,0.193)--(0.456,0.205)--(0.450,0.217)--(0.444,0.229)--(0.438,0.241)--(0.432,0.252)--(0.425,0.264)--(0.418,0.275)--(0.410,0.286)--(0.403,0.296)--(0.395,0.307)--(0.386,0.317)--(0.378,0.327)--(0.369,0.337)--(0.360,0.347)--(0.351,0.356)--(0.341,0.366)--(0.331,0.374)--(0.321,0.383)--(0.311,0.391)--(0.301,0.399)--(0.290,0.407)--(0.279,0.415)--(0.268,0.422)--(0.257,0.429)--(0.245,0.436)--(0.234,0.442)--(0.222,0.448)--(0.210,0.454)--(0.198,0.459)--(0.186,0.464)--(0.173,0.469)--(0.161,0.473)--(0.148,0.477)--(0.136,0.481)--(0.123,0.485)--(0.110,0.488)--(0.0972,0.490)--(0.0842,0.493)--(0.0712,0.495)--(0.0580,0.497)--(0.0449,0.498)--(0.0317,0.499)--(0.0185,0.500)--(0.00529,0.500)--(-0.00793,0.500)--(-0.0211,0.500)--(-0.0344,0.499)--(-0.0475,0.498)--(-0.0607,0.496)--(-0.0738,0.495)--(-0.0868,0.492)--(-0.0998,0.490)--(-0.113,0.487)--(-0.126,0.484)--(-0.138,0.480)--(-0.151,0.477)--(-0.164,0.473)--(-0.176,0.468)--(-0.188,0.463)--(-0.200,0.458)--(-0.213,0.453)--(-0.224,0.447)--(-0.236,0.441)--(-0.248,0.434)--(-0.259,0.428)--(-0.270,0.421)--(-0.281,0.413)--(-0.292,0.406)--(-0.303,0.398)--(-0.313,0.390)--(-0.323,0.381)--(-0.333,0.373)--(-0.343,0.364)--(-0.353,0.354)--(-0.362,0.345)--(-0.371,0.335)--(-0.380,0.325)--(-0.388,0.315)--(-0.396,0.305)--(-0.404,0.294)--(-0.412,0.284)--(-0.419,0.273)--(-0.426,0.261)--(-0.433,0.250);
 \draw [color=red,->,>=latex] (0,0) -- (1.285575219,1.532088886);
diff --git a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
index 1399869d5..4a92b4ea1 100644
--- a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
+++ b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
@@ -121,7 +121,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156037) -- (0,4.065868530);
+\draw [,->,>=latex] (0,-4.525156038) -- (0,4.065868530);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
diff --git a/src_phystricks/phystricksExerciceGraphesbis.py b/src_phystricks/phystricksExerciceGraphesbis.py
index a2299a277..94850866b 100644
--- a/src_phystricks/phystricksExerciceGraphesbis.py
+++ b/src_phystricks/phystricksExerciceGraphesbis.py
@@ -1,6 +1,6 @@
 from phystricks import *
 def ExerciceGraphesbis():
-    fig = GenericFigure("Exercicebis")
+    fig = GenericFigure("ExerciceGraphesbis")
 
     ssfig1 = fig.new_subfigure("","cosinus")
     pspict1 = ssfig1.new_pspicture("cosinus")
diff --git a/tex/exocorr/corrDS2010bis-0002.tex b/tex/exocorr/corrDS2010bis-0002.tex
index d09816c4d..dcde84b67 100644
--- a/tex/exocorr/corrDS2010bis-0002.tex
+++ b/tex/exocorr/corrDS2010bis-0002.tex
@@ -8,19 +8,19 @@
 	Chaque fonction a sa particularité qu'il faut reconnaître.
 	\begin{enumerate}
 		\item
-			La fonction $\cos(x)$ elle-même est celle qui vaut $1$ en $0$, qui s'annule en $\frac{ \pi }{2}$ et qui oscille. C'est donc le graphe \ref{LabelFigExercicebissscosinus}.
+			La fonction $\cos(x)$ elle-même est celle qui vaut $1$ en $0$, qui s'annule en $\frac{ \pi }{2}$ et qui oscille. C'est donc le graphe \ref{LabelFigExerciceGraphesbissscosinus}.
 		\item
-			La fonction $\cos(x+\frac{ \pi }{2})$ est la même que la fonction cosinus, mais décalée de $\frac{ \pi }{2}$ vers la gauche. C'est le graphe  \ref{LabelFigExercicebissssinus}.
+			La fonction $\cos(x+\frac{ \pi }{2})$ est la même que la fonction cosinus, mais décalée de $\frac{ \pi }{2}$ vers la gauche. C'est le graphe  \ref{LabelFigExerciceGraphesbissssinus}.
 		\item
-			La fonction $\cos( e^{x})$ est une fonction qui oscille de plus en plus vite parce que ce qui se trouve dans le cosinus (c'est à dire  $ e^{x}$) monte de plus en plus vite. Le graphe qui correspond est \ref{LabelFigExercicebissscosex}.
+			La fonction $\cos( e^{x})$ est une fonction qui oscille de plus en plus vite parce que ce qui se trouve dans le cosinus (c'est à dire  $ e^{x}$) monte de plus en plus vite. Le graphe qui correspond est \ref{LabelFigExerciceiGraphesbissscosex}.
 		\item
-			Le graphe de la fonction $\cos(x)+1$ est le même que celui de $\cos(x)$, mais décalé de $1$ vers le haut. C'est le graphe \ref{LabelFigExercicebissscosxplusun}.
+			Le graphe de la fonction $\cos(x)+1$ est le même que celui de $\cos(x)$, mais décalé de $1$ vers le haut. C'est le graphe \ref{LabelFigExerciceiGrphesbissscosxplusun}.
 		\item
-			La fonction $\cos(4x)$ oscille quatre fois plus vite que le cosinus (parce que $4x$ avance $4$ fois plus vite que $x$). C'est donc le graphe \ref{LabelFigExercicebissscosquattrox}.
+			La fonction $\cos(4x)$ oscille quatre fois plus vite que le cosinus (parce que $4x$ avance $4$ fois plus vite que $x$). C'est donc le graphe \ref{LabelFigExerciceGraphesbissscosquattrox}.
 		\item
-			La fonction $| \cos(x) |$ est la même que $\cos(x)$ sauf que partout où $\cos(x)$ est négatif, il devient positif. C'est le graphe \ref{LabelFigExercicebisssvalabsoluecosinus}
+			La fonction $| \cos(x) |$ est la même que $\cos(x)$ sauf que partout où $\cos(x)$ est négatif, il devient positif. C'est le graphe \ref{LabelFigExerciceiGraphesbisssvalabsoluecosinus}
 		\item
-			La fonction $\sqrt{\cos(x)}$ est reconnaissable au fait qu'elle n'est pas définie là où le cosinus est négatif. Il y a donc des «trous» dans son domaine. C'est le graphe \ref{LabelFigExercicebissssqrtcos}
+			La fonction $\sqrt{\cos(x)}$ est reconnaissable au fait qu'elle n'est pas définie là où le cosinus est négatif. Il y a donc des «trous» dans son domaine. C'est le graphe \ref{LabelFigExerciceiGraphesbissssqrtcos}
 	\end{enumerate}
 	
 
diff --git a/tex/exocorr/exoDS2010bis-0002.tex b/tex/exocorr/exoDS2010bis-0002.tex
index 5c08e83fc..bd5c908d4 100644
--- a/tex/exocorr/exoDS2010bis-0002.tex
+++ b/tex/exocorr/exoDS2010bis-0002.tex
@@ -5,7 +5,7 @@
 
 \begin{exercice}\label{exoDS2010bis-0002}
 
-	Indiquez le graphe correspondant à chacune des fonctions suivantes (figure \ref{LabelFigExercicebis}). Justifiez vos réponses
+	Indiquez le graphe correspondant à chacune des fonctions suivantes (figure \ref{LabelFigExerciceGraphebis}). Justifiez vos réponses
 \begin{multicols}{3}
   \begin{enumerate}
   \item $\cos (x) $ ;
@@ -18,8 +18,8 @@
   \end{enumerate}
 \end{multicols}
 
-\newcommand{\CaptionFigExercicebis}{Les graphes à considérer de la question \ref{exoDS2010bis-0002}.} 
-\input{auto/pictures_tex/Fig_Exercicebis.pstricks}
+\newcommand{\CaptionFigExerciceGrphesbis}{Les graphes à considérer de la question \ref{exoDS2010bis-0002}.} 
+\input{auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks}
 
 \corrref{DS2010bis-0002}
 \end{exercice}

From e4eaf350d4658e1c6ef1e18547fde84442fb17fe Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Mon, 12 Jun 2017 08:42:54 +0200
Subject: [PATCH 07/64] Typo : Gaphs -> Graphs

---
 tex/exocorr/exoDS2010bis-0002.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/tex/exocorr/exoDS2010bis-0002.tex b/tex/exocorr/exoDS2010bis-0002.tex
index bd5c908d4..9eca633dc 100644
--- a/tex/exocorr/exoDS2010bis-0002.tex
+++ b/tex/exocorr/exoDS2010bis-0002.tex
@@ -18,7 +18,7 @@
   \end{enumerate}
 \end{multicols}
 
-\newcommand{\CaptionFigExerciceGrphesbis}{Les graphes à considérer de la question \ref{exoDS2010bis-0002}.} 
+\newcommand{\CaptionFigExerciceGraphesbis}{Les graphes à considérer de la question \ref{exoDS2010bis-0002}.} 
 \input{auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks}
 
 \corrref{DS2010bis-0002}

From 5e2a49c51cc7e1de4b3b9241e0386592dbcad21b Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Mon, 12 Jun 2017 17:06:43 +0200
Subject: [PATCH 08/64] =?UTF-8?q?M=C3=A9thode=20des=20diff=C3=A9rences=20f?=
 =?UTF-8?q?inies=20en=20dimension=202=20:=20d=C3=A9but=20d'un=20exemple.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/frido/182_numerique.tex | 148 +++++++++++++++++++++++++++++++++---
 1 file changed, 136 insertions(+), 12 deletions(-)

diff --git a/tex/frido/182_numerique.tex b/tex/frido/182_numerique.tex
index 0306349e7..f79f05eb0 100644
--- a/tex/frido/182_numerique.tex
+++ b/tex/frido/182_numerique.tex
@@ -929,7 +929,7 @@ \subsection{M-matrice}
 \end{proof}
 
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Méthode des différences finies}
+\section{Méthode des différences finies de dimension un}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
 Soit une fonction \( u\colon \eR\to \eR\), et soit \( h>0\). Nous définissons les opérations suivantes (qui sont supposées approximer la dérivée \( u'(x)\) lorsqu'elle existe).
@@ -954,7 +954,7 @@ \section{Méthode des différences finies}
 
 Voici un lemme qui dit que tout cela n'est pas si mal, pourvu que \( u\) soit assez régulière.
 
-\begin{lemma}       \label{LEMooHWLGooDsgAIu}
+\begin{lemma}       \label{LEMooZECZooVKxOZZ}
     Soit un ouvert connexe \( \Omega\) de \( \eR\), soit \( x\in \Omega\) et \( h>0\) tel que \( \overline{ B(x,h) }\subset \Omega\).
     \begin{enumerate}
         \item
@@ -1179,7 +1179,7 @@ \subsubsection{Propriétés du système}
     \end{subproof}
 \end{proof}
 
-\begin{proposition}
+\begin{proposition}     \label{PROPooOQJVooJMTkVM}
     La matrice \( L_h\) est 
     \begin{enumerate}
         \item
@@ -1209,13 +1209,38 @@ \subsubsection{Propriétés du système}
 
 Cela étant rappelé, nous pouvons continuer.
 
+\begin{lemma}       \label{LEMooDXPRooOhwqSZ}
+    Soit \( \Omega=\mathopen] 0 , 1 \mathclose[\), soit \( N\in \eN\) et \( h=1/(N+1)\).  La solution \( w_h\colon \Omega_h \to \eR\) du problème discrétisé
+        \begin{subequations}        \label{SUBEQooFJKIooLvzMBG}
+            \begin{numcases}{}
+                -(D^-D^+w_h)(x_i)=1\\
+                w_h(0)=0\\
+                w_h(1)=0
+            \end{numcases}
+        \end{subequations}
+        pour tout \( x_i=ih\) (\( i=1,\ldots, N\)) donne les valeurs exactes des \( w(x_i)\) lorsque \( w\) est la solution de
+        \begin{subequations}        \label{SUBEQooCRFWooJegcUk}
+            \begin{numcases}{}
+                -w''(x)=1\\
+                w(0)=0\\
+                w(1)=0.
+            \end{numcases}
+        \end{subequations}
+\end{lemma}
+
+\begin{proof}
+    Un enseignement de la proposition \ref{PROPooOQJVooJMTkVM} est que le système \eqref{SUBEQooFJKIooLvzMBG} peut être écrit sous la forme d'un système linéaire \( L^0_hw_h=F_h\) où \( L_h^0\) est inversible. Il y a donc unicité de la solution.
+
+    D'autre part, la solution du système \eqref{SUBEQooCRFWooJegcUk} est \( w(x)=\frac{ 1 }{2}(x-x^2)\), qui est de classe \(  C^{\infty}\). Le lemme \ref{LEMooZECZooVKxOZZ}\ref{ITEMooRWUHooZJLKuL} dit que \( D^-D^+w=w''\). Donc les valeurs \( w(x_i)\) résolvent aussi le système \eqref{SUBEQooFJKIooLvzMBG}.
+\end{proof}
+
 \begin{lemma}[Quelque estimations]
-    La matrice \( L_h\) du problème sus-mentionné en \eqref{EQooEUHQooWHRelr} vérifie :
+    La matrice \( L_h\) du problème sus-mentionné en \eqref{EQooEUHQooWHRelr} vérifie\quext{Dans le CTES d'analyse numérique de Marseille, l'estimation donnée est \(  \| L_h^{-1} \|_{\infty}\leq \frac{1}{ 4 } \).} :
     \begin{enumerate}
         \item
-            \( \| L_h \|_{\infty}\leq \frac{4  }{ h^2 }+\| c \|_{\infty}\)
+            \( \| L_h \|_{\infty}\leq \frac{4 }{ h^2 }+\| c \|_{\infty}\)
         \item
-            \( \| L_h^{-1} \|_{\infty}\leq \frac{1}{ 4 }\).
+            \( \| L_h^{-1} \|_{\infty}\leq \frac{1}{ 8 }\).
     \end{enumerate}
 \end{lemma}
 
@@ -1239,15 +1264,114 @@ \subsubsection{Propriétés du système}
     \begin{equation}
         w(x)=-\frac{ 1 }{2}(x^2-x).
     \end{equation}
-    Le lemme \ref{LEMooFITMooBBBWGI}\ref{ITEMooSAWJooJUTWAb} dit que \( w'(x)=(D^0w)(x)\). Pour la même raison, \( D^-D^+w=w''\). Tentons d'écrire le système discrétisé de \eqref{SUBEQSooRENKooZaRjvL}. En comparant avec \eqref{EQooXJBWooRhCsLy}, il s'agit de poser \( \alpha=\beta=0\), \( c=0\) et \( f=1\). En particulier le vecteur \( F_h\) de \eqref{EQooMNTJooYPYoAj} est le vecteur \( \mtu_h\) (le vecteur dont toutes les composantes sont égales à \( 1\)). L'équation discrétisée est :
+    Le lemme \ref{LEMooDXPRooOhwqSZ} nous dit que la fonction \( w\) prise aux points \( x_i=ih\) donne les valeurs de \( w_h\).
+
+    La matrice \( L^0_h\) est une M-matrice et le vecteur \( w_h\) vérifie \( L_h^0w_h=\mtu\). Donc le théorème \ref{THOooWIFGooBQpddF} s'applique et 
+    \begin{equation}
+        \| (L_h^0)^{-1} \|\leq \| w_h \|_{\infty}=\frac{1}{ 8 }.
+    \end{equation}
+    L'obtention de \( 1/8\) n'est rien d'autre que la recherche du maximum (en valeur absolue) de la parabole \( x\mapsto (x-x^2)/2\) pour \( x\in \mathopen[ 0 , 1 \mathclose]\). Le maximum est atteint pour \( x=1/2\); calcul de dérivée et tout ça \ldots
+
+    Nous retournons maintenant à notre matrice originale \( L_h\). Nous avons
+    \begin{equation}
+        L_h-L_h^0=\diag(c_1,\ldots, c_n)\geq 0,
+    \end{equation}
+    et aussi
     \begin{equation}
-        L^0_hw_h=\mtu_h.
+        L_h^{-1}-(L_h^0)^{-1}=\underbrace{L_h^{-1}}_{\geq 0}\underbrace{(L_h^0-L_h)}_{\leq 0}\underbrace{(L_h^0)^{-1}}_{\geq 0}
     \end{equation}
-    Par construction de l'équation discrétisée, \(-D^-D^+w_h(x_i)=1\). Mais dans ce cas-ci nous avons les égalités
+    parce que \( L_h\) est une M-matrice. Donc tous les coefficients de \( L_h^{-1}-(L_h^0)^{-1}\) sont négatifs. Cela implique
     \begin{equation}
-       -(D^-D^+w)(x)=-w''(x)=1
+        L_h^{-1}\leq (L_h^0)^{-1}.
     \end{equation}
-    pour tout \( x\).
+    Mais nous savons que les coefficients de \( L_h^{-1}\) sont positifs, donc le maximum de ses coefficients en valeur absolue est plus petit que ceux de \( (L_h^0)^{-1}\), c'est à dire
+    \begin{equation}
+        \| L_h^{-1} \|_{\infty}\leq\| (L_h^0)^{-1} \|_{\infty}\leq\frac{1}{ 8 }.
+    \end{equation}
+    
 \end{proof}
-<++>
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Exemple}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit \( \Omega=\mathopen] 0 , 1 \mathclose[\) et une fonction \( u\colon \bar\Omega\to \eR\) de classe \( C^4\) vérifiant
+\begin{subequations}
+    \begin{numcases}{}
+        -u''(x)+u(x)=\sin(x)\\
+        u(0)=0\\
+        u(1)=0.
+    \end{numcases}
+\end{subequations}
+Nous allons écrire la méthode des différences finies pour \( h=1/4\). Nous posons donc les points
+\begin{subequations}
+    \begin{numcases}{}
+        x_0=0\\
+        x_1=1/4\\
+        x_2=1/2\\
+        x_3=3/4\\
+        x_4=1.
+    \end{numcases}
+\end{subequations}
+
+Vu que nous avons supposé \( u\) de classe \( C^4\), le lemme \ref{LEMooZECZooVKxOZZ}\ref{ITEMooRWUHooZJLKuL} nous donne\footnote{Nous ferions n'importe quoi pour ne pas écrire \( u''(x)=(D^-D^+u)(x)+o(h^2)\). Notez que vous faites ce que vous voulez : écrivez avec la notation «petit \( o\)» si cela vous chante.}
+\begin{equation}
+    u''(x)=(D^-D^+u)(x)+\alpha(h)
+\end{equation}
+avec \( \lim_{h\to 0} \alpha(h)/h=0\). L'équation discrétisée serait alors
+\begin{subequations}        \label{SYSTooNEQHooOWJSbT}
+    \begin{numcases}{}
+        -(D^-D^+u)(x)+u(x)=\sin(x)\\
+        u(0)=u(1)=0.
+    \end{numcases}
+\end{subequations}
+où nous n'avons pas précisé l'indice \( h\) au bas des opérateurs \( D^+\) et \( D^-\). Les équations \eqref{SYSTooNEQHooOWJSbT} ne doivent être posées que pour \( x_1\), \( x_2\) et \( x_3\) parce que les valeurs en \( x_0\) et \( x_4\) sont déjà connues.
+
+\begin{subproof}
+    \item[Pour \( x_1\)]
+        \begin{equation}
+            \frac{ u_2-2u_1+u_0 }{ h^2 }+u_1=\sin(x_1)
+        \end{equation}
+    \item[Pour \( x_2\)] 
+        \begin{equation}
+            \frac{ u_3-2u_2+u_1 }{ h^2 }+u_2=\sin(x_2)
+        \end{equation}
+    \item[Pour \( x_3\)]
+        \begin{equation}
+            \frac{ u_4-2u_3+u_2 }{ h^2 }+u_3=\sin(x_3).
+        \end{equation}
+\end{subproof}
+Nous tenons compte du fait que \( u_0=u_4=0\) et que \( h=1/4\) pour écrire le système
+\begin{equation}
+    \begin{pmatrix}
+        -31    &   16    &   0    \\
+        16    &   -31    &   16    \\
+        0    &   16    &   -31
+    \end{pmatrix}\begin{pmatrix}
+        u_1    \\ 
+        u_2    \\ 
+        u_3    
+    \end{pmatrix}=\begin{pmatrix}
+        s_1    \\ 
+        s_2    \\ 
+        s_3    
+    \end{pmatrix}
+\end{equation}A
+où les \( s_i\) sont des nombres parfaitement connus : par exemple \( s_1=\sin(x_1)=\sin(1/4)\simeq 0.247403959254523\).
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Méthode des différences finies de dimension deux}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Nous allons considérer le système
+\begin{equation}        
+     \begin{cases}
+         -\Delta u=f    &   \text{sur } \Omega\\
+         u=g            &   \text{sur } \partial\Omega
+     \end{cases}
+\end{equation}
+où \( \Omega=\mathopen] 0 , a \mathclose[\times \mathopen] 0 , b \mathclose[\).
+
+Nous discrétisons \( \Omega\) en mailles carrés de côté \( h\) : \( x_k=kkh\) et \( y_k=kh\).
+
 

From 0ca3c81cf2d84b5a33e500e67730782866f5be24 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Mon, 12 Jun 2017 17:07:34 +0200
Subject: [PATCH 09/64] Correction de quelque labels Exercicebis ->
 ExerciceGraphesbis

---
 tex/exocorr/corrDS2010bis-0002.tex | 11 +++++------
 tex/exocorr/exoDS2010bis-0002.tex  |  4 ++--
 2 files changed, 7 insertions(+), 8 deletions(-)

diff --git a/tex/exocorr/corrDS2010bis-0002.tex b/tex/exocorr/corrDS2010bis-0002.tex
index dcde84b67..481c9d31c 100644
--- a/tex/exocorr/corrDS2010bis-0002.tex
+++ b/tex/exocorr/corrDS2010bis-0002.tex
@@ -1,5 +1,5 @@
 % This is part of Exercices de mathématique pour SVT
-% Copyright (C) 2010
+% Copyright (C) 2010,2017
 %   Laurent Claessens et Carlotta Donadello
 % See the file fdl-1.3.txt for copying conditions.
 
@@ -12,16 +12,15 @@
 		\item
 			La fonction $\cos(x+\frac{ \pi }{2})$ est la même que la fonction cosinus, mais décalée de $\frac{ \pi }{2}$ vers la gauche. C'est le graphe  \ref{LabelFigExerciceGraphesbissssinus}.
 		\item
-			La fonction $\cos( e^{x})$ est une fonction qui oscille de plus en plus vite parce que ce qui se trouve dans le cosinus (c'est à dire  $ e^{x}$) monte de plus en plus vite. Le graphe qui correspond est \ref{LabelFigExerciceiGraphesbissscosex}.
+			La fonction $\cos( e^{x})$ est une fonction qui oscille de plus en plus vite parce que ce qui se trouve dans le cosinus (c'est à dire  $ e^{x}$) monte de plus en plus vite. Le graphe qui correspond est \ref{LabelFigExerciceGraphesbissscosex}.
 		\item
-			Le graphe de la fonction $\cos(x)+1$ est le même que celui de $\cos(x)$, mais décalé de $1$ vers le haut. C'est le graphe \ref{LabelFigExerciceiGrphesbissscosxplusun}.
+			Le graphe de la fonction $\cos(x)+1$ est le même que celui de $\cos(x)$, mais décalé de $1$ vers le haut. C'est le graphe \ref{LabelFigExerciceGraphesbissscosxplusun}.
 		\item
 			La fonction $\cos(4x)$ oscille quatre fois plus vite que le cosinus (parce que $4x$ avance $4$ fois plus vite que $x$). C'est donc le graphe \ref{LabelFigExerciceGraphesbissscosquattrox}.
 		\item
-			La fonction $| \cos(x) |$ est la même que $\cos(x)$ sauf que partout où $\cos(x)$ est négatif, il devient positif. C'est le graphe \ref{LabelFigExerciceiGraphesbisssvalabsoluecosinus}
+			La fonction $| \cos(x) |$ est la même que $\cos(x)$ sauf que partout où $\cos(x)$ est négatif, il devient positif. C'est le graphe \ref{LabelFigExerciceGraphesbisssvalabsoluecosinus}
 		\item
-			La fonction $\sqrt{\cos(x)}$ est reconnaissable au fait qu'elle n'est pas définie là où le cosinus est négatif. Il y a donc des «trous» dans son domaine. C'est le graphe \ref{LabelFigExerciceiGraphesbissssqrtcos}
+			La fonction $\sqrt{\cos(x)}$ est reconnaissable au fait qu'elle n'est pas définie là où le cosinus est négatif. Il y a donc des «trous» dans son domaine. C'est le graphe \ref{LabelFigExerciceGraphesbissssqrtcos}
 	\end{enumerate}
 	
-
 \end{corrige}
diff --git a/tex/exocorr/exoDS2010bis-0002.tex b/tex/exocorr/exoDS2010bis-0002.tex
index 9eca633dc..2101949d2 100644
--- a/tex/exocorr/exoDS2010bis-0002.tex
+++ b/tex/exocorr/exoDS2010bis-0002.tex
@@ -1,11 +1,11 @@
 % This is part of Exercices de mathématique pour SVT
-% Copyright (c) 2010
+% Copyright (c) 2010,2017
 %   Laurent Claessens et Carlotta Donadello
 % See the file fdl-1.3.txt for copying conditions.
 
 \begin{exercice}\label{exoDS2010bis-0002}
 
-	Indiquez le graphe correspondant à chacune des fonctions suivantes (figure \ref{LabelFigExerciceGraphebis}). Justifiez vos réponses
+	Indiquez le graphe correspondant à chacune des fonctions suivantes (figure \ref{LabelFigExerciceGraphesbis}). Justifiez vos réponses
 \begin{multicols}{3}
   \begin{enumerate}
   \item $\cos (x) $ ;

From 13327a767d5edde88762c96c99c4775beb088dbf Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 13 Jun 2017 10:56:15 +0200
Subject: [PATCH 10/64] =?UTF-8?q?Supprimer=20quelque=20lignes=20comment?=
 =?UTF-8?q?=C3=A9es?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 src_phystricks/figures_mazhe.py | 18 ------------------
 1 file changed, 18 deletions(-)

diff --git a/src_phystricks/figures_mazhe.py b/src_phystricks/figures_mazhe.py
index 8fd2c3c0b..f4be2061a 100755
--- a/src_phystricks/figures_mazhe.py
+++ b/src_phystricks/figures_mazhe.py
@@ -274,24 +274,6 @@
 from phystricksVANooZowSyO import VANooZowSyO
 from phystricksXTGooSFFtPu import XTGooSFFtPu
 
-
-# Naming inconstancies fixing.   February 26, 2016
-#   SurfaceDerive  replaced by BQXKooPqSEMN
-#   ArcCercleAngle replaced by ooIHLPooKLIxcH
-
-# Suppression août 2014
-#from figure_devoir1 import exercice1A1
-#from figure_devoir1 import exercice1A2
-#from figure_devoir1 import exercice4
-#from phystricksExerciceGraphes import ExerciceGraphes              # Remplacé par ACUooQwcDMZ
-#from phystricksIntCourbePolaire import IntCourbePolaire
-#from phystricksExoCourone import ExoCourone
-#from phystricksExampleChangementVariables import ExampleChangementVariables
-#from phystricksAIFsOQO import AIFsOQO  # Il n'est pas dans le fichier phystricksAIFsOQO.
-#from phystricksDessinExp import DessinExp
-#from phystricksDS2010exo1 import DS2010exo1
-#from phystricksDS2010bisExoGraph import DS2010bisExoGraph
-
 figures_list_1=[]
 figures_list_2=[]
 figures_list_3=[]

From 0888cde081f377e139859dc403f82a61e3d9cc6e Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 13 Jun 2017 14:38:18 +0200
Subject: [PATCH 11/64] (pictures creation) Compile the pictures the right
 number of times.

Some pictures were compiled only 1 times while they needed 2 passes.
---
 auto/pictures_tikz/tikzooIHLPooKLIxcH.md5 |   1 -
 auto/pictures_tikz/tikzooIHLPooKLIxcH.pdf | Bin 23739 -> 0 bytes
 src_phystricks/figures_mazhe.py           |   6 +++---
 3 files changed, 3 insertions(+), 4 deletions(-)
 delete mode 100644 auto/pictures_tikz/tikzooIHLPooKLIxcH.md5
 delete mode 100644 auto/pictures_tikz/tikzooIHLPooKLIxcH.pdf

diff --git a/auto/pictures_tikz/tikzooIHLPooKLIxcH.md5 b/auto/pictures_tikz/tikzooIHLPooKLIxcH.md5
deleted file mode 100644
index bda50dd84..000000000
--- a/auto/pictures_tikz/tikzooIHLPooKLIxcH.md5
+++ /dev/null
@@ -1 +0,0 @@
-\def \tikzexternallastkey {A5D14C9C9EE31D0FD05319C4B1FBC881}%
diff --git a/auto/pictures_tikz/tikzooIHLPooKLIxcH.pdf b/auto/pictures_tikz/tikzooIHLPooKLIxcH.pdf
deleted file mode 100644
index 838400f571cdb37265a3c6fed32b4fff9c40e3cd..0000000000000000000000000000000000000000
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diff --git a/src_phystricks/figures_mazhe.py b/src_phystricks/figures_mazhe.py
index f4be2061a..05db0eb19 100755
--- a/src_phystricks/figures_mazhe.py
+++ b/src_phystricks/figures_mazhe.py
@@ -516,8 +516,8 @@ def append_picture(fun,number):
 append_picture(PONXooXYjEot,2)
 
 append_picture(FXVooJYAfif,1)
-append_picture(VGZooJnvvZc,1)
-append_picture(LYORooNKDHqt,1)
+append_picture(VGZooJnvvZc,2)
+append_picture(LYORooNKDHqt,2)
 append_picture(TKXZooLwXzjS,1)
 append_picture(YQVHooYsGLHQ,1)
 append_picture(ZGUDooEsqCWQ,1)
@@ -545,7 +545,7 @@ def append_picture(fun,number):
 append_picture(TraceCycloide,1)
 append_picture(Osculateur,1)
 append_picture(JGuKEjH,1)
-append_picture(ExerciceGraphesbis,1)
+append_picture(ExerciceGraphesbis,2)
 append_picture(DGFSooWgbuuMoB,1)
 append_picture(ZOCNoowrfvQXsr,1)
 append_picture(UCDQooMCxpDszQ,1)

From 9fa45958064d2ea61e432c33699643242f266a80 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 13 Jun 2017 16:00:02 +0200
Subject: [PATCH 12/64] (picture generation) Fix the recall file of 'XOLB'.

One very small decimal was changed. I hope this is not the sign of some
non-deterministic behaviour under the hood.
---
 auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks     | 2 +-
 src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
index 4a92b4ea1..1399869d5 100644
--- a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
+++ b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
@@ -121,7 +121,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156038) -- (0,4.065868530);
+\draw [,->,>=latex] (0,-4.525156037) -- (0,4.065868530);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
diff --git a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
index 4a92b4ea1..1399869d5 100644
--- a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
+++ b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
@@ -121,7 +121,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156038) -- (0,4.065868530);
+\draw [,->,>=latex] (0,-4.525156037) -- (0,4.065868530);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);

From 70a4bf270ca7521ab05a4217a2f2730cae765367 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 15 Jun 2017 04:51:01 +0200
Subject: [PATCH 13/64] =?UTF-8?q?(num=C3=A9rique)=20Un=20exemple=20d'erreu?=
 =?UTF-8?q?r=20de=20cancellation=20dans=20la=20vie=20r=C3=A9elle.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Un résultat non déterministe par Sage lorsqu'on utilise des
approximations numériques sur une différence de deux nombres d'ordre
2200.
---
 "r\303\251serve.tex"        |  4 +++-
 tex/frido/145_numerique.tex | 10 ++++++++++
 tex/sage/sageSnip016.sage   |  3 +++
 3 files changed, 16 insertions(+), 1 deletion(-)
 create mode 100644 tex/sage/sageSnip016.sage

diff --git "a/r\303\251serve.tex" "b/r\303\251serve.tex"
index 029d57fc0..34bc09c99 100644
--- "a/r\303\251serve.tex"
+++ "b/r\303\251serve.tex"
@@ -9,8 +9,10 @@
 codeSnip_5.py
 codeSnip_6.py
 
-\lstinputlisting{tex/sage/sageSnip016.sage}
 \lstinputlisting{tex/sage/sageSnip017.sage}
+\lstinputlisting{tex/sage/sageSnip018.sage}
+\lstinputlisting{tex/sage/sageSnip019.sage}
+\lstinputlisting{tex/sage/sageSnip020.sage}
 
 \Exo{mazhe-0018}
 \Exo{mazhe-0019}
diff --git a/tex/frido/145_numerique.tex b/tex/frido/145_numerique.tex
index 1ba0733b9..f209ea79c 100644
--- a/tex/frido/145_numerique.tex
+++ b/tex/frido/145_numerique.tex
@@ -518,6 +518,16 @@ \section{Erreur de ``cancellation''}
 
 Les erreurs de cancellation ne se résolvent pas en augmentant la précision des nombres donnés.
 
+\begin{example}[Dans la vie réelle]
+    La préparation de l'exemple \ref{EXooJXIGooQtotMc} nous a porté à calculer la différence entre \( \exp(x)\) et \( f_{30}(x)\) où \( f_{30}\) est censée être une bonne approximation de l'exponentielle. Des erreurs de cancellation sont donc à craindre.
+
+Et en effet, le code suivant produit un résultat non déterministe :
+\lstinputlisting{tex/sage/sageSnip016.sage}
+
+Voir la question ici :\\ \url{https://ask.sagemath.org/question/37946/undeterministic-numerical-approximation/}
+
+\end{example}
+
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Erreur d'absorption}
 %---------------------------------------------------------------------------------------------------------------------------
diff --git a/tex/sage/sageSnip016.sage b/tex/sage/sageSnip016.sage
new file mode 100644
index 000000000..026433321
--- /dev/null
+++ b/tex/sage/sageSnip016.sage
@@ -0,0 +1,3 @@
+f=1/152444172305856930250752000000*x^28 + 1/10888869450418352160768000000*x^27 + 1/15511210043330985984000000*x^25 + 1/310224200866619719680000*x^24 + 1/25852016738884976640000*x^23 + 1/51090942171709440000*x^21 + 1/1216451004088320000*x^20 + 1/121645100408832000*x^19 + 1/355687428096000*x^17 + 1/10461394944000*x^16 + 1/1307674368000*x^15 + 1/6227020800*x^13 + 1/239500800*x^12 + 1/39916800*x^11 + 1/362880*x^9 + 1/20160*x^8 + 1/5040*x^7 + 1/120*x^5 + 1/12*x^4 + 1/6*x^3 + x - cos(x) + 2 -exp(x)
+a=numerical_approx(10)
+print(f(a))

From 595fdfde9f5af3a194ccb4581b22d13449148c66 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 15 Jun 2017 04:54:48 +0200
Subject: [PATCH 14/64] =?UTF-8?q?(num=C3=A9rique)=20mini=20mise=20en=20ord?=
 =?UTF-8?q?re=20des=20diff=C3=A9rents=20types=20d'erreurs=20de=20repr?=
 =?UTF-8?q?=C3=A9sentation=20num=C3=A9rique.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/frido/145_numerique.tex | 37 +++++++++++++++++++------------------
 1 file changed, 19 insertions(+), 18 deletions(-)

diff --git a/tex/frido/145_numerique.tex b/tex/frido/145_numerique.tex
index f209ea79c..58d449428 100644
--- a/tex/frido/145_numerique.tex
+++ b/tex/frido/145_numerique.tex
@@ -415,9 +415,9 @@ \subsection{Quelque bonnes règles}
 		Les opérations délicates sont l'addition et la soustraction. La multiplication et la division sont sans dangers, à part l'erreur de dépassement du maximum. Dans une multiplication, on perd au pire quelque chiffres significatifs, mais certainement les derniers, pas les premiers.
 \end{enumerate}
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Erreur de ``cancellation''}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Erreur de ``cancellation''}
+%---------------------------------------------------------------------------------------------------------------------------
 
 Lorsque deux nombres sont de même ordre de grandeur, avec plusieurs nombres significatifs identiques. La cancellation est le fait que, suite à la soustraction, tous les chiffres significatifs ou presque se sont simplifiés et qu'il ne reste plus que des chiffres non significatifs. 
 
@@ -529,38 +529,39 @@ \section{Erreur de ``cancellation''}
 \end{example}
 
 %--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Erreur d'absorption}
+\subsection{Calcul d'une dérivée}
 %---------------------------------------------------------------------------------------------------------------------------
 
-L'addition d'un nombre avec un nombre très différent peut faire perdre de l'information sur le plus petit. Par exemple avec \( 4\) chiffres significatifs,
+Pour calculer la dérivée de \( f\) en \( a\), il est loisible d'utiliser la formule
 \begin{equation}
-    0.5678\oplus 0.0001237=0.5679
+    f'(a)=\lim_{h\to 0} \frac{ f(a+h)-f(a) }{ h }.
 \end{equation}
-où nous avons perdu presque toute l'information du petit nombre.
+Le numérateur est alors sujet à une erreur d'absorption dans le calcul de \( a+h\) et ensuite une erreur de cancellation dans le calcul de la différence.
 
-Une situation particulièrement ennuyeuse est celle où justement c'est le petit nombre qui nous intéresse parce que le grand est censé se simplifier : 
+En utilisant la formule
 \begin{equation}
-    (0.0001327\oplus 0.5678)\ominus 0.5678=0.5679\ominus 0.5678=0.0001
+    f'(a)=\lim_{h\to 0} \frac{ f(a+h)-f(a-h) }{ 2h }
 \end{equation}
-qui ne possède qu'un seul chiffre significatif correct alors que voyant le calcul, la réponse aurait pu être trouvée.
+nous pouvons espérer avoir une erreur de cancellation plus petite.
 
-Moralité : si certains manipulations algébrique peuvent faire apparaître des simplifications avant de passer le calcul à la machine, il est bon de les effectuer.
 
 %--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Calcul d'une dérivée}
+\subsection{Erreur d'absorption}
 %---------------------------------------------------------------------------------------------------------------------------
 
-Pour calculer la dérivée de \( f\) en \( a\), il est loisible d'utiliser la formule
+L'addition d'un nombre avec un nombre très différent peut faire perdre de l'information sur le plus petit. Par exemple avec \( 4\) chiffres significatifs,
 \begin{equation}
-    f'(a)=\lim_{h\to 0} \frac{ f(a+h)-f(a) }{ h }.
+    0.5678\oplus 0.0001237=0.5679
 \end{equation}
-Le numérateur est alors sujet à une erreur d'absorption dans le calcul de \( a+h\) et ensuite une erreur de cancellation dans le calcul de la différence.
+où nous avons perdu presque toute l'information du petit nombre.
 
-En utilisant la formule
+Une situation particulièrement ennuyeuse est celle où justement c'est le petit nombre qui nous intéresse parce que le grand est censé se simplifier : 
 \begin{equation}
-    f'(a)=\lim_{h\to 0} \frac{ f(a+h)-f(a-h) }{ 2h }
+    (0.0001327\oplus 0.5678)\ominus 0.5678=0.5679\ominus 0.5678=0.0001
 \end{equation}
-nous pouvons espérer avoir une erreur de cancellation plus petite.
+qui ne possède qu'un seul chiffre significatif correct alors que voyant le calcul, la réponse aurait pu être trouvée.
+
+Moralité : si certains manipulations algébrique peuvent faire apparaître des simplifications avant de passer le calcul à la machine, il est bon de les effectuer.
 
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 \section{Conditionnement et stabilité}

From 8266a9a3e273888cff98fec28da33112fe8dd519 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 15 Jun 2017 15:36:48 +0200
Subject: [PATCH 15/64] (unit tests) Update the test pictures for the new
 syntax about polygon prameters.

---
 auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks    |  2 +-
 auto/pictures_tex/Fig_BEHTooWsdrys.pstricks   |  6 ++---
 auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks   |  8 +++----
 auto/pictures_tex/Fig_CbCartTui.pstricks      |  4 ++--
 auto/pictures_tex/Fig_CbCartTuiii.pstricks    |  4 ++--
 .../Fig_ChiSquaresQuantile.pstricks           |  2 +-
 auto/pictures_tex/Fig_CycloideA.pstricks      |  2 +-
 auto/pictures_tex/Fig_DZVooQZLUtf.pstricks    |  2 +-
 auto/pictures_tex/Fig_DessinLim.pstricks      |  4 ++--
 auto/pictures_tex/Fig_Differentielle.pstricks |  4 ++--
 auto/pictures_tex/Fig_DivergenceUn.pstricks   |  6 ++---
 auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks |  6 ++---
 .../Fig_ExerciceGraphesbis.pstricks           | 14 +++++------
 auto/pictures_tex/Fig_ExoXLVL.pstricks        | 24 +++++++++----------
 auto/pictures_tex/Fig_FWJuNhU.pstricks        | 24 +++++++++----------
 .../Fig_FonctionEtDeriveOM.pstricks           |  2 +-
 .../pictures_tex/Fig_IntegraleSimple.pstricks |  2 +-
 auto/pictures_tex/Fig_JJAooWpimYW.pstricks    |  2 +-
 auto/pictures_tex/Fig_JSLooFJWXtB.pstricks    |  2 +-
 auto/pictures_tex/Fig_KKRooHseDzC.pstricks    |  8 +++----
 auto/pictures_tex/Fig_MethodeNewton.pstricks  |  2 +-
 auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks |  2 +-
 auto/pictures_tex/Fig_Polirettangolo.pstricks | 24 -------------------
 auto/pictures_tex/Fig_RLuqsrr.pstricks        |  2 +-
 auto/pictures_tex/Fig_SenoTopologo.pstricks   |  2 +-
 auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks   |  6 ++---
 auto/pictures_tex/Fig_TVXooWoKkqV.pstricks    |  2 +-
 auto/pictures_tex/Fig_TWHooJjXEtS.pstricks    |  2 +-
 auto/pictures_tex/Fig_TangentSegment.pstricks |  2 +-
 auto/pictures_tex/Fig_TriangleUV.pstricks     |  4 +---
 auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks   |  2 +-
 auto/pictures_tex/Fig_WJBooMTAhtl.pstricks    |  2 +-
 auto/pictures_tex/Fig_WUYooCISzeB.pstricks    |  2 +-
 auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks |  2 +-
 lst_actu.py                                   |  3 +--
 .../Fig_XOLBooGcrjiwoU.pstricks.recall        |  2 +-
 src_phystricks/figures_mazhe.py               |  2 --
 src_phystricks/phystricksBEHTooWsdrys.py      |  7 +++---
 src_phystricks/phystricksBQXKooPqSEMN.py      |  8 +++----
 src_phystricks/phystricksCardioideexo.py      |  4 ++--
 src_phystricks/phystricksDivergenceUn.py      |  6 ++---
 src_phystricks/phystricksEJRsWXw.py           |  4 ++--
 src_phystricks/phystricksExPolygone.py        |  7 +++---
 src_phystricks/phystricksExoXLVL.py           | 18 +++++++-------
 src_phystricks/phystricksFCUEooTpEPFoeQ.py    |  4 ++--
 src_phystricks/phystricksFWJuNhU.py           | 18 +++++++-------
 src_phystricks/phystricksIntDeuxCarres.py     | 13 +++++-----
 src_phystricks/phystricksIsomCarre.py         |  2 +-
 src_phystricks/phystricksKKRooHseDzC.py       |  8 +++----
 src_phystricks/phystricksNOCGooYRHLCn.py      | 10 ++++----
 src_phystricks/phystricksPolirettangolo.py    | 15 ++++++------
 src_phystricks/phystricksTKXZooLwXzjS.py      |  7 +++---
 src_phystricks/phystricksTriangleUV.py        |  6 ++---
 src_phystricks/phystricksUneCellule.py        |  4 ++--
 src_phystricks/phystricksVWFLooPSrOqz.py      |  7 +++---
 src_phystricks/phystricksZTTooXtHkci.py       |  8 +++----
 56 files changed, 154 insertions(+), 193 deletions(-)

diff --git a/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks b/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
index 7b8366461..9e1c13cba 100644
--- a/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
+++ b/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
@@ -346,7 +346,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.548147074);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.548147075);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.020,0.1414)--(1.040,0.1990)--(1.061,0.2426)--(1.081,0.2788)--(1.101,0.3102)--(1.121,0.3382)--(1.141,0.3637)--(1.162,0.3871)--(1.182,0.4087)--(1.202,0.4290)--(1.222,0.4480)--(1.242,0.4659)--(1.263,0.4829)--(1.283,0.4991)--(1.303,0.5145)--(1.323,0.5292)--(1.343,0.5434)--(1.364,0.5569)--(1.384,0.5700)--(1.404,0.5825)--(1.424,0.5947)--(1.444,0.6064)--(1.465,0.6178)--(1.485,0.6287)--(1.505,0.6394)--(1.525,0.6497)--(1.545,0.6598)--(1.566,0.6696)--(1.586,0.6791)--(1.606,0.6883)--(1.626,0.6973)--(1.646,0.7061)--(1.667,0.7147)--(1.687,0.7231)--(1.707,0.7313)--(1.727,0.7393)--(1.747,0.7471)--(1.768,0.7548)--(1.788,0.7623)--(1.808,0.7696)--(1.828,0.7768)--(1.848,0.7838)--(1.869,0.7907)--(1.889,0.7975)--(1.909,0.8041)--(1.929,0.8107)--(1.949,0.8170)--(1.970,0.8233)--(1.990,0.8295)--(2.010,0.8356)--(2.030,0.8415)--(2.051,0.8474)--(2.071,0.8532)--(2.091,0.8588)--(2.111,0.8644)--(2.131,0.8699)--(2.152,0.8753)--(2.172,0.8806)--(2.192,0.8859)--(2.212,0.8910)--(2.232,0.8961)--(2.253,0.9011)--(2.273,0.9061)--(2.293,0.9109)--(2.313,0.9157)--(2.333,0.9205)--(2.354,0.9252)--(2.374,0.9298)--(2.394,0.9343)--(2.414,0.9388)--(2.434,0.9432)--(2.455,0.9476)--(2.475,0.9519)--(2.495,0.9562)--(2.515,0.9604)--(2.535,0.9645)--(2.556,0.9686)--(2.576,0.9727)--(2.596,0.9767)--(2.616,0.9807)--(2.636,0.9846)--(2.657,0.9884)--(2.677,0.9923)--(2.697,0.9961)--(2.717,0.9998)--(2.737,1.003)--(2.758,1.007)--(2.778,1.011)--(2.798,1.014)--(2.818,1.018)--(2.838,1.021)--(2.859,1.025)--(2.879,1.028)--(2.899,1.032)--(2.919,1.035)--(2.939,1.038)--(2.960,1.042)--(2.980,1.045)--(3.000,1.048);
diff --git a/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks b/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
index 694e73559..cc448317f 100644
--- a/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
+++ b/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
@@ -66,10 +66,8 @@
 %PSTRICKS CODE
 %DEFAULT
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
-\draw [] (1.60,-1.20) -- (3.07,-1.20);
-\draw [] (3.07,-1.20) -- (3.07,1.20);
-\draw [] (3.07,1.20) -- (1.60,1.20);
-\draw [] (1.60,1.20) -- (1.60,-1.20);
+
+
 \draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
 \draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
 \draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
diff --git a/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks b/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
index b84460edb..87e4831e7 100644
--- a/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
+++ b/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
@@ -100,10 +100,10 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,5.64) -- (6.00,5.64) -- (5.00,5.64) -- (5.00,5.64) -- (5.00,0) --  cycle;
-\draw [color=red] (5.00,0) -- (6.00,0);
-\draw [color=red] (6.00,0) -- (6.00,5.64);
-\draw [color=red] (6.00,5.64) -- (5.00,5.64);
-\draw [color=red] (5.00,5.64) -- (5.00,0);
+\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,5.64);
+\draw [color=red,style=dashed] (6.00,5.64) -- (5.00,5.64);
+\draw [color=red,style=dashed] (5.00,5.64) -- (5.00,0);
 \draw []  (5.000000000,5.638888889) node [rotate=0] {$\bullet$};
 \draw (5.441978850,6.211818713) node {$f(x)$};
 \draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
diff --git a/auto/pictures_tex/Fig_CbCartTui.pstricks b/auto/pictures_tex/Fig_CbCartTui.pstricks
index d7044745a..398a015fa 100644
--- a/auto/pictures_tex/Fig_CbCartTui.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTui.pstricks
@@ -103,8 +103,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.140000000,0) -- (4.140000000,0);
-\draw [,->,>=latex] (0,-3.972000000) -- (0,4.028000000);
+\draw [,->,>=latex] (-4.140000002,0) -- (4.140000002,0);
+\draw [,->,>=latex] (0,-3.972000001) -- (0,4.028000000);
 %DEFAULT
 \draw [color=blue] (-3.640,-3.472)--(-3.609,-3.440)--(-3.579,-3.407)--(-3.548,-3.375)--(-3.518,-3.343)--(-3.488,-3.310)--(-3.457,-3.278)--(-3.427,-3.245)--(-3.396,-3.213)--(-3.366,-3.180)--(-3.336,-3.148)--(-3.306,-3.115)--(-3.275,-3.083)--(-3.245,-3.050)--(-3.215,-3.018)--(-3.185,-2.985)--(-3.155,-2.953)--(-3.125,-2.920)--(-3.095,-2.887)--(-3.065,-2.855)--(-3.035,-2.822)--(-3.005,-2.789)--(-2.975,-2.756)--(-2.945,-2.723)--(-2.915,-2.691)--(-2.886,-2.658)--(-2.856,-2.625)--(-2.826,-2.592)--(-2.797,-2.559)--(-2.767,-2.526)--(-2.738,-2.493)--(-2.709,-2.459)--(-2.679,-2.426)--(-2.650,-2.393)--(-2.621,-2.360)--(-2.592,-2.326)--(-2.563,-2.293)--(-2.534,-2.259)--(-2.505,-2.226)--(-2.476,-2.192)--(-2.447,-2.158)--(-2.419,-2.124)--(-2.390,-2.090)--(-2.362,-2.056)--(-2.333,-2.022)--(-2.305,-1.988)--(-2.277,-1.954)--(-2.249,-1.919)--(-2.221,-1.885)--(-2.193,-1.850)--(-2.166,-1.815)--(-2.138,-1.780)--(-2.111,-1.745)--(-2.084,-1.709)--(-2.057,-1.674)--(-2.030,-1.638)--(-2.003,-1.602)--(-1.977,-1.566)--(-1.951,-1.529)--(-1.925,-1.492)--(-1.899,-1.455)--(-1.873,-1.418)--(-1.848,-1.380)--(-1.823,-1.342)--(-1.798,-1.304)--(-1.774,-1.265)--(-1.750,-1.225)--(-1.726,-1.185)--(-1.703,-1.144)--(-1.680,-1.103)--(-1.658,-1.061)--(-1.636,-1.018)--(-1.614,-0.9745)--(-1.593,-0.9298)--(-1.573,-0.8840)--(-1.554,-0.8371)--(-1.535,-0.7887)--(-1.517,-0.7389)--(-1.499,-0.6873)--(-1.483,-0.6338)--(-1.468,-0.5781)--(-1.454,-0.5199)--(-1.441,-0.4588)--(-1.429,-0.3943)--(-1.420,-0.3261)--(-1.411,-0.2534)--(-1.405,-0.1754)--(-1.401,-0.09136)--(-1.400,0)--(-1.402,0.1001)--(-1.406,0.2106)--(-1.415,0.3340)--(-1.428,0.4729)--(-1.447,0.6314)--(-1.472,0.8143)--(-1.504,1.029)--(-1.545,1.283)--(-1.598,1.591)--(-1.665,1.971)--(-1.750,2.450);
 \draw [,->,>=latex] (-3.505833333,-3.329618056) -- (-3.501040033,-3.324516656);
diff --git a/auto/pictures_tex/Fig_CbCartTuiii.pstricks b/auto/pictures_tex/Fig_CbCartTuiii.pstricks
index 40b0f31eb..95ddf09e7 100644
--- a/auto/pictures_tex/Fig_CbCartTuiii.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTuiii.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.499748255,0) -- (2.499748255,0);
-\draw [,->,>=latex] (0,-2.497734678) -- (0,2.497734678);
+\draw [,->,>=latex] (-2.499748256,0) -- (2.499748256,0);
+\draw [,->,>=latex] (0,-2.497734679) -- (0,2.497734679);
 %DEFAULT
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diff --git a/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks b/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
index c0b4c0635..ff609a685 100644
--- a/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
+++ b/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
@@ -96,7 +96,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (0,0) -- (9.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.384154941);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.384154943);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_CycloideA.pstricks b/auto/pictures_tex/Fig_CycloideA.pstricks
index 974207361..9f4df5aac 100644
--- a/auto/pictures_tex/Fig_CycloideA.pstricks
+++ b/auto/pictures_tex/Fig_CycloideA.pstricks
@@ -115,7 +115,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (13.06637061,0);
+\draw [,->,>=latex] (-0.5000000000,0) -- (13.06637062,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,2.499496542);
 %DEFAULT
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diff --git a/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks b/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
index 239b992f5..fe50e66aa 100644
--- a/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
+++ b/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
@@ -230,7 +230,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.023619130) -- (0,2.170406053);
+\draw [,->,>=latex] (0,-3.023619131) -- (0,2.170406053);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_DessinLim.pstricks b/auto/pictures_tex/Fig_DessinLim.pstricks
index 5d8fce5cf..1b95a76d9 100644
--- a/auto/pictures_tex/Fig_DessinLim.pstricks
+++ b/auto/pictures_tex/Fig_DessinLim.pstricks
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.800000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.800000000);
+\draw [,->,>=latex] (-0.5000000000,0) -- (2.800000001,0);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.800000001);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_Differentielle.pstricks b/auto/pictures_tex/Fig_Differentielle.pstricks
index b865a7791..c47af80f4 100644
--- a/auto/pictures_tex/Fig_Differentielle.pstricks
+++ b/auto/pictures_tex/Fig_Differentielle.pstricks
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.700000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.700000000);
+\draw [,->,>=latex] (-0.5000000000,0) -- (4.700000003,0);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.700000003);
 %DEFAULT
 \draw [style=dotted] (2.00,2.00) -- (4.00,2.00);
 \draw [style=dotted] (4.00,2.00) -- (4.00,4.00);
diff --git a/auto/pictures_tex/Fig_DivergenceUn.pstricks b/auto/pictures_tex/Fig_DivergenceUn.pstricks
index d20b21eaf..a42173e46 100644
--- a/auto/pictures_tex/Fig_DivergenceUn.pstricks
+++ b/auto/pictures_tex/Fig_DivergenceUn.pstricks
@@ -66,10 +66,8 @@
 %PSTRICKS CODE
 %DEFAULT
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-\draw [] (1.60,-1.20) -- (3.07,-1.20);
-\draw [] (3.07,-1.20) -- (3.07,1.20);
-\draw [] (3.07,1.20) -- (1.60,1.20);
-\draw [] (1.60,1.20) -- (1.60,-1.20);
+
+
 \draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
 \draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
 \draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
diff --git a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
index 2d0b6e358..5f13b244a 100644
--- a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
+++ b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
@@ -80,7 +80,7 @@
 \draw []  (2.086833187,1.759539037) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.410240275,2.935347274);
 \draw []  (1.735039751,1.464349294) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246914);
+\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246913);
 \draw []  (2.657517092,2.437667613) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.247852509,3.098210238);
 \draw []  (0.9681152512,1.020091151) node [rotate=0] {$\bullet$};
@@ -96,7 +96,7 @@
 \draw []  (2.896085542,4.007090670) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.671786357,3.458304371);
 \draw []  (-0.6188917196,1.057674546) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357359,1.183270382);
+\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357358,1.183270382);
 \draw [] (-3.00,0) -- (3.00,0);
 \draw [] (0,0) -- (3.83,3.21);
 \draw (0.7745541682,0.2495862383) node {\( \alpha\)};
@@ -104,7 +104,7 @@
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 \draw []  (1.915111108,1.606969024) node [rotate=0] {$\bullet$};
 \draw (2.186183463,1.329052136) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.38)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.456,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.821)(0.211,0.825)(0.323,0.838)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
 \draw (4.139119438,3.522532237) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
index e519f6799..f7088ee5d 100644
--- a/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
+++ b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
@@ -65,7 +65,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.200000000) -- (0,1.199647580);
 %DEFAULT
 
@@ -151,7 +151,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
 %DEFAULT
 
@@ -239,7 +239,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (2.856194490,0);
+\draw [,->,>=latex] (-2.070796328,0) -- (2.856194490,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,1.499748271);
 %DEFAULT
 
@@ -329,7 +329,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
 %DEFAULT
 
@@ -415,7 +415,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.199998598) -- (0,1.199904439);
 %DEFAULT
 
@@ -502,7 +502,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.599557429,0) -- (4.898229715,0);
+\draw [,->,>=latex] (-1.599557429,0) -- (4.898229717,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
 %DEFAULT
 
@@ -588,7 +588,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_ExoXLVL.pstricks b/auto/pictures_tex/Fig_ExoXLVL.pstricks
index 5007c3cbe..b213578f7 100644
--- a/auto/pictures_tex/Fig_ExoXLVL.pstricks
+++ b/auto/pictures_tex/Fig_ExoXLVL.pstricks
@@ -82,22 +82,22 @@
 \draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
 \draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
 %DEFAULT
-\draw [color=blue] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red] (0,2.50) -- (2.50,2.50);
-\draw [color=red] (2.50,2.50) -- (2.50,0);
-\draw [color=red] (2.50,0) -- (0,0);
-\draw [color=red] (0,0) -- (0,2.50);
+\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
+\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
+\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
+\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
+\draw [color=red,style=dashed] (0,2.50) -- (2.50,2.50);
+\draw [color=red,style=dashed] (2.50,2.50) -- (2.50,0);
+\draw [color=red,style=dashed] (2.50,0) -- (0,0);
+\draw [color=red,style=dashed] (0,0) -- (0,2.50);
 \draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
 \draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
 \draw [color=cyan] (-2.50,0) -- (0,0);
 \draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green] (0.100,-0.100) -- (0.100,-2.50);
+\draw [color=green,style=dashed] (0.100,-2.50) -- (2.50,-2.50);
+\draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
+\draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
+\draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
 \draw (-1.099672167,1.300000000) node {\( xy\)};
 \draw (1.673347000,1.250000000) node {\( x-y\)};
 \draw (-0.9705055000,-1.250000000) node {\( x^2y\)};
diff --git a/auto/pictures_tex/Fig_FWJuNhU.pstricks b/auto/pictures_tex/Fig_FWJuNhU.pstricks
index ef8cb70ee..c44266c42 100644
--- a/auto/pictures_tex/Fig_FWJuNhU.pstricks
+++ b/auto/pictures_tex/Fig_FWJuNhU.pstricks
@@ -82,22 +82,22 @@
 \draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
 \draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
 %DEFAULT
-\draw [color=blue] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red] (0,2.50) -- (2.50,2.50);
-\draw [color=red] (2.50,2.50) -- (2.50,0);
-\draw [color=red] (2.50,0) -- (0,0);
-\draw [color=red] (0,0) -- (0,2.50);
+\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
+\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
+\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
+\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
+\draw [color=red,style=dashed] (0,2.50) -- (2.50,2.50);
+\draw [color=red,style=dashed] (2.50,2.50) -- (2.50,0);
+\draw [color=red,style=dashed] (2.50,0) -- (0,0);
+\draw [color=red,style=dashed] (0,0) -- (0,2.50);
 \draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
 \draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
 \draw [color=cyan] (-2.50,0) -- (0,0);
 \draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green] (0.100,-0.100) -- (0.100,-2.50);
+\draw [color=green,style=dashed] (0.100,-2.50) -- (2.50,-2.50);
+\draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
+\draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
+\draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
 \draw (-1.099672167,1.300000000) node {\( xy\)};
 \draw (1.673347000,1.250000000) node {\( x-y\)};
 \draw (-0.9705055000,-1.250000000) node {\( x^2y\)};
diff --git a/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks b/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
index 68cd080cf..85c01337c 100644
--- a/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
+++ b/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
@@ -120,7 +120,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525262) -- (0,4.054798491);
+\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_IntegraleSimple.pstricks b/auto/pictures_tex/Fig_IntegraleSimple.pstricks
index a4e8199e1..5f46e6eb9 100644
--- a/auto/pictures_tex/Fig_IntegraleSimple.pstricks
+++ b/auto/pictures_tex/Fig_IntegraleSimple.pstricks
@@ -71,7 +71,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (6.783185307,0);
+\draw [,->,>=latex] (-2.070796328,0) -- (6.783185311,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
 %DEFAULT
 \draw []  (-1.570796327,0) node [rotate=0] {$\bullet$};
diff --git a/auto/pictures_tex/Fig_JJAooWpimYW.pstricks b/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
index 55e861180..00789f209 100644
--- a/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
+++ b/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
@@ -108,7 +108,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981634,0) -- (8.353981634,0);
+\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
 \draw [,->,>=latex] (0,-1.498867339) -- (0,1.499874128);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks b/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
index 17cebb6a9..8d281f24f 100644
--- a/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
+++ b/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
@@ -92,7 +92,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.212388980,0) -- (5.212388980,0);
+\draw [,->,>=latex] (-5.212388985,0) -- (5.212388985,0);
 \draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_KKRooHseDzC.pstricks b/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
index f12ee2ecf..597246ba4 100644
--- a/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
+++ b/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
@@ -100,10 +100,10 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.82) -- (6.00,2.82) -- (5.00,2.82) -- (5.00,2.82) -- (5.00,0) --  cycle;
-\draw [color=red] (5.00,0) -- (6.00,0);
-\draw [color=red] (6.00,0) -- (6.00,2.82);
-\draw [color=red] (6.00,2.82) -- (5.00,2.82);
-\draw [color=red] (5.00,2.82) -- (5.00,0);
+\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.82);
+\draw [color=red,style=dashed] (6.00,2.82) -- (5.00,2.82);
+\draw [color=red,style=dashed] (5.00,2.82) -- (5.00,0);
 \draw []  (5.000000000,2.819444444) node [rotate=0] {$\bullet$};
 \draw (5.441978850,3.392374269) node {$f(x)$};
 \draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
diff --git a/auto/pictures_tex/Fig_MethodeNewton.pstricks b/auto/pictures_tex/Fig_MethodeNewton.pstricks
index 3fdca5746..23469e25c 100644
--- a/auto/pictures_tex/Fig_MethodeNewton.pstricks
+++ b/auto/pictures_tex/Fig_MethodeNewton.pstricks
@@ -96,7 +96,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-2.000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-1.287500000) -- (0,4.400000000);
+\draw [,->,>=latex] (0,-1.287500000) -- (0,4.400000002);
 %DEFAULT
 
 \draw [color=blue] (-1.500,3.900)--(-1.424,3.713)--(-1.348,3.529)--(-1.273,3.349)--(-1.197,3.173)--(-1.121,3.001)--(-1.045,2.833)--(-0.9697,2.668)--(-0.8939,2.507)--(-0.8182,2.350)--(-0.7424,2.197)--(-0.6667,2.048)--(-0.5909,1.903)--(-0.5152,1.761)--(-0.4394,1.623)--(-0.3636,1.490)--(-0.2879,1.359)--(-0.2121,1.233)--(-0.1364,1.111)--(-0.06061,0.9921)--(0.01515,0.8773)--(0.09091,0.7664)--(0.1667,0.6593)--(0.2424,0.5560)--(0.3182,0.4565)--(0.3939,0.3608)--(0.4697,0.2690)--(0.5455,0.1810)--(0.6212,0.09682)--(0.6970,0.01647)--(0.7727,-0.06006)--(0.8485,-0.1328)--(0.9242,-0.2016)--(1.000,-0.2667)--(1.076,-0.3279)--(1.152,-0.3853)--(1.227,-0.4388)--(1.303,-0.4886)--(1.379,-0.5345)--(1.455,-0.5766)--(1.530,-0.6148)--(1.606,-0.6493)--(1.682,-0.6799)--(1.758,-0.7067)--(1.833,-0.7296)--(1.909,-0.7488)--(1.985,-0.7641)--(2.061,-0.7755)--(2.136,-0.7832)--(2.212,-0.7870)--(2.288,-0.7870)--(2.364,-0.7832)--(2.439,-0.7755)--(2.515,-0.7641)--(2.591,-0.7488)--(2.667,-0.7296)--(2.742,-0.7067)--(2.818,-0.6799)--(2.894,-0.6493)--(2.970,-0.6148)--(3.045,-0.5766)--(3.121,-0.5345)--(3.197,-0.4886)--(3.273,-0.4388)--(3.348,-0.3853)--(3.424,-0.3279)--(3.500,-0.2667)--(3.576,-0.2016)--(3.652,-0.1328)--(3.727,-0.06006)--(3.803,0.01647)--(3.879,0.09682)--(3.955,0.1810)--(4.030,0.2690)--(4.106,0.3608)--(4.182,0.4565)--(4.258,0.5560)--(4.333,0.6593)--(4.409,0.7664)--(4.485,0.8773)--(4.561,0.9921)--(4.636,1.111)--(4.712,1.233)--(4.788,1.359)--(4.864,1.490)--(4.939,1.623)--(5.015,1.761)--(5.091,1.903)--(5.167,2.048)--(5.242,2.197)--(5.318,2.350)--(5.394,2.507)--(5.470,2.668)--(5.545,2.833)--(5.621,3.001)--(5.697,3.173)--(5.773,3.349)--(5.849,3.529)--(5.924,3.713)--(6.000,3.900);
diff --git a/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks b/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
index 47759ee2f..152de5dbe 100644
--- a/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
+++ b/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
@@ -117,7 +117,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-4.900000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-2.813086866) -- (0,3.699326205);
+\draw [,->,>=latex] (0,-2.813086867) -- (0,3.699326205);
 %DEFAULT
 
 \draw [color=blue] (-4.4000,-2.3131)--(-4.3051,-1.7786)--(-4.2101,-1.3073)--(-4.1152,-0.89319)--(-4.0202,-0.53075)--(-3.9253,-0.21496)--(-3.8303,0.058716)--(-3.7354,0.29443)--(-3.6404,0.49595)--(-3.5455,0.66669)--(-3.4505,0.80977)--(-3.3556,0.92802)--(-3.2606,1.0240)--(-3.1657,1.1001)--(-3.0707,1.1584)--(-2.9758,1.2009)--(-2.8808,1.2293)--(-2.7859,1.2452)--(-2.6909,1.2501)--(-2.5960,1.2452)--(-2.5010,1.2319)--(-2.4061,1.2112)--(-2.3111,1.1841)--(-2.2162,1.1515)--(-2.1212,1.1143)--(-2.0263,1.0731)--(-1.9313,1.0287)--(-1.8364,0.98162)--(-1.7414,0.93253)--(-1.6465,0.88190)--(-1.5515,0.83018)--(-1.4566,0.77782)--(-1.3616,0.72520)--(-1.2667,0.67266)--(-1.1717,0.62052)--(-1.0768,0.56908)--(-0.98182,0.51859)--(-0.88687,0.46930)--(-0.79192,0.42143)--(-0.69697,0.37518)--(-0.60202,0.33071)--(-0.50707,0.28821)--(-0.41212,0.24782)--(-0.31717,0.20966)--(-0.22222,0.17388)--(-0.12727,0.14058)--(-0.032323,0.10985)--(0.062626,0.081805)--(0.15758,0.056515)--(0.25253,0.034060)--(0.34747,0.014511)--(0.44242,-0.0020696)--(0.53737,-0.015623)--(0.63232,-0.026098)--(0.72727,-0.033446)--(0.82222,-0.037624)--(0.91717,-0.038593)--(1.0121,-0.036315)--(1.1071,-0.030760)--(1.2020,-0.021896)--(1.2970,-0.0096977)--(1.3919,0.0058604)--(1.4869,0.024800)--(1.5818,0.047142)--(1.6768,0.072905)--(1.7717,0.10211)--(1.8667,0.13476)--(1.9616,0.17088)--(2.0566,0.21048)--(2.1515,0.25357)--(2.2465,0.30016)--(2.3414,0.35026)--(2.4364,0.40388)--(2.5313,0.46103)--(2.6263,0.52171)--(2.7212,0.58593)--(2.8162,0.65370)--(2.9111,0.72503)--(3.0061,0.79991)--(3.1010,0.87835)--(3.1960,0.96035)--(3.2909,1.0459)--(3.3859,1.1351)--(3.4808,1.2278)--(3.5758,1.3241)--(3.6707,1.4240)--(3.7657,1.5275)--(3.8606,1.6345)--(3.9556,1.7451)--(4.0505,1.8594)--(4.1455,1.9772)--(4.2404,2.0986)--(4.3354,2.2236)--(4.4303,2.3522)--(4.5253,2.4844)--(4.6202,2.6202)--(4.7151,2.7596)--(4.8101,2.9026)--(4.9051,3.0491)--(5.0000,3.1993);
diff --git a/auto/pictures_tex/Fig_Polirettangolo.pstricks b/auto/pictures_tex/Fig_Polirettangolo.pstricks
index f41ef73c0..de9a6ea44 100644
--- a/auto/pictures_tex/Fig_Polirettangolo.pstricks
+++ b/auto/pictures_tex/Fig_Polirettangolo.pstricks
@@ -94,38 +94,14 @@
 \draw [style=dotted] (1.50,2.00) -- (1.50,1.00);
 \draw [style=dotted] (1.50,1.00) -- (0,1.00);
 \draw [style=dotted] (0,1.00) -- (0,2.00);
-
-% declaring the keys in tikz
-\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
-         hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
-% setting the default values
-\tikzset{hatchspread=3pt,
-         hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.500,2.50) -- (2.00,2.50) -- (2.00,2.50) -- (2.00,2.00) -- (2.00,2.00) -- (0.500,2.00) -- (0.500,2.00) -- (0.500,2.50) --  cycle;
 \draw [style=dotted] (0.500,2.50) -- (2.00,2.50);
 \draw [style=dotted] (2.00,2.50) -- (2.00,2.00);
 \draw [style=dotted] (2.00,2.00) -- (0.500,2.00);
 \draw [style=dotted] (0.500,2.00) -- (0.500,2.50);
-
-% declaring the keys in tikz
-\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
-         hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
-% setting the default values
-\tikzset{hatchspread=3pt,
-         hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,1.00) -- (3.00,1.00) -- (3.00,1.00) -- (3.00,0) -- (3.00,0) -- (2.00,0) -- (2.00,0) -- (2.00,1.00) --  cycle;
 \draw [style=dotted] (2.00,1.00) -- (3.00,1.00);
 \draw [style=dotted] (3.00,1.00) -- (3.00,0);
 \draw [style=dotted] (3.00,0) -- (2.00,0);
 \draw [style=dotted] (2.00,0) -- (2.00,1.00);
-
-% declaring the keys in tikz
-\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
-         hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
-% setting the default values
-\tikzset{hatchspread=3pt,
-         hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,3.50) -- (3.50,3.50) -- (3.50,3.50) -- (3.50,1.50) -- (3.50,1.50) -- (2.00,1.50) -- (2.00,1.50) -- (2.00,3.50) --  cycle;
 \draw [style=dotted] (2.00,3.50) -- (3.50,3.50);
 \draw [style=dotted] (3.50,3.50) -- (3.50,1.50);
 \draw [style=dotted] (3.50,1.50) -- (2.00,1.50);
diff --git a/auto/pictures_tex/Fig_RLuqsrr.pstricks b/auto/pictures_tex/Fig_RLuqsrr.pstricks
index cdeaa0644..ae4cd3b5d 100644
--- a/auto/pictures_tex/Fig_RLuqsrr.pstricks
+++ b/auto/pictures_tex/Fig_RLuqsrr.pstricks
@@ -87,7 +87,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185307,0);
+\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185311,0);
 \draw [,->,>=latex] (0,-0.9138130496) -- (0,2.914169059);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_SenoTopologo.pstricks b/auto/pictures_tex/Fig_SenoTopologo.pstricks
index 6cb4cdd52..64c749052 100644
--- a/auto/pictures_tex/Fig_SenoTopologo.pstricks
+++ b/auto/pictures_tex/Fig_SenoTopologo.pstricks
@@ -64,7 +64,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-1.586160392) -- (0,2.773243567);
+\draw [,->,>=latex] (0,-1.586160393) -- (0,2.773243568);
 %DEFAULT
 
 \draw [color=blue] 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diff --git a/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks b/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
index b6179f2bd..92ab9d915 100644
--- a/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
+++ b/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
@@ -66,10 +66,8 @@
 %PSTRICKS CODE
 %DEFAULT
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
-\draw [] (1.60,-1.20) -- (3.07,-1.20);
-\draw [] (3.07,-1.20) -- (3.07,1.20);
-\draw [] (3.07,1.20) -- (1.60,1.20);
-\draw [] (1.60,1.20) -- (1.60,-1.20);
+
+
 \draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
 \draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
 \draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
diff --git a/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks b/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
index f19ce3bcb..92b9e2f6a 100644
--- a/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
+++ b/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
@@ -89,7 +89,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-2.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.6113094300) -- (0,2.527798435);
+\draw [,->,>=latex] (0,-0.6113094300) -- (0,2.527798437);
 %DEFAULT
 
 \draw [color=blue] (-2.000,1.127)--(-1.929,1.000)--(-1.859,0.8813)--(-1.788,0.7700)--(-1.717,0.6663)--(-1.646,0.5699)--(-1.576,0.4807)--(-1.505,0.3985)--(-1.434,0.3231)--(-1.364,0.2544)--(-1.293,0.1922)--(-1.222,0.1363)--(-1.152,0.08657)--(-1.081,0.04284)--(-1.010,0.004948)--(-0.9394,-0.02729)--(-0.8687,-0.05404)--(-0.7980,-0.07546)--(-0.7273,-0.09173)--(-0.6566,-0.1030)--(-0.5859,-0.1095)--(-0.5152,-0.1113)--(-0.4444,-0.1087)--(-0.3737,-0.1017)--(-0.3030,-0.09057)--(-0.2323,-0.07548)--(-0.1616,-0.05658)--(-0.09091,-0.03405)--(-0.02020,-0.008044)--(0.05051,0.02126)--(0.1212,0.05370)--(0.1919,0.08911)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5455,0.3047)--(0.6162,0.3544)--(0.6869,0.4058)--(0.7576,0.4589)--(0.8283,0.5134)--(0.8990,0.5692)--(0.9697,0.6261)--(1.040,0.6840)--(1.111,0.7426)--(1.182,0.8018)--(1.253,0.8615)--(1.323,0.9215)--(1.394,0.9815)--(1.465,1.042)--(1.535,1.101)--(1.606,1.161)--(1.677,1.219)--(1.747,1.277)--(1.818,1.335)--(1.889,1.391)--(1.960,1.445)--(2.030,1.499)--(2.101,1.551)--(2.172,1.601)--(2.242,1.649)--(2.313,1.695)--(2.384,1.739)--(2.455,1.780)--(2.525,1.819)--(2.596,1.855)--(2.667,1.888)--(2.737,1.918)--(2.808,1.945)--(2.879,1.968)--(2.949,1.988)--(3.020,2.004)--(3.091,2.016)--(3.162,2.024)--(3.232,2.028)--(3.303,2.027)--(3.374,2.022)--(3.444,2.011)--(3.515,1.996)--(3.586,1.976)--(3.657,1.951)--(3.727,1.920)--(3.798,1.883)--(3.869,1.841)--(3.939,1.792)--(4.010,1.738)--(4.081,1.677)--(4.151,1.610)--(4.222,1.536)--(4.293,1.455)--(4.364,1.368)--(4.434,1.273)--(4.505,1.171)--(4.576,1.062)--(4.646,0.9444)--(4.717,0.8194)--(4.788,0.6865)--(4.859,0.5454)--(4.929,0.3960)--(5.000,0.2381);
diff --git a/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks b/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
index 7657de6aa..42f5d021d 100644
--- a/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
+++ b/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
@@ -108,7 +108,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981634,0) -- (8.353981634,0);
+\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
 \draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_TangentSegment.pstricks b/auto/pictures_tex/Fig_TangentSegment.pstricks
index aa8ff1a14..49f2ac546 100644
--- a/auto/pictures_tex/Fig_TangentSegment.pstricks
+++ b/auto/pictures_tex/Fig_TangentSegment.pstricks
@@ -103,7 +103,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.703789489,0) -- (7.783185307,0);
+\draw [,->,>=latex] (-3.703789489,0) -- (7.783185311,0);
 \draw [,->,>=latex] (0,-3.068914101) -- (0,2.500000000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_TriangleUV.pstricks b/auto/pictures_tex/Fig_TriangleUV.pstricks
index 1a1899f26..f48a25681 100644
--- a/auto/pictures_tex/Fig_TriangleUV.pstricks
+++ b/auto/pictures_tex/Fig_TriangleUV.pstricks
@@ -90,9 +90,7 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,0) -- (0,3.00) -- (0,3.00) -- (3.00,0) -- (3.00,0) -- (0,0) --  cycle;
-\draw [] (0,0) -- (0,3.00);
-\draw [] (0,3.00) -- (3.00,0);
-\draw [] (3.00,0) -- (0,0);
+
 \draw [color=green,->,>=latex] (0,0) -- (1.000000000,0);
 \draw (1.000000000,-0.2059510000) node {\( e_u\)};
 \draw [color=red,->,>=latex] (0,0) -- (0,1.000000000);
diff --git a/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks b/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
index cd77c60af..fd2268f2a 100644
--- a/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
+++ b/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
@@ -120,7 +120,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525262) -- (0,4.054798491);
+\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
 %DEFAULT
 
 \draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
diff --git a/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks b/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
index 0b27d5051..bd5d82aaf 100644
--- a/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
+++ b/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
@@ -101,7 +101,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-5.212388980,0) -- (9.924777961,0);
-\draw [,->,>=latex] (0,-3.445045351) -- (0,3.317012041);
+\draw [,->,>=latex] (0,-3.445045352) -- (0,3.317012042);
 %DEFAULT
 
 \draw [color=black] (-4.712,0)--(-4.570,-0.1423)--(-4.427,-0.2817)--(-4.284,-0.4154)--(-4.141,-0.5406)--(-3.998,-0.6549)--(-3.856,-0.7558)--(-3.713,-0.8413)--(-3.570,-0.9096)--(-3.427,-0.9595)--(-3.284,-0.9898)--(-3.142,-1.000)--(-2.999,-0.9898)--(-2.856,-0.9595)--(-2.713,-0.9096)--(-2.570,-0.8413)--(-2.428,-0.7558)--(-2.285,-0.6549)--(-2.142,-0.5406)--(-1.999,-0.4154)--(-1.856,-0.2817)--(-1.714,-0.1423)--(-1.571,0)--(-1.428,0.1423)--(-1.285,0.2817)--(-1.142,0.4154)--(-0.9996,0.5406)--(-0.8568,0.6549)--(-0.7140,0.7558)--(-0.5712,0.8413)--(-0.4284,0.9096)--(-0.2856,0.9595)--(-0.1428,0.9898)--(0,1.000)--(0.1428,0.9898)--(0.2856,0.9595)--(0.4284,0.9096)--(0.5712,0.8413)--(0.7140,0.7558)--(0.8568,0.6549)--(0.9996,0.5406)--(1.142,0.4154)--(1.285,0.2817)--(1.428,0.1423)--(1.571,0)--(1.714,-0.1423)--(1.856,-0.2817)--(1.999,-0.4154)--(2.142,-0.5406)--(2.285,-0.6549)--(2.428,-0.7558)--(2.570,-0.8413)--(2.713,-0.9096)--(2.856,-0.9595)--(2.999,-0.9898)--(3.142,-1.000)--(3.284,-0.9898)--(3.427,-0.9595)--(3.570,-0.9096)--(3.713,-0.8413)--(3.856,-0.7558)--(3.998,-0.6549)--(4.141,-0.5406)--(4.284,-0.4154)--(4.427,-0.2817)--(4.570,-0.1423)--(4.712,0)--(4.855,0.1423)--(4.998,0.2817)--(5.141,0.4154)--(5.284,0.5406)--(5.426,0.6549)--(5.569,0.7558)--(5.712,0.8413)--(5.855,0.9096)--(5.998,0.9595)--(6.140,0.9898)--(6.283,1.000)--(6.426,0.9898)--(6.569,0.9595)--(6.712,0.9096)--(6.854,0.8413)--(6.997,0.7558)--(7.140,0.6549)--(7.283,0.5406)--(7.426,0.4154)--(7.568,0.2817)--(7.711,0.1423)--(7.854,0)--(7.997,-0.1423)--(8.140,-0.2817)--(8.282,-0.4154)--(8.425,-0.5406)--(8.568,-0.6549)--(8.711,-0.7558)--(8.854,-0.8413)--(8.996,-0.9096)--(9.139,-0.9595)--(9.282,-0.9898)--(9.425,-1.000);
diff --git a/auto/pictures_tex/Fig_WUYooCISzeB.pstricks b/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
index e22c05f29..bf8eb2063 100644
--- a/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
+++ b/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
@@ -62,7 +62,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.9918454936) -- (0,4.027599983);
+\draw [,->,>=latex] (0,-0.9918454937) -- (0,4.027599983);
 %DEFAULT
 
 \draw [color=red] (-0.8400,2.700)--(-0.8013,2.668)--(-0.7626,2.636)--(-0.7239,2.603)--(-0.6852,2.571)--(-0.6466,2.539)--(-0.6079,2.507)--(-0.5692,2.474)--(-0.5305,2.442)--(-0.4918,2.410)--(-0.4531,2.378)--(-0.4144,2.345)--(-0.3757,2.313)--(-0.3370,2.281)--(-0.2984,2.249)--(-0.2597,2.216)--(-0.2210,2.184)--(-0.1823,2.152)--(-0.1436,2.120)--(-0.1049,2.087)--(-0.06622,2.055)--(-0.02753,2.023)--(0.01116,1.991)--(0.04985,1.958)--(0.08854,1.926)--(0.1272,1.894)--(0.1659,1.862)--(0.2046,1.829)--(0.2433,1.797)--(0.2820,1.765)--(0.3207,1.733)--(0.3594,1.701)--(0.3980,1.668)--(0.4367,1.636)--(0.4754,1.604)--(0.5141,1.572)--(0.5528,1.539)--(0.5915,1.507)--(0.6302,1.475)--(0.6689,1.443)--(0.7076,1.410)--(0.7463,1.378)--(0.7849,1.346)--(0.8236,1.314)--(0.8623,1.281)--(0.9010,1.249)--(0.9397,1.217)--(0.9784,1.185)--(1.017,1.152)--(1.056,1.120)--(1.094,1.088)--(1.133,1.056)--(1.172,1.023)--(1.211,0.9912)--(1.249,0.9590)--(1.288,0.9268)--(1.327,0.8945)--(1.365,0.8623)--(1.404,0.8300)--(1.443,0.7978)--(1.481,0.7655)--(1.520,0.7333)--(1.559,0.7011)--(1.597,0.6688)--(1.636,0.6366)--(1.675,0.6043)--(1.713,0.5721)--(1.752,0.5399)--(1.791,0.5076)--(1.830,0.4754)--(1.868,0.4431)--(1.907,0.4109)--(1.946,0.3787)--(1.984,0.3464)--(2.023,0.3142)--(2.062,0.2819)--(2.100,0.2497)--(2.139,0.2175)--(2.178,0.1852)--(2.216,0.1530)--(2.255,0.1207)--(2.294,0.08849)--(2.332,0.05625)--(2.371,0.02401)--(2.410,-0.008233)--(2.449,-0.04047)--(2.487,-0.07271)--(2.526,-0.1050)--(2.565,-0.1372)--(2.603,-0.1694)--(2.642,-0.2017)--(2.681,-0.2339)--(2.719,-0.2662)--(2.758,-0.2984)--(2.797,-0.3306)--(2.835,-0.3629)--(2.874,-0.3951)--(2.913,-0.4274)--(2.952,-0.4596)--(2.990,-0.4918);
diff --git a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
index 1399869d5..4a92b4ea1 100644
--- a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
+++ b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
@@ -121,7 +121,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156037) -- (0,4.065868530);
+\draw [,->,>=latex] (0,-4.525156038) -- (0,4.065868530);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
diff --git a/lst_actu.py b/lst_actu.py
index 0f3130704..a6e2b959a 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -16,8 +16,7 @@
 myRequest.original_filename="mazhe.tex"
 
 myRequest.ok_filenames_list=["e_mazhe"]
-myRequest.ok_filenames_list.extend(["182_numerique"])
-myRequest.ok_filenames_list.extend(["<++>"])
+myRequest.ok_filenames_list.extend(["78_inversion_locale"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
diff --git a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
index 1399869d5..ac5ea3350 100644
--- a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
+++ b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
@@ -121,7 +121,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156037) -- (0,4.065868530);
+\draw [,->,>=latex] (0,-4.525156039) -- (0,4.065868530);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
diff --git a/src_phystricks/figures_mazhe.py b/src_phystricks/figures_mazhe.py
index 05db0eb19..93e25fa17 100755
--- a/src_phystricks/figures_mazhe.py
+++ b/src_phystricks/figures_mazhe.py
@@ -286,11 +286,9 @@ def append_picture(fun,number):
         figures_list_3.append(fun)
 
 
-
 append_picture(UUNEooCNVOOs,3)
 append_picture(NOCGooYRHLCn,3)
 append_picture(YQIDooBqpAdbIM,3)
-
 append_picture(SurfacePrimiteGeog,2)
 append_picture(XOLBooGcrjiwoU,2)
 append_picture(DynkinrjbHIu,2)
diff --git a/src_phystricks/phystricksBEHTooWsdrys.py b/src_phystricks/phystricksBEHTooWsdrys.py
index ed6a0eb66..5990ed1c3 100644
--- a/src_phystricks/phystricksBEHTooWsdrys.py
+++ b/src_phystricks/phystricksBEHTooWsdrys.py
@@ -17,11 +17,10 @@ def BEHTooWsdrys():
     cmx=F.pos_x[0]
     cMx=F.pos_x[2]
     rect=Rectangle( Point(cmx,My+0.2),Point(cMx,my-0.2) )
-    rect.parameters.filled()
-    rect.parameters.fill.color="cyan"
-    rect.parameters.style="none"
+    rect.filled()
+    rect.fill_parameters.color="cyan"
+    rect.edges_parameters.style="none"
 
     pspict.DrawGraphs(rect,F)
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksBQXKooPqSEMN.py b/src_phystricks/phystricksBQXKooPqSEMN.py
index 110d1d063..9c6793935 100644
--- a/src_phystricks/phystricksBQXKooPqSEMN.py
+++ b/src_phystricks/phystricksBQXKooPqSEMN.py
@@ -22,10 +22,10 @@ def BQXKooPqSEMN():                # ex SurfaceDerive  (February 2016)
         surface.parameters.hatched()
 	surface.parameters.hatch.color="blue"
 	rectangle=Rectangle(P,Q)
-	rectangle.parameters.hatched()
-	rectangle.parameters.hatch.color="red"
-	rectangle.parameters.color="red"
-	rectangle.parameters.style="dashed"
+	rectangle.hatched()
+	rectangle.hatch_parameters.color="red"
+	rectangle.edges_parameters.color="red"
+	rectangle.edges_parameters.style="dashed"
 	
 
 	pspict.DrawGraphs(surface,f,rectangle,P,Px,Q)
diff --git a/src_phystricks/phystricksCardioideexo.py b/src_phystricks/phystricksCardioideexo.py
index e598c0ab3..057221e0b 100644
--- a/src_phystricks/phystricksCardioideexo.py
+++ b/src_phystricks/phystricksCardioideexo.py
@@ -10,7 +10,7 @@ def Cardioideexo():
 	O=Point(0,0)
 	Q=curve.get_point(pi/4)
 	rect=Rectangle(O,P)
-	rect.parameters.color="lightgray"
+	rect.edges_parameters.color="lightgray"
 
 	C1=Circle(O,1)
 	C2=Circle(O,2)
@@ -19,7 +19,7 @@ def Cardioideexo():
 
 	seg=Segment(O,Q)
 	seg.parameters.style="dashed"
-	seg.parameters.color=rect.parameters.color
+	seg.parameters.color=rect.edges_parameters.color
 
 	pspict.DrawGraphs(C1,C2,curve,rect,seg,P,Q)
 	pspict.DrawDefaultAxes()
diff --git a/src_phystricks/phystricksDivergenceUn.py b/src_phystricks/phystricksDivergenceUn.py
index 6ed42ed6a..9f2730150 100644
--- a/src_phystricks/phystricksDivergenceUn.py
+++ b/src_phystricks/phystricksDivergenceUn.py
@@ -16,9 +16,9 @@ def DivergenceUn():
     cmx=F.pos_x[0]
     cMx=F.pos_x[2]
     rect=Rectangle( Point(cmx,My+0.2),Point(cMx,my-0.2) )
-    rect.parameters.filled()
-    rect.parameters.fill.color="cyan"
-    rect.parameters.style="none"
+    rect.filled()
+    rect.fill_parameters.color="cyan"
+    rect.edges_parameters.style="none"
 
     pspict.DrawGraphs(rect,F)
     pspict.dilatation(1)
diff --git a/src_phystricks/phystricksEJRsWXw.py b/src_phystricks/phystricksEJRsWXw.py
index 5811abdda..fccfd667e 100644
--- a/src_phystricks/phystricksEJRsWXw.py
+++ b/src_phystricks/phystricksEJRsWXw.py
@@ -10,8 +10,8 @@ def EJRsWXw():
     B=S+(2,2)
 
     trig=Polygon(S,A,B)
-    trig.parameters.hatched()
-    trig.parameters.hatch.color="green"
+    trig.hatched()
+    trig.hatch_parameters.color="green"
 
     s1=Segment(A,B)
     t1=Segment(S,A)
diff --git a/src_phystricks/phystricksExPolygone.py b/src_phystricks/phystricksExPolygone.py
index a2fbbea75..c3eaf0f04 100644
--- a/src_phystricks/phystricksExPolygone.py
+++ b/src_phystricks/phystricksExPolygone.py
@@ -8,9 +8,9 @@ def ExPolygone():
     D=Point(0,0)
 
     poly=Polygon( A,B,C,D )
-    poly.parameters.hatched()
-    poly.parameters.hatch.color="green"
-    poly.edge_model.parameters.color="blue"
+    poly.hatched()
+    poly.hatch_parameters.color="green"
+    poly.edges_parameters.color="blue"
 
     segments=[s.copy().dilatation(3.3) for s in poly.edges]
     for s in segments :
@@ -21,4 +21,3 @@ def ExPolygone():
     pspict.dilatation(1)
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksExoXLVL.py b/src_phystricks/phystricksExoXLVL.py
index 88c8aeefb..501c141b0 100644
--- a/src_phystricks/phystricksExoXLVL.py
+++ b/src_phystricks/phystricksExoXLVL.py
@@ -11,15 +11,15 @@ def ExoXLVL():
     C3=Rectangle( Point(0,0),Point(-l,-l) )
     C4=Rectangle( Point(dist,-dist),Point(l,-l) )
 
-    C1.parameters.color="blue"
-    C2.parameters.color="red"
-    C3.parameters.color="cyan"
-    C4.parameters.color="green"
-
-    C1.parameters.style="dashed"
-    C2.parameters.style=C1.parameters.style
-    C2.parameters.style=C1.parameters.style
-    C4.parameters.style=C1.parameters.style
+    C1.edges_parameters.color="blue"
+    C2.edges_parameters.color="red"
+    C3.edges_parameters.color="cyan"
+    C4.edges_parameters.color="green"
+
+    C1.edges_parameters.style="dashed"
+    C2.edges_parameters.style=C1.edges_parameters.style
+    C2.edges_parameters.style=C1.edges_parameters.style
+    C4.edges_parameters.style=C1.edges_parameters.style
 
     a1=C1.center()
     a1.parameters.symbol=""
diff --git a/src_phystricks/phystricksFCUEooTpEPFoeQ.py b/src_phystricks/phystricksFCUEooTpEPFoeQ.py
index 64b9aa5a9..45d0d0488 100644
--- a/src_phystricks/phystricksFCUEooTpEPFoeQ.py
+++ b/src_phystricks/phystricksFCUEooTpEPFoeQ.py
@@ -16,13 +16,13 @@ def FCUEooTpEPFoeQ():
     matrix.elements[5,2].text="0"
 
     squareDelta=matrix.square(   (1,1) , (2,2),pspict )
-    squareDelta.parameters.color="red"
+    squareDelta.edges_parameters.color="red"
     hD=squareDelta.edges[0].midpoint()
     hD.parameters.symbol=""
     hD.put_mark(0.1,angle=70,text="\( \Delta_k(A_2)\)",pspict=pspict)
 
     squareOmega=matrix.square(   (3,3) , (5,5),pspict )
-    squareOmega.parameters.color="blue"
+    squareOmega.edges_parameters.color="blue"
     hO=squareOmega.edges[2].midpoint()
     hO.parameters.symbol=""
     hO.put_mark(0.1,angle=-90,text="\( \Omega_{k+1}(A_2)\)",pspict=pspict)
diff --git a/src_phystricks/phystricksFWJuNhU.py b/src_phystricks/phystricksFWJuNhU.py
index 83faf8aeb..1bd124d24 100644
--- a/src_phystricks/phystricksFWJuNhU.py
+++ b/src_phystricks/phystricksFWJuNhU.py
@@ -13,15 +13,15 @@ def FWJuNhU():
     C3=Rectangle( Point(0,0),Point(-l,-l) )
     C4=Rectangle( Point(dist,-dist),Point(l,-l) )
 
-    C1.parameters.color="blue"
-    C2.parameters.color="red"
-    C3.parameters.color="cyan"
-    C4.parameters.color="green"
-
-    C1.parameters.style="dashed"
-    C2.parameters.style=C1.parameters.style
-    C2.parameters.style=C1.parameters.style
-    C4.parameters.style=C1.parameters.style
+    C1.edges_parameters.color="blue"
+    C2.edges_parameters.color="red"
+    C3.edges_parameters.color="cyan"
+    C4.edges_parameters.color="green"
+
+    C1.edges_parameters.style="dashed"
+    C2.edges_parameters.style=C1.edges_parameters.style
+    C2.edges_parameters.style=C1.edges_parameters.style
+    C4.edges_parameters.style=C1.edges_parameters.style
 
     a1=C1.center()
     a1.parameters.symbol=""
diff --git a/src_phystricks/phystricksIntDeuxCarres.py b/src_phystricks/phystricksIntDeuxCarres.py
index 19e48b709..0b07419e2 100644
--- a/src_phystricks/phystricksIntDeuxCarres.py
+++ b/src_phystricks/phystricksIntDeuxCarres.py
@@ -7,17 +7,16 @@ def IntDeuxCarres():
     rectangle1=Rectangle(Point(-c1,-c1),Point(c1,c1))
     rectangle2=Rectangle(Point(-c2,-c2),Point(c2,c2))
 
-    rectangle1.parameters.hatched()
-    rectangle1.parameters.hatch.color="green"
-    rectangle1.parameters.color="red"
+    rectangle1.hatched()
+    rectangle1.hatch_parameters.color="green"
+    rectangle1.edges_parameters.color="red"
 
-    rectangle2.parameters.color="red"
-    rectangle2.parameters.filled()
-    rectangle2.parameters.fill.color="white"
+    rectangle2.edges_parameters.color="red"
+    rectangle2.filled()
+    rectangle2.fill_parameters.color="white"
 
     pspict.DrawGraphs(rectangle1,rectangle2)
 
     pspict.dilatation(1)
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksIsomCarre.py b/src_phystricks/phystricksIsomCarre.py
index 7c86b8bdb..61370fafc 100644
--- a/src_phystricks/phystricksIsomCarre.py
+++ b/src_phystricks/phystricksIsomCarre.py
@@ -19,7 +19,7 @@ def IsomCarre():
     E.put_mark(0.1,45,"\( s\)",pspict=pspict)
     E.parameters.symbol=""
     Carre=Rectangle(A,C)
-    Carre.parameters.color="blue"
+    Carre.edges_parameters.color="blue"
 
     pspict.DrawGraphs(Carre,S,A,B,C,D,E)
     fig.conclude()
diff --git a/src_phystricks/phystricksKKRooHseDzC.py b/src_phystricks/phystricksKKRooHseDzC.py
index c82cf5025..96155ba44 100644
--- a/src_phystricks/phystricksKKRooHseDzC.py
+++ b/src_phystricks/phystricksKKRooHseDzC.py
@@ -26,10 +26,10 @@ def KKRooHseDzC():
     surface.parameters.hatch.color="blue"
 
     rectangle=Rectangle(P,Q)
-    rectangle.parameters.hatched()
-    rectangle.parameters.hatch.color="red"
-    rectangle.parameters.color="red"
-    rectangle.parameters.style="dashed"
+    rectangle.hatched()
+    rectangle.hatch_parameters.color="red"
+    rectangle.edges_parameters.color="red"
+    rectangle.edges_parameters.style="dashed"
 
     pspict.DrawGraphs(surface,f,rectangle,P,Px,Q)
     pspict.axes.no_graduation()
diff --git a/src_phystricks/phystricksNOCGooYRHLCn.py b/src_phystricks/phystricksNOCGooYRHLCn.py
index b6ea074d5..1261318ff 100644
--- a/src_phystricks/phystricksNOCGooYRHLCn.py
+++ b/src_phystricks/phystricksNOCGooYRHLCn.py
@@ -21,11 +21,12 @@ def NOCGooYRHLCn():
 	surface=SurfaceUnderFunction(f,0,x0)
 	surface.parameters.hatched()
 	surface.parameters.hatch.color="blue"
+
 	rectangle=Rectangle(P,Q)
-	rectangle.parameters.hatched()
-	rectangle.parameters.hatch.color="red"
-	rectangle.parameters.color="red"
-	rectangle.parameters.style="dashed"
+	rectangle.hatched()
+	rectangle.hatch_parameters.color="red"
+	rectangle.edges_parameters.color="red"
+	rectangle.edges_parameters.style="dashed"
 	
 
 	pspict.DrawGraphs(surface,f,rectangle,P,Px,Q)
@@ -34,4 +35,3 @@ def NOCGooYRHLCn():
 	pspict.dilatation(1)
 	fig.conclude()
 	fig.write_the_file()
-
diff --git a/src_phystricks/phystricksPolirettangolo.py b/src_phystricks/phystricksPolirettangolo.py
index bf52fbc51..a53359343 100644
--- a/src_phystricks/phystricksPolirettangolo.py
+++ b/src_phystricks/phystricksPolirettangolo.py
@@ -16,13 +16,15 @@ def Polirettangolo():
     R1=Rectangle(P2,P3)
     R2=Rectangle(P4,P5)
     R3=Rectangle(P6,P7)
-    R.parameters.hatched()
-    R.parameters.hatch.color="red"
+    R.hatched()
+    R.hatch_parameters.color="red"
+
     for rect in [R,R1,R2,R3]:
-        rect.edge_model.parameters.style="dotted"
-    R1.parameters=R.parameters.copy()
-    R2.parameters=R.parameters.copy()
-    R3.parameters=R.parameters.copy()
+        rect.edges_parameters.style="dotted"
+
+    for rect in [R1,R2,R3]:
+        rect.edges_parameters=R.edges_parameters.copy()
+        rect.hatch_parameters=R.hatch_parameters.copy()
 
     pspict.DrawGraphs(R,R1,R2,R3)
     
@@ -31,4 +33,3 @@ def Polirettangolo():
     pspict.DrawDefaultAxes()
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksTKXZooLwXzjS.py b/src_phystricks/phystricksTKXZooLwXzjS.py
index 68a1668a4..451f55302 100644
--- a/src_phystricks/phystricksTKXZooLwXzjS.py
+++ b/src_phystricks/phystricksTKXZooLwXzjS.py
@@ -18,11 +18,10 @@ def TKXZooLwXzjS():
     cmx=F.pos_x[0]
     cMx=F.pos_x[2]
     rect=Rectangle( Point(cmx,My+0.2),Point(cMx,my-0.2) )
-    rect.parameters.filled()
-    rect.parameters.fill.color="cyan"
-    rect.parameters.style="none"
+    rect.filled()
+    rect.fill_parameters.color="cyan"
+    rect.edges_parameters.style="none"
 
     pspict.DrawGraphs(rect,F)
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksTriangleUV.py b/src_phystricks/phystricksTriangleUV.py
index 72f7c0a6c..2778d2048 100644
--- a/src_phystricks/phystricksTriangleUV.py
+++ b/src_phystricks/phystricksTriangleUV.py
@@ -32,9 +32,9 @@ def TriangleUV():
     t.put_mark(0.1,n.advised_mark_angle(pspict),"\( T\)",pspict=pspict)
 
     Trig=Polygon(O,A,B)
-    Trig.parameters.style="none"
-    Trig.parameters.hatched()
-    Trig.parameters.hatch.color="blue"
+    Trig.edges_parameters.style="none"
+    Trig.hatched()
+    Trig.hatch_parameters.color="blue"
 
     pspict.axes.no_graduation()
     pspict.DrawGraphs(Trig,u,v,n,t)
diff --git a/src_phystricks/phystricksUneCellule.py b/src_phystricks/phystricksUneCellule.py
index 88e5e9e33..dbed6e649 100644
--- a/src_phystricks/phystricksUneCellule.py
+++ b/src_phystricks/phystricksUneCellule.py
@@ -52,8 +52,8 @@ def Sigma(dep,leng):
 
     cellule=Rectangle(Point(sigma1[3],sigma2[1]),
                                         Point(sigma1[4],sigma2[2])  )
-    cellule.parameters.filled()
-    cellule.parameters.fill.color="lightgray"
+    cellule.filled()
+    cellule.fill_parameters.color="lightgray"
     pspict.DrawGraphs(cellule)
 
     pspict.axes.no_graduation()
diff --git a/src_phystricks/phystricksVWFLooPSrOqz.py b/src_phystricks/phystricksVWFLooPSrOqz.py
index b1e3401f7..5ae456260 100644
--- a/src_phystricks/phystricksVWFLooPSrOqz.py
+++ b/src_phystricks/phystricksVWFLooPSrOqz.py
@@ -8,9 +8,9 @@ def VWFLooPSrOqz():
     D=Point(0,0)
 
     poly=Polygon( A,B,C,D )
-    poly.parameters.hatched()
-    poly.parameters.hatch.color="green"
-    poly.edge_model.parameters.color="blue"
+    poly.hatched()
+    poly.hatch_parameters.color="green"
+    poly.edges_parameters.color="blue"
 
     segments=[s.copy().dilatation(3.3) for s in poly.edges]
     for s in segments :
@@ -21,4 +21,3 @@ def VWFLooPSrOqz():
     pspict.dilatation(1)
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksZTTooXtHkci.py b/src_phystricks/phystricksZTTooXtHkci.py
index 7671d069c..79dde6599 100644
--- a/src_phystricks/phystricksZTTooXtHkci.py
+++ b/src_phystricks/phystricksZTTooXtHkci.py
@@ -18,8 +18,8 @@ def ZTTooXtHkci():
     D=Point(0,-1)
 
     regV=Polygon(A,B,C,D)
-    regV.parameters.hatched()
-    regV.parameters.hatch.color="red"
+    regV.hatched()
+    regV.hatch_parameters.color="red"
 
     K=Point(-2,2)
     L=Point(2,2)
@@ -27,8 +27,8 @@ def ZTTooXtHkci():
     N=Point(-1,1)
 
     regU=Polygon(K,L,M,N)
-    regU.parameters.hatched()
-    regU.parameters.hatch.color="blue"
+    regU.hatched()
+    regU.hatch_parameters.color="blue"
 
     pspicts[0].DrawGraphs(regV)
     pspicts[1].DrawGraphs(regU)

From 73e8ec5f13f8e7315278d415efddf2202d524293 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 15 Jun 2017 23:42:51 +0200
Subject: [PATCH 16/64] (pictures) Provide some new '.pstricks' files.

---
 auto/pictures_tex/Fig_ADUGmRRA.pstricks       |   6 +-
 auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks   |   8 +-
 auto/pictures_tex/Fig_ASHYooUVHkak.pstricks   |  16 +--
 .../Fig_AccumulationIsole.pstricks            |  12 +-
 auto/pictures_tex/Fig_AdhIntFr.pstricks       |  24 ++--
 auto/pictures_tex/Fig_BIFooDsvVHb.pstricks    |  14 +--
 auto/pictures_tex/Fig_BiaisOuPas.pstricks     |  26 ++---
 auto/pictures_tex/Fig_BoulePtLoin.pstricks    |  12 +-
 auto/pictures_tex/Fig_CFMooGzvfRP.pstricks    |  12 +-
 auto/pictures_tex/Fig_CSCii.pstricks          |  14 +--
 auto/pictures_tex/Fig_CSCiii.pstricks         |  32 ++---
 auto/pictures_tex/Fig_CSCiv.pstricks          |  24 ++--
 auto/pictures_tex/Fig_CSCv.pstricks           |  32 ++---
 auto/pictures_tex/Fig_Cardioideexo.pstricks   |  24 ++--
 auto/pictures_tex/Fig_CbCartTuii.pstricks     |   6 +-
 auto/pictures_tex/Fig_CercleTnu.pstricks      |  16 +--
 auto/pictures_tex/Fig_CercleTrigono.pstricks  |  26 ++---
 auto/pictures_tex/Fig_CheminFresnel.pstricks  |  16 +--
 auto/pictures_tex/Fig_ConeRevolution.pstricks |  14 +--
 auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks |  40 +++----
 auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks |  28 ++---
 auto/pictures_tex/Fig_DTIYKkP.pstricks        |  12 +-
 .../Fig_DefinitionCartesiennes.pstricks       |  42 +++----
 auto/pictures_tex/Fig_DisqueConv.pstricks     |   8 +-
 .../pictures_tex/Fig_DistanceEuclide.pstricks |  30 ++---
 auto/pictures_tex/Fig_DynkinpWjUbE.pstricks   |   4 +-
 auto/pictures_tex/Fig_DynkinrjbHIu.pstricks   |   4 +-
 auto/pictures_tex/Fig_ExoCUd.pstricks         |  20 ++--
 auto/pictures_tex/Fig_ExoParamCD.pstricks     |  26 ++---
 .../Fig_ExoUnSurxPolaire.pstricks             |  40 +++----
 auto/pictures_tex/Fig_FnCosApprox.pstricks    |  20 ++--
 auto/pictures_tex/Fig_GBnUivi.pstricks        |  22 ++--
 auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks   |  18 +--
 auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks   |  36 +++---
 .../Fig_Grapheunsurunmoinsx.pstricks          |  44 +++----
 auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks   |  24 ++--
 auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks   |   8 +-
 auto/pictures_tex/Fig_HLJooGDZnqF.pstricks    |  42 +++----
 auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks |  24 ++--
 auto/pictures_tex/Fig_IntEcourbe.pstricks     |  18 +--
 auto/pictures_tex/Fig_IntTrois.pstricks       |  28 ++---
 auto/pictures_tex/Fig_IntervalleUn.pstricks   |  22 ++--
 auto/pictures_tex/Fig_IsomCarre.pstricks      |  18 +--
 auto/pictures_tex/Fig_LAfWmaN.pstricks        |  22 ++--
 auto/pictures_tex/Fig_LBGooAdteCt.pstricks    |  28 ++---
 auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks   |  12 +-
 auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks |  28 ++---
 auto/pictures_tex/Fig_Laurin.pstricks         |  28 ++---
 auto/pictures_tex/Fig_MCQueGF.pstricks        |  28 ++---
 auto/pictures_tex/Fig_Mantisse.pstricks       |  16 +--
 auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks   |  24 ++--
 .../Fig_NiveauHyperboleDeux.pstricks          |  36 +++---
 .../Fig_ProjectionScalaire.pstricks           |  20 ++--
 auto/pictures_tex/Fig_QIZooQNQSJj.pstricks    |   8 +-
 auto/pictures_tex/Fig_QPcdHwP.pstricks        |   6 +-
 auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks |   6 +-
 auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks    |  20 ++--
 auto/pictures_tex/Fig_Refraction.pstricks     |  12 +-
 auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks |  22 ++--
 auto/pictures_tex/Fig_SolsSinpA.pstricks      |  16 +--
 .../Fig_SuiteInverseAlterne.pstricks          |  44 +++----
 .../Fig_SurfaceEntreCourbes.pstricks          |  36 +++---
 .../Fig_SurfacePrimiteGeog.pstricks           |  12 +-
 auto/pictures_tex/Fig_TZCISko.pstricks        |  10 +-
 .../pictures_tex/Fig_TgCercleTrigono.pstricks |  20 ++--
 auto/pictures_tex/Fig_TracerUn.pstricks       | 102 ++++++++--------
 auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks |  40 +++----
 auto/pictures_tex/Fig_UGCFooQoCihh.pstricks   |  42 +++----
 auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks   |  28 ++---
 auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks   |  40 +++----
 auto/pictures_tex/Fig_UYJooCWjLgK.pstricks    |  18 +--
 auto/pictures_tex/Fig_UneCellule.pstricks     |  44 +++----
 auto/pictures_tex/Fig_VANooZowSyO.pstricks    | 110 +++++++++---------
 auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks   |  22 ++--
 auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks   |  12 +-
 auto/pictures_tex/Fig_XJMooCQTlNL.pstricks    |  28 ++---
 auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks |  34 +++---
 auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks |  16 +--
 auto/pictures_tex/Fig_examssepti.pstricks     |  14 +--
 auto/pictures_tex/Fig_trigoWedd.pstricks      |  16 +--
 80 files changed, 966 insertions(+), 966 deletions(-)

diff --git a/auto/pictures_tex/Fig_ADUGmRRA.pstricks b/auto/pictures_tex/Fig_ADUGmRRA.pstricks
index ade88656d..3eafef08f 100644
--- a/auto/pictures_tex/Fig_ADUGmRRA.pstricks
+++ b/auto/pictures_tex/Fig_ADUGmRRA.pstricks
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.2059510000) node {\( \alpha_i\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2191818333) node {\( \alpha_{i+1}\)};
+\draw (0,0.20595) node {\( \alpha_i\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks b/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
index 914f91169..6fbdb6220 100644
--- a/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
+++ b/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
@@ -89,14 +89,14 @@
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-\draw (0.6519509490,0.1894063750) node {$\theta$};
+\draw (0.65195,0.18941) node {$\theta$};
 
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 \draw [] (0,0) -- (1.97,-0.347);
 \draw [] (0,0) -- (1.00,1.73);
-\draw (0.1002123789,1.140733404) node {$R$};
-\draw (2.429897999,-0.5535014753) node {$\theta_0$};
-\draw (1.314840167,2.145969095) node {$\theta_1$};
+\draw (0.10021,1.1407) node {$R$};
+\draw (2.4299,-0.55350) node {$\theta_0$};
+\draw (1.3148,2.1460) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks b/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks
index a50600833..4e8788b1b 100644
--- a/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks
+++ b/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks
@@ -72,18 +72,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.400000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.900000000);
+\draw [,->,>=latex] (-2.4000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.9000);
 %DEFAULT
 \draw [style=dashed] (-1.90,1.20) -- (3.00,1.20);
-\draw []  (0,1.200000000) node [rotate=0] {$\bullet$};
-\draw (0.2294391896,1.468157356) node {\( \delta\)};
+\draw []  (0,1.2000) node [rotate=0] {$\bullet$};
+\draw (0.22944,1.4682) node {\( \delta\)};
 \draw [] (-0.300,1.40) -- (0.300,1.00);
 \draw [] (1.70,1.40) -- (2.30,1.00);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3396268333) node {\( t_1\)};
-\draw []  (-1.500000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.500000000,-0.3396268333) node {\( t_2\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.33963) node {\( t_1\)};
+\draw []  (-1.5000,0) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.33963) node {\( t_2\)};
 \draw [style=dotted] (2.00,0) -- (2.00,1.20);
 \draw [style=dotted] (-1.50,0) -- (-1.50,1.20);
 \draw [] (-1.80,1.40) -- (-1.20,1.00);
diff --git a/auto/pictures_tex/Fig_AccumulationIsole.pstricks b/auto/pictures_tex/Fig_AccumulationIsole.pstricks
index a597d68e8..3b674d22b 100644
--- a/auto/pictures_tex/Fig_AccumulationIsole.pstricks
+++ b/auto/pictures_tex/Fig_AccumulationIsole.pstricks
@@ -79,12 +79,12 @@
 %DEFAULT
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 \draw [color=red] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.542514833,1.000000000) node {$P$};
-\draw [color=red]  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.544276167,0) node {$Q$};
-\draw []  (0.5000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (0.1654411323,0.1631599656) node {$S$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.5425,1.0000) node {$P$};
+\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-1.5443,0) node {$Q$};
+\draw []  (0.50000,0.50000) node [rotate=0] {$\bullet$};
+\draw (0.16544,0.16316) node {$S$};
 
 \draw [color=black] (1.200,1.000)--(1.200,1.013)--(1.198,1.025)--(1.196,1.038)--(1.194,1.050)--(1.190,1.062)--(1.186,1.074)--(1.181,1.086)--(1.175,1.097)--(1.168,1.108)--(1.161,1.119)--(1.153,1.129)--(1.145,1.138)--(1.136,1.147)--(1.126,1.155)--(1.116,1.163)--(1.105,1.170)--(1.094,1.176)--(1.083,1.182)--(1.071,1.187)--(1.059,1.191)--(1.047,1.194)--(1.035,1.197)--(1.022,1.199)--(1.010,1.200)--(0.9968,1.200)--(0.9842,1.199)--(0.9715,1.198)--(0.9590,1.196)--(0.9467,1.193)--(0.9346,1.189)--(0.9227,1.184)--(0.9112,1.179)--(0.9000,1.173)--(0.8892,1.167)--(0.8789,1.159)--(0.8690,1.151)--(0.8597,1.143)--(0.8509,1.133)--(0.8428,1.124)--(0.8353,1.113)--(0.8284,1.103)--(0.8222,1.092)--(0.8168,1.080)--(0.8121,1.068)--(0.8081,1.056)--(0.8049,1.044)--(0.8025,1.032)--(0.8009,1.019)--(0.8001,1.006)--(0.8001,0.9937)--(0.8009,0.9810)--(0.8025,0.9684)--(0.8049,0.9559)--(0.8081,0.9437)--(0.8121,0.9316)--(0.8168,0.9198)--(0.8222,0.9084)--(0.8284,0.8973)--(0.8353,0.8866)--(0.8428,0.8764)--(0.8509,0.8666)--(0.8597,0.8575)--(0.8690,0.8488)--(0.8789,0.8409)--(0.8892,0.8335)--(0.9000,0.8268)--(0.9112,0.8208)--(0.9227,0.8155)--(0.9346,0.8110)--(0.9467,0.8072)--(0.9590,0.8042)--(0.9715,0.8020)--(0.9842,0.8006)--(0.9968,0.8000)--(1.010,0.8002)--(1.022,0.8012)--(1.035,0.8030)--(1.047,0.8056)--(1.059,0.8090)--(1.071,0.8132)--(1.083,0.8181)--(1.094,0.8237)--(1.105,0.8301)--(1.116,0.8371)--(1.126,0.8448)--(1.136,0.8531)--(1.145,0.8620)--(1.153,0.8714)--(1.161,0.8814)--(1.168,0.8919)--(1.175,0.9028)--(1.181,0.9140)--(1.186,0.9257)--(1.190,0.9376)--(1.194,0.9498)--(1.196,0.9622)--(1.198,0.9747)--(1.200,0.9873)--(1.200,1.000);
 
diff --git a/auto/pictures_tex/Fig_AdhIntFr.pstricks b/auto/pictures_tex/Fig_AdhIntFr.pstricks
index 9da41f9c7..a1650bae7 100644
--- a/auto/pictures_tex/Fig_AdhIntFr.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFr.pstricks
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.000000000,0);
-\draw [,->,>=latex] (0,-0.7499836751) -- (0,4.250000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.0000,0);
+\draw [,->,>=latex] (0,-0.74998) -- (0,4.2500);
 %DEFAULT
 
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@@ -102,26 +102,26 @@
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 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
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-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
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diff --git a/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks b/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
index ce92d3c58..7ac338be2 100644
--- a/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
+++ b/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
@@ -72,18 +72,18 @@
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-\draw (-0.2960240000,0.7500000000) node {\( y\)};
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diff --git a/auto/pictures_tex/Fig_BiaisOuPas.pstricks b/auto/pictures_tex/Fig_BiaisOuPas.pstricks
index ab7e522b6..5a720b6dc 100644
--- a/auto/pictures_tex/Fig_BiaisOuPas.pstricks
+++ b/auto/pictures_tex/Fig_BiaisOuPas.pstricks
@@ -91,31 +91,31 @@
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-\draw [color=cyan,->,>=latex] (0,-0.5000000000) -- (1.250000000,-0.5000000000);
-\draw (0,-0.9247080000) node {\( I\)};
-\draw (-7.500000000,-0.3298256667) node {$ -3 $};
+\draw [color=cyan,->,>=latex] (0,-0.50000) -- (-1.2500,-0.50000);
+\draw [color=cyan,->,>=latex] (0,-0.50000) -- (1.2500,-0.50000);
+\draw (0,-0.92471) node {\( I\)};
+\draw (-7.5000,-0.32983) node {$ -3 $};
 \draw [] (-7.50,-0.100) -- (-7.50,0.100);
-\draw (-5.000000000,-0.3298256667) node {$ -2 $};
+\draw (-5.0000,-0.32983) node {$ -2 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-2.500000000,-0.3298256667) node {$ -1 $};
+\draw (-2.5000,-0.32983) node {$ -1 $};
 \draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (2.500000000,-0.3149246667) node {$ 1 $};
+\draw (2.5000,-0.31492) node {$ 1 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 2 $};
+\draw (5.0000,-0.31492) node {$ 2 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (7.500000000,-0.3149246667) node {$ 3 $};
+\draw (7.5000,-0.31492) node {$ 3 $};
 \draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (-0.2912498333,2.500000000) node {$ 1 $};
+\draw (-0.29125,2.5000) node {$ 1 $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.2912498333,5.000000000) node {$ 2 $};
+\draw (-0.29125,5.0000) node {$ 2 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_BoulePtLoin.pstricks b/auto/pictures_tex/Fig_BoulePtLoin.pstricks
index d7d8175e9..7e8a51e8e 100644
--- a/auto/pictures_tex/Fig_BoulePtLoin.pstricks
+++ b/auto/pictures_tex/Fig_BoulePtLoin.pstricks
@@ -79,14 +79,14 @@
 %DEFAULT
 \draw [style=dashed] (1.41,1.41) -- (1.94,1.94);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.3085997010,0.2907082010) node {$a$};
-\draw []  (1.414213562,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.050102275,1.642789729) node {$x$};
-\draw [,->,>=latex] (0,0) -- (1.414213562,1.414213562);
+\draw (-0.30860,0.29071) node {$a$};
+\draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.0501,1.6428) node {$x$};
+\draw [,->,>=latex] (0,0) -- (1.4142,1.4142);
 
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-\draw []  (1.944543648,1.944543648) node [rotate=0] {$\bullet$};
-\draw (1.944543648,1.519835648) node {$P$};
+\draw []  (1.9445,1.9445) node [rotate=0] {$\bullet$};
+\draw (1.9445,1.5198) node {$P$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks b/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
index f79c6c197..0eb3ad650 100644
--- a/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
+++ b/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
@@ -64,19 +64,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (2.00,0)--(2.00,0.127)--(1.98,0.253)--(1.96,0.379)--(1.94,0.502)--(1.90,0.624)--(1.86,0.743)--(1.81,0.860)--(1.75,0.972)--(1.68,1.08)--(1.61,1.19)--(1.53,1.29)--(1.45,1.38)--(1.36,1.47)--(1.26,1.55)--(1.16,1.63)--(1.05,1.70)--(0.945,1.76)--(0.831,1.82)--(0.714,1.87)--(0.594,1.91)--(0.472,1.94)--(0.347,1.97)--(0.222,1.99)--(0.0952,2.00)--(-0.0317,2.00)--(-0.158,1.99)--(-0.285,1.98)--(-0.410,1.96)--(-0.533,1.93)--(-0.654,1.89)--(-0.773,1.84)--(-0.888,1.79)--(-1.00,1.73)--(-1.11,1.67)--(-1.21,1.59)--(-1.31,1.51)--(-1.40,1.43)--(-1.49,1.33)--(-1.57,1.24)--(-1.65,1.13)--(-1.72,1.03)--(-1.78,0.916)--(-1.83,0.802)--(-1.88,0.684)--(-1.92,0.563)--(-1.95,0.441)--(-1.97,0.316)--(-1.99,0.190)--(-2.00,0.0635)--(-2.00,-0.0635)--(-1.99,-0.190)--(-1.97,-0.316)--(-1.95,-0.441)--(-1.92,-0.563)--(-1.88,-0.684)--(-1.83,-0.802)--(-1.78,-0.916)--(-1.72,-1.03)--(-1.65,-1.13)--(-1.57,-1.24)--(-1.49,-1.33)--(-1.40,-1.43)--(-1.31,-1.51)--(-1.21,-1.59)--(-1.11,-1.67)--(-1.00,-1.73)--(-0.888,-1.79)--(-0.773,-1.84)--(-0.654,-1.89)--(-0.533,-1.93)--(-0.410,-1.96)--(-0.285,-1.98)--(-0.158,-1.99)--(-0.0317,-2.00)--(0.0952,-2.00)--(0.222,-1.99)--(0.347,-1.97)--(0.472,-1.94)--(0.594,-1.91)--(0.714,-1.87)--(0.831,-1.82)--(0.945,-1.76)--(1.05,-1.70)--(1.16,-1.63)--(1.26,-1.55)--(1.36,-1.47)--(1.45,-1.38)--(1.53,-1.29)--(1.61,-1.19)--(1.68,-1.08)--(1.75,-0.972)--(1.81,-0.860)--(1.86,-0.743)--(1.90,-0.624)--(1.94,-0.502)--(1.96,-0.379)--(1.98,-0.253)--(2.00,-0.127)--(2.00,0);
 \draw [] (0,0) -- (1.73,1.00);
 \draw [] (0,0) -- (-1.73,1.00);
-\draw []  (1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (-1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.394880073,0.3118645226) node {\( \pi/6\)};
+\draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (-1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3949,0.31186) node {\( \pi/6\)};
 
 \draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw (-1.394880073,0.3118645226) node {\( \pi/6\)};
+\draw (-1.3949,0.31186) node {\( \pi/6\)};
 
 \draw [] (-0.433,0.250)--(-0.434,0.248)--(-0.436,0.245)--(-0.437,0.243)--(-0.438,0.241)--(-0.439,0.238)--(-0.441,0.236)--(-0.442,0.234)--(-0.443,0.231)--(-0.444,0.229)--(-0.446,0.227)--(-0.447,0.224)--(-0.448,0.222)--(-0.449,0.220)--(-0.450,0.217)--(-0.451,0.215)--(-0.453,0.213)--(-0.454,0.210)--(-0.455,0.208)--(-0.456,0.205)--(-0.457,0.203)--(-0.458,0.200)--(-0.459,0.198)--(-0.460,0.196)--(-0.461,0.193)--(-0.462,0.191)--(-0.463,0.188)--(-0.464,0.186)--(-0.465,0.183)--(-0.466,0.181)--(-0.467,0.178)--(-0.468,0.176)--(-0.469,0.173)--(-0.470,0.171)--(-0.471,0.169)--(-0.472,0.166)--(-0.473,0.164)--(-0.473,0.161)--(-0.474,0.159)--(-0.475,0.156)--(-0.476,0.154)--(-0.477,0.151)--(-0.477,0.148)--(-0.478,0.146)--(-0.479,0.143)--(-0.480,0.141)--(-0.480,0.138)--(-0.481,0.136)--(-0.482,0.133)--(-0.483,0.131)--(-0.483,0.128)--(-0.484,0.126)--(-0.485,0.123)--(-0.485,0.120)--(-0.486,0.118)--(-0.487,0.115)--(-0.487,0.113)--(-0.488,0.110)--(-0.488,0.108)--(-0.489,0.105)--(-0.489,0.102)--(-0.490,0.0998)--(-0.490,0.0972)--(-0.491,0.0946)--(-0.491,0.0920)--(-0.492,0.0894)--(-0.492,0.0868)--(-0.493,0.0842)--(-0.493,0.0816)--(-0.494,0.0790)--(-0.494,0.0764)--(-0.495,0.0738)--(-0.495,0.0712)--(-0.495,0.0685)--(-0.496,0.0659)--(-0.496,0.0633)--(-0.496,0.0607)--(-0.497,0.0580)--(-0.497,0.0554)--(-0.497,0.0528)--(-0.497,0.0502)--(-0.498,0.0475)--(-0.498,0.0449)--(-0.498,0.0423)--(-0.498,0.0396)--(-0.499,0.0370)--(-0.499,0.0344)--(-0.499,0.0317)--(-0.499,0.0291)--(-0.499,0.0264)--(-0.499,0.0238)--(-0.500,0.0211)--(-0.500,0.0185)--(-0.500,0.0159)--(-0.500,0.0132)--(-0.500,0.0106)--(-0.500,0.00793)--(-0.500,0.00529)--(-0.500,0.00264)--(-0.500,0);
 
diff --git a/auto/pictures_tex/Fig_CSCii.pstricks b/auto/pictures_tex/Fig_CSCii.pstricks
index 2607f006e..fe4d62089 100644
--- a/auto/pictures_tex/Fig_CSCii.pstricks
+++ b/auto/pictures_tex/Fig_CSCii.pstricks
@@ -71,21 +71,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.269774942,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.269774942) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.2698,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.2698) -- (0,1.5000);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.0317,0)--(0.0634,0.00201)--(0.0949,0.00452)--(0.126,0.00803)--(0.158,0.0125)--(0.188,0.0180)--(0.219,0.0244)--(0.249,0.0318)--(0.279,0.0401)--(0.308,0.0493)--(0.337,0.0594)--(0.365,0.0703)--(0.392,0.0821)--(0.419,0.0947)--(0.445,0.108)--(0.471,0.122)--(0.495,0.137)--(0.519,0.152)--(0.542,0.168)--(0.563,0.185)--(0.584,0.202)--(0.604,0.220)--(0.623,0.238)--(0.641,0.256)--(0.657,0.275)--(0.673,0.295)--(0.687,0.314)--(0.701,0.334)--(0.713,0.353)--(0.724,0.373)--(0.734,0.393)--(0.743,0.413)--(0.750,0.433)--(0.756,0.453)--(0.761,0.472)--(0.765,0.492)--(0.768,0.511)--(0.769,0.530)--(0.770,0.548)--(0.769,0.566)--(0.767,0.584)--(0.764,0.601)--(0.760,0.617)--(0.754,0.633)--(0.748,0.648)--(0.741,0.663)--(0.732,0.676)--(0.723,0.689)--(0.713,0.701)--(0.701,0.713)--(0.689,0.723)--(0.676,0.732)--(0.663,0.741)--(0.648,0.748)--(0.633,0.754)--(0.617,0.760)--(0.601,0.764)--(0.584,0.767)--(0.566,0.769)--(0.548,0.770)--(0.530,0.769)--(0.511,0.768)--(0.492,0.765)--(0.472,0.761)--(0.453,0.756)--(0.433,0.750)--(0.413,0.743)--(0.393,0.734)--(0.373,0.724)--(0.353,0.713)--(0.334,0.701)--(0.314,0.687)--(0.295,0.673)--(0.275,0.657)--(0.256,0.641)--(0.238,0.623)--(0.220,0.604)--(0.202,0.584)--(0.185,0.563)--(0.168,0.542)--(0.152,0.519)--(0.137,0.495)--(0.122,0.471)--(0.108,0.445)--(0.0947,0.419)--(0.0821,0.392)--(0.0703,0.365)--(0.0594,0.337)--(0.0493,0.308)--(0.0401,0.279)--(0.0318,0.249)--(0.0244,0.219)--(0.0180,0.188)--(0.0125,0.158)--(0.00803,0.126)--(0.00452,0.0949)--(0.00201,0.0634)--(0,0.0317)--(0,0);
 \draw [color=blue] (0,0)--(-0.0317,0)--(-0.0634,-0.00201)--(-0.0949,-0.00452)--(-0.126,-0.00803)--(-0.158,-0.0125)--(-0.188,-0.0180)--(-0.219,-0.0244)--(-0.249,-0.0318)--(-0.279,-0.0401)--(-0.308,-0.0493)--(-0.337,-0.0594)--(-0.365,-0.0703)--(-0.392,-0.0821)--(-0.419,-0.0947)--(-0.445,-0.108)--(-0.471,-0.122)--(-0.495,-0.137)--(-0.519,-0.152)--(-0.542,-0.168)--(-0.563,-0.185)--(-0.584,-0.202)--(-0.604,-0.220)--(-0.623,-0.238)--(-0.641,-0.256)--(-0.657,-0.275)--(-0.673,-0.295)--(-0.687,-0.314)--(-0.701,-0.334)--(-0.713,-0.353)--(-0.724,-0.373)--(-0.734,-0.393)--(-0.743,-0.413)--(-0.750,-0.433)--(-0.756,-0.453)--(-0.761,-0.472)--(-0.765,-0.492)--(-0.768,-0.511)--(-0.769,-0.530)--(-0.770,-0.548)--(-0.769,-0.566)--(-0.767,-0.584)--(-0.764,-0.601)--(-0.760,-0.617)--(-0.754,-0.633)--(-0.748,-0.648)--(-0.741,-0.663)--(-0.732,-0.676)--(-0.723,-0.689)--(-0.713,-0.701)--(-0.701,-0.713)--(-0.689,-0.723)--(-0.676,-0.732)--(-0.663,-0.741)--(-0.648,-0.748)--(-0.633,-0.754)--(-0.617,-0.760)--(-0.601,-0.764)--(-0.584,-0.767)--(-0.566,-0.769)--(-0.548,-0.770)--(-0.530,-0.769)--(-0.511,-0.768)--(-0.492,-0.765)--(-0.472,-0.761)--(-0.453,-0.756)--(-0.433,-0.750)--(-0.413,-0.743)--(-0.393,-0.734)--(-0.373,-0.724)--(-0.353,-0.713)--(-0.334,-0.701)--(-0.314,-0.687)--(-0.295,-0.673)--(-0.275,-0.657)--(-0.256,-0.641)--(-0.238,-0.623)--(-0.220,-0.604)--(-0.202,-0.584)--(-0.185,-0.563)--(-0.168,-0.542)--(-0.152,-0.519)--(-0.137,-0.495)--(-0.122,-0.471)--(-0.108,-0.445)--(-0.0947,-0.419)--(-0.0821,-0.392)--(-0.0703,-0.365)--(-0.0594,-0.337)--(-0.0493,-0.308)--(-0.0401,-0.279)--(-0.0318,-0.249)--(-0.0244,-0.219)--(-0.0180,-0.188)--(-0.0125,-0.158)--(-0.00803,-0.126)--(-0.00452,-0.0949)--(-0.00201,-0.0634)--(0,-0.0317)--(0,0);
 \draw [color=lightgray] (1.00,1.00) -- (1.00,0);
 \draw [color=lightgray] (1.00,1.00) -- (0,1.00);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CSCiii.pstricks b/auto/pictures_tex/Fig_CSCiii.pstricks
index 938e78081..ebdb1078f 100644
--- a/auto/pictures_tex/Fig_CSCiii.pstricks
+++ b/auto/pictures_tex/Fig_CSCiii.pstricks
@@ -41,15 +41,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.151591227,0) -- (3.499800004,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.673444770);
+\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.6734);
 %DEFAULT
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-\draw (1.500000000,-0.4207143333) node {$ \frac{1}{2} $};
+\draw (1.5000,-0.42071) node {$ \frac{1}{2} $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.3108333333,1.500000000) node {$ \frac{1}{2} $};
+\draw (-0.31083,1.5000) node {$ \frac{1}{2} $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
 %OTHER STUFF
 %END PSPICTURE
@@ -92,18 +92,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.238758171,0) -- (2.626823348,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.334460604);
+\draw [,->,>=latex] (-2.2388,0) -- (2.6268,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.3345);
 %DEFAULT
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-\draw (-1.500000000,-0.4207143333) node {$ -\frac{1}{20} $};
+\draw (-1.5000,-0.42071) node {$ -\frac{1}{20} $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (1.500000000,-0.4207143333) node {$ \frac{1}{20} $};
+\draw (1.5000,-0.42071) node {$ \frac{1}{20} $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (-0.3816666667,3.000000000) node {$ \frac{1}{10} $};
+\draw (-0.38167,3.0000) node {$ \frac{1}{10} $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -146,8 +146,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.151591227,0) -- (3.499800004,0);
-\draw [,->,>=latex] (0,-2.673444770) -- (0,2.673444770);
+\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
+\draw [,->,>=latex] (0,-2.6734) -- (0,2.6734);
 %DEFAULT
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@@ -159,13 +159,13 @@
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-\draw (1.500000000,-0.4207143333) node {$ \frac{1}{2} $};
+\draw (1.5000,-0.42071) node {$ \frac{1}{2} $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4527428333,-1.500000000) node {$ -\frac{1}{2} $};
+\draw (-0.45274,-1.5000) node {$ -\frac{1}{2} $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.3108333333,1.500000000) node {$ \frac{1}{2} $};
+\draw (-0.31083,1.5000) node {$ \frac{1}{2} $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CSCiv.pstricks b/auto/pictures_tex/Fig_CSCiv.pstricks
index 6266f7948..42c742b85 100644
--- a/auto/pictures_tex/Fig_CSCiv.pstricks
+++ b/auto/pictures_tex/Fig_CSCiv.pstricks
@@ -91,32 +91,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-5.487474933) -- (0,2.045853861);
+\draw [,->,>=latex] (-2.4935,0) -- (1.8801,0);
+\draw [,->,>=latex] (0,-5.4875) -- (0,2.0459);
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diff --git a/auto/pictures_tex/Fig_CSCv.pstricks b/auto/pictures_tex/Fig_CSCv.pstricks
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--- a/auto/pictures_tex/Fig_CSCv.pstricks
+++ b/auto/pictures_tex/Fig_CSCv.pstricks
@@ -65,28 +65,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185307,0);
-\draw [,->,>=latex] (0,-2.497483219) -- (0,1.624594384);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-2.4975) -- (0,1.6246);
 %DEFAULT
 
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 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -125,16 +125,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.6576218188,0) -- (1.028045149,0);
-\draw [,->,>=latex] (0,-1.597472712) -- (0,1.597472712);
+\draw [,->,>=latex] (-0.65762,0) -- (1.0280,0);
+\draw [,->,>=latex] (0,-1.5975) -- (0,1.5975);
 %DEFAULT
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Cardioideexo.pstricks b/auto/pictures_tex/Fig_Cardioideexo.pstricks
index 4ae6791fd..1b2d525b7 100644
--- a/auto/pictures_tex/Fig_Cardioideexo.pstricks
+++ b/auto/pictures_tex/Fig_Cardioideexo.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
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@@ -92,23 +92,23 @@
 \draw [color=lightgray] (1.00,0) -- (0,0);
 \draw [color=lightgray] (0,0) -- (0,1.00);
 \draw [color=lightgray,style=dashed] (0,0) -- (1.21,1.21);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.207106781,1.207106781) node [rotate=0] {$\bullet$};
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.2071,1.2071) node [rotate=0] {$\bullet$};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CbCartTuii.pstricks b/auto/pictures_tex/Fig_CbCartTuii.pstricks
index bd6a697f8..f3a5e95ac 100644
--- a/auto/pictures_tex/Fig_CbCartTuii.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTuii.pstricks
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.148869145) -- (0,1.148869145);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.1489) -- (0,1.1489);
 %DEFAULT
 \draw [color=blue] (2.00,0)--(1.99,0.126)--(1.97,0.247)--(1.93,0.358)--(1.87,0.456)--(1.81,0.535)--(1.72,0.595)--(1.63,0.633)--(1.53,0.649)--(1.42,0.644)--(1.30,0.619)--(1.17,0.578)--(1.05,0.523)--(0.921,0.459)--(0.795,0.389)--(0.673,0.318)--(0.556,0.249)--(0.446,0.186)--(0.345,0.130)--(0.255,0.0849)--(0.176,0.0500)--(0.111,0.0255)--(0.0603,0.0103)--(0.0246,0.00271)--(0.00453,0)--(0,0)--(0.0126,0)--(0.0405,-0.00571)--(0.0839,-0.0168)--(0.142,-0.0365)--(0.214,-0.0661)--(0.299,-0.106)--(0.394,-0.157)--(0.500,-0.217)--(0.614,-0.283)--(0.734,-0.354)--(0.858,-0.424)--(0.984,-0.492)--(1.11,-0.552)--(1.24,-0.600)--(1.36,-0.634)--(1.47,-0.649)--(1.58,-0.644)--(1.68,-0.617)--(1.77,-0.568)--(1.84,-0.498)--(1.90,-0.409)--(1.95,-0.304)--(1.98,-0.188)--(2.00,-0.0634)--(2.00,0.0634)--(1.98,0.188)--(1.95,0.304)--(1.90,0.409)--(1.84,0.498)--(1.77,0.568)--(1.68,0.617)--(1.58,0.644)--(1.47,0.649)--(1.36,0.634)--(1.24,0.600)--(1.11,0.552)--(0.984,0.492)--(0.858,0.424)--(0.734,0.354)--(0.614,0.283)--(0.500,0.217)--(0.394,0.157)--(0.299,0.106)--(0.214,0.0661)--(0.142,0.0365)--(0.0839,0.0168)--(0.0405,0.00571)--(0.0126,0)--(0,0)--(0.00453,0)--(0.0246,-0.00271)--(0.0603,-0.0103)--(0.111,-0.0255)--(0.176,-0.0500)--(0.255,-0.0849)--(0.345,-0.130)--(0.446,-0.186)--(0.556,-0.249)--(0.673,-0.318)--(0.795,-0.389)--(0.921,-0.459)--(1.05,-0.523)--(1.17,-0.578)--(1.30,-0.619)--(1.42,-0.644)--(1.53,-0.649)--(1.63,-0.633)--(1.72,-0.595)--(1.81,-0.535)--(1.87,-0.456)--(1.93,-0.358)--(1.97,-0.247)--(1.99,-0.126)--(2.00,0);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CercleTnu.pstricks b/auto/pictures_tex/Fig_CercleTnu.pstricks
index 91e3ce90f..3feef34ae 100644
--- a/auto/pictures_tex/Fig_CercleTnu.pstricks
+++ b/auto/pictures_tex/Fig_CercleTnu.pstricks
@@ -73,14 +73,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=green,->,>=latex] (1.732050808,1.000000000) -- (2.598076211,1.500000000);
-\draw (2.338533378,1.838383788) node {\( n\)};
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (1.232050808,1.866025404);
-\draw (1.655192262,2.124333237) node {\( e_{\theta}\)};
-\draw [color=green,->,>=latex] (-0.6840402866,1.879385242) -- (-1.026060430,2.819077862);
-\draw (-1.417511050,2.637895653) node {\( n\)};
-\draw [color=red,->,>=latex] (-0.6840402866,1.879385242) -- (-1.623732907,1.537365098);
-\draw (-1.889672784,1.927580718) node {\( e_{\theta}\)};
+\draw [color=green,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
+\draw (2.3385,1.8384) node {\( n\)};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
+\draw (1.6552,2.1243) node {\( e_{\theta}\)};
+\draw [color=green,->,>=latex] (-0.68404,1.8794) -- (-1.0261,2.8191);
+\draw (-1.4175,2.6379) node {\( n\)};
+\draw [color=red,->,>=latex] (-0.68404,1.8794) -- (-1.6237,1.5374);
+\draw (-1.8897,1.9276) node {\( e_{\theta}\)};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/auto/pictures_tex/Fig_CercleTrigono.pstricks b/auto/pictures_tex/Fig_CercleTrigono.pstricks
index 5097fdb00..391fc1759 100644
--- a/auto/pictures_tex/Fig_CercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_CercleTrigono.pstricks
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
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 \draw [color=brown,style=dotted] (1.73,1.00) -- (1.73,0);
 \draw [color=brown,style=dotted] (1.73,1.00) -- (0,1.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.732050808,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (-0.2000000000,0.5000000000) -- (-0.2000000000,0);
-\draw [,->,>=latex] (-0.2000000000,0.5000000000) -- (-0.2000000000,1.000000000);
-\draw (-0.7580386667,0.5000000000) node {$\sin(\theta)$};
-\draw [,->,>=latex] (0.8660254038,-0.2000000000) -- (0,-0.2000000000);
-\draw [,->,>=latex] (0.8660254038,-0.2000000000) -- (1.732050808,-0.2000000000);
-\draw (0.8660254038,-0.4824550000) node {$\cos(\theta)$};
-\draw [color=brown,->,>=latex] (0,0) -- (1.732050808,1.000000000);
-\draw (0.9161397128,0.2302636180) node {$\theta$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.7320,0) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,0);
+\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,1.0000);
+\draw (-0.75804,0.50000) node {$\sin(\theta)$};
+\draw [,->,>=latex] (0.86602,-0.20000) -- (0,-0.20000);
+\draw [,->,>=latex] (0.86602,-0.20000) -- (1.7320,-0.20000);
+\draw (0.86602,-0.48246) node {$\cos(\theta)$};
+\draw [color=brown,->,>=latex] (0,0) -- (1.7320,1.0000);
+\draw (0.91614,0.23026) node {$\theta$};
 
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-\draw (2.134373262,1.274708000) node {$P$};
+\draw (2.1344,1.2747) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CheminFresnel.pstricks b/auto/pictures_tex/Fig_CheminFresnel.pstricks
index 4172b519f..5589fe1bc 100644
--- a/auto/pictures_tex/Fig_CheminFresnel.pstricks
+++ b/auto/pictures_tex/Fig_CheminFresnel.pstricks
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.914213562);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.9142);
 %DEFAULT
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-\draw [,->,>=latex] (1.000000000,0) -- (1.010000000,0);
+\draw [,->,>=latex] (1.0000,0) -- (1.0100,0);
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-\draw [,->,>=latex] (1.847759065,0.7653668647) -- (1.843932231,0.7746056601);
+\draw [,->,>=latex] (1.8478,0.76537) -- (1.8439,0.77461);
 \draw [color=blue] (0,0)--(0.01428,0.01428)--(0.02857,0.02857)--(0.04285,0.04285)--(0.05714,0.05714)--(0.07142,0.07142)--(0.08571,0.08571)--(0.09999,0.09999)--(0.1143,0.1143)--(0.1286,0.1286)--(0.1428,0.1428)--(0.1571,0.1571)--(0.1714,0.1714)--(0.1857,0.1857)--(0.2000,0.2000)--(0.2143,0.2143)--(0.2286,0.2286)--(0.2428,0.2428)--(0.2571,0.2571)--(0.2714,0.2714)--(0.2857,0.2857)--(0.3000,0.3000)--(0.3143,0.3143)--(0.3286,0.3286)--(0.3428,0.3428)--(0.3571,0.3571)--(0.3714,0.3714)--(0.3857,0.3857)--(0.4000,0.4000)--(0.4143,0.4143)--(0.4286,0.4286)--(0.4428,0.4428)--(0.4571,0.4571)--(0.4714,0.4714)--(0.4857,0.4857)--(0.5000,0.5000)--(0.5143,0.5143)--(0.5285,0.5285)--(0.5428,0.5428)--(0.5571,0.5571)--(0.5714,0.5714)--(0.5857,0.5857)--(0.6000,0.6000)--(0.6143,0.6143)--(0.6285,0.6285)--(0.6428,0.6428)--(0.6571,0.6571)--(0.6714,0.6714)--(0.6857,0.6857)--(0.7000,0.7000)--(0.7142,0.7142)--(0.7285,0.7285)--(0.7428,0.7428)--(0.7571,0.7571)--(0.7714,0.7714)--(0.7857,0.7857)--(0.8000,0.8000)--(0.8142,0.8142)--(0.8285,0.8285)--(0.8428,0.8428)--(0.8571,0.8571)--(0.8714,0.8714)--(0.8857,0.8857)--(0.9000,0.9000)--(0.9142,0.9142)--(0.9285,0.9285)--(0.9428,0.9428)--(0.9571,0.9571)--(0.9714,0.9714)--(0.9857,0.9857)--(0.9999,0.9999)--(1.014,1.014)--(1.029,1.029)--(1.043,1.043)--(1.057,1.057)--(1.071,1.071)--(1.086,1.086)--(1.100,1.100)--(1.114,1.114)--(1.129,1.129)--(1.143,1.143)--(1.157,1.157)--(1.171,1.171)--(1.186,1.186)--(1.200,1.200)--(1.214,1.214)--(1.229,1.229)--(1.243,1.243)--(1.257,1.257)--(1.271,1.271)--(1.286,1.286)--(1.300,1.300)--(1.314,1.314)--(1.329,1.329)--(1.343,1.343)--(1.357,1.357)--(1.371,1.371)--(1.386,1.386)--(1.400,1.400)--(1.414,1.414);
-\draw [,->,>=latex] (0.7071067812,0.7071067812) -- (0.7141778490,0.7141778490);
-\draw (1.000000000,-0.2140621667) node {\( \gamma_1\)};
-\draw (2.113799352,0.9176973746) node {\( \gamma_2\)};
-\draw (0.4627437697,0.8918796260) node {\( \gamma_3\)};
+\draw [,->,>=latex] (0.70711,0.70711) -- (0.71418,0.71418);
+\draw (1.0000,-0.21406) node {\( \gamma_1\)};
+\draw (2.1138,0.91770) node {\( \gamma_2\)};
+\draw (0.46274,0.89188) node {\( \gamma_3\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ConeRevolution.pstricks b/auto/pictures_tex/Fig_ConeRevolution.pstricks
index c3099f553..1c9a11ba0 100644
--- a/auto/pictures_tex/Fig_ConeRevolution.pstricks
+++ b/auto/pictures_tex/Fig_ConeRevolution.pstricks
@@ -79,19 +79,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
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-\draw []  (2.000000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (2.538859511,3.253165678) node {$(R,h)$};
-\draw (0.7236718444,0.3359520608) node {$\alpha$};
+\draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw (2.5389,3.2532) node {$(R,h)$};
+\draw (0.72367,0.33595) node {$\alpha$};
 
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-\draw (2.000000000,-0.3257195000) node {$\mathit{R}$};
+\draw (2.0000,-0.32572) node {$\mathit{R}$};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.3027346667,3.000000000) node {$\mathit{h}$};
+\draw (-0.30273,3.0000) node {$\mathit{h}$};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks b/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
index 21700a099..7432ae77f 100644
--- a/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
+++ b/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
@@ -107,8 +107,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 
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@@ -116,35 +116,35 @@
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 \draw [] (5.000,0)--(4.994,0.1903)--(4.976,0.3798)--(4.946,0.5678)--(4.904,0.7534)--(4.850,0.9361)--(4.785,1.115)--(4.709,1.289)--(4.622,1.459)--(4.524,1.622)--(4.416,1.779)--(4.298,1.928)--(4.171,2.070)--(4.036,2.204)--(3.892,2.328)--(3.740,2.444)--(3.582,2.549)--(3.417,2.644)--(3.246,2.729)--(3.071,2.802)--(2.891,2.865)--(2.707,2.915)--(2.521,2.954)--(2.333,2.982)--(2.143,2.997)--(1.952,3.000)--(1.762,2.991)--(1.573,2.969)--(1.386,2.936)--(1.201,2.892)--(1.019,2.835)--(0.8410,2.767)--(0.6678,2.688)--(0.5000,2.598)--(0.3382,2.498)--(0.1832,2.387)--(0.03542,2.267)--(-0.1044,2.138)--(-0.2358,2.000)--(-0.3582,1.854)--(-0.4710,1.701)--(-0.5740,1.541)--(-0.6665,1.375)--(-0.7483,1.203)--(-0.8191,1.026)--(-0.8785,0.8452)--(-0.9263,0.6609)--(-0.9623,0.4740)--(-0.9864,0.2852)--(-0.9985,0.09518)--(-0.9985,-0.09518)--(-0.9864,-0.2852)--(-0.9623,-0.4740)--(-0.9263,-0.6609)--(-0.8785,-0.8452)--(-0.8191,-1.026)--(-0.7483,-1.203)--(-0.6665,-1.375)--(-0.5740,-1.541)--(-0.4710,-1.701)--(-0.3582,-1.854)--(-0.2358,-2.000)--(-0.1044,-2.138)--(0.03542,-2.267)--(0.1832,-2.387)--(0.3382,-2.498)--(0.5000,-2.598)--(0.6678,-2.688)--(0.8410,-2.767)--(1.019,-2.835)--(1.201,-2.892)--(1.386,-2.936)--(1.573,-2.969)--(1.762,-2.991)--(1.952,-3.000)--(2.143,-2.997)--(2.333,-2.982)--(2.521,-2.954)--(2.707,-2.915)--(2.891,-2.865)--(3.071,-2.802)--(3.246,-2.729)--(3.417,-2.644)--(3.582,-2.549)--(3.740,-2.444)--(3.892,-2.328)--(4.036,-2.204)--(4.171,-2.070)--(4.298,-1.928)--(4.416,-1.779)--(4.524,-1.622)--(4.622,-1.459)--(4.709,-1.289)--(4.785,-1.115)--(4.850,-0.9361)--(4.904,-0.7534)--(4.946,-0.5678)--(4.976,-0.3798)--(4.994,-0.1903)--(5.000,0);
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.327046523,0.2955320229) node {\( \lambda_1\)};
-\draw []  (2.000000000,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.672953477,1.118681539) node {\( \lambda_2\)};
-\draw []  (2.000000000,-1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.672953477,-1.118681539) node {\( \lambda_3\)};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-1.3270,0.29553) node {\( \lambda_1\)};
+\draw []  (2.0000,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.6730,1.1187) node {\( \lambda_2\)};
+\draw []  (2.0000,-1.4142) node [rotate=0] {$\bullet$};
+\draw (1.6730,-1.1187) node {\( \lambda_3\)};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks b/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
index dfb3ebc29..86b9be42e 100644
--- a/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
+++ b/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (8.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (6.000,0)--(5.996,0.1268)--(5.984,0.2532)--(5.964,0.3785)--(5.936,0.5023)--(5.900,0.6241)--(5.857,0.7433)--(5.806,0.8596)--(5.748,0.9724)--(5.682,1.081)--(5.611,1.186)--(5.532,1.286)--(5.447,1.380)--(5.357,1.469)--(5.261,1.552)--(5.160,1.629)--(5.054,1.699)--(4.945,1.763)--(4.831,1.819)--(4.714,1.868)--(4.594,1.910)--(4.471,1.944)--(4.347,1.970)--(4.222,1.988)--(4.095,1.998)--(3.968,2.000)--(3.841,1.994)--(3.715,1.980)--(3.590,1.958)--(3.467,1.928)--(3.346,1.890)--(3.227,1.845)--(3.112,1.792)--(3.000,1.732)--(2.892,1.665)--(2.789,1.592)--(2.690,1.512)--(2.597,1.425)--(2.509,1.334)--(2.428,1.236)--(2.353,1.134)--(2.284,1.027)--(2.222,0.9165)--(2.168,0.8019)--(2.121,0.6840)--(2.081,0.5635)--(2.049,0.4406)--(2.025,0.3160)--(2.009,0.1901)--(2.001,0.06346)--(2.001,-0.06346)--(2.009,-0.1901)--(2.025,-0.3160)--(2.049,-0.4406)--(2.081,-0.5635)--(2.121,-0.6840)--(2.168,-0.8019)--(2.222,-0.9165)--(2.284,-1.027)--(2.353,-1.134)--(2.428,-1.236)--(2.509,-1.334)--(2.597,-1.425)--(2.690,-1.512)--(2.789,-1.592)--(2.892,-1.665)--(3.000,-1.732)--(3.112,-1.792)--(3.227,-1.845)--(3.346,-1.890)--(3.467,-1.928)--(3.590,-1.958)--(3.715,-1.980)--(3.841,-1.994)--(3.968,-2.000)--(4.095,-1.998)--(4.222,-1.988)--(4.347,-1.970)--(4.471,-1.944)--(4.594,-1.910)--(4.714,-1.868)--(4.831,-1.819)--(4.945,-1.763)--(5.054,-1.699)--(5.160,-1.629)--(5.261,-1.552)--(5.357,-1.469)--(5.447,-1.380)--(5.532,-1.286)--(5.611,-1.186)--(5.682,-1.081)--(5.748,-0.9724)--(5.806,-0.8596)--(5.857,-0.7433)--(5.900,-0.6241)--(5.936,-0.5023)--(5.964,-0.3785)--(5.984,-0.2532)--(5.996,-0.1268)--(6.000,0);
@@ -104,23 +104,23 @@
 \draw [] (3.000,0)--(2.998,0.06342)--(2.992,0.1266)--(2.982,0.1893)--(2.968,0.2511)--(2.950,0.3120)--(2.928,0.3717)--(2.903,0.4298)--(2.874,0.4862)--(2.841,0.5406)--(2.805,0.5929)--(2.766,0.6428)--(2.724,0.6901)--(2.678,0.7346)--(2.631,0.7761)--(2.580,0.8146)--(2.527,0.8497)--(2.472,0.8815)--(2.415,0.9096)--(2.357,0.9342)--(2.297,0.9549)--(2.236,0.9718)--(2.174,0.9848)--(2.111,0.9938)--(2.048,0.9989)--(1.984,0.9999)--(1.921,0.9969)--(1.858,0.9898)--(1.795,0.9788)--(1.734,0.9638)--(1.673,0.9450)--(1.614,0.9224)--(1.556,0.8960)--(1.500,0.8660)--(1.446,0.8326)--(1.394,0.7958)--(1.345,0.7558)--(1.299,0.7127)--(1.255,0.6668)--(1.214,0.6182)--(1.176,0.5671)--(1.142,0.5137)--(1.111,0.4582)--(1.084,0.4009)--(1.060,0.3420)--(1.041,0.2817)--(1.025,0.2203)--(1.013,0.1580)--(1.005,0.09506)--(1.001,0.03173)--(1.001,-0.03173)--(1.005,-0.09506)--(1.013,-0.1580)--(1.025,-0.2203)--(1.041,-0.2817)--(1.060,-0.3420)--(1.084,-0.4009)--(1.111,-0.4582)--(1.142,-0.5137)--(1.176,-0.5671)--(1.214,-0.6182)--(1.255,-0.6668)--(1.299,-0.7127)--(1.345,-0.7558)--(1.394,-0.7958)--(1.446,-0.8326)--(1.500,-0.8660)--(1.556,-0.8960)--(1.614,-0.9224)--(1.673,-0.9450)--(1.734,-0.9638)--(1.795,-0.9788)--(1.858,-0.9898)--(1.921,-0.9969)--(1.984,-0.9999)--(2.048,-0.9989)--(2.111,-0.9938)--(2.174,-0.9848)--(2.236,-0.9718)--(2.297,-0.9549)--(2.357,-0.9342)--(2.415,-0.9096)--(2.472,-0.8815)--(2.527,-0.8497)--(2.580,-0.8146)--(2.631,-0.7761)--(2.678,-0.7346)--(2.724,-0.6901)--(2.766,-0.6428)--(2.805,-0.5929)--(2.841,-0.5406)--(2.874,-0.4862)--(2.903,-0.4298)--(2.928,-0.3717)--(2.950,-0.3120)--(2.968,-0.2511)--(2.982,-0.1893)--(2.992,-0.1266)--(2.998,-0.06342)--(3.000,0);
 
 \draw [] (8.000,0)--(7.996,0.1268)--(7.984,0.2532)--(7.964,0.3785)--(7.936,0.5023)--(7.900,0.6241)--(7.857,0.7433)--(7.806,0.8596)--(7.748,0.9724)--(7.682,1.081)--(7.611,1.186)--(7.532,1.286)--(7.447,1.380)--(7.357,1.469)--(7.261,1.552)--(7.160,1.629)--(7.054,1.699)--(6.945,1.763)--(6.831,1.819)--(6.714,1.868)--(6.594,1.910)--(6.471,1.944)--(6.347,1.970)--(6.222,1.988)--(6.095,1.998)--(5.968,2.000)--(5.841,1.994)--(5.715,1.980)--(5.590,1.958)--(5.467,1.928)--(5.346,1.890)--(5.227,1.845)--(5.112,1.792)--(5.000,1.732)--(4.892,1.665)--(4.789,1.592)--(4.690,1.512)--(4.597,1.425)--(4.509,1.334)--(4.428,1.236)--(4.353,1.134)--(4.284,1.027)--(4.222,0.9165)--(4.168,0.8019)--(4.121,0.6840)--(4.081,0.5635)--(4.049,0.4406)--(4.025,0.3160)--(4.009,0.1901)--(4.001,0.06346)--(4.001,-0.06346)--(4.009,-0.1901)--(4.025,-0.3160)--(4.049,-0.4406)--(4.081,-0.5635)--(4.121,-0.6840)--(4.168,-0.8019)--(4.222,-0.9165)--(4.284,-1.027)--(4.353,-1.134)--(4.428,-1.236)--(4.509,-1.334)--(4.597,-1.425)--(4.690,-1.512)--(4.789,-1.592)--(4.892,-1.665)--(5.000,-1.732)--(5.112,-1.792)--(5.227,-1.845)--(5.346,-1.890)--(5.467,-1.928)--(5.590,-1.958)--(5.715,-1.980)--(5.841,-1.994)--(5.968,-2.000)--(6.095,-1.998)--(6.222,-1.988)--(6.347,-1.970)--(6.471,-1.944)--(6.594,-1.910)--(6.714,-1.868)--(6.831,-1.819)--(6.945,-1.763)--(7.054,-1.699)--(7.160,-1.629)--(7.261,-1.552)--(7.357,-1.469)--(7.447,-1.380)--(7.532,-1.286)--(7.611,-1.186)--(7.682,-1.081)--(7.748,-0.9724)--(7.806,-0.8596)--(7.857,-0.7433)--(7.900,-0.6241)--(7.936,-0.5023)--(7.964,-0.3785)--(7.984,-0.2532)--(7.996,-0.1268)--(8.000,0);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.672953477,0.2955320229) node {\( \lambda_1\)};
-\draw []  (5.000000000,1.732050808) node [rotate=0] {$\bullet$};
-\draw (5.000000000,2.086161474) node {\( \lambda_2\)};
-\draw []  (5.000000000,-1.732050808) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-2.086161474) node {\( \lambda_3\)};
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.6730,0.29553) node {\( \lambda_1\)};
+\draw []  (5.0000,1.7320) node [rotate=0] {$\bullet$};
+\draw (5.0000,2.0862) node {\( \lambda_2\)};
+\draw []  (5.0000,-1.7320) node [rotate=0] {$\bullet$};
+\draw (5.0000,-2.0862) node {\( \lambda_3\)};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 3 $};
+\draw (6.0000,-0.31492) node {$ 3 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 4 $};
+\draw (8.0000,-0.31492) node {$ 4 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DTIYKkP.pstricks b/auto/pictures_tex/Fig_DTIYKkP.pstricks
index a87182c34..2166b0fcf 100644
--- a/auto/pictures_tex/Fig_DTIYKkP.pstricks
+++ b/auto/pictures_tex/Fig_DTIYKkP.pstricks
@@ -83,12 +83,12 @@
 %DEFAULT
 
 \draw [color=blue] (-5.000,0)--(-4.970,0.1206)--(-4.939,0.2400)--(-4.909,0.3581)--(-4.879,0.4751)--(-4.849,0.5908)--(-4.818,0.7052)--(-4.788,0.8185)--(-4.758,0.9305)--(-4.727,1.041)--(-4.697,1.151)--(-4.667,1.259)--(-4.636,1.366)--(-4.606,1.472)--(-4.576,1.577)--(-4.545,1.680)--(-4.515,1.783)--(-4.485,1.884)--(-4.455,1.983)--(-4.424,2.082)--(-4.394,2.179)--(-4.364,2.275)--(-4.333,2.370)--(-4.303,2.464)--(-4.273,2.556)--(-4.242,2.648)--(-4.212,2.738)--(-4.182,2.826)--(-4.151,2.914)--(-4.121,3.000)--(-4.091,3.085)--(-4.061,3.169)--(-4.030,3.252)--(-4.000,3.333)--(-3.970,3.414)--(-3.939,3.492)--(-3.909,3.570)--(-3.879,3.647)--(-3.848,3.722)--(-3.818,3.796)--(-3.788,3.869)--(-3.758,3.941)--(-3.727,4.011)--(-3.697,4.080)--(-3.667,4.148)--(-3.636,4.215)--(-3.606,4.280)--(-3.576,4.345)--(-3.545,4.408)--(-3.515,4.470)--(-3.485,4.530)--(-3.455,4.590)--(-3.424,4.648)--(-3.394,4.705)--(-3.364,4.760)--(-3.333,4.815)--(-3.303,4.868)--(-3.273,4.920)--(-3.242,4.971)--(-3.212,5.021)--(-3.182,5.069)--(-3.152,5.116)--(-3.121,5.162)--(-3.091,5.207)--(-3.061,5.250)--(-3.030,5.292)--(-3.000,5.333)--(-2.970,5.373)--(-2.939,5.412)--(-2.909,5.449)--(-2.879,5.485)--(-2.848,5.520)--(-2.818,5.554)--(-2.788,5.586)--(-2.758,5.617)--(-2.727,5.647)--(-2.697,5.676)--(-2.667,5.704)--(-2.636,5.730)--(-2.606,5.755)--(-2.576,5.779)--(-2.545,5.802)--(-2.515,5.823)--(-2.485,5.843)--(-2.455,5.862)--(-2.424,5.880)--(-2.394,5.897)--(-2.364,5.912)--(-2.333,5.926)--(-2.303,5.939)--(-2.273,5.950)--(-2.242,5.961)--(-2.212,5.970)--(-2.182,5.978)--(-2.152,5.985)--(-2.121,5.990)--(-2.091,5.995)--(-2.061,5.998)--(-2.030,5.999)--(-2.000,6.000);
-\draw [color=brown]  (-4.000000000,3.333333333) node [rotate=0] {$\bullet$};
-\draw (-4.727456667,3.635788333) node {$o=[\mtu]$};
-\draw (-1.197814000,6.000000000) node {$[\SO(2)]$};
-\draw [color=cyan,->,>=latex] (-4.500000000,1.833333333) -- (-2.500000000,2.333333333);
-\draw (-1.441613144,1.979900977) node {$[ e^{sE(w)} e^{xq_0}]$};
-\draw (-5.102939752,2.118687484) node {$[ e^{xq_0}]$};
+\draw [color=brown]  (-4.0000,3.3333) node [rotate=0] {$\bullet$};
+\draw (-4.7275,3.6358) node {$o=[\mtu]$};
+\draw (-1.1978,6.0000) node {$[\SO(2)]$};
+\draw [color=cyan,->,>=latex] (-4.5000,1.8333) -- (-2.5000,2.3333);
+\draw (-1.4416,1.9799) node {$[ e^{sE(w)} e^{xq_0}]$};
+\draw (-5.1029,2.1187) node {$[ e^{xq_0}]$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks b/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks
index 7914ba074..0ed1188bd 100644
--- a/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks
+++ b/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks
@@ -103,46 +103,46 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-2.0000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
 \draw [color=blue,style=dashed] (3.00,1.00) -- (3.00,0);
 \draw [color=blue,style=dashed] (3.00,1.00) -- (0,1.00);
-\draw (3.500387996,1.214077777) node {$(3,1)$};
-\draw [color=blue,->,>=latex] (0,0) -- (3.000000000,1.000000000);
+\draw (3.5004,1.2141) node {$(3,1)$};
+\draw [color=blue,->,>=latex] (0,0) -- (3.0000,1.0000);
 \draw [color=green,style=dashed] (-1.50,-2.50) -- (-1.50,0);
 \draw [color=green,style=dashed] (-1.50,-2.50) -- (0,-2.50);
-\draw (-2.524677076,-2.768204293) node {$(-1.5,-2.5)$};
-\draw [color=green,->,>=latex] (0,0) -- (-1.500000000,-2.500000000);
+\draw (-2.5247,-2.7682) node {$(-1.5,-2.5)$};
+\draw [color=green,->,>=latex] (0,0) -- (-1.5000,-2.5000);
 \draw [color=brown,style=dashed] (-1.00,2.50) -- (-1.00,0);
 \draw [color=brown,style=dashed] (-1.00,2.50) -- (0,2.50);
-\draw (-1.726512568,2.775302669) node {$(-1,2.5)$};
-\draw [color=brown,->,>=latex] (0,0) -- (-1.000000000,2.500000000);
+\draw (-1.7265,2.7753) node {$(-1,2.5)$};
+\draw [color=brown,->,>=latex] (0,0) -- (-1.0000,2.5000);
 \draw [color=cyan,style=dashed] (1.50,-1.00) -- (1.50,0);
 \draw [color=cyan,style=dashed] (1.50,-1.00) -- (0,-1.00);
-\draw (2.272578529,-1.237925020) node {$(1.5,-1)$};
-\draw [color=cyan,->,>=latex] (0,0) -- (1.500000000,-1.000000000);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (2.2726,-1.2379) node {$(1.5,-1)$};
+\draw [color=cyan,->,>=latex] (0,0) -- (1.5000,-1.0000);
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DisqueConv.pstricks b/auto/pictures_tex/Fig_DisqueConv.pstricks
index b5030e05d..f268bbcad 100644
--- a/auto/pictures_tex/Fig_DisqueConv.pstricks
+++ b/auto/pictures_tex/Fig_DisqueConv.pstricks
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
-\draw []  (2.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (1.765251155,1.823338322) node {$z_0$};
+\draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (1.7653,1.8233) node {$z_0$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/auto/pictures_tex/Fig_DistanceEuclide.pstricks b/auto/pictures_tex/Fig_DistanceEuclide.pstricks
index 2c172a3a5..de1bd21a0 100644
--- a/auto/pictures_tex/Fig_DistanceEuclide.pstricks
+++ b/auto/pictures_tex/Fig_DistanceEuclide.pstricks
@@ -91,31 +91,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
-\draw []  (1.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw (1.000000000,4.490142000) node {$(A_x,A_y)$};
-\draw []  (3.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.971096333,1.000000000) node {$(B_x,B_y)$};
-\draw []  (3.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw (3.355622034,4.336840034) node {$C$};
+\draw []  (1.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,4.4901) node {$(A_x,A_y)$};
+\draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.9711,1.0000) node {$(B_x,B_y)$};
+\draw []  (3.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (3.3556,4.3368) node {$C$};
 \draw [] (1.00,4.00) -- (3.00,1.00);
 \draw [] (1.00,4.00) -- (3.00,4.00);
 \draw [] (3.00,1.00) -- (3.00,4.00);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks b/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks
index 86046b293..aad544245 100644
--- a/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks
+++ b/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2644441667) node {\( 1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.26444) node {\( 1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks b/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks
index 1f78404fd..de90bcf5a 100644
--- a/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks
+++ b/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.2644441667) node {\( 1\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
+\draw (0,0.26444) node {\( 1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoCUd.pstricks b/auto/pictures_tex/Fig_ExoCUd.pstricks
index c0a8dea4b..271934b9a 100644
--- a/auto/pictures_tex/Fig_ExoCUd.pstricks
+++ b/auto/pictures_tex/Fig_ExoCUd.pstricks
@@ -75,22 +75,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,4.5000);
 %DEFAULT
 
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 \draw [color=gray,style=dashed] (1.00,-0.500) -- (1.00,4.00);
-\draw []  (0,2.560000000) node [rotate=0] {$\bullet$};
-\draw (0.3081560344,2.886194201) node {$y$};
-\draw []  (2.600000000,2.560000000) node [rotate=0] {$\bullet$};
-\draw []  (-0.6000000000,2.560000000) node [rotate=0] {$\bullet$};
-\draw []  (2.600000000,0) node [rotate=0] {$\bullet$};
-\draw (2.600000000,-0.4191818333) node {$x_+$};
-\draw []  (-0.6000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.6000000000,-0.4191818333) node {$x_-$};
+\draw []  (0,2.5600) node [rotate=0] {$\bullet$};
+\draw (0.30816,2.8862) node {$y$};
+\draw []  (2.6000,2.5600) node [rotate=0] {$\bullet$};
+\draw []  (-0.60000,2.5600) node [rotate=0] {$\bullet$};
+\draw []  (2.6000,0) node [rotate=0] {$\bullet$};
+\draw (2.6000,-0.41918) node {$x_+$};
+\draw []  (-0.60000,0) node [rotate=0] {$\bullet$};
+\draw (-0.60000,-0.41918) node {$x_-$};
 \draw [style=dashed] (2.60,2.56) -- (2.60,0);
 \draw [style=dashed] (-0.600,2.56) -- (-0.600,0);
 \draw [style=dotted] (2.60,2.56) -- (-0.600,2.56);
diff --git a/auto/pictures_tex/Fig_ExoParamCD.pstricks b/auto/pictures_tex/Fig_ExoParamCD.pstricks
index f3ed297b6..126669a8f 100644
--- a/auto/pictures_tex/Fig_ExoParamCD.pstricks
+++ b/auto/pictures_tex/Fig_ExoParamCD.pstricks
@@ -71,25 +71,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.190,0.285)--(0.380,0.568)--(0.568,0.845)--(0.753,1.11)--(0.936,1.37)--(1.11,1.62)--(1.29,1.85)--(1.46,2.07)--(1.62,2.27)--(1.78,2.44)--(1.93,2.60)--(2.07,2.73)--(2.20,2.83)--(2.33,2.92)--(2.44,2.97)--(2.55,3.00)--(2.64,3.00)--(2.73,2.97)--(2.80,2.92)--(2.86,2.83)--(2.92,2.73)--(2.95,2.60)--(2.98,2.44)--(3.00,2.27)--(3.00,2.07)--(2.99,1.85)--(2.97,1.62)--(2.94,1.37)--(2.89,1.11)--(2.83,0.845)--(2.77,0.568)--(2.69,0.285)--(2.60,0)--(2.50,-0.285)--(2.39,-0.568)--(2.27,-0.845)--(2.14,-1.11)--(2.00,-1.37)--(1.85,-1.62)--(1.70,-1.85)--(1.54,-2.07)--(1.37,-2.27)--(1.20,-2.44)--(1.03,-2.60)--(0.845,-2.73)--(0.661,-2.83)--(0.474,-2.92)--(0.285,-2.97)--(0.0952,-3.00)--(-0.0952,-3.00)--(-0.285,-2.97)--(-0.474,-2.92)--(-0.661,-2.83)--(-0.845,-2.73)--(-1.03,-2.60)--(-1.20,-2.44)--(-1.37,-2.27)--(-1.54,-2.07)--(-1.70,-1.85)--(-1.85,-1.62)--(-2.00,-1.37)--(-2.14,-1.11)--(-2.27,-0.845)--(-2.39,-0.568)--(-2.50,-0.285)--(-2.60,0)--(-2.69,0.285)--(-2.77,0.568)--(-2.83,0.845)--(-2.89,1.11)--(-2.94,1.37)--(-2.97,1.62)--(-2.99,1.85)--(-3.00,2.07)--(-3.00,2.27)--(-2.98,2.44)--(-2.95,2.60)--(-2.92,2.73)--(-2.86,2.83)--(-2.80,2.92)--(-2.73,2.97)--(-2.64,3.00)--(-2.55,3.00)--(-2.44,2.97)--(-2.33,2.92)--(-2.20,2.83)--(-2.07,2.73)--(-1.93,2.60)--(-1.78,2.44)--(-1.62,2.27)--(-1.46,2.07)--(-1.29,1.85)--(-1.11,1.62)--(-0.936,1.37)--(-0.753,1.11)--(-0.568,0.845)--(-0.380,0.568)--(-0.190,0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.01664100589,0.02496150883);
-\draw [,->,>=latex] (3.000000000,2.121320344) -- (3.000000000,2.091320344);
-\draw [,->,>=latex] (0,-3.000000000) -- (-0.03000000000,-3.000000000);
-\draw [,->,>=latex] (-3.000000000,2.121320344) -- (-3.000000000,2.151320344);
+\draw [,->,>=latex] (0,0) -- (0.016641,0.024962);
+\draw [,->,>=latex] (3.0000,2.1213) -- (3.0000,2.0913);
+\draw [,->,>=latex] (0,-3.0000) -- (-0.030000,-3.0000);
+\draw [,->,>=latex] (-3.0000,2.1213) -- (-3.0000,2.1513);
 \draw [color=red] (0,0)--(0.190,-0.285)--(0.380,-0.568)--(0.568,-0.845)--(0.753,-1.11)--(0.936,-1.37)--(1.11,-1.62)--(1.29,-1.85)--(1.46,-2.07)--(1.62,-2.27)--(1.78,-2.44)--(1.93,-2.60)--(2.07,-2.73)--(2.20,-2.83)--(2.33,-2.92)--(2.44,-2.97)--(2.55,-3.00)--(2.64,-3.00)--(2.73,-2.97)--(2.80,-2.92)--(2.86,-2.83)--(2.92,-2.73)--(2.95,-2.60)--(2.98,-2.44)--(3.00,-2.27)--(3.00,-2.07)--(2.99,-1.85)--(2.97,-1.62)--(2.94,-1.37)--(2.89,-1.11)--(2.83,-0.845)--(2.77,-0.568)--(2.69,-0.285)--(2.60,0)--(2.50,0.285)--(2.39,0.568)--(2.27,0.845)--(2.14,1.11)--(2.00,1.37)--(1.85,1.62)--(1.70,1.85)--(1.54,2.07)--(1.37,2.27)--(1.20,2.44)--(1.03,2.60)--(0.845,2.73)--(0.661,2.83)--(0.474,2.92)--(0.285,2.97)--(0.0952,3.00)--(-0.0952,3.00)--(-0.285,2.97)--(-0.474,2.92)--(-0.661,2.83)--(-0.845,2.73)--(-1.03,2.60)--(-1.20,2.44)--(-1.37,2.27)--(-1.54,2.07)--(-1.70,1.85)--(-1.85,1.62)--(-2.00,1.37)--(-2.14,1.11)--(-2.27,0.845)--(-2.39,0.568)--(-2.50,0.285)--(-2.60,0)--(-2.69,-0.285)--(-2.77,-0.568)--(-2.83,-0.845)--(-2.89,-1.11)--(-2.94,-1.37)--(-2.97,-1.62)--(-2.99,-1.85)--(-3.00,-2.07)--(-3.00,-2.27)--(-2.98,-2.44)--(-2.95,-2.60)--(-2.92,-2.73)--(-2.86,-2.83)--(-2.80,-2.92)--(-2.73,-2.97)--(-2.64,-3.00)--(-2.55,-3.00)--(-2.44,-2.97)--(-2.33,-2.92)--(-2.20,-2.83)--(-2.07,-2.73)--(-1.93,-2.60)--(-1.78,-2.44)--(-1.62,-2.27)--(-1.46,-2.07)--(-1.29,-1.85)--(-1.11,-1.62)--(-0.936,-1.37)--(-0.753,-1.11)--(-0.568,-0.845)--(-0.380,-0.568)--(-0.190,-0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.01664100589,-0.02496150883);
-\draw [,->,>=latex] (3.000000000,-2.121320344) -- (3.000000000,-2.091320344);
-\draw [,->,>=latex] (0,3.000000000) -- (-0.03000000000,3.000000000);
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw [,->,>=latex] (0,0) -- (0.016641,-0.024962);
+\draw [,->,>=latex] (3.0000,-2.1213) -- (3.0000,-2.0913);
+\draw [,->,>=latex] (0,3.0000) -- (-0.030000,3.0000);
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -1 $};
+\draw (-0.43316,-3.0000) node {$ -1 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks b/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks
index 48a9267a2..c9969ffe2 100644
--- a/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks
+++ b/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks
@@ -103,48 +103,48 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
 %DEFAULT
 
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-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
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 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
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 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
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 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
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+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_FnCosApprox.pstricks b/auto/pictures_tex/Fig_FnCosApprox.pstricks
index 32f38a6a3..890159f3e 100644
--- a/auto/pictures_tex/Fig_FnCosApprox.pstricks
+++ b/auto/pictures_tex/Fig_FnCosApprox.pstricks
@@ -91,24 +91,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185307,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=blue] (0,2.000)--(0.06347,1.999)--(0.1269,1.996)--(0.1904,1.991)--(0.2539,1.984)--(0.3173,1.975)--(0.3808,1.964)--(0.4443,1.951)--(0.5077,1.936)--(0.5712,1.919)--(0.6347,1.900)--(0.6981,1.879)--(0.7616,1.857)--(0.8251,1.832)--(0.8885,1.806)--(0.9520,1.778)--(1.015,1.748)--(1.079,1.716)--(1.142,1.683)--(1.206,1.647)--(1.269,1.611)--(1.333,1.572)--(1.396,1.532)--(1.460,1.491)--(1.523,1.447)--(1.587,1.403)--(1.650,1.357)--(1.714,1.310)--(1.777,1.261)--(1.841,1.211)--(1.904,1.160)--(1.967,1.108)--(2.031,1.054)--(2.094,1.000)--(2.158,0.9445)--(2.221,0.8881)--(2.285,0.8308)--(2.348,0.7727)--(2.412,0.7138)--(2.475,0.6541)--(2.539,0.5938)--(2.602,0.5330)--(2.666,0.4715)--(2.729,0.4096)--(2.793,0.3473)--(2.856,0.2846)--(2.919,0.2217)--(2.983,0.1585)--(3.046,0.09516)--(3.110,0.03173)--(3.173,-0.03173)--(3.237,-0.09516)--(3.300,-0.1585)--(3.364,-0.2217)--(3.427,-0.2846)--(3.491,-0.3473)--(3.554,-0.4096)--(3.618,-0.4715)--(3.681,-0.5330)--(3.745,-0.5938)--(3.808,-0.6541)--(3.871,-0.7138)--(3.935,-0.7727)--(3.998,-0.8308)--(4.062,-0.8881)--(4.125,-0.9445)--(4.189,-1.000)--(4.252,-1.054)--(4.316,-1.108)--(4.379,-1.160)--(4.443,-1.211)--(4.506,-1.261)--(4.570,-1.310)--(4.633,-1.357)--(4.697,-1.403)--(4.760,-1.447)--(4.823,-1.491)--(4.887,-1.532)--(4.950,-1.572)--(5.014,-1.611)--(5.077,-1.647)--(5.141,-1.683)--(5.204,-1.716)--(5.268,-1.748)--(5.331,-1.778)--(5.395,-1.806)--(5.458,-1.832)--(5.522,-1.857)--(5.585,-1.879)--(5.648,-1.900)--(5.712,-1.919)--(5.775,-1.936)--(5.839,-1.951)--(5.902,-1.964)--(5.966,-1.975)--(6.029,-1.984)--(6.093,-1.991)--(6.156,-1.996)--(6.220,-1.999)--(6.283,-2.000);
-\draw []  (1.570796327,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.799913701,1.661396050) node {$P$};
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{4} \, \pi $};
+\draw []  (1.5708,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.7999,1.6614) node {$P$};
+\draw (1.5708,-0.42071) node {$ \frac{1}{4} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (3.1416,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{4} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{4} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.2785761667) node {$ \pi $};
+\draw (6.2832,-0.27858) node {$ \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_GBnUivi.pstricks b/auto/pictures_tex/Fig_GBnUivi.pstricks
index c1f71fb15..447475bac 100644
--- a/auto/pictures_tex/Fig_GBnUivi.pstricks
+++ b/auto/pictures_tex/Fig_GBnUivi.pstricks
@@ -102,57 +102,57 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (0.09124983333,1.000000000) node {1};
+\draw (0.091250,1.0000) node {1};
 \draw [] (-0.250,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,1.25);
 \draw [] (0.250,1.25) -- (-0.250,1.25);
 \draw [] (-0.250,1.25) -- (-0.250,0.750);
-\draw (0.5912498333,1.000000000) node {2};
+\draw (0.59125,1.0000) node {2};
 \draw [] (0.250,0.750) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.750,1.25);
 \draw [] (0.750,1.25) -- (0.250,1.25);
 \draw [] (0.250,1.25) -- (0.250,0.750);
-\draw (1.091249833,1.000000000) node {3};
+\draw (1.0912,1.0000) node {3};
 \draw [] (0.750,0.750) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (1.25,1.25);
 \draw [] (1.25,1.25) -- (0.750,1.25);
 \draw [] (0.750,1.25) -- (0.750,0.750);
-\draw (1.591249833,1.000000000) node {4};
+\draw (1.5912,1.0000) node {4};
 \draw [] (1.25,0.750) -- (1.75,0.750);
 \draw [] (1.75,0.750) -- (1.75,1.25);
 \draw [] (1.75,1.25) -- (1.25,1.25);
 \draw [] (1.25,1.25) -- (1.25,0.750);
-\draw (2.091249833,1.000000000) node {7};
+\draw (2.0913,1.0000) node {7};
 \draw [] (1.75,0.750) -- (2.25,0.750);
 \draw [] (2.25,0.750) -- (2.25,1.25);
 \draw [] (2.25,1.25) -- (1.75,1.25);
 \draw [] (1.75,1.25) -- (1.75,0.750);
-\draw (2.591249833,1.000000000) node {8};
+\draw (2.5913,1.0000) node {8};
 \draw [] (2.25,0.750) -- (2.75,0.750);
 \draw [] (2.75,0.750) -- (2.75,1.25);
 \draw [] (2.75,1.25) -- (2.25,1.25);
 \draw [] (2.25,1.25) -- (2.25,0.750);
-\draw (0.09124983333,0.5000000000) node {3};
+\draw (0.091250,0.50000) node {3};
 \draw [] (-0.250,0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (-0.250,0.750);
 \draw [] (-0.250,0.750) -- (-0.250,0.250);
-\draw (0.5912498333,0.5000000000) node {5};
+\draw (0.59125,0.50000) node {5};
 \draw [] (0.250,0.250) -- (0.750,0.250);
 \draw [] (0.750,0.250) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,0.250);
-\draw (1.091249833,0.5000000000) node {6};
+\draw (1.0912,0.50000) node {6};
 \draw [] (0.750,0.250) -- (1.25,0.250);
 \draw [] (1.25,0.250) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.750,0.250);
-\draw (1.591249833,0.5000000000) node {9};
+\draw (1.5912,0.50000) node {9};
 \draw [] (1.25,0.250) -- (1.75,0.250);
 \draw [] (1.75,0.250) -- (1.75,0.750);
 \draw [] (1.75,0.750) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (1.25,0.250);
-\draw (0.1824996667,0) node {10};
+\draw (0.18250,0) node {10};
 \draw [] (-0.250,-0.250) -- (0.250,-0.250);
 \draw [] (0.250,-0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (-0.250,0.250);
diff --git a/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks b/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks
index a79abb2ce..02a0a4b23 100644
--- a/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks
+++ b/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks
@@ -93,20 +93,20 @@
 \draw [] (2.00,3.46) -- (4.00,0);
 \draw [] (4.00,0) -- (0,0);
 \draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.251784057,0.3649246667) node {$60$};
+\draw (3.2518,0.36492) node {$60$};
 
 \draw [color=red] (3.75,0.433)--(3.75,0.430)--(3.74,0.428)--(3.74,0.425)--(3.73,0.422)--(3.73,0.419)--(3.72,0.416)--(3.72,0.413)--(3.71,0.410)--(3.71,0.407)--(3.71,0.404)--(3.70,0.401)--(3.70,0.398)--(3.69,0.395)--(3.69,0.391)--(3.68,0.388)--(3.68,0.385)--(3.68,0.381)--(3.67,0.378)--(3.67,0.374)--(3.66,0.371)--(3.66,0.367)--(3.66,0.364)--(3.65,0.360)--(3.65,0.356)--(3.65,0.353)--(3.64,0.349)--(3.64,0.345)--(3.63,0.341)--(3.63,0.337)--(3.63,0.333)--(3.62,0.329)--(3.62,0.325)--(3.62,0.321)--(3.61,0.317)--(3.61,0.313)--(3.61,0.309)--(3.60,0.305)--(3.60,0.301)--(3.60,0.296)--(3.59,0.292)--(3.59,0.288)--(3.59,0.284)--(3.59,0.279)--(3.58,0.275)--(3.58,0.270)--(3.58,0.266)--(3.57,0.261)--(3.57,0.257)--(3.57,0.252)--(3.57,0.248)--(3.56,0.243)--(3.56,0.238)--(3.56,0.234)--(3.56,0.229)--(3.55,0.224)--(3.55,0.220)--(3.55,0.215)--(3.55,0.210)--(3.54,0.205)--(3.54,0.200)--(3.54,0.196)--(3.54,0.191)--(3.54,0.186)--(3.53,0.181)--(3.53,0.176)--(3.53,0.171)--(3.53,0.166)--(3.53,0.161)--(3.52,0.156)--(3.52,0.151)--(3.52,0.146)--(3.52,0.141)--(3.52,0.136)--(3.52,0.131)--(3.52,0.126)--(3.51,0.120)--(3.51,0.115)--(3.51,0.110)--(3.51,0.105)--(3.51,0.0998)--(3.51,0.0946)--(3.51,0.0894)--(3.51,0.0842)--(3.51,0.0790)--(3.51,0.0738)--(3.50,0.0685)--(3.50,0.0633)--(3.50,0.0580)--(3.50,0.0528)--(3.50,0.0475)--(3.50,0.0423)--(3.50,0.0370)--(3.50,0.0317)--(3.50,0.0264)--(3.50,0.0211)--(3.50,0.0159)--(3.50,0.0106)--(3.50,0.00529)--(3.50,0);
-\draw (2.311909189,2.234016645) node {$30$};
+\draw (2.3119,2.2340) node {$30$};
 
 \draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
-\draw []  (2.000000000,3.464101615) node [rotate=0] {$\bullet$};
-\draw (2.000000000,3.888809615) node {$A$};
+\draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
+\draw (2.0000,3.8888) node {$A$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.4475840000,0) node {$B$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.443490000,0) node {$C$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.4247080000) node {$H$};
+\draw (-0.44758,0) node {$B$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.4435,0) node {$C$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.42471) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks b/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks
index 78705cb3c..bbb7404a6 100644
--- a/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks
+++ b/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.875000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,7.851851852);
+\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
 %DEFAULT
 \draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
 \draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
@@ -98,22 +98,22 @@
 \draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
 
 \draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
-\draw []  (3.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-0.2785761667) node {$a$};
-\draw []  (0,1.500000000) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,1.500000000) node {$f(a)$};
-\draw []  (6.500000000,6.085648148) node [rotate=0] {$\bullet$};
-\draw []  (6.500000000,0) node [rotate=0] {$\bullet$};
-\draw (6.500000000,-0.2785761667) node {$x$};
-\draw []  (0,6.085648148) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,6.085648148) node {$f(x)$};
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (3.000000000,1.300000000);
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (6.500000000,1.300000000);
-\draw (4.750000000,0.9789703333) node {$x-a$};
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,6.085648148);
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,1.500000000);
-\draw (7.825596167,3.792824074) node {$f(x)-f(a)$};
+\draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.27858) node {$a$};
+\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
+\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.27858) node {$x$};
+\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.45521,6.0856) node {$f(x)$};
+\draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
+\draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
+\draw (4.7500,0.97897) node {$x-a$};
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
+\draw (7.8256,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks b/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks
index c29b0028a..61cae4865 100644
--- a/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks
+++ b/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks
@@ -107,53 +107,53 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-5.261904762) -- (0,5.261904762);
+\draw [,->,>=latex] (-4.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-5.2619) -- (0,5.2619);
 %DEFAULT
 
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 \draw [color=blue] (1.210,-4.762)--(1.258,-3.870)--(1.307,-3.260)--(1.355,-2.816)--(1.404,-2.478)--(1.452,-2.213)--(1.500,-1.999)--(1.549,-1.823)--(1.597,-1.675)--(1.645,-1.549)--(1.694,-1.441)--(1.742,-1.347)--(1.791,-1.265)--(1.839,-1.192)--(1.887,-1.127)--(1.936,-1.069)--(1.984,-1.016)--(2.033,-0.9685)--(2.081,-0.9251)--(2.129,-0.8855)--(2.178,-0.8491)--(2.226,-0.8156)--(2.274,-0.7847)--(2.323,-0.7560)--(2.371,-0.7293)--(2.420,-0.7044)--(2.468,-0.6812)--(2.516,-0.6595)--(2.565,-0.6391)--(2.613,-0.6199)--(2.662,-0.6019)--(2.710,-0.5848)--(2.758,-0.5687)--(2.807,-0.5535)--(2.855,-0.5391)--(2.903,-0.5254)--(2.952,-0.5123)--(3.000,-0.5000)--(3.049,-0.4881)--(3.097,-0.4769)--(3.145,-0.4661)--(3.194,-0.4558)--(3.242,-0.4460)--(3.290,-0.4366)--(3.339,-0.4276)--(3.387,-0.4189)--(3.436,-0.4106)--(3.484,-0.4026)--(3.532,-0.3949)--(3.581,-0.3875)--(3.629,-0.3803)--(3.678,-0.3735)--(3.726,-0.3668)--(3.774,-0.3604)--(3.823,-0.3543)--(3.871,-0.3483)--(3.919,-0.3425)--(3.968,-0.3369)--(4.016,-0.3315)--(4.065,-0.3263)--(4.113,-0.3212)--(4.161,-0.3163)--(4.210,-0.3115)--(4.258,-0.3069)--(4.307,-0.3024)--(4.355,-0.2981)--(4.403,-0.2938)--(4.452,-0.2897)--(4.500,-0.2857)--(4.548,-0.2818)--(4.597,-0.2780)--(4.645,-0.2743)--(4.694,-0.2707)--(4.742,-0.2672)--(4.790,-0.2638)--(4.839,-0.2605)--(4.887,-0.2573)--(4.936,-0.2541)--(4.984,-0.2510)--(5.032,-0.2480)--(5.081,-0.2451)--(5.129,-0.2422)--(5.177,-0.2394)--(5.226,-0.2366)--(5.274,-0.2340)--(5.323,-0.2313)--(5.371,-0.2288)--(5.419,-0.2263)--(5.468,-0.2238)--(5.516,-0.2214)--(5.565,-0.2191)--(5.613,-0.2168)--(5.661,-0.2145)--(5.710,-0.2123)--(5.758,-0.2102)--(5.806,-0.2081)--(5.855,-0.2060)--(5.903,-0.2039)--(5.952,-0.2020)--(6.000,-0.2000);
 \draw [style=dotted] (1.00,-4.76) -- (1.00,4.76);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
 \draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks b/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks
index 9af02e5bb..1d7103b76 100644
--- a/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks
+++ b/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.499883862);
+\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -56,10 +56,10 @@
 \draw [color=blue,style=solid] (1.000,1.635)--(1.051,1.562)--(1.101,1.497)--(1.152,1.439)--(1.202,1.390)--(1.253,1.348)--(1.303,1.316)--(1.354,1.293)--(1.404,1.279)--(1.455,1.273)--(1.505,1.276)--(1.556,1.288)--(1.606,1.307)--(1.657,1.334)--(1.707,1.367)--(1.758,1.406)--(1.808,1.450)--(1.859,1.498)--(1.909,1.549)--(1.960,1.603)--(2.010,1.658)--(2.061,1.713)--(2.111,1.768)--(2.162,1.821)--(2.212,1.873)--(2.263,1.921)--(2.313,1.965)--(2.364,2.006)--(2.414,2.041)--(2.465,2.071)--(2.515,2.096)--(2.566,2.116)--(2.616,2.129)--(2.667,2.137)--(2.717,2.139)--(2.768,2.136)--(2.818,2.128)--(2.869,2.116)--(2.919,2.100)--(2.970,2.081)--(3.020,2.059)--(3.071,2.035)--(3.121,2.010)--(3.172,1.985)--(3.222,1.960)--(3.273,1.937)--(3.323,1.915)--(3.374,1.896)--(3.424,1.881)--(3.475,1.870)--(3.525,1.863)--(3.576,1.861)--(3.626,1.864)--(3.677,1.873)--(3.727,1.888)--(3.778,1.908)--(3.828,1.934)--(3.879,1.965)--(3.929,2.002)--(3.980,2.043)--(4.030,2.088)--(4.081,2.137)--(4.131,2.189)--(4.182,2.242)--(4.232,2.297)--(4.283,2.353)--(4.333,2.408)--(4.384,2.461)--(4.434,2.512)--(4.485,2.559)--(4.535,2.602)--(4.586,2.640)--(4.636,2.672)--(4.687,2.697)--(4.737,2.715)--(4.788,2.725)--(4.838,2.727)--(4.889,2.719)--(4.939,2.703)--(4.990,2.678)--(5.040,2.644)--(5.091,2.602)--(5.141,2.550)--(5.192,2.491)--(5.242,2.424)--(5.293,2.351)--(5.343,2.271)--(5.394,2.186)--(5.444,2.097)--(5.495,2.005)--(5.545,1.911)--(5.596,1.816)--(5.646,1.722)--(5.697,1.629)--(5.747,1.538)--(5.798,1.452)--(5.849,1.370)--(5.899,1.294)--(5.950,1.225)--(6.000,1.165);
 \draw [color=magenta,style=dashed] (1.00,1.63) -- (1.00,4.84);
 \draw [color=magenta,style=dashed] (6.00,3.72) -- (6.00,1.16);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.2785761667) node {$a$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3267360000) node {$b$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.27858) node {$a$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.32674) node {$b$};
 \draw [style=dotted] (1.00,0) -- (1.00,4.84);
 
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@@ -108,8 +108,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.499883862,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -123,10 +123,10 @@
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 \draw [color=magenta,style=dashed] (1.16,6.00) -- (3.72,6.00);
 \draw [color=magenta,style=dashed] (4.84,1.00) -- (1.63,1.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2789780000,1.000000000) node {$c$};
-\draw []  (0,6.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2949888333,6.000000000) node {$d$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.27898,1.0000) node {$c$};
+\draw []  (0,6.0000) node [rotate=0] {$\bullet$};
+\draw (-0.29499,6.0000) node {$d$};
 \draw [style=dotted] (0,1.00) -- (4.84,1.00);
 \draw [style=dotted] (0,6.00) -- (1.16,6.00);
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diff --git a/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks b/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks
index 4b211b1e2..bf237b1bd 100644
--- a/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks
+++ b/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=red] (0,0)--(0.0159,0)--(0.0317,0)--(0.0476,0.00113)--(0.0634,0.00201)--(0.0792,0.00315)--(0.0951,0.00453)--(0.111,0.00616)--(0.127,0.00805)--(0.142,0.0102)--(0.158,0.0126)--(0.174,0.0152)--(0.189,0.0181)--(0.205,0.0212)--(0.220,0.0246)--(0.236,0.0282)--(0.251,0.0321)--(0.266,0.0362)--(0.282,0.0405)--(0.297,0.0451)--(0.312,0.0499)--(0.327,0.0550)--(0.342,0.0603)--(0.357,0.0658)--(0.372,0.0716)--(0.386,0.0776)--(0.401,0.0839)--(0.415,0.0904)--(0.430,0.0971)--(0.444,0.104)--(0.458,0.111)--(0.472,0.119)--(0.486,0.126)--(0.500,0.134)--(0.514,0.142)--(0.527,0.150)--(0.541,0.159)--(0.554,0.167)--(0.567,0.176)--(0.580,0.185)--(0.593,0.195)--(0.606,0.204)--(0.618,0.214)--(0.631,0.224)--(0.643,0.234)--(0.655,0.244)--(0.667,0.255)--(0.679,0.265)--(0.690,0.276)--(0.701,0.287)--(0.713,0.299)--(0.724,0.310)--(0.735,0.322)--(0.745,0.333)--(0.756,0.345)--(0.766,0.357)--(0.776,0.369)--(0.786,0.382)--(0.796,0.394)--(0.805,0.407)--(0.815,0.420)--(0.824,0.433)--(0.833,0.446)--(0.841,0.459)--(0.850,0.473)--(0.858,0.486)--(0.866,0.500)--(0.874,0.514)--(0.881,0.528)--(0.889,0.542)--(0.896,0.556)--(0.903,0.570)--(0.910,0.585)--(0.916,0.599)--(0.922,0.614)--(0.928,0.628)--(0.934,0.643)--(0.940,0.658)--(0.945,0.673)--(0.950,0.688)--(0.955,0.703)--(0.959,0.718)--(0.964,0.734)--(0.968,0.749)--(0.972,0.764)--(0.975,0.780)--(0.979,0.795)--(0.982,0.811)--(0.985,0.826)--(0.987,0.842)--(0.990,0.858)--(0.992,0.873)--(0.994,0.889)--(0.995,0.905)--(0.997,0.921)--(0.998,0.937)--(0.999,0.952)--(1.00,0.968)--(1.00,0.984)--(1.00,1.00);
@@ -80,8 +80,8 @@
 \draw [] (0,0) -- (0,0);
 \draw [] (1.00,1.00) -- (1.00,1.00);
 \draw [color=red] (-0.500,-0.500) -- (1.50,1.50);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.354646868,0.6631599656) node {$P$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3546,0.66316) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks b/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks
index 6a7b1f5d8..d8605c418 100644
--- a/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks
+++ b/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks
@@ -138,8 +138,8 @@
 \draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
 \draw [color=gray,style=solid] (-5.00,6.00) -- (2.00,6.00);
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,6.5000);
 %DEFAULT
 
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@@ -147,42 +147,42 @@
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 \draw [color=blue] (1.000,3.000)--(1.010,3.030)--(1.020,3.061)--(1.030,3.091)--(1.040,3.121)--(1.051,3.152)--(1.061,3.182)--(1.071,3.212)--(1.081,3.242)--(1.091,3.273)--(1.101,3.303)--(1.111,3.333)--(1.121,3.364)--(1.131,3.394)--(1.141,3.424)--(1.152,3.455)--(1.162,3.485)--(1.172,3.515)--(1.182,3.545)--(1.192,3.576)--(1.202,3.606)--(1.212,3.636)--(1.222,3.667)--(1.232,3.697)--(1.242,3.727)--(1.253,3.758)--(1.263,3.788)--(1.273,3.818)--(1.283,3.848)--(1.293,3.879)--(1.303,3.909)--(1.313,3.939)--(1.323,3.970)--(1.333,4.000)--(1.343,4.030)--(1.354,4.061)--(1.364,4.091)--(1.374,4.121)--(1.384,4.151)--(1.394,4.182)--(1.404,4.212)--(1.414,4.242)--(1.424,4.273)--(1.434,4.303)--(1.444,4.333)--(1.455,4.364)--(1.465,4.394)--(1.475,4.424)--(1.485,4.455)--(1.495,4.485)--(1.505,4.515)--(1.515,4.545)--(1.525,4.576)--(1.535,4.606)--(1.545,4.636)--(1.556,4.667)--(1.566,4.697)--(1.576,4.727)--(1.586,4.758)--(1.596,4.788)--(1.606,4.818)--(1.616,4.849)--(1.626,4.879)--(1.636,4.909)--(1.646,4.939)--(1.657,4.970)--(1.667,5.000)--(1.677,5.030)--(1.687,5.061)--(1.697,5.091)--(1.707,5.121)--(1.717,5.151)--(1.727,5.182)--(1.737,5.212)--(1.747,5.242)--(1.758,5.273)--(1.768,5.303)--(1.778,5.333)--(1.788,5.364)--(1.798,5.394)--(1.808,5.424)--(1.818,5.455)--(1.828,5.485)--(1.838,5.515)--(1.848,5.545)--(1.859,5.576)--(1.869,5.606)--(1.879,5.636)--(1.889,5.667)--(1.899,5.697)--(1.909,5.727)--(1.919,5.758)--(1.929,5.788)--(1.939,5.818)--(1.949,5.849)--(1.960,5.879)--(1.970,5.909)--(1.980,5.939)--(1.990,5.970)--(2.000,6.000);
-\draw []  (0,3.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,3.000000000) node [rotate=0] {$o$};
+\draw []  (1.0000,3.0000) node [rotate=0] {$o$};
 
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks b/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks
index 7974b5f8f..b7cae7f35 100644
--- a/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks
+++ b/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks
@@ -99,20 +99,20 @@
 
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 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.5086990000) node {\( \tilde \phi(a)\)};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.517164111,-0.3372565219) node {\( \tilde\phi(c)\)};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.5086990000) node {\( \tilde \phi(b)\)};
+\draw (0,-0.50870) node {\( \tilde \phi(a)\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.5172,-0.33726) node {\( \tilde\phi(c)\)};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.50870) node {\( \tilde \phi(b)\)};
 \draw [] (-4.00,0) -- (8.00,0);
-\draw []  (1.079008574,3.683965706) node [rotate=0] {$\bullet$};
-\draw (1.689894907,3.683965706) node {\( \tilde\phi(m)\)};
-\draw (-2.292780055,3.308699000) node {\( \tilde \phi(A)\)};
-\draw (6.660974700,5.471440700) node {\( \tilde \phi(B)\)};
-\draw []  (1.744520838,1.021916647) node [rotate=0] {$\bullet$};
-\draw (2.035770672,1.021916647) node {\( 0\)};
+\draw []  (1.0790,3.6840) node [rotate=0] {$\bullet$};
+\draw (1.6899,3.6840) node {\( \tilde\phi(m)\)};
+\draw (-2.2928,3.3087) node {\( \tilde \phi(A)\)};
+\draw (6.6610,5.4714) node {\( \tilde \phi(B)\)};
+\draw []  (1.7445,1.0219) node [rotate=0] {$\bullet$};
+\draw (2.0358,1.0219) node {\( 0\)};
 \draw [] (0.746,5.02) -- (2.08,-0.309);
-\draw (8.557386667,0) node {\( \tilde\phi(\mC)\)};
+\draw (8.5574,0) node {\( \tilde\phi(\mC)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntEcourbe.pstricks b/auto/pictures_tex/Fig_IntEcourbe.pstricks
index 7f740bca4..f9fb48682 100644
--- a/auto/pictures_tex/Fig_IntEcourbe.pstricks
+++ b/auto/pictures_tex/Fig_IntEcourbe.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
 %DEFAULT
 
 \draw [color=blue] (1.000,3.500)--(1.020,3.460)--(1.040,3.421)--(1.061,3.382)--(1.081,3.345)--(1.101,3.308)--(1.121,3.272)--(1.141,3.237)--(1.162,3.203)--(1.182,3.169)--(1.202,3.137)--(1.222,3.105)--(1.242,3.074)--(1.263,3.044)--(1.283,3.014)--(1.303,2.986)--(1.323,2.958)--(1.343,2.931)--(1.364,2.905)--(1.384,2.880)--(1.404,2.855)--(1.424,2.831)--(1.444,2.809)--(1.465,2.787)--(1.485,2.765)--(1.505,2.745)--(1.525,2.725)--(1.545,2.707)--(1.566,2.689)--(1.586,2.672)--(1.606,2.655)--(1.626,2.640)--(1.646,2.625)--(1.667,2.611)--(1.687,2.598)--(1.707,2.586)--(1.727,2.574)--(1.747,2.564)--(1.768,2.554)--(1.788,2.545)--(1.808,2.537)--(1.828,2.529)--(1.848,2.523)--(1.869,2.517)--(1.889,2.512)--(1.909,2.508)--(1.929,2.505)--(1.949,2.503)--(1.970,2.501)--(1.990,2.500)--(2.010,2.500)--(2.030,2.501)--(2.051,2.503)--(2.071,2.505)--(2.091,2.508)--(2.111,2.512)--(2.131,2.517)--(2.152,2.523)--(2.172,2.529)--(2.192,2.537)--(2.212,2.545)--(2.232,2.554)--(2.253,2.564)--(2.273,2.574)--(2.293,2.586)--(2.313,2.598)--(2.333,2.611)--(2.354,2.625)--(2.374,2.640)--(2.394,2.655)--(2.414,2.672)--(2.434,2.689)--(2.455,2.707)--(2.475,2.725)--(2.495,2.745)--(2.515,2.765)--(2.535,2.787)--(2.556,2.809)--(2.576,2.831)--(2.596,2.855)--(2.616,2.880)--(2.636,2.905)--(2.657,2.931)--(2.677,2.958)--(2.697,2.986)--(2.717,3.014)--(2.737,3.044)--(2.758,3.074)--(2.778,3.105)--(2.798,3.137)--(2.818,3.169)--(2.838,3.203)--(2.859,3.237)--(2.879,3.272)--(2.899,3.308)--(2.919,3.345)--(2.939,3.382)--(2.960,3.421)--(2.980,3.460)--(3.000,3.500);
@@ -103,19 +103,19 @@
 \draw [color=cyan] (3.00,3.50) -- (3.00,0.641);
 \draw [color=cyan] (3.00,0.641) -- (1.00,0.641);
 \draw [color=cyan] (1.00,0.641) -- (1.00,3.50);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntTrois.pstricks b/auto/pictures_tex/Fig_IntTrois.pstricks
index efe3a3fb1..654f2a3b0 100644
--- a/auto/pictures_tex/Fig_IntTrois.pstricks
+++ b/auto/pictures_tex/Fig_IntTrois.pstricks
@@ -37,19 +37,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \fill [color=lightgray] (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,0) --  cycle;
 \draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
 \draw [] (1.73,1.00) -- (1.73,2.00);
 \draw [color=red] (0,2.00) -- (2.00,2.00);
 \draw [color=red] (2.00,2.00) -- (2.00,0);
-\draw []  (1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.732050808,2.000000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.7320,2.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -84,8 +84,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [style=dotted] (0,0) -- (1.64,1.15);
 \draw [style=dotted] (0,0) -- (1.15,1.64);
@@ -94,14 +94,14 @@
 \draw [color=red] (2.00,2.00) -- (2.00,0);
 \draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
 \draw [] (1.64,1.15) -- (2.00,1.40);
-\draw []  (1.638304089,1.147152873) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,1.400415076) node [rotate=0] {$\bullet$};
-\draw []  (1.147152873,1.638304089) node [rotate=0] {$\bullet$};
-\draw []  (1.400415076,2.000000000) node [rotate=0] {$\bullet$};
+\draw []  (1.6383,1.1472) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,1.4004) node [rotate=0] {$\bullet$};
+\draw []  (1.1472,1.6383) node [rotate=0] {$\bullet$};
+\draw []  (1.4004,2.0000) node [rotate=0] {$\bullet$};
 \draw [] (1.15,1.64) -- (1.40,2.00);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntervalleUn.pstricks b/auto/pictures_tex/Fig_IntervalleUn.pstricks
index fb91c285c..27457d70b 100644
--- a/auto/pictures_tex/Fig_IntervalleUn.pstricks
+++ b/auto/pictures_tex/Fig_IntervalleUn.pstricks
@@ -82,18 +82,18 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (-1.50,0) -- (4.50,0);
-\draw []  (0.9000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.9000000000,-0.3785761667) node {$a$};
+\draw []  (0.90000,0) node [rotate=0] {$\bullet$};
+\draw (0.90000,-0.37858) node {$a$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.4149246667) node {$0$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-0.4149246667) node {$1$};
-\draw [,->,>=latex] (0.4500000000,0.3000000000) -- (0,0.3000000000);
-\draw [,->,>=latex] (0.4500000000,0.3000000000) -- (0.9000000000,0.3000000000);
-\draw (0.4500000000,0.6785761667) node {$a$};
-\draw [,->,>=latex] (1.950000000,0.3000000000) -- (0.9000000000,0.3000000000);
-\draw [,->,>=latex] (1.950000000,0.3000000000) -- (3.000000000,0.3000000000);
-\draw (1.950000000,0.7298256667) node {$1-a$};
+\draw (0,-0.41492) node {$0$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.41492) node {$1$};
+\draw [,->,>=latex] (0.45000,0.30000) -- (0,0.30000);
+\draw [,->,>=latex] (0.45000,0.30000) -- (0.90000,0.30000);
+\draw (0.45000,0.67858) node {$a$};
+\draw [,->,>=latex] (1.9500,0.30000) -- (0.90000,0.30000);
+\draw [,->,>=latex] (1.9500,0.30000) -- (3.0000,0.30000);
+\draw (1.9500,0.72983) node {$1-a$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_IsomCarre.pstricks b/auto/pictures_tex/Fig_IsomCarre.pstricks
index bc9176e86..52c75fa0f 100644
--- a/auto/pictures_tex/Fig_IsomCarre.pstricks
+++ b/auto/pictures_tex/Fig_IsomCarre.pstricks
@@ -90,15 +90,15 @@
 \draw [color=blue] (1.00,1.00) -- (-1.00,1.00);
 \draw [color=blue] (-1.00,1.00) -- (-1.00,-1.00);
 \draw [] (0,-1.50) -- (0,1.50);
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.207585845,1.195418678) node {\( A\)};
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.218294678,1.195418678) node {\( B\)};
-\draw []  (1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.214200678,-1.195418678) node {\( C\)};
-\draw []  (-1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.226875011,-1.195418678) node {\( D\)};
-\draw (0.1562573448,1.649286845) node {\( s\)};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.2076,1.1954) node {\( A\)};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.2183,1.1954) node {\( B\)};
+\draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (1.2142,-1.1954) node {\( C\)};
+\draw []  (-1.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.2269,-1.1954) node {\( D\)};
+\draw (0.15626,1.6493) node {\( s\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_LAfWmaN.pstricks b/auto/pictures_tex/Fig_LAfWmaN.pstricks
index 43ddf6e0c..45c4fc2fa 100644
--- a/auto/pictures_tex/Fig_LAfWmaN.pstricks
+++ b/auto/pictures_tex/Fig_LAfWmaN.pstricks
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.620000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.100000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.6200,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.1000);
 %DEFAULT
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@@ -100,23 +100,23 @@
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LBGooAdteCt.pstricks b/auto/pictures_tex/Fig_LBGooAdteCt.pstricks
index 516c4289e..5bc36bb3b 100644
--- a/auto/pictures_tex/Fig_LBGooAdteCt.pstricks
+++ b/auto/pictures_tex/Fig_LBGooAdteCt.pstricks
@@ -126,35 +126,35 @@
 \draw [color=gray,style=solid] (-5.00,4.00) -- (2.00,4.00);
 \draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
 
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-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks b/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks
index 1dd005af5..03400f4b5 100644
--- a/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks
+++ b/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -79,14 +79,14 @@
          hatchthickness=0.4pt}
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-\draw [,->,>=latex] (1.500000000,0.7500000000) -- (1.521213203,0.7712132034);
+\draw [,->,>=latex] (1.5000,0.75000) -- (1.5212,0.77121);
 \draw [color=green] (0,0) -- (0,0);
 \draw [color=green] (3.00,3.00) -- (3.00,3.00);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks b/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks
index c7390d669..d1387e358 100644
--- a/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks
+++ b/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks
@@ -100,37 +100,37 @@
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+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
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-\draw (3.000000000,-0.3149246667) node {$ 3 $};
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 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Laurin.pstricks b/auto/pictures_tex/Fig_Laurin.pstricks
index f96d4776d..bd70fac13 100644
--- a/auto/pictures_tex/Fig_Laurin.pstricks
+++ b/auto/pictures_tex/Fig_Laurin.pstricks
@@ -99,8 +99,8 @@
 %GRID
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-\draw [,->,>=latex] (0,-1.500000000) -- (0,7.889056099);
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+\draw [,->,>=latex] (0,-1.5000) -- (0,7.8891);
 %DEFAULT
 
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@@ -110,29 +110,29 @@
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-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
-\draw (-0.2912498333,7.000000000) node {$ 7 $};
+\draw (-0.29125,7.0000) node {$ 7 $};
 \draw [] (-0.100,7.00) -- (0.100,7.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MCQueGF.pstricks b/auto/pictures_tex/Fig_MCQueGF.pstricks
index 681ea85af..8594e6364 100644
--- a/auto/pictures_tex/Fig_MCQueGF.pstricks
+++ b/auto/pictures_tex/Fig_MCQueGF.pstricks
@@ -87,36 +87,36 @@
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 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
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-\draw (-0.2912498333,2.000000000) node {$ 2 $};
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Mantisse.pstricks b/auto/pictures_tex/Fig_Mantisse.pstricks
index 9669ab94c..1c275f207 100644
--- a/auto/pictures_tex/Fig_Mantisse.pstricks
+++ b/auto/pictures_tex/Fig_Mantisse.pstricks
@@ -83,8 +83,8 @@
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-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.498998999);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
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@@ -94,17 +94,17 @@
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks b/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks
index 039e10817..fc28bdea2 100644
--- a/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks
+++ b/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.249988663);
+\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -100,16 +100,16 @@
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          hatchthickness=0.4pt}
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-\draw [color=red] (5.00,0) -- (6.00,0);
-\draw [color=red] (6.00,0) -- (6.00,2.64);
-\draw [color=red] (6.00,2.64) -- (5.00,2.64);
-\draw [color=red] (5.00,2.64) -- (5.00,0);
-\draw []  (5.000000000,2.638888889) node [rotate=0] {$\bullet$};
-\draw (5.441978850,3.211818713) node {$f(x)$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.2785761667) node {$x$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3455708333) node {$x+\Delta x$};
+\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
+\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
+\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
+\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
+\draw (5.4420,3.2118) node {$f(x)$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.27858) node {$x$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.34557) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks b/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks
index 51c4c7dea..9dc95b22d 100644
--- a/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks
+++ b/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.662277660) -- (0,3.662277660);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
 %DEFAULT
 
 \draw [color=red] (-3.000,3.162)--(-2.939,3.105)--(-2.879,3.048)--(-2.818,2.990)--(-2.758,2.933)--(-2.697,2.876)--(-2.636,2.820)--(-2.576,2.763)--(-2.515,2.707)--(-2.455,2.650)--(-2.394,2.594)--(-2.333,2.539)--(-2.273,2.483)--(-2.212,2.428)--(-2.152,2.373)--(-2.091,2.318)--(-2.030,2.263)--(-1.970,2.209)--(-1.909,2.155)--(-1.848,2.102)--(-1.788,2.049)--(-1.727,1.996)--(-1.667,1.944)--(-1.606,1.892)--(-1.545,1.841)--(-1.485,1.790)--(-1.424,1.740)--(-1.364,1.691)--(-1.303,1.643)--(-1.242,1.595)--(-1.182,1.548)--(-1.121,1.502)--(-1.061,1.458)--(-1.000,1.414)--(-0.9394,1.372)--(-0.8788,1.331)--(-0.8182,1.292)--(-0.7576,1.255)--(-0.6970,1.219)--(-0.6364,1.185)--(-0.5758,1.154)--(-0.5152,1.125)--(-0.4545,1.098)--(-0.3939,1.075)--(-0.3333,1.054)--(-0.2727,1.037)--(-0.2121,1.022)--(-0.1515,1.011)--(-0.09091,1.004)--(-0.03030,1.000)--(0.03030,1.000)--(0.09091,1.004)--(0.1515,1.011)--(0.2121,1.022)--(0.2727,1.037)--(0.3333,1.054)--(0.3939,1.075)--(0.4545,1.098)--(0.5152,1.125)--(0.5758,1.154)--(0.6364,1.185)--(0.6970,1.219)--(0.7576,1.255)--(0.8182,1.292)--(0.8788,1.331)--(0.9394,1.372)--(1.000,1.414)--(1.061,1.458)--(1.121,1.502)--(1.182,1.548)--(1.242,1.595)--(1.303,1.643)--(1.364,1.691)--(1.424,1.740)--(1.485,1.790)--(1.545,1.841)--(1.606,1.892)--(1.667,1.944)--(1.727,1.996)--(1.788,2.049)--(1.848,2.102)--(1.909,2.155)--(1.970,2.209)--(2.030,2.263)--(2.091,2.318)--(2.152,2.373)--(2.212,2.428)--(2.273,2.483)--(2.333,2.539)--(2.394,2.594)--(2.455,2.650)--(2.515,2.707)--(2.576,2.763)--(2.636,2.820)--(2.697,2.876)--(2.758,2.933)--(2.818,2.990)--(2.879,3.048)--(2.939,3.105)--(3.000,3.162);
@@ -104,33 +104,33 @@
 \draw [color=red] (-3.000,-3.162)--(-2.939,-3.105)--(-2.879,-3.048)--(-2.818,-2.990)--(-2.758,-2.933)--(-2.697,-2.876)--(-2.636,-2.820)--(-2.576,-2.763)--(-2.515,-2.707)--(-2.455,-2.650)--(-2.394,-2.594)--(-2.333,-2.539)--(-2.273,-2.483)--(-2.212,-2.428)--(-2.152,-2.373)--(-2.091,-2.318)--(-2.030,-2.263)--(-1.970,-2.209)--(-1.909,-2.155)--(-1.848,-2.102)--(-1.788,-2.049)--(-1.727,-1.996)--(-1.667,-1.944)--(-1.606,-1.892)--(-1.545,-1.841)--(-1.485,-1.790)--(-1.424,-1.740)--(-1.364,-1.691)--(-1.303,-1.643)--(-1.242,-1.595)--(-1.182,-1.548)--(-1.121,-1.502)--(-1.061,-1.458)--(-1.000,-1.414)--(-0.9394,-1.372)--(-0.8788,-1.331)--(-0.8182,-1.292)--(-0.7576,-1.255)--(-0.6970,-1.219)--(-0.6364,-1.185)--(-0.5758,-1.154)--(-0.5152,-1.125)--(-0.4545,-1.098)--(-0.3939,-1.075)--(-0.3333,-1.054)--(-0.2727,-1.037)--(-0.2121,-1.022)--(-0.1515,-1.011)--(-0.09091,-1.004)--(-0.03030,-1.000)--(0.03030,-1.000)--(0.09091,-1.004)--(0.1515,-1.011)--(0.2121,-1.022)--(0.2727,-1.037)--(0.3333,-1.054)--(0.3939,-1.075)--(0.4545,-1.098)--(0.5152,-1.125)--(0.5758,-1.154)--(0.6364,-1.185)--(0.6970,-1.219)--(0.7576,-1.255)--(0.8182,-1.292)--(0.8788,-1.331)--(0.9394,-1.372)--(1.000,-1.414)--(1.061,-1.458)--(1.121,-1.502)--(1.182,-1.548)--(1.242,-1.595)--(1.303,-1.643)--(1.364,-1.691)--(1.424,-1.740)--(1.485,-1.790)--(1.545,-1.841)--(1.606,-1.892)--(1.667,-1.944)--(1.727,-1.996)--(1.788,-2.049)--(1.848,-2.102)--(1.909,-2.155)--(1.970,-2.209)--(2.030,-2.263)--(2.091,-2.318)--(2.152,-2.373)--(2.212,-2.428)--(2.273,-2.483)--(2.333,-2.539)--(2.394,-2.594)--(2.455,-2.650)--(2.515,-2.707)--(2.576,-2.763)--(2.636,-2.820)--(2.697,-2.876)--(2.758,-2.933)--(2.818,-2.990)--(2.879,-3.048)--(2.939,-3.105)--(3.000,-3.162);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.2106906781,0.8045813219) node {$R$};
-\draw []  (0,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.1931375115,-0.8045813219) node {$S$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.21069,0.80458) node {$R$};
+\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.19314,-0.80458) node {$S$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ProjectionScalaire.pstricks b/auto/pictures_tex/Fig_ProjectionScalaire.pstricks
index 89cf83d91..aa94342af 100644
--- a/auto/pictures_tex/Fig_ProjectionScalaire.pstricks
+++ b/auto/pictures_tex/Fig_ProjectionScalaire.pstricks
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
-\draw [color=blue,->,>=latex] (0,0) -- (1.500000000,2.000000000);
-\draw (1.877649201,2.336840034) node {$X$};
-\draw [color=blue,->,>=latex] (0,0) -- (2.500000000,0);
-\draw (2.500000000,0.3247080000) node {$Y$};
-\draw []  (1.500000000,0) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (0,0) -- (1.5000,2.0000);
+\draw (1.8776,2.3368) node {$X$};
+\draw [color=blue,->,>=latex] (0,0) -- (2.5000,0);
+\draw (2.5000,0.32471) node {$Y$};
+\draw []  (1.5000,0) node [rotate=0] {$\bullet$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (0.7500000000,-0.3000000000) -- (0,-0.3000000000);
-\draw [,->,>=latex] (0.7500000000,-0.3000000000) -- (1.500000000,-0.3000000000);
-\draw (0.7500000000,-0.6785761667) node {$x$};
+\draw [,->,>=latex] (0.75000,-0.30000) -- (0,-0.30000);
+\draw [,->,>=latex] (0.75000,-0.30000) -- (1.5000,-0.30000);
+\draw (0.75000,-0.67858) node {$x$};
 \draw [style=dotted] (1.50,2.00) -- (1.50,0);
 
 %OTHER STUFF
diff --git a/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks b/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks
index 8f4f807c0..3b06e3eda 100644
--- a/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks
+++ b/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks
@@ -78,12 +78,12 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (-1.411630648,1.580406821) node {\( A\)};
-\draw (-1.979984611,-0.7916472795) node {\( B\)};
-\draw (1.975890611,0.7916472795) node {\( C\)};
+\draw (-1.4116,1.5804) node {\( A\)};
+\draw (-1.9800,-0.79165) node {\( B\)};
+\draw (1.9759,0.79165) node {\( C\)};
 
 \draw [] (1.750,0)--(1.746,0.1110)--(1.736,0.2215)--(1.718,0.3312)--(1.694,0.4395)--(1.663,0.5461)--(1.625,0.6504)--(1.580,0.7521)--(1.529,0.8508)--(1.472,0.9461)--(1.409,1.038)--(1.341,1.125)--(1.267,1.208)--(1.187,1.286)--(1.103,1.358)--(1.015,1.426)--(0.9226,1.487)--(0.8265,1.543)--(0.7270,1.592)--(0.6245,1.635)--(0.5196,1.671)--(0.4126,1.701)--(0.3039,1.723)--(0.1940,1.739)--(0.08327,1.748)--(-0.02777,1.750)--(-0.1387,1.744)--(-0.2491,1.732)--(-0.3584,1.713)--(-0.4663,1.687)--(-0.5724,1.654)--(-0.6761,1.614)--(-0.7771,1.568)--(-0.8750,1.516)--(-0.9694,1.457)--(-1.060,1.393)--(-1.146,1.323)--(-1.228,1.247)--(-1.304,1.167)--(-1.376,1.082)--(-1.441,0.9924)--(-1.501,0.8989)--(-1.555,0.8019)--(-1.603,0.7016)--(-1.644,0.5985)--(-1.679,0.4930)--(-1.707,0.3855)--(-1.728,0.2765)--(-1.742,0.1663)--(-1.749,0.05552)--(-1.749,-0.05552)--(-1.742,-0.1663)--(-1.728,-0.2765)--(-1.707,-0.3855)--(-1.679,-0.4930)--(-1.644,-0.5985)--(-1.603,-0.7016)--(-1.555,-0.8019)--(-1.501,-0.8989)--(-1.441,-0.9924)--(-1.376,-1.082)--(-1.304,-1.167)--(-1.228,-1.247)--(-1.146,-1.323)--(-1.060,-1.393)--(-0.9694,-1.457)--(-0.8750,-1.516)--(-0.7771,-1.568)--(-0.6761,-1.614)--(-0.5724,-1.654)--(-0.4663,-1.687)--(-0.3584,-1.713)--(-0.2491,-1.732)--(-0.1387,-1.744)--(-0.02777,-1.750)--(0.08327,-1.748)--(0.1940,-1.739)--(0.3039,-1.723)--(0.4126,-1.701)--(0.5196,-1.671)--(0.6245,-1.635)--(0.7270,-1.592)--(0.8265,-1.543)--(0.9226,-1.487)--(1.015,-1.426)--(1.103,-1.358)--(1.187,-1.286)--(1.267,-1.208)--(1.341,-1.125)--(1.409,-1.038)--(1.472,-0.9461)--(1.529,-0.8508)--(1.580,-0.7521)--(1.625,-0.6504)--(1.663,-0.5461)--(1.694,-0.4395)--(1.718,-0.3312)--(1.736,-0.2215)--(1.746,-0.1110)--(1.750,0);
-\draw (-0.4228499726,-0.5520606245) node {\( H\)};
+\draw (-0.42285,-0.55206) node {\( H\)};
 \draw [] (-1.21,1.27) -- (-0.658,-0.239);
 \draw [] (-1.21,1.27) -- (-1.64,-0.599);
 \draw [] (-1.64,-0.599) -- (1.64,0.599);
diff --git a/auto/pictures_tex/Fig_QPcdHwP.pstricks b/auto/pictures_tex/Fig_QPcdHwP.pstricks
index 2142b819a..bc69167a5 100644
--- a/auto/pictures_tex/Fig_QPcdHwP.pstricks
+++ b/auto/pictures_tex/Fig_QPcdHwP.pstricks
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3059510000) node {\( \alpha_2\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,-0.3059510000) node {\( \alpha_4\)};
+\draw (0,-0.30595) node {\( \alpha_2\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,-0.30595) node {\( \alpha_4\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks b/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks
index 84f832b73..af9c3b2cf 100644
--- a/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks
+++ b/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (-2.10,0.700) -- (2.10,0.700);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3824550000) node {\( \pi(b_1)\)};
-\draw []  (0.7000000000,0.7000000000) node [rotate=0] {$\bullet$};
-\draw (0.7000000000,1.082455000) node {\( \pi(b_2)\)};
+\draw (0,-0.38245) node {\( \pi(b_1)\)};
+\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
+\draw (0.70000,1.0825) node {\( \pi(b_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks b/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks
index 1c4a0c331..9b902f047 100644
--- a/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks
+++ b/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks
@@ -92,26 +92,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-1.945688830) -- (0,1.945688830);
+\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-1.9457) -- (0,1.9457);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,0.059192)--(-5.8788,0.027135)--(-5.7576,0.0060480)--(-5.6364,0)--(-5.5152,0.011586)--(-5.3939,0.039487)--(-5.2727,0.080491)--(-5.1515,0.12891)--(-5.0303,0.17762)--(-4.9091,0.21910)--(-4.7879,0.24669)--(-4.6667,0.25565)--(-4.5455,0.24409)--(-4.4242,0.21334)--(-4.3030,0.16799)--(-4.1818,0.11529)--(-4.0606,0.064108)--(-3.9394,0.023666)--(-3.8182,0.0020298)--(-3.6970,0.0047735)--(-3.5758,0.033930)--(-3.4545,0.087448)--(-3.3333,0.15925)--(-3.2121,0.23994)--(-3.0909,0.31806)--(-2.9697,0.38179)--(-2.8485,0.42076)--(-2.7273,0.42785)--(-2.6061,0.40058)--(-2.4848,0.34187)--(-2.3636,0.26014)--(-2.2424,0.16851)--(-2.1212,0.083272)--(-2.0000,0.021790)--(-1.8788,0)--(-1.7576,0.030313)--(-1.6364,0.11884)--(-1.5152,0.26464)--(-1.3939,0.45879)--(-1.2727,0.68492)--(-1.1515,0.92065)--(-1.0303,1.1399)--(-0.90909,1.3159)--(-0.78788,1.4242)--(-0.66667,1.4457)--(-0.54545,1.3694)--(-0.42424,1.1936)--(-0.30303,0.92708)--(-0.18182,0.58774)--(-0.060606,0.20133)--(0.060606,-0.20133)--(0.18182,-0.58774)--(0.30303,-0.92708)--(0.42424,-1.1936)--(0.54545,-1.3694)--(0.66667,-1.4457)--(0.78788,-1.4242)--(0.90909,-1.3159)--(1.0303,-1.1399)--(1.1515,-0.92065)--(1.2727,-0.68492)--(1.3939,-0.45879)--(1.5152,-0.26464)--(1.6364,-0.11884)--(1.7576,-0.030313)--(1.8788,0)--(2.0000,-0.021790)--(2.1212,-0.083272)--(2.2424,-0.16851)--(2.3636,-0.26014)--(2.4848,-0.34187)--(2.6061,-0.40058)--(2.7273,-0.42785)--(2.8485,-0.42076)--(2.9697,-0.38179)--(3.0909,-0.31806)--(3.2121,-0.23994)--(3.3333,-0.15925)--(3.4545,-0.087448)--(3.5758,-0.033930)--(3.6970,-0.0047735)--(3.8182,-0.0020298)--(3.9394,-0.023666)--(4.0606,-0.064108)--(4.1818,-0.11529)--(4.3030,-0.16799)--(4.4242,-0.21334)--(4.5455,-0.24409)--(4.6667,-0.25565)--(4.7879,-0.24669)--(4.9091,-0.21910)--(5.0303,-0.17762)--(5.1515,-0.12891)--(5.2727,-0.080491)--(5.3939,-0.039487)--(5.5152,-0.011586)--(5.6364,0)--(5.7576,-0.0060480)--(5.8788,-0.027135)--(6.0000,-0.059192);
-\draw (-5.654866776,-0.3298256667) node {$ -6 \, \pi $};
+\draw (-5.6549,-0.32983) node {$ -6 \, \pi $};
 \draw [] (-5.66,-0.100) -- (-5.66,0.100);
-\draw (-3.769911184,-0.3298256667) node {$ -4 \, \pi $};
+\draw (-3.7699,-0.32983) node {$ -4 \, \pi $};
 \draw [] (-3.77,-0.100) -- (-3.77,0.100);
-\draw (-1.884955592,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-1.8850,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-1.88,-0.100) -- (-1.88,0.100);
-\draw (1.884955592,-0.3149246667) node {$ 2 \, \pi $};
+\draw (1.8850,-0.31492) node {$ 2 \, \pi $};
 \draw [] (1.88,-0.100) -- (1.88,0.100);
-\draw (3.769911184,-0.3149246667) node {$ 4 \, \pi $};
+\draw (3.7699,-0.31492) node {$ 4 \, \pi $};
 \draw [] (3.77,-0.100) -- (3.77,0.100);
-\draw (5.654866776,-0.3149246667) node {$ 6 \, \pi $};
+\draw (5.6549,-0.31492) node {$ 6 \, \pi $};
 \draw [] (5.66,-0.100) -- (5.66,0.100);
-\draw (-0.4527428333,-1.000000000) node {$ -\frac{1}{2} $};
+\draw (-0.45274,-1.0000) node {$ -\frac{1}{2} $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.3108333333,1.000000000) node {$ \frac{1}{2} $};
+\draw (-0.31083,1.0000) node {$ \frac{1}{2} $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Refraction.pstricks b/auto/pictures_tex/Fig_Refraction.pstricks
index a1b1e5e23..9e1e8d206 100644
--- a/auto/pictures_tex/Fig_Refraction.pstricks
+++ b/auto/pictures_tex/Fig_Refraction.pstricks
@@ -78,16 +78,16 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (-3.00,0) -- (3.00,0);
-\draw [,->,>=latex] (0,-2.000000000) -- (0,2.000000000);
-\draw (0.4665308333,2.000000000) node {$\overline{ N }$};
-\draw (0.3561818828,0.9457307663) node {$\theta_1$};
+\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
+\draw (0.46653,2.0000) node {$\overline{ N }$};
+\draw (0.35618,0.94573) node {$\theta_1$};
 
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-\draw (-0.4277057227,-0.7442762375) node {$\theta_2$};
+\draw (-0.42770,-0.74428) node {$\theta_2$};
 
 \draw [] (-0.447,-0.224)--(-0.445,-0.229)--(-0.442,-0.234)--(-0.439,-0.238)--(-0.437,-0.243)--(-0.434,-0.248)--(-0.431,-0.253)--(-0.428,-0.258)--(-0.425,-0.263)--(-0.422,-0.267)--(-0.419,-0.272)--(-0.416,-0.277)--(-0.413,-0.281)--(-0.410,-0.286)--(-0.407,-0.291)--(-0.404,-0.295)--(-0.400,-0.300)--(-0.397,-0.304)--(-0.393,-0.309)--(-0.390,-0.313)--(-0.386,-0.317)--(-0.383,-0.322)--(-0.379,-0.326)--(-0.376,-0.330)--(-0.372,-0.334)--(-0.368,-0.338)--(-0.364,-0.342)--(-0.360,-0.346)--(-0.357,-0.350)--(-0.353,-0.354)--(-0.349,-0.358)--(-0.345,-0.362)--(-0.341,-0.366)--(-0.336,-0.370)--(-0.332,-0.374)--(-0.328,-0.377)--(-0.324,-0.381)--(-0.320,-0.385)--(-0.315,-0.388)--(-0.311,-0.392)--(-0.306,-0.395)--(-0.302,-0.398)--(-0.298,-0.402)--(-0.293,-0.405)--(-0.289,-0.408)--(-0.284,-0.412)--(-0.279,-0.415)--(-0.275,-0.418)--(-0.270,-0.421)--(-0.265,-0.424)--(-0.260,-0.427)--(-0.256,-0.430)--(-0.251,-0.432)--(-0.246,-0.435)--(-0.241,-0.438)--(-0.236,-0.441)--(-0.231,-0.443)--(-0.226,-0.446)--(-0.221,-0.448)--(-0.216,-0.451)--(-0.211,-0.453)--(-0.206,-0.456)--(-0.201,-0.458)--(-0.196,-0.460)--(-0.191,-0.462)--(-0.186,-0.464)--(-0.180,-0.466)--(-0.175,-0.468)--(-0.170,-0.470)--(-0.165,-0.472)--(-0.159,-0.474)--(-0.154,-0.476)--(-0.149,-0.477)--(-0.143,-0.479)--(-0.138,-0.481)--(-0.133,-0.482)--(-0.127,-0.484)--(-0.122,-0.485)--(-0.116,-0.486)--(-0.111,-0.488)--(-0.105,-0.489)--(-0.100,-0.490)--(-0.0945,-0.491)--(-0.0890,-0.492)--(-0.0835,-0.493)--(-0.0780,-0.494)--(-0.0724,-0.495)--(-0.0669,-0.496)--(-0.0614,-0.496)--(-0.0558,-0.497)--(-0.0502,-0.497)--(-0.0447,-0.498)--(-0.0391,-0.498)--(-0.0335,-0.499)--(-0.0279,-0.499)--(-0.0224,-0.500)--(-0.0168,-0.500)--(-0.0112,-0.500)--(-0.00559,-0.500)--(0,-0.500);
-\draw [color=red,->,>=latex] (1.000000000,1.000000000) -- (0,0);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.264911064,-0.6324555320);
+\draw [color=red,->,>=latex] (1.0000,1.0000) -- (0,0);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.2649,-0.63246);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks b/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks
index 519222dec..d7811ff6e 100644
--- a/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks
+++ b/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks
@@ -88,20 +88,20 @@
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.3420201433,0.9396926208) node [rotate=0] {$\bullet$};
-\draw (0.5068918387,1.206207312) node {\( a\)};
-\draw []  (0.3420201433,-0.9396926208) node [rotate=0] {$\bullet$};
-\draw (0.4887470053,-1.254367145) node {\( b\)};
-\draw []  (2.923804400,0) node [rotate=0] {$\bullet$};
-\draw (3.484041733,0) node {\( m\)};
+\draw []  (0.34202,0.93969) node [rotate=0] {$\bullet$};
+\draw (0.50689,1.2062) node {\( a\)};
+\draw []  (0.34202,-0.93969) node [rotate=0] {$\bullet$};
+\draw (0.48875,-1.2544) node {\( b\)};
+\draw []  (2.9238,0) node [rotate=0] {$\bullet$};
+\draw (3.4840,0) node {\( m\)};
 \draw [] (-0.303,1.17) -- (3.57,-0.235);
 \draw [] (-0.303,-1.17) -- (3.57,0.235);
-\draw []  (0.8660254038,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (0.5885166564,0.3214238333) node {\( x\)};
-\draw []  (-0.5400320788,0.8416444344) node [rotate=0] {$\bullet$};
-\draw (-0.7270164945,1.088549488) node {\( c\)};
+\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
+\draw (0.58852,0.32142) node {\( x\)};
+\draw []  (-0.54003,0.84164) node [rotate=0] {$\bullet$};
+\draw (-0.72702,1.0885) node {\( c\)};
 \draw [] (3.44,-0.126) -- (-1.06,0.968);
-\draw (-1.234368978,-0.5351321720) node {\( \mC\)};
+\draw (-1.2344,-0.53513) node {\( \mC\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_SolsSinpA.pstricks b/auto/pictures_tex/Fig_SolsSinpA.pstricks
index 78520694e..b23b7aff3 100644
--- a/auto/pictures_tex/Fig_SolsSinpA.pstricks
+++ b/auto/pictures_tex/Fig_SolsSinpA.pstricks
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.641592654,0) -- (3.641592654,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-3.6416,0) -- (3.6416,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [color=red] (0,-1.000)--(0.03173,-0.9995)--(0.06347,-0.9980)--(0.09520,-0.9955)--(0.1269,-0.9920)--(0.1587,-0.9874)--(0.1904,-0.9819)--(0.2221,-0.9754)--(0.2539,-0.9679)--(0.2856,-0.9595)--(0.3173,-0.9501)--(0.3491,-0.9397)--(0.3808,-0.9284)--(0.4125,-0.9161)--(0.4443,-0.9029)--(0.4760,-0.8888)--(0.5077,-0.8738)--(0.5395,-0.8580)--(0.5712,-0.8413)--(0.6029,-0.8237)--(0.6347,-0.8053)--(0.6664,-0.7861)--(0.6981,-0.7660)--(0.7299,-0.7453)--(0.7616,-0.7237)--(0.7933,-0.7015)--(0.8251,-0.6785)--(0.8568,-0.6549)--(0.8885,-0.6306)--(0.9203,-0.6056)--(0.9520,-0.5801)--(0.9837,-0.5539)--(1.015,-0.5272)--(1.047,-0.5000)--(1.079,-0.4723)--(1.111,-0.4441)--(1.142,-0.4154)--(1.174,-0.3863)--(1.206,-0.3569)--(1.238,-0.3271)--(1.269,-0.2969)--(1.301,-0.2665)--(1.333,-0.2358)--(1.365,-0.2048)--(1.396,-0.1736)--(1.428,-0.1423)--(1.460,-0.1108)--(1.491,-0.07925)--(1.523,-0.04758)--(1.555,-0.01587)--(1.587,0.01587)--(1.618,0.04758)--(1.650,0.07925)--(1.682,0.1108)--(1.714,0.1423)--(1.745,0.1736)--(1.777,0.2048)--(1.809,0.2358)--(1.841,0.2665)--(1.872,0.2969)--(1.904,0.3271)--(1.936,0.3569)--(1.967,0.3863)--(1.999,0.4154)--(2.031,0.4441)--(2.063,0.4723)--(2.094,0.5000)--(2.126,0.5272)--(2.158,0.5539)--(2.190,0.5801)--(2.221,0.6056)--(2.253,0.6306)--(2.285,0.6549)--(2.317,0.6785)--(2.348,0.7015)--(2.380,0.7237)--(2.412,0.7453)--(2.443,0.7660)--(2.475,0.7861)--(2.507,0.8053)--(2.539,0.8237)--(2.570,0.8413)--(2.602,0.8580)--(2.634,0.8738)--(2.666,0.8888)--(2.697,0.9029)--(2.729,0.9161)--(2.761,0.9284)--(2.793,0.9397)--(2.824,0.9501)--(2.856,0.9595)--(2.888,0.9679)--(2.919,0.9754)--(2.951,0.9819)--(2.983,0.9874)--(3.015,0.9920)--(3.046,0.9955)--(3.078,0.9980)--(3.110,0.9995)--(3.142,1.000);
@@ -100,17 +100,17 @@
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-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks b/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks
index 8314eadf6..ad7c6baeb 100644
--- a/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks
+++ b/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,2.000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,2.0000);
 %DEFAULT
-\draw []  (1.000000000,-3.000000000) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-3.429825667) node {$-1$};
-\draw []  (2.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,1.982455000) node {$1/2$};
-\draw []  (3.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-1.482455000) node {$-1/3$};
-\draw []  (4.000000000,0.7500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,1.232455000) node {$1/4$};
-\draw []  (5.000000000,-0.6000000000) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-1.082455000) node {$-1/5$};
-\draw []  (6.000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (6.000000000,0.9824550000) node {$1/6$};
-\draw []  (7.000000000,-0.4285714286) node [rotate=0] {$\bullet$};
-\draw (7.000000000,-0.9110264286) node {$-1/7$};
-\draw []  (8.000000000,0.3750000000) node [rotate=0] {$\bullet$};
-\draw (8.000000000,0.8574550000) node {$1/8$};
-\draw []  (9.000000000,-0.3333333333) node [rotate=0] {$\bullet$};
-\draw (9.000000000,-0.8157883333) node {$-1/9$};
-\draw []  (10.00000000,0.3000000000) node [rotate=0] {$\bullet$};
-\draw (10.00000000,0.7824550000) node {$1/10$};
+\draw []  (1.0000,-3.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-3.4298) node {$-1$};
+\draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (2.0000,1.9825) node {$1/2$};
+\draw []  (3.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-1.4825) node {$-1/3$};
+\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2325) node {$1/4$};
+\draw []  (5.0000,-0.60000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-1.0825) node {$-1/5$};
+\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.98246) node {$1/6$};
+\draw []  (7.0000,-0.42857) node [rotate=0] {$\bullet$};
+\draw (7.0000,-0.91103) node {$-1/7$};
+\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.85746) node {$1/8$};
+\draw []  (9.0000,-0.33333) node [rotate=0] {$\bullet$};
+\draw (9.0000,-0.81579) node {$-1/9$};
+\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.78246) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks b/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks
index 1f89f1f46..3a69fd3a0 100644
--- a/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks
+++ b/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (-1.000,2.925)--(-0.9394,2.844)--(-0.8788,2.765)--(-0.8182,2.687)--(-0.7576,2.611)--(-0.6970,2.537)--(-0.6364,2.464)--(-0.5758,2.393)--(-0.5152,2.323)--(-0.4545,2.256)--(-0.3939,2.189)--(-0.3333,2.125)--(-0.2727,2.062)--(-0.2121,2.001)--(-0.1515,1.942)--(-0.09091,1.884)--(-0.03030,1.827)--(0.03030,1.773)--(0.09091,1.720)--(0.1515,1.669)--(0.2121,1.619)--(0.2727,1.571)--(0.3333,1.525)--(0.3939,1.480)--(0.4545,1.437)--(0.5152,1.396)--(0.5758,1.356)--(0.6364,1.318)--(0.6970,1.282)--(0.7576,1.247)--(0.8182,1.214)--(0.8788,1.183)--(0.9394,1.153)--(1.000,1.125)--(1.061,1.099)--(1.121,1.074)--(1.182,1.051)--(1.242,1.029)--(1.303,1.009)--(1.364,0.9911)--(1.424,0.9746)--(1.485,0.9597)--(1.545,0.9465)--(1.606,0.9349)--(1.667,0.9250)--(1.727,0.9167)--(1.788,0.9101)--(1.848,0.9052)--(1.909,0.9019)--(1.970,0.9002)--(2.030,0.9002)--(2.091,0.9019)--(2.152,0.9052)--(2.212,0.9101)--(2.273,0.9167)--(2.333,0.9250)--(2.394,0.9349)--(2.455,0.9465)--(2.515,0.9597)--(2.576,0.9746)--(2.636,0.9911)--(2.697,1.009)--(2.758,1.029)--(2.818,1.051)--(2.879,1.074)--(2.939,1.099)--(3.000,1.125)--(3.061,1.153)--(3.121,1.183)--(3.182,1.214)--(3.242,1.247)--(3.303,1.282)--(3.364,1.318)--(3.424,1.356)--(3.485,1.396)--(3.545,1.437)--(3.606,1.480)--(3.667,1.525)--(3.727,1.571)--(3.788,1.619)--(3.848,1.669)--(3.909,1.720)--(3.970,1.773)--(4.030,1.827)--(4.091,1.884)--(4.151,1.942)--(4.212,2.001)--(4.273,2.062)--(4.333,2.125)--(4.394,2.189)--(4.455,2.256)--(4.515,2.323)--(4.576,2.393)--(4.636,2.464)--(4.697,2.537)--(4.758,2.611)--(4.818,2.687)--(4.879,2.765)--(4.939,2.844)--(5.000,2.925);
@@ -53,10 +53,10 @@
 \draw [color=blue] (0.1743,0)--(0.2111,0)--(0.2480,0)--(0.2849,0)--(0.3218,0)--(0.3587,0)--(0.3956,0)--(0.4324,0)--(0.4693,0)--(0.5062,0)--(0.5431,0)--(0.5800,0)--(0.6169,0)--(0.6537,0)--(0.6906,0)--(0.7275,0)--(0.7644,0)--(0.8013,0)--(0.8382,0)--(0.8750,0)--(0.9119,0)--(0.9488,0)--(0.9857,0)--(1.023,0)--(1.059,0)--(1.096,0)--(1.133,0)--(1.170,0)--(1.207,0)--(1.244,0)--(1.281,0)--(1.318,0)--(1.355,0)--(1.391,0)--(1.428,0)--(1.465,0)--(1.502,0)--(1.539,0)--(1.576,0)--(1.613,0)--(1.650,0)--(1.686,0)--(1.723,0)--(1.760,0)--(1.797,0)--(1.834,0)--(1.871,0)--(1.908,0)--(1.945,0)--(1.982,0)--(2.018,0)--(2.055,0)--(2.092,0)--(2.129,0)--(2.166,0)--(2.203,0)--(2.240,0)--(2.277,0)--(2.314,0)--(2.350,0)--(2.387,0)--(2.424,0)--(2.461,0)--(2.498,0)--(2.535,0)--(2.572,0)--(2.609,0)--(2.645,0)--(2.682,0)--(2.719,0)--(2.756,0)--(2.793,0)--(2.830,0)--(2.867,0)--(2.904,0)--(2.941,0)--(2.977,0)--(3.014,0)--(3.051,0)--(3.088,0)--(3.125,0)--(3.162,0)--(3.199,0)--(3.236,0)--(3.272,0)--(3.309,0)--(3.346,0)--(3.383,0)--(3.420,0)--(3.457,0)--(3.494,0)--(3.531,0)--(3.568,0)--(3.604,0)--(3.641,0)--(3.678,0)--(3.715,0)--(3.752,0)--(3.789,0)--(3.826,0);
 \draw [] (0.174,0) -- (0.174,1.65);
 \draw [] (3.83,1.65) -- (3.83,0);
-\draw []  (0.1742581416,0) node [rotate=0] {$\bullet$};
-\draw (0.1742581416,-0.3785761667) node {$a$};
-\draw []  (3.825741858,0) node [rotate=0] {$\bullet$};
-\draw (3.825741858,-0.4267360000) node {$b$};
+\draw []  (0.17426,0) node [rotate=0] {$\bullet$};
+\draw (0.17426,-0.37858) node {$a$};
+\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
+\draw (3.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (-1.000,2.925)--(-0.9394,2.844)--(-0.8788,2.765)--(-0.8182,2.687)--(-0.7576,2.611)--(-0.6970,2.537)--(-0.6364,2.464)--(-0.5758,2.393)--(-0.5152,2.323)--(-0.4545,2.256)--(-0.3939,2.189)--(-0.3333,2.125)--(-0.2727,2.062)--(-0.2121,2.001)--(-0.1515,1.942)--(-0.09091,1.884)--(-0.03030,1.827)--(0.03030,1.773)--(0.09091,1.720)--(0.1515,1.669)--(0.2121,1.619)--(0.2727,1.571)--(0.3333,1.525)--(0.3939,1.480)--(0.4545,1.437)--(0.5152,1.396)--(0.5758,1.356)--(0.6364,1.318)--(0.6970,1.282)--(0.7576,1.247)--(0.8182,1.214)--(0.8788,1.183)--(0.9394,1.153)--(1.000,1.125)--(1.061,1.099)--(1.121,1.074)--(1.182,1.051)--(1.242,1.029)--(1.303,1.009)--(1.364,0.9911)--(1.424,0.9746)--(1.485,0.9597)--(1.545,0.9465)--(1.606,0.9349)--(1.667,0.9250)--(1.727,0.9167)--(1.788,0.9101)--(1.848,0.9052)--(1.909,0.9019)--(1.970,0.9002)--(2.030,0.9002)--(2.091,0.9019)--(2.152,0.9052)--(2.212,0.9101)--(2.273,0.9167)--(2.333,0.9250)--(2.394,0.9349)--(2.455,0.9465)--(2.515,0.9597)--(2.576,0.9746)--(2.636,0.9911)--(2.697,1.009)--(2.758,1.029)--(2.818,1.051)--(2.879,1.074)--(2.939,1.099)--(3.000,1.125)--(3.061,1.153)--(3.121,1.183)--(3.182,1.214)--(3.242,1.247)--(3.303,1.282)--(3.364,1.318)--(3.424,1.356)--(3.485,1.396)--(3.545,1.437)--(3.606,1.480)--(3.667,1.525)--(3.727,1.571)--(3.788,1.619)--(3.848,1.669)--(3.909,1.720)--(3.970,1.773)--(4.030,1.827)--(4.091,1.884)--(4.151,1.942)--(4.212,2.001)--(4.273,2.062)--(4.333,2.125)--(4.394,2.189)--(4.455,2.256)--(4.515,2.323)--(4.576,2.393)--(4.636,2.464)--(4.697,2.537)--(4.758,2.611)--(4.818,2.687)--(4.879,2.765)--(4.939,2.844)--(5.000,2.925);
@@ -107,10 +107,10 @@
 \draw [color=blue] (0.1743,0)--(0.2111,0)--(0.2480,0)--(0.2849,0)--(0.3218,0)--(0.3587,0)--(0.3956,0)--(0.4324,0)--(0.4693,0)--(0.5062,0)--(0.5431,0)--(0.5800,0)--(0.6169,0)--(0.6537,0)--(0.6906,0)--(0.7275,0)--(0.7644,0)--(0.8013,0)--(0.8382,0)--(0.8750,0)--(0.9119,0)--(0.9488,0)--(0.9857,0)--(1.023,0)--(1.059,0)--(1.096,0)--(1.133,0)--(1.170,0)--(1.207,0)--(1.244,0)--(1.281,0)--(1.318,0)--(1.355,0)--(1.391,0)--(1.428,0)--(1.465,0)--(1.502,0)--(1.539,0)--(1.576,0)--(1.613,0)--(1.650,0)--(1.686,0)--(1.723,0)--(1.760,0)--(1.797,0)--(1.834,0)--(1.871,0)--(1.908,0)--(1.945,0)--(1.982,0)--(2.018,0)--(2.055,0)--(2.092,0)--(2.129,0)--(2.166,0)--(2.203,0)--(2.240,0)--(2.277,0)--(2.314,0)--(2.350,0)--(2.387,0)--(2.424,0)--(2.461,0)--(2.498,0)--(2.535,0)--(2.572,0)--(2.609,0)--(2.645,0)--(2.682,0)--(2.719,0)--(2.756,0)--(2.793,0)--(2.830,0)--(2.867,0)--(2.904,0)--(2.941,0)--(2.977,0)--(3.014,0)--(3.051,0)--(3.088,0)--(3.125,0)--(3.162,0)--(3.199,0)--(3.236,0)--(3.272,0)--(3.309,0)--(3.346,0)--(3.383,0)--(3.420,0)--(3.457,0)--(3.494,0)--(3.531,0)--(3.568,0)--(3.604,0)--(3.641,0)--(3.678,0)--(3.715,0)--(3.752,0)--(3.789,0)--(3.826,0);
 \draw [] (0.174,0) -- (0.174,1.65);
 \draw [] (3.83,1.65) -- (3.83,0);
-\draw []  (0.1742581416,0) node [rotate=0] {$\bullet$};
-\draw (0.1742581416,-0.3785761667) node {$a$};
-\draw []  (3.825741858,0) node [rotate=0] {$\bullet$};
-\draw (3.825741858,-0.4267360000) node {$b$};
+\draw []  (0.17426,0) node [rotate=0] {$\bullet$};
+\draw (0.17426,-0.37858) node {$a$};
+\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
+\draw (3.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,8 +149,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (-1.000,2.925)--(-0.9394,2.844)--(-0.8788,2.765)--(-0.8182,2.687)--(-0.7576,2.611)--(-0.6970,2.537)--(-0.6364,2.464)--(-0.5758,2.393)--(-0.5152,2.323)--(-0.4545,2.256)--(-0.3939,2.189)--(-0.3333,2.125)--(-0.2727,2.062)--(-0.2121,2.001)--(-0.1515,1.942)--(-0.09091,1.884)--(-0.03030,1.827)--(0.03030,1.773)--(0.09091,1.720)--(0.1515,1.669)--(0.2121,1.619)--(0.2727,1.571)--(0.3333,1.525)--(0.3939,1.480)--(0.4545,1.437)--(0.5152,1.396)--(0.5758,1.356)--(0.6364,1.318)--(0.6970,1.282)--(0.7576,1.247)--(0.8182,1.214)--(0.8788,1.183)--(0.9394,1.153)--(1.000,1.125)--(1.061,1.099)--(1.121,1.074)--(1.182,1.051)--(1.242,1.029)--(1.303,1.009)--(1.364,0.9911)--(1.424,0.9746)--(1.485,0.9597)--(1.545,0.9465)--(1.606,0.9349)--(1.667,0.9250)--(1.727,0.9167)--(1.788,0.9101)--(1.848,0.9052)--(1.909,0.9019)--(1.970,0.9002)--(2.030,0.9002)--(2.091,0.9019)--(2.152,0.9052)--(2.212,0.9101)--(2.273,0.9167)--(2.333,0.9250)--(2.394,0.9349)--(2.455,0.9465)--(2.515,0.9597)--(2.576,0.9746)--(2.636,0.9911)--(2.697,1.009)--(2.758,1.029)--(2.818,1.051)--(2.879,1.074)--(2.939,1.099)--(3.000,1.125)--(3.061,1.153)--(3.121,1.183)--(3.182,1.214)--(3.242,1.247)--(3.303,1.282)--(3.364,1.318)--(3.424,1.356)--(3.485,1.396)--(3.545,1.437)--(3.606,1.480)--(3.667,1.525)--(3.727,1.571)--(3.788,1.619)--(3.848,1.669)--(3.909,1.720)--(3.970,1.773)--(4.030,1.827)--(4.091,1.884)--(4.151,1.942)--(4.212,2.001)--(4.273,2.062)--(4.333,2.125)--(4.394,2.189)--(4.455,2.256)--(4.515,2.323)--(4.576,2.393)--(4.636,2.464)--(4.697,2.537)--(4.758,2.611)--(4.818,2.687)--(4.879,2.765)--(4.939,2.844)--(5.000,2.925);
@@ -161,10 +161,10 @@
 \draw [color=blue] (0.1743,1.650)--(0.2111,1.680)--(0.2480,1.709)--(0.2849,1.738)--(0.3218,1.766)--(0.3587,1.794)--(0.3956,1.821)--(0.4324,1.847)--(0.4693,1.873)--(0.5062,1.898)--(0.5431,1.922)--(0.5800,1.946)--(0.6169,1.970)--(0.6537,1.992)--(0.6906,2.014)--(0.7275,2.036)--(0.7644,2.056)--(0.8013,2.077)--(0.8382,2.096)--(0.8750,2.115)--(0.9119,2.134)--(0.9488,2.151)--(0.9857,2.169)--(1.023,2.185)--(1.059,2.201)--(1.096,2.216)--(1.133,2.231)--(1.170,2.245)--(1.207,2.259)--(1.244,2.271)--(1.281,2.284)--(1.318,2.295)--(1.355,2.306)--(1.391,2.317)--(1.428,2.326)--(1.465,2.336)--(1.502,2.344)--(1.539,2.352)--(1.576,2.360)--(1.613,2.366)--(1.650,2.372)--(1.686,2.378)--(1.723,2.383)--(1.760,2.387)--(1.797,2.391)--(1.834,2.394)--(1.871,2.396)--(1.908,2.398)--(1.945,2.399)--(1.982,2.400)--(2.018,2.400)--(2.055,2.399)--(2.092,2.398)--(2.129,2.396)--(2.166,2.394)--(2.203,2.391)--(2.240,2.387)--(2.277,2.383)--(2.314,2.378)--(2.350,2.372)--(2.387,2.366)--(2.424,2.360)--(2.461,2.352)--(2.498,2.344)--(2.535,2.336)--(2.572,2.326)--(2.609,2.317)--(2.645,2.306)--(2.682,2.295)--(2.719,2.284)--(2.756,2.271)--(2.793,2.259)--(2.830,2.245)--(2.867,2.231)--(2.904,2.216)--(2.941,2.201)--(2.977,2.185)--(3.014,2.169)--(3.051,2.151)--(3.088,2.134)--(3.125,2.115)--(3.162,2.096)--(3.199,2.077)--(3.236,2.056)--(3.272,2.036)--(3.309,2.014)--(3.346,1.992)--(3.383,1.970)--(3.420,1.946)--(3.457,1.922)--(3.494,1.898)--(3.531,1.873)--(3.568,1.847)--(3.604,1.821)--(3.641,1.794)--(3.678,1.766)--(3.715,1.738)--(3.752,1.709)--(3.789,1.680)--(3.826,1.650);
 \draw [] (0.174,1.65) -- (0.174,1.65);
 \draw [] (3.83,1.65) -- (3.83,1.65);
-\draw []  (0.1742581416,0) node [rotate=0] {$\bullet$};
-\draw (0.1742581416,-0.3785761667) node {$a$};
-\draw []  (3.825741858,0) node [rotate=0] {$\bullet$};
-\draw (3.825741858,-0.4267360000) node {$b$};
+\draw []  (0.17426,0) node [rotate=0] {$\bullet$};
+\draw (0.17426,-0.37858) node {$a$};
+\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
+\draw (3.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks b/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks
index cc25d836a..fd9c3b21d 100644
--- a/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks
+++ b/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.400000000);
+\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
 %DEFAULT
 
 \draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
@@ -96,10 +96,10 @@
 \draw [color=blue] (3.000,0)--(3.030,0)--(3.061,0)--(3.091,0)--(3.121,0)--(3.152,0)--(3.182,0)--(3.212,0)--(3.242,0)--(3.273,0)--(3.303,0)--(3.333,0)--(3.364,0)--(3.394,0)--(3.424,0)--(3.455,0)--(3.485,0)--(3.515,0)--(3.545,0)--(3.576,0)--(3.606,0)--(3.636,0)--(3.667,0)--(3.697,0)--(3.727,0)--(3.758,0)--(3.788,0)--(3.818,0)--(3.848,0)--(3.879,0)--(3.909,0)--(3.939,0)--(3.970,0)--(4.000,0)--(4.030,0)--(4.061,0)--(4.091,0)--(4.121,0)--(4.151,0)--(4.182,0)--(4.212,0)--(4.242,0)--(4.273,0)--(4.303,0)--(4.333,0)--(4.364,0)--(4.394,0)--(4.424,0)--(4.455,0)--(4.485,0)--(4.515,0)--(4.545,0)--(4.576,0)--(4.606,0)--(4.636,0)--(4.667,0)--(4.697,0)--(4.727,0)--(4.758,0)--(4.788,0)--(4.818,0)--(4.849,0)--(4.879,0)--(4.909,0)--(4.939,0)--(4.970,0)--(5.000,0)--(5.030,0)--(5.061,0)--(5.091,0)--(5.121,0)--(5.151,0)--(5.182,0)--(5.212,0)--(5.242,0)--(5.273,0)--(5.303,0)--(5.333,0)--(5.364,0)--(5.394,0)--(5.424,0)--(5.455,0)--(5.485,0)--(5.515,0)--(5.545,0)--(5.576,0)--(5.606,0)--(5.636,0)--(5.667,0)--(5.697,0)--(5.727,0)--(5.758,0)--(5.788,0)--(5.818,0)--(5.849,0)--(5.879,0)--(5.909,0)--(5.939,0)--(5.970,0)--(6.000,0);
 \draw [style=dashed] (3.00,0) -- (3.00,3.00);
 \draw [style=dashed] (6.00,3.75) -- (6.00,0);
-\draw (3.000000000,-0.3785761667) node {$a$};
-\draw (6.000000000,-0.3785761667) node {$x$};
-\draw (8.355206667,3.900000000) node {$f(x)$};
-\draw (9.670153833,1.875000000) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (3.0000,-0.37858) node {$a$};
+\draw (6.0000,-0.37858) node {$x$};
+\draw (8.3552,3.9000) node {$f(x)$};
+\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/auto/pictures_tex/Fig_TZCISko.pstricks b/auto/pictures_tex/Fig_TZCISko.pstricks
index fa9218d09..068402609 100644
--- a/auto/pictures_tex/Fig_TZCISko.pstricks
+++ b/auto/pictures_tex/Fig_TZCISko.pstricks
@@ -71,19 +71,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-2.499989537) -- (0,2.499999895);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
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-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
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-\draw (-0.2912498333,2.000000000) node {$ 1 $};
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 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
index 54059faa9..4b392a788 100644
--- a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
@@ -79,26 +79,26 @@
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-\draw (0.3487948559,0.6970250798) node {$\varphi$};
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 \draw [color=cyan] (0.500,0)--(0.500,0.0132)--(0.499,0.0264)--(0.498,0.0396)--(0.497,0.0528)--(0.496,0.0659)--(0.494,0.0790)--(0.491,0.0920)--(0.489,0.105)--(0.486,0.118)--(0.483,0.131)--(0.479,0.143)--(0.475,0.156)--(0.471,0.169)--(0.466,0.181)--(0.461,0.193)--(0.456,0.205)--(0.450,0.217)--(0.444,0.229)--(0.438,0.241)--(0.432,0.252)--(0.425,0.264)--(0.418,0.275)--(0.410,0.286)--(0.403,0.296)--(0.395,0.307)--(0.386,0.317)--(0.378,0.327)--(0.369,0.337)--(0.360,0.347)--(0.351,0.356)--(0.341,0.366)--(0.331,0.374)--(0.321,0.383)--(0.311,0.391)--(0.301,0.399)--(0.290,0.407)--(0.279,0.415)--(0.268,0.422)--(0.257,0.429)--(0.245,0.436)--(0.234,0.442)--(0.222,0.448)--(0.210,0.454)--(0.198,0.459)--(0.186,0.464)--(0.173,0.469)--(0.161,0.473)--(0.148,0.477)--(0.136,0.481)--(0.123,0.485)--(0.110,0.488)--(0.0972,0.490)--(0.0842,0.493)--(0.0712,0.495)--(0.0580,0.497)--(0.0449,0.498)--(0.0317,0.499)--(0.0185,0.500)--(0.00529,0.500)--(-0.00793,0.500)--(-0.0211,0.500)--(-0.0344,0.499)--(-0.0475,0.498)--(-0.0607,0.496)--(-0.0738,0.495)--(-0.0868,0.492)--(-0.0998,0.490)--(-0.113,0.487)--(-0.126,0.484)--(-0.138,0.480)--(-0.151,0.477)--(-0.164,0.473)--(-0.176,0.468)--(-0.188,0.463)--(-0.200,0.458)--(-0.213,0.453)--(-0.224,0.447)--(-0.236,0.441)--(-0.248,0.434)--(-0.259,0.428)--(-0.270,0.421)--(-0.281,0.413)--(-0.292,0.406)--(-0.303,0.398)--(-0.313,0.390)--(-0.323,0.381)--(-0.333,0.373)--(-0.343,0.364)--(-0.353,0.354)--(-0.362,0.345)--(-0.371,0.335)--(-0.380,0.325)--(-0.388,0.315)--(-0.396,0.305)--(-0.404,0.294)--(-0.412,0.284)--(-0.419,0.273)--(-0.426,0.261)--(-0.433,0.250);
-\draw [color=red,->,>=latex] (0,0) -- (1.285575219,1.532088886);
-\draw [color=cyan,->,>=latex] (0,0) -- (-1.732050808,1.000000000);
+\draw [color=red,->,>=latex] (0,0) -- (1.2856,1.5321);
+\draw [color=cyan,->,>=latex] (0,0) -- (-1.7320,1.0000);
 \draw [] (2.00,3.27) -- (2.00,-2.04);
-\draw [color=red]  (2.000000000,2.383507185) node [rotate=0] {$\bullet$};
-\draw (2.697581667,2.383507185) node {$\tan(\theta)$};
-\draw [color=cyan]  (2.000000000,-1.154700538) node [rotate=0] {$\bullet$};
-\draw (2.726224000,-1.154700538) node {$\tan(\varphi)$};
+\draw [color=red]  (2.0000,2.3835) node [rotate=0] {$\bullet$};
+\draw (2.6976,2.3835) node {$\tan(\theta)$};
+\draw [color=cyan]  (2.0000,-1.1547) node [rotate=0] {$\bullet$};
+\draw (2.7262,-1.1547) node {$\tan(\varphi)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TracerUn.pstricks b/auto/pictures_tex/Fig_TracerUn.pstricks
index 0d477f127..5b92fa7bb 100644
--- a/auto/pictures_tex/Fig_TracerUn.pstricks
+++ b/auto/pictures_tex/Fig_TracerUn.pstricks
@@ -81,42 +81,42 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.900000000,0) -- (1.900000000,0);
-\draw [,->,>=latex] (0,-4.305593400) -- (0,4.305593400);
+\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
+\draw [,->,>=latex] (0,-4.3056) -- (0,4.3056);
 %DEFAULT
 
 \draw [color=blue] (-1.400,-3.806)--(-1.372,-3.729)--(-1.343,-3.652)--(-1.315,-3.575)--(-1.287,-3.498)--(-1.259,-3.421)--(-1.230,-3.344)--(-1.202,-3.267)--(-1.174,-3.191)--(-1.145,-3.114)--(-1.117,-3.037)--(-1.089,-2.960)--(-1.061,-2.883)--(-1.032,-2.806)--(-1.004,-2.729)--(-0.9758,-2.652)--(-0.9475,-2.576)--(-0.9192,-2.499)--(-0.8909,-2.422)--(-0.8626,-2.345)--(-0.8343,-2.268)--(-0.8061,-2.191)--(-0.7778,-2.114)--(-0.7495,-2.037)--(-0.7212,-1.960)--(-0.6929,-1.884)--(-0.6646,-1.807)--(-0.6364,-1.730)--(-0.6081,-1.653)--(-0.5798,-1.576)--(-0.5515,-1.499)--(-0.5232,-1.422)--(-0.4949,-1.345)--(-0.4667,-1.269)--(-0.4384,-1.192)--(-0.4101,-1.115)--(-0.3818,-1.038)--(-0.3535,-0.9610)--(-0.3253,-0.8841)--(-0.2970,-0.8072)--(-0.2687,-0.7304)--(-0.2404,-0.6535)--(-0.2121,-0.5766)--(-0.1838,-0.4997)--(-0.1556,-0.4228)--(-0.1273,-0.3460)--(-0.09899,-0.2691)--(-0.07071,-0.1922)--(-0.04242,-0.1153)--(-0.01414,-0.03844)--(0.01414,0.03844)--(0.04242,0.1153)--(0.07071,0.1922)--(0.09899,0.2691)--(0.1273,0.3460)--(0.1556,0.4228)--(0.1838,0.4997)--(0.2121,0.5766)--(0.2404,0.6535)--(0.2687,0.7304)--(0.2970,0.8072)--(0.3253,0.8841)--(0.3535,0.9610)--(0.3818,1.038)--(0.4101,1.115)--(0.4384,1.192)--(0.4667,1.269)--(0.4949,1.345)--(0.5232,1.422)--(0.5515,1.499)--(0.5798,1.576)--(0.6081,1.653)--(0.6364,1.730)--(0.6646,1.807)--(0.6929,1.884)--(0.7212,1.960)--(0.7495,2.037)--(0.7778,2.114)--(0.8061,2.191)--(0.8343,2.268)--(0.8626,2.345)--(0.8909,2.422)--(0.9192,2.499)--(0.9475,2.576)--(0.9758,2.652)--(1.004,2.729)--(1.032,2.806)--(1.061,2.883)--(1.089,2.960)--(1.117,3.037)--(1.145,3.114)--(1.174,3.191)--(1.202,3.267)--(1.230,3.344)--(1.259,3.421)--(1.287,3.498)--(1.315,3.575)--(1.343,3.652)--(1.372,3.729)--(1.400,3.806);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.4331593333,-4.200000000) node {$ -6 $};
+\draw (-0.43316,-4.2000) node {$ -6 $};
 \draw [] (-0.100,-4.20) -- (0.100,-4.20);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.2912498333,4.200000000) node {$ 6 $};
+\draw (-0.29125,4.2000) node {$ 6 $};
 \draw [] (-0.100,4.20) -- (0.100,4.20);
 %OTHER STUFF
 %END PSPICTURE
@@ -163,22 +163,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=blue] (-2.000,2.000)--(-1.960,1.960)--(-1.919,1.919)--(-1.879,1.879)--(-1.838,1.838)--(-1.798,1.798)--(-1.758,1.758)--(-1.717,1.717)--(-1.677,1.677)--(-1.636,1.636)--(-1.596,1.596)--(-1.556,1.556)--(-1.515,1.515)--(-1.475,1.475)--(-1.434,1.434)--(-1.394,1.394)--(-1.354,1.354)--(-1.313,1.313)--(-1.273,1.273)--(-1.232,1.232)--(-1.192,1.192)--(-1.152,1.152)--(-1.111,1.111)--(-1.071,1.071)--(-1.030,1.030)--(-0.9899,0.9899)--(-0.9495,0.9495)--(-0.9091,0.9091)--(-0.8687,0.8687)--(-0.8283,0.8283)--(-0.7879,0.7879)--(-0.7475,0.7475)--(-0.7071,0.7071)--(-0.6667,0.6667)--(-0.6263,0.6263)--(-0.5859,0.5859)--(-0.5455,0.5455)--(-0.5051,0.5051)--(-0.4646,0.4646)--(-0.4242,0.4242)--(-0.3838,0.3838)--(-0.3434,0.3434)--(-0.3030,0.3030)--(-0.2626,0.2626)--(-0.2222,0.2222)--(-0.1818,0.1818)--(-0.1414,0.1414)--(-0.1010,0.1010)--(-0.06061,0.06061)--(-0.02020,0.02020)--(0.02020,0.02020)--(0.06061,0.06061)--(0.1010,0.1010)--(0.1414,0.1414)--(0.1818,0.1818)--(0.2222,0.2222)--(0.2626,0.2626)--(0.3030,0.3030)--(0.3434,0.3434)--(0.3838,0.3838)--(0.4242,0.4242)--(0.4646,0.4646)--(0.5051,0.5051)--(0.5455,0.5455)--(0.5859,0.5859)--(0.6263,0.6263)--(0.6667,0.6667)--(0.7071,0.7071)--(0.7475,0.7475)--(0.7879,0.7879)--(0.8283,0.8283)--(0.8687,0.8687)--(0.9091,0.9091)--(0.9495,0.9495)--(0.9899,0.9899)--(1.030,1.030)--(1.071,1.071)--(1.111,1.111)--(1.152,1.152)--(1.192,1.192)--(1.232,1.232)--(1.273,1.273)--(1.313,1.313)--(1.354,1.354)--(1.394,1.394)--(1.434,1.434)--(1.475,1.475)--(1.515,1.515)--(1.556,1.556)--(1.596,1.596)--(1.636,1.636)--(1.677,1.677)--(1.717,1.717)--(1.758,1.758)--(1.798,1.798)--(1.838,1.838)--(1.879,1.879)--(1.919,1.919)--(1.960,1.960)--(2.000,2.000);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -253,38 +253,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.900000000,0) -- (1.900000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.800000000);
+\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.8000);
 %DEFAULT
 
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-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.2912498333,4.200000000) node {$ 6 $};
+\draw (-0.29125,4.2000) node {$ 6 $};
 \draw [] (-0.100,4.20) -- (0.100,4.20);
-\draw (-0.2912498333,4.900000000) node {$ 7 $};
+\draw (-0.29125,4.9000) node {$ 7 $};
 \draw [] (-0.100,4.90) -- (0.100,4.90);
-\draw (-0.2912498333,5.600000000) node {$ 8 $};
+\draw (-0.29125,5.6000) node {$ 8 $};
 \draw [] (-0.100,5.60) -- (0.100,5.60);
-\draw (-0.2912498333,6.300000000) node {$ 9 $};
+\draw (-0.29125,6.3000) node {$ 9 $};
 \draw [] (-0.100,6.30) -- (0.100,6.30);
 %OTHER STUFF
 %END PSPICTURE
@@ -335,26 +335,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.900000000,0) -- (1.900000000,0);
-\draw [,->,>=latex] (0,-2.600000000) -- (0,1.200000000);
+\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
+\draw [,->,>=latex] (0,-2.6000) -- (0,1.2000);
 %DEFAULT
 
 \draw [color=blue] (-1.400,-2.100)--(-1.372,-2.072)--(-1.343,-2.043)--(-1.315,-2.015)--(-1.287,-1.987)--(-1.259,-1.959)--(-1.230,-1.930)--(-1.202,-1.902)--(-1.174,-1.874)--(-1.145,-1.845)--(-1.117,-1.817)--(-1.089,-1.789)--(-1.061,-1.761)--(-1.032,-1.732)--(-1.004,-1.704)--(-0.9758,-1.676)--(-0.9475,-1.647)--(-0.9192,-1.619)--(-0.8909,-1.591)--(-0.8626,-1.563)--(-0.8343,-1.534)--(-0.8061,-1.506)--(-0.7778,-1.478)--(-0.7495,-1.449)--(-0.7212,-1.421)--(-0.6929,-1.393)--(-0.6646,-1.365)--(-0.6364,-1.336)--(-0.6081,-1.308)--(-0.5798,-1.280)--(-0.5515,-1.252)--(-0.5232,-1.223)--(-0.4949,-1.195)--(-0.4667,-1.167)--(-0.4384,-1.138)--(-0.4101,-1.110)--(-0.3818,-1.082)--(-0.3535,-1.054)--(-0.3253,-1.025)--(-0.2970,-0.9970)--(-0.2687,-0.9687)--(-0.2404,-0.9404)--(-0.2121,-0.9121)--(-0.1838,-0.8838)--(-0.1556,-0.8556)--(-0.1273,-0.8273)--(-0.09899,-0.7990)--(-0.07071,-0.7707)--(-0.04242,-0.7424)--(-0.01414,-0.7141)--(0.01414,-0.6859)--(0.04242,-0.6576)--(0.07071,-0.6293)--(0.09899,-0.6010)--(0.1273,-0.5727)--(0.1556,-0.5444)--(0.1838,-0.5162)--(0.2121,-0.4879)--(0.2404,-0.4596)--(0.2687,-0.4313)--(0.2970,-0.4030)--(0.3253,-0.3747)--(0.3535,-0.3465)--(0.3818,-0.3182)--(0.4101,-0.2899)--(0.4384,-0.2616)--(0.4667,-0.2333)--(0.4949,-0.2051)--(0.5232,-0.1768)--(0.5515,-0.1485)--(0.5798,-0.1202)--(0.6081,-0.09192)--(0.6364,-0.06364)--(0.6646,-0.03535)--(0.6929,-0.007071)--(0.7212,0.02121)--(0.7495,0.04949)--(0.7778,0.07778)--(0.8061,0.1061)--(0.8343,0.1343)--(0.8626,0.1626)--(0.8909,0.1909)--(0.9192,0.2192)--(0.9475,0.2475)--(0.9758,0.2758)--(1.004,0.3040)--(1.032,0.3323)--(1.061,0.3606)--(1.089,0.3889)--(1.117,0.4172)--(1.145,0.4455)--(1.174,0.4737)--(1.202,0.5020)--(1.230,0.5303)--(1.259,0.5586)--(1.287,0.5869)--(1.315,0.6152)--(1.343,0.6434)--(1.372,0.6717)--(1.400,0.7000);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks b/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks
index 32b28188c..23fd0b1bb 100644
--- a/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks
+++ b/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks
@@ -96,34 +96,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
 \draw [] (0,0) -- (5.00,5.00);
 
 \draw [color=blue] (0.1000,4.996)--(0.1495,4.594)--(0.1990,4.308)--(0.2485,4.086)--(0.2980,3.904)--(0.3475,3.750)--(0.3970,3.617)--(0.4465,3.500)--(0.4960,3.394)--(0.5455,3.299)--(0.5949,3.212)--(0.6444,3.133)--(0.6939,3.059)--(0.7434,2.990)--(0.7929,2.925)--(0.8424,2.865)--(0.8919,2.808)--(0.9414,2.754)--(0.9909,2.702)--(1.040,2.654)--(1.090,2.607)--(1.139,2.563)--(1.189,2.520)--(1.238,2.479)--(1.288,2.440)--(1.337,2.402)--(1.387,2.366)--(1.436,2.331)--(1.486,2.297)--(1.535,2.264)--(1.585,2.233)--(1.634,2.202)--(1.684,2.172)--(1.733,2.143)--(1.783,2.115)--(1.832,2.088)--(1.882,2.061)--(1.931,2.035)--(1.981,2.010)--(2.030,1.985)--(2.080,1.961)--(2.129,1.937)--(2.179,1.914)--(2.228,1.892)--(2.278,1.870)--(2.327,1.848)--(2.377,1.827)--(2.426,1.807)--(2.476,1.787)--(2.525,1.767)--(2.575,1.747)--(2.624,1.728)--(2.674,1.710)--(2.723,1.691)--(2.773,1.673)--(2.822,1.656)--(2.872,1.638)--(2.921,1.621)--(2.971,1.604)--(3.020,1.588)--(3.070,1.572)--(3.119,1.556)--(3.169,1.540)--(3.218,1.524)--(3.268,1.509)--(3.317,1.494)--(3.367,1.479)--(3.416,1.465)--(3.466,1.450)--(3.515,1.436)--(3.565,1.422)--(3.614,1.408)--(3.664,1.395)--(3.713,1.381)--(3.763,1.368)--(3.812,1.355)--(3.862,1.342)--(3.911,1.329)--(3.961,1.317)--(4.010,1.304)--(4.060,1.292)--(4.109,1.280)--(4.159,1.268)--(4.208,1.256)--(4.258,1.244)--(4.307,1.233)--(4.357,1.221)--(4.406,1.210)--(4.456,1.199)--(4.505,1.188)--(4.555,1.177)--(4.604,1.166)--(4.654,1.156)--(4.703,1.145)--(4.753,1.134)--(4.802,1.124)--(4.852,1.114)--(4.901,1.104)--(4.951,1.094)--(5.000,1.084);
-\draw []  (5.000000000,1.083709268) node [rotate=0] {$\bullet$};
-\draw (5.391795586,1.278156808) node {\( P_{ 0  }\)};
-\draw []  (1.083709268,1.083709268) node [rotate=0] {$\bullet$};
-\draw (0.7188450786,1.385324791) node {\( Q_{0}\)};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.3059510000) node {\( x_{ 0  }\)};
+\draw []  (5.0000,1.0837) node [rotate=0] {$\bullet$};
+\draw (5.3918,1.2782) node {\( P_{ 0  }\)};
+\draw []  (1.0837,1.0837) node [rotate=0] {$\bullet$};
+\draw (0.71885,1.3853) node {\( Q_{0}\)};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.30595) node {\( x_{ 0  }\)};
 \draw [color=cyan,style=dashed] (5.00,1.08) -- (1.08,1.08);
 \draw [color=red,style=dashed] (5.00,1.08) -- (5.00,0);
-\draw []  (1.083709268,2.612757516) node [rotate=0] {$\bullet$};
-\draw (1.356668643,2.949579495) node {\( P_{ 1  }\)};
-\draw []  (2.612757516,2.612757516) node [rotate=0] {$\bullet$};
-\draw (2.247893327,2.914373039) node {\( Q_{1}\)};
-\draw []  (1.083709268,0) node [rotate=0] {$\bullet$};
-\draw (1.083709268,-0.3059510000) node {\( x_{ 1  }\)};
+\draw []  (1.0837,2.6128) node [rotate=0] {$\bullet$};
+\draw (1.3567,2.9496) node {\( P_{ 1  }\)};
+\draw []  (2.6128,2.6128) node [rotate=0] {$\bullet$};
+\draw (2.2479,2.9144) node {\( Q_{1}\)};
+\draw []  (1.0837,0) node [rotate=0] {$\bullet$};
+\draw (1.0837,-0.30595) node {\( x_{ 1  }\)};
 \draw [color=cyan,style=dashed] (1.08,2.61) -- (2.61,2.61);
 \draw [color=red,style=dashed] (1.08,2.61) -- (1.08,0);
-\draw []  (2.612757516,1.732740997) node [rotate=0] {$\bullet$};
-\draw (2.975768946,1.995361763) node {\( P_{ 2  }\)};
-\draw []  (1.732740997,1.732740997) node [rotate=0] {$\bullet$};
-\draw (1.367876808,2.034356520) node {\( Q_{2}\)};
-\draw []  (2.612757516,0) node [rotate=0] {$\bullet$};
-\draw (2.612757516,-0.3059510000) node {\( x_{ 2  }\)};
+\draw []  (2.6128,1.7327) node [rotate=0] {$\bullet$};
+\draw (2.9758,1.9954) node {\( P_{ 2  }\)};
+\draw []  (1.7327,1.7327) node [rotate=0] {$\bullet$};
+\draw (1.3679,2.0344) node {\( Q_{2}\)};
+\draw []  (2.6128,0) node [rotate=0] {$\bullet$};
+\draw (2.6128,-0.30595) node {\( x_{ 2  }\)};
 \draw [color=cyan,style=dashed] (2.61,1.73) -- (1.73,1.73);
 \draw [color=red,style=dashed] (2.61,1.73) -- (2.61,0);
 \draw [] (1.00,-0.100) -- (1.00,0.100);
diff --git a/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks b/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks
index 519863fb1..a903ec27e 100644
--- a/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks
+++ b/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks
@@ -96,36 +96,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (9.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.206705665);
+\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.2067);
 %DEFAULT
-\draw []  (0,1.353352832) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,2.706705665) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,2.706705665) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,1.353352832) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (7.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (8.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (9.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (0,1.3534) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,2.7067) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,2.7067) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,1.3534) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks b/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks
index 5bf8d7b5f..003cc3578 100644
--- a/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks
+++ b/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks
@@ -100,34 +100,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 \draw [color=blue] (0,6.0000)--(0.10101,4.9025)--(0.20202,4.0057)--(0.30303,3.2730)--(0.40404,2.6743)--(0.50505,2.1851)--(0.60606,1.7854)--(0.70707,1.4588)--(0.80808,1.1920)--(0.90909,0.97392)--(1.0101,0.79577)--(1.1111,0.65021)--(1.2121,0.53127)--(1.3131,0.43409)--(1.4141,0.35469)--(1.5152,0.28981)--(1.6162,0.23679)--(1.7172,0.19348)--(1.8182,0.15809)--(1.9192,0.12917)--(2.0202,0.10554)--(2.1212,0.086236)--(2.2222,0.070462)--(2.3232,0.057573)--(2.4242,0.047041)--(2.5253,0.038437)--(2.6263,0.031406)--(2.7273,0.025661)--(2.8283,0.020967)--(2.9293,0.017132)--(3.0303,0.013998)--(3.1313,0.011437)--(3.2323,0.0093452)--(3.3333,0.0076358)--(3.4343,0.0062390)--(3.5354,0.0050978)--(3.6364,0.0041653)--(3.7374,0.0034034)--(3.8384,0.0027808)--(3.9394,0.0022722)--(4.0404,0.0018565)--(4.1414,0.0015169)--(4.2424,0.0012394)--(4.3434,0.0010127)--(4.4444,0)--(4.5455,0)--(4.6465,0)--(4.7475,0)--(4.8485,0)--(4.9495,0)--(5.0505,0)--(5.1515,0)--(5.2525,0)--(5.3535,0)--(5.4545,0)--(5.5556,0)--(5.6566,0)--(5.7576,0)--(5.8586,0)--(5.9596,0)--(6.0606,0)--(6.1616,0)--(6.2626,0)--(6.3636,0)--(6.4646,0)--(6.5657,0)--(6.6667,0)--(6.7677,0)--(6.8687,0)--(6.9697,0)--(7.0707,0)--(7.1717,0)--(7.2727,0)--(7.3737,0)--(7.4747,0)--(7.5758,0)--(7.6768,0)--(7.7778,0)--(7.8788,0)--(7.9798,0)--(8.0808,0)--(8.1818,0)--(8.2828,0)--(8.3838,0)--(8.4848,0)--(8.5859,0)--(8.6869,0)--(8.7879,0)--(8.8889,0)--(8.9899,0)--(9.0909,0)--(9.1919,0)--(9.2929,0)--(9.3939,0)--(9.4949,0)--(9.5960,0)--(9.6970,0)--(9.7980,0)--(9.8990,0)--(10.000,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,6.000000000) node {$ 2 $};
+\draw (-0.29125,6.0000) node {$ 2 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks b/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks
index ac724f702..71823fe52 100644
--- a/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks
+++ b/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks
@@ -36,15 +36,15 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -76,17 +76,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (-1.88,1.32);
-\draw [color=red,->,>=latex] (0,0) -- (-0.1169777784,0.3213938048);
-\draw [color=green,->,>=latex] (0,0) -- (-0.8830222216,-0.3213938048);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=red,->,>=latex] (0,0) -- (-0.11698,0.32139);
+\draw [color=green,->,>=latex] (0,0) -- (-0.88302,-0.32139);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks b/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks
index d9f5e78d1..bfe4147ab 100644
--- a/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks
+++ b/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks
@@ -85,15 +85,15 @@
 \draw [] (0,0) -- (3.46,2.00);
 \draw [] (-1.20,0) -- (13.2,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.1912498333,0.2881297474) node {\( 0\)};
-\draw []  (3.464101615,2.000000000) node [rotate=0] {$\bullet$};
-\draw (3.268077615,2.287267247) node {\( y\)};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.2785761667) node {\( x\)};
-\draw []  (12.00000000,0) node [rotate=0] {$\bullet$};
-\draw (12.00000000,-0.3140621667) node {\( xy\)};
-\draw []  (1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.540800974,1.288129747) node {\( 1\)};
+\draw (-0.19125,0.28813) node {\( 0\)};
+\draw []  (3.4641,2.0000) node [rotate=0] {$\bullet$};
+\draw (3.2681,2.2873) node {\( y\)};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.27858) node {\( x\)};
+\draw []  (12.000,0) node [rotate=0] {$\bullet$};
+\draw (12.000,-0.31406) node {\( xy\)};
+\draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.5408,1.2881) node {\( 1\)};
 \draw [style=dashed] (1.73,1.00) -- (6.00,0);
 \draw [style=dashed] (3.46,2.00) -- (12.0,0);
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UneCellule.pstricks b/auto/pictures_tex/Fig_UneCellule.pstricks
index 0d7681818..5b9b823d9 100644
--- a/auto/pictures_tex/Fig_UneCellule.pstricks
+++ b/auto/pictures_tex/Fig_UneCellule.pstricks
@@ -103,47 +103,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.700000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.4140621667) node {$a_1=y_{10}$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.41406) node {$a_1=y_{10}$};
 \draw [style=dotted] (1.00,0) -- (1.00,2.00);
 \draw [] (1.00,2.00) -- (1.00,5.00);
-\draw []  (2.200000000,0) node [rotate=0] {$\bullet$};
-\draw (2.200000000,-0.7140621667) node {$y_{11}$};
+\draw []  (2.2000,0) node [rotate=0] {$\bullet$};
+\draw (2.2000,-0.71406) node {$y_{11}$};
 \draw [style=dotted] (2.20,0) -- (2.20,2.00);
 \draw [] (2.20,2.00) -- (2.20,5.00);
-\draw []  (3.700000000,0) node [rotate=0] {$\bullet$};
-\draw (3.700000000,-0.4140621667) node {$y_{12}$};
+\draw []  (3.7000,0) node [rotate=0] {$\bullet$};
+\draw (3.7000,-0.41406) node {$y_{12}$};
 \draw [style=dotted] (3.70,0) -- (3.70,2.00);
 \draw [] (3.70,2.00) -- (3.70,5.00);
-\draw []  (4.200000000,0) node [rotate=0] {$\bullet$};
-\draw (4.200000000,-0.7140621667) node {$y_{13}$};
+\draw []  (4.2000,0) node [rotate=0] {$\bullet$};
+\draw (4.2000,-0.71406) node {$y_{13}$};
 \draw [style=dotted] (4.20,0) -- (4.20,2.00);
 \draw [] (4.20,2.00) -- (4.20,5.00);
-\draw []  (5.200000000,0) node [rotate=0] {$\bullet$};
-\draw (5.200000000,-0.4140621667) node {$y_{14}$};
+\draw []  (5.2000,0) node [rotate=0] {$\bullet$};
+\draw (5.2000,-0.41406) node {$y_{14}$};
 \draw [style=dotted] (5.20,0) -- (5.20,2.00);
 \draw [] (5.20,2.00) -- (5.20,5.00);
-\draw []  (6.200000000,0) node [rotate=0] {$\bullet$};
-\draw (6.200000000,-0.7622220000) node {$b_1=y_{15}$};
+\draw []  (6.2000,0) node [rotate=0] {$\bullet$};
+\draw (6.2000,-0.76222) node {$b_1=y_{15}$};
 \draw [style=dotted] (6.20,0) -- (6.20,2.00);
 \draw [] (6.20,2.00) -- (6.20,5.00);
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.9584071667,2.000000000) node {$a_2=y_{20}$};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.95841,2.0000) node {$a_2=y_{20}$};
 \draw [style=dotted] (0,2.00) -- (1.00,2.00);
 \draw [] (1.00,2.00) -- (6.20,2.00);
-\draw []  (0,2.500000000) node [rotate=0] {$\bullet$};
-\draw (-0.5394763333,2.500000000) node {$y_{21}$};
+\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
+\draw (-0.53948,2.5000) node {$y_{21}$};
 \draw [style=dotted] (0,2.50) -- (1.00,2.50);
 \draw [] (1.00,2.50) -- (6.20,2.50);
-\draw []  (0,4.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.5394763333,4.000000000) node {$y_{22}$};
+\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
+\draw (-0.53948,4.0000) node {$y_{22}$};
 \draw [style=dotted] (0,4.00) -- (1.00,4.00);
 \draw [] (1.00,4.00) -- (6.20,4.00);
-\draw []  (0,5.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.9402623333,5.000000000) node {$b_2=y_{23}$};
+\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.94026,5.0000) node {$b_2=y_{23}$};
 \draw [style=dotted] (0,5.00) -- (1.00,5.00);
 \draw [] (1.00,5.00) -- (6.20,5.00);
 \fill [color=lightgray] (4.20,4.00) -- (5.20,4.00) -- (5.20,4.00) -- (5.20,2.50) -- (5.20,2.50) -- (4.20,2.50) -- (4.20,2.50) -- (4.20,4.00) --  cycle;
diff --git a/auto/pictures_tex/Fig_VANooZowSyO.pstricks b/auto/pictures_tex/Fig_VANooZowSyO.pstricks
index 3d09f7478..653bbd4a8 100644
--- a/auto/pictures_tex/Fig_VANooZowSyO.pstricks
+++ b/auto/pictures_tex/Fig_VANooZowSyO.pstricks
@@ -57,22 +57,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.499496542);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.4995);
 %DEFAULT
 
 \draw [color=blue] (-2.199,-1.000)--(-2.155,-0.9980)--(-2.110,-0.9920)--(-2.066,-0.9819)--(-2.021,-0.9679)--(-1.977,-0.9501)--(-1.933,-0.9284)--(-1.888,-0.9029)--(-1.844,-0.8738)--(-1.799,-0.8413)--(-1.755,-0.8053)--(-1.710,-0.7660)--(-1.666,-0.7237)--(-1.622,-0.6785)--(-1.577,-0.6306)--(-1.533,-0.5801)--(-1.488,-0.5272)--(-1.444,-0.4723)--(-1.399,-0.4154)--(-1.355,-0.3569)--(-1.311,-0.2969)--(-1.266,-0.2358)--(-1.222,-0.1736)--(-1.177,-0.1108)--(-1.133,-0.04758)--(-1.088,0.01587)--(-1.044,0.07925)--(-0.9996,0.1423)--(-0.9552,0.2048)--(-0.9107,0.2665)--(-0.8663,0.3271)--(-0.8219,0.3863)--(-0.7775,0.4441)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6442,0.6056)--(-0.5998,0.6549)--(-0.5553,0.7015)--(-0.5109,0.7453)--(-0.4665,0.7861)--(-0.4221,0.8237)--(-0.3776,0.8580)--(-0.3332,0.8888)--(-0.2888,0.9161)--(-0.2443,0.9397)--(-0.1999,0.9595)--(-0.1555,0.9754)--(-0.1111,0.9874)--(-0.06664,0.9955)--(-0.02221,0.9995)--(0.02221,0.9995)--(0.06664,0.9955)--(0.1111,0.9874)--(0.1555,0.9754)--(0.1999,0.9595)--(0.2443,0.9397)--(0.2888,0.9161)--(0.3332,0.8888)--(0.3776,0.8580)--(0.4221,0.8237)--(0.4665,0.7861)--(0.5109,0.7453)--(0.5553,0.7015)--(0.5998,0.6549)--(0.6442,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7775,0.4441)--(0.8219,0.3863)--(0.8663,0.3271)--(0.9107,0.2665)--(0.9552,0.2048)--(0.9996,0.1423)--(1.044,0.07925)--(1.088,0.01587)--(1.133,-0.04758)--(1.177,-0.1108)--(1.222,-0.1736)--(1.266,-0.2358)--(1.311,-0.2969)--(1.355,-0.3569)--(1.399,-0.4154)--(1.444,-0.4723)--(1.488,-0.5272)--(1.533,-0.5801)--(1.577,-0.6306)--(1.622,-0.6785)--(1.666,-0.7237)--(1.710,-0.7660)--(1.755,-0.8053)--(1.799,-0.8413)--(1.844,-0.8738)--(1.888,-0.9029)--(1.933,-0.9284)--(1.977,-0.9501)--(2.021,-0.9679)--(2.066,-0.9819)--(2.110,-0.9920)--(2.155,-0.9980)--(2.199,-1.000);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -127,22 +127,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.499874128) -- (0,1.499874128);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.4999) -- (0,1.4999);
 %DEFAULT
 
 \draw [color=blue] (-2.199,0)--(-2.155,0.06342)--(-2.110,0.1266)--(-2.066,0.1893)--(-2.021,0.2511)--(-1.977,0.3120)--(-1.933,0.3717)--(-1.888,0.4298)--(-1.844,0.4862)--(-1.799,0.5406)--(-1.755,0.5929)--(-1.710,0.6428)--(-1.666,0.6901)--(-1.622,0.7346)--(-1.577,0.7761)--(-1.533,0.8146)--(-1.488,0.8497)--(-1.444,0.8815)--(-1.399,0.9096)--(-1.355,0.9342)--(-1.311,0.9549)--(-1.266,0.9718)--(-1.222,0.9848)--(-1.177,0.9938)--(-1.133,0.9989)--(-1.088,0.9999)--(-1.044,0.9969)--(-0.9996,0.9898)--(-0.9552,0.9788)--(-0.9107,0.9638)--(-0.8663,0.9450)--(-0.8219,0.9224)--(-0.7775,0.8960)--(-0.7330,0.8660)--(-0.6886,0.8326)--(-0.6442,0.7958)--(-0.5998,0.7558)--(-0.5553,0.7127)--(-0.5109,0.6668)--(-0.4665,0.6182)--(-0.4221,0.5671)--(-0.3776,0.5137)--(-0.3332,0.4582)--(-0.2888,0.4009)--(-0.2443,0.3420)--(-0.1999,0.2817)--(-0.1555,0.2203)--(-0.1111,0.1580)--(-0.06664,0.09506)--(-0.02221,0.03173)--(0.02221,-0.03173)--(0.06664,-0.09506)--(0.1111,-0.1580)--(0.1555,-0.2203)--(0.1999,-0.2817)--(0.2443,-0.3420)--(0.2888,-0.4009)--(0.3332,-0.4582)--(0.3776,-0.5137)--(0.4221,-0.5671)--(0.4665,-0.6182)--(0.5109,-0.6668)--(0.5553,-0.7127)--(0.5998,-0.7558)--(0.6442,-0.7958)--(0.6886,-0.8326)--(0.7330,-0.8660)--(0.7775,-0.8960)--(0.8219,-0.9224)--(0.8663,-0.9450)--(0.9107,-0.9638)--(0.9552,-0.9788)--(0.9996,-0.9898)--(1.044,-0.9969)--(1.088,-0.9999)--(1.133,-0.9989)--(1.177,-0.9938)--(1.222,-0.9848)--(1.266,-0.9718)--(1.311,-0.9549)--(1.355,-0.9342)--(1.399,-0.9096)--(1.444,-0.8815)--(1.488,-0.8497)--(1.533,-0.8146)--(1.577,-0.7761)--(1.622,-0.7346)--(1.666,-0.6901)--(1.710,-0.6428)--(1.755,-0.5929)--(1.799,-0.5406)--(1.844,-0.4862)--(1.888,-0.4298)--(1.933,-0.3717)--(1.977,-0.3120)--(2.021,-0.2511)--(2.066,-0.1893)--(2.110,-0.1266)--(2.155,-0.06342)--(2.199,0);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -197,22 +197,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.495766562) -- (0,1.499066424);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.4958) -- (0,1.4991);
 %DEFAULT
 
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@@ -271,24 +271,24 @@
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 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
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 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
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 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.3105776667,3.141592654) node {$ \pi $};
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 \draw [] (-0.100,3.14) -- (0.100,3.14);
 %OTHER STUFF
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@@ -343,22 +343,22 @@
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+\draw [,->,>=latex] (0,-1.4995) -- (0,1.5000);
 %DEFAULT
 
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 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
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 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
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 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
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 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
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@@ -413,22 +413,22 @@
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-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
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+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
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 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
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 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
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@@ -475,18 +475,18 @@
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 %AXES
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+\draw [,->,>=latex] (-1.5996,0) -- (1.5996,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4999);
 %DEFAULT
 
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-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
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 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
 %OTHER STUFF
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diff --git a/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks b/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks
index 1b44cbd98..1b0b6770c 100644
--- a/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks
+++ b/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks
@@ -91,30 +91,30 @@
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+\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
+\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
 %DEFAULT
 
 \draw [color=blue] (-0.9000,-2.025)--(-0.8818,-1.905)--(-0.8636,-1.789)--(-0.8455,-1.679)--(-0.8273,-1.573)--(-0.8091,-1.471)--(-0.7909,-1.374)--(-0.7727,-1.282)--(-0.7545,-1.193)--(-0.7364,-1.109)--(-0.7182,-1.029)--(-0.7000,-0.9528)--(-0.6818,-0.8804)--(-0.6636,-0.8119)--(-0.6455,-0.7470)--(-0.6273,-0.6856)--(-0.6091,-0.6277)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5364,-0.4286)--(-0.5182,-0.3865)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4455,-0.2455)--(-0.4273,-0.2167)--(-0.4091,-0.1902)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1238)--(-0.3364,-0.1057)--(-0.3182,-0.08948)--(-0.3000,-0.07500)--(-0.2818,-0.06217)--(-0.2636,-0.05090)--(-0.2455,-0.04108)--(-0.2273,-0.03261)--(-0.2091,-0.02539)--(-0.1909,-0.01933)--(-0.1727,-0.01431)--(-0.1545,-0.01025)--(-0.1364,-0.007044)--(-0.1182,-0.004585)--(-0.1000,-0.002778)--(-0.08182,-0.001521)--(-0.06364,0)--(-0.04545,0)--(-0.02727,0)--(-0.009091,0)--(0.009091,0)--(0.02727,0)--(0.04545,0)--(0.06364,0)--(0.08182,0.001521)--(0.1000,0.002778)--(0.1182,0.004585)--(0.1364,0.007044)--(0.1545,0.01025)--(0.1727,0.01431)--(0.1909,0.01933)--(0.2091,0.02539)--(0.2273,0.03261)--(0.2455,0.04108)--(0.2636,0.05090)--(0.2818,0.06217)--(0.3000,0.07500)--(0.3182,0.08948)--(0.3364,0.1057)--(0.3545,0.1238)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4091,0.1902)--(0.4273,0.2167)--(0.4455,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5182,0.3865)--(0.5364,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6091,0.6277)--(0.6273,0.6856)--(0.6455,0.7470)--(0.6636,0.8119)--(0.6818,0.8804)--(0.7000,0.9528)--(0.7182,1.029)--(0.7364,1.109)--(0.7545,1.193)--(0.7727,1.282)--(0.7909,1.374)--(0.8091,1.471)--(0.8273,1.573)--(0.8455,1.679)--(0.8636,1.789)--(0.8818,1.905)--(0.9000,2.025);
 
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-\draw (-1.200000000,-0.3298256667) node {$ -2 $};
+\draw (-1.2000,-0.32983) node {$ -2 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (-0.6000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.60000,-0.32983) node {$ -1 $};
 \draw [] (-0.600,-0.100) -- (-0.600,0.100);
-\draw (0.6000000000,-0.3149246667) node {$ 1 $};
+\draw (0.60000,-0.31492) node {$ 1 $};
 \draw [] (0.600,-0.100) -- (0.600,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 2 $};
+\draw (1.2000,-0.31492) node {$ 2 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (-0.4331593333,-2.400000000) node {$ -4 $};
+\draw (-0.43316,-2.4000) node {$ -4 $};
 \draw [] (-0.100,-2.40) -- (0.100,-2.40);
-\draw (-0.4331593333,-1.200000000) node {$ -2 $};
+\draw (-0.43316,-1.2000) node {$ -2 $};
 \draw [] (-0.100,-1.20) -- (0.100,-1.20);
-\draw (-0.2912498333,1.200000000) node {$ 2 $};
+\draw (-0.29125,1.2000) node {$ 2 $};
 \draw [] (-0.100,1.20) -- (0.100,1.20);
-\draw (-0.2912498333,2.400000000) node {$ 4 $};
+\draw (-0.29125,2.4000) node {$ 4 $};
 \draw [] (-0.100,2.40) -- (0.100,2.40);
-\draw (-0.2912498333,3.600000000) node {$ 6 $};
+\draw (-0.29125,3.6000) node {$ 6 $};
 \draw [] (-0.100,3.60) -- (0.100,3.60);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks b/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks
index 763cdabf1..accdd4394 100644
--- a/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks
+++ b/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.400000000);
+\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
 %DEFAULT
 
 \draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
@@ -96,10 +96,10 @@
 \draw [color=blue] (3.000,0)--(3.030,0)--(3.061,0)--(3.091,0)--(3.121,0)--(3.152,0)--(3.182,0)--(3.212,0)--(3.242,0)--(3.273,0)--(3.303,0)--(3.333,0)--(3.364,0)--(3.394,0)--(3.424,0)--(3.455,0)--(3.485,0)--(3.515,0)--(3.545,0)--(3.576,0)--(3.606,0)--(3.636,0)--(3.667,0)--(3.697,0)--(3.727,0)--(3.758,0)--(3.788,0)--(3.818,0)--(3.848,0)--(3.879,0)--(3.909,0)--(3.939,0)--(3.970,0)--(4.000,0)--(4.030,0)--(4.061,0)--(4.091,0)--(4.121,0)--(4.151,0)--(4.182,0)--(4.212,0)--(4.242,0)--(4.273,0)--(4.303,0)--(4.333,0)--(4.364,0)--(4.394,0)--(4.424,0)--(4.455,0)--(4.485,0)--(4.515,0)--(4.545,0)--(4.576,0)--(4.606,0)--(4.636,0)--(4.667,0)--(4.697,0)--(4.727,0)--(4.758,0)--(4.788,0)--(4.818,0)--(4.849,0)--(4.879,0)--(4.909,0)--(4.939,0)--(4.970,0)--(5.000,0)--(5.030,0)--(5.061,0)--(5.091,0)--(5.121,0)--(5.151,0)--(5.182,0)--(5.212,0)--(5.242,0)--(5.273,0)--(5.303,0)--(5.333,0)--(5.364,0)--(5.394,0)--(5.424,0)--(5.455,0)--(5.485,0)--(5.515,0)--(5.545,0)--(5.576,0)--(5.606,0)--(5.636,0)--(5.667,0)--(5.697,0)--(5.727,0)--(5.758,0)--(5.788,0)--(5.818,0)--(5.849,0)--(5.879,0)--(5.909,0)--(5.939,0)--(5.970,0)--(6.000,0);
 \draw [style=dashed] (3.00,0) -- (3.00,3.00);
 \draw [style=dashed] (6.00,3.75) -- (6.00,0);
-\draw (3.000000000,-0.3785761667) node {$a$};
-\draw (6.000000000,-0.3785761667) node {$x$};
-\draw (8.355206667,3.900000000) node {$f(x)$};
-\draw (9.670153833,1.875000000) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (3.0000,-0.37858) node {$a$};
+\draw (6.0000,-0.37858) node {$x$};
+\draw (8.3552,3.9000) node {$f(x)$};
+\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks b/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks
index 11ca058d3..a1dcf7284 100644
--- a/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks
+++ b/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks
@@ -84,34 +84,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-3.795836866) -- (0,3.795836866);
+\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-3.7958) -- (0,3.7958);
 %DEFAULT
 
 \draw [color=blue] (-6.000,-3.296)--(-5.879,-3.169)--(-5.758,-3.044)--(-5.636,-2.920)--(-5.515,-2.797)--(-5.394,-2.676)--(-5.273,-2.556)--(-5.151,-2.437)--(-5.030,-2.320)--(-4.909,-2.204)--(-4.788,-2.090)--(-4.667,-1.977)--(-4.545,-1.866)--(-4.424,-1.756)--(-4.303,-1.648)--(-4.182,-1.542)--(-4.061,-1.438)--(-3.939,-1.335)--(-3.818,-1.234)--(-3.697,-1.136)--(-3.576,-1.039)--(-3.455,-0.9440)--(-3.333,-0.8514)--(-3.212,-0.7609)--(-3.091,-0.6728)--(-2.970,-0.5870)--(-2.848,-0.5037)--(-2.727,-0.4229)--(-2.606,-0.3449)--(-2.485,-0.2697)--(-2.364,-0.1974)--(-2.242,-0.1283)--(-2.121,-0.06241)--(-2.000,0)--(-1.879,0.05873)--(-1.758,0.1135)--(-1.636,0.1642)--(-1.515,0.2103)--(-1.394,0.2516)--(-1.273,0.2876)--(-1.152,0.3179)--(-1.030,0.3417)--(-0.9091,0.3584)--(-0.7879,0.3670)--(-0.6667,0.3662)--(-0.5455,0.3544)--(-0.4242,0.3289)--(-0.3030,0.2859)--(-0.1818,0.2180)--(-0.06061,0.1060)--(0.06061,-0.1060)--(0.1818,-0.2180)--(0.3030,-0.2859)--(0.4242,-0.3289)--(0.5455,-0.3544)--(0.6667,-0.3662)--(0.7879,-0.3670)--(0.9091,-0.3584)--(1.030,-0.3417)--(1.152,-0.3179)--(1.273,-0.2876)--(1.394,-0.2516)--(1.515,-0.2103)--(1.636,-0.1642)--(1.758,-0.1135)--(1.879,-0.05873)--(2.000,0)--(2.121,0.06241)--(2.242,0.1283)--(2.364,0.1974)--(2.485,0.2697)--(2.606,0.3449)--(2.727,0.4229)--(2.848,0.5037)--(2.970,0.5870)--(3.091,0.6728)--(3.212,0.7609)--(3.333,0.8514)--(3.455,0.9440)--(3.576,1.039)--(3.697,1.136)--(3.818,1.234)--(3.939,1.335)--(4.061,1.438)--(4.182,1.542)--(4.303,1.648)--(4.424,1.756)--(4.545,1.866)--(4.667,1.977)--(4.788,2.090)--(4.909,2.204)--(5.030,2.320)--(5.151,2.437)--(5.273,2.556)--(5.394,2.676)--(5.515,2.797)--(5.636,2.920)--(5.758,3.044)--(5.879,3.169)--(6.000,3.296);
-\draw (-6.000000000,-0.3298256667) node {$ -3 $};
+\draw (-6.0000,-0.32983) node {$ -3 $};
 \draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -2 $};
+\draw (-4.0000,-0.32983) node {$ -2 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 3 $};
+\draw (6.0000,-0.31492) node {$ 3 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
index 4a92b4ea1..75b352840 100644
--- a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
+++ b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
@@ -120,40 +120,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156038) -- (0,4.065868530);
+\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-4.5252) -- (0,4.0659);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
-\draw (-6.000000000,-0.3298256667) node {$ -10 $};
+\draw (-6.0000,-0.32983) node {$ -10 $};
 \draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.800000000,-0.3298256667) node {$ -8 $};
+\draw (-4.8000,-0.32983) node {$ -8 $};
 \draw [] (-4.80,-0.100) -- (-4.80,0.100);
-\draw (-3.600000000,-0.3298256667) node {$ -6 $};
+\draw (-3.6000,-0.32983) node {$ -6 $};
 \draw [] (-3.60,-0.100) -- (-3.60,0.100);
-\draw (-2.400000000,-0.3298256667) node {$ -4 $};
+\draw (-2.4000,-0.32983) node {$ -4 $};
 \draw [] (-2.40,-0.100) -- (-2.40,0.100);
-\draw (-1.200000000,-0.3298256667) node {$ -2 $};
+\draw (-1.2000,-0.32983) node {$ -2 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 2 $};
+\draw (1.2000,-0.31492) node {$ 2 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 4 $};
+\draw (2.4000,-0.31492) node {$ 4 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 6 $};
+\draw (3.6000,-0.31492) node {$ 6 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (4.800000000,-0.3149246667) node {$ 8 $};
+\draw (4.8000,-0.31492) node {$ 8 $};
 \draw [] (4.80,-0.100) -- (4.80,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 10 $};
+\draw (6.0000,-0.31492) node {$ 10 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.5944095000,-4.500000000) node {$ -\frac{3}{200} $};
+\draw (-0.59441,-4.5000) node {$ -\frac{3}{200} $};
 \draw [] (-0.100,-4.50) -- (0.100,-4.50);
-\draw (-0.5944095000,-3.000000000) node {$ -\frac{1}{100} $};
+\draw (-0.59441,-3.0000) node {$ -\frac{1}{100} $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.5944095000,-1.500000000) node {$ -\frac{1}{200} $};
+\draw (-0.59441,-1.5000) node {$ -\frac{1}{200} $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.4525000000,1.500000000) node {$ \frac{1}{200} $};
+\draw (-0.45250,1.5000) node {$ \frac{1}{200} $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.4525000000,3.000000000) node {$ \frac{1}{100} $};
+\draw (-0.45250,3.0000) node {$ \frac{1}{100} $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks b/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks
index 965b615c5..9465f44db 100644
--- a/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks
+++ b/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks
@@ -75,21 +75,21 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.3179930509,-0.1489399780) node {\( A \)};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.328701884,-0.1539557273) node {\( B \)};
-\draw []  (2.500000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (2.500000000,3.324708000) node {\( O \)};
+\draw (-0.31799,-0.14894) node {\( A \)};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.3287,-0.15396) node {\( B \)};
+\draw []  (2.5000,3.0000) node [rotate=0] {$\bullet$};
+\draw (2.5000,3.3247) node {\( O \)};
 \draw [] (0,0) -- (5.00,0);
 \draw [] (5.00,0) -- (2.50,3.00);
 \draw [] (2.50,3.00) -- (0,0);
-\draw [color=red,->,>=latex] (0.3200921998,0.3841106398) -- (0.3124099870,0.3905124838);
+\draw [color=red,->,>=latex] (0.32009,0.38411) -- (0.31241,0.39051);
 
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-\draw [color=red,->,>=latex] (2.820092200,2.615889360) -- (2.827774413,2.622291204);
+\draw [color=red,->,>=latex] (2.8201,2.6159) -- (2.8278,2.6223);
 
 \draw [color=red] (2.180,2.616)--(2.185,2.611)--(2.191,2.607)--(2.196,2.603)--(2.202,2.599)--(2.208,2.594)--(2.213,2.590)--(2.219,2.586)--(2.225,2.582)--(2.231,2.579)--(2.237,2.575)--(2.243,2.571)--(2.249,2.568)--(2.255,2.564)--(2.261,2.561)--(2.267,2.557)--(2.273,2.554)--(2.280,2.551)--(2.286,2.548)--(2.292,2.545)--(2.299,2.542)--(2.305,2.539)--(2.312,2.537)--(2.318,2.534)--(2.325,2.532)--(2.331,2.529)--(2.338,2.527)--(2.345,2.525)--(2.351,2.523)--(2.358,2.521)--(2.365,2.519)--(2.372,2.517)--(2.378,2.515)--(2.385,2.513)--(2.392,2.512)--(2.399,2.510)--(2.406,2.509)--(2.413,2.508)--(2.420,2.507)--(2.427,2.505)--(2.434,2.504)--(2.440,2.504)--(2.447,2.503)--(2.454,2.502)--(2.461,2.501)--(2.468,2.501)--(2.475,2.501)--(2.482,2.500)--(2.489,2.500)--(2.496,2.500)--(2.504,2.500)--(2.511,2.500)--(2.518,2.500)--(2.525,2.501)--(2.532,2.501)--(2.539,2.501)--(2.546,2.502)--(2.553,2.503)--(2.560,2.504)--(2.566,2.504)--(2.573,2.505)--(2.580,2.507)--(2.587,2.508)--(2.594,2.509)--(2.601,2.510)--(2.608,2.512)--(2.615,2.513)--(2.622,2.515)--(2.628,2.517)--(2.635,2.519)--(2.642,2.521)--(2.649,2.523)--(2.655,2.525)--(2.662,2.527)--(2.669,2.529)--(2.675,2.532)--(2.682,2.534)--(2.688,2.537)--(2.695,2.539)--(2.701,2.542)--(2.708,2.545)--(2.714,2.548)--(2.720,2.551)--(2.727,2.554)--(2.733,2.557)--(2.739,2.561)--(2.745,2.564)--(2.751,2.568)--(2.757,2.571)--(2.763,2.575)--(2.769,2.579)--(2.775,2.582)--(2.781,2.586)--(2.787,2.590)--(2.792,2.594)--(2.798,2.599)--(2.804,2.603)--(2.809,2.607)--(2.815,2.611)--(2.820,2.616);
-\draw [color=red,->,>=latex] (4.500000000,0) -- (4.500000000,-0.01000000000);
+\draw [color=red,->,>=latex] (4.5000,0) -- (4.5000,-0.010000);
 
 \draw [color=red] (4.680,0.3841)--(4.677,0.3813)--(4.673,0.3784)--(4.670,0.3755)--(4.667,0.3725)--(4.663,0.3696)--(4.660,0.3666)--(4.657,0.3636)--(4.654,0.3605)--(4.650,0.3574)--(4.647,0.3543)--(4.644,0.3512)--(4.641,0.3480)--(4.638,0.3448)--(4.635,0.3416)--(4.632,0.3384)--(4.629,0.3351)--(4.626,0.3318)--(4.623,0.3285)--(4.620,0.3251)--(4.617,0.3218)--(4.614,0.3184)--(4.612,0.3149)--(4.609,0.3115)--(4.606,0.3080)--(4.603,0.3045)--(4.601,0.3010)--(4.598,0.2974)--(4.595,0.2939)--(4.593,0.2903)--(4.590,0.2867)--(4.588,0.2830)--(4.585,0.2794)--(4.583,0.2757)--(4.580,0.2720)--(4.578,0.2683)--(4.576,0.2645)--(4.573,0.2608)--(4.571,0.2570)--(4.569,0.2532)--(4.567,0.2493)--(4.564,0.2455)--(4.562,0.2416)--(4.560,0.2378)--(4.558,0.2339)--(4.556,0.2299)--(4.554,0.2260)--(4.552,0.2220)--(4.550,0.2181)--(4.548,0.2141)--(4.546,0.2101)--(4.544,0.2060)--(4.543,0.2020)--(4.541,0.1980)--(4.539,0.1939)--(4.537,0.1898)--(4.536,0.1857)--(4.534,0.1816)--(4.533,0.1775)--(4.531,0.1733)--(4.529,0.1692)--(4.528,0.1650)--(4.527,0.1608)--(4.525,0.1566)--(4.524,0.1524)--(4.522,0.1482)--(4.521,0.1439)--(4.520,0.1397)--(4.519,0.1354)--(4.518,0.1312)--(4.516,0.1269)--(4.515,0.1226)--(4.514,0.1183)--(4.513,0.1140)--(4.512,0.1097)--(4.511,0.1054)--(4.510,0.1011)--(4.509,0.09673)--(4.509,0.09238)--(4.508,0.08803)--(4.507,0.08367)--(4.506,0.07931)--(4.506,0.07493)--(4.505,0.07056)--(4.504,0.06617)--(4.504,0.06179)--(4.503,0.05739)--(4.503,0.05299)--(4.502,0.04859)--(4.502,0.04419)--(4.502,0.03978)--(4.501,0.03537)--(4.501,0.03095)--(4.501,0.02653)--(4.500,0.02212)--(4.500,0.01769)--(4.500,0.01327)--(4.500,0.008849)--(4.500,0.004424)--(4.500,0);
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_examssepti.pstricks b/auto/pictures_tex/Fig_examssepti.pstricks
index e8a2ef9ff..74c463784 100644
--- a/auto/pictures_tex/Fig_examssepti.pstricks
+++ b/auto/pictures_tex/Fig_examssepti.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 \draw [style=dotted] (-2.36,1.85) -- (-2.36,0);
 
@@ -95,12 +95,12 @@
 \draw [color=red] (-3.00,3.00)--(-2.95,2.91)--(-2.91,2.82)--(-2.86,2.73)--(-2.82,2.65)--(-2.77,2.56)--(-2.73,2.48)--(-2.68,2.40)--(-2.64,2.32)--(-2.59,2.24)--(-2.55,2.16)--(-2.50,2.08)--(-2.45,2.01)--(-2.41,1.93)--(-2.36,1.86)--(-2.32,1.79)--(-2.27,1.72)--(-2.23,1.65)--(-2.18,1.59)--(-2.14,1.52)--(-2.09,1.46)--(-2.05,1.39)--(-2.00,1.33)--(-1.95,1.27)--(-1.91,1.21)--(-1.86,1.16)--(-1.82,1.10)--(-1.77,1.05)--(-1.73,0.995)--(-1.68,0.943)--(-1.64,0.893)--(-1.59,0.844)--(-1.55,0.796)--(-1.50,0.750)--(-1.45,0.705)--(-1.41,0.662)--(-1.36,0.620)--(-1.32,0.579)--(-1.27,0.540)--(-1.23,0.502)--(-1.18,0.466)--(-1.14,0.430)--(-1.09,0.397)--(-1.05,0.364)--(-1.00,0.333)--(-0.955,0.304)--(-0.909,0.275)--(-0.864,0.249)--(-0.818,0.223)--(-0.773,0.199)--(-0.727,0.176)--(-0.682,0.155)--(-0.636,0.135)--(-0.591,0.116)--(-0.545,0.0992)--(-0.500,0.0833)--(-0.455,0.0689)--(-0.409,0.0558)--(-0.364,0.0441)--(-0.318,0.0337)--(-0.273,0.0248)--(-0.227,0.0172)--(-0.182,0.0110)--(-0.136,0.00620)--(-0.0909,0.00275)--(-0.0455,0)--(0,0)--(0.0455,0)--(0.0909,0.00275)--(0.136,0.00620)--(0.182,0.0110)--(0.227,0.0172)--(0.273,0.0248)--(0.318,0.0337)--(0.364,0.0441)--(0.409,0.0558)--(0.455,0.0689)--(0.500,0.0833)--(0.545,0.0992)--(0.591,0.116)--(0.636,0.135)--(0.682,0.155)--(0.727,0.176)--(0.773,0.199)--(0.818,0.223)--(0.864,0.249)--(0.909,0.275)--(0.955,0.304)--(1.00,0.333)--(1.05,0.364)--(1.09,0.397)--(1.14,0.430)--(1.18,0.466)--(1.23,0.502)--(1.27,0.540)--(1.32,0.579)--(1.36,0.620)--(1.41,0.662)--(1.45,0.705)--(1.50,0.750);
 
 \draw [color=blue] (-3.00,0)--(-2.95,0.520)--(-2.91,0.733)--(-2.86,0.894)--(-2.82,1.03)--(-2.77,1.15)--(-2.73,1.25)--(-2.68,1.34)--(-2.64,1.43)--(-2.59,1.51)--(-2.55,1.59)--(-2.50,1.66)--(-2.45,1.72)--(-2.41,1.79)--(-2.36,1.85)--(-2.32,1.90)--(-2.27,1.96)--(-2.23,2.01)--(-2.18,2.06)--(-2.14,2.11)--(-2.09,2.15)--(-2.05,2.19)--(-2.00,2.24)--(-1.95,2.28)--(-1.91,2.31)--(-1.86,2.35)--(-1.82,2.39)--(-1.77,2.42)--(-1.73,2.45)--(-1.68,2.48)--(-1.64,2.51)--(-1.59,2.54)--(-1.55,2.57)--(-1.50,2.60)--(-1.45,2.62)--(-1.41,2.65)--(-1.36,2.67)--(-1.32,2.69)--(-1.27,2.72)--(-1.23,2.74)--(-1.18,2.76)--(-1.14,2.78)--(-1.09,2.79)--(-1.05,2.81)--(-1.00,2.83)--(-0.955,2.84)--(-0.909,2.86)--(-0.864,2.87)--(-0.818,2.89)--(-0.773,2.90)--(-0.727,2.91)--(-0.682,2.92)--(-0.636,2.93)--(-0.591,2.94)--(-0.545,2.95)--(-0.500,2.96)--(-0.455,2.97)--(-0.409,2.97)--(-0.364,2.98)--(-0.318,2.98)--(-0.273,2.99)--(-0.227,2.99)--(-0.182,2.99)--(-0.136,3.00)--(-0.0909,3.00)--(-0.0455,3.00)--(0,3.00)--(0.0455,3.00)--(0.0909,3.00)--(0.136,3.00)--(0.182,2.99)--(0.227,2.99)--(0.273,2.99)--(0.318,2.98)--(0.364,2.98)--(0.409,2.97)--(0.455,2.97)--(0.500,2.96)--(0.545,2.95)--(0.591,2.94)--(0.636,2.93)--(0.682,2.92)--(0.727,2.91)--(0.773,2.90)--(0.818,2.89)--(0.864,2.87)--(0.909,2.86)--(0.955,2.84)--(1.00,2.83)--(1.05,2.81)--(1.09,2.79)--(1.14,2.78)--(1.18,2.76)--(1.23,2.74)--(1.27,2.72)--(1.32,2.69)--(1.36,2.67)--(1.41,2.65)--(1.45,2.62)--(1.50,2.60);
-\draw []  (-2.358454133,1.854101966) node [rotate=0] {$\bullet$};
-\draw []  (-2.358454133,0) node [rotate=0] {$\bullet$};
-\draw (-2.358454133,-0.2335035000) node {\( -x_0\)};
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (-2.3585,1.8541) node [rotate=0] {$\bullet$};
+\draw []  (-2.3585,0) node [rotate=0] {$\bullet$};
+\draw (-2.3585,-0.23350) node {\( -x_0\)};
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_trigoWedd.pstricks b/auto/pictures_tex/Fig_trigoWedd.pstricks
index ed0d4b55a..899d83cd8 100644
--- a/auto/pictures_tex/Fig_trigoWedd.pstricks
+++ b/auto/pictures_tex/Fig_trigoWedd.pstricks
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.8660254038,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (1.289871192,0.7559510000) node {$z_0$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.2140621667) node {$q$};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$1$};
+\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
+\draw (1.2899,0.75595) node {$z_0$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.21406) node {$q$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$1$};
 
 %OTHER STUFF
 %END PSPICTURE

From ab676d9e7bcac2b2a9422b7f57e96263dea7a52f Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Fri, 16 Jun 2017 09:09:11 +0200
Subject: [PATCH 17/64] (testing) Remove 'TestRecall.py' because it is going to
 be included in 'phystricks'.

---
 testing/TestRecall.py | 46 -------------------------------------------
 1 file changed, 46 deletions(-)
 delete mode 100755 testing/TestRecall.py

diff --git a/testing/TestRecall.py b/testing/TestRecall.py
deleted file mode 100755
index ceef7913f..000000000
--- a/testing/TestRecall.py
+++ /dev/null
@@ -1,46 +0,0 @@
-#! /usr/bin/python3
-# -*- coding: utf8 -*-
-
-
-# This script compares the files "*.pstricks" with the corresponding one 
-# "*.pstricks.recall" and prints a warning if they are not equal.
-
-# The directory passed as argument is the main directory of the
-# mazhe project.
-#
-# So, relative to the passed argument 
-# - '.pstricks' files to be checked are in `auto/pictures_tex`
-# - '.recall' files to be checked against are in `src_phystricks`
-
-import os
-
-def pstricks_files_iterator(directory):
-    for f in os.listdir(directory):
-        if f.endswith(".pstricks"):
-            yield os.path.join(directory,f)
-
-def wrong_file_list(directory):
-    """
-    return a tuple of lists
-    - the list of missing 'recall'
-    - the list of 'recall/pstricks' which do not match
-    """
-    src_phystricks_dir=os.path.join(directory,"src_phystricks")
-    auto_pictures_tex_dir=os.path.join(directory,"auto/pictures_tex")
-    wfl=[]  # wrong file list
-    mfl=[]  # missing file list
-    for filename in pstricks_files_iterator(auto_pictures_tex_dir):
-        with open(filename,'r') as f:
-            get_text=f.read()
-        try :
-            recall_filename=os.path.join(src_phystricks_dir,
-                                os.path.split(filename)[1]+".recall")
-            with open(recall_filename,'r') as f:
-                recall_text=f.read()
-        except FileNotFoundError as err :
-            mfl.append(filename)
-            recall_text=get_text    # we do not append it to the wfl list.
-
-        if get_text != recall_text :
-            wfl.append(filename)
-    return mfl,wfl

From 7f60d509c365674c63413e307f8d2228f28fe28e Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 17 Jun 2017 06:42:02 +0200
Subject: [PATCH 18/64] (picture testing) Use the module 'TestRecall' from
 'phystricks'.

---
 testing/test_recall.py | 26 +++++++++++++++-----------
 1 file changed, 15 insertions(+), 11 deletions(-)

diff --git a/testing/test_recall.py b/testing/test_recall.py
index ae0351041..67776bbdf 100755
--- a/testing/test_recall.py
+++ b/testing/test_recall.py
@@ -9,19 +9,23 @@
 # - '.recall' files to be checked against are in `src_phystricks`
 
 import sys
-from TestRecall import wrong_file_list
+import os
+import importlib
 
-directory=sys.argv[1]
+# Get the directory name of 'phystricks'
+pkg_loader = importlib.find_loader('phystricks')
+phystricks_dir=os.path.split(pkg_loader.path)[0]
+
+# Append to 'sys.path' the name of the directory in which is 'TestRecall.py'
+test_recall_file=os.path.join(phystricks_dir,"testing/recall_tests/TestRecall.py")
+sys.path.append(os.path.dirname(test_recall_file))
 
-try:
-    mfl,wfl=wrong_file_list(directory)
-except NotADirectoryError :
-    print("[test_recall.py] the passed directory does not exist")
-    raise
+# Import 'TestRecall'
+TestRecall = importlib.import_module("TestRecall")
 
+directory=sys.argv[1]
 
-for f in mfl:
-    print("missing recall : ",f)
-for f in wfl:
-    print("Wrong : ",f)
+pstricks_directory=os.path.join(directory,"auto/pictures_tex")
+recall_directory=os.path.join(directory,"src_phystricks")
 
+TestRecall.check_pictures(pstricks_directory,recall_directory)

From 18608b4f9e61de31610aa58630c5712c0c9123fb Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 20 Jun 2017 14:11:28 +0200
Subject: [PATCH 19/64] (testing) Add a small README.md

---
 testing/README.md | 4 ++++
 1 file changed, 4 insertions(+)
 create mode 100644 testing/README.md

diff --git a/testing/README.md b/testing/README.md
new file mode 100644
index 000000000..f3fa6e4da
--- /dev/null
+++ b/testing/README.md
@@ -0,0 +1,4 @@
+# Testing mazhe
+
+
+- You need `phystricks` to be in `$PYTHONPATH`

From 42b3e99da3840c1fcee35250cd32d1a77e116004 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 20 Jun 2017 14:12:03 +0200
Subject: [PATCH 20/64] (testing,pictures) Change the way we determine the
 directory in which search for 'TestRecall.py'.

In particular, use better 'importlib'
---
 testing/test_recall.py | 13 +++++++++----
 testing/testing.sh     |  2 ++
 2 files changed, 11 insertions(+), 4 deletions(-)

diff --git a/testing/test_recall.py b/testing/test_recall.py
index 67776bbdf..b2137c9b0 100755
--- a/testing/test_recall.py
+++ b/testing/test_recall.py
@@ -8,13 +8,18 @@
 # - '.pstricks' files to be checked are in `auto/pictures_tex`
 # - '.recall' files to be checked against are in `src_phystricks`
 
+
+# This script only works when the directory of 'phystricks' is in
+# $PYTHONPATH
+# The reason is that we need to import 'TestRecall'
+
 import sys
 import os
-import importlib
+import importlib.util
 
-# Get the directory name of 'phystricks'
-pkg_loader = importlib.find_loader('phystricks')
-phystricks_dir=os.path.split(pkg_loader.path)[0]
+# Get the path in which to find the module 'phystricks'
+spec=importlib.util.find_spec('phystricks')
+phystricks_dir=spec.submodule_search_locations[0]
 
 # Append to 'sys.path' the name of the directory in which is 'TestRecall.py'
 test_recall_file=os.path.join(phystricks_dir,"testing/recall_tests/TestRecall.py")
diff --git a/testing/testing.sh b/testing/testing.sh
index 094cf14fb..d72de039a 100755
--- a/testing/testing.sh
+++ b/testing/testing.sh
@@ -11,6 +11,8 @@
 
 # The fact that the '.pstricks' are equal to the '.recall'
 # is checked at each compilation in 'lst_frido.py'
+# For that, the directory of 'phystricks' must be in $PYTONPATH,
+# see test_recall.py
 
 
 # If everything goes well (not yet implemented) :

From 1d7ef03ce1ea1e9d71fb3dbd2b80192cf14b4b03 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 20 Jun 2017 15:38:14 +0200
Subject: [PATCH 21/64] (testing) Update the '.recall' files of many pictures.

---
 .../Fig_ACUooQwcDMZ.pstricks.recall           |   2 +-
 src_phystricks/Fig_ADUGmRRA.pstricks.recall   |   6 +-
 .../Fig_AMDUooZZUOqa.pstricks.recall          |   8 +-
 .../Fig_ASHYooUVHkak.pstricks.recall          |  16 +--
 .../Fig_AccumulationIsole.pstricks.recall     |  12 +-
 src_phystricks/Fig_AdhIntFr.pstricks.recall   |  24 ++--
 .../Fig_BIFooDsvVHb.pstricks.recall           |  14 +--
 src_phystricks/Fig_BiaisOuPas.pstricks.recall |  26 ++---
 .../Fig_BoulePtLoin.pstricks.recall           |  12 +-
 .../Fig_CFMooGzvfRP.pstricks.recall           |  12 +-
 src_phystricks/Fig_CSCii.pstricks.recall      |  14 +--
 src_phystricks/Fig_CSCiii.pstricks.recall     |  32 ++---
 src_phystricks/Fig_CSCiv.pstricks.recall      |  24 ++--
 src_phystricks/Fig_CSCv.pstricks.recall       |  32 ++---
 .../Fig_Cardioideexo.pstricks.recall          |  24 ++--
 src_phystricks/Fig_CbCartTui.pstricks.recall  |   4 +-
 src_phystricks/Fig_CbCartTuii.pstricks.recall |   6 +-
 .../Fig_CbCartTuiii.pstricks.recall           |   4 +-
 src_phystricks/Fig_CercleTnu.pstricks.recall  |  16 +--
 .../Fig_CercleTrigono.pstricks.recall         |  26 ++---
 .../Fig_CheminFresnel.pstricks.recall         |  16 +--
 .../Fig_ChiSquaresQuantile.pstricks.recall    |   2 +-
 .../Fig_ConeRevolution.pstricks.recall        |  14 +--
 src_phystricks/Fig_CycloideA.pstricks.recall  |   2 +-
 .../Fig_DNHRooqGtffLkd.pstricks.recall        |  40 +++----
 .../Fig_DNRRooJWRHgOCw.pstricks.recall        |  28 ++---
 src_phystricks/Fig_DTIYKkP.pstricks.recall    |  12 +-
 .../Fig_DZVooQZLUtf.pstricks.recall           |   2 +-
 ...Fig_DefinitionCartesiennes.pstricks.recall |  42 +++----
 src_phystricks/Fig_DessinLim.pstricks.recall  |   4 +-
 .../Fig_Differentielle.pstricks.recall        |   4 +-
 src_phystricks/Fig_DisqueConv.pstricks.recall |   8 +-
 .../Fig_DistanceEuclide.pstricks.recall       |  30 ++---
 .../Fig_DynkinpWjUbE.pstricks.recall          |   4 +-
 .../Fig_DynkinrjbHIu.pstricks.recall          |   4 +-
 .../Fig_ExerciceGraphesbis.pstricks.recall    |  14 +--
 src_phystricks/Fig_ExoCUd.pstricks.recall     |  20 ++--
 src_phystricks/Fig_ExoParamCD.pstricks.recall |  26 ++---
 .../Fig_ExoUnSurxPolaire.pstricks.recall      |  40 +++----
 .../Fig_FnCosApprox.pstricks.recall           |  20 ++--
 .../Fig_FonctionEtDeriveOM.pstricks.recall    |   2 +-
 src_phystricks/Fig_GBnUivi.pstricks.recall    |  22 ++--
 .../Fig_GVDJooYzMxLW.pstricks.recall          |  18 +--
 .../Fig_GWOYooRxHKSm.pstricks.recall          |  36 +++---
 .../Fig_Grapheunsurunmoinsx.pstricks.recall   |  44 +++----
 .../Fig_HCJPooHsaTgI.pstricks.recall          |  24 ++--
 .../Fig_HFAYooOrfMAA.pstricks.recall          |   8 +-
 .../Fig_HLJooGDZnqF.pstricks.recall           |  42 +++----
 .../Fig_IOCTooePeHGCXH.pstricks.recall        |  24 ++--
 src_phystricks/Fig_IntEcourbe.pstricks.recall |  18 +--
 src_phystricks/Fig_IntTrois.pstricks.recall   |  28 ++---
 .../Fig_IntegraleSimple.pstricks.recall       |   2 +-
 .../Fig_IntervalleUn.pstricks.recall          |  22 ++--
 src_phystricks/Fig_IsomCarre.pstricks.recall  |  18 +--
 .../Fig_JJAooWpimYW.pstricks.recall           |   2 +-
 .../Fig_JSLooFJWXtB.pstricks.recall           |   2 +-
 src_phystricks/Fig_LAfWmaN.pstricks.recall    |  22 ++--
 .../Fig_LBGooAdteCt.pstricks.recall           |  28 ++---
 .../Fig_LLVMooWOkvAB.pstricks.recall          |  12 +-
 .../Fig_LMHMooCscXNNdU.pstricks.recall        |  28 ++---
 src_phystricks/Fig_Laurin.pstricks.recall     |  28 ++---
 src_phystricks/Fig_MCQueGF.pstricks.recall    |  28 ++---
 src_phystricks/Fig_Mantisse.pstricks.recall   |  16 +--
 .../Fig_MethodeNewton.pstricks.recall         |   2 +-
 .../Fig_NiveauHyperboleDeux.pstricks.recall   |  36 +++---
 .../Fig_OQTEoodIwAPfZE.pstricks.recall        |   2 +-
 .../Fig_ProjectionScalaire.pstricks.recall    |  20 ++--
 .../Fig_QIZooQNQSJj.pstricks.recall           |   8 +-
 src_phystricks/Fig_QPcdHwP.pstricks.recall    |   6 +-
 .../Fig_QSKDooujUbDCsu.pstricks.recall        |   6 +-
 src_phystricks/Fig_RLuqsrr.pstricks.recall    |   2 +-
 .../Fig_RPNooQXxpZZ.pstricks.recall           |  20 ++--
 src_phystricks/Fig_Refraction.pstricks.recall |  12 +-
 .../Fig_SJAWooRDGzIkrj.pstricks.recall        |  22 ++--
 .../Fig_SenoTopologo.pstricks.recall          |   2 +-
 src_phystricks/Fig_SolsSinpA.pstricks.recall  |  16 +--
 .../Fig_SuiteInverseAlterne.pstricks.recall   |  44 +++----
 .../Fig_SurfaceEntreCourbes.pstricks.recall   |  36 +++---
 .../Fig_SurfacePrimiteGeog.pstricks.recall    |  12 +-
 .../Fig_TVXooWoKkqV.pstricks.recall           |   2 +-
 .../Fig_TWHooJjXEtS.pstricks.recall           |   2 +-
 src_phystricks/Fig_TZCISko.pstricks.recall    |  10 +-
 .../Fig_TangentSegment.pstricks.recall        |   2 +-
 .../Fig_TgCercleTrigono.pstricks.recall       |  20 ++--
 src_phystricks/Fig_TracerUn.pstricks.recall   | 102 ++++++++--------
 .../Fig_UEGEooHEDIJVPn.pstricks.recall        |  40 +++----
 .../Fig_UGCFooQoCihh.pstricks.recall          |  42 +++----
 .../Fig_UMEBooVTMyfD.pstricks.recall          |  28 ++---
 .../Fig_UUNEooCNVOOs.pstricks.recall          |  40 +++----
 .../Fig_UYJooCWjLgK.pstricks.recall           |  18 +--
 src_phystricks/Fig_UneCellule.pstricks.recall |  44 +++----
 .../Fig_VANooZowSyO.pstricks.recall           | 110 +++++++++---------
 .../Fig_VBOIooRHhKOH.pstricks.recall          |  22 ++--
 .../Fig_VSZRooRWgUGu.pstricks.recall          |  12 +-
 .../Fig_WIRAooTCcpOV.pstricks.recall          |   2 +-
 .../Fig_WJBooMTAhtl.pstricks.recall           |   2 +-
 .../Fig_WUYooCISzeB.pstricks.recall           |   2 +-
 .../Fig_XJMooCQTlNL.pstricks.recall           |  28 ++---
 .../Fig_XOLBooGcrjiwoU.pstricks.recall        |  34 +++---
 .../Fig_YQIDooBqpAdbIM.pstricks.recall        |  16 +--
 src_phystricks/Fig_examssepti.pstricks.recall |  14 +--
 src_phystricks/Fig_trigoWedd.pstricks.recall  |  16 +--
 102 files changed, 987 insertions(+), 987 deletions(-)

diff --git a/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall b/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
index 7b8366461..9e1c13cba 100644
--- a/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
+++ b/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
@@ -346,7 +346,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.548147074);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.548147075);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.020,0.1414)--(1.040,0.1990)--(1.061,0.2426)--(1.081,0.2788)--(1.101,0.3102)--(1.121,0.3382)--(1.141,0.3637)--(1.162,0.3871)--(1.182,0.4087)--(1.202,0.4290)--(1.222,0.4480)--(1.242,0.4659)--(1.263,0.4829)--(1.283,0.4991)--(1.303,0.5145)--(1.323,0.5292)--(1.343,0.5434)--(1.364,0.5569)--(1.384,0.5700)--(1.404,0.5825)--(1.424,0.5947)--(1.444,0.6064)--(1.465,0.6178)--(1.485,0.6287)--(1.505,0.6394)--(1.525,0.6497)--(1.545,0.6598)--(1.566,0.6696)--(1.586,0.6791)--(1.606,0.6883)--(1.626,0.6973)--(1.646,0.7061)--(1.667,0.7147)--(1.687,0.7231)--(1.707,0.7313)--(1.727,0.7393)--(1.747,0.7471)--(1.768,0.7548)--(1.788,0.7623)--(1.808,0.7696)--(1.828,0.7768)--(1.848,0.7838)--(1.869,0.7907)--(1.889,0.7975)--(1.909,0.8041)--(1.929,0.8107)--(1.949,0.8170)--(1.970,0.8233)--(1.990,0.8295)--(2.010,0.8356)--(2.030,0.8415)--(2.051,0.8474)--(2.071,0.8532)--(2.091,0.8588)--(2.111,0.8644)--(2.131,0.8699)--(2.152,0.8753)--(2.172,0.8806)--(2.192,0.8859)--(2.212,0.8910)--(2.232,0.8961)--(2.253,0.9011)--(2.273,0.9061)--(2.293,0.9109)--(2.313,0.9157)--(2.333,0.9205)--(2.354,0.9252)--(2.374,0.9298)--(2.394,0.9343)--(2.414,0.9388)--(2.434,0.9432)--(2.455,0.9476)--(2.475,0.9519)--(2.495,0.9562)--(2.515,0.9604)--(2.535,0.9645)--(2.556,0.9686)--(2.576,0.9727)--(2.596,0.9767)--(2.616,0.9807)--(2.636,0.9846)--(2.657,0.9884)--(2.677,0.9923)--(2.697,0.9961)--(2.717,0.9998)--(2.737,1.003)--(2.758,1.007)--(2.778,1.011)--(2.798,1.014)--(2.818,1.018)--(2.838,1.021)--(2.859,1.025)--(2.879,1.028)--(2.899,1.032)--(2.919,1.035)--(2.939,1.038)--(2.960,1.042)--(2.980,1.045)--(3.000,1.048);
diff --git a/src_phystricks/Fig_ADUGmRRA.pstricks.recall b/src_phystricks/Fig_ADUGmRRA.pstricks.recall
index ade88656d..3eafef08f 100644
--- a/src_phystricks/Fig_ADUGmRRA.pstricks.recall
+++ b/src_phystricks/Fig_ADUGmRRA.pstricks.recall
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.2059510000) node {\( \alpha_i\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2191818333) node {\( \alpha_{i+1}\)};
+\draw (0,0.20595) node {\( \alpha_i\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall b/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall
index 914f91169..6fbdb6220 100644
--- a/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall
+++ b/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall
@@ -89,14 +89,14 @@
 \draw [style=dashed] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
 
 \draw [color=blue,style=] (1.970,-0.3473)--(1.974,-0.3230)--(1.978,-0.2986)--(1.981,-0.2742)--(1.984,-0.2497)--(1.987,-0.2252)--(1.990,-0.2006)--(1.992,-0.1761)--(1.994,-0.1515)--(1.996,-0.1268)--(1.997,-0.1022)--(1.998,-0.07755)--(1.999,-0.05288)--(2.000,-0.02821)--(2.000,-0.003526)--(2.000,0.02116)--(1.999,0.04583)--(1.999,0.07050)--(1.998,0.09516)--(1.996,0.1198)--(1.995,0.1444)--(1.993,0.1690)--(1.991,0.1936)--(1.988,0.2182)--(1.985,0.2427)--(1.982,0.2672)--(1.979,0.2916)--(1.975,0.3160)--(1.971,0.3404)--(1.966,0.3646)--(1.962,0.3889)--(1.957,0.4131)--(1.952,0.4372)--(1.946,0.4612)--(1.940,0.4852)--(1.934,0.5091)--(1.928,0.5330)--(1.921,0.5567)--(1.914,0.5804)--(1.907,0.6039)--(1.899,0.6274)--(1.891,0.6508)--(1.883,0.6741)--(1.875,0.6973)--(1.866,0.7204)--(1.857,0.7433)--(1.847,0.7662)--(1.838,0.7889)--(1.828,0.8115)--(1.818,0.8340)--(1.807,0.8564)--(1.797,0.8786)--(1.786,0.9007)--(1.774,0.9227)--(1.763,0.9445)--(1.751,0.9662)--(1.739,0.9878)--(1.727,1.009)--(1.714,1.030)--(1.701,1.051)--(1.688,1.072)--(1.675,1.093)--(1.661,1.114)--(1.647,1.134)--(1.633,1.154)--(1.619,1.174)--(1.604,1.194)--(1.589,1.214)--(1.574,1.234)--(1.559,1.253)--(1.543,1.272)--(1.528,1.291)--(1.512,1.310)--(1.495,1.328)--(1.479,1.347)--(1.462,1.365)--(1.445,1.383)--(1.428,1.400)--(1.410,1.418)--(1.393,1.435)--(1.375,1.452)--(1.357,1.469)--(1.339,1.486)--(1.320,1.502)--(1.302,1.518)--(1.283,1.534)--(1.264,1.550)--(1.245,1.566)--(1.225,1.581)--(1.206,1.596)--(1.186,1.611)--(1.166,1.625)--(1.146,1.639)--(1.125,1.653)--(1.105,1.667)--(1.084,1.681)--(1.063,1.694)--(1.042,1.707)--(1.021,1.720)--(1.000,1.732);
-\draw (0.6519509490,0.1894063750) node {$\theta$};
+\draw (0.65195,0.18941) node {$\theta$};
 
 \draw [] (0.492,-0.0868)--(0.493,-0.0807)--(0.494,-0.0746)--(0.495,-0.0685)--(0.496,-0.0624)--(0.497,-0.0563)--(0.497,-0.0502)--(0.498,-0.0440)--(0.499,-0.0379)--(0.499,-0.0317)--(0.499,-0.0256)--(0.500,-0.0194)--(0.500,-0.0132)--(0.500,-0.00705)--(0.500,0)--(0.500,0.00529)--(0.500,0.0115)--(0.500,0.0176)--(0.499,0.0238)--(0.499,0.0300)--(0.499,0.0361)--(0.498,0.0423)--(0.498,0.0484)--(0.497,0.0545)--(0.496,0.0607)--(0.496,0.0668)--(0.495,0.0729)--(0.494,0.0790)--(0.493,0.0851)--(0.492,0.0912)--(0.490,0.0972)--(0.489,0.103)--(0.488,0.109)--(0.487,0.115)--(0.485,0.121)--(0.484,0.127)--(0.482,0.133)--(0.480,0.139)--(0.478,0.145)--(0.477,0.151)--(0.475,0.157)--(0.473,0.163)--(0.471,0.169)--(0.469,0.174)--(0.466,0.180)--(0.464,0.186)--(0.462,0.192)--(0.459,0.197)--(0.457,0.203)--(0.454,0.209)--(0.452,0.214)--(0.449,0.220)--(0.446,0.225)--(0.444,0.231)--(0.441,0.236)--(0.438,0.242)--(0.435,0.247)--(0.432,0.252)--(0.429,0.258)--(0.425,0.263)--(0.422,0.268)--(0.419,0.273)--(0.415,0.278)--(0.412,0.284)--(0.408,0.289)--(0.405,0.294)--(0.401,0.299)--(0.397,0.303)--(0.394,0.308)--(0.390,0.313)--(0.386,0.318)--(0.382,0.323)--(0.378,0.327)--(0.374,0.332)--(0.370,0.337)--(0.366,0.341)--(0.361,0.346)--(0.357,0.350)--(0.353,0.354)--(0.348,0.359)--(0.344,0.363)--(0.339,0.367)--(0.335,0.371)--(0.330,0.376)--(0.325,0.380)--(0.321,0.384)--(0.316,0.388)--(0.311,0.391)--(0.306,0.395)--(0.301,0.399)--(0.296,0.403)--(0.291,0.406)--(0.286,0.410)--(0.281,0.413)--(0.276,0.417)--(0.271,0.420)--(0.266,0.423)--(0.261,0.427)--(0.255,0.430)--(0.250,0.433);
 \draw [] (0,0) -- (1.97,-0.347);
 \draw [] (0,0) -- (1.00,1.73);
-\draw (0.1002123789,1.140733404) node {$R$};
-\draw (2.429897999,-0.5535014753) node {$\theta_0$};
-\draw (1.314840167,2.145969095) node {$\theta_1$};
+\draw (0.10021,1.1407) node {$R$};
+\draw (2.4299,-0.55350) node {$\theta_0$};
+\draw (1.3148,2.1460) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall b/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall
index a50600833..4e8788b1b 100644
--- a/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall
+++ b/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall
@@ -72,18 +72,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.400000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.900000000);
+\draw [,->,>=latex] (-2.4000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.9000);
 %DEFAULT
 \draw [style=dashed] (-1.90,1.20) -- (3.00,1.20);
-\draw []  (0,1.200000000) node [rotate=0] {$\bullet$};
-\draw (0.2294391896,1.468157356) node {\( \delta\)};
+\draw []  (0,1.2000) node [rotate=0] {$\bullet$};
+\draw (0.22944,1.4682) node {\( \delta\)};
 \draw [] (-0.300,1.40) -- (0.300,1.00);
 \draw [] (1.70,1.40) -- (2.30,1.00);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3396268333) node {\( t_1\)};
-\draw []  (-1.500000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.500000000,-0.3396268333) node {\( t_2\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.33963) node {\( t_1\)};
+\draw []  (-1.5000,0) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.33963) node {\( t_2\)};
 \draw [style=dotted] (2.00,0) -- (2.00,1.20);
 \draw [style=dotted] (-1.50,0) -- (-1.50,1.20);
 \draw [] (-1.80,1.40) -- (-1.20,1.00);
diff --git a/src_phystricks/Fig_AccumulationIsole.pstricks.recall b/src_phystricks/Fig_AccumulationIsole.pstricks.recall
index a597d68e8..3b674d22b 100644
--- a/src_phystricks/Fig_AccumulationIsole.pstricks.recall
+++ b/src_phystricks/Fig_AccumulationIsole.pstricks.recall
@@ -79,12 +79,12 @@
 %DEFAULT
 \fill [color=lightgray] (1.00,0) -- (0.998,0.0634) -- (0.992,0.127) -- (0.982,0.189) -- (0.968,0.251) -- (0.950,0.312) -- (0.928,0.372) -- (0.903,0.430) -- (0.874,0.486) -- (0.841,0.541) -- (0.805,0.593) -- (0.766,0.643) -- (0.724,0.690) -- (0.679,0.735) -- (0.631,0.776) -- (0.580,0.815) -- (0.527,0.850) -- (0.472,0.881) -- (0.415,0.910) -- (0.357,0.934) -- (0.297,0.955) -- (0.236,0.972) -- (0.174,0.985) -- (0.111,0.994) -- (0.0476,0.999) -- (-0.0159,1.00) -- (-0.0792,0.997) -- (-0.142,0.990) -- (-0.205,0.979) -- (-0.266,0.964) -- (-0.327,0.945) -- (-0.386,0.922) -- (-0.444,0.896) -- (-0.500,0.866) -- (-0.554,0.833) -- (-0.606,0.796) -- (-0.655,0.756) -- (-0.701,0.713) -- (-0.745,0.667) -- (-0.786,0.618) -- (-0.824,0.567) -- (-0.858,0.514) -- (-0.889,0.458) -- (-0.916,0.401) -- (-0.940,0.342) -- (-0.959,0.282) -- (-0.975,0.220) -- (-0.987,0.158) -- (-0.995,0.0951) -- (-1.00,0.0317) -- (-1.00,-0.0317) -- (-0.995,-0.0951) -- (-0.987,-0.158) -- (-0.975,-0.220) -- (-0.959,-0.282) -- (-0.940,-0.342) -- (-0.916,-0.401) -- (-0.889,-0.458) -- (-0.858,-0.514) -- (-0.824,-0.567) -- (-0.786,-0.618) -- (-0.745,-0.667) -- (-0.701,-0.713) -- (-0.655,-0.756) -- (-0.606,-0.796) -- (-0.554,-0.833) -- (-0.500,-0.866) -- (-0.444,-0.896) -- (-0.386,-0.922) -- (-0.327,-0.945) -- (-0.266,-0.964) -- (-0.205,-0.979) -- (-0.142,-0.990) -- (-0.0792,-0.997) -- (-0.0159,-1.00) -- (0.0476,-0.999) -- (0.111,-0.994) -- (0.174,-0.985) -- (0.236,-0.972) -- (0.297,-0.955) -- (0.357,-0.934) -- (0.415,-0.910) -- (0.472,-0.881) -- (0.527,-0.850) -- (0.580,-0.815) -- (0.631,-0.776) -- (0.679,-0.735) -- (0.724,-0.690) -- (0.766,-0.643) -- (0.805,-0.593) -- (0.841,-0.541) -- (0.874,-0.486) -- (0.903,-0.430) -- (0.928,-0.372) -- (0.950,-0.312) -- (0.968,-0.251) -- (0.982,-0.189) -- (0.992,-0.127) -- (0.998,-0.0634) -- (1.00,0) --  cycle;
 \draw [color=red] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.542514833,1.000000000) node {$P$};
-\draw [color=red]  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.544276167,0) node {$Q$};
-\draw []  (0.5000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (0.1654411323,0.1631599656) node {$S$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.5425,1.0000) node {$P$};
+\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-1.5443,0) node {$Q$};
+\draw []  (0.50000,0.50000) node [rotate=0] {$\bullet$};
+\draw (0.16544,0.16316) node {$S$};
 
 \draw [color=black] (1.200,1.000)--(1.200,1.013)--(1.198,1.025)--(1.196,1.038)--(1.194,1.050)--(1.190,1.062)--(1.186,1.074)--(1.181,1.086)--(1.175,1.097)--(1.168,1.108)--(1.161,1.119)--(1.153,1.129)--(1.145,1.138)--(1.136,1.147)--(1.126,1.155)--(1.116,1.163)--(1.105,1.170)--(1.094,1.176)--(1.083,1.182)--(1.071,1.187)--(1.059,1.191)--(1.047,1.194)--(1.035,1.197)--(1.022,1.199)--(1.010,1.200)--(0.9968,1.200)--(0.9842,1.199)--(0.9715,1.198)--(0.9590,1.196)--(0.9467,1.193)--(0.9346,1.189)--(0.9227,1.184)--(0.9112,1.179)--(0.9000,1.173)--(0.8892,1.167)--(0.8789,1.159)--(0.8690,1.151)--(0.8597,1.143)--(0.8509,1.133)--(0.8428,1.124)--(0.8353,1.113)--(0.8284,1.103)--(0.8222,1.092)--(0.8168,1.080)--(0.8121,1.068)--(0.8081,1.056)--(0.8049,1.044)--(0.8025,1.032)--(0.8009,1.019)--(0.8001,1.006)--(0.8001,0.9937)--(0.8009,0.9810)--(0.8025,0.9684)--(0.8049,0.9559)--(0.8081,0.9437)--(0.8121,0.9316)--(0.8168,0.9198)--(0.8222,0.9084)--(0.8284,0.8973)--(0.8353,0.8866)--(0.8428,0.8764)--(0.8509,0.8666)--(0.8597,0.8575)--(0.8690,0.8488)--(0.8789,0.8409)--(0.8892,0.8335)--(0.9000,0.8268)--(0.9112,0.8208)--(0.9227,0.8155)--(0.9346,0.8110)--(0.9467,0.8072)--(0.9590,0.8042)--(0.9715,0.8020)--(0.9842,0.8006)--(0.9968,0.8000)--(1.010,0.8002)--(1.022,0.8012)--(1.035,0.8030)--(1.047,0.8056)--(1.059,0.8090)--(1.071,0.8132)--(1.083,0.8181)--(1.094,0.8237)--(1.105,0.8301)--(1.116,0.8371)--(1.126,0.8448)--(1.136,0.8531)--(1.145,0.8620)--(1.153,0.8714)--(1.161,0.8814)--(1.168,0.8919)--(1.175,0.9028)--(1.181,0.9140)--(1.186,0.9257)--(1.190,0.9376)--(1.194,0.9498)--(1.196,0.9622)--(1.198,0.9747)--(1.200,0.9873)--(1.200,1.000);
 
diff --git a/src_phystricks/Fig_AdhIntFr.pstricks.recall b/src_phystricks/Fig_AdhIntFr.pstricks.recall
index 9da41f9c7..a1650bae7 100644
--- a/src_phystricks/Fig_AdhIntFr.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFr.pstricks.recall
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.000000000,0);
-\draw [,->,>=latex] (0,-0.7499836751) -- (0,4.250000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.0000,0);
+\draw [,->,>=latex] (0,-0.74998) -- (0,4.2500);
 %DEFAULT
 
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@@ -102,26 +102,26 @@
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 \draw [] (1.00,2.00) -- (1.00,2.00);
 \draw [] (4.00,2.00) -- (4.00,2.00);
-\draw []  (3.500000000,0.7500000000) node [rotate=0] {$\bullet$};
+\draw []  (3.5000,0.75000) node [rotate=0] {$\bullet$};
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall b/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall
index ce92d3c58..7ac338be2 100644
--- a/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall
+++ b/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall
@@ -72,18 +72,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-2.000000000) -- (0,2.000000000);
+\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
 %DEFAULT
 
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-\draw (1.012732295,0.2561455226) node {\( \theta\)};
+\draw (1.0127,0.25615) node {\( \theta\)};
 
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-\draw []  (1.299038106,0) node [rotate=0] {$\bullet$};
-\draw (1.299038106,-0.2785761667) node {\( x\)};
-\draw []  (0,0.7500000000) node [rotate=0] {$\bullet$};
-\draw (-0.2960240000,0.7500000000) node {\( y\)};
+\draw []  (1.2990,0) node [rotate=0] {$\bullet$};
+\draw (1.2990,-0.27858) node {\( x\)};
+\draw []  (0,0.75000) node [rotate=0] {$\bullet$};
+\draw (-0.29602,0.75000) node {\( y\)};
 \draw [] (0,0) -- (1.30,0.750);
 \draw [style=dashed] (1.30,0.750) -- (1.30,0);
 \draw [style=dashed] (1.30,0.750) -- (0,0.750);
diff --git a/src_phystricks/Fig_BiaisOuPas.pstricks.recall b/src_phystricks/Fig_BiaisOuPas.pstricks.recall
index ab7e522b6..5a720b6dc 100644
--- a/src_phystricks/Fig_BiaisOuPas.pstricks.recall
+++ b/src_phystricks/Fig_BiaisOuPas.pstricks.recall
@@ -91,31 +91,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.000000000,0) -- (8.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.468495195);
+\draw [,->,>=latex] (-8.0000,0) -- (8.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4685);
 %DEFAULT
 
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 \draw [color=red] (-7.500,0)--(-7.349,0)--(-7.197,0)--(-7.045,0)--(-6.894,0)--(-6.742,0)--(-6.591,0)--(-6.439,0)--(-6.288,0)--(-6.136,0)--(-5.985,0)--(-5.833,0)--(-5.682,0)--(-5.530,0)--(-5.379,0)--(-5.227,0)--(-5.076,0)--(-4.924,0)--(-4.773,0)--(-4.621,0)--(-4.470,0)--(-4.318,0)--(-4.167,0)--(-4.015,0)--(-3.864,0)--(-3.712,0)--(-3.561,0)--(-3.409,0)--(-3.258,0)--(-3.106,0)--(-2.955,0)--(-2.803,0)--(-2.652,0)--(-2.500,0)--(-2.348,0)--(-2.197,0)--(-2.045,0)--(-1.894,0.002101)--(-1.742,0.01038)--(-1.591,0.04270)--(-1.439,0.1462)--(-1.288,0.4164)--(-1.136,0.9870)--(-0.9848,1.947)--(-0.8333,3.197)--(-0.6818,4.369)--(-0.5303,4.969)--(-0.3788,4.702)--(-0.2273,3.703)--(-0.07576,2.427)--(0.07576,1.324)--(0.2273,0.6012)--(0.3788,0.2271)--(0.5303,0.07141)--(0.6818,0.01869)--(0.8333,0.004069)--(0.9848,0)--(1.136,0)--(1.288,0)--(1.439,0)--(1.591,0)--(1.742,0)--(1.894,0)--(2.045,0)--(2.197,0)--(2.348,0)--(2.500,0)--(2.652,0)--(2.803,0)--(2.955,0)--(3.106,0)--(3.258,0)--(3.409,0)--(3.561,0)--(3.712,0)--(3.864,0)--(4.015,0)--(4.167,0)--(4.318,0)--(4.470,0)--(4.621,0)--(4.773,0)--(4.924,0)--(5.076,0)--(5.227,0)--(5.379,0)--(5.530,0)--(5.682,0)--(5.833,0)--(5.985,0)--(6.136,0)--(6.288,0)--(6.439,0)--(6.591,0)--(6.742,0)--(6.894,0)--(7.045,0)--(7.197,0)--(7.349,0)--(7.500,0);
-\draw [color=cyan,->,>=latex] (0,-0.5000000000) -- (-1.250000000,-0.5000000000);
-\draw [color=cyan,->,>=latex] (0,-0.5000000000) -- (1.250000000,-0.5000000000);
-\draw (0,-0.9247080000) node {\( I\)};
-\draw (-7.500000000,-0.3298256667) node {$ -3 $};
+\draw [color=cyan,->,>=latex] (0,-0.50000) -- (-1.2500,-0.50000);
+\draw [color=cyan,->,>=latex] (0,-0.50000) -- (1.2500,-0.50000);
+\draw (0,-0.92471) node {\( I\)};
+\draw (-7.5000,-0.32983) node {$ -3 $};
 \draw [] (-7.50,-0.100) -- (-7.50,0.100);
-\draw (-5.000000000,-0.3298256667) node {$ -2 $};
+\draw (-5.0000,-0.32983) node {$ -2 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-2.500000000,-0.3298256667) node {$ -1 $};
+\draw (-2.5000,-0.32983) node {$ -1 $};
 \draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (2.500000000,-0.3149246667) node {$ 1 $};
+\draw (2.5000,-0.31492) node {$ 1 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 2 $};
+\draw (5.0000,-0.31492) node {$ 2 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (7.500000000,-0.3149246667) node {$ 3 $};
+\draw (7.5000,-0.31492) node {$ 3 $};
 \draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (-0.2912498333,2.500000000) node {$ 1 $};
+\draw (-0.29125,2.5000) node {$ 1 $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.2912498333,5.000000000) node {$ 2 $};
+\draw (-0.29125,5.0000) node {$ 2 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_BoulePtLoin.pstricks.recall b/src_phystricks/Fig_BoulePtLoin.pstricks.recall
index d7d8175e9..7e8a51e8e 100644
--- a/src_phystricks/Fig_BoulePtLoin.pstricks.recall
+++ b/src_phystricks/Fig_BoulePtLoin.pstricks.recall
@@ -79,14 +79,14 @@
 %DEFAULT
 \draw [style=dashed] (1.41,1.41) -- (1.94,1.94);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.3085997010,0.2907082010) node {$a$};
-\draw []  (1.414213562,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.050102275,1.642789729) node {$x$};
-\draw [,->,>=latex] (0,0) -- (1.414213562,1.414213562);
+\draw (-0.30860,0.29071) node {$a$};
+\draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.0501,1.6428) node {$x$};
+\draw [,->,>=latex] (0,0) -- (1.4142,1.4142);
 
 \draw [style=dotted] (2.414,1.414)--(2.412,1.478)--(2.406,1.541)--(2.396,1.603)--(2.382,1.665)--(2.364,1.726)--(2.343,1.786)--(2.317,1.844)--(2.288,1.900)--(2.255,1.955)--(2.219,2.007)--(2.180,2.057)--(2.138,2.104)--(2.093,2.149)--(2.045,2.190)--(1.994,2.229)--(1.941,2.264)--(1.886,2.296)--(1.830,2.324)--(1.771,2.348)--(1.711,2.369)--(1.650,2.386)--(1.588,2.399)--(1.525,2.408)--(1.462,2.413)--(1.398,2.414)--(1.335,2.411)--(1.272,2.404)--(1.209,2.393)--(1.148,2.378)--(1.087,2.359)--(1.028,2.337)--(0.9701,2.310)--(0.9142,2.280)--(0.8603,2.247)--(0.8086,2.210)--(0.7594,2.170)--(0.7127,2.127)--(0.6689,2.081)--(0.6282,2.032)--(0.5905,1.981)--(0.5562,1.928)--(0.5254,1.872)--(0.4981,1.815)--(0.4745,1.756)--(0.4547,1.696)--(0.4388,1.635)--(0.4268,1.572)--(0.4187,1.509)--(0.4147,1.446)--(0.4147,1.382)--(0.4187,1.319)--(0.4268,1.256)--(0.4388,1.194)--(0.4547,1.132)--(0.4745,1.072)--(0.4981,1.013)--(0.5254,0.9560)--(0.5562,0.9005)--(0.5905,0.8472)--(0.6282,0.7961)--(0.6689,0.7474)--(0.7127,0.7015)--(0.7594,0.6585)--(0.8086,0.6185)--(0.8603,0.5816)--(0.9142,0.5482)--(0.9701,0.5182)--(1.028,0.4919)--(1.087,0.4692)--(1.148,0.4504)--(1.209,0.4354)--(1.272,0.4244)--(1.335,0.4174)--(1.398,0.4143)--(1.462,0.4153)--(1.525,0.4204)--(1.588,0.4294)--(1.650,0.4424)--(1.711,0.4593)--(1.771,0.4801)--(1.830,0.5046)--(1.886,0.5328)--(1.941,0.5645)--(1.994,0.5996)--(2.045,0.6381)--(2.093,0.6796)--(2.138,0.7241)--(2.180,0.7714)--(2.219,0.8213)--(2.255,0.8736)--(2.288,0.9280)--(2.317,0.9844)--(2.343,1.043)--(2.364,1.102)--(2.382,1.163)--(2.396,1.225)--(2.406,1.288)--(2.412,1.351)--(2.414,1.414);
-\draw []  (1.944543648,1.944543648) node [rotate=0] {$\bullet$};
-\draw (1.944543648,1.519835648) node {$P$};
+\draw []  (1.9445,1.9445) node [rotate=0] {$\bullet$};
+\draw (1.9445,1.5198) node {$P$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall b/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall
index f79c6c197..0eb3ad650 100644
--- a/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall
+++ b/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall
@@ -64,19 +64,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (2.00,0)--(2.00,0.127)--(1.98,0.253)--(1.96,0.379)--(1.94,0.502)--(1.90,0.624)--(1.86,0.743)--(1.81,0.860)--(1.75,0.972)--(1.68,1.08)--(1.61,1.19)--(1.53,1.29)--(1.45,1.38)--(1.36,1.47)--(1.26,1.55)--(1.16,1.63)--(1.05,1.70)--(0.945,1.76)--(0.831,1.82)--(0.714,1.87)--(0.594,1.91)--(0.472,1.94)--(0.347,1.97)--(0.222,1.99)--(0.0952,2.00)--(-0.0317,2.00)--(-0.158,1.99)--(-0.285,1.98)--(-0.410,1.96)--(-0.533,1.93)--(-0.654,1.89)--(-0.773,1.84)--(-0.888,1.79)--(-1.00,1.73)--(-1.11,1.67)--(-1.21,1.59)--(-1.31,1.51)--(-1.40,1.43)--(-1.49,1.33)--(-1.57,1.24)--(-1.65,1.13)--(-1.72,1.03)--(-1.78,0.916)--(-1.83,0.802)--(-1.88,0.684)--(-1.92,0.563)--(-1.95,0.441)--(-1.97,0.316)--(-1.99,0.190)--(-2.00,0.0635)--(-2.00,-0.0635)--(-1.99,-0.190)--(-1.97,-0.316)--(-1.95,-0.441)--(-1.92,-0.563)--(-1.88,-0.684)--(-1.83,-0.802)--(-1.78,-0.916)--(-1.72,-1.03)--(-1.65,-1.13)--(-1.57,-1.24)--(-1.49,-1.33)--(-1.40,-1.43)--(-1.31,-1.51)--(-1.21,-1.59)--(-1.11,-1.67)--(-1.00,-1.73)--(-0.888,-1.79)--(-0.773,-1.84)--(-0.654,-1.89)--(-0.533,-1.93)--(-0.410,-1.96)--(-0.285,-1.98)--(-0.158,-1.99)--(-0.0317,-2.00)--(0.0952,-2.00)--(0.222,-1.99)--(0.347,-1.97)--(0.472,-1.94)--(0.594,-1.91)--(0.714,-1.87)--(0.831,-1.82)--(0.945,-1.76)--(1.05,-1.70)--(1.16,-1.63)--(1.26,-1.55)--(1.36,-1.47)--(1.45,-1.38)--(1.53,-1.29)--(1.61,-1.19)--(1.68,-1.08)--(1.75,-0.972)--(1.81,-0.860)--(1.86,-0.743)--(1.90,-0.624)--(1.94,-0.502)--(1.96,-0.379)--(1.98,-0.253)--(2.00,-0.127)--(2.00,0);
 \draw [] (0,0) -- (1.73,1.00);
 \draw [] (0,0) -- (-1.73,1.00);
-\draw []  (1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (-1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.394880073,0.3118645226) node {\( \pi/6\)};
+\draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (-1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3949,0.31186) node {\( \pi/6\)};
 
 \draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw (-1.394880073,0.3118645226) node {\( \pi/6\)};
+\draw (-1.3949,0.31186) node {\( \pi/6\)};
 
 \draw [] (-0.433,0.250)--(-0.434,0.248)--(-0.436,0.245)--(-0.437,0.243)--(-0.438,0.241)--(-0.439,0.238)--(-0.441,0.236)--(-0.442,0.234)--(-0.443,0.231)--(-0.444,0.229)--(-0.446,0.227)--(-0.447,0.224)--(-0.448,0.222)--(-0.449,0.220)--(-0.450,0.217)--(-0.451,0.215)--(-0.453,0.213)--(-0.454,0.210)--(-0.455,0.208)--(-0.456,0.205)--(-0.457,0.203)--(-0.458,0.200)--(-0.459,0.198)--(-0.460,0.196)--(-0.461,0.193)--(-0.462,0.191)--(-0.463,0.188)--(-0.464,0.186)--(-0.465,0.183)--(-0.466,0.181)--(-0.467,0.178)--(-0.468,0.176)--(-0.469,0.173)--(-0.470,0.171)--(-0.471,0.169)--(-0.472,0.166)--(-0.473,0.164)--(-0.473,0.161)--(-0.474,0.159)--(-0.475,0.156)--(-0.476,0.154)--(-0.477,0.151)--(-0.477,0.148)--(-0.478,0.146)--(-0.479,0.143)--(-0.480,0.141)--(-0.480,0.138)--(-0.481,0.136)--(-0.482,0.133)--(-0.483,0.131)--(-0.483,0.128)--(-0.484,0.126)--(-0.485,0.123)--(-0.485,0.120)--(-0.486,0.118)--(-0.487,0.115)--(-0.487,0.113)--(-0.488,0.110)--(-0.488,0.108)--(-0.489,0.105)--(-0.489,0.102)--(-0.490,0.0998)--(-0.490,0.0972)--(-0.491,0.0946)--(-0.491,0.0920)--(-0.492,0.0894)--(-0.492,0.0868)--(-0.493,0.0842)--(-0.493,0.0816)--(-0.494,0.0790)--(-0.494,0.0764)--(-0.495,0.0738)--(-0.495,0.0712)--(-0.495,0.0685)--(-0.496,0.0659)--(-0.496,0.0633)--(-0.496,0.0607)--(-0.497,0.0580)--(-0.497,0.0554)--(-0.497,0.0528)--(-0.497,0.0502)--(-0.498,0.0475)--(-0.498,0.0449)--(-0.498,0.0423)--(-0.498,0.0396)--(-0.499,0.0370)--(-0.499,0.0344)--(-0.499,0.0317)--(-0.499,0.0291)--(-0.499,0.0264)--(-0.499,0.0238)--(-0.500,0.0211)--(-0.500,0.0185)--(-0.500,0.0159)--(-0.500,0.0132)--(-0.500,0.0106)--(-0.500,0.00793)--(-0.500,0.00529)--(-0.500,0.00264)--(-0.500,0);
 
diff --git a/src_phystricks/Fig_CSCii.pstricks.recall b/src_phystricks/Fig_CSCii.pstricks.recall
index 2607f006e..fe4d62089 100644
--- a/src_phystricks/Fig_CSCii.pstricks.recall
+++ b/src_phystricks/Fig_CSCii.pstricks.recall
@@ -71,21 +71,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.269774942,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.269774942) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.2698,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.2698) -- (0,1.5000);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.0317,0)--(0.0634,0.00201)--(0.0949,0.00452)--(0.126,0.00803)--(0.158,0.0125)--(0.188,0.0180)--(0.219,0.0244)--(0.249,0.0318)--(0.279,0.0401)--(0.308,0.0493)--(0.337,0.0594)--(0.365,0.0703)--(0.392,0.0821)--(0.419,0.0947)--(0.445,0.108)--(0.471,0.122)--(0.495,0.137)--(0.519,0.152)--(0.542,0.168)--(0.563,0.185)--(0.584,0.202)--(0.604,0.220)--(0.623,0.238)--(0.641,0.256)--(0.657,0.275)--(0.673,0.295)--(0.687,0.314)--(0.701,0.334)--(0.713,0.353)--(0.724,0.373)--(0.734,0.393)--(0.743,0.413)--(0.750,0.433)--(0.756,0.453)--(0.761,0.472)--(0.765,0.492)--(0.768,0.511)--(0.769,0.530)--(0.770,0.548)--(0.769,0.566)--(0.767,0.584)--(0.764,0.601)--(0.760,0.617)--(0.754,0.633)--(0.748,0.648)--(0.741,0.663)--(0.732,0.676)--(0.723,0.689)--(0.713,0.701)--(0.701,0.713)--(0.689,0.723)--(0.676,0.732)--(0.663,0.741)--(0.648,0.748)--(0.633,0.754)--(0.617,0.760)--(0.601,0.764)--(0.584,0.767)--(0.566,0.769)--(0.548,0.770)--(0.530,0.769)--(0.511,0.768)--(0.492,0.765)--(0.472,0.761)--(0.453,0.756)--(0.433,0.750)--(0.413,0.743)--(0.393,0.734)--(0.373,0.724)--(0.353,0.713)--(0.334,0.701)--(0.314,0.687)--(0.295,0.673)--(0.275,0.657)--(0.256,0.641)--(0.238,0.623)--(0.220,0.604)--(0.202,0.584)--(0.185,0.563)--(0.168,0.542)--(0.152,0.519)--(0.137,0.495)--(0.122,0.471)--(0.108,0.445)--(0.0947,0.419)--(0.0821,0.392)--(0.0703,0.365)--(0.0594,0.337)--(0.0493,0.308)--(0.0401,0.279)--(0.0318,0.249)--(0.0244,0.219)--(0.0180,0.188)--(0.0125,0.158)--(0.00803,0.126)--(0.00452,0.0949)--(0.00201,0.0634)--(0,0.0317)--(0,0);
 \draw [color=blue] (0,0)--(-0.0317,0)--(-0.0634,-0.00201)--(-0.0949,-0.00452)--(-0.126,-0.00803)--(-0.158,-0.0125)--(-0.188,-0.0180)--(-0.219,-0.0244)--(-0.249,-0.0318)--(-0.279,-0.0401)--(-0.308,-0.0493)--(-0.337,-0.0594)--(-0.365,-0.0703)--(-0.392,-0.0821)--(-0.419,-0.0947)--(-0.445,-0.108)--(-0.471,-0.122)--(-0.495,-0.137)--(-0.519,-0.152)--(-0.542,-0.168)--(-0.563,-0.185)--(-0.584,-0.202)--(-0.604,-0.220)--(-0.623,-0.238)--(-0.641,-0.256)--(-0.657,-0.275)--(-0.673,-0.295)--(-0.687,-0.314)--(-0.701,-0.334)--(-0.713,-0.353)--(-0.724,-0.373)--(-0.734,-0.393)--(-0.743,-0.413)--(-0.750,-0.433)--(-0.756,-0.453)--(-0.761,-0.472)--(-0.765,-0.492)--(-0.768,-0.511)--(-0.769,-0.530)--(-0.770,-0.548)--(-0.769,-0.566)--(-0.767,-0.584)--(-0.764,-0.601)--(-0.760,-0.617)--(-0.754,-0.633)--(-0.748,-0.648)--(-0.741,-0.663)--(-0.732,-0.676)--(-0.723,-0.689)--(-0.713,-0.701)--(-0.701,-0.713)--(-0.689,-0.723)--(-0.676,-0.732)--(-0.663,-0.741)--(-0.648,-0.748)--(-0.633,-0.754)--(-0.617,-0.760)--(-0.601,-0.764)--(-0.584,-0.767)--(-0.566,-0.769)--(-0.548,-0.770)--(-0.530,-0.769)--(-0.511,-0.768)--(-0.492,-0.765)--(-0.472,-0.761)--(-0.453,-0.756)--(-0.433,-0.750)--(-0.413,-0.743)--(-0.393,-0.734)--(-0.373,-0.724)--(-0.353,-0.713)--(-0.334,-0.701)--(-0.314,-0.687)--(-0.295,-0.673)--(-0.275,-0.657)--(-0.256,-0.641)--(-0.238,-0.623)--(-0.220,-0.604)--(-0.202,-0.584)--(-0.185,-0.563)--(-0.168,-0.542)--(-0.152,-0.519)--(-0.137,-0.495)--(-0.122,-0.471)--(-0.108,-0.445)--(-0.0947,-0.419)--(-0.0821,-0.392)--(-0.0703,-0.365)--(-0.0594,-0.337)--(-0.0493,-0.308)--(-0.0401,-0.279)--(-0.0318,-0.249)--(-0.0244,-0.219)--(-0.0180,-0.188)--(-0.0125,-0.158)--(-0.00803,-0.126)--(-0.00452,-0.0949)--(-0.00201,-0.0634)--(0,-0.0317)--(0,0);
 \draw [color=lightgray] (1.00,1.00) -- (1.00,0);
 \draw [color=lightgray] (1.00,1.00) -- (0,1.00);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CSCiii.pstricks.recall b/src_phystricks/Fig_CSCiii.pstricks.recall
index 938e78081..ebdb1078f 100644
--- a/src_phystricks/Fig_CSCiii.pstricks.recall
+++ b/src_phystricks/Fig_CSCiii.pstricks.recall
@@ -41,15 +41,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.151591227,0) -- (3.499800004,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.673444770);
+\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.6734);
 %DEFAULT
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-\draw (1.500000000,-0.4207143333) node {$ \frac{1}{2} $};
+\draw (1.5000,-0.42071) node {$ \frac{1}{2} $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.3108333333,1.500000000) node {$ \frac{1}{2} $};
+\draw (-0.31083,1.5000) node {$ \frac{1}{2} $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
 %OTHER STUFF
 %END PSPICTURE
@@ -92,18 +92,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.238758171,0) -- (2.626823348,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.334460604);
+\draw [,->,>=latex] (-2.2388,0) -- (2.6268,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.3345);
 %DEFAULT
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-\draw (-1.500000000,-0.4207143333) node {$ -\frac{1}{20} $};
+\draw (-1.5000,-0.42071) node {$ -\frac{1}{20} $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (1.500000000,-0.4207143333) node {$ \frac{1}{20} $};
+\draw (1.5000,-0.42071) node {$ \frac{1}{20} $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (-0.3816666667,3.000000000) node {$ \frac{1}{10} $};
+\draw (-0.38167,3.0000) node {$ \frac{1}{10} $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -146,8 +146,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.151591227,0) -- (3.499800004,0);
-\draw [,->,>=latex] (0,-2.673444770) -- (0,2.673444770);
+\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
+\draw [,->,>=latex] (0,-2.6734) -- (0,2.6734);
 %DEFAULT
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@@ -159,13 +159,13 @@
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-\draw (1.500000000,-0.4207143333) node {$ \frac{1}{2} $};
+\draw (1.5000,-0.42071) node {$ \frac{1}{2} $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4527428333,-1.500000000) node {$ -\frac{1}{2} $};
+\draw (-0.45274,-1.5000) node {$ -\frac{1}{2} $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.3108333333,1.500000000) node {$ \frac{1}{2} $};
+\draw (-0.31083,1.5000) node {$ \frac{1}{2} $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CSCiv.pstricks.recall b/src_phystricks/Fig_CSCiv.pstricks.recall
index 6266f7948..42c742b85 100644
--- a/src_phystricks/Fig_CSCiv.pstricks.recall
+++ b/src_phystricks/Fig_CSCiv.pstricks.recall
@@ -91,32 +91,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.493518414,0) -- (1.880150393,0);
-\draw [,->,>=latex] (0,-5.487474933) -- (0,2.045853861);
+\draw [,->,>=latex] (-2.4935,0) -- (1.8801,0);
+\draw [,->,>=latex] (0,-5.4875) -- (0,2.0459);
 %DEFAULT
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+\draw (-2.0000,-0.32983) node {$ -2 $};
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
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-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
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-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
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-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
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-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
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-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
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 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CSCv.pstricks.recall b/src_phystricks/Fig_CSCv.pstricks.recall
index ed3393639..eafafce59 100644
--- a/src_phystricks/Fig_CSCv.pstricks.recall
+++ b/src_phystricks/Fig_CSCv.pstricks.recall
@@ -65,28 +65,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185307,0);
-\draw [,->,>=latex] (0,-2.497483219) -- (0,1.624594384);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-2.4975) -- (0,1.6246);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.06347,0.006032)--(0.1269,0.02401)--(0.1904,0.05356)--(0.2539,0.09410)--(0.3173,0.1448)--(0.3808,0.2046)--(0.4443,0.2724)--(0.5077,0.3466)--(0.5712,0.4258)--(0.6347,0.5083)--(0.6981,0.5924)--(0.7616,0.6762)--(0.8251,0.7578)--(0.8885,0.8354)--(0.9520,0.9071)--(1.015,0.9713)--(1.079,1.026)--(1.142,1.070)--(1.206,1.102)--(1.269,1.121)--(1.333,1.125)--(1.396,1.113)--(1.460,1.086)--(1.523,1.043)--(1.587,0.9836)--(1.650,0.9082)--(1.714,0.8172)--(1.777,0.7113)--(1.841,0.5915)--(1.904,0.4590)--(1.967,0.3151)--(2.031,0.1615)--(2.094,0)--(2.158,-0.1676)--(2.221,-0.3391)--(2.285,-0.5125)--(2.348,-0.6856)--(2.412,-0.8561)--(2.475,-1.022)--(2.539,-1.181)--(2.602,-1.330)--(2.666,-1.469)--(2.729,-1.595)--(2.793,-1.706)--(2.856,-1.801)--(2.919,-1.878)--(2.983,-1.938)--(3.046,-1.977)--(3.110,-1.997)--(3.173,-1.997)--(3.237,-1.977)--(3.300,-1.938)--(3.364,-1.878)--(3.427,-1.801)--(3.491,-1.706)--(3.554,-1.595)--(3.618,-1.469)--(3.681,-1.330)--(3.745,-1.181)--(3.808,-1.022)--(3.871,-0.8561)--(3.935,-0.6856)--(3.998,-0.5125)--(4.062,-0.3391)--(4.125,-0.1676)--(4.189,0)--(4.252,0.1615)--(4.316,0.3151)--(4.379,0.4590)--(4.443,0.5915)--(4.506,0.7113)--(4.570,0.8172)--(4.633,0.9082)--(4.697,0.9836)--(4.760,1.043)--(4.823,1.086)--(4.887,1.113)--(4.950,1.125)--(5.014,1.121)--(5.077,1.102)--(5.141,1.070)--(5.204,1.026)--(5.268,0.9713)--(5.331,0.9071)--(5.395,0.8354)--(5.458,0.7578)--(5.522,0.6762)--(5.585,0.5924)--(5.648,0.5083)--(5.712,0.4258)--(5.775,0.3466)--(5.839,0.2724)--(5.902,0.2046)--(5.966,0.1448)--(6.029,0.09410)--(6.093,0.05356)--(6.156,0.02401)--(6.220,0.006032)--(6.283,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -125,16 +125,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.6576218188,0) -- (1.028045149,0);
-\draw [,->,>=latex] (0,-1.597472712) -- (0,1.597472712);
+\draw [,->,>=latex] (-0.65762,0) -- (1.0280,0);
+\draw [,->,>=latex] (0,-1.5975) -- (0,1.5975);
 %DEFAULT
 \draw [color=blue] (0,0)--(0,0)--(0.002681,0)--(0.006020,0)--(0.01067,0)--(0.01661,0.001764)--(0.02381,0.003039)--(0.03224,0.004809)--(0.04185,0.007151)--(0.05259,0.01014)--(0.06442,0.01384)--(0.07727,0.01831)--(0.09108,0.02363)--(0.1058,0.02985)--(0.1213,0.03702)--(0.1376,0.04518)--(0.1545,0.05439)--(0.1720,0.06467)--(0.1900,0.07605)--(0.2084,0.08857)--(0.2270,0.1022)--(0.2459,0.1171)--(0.2649,0.1331)--(0.2840,0.1502)--(0.3029,0.1685)--(0.3217,0.1880)--(0.3401,0.2086)--(0.3582,0.2302)--(0.3759,0.2530)--(0.3929,0.2767)--(0.4094,0.3014)--(0.4250,0.3270)--(0.4399,0.3535)--(0.4538,0.3808)--(0.4667,0.4088)--(0.4786,0.4374)--(0.4894,0.4666)--(0.4989,0.4963)--(0.5072,0.5263)--(0.5141,0.5566)--(0.5198,0.5872)--(0.5240,0.6178)--(0.5267,0.6484)--(0.5280,0.6788)--(0.5279,0.7090)--(0.5262,0.7389)--(0.5230,0.7683)--(0.5183,0.7972)--(0.5121,0.8253)--(0.5044,0.8527)--(0.4952,0.8791)--(0.4846,0.9045)--(0.4726,0.9288)--(0.4593,0.9519)--(0.4446,0.9736)--(0.4287,0.9938)--(0.4116,1.013)--(0.3933,1.030)--(0.3741,1.045)--(0.3538,1.058)--(0.3327,1.070)--(0.3108,1.080)--(0.2883,1.087)--(0.2651,1.093)--(0.2415,1.096)--(0.2175,1.097)--(0.1933,1.096)--(0.1690,1.093)--(0.1446,1.087)--(0.1204,1.080)--(0.09640,1.069)--(0.07277,1.057)--(0.04963,1.042)--(0.02710,1.025)--(0.005317,1.005)--(-0.01561,0.9835)--(-0.03554,0.9596)--(-0.05437,0.9335)--(-0.07197,0.9053)--(-0.08824,0.8751)--(-0.1030,0.8429)--(-0.1163,0.8089)--(-0.1279,0.7730)--(-0.1377,0.7354)--(-0.1457,0.6962)--(-0.1517,0.6555)--(-0.1557,0.6135)--(-0.1576,0.5701)--(-0.1574,0.5256)--(-0.1549,0.4801)--(-0.1501,0.4337)--(-0.1430,0.3866)--(-0.1336,0.3389)--(-0.1217,0.2907)--(-0.1075,0.2421)--(-0.09083,0.1935)--(-0.07174,0.1447)--(-0.05022,0.09616)--(-0.02630,0.04787)--(0,0);
 \draw [color=blue] (0,0)--(-0.02630,-0.04787)--(-0.05022,-0.09616)--(-0.07174,-0.1447)--(-0.09083,-0.1935)--(-0.1075,-0.2421)--(-0.1217,-0.2907)--(-0.1336,-0.3389)--(-0.1430,-0.3866)--(-0.1501,-0.4337)--(-0.1549,-0.4801)--(-0.1574,-0.5256)--(-0.1576,-0.5701)--(-0.1557,-0.6135)--(-0.1517,-0.6555)--(-0.1457,-0.6962)--(-0.1377,-0.7354)--(-0.1279,-0.7730)--(-0.1163,-0.8089)--(-0.1030,-0.8429)--(-0.08824,-0.8751)--(-0.07197,-0.9053)--(-0.05437,-0.9335)--(-0.03554,-0.9596)--(-0.01561,-0.9835)--(0.005317,-1.005)--(0.02710,-1.025)--(0.04963,-1.042)--(0.07277,-1.057)--(0.09640,-1.069)--(0.1204,-1.080)--(0.1446,-1.087)--(0.1690,-1.093)--(0.1933,-1.096)--(0.2175,-1.097)--(0.2415,-1.096)--(0.2651,-1.093)--(0.2883,-1.087)--(0.3108,-1.080)--(0.3327,-1.070)--(0.3538,-1.058)--(0.3741,-1.045)--(0.3933,-1.030)--(0.4116,-1.013)--(0.4287,-0.9938)--(0.4446,-0.9736)--(0.4593,-0.9519)--(0.4726,-0.9288)--(0.4846,-0.9045)--(0.4952,-0.8791)--(0.5044,-0.8527)--(0.5121,-0.8253)--(0.5183,-0.7972)--(0.5230,-0.7683)--(0.5262,-0.7389)--(0.5279,-0.7090)--(0.5280,-0.6788)--(0.5267,-0.6484)--(0.5240,-0.6178)--(0.5198,-0.5872)--(0.5141,-0.5566)--(0.5072,-0.5263)--(0.4989,-0.4963)--(0.4894,-0.4666)--(0.4786,-0.4374)--(0.4667,-0.4088)--(0.4538,-0.3808)--(0.4399,-0.3535)--(0.4250,-0.3270)--(0.4094,-0.3014)--(0.3929,-0.2767)--(0.3759,-0.2530)--(0.3582,-0.2302)--(0.3401,-0.2086)--(0.3217,-0.1880)--(0.3029,-0.1685)--(0.2840,-0.1502)--(0.2649,-0.1331)--(0.2459,-0.1171)--(0.2270,-0.1022)--(0.2084,-0.08857)--(0.1900,-0.07605)--(0.1720,-0.06467)--(0.1545,-0.05439)--(0.1376,-0.04518)--(0.1213,-0.03702)--(0.1058,-0.02985)--(0.09108,-0.02363)--(0.07727,-0.01831)--(0.06442,-0.01384)--(0.05259,-0.01014)--(0.04185,-0.007151)--(0.03224,-0.004809)--(0.02381,-0.003039)--(0.01661,-0.001764)--(0.01067,0)--(0.006020,0)--(0.002681,0)--(0,0)--(0,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Cardioideexo.pstricks.recall b/src_phystricks/Fig_Cardioideexo.pstricks.recall
index 4ae6791fd..1b2d525b7 100644
--- a/src_phystricks/Fig_Cardioideexo.pstricks.recall
+++ b/src_phystricks/Fig_Cardioideexo.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=lightgray] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
@@ -92,23 +92,23 @@
 \draw [color=lightgray] (1.00,0) -- (0,0);
 \draw [color=lightgray] (0,0) -- (0,1.00);
 \draw [color=lightgray,style=dashed] (0,0) -- (1.21,1.21);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.207106781,1.207106781) node [rotate=0] {$\bullet$};
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.2071,1.2071) node [rotate=0] {$\bullet$};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CbCartTui.pstricks.recall b/src_phystricks/Fig_CbCartTui.pstricks.recall
index d7044745a..398a015fa 100644
--- a/src_phystricks/Fig_CbCartTui.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTui.pstricks.recall
@@ -103,8 +103,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.140000000,0) -- (4.140000000,0);
-\draw [,->,>=latex] (0,-3.972000000) -- (0,4.028000000);
+\draw [,->,>=latex] (-4.140000002,0) -- (4.140000002,0);
+\draw [,->,>=latex] (0,-3.972000001) -- (0,4.028000000);
 %DEFAULT
 \draw [color=blue] (-3.640,-3.472)--(-3.609,-3.440)--(-3.579,-3.407)--(-3.548,-3.375)--(-3.518,-3.343)--(-3.488,-3.310)--(-3.457,-3.278)--(-3.427,-3.245)--(-3.396,-3.213)--(-3.366,-3.180)--(-3.336,-3.148)--(-3.306,-3.115)--(-3.275,-3.083)--(-3.245,-3.050)--(-3.215,-3.018)--(-3.185,-2.985)--(-3.155,-2.953)--(-3.125,-2.920)--(-3.095,-2.887)--(-3.065,-2.855)--(-3.035,-2.822)--(-3.005,-2.789)--(-2.975,-2.756)--(-2.945,-2.723)--(-2.915,-2.691)--(-2.886,-2.658)--(-2.856,-2.625)--(-2.826,-2.592)--(-2.797,-2.559)--(-2.767,-2.526)--(-2.738,-2.493)--(-2.709,-2.459)--(-2.679,-2.426)--(-2.650,-2.393)--(-2.621,-2.360)--(-2.592,-2.326)--(-2.563,-2.293)--(-2.534,-2.259)--(-2.505,-2.226)--(-2.476,-2.192)--(-2.447,-2.158)--(-2.419,-2.124)--(-2.390,-2.090)--(-2.362,-2.056)--(-2.333,-2.022)--(-2.305,-1.988)--(-2.277,-1.954)--(-2.249,-1.919)--(-2.221,-1.885)--(-2.193,-1.850)--(-2.166,-1.815)--(-2.138,-1.780)--(-2.111,-1.745)--(-2.084,-1.709)--(-2.057,-1.674)--(-2.030,-1.638)--(-2.003,-1.602)--(-1.977,-1.566)--(-1.951,-1.529)--(-1.925,-1.492)--(-1.899,-1.455)--(-1.873,-1.418)--(-1.848,-1.380)--(-1.823,-1.342)--(-1.798,-1.304)--(-1.774,-1.265)--(-1.750,-1.225)--(-1.726,-1.185)--(-1.703,-1.144)--(-1.680,-1.103)--(-1.658,-1.061)--(-1.636,-1.018)--(-1.614,-0.9745)--(-1.593,-0.9298)--(-1.573,-0.8840)--(-1.554,-0.8371)--(-1.535,-0.7887)--(-1.517,-0.7389)--(-1.499,-0.6873)--(-1.483,-0.6338)--(-1.468,-0.5781)--(-1.454,-0.5199)--(-1.441,-0.4588)--(-1.429,-0.3943)--(-1.420,-0.3261)--(-1.411,-0.2534)--(-1.405,-0.1754)--(-1.401,-0.09136)--(-1.400,0)--(-1.402,0.1001)--(-1.406,0.2106)--(-1.415,0.3340)--(-1.428,0.4729)--(-1.447,0.6314)--(-1.472,0.8143)--(-1.504,1.029)--(-1.545,1.283)--(-1.598,1.591)--(-1.665,1.971)--(-1.750,2.450);
 \draw [,->,>=latex] (-3.505833333,-3.329618056) -- (-3.501040033,-3.324516656);
diff --git a/src_phystricks/Fig_CbCartTuii.pstricks.recall b/src_phystricks/Fig_CbCartTuii.pstricks.recall
index bd6a697f8..f3a5e95ac 100644
--- a/src_phystricks/Fig_CbCartTuii.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTuii.pstricks.recall
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.148869145) -- (0,1.148869145);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.1489) -- (0,1.1489);
 %DEFAULT
 \draw [color=blue] (2.00,0)--(1.99,0.126)--(1.97,0.247)--(1.93,0.358)--(1.87,0.456)--(1.81,0.535)--(1.72,0.595)--(1.63,0.633)--(1.53,0.649)--(1.42,0.644)--(1.30,0.619)--(1.17,0.578)--(1.05,0.523)--(0.921,0.459)--(0.795,0.389)--(0.673,0.318)--(0.556,0.249)--(0.446,0.186)--(0.345,0.130)--(0.255,0.0849)--(0.176,0.0500)--(0.111,0.0255)--(0.0603,0.0103)--(0.0246,0.00271)--(0.00453,0)--(0,0)--(0.0126,0)--(0.0405,-0.00571)--(0.0839,-0.0168)--(0.142,-0.0365)--(0.214,-0.0661)--(0.299,-0.106)--(0.394,-0.157)--(0.500,-0.217)--(0.614,-0.283)--(0.734,-0.354)--(0.858,-0.424)--(0.984,-0.492)--(1.11,-0.552)--(1.24,-0.600)--(1.36,-0.634)--(1.47,-0.649)--(1.58,-0.644)--(1.68,-0.617)--(1.77,-0.568)--(1.84,-0.498)--(1.90,-0.409)--(1.95,-0.304)--(1.98,-0.188)--(2.00,-0.0634)--(2.00,0.0634)--(1.98,0.188)--(1.95,0.304)--(1.90,0.409)--(1.84,0.498)--(1.77,0.568)--(1.68,0.617)--(1.58,0.644)--(1.47,0.649)--(1.36,0.634)--(1.24,0.600)--(1.11,0.552)--(0.984,0.492)--(0.858,0.424)--(0.734,0.354)--(0.614,0.283)--(0.500,0.217)--(0.394,0.157)--(0.299,0.106)--(0.214,0.0661)--(0.142,0.0365)--(0.0839,0.0168)--(0.0405,0.00571)--(0.0126,0)--(0,0)--(0.00453,0)--(0.0246,-0.00271)--(0.0603,-0.0103)--(0.111,-0.0255)--(0.176,-0.0500)--(0.255,-0.0849)--(0.345,-0.130)--(0.446,-0.186)--(0.556,-0.249)--(0.673,-0.318)--(0.795,-0.389)--(0.921,-0.459)--(1.05,-0.523)--(1.17,-0.578)--(1.30,-0.619)--(1.42,-0.644)--(1.53,-0.649)--(1.63,-0.633)--(1.72,-0.595)--(1.81,-0.535)--(1.87,-0.456)--(1.93,-0.358)--(1.97,-0.247)--(1.99,-0.126)--(2.00,0);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CbCartTuiii.pstricks.recall b/src_phystricks/Fig_CbCartTuiii.pstricks.recall
index 40b0f31eb..95ddf09e7 100644
--- a/src_phystricks/Fig_CbCartTuiii.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTuiii.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.499748255,0) -- (2.499748255,0);
-\draw [,->,>=latex] (0,-2.497734678) -- (0,2.497734678);
+\draw [,->,>=latex] (-2.499748256,0) -- (2.499748256,0);
+\draw [,->,>=latex] (0,-2.497734679) -- (0,2.497734679);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.253,0.379)--(0.502,0.743)--(0.743,1.08)--(0.972,1.38)--(1.19,1.63)--(1.38,1.82)--(1.55,1.94)--(1.70,2.00)--(1.82,1.98)--(1.91,1.89)--(1.97,1.73)--(2.00,1.51)--(1.99,1.24)--(1.96,0.916)--(1.89,0.563)--(1.79,0.190)--(1.67,-0.190)--(1.51,-0.563)--(1.33,-0.916)--(1.13,-1.24)--(0.916,-1.51)--(0.684,-1.73)--(0.441,-1.89)--(0.190,-1.98)--(-0.0635,-2.00)--(-0.316,-1.94)--(-0.563,-1.82)--(-0.802,-1.63)--(-1.03,-1.38)--(-1.24,-1.08)--(-1.43,-0.743)--(-1.59,-0.379)--(-1.73,0)--(-1.84,0.379)--(-1.93,0.743)--(-1.98,1.08)--(-2.00,1.38)--(-1.99,1.63)--(-1.94,1.82)--(-1.87,1.94)--(-1.76,2.00)--(-1.63,1.98)--(-1.47,1.89)--(-1.29,1.73)--(-1.08,1.51)--(-0.860,1.24)--(-0.624,0.916)--(-0.379,0.563)--(-0.127,0.190)--(0.127,-0.190)--(0.379,-0.563)--(0.624,-0.916)--(0.860,-1.24)--(1.08,-1.51)--(1.29,-1.73)--(1.47,-1.89)--(1.63,-1.98)--(1.76,-2.00)--(1.87,-1.94)--(1.94,-1.82)--(1.99,-1.63)--(2.00,-1.38)--(1.98,-1.08)--(1.93,-0.743)--(1.84,-0.379)--(1.73,0)--(1.59,0.379)--(1.43,0.743)--(1.24,1.08)--(1.03,1.38)--(0.802,1.63)--(0.563,1.82)--(0.316,1.94)--(0.0635,2.00)--(-0.190,1.98)--(-0.441,1.89)--(-0.684,1.73)--(-0.916,1.51)--(-1.13,1.24)--(-1.33,0.916)--(-1.51,0.563)--(-1.67,0.190)--(-1.79,-0.190)--(-1.89,-0.563)--(-1.96,-0.916)--(-1.99,-1.24)--(-2.00,-1.51)--(-1.97,-1.73)--(-1.91,-1.89)--(-1.82,-1.98)--(-1.70,-2.00)--(-1.55,-1.94)--(-1.38,-1.82)--(-1.19,-1.63)--(-0.972,-1.38)--(-0.743,-1.08)--(-0.502,-0.743)--(-0.253,-0.379)--(0,0);
 \draw (-2.000000000,-0.3298256667) node {$ -1 $};
diff --git a/src_phystricks/Fig_CercleTnu.pstricks.recall b/src_phystricks/Fig_CercleTnu.pstricks.recall
index 91e3ce90f..3feef34ae 100644
--- a/src_phystricks/Fig_CercleTnu.pstricks.recall
+++ b/src_phystricks/Fig_CercleTnu.pstricks.recall
@@ -73,14 +73,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=green,->,>=latex] (1.732050808,1.000000000) -- (2.598076211,1.500000000);
-\draw (2.338533378,1.838383788) node {\( n\)};
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (1.232050808,1.866025404);
-\draw (1.655192262,2.124333237) node {\( e_{\theta}\)};
-\draw [color=green,->,>=latex] (-0.6840402866,1.879385242) -- (-1.026060430,2.819077862);
-\draw (-1.417511050,2.637895653) node {\( n\)};
-\draw [color=red,->,>=latex] (-0.6840402866,1.879385242) -- (-1.623732907,1.537365098);
-\draw (-1.889672784,1.927580718) node {\( e_{\theta}\)};
+\draw [color=green,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
+\draw (2.3385,1.8384) node {\( n\)};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
+\draw (1.6552,2.1243) node {\( e_{\theta}\)};
+\draw [color=green,->,>=latex] (-0.68404,1.8794) -- (-1.0261,2.8191);
+\draw (-1.4175,2.6379) node {\( n\)};
+\draw [color=red,->,>=latex] (-0.68404,1.8794) -- (-1.6237,1.5374);
+\draw (-1.8897,1.9276) node {\( e_{\theta}\)};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/src_phystricks/Fig_CercleTrigono.pstricks.recall b/src_phystricks/Fig_CercleTrigono.pstricks.recall
index 5097fdb00..391fc1759 100644
--- a/src_phystricks/Fig_CercleTrigono.pstricks.recall
+++ b/src_phystricks/Fig_CercleTrigono.pstricks.recall
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
 \draw [color=brown,style=dotted] (1.73,1.00) -- (1.73,0);
 \draw [color=brown,style=dotted] (1.73,1.00) -- (0,1.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.732050808,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (-0.2000000000,0.5000000000) -- (-0.2000000000,0);
-\draw [,->,>=latex] (-0.2000000000,0.5000000000) -- (-0.2000000000,1.000000000);
-\draw (-0.7580386667,0.5000000000) node {$\sin(\theta)$};
-\draw [,->,>=latex] (0.8660254038,-0.2000000000) -- (0,-0.2000000000);
-\draw [,->,>=latex] (0.8660254038,-0.2000000000) -- (1.732050808,-0.2000000000);
-\draw (0.8660254038,-0.4824550000) node {$\cos(\theta)$};
-\draw [color=brown,->,>=latex] (0,0) -- (1.732050808,1.000000000);
-\draw (0.9161397128,0.2302636180) node {$\theta$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.7320,0) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,0);
+\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,1.0000);
+\draw (-0.75804,0.50000) node {$\sin(\theta)$};
+\draw [,->,>=latex] (0.86602,-0.20000) -- (0,-0.20000);
+\draw [,->,>=latex] (0.86602,-0.20000) -- (1.7320,-0.20000);
+\draw (0.86602,-0.48246) node {$\cos(\theta)$};
+\draw [color=brown,->,>=latex] (0,0) -- (1.7320,1.0000);
+\draw (0.91614,0.23026) node {$\theta$};
 
 \draw [] (0.400,0)--(0.400,0.00212)--(0.400,0.00423)--(0.400,0.00635)--(0.400,0.00846)--(0.400,0.0106)--(0.400,0.0127)--(0.400,0.0148)--(0.400,0.0169)--(0.400,0.0190)--(0.399,0.0211)--(0.399,0.0233)--(0.399,0.0254)--(0.399,0.0275)--(0.399,0.0296)--(0.399,0.0317)--(0.399,0.0338)--(0.398,0.0359)--(0.398,0.0380)--(0.398,0.0401)--(0.398,0.0422)--(0.398,0.0443)--(0.397,0.0464)--(0.397,0.0485)--(0.397,0.0506)--(0.397,0.0527)--(0.396,0.0548)--(0.396,0.0569)--(0.396,0.0590)--(0.395,0.0611)--(0.395,0.0632)--(0.395,0.0653)--(0.394,0.0674)--(0.394,0.0695)--(0.394,0.0715)--(0.393,0.0736)--(0.393,0.0757)--(0.392,0.0778)--(0.392,0.0798)--(0.392,0.0819)--(0.391,0.0840)--(0.391,0.0861)--(0.390,0.0881)--(0.390,0.0902)--(0.389,0.0922)--(0.389,0.0943)--(0.388,0.0964)--(0.388,0.0984)--(0.387,0.100)--(0.387,0.103)--(0.386,0.105)--(0.386,0.107)--(0.385,0.109)--(0.384,0.111)--(0.384,0.113)--(0.383,0.115)--(0.383,0.117)--(0.382,0.119)--(0.381,0.121)--(0.381,0.123)--(0.380,0.125)--(0.379,0.127)--(0.379,0.129)--(0.378,0.131)--(0.377,0.133)--(0.377,0.135)--(0.376,0.137)--(0.375,0.139)--(0.374,0.141)--(0.374,0.143)--(0.373,0.145)--(0.372,0.147)--(0.371,0.149)--(0.371,0.151)--(0.370,0.153)--(0.369,0.155)--(0.368,0.156)--(0.367,0.158)--(0.366,0.160)--(0.366,0.162)--(0.365,0.164)--(0.364,0.166)--(0.363,0.168)--(0.362,0.170)--(0.361,0.172)--(0.360,0.174)--(0.359,0.176)--(0.358,0.178)--(0.357,0.180)--(0.357,0.181)--(0.356,0.183)--(0.355,0.185)--(0.354,0.187)--(0.353,0.189)--(0.352,0.191)--(0.351,0.193)--(0.350,0.194)--(0.349,0.196)--(0.347,0.198)--(0.346,0.200);
-\draw (2.134373262,1.274708000) node {$P$};
+\draw (2.1344,1.2747) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CheminFresnel.pstricks.recall b/src_phystricks/Fig_CheminFresnel.pstricks.recall
index 4172b519f..5589fe1bc 100644
--- a/src_phystricks/Fig_CheminFresnel.pstricks.recall
+++ b/src_phystricks/Fig_CheminFresnel.pstricks.recall
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.914213562);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.9142);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.02020,0)--(0.04040,0)--(0.06061,0)--(0.08081,0)--(0.1010,0)--(0.1212,0)--(0.1414,0)--(0.1616,0)--(0.1818,0)--(0.2020,0)--(0.2222,0)--(0.2424,0)--(0.2626,0)--(0.2828,0)--(0.3030,0)--(0.3232,0)--(0.3434,0)--(0.3636,0)--(0.3838,0)--(0.4040,0)--(0.4242,0)--(0.4444,0)--(0.4646,0)--(0.4848,0)--(0.5051,0)--(0.5253,0)--(0.5455,0)--(0.5657,0)--(0.5859,0)--(0.6061,0)--(0.6263,0)--(0.6465,0)--(0.6667,0)--(0.6869,0)--(0.7071,0)--(0.7273,0)--(0.7475,0)--(0.7677,0)--(0.7879,0)--(0.8081,0)--(0.8283,0)--(0.8485,0)--(0.8687,0)--(0.8889,0)--(0.9091,0)--(0.9293,0)--(0.9495,0)--(0.9697,0)--(0.9899,0)--(1.010,0)--(1.030,0)--(1.051,0)--(1.071,0)--(1.091,0)--(1.111,0)--(1.131,0)--(1.152,0)--(1.172,0)--(1.192,0)--(1.212,0)--(1.232,0)--(1.253,0)--(1.273,0)--(1.293,0)--(1.313,0)--(1.333,0)--(1.354,0)--(1.374,0)--(1.394,0)--(1.414,0)--(1.434,0)--(1.455,0)--(1.475,0)--(1.495,0)--(1.515,0)--(1.535,0)--(1.556,0)--(1.576,0)--(1.596,0)--(1.616,0)--(1.636,0)--(1.657,0)--(1.677,0)--(1.697,0)--(1.717,0)--(1.737,0)--(1.758,0)--(1.778,0)--(1.798,0)--(1.818,0)--(1.838,0)--(1.859,0)--(1.879,0)--(1.899,0)--(1.919,0)--(1.939,0)--(1.960,0)--(1.980,0)--(2.000,0);
-\draw [,->,>=latex] (1.000000000,0) -- (1.010000000,0);
+\draw [,->,>=latex] (1.0000,0) -- (1.0100,0);
 \draw [color=blue] (2.000,0)--(2.000,0.01587)--(2.000,0.03173)--(1.999,0.04760)--(1.999,0.06346)--(1.998,0.07931)--(1.998,0.09516)--(1.997,0.1110)--(1.996,0.1268)--(1.995,0.1427)--(1.994,0.1585)--(1.992,0.1743)--(1.991,0.1901)--(1.989,0.2059)--(1.988,0.2217)--(1.986,0.2374)--(1.984,0.2532)--(1.982,0.2689)--(1.980,0.2846)--(1.977,0.3003)--(1.975,0.3160)--(1.972,0.3317)--(1.970,0.3473)--(1.967,0.3629)--(1.964,0.3785)--(1.961,0.3941)--(1.958,0.4096)--(1.954,0.4251)--(1.951,0.4406)--(1.947,0.4561)--(1.944,0.4715)--(1.940,0.4869)--(1.936,0.5023)--(1.932,0.5176)--(1.928,0.5330)--(1.923,0.5482)--(1.919,0.5635)--(1.914,0.5787)--(1.910,0.5938)--(1.905,0.6090)--(1.900,0.6241)--(1.895,0.6391)--(1.890,0.6541)--(1.885,0.6691)--(1.879,0.6840)--(1.874,0.6989)--(1.868,0.7138)--(1.863,0.7286)--(1.857,0.7433)--(1.851,0.7580)--(1.845,0.7727)--(1.839,0.7873)--(1.832,0.8019)--(1.826,0.8164)--(1.819,0.8308)--(1.813,0.8452)--(1.806,0.8596)--(1.799,0.8739)--(1.792,0.8881)--(1.785,0.9023)--(1.778,0.9165)--(1.770,0.9305)--(1.763,0.9445)--(1.755,0.9585)--(1.748,0.9724)--(1.740,0.9862)--(1.732,1.000)--(1.724,1.014)--(1.716,1.027)--(1.708,1.041)--(1.699,1.054)--(1.691,1.068)--(1.683,1.081)--(1.674,1.095)--(1.665,1.108)--(1.656,1.121)--(1.647,1.134)--(1.638,1.147)--(1.629,1.160)--(1.620,1.173)--(1.611,1.186)--(1.601,1.199)--(1.592,1.211)--(1.582,1.224)--(1.572,1.236)--(1.562,1.249)--(1.552,1.261)--(1.542,1.273)--(1.532,1.286)--(1.522,1.298)--(1.512,1.310)--(1.501,1.322)--(1.491,1.334)--(1.480,1.345)--(1.469,1.357)--(1.458,1.369)--(1.447,1.380)--(1.436,1.392)--(1.425,1.403)--(1.414,1.414);
-\draw [,->,>=latex] (1.847759065,0.7653668647) -- (1.843932231,0.7746056601);
+\draw [,->,>=latex] (1.8478,0.76537) -- (1.8439,0.77461);
 \draw [color=blue] (0,0)--(0.01428,0.01428)--(0.02857,0.02857)--(0.04285,0.04285)--(0.05714,0.05714)--(0.07142,0.07142)--(0.08571,0.08571)--(0.09999,0.09999)--(0.1143,0.1143)--(0.1286,0.1286)--(0.1428,0.1428)--(0.1571,0.1571)--(0.1714,0.1714)--(0.1857,0.1857)--(0.2000,0.2000)--(0.2143,0.2143)--(0.2286,0.2286)--(0.2428,0.2428)--(0.2571,0.2571)--(0.2714,0.2714)--(0.2857,0.2857)--(0.3000,0.3000)--(0.3143,0.3143)--(0.3286,0.3286)--(0.3428,0.3428)--(0.3571,0.3571)--(0.3714,0.3714)--(0.3857,0.3857)--(0.4000,0.4000)--(0.4143,0.4143)--(0.4286,0.4286)--(0.4428,0.4428)--(0.4571,0.4571)--(0.4714,0.4714)--(0.4857,0.4857)--(0.5000,0.5000)--(0.5143,0.5143)--(0.5285,0.5285)--(0.5428,0.5428)--(0.5571,0.5571)--(0.5714,0.5714)--(0.5857,0.5857)--(0.6000,0.6000)--(0.6143,0.6143)--(0.6285,0.6285)--(0.6428,0.6428)--(0.6571,0.6571)--(0.6714,0.6714)--(0.6857,0.6857)--(0.7000,0.7000)--(0.7142,0.7142)--(0.7285,0.7285)--(0.7428,0.7428)--(0.7571,0.7571)--(0.7714,0.7714)--(0.7857,0.7857)--(0.8000,0.8000)--(0.8142,0.8142)--(0.8285,0.8285)--(0.8428,0.8428)--(0.8571,0.8571)--(0.8714,0.8714)--(0.8857,0.8857)--(0.9000,0.9000)--(0.9142,0.9142)--(0.9285,0.9285)--(0.9428,0.9428)--(0.9571,0.9571)--(0.9714,0.9714)--(0.9857,0.9857)--(0.9999,0.9999)--(1.014,1.014)--(1.029,1.029)--(1.043,1.043)--(1.057,1.057)--(1.071,1.071)--(1.086,1.086)--(1.100,1.100)--(1.114,1.114)--(1.129,1.129)--(1.143,1.143)--(1.157,1.157)--(1.171,1.171)--(1.186,1.186)--(1.200,1.200)--(1.214,1.214)--(1.229,1.229)--(1.243,1.243)--(1.257,1.257)--(1.271,1.271)--(1.286,1.286)--(1.300,1.300)--(1.314,1.314)--(1.329,1.329)--(1.343,1.343)--(1.357,1.357)--(1.371,1.371)--(1.386,1.386)--(1.400,1.400)--(1.414,1.414);
-\draw [,->,>=latex] (0.7071067812,0.7071067812) -- (0.7141778490,0.7141778490);
-\draw (1.000000000,-0.2140621667) node {\( \gamma_1\)};
-\draw (2.113799352,0.9176973746) node {\( \gamma_2\)};
-\draw (0.4627437697,0.8918796260) node {\( \gamma_3\)};
+\draw [,->,>=latex] (0.70711,0.70711) -- (0.71418,0.71418);
+\draw (1.0000,-0.21406) node {\( \gamma_1\)};
+\draw (2.1138,0.91770) node {\( \gamma_2\)};
+\draw (0.46274,0.89188) node {\( \gamma_3\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall b/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
index c0b4c0635..ff609a685 100644
--- a/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
+++ b/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
@@ -96,7 +96,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (0,0) -- (9.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.384154941);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.384154943);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.045226,0)--(0.090452,0)--(0.13568,0.0021725)--(0.18090,0.0063676)--(0.22613,0.014417)--(0.27136,0.027725)--(0.31658,0.047635)--(0.36181,0.075363)--(0.40704,0.11195)--(0.45226,0.15824)--(0.49749,0.21486)--(0.54271,0.28221)--(0.58794,0.36049)--(0.63317,0.44967)--(0.67839,0.54955)--(0.72362,0.65977)--(0.76884,0.77977)--(0.81407,0.90892)--(0.85930,1.0464)--(0.90452,1.1915)--(0.94975,1.3431)--(0.99498,1.5003)--(1.0402,1.6621)--(1.0854,1.8275)--(1.1307,1.9955)--(1.1759,2.1649)--(1.2211,2.3349)--(1.2663,2.5044)--(1.3116,2.6726)--(1.3568,2.8385)--(1.4020,3.0014)--(1.4472,3.1604)--(1.4925,3.3148)--(1.5377,3.4640)--(1.5829,3.6075)--(1.6281,3.7446)--(1.6734,3.8749)--(1.7186,3.9981)--(1.7638,4.1138)--(1.8090,4.2217)--(1.8543,4.3217)--(1.8995,4.4134)--(1.9447,4.4969)--(1.9900,4.5721)--(2.0352,4.6390)--(2.0804,4.6975)--(2.1256,4.7478)--(2.1709,4.7899)--(2.2161,4.8241)--(2.2613,4.8503)--(2.3065,4.8690)--(2.3518,4.8802)--(2.3970,4.8842)--(2.4422,4.8812)--(2.4874,4.8715)--(2.5327,4.8555)--(2.5779,4.8333)--(2.6231,4.8053)--(2.6683,4.7718)--(2.7136,4.7330)--(2.7588,4.6894)--(2.8040,4.6412)--(2.8492,4.5887)--(2.8945,4.5322)--(2.9397,4.4721)--(2.9849,4.4086)--(3.0302,4.3419)--(3.0754,4.2725)--(3.1206,4.2006)--(3.1658,4.1264)--(3.2111,4.0502)--(3.2563,3.9722)--(3.3015,3.8928)--(3.3467,3.8121)--(3.3920,3.7303)--(3.4372,3.6477)--(3.4824,3.5644)--(3.5276,3.4807)--(3.5729,3.3968)--(3.6181,3.3127)--(3.6633,3.2287)--(3.7085,3.1449)--(3.7538,3.0614)--(3.7990,2.9785)--(3.8442,2.8961)--(3.8894,2.8145)--(3.9347,2.7337)--(3.9799,2.6538)--(4.0251,2.5749)--(4.0704,2.4971)--(4.1156,2.4204)--(4.1608,2.3450)--(4.2060,2.2708)--(4.2513,2.1980)--(4.2965,2.1266)--(4.3417,2.0565)--(4.3869,1.9879)--(4.4322,1.9208)--(4.4774,1.8552)--(4.5226,1.7910)--(4.5678,1.7284)--(4.6131,1.6674)--(4.6583,1.6079)--(4.7035,1.5499)--(4.7487,1.4934)--(4.7940,1.4385)--(4.8392,1.3851)--(4.8844,1.3333)--(4.9296,1.2829)--(4.9749,1.2340)--(5.0201,1.1866)--(5.0653,1.1406)--(5.1105,1.0961)--(5.1558,1.0530)--(5.2010,1.0113)--(5.2462,0.97088)--(5.2915,0.93184)--(5.3367,0.89411)--(5.3819,0.85766)--(5.4271,0.82246)--(5.4724,0.78849)--(5.5176,0.75571)--(5.5628,0.72411)--(5.6080,0.69364)--(5.6533,0.66428)--(5.6985,0.63600)--(5.7437,0.60877)--(5.7889,0.58256)--(5.8342,0.55735)--(5.8794,0.53310)--(5.9246,0.50978)--(5.9698,0.48737)--(6.0151,0.46583)--(6.0603,0.44515)--(6.1055,0.42529)--(6.1508,0.40623)--(6.1960,0.38794)--(6.2412,0.37039)--(6.2864,0.35357)--(6.3317,0.33743)--(6.3769,0.32197)--(6.4221,0.30715)--(6.4673,0.29296)--(6.5126,0.27937)--(6.5578,0.26636)--(6.6030,0.25390)--(6.6482,0.24199)--(6.6935,0.23059)--(6.7387,0.21968)--(6.7839,0.20926)--(6.8291,0.19929)--(6.8744,0.18977)--(6.9196,0.18067)--(6.9648,0.17197)--(7.0100,0.16367)--(7.0553,0.15574)--(7.1005,0.14817)--(7.1457,0.14095)--(7.1910,0.13406)--(7.2362,0.12748)--(7.2814,0.12121)--(7.3266,0.11523)--(7.3719,0.10952)--(7.4171,0.10409)--(7.4623,0.098904)--(7.5075,0.093967)--(7.5528,0.089264)--(7.5980,0.084783)--(7.6432,0.080516)--(7.6884,0.076453)--(7.7337,0.072586)--(7.7789,0.068904)--(7.8241,0.065400)--(7.8693,0.062066)--(7.9146,0.058895)--(7.9598,0.055878)--(8.0050,0.053008)--(8.0502,0.050280)--(8.0955,0.047686)--(8.1407,0.045220)--(8.1859,0.042877)--(8.2312,0.040650)--(8.2764,0.038534)--(8.3216,0.036523)--(8.3668,0.034614)--(8.4121,0.032801)--(8.4573,0.031078)--(8.5025,0.029443)--(8.5477,0.027891)--(8.5930,0.026418)--(8.6382,0.025020)--(8.6834,0.023693)--(8.7286,0.022434)--(8.7739,0.021240)--(8.8191,0.020107)--(8.8643,0.019033)--(8.9095,0.018014)--(8.9548,0.017047)--(9.0000,0.016132);
diff --git a/src_phystricks/Fig_ConeRevolution.pstricks.recall b/src_phystricks/Fig_ConeRevolution.pstricks.recall
index c3099f553..1c9a11ba0 100644
--- a/src_phystricks/Fig_ConeRevolution.pstricks.recall
+++ b/src_phystricks/Fig_ConeRevolution.pstricks.recall
@@ -79,19 +79,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.02020,0.03030)--(0.04040,0.06061)--(0.06061,0.09091)--(0.08081,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5152)--(0.3636,0.5455)--(0.3838,0.5758)--(0.4040,0.6061)--(0.4242,0.6364)--(0.4444,0.6667)--(0.4646,0.6970)--(0.4848,0.7273)--(0.5051,0.7576)--(0.5253,0.7879)--(0.5455,0.8182)--(0.5657,0.8485)--(0.5859,0.8788)--(0.6061,0.9091)--(0.6263,0.9394)--(0.6465,0.9697)--(0.6667,1.000)--(0.6869,1.030)--(0.7071,1.061)--(0.7273,1.091)--(0.7475,1.121)--(0.7677,1.152)--(0.7879,1.182)--(0.8081,1.212)--(0.8283,1.242)--(0.8485,1.273)--(0.8687,1.303)--(0.8889,1.333)--(0.9091,1.364)--(0.9293,1.394)--(0.9495,1.424)--(0.9697,1.455)--(0.9899,1.485)--(1.010,1.515)--(1.030,1.545)--(1.051,1.576)--(1.071,1.606)--(1.091,1.636)--(1.111,1.667)--(1.131,1.697)--(1.152,1.727)--(1.172,1.758)--(1.192,1.788)--(1.212,1.818)--(1.232,1.848)--(1.253,1.879)--(1.273,1.909)--(1.293,1.939)--(1.313,1.970)--(1.333,2.000)--(1.354,2.030)--(1.374,2.061)--(1.394,2.091)--(1.414,2.121)--(1.434,2.152)--(1.455,2.182)--(1.475,2.212)--(1.495,2.242)--(1.515,2.273)--(1.535,2.303)--(1.556,2.333)--(1.576,2.364)--(1.596,2.394)--(1.616,2.424)--(1.636,2.455)--(1.657,2.485)--(1.677,2.515)--(1.697,2.545)--(1.717,2.576)--(1.737,2.606)--(1.758,2.636)--(1.778,2.667)--(1.798,2.697)--(1.818,2.727)--(1.838,2.758)--(1.859,2.788)--(1.879,2.818)--(1.899,2.848)--(1.919,2.879)--(1.939,2.909)--(1.960,2.939)--(1.980,2.970)--(2.000,3.000);
-\draw []  (2.000000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (2.538859511,3.253165678) node {$(R,h)$};
-\draw (0.7236718444,0.3359520608) node {$\alpha$};
+\draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw (2.5389,3.2532) node {$(R,h)$};
+\draw (0.72367,0.33595) node {$\alpha$};
 
 \draw [color=red] (0.500,0)--(0.500,0.00496)--(0.500,0.00993)--(0.500,0.0149)--(0.500,0.0198)--(0.499,0.0248)--(0.499,0.0298)--(0.499,0.0347)--(0.498,0.0397)--(0.498,0.0446)--(0.498,0.0496)--(0.497,0.0545)--(0.496,0.0594)--(0.496,0.0643)--(0.495,0.0693)--(0.494,0.0742)--(0.494,0.0791)--(0.493,0.0840)--(0.492,0.0889)--(0.491,0.0938)--(0.490,0.0986)--(0.489,0.103)--(0.488,0.108)--(0.487,0.113)--(0.486,0.118)--(0.485,0.123)--(0.483,0.128)--(0.482,0.132)--(0.481,0.137)--(0.479,0.142)--(0.478,0.147)--(0.477,0.151)--(0.475,0.156)--(0.473,0.161)--(0.472,0.166)--(0.470,0.170)--(0.468,0.175)--(0.467,0.180)--(0.465,0.184)--(0.463,0.189)--(0.461,0.193)--(0.459,0.198)--(0.457,0.202)--(0.455,0.207)--(0.453,0.212)--(0.451,0.216)--(0.449,0.220)--(0.447,0.225)--(0.444,0.229)--(0.442,0.234)--(0.440,0.238)--(0.437,0.242)--(0.435,0.247)--(0.432,0.251)--(0.430,0.255)--(0.427,0.260)--(0.425,0.264)--(0.422,0.268)--(0.419,0.272)--(0.417,0.276)--(0.414,0.281)--(0.411,0.285)--(0.408,0.289)--(0.405,0.293)--(0.402,0.297)--(0.399,0.301)--(0.396,0.305)--(0.393,0.309)--(0.390,0.312)--(0.387,0.316)--(0.384,0.320)--(0.381,0.324)--(0.378,0.328)--(0.374,0.331)--(0.371,0.335)--(0.368,0.339)--(0.364,0.342)--(0.361,0.346)--(0.357,0.350)--(0.354,0.353)--(0.350,0.357)--(0.347,0.360)--(0.343,0.364)--(0.340,0.367)--(0.336,0.370)--(0.332,0.374)--(0.329,0.377)--(0.325,0.380)--(0.321,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.306,0.396)--(0.302,0.399)--(0.298,0.402)--(0.294,0.405)--(0.290,0.408)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416);
-\draw (2.000000000,-0.3257195000) node {$\mathit{R}$};
+\draw (2.0000,-0.32572) node {$\mathit{R}$};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.3027346667,3.000000000) node {$\mathit{h}$};
+\draw (-0.30273,3.0000) node {$\mathit{h}$};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CycloideA.pstricks.recall b/src_phystricks/Fig_CycloideA.pstricks.recall
index 974207361..9f4df5aac 100644
--- a/src_phystricks/Fig_CycloideA.pstricks.recall
+++ b/src_phystricks/Fig_CycloideA.pstricks.recall
@@ -115,7 +115,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (13.06637061,0);
+\draw [,->,>=latex] (-0.5000000000,0) -- (13.06637062,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,2.499496542);
 %DEFAULT
 \draw [color=blue] (0,0)--(0,0.0080452)--(0.0027181,0.032051)--(0.0091366,0.071632)--(0.021535,0.12615)--(0.041757,0.19473)--(0.071519,0.27627)--(0.11238,0.36945)--(0.16574,0.47277)--(0.23277,0.58459)--(0.31443,0.70308)--(0.41146,0.82635)--(0.52433,0.95242)--(0.65327,1.0793)--(0.79826,1.2048)--(0.95899,1.3271)--(1.1349,1.4441)--(1.3253,1.5539)--(1.5290,1.6549)--(1.7450,1.7453)--(1.9716,1.8237)--(2.2074,1.8888)--(2.4505,1.9397)--(2.6992,1.9754)--(2.9513,1.9955)--(3.2051,1.9995)--(3.4583,1.9874)--(3.7089,1.9595)--(3.9551,1.9161)--(4.1947,1.8580)--(4.4261,1.7861)--(4.6476,1.7015)--(4.8576,1.6056)--(5.0548,1.5000)--(5.2381,1.3863)--(5.4065,1.2665)--(5.5594,1.1423)--(5.6964,1.0159)--(5.8173,0.88916)--(5.9222,0.76424)--(6.0115,0.64311)--(6.0857,0.52773)--(6.1458,0.41994)--(6.1927,0.32149)--(6.2278,0.23396)--(6.2526,0.15875)--(6.2687,0.097073)--(6.2779,0.049929)--(6.2820,0.018071)--(6.2831,0.0020133)--(6.2832,0.0020133)--(6.2843,0.018071)--(6.2885,0.049929)--(6.2977,0.097073)--(6.3137,0.15875)--(6.3385,0.23396)--(6.3737,0.32149)--(6.4206,0.41994)--(6.4807,0.52773)--(6.5549,0.64311)--(6.6442,0.76424)--(6.7491,0.88916)--(6.8700,1.0159)--(7.0070,1.1423)--(7.1599,1.2665)--(7.3283,1.3863)--(7.5116,1.5000)--(7.7087,1.6056)--(7.9188,1.7015)--(8.1402,1.7861)--(8.3716,1.8580)--(8.6113,1.9161)--(8.8575,1.9595)--(9.1081,1.9874)--(9.3613,1.9995)--(9.6150,1.9955)--(9.8672,1.9754)--(10.116,1.9397)--(10.359,1.8888)--(10.595,1.8237)--(10.821,1.7453)--(11.037,1.6549)--(11.241,1.5539)--(11.431,1.4441)--(11.607,1.3271)--(11.768,1.2048)--(11.913,1.0793)--(12.042,0.95242)--(12.155,0.82635)--(12.252,0.70308)--(12.334,0.58459)--(12.401,0.47277)--(12.454,0.36945)--(12.495,0.27627)--(12.525,0.19473)--(12.545,0.12615)--(12.557,0.071632)--(12.564,0.032051)--(12.566,0.0080452)--(12.566,0);
diff --git a/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall b/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall
index 21700a099..7432ae77f 100644
--- a/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall
+++ b/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall
@@ -107,8 +107,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
@@ -116,35 +116,35 @@
 \draw [] (3.000,0)--(2.996,0.1268)--(2.984,0.2532)--(2.964,0.3785)--(2.936,0.5023)--(2.900,0.6241)--(2.857,0.7433)--(2.806,0.8596)--(2.748,0.9724)--(2.682,1.081)--(2.611,1.186)--(2.532,1.286)--(2.447,1.380)--(2.357,1.469)--(2.261,1.552)--(2.160,1.629)--(2.054,1.699)--(1.945,1.763)--(1.831,1.819)--(1.714,1.868)--(1.594,1.910)--(1.472,1.944)--(1.347,1.970)--(1.222,1.988)--(1.095,1.998)--(0.9683,2.000)--(0.8415,1.994)--(0.7154,1.980)--(0.5904,1.958)--(0.4671,1.928)--(0.3459,1.890)--(0.2273,1.845)--(0.1119,1.792)--(0,1.732)--(-0.1078,1.665)--(-0.2112,1.592)--(-0.3097,1.512)--(-0.4030,1.425)--(-0.4905,1.334)--(-0.5721,1.236)--(-0.6474,1.134)--(-0.7160,1.027)--(-0.7777,0.9165)--(-0.8322,0.8019)--(-0.8794,0.6840)--(-0.9190,0.5635)--(-0.9509,0.4406)--(-0.9749,0.3160)--(-0.9909,0.1901)--(-0.9990,0.06346)--(-0.9990,-0.06346)--(-0.9909,-0.1901)--(-0.9749,-0.3160)--(-0.9509,-0.4406)--(-0.9190,-0.5635)--(-0.8794,-0.6840)--(-0.8322,-0.8019)--(-0.7777,-0.9165)--(-0.7160,-1.027)--(-0.6474,-1.134)--(-0.5721,-1.236)--(-0.4905,-1.334)--(-0.4030,-1.425)--(-0.3097,-1.512)--(-0.2112,-1.592)--(-0.1078,-1.665)--(0,-1.732)--(0.1119,-1.792)--(0.2273,-1.845)--(0.3459,-1.890)--(0.4671,-1.928)--(0.5904,-1.958)--(0.7154,-1.980)--(0.8415,-1.994)--(0.9683,-2.000)--(1.095,-1.998)--(1.222,-1.988)--(1.347,-1.970)--(1.472,-1.944)--(1.594,-1.910)--(1.714,-1.868)--(1.831,-1.819)--(1.945,-1.763)--(2.054,-1.699)--(2.160,-1.629)--(2.261,-1.552)--(2.357,-1.469)--(2.447,-1.380)--(2.532,-1.286)--(2.611,-1.186)--(2.682,-1.081)--(2.748,-0.9724)--(2.806,-0.8596)--(2.857,-0.7433)--(2.900,-0.6241)--(2.936,-0.5023)--(2.964,-0.3785)--(2.984,-0.2532)--(2.996,-0.1268)--(3.000,0);
 
 \draw [] (5.000,0)--(4.994,0.1903)--(4.976,0.3798)--(4.946,0.5678)--(4.904,0.7534)--(4.850,0.9361)--(4.785,1.115)--(4.709,1.289)--(4.622,1.459)--(4.524,1.622)--(4.416,1.779)--(4.298,1.928)--(4.171,2.070)--(4.036,2.204)--(3.892,2.328)--(3.740,2.444)--(3.582,2.549)--(3.417,2.644)--(3.246,2.729)--(3.071,2.802)--(2.891,2.865)--(2.707,2.915)--(2.521,2.954)--(2.333,2.982)--(2.143,2.997)--(1.952,3.000)--(1.762,2.991)--(1.573,2.969)--(1.386,2.936)--(1.201,2.892)--(1.019,2.835)--(0.8410,2.767)--(0.6678,2.688)--(0.5000,2.598)--(0.3382,2.498)--(0.1832,2.387)--(0.03542,2.267)--(-0.1044,2.138)--(-0.2358,2.000)--(-0.3582,1.854)--(-0.4710,1.701)--(-0.5740,1.541)--(-0.6665,1.375)--(-0.7483,1.203)--(-0.8191,1.026)--(-0.8785,0.8452)--(-0.9263,0.6609)--(-0.9623,0.4740)--(-0.9864,0.2852)--(-0.9985,0.09518)--(-0.9985,-0.09518)--(-0.9864,-0.2852)--(-0.9623,-0.4740)--(-0.9263,-0.6609)--(-0.8785,-0.8452)--(-0.8191,-1.026)--(-0.7483,-1.203)--(-0.6665,-1.375)--(-0.5740,-1.541)--(-0.4710,-1.701)--(-0.3582,-1.854)--(-0.2358,-2.000)--(-0.1044,-2.138)--(0.03542,-2.267)--(0.1832,-2.387)--(0.3382,-2.498)--(0.5000,-2.598)--(0.6678,-2.688)--(0.8410,-2.767)--(1.019,-2.835)--(1.201,-2.892)--(1.386,-2.936)--(1.573,-2.969)--(1.762,-2.991)--(1.952,-3.000)--(2.143,-2.997)--(2.333,-2.982)--(2.521,-2.954)--(2.707,-2.915)--(2.891,-2.865)--(3.071,-2.802)--(3.246,-2.729)--(3.417,-2.644)--(3.582,-2.549)--(3.740,-2.444)--(3.892,-2.328)--(4.036,-2.204)--(4.171,-2.070)--(4.298,-1.928)--(4.416,-1.779)--(4.524,-1.622)--(4.622,-1.459)--(4.709,-1.289)--(4.785,-1.115)--(4.850,-0.9361)--(4.904,-0.7534)--(4.946,-0.5678)--(4.976,-0.3798)--(4.994,-0.1903)--(5.000,0);
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.327046523,0.2955320229) node {\( \lambda_1\)};
-\draw []  (2.000000000,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.672953477,1.118681539) node {\( \lambda_2\)};
-\draw []  (2.000000000,-1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.672953477,-1.118681539) node {\( \lambda_3\)};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-1.3270,0.29553) node {\( \lambda_1\)};
+\draw []  (2.0000,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.6730,1.1187) node {\( \lambda_2\)};
+\draw []  (2.0000,-1.4142) node [rotate=0] {$\bullet$};
+\draw (1.6730,-1.1187) node {\( \lambda_3\)};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall b/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall
index dfb3ebc29..86b9be42e 100644
--- a/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall
+++ b/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (8.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (6.000,0)--(5.996,0.1268)--(5.984,0.2532)--(5.964,0.3785)--(5.936,0.5023)--(5.900,0.6241)--(5.857,0.7433)--(5.806,0.8596)--(5.748,0.9724)--(5.682,1.081)--(5.611,1.186)--(5.532,1.286)--(5.447,1.380)--(5.357,1.469)--(5.261,1.552)--(5.160,1.629)--(5.054,1.699)--(4.945,1.763)--(4.831,1.819)--(4.714,1.868)--(4.594,1.910)--(4.471,1.944)--(4.347,1.970)--(4.222,1.988)--(4.095,1.998)--(3.968,2.000)--(3.841,1.994)--(3.715,1.980)--(3.590,1.958)--(3.467,1.928)--(3.346,1.890)--(3.227,1.845)--(3.112,1.792)--(3.000,1.732)--(2.892,1.665)--(2.789,1.592)--(2.690,1.512)--(2.597,1.425)--(2.509,1.334)--(2.428,1.236)--(2.353,1.134)--(2.284,1.027)--(2.222,0.9165)--(2.168,0.8019)--(2.121,0.6840)--(2.081,0.5635)--(2.049,0.4406)--(2.025,0.3160)--(2.009,0.1901)--(2.001,0.06346)--(2.001,-0.06346)--(2.009,-0.1901)--(2.025,-0.3160)--(2.049,-0.4406)--(2.081,-0.5635)--(2.121,-0.6840)--(2.168,-0.8019)--(2.222,-0.9165)--(2.284,-1.027)--(2.353,-1.134)--(2.428,-1.236)--(2.509,-1.334)--(2.597,-1.425)--(2.690,-1.512)--(2.789,-1.592)--(2.892,-1.665)--(3.000,-1.732)--(3.112,-1.792)--(3.227,-1.845)--(3.346,-1.890)--(3.467,-1.928)--(3.590,-1.958)--(3.715,-1.980)--(3.841,-1.994)--(3.968,-2.000)--(4.095,-1.998)--(4.222,-1.988)--(4.347,-1.970)--(4.471,-1.944)--(4.594,-1.910)--(4.714,-1.868)--(4.831,-1.819)--(4.945,-1.763)--(5.054,-1.699)--(5.160,-1.629)--(5.261,-1.552)--(5.357,-1.469)--(5.447,-1.380)--(5.532,-1.286)--(5.611,-1.186)--(5.682,-1.081)--(5.748,-0.9724)--(5.806,-0.8596)--(5.857,-0.7433)--(5.900,-0.6241)--(5.936,-0.5023)--(5.964,-0.3785)--(5.984,-0.2532)--(5.996,-0.1268)--(6.000,0);
@@ -104,23 +104,23 @@
 \draw [] (3.000,0)--(2.998,0.06342)--(2.992,0.1266)--(2.982,0.1893)--(2.968,0.2511)--(2.950,0.3120)--(2.928,0.3717)--(2.903,0.4298)--(2.874,0.4862)--(2.841,0.5406)--(2.805,0.5929)--(2.766,0.6428)--(2.724,0.6901)--(2.678,0.7346)--(2.631,0.7761)--(2.580,0.8146)--(2.527,0.8497)--(2.472,0.8815)--(2.415,0.9096)--(2.357,0.9342)--(2.297,0.9549)--(2.236,0.9718)--(2.174,0.9848)--(2.111,0.9938)--(2.048,0.9989)--(1.984,0.9999)--(1.921,0.9969)--(1.858,0.9898)--(1.795,0.9788)--(1.734,0.9638)--(1.673,0.9450)--(1.614,0.9224)--(1.556,0.8960)--(1.500,0.8660)--(1.446,0.8326)--(1.394,0.7958)--(1.345,0.7558)--(1.299,0.7127)--(1.255,0.6668)--(1.214,0.6182)--(1.176,0.5671)--(1.142,0.5137)--(1.111,0.4582)--(1.084,0.4009)--(1.060,0.3420)--(1.041,0.2817)--(1.025,0.2203)--(1.013,0.1580)--(1.005,0.09506)--(1.001,0.03173)--(1.001,-0.03173)--(1.005,-0.09506)--(1.013,-0.1580)--(1.025,-0.2203)--(1.041,-0.2817)--(1.060,-0.3420)--(1.084,-0.4009)--(1.111,-0.4582)--(1.142,-0.5137)--(1.176,-0.5671)--(1.214,-0.6182)--(1.255,-0.6668)--(1.299,-0.7127)--(1.345,-0.7558)--(1.394,-0.7958)--(1.446,-0.8326)--(1.500,-0.8660)--(1.556,-0.8960)--(1.614,-0.9224)--(1.673,-0.9450)--(1.734,-0.9638)--(1.795,-0.9788)--(1.858,-0.9898)--(1.921,-0.9969)--(1.984,-0.9999)--(2.048,-0.9989)--(2.111,-0.9938)--(2.174,-0.9848)--(2.236,-0.9718)--(2.297,-0.9549)--(2.357,-0.9342)--(2.415,-0.9096)--(2.472,-0.8815)--(2.527,-0.8497)--(2.580,-0.8146)--(2.631,-0.7761)--(2.678,-0.7346)--(2.724,-0.6901)--(2.766,-0.6428)--(2.805,-0.5929)--(2.841,-0.5406)--(2.874,-0.4862)--(2.903,-0.4298)--(2.928,-0.3717)--(2.950,-0.3120)--(2.968,-0.2511)--(2.982,-0.1893)--(2.992,-0.1266)--(2.998,-0.06342)--(3.000,0);
 
 \draw [] (8.000,0)--(7.996,0.1268)--(7.984,0.2532)--(7.964,0.3785)--(7.936,0.5023)--(7.900,0.6241)--(7.857,0.7433)--(7.806,0.8596)--(7.748,0.9724)--(7.682,1.081)--(7.611,1.186)--(7.532,1.286)--(7.447,1.380)--(7.357,1.469)--(7.261,1.552)--(7.160,1.629)--(7.054,1.699)--(6.945,1.763)--(6.831,1.819)--(6.714,1.868)--(6.594,1.910)--(6.471,1.944)--(6.347,1.970)--(6.222,1.988)--(6.095,1.998)--(5.968,2.000)--(5.841,1.994)--(5.715,1.980)--(5.590,1.958)--(5.467,1.928)--(5.346,1.890)--(5.227,1.845)--(5.112,1.792)--(5.000,1.732)--(4.892,1.665)--(4.789,1.592)--(4.690,1.512)--(4.597,1.425)--(4.509,1.334)--(4.428,1.236)--(4.353,1.134)--(4.284,1.027)--(4.222,0.9165)--(4.168,0.8019)--(4.121,0.6840)--(4.081,0.5635)--(4.049,0.4406)--(4.025,0.3160)--(4.009,0.1901)--(4.001,0.06346)--(4.001,-0.06346)--(4.009,-0.1901)--(4.025,-0.3160)--(4.049,-0.4406)--(4.081,-0.5635)--(4.121,-0.6840)--(4.168,-0.8019)--(4.222,-0.9165)--(4.284,-1.027)--(4.353,-1.134)--(4.428,-1.236)--(4.509,-1.334)--(4.597,-1.425)--(4.690,-1.512)--(4.789,-1.592)--(4.892,-1.665)--(5.000,-1.732)--(5.112,-1.792)--(5.227,-1.845)--(5.346,-1.890)--(5.467,-1.928)--(5.590,-1.958)--(5.715,-1.980)--(5.841,-1.994)--(5.968,-2.000)--(6.095,-1.998)--(6.222,-1.988)--(6.347,-1.970)--(6.471,-1.944)--(6.594,-1.910)--(6.714,-1.868)--(6.831,-1.819)--(6.945,-1.763)--(7.054,-1.699)--(7.160,-1.629)--(7.261,-1.552)--(7.357,-1.469)--(7.447,-1.380)--(7.532,-1.286)--(7.611,-1.186)--(7.682,-1.081)--(7.748,-0.9724)--(7.806,-0.8596)--(7.857,-0.7433)--(7.900,-0.6241)--(7.936,-0.5023)--(7.964,-0.3785)--(7.984,-0.2532)--(7.996,-0.1268)--(8.000,0);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.672953477,0.2955320229) node {\( \lambda_1\)};
-\draw []  (5.000000000,1.732050808) node [rotate=0] {$\bullet$};
-\draw (5.000000000,2.086161474) node {\( \lambda_2\)};
-\draw []  (5.000000000,-1.732050808) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-2.086161474) node {\( \lambda_3\)};
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.6730,0.29553) node {\( \lambda_1\)};
+\draw []  (5.0000,1.7320) node [rotate=0] {$\bullet$};
+\draw (5.0000,2.0862) node {\( \lambda_2\)};
+\draw []  (5.0000,-1.7320) node [rotate=0] {$\bullet$};
+\draw (5.0000,-2.0862) node {\( \lambda_3\)};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 3 $};
+\draw (6.0000,-0.31492) node {$ 3 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 4 $};
+\draw (8.0000,-0.31492) node {$ 4 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DTIYKkP.pstricks.recall b/src_phystricks/Fig_DTIYKkP.pstricks.recall
index a87182c34..2166b0fcf 100644
--- a/src_phystricks/Fig_DTIYKkP.pstricks.recall
+++ b/src_phystricks/Fig_DTIYKkP.pstricks.recall
@@ -83,12 +83,12 @@
 %DEFAULT
 
 \draw [color=blue] (-5.000,0)--(-4.970,0.1206)--(-4.939,0.2400)--(-4.909,0.3581)--(-4.879,0.4751)--(-4.849,0.5908)--(-4.818,0.7052)--(-4.788,0.8185)--(-4.758,0.9305)--(-4.727,1.041)--(-4.697,1.151)--(-4.667,1.259)--(-4.636,1.366)--(-4.606,1.472)--(-4.576,1.577)--(-4.545,1.680)--(-4.515,1.783)--(-4.485,1.884)--(-4.455,1.983)--(-4.424,2.082)--(-4.394,2.179)--(-4.364,2.275)--(-4.333,2.370)--(-4.303,2.464)--(-4.273,2.556)--(-4.242,2.648)--(-4.212,2.738)--(-4.182,2.826)--(-4.151,2.914)--(-4.121,3.000)--(-4.091,3.085)--(-4.061,3.169)--(-4.030,3.252)--(-4.000,3.333)--(-3.970,3.414)--(-3.939,3.492)--(-3.909,3.570)--(-3.879,3.647)--(-3.848,3.722)--(-3.818,3.796)--(-3.788,3.869)--(-3.758,3.941)--(-3.727,4.011)--(-3.697,4.080)--(-3.667,4.148)--(-3.636,4.215)--(-3.606,4.280)--(-3.576,4.345)--(-3.545,4.408)--(-3.515,4.470)--(-3.485,4.530)--(-3.455,4.590)--(-3.424,4.648)--(-3.394,4.705)--(-3.364,4.760)--(-3.333,4.815)--(-3.303,4.868)--(-3.273,4.920)--(-3.242,4.971)--(-3.212,5.021)--(-3.182,5.069)--(-3.152,5.116)--(-3.121,5.162)--(-3.091,5.207)--(-3.061,5.250)--(-3.030,5.292)--(-3.000,5.333)--(-2.970,5.373)--(-2.939,5.412)--(-2.909,5.449)--(-2.879,5.485)--(-2.848,5.520)--(-2.818,5.554)--(-2.788,5.586)--(-2.758,5.617)--(-2.727,5.647)--(-2.697,5.676)--(-2.667,5.704)--(-2.636,5.730)--(-2.606,5.755)--(-2.576,5.779)--(-2.545,5.802)--(-2.515,5.823)--(-2.485,5.843)--(-2.455,5.862)--(-2.424,5.880)--(-2.394,5.897)--(-2.364,5.912)--(-2.333,5.926)--(-2.303,5.939)--(-2.273,5.950)--(-2.242,5.961)--(-2.212,5.970)--(-2.182,5.978)--(-2.152,5.985)--(-2.121,5.990)--(-2.091,5.995)--(-2.061,5.998)--(-2.030,5.999)--(-2.000,6.000);
-\draw [color=brown]  (-4.000000000,3.333333333) node [rotate=0] {$\bullet$};
-\draw (-4.727456667,3.635788333) node {$o=[\mtu]$};
-\draw (-1.197814000,6.000000000) node {$[\SO(2)]$};
-\draw [color=cyan,->,>=latex] (-4.500000000,1.833333333) -- (-2.500000000,2.333333333);
-\draw (-1.441613144,1.979900977) node {$[ e^{sE(w)} e^{xq_0}]$};
-\draw (-5.102939752,2.118687484) node {$[ e^{xq_0}]$};
+\draw [color=brown]  (-4.0000,3.3333) node [rotate=0] {$\bullet$};
+\draw (-4.7275,3.6358) node {$o=[\mtu]$};
+\draw (-1.1978,6.0000) node {$[\SO(2)]$};
+\draw [color=cyan,->,>=latex] (-4.5000,1.8333) -- (-2.5000,2.3333);
+\draw (-1.4416,1.9799) node {$[ e^{sE(w)} e^{xq_0}]$};
+\draw (-5.1029,2.1187) node {$[ e^{xq_0}]$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall b/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
index 239b992f5..fe50e66aa 100644
--- a/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
+++ b/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
@@ -230,7 +230,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.023619130) -- (0,2.170406053);
+\draw [,->,>=latex] (0,-3.023619131) -- (0,2.170406053);
 %DEFAULT
 
 \draw [color=blue] (0.01000,-2.524)--(0.05030,-1.393)--(0.09061,-0.9809)--(0.1309,-0.7233)--(0.1712,-0.5354)--(0.2115,-0.3874)--(0.2518,-0.2653)--(0.2921,-0.1614)--(0.3324,-0.07094)--(0.3727,0.009164)--(0.4130,0.08104)--(0.4533,0.1462)--(0.4936,0.2058)--(0.5339,0.2608)--(0.5742,0.3117)--(0.6145,0.3592)--(0.6548,0.4037)--(0.6952,0.4455)--(0.7355,0.4849)--(0.7758,0.5223)--(0.8161,0.5577)--(0.8564,0.5915)--(0.8967,0.6236)--(0.9370,0.6544)--(0.9773,0.6839)--(1.018,0.7122)--(1.058,0.7394)--(1.098,0.7656)--(1.138,0.7908)--(1.179,0.8151)--(1.219,0.8387)--(1.259,0.8614)--(1.300,0.8835)--(1.340,0.9049)--(1.380,0.9256)--(1.421,0.9458)--(1.461,0.9653)--(1.501,0.9844)--(1.542,1.003)--(1.582,1.021)--(1.622,1.039)--(1.662,1.056)--(1.703,1.073)--(1.743,1.089)--(1.783,1.105)--(1.824,1.121)--(1.864,1.136)--(1.904,1.151)--(1.945,1.166)--(1.985,1.180)--(2.025,1.194)--(2.065,1.208)--(2.106,1.221)--(2.146,1.235)--(2.186,1.248)--(2.227,1.260)--(2.267,1.273)--(2.307,1.285)--(2.348,1.297)--(2.388,1.309)--(2.428,1.321)--(2.468,1.333)--(2.509,1.344)--(2.549,1.355)--(2.589,1.366)--(2.630,1.377)--(2.670,1.387)--(2.710,1.398)--(2.751,1.408)--(2.791,1.418)--(2.831,1.428)--(2.872,1.438)--(2.912,1.448)--(2.952,1.458)--(2.992,1.467)--(3.033,1.477)--(3.073,1.486)--(3.113,1.495)--(3.154,1.504)--(3.194,1.513)--(3.234,1.522)--(3.275,1.530)--(3.315,1.539)--(3.355,1.547)--(3.395,1.556)--(3.436,1.564)--(3.476,1.572)--(3.516,1.580)--(3.557,1.588)--(3.597,1.596)--(3.637,1.604)--(3.678,1.612)--(3.718,1.619)--(3.758,1.627)--(3.798,1.634)--(3.839,1.642)--(3.879,1.649)--(3.919,1.656)--(3.960,1.663)--(4.000,1.670);
diff --git a/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall b/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall
index 7914ba074..0ed1188bd 100644
--- a/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall
+++ b/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall
@@ -103,46 +103,46 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-2.0000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
 \draw [color=blue,style=dashed] (3.00,1.00) -- (3.00,0);
 \draw [color=blue,style=dashed] (3.00,1.00) -- (0,1.00);
-\draw (3.500387996,1.214077777) node {$(3,1)$};
-\draw [color=blue,->,>=latex] (0,0) -- (3.000000000,1.000000000);
+\draw (3.5004,1.2141) node {$(3,1)$};
+\draw [color=blue,->,>=latex] (0,0) -- (3.0000,1.0000);
 \draw [color=green,style=dashed] (-1.50,-2.50) -- (-1.50,0);
 \draw [color=green,style=dashed] (-1.50,-2.50) -- (0,-2.50);
-\draw (-2.524677076,-2.768204293) node {$(-1.5,-2.5)$};
-\draw [color=green,->,>=latex] (0,0) -- (-1.500000000,-2.500000000);
+\draw (-2.5247,-2.7682) node {$(-1.5,-2.5)$};
+\draw [color=green,->,>=latex] (0,0) -- (-1.5000,-2.5000);
 \draw [color=brown,style=dashed] (-1.00,2.50) -- (-1.00,0);
 \draw [color=brown,style=dashed] (-1.00,2.50) -- (0,2.50);
-\draw (-1.726512568,2.775302669) node {$(-1,2.5)$};
-\draw [color=brown,->,>=latex] (0,0) -- (-1.000000000,2.500000000);
+\draw (-1.7265,2.7753) node {$(-1,2.5)$};
+\draw [color=brown,->,>=latex] (0,0) -- (-1.0000,2.5000);
 \draw [color=cyan,style=dashed] (1.50,-1.00) -- (1.50,0);
 \draw [color=cyan,style=dashed] (1.50,-1.00) -- (0,-1.00);
-\draw (2.272578529,-1.237925020) node {$(1.5,-1)$};
-\draw [color=cyan,->,>=latex] (0,0) -- (1.500000000,-1.000000000);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (2.2726,-1.2379) node {$(1.5,-1)$};
+\draw [color=cyan,->,>=latex] (0,0) -- (1.5000,-1.0000);
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DessinLim.pstricks.recall b/src_phystricks/Fig_DessinLim.pstricks.recall
index 5d8fce5cf..1b95a76d9 100644
--- a/src_phystricks/Fig_DessinLim.pstricks.recall
+++ b/src_phystricks/Fig_DessinLim.pstricks.recall
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.800000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.800000000);
+\draw [,->,>=latex] (-0.5000000000,0) -- (2.800000001,0);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.800000001);
 %DEFAULT
 
 \draw [color=blue] (2.300,0)--(2.300,0.03649)--(2.299,0.07297)--(2.297,0.1094)--(2.295,0.1459)--(2.293,0.1823)--(2.290,0.2186)--(2.286,0.2549)--(2.281,0.2912)--(2.277,0.3273)--(2.271,0.3634)--(2.265,0.3994)--(2.258,0.4353)--(2.251,0.4711)--(2.243,0.5067)--(2.235,0.5422)--(2.226,0.5776)--(2.217,0.6129)--(2.207,0.6480)--(2.196,0.6829)--(2.185,0.7177)--(2.173,0.7523)--(2.161,0.7866)--(2.149,0.8208)--(2.135,0.8548)--(2.121,0.8886)--(2.107,0.9221)--(2.092,0.9555)--(2.077,0.9885)--(2.061,1.021)--(2.044,1.054)--(2.027,1.086)--(2.010,1.118)--(1.992,1.150)--(1.973,1.181)--(1.954,1.213)--(1.935,1.243)--(1.915,1.274)--(1.894,1.304)--(1.874,1.334)--(1.852,1.364)--(1.830,1.393)--(1.808,1.422)--(1.785,1.450)--(1.762,1.478)--(1.738,1.506)--(1.714,1.534)--(1.690,1.561)--(1.665,1.587)--(1.639,1.613)--(1.613,1.639)--(1.587,1.665)--(1.561,1.690)--(1.534,1.714)--(1.506,1.738)--(1.478,1.762)--(1.450,1.785)--(1.422,1.808)--(1.393,1.830)--(1.364,1.852)--(1.334,1.874)--(1.304,1.894)--(1.274,1.915)--(1.243,1.935)--(1.213,1.954)--(1.181,1.973)--(1.150,1.992)--(1.118,2.010)--(1.086,2.027)--(1.054,2.044)--(1.021,2.061)--(0.9885,2.077)--(0.9555,2.092)--(0.9221,2.107)--(0.8886,2.121)--(0.8548,2.135)--(0.8208,2.149)--(0.7866,2.161)--(0.7523,2.173)--(0.7177,2.185)--(0.6829,2.196)--(0.6480,2.207)--(0.6129,2.217)--(0.5776,2.226)--(0.5422,2.235)--(0.5067,2.243)--(0.4711,2.251)--(0.4353,2.258)--(0.3994,2.265)--(0.3634,2.271)--(0.3273,2.277)--(0.2912,2.281)--(0.2549,2.286)--(0.2186,2.290)--(0.1823,2.293)--(0.1459,2.295)--(0.1094,2.297)--(0.07297,2.299)--(0.03649,2.300)--(0,2.300);
diff --git a/src_phystricks/Fig_Differentielle.pstricks.recall b/src_phystricks/Fig_Differentielle.pstricks.recall
index b865a7791..c47af80f4 100644
--- a/src_phystricks/Fig_Differentielle.pstricks.recall
+++ b/src_phystricks/Fig_Differentielle.pstricks.recall
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.700000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.700000000);
+\draw [,->,>=latex] (-0.5000000000,0) -- (4.700000003,0);
+\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.700000003);
 %DEFAULT
 \draw [style=dotted] (2.00,2.00) -- (4.00,2.00);
 \draw [style=dotted] (4.00,2.00) -- (4.00,4.00);
diff --git a/src_phystricks/Fig_DisqueConv.pstricks.recall b/src_phystricks/Fig_DisqueConv.pstricks.recall
index b5030e05d..f268bbcad 100644
--- a/src_phystricks/Fig_DisqueConv.pstricks.recall
+++ b/src_phystricks/Fig_DisqueConv.pstricks.recall
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
-\draw []  (2.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (1.765251155,1.823338322) node {$z_0$};
+\draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (1.7653,1.8233) node {$z_0$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/src_phystricks/Fig_DistanceEuclide.pstricks.recall b/src_phystricks/Fig_DistanceEuclide.pstricks.recall
index 2c172a3a5..de1bd21a0 100644
--- a/src_phystricks/Fig_DistanceEuclide.pstricks.recall
+++ b/src_phystricks/Fig_DistanceEuclide.pstricks.recall
@@ -91,31 +91,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
-\draw []  (1.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw (1.000000000,4.490142000) node {$(A_x,A_y)$};
-\draw []  (3.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.971096333,1.000000000) node {$(B_x,B_y)$};
-\draw []  (3.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw (3.355622034,4.336840034) node {$C$};
+\draw []  (1.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,4.4901) node {$(A_x,A_y)$};
+\draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.9711,1.0000) node {$(B_x,B_y)$};
+\draw []  (3.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (3.3556,4.3368) node {$C$};
 \draw [] (1.00,4.00) -- (3.00,1.00);
 \draw [] (1.00,4.00) -- (3.00,4.00);
 \draw [] (3.00,1.00) -- (3.00,4.00);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall b/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall
index 86046b293..aad544245 100644
--- a/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall
+++ b/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2644441667) node {\( 1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.26444) node {\( 1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall b/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall
index 1f78404fd..de90bcf5a 100644
--- a/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall
+++ b/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.2644441667) node {\( 1\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
+\draw (0,0.26444) node {\( 1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
index e519f6799..f7088ee5d 100644
--- a/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
+++ b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
@@ -65,7 +65,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.200000000) -- (0,1.199647580);
 %DEFAULT
 
@@ -151,7 +151,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
 %DEFAULT
 
@@ -239,7 +239,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (2.856194490,0);
+\draw [,->,>=latex] (-2.070796328,0) -- (2.856194490,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,1.499748271);
 %DEFAULT
 
@@ -329,7 +329,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
 %DEFAULT
 
@@ -415,7 +415,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.199998598) -- (0,1.199904439);
 %DEFAULT
 
@@ -502,7 +502,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.599557429,0) -- (4.898229715,0);
+\draw [,->,>=latex] (-1.599557429,0) -- (4.898229717,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
 %DEFAULT
 
@@ -588,7 +588,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (3.798672286,0);
+\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
 \draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
 %DEFAULT
 
diff --git a/src_phystricks/Fig_ExoCUd.pstricks.recall b/src_phystricks/Fig_ExoCUd.pstricks.recall
index c0a8dea4b..271934b9a 100644
--- a/src_phystricks/Fig_ExoCUd.pstricks.recall
+++ b/src_phystricks/Fig_ExoCUd.pstricks.recall
@@ -75,22 +75,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=red] (-1.000,4.000)--(-0.9798,3.920)--(-0.9596,3.840)--(-0.9394,3.761)--(-0.9192,3.683)--(-0.8990,3.606)--(-0.8788,3.530)--(-0.8586,3.454)--(-0.8384,3.380)--(-0.8182,3.306)--(-0.7980,3.233)--(-0.7778,3.160)--(-0.7576,3.089)--(-0.7374,3.018)--(-0.7172,2.949)--(-0.6970,2.880)--(-0.6768,2.812)--(-0.6566,2.744)--(-0.6364,2.678)--(-0.6162,2.612)--(-0.5960,2.547)--(-0.5758,2.483)--(-0.5556,2.420)--(-0.5354,2.357)--(-0.5152,2.296)--(-0.4949,2.235)--(-0.4747,2.175)--(-0.4545,2.116)--(-0.4343,2.057)--(-0.4141,2.000)--(-0.3939,1.943)--(-0.3737,1.887)--(-0.3535,1.832)--(-0.3333,1.778)--(-0.3131,1.724)--(-0.2929,1.672)--(-0.2727,1.620)--(-0.2525,1.569)--(-0.2323,1.519)--(-0.2121,1.469)--(-0.1919,1.421)--(-0.1717,1.373)--(-0.1515,1.326)--(-0.1313,1.280)--(-0.1111,1.235)--(-0.09091,1.190)--(-0.07071,1.146)--(-0.05051,1.104)--(-0.03030,1.062)--(-0.01010,1.020)--(0.01010,0.9799)--(0.03030,0.9403)--(0.05051,0.9015)--(0.07071,0.8636)--(0.09091,0.8264)--(0.1111,0.7901)--(0.1313,0.7546)--(0.1515,0.7199)--(0.1717,0.6861)--(0.1919,0.6530)--(0.2121,0.6208)--(0.2323,0.5893)--(0.2525,0.5587)--(0.2727,0.5289)--(0.2929,0.5000)--(0.3131,0.4718)--(0.3333,0.4444)--(0.3535,0.4179)--(0.3737,0.3922)--(0.3939,0.3673)--(0.4141,0.3432)--(0.4343,0.3200)--(0.4545,0.2975)--(0.4747,0.2759)--(0.4949,0.2551)--(0.5152,0.2351)--(0.5354,0.2159)--(0.5556,0.1975)--(0.5758,0.1800)--(0.5960,0.1632)--(0.6162,0.1473)--(0.6364,0.1322)--(0.6566,0.1179)--(0.6768,0.1045)--(0.6970,0.09183)--(0.7172,0.07999)--(0.7374,0.06897)--(0.7576,0.05877)--(0.7778,0.04938)--(0.7980,0.04081)--(0.8182,0.03306)--(0.8384,0.02612)--(0.8586,0.02000)--(0.8788,0.01469)--(0.8990,0.01020)--(0.9192,0.006530)--(0.9394,0.003673)--(0.9596,0.001632)--(0.9798,0)--(1.000,0);
 
 \draw [color=blue] (1.000,0)--(1.020,0)--(1.040,0.001632)--(1.061,0.003673)--(1.081,0.006530)--(1.101,0.01020)--(1.121,0.01469)--(1.141,0.02000)--(1.162,0.02612)--(1.182,0.03306)--(1.202,0.04081)--(1.222,0.04938)--(1.242,0.05877)--(1.263,0.06897)--(1.283,0.07999)--(1.303,0.09183)--(1.323,0.1045)--(1.343,0.1179)--(1.364,0.1322)--(1.384,0.1473)--(1.404,0.1632)--(1.424,0.1800)--(1.444,0.1975)--(1.465,0.2159)--(1.485,0.2351)--(1.505,0.2551)--(1.525,0.2759)--(1.545,0.2975)--(1.566,0.3200)--(1.586,0.3432)--(1.606,0.3673)--(1.626,0.3922)--(1.646,0.4179)--(1.667,0.4444)--(1.687,0.4718)--(1.707,0.5000)--(1.727,0.5289)--(1.747,0.5587)--(1.768,0.5893)--(1.788,0.6208)--(1.808,0.6530)--(1.828,0.6861)--(1.848,0.7199)--(1.869,0.7546)--(1.889,0.7901)--(1.909,0.8264)--(1.929,0.8636)--(1.949,0.9015)--(1.970,0.9403)--(1.990,0.9799)--(2.010,1.020)--(2.030,1.062)--(2.051,1.104)--(2.071,1.146)--(2.091,1.190)--(2.111,1.235)--(2.131,1.280)--(2.152,1.326)--(2.172,1.373)--(2.192,1.421)--(2.212,1.469)--(2.232,1.519)--(2.253,1.569)--(2.273,1.620)--(2.293,1.672)--(2.313,1.724)--(2.333,1.778)--(2.354,1.832)--(2.374,1.887)--(2.394,1.943)--(2.414,2.000)--(2.434,2.057)--(2.455,2.116)--(2.475,2.175)--(2.495,2.235)--(2.515,2.296)--(2.535,2.357)--(2.556,2.420)--(2.576,2.483)--(2.596,2.547)--(2.616,2.612)--(2.636,2.678)--(2.657,2.744)--(2.677,2.812)--(2.697,2.880)--(2.717,2.949)--(2.737,3.018)--(2.758,3.089)--(2.778,3.160)--(2.798,3.233)--(2.818,3.306)--(2.838,3.380)--(2.859,3.454)--(2.879,3.530)--(2.899,3.606)--(2.919,3.683)--(2.939,3.761)--(2.960,3.840)--(2.980,3.920)--(3.000,4.000);
 \draw [color=gray,style=dashed] (1.00,-0.500) -- (1.00,4.00);
-\draw []  (0,2.560000000) node [rotate=0] {$\bullet$};
-\draw (0.3081560344,2.886194201) node {$y$};
-\draw []  (2.600000000,2.560000000) node [rotate=0] {$\bullet$};
-\draw []  (-0.6000000000,2.560000000) node [rotate=0] {$\bullet$};
-\draw []  (2.600000000,0) node [rotate=0] {$\bullet$};
-\draw (2.600000000,-0.4191818333) node {$x_+$};
-\draw []  (-0.6000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.6000000000,-0.4191818333) node {$x_-$};
+\draw []  (0,2.5600) node [rotate=0] {$\bullet$};
+\draw (0.30816,2.8862) node {$y$};
+\draw []  (2.6000,2.5600) node [rotate=0] {$\bullet$};
+\draw []  (-0.60000,2.5600) node [rotate=0] {$\bullet$};
+\draw []  (2.6000,0) node [rotate=0] {$\bullet$};
+\draw (2.6000,-0.41918) node {$x_+$};
+\draw []  (-0.60000,0) node [rotate=0] {$\bullet$};
+\draw (-0.60000,-0.41918) node {$x_-$};
 \draw [style=dashed] (2.60,2.56) -- (2.60,0);
 \draw [style=dashed] (-0.600,2.56) -- (-0.600,0);
 \draw [style=dotted] (2.60,2.56) -- (-0.600,2.56);
diff --git a/src_phystricks/Fig_ExoParamCD.pstricks.recall b/src_phystricks/Fig_ExoParamCD.pstricks.recall
index f3ed297b6..126669a8f 100644
--- a/src_phystricks/Fig_ExoParamCD.pstricks.recall
+++ b/src_phystricks/Fig_ExoParamCD.pstricks.recall
@@ -71,25 +71,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.190,0.285)--(0.380,0.568)--(0.568,0.845)--(0.753,1.11)--(0.936,1.37)--(1.11,1.62)--(1.29,1.85)--(1.46,2.07)--(1.62,2.27)--(1.78,2.44)--(1.93,2.60)--(2.07,2.73)--(2.20,2.83)--(2.33,2.92)--(2.44,2.97)--(2.55,3.00)--(2.64,3.00)--(2.73,2.97)--(2.80,2.92)--(2.86,2.83)--(2.92,2.73)--(2.95,2.60)--(2.98,2.44)--(3.00,2.27)--(3.00,2.07)--(2.99,1.85)--(2.97,1.62)--(2.94,1.37)--(2.89,1.11)--(2.83,0.845)--(2.77,0.568)--(2.69,0.285)--(2.60,0)--(2.50,-0.285)--(2.39,-0.568)--(2.27,-0.845)--(2.14,-1.11)--(2.00,-1.37)--(1.85,-1.62)--(1.70,-1.85)--(1.54,-2.07)--(1.37,-2.27)--(1.20,-2.44)--(1.03,-2.60)--(0.845,-2.73)--(0.661,-2.83)--(0.474,-2.92)--(0.285,-2.97)--(0.0952,-3.00)--(-0.0952,-3.00)--(-0.285,-2.97)--(-0.474,-2.92)--(-0.661,-2.83)--(-0.845,-2.73)--(-1.03,-2.60)--(-1.20,-2.44)--(-1.37,-2.27)--(-1.54,-2.07)--(-1.70,-1.85)--(-1.85,-1.62)--(-2.00,-1.37)--(-2.14,-1.11)--(-2.27,-0.845)--(-2.39,-0.568)--(-2.50,-0.285)--(-2.60,0)--(-2.69,0.285)--(-2.77,0.568)--(-2.83,0.845)--(-2.89,1.11)--(-2.94,1.37)--(-2.97,1.62)--(-2.99,1.85)--(-3.00,2.07)--(-3.00,2.27)--(-2.98,2.44)--(-2.95,2.60)--(-2.92,2.73)--(-2.86,2.83)--(-2.80,2.92)--(-2.73,2.97)--(-2.64,3.00)--(-2.55,3.00)--(-2.44,2.97)--(-2.33,2.92)--(-2.20,2.83)--(-2.07,2.73)--(-1.93,2.60)--(-1.78,2.44)--(-1.62,2.27)--(-1.46,2.07)--(-1.29,1.85)--(-1.11,1.62)--(-0.936,1.37)--(-0.753,1.11)--(-0.568,0.845)--(-0.380,0.568)--(-0.190,0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.01664100589,0.02496150883);
-\draw [,->,>=latex] (3.000000000,2.121320344) -- (3.000000000,2.091320344);
-\draw [,->,>=latex] (0,-3.000000000) -- (-0.03000000000,-3.000000000);
-\draw [,->,>=latex] (-3.000000000,2.121320344) -- (-3.000000000,2.151320344);
+\draw [,->,>=latex] (0,0) -- (0.016641,0.024962);
+\draw [,->,>=latex] (3.0000,2.1213) -- (3.0000,2.0913);
+\draw [,->,>=latex] (0,-3.0000) -- (-0.030000,-3.0000);
+\draw [,->,>=latex] (-3.0000,2.1213) -- (-3.0000,2.1513);
 \draw [color=red] (0,0)--(0.190,-0.285)--(0.380,-0.568)--(0.568,-0.845)--(0.753,-1.11)--(0.936,-1.37)--(1.11,-1.62)--(1.29,-1.85)--(1.46,-2.07)--(1.62,-2.27)--(1.78,-2.44)--(1.93,-2.60)--(2.07,-2.73)--(2.20,-2.83)--(2.33,-2.92)--(2.44,-2.97)--(2.55,-3.00)--(2.64,-3.00)--(2.73,-2.97)--(2.80,-2.92)--(2.86,-2.83)--(2.92,-2.73)--(2.95,-2.60)--(2.98,-2.44)--(3.00,-2.27)--(3.00,-2.07)--(2.99,-1.85)--(2.97,-1.62)--(2.94,-1.37)--(2.89,-1.11)--(2.83,-0.845)--(2.77,-0.568)--(2.69,-0.285)--(2.60,0)--(2.50,0.285)--(2.39,0.568)--(2.27,0.845)--(2.14,1.11)--(2.00,1.37)--(1.85,1.62)--(1.70,1.85)--(1.54,2.07)--(1.37,2.27)--(1.20,2.44)--(1.03,2.60)--(0.845,2.73)--(0.661,2.83)--(0.474,2.92)--(0.285,2.97)--(0.0952,3.00)--(-0.0952,3.00)--(-0.285,2.97)--(-0.474,2.92)--(-0.661,2.83)--(-0.845,2.73)--(-1.03,2.60)--(-1.20,2.44)--(-1.37,2.27)--(-1.54,2.07)--(-1.70,1.85)--(-1.85,1.62)--(-2.00,1.37)--(-2.14,1.11)--(-2.27,0.845)--(-2.39,0.568)--(-2.50,0.285)--(-2.60,0)--(-2.69,-0.285)--(-2.77,-0.568)--(-2.83,-0.845)--(-2.89,-1.11)--(-2.94,-1.37)--(-2.97,-1.62)--(-2.99,-1.85)--(-3.00,-2.07)--(-3.00,-2.27)--(-2.98,-2.44)--(-2.95,-2.60)--(-2.92,-2.73)--(-2.86,-2.83)--(-2.80,-2.92)--(-2.73,-2.97)--(-2.64,-3.00)--(-2.55,-3.00)--(-2.44,-2.97)--(-2.33,-2.92)--(-2.20,-2.83)--(-2.07,-2.73)--(-1.93,-2.60)--(-1.78,-2.44)--(-1.62,-2.27)--(-1.46,-2.07)--(-1.29,-1.85)--(-1.11,-1.62)--(-0.936,-1.37)--(-0.753,-1.11)--(-0.568,-0.845)--(-0.380,-0.568)--(-0.190,-0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.01664100589,-0.02496150883);
-\draw [,->,>=latex] (3.000000000,-2.121320344) -- (3.000000000,-2.091320344);
-\draw [,->,>=latex] (0,3.000000000) -- (-0.03000000000,3.000000000);
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw [,->,>=latex] (0,0) -- (0.016641,-0.024962);
+\draw [,->,>=latex] (3.0000,-2.1213) -- (3.0000,-2.0913);
+\draw [,->,>=latex] (0,3.0000) -- (-0.030000,3.0000);
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -1 $};
+\draw (-0.43316,-3.0000) node {$ -1 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall b/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall
index 48a9267a2..c9969ffe2 100644
--- a/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall
+++ b/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall
@@ -103,48 +103,48 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=blue] (-5.000,-0.2000)--(-4.952,-0.2019)--(-4.904,-0.2039)--(-4.856,-0.2059)--(-4.808,-0.2080)--(-4.760,-0.2101)--(-4.712,-0.2122)--(-4.664,-0.2144)--(-4.616,-0.2166)--(-4.568,-0.2189)--(-4.520,-0.2212)--(-4.472,-0.2236)--(-4.424,-0.2260)--(-4.376,-0.2285)--(-4.328,-0.2310)--(-4.280,-0.2336)--(-4.232,-0.2363)--(-4.184,-0.2390)--(-4.136,-0.2418)--(-4.088,-0.2446)--(-4.040,-0.2475)--(-3.992,-0.2505)--(-3.944,-0.2535)--(-3.896,-0.2566)--(-3.848,-0.2598)--(-3.801,-0.2631)--(-3.753,-0.2665)--(-3.705,-0.2699)--(-3.657,-0.2735)--(-3.609,-0.2771)--(-3.561,-0.2808)--(-3.513,-0.2847)--(-3.465,-0.2886)--(-3.417,-0.2927)--(-3.369,-0.2969)--(-3.321,-0.3011)--(-3.273,-0.3056)--(-3.225,-0.3101)--(-3.177,-0.3148)--(-3.129,-0.3196)--(-3.081,-0.3246)--(-3.033,-0.3297)--(-2.985,-0.3350)--(-2.937,-0.3405)--(-2.889,-0.3462)--(-2.841,-0.3520)--(-2.793,-0.3580)--(-2.745,-0.3643)--(-2.697,-0.3708)--(-2.649,-0.3775)--(-2.601,-0.3845)--(-2.553,-0.3917)--(-2.505,-0.3992)--(-2.457,-0.4070)--(-2.409,-0.4151)--(-2.361,-0.4235)--(-2.313,-0.4323)--(-2.265,-0.4415)--(-2.217,-0.4510)--(-2.169,-0.4610)--(-2.121,-0.4714)--(-2.073,-0.4823)--(-2.025,-0.4938)--(-1.977,-0.5057)--(-1.929,-0.5183)--(-1.881,-0.5315)--(-1.833,-0.5455)--(-1.785,-0.5601)--(-1.737,-0.5756)--(-1.689,-0.5919)--(-1.641,-0.6092)--(-1.593,-0.6276)--(-1.545,-0.6471)--(-1.497,-0.6678)--(-1.449,-0.6899)--(-1.402,-0.7135)--(-1.354,-0.7388)--(-1.306,-0.7660)--(-1.258,-0.7952)--(-1.210,-0.8267)--(-1.162,-0.8609)--(-1.114,-0.8980)--(-1.066,-0.9384)--(-1.018,-0.9826)--(-0.9697,-1.031)--(-0.9217,-1.085)--(-0.8737,-1.145)--(-0.8258,-1.211)--(-0.7778,-1.286)--(-0.7298,-1.370)--(-0.6818,-1.467)--(-0.6338,-1.578)--(-0.5859,-1.707)--(-0.5379,-1.859)--(-0.4899,-2.041)--(-0.4419,-2.263)--(-0.3939,-2.538)--(-0.3460,-2.891)--(-0.2980,-3.356)--(-0.2500,-4.000);
 
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-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
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-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_FnCosApprox.pstricks.recall b/src_phystricks/Fig_FnCosApprox.pstricks.recall
index 32f38a6a3..890159f3e 100644
--- a/src_phystricks/Fig_FnCosApprox.pstricks.recall
+++ b/src_phystricks/Fig_FnCosApprox.pstricks.recall
@@ -91,24 +91,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185307,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=blue] (0,2.000)--(0.06347,1.999)--(0.1269,1.996)--(0.1904,1.991)--(0.2539,1.984)--(0.3173,1.975)--(0.3808,1.964)--(0.4443,1.951)--(0.5077,1.936)--(0.5712,1.919)--(0.6347,1.900)--(0.6981,1.879)--(0.7616,1.857)--(0.8251,1.832)--(0.8885,1.806)--(0.9520,1.778)--(1.015,1.748)--(1.079,1.716)--(1.142,1.683)--(1.206,1.647)--(1.269,1.611)--(1.333,1.572)--(1.396,1.532)--(1.460,1.491)--(1.523,1.447)--(1.587,1.403)--(1.650,1.357)--(1.714,1.310)--(1.777,1.261)--(1.841,1.211)--(1.904,1.160)--(1.967,1.108)--(2.031,1.054)--(2.094,1.000)--(2.158,0.9445)--(2.221,0.8881)--(2.285,0.8308)--(2.348,0.7727)--(2.412,0.7138)--(2.475,0.6541)--(2.539,0.5938)--(2.602,0.5330)--(2.666,0.4715)--(2.729,0.4096)--(2.793,0.3473)--(2.856,0.2846)--(2.919,0.2217)--(2.983,0.1585)--(3.046,0.09516)--(3.110,0.03173)--(3.173,-0.03173)--(3.237,-0.09516)--(3.300,-0.1585)--(3.364,-0.2217)--(3.427,-0.2846)--(3.491,-0.3473)--(3.554,-0.4096)--(3.618,-0.4715)--(3.681,-0.5330)--(3.745,-0.5938)--(3.808,-0.6541)--(3.871,-0.7138)--(3.935,-0.7727)--(3.998,-0.8308)--(4.062,-0.8881)--(4.125,-0.9445)--(4.189,-1.000)--(4.252,-1.054)--(4.316,-1.108)--(4.379,-1.160)--(4.443,-1.211)--(4.506,-1.261)--(4.570,-1.310)--(4.633,-1.357)--(4.697,-1.403)--(4.760,-1.447)--(4.823,-1.491)--(4.887,-1.532)--(4.950,-1.572)--(5.014,-1.611)--(5.077,-1.647)--(5.141,-1.683)--(5.204,-1.716)--(5.268,-1.748)--(5.331,-1.778)--(5.395,-1.806)--(5.458,-1.832)--(5.522,-1.857)--(5.585,-1.879)--(5.648,-1.900)--(5.712,-1.919)--(5.775,-1.936)--(5.839,-1.951)--(5.902,-1.964)--(5.966,-1.975)--(6.029,-1.984)--(6.093,-1.991)--(6.156,-1.996)--(6.220,-1.999)--(6.283,-2.000);
-\draw []  (1.570796327,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.799913701,1.661396050) node {$P$};
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{4} \, \pi $};
+\draw []  (1.5708,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.7999,1.6614) node {$P$};
+\draw (1.5708,-0.42071) node {$ \frac{1}{4} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (3.1416,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{4} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{4} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.2785761667) node {$ \pi $};
+\draw (6.2832,-0.27858) node {$ \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall b/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
index 68cd080cf..85c01337c 100644
--- a/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
+++ b/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
@@ -120,7 +120,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525262) -- (0,4.054798491);
+\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
 %DEFAULT
 
 \draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
diff --git a/src_phystricks/Fig_GBnUivi.pstricks.recall b/src_phystricks/Fig_GBnUivi.pstricks.recall
index c1f71fb15..447475bac 100644
--- a/src_phystricks/Fig_GBnUivi.pstricks.recall
+++ b/src_phystricks/Fig_GBnUivi.pstricks.recall
@@ -102,57 +102,57 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (0.09124983333,1.000000000) node {1};
+\draw (0.091250,1.0000) node {1};
 \draw [] (-0.250,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,1.25);
 \draw [] (0.250,1.25) -- (-0.250,1.25);
 \draw [] (-0.250,1.25) -- (-0.250,0.750);
-\draw (0.5912498333,1.000000000) node {2};
+\draw (0.59125,1.0000) node {2};
 \draw [] (0.250,0.750) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.750,1.25);
 \draw [] (0.750,1.25) -- (0.250,1.25);
 \draw [] (0.250,1.25) -- (0.250,0.750);
-\draw (1.091249833,1.000000000) node {3};
+\draw (1.0912,1.0000) node {3};
 \draw [] (0.750,0.750) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (1.25,1.25);
 \draw [] (1.25,1.25) -- (0.750,1.25);
 \draw [] (0.750,1.25) -- (0.750,0.750);
-\draw (1.591249833,1.000000000) node {4};
+\draw (1.5912,1.0000) node {4};
 \draw [] (1.25,0.750) -- (1.75,0.750);
 \draw [] (1.75,0.750) -- (1.75,1.25);
 \draw [] (1.75,1.25) -- (1.25,1.25);
 \draw [] (1.25,1.25) -- (1.25,0.750);
-\draw (2.091249833,1.000000000) node {7};
+\draw (2.0913,1.0000) node {7};
 \draw [] (1.75,0.750) -- (2.25,0.750);
 \draw [] (2.25,0.750) -- (2.25,1.25);
 \draw [] (2.25,1.25) -- (1.75,1.25);
 \draw [] (1.75,1.25) -- (1.75,0.750);
-\draw (2.591249833,1.000000000) node {8};
+\draw (2.5913,1.0000) node {8};
 \draw [] (2.25,0.750) -- (2.75,0.750);
 \draw [] (2.75,0.750) -- (2.75,1.25);
 \draw [] (2.75,1.25) -- (2.25,1.25);
 \draw [] (2.25,1.25) -- (2.25,0.750);
-\draw (0.09124983333,0.5000000000) node {3};
+\draw (0.091250,0.50000) node {3};
 \draw [] (-0.250,0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (-0.250,0.750);
 \draw [] (-0.250,0.750) -- (-0.250,0.250);
-\draw (0.5912498333,0.5000000000) node {5};
+\draw (0.59125,0.50000) node {5};
 \draw [] (0.250,0.250) -- (0.750,0.250);
 \draw [] (0.750,0.250) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,0.250);
-\draw (1.091249833,0.5000000000) node {6};
+\draw (1.0912,0.50000) node {6};
 \draw [] (0.750,0.250) -- (1.25,0.250);
 \draw [] (1.25,0.250) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.750,0.250);
-\draw (1.591249833,0.5000000000) node {9};
+\draw (1.5912,0.50000) node {9};
 \draw [] (1.25,0.250) -- (1.75,0.250);
 \draw [] (1.75,0.250) -- (1.75,0.750);
 \draw [] (1.75,0.750) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (1.25,0.250);
-\draw (0.1824996667,0) node {10};
+\draw (0.18250,0) node {10};
 \draw [] (-0.250,-0.250) -- (0.250,-0.250);
 \draw [] (0.250,-0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (-0.250,0.250);
diff --git a/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall b/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall
index a79abb2ce..02a0a4b23 100644
--- a/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall
+++ b/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall
@@ -93,20 +93,20 @@
 \draw [] (2.00,3.46) -- (4.00,0);
 \draw [] (4.00,0) -- (0,0);
 \draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.251784057,0.3649246667) node {$60$};
+\draw (3.2518,0.36492) node {$60$};
 
 \draw [color=red] (3.75,0.433)--(3.75,0.430)--(3.74,0.428)--(3.74,0.425)--(3.73,0.422)--(3.73,0.419)--(3.72,0.416)--(3.72,0.413)--(3.71,0.410)--(3.71,0.407)--(3.71,0.404)--(3.70,0.401)--(3.70,0.398)--(3.69,0.395)--(3.69,0.391)--(3.68,0.388)--(3.68,0.385)--(3.68,0.381)--(3.67,0.378)--(3.67,0.374)--(3.66,0.371)--(3.66,0.367)--(3.66,0.364)--(3.65,0.360)--(3.65,0.356)--(3.65,0.353)--(3.64,0.349)--(3.64,0.345)--(3.63,0.341)--(3.63,0.337)--(3.63,0.333)--(3.62,0.329)--(3.62,0.325)--(3.62,0.321)--(3.61,0.317)--(3.61,0.313)--(3.61,0.309)--(3.60,0.305)--(3.60,0.301)--(3.60,0.296)--(3.59,0.292)--(3.59,0.288)--(3.59,0.284)--(3.59,0.279)--(3.58,0.275)--(3.58,0.270)--(3.58,0.266)--(3.57,0.261)--(3.57,0.257)--(3.57,0.252)--(3.57,0.248)--(3.56,0.243)--(3.56,0.238)--(3.56,0.234)--(3.56,0.229)--(3.55,0.224)--(3.55,0.220)--(3.55,0.215)--(3.55,0.210)--(3.54,0.205)--(3.54,0.200)--(3.54,0.196)--(3.54,0.191)--(3.54,0.186)--(3.53,0.181)--(3.53,0.176)--(3.53,0.171)--(3.53,0.166)--(3.53,0.161)--(3.52,0.156)--(3.52,0.151)--(3.52,0.146)--(3.52,0.141)--(3.52,0.136)--(3.52,0.131)--(3.52,0.126)--(3.51,0.120)--(3.51,0.115)--(3.51,0.110)--(3.51,0.105)--(3.51,0.0998)--(3.51,0.0946)--(3.51,0.0894)--(3.51,0.0842)--(3.51,0.0790)--(3.51,0.0738)--(3.50,0.0685)--(3.50,0.0633)--(3.50,0.0580)--(3.50,0.0528)--(3.50,0.0475)--(3.50,0.0423)--(3.50,0.0370)--(3.50,0.0317)--(3.50,0.0264)--(3.50,0.0211)--(3.50,0.0159)--(3.50,0.0106)--(3.50,0.00529)--(3.50,0);
-\draw (2.311909189,2.234016645) node {$30$};
+\draw (2.3119,2.2340) node {$30$};
 
 \draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
-\draw []  (2.000000000,3.464101615) node [rotate=0] {$\bullet$};
-\draw (2.000000000,3.888809615) node {$A$};
+\draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
+\draw (2.0000,3.8888) node {$A$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.4475840000,0) node {$B$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.443490000,0) node {$C$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.4247080000) node {$H$};
+\draw (-0.44758,0) node {$B$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.4435,0) node {$C$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.42471) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall b/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall
index 78705cb3c..bbb7404a6 100644
--- a/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall
+++ b/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.875000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,7.851851852);
+\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
 %DEFAULT
 \draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
 \draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
@@ -98,22 +98,22 @@
 \draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
 
 \draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
-\draw []  (3.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-0.2785761667) node {$a$};
-\draw []  (0,1.500000000) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,1.500000000) node {$f(a)$};
-\draw []  (6.500000000,6.085648148) node [rotate=0] {$\bullet$};
-\draw []  (6.500000000,0) node [rotate=0] {$\bullet$};
-\draw (6.500000000,-0.2785761667) node {$x$};
-\draw []  (0,6.085648148) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,6.085648148) node {$f(x)$};
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (3.000000000,1.300000000);
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (6.500000000,1.300000000);
-\draw (4.750000000,0.9789703333) node {$x-a$};
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,6.085648148);
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,1.500000000);
-\draw (7.825596167,3.792824074) node {$f(x)-f(a)$};
+\draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.27858) node {$a$};
+\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
+\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.27858) node {$x$};
+\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.45521,6.0856) node {$f(x)$};
+\draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
+\draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
+\draw (4.7500,0.97897) node {$x-a$};
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
+\draw (7.8256,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall b/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall
index c29b0028a..61cae4865 100644
--- a/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall
+++ b/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall
@@ -107,53 +107,53 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-5.261904762) -- (0,5.261904762);
+\draw [,->,>=latex] (-4.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-5.2619) -- (0,5.2619);
 %DEFAULT
 
 \draw [color=red] (-4.000,0.2000)--(-3.952,0.2020)--(-3.903,0.2039)--(-3.855,0.2060)--(-3.806,0.2081)--(-3.758,0.2102)--(-3.710,0.2123)--(-3.661,0.2145)--(-3.613,0.2168)--(-3.565,0.2191)--(-3.516,0.2214)--(-3.468,0.2238)--(-3.419,0.2263)--(-3.371,0.2288)--(-3.323,0.2313)--(-3.274,0.2340)--(-3.226,0.2366)--(-3.177,0.2394)--(-3.129,0.2422)--(-3.081,0.2451)--(-3.032,0.2480)--(-2.984,0.2510)--(-2.936,0.2541)--(-2.887,0.2573)--(-2.839,0.2605)--(-2.790,0.2638)--(-2.742,0.2672)--(-2.694,0.2707)--(-2.645,0.2743)--(-2.597,0.2780)--(-2.548,0.2818)--(-2.500,0.2857)--(-2.452,0.2897)--(-2.403,0.2938)--(-2.355,0.2981)--(-2.307,0.3024)--(-2.258,0.3069)--(-2.210,0.3115)--(-2.161,0.3163)--(-2.113,0.3212)--(-2.065,0.3263)--(-2.016,0.3315)--(-1.968,0.3369)--(-1.919,0.3425)--(-1.871,0.3483)--(-1.823,0.3543)--(-1.774,0.3604)--(-1.726,0.3668)--(-1.678,0.3735)--(-1.629,0.3803)--(-1.581,0.3875)--(-1.532,0.3949)--(-1.484,0.4026)--(-1.436,0.4106)--(-1.387,0.4189)--(-1.339,0.4276)--(-1.291,0.4366)--(-1.242,0.4460)--(-1.194,0.4558)--(-1.145,0.4661)--(-1.097,0.4769)--(-1.049,0.4881)--(-1.000,0.5000)--(-0.9518,0.5123)--(-0.9034,0.5254)--(-0.8550,0.5391)--(-0.8067,0.5535)--(-0.7583,0.5687)--(-0.7099,0.5848)--(-0.6615,0.6019)--(-0.6131,0.6199)--(-0.5648,0.6391)--(-0.5164,0.6595)--(-0.4680,0.6812)--(-0.4196,0.7044)--(-0.3712,0.7293)--(-0.3228,0.7560)--(-0.2744,0.7847)--(-0.2261,0.8156)--(-0.1777,0.8491)--(-0.1293,0.8855)--(-0.08091,0.9251)--(-0.03253,0.9685)--(0.01586,1.016)--(0.06424,1.069)--(0.1126,1.127)--(0.1610,1.192)--(0.2094,1.265)--(0.2578,1.347)--(0.3062,1.441)--(0.3545,1.549)--(0.4029,1.675)--(0.4513,1.823)--(0.4997,1.999)--(0.5481,2.213)--(0.5965,2.478)--(0.6449,2.816)--(0.6932,3.260)--(0.7416,3.870)--(0.7900,4.762);
 
 \draw [color=blue] (1.210,-4.762)--(1.258,-3.870)--(1.307,-3.260)--(1.355,-2.816)--(1.404,-2.478)--(1.452,-2.213)--(1.500,-1.999)--(1.549,-1.823)--(1.597,-1.675)--(1.645,-1.549)--(1.694,-1.441)--(1.742,-1.347)--(1.791,-1.265)--(1.839,-1.192)--(1.887,-1.127)--(1.936,-1.069)--(1.984,-1.016)--(2.033,-0.9685)--(2.081,-0.9251)--(2.129,-0.8855)--(2.178,-0.8491)--(2.226,-0.8156)--(2.274,-0.7847)--(2.323,-0.7560)--(2.371,-0.7293)--(2.420,-0.7044)--(2.468,-0.6812)--(2.516,-0.6595)--(2.565,-0.6391)--(2.613,-0.6199)--(2.662,-0.6019)--(2.710,-0.5848)--(2.758,-0.5687)--(2.807,-0.5535)--(2.855,-0.5391)--(2.903,-0.5254)--(2.952,-0.5123)--(3.000,-0.5000)--(3.049,-0.4881)--(3.097,-0.4769)--(3.145,-0.4661)--(3.194,-0.4558)--(3.242,-0.4460)--(3.290,-0.4366)--(3.339,-0.4276)--(3.387,-0.4189)--(3.436,-0.4106)--(3.484,-0.4026)--(3.532,-0.3949)--(3.581,-0.3875)--(3.629,-0.3803)--(3.678,-0.3735)--(3.726,-0.3668)--(3.774,-0.3604)--(3.823,-0.3543)--(3.871,-0.3483)--(3.919,-0.3425)--(3.968,-0.3369)--(4.016,-0.3315)--(4.065,-0.3263)--(4.113,-0.3212)--(4.161,-0.3163)--(4.210,-0.3115)--(4.258,-0.3069)--(4.307,-0.3024)--(4.355,-0.2981)--(4.403,-0.2938)--(4.452,-0.2897)--(4.500,-0.2857)--(4.548,-0.2818)--(4.597,-0.2780)--(4.645,-0.2743)--(4.694,-0.2707)--(4.742,-0.2672)--(4.790,-0.2638)--(4.839,-0.2605)--(4.887,-0.2573)--(4.936,-0.2541)--(4.984,-0.2510)--(5.032,-0.2480)--(5.081,-0.2451)--(5.129,-0.2422)--(5.177,-0.2394)--(5.226,-0.2366)--(5.274,-0.2340)--(5.323,-0.2313)--(5.371,-0.2288)--(5.419,-0.2263)--(5.468,-0.2238)--(5.516,-0.2214)--(5.565,-0.2191)--(5.613,-0.2168)--(5.661,-0.2145)--(5.710,-0.2123)--(5.758,-0.2102)--(5.806,-0.2081)--(5.855,-0.2060)--(5.903,-0.2039)--(5.952,-0.2020)--(6.000,-0.2000);
 \draw [style=dotted] (1.00,-4.76) -- (1.00,4.76);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
 \draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall b/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall
index 9af02e5bb..1d7103b76 100644
--- a/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall
+++ b/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.499883862);
+\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -56,10 +56,10 @@
 \draw [color=blue,style=solid] (1.000,1.635)--(1.051,1.562)--(1.101,1.497)--(1.152,1.439)--(1.202,1.390)--(1.253,1.348)--(1.303,1.316)--(1.354,1.293)--(1.404,1.279)--(1.455,1.273)--(1.505,1.276)--(1.556,1.288)--(1.606,1.307)--(1.657,1.334)--(1.707,1.367)--(1.758,1.406)--(1.808,1.450)--(1.859,1.498)--(1.909,1.549)--(1.960,1.603)--(2.010,1.658)--(2.061,1.713)--(2.111,1.768)--(2.162,1.821)--(2.212,1.873)--(2.263,1.921)--(2.313,1.965)--(2.364,2.006)--(2.414,2.041)--(2.465,2.071)--(2.515,2.096)--(2.566,2.116)--(2.616,2.129)--(2.667,2.137)--(2.717,2.139)--(2.768,2.136)--(2.818,2.128)--(2.869,2.116)--(2.919,2.100)--(2.970,2.081)--(3.020,2.059)--(3.071,2.035)--(3.121,2.010)--(3.172,1.985)--(3.222,1.960)--(3.273,1.937)--(3.323,1.915)--(3.374,1.896)--(3.424,1.881)--(3.475,1.870)--(3.525,1.863)--(3.576,1.861)--(3.626,1.864)--(3.677,1.873)--(3.727,1.888)--(3.778,1.908)--(3.828,1.934)--(3.879,1.965)--(3.929,2.002)--(3.980,2.043)--(4.030,2.088)--(4.081,2.137)--(4.131,2.189)--(4.182,2.242)--(4.232,2.297)--(4.283,2.353)--(4.333,2.408)--(4.384,2.461)--(4.434,2.512)--(4.485,2.559)--(4.535,2.602)--(4.586,2.640)--(4.636,2.672)--(4.687,2.697)--(4.737,2.715)--(4.788,2.725)--(4.838,2.727)--(4.889,2.719)--(4.939,2.703)--(4.990,2.678)--(5.040,2.644)--(5.091,2.602)--(5.141,2.550)--(5.192,2.491)--(5.242,2.424)--(5.293,2.351)--(5.343,2.271)--(5.394,2.186)--(5.444,2.097)--(5.495,2.005)--(5.545,1.911)--(5.596,1.816)--(5.646,1.722)--(5.697,1.629)--(5.747,1.538)--(5.798,1.452)--(5.849,1.370)--(5.899,1.294)--(5.950,1.225)--(6.000,1.165);
 \draw [color=magenta,style=dashed] (1.00,1.63) -- (1.00,4.84);
 \draw [color=magenta,style=dashed] (6.00,3.72) -- (6.00,1.16);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.2785761667) node {$a$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3267360000) node {$b$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.27858) node {$a$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.32674) node {$b$};
 \draw [style=dotted] (1.00,0) -- (1.00,4.84);
 
 \draw [color=blue] (1.000,4.841)--(1.051,4.868)--(1.101,4.892)--(1.152,4.913)--(1.202,4.933)--(1.253,4.950)--(1.303,4.964)--(1.354,4.977)--(1.404,4.986)--(1.455,4.993)--(1.505,4.998)--(1.556,5.000)--(1.606,4.999)--(1.657,4.996)--(1.707,4.991)--(1.758,4.983)--(1.808,4.972)--(1.859,4.959)--(1.909,4.943)--(1.960,4.925)--(2.010,4.905)--(2.061,4.882)--(2.111,4.858)--(2.162,4.831)--(2.212,4.801)--(2.263,4.770)--(2.313,4.737)--(2.364,4.702)--(2.414,4.665)--(2.465,4.626)--(2.515,4.586)--(2.566,4.545)--(2.616,4.502)--(2.667,4.457)--(2.717,4.412)--(2.768,4.365)--(2.818,4.318)--(2.869,4.270)--(2.919,4.221)--(2.970,4.171)--(3.020,4.121)--(3.071,4.071)--(3.121,4.020)--(3.172,3.970)--(3.222,3.919)--(3.273,3.869)--(3.323,3.819)--(3.374,3.770)--(3.424,3.721)--(3.475,3.673)--(3.525,3.626)--(3.576,3.579)--(3.626,3.534)--(3.677,3.490)--(3.727,3.447)--(3.778,3.406)--(3.828,3.366)--(3.879,3.328)--(3.929,3.291)--(3.980,3.257)--(4.030,3.224)--(4.081,3.193)--(4.131,3.164)--(4.182,3.137)--(4.232,3.113)--(4.283,3.091)--(4.333,3.071)--(4.384,3.053)--(4.434,3.038)--(4.485,3.026)--(4.535,3.016)--(4.586,3.008)--(4.636,3.003)--(4.687,3.000)--(4.737,3.000)--(4.788,3.003)--(4.838,3.008)--(4.889,3.016)--(4.939,3.026)--(4.990,3.038)--(5.040,3.053)--(5.091,3.071)--(5.141,3.091)--(5.192,3.113)--(5.242,3.137)--(5.293,3.164)--(5.343,3.193)--(5.394,3.223)--(5.444,3.256)--(5.495,3.291)--(5.545,3.327)--(5.596,3.366)--(5.646,3.405)--(5.697,3.447)--(5.747,3.490)--(5.798,3.534)--(5.849,3.579)--(5.899,3.625)--(5.950,3.672)--(6.000,3.721);
@@ -108,8 +108,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.499883862,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -123,10 +123,10 @@
 \draw [color=blue] (1.635,1.000)--(1.562,1.051)--(1.497,1.101)--(1.439,1.152)--(1.390,1.202)--(1.348,1.253)--(1.316,1.303)--(1.293,1.354)--(1.279,1.404)--(1.273,1.455)--(1.276,1.505)--(1.288,1.556)--(1.307,1.606)--(1.334,1.657)--(1.367,1.707)--(1.406,1.758)--(1.450,1.808)--(1.498,1.859)--(1.549,1.909)--(1.603,1.960)--(1.658,2.010)--(1.713,2.061)--(1.768,2.111)--(1.821,2.162)--(1.873,2.212)--(1.921,2.263)--(1.965,2.313)--(2.006,2.364)--(2.041,2.414)--(2.071,2.465)--(2.096,2.515)--(2.116,2.566)--(2.129,2.616)--(2.137,2.667)--(2.139,2.717)--(2.136,2.768)--(2.128,2.818)--(2.116,2.869)--(2.100,2.919)--(2.081,2.970)--(2.059,3.020)--(2.035,3.071)--(2.010,3.121)--(1.985,3.172)--(1.960,3.222)--(1.937,3.273)--(1.915,3.323)--(1.896,3.374)--(1.881,3.424)--(1.870,3.475)--(1.863,3.525)--(1.861,3.576)--(1.864,3.626)--(1.873,3.677)--(1.888,3.727)--(1.908,3.778)--(1.934,3.828)--(1.965,3.879)--(2.002,3.929)--(2.043,3.980)--(2.088,4.030)--(2.137,4.081)--(2.189,4.131)--(2.242,4.182)--(2.297,4.232)--(2.353,4.283)--(2.408,4.333)--(2.461,4.384)--(2.512,4.434)--(2.559,4.485)--(2.602,4.535)--(2.640,4.586)--(2.672,4.636)--(2.697,4.687)--(2.715,4.737)--(2.725,4.788)--(2.727,4.838)--(2.719,4.889)--(2.703,4.939)--(2.678,4.990)--(2.644,5.040)--(2.602,5.091)--(2.550,5.141)--(2.491,5.192)--(2.424,5.242)--(2.351,5.293)--(2.271,5.343)--(2.186,5.394)--(2.097,5.444)--(2.005,5.495)--(1.911,5.545)--(1.816,5.596)--(1.722,5.646)--(1.629,5.697)--(1.538,5.747)--(1.452,5.798)--(1.370,5.849)--(1.294,5.899)--(1.225,5.950)--(1.165,6.000);
 \draw [color=magenta,style=dashed] (1.16,6.00) -- (3.72,6.00);
 \draw [color=magenta,style=dashed] (4.84,1.00) -- (1.63,1.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2789780000,1.000000000) node {$c$};
-\draw []  (0,6.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2949888333,6.000000000) node {$d$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.27898,1.0000) node {$c$};
+\draw []  (0,6.0000) node [rotate=0] {$\bullet$};
+\draw (-0.29499,6.0000) node {$d$};
 \draw [style=dotted] (0,1.00) -- (4.84,1.00);
 \draw [style=dotted] (0,6.00) -- (1.16,6.00);
 \draw [color=blue,style=solid] (4.841,1.000)--(4.868,1.051)--(4.892,1.101)--(4.913,1.152)--(4.933,1.202)--(4.950,1.253)--(4.964,1.303)--(4.977,1.354)--(4.986,1.404)--(4.993,1.455)--(4.998,1.505)--(5.000,1.556)--(4.999,1.606)--(4.996,1.657)--(4.991,1.707)--(4.983,1.758)--(4.972,1.808)--(4.959,1.859)--(4.943,1.909)--(4.925,1.960)--(4.905,2.010)--(4.882,2.061)--(4.858,2.111)--(4.831,2.162)--(4.801,2.212)--(4.770,2.263)--(4.737,2.313)--(4.702,2.364)--(4.665,2.414)--(4.626,2.465)--(4.586,2.515)--(4.545,2.566)--(4.502,2.616)--(4.457,2.667)--(4.412,2.717)--(4.365,2.768)--(4.318,2.818)--(4.270,2.869)--(4.221,2.919)--(4.171,2.970)--(4.121,3.020)--(4.071,3.071)--(4.020,3.121)--(3.970,3.172)--(3.919,3.222)--(3.869,3.273)--(3.819,3.323)--(3.770,3.374)--(3.721,3.424)--(3.673,3.475)--(3.626,3.525)--(3.579,3.576)--(3.534,3.626)--(3.490,3.677)--(3.447,3.727)--(3.406,3.778)--(3.366,3.828)--(3.328,3.879)--(3.291,3.929)--(3.257,3.980)--(3.224,4.030)--(3.193,4.081)--(3.164,4.131)--(3.137,4.182)--(3.113,4.232)--(3.091,4.283)--(3.071,4.333)--(3.053,4.384)--(3.038,4.434)--(3.026,4.485)--(3.016,4.535)--(3.008,4.586)--(3.003,4.636)--(3.000,4.687)--(3.000,4.737)--(3.003,4.788)--(3.008,4.838)--(3.016,4.889)--(3.026,4.939)--(3.038,4.990)--(3.053,5.040)--(3.071,5.091)--(3.091,5.141)--(3.113,5.192)--(3.137,5.242)--(3.164,5.293)--(3.193,5.343)--(3.223,5.394)--(3.256,5.444)--(3.291,5.495)--(3.327,5.545)--(3.366,5.596)--(3.405,5.646)--(3.447,5.697)--(3.490,5.747)--(3.534,5.798)--(3.579,5.849)--(3.625,5.899)--(3.672,5.950)--(3.721,6.000);
diff --git a/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall b/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall
index 4b211b1e2..bf237b1bd 100644
--- a/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall
+++ b/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=red] (0,0)--(0.0159,0)--(0.0317,0)--(0.0476,0.00113)--(0.0634,0.00201)--(0.0792,0.00315)--(0.0951,0.00453)--(0.111,0.00616)--(0.127,0.00805)--(0.142,0.0102)--(0.158,0.0126)--(0.174,0.0152)--(0.189,0.0181)--(0.205,0.0212)--(0.220,0.0246)--(0.236,0.0282)--(0.251,0.0321)--(0.266,0.0362)--(0.282,0.0405)--(0.297,0.0451)--(0.312,0.0499)--(0.327,0.0550)--(0.342,0.0603)--(0.357,0.0658)--(0.372,0.0716)--(0.386,0.0776)--(0.401,0.0839)--(0.415,0.0904)--(0.430,0.0971)--(0.444,0.104)--(0.458,0.111)--(0.472,0.119)--(0.486,0.126)--(0.500,0.134)--(0.514,0.142)--(0.527,0.150)--(0.541,0.159)--(0.554,0.167)--(0.567,0.176)--(0.580,0.185)--(0.593,0.195)--(0.606,0.204)--(0.618,0.214)--(0.631,0.224)--(0.643,0.234)--(0.655,0.244)--(0.667,0.255)--(0.679,0.265)--(0.690,0.276)--(0.701,0.287)--(0.713,0.299)--(0.724,0.310)--(0.735,0.322)--(0.745,0.333)--(0.756,0.345)--(0.766,0.357)--(0.776,0.369)--(0.786,0.382)--(0.796,0.394)--(0.805,0.407)--(0.815,0.420)--(0.824,0.433)--(0.833,0.446)--(0.841,0.459)--(0.850,0.473)--(0.858,0.486)--(0.866,0.500)--(0.874,0.514)--(0.881,0.528)--(0.889,0.542)--(0.896,0.556)--(0.903,0.570)--(0.910,0.585)--(0.916,0.599)--(0.922,0.614)--(0.928,0.628)--(0.934,0.643)--(0.940,0.658)--(0.945,0.673)--(0.950,0.688)--(0.955,0.703)--(0.959,0.718)--(0.964,0.734)--(0.968,0.749)--(0.972,0.764)--(0.975,0.780)--(0.979,0.795)--(0.982,0.811)--(0.985,0.826)--(0.987,0.842)--(0.990,0.858)--(0.992,0.873)--(0.994,0.889)--(0.995,0.905)--(0.997,0.921)--(0.998,0.937)--(0.999,0.952)--(1.00,0.968)--(1.00,0.984)--(1.00,1.00);
@@ -80,8 +80,8 @@
 \draw [] (0,0) -- (0,0);
 \draw [] (1.00,1.00) -- (1.00,1.00);
 \draw [color=red] (-0.500,-0.500) -- (1.50,1.50);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.354646868,0.6631599656) node {$P$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3546,0.66316) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall b/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall
index 6a7b1f5d8..d8605c418 100644
--- a/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall
+++ b/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall
@@ -138,8 +138,8 @@
 \draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
 \draw [color=gray,style=solid] (-5.00,6.00) -- (2.00,6.00);
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,6.5000);
 %DEFAULT
 
 \draw [color=blue] (-5.000,-2.500)--(-4.950,-2.475)--(-4.899,-2.449)--(-4.849,-2.424)--(-4.798,-2.399)--(-4.747,-2.374)--(-4.697,-2.348)--(-4.646,-2.323)--(-4.596,-2.298)--(-4.545,-2.273)--(-4.495,-2.247)--(-4.444,-2.222)--(-4.394,-2.197)--(-4.343,-2.172)--(-4.293,-2.146)--(-4.242,-2.121)--(-4.192,-2.096)--(-4.141,-2.071)--(-4.091,-2.045)--(-4.040,-2.020)--(-3.990,-1.995)--(-3.939,-1.970)--(-3.889,-1.944)--(-3.838,-1.919)--(-3.788,-1.894)--(-3.737,-1.869)--(-3.687,-1.843)--(-3.636,-1.818)--(-3.586,-1.793)--(-3.535,-1.768)--(-3.485,-1.742)--(-3.434,-1.717)--(-3.384,-1.692)--(-3.333,-1.667)--(-3.283,-1.641)--(-3.232,-1.616)--(-3.182,-1.591)--(-3.131,-1.566)--(-3.081,-1.540)--(-3.030,-1.515)--(-2.980,-1.490)--(-2.929,-1.465)--(-2.879,-1.439)--(-2.828,-1.414)--(-2.778,-1.389)--(-2.727,-1.364)--(-2.677,-1.338)--(-2.626,-1.313)--(-2.576,-1.288)--(-2.525,-1.263)--(-2.475,-1.237)--(-2.424,-1.212)--(-2.374,-1.187)--(-2.323,-1.162)--(-2.273,-1.136)--(-2.222,-1.111)--(-2.172,-1.086)--(-2.121,-1.061)--(-2.071,-1.035)--(-2.020,-1.010)--(-1.970,-0.9848)--(-1.919,-0.9596)--(-1.869,-0.9343)--(-1.818,-0.9091)--(-1.768,-0.8838)--(-1.717,-0.8586)--(-1.667,-0.8333)--(-1.616,-0.8081)--(-1.566,-0.7828)--(-1.515,-0.7576)--(-1.465,-0.7323)--(-1.414,-0.7071)--(-1.364,-0.6818)--(-1.313,-0.6566)--(-1.263,-0.6313)--(-1.212,-0.6061)--(-1.162,-0.5808)--(-1.111,-0.5556)--(-1.061,-0.5303)--(-1.010,-0.5051)--(-0.9596,-0.4798)--(-0.9091,-0.4545)--(-0.8586,-0.4293)--(-0.8081,-0.4040)--(-0.7576,-0.3788)--(-0.7071,-0.3535)--(-0.6566,-0.3283)--(-0.6061,-0.3030)--(-0.5556,-0.2778)--(-0.5051,-0.2525)--(-0.4545,-0.2273)--(-0.4040,-0.2020)--(-0.3535,-0.1768)--(-0.3030,-0.1515)--(-0.2525,-0.1263)--(-0.2020,-0.1010)--(-0.1515,-0.07576)--(-0.1010,-0.05051)--(-0.05051,-0.02525)--(0,0);
@@ -147,42 +147,42 @@
 \draw [color=blue] (0,3.000)--(0.01010,2.970)--(0.02020,2.939)--(0.03030,2.909)--(0.04040,2.879)--(0.05051,2.848)--(0.06061,2.818)--(0.07071,2.788)--(0.08081,2.758)--(0.09091,2.727)--(0.1010,2.697)--(0.1111,2.667)--(0.1212,2.636)--(0.1313,2.606)--(0.1414,2.576)--(0.1515,2.545)--(0.1616,2.515)--(0.1717,2.485)--(0.1818,2.455)--(0.1919,2.424)--(0.2020,2.394)--(0.2121,2.364)--(0.2222,2.333)--(0.2323,2.303)--(0.2424,2.273)--(0.2525,2.242)--(0.2626,2.212)--(0.2727,2.182)--(0.2828,2.152)--(0.2929,2.121)--(0.3030,2.091)--(0.3131,2.061)--(0.3232,2.030)--(0.3333,2.000)--(0.3434,1.970)--(0.3535,1.939)--(0.3636,1.909)--(0.3737,1.879)--(0.3838,1.848)--(0.3939,1.818)--(0.4040,1.788)--(0.4141,1.758)--(0.4242,1.727)--(0.4343,1.697)--(0.4444,1.667)--(0.4545,1.636)--(0.4646,1.606)--(0.4747,1.576)--(0.4848,1.545)--(0.4949,1.515)--(0.5051,1.485)--(0.5152,1.455)--(0.5253,1.424)--(0.5354,1.394)--(0.5455,1.364)--(0.5556,1.333)--(0.5657,1.303)--(0.5758,1.273)--(0.5859,1.242)--(0.5960,1.212)--(0.6061,1.182)--(0.6162,1.152)--(0.6263,1.121)--(0.6364,1.091)--(0.6465,1.061)--(0.6566,1.030)--(0.6667,1.000)--(0.6768,0.9697)--(0.6869,0.9394)--(0.6970,0.9091)--(0.7071,0.8788)--(0.7172,0.8485)--(0.7273,0.8182)--(0.7374,0.7879)--(0.7475,0.7576)--(0.7576,0.7273)--(0.7677,0.6970)--(0.7778,0.6667)--(0.7879,0.6364)--(0.7980,0.6061)--(0.8081,0.5758)--(0.8182,0.5455)--(0.8283,0.5152)--(0.8384,0.4848)--(0.8485,0.4545)--(0.8586,0.4242)--(0.8687,0.3939)--(0.8788,0.3636)--(0.8889,0.3333)--(0.8990,0.3030)--(0.9091,0.2727)--(0.9192,0.2424)--(0.9293,0.2121)--(0.9394,0.1818)--(0.9495,0.1515)--(0.9596,0.1212)--(0.9697,0.09091)--(0.9798,0.06061)--(0.9899,0.03030)--(1.000,0);
 
 \draw [color=blue] (1.000,3.000)--(1.010,3.030)--(1.020,3.061)--(1.030,3.091)--(1.040,3.121)--(1.051,3.152)--(1.061,3.182)--(1.071,3.212)--(1.081,3.242)--(1.091,3.273)--(1.101,3.303)--(1.111,3.333)--(1.121,3.364)--(1.131,3.394)--(1.141,3.424)--(1.152,3.455)--(1.162,3.485)--(1.172,3.515)--(1.182,3.545)--(1.192,3.576)--(1.202,3.606)--(1.212,3.636)--(1.222,3.667)--(1.232,3.697)--(1.242,3.727)--(1.253,3.758)--(1.263,3.788)--(1.273,3.818)--(1.283,3.848)--(1.293,3.879)--(1.303,3.909)--(1.313,3.939)--(1.323,3.970)--(1.333,4.000)--(1.343,4.030)--(1.354,4.061)--(1.364,4.091)--(1.374,4.121)--(1.384,4.151)--(1.394,4.182)--(1.404,4.212)--(1.414,4.242)--(1.424,4.273)--(1.434,4.303)--(1.444,4.333)--(1.455,4.364)--(1.465,4.394)--(1.475,4.424)--(1.485,4.455)--(1.495,4.485)--(1.505,4.515)--(1.515,4.545)--(1.525,4.576)--(1.535,4.606)--(1.545,4.636)--(1.556,4.667)--(1.566,4.697)--(1.576,4.727)--(1.586,4.758)--(1.596,4.788)--(1.606,4.818)--(1.616,4.849)--(1.626,4.879)--(1.636,4.909)--(1.646,4.939)--(1.657,4.970)--(1.667,5.000)--(1.677,5.030)--(1.687,5.061)--(1.697,5.091)--(1.707,5.121)--(1.717,5.151)--(1.727,5.182)--(1.737,5.212)--(1.747,5.242)--(1.758,5.273)--(1.768,5.303)--(1.778,5.333)--(1.788,5.364)--(1.798,5.394)--(1.808,5.424)--(1.818,5.455)--(1.828,5.485)--(1.838,5.515)--(1.848,5.545)--(1.859,5.576)--(1.869,5.606)--(1.879,5.636)--(1.889,5.667)--(1.899,5.697)--(1.909,5.727)--(1.919,5.758)--(1.929,5.788)--(1.939,5.818)--(1.949,5.849)--(1.960,5.879)--(1.970,5.909)--(1.980,5.939)--(1.990,5.970)--(2.000,6.000);
-\draw []  (0,3.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,3.000000000) node [rotate=0] {$o$};
+\draw []  (1.0000,3.0000) node [rotate=0] {$o$};
 
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall b/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall
index 7974b5f8f..b7cae7f35 100644
--- a/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall
+++ b/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall
@@ -99,20 +99,20 @@
 
 \draw [] (7.000,3.000)--(6.994,3.190)--(6.976,3.380)--(6.946,3.568)--(6.904,3.753)--(6.850,3.936)--(6.785,4.115)--(6.709,4.289)--(6.622,4.459)--(6.524,4.622)--(6.416,4.779)--(6.298,4.928)--(6.171,5.070)--(6.036,5.204)--(5.892,5.328)--(5.740,5.444)--(5.582,5.549)--(5.417,5.644)--(5.246,5.729)--(5.071,5.802)--(4.891,5.865)--(4.707,5.915)--(4.521,5.954)--(4.333,5.982)--(4.143,5.997)--(3.952,6.000)--(3.762,5.991)--(3.573,5.969)--(3.386,5.936)--(3.201,5.892)--(3.019,5.835)--(2.841,5.767)--(2.668,5.688)--(2.500,5.598)--(2.338,5.498)--(2.183,5.387)--(2.035,5.267)--(1.896,5.138)--(1.764,5.000)--(1.642,4.854)--(1.529,4.701)--(1.426,4.541)--(1.333,4.375)--(1.252,4.203)--(1.181,4.026)--(1.122,3.845)--(1.074,3.661)--(1.038,3.474)--(1.014,3.285)--(1.002,3.095)--(1.002,2.905)--(1.014,2.715)--(1.038,2.526)--(1.074,2.339)--(1.122,2.155)--(1.181,1.974)--(1.252,1.797)--(1.333,1.625)--(1.426,1.459)--(1.529,1.299)--(1.642,1.146)--(1.764,0.9997)--(1.896,0.8619)--(2.035,0.7327)--(2.183,0.6127)--(2.338,0.5023)--(2.500,0.4019)--(2.668,0.3120)--(2.841,0.2329)--(3.019,0.1650)--(3.201,0.1085)--(3.386,0.06359)--(3.573,0.03054)--(3.762,0.009436)--(3.952,0)--(4.143,0.003398)--(4.333,0.01848)--(4.521,0.04558)--(4.707,0.08457)--(4.891,0.1353)--(5.071,0.1976)--(5.246,0.2711)--(5.417,0.3556)--(5.582,0.4508)--(5.740,0.5563)--(5.892,0.6716)--(6.036,0.7962)--(6.171,0.9298)--(6.298,1.072)--(6.416,1.221)--(6.524,1.378)--(6.622,1.541)--(6.709,1.711)--(6.785,1.885)--(6.850,2.064)--(6.904,2.247)--(6.946,2.432)--(6.976,2.620)--(6.994,2.810)--(7.000,3.000);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.5086990000) node {\( \tilde \phi(a)\)};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.517164111,-0.3372565219) node {\( \tilde\phi(c)\)};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.5086990000) node {\( \tilde \phi(b)\)};
+\draw (0,-0.50870) node {\( \tilde \phi(a)\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.5172,-0.33726) node {\( \tilde\phi(c)\)};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.50870) node {\( \tilde \phi(b)\)};
 \draw [] (-4.00,0) -- (8.00,0);
-\draw []  (1.079008574,3.683965706) node [rotate=0] {$\bullet$};
-\draw (1.689894907,3.683965706) node {\( \tilde\phi(m)\)};
-\draw (-2.292780055,3.308699000) node {\( \tilde \phi(A)\)};
-\draw (6.660974700,5.471440700) node {\( \tilde \phi(B)\)};
-\draw []  (1.744520838,1.021916647) node [rotate=0] {$\bullet$};
-\draw (2.035770672,1.021916647) node {\( 0\)};
+\draw []  (1.0790,3.6840) node [rotate=0] {$\bullet$};
+\draw (1.6899,3.6840) node {\( \tilde\phi(m)\)};
+\draw (-2.2928,3.3087) node {\( \tilde \phi(A)\)};
+\draw (6.6610,5.4714) node {\( \tilde \phi(B)\)};
+\draw []  (1.7445,1.0219) node [rotate=0] {$\bullet$};
+\draw (2.0358,1.0219) node {\( 0\)};
 \draw [] (0.746,5.02) -- (2.08,-0.309);
-\draw (8.557386667,0) node {\( \tilde\phi(\mC)\)};
+\draw (8.5574,0) node {\( \tilde\phi(\mC)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IntEcourbe.pstricks.recall b/src_phystricks/Fig_IntEcourbe.pstricks.recall
index 7f740bca4..f9fb48682 100644
--- a/src_phystricks/Fig_IntEcourbe.pstricks.recall
+++ b/src_phystricks/Fig_IntEcourbe.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
 %DEFAULT
 
 \draw [color=blue] (1.000,3.500)--(1.020,3.460)--(1.040,3.421)--(1.061,3.382)--(1.081,3.345)--(1.101,3.308)--(1.121,3.272)--(1.141,3.237)--(1.162,3.203)--(1.182,3.169)--(1.202,3.137)--(1.222,3.105)--(1.242,3.074)--(1.263,3.044)--(1.283,3.014)--(1.303,2.986)--(1.323,2.958)--(1.343,2.931)--(1.364,2.905)--(1.384,2.880)--(1.404,2.855)--(1.424,2.831)--(1.444,2.809)--(1.465,2.787)--(1.485,2.765)--(1.505,2.745)--(1.525,2.725)--(1.545,2.707)--(1.566,2.689)--(1.586,2.672)--(1.606,2.655)--(1.626,2.640)--(1.646,2.625)--(1.667,2.611)--(1.687,2.598)--(1.707,2.586)--(1.727,2.574)--(1.747,2.564)--(1.768,2.554)--(1.788,2.545)--(1.808,2.537)--(1.828,2.529)--(1.848,2.523)--(1.869,2.517)--(1.889,2.512)--(1.909,2.508)--(1.929,2.505)--(1.949,2.503)--(1.970,2.501)--(1.990,2.500)--(2.010,2.500)--(2.030,2.501)--(2.051,2.503)--(2.071,2.505)--(2.091,2.508)--(2.111,2.512)--(2.131,2.517)--(2.152,2.523)--(2.172,2.529)--(2.192,2.537)--(2.212,2.545)--(2.232,2.554)--(2.253,2.564)--(2.273,2.574)--(2.293,2.586)--(2.313,2.598)--(2.333,2.611)--(2.354,2.625)--(2.374,2.640)--(2.394,2.655)--(2.414,2.672)--(2.434,2.689)--(2.455,2.707)--(2.475,2.725)--(2.495,2.745)--(2.515,2.765)--(2.535,2.787)--(2.556,2.809)--(2.576,2.831)--(2.596,2.855)--(2.616,2.880)--(2.636,2.905)--(2.657,2.931)--(2.677,2.958)--(2.697,2.986)--(2.717,3.014)--(2.737,3.044)--(2.758,3.074)--(2.778,3.105)--(2.798,3.137)--(2.818,3.169)--(2.838,3.203)--(2.859,3.237)--(2.879,3.272)--(2.899,3.308)--(2.919,3.345)--(2.939,3.382)--(2.960,3.421)--(2.980,3.460)--(3.000,3.500);
@@ -103,19 +103,19 @@
 \draw [color=cyan] (3.00,3.50) -- (3.00,0.641);
 \draw [color=cyan] (3.00,0.641) -- (1.00,0.641);
 \draw [color=cyan] (1.00,0.641) -- (1.00,3.50);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IntTrois.pstricks.recall b/src_phystricks/Fig_IntTrois.pstricks.recall
index efe3a3fb1..654f2a3b0 100644
--- a/src_phystricks/Fig_IntTrois.pstricks.recall
+++ b/src_phystricks/Fig_IntTrois.pstricks.recall
@@ -37,19 +37,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \fill [color=lightgray] (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,0) --  cycle;
 \draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
 \draw [] (1.73,1.00) -- (1.73,2.00);
 \draw [color=red] (0,2.00) -- (2.00,2.00);
 \draw [color=red] (2.00,2.00) -- (2.00,0);
-\draw []  (1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.732050808,2.000000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.7320,2.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -84,8 +84,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [style=dotted] (0,0) -- (1.64,1.15);
 \draw [style=dotted] (0,0) -- (1.15,1.64);
@@ -94,14 +94,14 @@
 \draw [color=red] (2.00,2.00) -- (2.00,0);
 \draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
 \draw [] (1.64,1.15) -- (2.00,1.40);
-\draw []  (1.638304089,1.147152873) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,1.400415076) node [rotate=0] {$\bullet$};
-\draw []  (1.147152873,1.638304089) node [rotate=0] {$\bullet$};
-\draw []  (1.400415076,2.000000000) node [rotate=0] {$\bullet$};
+\draw []  (1.6383,1.1472) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,1.4004) node [rotate=0] {$\bullet$};
+\draw []  (1.1472,1.6383) node [rotate=0] {$\bullet$};
+\draw []  (1.4004,2.0000) node [rotate=0] {$\bullet$};
 \draw [] (1.15,1.64) -- (1.40,2.00);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IntegraleSimple.pstricks.recall b/src_phystricks/Fig_IntegraleSimple.pstricks.recall
index a4e8199e1..5f46e6eb9 100644
--- a/src_phystricks/Fig_IntegraleSimple.pstricks.recall
+++ b/src_phystricks/Fig_IntegraleSimple.pstricks.recall
@@ -71,7 +71,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (6.783185307,0);
+\draw [,->,>=latex] (-2.070796328,0) -- (6.783185311,0);
 \draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
 %DEFAULT
 \draw []  (-1.570796327,0) node [rotate=0] {$\bullet$};
diff --git a/src_phystricks/Fig_IntervalleUn.pstricks.recall b/src_phystricks/Fig_IntervalleUn.pstricks.recall
index fb91c285c..27457d70b 100644
--- a/src_phystricks/Fig_IntervalleUn.pstricks.recall
+++ b/src_phystricks/Fig_IntervalleUn.pstricks.recall
@@ -82,18 +82,18 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (-1.50,0) -- (4.50,0);
-\draw []  (0.9000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.9000000000,-0.3785761667) node {$a$};
+\draw []  (0.90000,0) node [rotate=0] {$\bullet$};
+\draw (0.90000,-0.37858) node {$a$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.4149246667) node {$0$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-0.4149246667) node {$1$};
-\draw [,->,>=latex] (0.4500000000,0.3000000000) -- (0,0.3000000000);
-\draw [,->,>=latex] (0.4500000000,0.3000000000) -- (0.9000000000,0.3000000000);
-\draw (0.4500000000,0.6785761667) node {$a$};
-\draw [,->,>=latex] (1.950000000,0.3000000000) -- (0.9000000000,0.3000000000);
-\draw [,->,>=latex] (1.950000000,0.3000000000) -- (3.000000000,0.3000000000);
-\draw (1.950000000,0.7298256667) node {$1-a$};
+\draw (0,-0.41492) node {$0$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.41492) node {$1$};
+\draw [,->,>=latex] (0.45000,0.30000) -- (0,0.30000);
+\draw [,->,>=latex] (0.45000,0.30000) -- (0.90000,0.30000);
+\draw (0.45000,0.67858) node {$a$};
+\draw [,->,>=latex] (1.9500,0.30000) -- (0.90000,0.30000);
+\draw [,->,>=latex] (1.9500,0.30000) -- (3.0000,0.30000);
+\draw (1.9500,0.72983) node {$1-a$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IsomCarre.pstricks.recall b/src_phystricks/Fig_IsomCarre.pstricks.recall
index bc9176e86..52c75fa0f 100644
--- a/src_phystricks/Fig_IsomCarre.pstricks.recall
+++ b/src_phystricks/Fig_IsomCarre.pstricks.recall
@@ -90,15 +90,15 @@
 \draw [color=blue] (1.00,1.00) -- (-1.00,1.00);
 \draw [color=blue] (-1.00,1.00) -- (-1.00,-1.00);
 \draw [] (0,-1.50) -- (0,1.50);
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.207585845,1.195418678) node {\( A\)};
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.218294678,1.195418678) node {\( B\)};
-\draw []  (1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.214200678,-1.195418678) node {\( C\)};
-\draw []  (-1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.226875011,-1.195418678) node {\( D\)};
-\draw (0.1562573448,1.649286845) node {\( s\)};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.2076,1.1954) node {\( A\)};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.2183,1.1954) node {\( B\)};
+\draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (1.2142,-1.1954) node {\( C\)};
+\draw []  (-1.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.2269,-1.1954) node {\( D\)};
+\draw (0.15626,1.6493) node {\( s\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_JJAooWpimYW.pstricks.recall b/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
index 55e861180..00789f209 100644
--- a/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
+++ b/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
@@ -108,7 +108,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981634,0) -- (8.353981634,0);
+\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
 \draw [,->,>=latex] (0,-1.498867339) -- (0,1.499874128);
 %DEFAULT
 
diff --git a/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall b/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
index 17cebb6a9..8d281f24f 100644
--- a/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
+++ b/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
@@ -92,7 +92,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.212388980,0) -- (5.212388980,0);
+\draw [,->,>=latex] (-5.212388985,0) -- (5.212388985,0);
 \draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
 %DEFAULT
 
diff --git a/src_phystricks/Fig_LAfWmaN.pstricks.recall b/src_phystricks/Fig_LAfWmaN.pstricks.recall
index 43ddf6e0c..45c4fc2fa 100644
--- a/src_phystricks/Fig_LAfWmaN.pstricks.recall
+++ b/src_phystricks/Fig_LAfWmaN.pstricks.recall
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.620000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.100000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.6200,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.1000);
 %DEFAULT
 \draw [color=blue] (2.000,-2.000)--(1.896,-1.974)--(1.795,-1.947)--(1.697,-1.921)--(1.602,-1.895)--(1.509,-1.869)--(1.419,-1.842)--(1.332,-1.816)--(1.248,-1.790)--(1.166,-1.764)--(1.087,-1.737)--(1.011,-1.711)--(0.9380,-1.685)--(0.8675,-1.659)--(0.7997,-1.632)--(0.7346,-1.606)--(0.6723,-1.580)--(0.6128,-1.554)--(0.5560,-1.527)--(0.5020,-1.501)--(0.4508,-1.475)--(0.4023,-1.448)--(0.3565,-1.422)--(0.3136,-1.396)--(0.2734,-1.370)--(0.2359,-1.343)--(0.2012,-1.317)--(0.1693,-1.291)--(0.1401,-1.265)--(0.1137,-1.238)--(0.08999,-1.212)--(0.06909,-1.186)--(0.05094,-1.160)--(0.03556,-1.133)--(0.02293,-1.107)--(0.01306,-1.081)--(0.005950,-1.055)--(0.001600,-1.028)--(0,-1.002)--(0.001175,-0.9758)--(0.005102,-0.9495)--(0.01179,-0.9232)--(0.02123,-0.8970)--(0.03343,-0.8707)--(0.04840,-0.8444)--(0.06612,-0.8182)--(0.08660,-0.7919)--(0.1098,-0.7657)--(0.1358,-0.7394)--(0.1646,-0.7131)--(0.1961,-0.6869)--(0.2304,-0.6606)--(0.2674,-0.6343)--(0.3072,-0.6081)--(0.3498,-0.5818)--(0.3951,-0.5556)--(0.4431,-0.5293)--(0.4940,-0.5030)--(0.5475,-0.4768)--(0.6039,-0.4505)--(0.6630,-0.4242)--(0.7249,-0.3980)--(0.7895,-0.3717)--(0.8569,-0.3455)--(0.9270,-0.3192)--(0.9999,-0.2929)--(1.076,-0.2667)--(1.154,-0.2404)--(1.235,-0.2141)--(1.319,-0.1879)--(1.406,-0.1616)--(1.495,-0.1354)--(1.587,-0.1091)--(1.682,-0.08283)--(1.780,-0.05657)--(1.881,-0.03030)--(1.984,-0.004040)--(2.090,0.02222)--(2.199,0.04848)--(2.310,0.07475)--(2.424,0.1010)--(2.541,0.1273)--(2.661,0.1535)--(2.784,0.1798)--(2.909,0.2061)--(3.037,0.2323)--(3.168,0.2586)--(3.302,0.2848)--(3.438,0.3111)--(3.577,0.3374)--(3.719,0.3636)--(3.864,0.3899)--(4.011,0.4162)--(4.161,0.4424)--(4.314,0.4687)--(4.470,0.4949)--(4.628,0.5212)--(4.789,0.5475)--(4.953,0.5737)--(5.120,0.6000);
 \draw [color=blue] (0,-2.000)--(0.3230,-1.974)--(0.4553,-1.947)--(0.5558,-1.921)--(0.6397,-1.895)--(0.7127,-1.869)--(0.7781,-1.842)--(0.8376,-1.816)--(0.8923,-1.790)--(0.9432,-1.764)--(0.9907,-1.737)--(1.035,-1.711)--(1.078,-1.685)--(1.118,-1.659)--(1.156,-1.632)--(1.192,-1.606)--(1.226,-1.580)--(1.260,-1.554)--(1.291,-1.527)--(1.322,-1.501)--(1.351,-1.475)--(1.379,-1.448)--(1.406,-1.422)--(1.432,-1.396)--(1.457,-1.370)--(1.482,-1.343)--(1.505,-1.317)--(1.528,-1.291)--(1.549,-1.265)--(1.570,-1.238)--(1.591,-1.212)--(1.611,-1.186)--(1.630,-1.160)--(1.648,-1.133)--(1.666,-1.107)--(1.683,-1.081)--(1.699,-1.055)--(1.715,-1.028)--(1.731,-1.002)--(1.746,-0.9758)--(1.760,-0.9495)--(1.774,-0.9232)--(1.788,-0.8970)--(1.801,-0.8707)--(1.813,-0.8444)--(1.825,-0.8182)--(1.837,-0.7919)--(1.848,-0.7657)--(1.858,-0.7394)--(1.869,-0.7131)--(1.878,-0.6869)--(1.888,-0.6606)--(1.897,-0.6343)--(1.905,-0.6081)--(1.913,-0.5818)--(1.921,-0.5556)--(1.929,-0.5293)--(1.936,-0.5030)--(1.942,-0.4768)--(1.949,-0.4505)--(1.954,-0.4242)--(1.960,-0.3980)--(1.965,-0.3717)--(1.970,-0.3455)--(1.974,-0.3192)--(1.978,-0.2929)--(1.982,-0.2667)--(1.986,-0.2404)--(1.989,-0.2141)--(1.991,-0.1879)--(1.993,-0.1616)--(1.995,-0.1354)--(1.997,-0.1091)--(1.998,-0.08283)--(1.999,-0.05657)--(2.000,-0.03030)--(2.000,-0.004040)--(2.000,0.02222)--(1.999,0.04848)--(1.999,0.07475)--(1.997,0.1010)--(1.996,0.1273)--(1.994,0.1535)--(1.992,0.1798)--(1.989,0.2061)--(1.986,0.2323)--(1.983,0.2586)--(1.980,0.2848)--(1.976,0.3111)--(1.971,0.3374)--(1.967,0.3636)--(1.962,0.3899)--(1.956,0.4162)--(1.950,0.4424)--(1.944,0.4687)--(1.938,0.4949)--(1.931,0.5212)--(1.924,0.5475)--(1.916,0.5737)--(1.908,0.6000);
@@ -100,23 +100,23 @@
 \draw [color=blue] (-1.000,0.5000)--(-0.9394,0.5000)--(-0.8788,0.5000)--(-0.8182,0.5000)--(-0.7576,0.5000)--(-0.6970,0.5000)--(-0.6364,0.5000)--(-0.5758,0.5000)--(-0.5152,0.5000)--(-0.4545,0.5000)--(-0.3939,0.5000)--(-0.3333,0.5000)--(-0.2727,0.5000)--(-0.2121,0.5000)--(-0.1515,0.5000)--(-0.09091,0.5000)--(-0.03030,0.5000)--(0.03030,0.5000)--(0.09091,0.5000)--(0.1515,0.5000)--(0.2121,0.5000)--(0.2727,0.5000)--(0.3333,0.5000)--(0.3939,0.5000)--(0.4545,0.5000)--(0.5152,0.5000)--(0.5758,0.5000)--(0.6364,0.5000)--(0.6970,0.5000)--(0.7576,0.5000)--(0.8182,0.5000)--(0.8788,0.5000)--(0.9394,0.5000)--(1.000,0.5000)--(1.061,0.5000)--(1.121,0.5000)--(1.182,0.5000)--(1.242,0.5000)--(1.303,0.5000)--(1.364,0.5000)--(1.424,0.5000)--(1.485,0.5000)--(1.545,0.5000)--(1.606,0.5000)--(1.667,0.5000)--(1.727,0.5000)--(1.788,0.5000)--(1.848,0.5000)--(1.909,0.5000)--(1.970,0.5000)--(2.030,0.5000)--(2.091,0.5000)--(2.152,0.5000)--(2.212,0.5000)--(2.273,0.5000)--(2.333,0.5000)--(2.394,0.5000)--(2.455,0.5000)--(2.515,0.5000)--(2.576,0.5000)--(2.636,0.5000)--(2.697,0.5000)--(2.758,0.5000)--(2.818,0.5000)--(2.879,0.5000)--(2.939,0.5000)--(3.000,0.5000)--(3.061,0.5000)--(3.121,0.5000)--(3.182,0.5000)--(3.242,0.5000)--(3.303,0.5000)--(3.364,0.5000)--(3.424,0.5000)--(3.485,0.5000)--(3.545,0.5000)--(3.606,0.5000)--(3.667,0.5000)--(3.727,0.5000)--(3.788,0.5000)--(3.848,0.5000)--(3.909,0.5000)--(3.970,0.5000)--(4.030,0.5000)--(4.091,0.5000)--(4.151,0.5000)--(4.212,0.5000)--(4.273,0.5000)--(4.333,0.5000)--(4.394,0.5000)--(4.455,0.5000)--(4.515,0.5000)--(4.576,0.5000)--(4.636,0.5000)--(4.697,0.5000)--(4.758,0.5000)--(4.818,0.5000)--(4.879,0.5000)--(4.939,0.5000)--(5.000,0.5000);
 
 \draw [color=blue] (-1.000,-1.500)--(-0.9394,-1.500)--(-0.8788,-1.500)--(-0.8182,-1.500)--(-0.7576,-1.500)--(-0.6970,-1.500)--(-0.6364,-1.500)--(-0.5758,-1.500)--(-0.5152,-1.500)--(-0.4545,-1.500)--(-0.3939,-1.500)--(-0.3333,-1.500)--(-0.2727,-1.500)--(-0.2121,-1.500)--(-0.1515,-1.500)--(-0.09091,-1.500)--(-0.03030,-1.500)--(0.03030,-1.500)--(0.09091,-1.500)--(0.1515,-1.500)--(0.2121,-1.500)--(0.2727,-1.500)--(0.3333,-1.500)--(0.3939,-1.500)--(0.4545,-1.500)--(0.5152,-1.500)--(0.5758,-1.500)--(0.6364,-1.500)--(0.6970,-1.500)--(0.7576,-1.500)--(0.8182,-1.500)--(0.8788,-1.500)--(0.9394,-1.500)--(1.000,-1.500)--(1.061,-1.500)--(1.121,-1.500)--(1.182,-1.500)--(1.242,-1.500)--(1.303,-1.500)--(1.364,-1.500)--(1.424,-1.500)--(1.485,-1.500)--(1.545,-1.500)--(1.606,-1.500)--(1.667,-1.500)--(1.727,-1.500)--(1.788,-1.500)--(1.848,-1.500)--(1.909,-1.500)--(1.970,-1.500)--(2.030,-1.500)--(2.091,-1.500)--(2.152,-1.500)--(2.212,-1.500)--(2.273,-1.500)--(2.333,-1.500)--(2.394,-1.500)--(2.455,-1.500)--(2.515,-1.500)--(2.576,-1.500)--(2.636,-1.500)--(2.697,-1.500)--(2.758,-1.500)--(2.818,-1.500)--(2.879,-1.500)--(2.939,-1.500)--(3.000,-1.500)--(3.061,-1.500)--(3.121,-1.500)--(3.182,-1.500)--(3.242,-1.500)--(3.303,-1.500)--(3.364,-1.500)--(3.424,-1.500)--(3.485,-1.500)--(3.545,-1.500)--(3.606,-1.500)--(3.667,-1.500)--(3.727,-1.500)--(3.788,-1.500)--(3.848,-1.500)--(3.909,-1.500)--(3.970,-1.500)--(4.030,-1.500)--(4.091,-1.500)--(4.151,-1.500)--(4.212,-1.500)--(4.273,-1.500)--(4.333,-1.500)--(4.394,-1.500)--(4.455,-1.500)--(4.515,-1.500)--(4.576,-1.500)--(4.636,-1.500)--(4.697,-1.500)--(4.758,-1.500)--(4.818,-1.500)--(4.879,-1.500)--(4.939,-1.500)--(5.000,-1.500);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LBGooAdteCt.pstricks.recall b/src_phystricks/Fig_LBGooAdteCt.pstricks.recall
index 516c4289e..5bc36bb3b 100644
--- a/src_phystricks/Fig_LBGooAdteCt.pstricks.recall
+++ b/src_phystricks/Fig_LBGooAdteCt.pstricks.recall
@@ -126,35 +126,35 @@
 \draw [color=gray,style=solid] (-5.00,4.00) -- (2.00,4.00);
 \draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
 
 \draw [color=blue] (-5.000,0.006738)--(-4.934,0.007200)--(-4.867,0.007694)--(-4.801,0.008221)--(-4.735,0.008785)--(-4.668,0.009388)--(-4.602,0.01003)--(-4.536,0.01072)--(-4.469,0.01145)--(-4.403,0.01224)--(-4.337,0.01308)--(-4.270,0.01398)--(-4.204,0.01493)--(-4.138,0.01596)--(-4.071,0.01705)--(-4.005,0.01822)--(-3.939,0.01947)--(-3.872,0.02081)--(-3.806,0.02223)--(-3.740,0.02376)--(-3.673,0.02539)--(-3.607,0.02713)--(-3.541,0.02899)--(-3.474,0.03098)--(-3.408,0.03310)--(-3.342,0.03537)--(-3.275,0.03780)--(-3.209,0.04039)--(-3.143,0.04316)--(-3.076,0.04612)--(-3.010,0.04928)--(-2.944,0.05266)--(-2.878,0.05628)--(-2.811,0.06014)--(-2.745,0.06426)--(-2.678,0.06867)--(-2.612,0.07337)--(-2.546,0.07841)--(-2.480,0.08378)--(-2.413,0.08953)--(-2.347,0.09567)--(-2.281,0.1022)--(-2.214,0.1092)--(-2.148,0.1167)--(-2.082,0.1247)--(-2.015,0.1333)--(-1.949,0.1424)--(-1.883,0.1522)--(-1.816,0.1626)--(-1.750,0.1738)--(-1.684,0.1857)--(-1.617,0.1984)--(-1.551,0.2121)--(-1.485,0.2266)--(-1.418,0.2421)--(-1.352,0.2587)--(-1.286,0.2765)--(-1.219,0.2954)--(-1.153,0.3157)--(-1.087,0.3374)--(-1.020,0.3605)--(-0.9540,0.3852)--(-0.8876,0.4116)--(-0.8213,0.4399)--(-0.7550,0.4700)--(-0.6886,0.5023)--(-0.6223,0.5367)--(-0.5560,0.5735)--(-0.4897,0.6128)--(-0.4233,0.6549)--(-0.3570,0.6998)--(-0.2907,0.7478)--(-0.2243,0.7990)--(-0.1580,0.8538)--(-0.09169,0.9124)--(-0.02536,0.9750)--(0.04097,1.042)--(0.1073,1.113)--(0.1736,1.190)--(0.2400,1.271)--(0.3063,1.358)--(0.3726,1.452)--(0.4389,1.551)--(0.5053,1.657)--(0.5716,1.771)--(0.6379,1.893)--(0.7043,2.022)--(0.7706,2.161)--(0.8369,2.309)--(0.9032,2.468)--(0.9696,2.637)--(1.036,2.818)--(1.102,3.011)--(1.169,3.217)--(1.235,3.438)--(1.301,3.674)--(1.368,3.926)--(1.434,4.195)--(1.500,4.483)--(1.567,4.790);
 
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall b/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall
index 1dd005af5..03400f4b5 100644
--- a/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall
+++ b/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -79,14 +79,14 @@
          hatchthickness=0.4pt}
 \fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,0) -- (0.0303,0.0303) -- (0.0606,0.0606) -- (0.0909,0.0909) -- (0.121,0.121) -- (0.152,0.152) -- (0.182,0.182) -- (0.212,0.212) -- (0.242,0.242) -- (0.273,0.273) -- (0.303,0.303) -- (0.333,0.333) -- (0.364,0.364) -- (0.394,0.394) -- (0.424,0.424) -- (0.455,0.455) -- (0.485,0.485) -- (0.515,0.515) -- (0.545,0.545) -- (0.576,0.576) -- (0.606,0.606) -- (0.636,0.636) -- (0.667,0.667) -- (0.697,0.697) -- (0.727,0.727) -- (0.758,0.758) -- (0.788,0.788) -- (0.818,0.818) -- (0.849,0.849) -- (0.879,0.879) -- (0.909,0.909) -- (0.939,0.939) -- (0.970,0.970) -- (1.00,1.00) -- (1.03,1.03) -- (1.06,1.06) -- (1.09,1.09) -- (1.12,1.12) -- (1.15,1.15) -- (1.18,1.18) -- (1.21,1.21) -- (1.24,1.24) -- (1.27,1.27) -- (1.30,1.30) -- (1.33,1.33) -- (1.36,1.36) -- (1.39,1.39) -- (1.42,1.42) -- (1.45,1.45) -- (1.48,1.48) -- (1.52,1.52) -- (1.55,1.55) -- (1.58,1.58) -- (1.61,1.61) -- (1.64,1.64) -- (1.67,1.67) -- (1.70,1.70) -- (1.73,1.73) -- (1.76,1.76) -- (1.79,1.79) -- (1.82,1.82) -- (1.85,1.85) -- (1.88,1.88) -- (1.91,1.91) -- (1.94,1.94) -- (1.97,1.97) -- (2.00,2.00) -- (2.03,2.03) -- (2.06,2.06) -- (2.09,2.09) -- (2.12,2.12) -- (2.15,2.15) -- (2.18,2.18) -- (2.21,2.21) -- (2.24,2.24) -- (2.27,2.27) -- (2.30,2.30) -- (2.33,2.33) -- (2.36,2.36) -- (2.39,2.39) -- (2.42,2.42) -- (2.45,2.45) -- (2.48,2.48) -- (2.52,2.52) -- (2.55,2.55) -- (2.58,2.58) -- (2.61,2.61) -- (2.64,2.64) -- (2.67,2.67) -- (2.70,2.70) -- (2.73,2.73) -- (2.76,2.76) -- (2.79,2.79) -- (2.82,2.82) -- (2.85,2.85) -- (2.88,2.88) -- (2.91,2.91) -- (2.94,2.94) -- (2.97,2.97) -- (3.00,3.00) -- (3.00,3.00) -- (3.00,3.00) -- (3.00,3.00) -- (2.97,2.94) -- (2.94,2.88) -- (2.91,2.82) -- (2.88,2.76) -- (2.85,2.70) -- (2.82,2.65) -- (2.79,2.59) -- (2.76,2.53) -- (2.73,2.48) -- (2.70,2.42) -- (2.67,2.37) -- (2.64,2.32) -- (2.61,2.26) -- (2.58,2.21) -- (2.55,2.16) -- (2.52,2.11) -- (2.48,2.06) -- (2.45,2.01) -- (2.42,1.96) -- (2.39,1.91) -- (2.36,1.86) -- (2.33,1.81) -- (2.30,1.77) -- (2.27,1.72) -- (2.24,1.68) -- (2.21,1.63) -- (2.18,1.59) -- (2.15,1.54) -- (2.12,1.50) -- (2.09,1.46) -- (2.06,1.42) -- (2.03,1.37) -- (2.00,1.33) -- (1.97,1.29) -- (1.94,1.25) -- (1.91,1.21) -- (1.88,1.18) -- (1.85,1.14) -- (1.82,1.10) -- (1.79,1.07) -- (1.76,1.03) -- (1.73,0.995) -- (1.70,0.960) -- (1.67,0.926) -- (1.64,0.893) -- (1.61,0.860) -- (1.58,0.828) -- (1.55,0.796) -- (1.52,0.765) -- (1.48,0.735) -- (1.45,0.705) -- (1.42,0.676) -- (1.39,0.648) -- (1.36,0.620) -- (1.33,0.593) -- (1.30,0.566) -- (1.27,0.540) -- (1.24,0.515) -- (1.21,0.490) -- (1.18,0.466) -- (1.15,0.442) -- (1.12,0.419) -- (1.09,0.397) -- (1.06,0.375) -- (1.03,0.354) -- (1.00,0.333) -- (0.970,0.313) -- (0.939,0.294) -- (0.909,0.275) -- (0.879,0.257) -- (0.849,0.240) -- (0.818,0.223) -- (0.788,0.207) -- (0.758,0.191) -- (0.727,0.176) -- (0.697,0.162) -- (0.667,0.148) -- (0.636,0.135) -- (0.606,0.122) -- (0.576,0.110) -- (0.545,0.0992) -- (0.515,0.0885) -- (0.485,0.0784) -- (0.455,0.0689) -- (0.424,0.0600) -- (0.394,0.0517) -- (0.364,0.0441) -- (0.333,0.0370) -- (0.303,0.0306) -- (0.273,0.0248) -- (0.242,0.0196) -- (0.212,0.0150) -- (0.182,0.0110) -- (0.152,0.00765) -- (0.121,0.00490) -- (0.0909,0.00275) -- (0.0606,0.00122) -- (0.0303,0) -- (0,0) -- (0,0) -- (0,0) --  cycle;
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-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.478786797,1.478786797);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.4788,1.4788);
 \draw [color=green] (0,0)--(0.0303,0)--(0.0606,0.00122)--(0.0909,0.00275)--(0.121,0.00490)--(0.152,0.00765)--(0.182,0.0110)--(0.212,0.0150)--(0.242,0.0196)--(0.273,0.0248)--(0.303,0.0306)--(0.333,0.0370)--(0.364,0.0441)--(0.394,0.0517)--(0.424,0.0600)--(0.455,0.0689)--(0.485,0.0784)--(0.515,0.0885)--(0.545,0.0992)--(0.576,0.110)--(0.606,0.122)--(0.636,0.135)--(0.667,0.148)--(0.697,0.162)--(0.727,0.176)--(0.758,0.191)--(0.788,0.207)--(0.818,0.223)--(0.849,0.240)--(0.879,0.257)--(0.909,0.275)--(0.939,0.294)--(0.970,0.313)--(1.00,0.333)--(1.03,0.354)--(1.06,0.375)--(1.09,0.397)--(1.12,0.419)--(1.15,0.442)--(1.18,0.466)--(1.21,0.490)--(1.24,0.515)--(1.27,0.540)--(1.30,0.566)--(1.33,0.593)--(1.36,0.620)--(1.39,0.648)--(1.42,0.676)--(1.45,0.705)--(1.48,0.735)--(1.52,0.765)--(1.55,0.796)--(1.58,0.828)--(1.61,0.860)--(1.64,0.893)--(1.67,0.926)--(1.70,0.960)--(1.73,0.995)--(1.76,1.03)--(1.79,1.07)--(1.82,1.10)--(1.85,1.14)--(1.88,1.18)--(1.91,1.21)--(1.94,1.25)--(1.97,1.29)--(2.00,1.33)--(2.03,1.37)--(2.06,1.42)--(2.09,1.46)--(2.12,1.50)--(2.15,1.54)--(2.18,1.59)--(2.21,1.63)--(2.24,1.68)--(2.27,1.72)--(2.30,1.77)--(2.33,1.81)--(2.36,1.86)--(2.39,1.91)--(2.42,1.96)--(2.45,2.01)--(2.48,2.06)--(2.52,2.11)--(2.55,2.16)--(2.58,2.21)--(2.61,2.26)--(2.64,2.32)--(2.67,2.37)--(2.70,2.42)--(2.73,2.48)--(2.76,2.53)--(2.79,2.59)--(2.82,2.65)--(2.85,2.70)--(2.88,2.76)--(2.91,2.82)--(2.94,2.88)--(2.97,2.94)--(3.00,3.00);
-\draw [,->,>=latex] (1.500000000,0.7500000000) -- (1.521213203,0.7712132034);
+\draw [,->,>=latex] (1.5000,0.75000) -- (1.5212,0.77121);
 \draw [color=green] (0,0) -- (0,0);
 \draw [color=green] (3.00,3.00) -- (3.00,3.00);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall b/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall
index c7390d669..d1387e358 100644
--- a/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall
+++ b/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall
@@ -100,37 +100,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=blue] (-5.000,1.857)--(-4.939,1.856)--(-4.879,1.855)--(-4.818,1.853)--(-4.758,1.852)--(-4.697,1.851)--(-4.636,1.849)--(-4.576,1.848)--(-4.515,1.847)--(-4.455,1.845)--(-4.394,1.844)--(-4.333,1.842)--(-4.273,1.841)--(-4.212,1.839)--(-4.151,1.837)--(-4.091,1.836)--(-4.030,1.834)--(-3.970,1.832)--(-3.909,1.831)--(-3.848,1.829)--(-3.788,1.827)--(-3.727,1.825)--(-3.667,1.824)--(-3.606,1.822)--(-3.545,1.820)--(-3.485,1.818)--(-3.424,1.816)--(-3.364,1.814)--(-3.303,1.811)--(-3.242,1.809)--(-3.182,1.807)--(-3.121,1.805)--(-3.061,1.802)--(-3.000,1.800)--(-2.939,1.798)--(-2.879,1.795)--(-2.818,1.792)--(-2.758,1.790)--(-2.697,1.787)--(-2.636,1.784)--(-2.576,1.781)--(-2.515,1.779)--(-2.455,1.776)--(-2.394,1.772)--(-2.333,1.769)--(-2.273,1.766)--(-2.212,1.763)--(-2.152,1.759)--(-2.091,1.756)--(-2.030,1.752)--(-1.970,1.748)--(-1.909,1.744)--(-1.848,1.740)--(-1.788,1.736)--(-1.727,1.732)--(-1.667,1.727)--(-1.606,1.723)--(-1.545,1.718)--(-1.485,1.713)--(-1.424,1.708)--(-1.364,1.703)--(-1.303,1.697)--(-1.242,1.692)--(-1.182,1.686)--(-1.121,1.680)--(-1.061,1.673)--(-1.000,1.667)--(-0.9394,1.660)--(-0.8788,1.653)--(-0.8182,1.645)--(-0.7576,1.637)--(-0.6970,1.629)--(-0.6364,1.621)--(-0.5758,1.612)--(-0.5152,1.602)--(-0.4545,1.593)--(-0.3939,1.582)--(-0.3333,1.571)--(-0.2727,1.560)--(-0.2121,1.548)--(-0.1515,1.535)--(-0.09091,1.522)--(-0.03030,1.507)--(0.03030,1.492)--(0.09091,1.476)--(0.1515,1.459)--(0.2121,1.441)--(0.2727,1.421)--(0.3333,1.400)--(0.3939,1.377)--(0.4545,1.353)--(0.5152,1.327)--(0.5758,1.298)--(0.6364,1.267)--(0.6970,1.233)--(0.7576,1.195)--(0.8182,1.154)--(0.8788,1.108)--(0.9394,1.057)--(1.000,1.000);
 
 \draw [color=blue] (1.000,1.000)--(1.040,0.9612)--(1.081,0.9252)--(1.121,0.8919)--(1.162,0.8609)--(1.202,0.8319)--(1.242,0.8049)--(1.283,0.7795)--(1.323,0.7557)--(1.364,0.7333)--(1.404,0.7122)--(1.444,0.6923)--(1.485,0.6735)--(1.525,0.6556)--(1.566,0.6387)--(1.606,0.6226)--(1.646,0.6074)--(1.687,0.5928)--(1.727,0.5789)--(1.768,0.5657)--(1.808,0.5531)--(1.848,0.5410)--(1.889,0.5294)--(1.929,0.5183)--(1.970,0.5077)--(2.010,0.4975)--(2.051,0.4877)--(2.091,0.4783)--(2.131,0.4692)--(2.172,0.4605)--(2.212,0.4521)--(2.253,0.4439)--(2.293,0.4361)--(2.333,0.4286)--(2.374,0.4213)--(2.414,0.4142)--(2.455,0.4074)--(2.495,0.4008)--(2.535,0.3944)--(2.576,0.3882)--(2.616,0.3822)--(2.657,0.3764)--(2.697,0.3708)--(2.737,0.3653)--(2.778,0.3600)--(2.818,0.3548)--(2.859,0.3498)--(2.899,0.3449)--(2.939,0.3402)--(2.980,0.3356)--(3.020,0.3311)--(3.061,0.3267)--(3.101,0.3225)--(3.141,0.3183)--(3.182,0.3143)--(3.222,0.3103)--(3.263,0.3065)--(3.303,0.3028)--(3.343,0.2991)--(3.384,0.2955)--(3.424,0.2920)--(3.465,0.2886)--(3.505,0.2853)--(3.545,0.2821)--(3.586,0.2789)--(3.626,0.2758)--(3.667,0.2727)--(3.707,0.2698)--(3.747,0.2668)--(3.788,0.2640)--(3.828,0.2612)--(3.869,0.2585)--(3.909,0.2558)--(3.949,0.2532)--(3.990,0.2506)--(4.030,0.2481)--(4.071,0.2457)--(4.111,0.2432)--(4.151,0.2409)--(4.192,0.2386)--(4.232,0.2363)--(4.273,0.2340)--(4.313,0.2318)--(4.354,0.2297)--(4.394,0.2276)--(4.434,0.2255)--(4.475,0.2235)--(4.515,0.2215)--(4.556,0.2195)--(4.596,0.2176)--(4.636,0.2157)--(4.677,0.2138)--(4.717,0.2120)--(4.758,0.2102)--(4.798,0.2084)--(4.838,0.2067)--(4.879,0.2050)--(4.919,0.2033)--(4.960,0.2016)--(5.000,0.2000);
 \draw [style=dashed] (-5.00,2.00) -- (5.00,2.00);
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Laurin.pstricks.recall b/src_phystricks/Fig_Laurin.pstricks.recall
index f96d4776d..bd70fac13 100644
--- a/src_phystricks/Fig_Laurin.pstricks.recall
+++ b/src_phystricks/Fig_Laurin.pstricks.recall
@@ -99,8 +99,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,7.889056099);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,7.8891);
 %DEFAULT
 
 \draw [color=blue] (-2.000,0.1353)--(-1.960,0.1409)--(-1.919,0.1467)--(-1.879,0.1528)--(-1.838,0.1591)--(-1.798,0.1656)--(-1.758,0.1725)--(-1.717,0.1796)--(-1.677,0.1870)--(-1.636,0.1947)--(-1.596,0.2027)--(-1.556,0.2111)--(-1.515,0.2198)--(-1.475,0.2288)--(-1.434,0.2383)--(-1.394,0.2481)--(-1.354,0.2583)--(-1.313,0.2690)--(-1.273,0.2801)--(-1.232,0.2916)--(-1.192,0.3036)--(-1.152,0.3162)--(-1.111,0.3292)--(-1.071,0.3428)--(-1.030,0.3569)--(-0.9899,0.3716)--(-0.9495,0.3869)--(-0.9091,0.4029)--(-0.8687,0.4195)--(-0.8283,0.4368)--(-0.7879,0.4548)--(-0.7475,0.4736)--(-0.7071,0.4931)--(-0.6667,0.5134)--(-0.6263,0.5346)--(-0.5859,0.5566)--(-0.5455,0.5796)--(-0.5051,0.6035)--(-0.4646,0.6284)--(-0.4242,0.6543)--(-0.3838,0.6812)--(-0.3434,0.7093)--(-0.3030,0.7386)--(-0.2626,0.7690)--(-0.2222,0.8007)--(-0.1818,0.8338)--(-0.1414,0.8681)--(-0.1010,0.9039)--(-0.06061,0.9412)--(-0.02020,0.9800)--(0.02020,1.020)--(0.06061,1.062)--(0.1010,1.106)--(0.1414,1.152)--(0.1818,1.199)--(0.2222,1.249)--(0.2626,1.300)--(0.3030,1.354)--(0.3434,1.410)--(0.3838,1.468)--(0.4242,1.528)--(0.4646,1.591)--(0.5051,1.657)--(0.5455,1.725)--(0.5859,1.797)--(0.6263,1.871)--(0.6667,1.948)--(0.7071,2.028)--(0.7475,2.112)--(0.7879,2.199)--(0.8283,2.289)--(0.8687,2.384)--(0.9091,2.482)--(0.9495,2.584)--(0.9899,2.691)--(1.030,2.802)--(1.071,2.917)--(1.111,3.038)--(1.152,3.163)--(1.192,3.293)--(1.232,3.429)--(1.273,3.571)--(1.313,3.718)--(1.354,3.871)--(1.394,4.031)--(1.434,4.197)--(1.475,4.370)--(1.515,4.550)--(1.556,4.738)--(1.596,4.933)--(1.636,5.136)--(1.677,5.348)--(1.717,5.569)--(1.758,5.798)--(1.798,6.037)--(1.838,6.286)--(1.879,6.546)--(1.919,6.815)--(1.960,7.096)--(2.000,7.389);
@@ -110,29 +110,29 @@
 \draw [color=green] (-2.000,-1.000)--(-1.960,-0.9596)--(-1.919,-0.9192)--(-1.879,-0.8788)--(-1.838,-0.8384)--(-1.798,-0.7980)--(-1.758,-0.7576)--(-1.717,-0.7172)--(-1.677,-0.6768)--(-1.636,-0.6364)--(-1.596,-0.5960)--(-1.556,-0.5556)--(-1.515,-0.5152)--(-1.475,-0.4747)--(-1.434,-0.4343)--(-1.394,-0.3939)--(-1.354,-0.3535)--(-1.313,-0.3131)--(-1.273,-0.2727)--(-1.232,-0.2323)--(-1.192,-0.1919)--(-1.152,-0.1515)--(-1.111,-0.1111)--(-1.071,-0.07071)--(-1.030,-0.03030)--(-0.9899,0.01010)--(-0.9495,0.05051)--(-0.9091,0.09091)--(-0.8687,0.1313)--(-0.8283,0.1717)--(-0.7879,0.2121)--(-0.7475,0.2525)--(-0.7071,0.2929)--(-0.6667,0.3333)--(-0.6263,0.3737)--(-0.5859,0.4141)--(-0.5455,0.4545)--(-0.5051,0.4949)--(-0.4646,0.5354)--(-0.4242,0.5758)--(-0.3838,0.6162)--(-0.3434,0.6566)--(-0.3030,0.6970)--(-0.2626,0.7374)--(-0.2222,0.7778)--(-0.1818,0.8182)--(-0.1414,0.8586)--(-0.1010,0.8990)--(-0.06061,0.9394)--(-0.02020,0.9798)--(0.02020,1.020)--(0.06061,1.061)--(0.1010,1.101)--(0.1414,1.141)--(0.1818,1.182)--(0.2222,1.222)--(0.2626,1.263)--(0.3030,1.303)--(0.3434,1.343)--(0.3838,1.384)--(0.4242,1.424)--(0.4646,1.465)--(0.5051,1.505)--(0.5455,1.545)--(0.5859,1.586)--(0.6263,1.626)--(0.6667,1.667)--(0.7071,1.707)--(0.7475,1.747)--(0.7879,1.788)--(0.8283,1.828)--(0.8687,1.869)--(0.9091,1.909)--(0.9495,1.949)--(0.9899,1.990)--(1.030,2.030)--(1.071,2.071)--(1.111,2.111)--(1.152,2.152)--(1.192,2.192)--(1.232,2.232)--(1.273,2.273)--(1.313,2.313)--(1.354,2.354)--(1.394,2.394)--(1.434,2.434)--(1.475,2.475)--(1.515,2.515)--(1.556,2.556)--(1.596,2.596)--(1.636,2.636)--(1.677,2.677)--(1.717,2.717)--(1.758,2.758)--(1.798,2.798)--(1.838,2.838)--(1.879,2.879)--(1.919,2.919)--(1.960,2.960)--(2.000,3.000);
 
 \draw [color=red] (-2.000,1.000)--(-1.960,0.9604)--(-1.919,0.9225)--(-1.879,0.8861)--(-1.838,0.8514)--(-1.798,0.8184)--(-1.758,0.7870)--(-1.717,0.7572)--(-1.677,0.7290)--(-1.636,0.7025)--(-1.596,0.6776)--(-1.556,0.6543)--(-1.515,0.6327)--(-1.475,0.6127)--(-1.434,0.5943)--(-1.394,0.5776)--(-1.354,0.5625)--(-1.313,0.5490)--(-1.273,0.5372)--(-1.232,0.5270)--(-1.192,0.5184)--(-1.152,0.5115)--(-1.111,0.5062)--(-1.071,0.5025)--(-1.030,0.5005)--(-0.9899,0.5001)--(-0.9495,0.5013)--(-0.9091,0.5041)--(-0.8687,0.5086)--(-0.8283,0.5147)--(-0.7879,0.5225)--(-0.7475,0.5319)--(-0.7071,0.5429)--(-0.6667,0.5556)--(-0.6263,0.5698)--(-0.5859,0.5858)--(-0.5455,0.6033)--(-0.5051,0.6225)--(-0.4646,0.6433)--(-0.4242,0.6657)--(-0.3838,0.6898)--(-0.3434,0.7155)--(-0.3030,0.7429)--(-0.2626,0.7719)--(-0.2222,0.8025)--(-0.1818,0.8347)--(-0.1414,0.8686)--(-0.1010,0.9041)--(-0.06061,0.9412)--(-0.02020,0.9800)--(0.02020,1.020)--(0.06061,1.062)--(0.1010,1.106)--(0.1414,1.151)--(0.1818,1.198)--(0.2222,1.247)--(0.2626,1.297)--(0.3030,1.349)--(0.3434,1.402)--(0.3838,1.458)--(0.4242,1.514)--(0.4646,1.573)--(0.5051,1.633)--(0.5455,1.694)--(0.5859,1.757)--(0.6263,1.822)--(0.6667,1.889)--(0.7071,1.957)--(0.7475,2.027)--(0.7879,2.098)--(0.8283,2.171)--(0.8687,2.246)--(0.9091,2.322)--(0.9495,2.400)--(0.9899,2.480)--(1.030,2.561)--(1.071,2.644)--(1.111,2.728)--(1.152,2.815)--(1.192,2.902)--(1.232,2.992)--(1.273,3.083)--(1.313,3.175)--(1.354,3.270)--(1.394,3.365)--(1.434,3.463)--(1.475,3.562)--(1.515,3.663)--(1.556,3.765)--(1.596,3.870)--(1.636,3.975)--(1.677,4.083)--(1.717,4.192)--(1.758,4.302)--(1.798,4.414)--(1.838,4.528)--(1.879,4.644)--(1.919,4.761)--(1.960,4.880)--(2.000,5.000);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
-\draw (-0.2912498333,7.000000000) node {$ 7 $};
+\draw (-0.29125,7.0000) node {$ 7 $};
 \draw [] (-0.100,7.00) -- (0.100,7.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MCQueGF.pstricks.recall b/src_phystricks/Fig_MCQueGF.pstricks.recall
index 681ea85af..8594e6364 100644
--- a/src_phystricks/Fig_MCQueGF.pstricks.recall
+++ b/src_phystricks/Fig_MCQueGF.pstricks.recall
@@ -87,36 +87,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
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-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Mantisse.pstricks.recall b/src_phystricks/Fig_Mantisse.pstricks.recall
index 9669ab94c..1c275f207 100644
--- a/src_phystricks/Fig_Mantisse.pstricks.recall
+++ b/src_phystricks/Fig_Mantisse.pstricks.recall
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.498998999);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4990);
 %DEFAULT
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@@ -94,17 +94,17 @@
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MethodeNewton.pstricks.recall b/src_phystricks/Fig_MethodeNewton.pstricks.recall
index 3fdca5746..23469e25c 100644
--- a/src_phystricks/Fig_MethodeNewton.pstricks.recall
+++ b/src_phystricks/Fig_MethodeNewton.pstricks.recall
@@ -96,7 +96,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-2.000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-1.287500000) -- (0,4.400000000);
+\draw [,->,>=latex] (0,-1.287500000) -- (0,4.400000002);
 %DEFAULT
 
 \draw [color=blue] (-1.500,3.900)--(-1.424,3.713)--(-1.348,3.529)--(-1.273,3.349)--(-1.197,3.173)--(-1.121,3.001)--(-1.045,2.833)--(-0.9697,2.668)--(-0.8939,2.507)--(-0.8182,2.350)--(-0.7424,2.197)--(-0.6667,2.048)--(-0.5909,1.903)--(-0.5152,1.761)--(-0.4394,1.623)--(-0.3636,1.490)--(-0.2879,1.359)--(-0.2121,1.233)--(-0.1364,1.111)--(-0.06061,0.9921)--(0.01515,0.8773)--(0.09091,0.7664)--(0.1667,0.6593)--(0.2424,0.5560)--(0.3182,0.4565)--(0.3939,0.3608)--(0.4697,0.2690)--(0.5455,0.1810)--(0.6212,0.09682)--(0.6970,0.01647)--(0.7727,-0.06006)--(0.8485,-0.1328)--(0.9242,-0.2016)--(1.000,-0.2667)--(1.076,-0.3279)--(1.152,-0.3853)--(1.227,-0.4388)--(1.303,-0.4886)--(1.379,-0.5345)--(1.455,-0.5766)--(1.530,-0.6148)--(1.606,-0.6493)--(1.682,-0.6799)--(1.758,-0.7067)--(1.833,-0.7296)--(1.909,-0.7488)--(1.985,-0.7641)--(2.061,-0.7755)--(2.136,-0.7832)--(2.212,-0.7870)--(2.288,-0.7870)--(2.364,-0.7832)--(2.439,-0.7755)--(2.515,-0.7641)--(2.591,-0.7488)--(2.667,-0.7296)--(2.742,-0.7067)--(2.818,-0.6799)--(2.894,-0.6493)--(2.970,-0.6148)--(3.045,-0.5766)--(3.121,-0.5345)--(3.197,-0.4886)--(3.273,-0.4388)--(3.348,-0.3853)--(3.424,-0.3279)--(3.500,-0.2667)--(3.576,-0.2016)--(3.652,-0.1328)--(3.727,-0.06006)--(3.803,0.01647)--(3.879,0.09682)--(3.955,0.1810)--(4.030,0.2690)--(4.106,0.3608)--(4.182,0.4565)--(4.258,0.5560)--(4.333,0.6593)--(4.409,0.7664)--(4.485,0.8773)--(4.561,0.9921)--(4.636,1.111)--(4.712,1.233)--(4.788,1.359)--(4.864,1.490)--(4.939,1.623)--(5.015,1.761)--(5.091,1.903)--(5.167,2.048)--(5.242,2.197)--(5.318,2.350)--(5.394,2.507)--(5.470,2.668)--(5.545,2.833)--(5.621,3.001)--(5.697,3.173)--(5.773,3.349)--(5.849,3.529)--(5.924,3.713)--(6.000,3.900);
diff --git a/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall b/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall
index 51c4c7dea..9dc95b22d 100644
--- a/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall
+++ b/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.662277660) -- (0,3.662277660);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
 %DEFAULT
 
 \draw [color=red] (-3.000,3.162)--(-2.939,3.105)--(-2.879,3.048)--(-2.818,2.990)--(-2.758,2.933)--(-2.697,2.876)--(-2.636,2.820)--(-2.576,2.763)--(-2.515,2.707)--(-2.455,2.650)--(-2.394,2.594)--(-2.333,2.539)--(-2.273,2.483)--(-2.212,2.428)--(-2.152,2.373)--(-2.091,2.318)--(-2.030,2.263)--(-1.970,2.209)--(-1.909,2.155)--(-1.848,2.102)--(-1.788,2.049)--(-1.727,1.996)--(-1.667,1.944)--(-1.606,1.892)--(-1.545,1.841)--(-1.485,1.790)--(-1.424,1.740)--(-1.364,1.691)--(-1.303,1.643)--(-1.242,1.595)--(-1.182,1.548)--(-1.121,1.502)--(-1.061,1.458)--(-1.000,1.414)--(-0.9394,1.372)--(-0.8788,1.331)--(-0.8182,1.292)--(-0.7576,1.255)--(-0.6970,1.219)--(-0.6364,1.185)--(-0.5758,1.154)--(-0.5152,1.125)--(-0.4545,1.098)--(-0.3939,1.075)--(-0.3333,1.054)--(-0.2727,1.037)--(-0.2121,1.022)--(-0.1515,1.011)--(-0.09091,1.004)--(-0.03030,1.000)--(0.03030,1.000)--(0.09091,1.004)--(0.1515,1.011)--(0.2121,1.022)--(0.2727,1.037)--(0.3333,1.054)--(0.3939,1.075)--(0.4545,1.098)--(0.5152,1.125)--(0.5758,1.154)--(0.6364,1.185)--(0.6970,1.219)--(0.7576,1.255)--(0.8182,1.292)--(0.8788,1.331)--(0.9394,1.372)--(1.000,1.414)--(1.061,1.458)--(1.121,1.502)--(1.182,1.548)--(1.242,1.595)--(1.303,1.643)--(1.364,1.691)--(1.424,1.740)--(1.485,1.790)--(1.545,1.841)--(1.606,1.892)--(1.667,1.944)--(1.727,1.996)--(1.788,2.049)--(1.848,2.102)--(1.909,2.155)--(1.970,2.209)--(2.030,2.263)--(2.091,2.318)--(2.152,2.373)--(2.212,2.428)--(2.273,2.483)--(2.333,2.539)--(2.394,2.594)--(2.455,2.650)--(2.515,2.707)--(2.576,2.763)--(2.636,2.820)--(2.697,2.876)--(2.758,2.933)--(2.818,2.990)--(2.879,3.048)--(2.939,3.105)--(3.000,3.162);
@@ -104,33 +104,33 @@
 \draw [color=red] (-3.000,-3.162)--(-2.939,-3.105)--(-2.879,-3.048)--(-2.818,-2.990)--(-2.758,-2.933)--(-2.697,-2.876)--(-2.636,-2.820)--(-2.576,-2.763)--(-2.515,-2.707)--(-2.455,-2.650)--(-2.394,-2.594)--(-2.333,-2.539)--(-2.273,-2.483)--(-2.212,-2.428)--(-2.152,-2.373)--(-2.091,-2.318)--(-2.030,-2.263)--(-1.970,-2.209)--(-1.909,-2.155)--(-1.848,-2.102)--(-1.788,-2.049)--(-1.727,-1.996)--(-1.667,-1.944)--(-1.606,-1.892)--(-1.545,-1.841)--(-1.485,-1.790)--(-1.424,-1.740)--(-1.364,-1.691)--(-1.303,-1.643)--(-1.242,-1.595)--(-1.182,-1.548)--(-1.121,-1.502)--(-1.061,-1.458)--(-1.000,-1.414)--(-0.9394,-1.372)--(-0.8788,-1.331)--(-0.8182,-1.292)--(-0.7576,-1.255)--(-0.6970,-1.219)--(-0.6364,-1.185)--(-0.5758,-1.154)--(-0.5152,-1.125)--(-0.4545,-1.098)--(-0.3939,-1.075)--(-0.3333,-1.054)--(-0.2727,-1.037)--(-0.2121,-1.022)--(-0.1515,-1.011)--(-0.09091,-1.004)--(-0.03030,-1.000)--(0.03030,-1.000)--(0.09091,-1.004)--(0.1515,-1.011)--(0.2121,-1.022)--(0.2727,-1.037)--(0.3333,-1.054)--(0.3939,-1.075)--(0.4545,-1.098)--(0.5152,-1.125)--(0.5758,-1.154)--(0.6364,-1.185)--(0.6970,-1.219)--(0.7576,-1.255)--(0.8182,-1.292)--(0.8788,-1.331)--(0.9394,-1.372)--(1.000,-1.414)--(1.061,-1.458)--(1.121,-1.502)--(1.182,-1.548)--(1.242,-1.595)--(1.303,-1.643)--(1.364,-1.691)--(1.424,-1.740)--(1.485,-1.790)--(1.545,-1.841)--(1.606,-1.892)--(1.667,-1.944)--(1.727,-1.996)--(1.788,-2.049)--(1.848,-2.102)--(1.909,-2.155)--(1.970,-2.209)--(2.030,-2.263)--(2.091,-2.318)--(2.152,-2.373)--(2.212,-2.428)--(2.273,-2.483)--(2.333,-2.539)--(2.394,-2.594)--(2.455,-2.650)--(2.515,-2.707)--(2.576,-2.763)--(2.636,-2.820)--(2.697,-2.876)--(2.758,-2.933)--(2.818,-2.990)--(2.879,-3.048)--(2.939,-3.105)--(3.000,-3.162);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.2106906781,0.8045813219) node {$R$};
-\draw []  (0,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.1931375115,-0.8045813219) node {$S$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.21069,0.80458) node {$R$};
+\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.19314,-0.80458) node {$S$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall b/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
index 47759ee2f..152de5dbe 100644
--- a/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
+++ b/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
@@ -117,7 +117,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-4.900000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-2.813086866) -- (0,3.699326205);
+\draw [,->,>=latex] (0,-2.813086867) -- (0,3.699326205);
 %DEFAULT
 
 \draw [color=blue] (-4.4000,-2.3131)--(-4.3051,-1.7786)--(-4.2101,-1.3073)--(-4.1152,-0.89319)--(-4.0202,-0.53075)--(-3.9253,-0.21496)--(-3.8303,0.058716)--(-3.7354,0.29443)--(-3.6404,0.49595)--(-3.5455,0.66669)--(-3.4505,0.80977)--(-3.3556,0.92802)--(-3.2606,1.0240)--(-3.1657,1.1001)--(-3.0707,1.1584)--(-2.9758,1.2009)--(-2.8808,1.2293)--(-2.7859,1.2452)--(-2.6909,1.2501)--(-2.5960,1.2452)--(-2.5010,1.2319)--(-2.4061,1.2112)--(-2.3111,1.1841)--(-2.2162,1.1515)--(-2.1212,1.1143)--(-2.0263,1.0731)--(-1.9313,1.0287)--(-1.8364,0.98162)--(-1.7414,0.93253)--(-1.6465,0.88190)--(-1.5515,0.83018)--(-1.4566,0.77782)--(-1.3616,0.72520)--(-1.2667,0.67266)--(-1.1717,0.62052)--(-1.0768,0.56908)--(-0.98182,0.51859)--(-0.88687,0.46930)--(-0.79192,0.42143)--(-0.69697,0.37518)--(-0.60202,0.33071)--(-0.50707,0.28821)--(-0.41212,0.24782)--(-0.31717,0.20966)--(-0.22222,0.17388)--(-0.12727,0.14058)--(-0.032323,0.10985)--(0.062626,0.081805)--(0.15758,0.056515)--(0.25253,0.034060)--(0.34747,0.014511)--(0.44242,-0.0020696)--(0.53737,-0.015623)--(0.63232,-0.026098)--(0.72727,-0.033446)--(0.82222,-0.037624)--(0.91717,-0.038593)--(1.0121,-0.036315)--(1.1071,-0.030760)--(1.2020,-0.021896)--(1.2970,-0.0096977)--(1.3919,0.0058604)--(1.4869,0.024800)--(1.5818,0.047142)--(1.6768,0.072905)--(1.7717,0.10211)--(1.8667,0.13476)--(1.9616,0.17088)--(2.0566,0.21048)--(2.1515,0.25357)--(2.2465,0.30016)--(2.3414,0.35026)--(2.4364,0.40388)--(2.5313,0.46103)--(2.6263,0.52171)--(2.7212,0.58593)--(2.8162,0.65370)--(2.9111,0.72503)--(3.0061,0.79991)--(3.1010,0.87835)--(3.1960,0.96035)--(3.2909,1.0459)--(3.3859,1.1351)--(3.4808,1.2278)--(3.5758,1.3241)--(3.6707,1.4240)--(3.7657,1.5275)--(3.8606,1.6345)--(3.9556,1.7451)--(4.0505,1.8594)--(4.1455,1.9772)--(4.2404,2.0986)--(4.3354,2.2236)--(4.4303,2.3522)--(4.5253,2.4844)--(4.6202,2.6202)--(4.7151,2.7596)--(4.8101,2.9026)--(4.9051,3.0491)--(5.0000,3.1993);
diff --git a/src_phystricks/Fig_ProjectionScalaire.pstricks.recall b/src_phystricks/Fig_ProjectionScalaire.pstricks.recall
index 89cf83d91..aa94342af 100644
--- a/src_phystricks/Fig_ProjectionScalaire.pstricks.recall
+++ b/src_phystricks/Fig_ProjectionScalaire.pstricks.recall
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
-\draw [color=blue,->,>=latex] (0,0) -- (1.500000000,2.000000000);
-\draw (1.877649201,2.336840034) node {$X$};
-\draw [color=blue,->,>=latex] (0,0) -- (2.500000000,0);
-\draw (2.500000000,0.3247080000) node {$Y$};
-\draw []  (1.500000000,0) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (0,0) -- (1.5000,2.0000);
+\draw (1.8776,2.3368) node {$X$};
+\draw [color=blue,->,>=latex] (0,0) -- (2.5000,0);
+\draw (2.5000,0.32471) node {$Y$};
+\draw []  (1.5000,0) node [rotate=0] {$\bullet$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (0.7500000000,-0.3000000000) -- (0,-0.3000000000);
-\draw [,->,>=latex] (0.7500000000,-0.3000000000) -- (1.500000000,-0.3000000000);
-\draw (0.7500000000,-0.6785761667) node {$x$};
+\draw [,->,>=latex] (0.75000,-0.30000) -- (0,-0.30000);
+\draw [,->,>=latex] (0.75000,-0.30000) -- (1.5000,-0.30000);
+\draw (0.75000,-0.67858) node {$x$};
 \draw [style=dotted] (1.50,2.00) -- (1.50,0);
 
 %OTHER STUFF
diff --git a/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall b/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall
index 8f4f807c0..3b06e3eda 100644
--- a/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall
+++ b/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall
@@ -78,12 +78,12 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (-1.411630648,1.580406821) node {\( A\)};
-\draw (-1.979984611,-0.7916472795) node {\( B\)};
-\draw (1.975890611,0.7916472795) node {\( C\)};
+\draw (-1.4116,1.5804) node {\( A\)};
+\draw (-1.9800,-0.79165) node {\( B\)};
+\draw (1.9759,0.79165) node {\( C\)};
 
 \draw [] (1.750,0)--(1.746,0.1110)--(1.736,0.2215)--(1.718,0.3312)--(1.694,0.4395)--(1.663,0.5461)--(1.625,0.6504)--(1.580,0.7521)--(1.529,0.8508)--(1.472,0.9461)--(1.409,1.038)--(1.341,1.125)--(1.267,1.208)--(1.187,1.286)--(1.103,1.358)--(1.015,1.426)--(0.9226,1.487)--(0.8265,1.543)--(0.7270,1.592)--(0.6245,1.635)--(0.5196,1.671)--(0.4126,1.701)--(0.3039,1.723)--(0.1940,1.739)--(0.08327,1.748)--(-0.02777,1.750)--(-0.1387,1.744)--(-0.2491,1.732)--(-0.3584,1.713)--(-0.4663,1.687)--(-0.5724,1.654)--(-0.6761,1.614)--(-0.7771,1.568)--(-0.8750,1.516)--(-0.9694,1.457)--(-1.060,1.393)--(-1.146,1.323)--(-1.228,1.247)--(-1.304,1.167)--(-1.376,1.082)--(-1.441,0.9924)--(-1.501,0.8989)--(-1.555,0.8019)--(-1.603,0.7016)--(-1.644,0.5985)--(-1.679,0.4930)--(-1.707,0.3855)--(-1.728,0.2765)--(-1.742,0.1663)--(-1.749,0.05552)--(-1.749,-0.05552)--(-1.742,-0.1663)--(-1.728,-0.2765)--(-1.707,-0.3855)--(-1.679,-0.4930)--(-1.644,-0.5985)--(-1.603,-0.7016)--(-1.555,-0.8019)--(-1.501,-0.8989)--(-1.441,-0.9924)--(-1.376,-1.082)--(-1.304,-1.167)--(-1.228,-1.247)--(-1.146,-1.323)--(-1.060,-1.393)--(-0.9694,-1.457)--(-0.8750,-1.516)--(-0.7771,-1.568)--(-0.6761,-1.614)--(-0.5724,-1.654)--(-0.4663,-1.687)--(-0.3584,-1.713)--(-0.2491,-1.732)--(-0.1387,-1.744)--(-0.02777,-1.750)--(0.08327,-1.748)--(0.1940,-1.739)--(0.3039,-1.723)--(0.4126,-1.701)--(0.5196,-1.671)--(0.6245,-1.635)--(0.7270,-1.592)--(0.8265,-1.543)--(0.9226,-1.487)--(1.015,-1.426)--(1.103,-1.358)--(1.187,-1.286)--(1.267,-1.208)--(1.341,-1.125)--(1.409,-1.038)--(1.472,-0.9461)--(1.529,-0.8508)--(1.580,-0.7521)--(1.625,-0.6504)--(1.663,-0.5461)--(1.694,-0.4395)--(1.718,-0.3312)--(1.736,-0.2215)--(1.746,-0.1110)--(1.750,0);
-\draw (-0.4228499726,-0.5520606245) node {\( H\)};
+\draw (-0.42285,-0.55206) node {\( H\)};
 \draw [] (-1.21,1.27) -- (-0.658,-0.239);
 \draw [] (-1.21,1.27) -- (-1.64,-0.599);
 \draw [] (-1.64,-0.599) -- (1.64,0.599);
diff --git a/src_phystricks/Fig_QPcdHwP.pstricks.recall b/src_phystricks/Fig_QPcdHwP.pstricks.recall
index 2142b819a..bc69167a5 100644
--- a/src_phystricks/Fig_QPcdHwP.pstricks.recall
+++ b/src_phystricks/Fig_QPcdHwP.pstricks.recall
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3059510000) node {\( \alpha_2\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,-0.3059510000) node {\( \alpha_4\)};
+\draw (0,-0.30595) node {\( \alpha_2\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,-0.30595) node {\( \alpha_4\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall b/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall
index 84f832b73..af9c3b2cf 100644
--- a/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall
+++ b/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (-2.10,0.700) -- (2.10,0.700);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3824550000) node {\( \pi(b_1)\)};
-\draw []  (0.7000000000,0.7000000000) node [rotate=0] {$\bullet$};
-\draw (0.7000000000,1.082455000) node {\( \pi(b_2)\)};
+\draw (0,-0.38245) node {\( \pi(b_1)\)};
+\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
+\draw (0.70000,1.0825) node {\( \pi(b_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_RLuqsrr.pstricks.recall b/src_phystricks/Fig_RLuqsrr.pstricks.recall
index cdeaa0644..ae4cd3b5d 100644
--- a/src_phystricks/Fig_RLuqsrr.pstricks.recall
+++ b/src_phystricks/Fig_RLuqsrr.pstricks.recall
@@ -87,7 +87,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185307,0);
+\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185311,0);
 \draw [,->,>=latex] (0,-0.9138130496) -- (0,2.914169059);
 %DEFAULT
 
diff --git a/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall b/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall
index 1c4a0c331..9b902f047 100644
--- a/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall
+++ b/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall
@@ -92,26 +92,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-1.945688830) -- (0,1.945688830);
+\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-1.9457) -- (0,1.9457);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,0.059192)--(-5.8788,0.027135)--(-5.7576,0.0060480)--(-5.6364,0)--(-5.5152,0.011586)--(-5.3939,0.039487)--(-5.2727,0.080491)--(-5.1515,0.12891)--(-5.0303,0.17762)--(-4.9091,0.21910)--(-4.7879,0.24669)--(-4.6667,0.25565)--(-4.5455,0.24409)--(-4.4242,0.21334)--(-4.3030,0.16799)--(-4.1818,0.11529)--(-4.0606,0.064108)--(-3.9394,0.023666)--(-3.8182,0.0020298)--(-3.6970,0.0047735)--(-3.5758,0.033930)--(-3.4545,0.087448)--(-3.3333,0.15925)--(-3.2121,0.23994)--(-3.0909,0.31806)--(-2.9697,0.38179)--(-2.8485,0.42076)--(-2.7273,0.42785)--(-2.6061,0.40058)--(-2.4848,0.34187)--(-2.3636,0.26014)--(-2.2424,0.16851)--(-2.1212,0.083272)--(-2.0000,0.021790)--(-1.8788,0)--(-1.7576,0.030313)--(-1.6364,0.11884)--(-1.5152,0.26464)--(-1.3939,0.45879)--(-1.2727,0.68492)--(-1.1515,0.92065)--(-1.0303,1.1399)--(-0.90909,1.3159)--(-0.78788,1.4242)--(-0.66667,1.4457)--(-0.54545,1.3694)--(-0.42424,1.1936)--(-0.30303,0.92708)--(-0.18182,0.58774)--(-0.060606,0.20133)--(0.060606,-0.20133)--(0.18182,-0.58774)--(0.30303,-0.92708)--(0.42424,-1.1936)--(0.54545,-1.3694)--(0.66667,-1.4457)--(0.78788,-1.4242)--(0.90909,-1.3159)--(1.0303,-1.1399)--(1.1515,-0.92065)--(1.2727,-0.68492)--(1.3939,-0.45879)--(1.5152,-0.26464)--(1.6364,-0.11884)--(1.7576,-0.030313)--(1.8788,0)--(2.0000,-0.021790)--(2.1212,-0.083272)--(2.2424,-0.16851)--(2.3636,-0.26014)--(2.4848,-0.34187)--(2.6061,-0.40058)--(2.7273,-0.42785)--(2.8485,-0.42076)--(2.9697,-0.38179)--(3.0909,-0.31806)--(3.2121,-0.23994)--(3.3333,-0.15925)--(3.4545,-0.087448)--(3.5758,-0.033930)--(3.6970,-0.0047735)--(3.8182,-0.0020298)--(3.9394,-0.023666)--(4.0606,-0.064108)--(4.1818,-0.11529)--(4.3030,-0.16799)--(4.4242,-0.21334)--(4.5455,-0.24409)--(4.6667,-0.25565)--(4.7879,-0.24669)--(4.9091,-0.21910)--(5.0303,-0.17762)--(5.1515,-0.12891)--(5.2727,-0.080491)--(5.3939,-0.039487)--(5.5152,-0.011586)--(5.6364,0)--(5.7576,-0.0060480)--(5.8788,-0.027135)--(6.0000,-0.059192);
-\draw (-5.654866776,-0.3298256667) node {$ -6 \, \pi $};
+\draw (-5.6549,-0.32983) node {$ -6 \, \pi $};
 \draw [] (-5.66,-0.100) -- (-5.66,0.100);
-\draw (-3.769911184,-0.3298256667) node {$ -4 \, \pi $};
+\draw (-3.7699,-0.32983) node {$ -4 \, \pi $};
 \draw [] (-3.77,-0.100) -- (-3.77,0.100);
-\draw (-1.884955592,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-1.8850,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-1.88,-0.100) -- (-1.88,0.100);
-\draw (1.884955592,-0.3149246667) node {$ 2 \, \pi $};
+\draw (1.8850,-0.31492) node {$ 2 \, \pi $};
 \draw [] (1.88,-0.100) -- (1.88,0.100);
-\draw (3.769911184,-0.3149246667) node {$ 4 \, \pi $};
+\draw (3.7699,-0.31492) node {$ 4 \, \pi $};
 \draw [] (3.77,-0.100) -- (3.77,0.100);
-\draw (5.654866776,-0.3149246667) node {$ 6 \, \pi $};
+\draw (5.6549,-0.31492) node {$ 6 \, \pi $};
 \draw [] (5.66,-0.100) -- (5.66,0.100);
-\draw (-0.4527428333,-1.000000000) node {$ -\frac{1}{2} $};
+\draw (-0.45274,-1.0000) node {$ -\frac{1}{2} $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.3108333333,1.000000000) node {$ \frac{1}{2} $};
+\draw (-0.31083,1.0000) node {$ \frac{1}{2} $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Refraction.pstricks.recall b/src_phystricks/Fig_Refraction.pstricks.recall
index a1b1e5e23..9e1e8d206 100644
--- a/src_phystricks/Fig_Refraction.pstricks.recall
+++ b/src_phystricks/Fig_Refraction.pstricks.recall
@@ -78,16 +78,16 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (-3.00,0) -- (3.00,0);
-\draw [,->,>=latex] (0,-2.000000000) -- (0,2.000000000);
-\draw (0.4665308333,2.000000000) node {$\overline{ N }$};
-\draw (0.3561818828,0.9457307663) node {$\theta_1$};
+\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
+\draw (0.46653,2.0000) node {$\overline{ N }$};
+\draw (0.35618,0.94573) node {$\theta_1$};
 
 \draw [] (0.354,0.354)--(0.351,0.356)--(0.348,0.359)--(0.345,0.362)--(0.342,0.365)--(0.339,0.367)--(0.336,0.370)--(0.333,0.373)--(0.330,0.375)--(0.327,0.378)--(0.324,0.380)--(0.321,0.383)--(0.318,0.386)--(0.315,0.388)--(0.312,0.391)--(0.309,0.393)--(0.306,0.395)--(0.303,0.398)--(0.300,0.400)--(0.296,0.403)--(0.293,0.405)--(0.290,0.407)--(0.287,0.410)--(0.284,0.412)--(0.280,0.414)--(0.277,0.416)--(0.274,0.418)--(0.270,0.421)--(0.267,0.423)--(0.264,0.425)--(0.260,0.427)--(0.257,0.429)--(0.253,0.431)--(0.250,0.433)--(0.247,0.435)--(0.243,0.437)--(0.240,0.439)--(0.236,0.441)--(0.233,0.443)--(0.229,0.444)--(0.226,0.446)--(0.222,0.448)--(0.218,0.450)--(0.215,0.451)--(0.211,0.453)--(0.208,0.455)--(0.204,0.456)--(0.200,0.458)--(0.197,0.460)--(0.193,0.461)--(0.190,0.463)--(0.186,0.464)--(0.182,0.466)--(0.178,0.467)--(0.175,0.468)--(0.171,0.470)--(0.167,0.471)--(0.164,0.473)--(0.160,0.474)--(0.156,0.475)--(0.152,0.476)--(0.148,0.477)--(0.145,0.479)--(0.141,0.480)--(0.137,0.481)--(0.133,0.482)--(0.129,0.483)--(0.126,0.484)--(0.122,0.485)--(0.118,0.486)--(0.114,0.487)--(0.110,0.488)--(0.106,0.489)--(0.102,0.489)--(0.0985,0.490)--(0.0946,0.491)--(0.0907,0.492)--(0.0868,0.492)--(0.0829,0.493)--(0.0790,0.494)--(0.0751,0.494)--(0.0712,0.495)--(0.0672,0.495)--(0.0633,0.496)--(0.0594,0.496)--(0.0554,0.497)--(0.0515,0.497)--(0.0475,0.498)--(0.0436,0.498)--(0.0396,0.498)--(0.0357,0.499)--(0.0317,0.499)--(0.0278,0.499)--(0.0238,0.499)--(0.0198,0.500)--(0.0159,0.500)--(0.0119,0.500)--(0.00793,0.500)--(0.00397,0.500)--(0,0.500);
-\draw (-0.4277057227,-0.7442762375) node {$\theta_2$};
+\draw (-0.42770,-0.74428) node {$\theta_2$};
 
 \draw [] (-0.447,-0.224)--(-0.445,-0.229)--(-0.442,-0.234)--(-0.439,-0.238)--(-0.437,-0.243)--(-0.434,-0.248)--(-0.431,-0.253)--(-0.428,-0.258)--(-0.425,-0.263)--(-0.422,-0.267)--(-0.419,-0.272)--(-0.416,-0.277)--(-0.413,-0.281)--(-0.410,-0.286)--(-0.407,-0.291)--(-0.404,-0.295)--(-0.400,-0.300)--(-0.397,-0.304)--(-0.393,-0.309)--(-0.390,-0.313)--(-0.386,-0.317)--(-0.383,-0.322)--(-0.379,-0.326)--(-0.376,-0.330)--(-0.372,-0.334)--(-0.368,-0.338)--(-0.364,-0.342)--(-0.360,-0.346)--(-0.357,-0.350)--(-0.353,-0.354)--(-0.349,-0.358)--(-0.345,-0.362)--(-0.341,-0.366)--(-0.336,-0.370)--(-0.332,-0.374)--(-0.328,-0.377)--(-0.324,-0.381)--(-0.320,-0.385)--(-0.315,-0.388)--(-0.311,-0.392)--(-0.306,-0.395)--(-0.302,-0.398)--(-0.298,-0.402)--(-0.293,-0.405)--(-0.289,-0.408)--(-0.284,-0.412)--(-0.279,-0.415)--(-0.275,-0.418)--(-0.270,-0.421)--(-0.265,-0.424)--(-0.260,-0.427)--(-0.256,-0.430)--(-0.251,-0.432)--(-0.246,-0.435)--(-0.241,-0.438)--(-0.236,-0.441)--(-0.231,-0.443)--(-0.226,-0.446)--(-0.221,-0.448)--(-0.216,-0.451)--(-0.211,-0.453)--(-0.206,-0.456)--(-0.201,-0.458)--(-0.196,-0.460)--(-0.191,-0.462)--(-0.186,-0.464)--(-0.180,-0.466)--(-0.175,-0.468)--(-0.170,-0.470)--(-0.165,-0.472)--(-0.159,-0.474)--(-0.154,-0.476)--(-0.149,-0.477)--(-0.143,-0.479)--(-0.138,-0.481)--(-0.133,-0.482)--(-0.127,-0.484)--(-0.122,-0.485)--(-0.116,-0.486)--(-0.111,-0.488)--(-0.105,-0.489)--(-0.100,-0.490)--(-0.0945,-0.491)--(-0.0890,-0.492)--(-0.0835,-0.493)--(-0.0780,-0.494)--(-0.0724,-0.495)--(-0.0669,-0.496)--(-0.0614,-0.496)--(-0.0558,-0.497)--(-0.0502,-0.497)--(-0.0447,-0.498)--(-0.0391,-0.498)--(-0.0335,-0.499)--(-0.0279,-0.499)--(-0.0224,-0.500)--(-0.0168,-0.500)--(-0.0112,-0.500)--(-0.00559,-0.500)--(0,-0.500);
-\draw [color=red,->,>=latex] (1.000000000,1.000000000) -- (0,0);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.264911064,-0.6324555320);
+\draw [color=red,->,>=latex] (1.0000,1.0000) -- (0,0);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.2649,-0.63246);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall b/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall
index 519222dec..d7811ff6e 100644
--- a/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall
+++ b/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall
@@ -88,20 +88,20 @@
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.3420201433,0.9396926208) node [rotate=0] {$\bullet$};
-\draw (0.5068918387,1.206207312) node {\( a\)};
-\draw []  (0.3420201433,-0.9396926208) node [rotate=0] {$\bullet$};
-\draw (0.4887470053,-1.254367145) node {\( b\)};
-\draw []  (2.923804400,0) node [rotate=0] {$\bullet$};
-\draw (3.484041733,0) node {\( m\)};
+\draw []  (0.34202,0.93969) node [rotate=0] {$\bullet$};
+\draw (0.50689,1.2062) node {\( a\)};
+\draw []  (0.34202,-0.93969) node [rotate=0] {$\bullet$};
+\draw (0.48875,-1.2544) node {\( b\)};
+\draw []  (2.9238,0) node [rotate=0] {$\bullet$};
+\draw (3.4840,0) node {\( m\)};
 \draw [] (-0.303,1.17) -- (3.57,-0.235);
 \draw [] (-0.303,-1.17) -- (3.57,0.235);
-\draw []  (0.8660254038,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (0.5885166564,0.3214238333) node {\( x\)};
-\draw []  (-0.5400320788,0.8416444344) node [rotate=0] {$\bullet$};
-\draw (-0.7270164945,1.088549488) node {\( c\)};
+\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
+\draw (0.58852,0.32142) node {\( x\)};
+\draw []  (-0.54003,0.84164) node [rotate=0] {$\bullet$};
+\draw (-0.72702,1.0885) node {\( c\)};
 \draw [] (3.44,-0.126) -- (-1.06,0.968);
-\draw (-1.234368978,-0.5351321720) node {\( \mC\)};
+\draw (-1.2344,-0.53513) node {\( \mC\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_SenoTopologo.pstricks.recall b/src_phystricks/Fig_SenoTopologo.pstricks.recall
index 6cb4cdd52..64c749052 100644
--- a/src_phystricks/Fig_SenoTopologo.pstricks.recall
+++ b/src_phystricks/Fig_SenoTopologo.pstricks.recall
@@ -64,7 +64,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-1.586160392) -- (0,2.773243567);
+\draw [,->,>=latex] (0,-1.586160393) -- (0,2.773243568);
 %DEFAULT
 
 \draw [color=blue] 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diff --git a/src_phystricks/Fig_SolsSinpA.pstricks.recall b/src_phystricks/Fig_SolsSinpA.pstricks.recall
index 78520694e..b23b7aff3 100644
--- a/src_phystricks/Fig_SolsSinpA.pstricks.recall
+++ b/src_phystricks/Fig_SolsSinpA.pstricks.recall
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.641592654,0) -- (3.641592654,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-3.6416,0) -- (3.6416,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
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@@ -100,17 +100,17 @@
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-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall b/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall
index 8314eadf6..ad7c6baeb 100644
--- a/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall
+++ b/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,2.000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,2.0000);
 %DEFAULT
-\draw []  (1.000000000,-3.000000000) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-3.429825667) node {$-1$};
-\draw []  (2.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,1.982455000) node {$1/2$};
-\draw []  (3.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-1.482455000) node {$-1/3$};
-\draw []  (4.000000000,0.7500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,1.232455000) node {$1/4$};
-\draw []  (5.000000000,-0.6000000000) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-1.082455000) node {$-1/5$};
-\draw []  (6.000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (6.000000000,0.9824550000) node {$1/6$};
-\draw []  (7.000000000,-0.4285714286) node [rotate=0] {$\bullet$};
-\draw (7.000000000,-0.9110264286) node {$-1/7$};
-\draw []  (8.000000000,0.3750000000) node [rotate=0] {$\bullet$};
-\draw (8.000000000,0.8574550000) node {$1/8$};
-\draw []  (9.000000000,-0.3333333333) node [rotate=0] {$\bullet$};
-\draw (9.000000000,-0.8157883333) node {$-1/9$};
-\draw []  (10.00000000,0.3000000000) node [rotate=0] {$\bullet$};
-\draw (10.00000000,0.7824550000) node {$1/10$};
+\draw []  (1.0000,-3.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-3.4298) node {$-1$};
+\draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (2.0000,1.9825) node {$1/2$};
+\draw []  (3.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-1.4825) node {$-1/3$};
+\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2325) node {$1/4$};
+\draw []  (5.0000,-0.60000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-1.0825) node {$-1/5$};
+\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.98246) node {$1/6$};
+\draw []  (7.0000,-0.42857) node [rotate=0] {$\bullet$};
+\draw (7.0000,-0.91103) node {$-1/7$};
+\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.85746) node {$1/8$};
+\draw []  (9.0000,-0.33333) node [rotate=0] {$\bullet$};
+\draw (9.0000,-0.81579) node {$-1/9$};
+\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.78246) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall b/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall
index 1f89f1f46..3a69fd3a0 100644
--- a/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall
+++ b/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (-1.000,2.925)--(-0.9394,2.844)--(-0.8788,2.765)--(-0.8182,2.687)--(-0.7576,2.611)--(-0.6970,2.537)--(-0.6364,2.464)--(-0.5758,2.393)--(-0.5152,2.323)--(-0.4545,2.256)--(-0.3939,2.189)--(-0.3333,2.125)--(-0.2727,2.062)--(-0.2121,2.001)--(-0.1515,1.942)--(-0.09091,1.884)--(-0.03030,1.827)--(0.03030,1.773)--(0.09091,1.720)--(0.1515,1.669)--(0.2121,1.619)--(0.2727,1.571)--(0.3333,1.525)--(0.3939,1.480)--(0.4545,1.437)--(0.5152,1.396)--(0.5758,1.356)--(0.6364,1.318)--(0.6970,1.282)--(0.7576,1.247)--(0.8182,1.214)--(0.8788,1.183)--(0.9394,1.153)--(1.000,1.125)--(1.061,1.099)--(1.121,1.074)--(1.182,1.051)--(1.242,1.029)--(1.303,1.009)--(1.364,0.9911)--(1.424,0.9746)--(1.485,0.9597)--(1.545,0.9465)--(1.606,0.9349)--(1.667,0.9250)--(1.727,0.9167)--(1.788,0.9101)--(1.848,0.9052)--(1.909,0.9019)--(1.970,0.9002)--(2.030,0.9002)--(2.091,0.9019)--(2.152,0.9052)--(2.212,0.9101)--(2.273,0.9167)--(2.333,0.9250)--(2.394,0.9349)--(2.455,0.9465)--(2.515,0.9597)--(2.576,0.9746)--(2.636,0.9911)--(2.697,1.009)--(2.758,1.029)--(2.818,1.051)--(2.879,1.074)--(2.939,1.099)--(3.000,1.125)--(3.061,1.153)--(3.121,1.183)--(3.182,1.214)--(3.242,1.247)--(3.303,1.282)--(3.364,1.318)--(3.424,1.356)--(3.485,1.396)--(3.545,1.437)--(3.606,1.480)--(3.667,1.525)--(3.727,1.571)--(3.788,1.619)--(3.848,1.669)--(3.909,1.720)--(3.970,1.773)--(4.030,1.827)--(4.091,1.884)--(4.151,1.942)--(4.212,2.001)--(4.273,2.062)--(4.333,2.125)--(4.394,2.189)--(4.455,2.256)--(4.515,2.323)--(4.576,2.393)--(4.636,2.464)--(4.697,2.537)--(4.758,2.611)--(4.818,2.687)--(4.879,2.765)--(4.939,2.844)--(5.000,2.925);
@@ -53,10 +53,10 @@
 \draw [color=blue] (0.1743,0)--(0.2111,0)--(0.2480,0)--(0.2849,0)--(0.3218,0)--(0.3587,0)--(0.3956,0)--(0.4324,0)--(0.4693,0)--(0.5062,0)--(0.5431,0)--(0.5800,0)--(0.6169,0)--(0.6537,0)--(0.6906,0)--(0.7275,0)--(0.7644,0)--(0.8013,0)--(0.8382,0)--(0.8750,0)--(0.9119,0)--(0.9488,0)--(0.9857,0)--(1.023,0)--(1.059,0)--(1.096,0)--(1.133,0)--(1.170,0)--(1.207,0)--(1.244,0)--(1.281,0)--(1.318,0)--(1.355,0)--(1.391,0)--(1.428,0)--(1.465,0)--(1.502,0)--(1.539,0)--(1.576,0)--(1.613,0)--(1.650,0)--(1.686,0)--(1.723,0)--(1.760,0)--(1.797,0)--(1.834,0)--(1.871,0)--(1.908,0)--(1.945,0)--(1.982,0)--(2.018,0)--(2.055,0)--(2.092,0)--(2.129,0)--(2.166,0)--(2.203,0)--(2.240,0)--(2.277,0)--(2.314,0)--(2.350,0)--(2.387,0)--(2.424,0)--(2.461,0)--(2.498,0)--(2.535,0)--(2.572,0)--(2.609,0)--(2.645,0)--(2.682,0)--(2.719,0)--(2.756,0)--(2.793,0)--(2.830,0)--(2.867,0)--(2.904,0)--(2.941,0)--(2.977,0)--(3.014,0)--(3.051,0)--(3.088,0)--(3.125,0)--(3.162,0)--(3.199,0)--(3.236,0)--(3.272,0)--(3.309,0)--(3.346,0)--(3.383,0)--(3.420,0)--(3.457,0)--(3.494,0)--(3.531,0)--(3.568,0)--(3.604,0)--(3.641,0)--(3.678,0)--(3.715,0)--(3.752,0)--(3.789,0)--(3.826,0);
 \draw [] (0.174,0) -- (0.174,1.65);
 \draw [] (3.83,1.65) -- (3.83,0);
-\draw []  (0.1742581416,0) node [rotate=0] {$\bullet$};
-\draw (0.1742581416,-0.3785761667) node {$a$};
-\draw []  (3.825741858,0) node [rotate=0] {$\bullet$};
-\draw (3.825741858,-0.4267360000) node {$b$};
+\draw []  (0.17426,0) node [rotate=0] {$\bullet$};
+\draw (0.17426,-0.37858) node {$a$};
+\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
+\draw (3.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (-1.000,2.925)--(-0.9394,2.844)--(-0.8788,2.765)--(-0.8182,2.687)--(-0.7576,2.611)--(-0.6970,2.537)--(-0.6364,2.464)--(-0.5758,2.393)--(-0.5152,2.323)--(-0.4545,2.256)--(-0.3939,2.189)--(-0.3333,2.125)--(-0.2727,2.062)--(-0.2121,2.001)--(-0.1515,1.942)--(-0.09091,1.884)--(-0.03030,1.827)--(0.03030,1.773)--(0.09091,1.720)--(0.1515,1.669)--(0.2121,1.619)--(0.2727,1.571)--(0.3333,1.525)--(0.3939,1.480)--(0.4545,1.437)--(0.5152,1.396)--(0.5758,1.356)--(0.6364,1.318)--(0.6970,1.282)--(0.7576,1.247)--(0.8182,1.214)--(0.8788,1.183)--(0.9394,1.153)--(1.000,1.125)--(1.061,1.099)--(1.121,1.074)--(1.182,1.051)--(1.242,1.029)--(1.303,1.009)--(1.364,0.9911)--(1.424,0.9746)--(1.485,0.9597)--(1.545,0.9465)--(1.606,0.9349)--(1.667,0.9250)--(1.727,0.9167)--(1.788,0.9101)--(1.848,0.9052)--(1.909,0.9019)--(1.970,0.9002)--(2.030,0.9002)--(2.091,0.9019)--(2.152,0.9052)--(2.212,0.9101)--(2.273,0.9167)--(2.333,0.9250)--(2.394,0.9349)--(2.455,0.9465)--(2.515,0.9597)--(2.576,0.9746)--(2.636,0.9911)--(2.697,1.009)--(2.758,1.029)--(2.818,1.051)--(2.879,1.074)--(2.939,1.099)--(3.000,1.125)--(3.061,1.153)--(3.121,1.183)--(3.182,1.214)--(3.242,1.247)--(3.303,1.282)--(3.364,1.318)--(3.424,1.356)--(3.485,1.396)--(3.545,1.437)--(3.606,1.480)--(3.667,1.525)--(3.727,1.571)--(3.788,1.619)--(3.848,1.669)--(3.909,1.720)--(3.970,1.773)--(4.030,1.827)--(4.091,1.884)--(4.151,1.942)--(4.212,2.001)--(4.273,2.062)--(4.333,2.125)--(4.394,2.189)--(4.455,2.256)--(4.515,2.323)--(4.576,2.393)--(4.636,2.464)--(4.697,2.537)--(4.758,2.611)--(4.818,2.687)--(4.879,2.765)--(4.939,2.844)--(5.000,2.925);
@@ -107,10 +107,10 @@
 \draw [color=blue] (0.1743,0)--(0.2111,0)--(0.2480,0)--(0.2849,0)--(0.3218,0)--(0.3587,0)--(0.3956,0)--(0.4324,0)--(0.4693,0)--(0.5062,0)--(0.5431,0)--(0.5800,0)--(0.6169,0)--(0.6537,0)--(0.6906,0)--(0.7275,0)--(0.7644,0)--(0.8013,0)--(0.8382,0)--(0.8750,0)--(0.9119,0)--(0.9488,0)--(0.9857,0)--(1.023,0)--(1.059,0)--(1.096,0)--(1.133,0)--(1.170,0)--(1.207,0)--(1.244,0)--(1.281,0)--(1.318,0)--(1.355,0)--(1.391,0)--(1.428,0)--(1.465,0)--(1.502,0)--(1.539,0)--(1.576,0)--(1.613,0)--(1.650,0)--(1.686,0)--(1.723,0)--(1.760,0)--(1.797,0)--(1.834,0)--(1.871,0)--(1.908,0)--(1.945,0)--(1.982,0)--(2.018,0)--(2.055,0)--(2.092,0)--(2.129,0)--(2.166,0)--(2.203,0)--(2.240,0)--(2.277,0)--(2.314,0)--(2.350,0)--(2.387,0)--(2.424,0)--(2.461,0)--(2.498,0)--(2.535,0)--(2.572,0)--(2.609,0)--(2.645,0)--(2.682,0)--(2.719,0)--(2.756,0)--(2.793,0)--(2.830,0)--(2.867,0)--(2.904,0)--(2.941,0)--(2.977,0)--(3.014,0)--(3.051,0)--(3.088,0)--(3.125,0)--(3.162,0)--(3.199,0)--(3.236,0)--(3.272,0)--(3.309,0)--(3.346,0)--(3.383,0)--(3.420,0)--(3.457,0)--(3.494,0)--(3.531,0)--(3.568,0)--(3.604,0)--(3.641,0)--(3.678,0)--(3.715,0)--(3.752,0)--(3.789,0)--(3.826,0);
 \draw [] (0.174,0) -- (0.174,1.65);
 \draw [] (3.83,1.65) -- (3.83,0);
-\draw []  (0.1742581416,0) node [rotate=0] {$\bullet$};
-\draw (0.1742581416,-0.3785761667) node {$a$};
-\draw []  (3.825741858,0) node [rotate=0] {$\bullet$};
-\draw (3.825741858,-0.4267360000) node {$b$};
+\draw []  (0.17426,0) node [rotate=0] {$\bullet$};
+\draw (0.17426,-0.37858) node {$a$};
+\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
+\draw (3.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,8 +149,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (-1.000,2.925)--(-0.9394,2.844)--(-0.8788,2.765)--(-0.8182,2.687)--(-0.7576,2.611)--(-0.6970,2.537)--(-0.6364,2.464)--(-0.5758,2.393)--(-0.5152,2.323)--(-0.4545,2.256)--(-0.3939,2.189)--(-0.3333,2.125)--(-0.2727,2.062)--(-0.2121,2.001)--(-0.1515,1.942)--(-0.09091,1.884)--(-0.03030,1.827)--(0.03030,1.773)--(0.09091,1.720)--(0.1515,1.669)--(0.2121,1.619)--(0.2727,1.571)--(0.3333,1.525)--(0.3939,1.480)--(0.4545,1.437)--(0.5152,1.396)--(0.5758,1.356)--(0.6364,1.318)--(0.6970,1.282)--(0.7576,1.247)--(0.8182,1.214)--(0.8788,1.183)--(0.9394,1.153)--(1.000,1.125)--(1.061,1.099)--(1.121,1.074)--(1.182,1.051)--(1.242,1.029)--(1.303,1.009)--(1.364,0.9911)--(1.424,0.9746)--(1.485,0.9597)--(1.545,0.9465)--(1.606,0.9349)--(1.667,0.9250)--(1.727,0.9167)--(1.788,0.9101)--(1.848,0.9052)--(1.909,0.9019)--(1.970,0.9002)--(2.030,0.9002)--(2.091,0.9019)--(2.152,0.9052)--(2.212,0.9101)--(2.273,0.9167)--(2.333,0.9250)--(2.394,0.9349)--(2.455,0.9465)--(2.515,0.9597)--(2.576,0.9746)--(2.636,0.9911)--(2.697,1.009)--(2.758,1.029)--(2.818,1.051)--(2.879,1.074)--(2.939,1.099)--(3.000,1.125)--(3.061,1.153)--(3.121,1.183)--(3.182,1.214)--(3.242,1.247)--(3.303,1.282)--(3.364,1.318)--(3.424,1.356)--(3.485,1.396)--(3.545,1.437)--(3.606,1.480)--(3.667,1.525)--(3.727,1.571)--(3.788,1.619)--(3.848,1.669)--(3.909,1.720)--(3.970,1.773)--(4.030,1.827)--(4.091,1.884)--(4.151,1.942)--(4.212,2.001)--(4.273,2.062)--(4.333,2.125)--(4.394,2.189)--(4.455,2.256)--(4.515,2.323)--(4.576,2.393)--(4.636,2.464)--(4.697,2.537)--(4.758,2.611)--(4.818,2.687)--(4.879,2.765)--(4.939,2.844)--(5.000,2.925);
@@ -161,10 +161,10 @@
 \draw [color=blue] (0.1743,1.650)--(0.2111,1.680)--(0.2480,1.709)--(0.2849,1.738)--(0.3218,1.766)--(0.3587,1.794)--(0.3956,1.821)--(0.4324,1.847)--(0.4693,1.873)--(0.5062,1.898)--(0.5431,1.922)--(0.5800,1.946)--(0.6169,1.970)--(0.6537,1.992)--(0.6906,2.014)--(0.7275,2.036)--(0.7644,2.056)--(0.8013,2.077)--(0.8382,2.096)--(0.8750,2.115)--(0.9119,2.134)--(0.9488,2.151)--(0.9857,2.169)--(1.023,2.185)--(1.059,2.201)--(1.096,2.216)--(1.133,2.231)--(1.170,2.245)--(1.207,2.259)--(1.244,2.271)--(1.281,2.284)--(1.318,2.295)--(1.355,2.306)--(1.391,2.317)--(1.428,2.326)--(1.465,2.336)--(1.502,2.344)--(1.539,2.352)--(1.576,2.360)--(1.613,2.366)--(1.650,2.372)--(1.686,2.378)--(1.723,2.383)--(1.760,2.387)--(1.797,2.391)--(1.834,2.394)--(1.871,2.396)--(1.908,2.398)--(1.945,2.399)--(1.982,2.400)--(2.018,2.400)--(2.055,2.399)--(2.092,2.398)--(2.129,2.396)--(2.166,2.394)--(2.203,2.391)--(2.240,2.387)--(2.277,2.383)--(2.314,2.378)--(2.350,2.372)--(2.387,2.366)--(2.424,2.360)--(2.461,2.352)--(2.498,2.344)--(2.535,2.336)--(2.572,2.326)--(2.609,2.317)--(2.645,2.306)--(2.682,2.295)--(2.719,2.284)--(2.756,2.271)--(2.793,2.259)--(2.830,2.245)--(2.867,2.231)--(2.904,2.216)--(2.941,2.201)--(2.977,2.185)--(3.014,2.169)--(3.051,2.151)--(3.088,2.134)--(3.125,2.115)--(3.162,2.096)--(3.199,2.077)--(3.236,2.056)--(3.272,2.036)--(3.309,2.014)--(3.346,1.992)--(3.383,1.970)--(3.420,1.946)--(3.457,1.922)--(3.494,1.898)--(3.531,1.873)--(3.568,1.847)--(3.604,1.821)--(3.641,1.794)--(3.678,1.766)--(3.715,1.738)--(3.752,1.709)--(3.789,1.680)--(3.826,1.650);
 \draw [] (0.174,1.65) -- (0.174,1.65);
 \draw [] (3.83,1.65) -- (3.83,1.65);
-\draw []  (0.1742581416,0) node [rotate=0] {$\bullet$};
-\draw (0.1742581416,-0.3785761667) node {$a$};
-\draw []  (3.825741858,0) node [rotate=0] {$\bullet$};
-\draw (3.825741858,-0.4267360000) node {$b$};
+\draw []  (0.17426,0) node [rotate=0] {$\bullet$};
+\draw (0.17426,-0.37858) node {$a$};
+\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
+\draw (3.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall b/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall
index cc25d836a..fd9c3b21d 100644
--- a/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall
+++ b/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.400000000);
+\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
 %DEFAULT
 
 \draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
@@ -96,10 +96,10 @@
 \draw [color=blue] (3.000,0)--(3.030,0)--(3.061,0)--(3.091,0)--(3.121,0)--(3.152,0)--(3.182,0)--(3.212,0)--(3.242,0)--(3.273,0)--(3.303,0)--(3.333,0)--(3.364,0)--(3.394,0)--(3.424,0)--(3.455,0)--(3.485,0)--(3.515,0)--(3.545,0)--(3.576,0)--(3.606,0)--(3.636,0)--(3.667,0)--(3.697,0)--(3.727,0)--(3.758,0)--(3.788,0)--(3.818,0)--(3.848,0)--(3.879,0)--(3.909,0)--(3.939,0)--(3.970,0)--(4.000,0)--(4.030,0)--(4.061,0)--(4.091,0)--(4.121,0)--(4.151,0)--(4.182,0)--(4.212,0)--(4.242,0)--(4.273,0)--(4.303,0)--(4.333,0)--(4.364,0)--(4.394,0)--(4.424,0)--(4.455,0)--(4.485,0)--(4.515,0)--(4.545,0)--(4.576,0)--(4.606,0)--(4.636,0)--(4.667,0)--(4.697,0)--(4.727,0)--(4.758,0)--(4.788,0)--(4.818,0)--(4.849,0)--(4.879,0)--(4.909,0)--(4.939,0)--(4.970,0)--(5.000,0)--(5.030,0)--(5.061,0)--(5.091,0)--(5.121,0)--(5.151,0)--(5.182,0)--(5.212,0)--(5.242,0)--(5.273,0)--(5.303,0)--(5.333,0)--(5.364,0)--(5.394,0)--(5.424,0)--(5.455,0)--(5.485,0)--(5.515,0)--(5.545,0)--(5.576,0)--(5.606,0)--(5.636,0)--(5.667,0)--(5.697,0)--(5.727,0)--(5.758,0)--(5.788,0)--(5.818,0)--(5.849,0)--(5.879,0)--(5.909,0)--(5.939,0)--(5.970,0)--(6.000,0);
 \draw [style=dashed] (3.00,0) -- (3.00,3.00);
 \draw [style=dashed] (6.00,3.75) -- (6.00,0);
-\draw (3.000000000,-0.3785761667) node {$a$};
-\draw (6.000000000,-0.3785761667) node {$x$};
-\draw (8.355206667,3.900000000) node {$f(x)$};
-\draw (9.670153833,1.875000000) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (3.0000,-0.37858) node {$a$};
+\draw (6.0000,-0.37858) node {$x$};
+\draw (8.3552,3.9000) node {$f(x)$};
+\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall b/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
index f19ce3bcb..92b9e2f6a 100644
--- a/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
+++ b/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
@@ -89,7 +89,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-2.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.6113094300) -- (0,2.527798435);
+\draw [,->,>=latex] (0,-0.6113094300) -- (0,2.527798437);
 %DEFAULT
 
 \draw [color=blue] (-2.000,1.127)--(-1.929,1.000)--(-1.859,0.8813)--(-1.788,0.7700)--(-1.717,0.6663)--(-1.646,0.5699)--(-1.576,0.4807)--(-1.505,0.3985)--(-1.434,0.3231)--(-1.364,0.2544)--(-1.293,0.1922)--(-1.222,0.1363)--(-1.152,0.08657)--(-1.081,0.04284)--(-1.010,0.004948)--(-0.9394,-0.02729)--(-0.8687,-0.05404)--(-0.7980,-0.07546)--(-0.7273,-0.09173)--(-0.6566,-0.1030)--(-0.5859,-0.1095)--(-0.5152,-0.1113)--(-0.4444,-0.1087)--(-0.3737,-0.1017)--(-0.3030,-0.09057)--(-0.2323,-0.07548)--(-0.1616,-0.05658)--(-0.09091,-0.03405)--(-0.02020,-0.008044)--(0.05051,0.02126)--(0.1212,0.05370)--(0.1919,0.08911)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5455,0.3047)--(0.6162,0.3544)--(0.6869,0.4058)--(0.7576,0.4589)--(0.8283,0.5134)--(0.8990,0.5692)--(0.9697,0.6261)--(1.040,0.6840)--(1.111,0.7426)--(1.182,0.8018)--(1.253,0.8615)--(1.323,0.9215)--(1.394,0.9815)--(1.465,1.042)--(1.535,1.101)--(1.606,1.161)--(1.677,1.219)--(1.747,1.277)--(1.818,1.335)--(1.889,1.391)--(1.960,1.445)--(2.030,1.499)--(2.101,1.551)--(2.172,1.601)--(2.242,1.649)--(2.313,1.695)--(2.384,1.739)--(2.455,1.780)--(2.525,1.819)--(2.596,1.855)--(2.667,1.888)--(2.737,1.918)--(2.808,1.945)--(2.879,1.968)--(2.949,1.988)--(3.020,2.004)--(3.091,2.016)--(3.162,2.024)--(3.232,2.028)--(3.303,2.027)--(3.374,2.022)--(3.444,2.011)--(3.515,1.996)--(3.586,1.976)--(3.657,1.951)--(3.727,1.920)--(3.798,1.883)--(3.869,1.841)--(3.939,1.792)--(4.010,1.738)--(4.081,1.677)--(4.151,1.610)--(4.222,1.536)--(4.293,1.455)--(4.364,1.368)--(4.434,1.273)--(4.505,1.171)--(4.576,1.062)--(4.646,0.9444)--(4.717,0.8194)--(4.788,0.6865)--(4.859,0.5454)--(4.929,0.3960)--(5.000,0.2381);
diff --git a/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall b/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
index 7657de6aa..42f5d021d 100644
--- a/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
+++ b/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
@@ -108,7 +108,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981634,0) -- (8.353981634,0);
+\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
 \draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
 %DEFAULT
 
diff --git a/src_phystricks/Fig_TZCISko.pstricks.recall b/src_phystricks/Fig_TZCISko.pstricks.recall
index fa9218d09..068402609 100644
--- a/src_phystricks/Fig_TZCISko.pstricks.recall
+++ b/src_phystricks/Fig_TZCISko.pstricks.recall
@@ -71,19 +71,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-2.499989537) -- (0,2.499999895);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
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-\draw (10.00000000,-0.3149246667) node {$ 1 $};
+\draw (10.000,-0.31492) node {$ 1 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangentSegment.pstricks.recall b/src_phystricks/Fig_TangentSegment.pstricks.recall
index aa8ff1a14..49f2ac546 100644
--- a/src_phystricks/Fig_TangentSegment.pstricks.recall
+++ b/src_phystricks/Fig_TangentSegment.pstricks.recall
@@ -103,7 +103,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.703789489,0) -- (7.783185307,0);
+\draw [,->,>=latex] (-3.703789489,0) -- (7.783185311,0);
 \draw [,->,>=latex] (0,-3.068914101) -- (0,2.500000000);
 %DEFAULT
 
diff --git a/src_phystricks/Fig_TgCercleTrigono.pstricks.recall b/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
index 54059faa9..4b392a788 100644
--- a/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
+++ b/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.539252469) -- (0,3.768059116);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5393) -- (0,3.7681);
 %DEFAULT
 \draw [color=red,style=dashed] (0,0) -- (2.00,2.38);
 \draw [color=cyan,style=dashed] (0,0) -- (2.00,-1.15);
 
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-\draw (0.7565851552,0.3380451309) node {$\theta$};
+\draw (0.75659,0.33805) node {$\theta$};
 
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-\draw (0.3487948559,0.6970250798) node {$\varphi$};
+\draw (0.34879,0.69703) node {$\varphi$};
 
 \draw [color=cyan] (0.500,0)--(0.500,0.0132)--(0.499,0.0264)--(0.498,0.0396)--(0.497,0.0528)--(0.496,0.0659)--(0.494,0.0790)--(0.491,0.0920)--(0.489,0.105)--(0.486,0.118)--(0.483,0.131)--(0.479,0.143)--(0.475,0.156)--(0.471,0.169)--(0.466,0.181)--(0.461,0.193)--(0.456,0.205)--(0.450,0.217)--(0.444,0.229)--(0.438,0.241)--(0.432,0.252)--(0.425,0.264)--(0.418,0.275)--(0.410,0.286)--(0.403,0.296)--(0.395,0.307)--(0.386,0.317)--(0.378,0.327)--(0.369,0.337)--(0.360,0.347)--(0.351,0.356)--(0.341,0.366)--(0.331,0.374)--(0.321,0.383)--(0.311,0.391)--(0.301,0.399)--(0.290,0.407)--(0.279,0.415)--(0.268,0.422)--(0.257,0.429)--(0.245,0.436)--(0.234,0.442)--(0.222,0.448)--(0.210,0.454)--(0.198,0.459)--(0.186,0.464)--(0.173,0.469)--(0.161,0.473)--(0.148,0.477)--(0.136,0.481)--(0.123,0.485)--(0.110,0.488)--(0.0972,0.490)--(0.0842,0.493)--(0.0712,0.495)--(0.0580,0.497)--(0.0449,0.498)--(0.0317,0.499)--(0.0185,0.500)--(0.00529,0.500)--(-0.00793,0.500)--(-0.0211,0.500)--(-0.0344,0.499)--(-0.0475,0.498)--(-0.0607,0.496)--(-0.0738,0.495)--(-0.0868,0.492)--(-0.0998,0.490)--(-0.113,0.487)--(-0.126,0.484)--(-0.138,0.480)--(-0.151,0.477)--(-0.164,0.473)--(-0.176,0.468)--(-0.188,0.463)--(-0.200,0.458)--(-0.213,0.453)--(-0.224,0.447)--(-0.236,0.441)--(-0.248,0.434)--(-0.259,0.428)--(-0.270,0.421)--(-0.281,0.413)--(-0.292,0.406)--(-0.303,0.398)--(-0.313,0.390)--(-0.323,0.381)--(-0.333,0.373)--(-0.343,0.364)--(-0.353,0.354)--(-0.362,0.345)--(-0.371,0.335)--(-0.380,0.325)--(-0.388,0.315)--(-0.396,0.305)--(-0.404,0.294)--(-0.412,0.284)--(-0.419,0.273)--(-0.426,0.261)--(-0.433,0.250);
-\draw [color=red,->,>=latex] (0,0) -- (1.285575219,1.532088886);
-\draw [color=cyan,->,>=latex] (0,0) -- (-1.732050808,1.000000000);
+\draw [color=red,->,>=latex] (0,0) -- (1.2856,1.5321);
+\draw [color=cyan,->,>=latex] (0,0) -- (-1.7320,1.0000);
 \draw [] (2.00,3.27) -- (2.00,-2.04);
-\draw [color=red]  (2.000000000,2.383507185) node [rotate=0] {$\bullet$};
-\draw (2.697581667,2.383507185) node {$\tan(\theta)$};
-\draw [color=cyan]  (2.000000000,-1.154700538) node [rotate=0] {$\bullet$};
-\draw (2.726224000,-1.154700538) node {$\tan(\varphi)$};
+\draw [color=red]  (2.0000,2.3835) node [rotate=0] {$\bullet$};
+\draw (2.6976,2.3835) node {$\tan(\theta)$};
+\draw [color=cyan]  (2.0000,-1.1547) node [rotate=0] {$\bullet$};
+\draw (2.7262,-1.1547) node {$\tan(\varphi)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TracerUn.pstricks.recall b/src_phystricks/Fig_TracerUn.pstricks.recall
index 0d477f127..5b92fa7bb 100644
--- a/src_phystricks/Fig_TracerUn.pstricks.recall
+++ b/src_phystricks/Fig_TracerUn.pstricks.recall
@@ -81,42 +81,42 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.900000000,0) -- (1.900000000,0);
-\draw [,->,>=latex] (0,-4.305593400) -- (0,4.305593400);
+\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
+\draw [,->,>=latex] (0,-4.3056) -- (0,4.3056);
 %DEFAULT
 
 \draw [color=blue] (-1.400,-3.806)--(-1.372,-3.729)--(-1.343,-3.652)--(-1.315,-3.575)--(-1.287,-3.498)--(-1.259,-3.421)--(-1.230,-3.344)--(-1.202,-3.267)--(-1.174,-3.191)--(-1.145,-3.114)--(-1.117,-3.037)--(-1.089,-2.960)--(-1.061,-2.883)--(-1.032,-2.806)--(-1.004,-2.729)--(-0.9758,-2.652)--(-0.9475,-2.576)--(-0.9192,-2.499)--(-0.8909,-2.422)--(-0.8626,-2.345)--(-0.8343,-2.268)--(-0.8061,-2.191)--(-0.7778,-2.114)--(-0.7495,-2.037)--(-0.7212,-1.960)--(-0.6929,-1.884)--(-0.6646,-1.807)--(-0.6364,-1.730)--(-0.6081,-1.653)--(-0.5798,-1.576)--(-0.5515,-1.499)--(-0.5232,-1.422)--(-0.4949,-1.345)--(-0.4667,-1.269)--(-0.4384,-1.192)--(-0.4101,-1.115)--(-0.3818,-1.038)--(-0.3535,-0.9610)--(-0.3253,-0.8841)--(-0.2970,-0.8072)--(-0.2687,-0.7304)--(-0.2404,-0.6535)--(-0.2121,-0.5766)--(-0.1838,-0.4997)--(-0.1556,-0.4228)--(-0.1273,-0.3460)--(-0.09899,-0.2691)--(-0.07071,-0.1922)--(-0.04242,-0.1153)--(-0.01414,-0.03844)--(0.01414,0.03844)--(0.04242,0.1153)--(0.07071,0.1922)--(0.09899,0.2691)--(0.1273,0.3460)--(0.1556,0.4228)--(0.1838,0.4997)--(0.2121,0.5766)--(0.2404,0.6535)--(0.2687,0.7304)--(0.2970,0.8072)--(0.3253,0.8841)--(0.3535,0.9610)--(0.3818,1.038)--(0.4101,1.115)--(0.4384,1.192)--(0.4667,1.269)--(0.4949,1.345)--(0.5232,1.422)--(0.5515,1.499)--(0.5798,1.576)--(0.6081,1.653)--(0.6364,1.730)--(0.6646,1.807)--(0.6929,1.884)--(0.7212,1.960)--(0.7495,2.037)--(0.7778,2.114)--(0.8061,2.191)--(0.8343,2.268)--(0.8626,2.345)--(0.8909,2.422)--(0.9192,2.499)--(0.9475,2.576)--(0.9758,2.652)--(1.004,2.729)--(1.032,2.806)--(1.061,2.883)--(1.089,2.960)--(1.117,3.037)--(1.145,3.114)--(1.174,3.191)--(1.202,3.267)--(1.230,3.344)--(1.259,3.421)--(1.287,3.498)--(1.315,3.575)--(1.343,3.652)--(1.372,3.729)--(1.400,3.806);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.4331593333,-4.200000000) node {$ -6 $};
+\draw (-0.43316,-4.2000) node {$ -6 $};
 \draw [] (-0.100,-4.20) -- (0.100,-4.20);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.2912498333,4.200000000) node {$ 6 $};
+\draw (-0.29125,4.2000) node {$ 6 $};
 \draw [] (-0.100,4.20) -- (0.100,4.20);
 %OTHER STUFF
 %END PSPICTURE
@@ -163,22 +163,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=blue] (-2.000,2.000)--(-1.960,1.960)--(-1.919,1.919)--(-1.879,1.879)--(-1.838,1.838)--(-1.798,1.798)--(-1.758,1.758)--(-1.717,1.717)--(-1.677,1.677)--(-1.636,1.636)--(-1.596,1.596)--(-1.556,1.556)--(-1.515,1.515)--(-1.475,1.475)--(-1.434,1.434)--(-1.394,1.394)--(-1.354,1.354)--(-1.313,1.313)--(-1.273,1.273)--(-1.232,1.232)--(-1.192,1.192)--(-1.152,1.152)--(-1.111,1.111)--(-1.071,1.071)--(-1.030,1.030)--(-0.9899,0.9899)--(-0.9495,0.9495)--(-0.9091,0.9091)--(-0.8687,0.8687)--(-0.8283,0.8283)--(-0.7879,0.7879)--(-0.7475,0.7475)--(-0.7071,0.7071)--(-0.6667,0.6667)--(-0.6263,0.6263)--(-0.5859,0.5859)--(-0.5455,0.5455)--(-0.5051,0.5051)--(-0.4646,0.4646)--(-0.4242,0.4242)--(-0.3838,0.3838)--(-0.3434,0.3434)--(-0.3030,0.3030)--(-0.2626,0.2626)--(-0.2222,0.2222)--(-0.1818,0.1818)--(-0.1414,0.1414)--(-0.1010,0.1010)--(-0.06061,0.06061)--(-0.02020,0.02020)--(0.02020,0.02020)--(0.06061,0.06061)--(0.1010,0.1010)--(0.1414,0.1414)--(0.1818,0.1818)--(0.2222,0.2222)--(0.2626,0.2626)--(0.3030,0.3030)--(0.3434,0.3434)--(0.3838,0.3838)--(0.4242,0.4242)--(0.4646,0.4646)--(0.5051,0.5051)--(0.5455,0.5455)--(0.5859,0.5859)--(0.6263,0.6263)--(0.6667,0.6667)--(0.7071,0.7071)--(0.7475,0.7475)--(0.7879,0.7879)--(0.8283,0.8283)--(0.8687,0.8687)--(0.9091,0.9091)--(0.9495,0.9495)--(0.9899,0.9899)--(1.030,1.030)--(1.071,1.071)--(1.111,1.111)--(1.152,1.152)--(1.192,1.192)--(1.232,1.232)--(1.273,1.273)--(1.313,1.313)--(1.354,1.354)--(1.394,1.394)--(1.434,1.434)--(1.475,1.475)--(1.515,1.515)--(1.556,1.556)--(1.596,1.596)--(1.636,1.636)--(1.677,1.677)--(1.717,1.717)--(1.758,1.758)--(1.798,1.798)--(1.838,1.838)--(1.879,1.879)--(1.919,1.919)--(1.960,1.960)--(2.000,2.000);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -253,38 +253,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.900000000,0) -- (1.900000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.800000000);
+\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.8000);
 %DEFAULT
 
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-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.2912498333,4.200000000) node {$ 6 $};
+\draw (-0.29125,4.2000) node {$ 6 $};
 \draw [] (-0.100,4.20) -- (0.100,4.20);
-\draw (-0.2912498333,4.900000000) node {$ 7 $};
+\draw (-0.29125,4.9000) node {$ 7 $};
 \draw [] (-0.100,4.90) -- (0.100,4.90);
-\draw (-0.2912498333,5.600000000) node {$ 8 $};
+\draw (-0.29125,5.6000) node {$ 8 $};
 \draw [] (-0.100,5.60) -- (0.100,5.60);
-\draw (-0.2912498333,6.300000000) node {$ 9 $};
+\draw (-0.29125,6.3000) node {$ 9 $};
 \draw [] (-0.100,6.30) -- (0.100,6.30);
 %OTHER STUFF
 %END PSPICTURE
@@ -335,26 +335,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.900000000,0) -- (1.900000000,0);
-\draw [,->,>=latex] (0,-2.600000000) -- (0,1.200000000);
+\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
+\draw [,->,>=latex] (0,-2.6000) -- (0,1.2000);
 %DEFAULT
 
 \draw [color=blue] (-1.400,-2.100)--(-1.372,-2.072)--(-1.343,-2.043)--(-1.315,-2.015)--(-1.287,-1.987)--(-1.259,-1.959)--(-1.230,-1.930)--(-1.202,-1.902)--(-1.174,-1.874)--(-1.145,-1.845)--(-1.117,-1.817)--(-1.089,-1.789)--(-1.061,-1.761)--(-1.032,-1.732)--(-1.004,-1.704)--(-0.9758,-1.676)--(-0.9475,-1.647)--(-0.9192,-1.619)--(-0.8909,-1.591)--(-0.8626,-1.563)--(-0.8343,-1.534)--(-0.8061,-1.506)--(-0.7778,-1.478)--(-0.7495,-1.449)--(-0.7212,-1.421)--(-0.6929,-1.393)--(-0.6646,-1.365)--(-0.6364,-1.336)--(-0.6081,-1.308)--(-0.5798,-1.280)--(-0.5515,-1.252)--(-0.5232,-1.223)--(-0.4949,-1.195)--(-0.4667,-1.167)--(-0.4384,-1.138)--(-0.4101,-1.110)--(-0.3818,-1.082)--(-0.3535,-1.054)--(-0.3253,-1.025)--(-0.2970,-0.9970)--(-0.2687,-0.9687)--(-0.2404,-0.9404)--(-0.2121,-0.9121)--(-0.1838,-0.8838)--(-0.1556,-0.8556)--(-0.1273,-0.8273)--(-0.09899,-0.7990)--(-0.07071,-0.7707)--(-0.04242,-0.7424)--(-0.01414,-0.7141)--(0.01414,-0.6859)--(0.04242,-0.6576)--(0.07071,-0.6293)--(0.09899,-0.6010)--(0.1273,-0.5727)--(0.1556,-0.5444)--(0.1838,-0.5162)--(0.2121,-0.4879)--(0.2404,-0.4596)--(0.2687,-0.4313)--(0.2970,-0.4030)--(0.3253,-0.3747)--(0.3535,-0.3465)--(0.3818,-0.3182)--(0.4101,-0.2899)--(0.4384,-0.2616)--(0.4667,-0.2333)--(0.4949,-0.2051)--(0.5232,-0.1768)--(0.5515,-0.1485)--(0.5798,-0.1202)--(0.6081,-0.09192)--(0.6364,-0.06364)--(0.6646,-0.03535)--(0.6929,-0.007071)--(0.7212,0.02121)--(0.7495,0.04949)--(0.7778,0.07778)--(0.8061,0.1061)--(0.8343,0.1343)--(0.8626,0.1626)--(0.8909,0.1909)--(0.9192,0.2192)--(0.9475,0.2475)--(0.9758,0.2758)--(1.004,0.3040)--(1.032,0.3323)--(1.061,0.3606)--(1.089,0.3889)--(1.117,0.4172)--(1.145,0.4455)--(1.174,0.4737)--(1.202,0.5020)--(1.230,0.5303)--(1.259,0.5586)--(1.287,0.5869)--(1.315,0.6152)--(1.343,0.6434)--(1.372,0.6717)--(1.400,0.7000);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall b/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall
index 32b28188c..23fd0b1bb 100644
--- a/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall
+++ b/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall
@@ -96,34 +96,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
 \draw [] (0,0) -- (5.00,5.00);
 
 \draw [color=blue] (0.1000,4.996)--(0.1495,4.594)--(0.1990,4.308)--(0.2485,4.086)--(0.2980,3.904)--(0.3475,3.750)--(0.3970,3.617)--(0.4465,3.500)--(0.4960,3.394)--(0.5455,3.299)--(0.5949,3.212)--(0.6444,3.133)--(0.6939,3.059)--(0.7434,2.990)--(0.7929,2.925)--(0.8424,2.865)--(0.8919,2.808)--(0.9414,2.754)--(0.9909,2.702)--(1.040,2.654)--(1.090,2.607)--(1.139,2.563)--(1.189,2.520)--(1.238,2.479)--(1.288,2.440)--(1.337,2.402)--(1.387,2.366)--(1.436,2.331)--(1.486,2.297)--(1.535,2.264)--(1.585,2.233)--(1.634,2.202)--(1.684,2.172)--(1.733,2.143)--(1.783,2.115)--(1.832,2.088)--(1.882,2.061)--(1.931,2.035)--(1.981,2.010)--(2.030,1.985)--(2.080,1.961)--(2.129,1.937)--(2.179,1.914)--(2.228,1.892)--(2.278,1.870)--(2.327,1.848)--(2.377,1.827)--(2.426,1.807)--(2.476,1.787)--(2.525,1.767)--(2.575,1.747)--(2.624,1.728)--(2.674,1.710)--(2.723,1.691)--(2.773,1.673)--(2.822,1.656)--(2.872,1.638)--(2.921,1.621)--(2.971,1.604)--(3.020,1.588)--(3.070,1.572)--(3.119,1.556)--(3.169,1.540)--(3.218,1.524)--(3.268,1.509)--(3.317,1.494)--(3.367,1.479)--(3.416,1.465)--(3.466,1.450)--(3.515,1.436)--(3.565,1.422)--(3.614,1.408)--(3.664,1.395)--(3.713,1.381)--(3.763,1.368)--(3.812,1.355)--(3.862,1.342)--(3.911,1.329)--(3.961,1.317)--(4.010,1.304)--(4.060,1.292)--(4.109,1.280)--(4.159,1.268)--(4.208,1.256)--(4.258,1.244)--(4.307,1.233)--(4.357,1.221)--(4.406,1.210)--(4.456,1.199)--(4.505,1.188)--(4.555,1.177)--(4.604,1.166)--(4.654,1.156)--(4.703,1.145)--(4.753,1.134)--(4.802,1.124)--(4.852,1.114)--(4.901,1.104)--(4.951,1.094)--(5.000,1.084);
-\draw []  (5.000000000,1.083709268) node [rotate=0] {$\bullet$};
-\draw (5.391795586,1.278156808) node {\( P_{ 0  }\)};
-\draw []  (1.083709268,1.083709268) node [rotate=0] {$\bullet$};
-\draw (0.7188450786,1.385324791) node {\( Q_{0}\)};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.3059510000) node {\( x_{ 0  }\)};
+\draw []  (5.0000,1.0837) node [rotate=0] {$\bullet$};
+\draw (5.3918,1.2782) node {\( P_{ 0  }\)};
+\draw []  (1.0837,1.0837) node [rotate=0] {$\bullet$};
+\draw (0.71885,1.3853) node {\( Q_{0}\)};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.30595) node {\( x_{ 0  }\)};
 \draw [color=cyan,style=dashed] (5.00,1.08) -- (1.08,1.08);
 \draw [color=red,style=dashed] (5.00,1.08) -- (5.00,0);
-\draw []  (1.083709268,2.612757516) node [rotate=0] {$\bullet$};
-\draw (1.356668643,2.949579495) node {\( P_{ 1  }\)};
-\draw []  (2.612757516,2.612757516) node [rotate=0] {$\bullet$};
-\draw (2.247893327,2.914373039) node {\( Q_{1}\)};
-\draw []  (1.083709268,0) node [rotate=0] {$\bullet$};
-\draw (1.083709268,-0.3059510000) node {\( x_{ 1  }\)};
+\draw []  (1.0837,2.6128) node [rotate=0] {$\bullet$};
+\draw (1.3567,2.9496) node {\( P_{ 1  }\)};
+\draw []  (2.6128,2.6128) node [rotate=0] {$\bullet$};
+\draw (2.2479,2.9144) node {\( Q_{1}\)};
+\draw []  (1.0837,0) node [rotate=0] {$\bullet$};
+\draw (1.0837,-0.30595) node {\( x_{ 1  }\)};
 \draw [color=cyan,style=dashed] (1.08,2.61) -- (2.61,2.61);
 \draw [color=red,style=dashed] (1.08,2.61) -- (1.08,0);
-\draw []  (2.612757516,1.732740997) node [rotate=0] {$\bullet$};
-\draw (2.975768946,1.995361763) node {\( P_{ 2  }\)};
-\draw []  (1.732740997,1.732740997) node [rotate=0] {$\bullet$};
-\draw (1.367876808,2.034356520) node {\( Q_{2}\)};
-\draw []  (2.612757516,0) node [rotate=0] {$\bullet$};
-\draw (2.612757516,-0.3059510000) node {\( x_{ 2  }\)};
+\draw []  (2.6128,1.7327) node [rotate=0] {$\bullet$};
+\draw (2.9758,1.9954) node {\( P_{ 2  }\)};
+\draw []  (1.7327,1.7327) node [rotate=0] {$\bullet$};
+\draw (1.3679,2.0344) node {\( Q_{2}\)};
+\draw []  (2.6128,0) node [rotate=0] {$\bullet$};
+\draw (2.6128,-0.30595) node {\( x_{ 2  }\)};
 \draw [color=cyan,style=dashed] (2.61,1.73) -- (1.73,1.73);
 \draw [color=red,style=dashed] (2.61,1.73) -- (2.61,0);
 \draw [] (1.00,-0.100) -- (1.00,0.100);
diff --git a/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall b/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall
index 519863fb1..a903ec27e 100644
--- a/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall
+++ b/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall
@@ -96,36 +96,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (9.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.206705665);
+\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.2067);
 %DEFAULT
-\draw []  (0,1.353352832) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,2.706705665) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,2.706705665) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,1.353352832) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (7.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (8.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (9.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (0,1.3534) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,2.7067) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,2.7067) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,1.3534) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall b/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall
index 5bf8d7b5f..003cc3578 100644
--- a/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall
+++ b/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall
@@ -100,34 +100,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 \draw [color=blue] (0,6.0000)--(0.10101,4.9025)--(0.20202,4.0057)--(0.30303,3.2730)--(0.40404,2.6743)--(0.50505,2.1851)--(0.60606,1.7854)--(0.70707,1.4588)--(0.80808,1.1920)--(0.90909,0.97392)--(1.0101,0.79577)--(1.1111,0.65021)--(1.2121,0.53127)--(1.3131,0.43409)--(1.4141,0.35469)--(1.5152,0.28981)--(1.6162,0.23679)--(1.7172,0.19348)--(1.8182,0.15809)--(1.9192,0.12917)--(2.0202,0.10554)--(2.1212,0.086236)--(2.2222,0.070462)--(2.3232,0.057573)--(2.4242,0.047041)--(2.5253,0.038437)--(2.6263,0.031406)--(2.7273,0.025661)--(2.8283,0.020967)--(2.9293,0.017132)--(3.0303,0.013998)--(3.1313,0.011437)--(3.2323,0.0093452)--(3.3333,0.0076358)--(3.4343,0.0062390)--(3.5354,0.0050978)--(3.6364,0.0041653)--(3.7374,0.0034034)--(3.8384,0.0027808)--(3.9394,0.0022722)--(4.0404,0.0018565)--(4.1414,0.0015169)--(4.2424,0.0012394)--(4.3434,0.0010127)--(4.4444,0)--(4.5455,0)--(4.6465,0)--(4.7475,0)--(4.8485,0)--(4.9495,0)--(5.0505,0)--(5.1515,0)--(5.2525,0)--(5.3535,0)--(5.4545,0)--(5.5556,0)--(5.6566,0)--(5.7576,0)--(5.8586,0)--(5.9596,0)--(6.0606,0)--(6.1616,0)--(6.2626,0)--(6.3636,0)--(6.4646,0)--(6.5657,0)--(6.6667,0)--(6.7677,0)--(6.8687,0)--(6.9697,0)--(7.0707,0)--(7.1717,0)--(7.2727,0)--(7.3737,0)--(7.4747,0)--(7.5758,0)--(7.6768,0)--(7.7778,0)--(7.8788,0)--(7.9798,0)--(8.0808,0)--(8.1818,0)--(8.2828,0)--(8.3838,0)--(8.4848,0)--(8.5859,0)--(8.6869,0)--(8.7879,0)--(8.8889,0)--(8.9899,0)--(9.0909,0)--(9.1919,0)--(9.2929,0)--(9.3939,0)--(9.4949,0)--(9.5960,0)--(9.6970,0)--(9.7980,0)--(9.8990,0)--(10.000,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,6.000000000) node {$ 2 $};
+\draw (-0.29125,6.0000) node {$ 2 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall b/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall
index ac724f702..71823fe52 100644
--- a/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall
+++ b/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall
@@ -36,15 +36,15 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -76,17 +76,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (-1.88,1.32);
-\draw [color=red,->,>=latex] (0,0) -- (-0.1169777784,0.3213938048);
-\draw [color=green,->,>=latex] (0,0) -- (-0.8830222216,-0.3213938048);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=red,->,>=latex] (0,0) -- (-0.11698,0.32139);
+\draw [color=green,->,>=latex] (0,0) -- (-0.88302,-0.32139);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall b/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall
index d9f5e78d1..bfe4147ab 100644
--- a/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall
+++ b/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall
@@ -85,15 +85,15 @@
 \draw [] (0,0) -- (3.46,2.00);
 \draw [] (-1.20,0) -- (13.2,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.1912498333,0.2881297474) node {\( 0\)};
-\draw []  (3.464101615,2.000000000) node [rotate=0] {$\bullet$};
-\draw (3.268077615,2.287267247) node {\( y\)};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.2785761667) node {\( x\)};
-\draw []  (12.00000000,0) node [rotate=0] {$\bullet$};
-\draw (12.00000000,-0.3140621667) node {\( xy\)};
-\draw []  (1.732050808,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.540800974,1.288129747) node {\( 1\)};
+\draw (-0.19125,0.28813) node {\( 0\)};
+\draw []  (3.4641,2.0000) node [rotate=0] {$\bullet$};
+\draw (3.2681,2.2873) node {\( y\)};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.27858) node {\( x\)};
+\draw []  (12.000,0) node [rotate=0] {$\bullet$};
+\draw (12.000,-0.31406) node {\( xy\)};
+\draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.5408,1.2881) node {\( 1\)};
 \draw [style=dashed] (1.73,1.00) -- (6.00,0);
 \draw [style=dashed] (3.46,2.00) -- (12.0,0);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UneCellule.pstricks.recall b/src_phystricks/Fig_UneCellule.pstricks.recall
index 0d7681818..5b9b823d9 100644
--- a/src_phystricks/Fig_UneCellule.pstricks.recall
+++ b/src_phystricks/Fig_UneCellule.pstricks.recall
@@ -103,47 +103,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.700000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.4140621667) node {$a_1=y_{10}$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.41406) node {$a_1=y_{10}$};
 \draw [style=dotted] (1.00,0) -- (1.00,2.00);
 \draw [] (1.00,2.00) -- (1.00,5.00);
-\draw []  (2.200000000,0) node [rotate=0] {$\bullet$};
-\draw (2.200000000,-0.7140621667) node {$y_{11}$};
+\draw []  (2.2000,0) node [rotate=0] {$\bullet$};
+\draw (2.2000,-0.71406) node {$y_{11}$};
 \draw [style=dotted] (2.20,0) -- (2.20,2.00);
 \draw [] (2.20,2.00) -- (2.20,5.00);
-\draw []  (3.700000000,0) node [rotate=0] {$\bullet$};
-\draw (3.700000000,-0.4140621667) node {$y_{12}$};
+\draw []  (3.7000,0) node [rotate=0] {$\bullet$};
+\draw (3.7000,-0.41406) node {$y_{12}$};
 \draw [style=dotted] (3.70,0) -- (3.70,2.00);
 \draw [] (3.70,2.00) -- (3.70,5.00);
-\draw []  (4.200000000,0) node [rotate=0] {$\bullet$};
-\draw (4.200000000,-0.7140621667) node {$y_{13}$};
+\draw []  (4.2000,0) node [rotate=0] {$\bullet$};
+\draw (4.2000,-0.71406) node {$y_{13}$};
 \draw [style=dotted] (4.20,0) -- (4.20,2.00);
 \draw [] (4.20,2.00) -- (4.20,5.00);
-\draw []  (5.200000000,0) node [rotate=0] {$\bullet$};
-\draw (5.200000000,-0.4140621667) node {$y_{14}$};
+\draw []  (5.2000,0) node [rotate=0] {$\bullet$};
+\draw (5.2000,-0.41406) node {$y_{14}$};
 \draw [style=dotted] (5.20,0) -- (5.20,2.00);
 \draw [] (5.20,2.00) -- (5.20,5.00);
-\draw []  (6.200000000,0) node [rotate=0] {$\bullet$};
-\draw (6.200000000,-0.7622220000) node {$b_1=y_{15}$};
+\draw []  (6.2000,0) node [rotate=0] {$\bullet$};
+\draw (6.2000,-0.76222) node {$b_1=y_{15}$};
 \draw [style=dotted] (6.20,0) -- (6.20,2.00);
 \draw [] (6.20,2.00) -- (6.20,5.00);
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.9584071667,2.000000000) node {$a_2=y_{20}$};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.95841,2.0000) node {$a_2=y_{20}$};
 \draw [style=dotted] (0,2.00) -- (1.00,2.00);
 \draw [] (1.00,2.00) -- (6.20,2.00);
-\draw []  (0,2.500000000) node [rotate=0] {$\bullet$};
-\draw (-0.5394763333,2.500000000) node {$y_{21}$};
+\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
+\draw (-0.53948,2.5000) node {$y_{21}$};
 \draw [style=dotted] (0,2.50) -- (1.00,2.50);
 \draw [] (1.00,2.50) -- (6.20,2.50);
-\draw []  (0,4.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.5394763333,4.000000000) node {$y_{22}$};
+\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
+\draw (-0.53948,4.0000) node {$y_{22}$};
 \draw [style=dotted] (0,4.00) -- (1.00,4.00);
 \draw [] (1.00,4.00) -- (6.20,4.00);
-\draw []  (0,5.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.9402623333,5.000000000) node {$b_2=y_{23}$};
+\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.94026,5.0000) node {$b_2=y_{23}$};
 \draw [style=dotted] (0,5.00) -- (1.00,5.00);
 \draw [] (1.00,5.00) -- (6.20,5.00);
 \fill [color=lightgray] (4.20,4.00) -- (5.20,4.00) -- (5.20,4.00) -- (5.20,2.50) -- (5.20,2.50) -- (4.20,2.50) -- (4.20,2.50) -- (4.20,4.00) --  cycle;
diff --git a/src_phystricks/Fig_VANooZowSyO.pstricks.recall b/src_phystricks/Fig_VANooZowSyO.pstricks.recall
index 3d09f7478..653bbd4a8 100644
--- a/src_phystricks/Fig_VANooZowSyO.pstricks.recall
+++ b/src_phystricks/Fig_VANooZowSyO.pstricks.recall
@@ -57,22 +57,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.499496542);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.4995);
 %DEFAULT
 
 \draw [color=blue] (-2.199,-1.000)--(-2.155,-0.9980)--(-2.110,-0.9920)--(-2.066,-0.9819)--(-2.021,-0.9679)--(-1.977,-0.9501)--(-1.933,-0.9284)--(-1.888,-0.9029)--(-1.844,-0.8738)--(-1.799,-0.8413)--(-1.755,-0.8053)--(-1.710,-0.7660)--(-1.666,-0.7237)--(-1.622,-0.6785)--(-1.577,-0.6306)--(-1.533,-0.5801)--(-1.488,-0.5272)--(-1.444,-0.4723)--(-1.399,-0.4154)--(-1.355,-0.3569)--(-1.311,-0.2969)--(-1.266,-0.2358)--(-1.222,-0.1736)--(-1.177,-0.1108)--(-1.133,-0.04758)--(-1.088,0.01587)--(-1.044,0.07925)--(-0.9996,0.1423)--(-0.9552,0.2048)--(-0.9107,0.2665)--(-0.8663,0.3271)--(-0.8219,0.3863)--(-0.7775,0.4441)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6442,0.6056)--(-0.5998,0.6549)--(-0.5553,0.7015)--(-0.5109,0.7453)--(-0.4665,0.7861)--(-0.4221,0.8237)--(-0.3776,0.8580)--(-0.3332,0.8888)--(-0.2888,0.9161)--(-0.2443,0.9397)--(-0.1999,0.9595)--(-0.1555,0.9754)--(-0.1111,0.9874)--(-0.06664,0.9955)--(-0.02221,0.9995)--(0.02221,0.9995)--(0.06664,0.9955)--(0.1111,0.9874)--(0.1555,0.9754)--(0.1999,0.9595)--(0.2443,0.9397)--(0.2888,0.9161)--(0.3332,0.8888)--(0.3776,0.8580)--(0.4221,0.8237)--(0.4665,0.7861)--(0.5109,0.7453)--(0.5553,0.7015)--(0.5998,0.6549)--(0.6442,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7775,0.4441)--(0.8219,0.3863)--(0.8663,0.3271)--(0.9107,0.2665)--(0.9552,0.2048)--(0.9996,0.1423)--(1.044,0.07925)--(1.088,0.01587)--(1.133,-0.04758)--(1.177,-0.1108)--(1.222,-0.1736)--(1.266,-0.2358)--(1.311,-0.2969)--(1.355,-0.3569)--(1.399,-0.4154)--(1.444,-0.4723)--(1.488,-0.5272)--(1.533,-0.5801)--(1.577,-0.6306)--(1.622,-0.6785)--(1.666,-0.7237)--(1.710,-0.7660)--(1.755,-0.8053)--(1.799,-0.8413)--(1.844,-0.8738)--(1.888,-0.9029)--(1.933,-0.9284)--(1.977,-0.9501)--(2.021,-0.9679)--(2.066,-0.9819)--(2.110,-0.9920)--(2.155,-0.9980)--(2.199,-1.000);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -127,22 +127,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.499874128) -- (0,1.499874128);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.4999) -- (0,1.4999);
 %DEFAULT
 
 \draw [color=blue] (-2.199,0)--(-2.155,0.06342)--(-2.110,0.1266)--(-2.066,0.1893)--(-2.021,0.2511)--(-1.977,0.3120)--(-1.933,0.3717)--(-1.888,0.4298)--(-1.844,0.4862)--(-1.799,0.5406)--(-1.755,0.5929)--(-1.710,0.6428)--(-1.666,0.6901)--(-1.622,0.7346)--(-1.577,0.7761)--(-1.533,0.8146)--(-1.488,0.8497)--(-1.444,0.8815)--(-1.399,0.9096)--(-1.355,0.9342)--(-1.311,0.9549)--(-1.266,0.9718)--(-1.222,0.9848)--(-1.177,0.9938)--(-1.133,0.9989)--(-1.088,0.9999)--(-1.044,0.9969)--(-0.9996,0.9898)--(-0.9552,0.9788)--(-0.9107,0.9638)--(-0.8663,0.9450)--(-0.8219,0.9224)--(-0.7775,0.8960)--(-0.7330,0.8660)--(-0.6886,0.8326)--(-0.6442,0.7958)--(-0.5998,0.7558)--(-0.5553,0.7127)--(-0.5109,0.6668)--(-0.4665,0.6182)--(-0.4221,0.5671)--(-0.3776,0.5137)--(-0.3332,0.4582)--(-0.2888,0.4009)--(-0.2443,0.3420)--(-0.1999,0.2817)--(-0.1555,0.2203)--(-0.1111,0.1580)--(-0.06664,0.09506)--(-0.02221,0.03173)--(0.02221,-0.03173)--(0.06664,-0.09506)--(0.1111,-0.1580)--(0.1555,-0.2203)--(0.1999,-0.2817)--(0.2443,-0.3420)--(0.2888,-0.4009)--(0.3332,-0.4582)--(0.3776,-0.5137)--(0.4221,-0.5671)--(0.4665,-0.6182)--(0.5109,-0.6668)--(0.5553,-0.7127)--(0.5998,-0.7558)--(0.6442,-0.7958)--(0.6886,-0.8326)--(0.7330,-0.8660)--(0.7775,-0.8960)--(0.8219,-0.9224)--(0.8663,-0.9450)--(0.9107,-0.9638)--(0.9552,-0.9788)--(0.9996,-0.9898)--(1.044,-0.9969)--(1.088,-0.9999)--(1.133,-0.9989)--(1.177,-0.9938)--(1.222,-0.9848)--(1.266,-0.9718)--(1.311,-0.9549)--(1.355,-0.9342)--(1.399,-0.9096)--(1.444,-0.8815)--(1.488,-0.8497)--(1.533,-0.8146)--(1.577,-0.7761)--(1.622,-0.7346)--(1.666,-0.6901)--(1.710,-0.6428)--(1.755,-0.5929)--(1.799,-0.5406)--(1.844,-0.4862)--(1.888,-0.4298)--(1.933,-0.3717)--(1.977,-0.3120)--(2.021,-0.2511)--(2.066,-0.1893)--(2.110,-0.1266)--(2.155,-0.06342)--(2.199,0);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -197,22 +197,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.495766562) -- (0,1.499066424);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.4958) -- (0,1.4991);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -271,24 +271,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.3105776667,3.141592654) node {$ \pi $};
+\draw (-0.31058,3.1416) node {$ \pi $};
 \draw [] (-0.100,3.14) -- (0.100,3.14);
 %OTHER STUFF
 %END PSPICTURE
@@ -343,22 +343,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-1.499496542) -- (0,1.500000000);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-1.4995) -- (0,1.5000);
 %DEFAULT
 
 \draw [color=blue] (-2.199,1.000)--(-2.155,0.9679)--(-2.110,0.8738)--(-2.066,0.7237)--(-2.021,0.5272)--(-1.977,0.2969)--(-1.933,0.04758)--(-1.888,-0.2048)--(-1.844,-0.4441)--(-1.799,-0.6549)--(-1.755,-0.8237)--(-1.710,-0.9397)--(-1.666,-0.9955)--(-1.622,-0.9874)--(-1.577,-0.9161)--(-1.533,-0.7861)--(-1.488,-0.6056)--(-1.444,-0.3863)--(-1.399,-0.1423)--(-1.355,0.1108)--(-1.311,0.3569)--(-1.266,0.5801)--(-1.222,0.7660)--(-1.177,0.9029)--(-1.133,0.9819)--(-1.088,0.9980)--(-1.044,0.9501)--(-0.9996,0.8413)--(-0.9552,0.6785)--(-0.9107,0.4723)--(-0.8663,0.2358)--(-0.8219,-0.01587)--(-0.7775,-0.2665)--(-0.7330,-0.5000)--(-0.6886,-0.7015)--(-0.6442,-0.8580)--(-0.5998,-0.9595)--(-0.5553,-0.9995)--(-0.5109,-0.9754)--(-0.4665,-0.8888)--(-0.4221,-0.7453)--(-0.3776,-0.5539)--(-0.3332,-0.3271)--(-0.2888,-0.07925)--(-0.2443,0.1736)--(-0.1999,0.4154)--(-0.1555,0.6306)--(-0.1111,0.8053)--(-0.06664,0.9284)--(-0.02221,0.9920)--(0.02221,0.9920)--(0.06664,0.9284)--(0.1111,0.8053)--(0.1555,0.6306)--(0.1999,0.4154)--(0.2443,0.1736)--(0.2888,-0.07925)--(0.3332,-0.3271)--(0.3776,-0.5539)--(0.4221,-0.7453)--(0.4665,-0.8888)--(0.5109,-0.9754)--(0.5553,-0.9995)--(0.5998,-0.9595)--(0.6442,-0.8580)--(0.6886,-0.7015)--(0.7330,-0.5000)--(0.7775,-0.2665)--(0.8219,-0.01587)--(0.8663,0.2358)--(0.9107,0.4723)--(0.9552,0.6785)--(0.9996,0.8413)--(1.044,0.9501)--(1.088,0.9980)--(1.133,0.9819)--(1.177,0.9029)--(1.222,0.7660)--(1.266,0.5801)--(1.311,0.3569)--(1.355,0.1108)--(1.399,-0.1423)--(1.444,-0.3863)--(1.488,-0.6056)--(1.533,-0.7861)--(1.577,-0.9161)--(1.622,-0.9874)--(1.666,-0.9955)--(1.710,-0.9397)--(1.755,-0.8237)--(1.799,-0.6549)--(1.844,-0.4441)--(1.888,-0.2048)--(1.933,0.04758)--(1.977,0.2969)--(2.021,0.5272)--(2.066,0.7237)--(2.110,0.8738)--(2.155,0.9679)--(2.199,1.000);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -413,22 +413,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114858,0) -- (2.699114858,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
 %DEFAULT
 
 \draw [color=blue] (-2.199,1.000)--(-2.155,0.9980)--(-2.110,0.9920)--(-2.066,0.9819)--(-2.021,0.9679)--(-1.977,0.9501)--(-1.933,0.9284)--(-1.888,0.9029)--(-1.844,0.8738)--(-1.799,0.8413)--(-1.755,0.8053)--(-1.710,0.7660)--(-1.666,0.7237)--(-1.622,0.6785)--(-1.577,0.6306)--(-1.533,0.5801)--(-1.488,0.5272)--(-1.444,0.4723)--(-1.399,0.4154)--(-1.355,0.3569)--(-1.311,0.2969)--(-1.266,0.2358)--(-1.222,0.1736)--(-1.177,0.1108)--(-1.133,0.04758)--(-1.088,0.01587)--(-1.044,0.07925)--(-0.9996,0.1423)--(-0.9552,0.2048)--(-0.9107,0.2665)--(-0.8663,0.3271)--(-0.8219,0.3863)--(-0.7775,0.4441)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6442,0.6056)--(-0.5998,0.6549)--(-0.5553,0.7015)--(-0.5109,0.7453)--(-0.4665,0.7861)--(-0.4221,0.8237)--(-0.3776,0.8580)--(-0.3332,0.8888)--(-0.2888,0.9161)--(-0.2443,0.9397)--(-0.1999,0.9595)--(-0.1555,0.9754)--(-0.1111,0.9874)--(-0.06664,0.9955)--(-0.02221,0.9995)--(0.02221,0.9995)--(0.06664,0.9955)--(0.1111,0.9874)--(0.1555,0.9754)--(0.1999,0.9595)--(0.2443,0.9397)--(0.2888,0.9161)--(0.3332,0.8888)--(0.3776,0.8580)--(0.4221,0.8237)--(0.4665,0.7861)--(0.5109,0.7453)--(0.5553,0.7015)--(0.5998,0.6549)--(0.6442,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7775,0.4441)--(0.8219,0.3863)--(0.8663,0.3271)--(0.9107,0.2665)--(0.9552,0.2048)--(0.9996,0.1423)--(1.044,0.07925)--(1.088,0.01587)--(1.133,0.04758)--(1.177,0.1108)--(1.222,0.1736)--(1.266,0.2358)--(1.311,0.2969)--(1.355,0.3569)--(1.399,0.4154)--(1.444,0.4723)--(1.488,0.5272)--(1.533,0.5801)--(1.577,0.6306)--(1.622,0.6785)--(1.666,0.7237)--(1.710,0.7660)--(1.755,0.8053)--(1.799,0.8413)--(1.844,0.8738)--(1.888,0.9029)--(1.933,0.9284)--(1.977,0.9501)--(2.021,0.9679)--(2.066,0.9819)--(2.110,0.9920)--(2.155,0.9980)--(2.199,1.000);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
 %OTHER STUFF
 %END PSPICTURE
@@ -475,18 +475,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.599557429,0) -- (1.599557429,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.499937062);
+\draw [,->,>=latex] (-1.5996,0) -- (1.5996,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4999);
 %DEFAULT
 
 \draw [color=blue] (-1.100,0)--(-1.077,0.1781)--(-1.055,0.2518)--(-1.033,0.3083)--(-1.011,0.3558)--(-0.9885,0.3975)--(-0.9663,0.4350)--(-0.9441,0.4694)--(-0.9219,0.5011)--(-0.8996,0.5308)--(-0.8774,0.5586)--(-0.8552,0.5848)--(-0.8330,0.6096)--(-0.8108,0.6332)--(-0.7886,0.6556)--(-0.7664,0.6769)--(-0.7441,0.6973)--(-0.7219,0.7167)--(-0.6997,0.7353)--(-0.6775,0.7530)--(-0.6553,0.7700)--(-0.6331,0.7862)--(-0.6109,0.8017)--(-0.5887,0.8166)--(-0.5664,0.8307)--(-0.5442,0.8442)--(-0.5220,0.8571)--(-0.4998,0.8693)--(-0.4776,0.8810)--(-0.4554,0.8921)--(-0.4332,0.9025)--(-0.4109,0.9125)--(-0.3887,0.9218)--(-0.3665,0.9306)--(-0.3443,0.9389)--(-0.3221,0.9466)--(-0.2999,0.9537)--(-0.2777,0.9604)--(-0.2555,0.9665)--(-0.2332,0.9721)--(-0.2110,0.9772)--(-0.1888,0.9818)--(-0.1666,0.9858)--(-0.1444,0.9893)--(-0.1222,0.9924)--(-0.09996,0.9949)--(-0.07775,0.9969)--(-0.05553,0.9984)--(-0.03332,0.9994)--(-0.01111,0.9999)--(0.01111,0.9999)--(0.03332,0.9994)--(0.05553,0.9984)--(0.07775,0.9969)--(0.09996,0.9949)--(0.1222,0.9924)--(0.1444,0.9893)--(0.1666,0.9858)--(0.1888,0.9818)--(0.2110,0.9772)--(0.2332,0.9721)--(0.2555,0.9665)--(0.2777,0.9604)--(0.2999,0.9537)--(0.3221,0.9466)--(0.3443,0.9389)--(0.3665,0.9306)--(0.3887,0.9218)--(0.4109,0.9125)--(0.4332,0.9025)--(0.4554,0.8921)--(0.4776,0.8810)--(0.4998,0.8693)--(0.5220,0.8571)--(0.5442,0.8442)--(0.5664,0.8307)--(0.5887,0.8166)--(0.6109,0.8017)--(0.6331,0.7862)--(0.6553,0.7700)--(0.6775,0.7530)--(0.6997,0.7353)--(0.7219,0.7167)--(0.7441,0.6973)--(0.7664,0.6769)--(0.7886,0.6556)--(0.8108,0.6332)--(0.8330,0.6096)--(0.8552,0.5848)--(0.8774,0.5586)--(0.8996,0.5308)--(0.9219,0.5011)--(0.9441,0.4694)--(0.9663,0.4350)--(0.9885,0.3975)--(1.011,0.3558)--(1.033,0.3083)--(1.055,0.2518)--(1.077,0.1781)--(1.100,0);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall b/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall
index 1b44cbd98..1b0b6770c 100644
--- a/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall
+++ b/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall
@@ -91,30 +91,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.400000000,0) -- (1.400000000,0);
-\draw [,->,>=latex] (0,-2.525000000) -- (0,4.550000000);
+\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
+\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
 %DEFAULT
 
 \draw [color=blue] (-0.9000,-2.025)--(-0.8818,-1.905)--(-0.8636,-1.789)--(-0.8455,-1.679)--(-0.8273,-1.573)--(-0.8091,-1.471)--(-0.7909,-1.374)--(-0.7727,-1.282)--(-0.7545,-1.193)--(-0.7364,-1.109)--(-0.7182,-1.029)--(-0.7000,-0.9528)--(-0.6818,-0.8804)--(-0.6636,-0.8119)--(-0.6455,-0.7470)--(-0.6273,-0.6856)--(-0.6091,-0.6277)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5364,-0.4286)--(-0.5182,-0.3865)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4455,-0.2455)--(-0.4273,-0.2167)--(-0.4091,-0.1902)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1238)--(-0.3364,-0.1057)--(-0.3182,-0.08948)--(-0.3000,-0.07500)--(-0.2818,-0.06217)--(-0.2636,-0.05090)--(-0.2455,-0.04108)--(-0.2273,-0.03261)--(-0.2091,-0.02539)--(-0.1909,-0.01933)--(-0.1727,-0.01431)--(-0.1545,-0.01025)--(-0.1364,-0.007044)--(-0.1182,-0.004585)--(-0.1000,-0.002778)--(-0.08182,-0.001521)--(-0.06364,0)--(-0.04545,0)--(-0.02727,0)--(-0.009091,0)--(0.009091,0)--(0.02727,0)--(0.04545,0)--(0.06364,0)--(0.08182,0.001521)--(0.1000,0.002778)--(0.1182,0.004585)--(0.1364,0.007044)--(0.1545,0.01025)--(0.1727,0.01431)--(0.1909,0.01933)--(0.2091,0.02539)--(0.2273,0.03261)--(0.2455,0.04108)--(0.2636,0.05090)--(0.2818,0.06217)--(0.3000,0.07500)--(0.3182,0.08948)--(0.3364,0.1057)--(0.3545,0.1238)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4091,0.1902)--(0.4273,0.2167)--(0.4455,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5182,0.3865)--(0.5364,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6091,0.6277)--(0.6273,0.6856)--(0.6455,0.7470)--(0.6636,0.8119)--(0.6818,0.8804)--(0.7000,0.9528)--(0.7182,1.029)--(0.7364,1.109)--(0.7545,1.193)--(0.7727,1.282)--(0.7909,1.374)--(0.8091,1.471)--(0.8273,1.573)--(0.8455,1.679)--(0.8636,1.789)--(0.8818,1.905)--(0.9000,2.025);
 
 \draw [color=red] (-0.9000,4.050)--(-0.8818,3.888)--(-0.8636,3.729)--(-0.8455,3.574)--(-0.8273,3.422)--(-0.8091,3.273)--(-0.7909,3.128)--(-0.7727,2.986)--(-0.7545,2.847)--(-0.7364,2.711)--(-0.7182,2.579)--(-0.7000,2.450)--(-0.6818,2.324)--(-0.6636,2.202)--(-0.6455,2.083)--(-0.6273,1.967)--(-0.6091,1.855)--(-0.5909,1.746)--(-0.5727,1.640)--(-0.5545,1.538)--(-0.5364,1.438)--(-0.5182,1.343)--(-0.5000,1.250)--(-0.4818,1.161)--(-0.4636,1.075)--(-0.4455,0.9921)--(-0.4273,0.9128)--(-0.4091,0.8368)--(-0.3909,0.7641)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3364,0.5657)--(-0.3182,0.5062)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2455,0.3012)--(-0.2273,0.2583)--(-0.2091,0.2186)--(-0.1909,0.1822)--(-0.1727,0.1492)--(-0.1545,0.1194)--(-0.1364,0.09298)--(-0.1182,0.06983)--(-0.1000,0.05000)--(-0.08182,0.03347)--(-0.06364,0.02025)--(-0.04545,0.01033)--(-0.02727,0.003719)--(-0.009091,0)--(0.009091,0)--(0.02727,0.003719)--(0.04545,0.01033)--(0.06364,0.02025)--(0.08182,0.03347)--(0.1000,0.05000)--(0.1182,0.06983)--(0.1364,0.09298)--(0.1545,0.1194)--(0.1727,0.1492)--(0.1909,0.1822)--(0.2091,0.2186)--(0.2273,0.2583)--(0.2455,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3182,0.5062)--(0.3364,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7641)--(0.4091,0.8368)--(0.4273,0.9128)--(0.4455,0.9921)--(0.4636,1.075)--(0.4818,1.161)--(0.5000,1.250)--(0.5182,1.343)--(0.5364,1.438)--(0.5545,1.538)--(0.5727,1.640)--(0.5909,1.746)--(0.6091,1.855)--(0.6273,1.967)--(0.6455,2.083)--(0.6636,2.202)--(0.6818,2.324)--(0.7000,2.450)--(0.7182,2.579)--(0.7364,2.711)--(0.7545,2.847)--(0.7727,2.986)--(0.7909,3.128)--(0.8091,3.273)--(0.8273,3.422)--(0.8455,3.574)--(0.8636,3.729)--(0.8818,3.888)--(0.9000,4.050);
-\draw (-1.200000000,-0.3298256667) node {$ -2 $};
+\draw (-1.2000,-0.32983) node {$ -2 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (-0.6000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.60000,-0.32983) node {$ -1 $};
 \draw [] (-0.600,-0.100) -- (-0.600,0.100);
-\draw (0.6000000000,-0.3149246667) node {$ 1 $};
+\draw (0.60000,-0.31492) node {$ 1 $};
 \draw [] (0.600,-0.100) -- (0.600,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 2 $};
+\draw (1.2000,-0.31492) node {$ 2 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (-0.4331593333,-2.400000000) node {$ -4 $};
+\draw (-0.43316,-2.4000) node {$ -4 $};
 \draw [] (-0.100,-2.40) -- (0.100,-2.40);
-\draw (-0.4331593333,-1.200000000) node {$ -2 $};
+\draw (-0.43316,-1.2000) node {$ -2 $};
 \draw [] (-0.100,-1.20) -- (0.100,-1.20);
-\draw (-0.2912498333,1.200000000) node {$ 2 $};
+\draw (-0.29125,1.2000) node {$ 2 $};
 \draw [] (-0.100,1.20) -- (0.100,1.20);
-\draw (-0.2912498333,2.400000000) node {$ 4 $};
+\draw (-0.29125,2.4000) node {$ 4 $};
 \draw [] (-0.100,2.40) -- (0.100,2.40);
-\draw (-0.2912498333,3.600000000) node {$ 6 $};
+\draw (-0.29125,3.6000) node {$ 6 $};
 \draw [] (-0.100,3.60) -- (0.100,3.60);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall b/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall
index 763cdabf1..accdd4394 100644
--- a/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall
+++ b/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.400000000);
+\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
 %DEFAULT
 
 \draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
@@ -96,10 +96,10 @@
 \draw [color=blue] (3.000,0)--(3.030,0)--(3.061,0)--(3.091,0)--(3.121,0)--(3.152,0)--(3.182,0)--(3.212,0)--(3.242,0)--(3.273,0)--(3.303,0)--(3.333,0)--(3.364,0)--(3.394,0)--(3.424,0)--(3.455,0)--(3.485,0)--(3.515,0)--(3.545,0)--(3.576,0)--(3.606,0)--(3.636,0)--(3.667,0)--(3.697,0)--(3.727,0)--(3.758,0)--(3.788,0)--(3.818,0)--(3.848,0)--(3.879,0)--(3.909,0)--(3.939,0)--(3.970,0)--(4.000,0)--(4.030,0)--(4.061,0)--(4.091,0)--(4.121,0)--(4.151,0)--(4.182,0)--(4.212,0)--(4.242,0)--(4.273,0)--(4.303,0)--(4.333,0)--(4.364,0)--(4.394,0)--(4.424,0)--(4.455,0)--(4.485,0)--(4.515,0)--(4.545,0)--(4.576,0)--(4.606,0)--(4.636,0)--(4.667,0)--(4.697,0)--(4.727,0)--(4.758,0)--(4.788,0)--(4.818,0)--(4.849,0)--(4.879,0)--(4.909,0)--(4.939,0)--(4.970,0)--(5.000,0)--(5.030,0)--(5.061,0)--(5.091,0)--(5.121,0)--(5.151,0)--(5.182,0)--(5.212,0)--(5.242,0)--(5.273,0)--(5.303,0)--(5.333,0)--(5.364,0)--(5.394,0)--(5.424,0)--(5.455,0)--(5.485,0)--(5.515,0)--(5.545,0)--(5.576,0)--(5.606,0)--(5.636,0)--(5.667,0)--(5.697,0)--(5.727,0)--(5.758,0)--(5.788,0)--(5.818,0)--(5.849,0)--(5.879,0)--(5.909,0)--(5.939,0)--(5.970,0)--(6.000,0);
 \draw [style=dashed] (3.00,0) -- (3.00,3.00);
 \draw [style=dashed] (6.00,3.75) -- (6.00,0);
-\draw (3.000000000,-0.3785761667) node {$a$};
-\draw (6.000000000,-0.3785761667) node {$x$};
-\draw (8.355206667,3.900000000) node {$f(x)$};
-\draw (9.670153833,1.875000000) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (3.0000,-0.37858) node {$a$};
+\draw (6.0000,-0.37858) node {$x$};
+\draw (8.3552,3.9000) node {$f(x)$};
+\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall b/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
index cd77c60af..fd2268f2a 100644
--- a/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
+++ b/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
@@ -120,7 +120,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525262) -- (0,4.054798491);
+\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
 %DEFAULT
 
 \draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
diff --git a/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall b/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
index 0b27d5051..bd5d82aaf 100644
--- a/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
+++ b/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
@@ -101,7 +101,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-5.212388980,0) -- (9.924777961,0);
-\draw [,->,>=latex] (0,-3.445045351) -- (0,3.317012041);
+\draw [,->,>=latex] (0,-3.445045352) -- (0,3.317012042);
 %DEFAULT
 
 \draw [color=black] (-4.712,0)--(-4.570,-0.1423)--(-4.427,-0.2817)--(-4.284,-0.4154)--(-4.141,-0.5406)--(-3.998,-0.6549)--(-3.856,-0.7558)--(-3.713,-0.8413)--(-3.570,-0.9096)--(-3.427,-0.9595)--(-3.284,-0.9898)--(-3.142,-1.000)--(-2.999,-0.9898)--(-2.856,-0.9595)--(-2.713,-0.9096)--(-2.570,-0.8413)--(-2.428,-0.7558)--(-2.285,-0.6549)--(-2.142,-0.5406)--(-1.999,-0.4154)--(-1.856,-0.2817)--(-1.714,-0.1423)--(-1.571,0)--(-1.428,0.1423)--(-1.285,0.2817)--(-1.142,0.4154)--(-0.9996,0.5406)--(-0.8568,0.6549)--(-0.7140,0.7558)--(-0.5712,0.8413)--(-0.4284,0.9096)--(-0.2856,0.9595)--(-0.1428,0.9898)--(0,1.000)--(0.1428,0.9898)--(0.2856,0.9595)--(0.4284,0.9096)--(0.5712,0.8413)--(0.7140,0.7558)--(0.8568,0.6549)--(0.9996,0.5406)--(1.142,0.4154)--(1.285,0.2817)--(1.428,0.1423)--(1.571,0)--(1.714,-0.1423)--(1.856,-0.2817)--(1.999,-0.4154)--(2.142,-0.5406)--(2.285,-0.6549)--(2.428,-0.7558)--(2.570,-0.8413)--(2.713,-0.9096)--(2.856,-0.9595)--(2.999,-0.9898)--(3.142,-1.000)--(3.284,-0.9898)--(3.427,-0.9595)--(3.570,-0.9096)--(3.713,-0.8413)--(3.856,-0.7558)--(3.998,-0.6549)--(4.141,-0.5406)--(4.284,-0.4154)--(4.427,-0.2817)--(4.570,-0.1423)--(4.712,0)--(4.855,0.1423)--(4.998,0.2817)--(5.141,0.4154)--(5.284,0.5406)--(5.426,0.6549)--(5.569,0.7558)--(5.712,0.8413)--(5.855,0.9096)--(5.998,0.9595)--(6.140,0.9898)--(6.283,1.000)--(6.426,0.9898)--(6.569,0.9595)--(6.712,0.9096)--(6.854,0.8413)--(6.997,0.7558)--(7.140,0.6549)--(7.283,0.5406)--(7.426,0.4154)--(7.568,0.2817)--(7.711,0.1423)--(7.854,0)--(7.997,-0.1423)--(8.140,-0.2817)--(8.282,-0.4154)--(8.425,-0.5406)--(8.568,-0.6549)--(8.711,-0.7558)--(8.854,-0.8413)--(8.996,-0.9096)--(9.139,-0.9595)--(9.282,-0.9898)--(9.425,-1.000);
diff --git a/src_phystricks/Fig_WUYooCISzeB.pstricks.recall b/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
index e22c05f29..bf8eb2063 100644
--- a/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
+++ b/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
@@ -62,7 +62,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.9918454936) -- (0,4.027599983);
+\draw [,->,>=latex] (0,-0.9918454937) -- (0,4.027599983);
 %DEFAULT
 
 \draw [color=red] (-0.8400,2.700)--(-0.8013,2.668)--(-0.7626,2.636)--(-0.7239,2.603)--(-0.6852,2.571)--(-0.6466,2.539)--(-0.6079,2.507)--(-0.5692,2.474)--(-0.5305,2.442)--(-0.4918,2.410)--(-0.4531,2.378)--(-0.4144,2.345)--(-0.3757,2.313)--(-0.3370,2.281)--(-0.2984,2.249)--(-0.2597,2.216)--(-0.2210,2.184)--(-0.1823,2.152)--(-0.1436,2.120)--(-0.1049,2.087)--(-0.06622,2.055)--(-0.02753,2.023)--(0.01116,1.991)--(0.04985,1.958)--(0.08854,1.926)--(0.1272,1.894)--(0.1659,1.862)--(0.2046,1.829)--(0.2433,1.797)--(0.2820,1.765)--(0.3207,1.733)--(0.3594,1.701)--(0.3980,1.668)--(0.4367,1.636)--(0.4754,1.604)--(0.5141,1.572)--(0.5528,1.539)--(0.5915,1.507)--(0.6302,1.475)--(0.6689,1.443)--(0.7076,1.410)--(0.7463,1.378)--(0.7849,1.346)--(0.8236,1.314)--(0.8623,1.281)--(0.9010,1.249)--(0.9397,1.217)--(0.9784,1.185)--(1.017,1.152)--(1.056,1.120)--(1.094,1.088)--(1.133,1.056)--(1.172,1.023)--(1.211,0.9912)--(1.249,0.9590)--(1.288,0.9268)--(1.327,0.8945)--(1.365,0.8623)--(1.404,0.8300)--(1.443,0.7978)--(1.481,0.7655)--(1.520,0.7333)--(1.559,0.7011)--(1.597,0.6688)--(1.636,0.6366)--(1.675,0.6043)--(1.713,0.5721)--(1.752,0.5399)--(1.791,0.5076)--(1.830,0.4754)--(1.868,0.4431)--(1.907,0.4109)--(1.946,0.3787)--(1.984,0.3464)--(2.023,0.3142)--(2.062,0.2819)--(2.100,0.2497)--(2.139,0.2175)--(2.178,0.1852)--(2.216,0.1530)--(2.255,0.1207)--(2.294,0.08849)--(2.332,0.05625)--(2.371,0.02401)--(2.410,-0.008233)--(2.449,-0.04047)--(2.487,-0.07271)--(2.526,-0.1050)--(2.565,-0.1372)--(2.603,-0.1694)--(2.642,-0.2017)--(2.681,-0.2339)--(2.719,-0.2662)--(2.758,-0.2984)--(2.797,-0.3306)--(2.835,-0.3629)--(2.874,-0.3951)--(2.913,-0.4274)--(2.952,-0.4596)--(2.990,-0.4918);
diff --git a/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall b/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall
index 11ca058d3..a1dcf7284 100644
--- a/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall
+++ b/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall
@@ -84,34 +84,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-3.795836866) -- (0,3.795836866);
+\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-3.7958) -- (0,3.7958);
 %DEFAULT
 
 \draw [color=blue] (-6.000,-3.296)--(-5.879,-3.169)--(-5.758,-3.044)--(-5.636,-2.920)--(-5.515,-2.797)--(-5.394,-2.676)--(-5.273,-2.556)--(-5.151,-2.437)--(-5.030,-2.320)--(-4.909,-2.204)--(-4.788,-2.090)--(-4.667,-1.977)--(-4.545,-1.866)--(-4.424,-1.756)--(-4.303,-1.648)--(-4.182,-1.542)--(-4.061,-1.438)--(-3.939,-1.335)--(-3.818,-1.234)--(-3.697,-1.136)--(-3.576,-1.039)--(-3.455,-0.9440)--(-3.333,-0.8514)--(-3.212,-0.7609)--(-3.091,-0.6728)--(-2.970,-0.5870)--(-2.848,-0.5037)--(-2.727,-0.4229)--(-2.606,-0.3449)--(-2.485,-0.2697)--(-2.364,-0.1974)--(-2.242,-0.1283)--(-2.121,-0.06241)--(-2.000,0)--(-1.879,0.05873)--(-1.758,0.1135)--(-1.636,0.1642)--(-1.515,0.2103)--(-1.394,0.2516)--(-1.273,0.2876)--(-1.152,0.3179)--(-1.030,0.3417)--(-0.9091,0.3584)--(-0.7879,0.3670)--(-0.6667,0.3662)--(-0.5455,0.3544)--(-0.4242,0.3289)--(-0.3030,0.2859)--(-0.1818,0.2180)--(-0.06061,0.1060)--(0.06061,-0.1060)--(0.1818,-0.2180)--(0.3030,-0.2859)--(0.4242,-0.3289)--(0.5455,-0.3544)--(0.6667,-0.3662)--(0.7879,-0.3670)--(0.9091,-0.3584)--(1.030,-0.3417)--(1.152,-0.3179)--(1.273,-0.2876)--(1.394,-0.2516)--(1.515,-0.2103)--(1.636,-0.1642)--(1.758,-0.1135)--(1.879,-0.05873)--(2.000,0)--(2.121,0.06241)--(2.242,0.1283)--(2.364,0.1974)--(2.485,0.2697)--(2.606,0.3449)--(2.727,0.4229)--(2.848,0.5037)--(2.970,0.5870)--(3.091,0.6728)--(3.212,0.7609)--(3.333,0.8514)--(3.455,0.9440)--(3.576,1.039)--(3.697,1.136)--(3.818,1.234)--(3.939,1.335)--(4.061,1.438)--(4.182,1.542)--(4.303,1.648)--(4.424,1.756)--(4.545,1.866)--(4.667,1.977)--(4.788,2.090)--(4.909,2.204)--(5.030,2.320)--(5.151,2.437)--(5.273,2.556)--(5.394,2.676)--(5.515,2.797)--(5.636,2.920)--(5.758,3.044)--(5.879,3.169)--(6.000,3.296);
-\draw (-6.000000000,-0.3298256667) node {$ -3 $};
+\draw (-6.0000,-0.32983) node {$ -3 $};
 \draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -2 $};
+\draw (-4.0000,-0.32983) node {$ -2 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 3 $};
+\draw (6.0000,-0.31492) node {$ 3 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
index ac5ea3350..75b352840 100644
--- a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
+++ b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
@@ -120,40 +120,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.525156039) -- (0,4.065868530);
+\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-4.5252) -- (0,4.0659);
 %DEFAULT
 
 \draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
-\draw (-6.000000000,-0.3298256667) node {$ -10 $};
+\draw (-6.0000,-0.32983) node {$ -10 $};
 \draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.800000000,-0.3298256667) node {$ -8 $};
+\draw (-4.8000,-0.32983) node {$ -8 $};
 \draw [] (-4.80,-0.100) -- (-4.80,0.100);
-\draw (-3.600000000,-0.3298256667) node {$ -6 $};
+\draw (-3.6000,-0.32983) node {$ -6 $};
 \draw [] (-3.60,-0.100) -- (-3.60,0.100);
-\draw (-2.400000000,-0.3298256667) node {$ -4 $};
+\draw (-2.4000,-0.32983) node {$ -4 $};
 \draw [] (-2.40,-0.100) -- (-2.40,0.100);
-\draw (-1.200000000,-0.3298256667) node {$ -2 $};
+\draw (-1.2000,-0.32983) node {$ -2 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 2 $};
+\draw (1.2000,-0.31492) node {$ 2 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 4 $};
+\draw (2.4000,-0.31492) node {$ 4 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 6 $};
+\draw (3.6000,-0.31492) node {$ 6 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (4.800000000,-0.3149246667) node {$ 8 $};
+\draw (4.8000,-0.31492) node {$ 8 $};
 \draw [] (4.80,-0.100) -- (4.80,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 10 $};
+\draw (6.0000,-0.31492) node {$ 10 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.5944095000,-4.500000000) node {$ -\frac{3}{200} $};
+\draw (-0.59441,-4.5000) node {$ -\frac{3}{200} $};
 \draw [] (-0.100,-4.50) -- (0.100,-4.50);
-\draw (-0.5944095000,-3.000000000) node {$ -\frac{1}{100} $};
+\draw (-0.59441,-3.0000) node {$ -\frac{1}{100} $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.5944095000,-1.500000000) node {$ -\frac{1}{200} $};
+\draw (-0.59441,-1.5000) node {$ -\frac{1}{200} $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.4525000000,1.500000000) node {$ \frac{1}{200} $};
+\draw (-0.45250,1.5000) node {$ \frac{1}{200} $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.4525000000,3.000000000) node {$ \frac{1}{100} $};
+\draw (-0.45250,3.0000) node {$ \frac{1}{100} $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall b/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall
index 965b615c5..9465f44db 100644
--- a/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall
+++ b/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall
@@ -75,21 +75,21 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.3179930509,-0.1489399780) node {\( A \)};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.328701884,-0.1539557273) node {\( B \)};
-\draw []  (2.500000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (2.500000000,3.324708000) node {\( O \)};
+\draw (-0.31799,-0.14894) node {\( A \)};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.3287,-0.15396) node {\( B \)};
+\draw []  (2.5000,3.0000) node [rotate=0] {$\bullet$};
+\draw (2.5000,3.3247) node {\( O \)};
 \draw [] (0,0) -- (5.00,0);
 \draw [] (5.00,0) -- (2.50,3.00);
 \draw [] (2.50,3.00) -- (0,0);
-\draw [color=red,->,>=latex] (0.3200921998,0.3841106398) -- (0.3124099870,0.3905124838);
+\draw [color=red,->,>=latex] (0.32009,0.38411) -- (0.31241,0.39051);
 
 \draw [color=red] (0.500,0)--(0.500,0.00442)--(0.500,0.00885)--(0.500,0.0133)--(0.500,0.0177)--(0.500,0.0221)--(0.499,0.0265)--(0.499,0.0310)--(0.499,0.0354)--(0.498,0.0398)--(0.498,0.0442)--(0.498,0.0486)--(0.497,0.0530)--(0.497,0.0574)--(0.496,0.0618)--(0.496,0.0662)--(0.495,0.0706)--(0.494,0.0749)--(0.494,0.0793)--(0.493,0.0837)--(0.492,0.0880)--(0.491,0.0924)--(0.491,0.0967)--(0.490,0.101)--(0.489,0.105)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.123)--(0.484,0.127)--(0.482,0.131)--(0.481,0.135)--(0.480,0.140)--(0.479,0.144)--(0.478,0.148)--(0.476,0.152)--(0.475,0.157)--(0.473,0.161)--(0.472,0.165)--(0.471,0.169)--(0.469,0.173)--(0.467,0.177)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.194)--(0.459,0.198)--(0.457,0.202)--(0.456,0.206)--(0.454,0.210)--(0.452,0.214)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.230)--(0.442,0.234)--(0.440,0.238)--(0.438,0.242)--(0.436,0.245)--(0.433,0.249)--(0.431,0.253)--(0.429,0.257)--(0.427,0.261)--(0.424,0.265)--(0.422,0.268)--(0.420,0.272)--(0.417,0.276)--(0.415,0.279)--(0.412,0.283)--(0.410,0.287)--(0.407,0.290)--(0.405,0.294)--(0.402,0.297)--(0.399,0.301)--(0.397,0.305)--(0.394,0.308)--(0.391,0.311)--(0.388,0.315)--(0.386,0.318)--(0.383,0.322)--(0.380,0.325)--(0.377,0.328)--(0.374,0.332)--(0.371,0.335)--(0.368,0.338)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.353,0.354)--(0.350,0.357)--(0.346,0.361)--(0.343,0.364)--(0.340,0.367)--(0.337,0.370)--(0.333,0.373)--(0.330,0.375)--(0.327,0.378)--(0.323,0.381)--(0.320,0.384);
-\draw [color=red,->,>=latex] (2.820092200,2.615889360) -- (2.827774413,2.622291204);
+\draw [color=red,->,>=latex] (2.8201,2.6159) -- (2.8278,2.6223);
 
 \draw [color=red] (2.180,2.616)--(2.185,2.611)--(2.191,2.607)--(2.196,2.603)--(2.202,2.599)--(2.208,2.594)--(2.213,2.590)--(2.219,2.586)--(2.225,2.582)--(2.231,2.579)--(2.237,2.575)--(2.243,2.571)--(2.249,2.568)--(2.255,2.564)--(2.261,2.561)--(2.267,2.557)--(2.273,2.554)--(2.280,2.551)--(2.286,2.548)--(2.292,2.545)--(2.299,2.542)--(2.305,2.539)--(2.312,2.537)--(2.318,2.534)--(2.325,2.532)--(2.331,2.529)--(2.338,2.527)--(2.345,2.525)--(2.351,2.523)--(2.358,2.521)--(2.365,2.519)--(2.372,2.517)--(2.378,2.515)--(2.385,2.513)--(2.392,2.512)--(2.399,2.510)--(2.406,2.509)--(2.413,2.508)--(2.420,2.507)--(2.427,2.505)--(2.434,2.504)--(2.440,2.504)--(2.447,2.503)--(2.454,2.502)--(2.461,2.501)--(2.468,2.501)--(2.475,2.501)--(2.482,2.500)--(2.489,2.500)--(2.496,2.500)--(2.504,2.500)--(2.511,2.500)--(2.518,2.500)--(2.525,2.501)--(2.532,2.501)--(2.539,2.501)--(2.546,2.502)--(2.553,2.503)--(2.560,2.504)--(2.566,2.504)--(2.573,2.505)--(2.580,2.507)--(2.587,2.508)--(2.594,2.509)--(2.601,2.510)--(2.608,2.512)--(2.615,2.513)--(2.622,2.515)--(2.628,2.517)--(2.635,2.519)--(2.642,2.521)--(2.649,2.523)--(2.655,2.525)--(2.662,2.527)--(2.669,2.529)--(2.675,2.532)--(2.682,2.534)--(2.688,2.537)--(2.695,2.539)--(2.701,2.542)--(2.708,2.545)--(2.714,2.548)--(2.720,2.551)--(2.727,2.554)--(2.733,2.557)--(2.739,2.561)--(2.745,2.564)--(2.751,2.568)--(2.757,2.571)--(2.763,2.575)--(2.769,2.579)--(2.775,2.582)--(2.781,2.586)--(2.787,2.590)--(2.792,2.594)--(2.798,2.599)--(2.804,2.603)--(2.809,2.607)--(2.815,2.611)--(2.820,2.616);
-\draw [color=red,->,>=latex] (4.500000000,0) -- (4.500000000,-0.01000000000);
+\draw [color=red,->,>=latex] (4.5000,0) -- (4.5000,-0.010000);
 
 \draw [color=red] (4.680,0.3841)--(4.677,0.3813)--(4.673,0.3784)--(4.670,0.3755)--(4.667,0.3725)--(4.663,0.3696)--(4.660,0.3666)--(4.657,0.3636)--(4.654,0.3605)--(4.650,0.3574)--(4.647,0.3543)--(4.644,0.3512)--(4.641,0.3480)--(4.638,0.3448)--(4.635,0.3416)--(4.632,0.3384)--(4.629,0.3351)--(4.626,0.3318)--(4.623,0.3285)--(4.620,0.3251)--(4.617,0.3218)--(4.614,0.3184)--(4.612,0.3149)--(4.609,0.3115)--(4.606,0.3080)--(4.603,0.3045)--(4.601,0.3010)--(4.598,0.2974)--(4.595,0.2939)--(4.593,0.2903)--(4.590,0.2867)--(4.588,0.2830)--(4.585,0.2794)--(4.583,0.2757)--(4.580,0.2720)--(4.578,0.2683)--(4.576,0.2645)--(4.573,0.2608)--(4.571,0.2570)--(4.569,0.2532)--(4.567,0.2493)--(4.564,0.2455)--(4.562,0.2416)--(4.560,0.2378)--(4.558,0.2339)--(4.556,0.2299)--(4.554,0.2260)--(4.552,0.2220)--(4.550,0.2181)--(4.548,0.2141)--(4.546,0.2101)--(4.544,0.2060)--(4.543,0.2020)--(4.541,0.1980)--(4.539,0.1939)--(4.537,0.1898)--(4.536,0.1857)--(4.534,0.1816)--(4.533,0.1775)--(4.531,0.1733)--(4.529,0.1692)--(4.528,0.1650)--(4.527,0.1608)--(4.525,0.1566)--(4.524,0.1524)--(4.522,0.1482)--(4.521,0.1439)--(4.520,0.1397)--(4.519,0.1354)--(4.518,0.1312)--(4.516,0.1269)--(4.515,0.1226)--(4.514,0.1183)--(4.513,0.1140)--(4.512,0.1097)--(4.511,0.1054)--(4.510,0.1011)--(4.509,0.09673)--(4.509,0.09238)--(4.508,0.08803)--(4.507,0.08367)--(4.506,0.07931)--(4.506,0.07493)--(4.505,0.07056)--(4.504,0.06617)--(4.504,0.06179)--(4.503,0.05739)--(4.503,0.05299)--(4.502,0.04859)--(4.502,0.04419)--(4.502,0.03978)--(4.501,0.03537)--(4.501,0.03095)--(4.501,0.02653)--(4.500,0.02212)--(4.500,0.01769)--(4.500,0.01327)--(4.500,0.008849)--(4.500,0.004424)--(4.500,0);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_examssepti.pstricks.recall b/src_phystricks/Fig_examssepti.pstricks.recall
index e8a2ef9ff..74c463784 100644
--- a/src_phystricks/Fig_examssepti.pstricks.recall
+++ b/src_phystricks/Fig_examssepti.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 \draw [style=dotted] (-2.36,1.85) -- (-2.36,0);
 
@@ -95,12 +95,12 @@
 \draw [color=red] (-3.00,3.00)--(-2.95,2.91)--(-2.91,2.82)--(-2.86,2.73)--(-2.82,2.65)--(-2.77,2.56)--(-2.73,2.48)--(-2.68,2.40)--(-2.64,2.32)--(-2.59,2.24)--(-2.55,2.16)--(-2.50,2.08)--(-2.45,2.01)--(-2.41,1.93)--(-2.36,1.86)--(-2.32,1.79)--(-2.27,1.72)--(-2.23,1.65)--(-2.18,1.59)--(-2.14,1.52)--(-2.09,1.46)--(-2.05,1.39)--(-2.00,1.33)--(-1.95,1.27)--(-1.91,1.21)--(-1.86,1.16)--(-1.82,1.10)--(-1.77,1.05)--(-1.73,0.995)--(-1.68,0.943)--(-1.64,0.893)--(-1.59,0.844)--(-1.55,0.796)--(-1.50,0.750)--(-1.45,0.705)--(-1.41,0.662)--(-1.36,0.620)--(-1.32,0.579)--(-1.27,0.540)--(-1.23,0.502)--(-1.18,0.466)--(-1.14,0.430)--(-1.09,0.397)--(-1.05,0.364)--(-1.00,0.333)--(-0.955,0.304)--(-0.909,0.275)--(-0.864,0.249)--(-0.818,0.223)--(-0.773,0.199)--(-0.727,0.176)--(-0.682,0.155)--(-0.636,0.135)--(-0.591,0.116)--(-0.545,0.0992)--(-0.500,0.0833)--(-0.455,0.0689)--(-0.409,0.0558)--(-0.364,0.0441)--(-0.318,0.0337)--(-0.273,0.0248)--(-0.227,0.0172)--(-0.182,0.0110)--(-0.136,0.00620)--(-0.0909,0.00275)--(-0.0455,0)--(0,0)--(0.0455,0)--(0.0909,0.00275)--(0.136,0.00620)--(0.182,0.0110)--(0.227,0.0172)--(0.273,0.0248)--(0.318,0.0337)--(0.364,0.0441)--(0.409,0.0558)--(0.455,0.0689)--(0.500,0.0833)--(0.545,0.0992)--(0.591,0.116)--(0.636,0.135)--(0.682,0.155)--(0.727,0.176)--(0.773,0.199)--(0.818,0.223)--(0.864,0.249)--(0.909,0.275)--(0.955,0.304)--(1.00,0.333)--(1.05,0.364)--(1.09,0.397)--(1.14,0.430)--(1.18,0.466)--(1.23,0.502)--(1.27,0.540)--(1.32,0.579)--(1.36,0.620)--(1.41,0.662)--(1.45,0.705)--(1.50,0.750);
 
 \draw [color=blue] (-3.00,0)--(-2.95,0.520)--(-2.91,0.733)--(-2.86,0.894)--(-2.82,1.03)--(-2.77,1.15)--(-2.73,1.25)--(-2.68,1.34)--(-2.64,1.43)--(-2.59,1.51)--(-2.55,1.59)--(-2.50,1.66)--(-2.45,1.72)--(-2.41,1.79)--(-2.36,1.85)--(-2.32,1.90)--(-2.27,1.96)--(-2.23,2.01)--(-2.18,2.06)--(-2.14,2.11)--(-2.09,2.15)--(-2.05,2.19)--(-2.00,2.24)--(-1.95,2.28)--(-1.91,2.31)--(-1.86,2.35)--(-1.82,2.39)--(-1.77,2.42)--(-1.73,2.45)--(-1.68,2.48)--(-1.64,2.51)--(-1.59,2.54)--(-1.55,2.57)--(-1.50,2.60)--(-1.45,2.62)--(-1.41,2.65)--(-1.36,2.67)--(-1.32,2.69)--(-1.27,2.72)--(-1.23,2.74)--(-1.18,2.76)--(-1.14,2.78)--(-1.09,2.79)--(-1.05,2.81)--(-1.00,2.83)--(-0.955,2.84)--(-0.909,2.86)--(-0.864,2.87)--(-0.818,2.89)--(-0.773,2.90)--(-0.727,2.91)--(-0.682,2.92)--(-0.636,2.93)--(-0.591,2.94)--(-0.545,2.95)--(-0.500,2.96)--(-0.455,2.97)--(-0.409,2.97)--(-0.364,2.98)--(-0.318,2.98)--(-0.273,2.99)--(-0.227,2.99)--(-0.182,2.99)--(-0.136,3.00)--(-0.0909,3.00)--(-0.0455,3.00)--(0,3.00)--(0.0455,3.00)--(0.0909,3.00)--(0.136,3.00)--(0.182,2.99)--(0.227,2.99)--(0.273,2.99)--(0.318,2.98)--(0.364,2.98)--(0.409,2.97)--(0.455,2.97)--(0.500,2.96)--(0.545,2.95)--(0.591,2.94)--(0.636,2.93)--(0.682,2.92)--(0.727,2.91)--(0.773,2.90)--(0.818,2.89)--(0.864,2.87)--(0.909,2.86)--(0.955,2.84)--(1.00,2.83)--(1.05,2.81)--(1.09,2.79)--(1.14,2.78)--(1.18,2.76)--(1.23,2.74)--(1.27,2.72)--(1.32,2.69)--(1.36,2.67)--(1.41,2.65)--(1.45,2.62)--(1.50,2.60);
-\draw []  (-2.358454133,1.854101966) node [rotate=0] {$\bullet$};
-\draw []  (-2.358454133,0) node [rotate=0] {$\bullet$};
-\draw (-2.358454133,-0.2335035000) node {\( -x_0\)};
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (-2.3585,1.8541) node [rotate=0] {$\bullet$};
+\draw []  (-2.3585,0) node [rotate=0] {$\bullet$};
+\draw (-2.3585,-0.23350) node {\( -x_0\)};
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_trigoWedd.pstricks.recall b/src_phystricks/Fig_trigoWedd.pstricks.recall
index ed0d4b55a..899d83cd8 100644
--- a/src_phystricks/Fig_trigoWedd.pstricks.recall
+++ b/src_phystricks/Fig_trigoWedd.pstricks.recall
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.8660254038,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (1.289871192,0.7559510000) node {$z_0$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.2140621667) node {$q$};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$1$};
+\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
+\draw (1.2899,0.75595) node {$z_0$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.21406) node {$q$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$1$};
 
 %OTHER STUFF
 %END PSPICTURE

From b2152e3a3cccedc58ddfb4b476e428193a41ebef Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 20 Jun 2017 17:08:27 +0200
Subject: [PATCH 22/64] =?UTF-8?q?(num=C3=A9rique)=20Propri=C3=A9t=C3=A9s?=
 =?UTF-8?q?=20de=20la=20matrice=20du=20syst=C3=A8me=20aux=20diff=C3=A9renc?=
 =?UTF-8?q?es=20finies.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/frido/159_numerique.tex |   4 +-
 tex/frido/182_numerique.tex | 137 +++++++++++++++++++++++++++++++++++-
 2 files changed, 137 insertions(+), 4 deletions(-)

diff --git a/tex/frido/159_numerique.tex b/tex/frido/159_numerique.tex
index 84b9e5248..8b4d0bb9c 100644
--- a/tex/frido/159_numerique.tex
+++ b/tex/frido/159_numerique.tex
@@ -942,9 +942,9 @@ \subsection{Le théorème}
     Donc  : \( (A_k)_{ii}\neq 0\) pour \( i\leq k\) et \( \det\big( \Omega_{k+1}(A_k) \big)\neq 0\).
 
     Pour fixer les idées, voici une image de \( k=2\) :
-    \[
+    \begin{equation}
     \input{auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks}
-    \]
+    \end{equation}
     
     Étant donné que \( \det\big( \Omega_{k+1}(A_k) \big)\neq 0\), parmi les nombres \( (A_k)_{i,k+1}\) (\( i\geq k+1\)), au moins un est non nul et nous posons \( r_{k+1}\) tel que \( | (A_k)_{r_{k+1},k+1} | \) soit maximum parmi ces éléments.
 
diff --git a/tex/frido/182_numerique.tex b/tex/frido/182_numerique.tex
index f79f05eb0..4b738fc42 100644
--- a/tex/frido/182_numerique.tex
+++ b/tex/frido/182_numerique.tex
@@ -1364,7 +1364,7 @@ \section{Méthode des différences finies de dimension deux}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
 Nous allons considérer le système
-\begin{equation}        
+\begin{equation}                \label{SYSooTANLooRgnIMp}
      \begin{cases}
          -\Delta u=f    &   \text{sur } \Omega\\
          u=g            &   \text{sur } \partial\Omega
@@ -1372,6 +1372,139 @@ \section{Méthode des différences finies de dimension deux}
 \end{equation}
 où \( \Omega=\mathopen] 0 , a \mathclose[\times \mathopen] 0 , b \mathclose[\).
 
-Nous discrétisons \( \Omega\) en mailles carrés de côté \( h\) : \( x_k=kkh\) et \( y_k=kh\).
+\begin{remark}
+    Pourquoi un signe moins devant le laplacien ? Pour avoir le corollaire \ref{CORooZFBXooVGuhQD} qui dira que la matrice correspondant aux différences finies appliquées à ce système est une M-matrice. Sinon, c'est la matrice \(-L_h\) qui en serait une.
+\end{remark}
+
+Nous discrétisons \( \Omega\) en mailles carrés de côté \( h\) : \( x_k=kkh\) et \( y_k=kh\). L'opération de dérivée partielle \( \partial_x\) est discrétisée par
+\begin{equation}
+    (D_x^+u)(x,y)=\frac{ u(x+h)-u(x,y) }{ h }
+\end{equation}
+ou
+\begin{equation}
+    (D_x^-u)(x,y)=\frac{ u(x,y)-u(x-h,y) }{ h }
+\end{equation}
+ou
+\begin{equation}
+    (D^0_xu)(x,y)=\frac{ u(x+h,y)-u(x-h,y) }{ 2h }
+\end{equation}
+où le \( h\) est sous-entendu dans les opérateurs \( D^0\), \( D^+\) et \( D^-\).
+
+La dérivée partielle seconde \( \partial^2_xu\) peut être approximée par toutes les combinaisons imaginable, par exemple
+\begin{equation}
+    (D^-_xD^+_xu)(x,y)=\frac{ u(x+h,y)-2u(x,y)+u(x-h,y) }{ h^2 }.
+\end{equation}
+Pour évaluer la différence entre \( (\partial^2_xu)(x,y)\) et \( (D^-D^+u)(x,y)\), il est possible de faire du Taylor en deux dimension, mais nous pouvons également recycler ce qui a été fait. Nous posons \( u_y(x)=u(x,y)\) et alors \( (\partial_x^2u)(x,y)=u_y''(x)\) et le lemme \ref{LEMooZECZooVKxOZZ}\ref{ITEMooRWUHooZJLKuL} donne, si \( u_y\) est de classe \( C^4\),
+\begin{equation}
+    | u_y''(x)-D^-D^+u_y(x) |\leq \frac{1}{ 12 }h^2\| u_y^{(4)} \|_{\infty}.
+\end{equation}
+Là, les opérateurs \( D^+\) et \( D^-\) sont ceux à une dimension. Mais nous avons \( (D^-D^+u_y)(x)=(D^-D^+u)(x,y)\) (à droite ce sont les opérateurs à deux dimension), donc
+\begin{equation}
+    \big| (\partial^2_xu)(x,y)-(D^-D^+u)(x,y) \big|\leq \frac{1}{ 12 }h^2\| \partial^4_xu \|_{\infty}
+\end{equation}
+et nous pouvons écrire
+\begin{equation}
+    (\partial^2_xu)(x,y)=(D^-D^+u)(x,y)+h^2R(x,y,h)
+\end{equation}
+où \( R\) est une fonction qui dépend de \( x\), \( y\) et \( h\), mais aussi de \( u\). Le point important est que \( R\) soit majoré par une quantité indépendante de \( h\), de telle sorte que nous ayons quelque garanties que négliger ce terme soit une bonne approximation lorsque \( h\to 0\).
+
+Au niveau de la discrétisation, nous considérons \( x_i\) avec \( i=0,\ldots, N_x\) et \( y_j\) avec \( j=0,\ldots, N_y\). La discrétisation de \( -(\Delta u)(x,y)=f(x,y)\) donne, pour \( i=1,\ldots, N_x-1\) et \( j=1,\ldots, N_y-1\),
+\begin{equation}        \label{EQooPWXBooPimUrU}
+    \frac{1}{ h^2 }(-u_{i+1,j}+4u_{ij}-u_{i-1,j}-u_{i,j+1}+u_{i,j-1})=f_{ij}.
+\end{equation}
+Les équations avec \( i\) ou \( j\) valant \( 0\) ou \( N_x\), \( N_y\) sont les valeurs au bords.
+
+Les équations \eqref{EQooPWXBooPimUrU} forment un système d'équations linéaires à résoudre. Certaines peuvent être simplifiées parce qu'elles «touchent» le bord. Nous verrons cela un peu plus tard.
+
+Nous allons d'abord numéroter correctement les équations de façon à ne pas avoir deux mais un seul indice. Notre fonction de numérotation sera
+\begin{equation}
+    \varphi(i,j)=(j-1)(N_x-1)+i
+\end{equation}
+avec \( i=1,\ldots, N_x-1\) et \( j=1,\ldots, N_y-1\). Cela correspond à numéroter les points de l'intérieur du quadrillage ligne par ligne en bas en haut et de gauche à droite. Avec cela les équations \eqref{EQooPWXBooPimUrU} vont être numérotées par un seul indice \( I\) allant de \( \varphi(1,1)=1\) à \( \varphi(N_x-1,N_y-1)=(N_x-1)(N_y-1)\).
+
+Si \( I=\varphi(i,j)\) alors nous avons vite
+\begin{subequations}
+    \begin{align}
+        \varphi(i+1,j)&=I+1\\
+        \varphi(i,j+1)&=I+N_x-1\\
+        \varphi(i-1,j)&=I-1\\
+        \varphi(i,j-1)&=I-N_x+1.
+    \end{align}
+\end{subequations}
+Nous posons \( U_I=u_{\varphi^{-1}(I)}\), et l'équation \eqref{EQooPWXBooPimUrU} devient 
+\begin{equation}
+    \frac{1}{ h^2 }(-U_{I+1}+4U_I-U_{I-1}-U_{I+N_x-1}-U_{I-N_x+1})=f_I.
+\end{equation}
+Pour savoir la matrice représentant ce système, nous devons simplifier les équations qui doivent l'être. Par exemple avec \( I=1\), le terme \( U_{I-1}=U_0\) vaut \( u_{0,1}=f_01\). Ce n'est donc pas réellement une inconnue de notre problème.
 
+Nous voulons mettre les équations sous la forme du système
+\begin{equation}
+    L_hU=F.
+\end{equation}
+Sur la ligne numéro \( I\) de \( L_h\), les éléments non nuls sont :
+\begin{subequations}        \label{SUBEQQooSRQNooYrCNhj}
+    \begin{align}
+        L_{I,I}=4\\
+        L_{I,I+1}=-1\\
+        L_{I,I-1}=-1\\
+        L_{I,I+N_x-1}=-1\\
+        L_{I,I-N_x+1}=-1
+    \end{align}
+\end{subequations}
+pour peu qu'ils existent. Par exemple pour \( I=1\), il n'y a pas d'éléments \( L_{I,I-1}\). Les indices \( I\) et \( J\) de \( L_{I,J}\) vont de \( 1\) à \( \varphi(N_x-1,N_y-1)=(N_y-1)(N_x-1)\).
+
+Voici un dessin de notre situation :
+
+\begin{center}
+   \input{auto/pictures_tex/Fig_GMRNooCNBpIl.pstricks}
+\end{center}
+
+À chaque élément du quadrillage correspond une équation. 
+\begin{itemize}
+    \item 
+        Aux points simples sur le bord, correspondent des équations triviales parce que la fonction \(u \) y est directement donnée par les conditions aux bords.
+    \item
+        Aux points étoilés entourés en traits continus correspondent des équations «incomplètes» parce que certains termes de l'équation \eqref{EQooPWXBooPimUrU} sont donnés par les conditions aux bords. Elle correspondent aussi aux lignes incomplète de la matrice \( L_h\) où certains éléments donnés en \eqref{SUBEQQooSRQNooYrCNhj} n'existent pas.
 
+        Le membre de droite de ces équations est par contre enrichi de ce qui à gauche est «donné».
+
+    \item
+        Au points étoilés du centre entourés en traits discontinus correspondent des équations complètes.
+
+\end{itemize}
+
+Notons que \( f_{00}\) ne joue aucun rôle dans notre histoire parce que dans les équations \eqref{EQooPWXBooPimUrU}, chaque point \( (i,j)\) du maillage n'est liée qu'aux quatre points situés «à côté».
+
+
+\begin{proposition} \label{PROPooWGTRooVjWhYY}
+    La matrice \(L_h\) est irréductible et à diagonale dominante\footnote{Définition \ref{DEFooLSUTooHuXabV}.}.
+\end{proposition}
+
+\begin{proof}
+    Une matrice \( n\times n\) dont les deux premières diagonales sont entièrement composées d'éléments non nuls est toujours irréductible. En effet, la première lie l'élément \( (1,2)\) à l'élément \( (n-1,n)\) et donc permet de dire que tous les \( i<j\) sont connectés.
+
+    La seconde diagonale lie l'élément \( (n,n-1)\) à l'élément \( (2,1)\).
+
+    En ce qui concerne la dominance de la diagonale, il faut sommer sur les lignes. Or chaque ligne contient (en valeur absolue) un \( 4\) sur la diagonale et \( 4\) éléments qui valent \( 1\). D'où
+    \begin{equation}
+        | L_{II} |\geq \sum_{J\neq I}| L_{IJ} |.
+    \end{equation}
+\end{proof}
+
+Notons que \( L_h\) est le plus souvent à diagonale fortement dominante parce que dès qu'une ligne est «incomplète», la diagonale domine.
+
+\begin{corollary}       \label{CORooZFBXooVGuhQD}
+    La matrice \( L_h\) est
+    \begin{enumerate}
+        \item
+            Une M-matrice
+        \item
+            inversible avec \( L_{h}>0\)
+    \end{enumerate}
+\end{corollary}
+
+\begin{proof}
+    D'après ce que nous venons de voir (proposition \ref{PROPooWGTRooVjWhYY}), le théorème \ref{THOooLZGSooSevggj} fonctionne et \( L_h\) est une M-matrice\footnote{Notons que c'est ici que nous sommes content d'avoir posé \( -\Delta u=f\) dans le système \eqref{SYSooTANLooRgnIMp}, avec un signe négatif devant le laplacien. Sinon tous le signes auraient changé et la matrice \( -L_h\) aurait été une M-matrice au lieu de \( L_h\).}.
+
+    La proposition \ref{PROPooZDMQooIZAbKK} nous assure qu'une M-matrice irréductible est d'inverse strictement positif. Donc \( L_h^{-1}>0\).
+\end{proof}

From ddd66aa7f6ff309a5ce2b8e992136ea1f026f504 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 21 Jun 2017 00:30:21 +0200
Subject: [PATCH 23/64] (testing) Provide a small script that creates the LaTeX
 code for including pictures from a file containing their names.

Utility for debugging.
---
 testing/fig_list_to_tex_test.py | 27 +++++++++++++++++++++++++++
 1 file changed, 27 insertions(+)
 create mode 100755 testing/fig_list_to_tex_test.py

diff --git a/testing/fig_list_to_tex_test.py b/testing/fig_list_to_tex_test.py
new file mode 100755
index 000000000..21b7b7d79
--- /dev/null
+++ b/testing/fig_list_to_tex_test.py
@@ -0,0 +1,27 @@
+#! /usr/bin/python3
+# -*- coding: utf8 -*-
+
+# read a file containing lines like
+#TKXZooLwXzjS
+#GMIooJvcCXg
+#DivergenceUn
+# and prints a sequence of lines to be included in LaTeX for test.
+
+import sys
+
+skel="""
+NAME
+\\begin{center}
+    \\newcommand{\CaptionFigNAME}{NAME}
+    \input{auto/pictures_tex/Fig_NAME.pstricks}
+\end{center}
+
+\clearpage
+
+"""
+
+with open(sys.argv[1],'r') as f:
+    for n in f.readlines() :
+        name=n[:-1]
+        print(skel.replace("NAME",name))
+

From 2f36845819900399a2277ed475f1b3f2ff5ecbc0 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 21 Jun 2017 00:31:41 +0200
Subject: [PATCH 24/64] (testing) Make 'testing/testing.sh' recall the name of
 the log file at the end of execution.

---
 testing/testing.sh | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/testing/testing.sh b/testing/testing.sh
index d72de039a..9d081f2ca 100755
--- a/testing/testing.sh
+++ b/testing/testing.sh
@@ -118,9 +118,9 @@ wait
 cd $CLONE_DIR
 
 
-echo "log file : $LOG_FILE "
 echo "----------------"
 
 cat  $LOG_FILE 
 
 echo "--------------------"
+echo "Find all the results in $LOG_FILE"

From 6854bd539d8ec0e5592e863a40505529754b15d3 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 21 Jun 2017 00:32:31 +0200
Subject: [PATCH 25/64] (exercice) typo

---
 tex/exocorr/corr0045.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/tex/exocorr/corr0045.tex b/tex/exocorr/corr0045.tex
index 81e656563..dff3d35b1 100644
--- a/tex/exocorr/corr0045.tex
+++ b/tex/exocorr/corr0045.tex
@@ -1,5 +1,5 @@
 % This is part of Exercices et corrigés de CdI-1
-% Copyright (c) 2011,2016
+% Copyright (c) 2011,2016-2017
 %   Laurent Claessens
 % See the file fdl-1.3.txt for copying conditions.
 
@@ -9,7 +9,7 @@
 \begin{equation*}
   f(x,y) =
   \begin{cases}
-      xy & \text{si} x < 0,y > 0\\
+      xy & \text{si } x < 0,y > 0\\
       x-y & \text{si } x \geq 0,y \geq 0\\
       x^2y & \text{si } x > 0,y < 0\\
     x+y & \text{sinon.}

From f69a2d02d295b260e5b7eae98bd882fa1317d06b Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 21 Jun 2017 00:33:05 +0200
Subject: [PATCH 26/64] (pictures) Debug some pictures.

- Fix some pictures to fit the author's intention
- the compiled pictures
- the corresponding recall files.
---
 auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks    | 110 ++---
 auto/pictures_tex/Fig_ADUGmRRB.pstricks       |   6 +-
 auto/pictures_tex/Fig_ADUGmRRC.pstricks       |   6 +-
 auto/pictures_tex/Fig_AIFsOQO.pstricks        |  16 +-
 auto/pictures_tex/Fig_ALIzHFm.pstricks        |   8 +-
 auto/pictures_tex/Fig_AdhIntFrDeux.pstricks   |  28 +-
 auto/pictures_tex/Fig_AdhIntFrSix.pstricks    |  10 +-
 auto/pictures_tex/Fig_AdhIntFrTrois.pstricks  |  20 +-
 auto/pictures_tex/Fig_AireParabole.pstricks   |  26 +-
 auto/pictures_tex/Fig_BEHTooWsdrys.pstricks   |  98 ++--
 auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks   |  16 +-
 auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks   |  34 +-
 auto/pictures_tex/Fig_Bateau.pstricks         |  40 +-
 auto/pictures_tex/Fig_CELooGVvzMc.pstricks    |  24 +-
 auto/pictures_tex/Fig_CMMAooQegASg.pstricks   |   4 +-
 auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks   |  36 +-
 auto/pictures_tex/Fig_CSCvi.pstricks          |  40 +-
 auto/pictures_tex/Fig_CURGooXvruWV.pstricks   |  12 +-
 auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks |  12 +-
 auto/pictures_tex/Fig_CbCartTui.pstricks      |  62 +--
 auto/pictures_tex/Fig_CbCartTuiii.pstricks    |  12 +-
 .../pictures_tex/Fig_CercleImplicite.pstricks |  20 +-
 .../pictures_tex/Fig_ChampGraviation.pstricks | 440 ++++++++---------
 auto/pictures_tex/Fig_ChiSquared.pstricks     |  20 +-
 .../Fig_ChiSquaresQuantile.pstricks           |  20 +-
 auto/pictures_tex/Fig_ChoixInfini.pstricks    |  38 +-
 auto/pictures_tex/Fig_CoinPasVar.pstricks     |  16 +-
 auto/pictures_tex/Fig_ContourGreen.pstricks   |  16 +-
 auto/pictures_tex/Fig_ContourSqL.pstricks     |  12 +-
 .../Fig_ContourTgNDivergence.pstricks         |  48 +-
 auto/pictures_tex/Fig_CoordPolaires.pstricks  |  18 +-
 auto/pictures_tex/Fig_CornetGlace.pstricks    |  10 +-
 .../Fig_CourbeRectifiable.pstricks            |  20 +-
 auto/pictures_tex/Fig_CouroneExam.pstricks    |  12 +-
 .../Fig_CurvilignesPolaires.pstricks          |  16 +-
 auto/pictures_tex/Fig_CycloideA.pstricks      |  34 +-
 auto/pictures_tex/Fig_DDCTooYscVzA.pstricks   |  48 +-
 auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks |   4 +-
 auto/pictures_tex/Fig_DZVooQZLUtf.pstricks    | 174 +++----
 .../pictures_tex/Fig_DerivTangenteOM.pstricks |  36 +-
 auto/pictures_tex/Fig_DessinLim.pstricks      |  24 +-
 auto/pictures_tex/Fig_DeuxCercles.pstricks    |   8 +-
 auto/pictures_tex/Fig_Differentielle.pstricks |  42 +-
 .../Fig_DistanceEnsemble.pstricks             |  14 +-
 auto/pictures_tex/Fig_DivergenceDeux.pstricks | 448 +++++++++---------
 .../pictures_tex/Fig_DivergenceTrois.pstricks |  82 ++--
 auto/pictures_tex/Fig_DivergenceUn.pstricks   |  98 ++--
 auto/pictures_tex/Fig_DynkinNUtPJx.pstricks   |  58 +--
 auto/pictures_tex/Fig_DynkinqlgIQl.pstricks   |   4 +-
 auto/pictures_tex/Fig_EELKooMwkockxB.pstricks |  32 +-
 auto/pictures_tex/Fig_EHDooGDwfjC.pstricks    |  18 +-
 auto/pictures_tex/Fig_EJRsWXw.pstricks        |  12 +-
 auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks |  50 +-
 auto/pictures_tex/Fig_ExPolygone.pstricks     |  20 +-
 auto/pictures_tex/Fig_ExSinLarge.pstricks     |  16 +-
 .../Fig_ExampleIntegration.pstricks           |  16 +-
 .../Fig_ExampleIntegrationdeux.pstricks       |  40 +-
 .../pictures_tex/Fig_ExempleArcParam.pstricks |  36 +-
 auto/pictures_tex/Fig_ExempleNonRang.pstricks |   4 +-
 .../Fig_ExerciceGraphesbis.pstricks           | 168 +++----
 auto/pictures_tex/Fig_ExoMagnetique.pstricks  |  12 +-
 auto/pictures_tex/Fig_ExoPolaire.pstricks     |  12 +-
 auto/pictures_tex/Fig_ExoProjection.pstricks  |  24 +-
 auto/pictures_tex/Fig_ExoVarj.pstricks        |   8 +-
 auto/pictures_tex/Fig_ExoXLVL.pstricks        |  36 +-
 auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks |  56 +--
 auto/pictures_tex/Fig_FGRooDhFkch.pstricks    |  38 +-
 auto/pictures_tex/Fig_FGWjJBX.pstricks        |  20 +-
 auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks   |  12 +-
 auto/pictures_tex/Fig_FWJuNhU.pstricks        |  12 +-
 auto/pictures_tex/Fig_FXVooJYAfif.pstricks    |   4 +-
 .../Fig_FonctionEtDeriveOM.pstricks           |  32 +-
 .../Fig_FonctionXtroisOM.pstricks             |  22 +-
 auto/pictures_tex/Fig_GCNooKEbjWB.pstricks    |  14 +-
 auto/pictures_tex/Fig_GMIooJvcCXg.pstricks    |  50 +-
 auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks |  18 +-
 auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks   |  16 +-
 auto/pictures_tex/Fig_HNxitLj.pstricks        |  32 +-
 auto/pictures_tex/Fig_HasseAGdfdy.pstricks    |  22 +-
 auto/pictures_tex/Fig_IWuPxFc.pstricks        |  30 +-
 auto/pictures_tex/Fig_IYAvSvI.pstricks        |   8 +-
 auto/pictures_tex/Fig_IntBoutCercle.pstricks  |   8 +-
 auto/pictures_tex/Fig_IntRectangle.pstricks   |  10 +-
 auto/pictures_tex/Fig_IntTriangle.pstricks    |  12 +-
 .../pictures_tex/Fig_IntegraleSimple.pstricks |  12 +-
 auto/pictures_tex/Fig_JGuKEjH.pstricks        |   6 +-
 auto/pictures_tex/Fig_JJAooWpimYW.pstricks    |  28 +-
 auto/pictures_tex/Fig_JSLooFJWXtB.pstricks    |  20 +-
 auto/pictures_tex/Fig_JWINooSfKCeA.pstricks   |  18 +-
 auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks   |  36 +-
 auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks |  12 +-
 auto/pictures_tex/Fig_KKLooMbjxdI.pstricks    |  12 +-
 auto/pictures_tex/Fig_KKRooHseDzC.pstricks    |  16 +-
 auto/pictures_tex/Fig_KScolorD.pstricks       |   8 +-
 auto/pictures_tex/Fig_LEJNDxI.pstricks        |  18 +-
 auto/pictures_tex/Fig_LesSpheres.pstricks     |  36 +-
 auto/pictures_tex/Fig_LesSubFigures.pstricks  |  48 +-
 .../pictures_tex/Fig_LesSubFiguresOM.pstricks |  48 +-
 auto/pictures_tex/Fig_MCKyvdk.pstricks        |  16 +-
 auto/pictures_tex/Fig_MNICGhR.pstricks        |  18 +-
 auto/pictures_tex/Fig_MaxVraissLp.pstricks    |  22 +-
 auto/pictures_tex/Fig_MethodeChemin.pstricks  |   8 +-
 auto/pictures_tex/Fig_MethodeNewton.pstricks  |  34 +-
 auto/pictures_tex/Fig_MomentForce.pstricks    |  12 +-
 auto/pictures_tex/Fig_MoulinEau.pstricks      |  40 +-
 auto/pictures_tex/Fig_NEtAchr.pstricks        |  24 +-
 auto/pictures_tex/Fig_NWDooOObSHB.pstricks    |  16 +-
 .../pictures_tex/Fig_NiveauHyperbole.pstricks |  36 +-
 auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks |  32 +-
 auto/pictures_tex/Fig_Osculateur.pstricks     |  12 +-
 auto/pictures_tex/Fig_PHTVjfk.pstricks        |  20 +-
 auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks |   4 +-
 auto/pictures_tex/Fig_PONXooXYjEot.pstricks   |  16 +-
 auto/pictures_tex/Fig_PVJooJDyNAg.pstricks    |  44 +-
 .../Fig_ParallelogrammeOM.pstricks            |  16 +-
 auto/pictures_tex/Fig_ParamTangente.pstricks  |  36 +-
 auto/pictures_tex/Fig_PartieEntiere.pstricks  |  24 +-
 auto/pictures_tex/Fig_Polirettangolo.pstricks |  44 +-
 auto/pictures_tex/Fig_ProjPoly.pstricks       |  50 +-
 auto/pictures_tex/Fig_QCb.pstricks            |  12 +-
 auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks |  22 +-
 auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks   |  36 +-
 auto/pictures_tex/Fig_QQa.pstricks            |  24 +-
 auto/pictures_tex/Fig_QXyVaKD.pstricks        |   8 +-
 auto/pictures_tex/Fig_QuelCote.pstricks       |  20 +-
 auto/pictures_tex/Fig_RGjjpwF.pstricks        |  18 +-
 auto/pictures_tex/Fig_RLuqsrr.pstricks        |  16 +-
 auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks   |  14 +-
 auto/pictures_tex/Fig_RQsQKTl.pstricks        |   4 +-
 .../Fig_RegioniPrimoeSecondoTipo.pstricks     |  32 +-
 auto/pictures_tex/Fig_SBTooEasQsT.pstricks    |  32 +-
 auto/pictures_tex/Fig_SFdgHdO.pstricks        |  16 +-
 auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks   | 440 ++++++++---------
 auto/pictures_tex/Fig_STdyNTH.pstricks        |  18 +-
 auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks |  40 +-
 auto/pictures_tex/Fig_SenoTopologo.pstricks   |   4 +-
 auto/pictures_tex/Fig_SolsEqDiffSin.pstricks  |  12 +-
 auto/pictures_tex/Fig_SpiraleLimite.pstricks  |   4 +-
 .../Fig_SubfiguresCDUTraceCycloide.pstricks   | 118 ++---
 auto/pictures_tex/Fig_SuiteUnSurn.pstricks    |  44 +-
 auto/pictures_tex/Fig_SurfaceCercle.pstricks  |   4 +-
 .../Fig_SurfaceHorizVerti.pstricks            |  24 +-
 auto/pictures_tex/Fig_TGdUoZR.pstricks        |  14 +-
 auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks |   6 +-
 auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks   |  98 ++--
 auto/pictures_tex/Fig_TVXooWoKkqV.pstricks    |  22 +-
 auto/pictures_tex/Fig_TWHooJjXEtS.pstricks    |  28 +-
 auto/pictures_tex/Fig_TangentSegment.pstricks |  38 +-
 auto/pictures_tex/Fig_TangenteDetail.pstricks |  40 +-
 .../Fig_TangenteDetailOM.pstricks             |  40 +-
 .../Fig_TangenteQuestion.pstricks             |   8 +-
 .../Fig_TangenteQuestionOM.pstricks           |   8 +-
 auto/pictures_tex/Fig_ToreRevolution.pstricks |  14 +-
 auto/pictures_tex/Fig_Trajs.pstricks          |  24 +-
 .../Fig_TriangleRectangle.pstricks            |  18 +-
 auto/pictures_tex/Fig_TriangleUV.pstricks     |  20 +-
 auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks |   4 +-
 auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks   |  62 +--
 auto/pictures_tex/Fig_UNVooMsXxHa.pstricks    |  40 +-
 auto/pictures_tex/Fig_UQZooGFLNEq.pstricks    |  28 +-
 auto/pictures_tex/Fig_UZGooBzlYxr.pstricks    |  40 +-
 auto/pictures_tex/Fig_UnSurxInt.pstricks      |  28 +-
 auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks   |  16 +-
 auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks   |  10 +-
 auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks   |  20 +-
 auto/pictures_tex/Fig_WHCooNZAmYB.pstricks    |  14 +-
 auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks   |  32 +-
 auto/pictures_tex/Fig_WJBooMTAhtl.pstricks    |  24 +-
 auto/pictures_tex/Fig_WUYooCISzeB.pstricks    |  80 ++--
 auto/pictures_tex/Fig_XTGooSFFtPu.pstricks    |  50 +-
 auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks   |  14 +-
 auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks   | 448 +++++++++---------
 auto/pictures_tex/Fig_YWxOAkh.pstricks        |   8 +-
 auto/pictures_tex/Fig_YYECooQlnKtD.pstricks   |  48 +-
 auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks   |  82 ++--
 auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks |   8 +-
 auto/pictures_tex/Fig_ZTTooXtHkci.pstricks    |  28 +-
 auto/pictures_tex/Fig_examsseptii.pstricks    |  30 +-
 auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks |   8 +-
 auto/pictures_tex/Fig_ratrap.pstricks         |  20 +-
 auto/pictures_tex/Fig_senotopologo.pstricks   |  24 +-
 .../Fig_ACUooQwcDMZ.pstricks.recall           | 110 ++---
 src_phystricks/Fig_ADUGmRRB.pstricks.recall   |   6 +-
 src_phystricks/Fig_ADUGmRRC.pstricks.recall   |   6 +-
 src_phystricks/Fig_ALIzHFm.pstricks.recall    |   8 +-
 .../Fig_AdhIntFrDeux.pstricks.recall          |  28 +-
 .../Fig_AdhIntFrSix.pstricks.recall           |  10 +-
 .../Fig_AdhIntFrTrois.pstricks.recall         |  20 +-
 .../Fig_AireParabole.pstricks.recall          |  26 +-
 .../Fig_BEHTooWsdrys.pstricks.recall          | 104 ++--
 .../Fig_BNHLooLDxdPA.pstricks.recall          |  16 +-
 .../Fig_BQXKooPqSEMN.pstricks.recall          |  36 +-
 src_phystricks/Fig_Bateau.pstricks.recall     |  40 +-
 .../Fig_CELooGVvzMc.pstricks.recall           |  24 +-
 .../Fig_CMMAooQegASg.pstricks.recall          |   4 +-
 .../Fig_CQIXooBEDnfK.pstricks.recall          |  36 +-
 src_phystricks/Fig_CSCvi.pstricks.recall      |  40 +-
 .../Fig_CURGooXvruWV.pstricks.recall          |  12 +-
 .../Fig_CWKJooppMsZXjw.pstricks.recall        |  12 +-
 src_phystricks/Fig_CbCartTui.pstricks.recall  |  62 +--
 .../Fig_CbCartTuiii.pstricks.recall           |  12 +-
 .../Fig_CercleImplicite.pstricks.recall       |  20 +-
 .../Fig_ChampGraviation.pstricks.recall       | 440 ++++++++---------
 src_phystricks/Fig_ChiSquared.pstricks.recall |  20 +-
 .../Fig_ChiSquaresQuantile.pstricks.recall    |  20 +-
 .../Fig_ChoixInfini.pstricks.recall           |  38 +-
 src_phystricks/Fig_CoinPasVar.pstricks.recall |  16 +-
 .../Fig_ContourGreen.pstricks.recall          |  16 +-
 src_phystricks/Fig_ContourSqL.pstricks.recall |  12 +-
 .../Fig_ContourTgNDivergence.pstricks.recall  |  48 +-
 .../Fig_CoordPolaires.pstricks.recall         |  18 +-
 .../Fig_CornetGlace.pstricks.recall           |  10 +-
 .../Fig_CourbeRectifiable.pstricks.recall     |  20 +-
 .../Fig_CouroneExam.pstricks.recall           |  12 +-
 .../Fig_CurvilignesPolaires.pstricks.recall   |  16 +-
 src_phystricks/Fig_CycloideA.pstricks.recall  |  34 +-
 .../Fig_DDCTooYscVzA.pstricks.recall          |  48 +-
 .../Fig_DGFSooWgbuuMoB.pstricks.recall        |   4 +-
 .../Fig_DZVooQZLUtf.pstricks.recall           | 174 +++----
 .../Fig_DerivTangenteOM.pstricks.recall       |  36 +-
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 .../Fig_Differentielle.pstricks.recall        |  42 +-
 .../Fig_DistanceEnsemble.pstricks.recall      |  14 +-
 .../Fig_DivergenceDeux.pstricks.recall        | 448 +++++++++---------
 .../Fig_DivergenceTrois.pstricks.recall       |  82 ++--
 .../Fig_DivergenceUn.pstricks.recall          | 104 ++--
 .../Fig_DynkinNUtPJx.pstricks.recall          |  58 +--
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 .../Fig_ExerciceGraphesbis.pstricks.recall    | 168 +++----
 .../Fig_ExoMagnetique.pstricks.recall         |  12 +-
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 src_phystricks/Fig_ExoVarj.pstricks.recall    |   8 +-
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 src_phystricks/Fig_UnSurxInt.pstricks.recall  |  28 +-
 .../Fig_VDFMooHMmFZr.pstricks.recall          |  16 +-
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 .../Fig_WJBooMTAhtl.pstricks.recall           |  24 +-
 .../Fig_WUYooCISzeB.pstricks.recall           |  80 ++--
 .../Fig_XTGooSFFtPu.pstricks.recall           |  50 +-
 .../Fig_YHJYooTEXLLn.pstricks.recall          |  14 +-
 .../Fig_YQVHooYsGLHQ.pstricks.recall          | 448 +++++++++---------
 src_phystricks/Fig_YWxOAkh.pstricks.recall    |   8 +-
 .../Fig_YYECooQlnKtD.pstricks.recall          |  48 +-
 .../Fig_ZGUDooEsqCWQ.pstricks.recall          |  82 ++--
 .../Fig_ZTTooXtHkci.pstricks.recall           |  28 +-
 .../Fig_examsseptii.pstricks.recall           |  30 +-
 .../Fig_ooIHLPooKLIxcH.pstricks.recall        |   8 +-
 src_phystricks/Fig_ratrap.pstricks.recall     |  20 +-
 .../Fig_senotopologo.pstricks.recall          |  24 +-
 src_phystricks/phystricksBQXKooPqSEMN.py      |   2 +-
 src_phystricks/phystricksExoXLVL.py           |  10 +-
 src_phystricks/phystricksPolirettangolo.py    |   1 +
 359 files changed, 6699 insertions(+), 6634 deletions(-)

diff --git a/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks b/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
index 9e1c13cba..974934d32 100644
--- a/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
+++ b/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
@@ -49,20 +49,20 @@
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 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.193147181);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -109,26 +109,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.193147181);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
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+\draw (-2.0000,-0.32983) node {$ -2 $};
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
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 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -175,22 +175,22 @@
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 %AXES
-\draw [,->,>=latex] (-1.364664717,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.598612289);
+\draw [,->,>=latex] (-1.3647,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.5986);
 %DEFAULT
 
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -233,20 +233,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,2.193147181);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,2.1931);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -289,20 +289,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -345,18 +345,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.548147075);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5481);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -411,30 +411,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,1.886294361);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,1.8863);
 %DEFAULT
 
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-\draw (2.000000000,-0.3149246667) node {$ 2 $};
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 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
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 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
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 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ADUGmRRB.pstricks b/auto/pictures_tex/Fig_ADUGmRRB.pstricks
index 9435f083e..70dd143f3 100644
--- a/auto/pictures_tex/Fig_ADUGmRRB.pstricks
+++ b/auto/pictures_tex/Fig_ADUGmRRB.pstricks
@@ -72,9 +72,9 @@
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-\draw (0,0.2059510000) node {\( \alpha_i\)};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,0.2191818333) node {\( \alpha_{i+1}\)};
+\draw (0,0.20595) node {\( \alpha_i\)};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ADUGmRRC.pstricks b/auto/pictures_tex/Fig_ADUGmRRC.pstricks
index 9f961ca53..8ffd11439 100644
--- a/auto/pictures_tex/Fig_ADUGmRRC.pstricks
+++ b/auto/pictures_tex/Fig_ADUGmRRC.pstricks
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.2059510000) node {\( \alpha_i\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2191818333) node {\( \alpha_{i+1}\)};
+\draw (0,0.20595) node {\( \alpha_i\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_AIFsOQO.pstricks b/auto/pictures_tex/Fig_AIFsOQO.pstricks
index fc179c531..598522bfa 100644
--- a/auto/pictures_tex/Fig_AIFsOQO.pstricks
+++ b/auto/pictures_tex/Fig_AIFsOQO.pstricks
@@ -66,32 +66,32 @@
 %OTHER STUFF
 %PSTRICKS CODE
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+\draw (0,0.50000) node {};
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-\draw (0.5000000000,0.5000000000) node {};
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-\draw (1.000000000,0.5000000000) node {};
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-\draw (1.500000000,0.5000000000) node {};
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 \draw [] (1.75,0.750) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (1.25,0.250);
-\draw (2.000000000,0.5000000000) node {};
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 \draw [] (1.75,0.750) -- (1.75,0.250);
-\draw (2.500000000,0.5000000000) node {};
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@@ -101,12 +101,12 @@
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 \draw [] (0.750,0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (0.250,-0.250);
-\draw (1.000000000,0) node {};
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 \draw [] (0.750,-0.250) -- (1.25,-0.250);
 \draw [] (1.25,-0.250) -- (1.25,0.250);
 \draw [] (1.25,0.250) -- (0.750,0.250);
diff --git a/auto/pictures_tex/Fig_ALIzHFm.pstricks b/auto/pictures_tex/Fig_ALIzHFm.pstricks
index 5d71b3b11..e5623dafb 100644
--- a/auto/pictures_tex/Fig_ALIzHFm.pstricks
+++ b/auto/pictures_tex/Fig_ALIzHFm.pstricks
@@ -75,10 +75,10 @@
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-\draw []  (0.5000000000,0.8660254038) node [rotate=0] {$\bullet$};
-\draw (0.7640381667,1.145181485) node {\( z_1\)};
-\draw []  (-0.5000000000,-0.8660254038) node [rotate=0] {$\bullet$};
-\draw (-0.7640381667,-1.145181485) node {\( z_2\)};
+\draw []  (0.50000,0.86602) node [rotate=0] {$\bullet$};
+\draw (0.76404,1.1452) node {\( z_1\)};
+\draw []  (-0.50000,-0.86602) node [rotate=0] {$\bullet$};
+\draw (-0.76404,-1.1452) node {\( z_2\)};
 \draw [] (2.67,-0.384) -- (-1.67,2.12);
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 \draw [style=dotted] (2.17,-1.25) -- (-2.17,1.25);
diff --git a/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks b/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks
index ac5b69a94..ff57c58d7 100644
--- a/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks
@@ -91,8 +91,8 @@
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-\draw [,->,>=latex] (0,-1.500000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-1.0000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
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@@ -110,30 +110,30 @@
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-\draw (-0.2912498333,2.000000000) node {$ 2 $};
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-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_AdhIntFrSix.pstricks b/auto/pictures_tex/Fig_AdhIntFrSix.pstricks
index a85c4e695..855693c3c 100644
--- a/auto/pictures_tex/Fig_AdhIntFrSix.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFrSix.pstricks
@@ -67,8 +67,8 @@
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 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.100000000,0) -- (4.500000000,0);
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+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
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@@ -170,12 +170,12 @@
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-\draw []  (0,3.200000000) node [rotate=0] {$\bullet$};
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-\draw (4.000000000,-0.3149246667) node {$ 1 $};
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 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,4.000000000) node {$ 1 $};
+\draw (-0.29125,4.0000) node {$ 1 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks b/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks
index e6c167ab0..0ccc5fe2d 100644
--- a/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks
@@ -79,8 +79,8 @@
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+\draw [,->,>=latex] (-4.5000,0) -- (8.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
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@@ -129,22 +129,22 @@
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-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_AireParabole.pstricks b/auto/pictures_tex/Fig_AireParabole.pstricks
index 4edb09d9e..69294f452 100644
--- a/auto/pictures_tex/Fig_AireParabole.pstricks
+++ b/auto/pictures_tex/Fig_AireParabole.pstricks
@@ -91,33 +91,33 @@
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks b/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
index cc448317f..a64d6dad8 100644
--- a/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
+++ b/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
@@ -68,55 +68,55 @@
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
 
 
-\draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
-\draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
-\draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
-\draw [,->,>=latex] (1.600000000,0) -- (2.225000000,0);
-\draw [,->,>=latex] (1.600000000,0.3333333333) -- (2.225000000,0.3333333333);
-\draw [,->,>=latex] (1.600000000,0.6666666667) -- (2.225000000,0.6666666667);
-\draw [,->,>=latex] (1.600000000,1.000000000) -- (2.225000000,1.000000000);
-\draw [,->,>=latex] (2.333333333,-1.000000000) -- (2.761904762,-1.000000000);
-\draw [,->,>=latex] (2.333333333,-0.6666666667) -- (2.761904762,-0.6666666667);
-\draw [,->,>=latex] (2.333333333,-0.3333333333) -- (2.761904762,-0.3333333333);
-\draw [,->,>=latex] (2.333333333,0) -- (2.761904762,0);
-\draw [,->,>=latex] (2.333333333,0.3333333333) -- (2.761904762,0.3333333333);
-\draw [,->,>=latex] (2.333333333,0.6666666667) -- (2.761904762,0.6666666667);
-\draw [,->,>=latex] (2.333333333,1.000000000) -- (2.761904762,1.000000000);
-\draw [,->,>=latex] (3.066666667,-1.000000000) -- (3.392753623,-1.000000000);
-\draw [,->,>=latex] (3.066666667,-0.6666666667) -- (3.392753623,-0.6666666667);
-\draw [,->,>=latex] (3.066666667,-0.3333333333) -- (3.392753623,-0.3333333333);
-\draw [,->,>=latex] (3.066666667,0) -- (3.392753623,0);
-\draw [,->,>=latex] (3.066666667,0.3333333333) -- (3.392753623,0.3333333333);
-\draw [,->,>=latex] (3.066666667,0.6666666667) -- (3.392753623,0.6666666667);
-\draw [,->,>=latex] (3.066666667,1.000000000) -- (3.392753623,1.000000000);
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-\draw [,->,>=latex] (3.800000000,0.3333333333) -- (4.063157895,0.3333333333);
-\draw [,->,>=latex] (3.800000000,0.6666666667) -- (4.063157895,0.6666666667);
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-\draw [,->,>=latex] (4.533333333,0) -- (4.753921569,0);
-\draw [,->,>=latex] (4.533333333,0.3333333333) -- (4.753921569,0.3333333333);
-\draw [,->,>=latex] (4.533333333,0.6666666667) -- (4.753921569,0.6666666667);
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-\draw [,->,>=latex] (5.266666667,0.6666666667) -- (5.456540084,0.6666666667);
-\draw [,->,>=latex] (5.266666667,1.000000000) -- (5.456540084,1.000000000);
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-\draw [,->,>=latex] (6.000000000,-0.6666666667) -- (6.166666667,-0.6666666667);
-\draw [,->,>=latex] (6.000000000,-0.3333333333) -- (6.166666667,-0.3333333333);
-\draw [,->,>=latex] (6.000000000,0) -- (6.166666667,0);
-\draw [,->,>=latex] (6.000000000,0.3333333333) -- (6.166666667,0.3333333333);
-\draw [,->,>=latex] (6.000000000,0.6666666667) -- (6.166666667,0.6666666667);
-\draw [,->,>=latex] (6.000000000,1.000000000) -- (6.166666667,1.000000000);
+\draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
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+\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
+\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
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+\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
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+\draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
+\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
+\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
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+\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
+\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
+\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
+\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
+\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
+\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
+\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
+\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
+\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
+\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
+\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
+\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
+\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
+\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
+\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
+\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
+\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
+\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
+\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
+\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
+\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
+\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
+\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks b/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks
index b0e1c2e24..a72e93213 100644
--- a/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks
+++ b/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.000000000,0);
-\draw (3.308599701,-0.2907082010) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.000000000,2.000000000);
-\draw (2.000000000,2.426736000) node {$b$};
-\draw []  (5.000000000,2.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (0,0) -- (3.0000,0);
+\draw (3.3086,-0.29071) node {$a$};
+\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw (2.0000,2.4267) node {$b$};
+\draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
 \draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
 \draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
 \draw [style=dashed] (2.00,2.00) -- (2.00,0);
-\draw (2.305148833,1.000000000) node {$h$};
-\draw (0.8091529067,0.3191815970) node {$\theta$};
+\draw (2.3051,1.0000) node {$h$};
+\draw (0.80915,0.31918) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks b/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
index 87e4831e7..3f5418461 100644
--- a/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
+++ b/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.249988663);
+\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
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+\draw [] (5.00,2.64) -- (5.00,0);
 
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+\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3485,0.6490)--(-0.2727,0.7215)--(-0.1970,0.7928)--(-0.1212,0.8628)--(-0.04545,0.9316)--(0.03030,0.9991)--(0.1061,1.065)--(0.1818,1.130)--(0.2576,1.194)--(0.3333,1.256)--(0.4091,1.317)--(0.4848,1.377)--(0.5606,1.436)--(0.6364,1.493)--(0.7121,1.549)--(0.7879,1.604)--(0.8636,1.657)--(0.9394,1.709)--(1.015,1.760)--(1.091,1.810)--(1.167,1.858)--(1.242,1.905)--(1.318,1.951)--(1.394,1.995)--(1.470,2.039)--(1.545,2.081)--(1.621,2.121)--(1.697,2.161)--(1.773,2.199)--(1.848,2.236)--(1.924,2.271)--(2.000,2.306)--(2.076,2.339)--(2.152,2.370)--(2.227,2.401)--(2.303,2.430)--(2.379,2.458)--(2.455,2.485)--(2.530,2.510)--(2.606,2.534)--(2.682,2.557)--(2.758,2.578)--(2.833,2.599)--(2.909,2.618)--(2.985,2.635)--(3.061,2.652)--(3.136,2.667)--(3.212,2.681)--(3.288,2.694)--(3.364,2.705)--(3.439,2.715)--(3.515,2.724)--(3.591,2.731)--(3.667,2.738)--(3.742,2.743)--(3.818,2.746)--(3.894,2.749)--(3.970,2.750)--(4.045,2.750)--(4.121,2.748)--(4.197,2.746)--(4.273,2.742)--(4.349,2.737)--(4.424,2.730)--(4.500,2.722)--(4.576,2.713)--(4.651,2.703)--(4.727,2.691)--(4.803,2.678)--(4.879,2.664)--(4.955,2.649)--(5.030,2.632)--(5.106,2.614)--(5.182,2.595)--(5.258,2.574)--(5.333,2.552)--(5.409,2.529)--(5.485,2.505)--(5.561,2.479)--(5.636,2.452)--(5.712,2.424)--(5.788,2.395)--(5.864,2.364)--(5.939,2.332)--(6.015,2.299)--(6.091,2.264)--(6.167,2.228)--(6.242,2.191)--(6.318,2.153)--(6.394,2.113)--(6.470,2.072)--(6.545,2.030)--(6.621,1.987)--(6.697,1.942)--(6.773,1.896)--(6.849,1.848)--(6.924,1.800)--(7.000,1.750);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,5.64) -- (6.00,5.64) -- (5.00,5.64) -- (5.00,5.64) -- (5.00,0) --  cycle;
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
 \draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,5.64);
-\draw [color=red,style=dashed] (6.00,5.64) -- (5.00,5.64);
-\draw [color=red,style=dashed] (5.00,5.64) -- (5.00,0);
-\draw []  (5.000000000,5.638888889) node [rotate=0] {$\bullet$};
-\draw (5.441978850,6.211818713) node {$f(x)$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.2785761667) node {$x$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3455708333) node {$x+\Delta x$};
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
+\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
+\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
+\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
+\draw (5.4420,3.2118) node {$f(x)$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.27858) node {$x$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.34557) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Bateau.pstricks b/auto/pictures_tex/Fig_Bateau.pstricks
index d6ad0aca5..117a93c45 100644
--- a/auto/pictures_tex/Fig_Bateau.pstricks
+++ b/auto/pictures_tex/Fig_Bateau.pstricks
@@ -98,28 +98,28 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (-1.00,0) -- (5.00,0);
-\draw []  (0,2.500000000) node [rotate=0] {$\bullet$};
-\draw (0,2.924708000) node {$A$};
-\draw [,->,>=latex] (2.000000000,2.500000000) -- (0,2.500000000);
-\draw [,->,>=latex] (2.000000000,2.500000000) -- (4.000000000,2.500000000);
-\draw (2.000000000,2.725719500) node {$\unit{4}{\kilo\meter}$};
-\draw [,->,>=latex] (4.200000000,2.250000000) -- (4.200000000,4.500000000);
-\draw [,->,>=latex] (4.200000000,2.250000000) -- (4.200000000,0);
-\draw (4.671424833,2.250000000) node {$\unit{9}{\kilo\meter}$};
-\draw [,->,>=latex] (0,1.250000000) -- (0,2.500000000);
-\draw [,->,>=latex] (0,1.250000000) -- (0,0);
-\draw (-0.4714248333,1.250000000) node {$\unit{3}{\kilo\meter}$};
-\draw []  (1.428571429,0) node [rotate=0] {$\bullet$};
-\draw (1.735248463,0.3368400344) node {$I$};
+\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
+\draw (0,2.9247) node {$A$};
+\draw [,->,>=latex] (2.0000,2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (2.0000,2.5000) -- (4.0000,2.5000);
+\draw (2.0000,2.7257) node {$\unit{4}{\kilo\meter}$};
+\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,4.5000);
+\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,0);
+\draw (4.6714,2.2500) node {$\unit{9}{\kilo\meter}$};
+\draw [,->,>=latex] (0,1.2500) -- (0,2.5000);
+\draw [,->,>=latex] (0,1.2500) -- (0,0);
+\draw (-0.47143,1.2500) node {$\unit{3}{\kilo\meter}$};
+\draw []  (1.4286,0) node [rotate=0] {$\bullet$};
+\draw (1.7352,0.33684) node {$I$};
 \draw [color=brown,style=dashed] (4.00,4.50) -- (4.00,-4.50);
 \draw [color=blue,style=dashed] (0,2.50) -- (4.00,-4.50);
-\draw []  (4.000000000,4.500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,4.924708000) node {$B$};
-\draw []  (4.000000000,-4.500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-4.940774333) node {$B'$};
-\draw [,->,>=latex] (0.7142857143,-0.2000000000) -- (0,-0.2000000000);
-\draw [,->,>=latex] (0.7142857143,-0.2000000000) -- (1.428571429,-0.2000000000);
-\draw (0.7142857143,-0.4257195000) node {$x\kilo\meter$};
+\draw []  (4.0000,4.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,4.9247) node {$B$};
+\draw []  (4.0000,-4.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-4.9408) node {$B'$};
+\draw [,->,>=latex] (0.71429,-0.20000) -- (0,-0.20000);
+\draw [,->,>=latex] (0.71429,-0.20000) -- (1.4286,-0.20000);
+\draw (0.71429,-0.42572) node {$x\kilo\meter$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CELooGVvzMc.pstricks b/auto/pictures_tex/Fig_CELooGVvzMc.pstricks
index a9d6c9369..3ceb434c7 100644
--- a/auto/pictures_tex/Fig_CELooGVvzMc.pstricks
+++ b/auto/pictures_tex/Fig_CELooGVvzMc.pstricks
@@ -80,33 +80,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.790000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.790000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.7900,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.7900);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.02323,0)--(0.04646,0.002159)--(0.06970,0.004858)--(0.09293,0.008636)--(0.1162,0.01349)--(0.1394,0.01943)--(0.1626,0.02645)--(0.1859,0.03454)--(0.2091,0.04372)--(0.2323,0.05397)--(0.2556,0.06531)--(0.2788,0.07772)--(0.3020,0.09122)--(0.3253,0.1058)--(0.3485,0.1214)--(0.3717,0.1382)--(0.3950,0.1560)--(0.4182,0.1749)--(0.4414,0.1948)--(0.4646,0.2159)--(0.4879,0.2380)--(0.5111,0.2612)--(0.5343,0.2855)--(0.5576,0.3109)--(0.5808,0.3373)--(0.6040,0.3649)--(0.6273,0.3935)--(0.6505,0.4232)--(0.6737,0.4539)--(0.6970,0.4858)--(0.7202,0.5187)--(0.7434,0.5527)--(0.7667,0.5878)--(0.7899,0.6239)--(0.8131,0.6612)--(0.8364,0.6995)--(0.8596,0.7389)--(0.8828,0.7794)--(0.9061,0.8209)--(0.9293,0.8636)--(0.9525,0.9073)--(0.9758,0.9521)--(0.9990,0.9980)--(1.022,1.045)--(1.045,1.093)--(1.069,1.142)--(1.092,1.192)--(1.115,1.244)--(1.138,1.296)--(1.162,1.349)--(1.185,1.404)--(1.208,1.459)--(1.231,1.516)--(1.255,1.574)--(1.278,1.633)--(1.301,1.693)--(1.324,1.754)--(1.347,1.816)--(1.371,1.879)--(1.394,1.943)--(1.417,2.008)--(1.440,2.075)--(1.464,2.142)--(1.487,2.211)--(1.510,2.280)--(1.533,2.351)--(1.557,2.423)--(1.580,2.496)--(1.603,2.570)--(1.626,2.645)--(1.649,2.721)--(1.673,2.798)--(1.696,2.876)--(1.719,2.956)--(1.742,3.036)--(1.766,3.118)--(1.789,3.200)--(1.812,3.284)--(1.835,3.369)--(1.859,3.454)--(1.882,3.541)--(1.905,3.629)--(1.928,3.718)--(1.952,3.808)--(1.975,3.900)--(1.998,3.992)--(2.021,4.085)--(2.044,4.180)--(2.068,4.275)--(2.091,4.372)--(2.114,4.470)--(2.137,4.568)--(2.161,4.668)--(2.184,4.769)--(2.207,4.871)--(2.230,4.974)--(2.254,5.078)--(2.277,5.184)--(2.300,5.290);
 
 \draw [color=blue] (0,0)--(0.05343,0.2312)--(0.1069,0.3269)--(0.1603,0.4004)--(0.2137,0.4623)--(0.2672,0.5169)--(0.3206,0.5662)--(0.3740,0.6116)--(0.4275,0.6538)--(0.4809,0.6935)--(0.5343,0.7310)--(0.5878,0.7667)--(0.6412,0.8008)--(0.6946,0.8335)--(0.7481,0.8649)--(0.8015,0.8953)--(0.8549,0.9246)--(0.9084,0.9531)--(0.9618,0.9807)--(1.015,1.008)--(1.069,1.034)--(1.122,1.059)--(1.176,1.084)--(1.229,1.109)--(1.282,1.132)--(1.336,1.156)--(1.389,1.179)--(1.443,1.201)--(1.496,1.223)--(1.550,1.245)--(1.603,1.266)--(1.656,1.287)--(1.710,1.308)--(1.763,1.328)--(1.817,1.348)--(1.870,1.368)--(1.924,1.387)--(1.977,1.406)--(2.031,1.425)--(2.084,1.444)--(2.137,1.462)--(2.191,1.480)--(2.244,1.498)--(2.298,1.516)--(2.351,1.533)--(2.405,1.551)--(2.458,1.568)--(2.511,1.585)--(2.565,1.602)--(2.618,1.618)--(2.672,1.635)--(2.725,1.651)--(2.779,1.667)--(2.832,1.683)--(2.885,1.699)--(2.939,1.714)--(2.992,1.730)--(3.046,1.745)--(3.099,1.760)--(3.153,1.776)--(3.206,1.791)--(3.259,1.805)--(3.313,1.820)--(3.366,1.835)--(3.420,1.849)--(3.473,1.864)--(3.527,1.878)--(3.580,1.892)--(3.634,1.906)--(3.687,1.920)--(3.740,1.934)--(3.794,1.948)--(3.847,1.961)--(3.901,1.975)--(3.954,1.988)--(4.008,2.002)--(4.061,2.015)--(4.114,2.028)--(4.168,2.042)--(4.221,2.055)--(4.275,2.068)--(4.328,2.080)--(4.382,2.093)--(4.435,2.106)--(4.488,2.119)--(4.542,2.131)--(4.595,2.144)--(4.649,2.156)--(4.702,2.168)--(4.756,2.181)--(4.809,2.193)--(4.863,2.205)--(4.916,2.217)--(4.969,2.229)--(5.023,2.241)--(5.076,2.253)--(5.130,2.265)--(5.183,2.277)--(5.237,2.288)--(5.290,2.300);
 \draw [style=dashed] (0,0) -- (3.79,3.79);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CMMAooQegASg.pstricks b/auto/pictures_tex/Fig_CMMAooQegASg.pstricks
index 014c6bf78..6b8a9b57d 100644
--- a/auto/pictures_tex/Fig_CMMAooQegASg.pstricks
+++ b/auto/pictures_tex/Fig_CMMAooQegASg.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.499897967) -- (0,2.499897967);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks b/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks
index 574d6a7fe..90633a3ea 100644
--- a/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks
+++ b/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks
@@ -95,8 +95,8 @@
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+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
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@@ -108,33 +108,33 @@
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-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (0.78677,0.19542) node {$P$};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-0.78501,0.23090) node {$Q$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
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-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
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 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
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-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
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 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CSCvi.pstricks b/auto/pictures_tex/Fig_CSCvi.pstricks
index 10e26577f..8a0a3f7b3 100644
--- a/auto/pictures_tex/Fig_CSCvi.pstricks
+++ b/auto/pictures_tex/Fig_CSCvi.pstricks
@@ -61,29 +61,29 @@
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 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (2.070796327,0);
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+\draw [,->,>=latex] (-2.0708,0) -- (2.0708,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4332);
 %DEFAULT
 
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-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
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 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
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-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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-\draw (-0.2912498333,2.000000000) node {$ 2 $};
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-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
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 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
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@@ -142,23 +142,23 @@
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-\draw [,->,>=latex] (0,-5.043736533) -- (0,0.8002358951);
+\draw [,->,>=latex] (-0.50000,0) -- (2.4211,0);
+\draw [,->,>=latex] (0,-5.0437) -- (0,0.80024);
 %DEFAULT
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-\draw (2.000000000,-0.3149246667) node {$ 2 $};
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-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
 \draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
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 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
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 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
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diff --git a/auto/pictures_tex/Fig_CURGooXvruWV.pstricks b/auto/pictures_tex/Fig_CURGooXvruWV.pstricks
index 21cc5cf13..8d44d1bc7 100644
--- a/auto/pictures_tex/Fig_CURGooXvruWV.pstricks
+++ b/auto/pictures_tex/Fig_CURGooXvruWV.pstricks
@@ -71,8 +71,8 @@
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-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -90,13 +90,13 @@
 \draw [color=green] (0,2.000)--(0.02020,2.000)--(0.04040,2.000)--(0.06061,2.000)--(0.08081,2.000)--(0.1010,2.000)--(0.1212,2.000)--(0.1414,2.000)--(0.1616,2.000)--(0.1818,2.000)--(0.2020,2.000)--(0.2222,2.000)--(0.2424,2.000)--(0.2626,2.000)--(0.2828,2.000)--(0.3030,2.000)--(0.3232,2.000)--(0.3434,2.000)--(0.3636,2.000)--(0.3838,2.000)--(0.4040,2.000)--(0.4242,2.000)--(0.4444,2.000)--(0.4646,2.000)--(0.4848,2.000)--(0.5051,2.000)--(0.5253,2.000)--(0.5455,2.000)--(0.5657,2.000)--(0.5859,2.000)--(0.6061,2.000)--(0.6263,2.000)--(0.6465,2.000)--(0.6667,2.000)--(0.6869,2.000)--(0.7071,2.000)--(0.7273,2.000)--(0.7475,2.000)--(0.7677,2.000)--(0.7879,2.000)--(0.8081,2.000)--(0.8283,2.000)--(0.8485,2.000)--(0.8687,2.000)--(0.8889,2.000)--(0.9091,2.000)--(0.9293,2.000)--(0.9495,2.000)--(0.9697,2.000)--(0.9899,2.000)--(1.010,2.000)--(1.030,2.000)--(1.051,2.000)--(1.071,2.000)--(1.091,2.000)--(1.111,2.000)--(1.131,2.000)--(1.152,2.000)--(1.172,2.000)--(1.192,2.000)--(1.212,2.000)--(1.232,2.000)--(1.253,2.000)--(1.273,2.000)--(1.293,2.000)--(1.313,2.000)--(1.333,2.000)--(1.354,2.000)--(1.374,2.000)--(1.394,2.000)--(1.414,2.000)--(1.434,2.000)--(1.455,2.000)--(1.475,2.000)--(1.495,2.000)--(1.515,2.000)--(1.535,2.000)--(1.556,2.000)--(1.576,2.000)--(1.596,2.000)--(1.616,2.000)--(1.636,2.000)--(1.657,2.000)--(1.677,2.000)--(1.697,2.000)--(1.717,2.000)--(1.737,2.000)--(1.758,2.000)--(1.778,2.000)--(1.798,2.000)--(1.818,2.000)--(1.838,2.000)--(1.859,2.000)--(1.879,2.000)--(1.899,2.000)--(1.919,2.000)--(1.939,2.000)--(1.960,2.000)--(1.980,2.000)--(2.000,2.000);
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
index 4c1ce07d4..b289ededc 100644
--- a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
+++ b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
@@ -68,14 +68,14 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (1.000000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,0.5000000000);
+\draw [,->,>=latex] (-3.5000,0) -- (1.0000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,0.50000);
 %DEFAULT
-\draw [,->,>=latex] (-3.000000000,-1.000000000) -- (-3.000000000,0);
+\draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0);
 \draw [] (0,0) -- (-3.00,-1.00);
-\draw []  (-3.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (-3.828301160,-1.204882387) node {\( -x+\lambda i\)};
-\draw (0.5312334950,-0.7091140396) node {\( \arg(z)\)};
+\draw []  (-3.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (-3.8283,-1.2049) node {\( -x+\lambda i\)};
+\draw (0.53123,-0.70911) node {\( \arg(z)\)};
 
 \draw [] (-0.474,-0.158)--(-0.470,-0.172)--(-0.465,-0.185)--(-0.459,-0.198)--(-0.453,-0.211)--(-0.447,-0.224)--(-0.441,-0.236)--(-0.434,-0.249)--(-0.426,-0.261)--(-0.419,-0.273)--(-0.411,-0.285)--(-0.403,-0.297)--(-0.394,-0.308)--(-0.385,-0.319)--(-0.376,-0.330)--(-0.366,-0.340)--(-0.356,-0.351)--(-0.346,-0.361)--(-0.336,-0.370)--(-0.325,-0.380)--(-0.314,-0.389)--(-0.303,-0.398)--(-0.292,-0.406)--(-0.280,-0.414)--(-0.268,-0.422)--(-0.256,-0.430)--(-0.243,-0.437)--(-0.231,-0.444)--(-0.218,-0.450)--(-0.205,-0.456)--(-0.192,-0.462)--(-0.179,-0.467)--(-0.166,-0.472)--(-0.152,-0.476)--(-0.138,-0.480)--(-0.125,-0.484)--(-0.111,-0.488)--(-0.0970,-0.491)--(-0.0830,-0.493)--(-0.0689,-0.495)--(-0.0547,-0.497)--(-0.0406,-0.498)--(-0.0264,-0.499)--(-0.0121,-0.500)--(0.00211,-0.500)--(0.0163,-0.500)--(0.0306,-0.499)--(0.0448,-0.498)--(0.0589,-0.497)--(0.0731,-0.495)--(0.0871,-0.492)--(0.101,-0.490)--(0.115,-0.487)--(0.129,-0.483)--(0.143,-0.479)--(0.156,-0.475)--(0.170,-0.470)--(0.183,-0.465)--(0.196,-0.460)--(0.209,-0.454)--(0.222,-0.448)--(0.235,-0.442)--(0.247,-0.435)--(0.259,-0.427)--(0.271,-0.420)--(0.283,-0.412)--(0.295,-0.404)--(0.306,-0.395)--(0.317,-0.386)--(0.328,-0.377)--(0.339,-0.368)--(0.349,-0.358)--(0.359,-0.348)--(0.369,-0.337)--(0.379,-0.327)--(0.388,-0.316)--(0.396,-0.305)--(0.405,-0.293)--(0.413,-0.282)--(0.421,-0.270)--(0.429,-0.258)--(0.436,-0.245)--(0.443,-0.233)--(0.449,-0.220)--(0.455,-0.207)--(0.461,-0.194)--(0.466,-0.181)--(0.471,-0.168)--(0.476,-0.154)--(0.480,-0.141)--(0.484,-0.127)--(0.487,-0.113)--(0.490,-0.0990)--(0.493,-0.0850)--(0.495,-0.0710)--(0.497,-0.0568)--(0.498,-0.0427)--(0.499,-0.0285)--(0.500,-0.0142)--(0.500,0);
 
diff --git a/auto/pictures_tex/Fig_CbCartTui.pstricks b/auto/pictures_tex/Fig_CbCartTui.pstricks
index 398a015fa..ad4c00516 100644
--- a/auto/pictures_tex/Fig_CbCartTui.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTui.pstricks
@@ -103,60 +103,60 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.140000002,0) -- (4.140000002,0);
-\draw [,->,>=latex] (0,-3.972000001) -- (0,4.028000000);
+\draw [,->,>=latex] (-4.1400,0) -- (4.1400,0);
+\draw [,->,>=latex] (0,-3.9720) -- (0,4.0280);
 %DEFAULT
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-\draw [,->,>=latex] (-1.750000000,-1.225000000) -- (-1.746398530,-1.218997550);
-\draw [,->,>=latex] (-1.490000000,0.9385714286) -- (-1.491054412,0.9454915598);
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+\draw [,->,>=latex] (-2.3333,-2.0222) -- (-2.3289,-2.0168);
+\draw [,->,>=latex] (-1.7500,-1.2250) -- (-1.7464,-1.2190);
+\draw [,->,>=latex] (-1.4900,0.93857) -- (-1.4911,0.94549);
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-\draw [,->,>=latex] (1.586666667,2.364444444) -- (1.585193677,2.357601178);
-\draw [,->,>=latex] (1.435000000,1.653750000) -- (1.433669847,1.646877541);
+\draw [,->,>=latex] (1.5867,2.3644) -- (1.5852,2.3576);
+\draw [,->,>=latex] (1.4350,1.6537) -- (1.4337,1.6469);
 \draw [color=blue] (1.400,1.400)--(1.401,1.375)--(1.404,1.356)--(1.409,1.342)--(1.416,1.332)--(1.424,1.326)--(1.433,1.323)--(1.444,1.323)--(1.455,1.326)--(1.468,1.331)--(1.481,1.338)--(1.496,1.347)--(1.511,1.357)--(1.527,1.369)--(1.543,1.382)--(1.560,1.396)--(1.578,1.411)--(1.596,1.427)--(1.614,1.444)--(1.633,1.461)--(1.653,1.480)--(1.673,1.499)--(1.693,1.518)--(1.713,1.539)--(1.734,1.559)--(1.755,1.580)--(1.777,1.602)--(1.798,1.624)--(1.820,1.646)--(1.843,1.669)--(1.865,1.692)--(1.888,1.715)--(1.910,1.738)--(1.933,1.762)--(1.957,1.786)--(1.980,1.810)--(2.003,1.834)--(2.027,1.859)--(2.051,1.884)--(2.075,1.909)--(2.099,1.934)--(2.123,1.959)--(2.147,1.984)--(2.172,2.010)--(2.196,2.035)--(2.221,2.061)--(2.246,2.087)--(2.271,2.113)--(2.296,2.139)--(2.321,2.165)--(2.346,2.191)--(2.371,2.217)--(2.396,2.243)--(2.422,2.270)--(2.447,2.296)--(2.473,2.323)--(2.498,2.350)--(2.524,2.376)--(2.550,2.403)--(2.576,2.430)--(2.601,2.457)--(2.627,2.484)--(2.653,2.511)--(2.679,2.538)--(2.705,2.565)--(2.731,2.592)--(2.758,2.619)--(2.784,2.646)--(2.810,2.673)--(2.836,2.700)--(2.863,2.728)--(2.889,2.755)--(2.915,2.782)--(2.942,2.810)--(2.968,2.837)--(2.995,2.864)--(3.021,2.892)--(3.048,2.919)--(3.075,2.947)--(3.101,2.974)--(3.128,3.002)--(3.155,3.029)--(3.181,3.057)--(3.208,3.084)--(3.235,3.112)--(3.262,3.140)--(3.289,3.167)--(3.316,3.195)--(3.343,3.223)--(3.369,3.250)--(3.396,3.278)--(3.423,3.306)--(3.450,3.333)--(3.477,3.361)--(3.504,3.389)--(3.532,3.417)--(3.559,3.445)--(3.586,3.472)--(3.613,3.500)--(3.640,3.528);
-\draw [,->,>=latex] (1.423333333,1.326111111) -- (1.429556155,1.322905415);
-\draw [,->,>=latex] (1.750000000,1.575000000) -- (1.754949747,1.579949747);
-\draw [,->,>=latex] (2.333333333,2.177777778) -- (2.338181056,2.182827489);
-\draw (-3.500000000,-0.3298256667) node {$ -5 $};
+\draw [,->,>=latex] (1.4233,1.3261) -- (1.4296,1.3229);
+\draw [,->,>=latex] (1.7500,1.5750) -- (1.7549,1.5800);
+\draw [,->,>=latex] (2.3333,2.1778) -- (2.3382,2.1828);
+\draw (-3.5000,-0.32983) node {$ -5 $};
 \draw [] (-3.50,-0.100) -- (-3.50,0.100);
-\draw (-2.800000000,-0.3298256667) node {$ -4 $};
+\draw (-2.8000,-0.32983) node {$ -4 $};
 \draw [] (-2.80,-0.100) -- (-2.80,0.100);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CbCartTuiii.pstricks b/auto/pictures_tex/Fig_CbCartTuiii.pstricks
index 95ddf09e7..8b557fe13 100644
--- a/auto/pictures_tex/Fig_CbCartTuiii.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTuiii.pstricks
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.499748256,0) -- (2.499748256,0);
-\draw [,->,>=latex] (0,-2.497734679) -- (0,2.497734679);
+\draw [,->,>=latex] (-2.4997,0) -- (2.4997,0);
+\draw [,->,>=latex] (0,-2.4977) -- (0,2.4977);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.253,0.379)--(0.502,0.743)--(0.743,1.08)--(0.972,1.38)--(1.19,1.63)--(1.38,1.82)--(1.55,1.94)--(1.70,2.00)--(1.82,1.98)--(1.91,1.89)--(1.97,1.73)--(2.00,1.51)--(1.99,1.24)--(1.96,0.916)--(1.89,0.563)--(1.79,0.190)--(1.67,-0.190)--(1.51,-0.563)--(1.33,-0.916)--(1.13,-1.24)--(0.916,-1.51)--(0.684,-1.73)--(0.441,-1.89)--(0.190,-1.98)--(-0.0635,-2.00)--(-0.316,-1.94)--(-0.563,-1.82)--(-0.802,-1.63)--(-1.03,-1.38)--(-1.24,-1.08)--(-1.43,-0.743)--(-1.59,-0.379)--(-1.73,0)--(-1.84,0.379)--(-1.93,0.743)--(-1.98,1.08)--(-2.00,1.38)--(-1.99,1.63)--(-1.94,1.82)--(-1.87,1.94)--(-1.76,2.00)--(-1.63,1.98)--(-1.47,1.89)--(-1.29,1.73)--(-1.08,1.51)--(-0.860,1.24)--(-0.624,0.916)--(-0.379,0.563)--(-0.127,0.190)--(0.127,-0.190)--(0.379,-0.563)--(0.624,-0.916)--(0.860,-1.24)--(1.08,-1.51)--(1.29,-1.73)--(1.47,-1.89)--(1.63,-1.98)--(1.76,-2.00)--(1.87,-1.94)--(1.94,-1.82)--(1.99,-1.63)--(2.00,-1.38)--(1.98,-1.08)--(1.93,-0.743)--(1.84,-0.379)--(1.73,0)--(1.59,0.379)--(1.43,0.743)--(1.24,1.08)--(1.03,1.38)--(0.802,1.63)--(0.563,1.82)--(0.316,1.94)--(0.0635,2.00)--(-0.190,1.98)--(-0.441,1.89)--(-0.684,1.73)--(-0.916,1.51)--(-1.13,1.24)--(-1.33,0.916)--(-1.51,0.563)--(-1.67,0.190)--(-1.79,-0.190)--(-1.89,-0.563)--(-1.96,-0.916)--(-1.99,-1.24)--(-2.00,-1.51)--(-1.97,-1.73)--(-1.91,-1.89)--(-1.82,-1.98)--(-1.70,-2.00)--(-1.55,-1.94)--(-1.38,-1.82)--(-1.19,-1.63)--(-0.972,-1.38)--(-0.743,-1.08)--(-0.502,-0.743)--(-0.253,-0.379)--(0,0);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CercleImplicite.pstricks b/auto/pictures_tex/Fig_CercleImplicite.pstricks
index de144d561..65090a7cf 100644
--- a/auto/pictures_tex/Fig_CercleImplicite.pstricks
+++ b/auto/pictures_tex/Fig_CercleImplicite.pstricks
@@ -79,19 +79,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
-\draw []  (1.414213562,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.768860430,1.751053597) node {$P$};
-\draw []  (1.414213562,-1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.815841763,-1.767119930) node {\( P'\)};
-\draw []  (-2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-2.356408201,0.3723262010) node {\( Q\)};
-\draw []  (1.414213562,0) node [rotate=0] {$\bullet$};
-\draw (1.097777861,-0.2907082010) node {\( x\)};
+\draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.7689,1.7511) node {$P$};
+\draw []  (1.4142,-1.4142) node [rotate=0] {$\bullet$};
+\draw (1.8158,-1.7671) node {\( P'\)};
+\draw []  (-2.0000,0) node [rotate=0] {$\bullet$};
+\draw (-2.3564,0.37233) node {\( Q\)};
+\draw []  (1.4142,0) node [rotate=0] {$\bullet$};
+\draw (1.0978,-0.29071) node {\( x\)};
 \draw [color=red,style=dotted] (1.41,1.41) -- (1.41,-1.41);
 
 %OTHER STUFF
diff --git a/auto/pictures_tex/Fig_ChampGraviation.pstricks b/auto/pictures_tex/Fig_ChampGraviation.pstricks
index f49509765..518a4dcc8 100644
--- a/auto/pictures_tex/Fig_ChampGraviation.pstricks
+++ b/auto/pictures_tex/Fig_ChampGraviation.pstricks
@@ -65,226 +65,226 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.000000000,-4.000000000) -- (-4.044194174,-4.044194174);
-\draw [,->,>=latex] (-4.000000000,-3.428571429) -- (-4.054711138,-3.475466689);
-\draw [,->,>=latex] (-4.000000000,-2.857142857) -- (-4.067352939,-2.905252100);
-\draw [,->,>=latex] (-4.000000000,-2.285714286) -- (-4.081815219,-2.332465840);
-\draw [,->,>=latex] (-4.000000000,-1.714285714) -- (-4.097064885,-1.755884951);
-\draw [,->,>=latex] (-4.000000000,-1.142857143) -- (-4.111119513,-1.174605575);
-\draw [,->,>=latex] (-4.000000000,-0.5714285714) -- (-4.121268813,-0.5887526876);
-\draw [,->,>=latex] (-4.000000000,0) -- (-4.125000000,0);
-\draw [,->,>=latex] (-4.000000000,0.5714285714) -- (-4.121268813,0.5887526876);
-\draw [,->,>=latex] (-4.000000000,1.142857143) -- (-4.111119513,1.174605575);
-\draw [,->,>=latex] (-4.000000000,1.714285714) -- (-4.097064885,1.755884951);
-\draw [,->,>=latex] (-4.000000000,2.285714286) -- (-4.081815219,2.332465840);
-\draw [,->,>=latex] (-4.000000000,2.857142857) -- (-4.067352939,2.905252100);
-\draw [,->,>=latex] (-4.000000000,3.428571429) -- (-4.054711138,3.475466689);
-\draw [,->,>=latex] (-4.000000000,4.000000000) -- (-4.044194174,4.044194174);
-\draw [,->,>=latex] (-3.428571429,-4.000000000) -- (-3.475466689,-4.054711138);
-\draw [,->,>=latex] (-3.428571429,-3.428571429) -- (-3.488724610,-3.488724610);
-\draw [,->,>=latex] (-3.428571429,-2.857142857) -- (-3.505708401,-2.921423668);
-\draw [,->,>=latex] (-3.428571429,-2.285714286) -- (-3.526577353,-2.351051568);
-\draw [,->,>=latex] (-3.428571429,-1.714285714) -- (-3.550312907,-1.775156454);
-\draw [,->,>=latex] (-3.428571429,-1.142857143) -- (-3.573838559,-1.191279520);
-\draw [,->,>=latex] (-3.428571429,-0.5714285714) -- (-3.591859612,-0.5986432687);
-\draw [,->,>=latex] (-3.428571429,0) -- (-3.598710317,0);
-\draw [,->,>=latex] (-3.428571429,0.5714285714) -- (-3.591859612,0.5986432687);
-\draw [,->,>=latex] (-3.428571429,1.142857143) -- (-3.573838559,1.191279520);
-\draw [,->,>=latex] (-3.428571429,1.714285714) -- (-3.550312907,1.775156454);
-\draw [,->,>=latex] (-3.428571429,2.285714286) -- (-3.526577353,2.351051568);
-\draw [,->,>=latex] (-3.428571429,2.857142857) -- (-3.505708401,2.921423668);
-\draw [,->,>=latex] (-3.428571429,3.428571429) -- (-3.488724610,3.488724610);
-\draw [,->,>=latex] (-3.428571429,4.000000000) -- (-3.475466689,4.054711138);
-\draw [,->,>=latex] (-2.857142857,-4.000000000) -- (-2.905252100,-4.067352939);
-\draw [,->,>=latex] (-2.857142857,-3.428571429) -- (-2.921423668,-3.505708401);
-\draw [,->,>=latex] (-2.857142857,-2.857142857) -- (-2.943763438,-2.943763438);
-\draw [,->,>=latex] (-2.857142857,-2.285714286) -- (-2.973797039,-2.379037631);
-\draw [,->,>=latex] (-2.857142857,-1.714285714) -- (-3.011617686,-1.806970611);
-\draw [,->,>=latex] (-2.857142857,-1.142857143) -- (-3.053243538,-1.221297415);
-\draw [,->,>=latex] (-2.857142857,-0.5714285714) -- (-3.088145036,-0.6176290071);
-\draw [,->,>=latex] (-2.857142857,0) -- (-3.102142857,0);
-\draw [,->,>=latex] (-2.857142857,0.5714285714) -- (-3.088145036,0.6176290071);
-\draw [,->,>=latex] (-2.857142857,1.142857143) -- (-3.053243538,1.221297415);
-\draw [,->,>=latex] (-2.857142857,1.714285714) -- (-3.011617686,1.806970611);
-\draw [,->,>=latex] (-2.857142857,2.285714286) -- (-2.973797039,2.379037631);
-\draw [,->,>=latex] (-2.857142857,2.857142857) -- (-2.943763438,2.943763438);
-\draw [,->,>=latex] (-2.857142857,3.428571429) -- (-2.921423668,3.505708401);
-\draw [,->,>=latex] (-2.857142857,4.000000000) -- (-2.905252100,4.067352939);
-\draw [,->,>=latex] (-2.285714286,-4.000000000) -- (-2.332465840,-4.081815219);
-\draw [,->,>=latex] (-2.285714286,-3.428571429) -- (-2.351051568,-3.526577353);
-\draw [,->,>=latex] (-2.285714286,-2.857142857) -- (-2.379037631,-2.973797039);
-\draw [,->,>=latex] (-2.285714286,-2.285714286) -- (-2.421058943,-2.421058943);
-\draw [,->,>=latex] (-2.285714286,-1.714285714) -- (-2.481714286,-1.861285714);
-\draw [,->,>=latex] (-2.285714286,-1.142857143) -- (-2.559632613,-1.279816306);
-\draw [,->,>=latex] (-2.285714286,-0.5714285714) -- (-2.635250922,-0.6588127304);
-\draw [,->,>=latex] (-2.285714286,0) -- (-2.668526786,0);
-\draw [,->,>=latex] (-2.285714286,0.5714285714) -- (-2.635250922,0.6588127304);
-\draw [,->,>=latex] (-2.285714286,1.142857143) -- (-2.559632613,1.279816306);
-\draw [,->,>=latex] (-2.285714286,1.714285714) -- (-2.481714286,1.861285714);
-\draw [,->,>=latex] (-2.285714286,2.285714286) -- (-2.421058943,2.421058943);
-\draw [,->,>=latex] (-2.285714286,2.857142857) -- (-2.379037631,2.973797039);
-\draw [,->,>=latex] (-2.285714286,3.428571429) -- (-2.351051568,3.526577353);
-\draw [,->,>=latex] (-2.285714286,4.000000000) -- (-2.332465840,4.081815219);
-\draw [,->,>=latex] (-1.714285714,-4.000000000) -- (-1.755884951,-4.097064885);
-\draw [,->,>=latex] (-1.714285714,-3.428571429) -- (-1.775156454,-3.550312907);
-\draw [,->,>=latex] (-1.714285714,-2.857142857) -- (-1.806970611,-3.011617686);
-\draw [,->,>=latex] (-1.714285714,-2.285714286) -- (-1.861285714,-2.481714286);
-\draw [,->,>=latex] (-1.714285714,-1.714285714) -- (-1.954898438,-1.954898438);
-\draw [,->,>=latex] (-1.714285714,-1.142857143) -- (-2.106309411,-1.404206274);
-\draw [,->,>=latex] (-1.714285714,-0.5714285714) -- (-2.295354234,-0.7651180781);
-\draw [,->,>=latex] (-1.714285714,0) -- (-2.394841270,0);
-\draw [,->,>=latex] (-1.714285714,0.5714285714) -- (-2.295354234,0.7651180781);
-\draw [,->,>=latex] (-1.714285714,1.142857143) -- (-2.106309411,1.404206274);
-\draw [,->,>=latex] (-1.714285714,1.714285714) -- (-1.954898438,1.954898438);
-\draw [,->,>=latex] (-1.714285714,2.285714286) -- (-1.861285714,2.481714286);
-\draw [,->,>=latex] (-1.714285714,2.857142857) -- (-1.806970611,3.011617686);
-\draw [,->,>=latex] (-1.714285714,3.428571429) -- (-1.775156454,3.550312907);
-\draw [,->,>=latex] (-1.714285714,4.000000000) -- (-1.755884951,4.097064885);
-\draw [,->,>=latex] (-1.142857143,-4.000000000) -- (-1.174605575,-4.111119513);
-\draw [,->,>=latex] (-1.142857143,-3.428571429) -- (-1.191279520,-3.573838559);
-\draw [,->,>=latex] (-1.142857143,-2.857142857) -- (-1.221297415,-3.053243538);
-\draw [,->,>=latex] (-1.142857143,-2.285714286) -- (-1.279816306,-2.559632613);
-\draw [,->,>=latex] (-1.142857143,-1.714285714) -- (-1.404206274,-2.106309411);
-\draw [,->,>=latex] (-1.142857143,-1.142857143) -- (-1.684235772,-1.684235772);
-\draw [,->,>=latex] (-1.142857143,-0.5714285714) -- (-2.238530452,-1.119265226);
-\draw [,->,>=latex] (-1.142857143,0) -- (-2.674107143,0);
-\draw [,->,>=latex] (-1.142857143,0.5714285714) -- (-2.238530452,1.119265226);
-\draw [,->,>=latex] (-1.142857143,1.142857143) -- (-1.684235772,1.684235772);
-\draw [,->,>=latex] (-1.142857143,1.714285714) -- (-1.404206274,2.106309411);
-\draw [,->,>=latex] (-1.142857143,2.285714286) -- (-1.279816306,2.559632613);
-\draw [,->,>=latex] (-1.142857143,2.857142857) -- (-1.221297415,3.053243538);
-\draw [,->,>=latex] (-1.142857143,3.428571429) -- (-1.191279520,3.573838559);
-\draw [,->,>=latex] (-1.142857143,4.000000000) -- (-1.174605575,4.111119513);
-\draw [,->,>=latex] (-0.5714285714,-4.000000000) -- (-0.5887526876,-4.121268813);
-\draw [,->,>=latex] (-0.5714285714,-3.428571429) -- (-0.5986432687,-3.591859612);
-\draw [,->,>=latex] (-0.5714285714,-2.857142857) -- (-0.6176290071,-3.088145036);
-\draw [,->,>=latex] (-0.5714285714,-2.285714286) -- (-0.6588127304,-2.635250922);
-\draw [,->,>=latex] (-0.5714285714,-1.714285714) -- (-0.7651180781,-2.295354234);
-\draw [,->,>=latex] (-0.5714285714,-1.142857143) -- (-1.119265226,-2.238530452);
-\draw [,->,>=latex] (-0.5714285714,-0.5714285714) -- (-2.736943089,-2.736943089);
-\draw [,->,>=latex] (-0.5714285714,0.5714285714) -- (-2.736943089,2.736943089);
-\draw [,->,>=latex] (-0.5714285714,1.142857143) -- (-1.119265226,2.238530452);
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 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ChiSquared.pstricks b/auto/pictures_tex/Fig_ChiSquared.pstricks
index 758660882..77a4f4ecd 100644
--- a/auto/pictures_tex/Fig_ChiSquared.pstricks
+++ b/auto/pictures_tex/Fig_ChiSquared.pstricks
@@ -95,26 +95,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (15.50000000,0);
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+\draw [,->,>=latex] (0,-0.50000) -- (0,5.3819);
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-\draw (10.00000000,-0.3149246667) node {$ 20 $};
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 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (12.50000000,-0.3149246667) node {$ 25 $};
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 \draw [] (12.5,-0.100) -- (12.5,0.100);
-\draw (15.00000000,-0.3149246667) node {$ 30 $};
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 \draw [] (15.0,-0.100) -- (15.0,0.100);
-\draw (-0.3816666667,2.500000000) node {$ \frac{1}{20} $};
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 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.3816666667,5.000000000) node {$ \frac{1}{10} $};
+\draw (-0.38167,5.0000) node {$ \frac{1}{10} $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks b/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
index ff609a685..8e28a13e5 100644
--- a/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
+++ b/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
@@ -95,8 +95,8 @@
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+\draw [,->,>=latex] (0,-0.50000) -- (0,5.3842);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.045226,0)--(0.090452,0)--(0.13568,0.0021725)--(0.18090,0.0063676)--(0.22613,0.014417)--(0.27136,0.027725)--(0.31658,0.047635)--(0.36181,0.075363)--(0.40704,0.11195)--(0.45226,0.15824)--(0.49749,0.21486)--(0.54271,0.28221)--(0.58794,0.36049)--(0.63317,0.44967)--(0.67839,0.54955)--(0.72362,0.65977)--(0.76884,0.77977)--(0.81407,0.90892)--(0.85930,1.0464)--(0.90452,1.1915)--(0.94975,1.3431)--(0.99498,1.5003)--(1.0402,1.6621)--(1.0854,1.8275)--(1.1307,1.9955)--(1.1759,2.1649)--(1.2211,2.3349)--(1.2663,2.5044)--(1.3116,2.6726)--(1.3568,2.8385)--(1.4020,3.0014)--(1.4472,3.1604)--(1.4925,3.3148)--(1.5377,3.4640)--(1.5829,3.6075)--(1.6281,3.7446)--(1.6734,3.8749)--(1.7186,3.9981)--(1.7638,4.1138)--(1.8090,4.2217)--(1.8543,4.3217)--(1.8995,4.4134)--(1.9447,4.4969)--(1.9900,4.5721)--(2.0352,4.6390)--(2.0804,4.6975)--(2.1256,4.7478)--(2.1709,4.7899)--(2.2161,4.8241)--(2.2613,4.8503)--(2.3065,4.8690)--(2.3518,4.8802)--(2.3970,4.8842)--(2.4422,4.8812)--(2.4874,4.8715)--(2.5327,4.8555)--(2.5779,4.8333)--(2.6231,4.8053)--(2.6683,4.7718)--(2.7136,4.7330)--(2.7588,4.6894)--(2.8040,4.6412)--(2.8492,4.5887)--(2.8945,4.5322)--(2.9397,4.4721)--(2.9849,4.4086)--(3.0302,4.3419)--(3.0754,4.2725)--(3.1206,4.2006)--(3.1658,4.1264)--(3.2111,4.0502)--(3.2563,3.9722)--(3.3015,3.8928)--(3.3467,3.8121)--(3.3920,3.7303)--(3.4372,3.6477)--(3.4824,3.5644)--(3.5276,3.4807)--(3.5729,3.3968)--(3.6181,3.3127)--(3.6633,3.2287)--(3.7085,3.1449)--(3.7538,3.0614)--(3.7990,2.9785)--(3.8442,2.8961)--(3.8894,2.8145)--(3.9347,2.7337)--(3.9799,2.6538)--(4.0251,2.5749)--(4.0704,2.4971)--(4.1156,2.4204)--(4.1608,2.3450)--(4.2060,2.2708)--(4.2513,2.1980)--(4.2965,2.1266)--(4.3417,2.0565)--(4.3869,1.9879)--(4.4322,1.9208)--(4.4774,1.8552)--(4.5226,1.7910)--(4.5678,1.7284)--(4.6131,1.6674)--(4.6583,1.6079)--(4.7035,1.5499)--(4.7487,1.4934)--(4.7940,1.4385)--(4.8392,1.3851)--(4.8844,1.3333)--(4.9296,1.2829)--(4.9749,1.2340)--(5.0201,1.1866)--(5.0653,1.1406)--(5.1105,1.0961)--(5.1558,1.0530)--(5.2010,1.0113)--(5.2462,0.97088)--(5.2915,0.93184)--(5.3367,0.89411)--(5.3819,0.85766)--(5.4271,0.82246)--(5.4724,0.78849)--(5.5176,0.75571)--(5.5628,0.72411)--(5.6080,0.69364)--(5.6533,0.66428)--(5.6985,0.63600)--(5.7437,0.60877)--(5.7889,0.58256)--(5.8342,0.55735)--(5.8794,0.53310)--(5.9246,0.50978)--(5.9698,0.48737)--(6.0151,0.46583)--(6.0603,0.44515)--(6.1055,0.42529)--(6.1508,0.40623)--(6.1960,0.38794)--(6.2412,0.37039)--(6.2864,0.35357)--(6.3317,0.33743)--(6.3769,0.32197)--(6.4221,0.30715)--(6.4673,0.29296)--(6.5126,0.27937)--(6.5578,0.26636)--(6.6030,0.25390)--(6.6482,0.24199)--(6.6935,0.23059)--(6.7387,0.21968)--(6.7839,0.20926)--(6.8291,0.19929)--(6.8744,0.18977)--(6.9196,0.18067)--(6.9648,0.17197)--(7.0100,0.16367)--(7.0553,0.15574)--(7.1005,0.14817)--(7.1457,0.14095)--(7.1910,0.13406)--(7.2362,0.12748)--(7.2814,0.12121)--(7.3266,0.11523)--(7.3719,0.10952)--(7.4171,0.10409)--(7.4623,0.098904)--(7.5075,0.093967)--(7.5528,0.089264)--(7.5980,0.084783)--(7.6432,0.080516)--(7.6884,0.076453)--(7.7337,0.072586)--(7.7789,0.068904)--(7.8241,0.065400)--(7.8693,0.062066)--(7.9146,0.058895)--(7.9598,0.055878)--(8.0050,0.053008)--(8.0502,0.050280)--(8.0955,0.047686)--(8.1407,0.045220)--(8.1859,0.042877)--(8.2312,0.040650)--(8.2764,0.038534)--(8.3216,0.036523)--(8.3668,0.034614)--(8.4121,0.032801)--(8.4573,0.031078)--(8.5025,0.029443)--(8.5477,0.027891)--(8.5930,0.026418)--(8.6382,0.025020)--(8.6834,0.023693)--(8.7286,0.022434)--(8.7739,0.021240)--(8.8191,0.020107)--(8.8643,0.019033)--(8.9095,0.018014)--(8.9548,0.017047)--(9.0000,0.016132);
@@ -124,21 +124,21 @@
 \draw [color=blue] (4.8000,0)--(4.8424,0)--(4.8848,0)--(4.9273,0)--(4.9697,0)--(5.0121,0)--(5.0545,0)--(5.0970,0)--(5.1394,0)--(5.1818,0)--(5.2242,0)--(5.2667,0)--(5.3091,0)--(5.3515,0)--(5.3939,0)--(5.4364,0)--(5.4788,0)--(5.5212,0)--(5.5636,0)--(5.6061,0)--(5.6485,0)--(5.6909,0)--(5.7333,0)--(5.7758,0)--(5.8182,0)--(5.8606,0)--(5.9030,0)--(5.9455,0)--(5.9879,0)--(6.0303,0)--(6.0727,0)--(6.1152,0)--(6.1576,0)--(6.2000,0)--(6.2424,0)--(6.2849,0)--(6.3273,0)--(6.3697,0)--(6.4121,0)--(6.4545,0)--(6.4970,0)--(6.5394,0)--(6.5818,0)--(6.6242,0)--(6.6667,0)--(6.7091,0)--(6.7515,0)--(6.7939,0)--(6.8364,0)--(6.8788,0)--(6.9212,0)--(6.9636,0)--(7.0061,0)--(7.0485,0)--(7.0909,0)--(7.1333,0)--(7.1758,0)--(7.2182,0)--(7.2606,0)--(7.3030,0)--(7.3455,0)--(7.3879,0)--(7.4303,0)--(7.4727,0)--(7.5152,0)--(7.5576,0)--(7.6000,0)--(7.6424,0)--(7.6848,0)--(7.7273,0)--(7.7697,0)--(7.8121,0)--(7.8545,0)--(7.8970,0)--(7.9394,0)--(7.9818,0)--(8.0242,0)--(8.0667,0)--(8.1091,0)--(8.1515,0)--(8.1939,0)--(8.2364,0)--(8.2788,0)--(8.3212,0)--(8.3636,0)--(8.4061,0)--(8.4485,0)--(8.4909,0)--(8.5333,0)--(8.5758,0)--(8.6182,0)--(8.6606,0)--(8.7030,0)--(8.7455,0)--(8.7879,0)--(8.8303,0)--(8.8727,0)--(8.9151,0)--(8.9576,0)--(9.0000,0);
 \draw [] (4.80,0) -- (4.80,1.43);
 \draw [] (9.00,0.0161) -- (9.00,0);
-\draw (1.500000000,-0.3149246667) node {$ 5 $};
+\draw (1.5000,-0.31492) node {$ 5 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 10 $};
+\draw (3.0000,-0.31492) node {$ 10 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.500000000,-0.3149246667) node {$ 15 $};
+\draw (4.5000,-0.31492) node {$ 15 $};
 \draw [] (4.50,-0.100) -- (4.50,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 20 $};
+\draw (6.0000,-0.31492) node {$ 20 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.500000000,-0.3149246667) node {$ 25 $};
+\draw (7.5000,-0.31492) node {$ 25 $};
 \draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 30 $};
+\draw (9.0000,-0.31492) node {$ 30 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.3816666667,2.500000000) node {$ \frac{1}{20} $};
+\draw (-0.38167,2.5000) node {$ \frac{1}{20} $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.3816666667,5.000000000) node {$ \frac{1}{10} $};
+\draw (-0.38167,5.0000) node {$ \frac{1}{10} $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ChoixInfini.pstricks b/auto/pictures_tex/Fig_ChoixInfini.pstricks
index 07e01a9e1..40c07f5d0 100644
--- a/auto/pictures_tex/Fig_ChoixInfini.pstricks
+++ b/auto/pictures_tex/Fig_ChoixInfini.pstricks
@@ -57,24 +57,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
 %DEFAULT
 \draw [color=blue] (-3.00,1.00) -- (3.00,1.00);
-\draw [color=blue]  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw [color=blue]  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -121,22 +121,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [color=blue] (-2.00,0) -- (2.00,2.00);
-\draw [color=blue]  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw [color=blue]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CoinPasVar.pstricks b/auto/pictures_tex/Fig_CoinPasVar.pstricks
index 92a1bf5a8..471cee114 100644
--- a/auto/pictures_tex/Fig_CoinPasVar.pstricks
+++ b/auto/pictures_tex/Fig_CoinPasVar.pstricks
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [color=blue] (-2.00,0) -- (0,2.00);
 \draw [color=blue] (2.00,0) -- (0,2.00);
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (0.2372415115,2.195418678) node {\( N\)};
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.215780345,1.210337511) node {\( t_1\)};
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.215780345,1.210337511) node {\( t_2\)};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (0.23724,2.1954) node {\( N\)};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.2158,1.2103) node {\( t_1\)};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.2158,1.2103) node {\( t_2\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ContourGreen.pstricks b/auto/pictures_tex/Fig_ContourGreen.pstricks
index 8fe9b877d..6990892da 100644
--- a/auto/pictures_tex/Fig_ContourGreen.pstricks
+++ b/auto/pictures_tex/Fig_ContourGreen.pstricks
@@ -66,14 +66,14 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.246812693,0.6811368787) -- (1.245451007,0.6910437354);
-\draw [,->,>=latex] (0.3575977601,1.218773069) -- (0.3480435445,1.215820618);
-\draw [,->,>=latex] (-0.2777141985,0.5083524306) -- (-0.2808616227,0.4988606594);
-\draw [,->,>=latex] (-0.7003261911,0.2054813022) -- (-0.7094196719,0.2013209407);
-\draw [,->,>=latex] (-1.246812693,-0.6811368787) -- (-1.245451007,-0.6910437354);
-\draw [,->,>=latex] (-0.3575977601,-1.218773069) -- (-0.3480435445,-1.215820618);
-\draw [,->,>=latex] (0.2777141985,-0.5083524306) -- (0.2808616227,-0.4988606594);
-\draw [,->,>=latex] (0.7003261912,-0.2054813022) -- (0.7094196719,-0.2013209407);
+\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
+\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
+\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
+\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
+\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
+\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
+\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
+\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ContourSqL.pstricks b/auto/pictures_tex/Fig_ContourSqL.pstricks
index 99087b680..766a75511 100644
--- a/auto/pictures_tex/Fig_ContourSqL.pstricks
+++ b/auto/pictures_tex/Fig_ContourSqL.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -79,14 +79,14 @@
          hatchthickness=0.4pt}
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 \draw [] (0,0) -- (0,0);
 \draw [] (3.00,3.00) -- (3.00,3.00);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks b/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks
index ea9d6a1ea..e2b22fef2 100644
--- a/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks
+++ b/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks
@@ -65,31 +65,31 @@
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 %END PSPICTURE
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 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CoordPolaires.pstricks b/auto/pictures_tex/Fig_CoordPolaires.pstricks
index 37c5a05be..827399ee2 100644
--- a/auto/pictures_tex/Fig_CoordPolaires.pstricks
+++ b/auto/pictures_tex/Fig_CoordPolaires.pstricks
@@ -83,20 +83,20 @@
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+\draw (0.68452,0.41391) node {$\theta$};
 
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-\draw (0.2337087285,1.168018886) node {$r$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (0.23371,1.1680) node {$r$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CornetGlace.pstricks b/auto/pictures_tex/Fig_CornetGlace.pstricks
index 38fbfdeac..a8a07a6d4 100644
--- a/auto/pictures_tex/Fig_CornetGlace.pstricks
+++ b/auto/pictures_tex/Fig_CornetGlace.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.700000000,0) -- (1.700000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.700000000);
+\draw [,->,>=latex] (-1.7000,0) -- (1.7000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.7000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -87,11 +87,11 @@
 \draw [color=blue] (-1.200,1.200)--(-1.188,1.188)--(-1.176,1.176)--(-1.164,1.164)--(-1.152,1.152)--(-1.139,1.139)--(-1.127,1.127)--(-1.115,1.115)--(-1.103,1.103)--(-1.091,1.091)--(-1.079,1.079)--(-1.067,1.067)--(-1.055,1.055)--(-1.042,1.042)--(-1.030,1.030)--(-1.018,1.018)--(-1.006,1.006)--(-0.9939,0.9939)--(-0.9818,0.9818)--(-0.9697,0.9697)--(-0.9576,0.9576)--(-0.9455,0.9455)--(-0.9333,0.9333)--(-0.9212,0.9212)--(-0.9091,0.9091)--(-0.8970,0.8970)--(-0.8848,0.8848)--(-0.8727,0.8727)--(-0.8606,0.8606)--(-0.8485,0.8485)--(-0.8364,0.8364)--(-0.8242,0.8242)--(-0.8121,0.8121)--(-0.8000,0.8000)--(-0.7879,0.7879)--(-0.7758,0.7758)--(-0.7636,0.7636)--(-0.7515,0.7515)--(-0.7394,0.7394)--(-0.7273,0.7273)--(-0.7151,0.7151)--(-0.7030,0.7030)--(-0.6909,0.6909)--(-0.6788,0.6788)--(-0.6667,0.6667)--(-0.6545,0.6545)--(-0.6424,0.6424)--(-0.6303,0.6303)--(-0.6182,0.6182)--(-0.6061,0.6061)--(-0.5939,0.5939)--(-0.5818,0.5818)--(-0.5697,0.5697)--(-0.5576,0.5576)--(-0.5455,0.5455)--(-0.5333,0.5333)--(-0.5212,0.5212)--(-0.5091,0.5091)--(-0.4970,0.4970)--(-0.4848,0.4848)--(-0.4727,0.4727)--(-0.4606,0.4606)--(-0.4485,0.4485)--(-0.4364,0.4364)--(-0.4242,0.4242)--(-0.4121,0.4121)--(-0.4000,0.4000)--(-0.3879,0.3879)--(-0.3758,0.3758)--(-0.3636,0.3636)--(-0.3515,0.3515)--(-0.3394,0.3394)--(-0.3273,0.3273)--(-0.3152,0.3152)--(-0.3030,0.3030)--(-0.2909,0.2909)--(-0.2788,0.2788)--(-0.2667,0.2667)--(-0.2545,0.2545)--(-0.2424,0.2424)--(-0.2303,0.2303)--(-0.2182,0.2182)--(-0.2061,0.2061)--(-0.1939,0.1939)--(-0.1818,0.1818)--(-0.1697,0.1697)--(-0.1576,0.1576)--(-0.1455,0.1455)--(-0.1333,0.1333)--(-0.1212,0.1212)--(-0.1091,0.1091)--(-0.09697,0.09697)--(-0.08485,0.08485)--(-0.07273,0.07273)--(-0.06061,0.06061)--(-0.04848,0.04848)--(-0.03636,0.03636)--(-0.02424,0.02424)--(-0.01212,0.01212)--(0,0);
 
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CourbeRectifiable.pstricks b/auto/pictures_tex/Fig_CourbeRectifiable.pstricks
index dbfe50a3c..c87a9cf28 100644
--- a/auto/pictures_tex/Fig_CourbeRectifiable.pstricks
+++ b/auto/pictures_tex/Fig_CourbeRectifiable.pstricks
@@ -89,16 +89,16 @@
 \draw [color=red] (-13.6,1.40) -- (-12.3,0);
 \draw [color=red] (-12.3,0) -- (-10.2,-1.40);
 \draw [color=red] (-10.2,-1.40) -- (-7.56,0);
-\draw []  (-14.00000000,0) node [rotate=0] {$\bullet$};
-\draw (-14.59160333,0) node {$\gamma(t_{0})$};
-\draw []  (-13.56477390,1.400000000) node [rotate=0] {$\bullet$};
-\draw (-13.56477390,1.782455000) node {$\gamma(t_{1})$};
-\draw []  (-12.28615587,0) node [rotate=0] {$\bullet$};
-\draw (-12.83675930,-0.3037767567) node {$\gamma(t_{2})$};
-\draw []  (-10.24364416,-1.400000000) node [rotate=0] {$\bullet$};
-\draw (-10.24364416,-1.782455000) node {$\gamma(t_{3})$};
-\draw []  (-7.564232282,0) node [rotate=0] {$\bullet$};
-\draw (-8.075495931,0.3427092006) node {$\gamma(t_{4})$};
+\draw []  (-14.000,0) node [rotate=0] {$\bullet$};
+\draw (-14.592,0) node {$\gamma(t_{0})$};
+\draw []  (-13.565,1.4000) node [rotate=0] {$\bullet$};
+\draw (-13.565,1.7825) node {$\gamma(t_{1})$};
+\draw []  (-12.286,0) node [rotate=0] {$\bullet$};
+\draw (-12.837,-0.30378) node {$\gamma(t_{2})$};
+\draw []  (-10.244,-1.4000) node [rotate=0] {$\bullet$};
+\draw (-10.244,-1.7825) node {$\gamma(t_{3})$};
+\draw []  (-7.5642,0) node [rotate=0] {$\bullet$};
+\draw (-8.0755,0.34271) node {$\gamma(t_{4})$};
 \draw [color=blue] (-14.000,0)--(-13.999,0.088794)--(-13.997,0.17723)--(-13.994,0.26495)--(-13.989,0.35161)--(-13.982,0.43685)--(-13.974,0.52033)--(-13.965,0.60171)--(-13.954,0.68068)--(-13.942,0.75690)--(-13.929,0.83007)--(-13.914,0.89990)--(-13.897,0.96611)--(-13.879,1.0284)--(-13.860,1.0866)--(-13.840,1.1404)--(-13.818,1.1896)--(-13.794,1.2340)--(-13.769,1.2735)--(-13.743,1.3078)--(-13.715,1.3369)--(-13.686,1.3605)--(-13.656,1.3787)--(-13.624,1.3914)--(-13.591,1.3984)--(-13.556,1.3998)--(-13.520,1.3956)--(-13.483,1.3857)--(-13.444,1.3703)--(-13.404,1.3494)--(-13.362,1.3230)--(-13.319,1.2913)--(-13.275,1.2544)--(-13.229,1.2124)--(-13.182,1.1656)--(-13.134,1.1141)--(-13.085,1.0581)--(-13.034,0.99777)--(-12.981,0.93348)--(-12.928,0.86542)--(-12.873,0.79388)--(-12.816,0.71915)--(-12.759,0.64152)--(-12.700,0.56130)--(-12.640,0.47883)--(-12.578,0.39443)--(-12.516,0.30843)--(-12.452,0.22120)--(-12.386,0.13308)--(-12.320,0.044419)--(-12.252,-0.044419)--(-12.183,-0.13308)--(-12.113,-0.22120)--(-12.041,-0.30843)--(-11.968,-0.39443)--(-11.895,-0.47883)--(-11.819,-0.56130)--(-11.743,-0.64152)--(-11.665,-0.71915)--(-11.587,-0.79388)--(-11.507,-0.86542)--(-11.425,-0.93348)--(-11.343,-0.99777)--(-11.260,-1.0581)--(-11.175,-1.1141)--(-11.089,-1.1656)--(-11.002,-1.2124)--(-10.914,-1.2544)--(-10.825,-1.2913)--(-10.735,-1.3230)--(-10.644,-1.3494)--(-10.551,-1.3703)--(-10.458,-1.3857)--(-10.363,-1.3956)--(-10.268,-1.3998)--(-10.171,-1.3984)--(-10.073,-1.3914)--(-9.9746,-1.3787)--(-9.8749,-1.3605)--(-9.7742,-1.3369)--(-9.6724,-1.3078)--(-9.5697,-1.2735)--(-9.4660,-1.2340)--(-9.3613,-1.1896)--(-9.2557,-1.1404)--(-9.1491,-1.0866)--(-9.0416,-1.0284)--(-8.9332,-0.96611)--(-8.8239,-0.89990)--(-8.7136,-0.83007)--(-8.6025,-0.75690)--(-8.4905,-0.68068)--(-8.3776,-0.60171)--(-8.2639,-0.52033)--(-8.1493,-0.43685)--(-8.0339,-0.35161)--(-7.9177,-0.26495)--(-7.8007,-0.17723)--(-7.6828,-0.088794)--(-7.5642,0);
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CouroneExam.pstricks b/auto/pictures_tex/Fig_CouroneExam.pstricks
index e90d72afd..33009d868 100644
--- a/auto/pictures_tex/Fig_CouroneExam.pstricks
+++ b/auto/pictures_tex/Fig_CouroneExam.pstricks
@@ -71,21 +71,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
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 \draw [color=blue] (1.00,0) -- (2.00,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks b/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks
index 2ab5032ab..401de7a4e 100644
--- a/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks
+++ b/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks
@@ -46,10 +46,10 @@
 
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-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (2.598076211,1.500000000);
-\draw (2.286902711,1.865758621) node {$e_{r}$};
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (1.232050808,1.866025404);
-\draw (1.607516675,2.186465271) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
+\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
+\draw (1.6075,2.1865) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -91,10 +91,10 @@
 
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-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-2.349231552,-0.8550503583);
-\draw (-2.085452009,-1.242909145) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-1.067518788,-1.452722836);
-\draw (-0.6920529202,-1.132282968) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
+\draw (-2.0855,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
+\draw (-0.69205,-1.1323) node {$e_{\theta}$};
 %END PSPICTURE
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diff --git a/auto/pictures_tex/Fig_CycloideA.pstricks b/auto/pictures_tex/Fig_CycloideA.pstricks
index 9f4df5aac..22616691c 100644
--- a/auto/pictures_tex/Fig_CycloideA.pstricks
+++ b/auto/pictures_tex/Fig_CycloideA.pstricks
@@ -115,39 +115,39 @@
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 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (13.06637062,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.499496542);
+\draw [,->,>=latex] (-0.50000,0) -- (13.066,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.4995);
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diff --git a/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks b/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks
index 03ab6c0ac..bb8a4f9fd 100644
--- a/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks
+++ b/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks
@@ -65,31 +65,31 @@
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diff --git a/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks b/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks
index 421e7c1d6..28b9aef27 100644
--- a/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks
+++ b/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks
@@ -60,8 +60,8 @@
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diff --git a/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks b/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
index fe50e66aa..aca707323 100644
--- a/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
+++ b/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
@@ -53,28 +53,28 @@
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+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
@@ -141,32 +141,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.723619130) -- (0,1.470406053);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.7236) -- (0,1.4704);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
 %OTHER STUFF
 %END PSPICTURE
@@ -229,32 +229,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.023619131) -- (0,2.170406053);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.0236) -- (0,2.1704);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
 %OTHER STUFF
 %END PSPICTURE
@@ -301,20 +301,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.324187016);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.3242);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.030,0.1209)--(1.061,0.1698)--(1.091,0.2065)--(1.121,0.2368)--(1.152,0.2629)--(1.182,0.2861)--(1.212,0.3070)--(1.242,0.3261)--(1.273,0.3438)--(1.303,0.3601)--(1.333,0.3755)--(1.364,0.3898)--(1.394,0.4034)--(1.424,0.4163)--(1.455,0.4285)--(1.485,0.4401)--(1.515,0.4512)--(1.545,0.4618)--(1.576,0.4720)--(1.606,0.4818)--(1.636,0.4912)--(1.667,0.5003)--(1.697,0.5090)--(1.727,0.5175)--(1.758,0.5257)--(1.788,0.5336)--(1.818,0.5412)--(1.848,0.5487)--(1.879,0.5559)--(1.909,0.5629)--(1.939,0.5697)--(1.970,0.5763)--(2.000,0.5828)--(2.030,0.5891)--(2.061,0.5952)--(2.091,0.6012)--(2.121,0.6070)--(2.152,0.6127)--(2.182,0.6183)--(2.212,0.6237)--(2.242,0.6291)--(2.273,0.6343)--(2.303,0.6394)--(2.333,0.6443)--(2.364,0.6492)--(2.394,0.6540)--(2.424,0.6587)--(2.455,0.6633)--(2.485,0.6678)--(2.515,0.6723)--(2.545,0.6766)--(2.576,0.6809)--(2.606,0.6851)--(2.636,0.6892)--(2.667,0.6933)--(2.697,0.6972)--(2.727,0.7012)--(2.758,0.7050)--(2.788,0.7088)--(2.818,0.7125)--(2.848,0.7162)--(2.879,0.7198)--(2.909,0.7234)--(2.939,0.7269)--(2.970,0.7303)--(3.000,0.7337)--(3.030,0.7371)--(3.061,0.7403)--(3.091,0.7436)--(3.121,0.7468)--(3.152,0.7500)--(3.182,0.7531)--(3.212,0.7562)--(3.242,0.7592)--(3.273,0.7622)--(3.303,0.7652)--(3.333,0.7681)--(3.364,0.7710)--(3.394,0.7738)--(3.424,0.7766)--(3.455,0.7794)--(3.485,0.7821)--(3.515,0.7848)--(3.545,0.7875)--(3.576,0.7902)--(3.606,0.7928)--(3.636,0.7953)--(3.667,0.7979)--(3.697,0.8004)--(3.727,0.8029)--(3.758,0.8054)--(3.788,0.8078)--(3.818,0.8102)--(3.848,0.8126)--(3.879,0.8150)--(3.909,0.8173)--(3.939,0.8196)--(3.970,0.8219)--(4.000,0.8242);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -381,34 +381,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.490000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-3.723619130) -- (0,1.552854178);
+\draw [,->,>=latex] (-1.4900,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-3.7236) -- (0,1.5529);
 %DEFAULT
 
 \draw [color=blue] (-0.9900,-3.224)--(-0.9446,-2.026)--(-0.8993,-1.607)--(-0.8539,-1.347)--(-0.8086,-1.157)--(-0.7632,-1.008)--(-0.7179,-0.8858)--(-0.6725,-0.7814)--(-0.6272,-0.6906)--(-0.5818,-0.6103)--(-0.5365,-0.5382)--(-0.4911,-0.4729)--(-0.4458,-0.4131)--(-0.4004,-0.3581)--(-0.3550,-0.3070)--(-0.3097,-0.2594)--(-0.2643,-0.2149)--(-0.2190,-0.1730)--(-0.1736,-0.1335)--(-0.1283,-0.09610)--(-0.08293,-0.06060)--(-0.03758,-0.02681)--(0.007778,0.005423)--(0.05313,0.03624)--(0.09848,0.06575)--(0.1438,0.09407)--(0.1892,0.1213)--(0.2345,0.1475)--(0.2799,0.1727)--(0.3253,0.1971)--(0.3706,0.2207)--(0.4160,0.2435)--(0.4613,0.2655)--(0.5067,0.2869)--(0.5520,0.3077)--(0.5974,0.3279)--(0.6427,0.3475)--(0.6881,0.3665)--(0.7334,0.3851)--(0.7788,0.4032)--(0.8241,0.4208)--(0.8695,0.4380)--(0.9148,0.4547)--(0.9602,0.4711)--(1.006,0.4871)--(1.051,0.5028)--(1.096,0.5181)--(1.142,0.5331)--(1.187,0.5478)--(1.232,0.5621)--(1.278,0.5762)--(1.323,0.5900)--(1.368,0.6035)--(1.414,0.6168)--(1.459,0.6299)--(1.504,0.6426)--(1.550,0.6552)--(1.595,0.6675)--(1.641,0.6797)--(1.686,0.6916)--(1.731,0.7033)--(1.777,0.7149)--(1.822,0.7262)--(1.867,0.7374)--(1.913,0.7483)--(1.958,0.7592)--(2.003,0.7698)--(2.049,0.7803)--(2.094,0.7906)--(2.139,0.8008)--(2.185,0.8109)--(2.230,0.8208)--(2.275,0.8305)--(2.321,0.8401)--(2.366,0.8496)--(2.412,0.8590)--(2.457,0.8683)--(2.502,0.8774)--(2.548,0.8864)--(2.593,0.8953)--(2.638,0.9041)--(2.684,0.9127)--(2.729,0.9213)--(2.774,0.9298)--(2.820,0.9381)--(2.865,0.9464)--(2.910,0.9546)--(2.956,0.9626)--(3.001,0.9706)--(3.046,0.9785)--(3.092,0.9863)--(3.137,0.9940)--(3.183,1.002)--(3.228,1.009)--(3.273,1.017)--(3.319,1.024)--(3.364,1.031)--(3.409,1.039)--(3.455,1.046)--(3.500,1.053);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
 %OTHER STUFF
 %END PSPICTURE
@@ -471,38 +471,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.250000000,0) -- (2.250000000,0);
-\draw [,->,>=latex] (0,-3.772532402) -- (0,1.782807025);
+\draw [,->,>=latex] (-2.2500,0) -- (2.2500,0);
+\draw [,->,>=latex] (0,-3.7725) -- (0,1.7828);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
 %OTHER STUFF
 %END PSPICTURE
@@ -565,36 +565,36 @@
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 %AXES
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-\draw [,->,>=latex] (0,-3.546698315) -- (0,1.269028602);
+\draw [,->,>=latex] (-2.6000,0) -- (2.6000,0);
+\draw [,->,>=latex] (0,-3.5467) -- (0,1.2690);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DerivTangenteOM.pstricks b/auto/pictures_tex/Fig_DerivTangenteOM.pstricks
index 8bfad4bc6..a0c5b4ece 100644
--- a/auto/pictures_tex/Fig_DerivTangenteOM.pstricks
+++ b/auto/pictures_tex/Fig_DerivTangenteOM.pstricks
@@ -87,8 +87,8 @@
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 %PSTRICKS CODE
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+\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
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 \draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
 \draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
@@ -98,22 +98,22 @@
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-\draw (6.500000000,-0.2785761667) node {$x$};
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-\draw (-0.4552066667,6.085648148) node {$f(x)$};
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (3.000000000,1.300000000);
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (6.500000000,1.300000000);
-\draw (4.750000000,0.9789703333) node {$x-a$};
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,6.085648148);
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,1.500000000);
-\draw (7.825596167,3.792824074) node {$f(x)-f(a)$};
+\draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.27858) node {$a$};
+\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
+\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.27858) node {$x$};
+\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.45521,6.0856) node {$f(x)$};
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+\draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
+\draw (4.7500,0.97897) node {$x-a$};
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
+\draw (7.8256,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DessinLim.pstricks b/auto/pictures_tex/Fig_DessinLim.pstricks
index 1b95a76d9..59d0386bb 100644
--- a/auto/pictures_tex/Fig_DessinLim.pstricks
+++ b/auto/pictures_tex/Fig_DessinLim.pstricks
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.800000001,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.800000001);
+\draw [,->,>=latex] (-0.50000,0) -- (2.8000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.8000);
 %DEFAULT
 
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@@ -96,16 +96,16 @@
 \draw [style=dashed] (0,1.63) -- (1.63,1.63);
 \draw [style=dashed] (1.63,0) -- (1.63,1.63);
 \draw [] (2.30,2.30) -- (2.30,0);
-\draw []  (0,1.626345597) node [rotate=0] {$\bullet$};
-\draw (-0.5715993333,1.626345597) node {\( \sin(x)\)};
-\draw []  (1.626345597,0) node [rotate=0] {$\bullet$};
-\draw (1.626345597,-0.2824550000) node {\( \cos(x)\)};
-\draw []  (2.300000000,0) node [rotate=0] {$\bullet$};
-\draw (2.486875167,-0.2113105404) node {\( A\)};
-\draw []  (2.300000000,2.300000000) node [rotate=0] {$\bullet$};
-\draw (2.531995833,2.300000000) node {\( T\)};
-\draw []  (1.626345597,1.626345597) node [rotate=0] {$\bullet$};
-\draw (2.068860430,1.626345597) node {\( P\)};
+\draw []  (0,1.6263) node [rotate=0] {$\bullet$};
+\draw (-0.57160,1.6263) node {\( \sin(x)\)};
+\draw []  (1.6263,0) node [rotate=0] {$\bullet$};
+\draw (1.6263,-0.28245) node {\( \cos(x)\)};
+\draw []  (2.3000,0) node [rotate=0] {$\bullet$};
+\draw (2.4869,-0.21131) node {\( A\)};
+\draw []  (2.3000,2.3000) node [rotate=0] {$\bullet$};
+\draw (2.5320,2.3000) node {\( T\)};
+\draw []  (1.6263,1.6263) node [rotate=0] {$\bullet$};
+\draw (2.0689,1.6263) node {\( P\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DeuxCercles.pstricks b/auto/pictures_tex/Fig_DeuxCercles.pstricks
index 0433aa39d..b1136c0f4 100644
--- a/auto/pictures_tex/Fig_DeuxCercles.pstricks
+++ b/auto/pictures_tex/Fig_DeuxCercles.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -83,9 +83,9 @@
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-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Differentielle.pstricks b/auto/pictures_tex/Fig_Differentielle.pstricks
index c47af80f4..00d72fbeb 100644
--- a/auto/pictures_tex/Fig_Differentielle.pstricks
+++ b/auto/pictures_tex/Fig_Differentielle.pstricks
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.700000003,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.700000003);
+\draw [,->,>=latex] (-0.50000,0) -- (4.7000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.7000);
 %DEFAULT
 \draw [style=dotted] (2.00,2.00) -- (4.00,2.00);
 \draw [style=dotted] (4.00,2.00) -- (4.00,4.00);
@@ -100,25 +100,25 @@
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-\draw (3.000000000,1.073264000) node {$h$};
-\draw []  (2.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (1.299076276,2.536008391) node {$f(a)$};
-\draw []  (4.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,3.386294361) node [rotate=0] {$\bullet$};
-\draw (4.850181414,2.708864614) node {$f(x)$};
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+\draw [,->,>=latex] (3.0000,1.5000) -- (4.0000,1.5000);
+\draw (3.0000,1.0733) node {$h$};
+\draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (1.2991,2.5360) node {$f(a)$};
+\draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,3.3863) node [rotate=0] {$\bullet$};
+\draw (4.8502,2.7089) node {$f(x)$};
+\draw []  (4.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.37858) node {$a$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.37858) node {$x$};
 \draw [style=dotted] (2.00,2.00) -- (2.00,0);
 \draw [style=dotted] (4.00,2.00) -- (4.00,0);
 
diff --git a/auto/pictures_tex/Fig_DistanceEnsemble.pstricks b/auto/pictures_tex/Fig_DistanceEnsemble.pstricks
index 581792965..fb287b1fd 100644
--- a/auto/pictures_tex/Fig_DistanceEnsemble.pstricks
+++ b/auto/pictures_tex/Fig_DistanceEnsemble.pstricks
@@ -77,18 +77,18 @@
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 %PSTRICKS CODE
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-\draw (1.763220763,1.751053597) node {$A$};
+\draw (1.7632,1.7511) node {$A$};
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 \draw [style=dotted] (-3.98,-2.30) -- (-0.347,1.97);
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-\draw []  (0,-1.400000000) node [rotate=0] {$\bullet$};
-\draw []  (-0.3472963553,1.969615506) node [rotate=0] {$\bullet$};
-\draw []  (-3.983716857,-2.300000000) node [rotate=0] {$\bullet$};
-\draw (-4.388020524,-2.300000000) node {$x$};
+\draw []  (-1.7320,-1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.7320,-1.6141) node {$p$};
+\draw []  (0,-1.4000) node [rotate=0] {$\bullet$};
+\draw []  (-0.34729,1.9696) node [rotate=0] {$\bullet$};
+\draw []  (-3.9837,-2.3000) node [rotate=0] {$\bullet$};
+\draw (-4.3880,-2.3000) node {$x$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DivergenceDeux.pstricks b/auto/pictures_tex/Fig_DivergenceDeux.pstricks
index cc1aa1379..1b692ca44 100644
--- a/auto/pictures_tex/Fig_DivergenceDeux.pstricks
+++ b/auto/pictures_tex/Fig_DivergenceDeux.pstricks
@@ -65,230 +65,230 @@
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diff --git a/auto/pictures_tex/Fig_DivergenceTrois.pstricks b/auto/pictures_tex/Fig_DivergenceTrois.pstricks
index 762748ea8..8a25fb77c 100644
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+\draw [,->,>=latex] (-0.42407,0.90563) -- (-0.84813,1.8113);
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+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2338,2.3504);
+\draw [,->,>=latex] (-1.2368,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7317,2.0120);
+\draw [,->,>=latex] (-1.6150,1.4940) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1405,1.5701);
+\draw [,->,>=latex] (-1.9100,1.0917) -- (-2.3047,1.3172);
+\draw [,->,>=latex] (-2.0215,0.86804) -- (-2.4392,1.0474);
+\draw [,->,>=latex] (-2.1069,0.63322) -- (-2.5422,0.76405);
+\draw [,->,>=latex] (-2.1651,0.39022) -- (-2.6125,0.47085);
+\draw [,->,>=latex] (-2.1954,0.14221) -- (-2.6490,0.17159);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DivergenceUn.pstricks b/auto/pictures_tex/Fig_DivergenceUn.pstricks
index a42173e46..8f1c5afca 100644
--- a/auto/pictures_tex/Fig_DivergenceUn.pstricks
+++ b/auto/pictures_tex/Fig_DivergenceUn.pstricks
@@ -68,55 +68,55 @@
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
 
 
-\draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
-\draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
-\draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
-\draw [,->,>=latex] (1.600000000,0) -- (2.225000000,0);
-\draw [,->,>=latex] (1.600000000,0.3333333333) -- (2.225000000,0.3333333333);
-\draw [,->,>=latex] (1.600000000,0.6666666667) -- (2.225000000,0.6666666667);
-\draw [,->,>=latex] (1.600000000,1.000000000) -- (2.225000000,1.000000000);
-\draw [,->,>=latex] (2.333333333,-1.000000000) -- (2.761904762,-1.000000000);
-\draw [,->,>=latex] (2.333333333,-0.6666666667) -- (2.761904762,-0.6666666667);
-\draw [,->,>=latex] (2.333333333,-0.3333333333) -- (2.761904762,-0.3333333333);
-\draw [,->,>=latex] (2.333333333,0) -- (2.761904762,0);
-\draw [,->,>=latex] (2.333333333,0.3333333333) -- (2.761904762,0.3333333333);
-\draw [,->,>=latex] (2.333333333,0.6666666667) -- (2.761904762,0.6666666667);
-\draw [,->,>=latex] (2.333333333,1.000000000) -- (2.761904762,1.000000000);
-\draw [,->,>=latex] (3.066666667,-1.000000000) -- (3.392753623,-1.000000000);
-\draw [,->,>=latex] (3.066666667,-0.6666666667) -- (3.392753623,-0.6666666667);
-\draw [,->,>=latex] (3.066666667,-0.3333333333) -- (3.392753623,-0.3333333333);
-\draw [,->,>=latex] (3.066666667,0) -- (3.392753623,0);
-\draw [,->,>=latex] (3.066666667,0.3333333333) -- (3.392753623,0.3333333333);
-\draw [,->,>=latex] (3.066666667,0.6666666667) -- (3.392753623,0.6666666667);
-\draw [,->,>=latex] (3.066666667,1.000000000) -- (3.392753623,1.000000000);
-\draw [,->,>=latex] (3.800000000,-1.000000000) -- (4.063157895,-1.000000000);
-\draw [,->,>=latex] (3.800000000,-0.6666666667) -- (4.063157895,-0.6666666667);
-\draw [,->,>=latex] (3.800000000,-0.3333333333) -- (4.063157895,-0.3333333333);
-\draw [,->,>=latex] (3.800000000,0) -- (4.063157895,0);
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-\draw [,->,>=latex] (3.800000000,0.6666666667) -- (4.063157895,0.6666666667);
-\draw [,->,>=latex] (3.800000000,1.000000000) -- (4.063157895,1.000000000);
-\draw [,->,>=latex] (4.533333333,-1.000000000) -- (4.753921569,-1.000000000);
-\draw [,->,>=latex] (4.533333333,-0.6666666667) -- (4.753921569,-0.6666666667);
-\draw [,->,>=latex] (4.533333333,-0.3333333333) -- (4.753921569,-0.3333333333);
-\draw [,->,>=latex] (4.533333333,0) -- (4.753921569,0);
-\draw [,->,>=latex] (4.533333333,0.3333333333) -- (4.753921569,0.3333333333);
-\draw [,->,>=latex] (4.533333333,0.6666666667) -- (4.753921569,0.6666666667);
-\draw [,->,>=latex] (4.533333333,1.000000000) -- (4.753921569,1.000000000);
-\draw [,->,>=latex] (5.266666667,-1.000000000) -- (5.456540084,-1.000000000);
-\draw [,->,>=latex] (5.266666667,-0.6666666667) -- (5.456540084,-0.6666666667);
-\draw [,->,>=latex] (5.266666667,-0.3333333333) -- (5.456540084,-0.3333333333);
-\draw [,->,>=latex] (5.266666667,0) -- (5.456540084,0);
-\draw [,->,>=latex] (5.266666667,0.3333333333) -- (5.456540084,0.3333333333);
-\draw [,->,>=latex] (5.266666667,0.6666666667) -- (5.456540084,0.6666666667);
-\draw [,->,>=latex] (5.266666667,1.000000000) -- (5.456540084,1.000000000);
-\draw [,->,>=latex] (6.000000000,-1.000000000) -- (6.166666667,-1.000000000);
-\draw [,->,>=latex] (6.000000000,-0.6666666667) -- (6.166666667,-0.6666666667);
-\draw [,->,>=latex] (6.000000000,-0.3333333333) -- (6.166666667,-0.3333333333);
-\draw [,->,>=latex] (6.000000000,0) -- (6.166666667,0);
-\draw [,->,>=latex] (6.000000000,0.3333333333) -- (6.166666667,0.3333333333);
-\draw [,->,>=latex] (6.000000000,0.6666666667) -- (6.166666667,0.6666666667);
-\draw [,->,>=latex] (6.000000000,1.000000000) -- (6.166666667,1.000000000);
+\draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
+\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
+\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
+\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
+\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
+\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
+\draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
+\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
+\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
+\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
+\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
+\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
+\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
+\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
+\draw [,->,>=latex] (3.0667,-0.33333) -- (3.3928,-0.33333);
+\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
+\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
+\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
+\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
+\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
+\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
+\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
+\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
+\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
+\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
+\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
+\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
+\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
+\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
+\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
+\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
+\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
+\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
+\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
+\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
+\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
+\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
+\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks b/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks
index 7abd61204..40c268f53 100644
--- a/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks
+++ b/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks
@@ -44,12 +44,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3149246667) node {\(1\)};
+\draw (0,0.31492) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -89,12 +89,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.3149246667) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.31492) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -134,12 +134,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.3149246667) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.31492) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -179,12 +179,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw (3.232671190,-0.2436539771) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw (3.2327,-0.24365) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -224,12 +224,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw (3.232671190,0.7563460229) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw (3.2327,0.75635) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks b/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks
index 5b742a721..569d4db5b 100644
--- a/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks
+++ b/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2644441667) node {\( 2\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.26444) node {\( 2\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks b/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks
index b6c9314d7..ad2371f2a 100644
--- a/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks
+++ b/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks
@@ -95,27 +95,27 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,0) -- (8.00,2.00);
-\draw []  (1.600000000,0.4000000000) node [rotate=0] {$\bullet$};
-\draw (1.744974792,0.1273953333) node {\( a\)};
-\draw []  (4.800000000,1.200000000) node [rotate=0] {$\bullet$};
-\draw (4.926829958,0.8792355000) node {\( b\)};
-\draw []  (6.400000000,1.600000000) node [rotate=0] {$\bullet$};
-\draw (6.527485125,1.327395333) node {\( c\)};
-\draw []  (4.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (4.152810792,0.7273953333) node {\( x\)};
+\draw []  (1.6000,0.40000) node [rotate=0] {$\bullet$};
+\draw (1.7450,0.12740) node {\( a\)};
+\draw []  (4.8000,1.2000) node [rotate=0] {$\bullet$};
+\draw (4.9268,0.87924) node {\( b\)};
+\draw []  (6.4000,1.6000) node [rotate=0] {$\bullet$};
+\draw (6.5275,1.3274) node {\( c\)};
+\draw []  (4.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (4.1528,0.72740) node {\( x\)};
 \draw [] (1.60,0.400) -- (2.60,3.40);
 \draw [] (4.80,1.20) -- (2.60,3.40);
-\draw []  (2.200000000,2.200000000) node [rotate=0] {$\bullet$};
-\draw (1.918443007,2.377307720) node {\( p\)};
-\draw []  (4.066666667,1.933333333) node [rotate=0] {$\bullet$};
-\draw (4.335311801,2.133267258) node {\( q\)};
+\draw []  (2.2000,2.2000) node [rotate=0] {$\bullet$};
+\draw (1.9184,2.3773) node {\( p\)};
+\draw []  (4.0667,1.9333) node [rotate=0] {$\bullet$};
+\draw (4.3353,2.1333) node {\( q\)};
 \draw [] (6.40,1.60) -- (0.940,2.38);
-\draw []  (2.600000000,3.400000000) node [rotate=0] {$\bullet$};
-\draw (2.881727355,3.637447733) node {\( m\)};
+\draw []  (2.6000,3.4000) node [rotate=0] {$\bullet$};
+\draw (2.8817,3.6374) node {\( m\)};
 \draw [] (4.07,1.93) -- (1.60,0.400);
 \draw [] (2.20,2.20) -- (4.80,1.20);
-\draw []  (3.618181818,1.654545455) node [rotate=0] {$\bullet$};
-\draw (3.326753965,1.492795736) node {\( n\)};
+\draw []  (3.6182,1.6545) node [rotate=0] {$\bullet$};
+\draw (3.3268,1.4928) node {\( n\)};
 \draw [] (2.60,3.40) -- (4.00,1.00);
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks b/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks
index 3a2443411..61e7d4c9d 100644
--- a/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks
+++ b/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks
@@ -83,15 +83,15 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.3247080000) node {\( A\)};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.289005356,0.2661293562) node {\( B\)};
-\draw []  (12.00000000,-3.000000000) node [rotate=0] {$\bullet$};
-\draw (12.28491136,-3.266129356) node {\( C\)};
-\draw []  (4.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.690526810,-1.266129356) node {\( K\)};
-\draw []  (1.333333333,0) node [rotate=0] {$\bullet$};
-\draw (1.333333333,0.3247080000) node {\( L\)};
+\draw (0,0.32471) node {\( A\)};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.2890,0.26613) node {\( B\)};
+\draw []  (12.000,-3.0000) node [rotate=0] {$\bullet$};
+\draw (12.285,-3.2661) node {\( C\)};
+\draw []  (4.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (3.6905,-1.2661) node {\( K\)};
+\draw []  (1.3333,0) node [rotate=0] {$\bullet$};
+\draw (1.3333,0.32471) node {\( L\)};
 \draw [] (0,0) -- (4.00,0);
 \draw [] (0,0) -- (12.0,-3.00);
 \draw [style=dashed] (12.0,-3.00) -- (4.00,0);
diff --git a/auto/pictures_tex/Fig_EJRsWXw.pstricks b/auto/pictures_tex/Fig_EJRsWXw.pstricks
index 810f409ea..c26000343 100644
--- a/auto/pictures_tex/Fig_EJRsWXw.pstricks
+++ b/auto/pictures_tex/Fig_EJRsWXw.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -88,10 +88,10 @@
 \draw [color=green] (-1.00,1.00) -- (-2.00,2.00);
 \draw [color=green] (-1.00,1.00) -- (1.00,3.00);
 \draw [color=red] (-2.00,2.00) -- (1.00,3.00);
-\draw (1.500000000,-0.2785761667) node {\( x\)};
-\draw (1.500000000,-0.2785761667) node {\( x\)};
-\draw (0.2659030000,3.500000000) node {\( t\)};
-\draw (0.2659030000,3.500000000) node {\( t\)};
+\draw (1.5000,-0.27858) node {\( x\)};
+\draw (1.5000,-0.27858) node {\( x\)};
+\draw (0.26590,3.5000) node {\( t\)};
+\draw (0.26590,3.5000) node {\( t\)};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
index 5f13b244a..31c8bb72e 100644
--- a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
+++ b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
@@ -77,35 +77,35 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (2.086833187,1.759539037) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.410240275,2.935347274);
-\draw []  (1.735039751,1.464349294) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246913);
-\draw []  (2.657517092,2.437667613) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.247852509,3.098210238);
-\draw []  (0.9681152512,1.020091151) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.2150970834,0.5534262785);
-\draw []  (2.853656194,2.793571599) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.155827251,3.175606475);
-\draw []  (0.5835590482,0.8887339218) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.1548566875,0.6574931816);
-\draw []  (3.013566327,3.308802930) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.999711929,3.287338346);
-\draw []  (0.04838713566,0.8207221648) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.07193141522,0.8306387974);
-\draw []  (2.896085542,4.007090670) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.671786357,3.458304371);
-\draw []  (-0.6188917196,1.057674546) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357358,1.183270382);
+\draw []  (2.0868,1.7595) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.4102,2.9353);
+\draw []  (1.7350,1.4643) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.34725,0.36521);
+\draw []  (2.6575,2.4377) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.2478,3.0982);
+\draw []  (0.96811,1.0201) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.21509,0.55343);
+\draw []  (2.8537,2.7936) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.1558,3.1756);
+\draw []  (0.58356,0.88873) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.15485,0.65749);
+\draw []  (3.0136,3.3088) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.9997,3.2873);
+\draw []  (0.048375,0.82073) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.071918,0.83064);
+\draw []  (2.8961,4.0071) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.6718,3.4583);
+\draw []  (-0.61890,1.0577) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (-0.039497,1.1833);
 \draw [] (-3.00,0) -- (3.00,0);
 \draw [] (0,0) -- (3.83,3.21);
-\draw (0.7745541682,0.2495862383) node {\( \alpha\)};
+\draw (0.77455,0.24959) node {\( \alpha\)};
 
 \draw [] (0.500,0)--(0.500,0.00353)--(0.500,0.00705)--(0.500,0.0106)--(0.500,0.0141)--(0.500,0.0176)--(0.500,0.0211)--(0.499,0.0247)--(0.499,0.0282)--(0.499,0.0317)--(0.499,0.0352)--(0.499,0.0387)--(0.498,0.0423)--(0.498,0.0458)--(0.498,0.0493)--(0.497,0.0528)--(0.497,0.0563)--(0.496,0.0598)--(0.496,0.0633)--(0.496,0.0668)--(0.495,0.0703)--(0.495,0.0738)--(0.494,0.0773)--(0.493,0.0807)--(0.493,0.0842)--(0.492,0.0877)--(0.492,0.0912)--(0.491,0.0946)--(0.490,0.0981)--(0.490,0.102)--(0.489,0.105)--(0.488,0.108)--(0.487,0.112)--(0.487,0.115)--(0.486,0.119)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.132)--(0.481,0.136)--(0.480,0.139)--(0.479,0.143)--(0.478,0.146)--(0.477,0.149)--(0.476,0.153)--(0.475,0.156)--(0.474,0.159)--(0.473,0.163)--(0.472,0.166)--(0.470,0.169)--(0.469,0.173)--(0.468,0.176)--(0.467,0.179)--(0.465,0.183)--(0.464,0.186)--(0.463,0.189)--(0.462,0.192)--(0.460,0.196)--(0.459,0.199)--(0.457,0.202)--(0.456,0.205)--(0.454,0.209)--(0.453,0.212)--(0.451,0.215)--(0.450,0.218)--(0.448,0.221)--(0.447,0.224)--(0.445,0.228)--(0.444,0.231)--(0.442,0.234)--(0.440,0.237)--(0.439,0.240)--(0.437,0.243)--(0.435,0.246)--(0.433,0.249)--(0.432,0.252)--(0.430,0.255)--(0.428,0.258)--(0.426,0.261)--(0.424,0.264)--(0.423,0.267)--(0.421,0.270)--(0.419,0.273)--(0.417,0.276)--(0.415,0.279)--(0.413,0.282)--(0.411,0.285)--(0.409,0.288)--(0.407,0.291)--(0.405,0.294)--(0.403,0.296)--(0.401,0.299)--(0.398,0.302)--(0.396,0.305)--(0.394,0.308)--(0.392,0.310)--(0.390,0.313)--(0.388,0.316)--(0.385,0.319)--(0.383,0.321);
-\draw []  (1.915111108,1.606969024) node [rotate=0] {$\bullet$};
-\draw (2.186183463,1.329052136) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.38)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.456,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.821)(0.211,0.825)(0.323,0.838)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
-\draw (4.139119438,3.522532237) node {\( \ell_p\)};
+\draw []  (1.9151,1.6070) node [rotate=0] {$\bullet$};
+\draw (2.1862,1.3291) node {\( P\)};
+\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw (4.1391,3.5225) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExPolygone.pstricks b/auto/pictures_tex/Fig_ExPolygone.pstricks
index 9de4be9a0..c729554c1 100644
--- a/auto/pictures_tex/Fig_ExPolygone.pstricks
+++ b/auto/pictures_tex/Fig_ExPolygone.pstricks
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.650000000,0) -- (3.650000000,0);
-\draw [,->,>=latex] (0,-2.650000000) -- (0,2.650000000);
+\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
+\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
 %DEFAULT
 \draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
 \draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
@@ -102,21 +102,21 @@
 \draw [color=blue] (2.00,0) -- (1.00,-1.00);
 \draw [color=blue] (1.00,-1.00) -- (0,0);
 \draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExSinLarge.pstricks b/auto/pictures_tex/Fig_ExSinLarge.pstricks
index 8cdc23e1e..306d962f8 100644
--- a/auto/pictures_tex/Fig_ExSinLarge.pstricks
+++ b/auto/pictures_tex/Fig_ExSinLarge.pstricks
@@ -75,25 +75,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.641592654,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499874128);
+\draw [,->,>=latex] (-0.50000,0) -- (3.6416,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4999);
 %DEFAULT
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-\draw (3.000000000,-0.3149246667) node {$ 3 $};
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-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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-\draw (-0.2912498333,2.000000000) node {$ 2 $};
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diff --git a/auto/pictures_tex/Fig_ExampleIntegration.pstricks b/auto/pictures_tex/Fig_ExampleIntegration.pstricks
index f5c2ea7a1..63012a8e2 100644
--- a/auto/pictures_tex/Fig_ExampleIntegration.pstricks
+++ b/auto/pictures_tex/Fig_ExampleIntegration.pstricks
@@ -79,8 +79,8 @@
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diff --git a/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks b/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks
index d674e887c..8e4adfdc0 100644
--- a/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks
+++ b/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks
@@ -103,8 +103,8 @@
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 % declaring the keys in tikz
@@ -136,41 +136,41 @@
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-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExempleArcParam.pstricks b/auto/pictures_tex/Fig_ExempleArcParam.pstricks
index 0ff051d15..3f874e1c9 100644
--- a/auto/pictures_tex/Fig_ExempleArcParam.pstricks
+++ b/auto/pictures_tex/Fig_ExempleArcParam.pstricks
@@ -65,29 +65,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.899823779,0) -- (1.899823779,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.898229715);
+\draw [,->,>=latex] (-1.8998,0) -- (1.8998,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.8982);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.08879,0.04443)--(0.1772,0.08885)--(0.2650,0.1333)--(0.3516,0.1777)--(0.4368,0.2221)--(0.5203,0.2666)--(0.6017,0.3110)--(0.6807,0.3554)--(0.7569,0.3998)--(0.8301,0.4443)--(0.8999,0.4887)--(0.9661,0.5331)--(1.028,0.5775)--(1.087,0.6220)--(1.140,0.6664)--(1.190,0.7108)--(1.234,0.7552)--(1.273,0.7997)--(1.308,0.8441)--(1.337,0.8885)--(1.361,0.9330)--(1.379,0.9774)--(1.391,1.022)--(1.398,1.066)--(1.400,1.111)--(1.396,1.155)--(1.386,1.200)--(1.370,1.244)--(1.349,1.288)--(1.323,1.333)--(1.291,1.377)--(1.254,1.422)--(1.212,1.466)--(1.166,1.510)--(1.114,1.555)--(1.058,1.599)--(0.9978,1.644)--(0.9335,1.688)--(0.8654,1.733)--(0.7939,1.777)--(0.7191,1.821)--(0.6415,1.866)--(0.5613,1.910)--(0.4788,1.955)--(0.3944,1.999)--(0.3084,2.044)--(0.2212,2.088)--(0.1331,2.132)--(0.04442,2.177)--(-0.04442,2.221)--(-0.1331,2.266)--(-0.2212,2.310)--(-0.3084,2.355)--(-0.3944,2.399)--(-0.4788,2.443)--(-0.5613,2.488)--(-0.6415,2.532)--(-0.7191,2.577)--(-0.7939,2.621)--(-0.8654,2.666)--(-0.9335,2.710)--(-0.9978,2.754)--(-1.058,2.799)--(-1.114,2.843)--(-1.166,2.888)--(-1.212,2.932)--(-1.254,2.977)--(-1.291,3.021)--(-1.323,3.065)--(-1.349,3.110)--(-1.370,3.154)--(-1.386,3.199)--(-1.396,3.243)--(-1.400,3.288)--(-1.398,3.332)--(-1.391,3.376)--(-1.379,3.421)--(-1.361,3.465)--(-1.337,3.510)--(-1.308,3.554)--(-1.273,3.599)--(-1.234,3.643)--(-1.190,3.687)--(-1.140,3.732)--(-1.087,3.776)--(-1.028,3.821)--(-0.9661,3.865)--(-0.8999,3.910)--(-0.8301,3.954)--(-0.7569,3.998)--(-0.6807,4.043)--(-0.6017,4.087)--(-0.5203,4.132)--(-0.4368,4.176)--(-0.3516,4.221)--(-0.2650,4.265)--(-0.1772,4.309)--(-0.08879,4.354)--(0,4.398);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.2912498333,4.200000000) node {$ 6 $};
+\draw (-0.29125,4.2000) node {$ 6 $};
 \draw [] (-0.100,4.20) -- (0.100,4.20);
 %OTHER STUFF
 %END PSPICTURE
@@ -126,17 +126,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.999811192,0) -- (1.999811192,0);
-\draw [,->,>=latex] (0,-1.999811192) -- (0,1.999811192);
+\draw [,->,>=latex] (-1.9998,0) -- (1.9998,0);
+\draw [,->,>=latex] (0,-1.9998) -- (0,1.9998);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.190,-0.0951)--(0.377,-0.190)--(0.557,-0.284)--(0.729,-0.377)--(0.889,-0.468)--(1.04,-0.557)--(1.16,-0.645)--(1.27,-0.729)--(1.36,-0.811)--(1.43,-0.889)--(1.48,-0.964)--(1.50,-1.04)--(1.50,-1.10)--(1.47,-1.16)--(1.42,-1.22)--(1.34,-1.27)--(1.25,-1.32)--(1.13,-1.36)--(1.00,-1.40)--(0.851,-1.43)--(0.687,-1.46)--(0.513,-1.48)--(0.330,-1.49)--(0.143,-1.50)--(-0.0476,-1.50)--(-0.237,-1.50)--(-0.423,-1.48)--(-0.601,-1.47)--(-0.771,-1.45)--(-0.927,-1.42)--(-1.07,-1.38)--(-1.19,-1.34)--(-1.30,-1.30)--(-1.38,-1.25)--(-1.45,-1.19)--(-1.48,-1.13)--(-1.50,-1.07)--(-1.49,-1.00)--(-1.46,-0.927)--(-1.40,-0.851)--(-1.32,-0.771)--(-1.22,-0.687)--(-1.10,-0.601)--(-0.964,-0.513)--(-0.811,-0.423)--(-0.645,-0.330)--(-0.468,-0.237)--(-0.284,-0.143)--(-0.0951,-0.0476)--(0.0951,0.0476)--(0.284,0.143)--(0.468,0.237)--(0.645,0.330)--(0.811,0.423)--(0.964,0.513)--(1.10,0.601)--(1.22,0.687)--(1.32,0.771)--(1.40,0.851)--(1.46,0.927)--(1.49,1.00)--(1.50,1.07)--(1.48,1.13)--(1.45,1.19)--(1.38,1.25)--(1.30,1.30)--(1.19,1.34)--(1.07,1.38)--(0.927,1.42)--(0.771,1.45)--(0.601,1.47)--(0.423,1.48)--(0.237,1.50)--(0.0476,1.50)--(-0.143,1.50)--(-0.330,1.49)--(-0.513,1.48)--(-0.687,1.46)--(-0.851,1.43)--(-1.00,1.40)--(-1.13,1.36)--(-1.25,1.32)--(-1.34,1.27)--(-1.42,1.22)--(-1.47,1.16)--(-1.50,1.10)--(-1.50,1.04)--(-1.48,0.964)--(-1.43,0.889)--(-1.36,0.811)--(-1.27,0.729)--(-1.16,0.645)--(-1.04,0.557)--(-0.889,0.468)--(-0.729,0.377)--(-0.557,0.284)--(-0.377,0.190)--(-0.190,0.0951)--(0,0);
-\draw (-1.500000000,-0.3298256667) node {$ -1 $};
+\draw (-1.5000,-0.32983) node {$ -1 $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (1.500000000,-0.3149246667) node {$ 1 $};
+\draw (1.5000,-0.31492) node {$ 1 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (-0.4331593333,-1.500000000) node {$ -1 $};
+\draw (-0.43316,-1.5000) node {$ -1 $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.2912498333,1.500000000) node {$ 1 $};
+\draw (-0.29125,1.5000) node {$ 1 $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExempleNonRang.pstricks b/auto/pictures_tex/Fig_ExempleNonRang.pstricks
index 446d77124..89185f4ea 100644
--- a/auto/pictures_tex/Fig_ExempleNonRang.pstricks
+++ b/auto/pictures_tex/Fig_ExempleNonRang.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.328427125) -- (0,3.328427125);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.3284) -- (0,3.3284);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.04040,0.002871)--(0.08081,0.008121)--(0.1212,0.01492)--(0.1616,0.02297)--(0.2020,0.03210)--(0.2424,0.04220)--(0.2828,0.05318)--(0.3232,0.06497)--(0.3636,0.07753)--(0.4040,0.09080)--(0.4444,0.1048)--(0.4848,0.1194)--(0.5253,0.1346)--(0.5657,0.1504)--(0.6061,0.1668)--(0.6465,0.1838)--(0.6869,0.2013)--(0.7273,0.2193)--(0.7677,0.2378)--(0.8081,0.2568)--(0.8485,0.2763)--(0.8889,0.2963)--(0.9293,0.3167)--(0.9697,0.3376)--(1.010,0.3589)--(1.051,0.3807)--(1.091,0.4028)--(1.131,0.4254)--(1.172,0.4484)--(1.212,0.4718)--(1.253,0.4956)--(1.293,0.5198)--(1.333,0.5443)--(1.374,0.5693)--(1.414,0.5946)--(1.455,0.6202)--(1.495,0.6462)--(1.535,0.6726)--(1.576,0.6993)--(1.616,0.7264)--(1.657,0.7538)--(1.697,0.7816)--(1.737,0.8096)--(1.778,0.8381)--(1.818,0.8668)--(1.859,0.8958)--(1.899,0.9252)--(1.939,0.9549)--(1.980,0.9849)--(2.020,1.015)--(2.061,1.046)--(2.101,1.077)--(2.141,1.108)--(2.182,1.139)--(2.222,1.171)--(2.263,1.203)--(2.303,1.236)--(2.343,1.268)--(2.384,1.301)--(2.424,1.335)--(2.465,1.368)--(2.505,1.402)--(2.545,1.436)--(2.586,1.470)--(2.626,1.505)--(2.667,1.540)--(2.707,1.575)--(2.747,1.610)--(2.788,1.646)--(2.828,1.682)--(2.869,1.718)--(2.909,1.754)--(2.949,1.791)--(2.990,1.828)--(3.030,1.865)--(3.071,1.902)--(3.111,1.940)--(3.152,1.978)--(3.192,2.016)--(3.232,2.055)--(3.273,2.093)--(3.313,2.132)--(3.354,2.171)--(3.394,2.211)--(3.434,2.250)--(3.475,2.290)--(3.515,2.330)--(3.556,2.370)--(3.596,2.411)--(3.636,2.452)--(3.677,2.493)--(3.717,2.534)--(3.758,2.575)--(3.798,2.617)--(3.838,2.659)--(3.879,2.701)--(3.919,2.743)--(3.960,2.786)--(4.000,2.828);
diff --git a/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
index f7088ee5d..637a3a657 100644
--- a/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
+++ b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
@@ -65,30 +65,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.200000000) -- (0,1.199647580);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.2000) -- (0,1.1996);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -151,28 +151,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -239,34 +239,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796328,0) -- (2.856194490,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.499748271);
+\draw [,->,>=latex] (-2.0708,0) -- (2.8562,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4997);
 %DEFAULT
 
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-\draw (-2.000000000,-0.3298256667) node {$ -4 $};
+\draw (-2.0000,-0.32983) node {$ -4 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.500000000,-0.3298256667) node {$ -3 $};
+\draw (-1.5000,-0.32983) node {$ -3 $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -2 $};
+\draw (-1.0000,-0.32983) node {$ -2 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.5000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.50000,-0.32983) node {$ -1 $};
 \draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.5000000000,-0.3149246667) node {$ 1 $};
+\draw (0.50000,-0.31492) node {$ 1 $};
 \draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 2 $};
+\draw (1.0000,-0.31492) node {$ 2 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.500000000,-0.3149246667) node {$ 3 $};
+\draw (1.5000,-0.31492) node {$ 3 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 4 $};
+\draw (2.0000,-0.31492) node {$ 4 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.500000000,-0.3149246667) node {$ 5 $};
+\draw (2.5000,-0.31492) node {$ 5 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (-0.4331593333,-0.5000000000) node {$ -1 $};
+\draw (-0.43316,-0.50000) node {$ -1 $};
 \draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.2912498333,0.5000000000) node {$ 1 $};
+\draw (-0.29125,0.50000) node {$ 1 $};
 \draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -329,30 +329,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -415,31 +415,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.199998598) -- (0,1.199904439);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.2000) -- (0,1.1999);
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+\draw (2.8000,-0.31492) node {$ 4 $};
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-\draw (3.500000000,-0.3149246667) node {$ 5 $};
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 %OTHER STUFF
 %END PSPICTURE
@@ -502,30 +502,30 @@
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 %AXES
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-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
+\draw [,->,>=latex] (-1.5996,0) -- (4.8982,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
 %DEFAULT
 
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-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (4.200000000,-0.3149246667) node {$ 6 $};
+\draw (4.2000,-0.31492) node {$ 6 $};
 \draw [] (4.20,-0.100) -- (4.20,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -588,30 +588,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
 %DEFAULT
 
 \draw [color=blue] (-2.199,0.7000)--(-2.144,0.6650)--(-2.088,0.5637)--(-2.033,0.4060)--(-1.977,0.2078)--(-1.921,-0.01111)--(-1.866,-0.2289)--(-1.810,-0.4239)--(-1.755,-0.5766)--(-1.699,-0.6716)--(-1.644,-0.6996)--(-1.588,-0.6578)--(-1.533,-0.5502)--(-1.477,-0.3877)--(-1.422,-0.1865)--(-1.366,0.03331)--(-1.311,0.2498)--(-1.255,0.4414)--(-1.200,0.5889)--(-1.144,0.6776)--(-1.088,0.6986)--(-1.033,0.6499)--(-0.9774,0.5362)--(-0.9219,0.3691)--(-0.8663,0.1650)--(-0.8108,-0.05547)--(-0.7552,-0.2704)--(-0.6997,-0.4584)--(-0.6442,-0.6006)--(-0.5887,-0.6828)--(-0.5331,-0.6968)--(-0.4776,-0.6413)--(-0.4221,-0.5217)--(-0.3665,-0.3500)--(-0.3110,-0.1434)--(-0.2555,0.07759)--(-0.1999,0.2908)--(-0.1444,0.4750)--(-0.08885,0.6117)--(-0.03332,0.6873)--(0.02221,0.6944)--(0.07775,0.6320)--(0.1333,0.5066)--(0.1888,0.3306)--(0.2443,0.1216)--(0.2999,-0.09962)--(0.3554,-0.3108)--(0.4109,-0.4910)--(0.4665,-0.6222)--(0.5220,-0.6912)--(0.5775,-0.6912)--(0.6331,-0.6222)--(0.6886,-0.4910)--(0.7441,-0.3108)--(0.7997,-0.09962)--(0.8552,0.1216)--(0.9107,0.3306)--(0.9663,0.5066)--(1.022,0.6320)--(1.077,0.6944)--(1.133,0.6873)--(1.188,0.6117)--(1.244,0.4750)--(1.299,0.2908)--(1.355,0.07759)--(1.411,-0.1434)--(1.466,-0.3500)--(1.522,-0.5217)--(1.577,-0.6413)--(1.633,-0.6968)--(1.688,-0.6828)--(1.744,-0.6006)--(1.799,-0.4584)--(1.855,-0.2704)--(1.910,-0.05547)--(1.966,0.1650)--(2.021,0.3691)--(2.077,0.5362)--(2.132,0.6499)--(2.188,0.6986)--(2.244,0.6776)--(2.299,0.5889)--(2.355,0.4414)--(2.410,0.2498)--(2.466,0.03331)--(2.521,-0.1865)--(2.577,-0.3877)--(2.632,-0.5502)--(2.688,-0.6578)--(2.743,-0.6996)--(2.799,-0.6716)--(2.854,-0.5766)--(2.910,-0.4239)--(2.965,-0.2289)--(3.021,-0.01111)--(3.077,0.2078)--(3.132,0.4060)--(3.188,0.5637)--(3.243,0.6650)--(3.299,0.7000);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoMagnetique.pstricks b/auto/pictures_tex/Fig_ExoMagnetique.pstricks
index fe440b24b..6466d25a8 100644
--- a/auto/pictures_tex/Fig_ExoMagnetique.pstricks
+++ b/auto/pictures_tex/Fig_ExoMagnetique.pstricks
@@ -78,12 +78,12 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [style=dashed] (0,-2.00) -- (0,2.00);
-\draw [color=red,->,>=latex] (0,0) -- (0,1.000000000);
-\draw (0.3945450000,1.000000000) node {$I$};
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-2.000000000,1.000000000);
-\draw (-1.763589810,0.7318426438) node {$d$};
-\draw []  (-2.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-2.706459690,1.323876356) node {$(r,\theta,z)$};
+\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
+\draw (0.39455,1.0000) node {$I$};
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-2.0000,1.0000);
+\draw (-1.7636,0.73184) node {$d$};
+\draw []  (-2.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-2.7065,1.3239) node {$(r,\theta,z)$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoPolaire.pstricks b/auto/pictures_tex/Fig_ExoPolaire.pstricks
index 853914bfa..6980c9bb4 100644
--- a/auto/pictures_tex/Fig_ExoPolaire.pstricks
+++ b/auto/pictures_tex/Fig_ExoPolaire.pstricks
@@ -75,15 +75,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.232050808,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.2321,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
 %DEFAULT
-\draw (2.389616308,1.000000000) node {$(\sqrt{3},1)$};
-\draw (0.6579799038,0.8865436211) node {$l$};
-\draw (1.012732295,0.2561455226) node {$\theta$};
+\draw (2.3896,1.0000) node {$(\sqrt{3},1)$};
+\draw (0.65798,0.88654) node {$l$};
+\draw (1.0127,0.25615) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw [,->,>=latex] (0,0) -- (1.732050808,1.000000000);
+\draw [,->,>=latex] (0,0) -- (1.7320,1.0000);
 \draw [] (1.00,-0.100) -- (1.00,0.100);
 \draw [] (2.00,-0.100) -- (2.00,0.100);
 \draw [] (-0.100,1.00) -- (0.100,1.00);
diff --git a/auto/pictures_tex/Fig_ExoProjection.pstricks b/auto/pictures_tex/Fig_ExoProjection.pstricks
index 2c166c030..7d7fed300 100644
--- a/auto/pictures_tex/Fig_ExoProjection.pstricks
+++ b/auto/pictures_tex/Fig_ExoProjection.pstricks
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.600000000,0) -- (3.300000000,0);
-\draw [,->,>=latex] (0,-1.550000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-2.6000,0) -- (3.3000,0);
+\draw [,->,>=latex] (0,-1.5500) -- (0,3.5000);
 %DEFAULT
 \draw [style=dashed] (2.80,1.40) -- (-2.10,-1.05);
-\draw []  (2.400000000,1.200000000) node [rotate=0] {$\bullet$};
-\draw (3.246325678,0.9435588219) node {$\pr_w(A)$};
-\draw [color=red,->,>=latex] (1.500000000,3.000000000) -- (2.400000000,1.200000000);
-\draw [color=blue,->,>=latex] (0,0) -- (1.400000000,0.7000000000);
-\draw (1.071598050,1.008389500) node {$w$};
-\draw []  (-1.000000000,-0.5000000000) node [rotate=0] {$\bullet$};
-\draw (-1.390882667,-0.7824550000) node {$P(\lambda)$};
-\draw [color=cyan,->,>=latex] (1.500000000,3.000000000) -- (-1.000000000,-0.5000000000);
-\draw []  (1.500000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (1.500000000,3.424708000) node {$A$};
+\draw []  (2.4000,1.2000) node [rotate=0] {$\bullet$};
+\draw (3.2463,0.94356) node {$\pr_w(A)$};
+\draw [color=red,->,>=latex] (1.5000,3.0000) -- (2.4000,1.2000);
+\draw [color=blue,->,>=latex] (0,0) -- (1.4000,0.70000);
+\draw (1.0716,1.0084) node {$w$};
+\draw []  (-1.0000,-0.50000) node [rotate=0] {$\bullet$};
+\draw (-1.3909,-0.78246) node {$P(\lambda)$};
+\draw [color=cyan,->,>=latex] (1.5000,3.0000) -- (-1.0000,-0.50000);
+\draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
+\draw (1.5000,3.4247) node {$A$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoVarj.pstricks b/auto/pictures_tex/Fig_ExoVarj.pstricks
index 5fc4b4037..9f183c7db 100644
--- a/auto/pictures_tex/Fig_ExoVarj.pstricks
+++ b/auto/pictures_tex/Fig_ExoVarj.pstricks
@@ -71,16 +71,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-1.999875734) -- (0,1.999875734);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.9999) -- (0,1.9999);
 %DEFAULT
 
 \draw [color=blue] (-3.00,1.46)--(-2.94,1.47)--(-2.88,1.48)--(-2.82,1.49)--(-2.76,1.49)--(-2.70,1.50)--(-2.64,1.50)--(-2.58,1.50)--(-2.52,1.50)--(-2.45,1.49)--(-2.39,1.49)--(-2.33,1.48)--(-2.27,1.47)--(-2.21,1.46)--(-2.15,1.45)--(-2.09,1.44)--(-2.03,1.42)--(-1.97,1.41)--(-1.91,1.39)--(-1.85,1.37)--(-1.79,1.34)--(-1.73,1.32)--(-1.67,1.29)--(-1.61,1.27)--(-1.55,1.24)--(-1.48,1.21)--(-1.42,1.18)--(-1.36,1.14)--(-1.30,1.11)--(-1.24,1.07)--(-1.18,1.03)--(-1.12,0.990)--(-1.06,0.947)--(-1.00,0.904)--(-0.939,0.858)--(-0.879,0.812)--(-0.818,0.763)--(-0.758,0.713)--(-0.697,0.662)--(-0.636,0.609)--(-0.576,0.556)--(-0.515,0.501)--(-0.455,0.444)--(-0.394,0.387)--(-0.333,0.329)--(-0.273,0.271)--(-0.212,0.211)--(-0.152,0.151)--(-0.0909,0.0908)--(-0.0303,0.0303)--(0.0303,0.0303)--(0.0909,0.0908)--(0.152,0.151)--(0.212,0.211)--(0.273,0.271)--(0.333,0.329)--(0.394,0.387)--(0.455,0.444)--(0.515,0.501)--(0.576,0.556)--(0.636,0.609)--(0.697,0.662)--(0.758,0.713)--(0.818,0.763)--(0.879,0.812)--(0.939,0.858)--(1.00,0.904)--(1.06,0.947)--(1.12,0.990)--(1.18,1.03)--(1.24,1.07)--(1.30,1.11)--(1.36,1.14)--(1.42,1.18)--(1.48,1.21)--(1.55,1.24)--(1.61,1.27)--(1.67,1.29)--(1.73,1.32)--(1.79,1.34)--(1.85,1.37)--(1.91,1.39)--(1.97,1.41)--(2.03,1.42)--(2.09,1.44)--(2.15,1.45)--(2.21,1.46)--(2.27,1.47)--(2.33,1.48)--(2.39,1.49)--(2.45,1.49)--(2.52,1.50)--(2.58,1.50)--(2.64,1.50)--(2.70,1.50)--(2.76,1.49)--(2.82,1.49)--(2.88,1.48)--(2.94,1.47)--(3.00,1.46);
 
 \draw [color=red] (-3.00,-1.46)--(-2.94,-1.47)--(-2.88,-1.48)--(-2.82,-1.49)--(-2.76,-1.49)--(-2.70,-1.50)--(-2.64,-1.50)--(-2.58,-1.50)--(-2.52,-1.50)--(-2.45,-1.49)--(-2.39,-1.49)--(-2.33,-1.48)--(-2.27,-1.47)--(-2.21,-1.46)--(-2.15,-1.45)--(-2.09,-1.44)--(-2.03,-1.42)--(-1.97,-1.41)--(-1.91,-1.39)--(-1.85,-1.37)--(-1.79,-1.34)--(-1.73,-1.32)--(-1.67,-1.29)--(-1.61,-1.27)--(-1.55,-1.24)--(-1.48,-1.21)--(-1.42,-1.18)--(-1.36,-1.14)--(-1.30,-1.11)--(-1.24,-1.07)--(-1.18,-1.03)--(-1.12,-0.990)--(-1.06,-0.947)--(-1.00,-0.904)--(-0.939,-0.858)--(-0.879,-0.812)--(-0.818,-0.763)--(-0.758,-0.713)--(-0.697,-0.662)--(-0.636,-0.609)--(-0.576,-0.556)--(-0.515,-0.501)--(-0.455,-0.444)--(-0.394,-0.387)--(-0.333,-0.329)--(-0.273,-0.271)--(-0.212,-0.211)--(-0.152,-0.151)--(-0.0909,-0.0908)--(-0.0303,-0.0303)--(0.0303,-0.0303)--(0.0909,-0.0908)--(0.152,-0.151)--(0.212,-0.211)--(0.273,-0.271)--(0.333,-0.329)--(0.394,-0.387)--(0.455,-0.444)--(0.515,-0.501)--(0.576,-0.556)--(0.636,-0.609)--(0.697,-0.662)--(0.758,-0.713)--(0.818,-0.763)--(0.879,-0.812)--(0.939,-0.858)--(1.00,-0.904)--(1.06,-0.947)--(1.12,-0.990)--(1.18,-1.03)--(1.24,-1.07)--(1.30,-1.11)--(1.36,-1.14)--(1.42,-1.18)--(1.48,-1.21)--(1.55,-1.24)--(1.61,-1.27)--(1.67,-1.29)--(1.73,-1.32)--(1.79,-1.34)--(1.85,-1.37)--(1.91,-1.39)--(1.97,-1.41)--(2.03,-1.42)--(2.09,-1.44)--(2.15,-1.45)--(2.21,-1.46)--(2.27,-1.47)--(2.33,-1.48)--(2.39,-1.49)--(2.45,-1.49)--(2.52,-1.50)--(2.58,-1.50)--(2.64,-1.50)--(2.70,-1.50)--(2.76,-1.49)--(2.82,-1.49)--(2.88,-1.48)--(2.94,-1.47)--(3.00,-1.46);
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoXLVL.pstricks b/auto/pictures_tex/Fig_ExoXLVL.pstricks
index b213578f7..9dd1f83e3 100644
--- a/auto/pictures_tex/Fig_ExoXLVL.pstricks
+++ b/auto/pictures_tex/Fig_ExoXLVL.pstricks
@@ -79,29 +79,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
 \draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
 \draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
 \draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
 \draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red,style=dashed] (0,2.50) -- (2.50,2.50);
-\draw [color=red,style=dashed] (2.50,2.50) -- (2.50,0);
-\draw [color=red,style=dashed] (2.50,0) -- (0,0);
-\draw [color=red,style=dashed] (0,0) -- (0,2.50);
-\draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
-\draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
-\draw [color=cyan] (-2.50,0) -- (0,0);
-\draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green,style=dashed] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
-\draw (-1.099672167,1.300000000) node {\( xy\)};
-\draw (1.673347000,1.250000000) node {\( x-y\)};
-\draw (-0.9705055000,-1.250000000) node {\( x^2y\)};
-\draw (1.723347000,-1.300000000) node {\( x+y\)};
+\draw [color=red] (0,2.50) -- (2.50,2.50);
+\draw [color=red] (2.50,2.50) -- (2.50,0);
+\draw [color=red] (2.50,0) -- (0,0);
+\draw [color=red] (0,0) -- (0,2.50);
+\draw [color=cyan,style=dashed] (-0.100,-2.50) -- (-2.50,-2.50);
+\draw [color=cyan,style=dashed] (-2.50,-2.50) -- (-2.50,-0.100);
+\draw [color=cyan,style=dashed] (-2.50,-0.100) -- (-0.100,-0.100);
+\draw [color=cyan,style=dashed] (-0.100,-0.100) -- (-0.100,-2.50);
+\draw [color=green] (0,-2.50) -- (2.50,-2.50);
+\draw [color=green] (2.50,-2.50) -- (2.50,0);
+\draw [color=green] (2.50,0) -- (0,0);
+\draw [color=green] (0,0) -- (0,-2.50);
+\draw (-1.0997,1.3000) node {\( xy\)};
+\draw (1.6733,1.2500) node {\( x-y\)};
+\draw (-1.0205,-1.3000) node {\( x^2y\)};
+\draw (1.6733,-1.2500) node {\( x+y\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks b/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks
index 042e0423e..aacb17a58 100644
--- a/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks
+++ b/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks
@@ -99,7 +99,7 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (1.456249167,-1.184375000) node {$
+\draw (1.4562,-1.1844) node {$
         \begin{pmatrix}
         \phantom{
         \begin{matrix}
@@ -108,31 +108,31 @@
         }
         \end{pmatrix}$
         };
-\draw (0.2912498333,-0.2368750000) node {*};
-\draw (0.8737495000,-0.2368750000) node {*};
-\draw (1.456249167,-0.2368750000) node {*};
-\draw (2.038748833,-0.2368750000) node {*};
-\draw (2.621248500,-0.2368750000) node {*};
-\draw (0.2912498333,-0.7106250000) node {0};
-\draw (0.8737495000,-0.7106250000) node {*};
-\draw (1.456249167,-0.7106250000) node {*};
-\draw (2.038748833,-0.7106250000) node {*};
-\draw (2.621248500,-0.7106250000) node {*};
-\draw (0.2912498333,-1.184375000) node {0};
-\draw (0.8737495000,-1.184375000) node {0};
-\draw (1.456249167,-1.184375000) node {*};
-\draw (2.038748833,-1.184375000) node {*};
-\draw (2.621248500,-1.184375000) node {*};
-\draw (0.2912498333,-1.658125000) node {0};
-\draw (0.8737495000,-1.658125000) node {0};
-\draw (1.456249167,-1.658125000) node {*};
-\draw (2.038748833,-1.658125000) node {*};
-\draw (2.621248500,-1.658125000) node {*};
-\draw (0.2912498333,-2.131875000) node {0};
-\draw (0.8737495000,-2.131875000) node {0};
-\draw (1.456249167,-2.131875000) node {*};
-\draw (2.038748833,-2.131875000) node {*};
-\draw (2.621248500,-2.131875000) node {*};
+\draw (0.29125,-0.23688) node {*};
+\draw (0.87375,-0.23688) node {*};
+\draw (1.4562,-0.23688) node {*};
+\draw (2.0387,-0.23688) node {*};
+\draw (2.6213,-0.23688) node {*};
+\draw (0.29125,-0.71062) node {0};
+\draw (0.87375,-0.71062) node {*};
+\draw (1.4562,-0.71062) node {*};
+\draw (2.0387,-0.71062) node {*};
+\draw (2.6213,-0.71062) node {*};
+\draw (0.29125,-1.1844) node {0};
+\draw (0.87375,-1.1844) node {0};
+\draw (1.4562,-1.1844) node {*};
+\draw (2.0387,-1.1844) node {*};
+\draw (2.6213,-1.1844) node {*};
+\draw (0.29125,-1.6581) node {0};
+\draw (0.87375,-1.6581) node {0};
+\draw (1.4562,-1.6581) node {*};
+\draw (2.0387,-1.6581) node {*};
+\draw (2.6213,-1.6581) node {*};
+\draw (0.29125,-2.1319) node {0};
+\draw (0.87375,-2.1319) node {0};
+\draw (1.4562,-2.1319) node {*};
+\draw (2.0387,-2.1319) node {*};
+\draw (2.6213,-2.1319) node {*};
 \draw [color=red] (0.100,-0.0500) -- (1.06,-0.0500);
 \draw [color=red] (1.06,-0.0500) -- (1.06,-0.898);
 \draw [color=red] (1.06,-0.898) -- (0.100,-0.898);
@@ -141,8 +141,8 @@
 \draw [color=blue] (2.81,-0.997) -- (2.81,-2.32);
 \draw [color=blue] (2.81,-2.32) -- (1.27,-2.32);
 \draw [color=blue] (1.27,-2.32) -- (1.27,-0.997);
-\draw (1.212378848,0.2264242621) node {\( \Delta_k(A_2)\)};
-\draw (2.038748833,-2.601205000) node {\( \Omega_{k+1}(A_2)\)};
+\draw (1.2124,0.22642) node {\( \Delta_k(A_2)\)};
+\draw (2.0387,-2.6012) node {\( \Omega_{k+1}(A_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_FGRooDhFkch.pstricks b/auto/pictures_tex/Fig_FGRooDhFkch.pstricks
index 7b0b588b1..f4993c771 100644
--- a/auto/pictures_tex/Fig_FGRooDhFkch.pstricks
+++ b/auto/pictures_tex/Fig_FGRooDhFkch.pstricks
@@ -62,6 +62,10 @@
 \immediate\write\writeOfphystricks{totalheightof2f9abdf237016df2db0c349fe50e30dbad8bcb6c:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ 1 $}}%
 \immediate\write\writeOfphystricks{widthof2f9abdf237016df2db0c349fe50e30dbad8bcb6c:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$ -\pi $}}%
+\immediate\write\writeOfphystricks{totalheightofae0a42324905a92dfd1fa03440abc5b447c8e6b8:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$ -\pi $}}%
+\immediate\write\writeOfphystricks{widthofae0a42324905a92dfd1fa03440abc5b447c8e6b8:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -\frac{1}{2} \, \pi $}}%
 \immediate\write\writeOfphystricks{totalheightof8244d948eb981a434cc8bc83b6ec2b87e2ede7b5:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ -\frac{1}{2} \, \pi $}}%
@@ -70,6 +74,10 @@
 \immediate\write\writeOfphystricks{totalheightof85c8661aa83890a29f3568c5ed92d1300142f127:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ \frac{1}{2} \, \pi $}}%
 \immediate\write\writeOfphystricks{widthof85c8661aa83890a29f3568c5ed92d1300142f127:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$ \pi $}}%
+\immediate\write\writeOfphystricks{totalheightof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$ \pi $}}%
+\immediate\write\writeOfphystricks{widthof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
 %CLOSE_WRITE_AND_LABEL
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
@@ -78,31 +86,39 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-3.00,-1.57) -- (-3.00,1.57);
-\draw [color=gray,style=solid] (0,-1.57) -- (0,1.57);
-\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,1.57);
-\draw [color=gray,style=dotted] (-1.50,-1.57) -- (-1.50,1.57);
-\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,1.57);
+\draw [color=gray,style=solid] (-3.00,-3.14) -- (-3.00,3.14);
+\draw [color=gray,style=solid] (0,-3.14) -- (0,3.14);
+\draw [color=gray,style=solid] (3.00,-3.14) -- (3.00,3.14);
+\draw [color=gray,style=dotted] (-1.50,-3.14) -- (-1.50,3.14);
+\draw [color=gray,style=dotted] (1.50,-3.14) -- (1.50,3.14);
+\draw [color=gray,style=dotted] (-3.00,-2.36) -- (3.00,-2.36);
 \draw [color=gray,style=dotted] (-3.00,-0.785) -- (3.00,-0.785);
 \draw [color=gray,style=dotted] (-3.00,0.785) -- (3.00,0.785);
+\draw [color=gray,style=dotted] (-3.00,2.36) -- (3.00,2.36);
+\draw [color=gray,style=solid] (-3.00,-3.14) -- (3.00,-3.14);
 \draw [color=gray,style=solid] (-3.00,-1.57) -- (3.00,-1.57);
 \draw [color=gray,style=solid] (-3.00,0) -- (3.00,0);
 \draw [color=gray,style=solid] (-3.00,1.57) -- (3.00,1.57);
+\draw [color=gray,style=solid] (-3.00,3.14) -- (3.00,3.14);
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-2.070796327) -- (0,2.070796327);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.6416) -- (0,3.6416);
 %DEFAULT
 
 \draw [color=blue] (-3.000,-1.571)--(-2.939,-1.369)--(-2.879,-1.286)--(-2.818,-1.221)--(-2.758,-1.166)--(-2.697,-1.117)--(-2.636,-1.073)--(-2.576,-1.033)--(-2.515,-0.9943)--(-2.455,-0.9582)--(-2.394,-0.9239)--(-2.333,-0.8911)--(-2.273,-0.8596)--(-2.212,-0.8292)--(-2.152,-0.7997)--(-2.091,-0.7712)--(-2.030,-0.7434)--(-1.970,-0.7163)--(-1.909,-0.6898)--(-1.848,-0.6639)--(-1.788,-0.6385)--(-1.727,-0.6135)--(-1.667,-0.5890)--(-1.606,-0.5649)--(-1.545,-0.5412)--(-1.485,-0.5178)--(-1.424,-0.4947)--(-1.364,-0.4719)--(-1.303,-0.4493)--(-1.242,-0.4270)--(-1.182,-0.4049)--(-1.121,-0.3830)--(-1.061,-0.3613)--(-1.000,-0.3398)--(-0.9394,-0.3185)--(-0.8788,-0.2973)--(-0.8182,-0.2762)--(-0.7576,-0.2553)--(-0.6970,-0.2345)--(-0.6364,-0.2137)--(-0.5758,-0.1931)--(-0.5152,-0.1726)--(-0.4545,-0.1521)--(-0.3939,-0.1317)--(-0.3333,-0.1113)--(-0.2727,-0.09103)--(-0.2121,-0.07077)--(-0.1515,-0.05053)--(-0.09091,-0.03031)--(-0.03030,-0.01010)--(0.03030,0.01010)--(0.09091,0.03031)--(0.1515,0.05053)--(0.2121,0.07077)--(0.2727,0.09103)--(0.3333,0.1113)--(0.3939,0.1317)--(0.4545,0.1521)--(0.5152,0.1726)--(0.5758,0.1931)--(0.6364,0.2137)--(0.6970,0.2345)--(0.7576,0.2553)--(0.8182,0.2762)--(0.8788,0.2973)--(0.9394,0.3185)--(1.000,0.3398)--(1.061,0.3613)--(1.121,0.3830)--(1.182,0.4049)--(1.242,0.4270)--(1.303,0.4493)--(1.364,0.4719)--(1.424,0.4947)--(1.485,0.5178)--(1.545,0.5412)--(1.606,0.5649)--(1.667,0.5890)--(1.727,0.6135)--(1.788,0.6385)--(1.848,0.6639)--(1.909,0.6898)--(1.970,0.7163)--(2.030,0.7434)--(2.091,0.7712)--(2.152,0.7997)--(2.212,0.8292)--(2.273,0.8596)--(2.333,0.8911)--(2.394,0.9239)--(2.455,0.9582)--(2.515,0.9943)--(2.576,1.033)--(2.636,1.073)--(2.697,1.117)--(2.758,1.166)--(2.818,1.221)--(2.879,1.286)--(2.939,1.369)--(3.000,1.571);
 
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.5937365000,-1.570796327) node {$ -\frac{1}{2} \, \pi $};
+\draw (-0.45249,-3.1416) node {$ -\pi $};
+\draw [] (-0.100,-3.14) -- (0.100,-3.14);
+\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.4518270000,1.570796327) node {$ \frac{1}{2} \, \pi $};
+\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
 \draw [] (-0.100,1.57) -- (0.100,1.57);
+\draw (-0.31058,3.1416) node {$ \pi $};
+\draw [] (-0.100,3.14) -- (0.100,3.14);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_FGWjJBX.pstricks b/auto/pictures_tex/Fig_FGWjJBX.pstricks
index 1af925b5d..78a04f112 100644
--- a/auto/pictures_tex/Fig_FGWjJBX.pstricks
+++ b/auto/pictures_tex/Fig_FGWjJBX.pstricks
@@ -88,16 +88,16 @@
 \draw [] (3.00,0) -- (4.00,0.500);
 \draw [] (3.00,0) -- (4.00,-0.500);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.3059510000) node {\( \alpha_1\)};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,0.3059510000) node {\( \alpha_2\)};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,0.3215388333) node {\( \alpha_{l-2}\)};
-\draw []  (4.000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.8215388333) node {\( \alpha_{l-1}\)};
-\draw []  (4.000000000,-0.5000000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.1916921667) node {\( \alpha_l\)};
+\draw (0,0.30595) node {\( \alpha_1\)};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.30595) node {\( \alpha_2\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,0.32154) node {\( \alpha_{l-2}\)};
+\draw []  (4.0000,0.50000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.82154) node {\( \alpha_{l-1}\)};
+\draw []  (4.0000,-0.50000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.19169) node {\( \alpha_l\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks b/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks
index 33fb116ff..1957bbd5f 100644
--- a/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks
+++ b/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks
@@ -75,12 +75,12 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=blue] (3.000,0)--(2.994,0.06342)--(2.976,0.1266)--(2.946,0.1893)--(2.904,0.2511)--(2.850,0.3120)--(2.785,0.3717)--(2.709,0.4298)--(2.622,0.4862)--(2.524,0.5406)--(2.416,0.5929)--(2.298,0.6428)--(2.171,0.6901)--(2.036,0.7346)--(1.892,0.7761)--(1.740,0.8146)--(1.582,0.8497)--(1.417,0.8815)--(1.246,0.9096)--(1.071,0.9342)--(0.8908,0.9549)--(0.7073,0.9718)--(0.5209,0.9848)--(0.3325,0.9938)--(0.1427,0.9989)--(-0.04760,0.9999)--(-0.2377,0.9969)--(-0.4269,0.9898)--(-0.6144,0.9788)--(-0.7994,0.9638)--(-0.9812,0.9450)--(-1.159,0.9224)--(-1.332,0.8960)--(-1.500,0.8660)--(-1.662,0.8326)--(-1.817,0.7958)--(-1.965,0.7558)--(-2.104,0.7127)--(-2.236,0.6668)--(-2.358,0.6182)--(-2.471,0.5671)--(-2.574,0.5137)--(-2.667,0.4582)--(-2.748,0.4009)--(-2.819,0.3420)--(-2.878,0.2817)--(-2.926,0.2203)--(-2.962,0.1580)--(-2.986,0.09506)--(-2.999,0.03173)--(-2.999,-0.03173)--(-2.986,-0.09506)--(-2.962,-0.1580)--(-2.926,-0.2203)--(-2.878,-0.2817)--(-2.819,-0.3420)--(-2.748,-0.4009)--(-2.667,-0.4582)--(-2.574,-0.5137)--(-2.471,-0.5671)--(-2.358,-0.6182)--(-2.236,-0.6668)--(-2.104,-0.7127)--(-1.965,-0.7558)--(-1.817,-0.7958)--(-1.662,-0.8326)--(-1.500,-0.8660)--(-1.332,-0.8960)--(-1.159,-0.9224)--(-0.9812,-0.9450)--(-0.7994,-0.9638)--(-0.6144,-0.9788)--(-0.4269,-0.9898)--(-0.2377,-0.9969)--(-0.04760,-0.9999)--(0.1427,-0.9989)--(0.3325,-0.9938)--(0.5209,-0.9848)--(0.7073,-0.9718)--(0.8908,-0.9549)--(1.071,-0.9342)--(1.246,-0.9096)--(1.417,-0.8815)--(1.582,-0.8497)--(1.740,-0.8146)--(1.892,-0.7761)--(2.036,-0.7346)--(2.171,-0.6901)--(2.298,-0.6428)--(2.416,-0.5929)--(2.524,-0.5406)--(2.622,-0.4862)--(2.709,-0.4298)--(2.785,-0.3717)--(2.850,-0.3120)--(2.904,-0.2511)--(2.946,-0.1893)--(2.976,-0.1266)--(2.994,-0.06342)--(3.000,0);
-\draw []  (2.121320344,0.7071067812) node [rotate=0] {$\bullet$};
-\draw (2.011554297,0.9856083399) node {\( x\)};
-\draw [,->,>=latex] (2.121320344,0.7071067812) -- (2.437548110,1.655790079);
-\draw (1.761496450,1.901490632) node {\( \nabla q(x)\)};
-\draw [,->,>=latex] (2.121320344,0.7071067812) -- (2.148632243,-0.2925201793);
-\draw (2.589736468,-0.1623497994) node {\( Ax\)};
+\draw []  (2.1213,0.70711) node [rotate=0] {$\bullet$};
+\draw (2.0116,0.98561) node {\( x\)};
+\draw [,->,>=latex] (2.1213,0.70711) -- (2.4375,1.6558);
+\draw (1.7615,1.9015) node {\( \nabla q(x)\)};
+\draw [,->,>=latex] (2.1213,0.70711) -- (2.1486,-0.29252);
+\draw (2.5897,-0.16235) node {\( Ax\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_FWJuNhU.pstricks b/auto/pictures_tex/Fig_FWJuNhU.pstricks
index c44266c42..c5cad1966 100644
--- a/auto/pictures_tex/Fig_FWJuNhU.pstricks
+++ b/auto/pictures_tex/Fig_FWJuNhU.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
 \draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
 \draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
@@ -98,10 +98,10 @@
 \draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
 \draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
 \draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
-\draw (-1.099672167,1.300000000) node {\( xy\)};
-\draw (1.673347000,1.250000000) node {\( x-y\)};
-\draw (-0.9705055000,-1.250000000) node {\( x^2y\)};
-\draw (1.723347000,-1.300000000) node {\( x+y\)};
+\draw (-1.0997,1.3000) node {\( xy\)};
+\draw (1.6733,1.2500) node {\( x-y\)};
+\draw (-0.97051,-1.2500) node {\( x^2y\)};
+\draw (1.7233,-1.3000) node {\( x+y\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_FXVooJYAfif.pstricks b/auto/pictures_tex/Fig_FXVooJYAfif.pstricks
index e272810db..d5c572a19 100644
--- a/auto/pictures_tex/Fig_FXVooJYAfif.pstricks
+++ b/auto/pictures_tex/Fig_FXVooJYAfif.pstricks
@@ -60,8 +60,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-4.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [color=blue] (-4.000,-0.8440)--(-3.919,-0.8410)--(-3.838,-0.8377)--(-3.758,-0.8344)--(-3.677,-0.8309)--(-3.596,-0.8273)--(-3.515,-0.8236)--(-3.434,-0.8196)--(-3.354,-0.8155)--(-3.273,-0.8112)--(-3.192,-0.8067)--(-3.111,-0.8020)--(-3.030,-0.7971)--(-2.949,-0.7919)--(-2.869,-0.7865)--(-2.788,-0.7807)--(-2.707,-0.7747)--(-2.626,-0.7684)--(-2.545,-0.7617)--(-2.465,-0.7546)--(-2.384,-0.7471)--(-2.303,-0.7392)--(-2.222,-0.7308)--(-2.141,-0.7219)--(-2.061,-0.7124)--(-1.980,-0.7022)--(-1.899,-0.6914)--(-1.818,-0.6799)--(-1.737,-0.6675)--(-1.657,-0.6543)--(-1.576,-0.6400)--(-1.495,-0.6247)--(-1.414,-0.6082)--(-1.333,-0.5903)--(-1.253,-0.5711)--(-1.172,-0.5502)--(-1.091,-0.5277)--(-1.010,-0.5032)--(-0.9293,-0.4767)--(-0.8485,-0.4479)--(-0.7677,-0.4168)--(-0.6869,-0.3832)--(-0.6061,-0.3469)--(-0.5253,-0.3079)--(-0.4444,-0.2663)--(-0.3636,-0.2220)--(-0.2828,-0.1755)--(-0.2020,-0.1269)--(-0.1212,-0.07679)--(-0.04040,-0.02571)--(0.04040,0.02571)--(0.1212,0.07679)--(0.2020,0.1269)--(0.2828,0.1755)--(0.3636,0.2220)--(0.4444,0.2663)--(0.5253,0.3079)--(0.6061,0.3469)--(0.6869,0.3832)--(0.7677,0.4168)--(0.8485,0.4479)--(0.9293,0.4767)--(1.010,0.5032)--(1.091,0.5277)--(1.172,0.5502)--(1.253,0.5711)--(1.333,0.5903)--(1.414,0.6082)--(1.495,0.6247)--(1.576,0.6400)--(1.657,0.6543)--(1.737,0.6675)--(1.818,0.6799)--(1.899,0.6914)--(1.980,0.7022)--(2.061,0.7124)--(2.141,0.7219)--(2.222,0.7308)--(2.303,0.7392)--(2.384,0.7471)--(2.465,0.7546)--(2.545,0.7617)--(2.626,0.7684)--(2.707,0.7747)--(2.788,0.7807)--(2.869,0.7865)--(2.949,0.7919)--(3.030,0.7971)--(3.111,0.8020)--(3.192,0.8067)--(3.273,0.8112)--(3.354,0.8155)--(3.434,0.8196)--(3.515,0.8236)--(3.596,0.8273)--(3.677,0.8309)--(3.758,0.8344)--(3.838,0.8377)--(3.919,0.8410)--(4.000,0.8440);
diff --git a/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks b/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
index 85c01337c..5fcac2505 100644
--- a/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
+++ b/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
@@ -119,40 +119,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
+\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
 %DEFAULT
 
 \draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
 
 \draw [color=red] (-3.500,3.555)--(-3.429,3.500)--(-3.359,3.406)--(-3.288,3.277)--(-3.217,3.114)--(-3.146,2.921)--(-3.076,2.702)--(-3.005,2.459)--(-2.934,2.198)--(-2.864,1.921)--(-2.793,1.632)--(-2.722,1.337)--(-2.652,1.038)--(-2.581,0.7401)--(-2.510,0.4468)--(-2.439,0.1618)--(-2.369,-0.1114)--(-2.298,-0.3696)--(-2.227,-0.6099)--(-2.157,-0.8297)--(-2.086,-1.027)--(-2.015,-1.199)--(-1.944,-1.346)--(-1.874,-1.466)--(-1.803,-1.558)--(-1.732,-1.622)--(-1.662,-1.658)--(-1.591,-1.667)--(-1.520,-1.650)--(-1.449,-1.608)--(-1.379,-1.542)--(-1.308,-1.456)--(-1.237,-1.350)--(-1.167,-1.228)--(-1.096,-1.092)--(-1.025,-0.9453)--(-0.9545,-0.7902)--(-0.8838,-0.6299)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.002540)--(-0.5303,0.1441)--(-0.4596,0.2739)--(-0.3889,0.3896)--(-0.3182,0.4892)--(-0.2475,0.5710)--(-0.1768,0.6336)--(-0.1061,0.6760)--(-0.03535,0.6973)--(0.03535,0.6973)--(0.1061,0.6760)--(0.1768,0.6336)--(0.2475,0.5710)--(0.3182,0.4892)--(0.3889,0.3896)--(0.4596,0.2739)--(0.5303,0.1441)--(0.6010,0.002540)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6299)--(0.9545,-0.7902)--(1.025,-0.9453)--(1.096,-1.092)--(1.167,-1.228)--(1.237,-1.350)--(1.308,-1.456)--(1.379,-1.542)--(1.449,-1.608)--(1.520,-1.650)--(1.591,-1.667)--(1.662,-1.658)--(1.732,-1.622)--(1.803,-1.558)--(1.874,-1.466)--(1.944,-1.346)--(2.015,-1.199)--(2.086,-1.027)--(2.157,-0.8297)--(2.227,-0.6099)--(2.298,-0.3696)--(2.369,-0.1114)--(2.439,0.1618)--(2.510,0.4468)--(2.581,0.7401)--(2.652,1.038)--(2.722,1.337)--(2.793,1.632)--(2.864,1.921)--(2.934,2.198)--(3.005,2.459)--(3.076,2.702)--(3.146,2.921)--(3.217,3.114)--(3.288,3.277)--(3.359,3.406)--(3.429,3.500)--(3.500,3.555);
-\draw (-3.298672286,-0.4207143333) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.199114858,-0.3210296667) node {$-\mathit{\pi}$};
+\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
 \draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.099557429,-0.4207143333) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.099557429,-0.4207143333) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.199114858,-0.2785761667) node {$\mathit{\pi}$};
+\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
 \draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.298672286,-0.4207143333) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks b/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks
index 81ec9b593..8c1bda278 100644
--- a/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks
+++ b/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks
@@ -91,30 +91,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.400000000,0) -- (1.400000000,0);
-\draw [,->,>=latex] (0,-2.525000000) -- (0,4.550000000);
+\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
+\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
 %DEFAULT
 
 \draw [color=blue] (-0.9000,-2.025)--(-0.8818,-1.905)--(-0.8636,-1.789)--(-0.8455,-1.679)--(-0.8273,-1.573)--(-0.8091,-1.471)--(-0.7909,-1.374)--(-0.7727,-1.282)--(-0.7545,-1.193)--(-0.7364,-1.109)--(-0.7182,-1.029)--(-0.7000,-0.9528)--(-0.6818,-0.8804)--(-0.6636,-0.8119)--(-0.6455,-0.7470)--(-0.6273,-0.6856)--(-0.6091,-0.6277)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5364,-0.4286)--(-0.5182,-0.3865)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4455,-0.2455)--(-0.4273,-0.2167)--(-0.4091,-0.1902)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1238)--(-0.3364,-0.1057)--(-0.3182,-0.08948)--(-0.3000,-0.07500)--(-0.2818,-0.06217)--(-0.2636,-0.05090)--(-0.2455,-0.04108)--(-0.2273,-0.03261)--(-0.2091,-0.02539)--(-0.1909,-0.01933)--(-0.1727,-0.01431)--(-0.1545,-0.01025)--(-0.1364,-0.007044)--(-0.1182,-0.004585)--(-0.1000,-0.002778)--(-0.08182,-0.001521)--(-0.06364,0)--(-0.04545,0)--(-0.02727,0)--(-0.009091,0)--(0.009091,0)--(0.02727,0)--(0.04545,0)--(0.06364,0)--(0.08182,0.001521)--(0.1000,0.002778)--(0.1182,0.004585)--(0.1364,0.007044)--(0.1545,0.01025)--(0.1727,0.01431)--(0.1909,0.01933)--(0.2091,0.02539)--(0.2273,0.03261)--(0.2455,0.04108)--(0.2636,0.05090)--(0.2818,0.06217)--(0.3000,0.07500)--(0.3182,0.08948)--(0.3364,0.1057)--(0.3545,0.1238)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4091,0.1902)--(0.4273,0.2167)--(0.4455,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5182,0.3865)--(0.5364,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6091,0.6277)--(0.6273,0.6856)--(0.6455,0.7470)--(0.6636,0.8119)--(0.6818,0.8804)--(0.7000,0.9528)--(0.7182,1.029)--(0.7364,1.109)--(0.7545,1.193)--(0.7727,1.282)--(0.7909,1.374)--(0.8091,1.471)--(0.8273,1.573)--(0.8455,1.679)--(0.8636,1.789)--(0.8818,1.905)--(0.9000,2.025);
 
 \draw [color=red] (-0.9000,4.050)--(-0.8818,3.888)--(-0.8636,3.729)--(-0.8455,3.574)--(-0.8273,3.422)--(-0.8091,3.273)--(-0.7909,3.128)--(-0.7727,2.986)--(-0.7545,2.847)--(-0.7364,2.711)--(-0.7182,2.579)--(-0.7000,2.450)--(-0.6818,2.324)--(-0.6636,2.202)--(-0.6455,2.083)--(-0.6273,1.967)--(-0.6091,1.855)--(-0.5909,1.746)--(-0.5727,1.640)--(-0.5545,1.538)--(-0.5364,1.438)--(-0.5182,1.343)--(-0.5000,1.250)--(-0.4818,1.161)--(-0.4636,1.075)--(-0.4455,0.9921)--(-0.4273,0.9128)--(-0.4091,0.8368)--(-0.3909,0.7641)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3364,0.5657)--(-0.3182,0.5062)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2455,0.3012)--(-0.2273,0.2583)--(-0.2091,0.2186)--(-0.1909,0.1822)--(-0.1727,0.1492)--(-0.1545,0.1194)--(-0.1364,0.09298)--(-0.1182,0.06983)--(-0.1000,0.05000)--(-0.08182,0.03347)--(-0.06364,0.02025)--(-0.04545,0.01033)--(-0.02727,0.003719)--(-0.009091,0)--(0.009091,0)--(0.02727,0.003719)--(0.04545,0.01033)--(0.06364,0.02025)--(0.08182,0.03347)--(0.1000,0.05000)--(0.1182,0.06983)--(0.1364,0.09298)--(0.1545,0.1194)--(0.1727,0.1492)--(0.1909,0.1822)--(0.2091,0.2186)--(0.2273,0.2583)--(0.2455,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3182,0.5062)--(0.3364,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7641)--(0.4091,0.8368)--(0.4273,0.9128)--(0.4455,0.9921)--(0.4636,1.075)--(0.4818,1.161)--(0.5000,1.250)--(0.5182,1.343)--(0.5364,1.438)--(0.5545,1.538)--(0.5727,1.640)--(0.5909,1.746)--(0.6091,1.855)--(0.6273,1.967)--(0.6455,2.083)--(0.6636,2.202)--(0.6818,2.324)--(0.7000,2.450)--(0.7182,2.579)--(0.7364,2.711)--(0.7545,2.847)--(0.7727,2.986)--(0.7909,3.128)--(0.8091,3.273)--(0.8273,3.422)--(0.8455,3.574)--(0.8636,3.729)--(0.8818,3.888)--(0.9000,4.050);
-\draw (-1.200000000,-0.3298256667) node {$ -2 $};
+\draw (-1.2000,-0.32983) node {$ -2 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (-0.6000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.60000,-0.32983) node {$ -1 $};
 \draw [] (-0.600,-0.100) -- (-0.600,0.100);
-\draw (0.6000000000,-0.3149246667) node {$ 1 $};
+\draw (0.60000,-0.31492) node {$ 1 $};
 \draw [] (0.600,-0.100) -- (0.600,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 2 $};
+\draw (1.2000,-0.31492) node {$ 2 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (-0.4331593333,-2.400000000) node {$ -4 $};
+\draw (-0.43316,-2.4000) node {$ -4 $};
 \draw [] (-0.100,-2.40) -- (0.100,-2.40);
-\draw (-0.4331593333,-1.200000000) node {$ -2 $};
+\draw (-0.43316,-1.2000) node {$ -2 $};
 \draw [] (-0.100,-1.20) -- (0.100,-1.20);
-\draw (-0.2912498333,1.200000000) node {$ 2 $};
+\draw (-0.29125,1.2000) node {$ 2 $};
 \draw [] (-0.100,1.20) -- (0.100,1.20);
-\draw (-0.2912498333,2.400000000) node {$ 4 $};
+\draw (-0.29125,2.4000) node {$ 4 $};
 \draw [] (-0.100,2.40) -- (0.100,2.40);
-\draw (-0.2912498333,3.600000000) node {$ 6 $};
+\draw (-0.29125,3.6000) node {$ 6 $};
 \draw [] (-0.100,3.60) -- (0.100,3.60);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks b/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks
index 2a7e1e559..115bccffb 100644
--- a/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks
+++ b/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks
@@ -70,22 +70,22 @@
 \draw [color=gray,style=solid] (0,0) -- (3.00,0);
 \draw [color=gray,style=solid] (0,3.00) -- (3.00,3.00);
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.0152,0.0152)--(0.0303,0.0303)--(0.0455,0.0455)--(0.0606,0.0606)--(0.0758,0.0758)--(0.0909,0.0909)--(0.106,0.106)--(0.121,0.121)--(0.136,0.136)--(0.152,0.152)--(0.167,0.167)--(0.182,0.182)--(0.197,0.197)--(0.212,0.212)--(0.227,0.227)--(0.242,0.242)--(0.258,0.258)--(0.273,0.273)--(0.288,0.288)--(0.303,0.303)--(0.318,0.318)--(0.333,0.333)--(0.348,0.348)--(0.364,0.364)--(0.379,0.379)--(0.394,0.394)--(0.409,0.409)--(0.424,0.424)--(0.439,0.439)--(0.455,0.455)--(0.470,0.470)--(0.485,0.485)--(0.500,0.500)--(0.515,0.515)--(0.530,0.530)--(0.545,0.545)--(0.561,0.561)--(0.576,0.576)--(0.591,0.591)--(0.606,0.606)--(0.621,0.621)--(0.636,0.636)--(0.651,0.651)--(0.667,0.667)--(0.682,0.682)--(0.697,0.697)--(0.712,0.712)--(0.727,0.727)--(0.742,0.742)--(0.758,0.758)--(0.773,0.773)--(0.788,0.788)--(0.803,0.803)--(0.818,0.818)--(0.833,0.833)--(0.849,0.849)--(0.864,0.864)--(0.879,0.879)--(0.894,0.894)--(0.909,0.909)--(0.924,0.924)--(0.939,0.939)--(0.955,0.955)--(0.970,0.970)--(0.985,0.985)--(1.00,1.00)--(1.02,1.02)--(1.03,1.03)--(1.05,1.05)--(1.06,1.06)--(1.08,1.08)--(1.09,1.09)--(1.11,1.11)--(1.12,1.12)--(1.14,1.14)--(1.15,1.15)--(1.17,1.17)--(1.18,1.18)--(1.20,1.20)--(1.21,1.21)--(1.23,1.23)--(1.24,1.24)--(1.26,1.26)--(1.27,1.27)--(1.29,1.29)--(1.30,1.30)--(1.32,1.32)--(1.33,1.33)--(1.35,1.35)--(1.36,1.36)--(1.38,1.38)--(1.39,1.39)--(1.41,1.41)--(1.42,1.42)--(1.44,1.44)--(1.45,1.45)--(1.47,1.47)--(1.48,1.48)--(1.50,1.50);
 
 \draw [color=blue] (1.50,3.00)--(1.52,2.98)--(1.53,2.97)--(1.55,2.95)--(1.56,2.94)--(1.58,2.92)--(1.59,2.91)--(1.61,2.89)--(1.62,2.88)--(1.64,2.86)--(1.65,2.85)--(1.67,2.83)--(1.68,2.82)--(1.70,2.80)--(1.71,2.79)--(1.73,2.77)--(1.74,2.76)--(1.76,2.74)--(1.77,2.73)--(1.79,2.71)--(1.80,2.70)--(1.82,2.68)--(1.83,2.67)--(1.85,2.65)--(1.86,2.64)--(1.88,2.62)--(1.89,2.61)--(1.91,2.59)--(1.92,2.58)--(1.94,2.56)--(1.95,2.55)--(1.97,2.53)--(1.98,2.52)--(2.00,2.50)--(2.02,2.48)--(2.03,2.47)--(2.05,2.45)--(2.06,2.44)--(2.08,2.42)--(2.09,2.41)--(2.11,2.39)--(2.12,2.38)--(2.14,2.36)--(2.15,2.35)--(2.17,2.33)--(2.18,2.32)--(2.20,2.30)--(2.21,2.29)--(2.23,2.27)--(2.24,2.26)--(2.26,2.24)--(2.27,2.23)--(2.29,2.21)--(2.30,2.20)--(2.32,2.18)--(2.33,2.17)--(2.35,2.15)--(2.36,2.14)--(2.38,2.12)--(2.39,2.11)--(2.41,2.09)--(2.42,2.08)--(2.44,2.06)--(2.45,2.05)--(2.47,2.03)--(2.48,2.02)--(2.50,2.00)--(2.52,1.98)--(2.53,1.97)--(2.55,1.95)--(2.56,1.94)--(2.58,1.92)--(2.59,1.91)--(2.61,1.89)--(2.62,1.88)--(2.64,1.86)--(2.65,1.85)--(2.67,1.83)--(2.68,1.82)--(2.70,1.80)--(2.71,1.79)--(2.73,1.77)--(2.74,1.76)--(2.76,1.74)--(2.77,1.73)--(2.79,1.71)--(2.80,1.70)--(2.82,1.68)--(2.83,1.67)--(2.85,1.65)--(2.86,1.64)--(2.88,1.62)--(2.89,1.61)--(2.91,1.59)--(2.92,1.58)--(2.94,1.56)--(2.95,1.55)--(2.97,1.53)--(2.98,1.52)--(3.00,1.50);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw []  (1.500000000,1.500000000) node [rotate=0] {$o$};
-\draw []  (1.500000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,1.500000000) node [rotate=0] {$\bullet$};
+\draw []  (1.5000,1.5000) node [rotate=0] {$o$};
+\draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
 \draw [style=dashed] (1.50,1.50) -- (1.50,3.00);
 
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks b/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks
index 545b53b85..83e1fe1b2 100644
--- a/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks
+++ b/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks
@@ -83,6 +83,10 @@
 \immediate\write\writeOfphystricks{totalheightof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ \pi $}}%
 \immediate\write\writeOfphystricks{widthof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$ \frac{3}{2} \, \pi $}}%
+\immediate\write\writeOfphystricks{totalheightofe3fecee9a68c3a0c0d73e0eaa33c6afc47ee8bde:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$ \frac{3}{2} \, \pi $}}%
+\immediate\write\writeOfphystricks{widthofe3fecee9a68c3a0c0d73e0eaa33c6afc47ee8bde:\the\lengthOfforphystricks-}
 %CLOSE_WRITE_AND_LABEL
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
@@ -91,49 +95,53 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-1.00,-1.57) -- (-1.00,3.14);
-\draw [color=gray,style=solid] (0,-1.57) -- (0,3.14);
-\draw [color=gray,style=solid] (1.00,-1.57) -- (1.00,3.14);
-\draw [color=gray,style=solid] (2.00,-1.57) -- (2.00,3.14);
-\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,3.14);
-\draw [color=gray,style=solid] (4.00,-1.57) -- (4.00,3.14);
-\draw [color=gray,style=dotted] (-0.500,-1.57) -- (-0.500,3.14);
-\draw [color=gray,style=dotted] (0.500,-1.57) -- (0.500,3.14);
-\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,3.14);
-\draw [color=gray,style=dotted] (2.50,-1.57) -- (2.50,3.14);
-\draw [color=gray,style=dotted] (3.50,-1.57) -- (3.50,3.14);
+\draw [color=gray,style=solid] (-1.00,-1.57) -- (-1.00,4.71);
+\draw [color=gray,style=solid] (0,-1.57) -- (0,4.71);
+\draw [color=gray,style=solid] (1.00,-1.57) -- (1.00,4.71);
+\draw [color=gray,style=solid] (2.00,-1.57) -- (2.00,4.71);
+\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,4.71);
+\draw [color=gray,style=solid] (4.00,-1.57) -- (4.00,4.71);
+\draw [color=gray,style=dotted] (-0.500,-1.57) -- (-0.500,4.71);
+\draw [color=gray,style=dotted] (0.500,-1.57) -- (0.500,4.71);
+\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,4.71);
+\draw [color=gray,style=dotted] (2.50,-1.57) -- (2.50,4.71);
+\draw [color=gray,style=dotted] (3.50,-1.57) -- (3.50,4.71);
 \draw [color=gray,style=dotted] (-1.00,-0.785) -- (4.00,-0.785);
 \draw [color=gray,style=dotted] (-1.00,0.785) -- (4.00,0.785);
 \draw [color=gray,style=dotted] (-1.00,2.36) -- (4.00,2.36);
+\draw [color=gray,style=dotted] (-1.00,3.93) -- (4.00,3.93);
 \draw [color=gray,style=solid] (-1.00,-1.57) -- (4.00,-1.57);
 \draw [color=gray,style=solid] (-1.00,0) -- (4.00,0);
 \draw [color=gray,style=solid] (-1.00,1.57) -- (4.00,1.57);
 \draw [color=gray,style=solid] (-1.00,3.14) -- (4.00,3.14);
+\draw [color=gray,style=solid] (-1.00,4.71) -- (4.00,4.71);
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-2.070796327) -- (0,3.641592654);
+\draw [,->,>=latex] (-1.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-2.0708) -- (0,5.2124);
 %DEFAULT
 
 \draw [color=blue] (0,1.000)--(0.03173,0.9995)--(0.06347,0.9980)--(0.09520,0.9955)--(0.1269,0.9920)--(0.1587,0.9874)--(0.1904,0.9819)--(0.2221,0.9754)--(0.2539,0.9679)--(0.2856,0.9595)--(0.3173,0.9501)--(0.3491,0.9397)--(0.3808,0.9284)--(0.4125,0.9161)--(0.4443,0.9029)--(0.4760,0.8888)--(0.5077,0.8738)--(0.5395,0.8580)--(0.5712,0.8413)--(0.6029,0.8237)--(0.6347,0.8053)--(0.6664,0.7861)--(0.6981,0.7660)--(0.7299,0.7453)--(0.7616,0.7237)--(0.7933,0.7015)--(0.8251,0.6785)--(0.8568,0.6549)--(0.8885,0.6306)--(0.9203,0.6056)--(0.9520,0.5801)--(0.9837,0.5539)--(1.015,0.5272)--(1.047,0.5000)--(1.079,0.4723)--(1.111,0.4441)--(1.142,0.4154)--(1.174,0.3863)--(1.206,0.3569)--(1.238,0.3271)--(1.269,0.2969)--(1.301,0.2665)--(1.333,0.2358)--(1.365,0.2048)--(1.396,0.1736)--(1.428,0.1423)--(1.460,0.1108)--(1.491,0.07925)--(1.523,0.04758)--(1.555,0.01587)--(1.587,-0.01587)--(1.618,-0.04758)--(1.650,-0.07925)--(1.682,-0.1108)--(1.714,-0.1423)--(1.745,-0.1736)--(1.777,-0.2048)--(1.809,-0.2358)--(1.841,-0.2665)--(1.872,-0.2969)--(1.904,-0.3271)--(1.936,-0.3569)--(1.967,-0.3863)--(1.999,-0.4154)--(2.031,-0.4441)--(2.063,-0.4723)--(2.094,-0.5000)--(2.126,-0.5272)--(2.158,-0.5539)--(2.190,-0.5801)--(2.221,-0.6056)--(2.253,-0.6306)--(2.285,-0.6549)--(2.317,-0.6785)--(2.348,-0.7015)--(2.380,-0.7237)--(2.412,-0.7453)--(2.443,-0.7660)--(2.475,-0.7861)--(2.507,-0.8053)--(2.539,-0.8237)--(2.570,-0.8413)--(2.602,-0.8580)--(2.634,-0.8738)--(2.666,-0.8888)--(2.697,-0.9029)--(2.729,-0.9161)--(2.761,-0.9284)--(2.793,-0.9397)--(2.824,-0.9501)--(2.856,-0.9595)--(2.888,-0.9679)--(2.919,-0.9754)--(2.951,-0.9819)--(2.983,-0.9874)--(3.015,-0.9920)--(3.046,-0.9955)--(3.078,-0.9980)--(3.110,-0.9995)--(3.142,-1.000);
 
 \draw [color=blue] (-1.000,3.142)--(-0.9798,2.940)--(-0.9596,2.856)--(-0.9394,2.792)--(-0.9192,2.737)--(-0.8990,2.688)--(-0.8788,2.644)--(-0.8586,2.603)--(-0.8384,2.565)--(-0.8182,2.529)--(-0.7980,2.495)--(-0.7778,2.462)--(-0.7576,2.430)--(-0.7374,2.400)--(-0.7172,2.371)--(-0.6970,2.342)--(-0.6768,2.314)--(-0.6566,2.287)--(-0.6364,2.261)--(-0.6162,2.235)--(-0.5960,2.209)--(-0.5758,2.184)--(-0.5556,2.160)--(-0.5354,2.136)--(-0.5152,2.112)--(-0.4949,2.089)--(-0.4747,2.065)--(-0.4545,2.043)--(-0.4343,2.020)--(-0.4141,1.998)--(-0.3939,1.976)--(-0.3737,1.954)--(-0.3535,1.932)--(-0.3333,1.911)--(-0.3131,1.889)--(-0.2929,1.868)--(-0.2727,1.847)--(-0.2525,1.826)--(-0.2323,1.805)--(-0.2121,1.785)--(-0.1919,1.764)--(-0.1717,1.743)--(-0.1515,1.723)--(-0.1313,1.702)--(-0.1111,1.682)--(-0.09091,1.662)--(-0.07071,1.642)--(-0.05051,1.621)--(-0.03030,1.601)--(-0.01010,1.581)--(0.01010,1.561)--(0.03030,1.540)--(0.05051,1.520)--(0.07071,1.500)--(0.09091,1.480)--(0.1111,1.459)--(0.1313,1.439)--(0.1515,1.419)--(0.1717,1.398)--(0.1919,1.378)--(0.2121,1.357)--(0.2323,1.336)--(0.2525,1.316)--(0.2727,1.295)--(0.2929,1.274)--(0.3131,1.252)--(0.3333,1.231)--(0.3535,1.209)--(0.3737,1.188)--(0.3939,1.166)--(0.4141,1.144)--(0.4343,1.121)--(0.4545,1.099)--(0.4747,1.076)--(0.4949,1.053)--(0.5152,1.030)--(0.5354,1.006)--(0.5556,0.9818)--(0.5758,0.9573)--(0.5960,0.9323)--(0.6162,0.9069)--(0.6364,0.8810)--(0.6566,0.8545)--(0.6768,0.8274)--(0.6970,0.7996)--(0.7172,0.7711)--(0.7374,0.7416)--(0.7576,0.7112)--(0.7778,0.6797)--(0.7980,0.6469)--(0.8182,0.6126)--(0.8384,0.5765)--(0.8586,0.5383)--(0.8788,0.4975)--(0.8990,0.4533)--(0.9192,0.4048)--(0.9394,0.3499)--(0.9596,0.2852)--(0.9798,0.2013)--(1.000,0);
 
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.5937365000,-1.570796327) node {$ -\frac{1}{2} \, \pi $};
+\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.4518270000,1.570796327) node {$ \frac{1}{2} \, \pi $};
+\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
 \draw [] (-0.100,1.57) -- (0.100,1.57);
-\draw (-0.3105776667,3.141592654) node {$ \pi $};
+\draw (-0.31058,3.1416) node {$ \pi $};
 \draw [] (-0.100,3.14) -- (0.100,3.14);
+\draw (-0.45183,4.7124) node {$ \frac{3}{2} \, \pi $};
+\draw [] (-0.100,4.71) -- (0.100,4.71);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks b/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks
index 4dc7f5db6..b6547c9ce 100644
--- a/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks
+++ b/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks
@@ -83,19 +83,19 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.4240027746,-0.1283777105) node {\(A\)};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.281992727,-0.3611696314) node {\(B\)};
-\draw []  (6.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw (6.260698653,4.369932668) node {\(C\)};
+\draw (-0.42400,-0.12838) node {\(A\)};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.2820,-0.36117) node {\(B\)};
+\draw []  (6.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (6.2607,4.3699) node {\(C\)};
 \draw [] (0,0) -- (5.00,0);
 \draw [] (5.00,0) -- (6.00,4.00);
 \draw [] (6.00,4.00) -- (0,0);
 \draw [] (0,0) -- (5.60,2.40);
-\draw []  (4.480000000,1.920000000) node [rotate=0] {$\bullet$};
-\draw (4.725314693,1.611462994) node {\( N\)};
-\draw []  (5.600000000,2.400000000) node [rotate=0] {$\bullet$};
-\draw (5.936543333,2.226784875) node {\( P\)};
+\draw []  (4.4800,1.9200) node [rotate=0] {$\bullet$};
+\draw (4.7253,1.6115) node {\( N\)};
+\draw []  (5.6000,2.4000) node [rotate=0] {$\bullet$};
+\draw (5.9365,2.2268) node {\( P\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks b/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks
index 201617687..feb15aa45 100644
--- a/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks
+++ b/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks
@@ -46,10 +46,10 @@
 
 \draw [color=brown,style=dashed] (2.000,0)--(2.000,0.02116)--(2.000,0.04231)--(1.999,0.06346)--(1.998,0.08460)--(1.997,0.1057)--(1.996,0.1268)--(1.995,0.1480)--(1.993,0.1690)--(1.991,0.1901)--(1.989,0.2112)--(1.986,0.2322)--(1.984,0.2532)--(1.981,0.2742)--(1.978,0.2951)--(1.975,0.3160)--(1.971,0.3369)--(1.968,0.3577)--(1.964,0.3785)--(1.960,0.3993)--(1.955,0.4200)--(1.951,0.4406)--(1.946,0.4612)--(1.941,0.4818)--(1.936,0.5023)--(1.930,0.5227)--(1.925,0.5431)--(1.919,0.5635)--(1.913,0.5837)--(1.907,0.6039)--(1.900,0.6241)--(1.893,0.6441)--(1.887,0.6641)--(1.879,0.6840)--(1.872,0.7039)--(1.865,0.7236)--(1.857,0.7433)--(1.849,0.7629)--(1.841,0.7824)--(1.832,0.8019)--(1.824,0.8212)--(1.815,0.8404)--(1.806,0.8596)--(1.797,0.8786)--(1.787,0.8976)--(1.778,0.9165)--(1.768,0.9352)--(1.758,0.9538)--(1.748,0.9724)--(1.737,0.9908)--(1.727,1.009)--(1.716,1.027)--(1.705,1.045)--(1.694,1.063)--(1.683,1.081)--(1.671,1.099)--(1.659,1.117)--(1.647,1.134)--(1.635,1.151)--(1.623,1.169)--(1.611,1.186)--(1.598,1.203)--(1.585,1.220)--(1.572,1.236)--(1.559,1.253)--(1.546,1.269)--(1.532,1.286)--(1.518,1.302)--(1.505,1.318)--(1.491,1.334)--(1.476,1.349)--(1.462,1.365)--(1.447,1.380)--(1.433,1.395)--(1.418,1.410)--(1.403,1.425)--(1.388,1.440)--(1.372,1.455)--(1.357,1.469)--(1.341,1.483)--(1.326,1.498)--(1.310,1.512)--(1.294,1.525)--(1.277,1.539)--(1.261,1.552)--(1.245,1.566)--(1.228,1.579)--(1.211,1.592)--(1.194,1.604)--(1.177,1.617)--(1.160,1.629)--(1.143,1.641)--(1.125,1.653)--(1.108,1.665)--(1.090,1.677)--(1.072,1.688)--(1.054,1.699)--(1.036,1.711)--(1.018,1.721)--(1.000,1.732);
 \draw [color=brown,style=dashed] (1.30,0.750) -- (3.03,1.75);
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (2.598076211,1.500000000);
-\draw (2.286902711,1.865758621) node {$e_{r}$};
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (1.232050808,1.866025404);
-\draw (1.607516675,2.186465271) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
+\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
+\draw (1.6075,2.1865) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -91,10 +91,10 @@
 
 \draw [color=brown,style=dashed] (-1.477,0.2605)--(-1.480,0.2448)--(-1.482,0.2292)--(-1.485,0.2135)--(-1.487,0.1978)--(-1.489,0.1820)--(-1.491,0.1663)--(-1.492,0.1505)--(-1.494,0.1347)--(-1.495,0.1189)--(-1.496,0.1031)--(-1.497,0.08722)--(-1.498,0.07137)--(-1.499,0.05552)--(-1.499,0.03966)--(-1.500,0.02380)--(-1.500,0.007933)--(-1.500,-0.007933)--(-1.500,-0.02380)--(-1.499,-0.03966)--(-1.499,-0.05552)--(-1.498,-0.07137)--(-1.497,-0.08722)--(-1.496,-0.1031)--(-1.495,-0.1189)--(-1.494,-0.1347)--(-1.492,-0.1505)--(-1.491,-0.1663)--(-1.489,-0.1820)--(-1.487,-0.1978)--(-1.485,-0.2135)--(-1.482,-0.2292)--(-1.480,-0.2448)--(-1.477,-0.2605)--(-1.474,-0.2761)--(-1.471,-0.2917)--(-1.468,-0.3072)--(-1.465,-0.3227)--(-1.461,-0.3382)--(-1.458,-0.3536)--(-1.454,-0.3690)--(-1.450,-0.3844)--(-1.446,-0.3997)--(-1.441,-0.4150)--(-1.437,-0.4302)--(-1.432,-0.4454)--(-1.428,-0.4605)--(-1.423,-0.4756)--(-1.417,-0.4906)--(-1.412,-0.5056)--(-1.407,-0.5205)--(-1.401,-0.5353)--(-1.395,-0.5501)--(-1.390,-0.5648)--(-1.384,-0.5795)--(-1.377,-0.5941)--(-1.371,-0.6087)--(-1.364,-0.6231)--(-1.358,-0.6375)--(-1.351,-0.6518)--(-1.344,-0.6661)--(-1.337,-0.6803)--(-1.330,-0.6944)--(-1.322,-0.7084)--(-1.315,-0.7224)--(-1.307,-0.7362)--(-1.299,-0.7500)--(-1.291,-0.7637)--(-1.283,-0.7773)--(-1.275,-0.7908)--(-1.266,-0.8043)--(-1.258,-0.8176)--(-1.249,-0.8309)--(-1.240,-0.8440)--(-1.231,-0.8571)--(-1.222,-0.8701)--(-1.213,-0.8830)--(-1.203,-0.8957)--(-1.194,-0.9084)--(-1.184,-0.9210)--(-1.174,-0.9335)--(-1.164,-0.9458)--(-1.154,-0.9581)--(-1.144,-0.9702)--(-1.134,-0.9823)--(-1.123,-0.9942)--(-1.113,-1.006)--(-1.102,-1.018)--(-1.091,-1.029)--(-1.080,-1.041)--(-1.069,-1.052)--(-1.058,-1.063)--(-1.047,-1.075)--(-1.035,-1.086)--(-1.024,-1.096)--(-1.012,-1.107)--(-1.000,-1.118)--(-0.9883,-1.128)--(-0.9763,-1.139)--(-0.9642,-1.149);
 \draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-2.349231552,-0.8550503583);
-\draw (-2.085452009,-1.242909145) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-1.067518788,-1.452722836);
-\draw (-0.6920529202,-1.132282968) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
+\draw (-2.0855,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
+\draw (-0.69205,-1.1323) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_HNxitLj.pstricks b/auto/pictures_tex/Fig_HNxitLj.pstricks
index f80923d7c..ea0ee8ecf 100644
--- a/auto/pictures_tex/Fig_HNxitLj.pstricks
+++ b/auto/pictures_tex/Fig_HNxitLj.pstricks
@@ -75,26 +75,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (-1.000000000,0) node [rotate=0] {$\diamondsuit$};
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\diamondsuit$};
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\diamondsuit$};
-\draw []  (-1.000000000,-1.000000000) node [rotate=0] {$\diamondsuit$};
-\draw (1.500000000,-0.3824895000) node {\( \sA^*_{\sH}\)};
-\draw (1.500000000,-0.3824895000) node {\( \sA^*_{\sH}\)};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (-1.0000,0) node [rotate=0] {$\diamondsuit$};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
+\draw []  (-1.0000,-1.0000) node [rotate=0] {$\diamondsuit$};
+\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
+\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_HasseAGdfdy.pstricks b/auto/pictures_tex/Fig_HasseAGdfdy.pstricks
index 969e6fbb9..95b7c5107 100644
--- a/auto/pictures_tex/Fig_HasseAGdfdy.pstricks
+++ b/auto/pictures_tex/Fig_HasseAGdfdy.pstricks
@@ -87,17 +87,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.2785761667) node {\( \alpha\)};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3622220000) node {\( \beta\)};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.3140621667) node {\( \gamma\)};
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (0,2.278576167) node {\( a\)};
-\draw []  (2.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,2.326736000) node {\( b\)};
-\draw []  (4.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,2.278576167) node {\( c\)};
+\draw (0,-0.27858) node {\( \alpha\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.36222) node {\( \beta\)};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.31406) node {\( \gamma\)};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (0,2.2786) node {\( a\)};
+\draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,2.3267) node {\( b\)};
+\draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,2.2786) node {\( c\)};
 \draw [] (0,0) -- (0,2.00);
 \draw [] (2.00,0) -- (2.00,2.00);
 \draw [] (4.00,0) -- (4.00,2.00);
diff --git a/auto/pictures_tex/Fig_IWuPxFc.pstricks b/auto/pictures_tex/Fig_IWuPxFc.pstricks
index 5ee6a5c4e..a40e3d90c 100644
--- a/auto/pictures_tex/Fig_IWuPxFc.pstricks
+++ b/auto/pictures_tex/Fig_IWuPxFc.pstricks
@@ -95,39 +95,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.498873925,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-3.145689210) -- (0,3.145689210);
+\draw [,->,>=latex] (-2.4989,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-3.1457) -- (0,3.1457);
 %DEFAULT
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diff --git a/auto/pictures_tex/Fig_IYAvSvI.pstricks b/auto/pictures_tex/Fig_IYAvSvI.pstricks
index fd1a6616f..9bf4d113f 100644
--- a/auto/pictures_tex/Fig_IYAvSvI.pstricks
+++ b/auto/pictures_tex/Fig_IYAvSvI.pstricks
@@ -78,22 +78,22 @@
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diff --git a/auto/pictures_tex/Fig_IntBoutCercle.pstricks b/auto/pictures_tex/Fig_IntBoutCercle.pstricks
index a01592ed6..01492c84e 100644
--- a/auto/pictures_tex/Fig_IntBoutCercle.pstricks
+++ b/auto/pictures_tex/Fig_IntBoutCercle.pstricks
@@ -67,8 +67,8 @@
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-\draw (1.354646868,0.6631599656) node {$P$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
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diff --git a/auto/pictures_tex/Fig_IntRectangle.pstricks b/auto/pictures_tex/Fig_IntRectangle.pstricks
index a3d7d1732..1ac58ccc5 100644
--- a/auto/pictures_tex/Fig_IntRectangle.pstricks
+++ b/auto/pictures_tex/Fig_IntRectangle.pstricks
@@ -71,8 +71,8 @@
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@@ -81,11 +81,11 @@
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntTriangle.pstricks b/auto/pictures_tex/Fig_IntTriangle.pstricks
index 5dc2cfc1d..5831092a4 100644
--- a/auto/pictures_tex/Fig_IntTriangle.pstricks
+++ b/auto/pictures_tex/Fig_IntTriangle.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -90,13 +90,13 @@
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntegraleSimple.pstricks b/auto/pictures_tex/Fig_IntegraleSimple.pstricks
index 5f46e6eb9..50af4c229 100644
--- a/auto/pictures_tex/Fig_IntegraleSimple.pstricks
+++ b/auto/pictures_tex/Fig_IntegraleSimple.pstricks
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796328,0) -- (6.783185311,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
+\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
 %DEFAULT
-\draw []  (-1.570796327,0) node [rotate=0] {$\bullet$};
-\draw (-1.570796327,-0.2785761667) node {$a$};
-\draw []  (6.283185307,0) node [rotate=0] {$\bullet$};
-\draw (6.283185307,-0.3267360000) node {$b$};
+\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
+\draw (-1.5708,-0.27858) node {$a$};
+\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
+\draw (6.2832,-0.32674) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/auto/pictures_tex/Fig_JGuKEjH.pstricks b/auto/pictures_tex/Fig_JGuKEjH.pstricks
index f879ed6f9..a14c0796d 100644
--- a/auto/pictures_tex/Fig_JGuKEjH.pstricks
+++ b/auto/pictures_tex/Fig_JGuKEjH.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
@@ -75,7 +75,7 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,0) -- (-0.643,0.766) -- (-0.643,0.766) -- (-0.669,0.743) -- (-0.695,0.719) -- (-0.719,0.695) -- (-0.743,0.669) -- (-0.766,0.643) -- (-0.788,0.616) -- (-0.809,0.588) -- (-0.829,0.559) -- (-0.848,0.530) -- (-0.866,0.500) -- (-0.883,0.470) -- (-0.899,0.438) -- (-0.914,0.407) -- (-0.927,0.374) -- (-0.940,0.342) -- (-0.951,0.309) -- (-0.961,0.276) -- (-0.970,0.242) -- (-0.978,0.208) -- (-0.985,0.174) -- (-0.990,0.139) -- (-0.995,0.105) -- (-0.998,0.0698) -- (-0.999,0.0349) -- (-1.00,0) -- (-0.999,-0.0349) -- (-0.998,-0.0698) -- (-0.995,-0.105) -- (-0.990,-0.139) -- (-0.985,-0.174) -- (-0.978,-0.208) -- (-0.970,-0.242) -- (-0.961,-0.276) -- (-0.951,-0.309) -- (-0.940,-0.342) -- (-0.927,-0.374) -- (-0.914,-0.407) -- (-0.899,-0.438) -- (-0.883,-0.469) -- (-0.866,-0.500) -- (-0.848,-0.530) -- (-0.829,-0.559) -- (-0.809,-0.588) -- (-0.788,-0.616) -- (-0.766,-0.643) -- (-0.743,-0.669) -- (-0.719,-0.695) -- (-0.695,-0.719) -- (-0.669,-0.743) -- (1.00,0) -- (-0.643,-0.766) --  cycle;
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
 \draw [color=lightgray] (1.00,0) -- (-0.643,0.766);
 \draw [color=lightgray] (1.00,0) -- (-0.643,-0.766);
 
diff --git a/auto/pictures_tex/Fig_JJAooWpimYW.pstricks b/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
index 00789f209..9f0dd8da9 100644
--- a/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
+++ b/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
@@ -108,34 +108,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
-\draw [,->,>=latex] (0,-1.498867339) -- (0,1.499874128);
+\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
+\draw [,->,>=latex] (0,-1.4989) -- (0,1.4999);
 %DEFAULT
 
 \draw [color=blue] (-7.854,0)--(-7.695,0.1580)--(-7.537,0.3120)--(-7.378,0.4582)--(-7.219,0.5929)--(-7.061,0.7127)--(-6.902,0.8146)--(-6.743,0.8960)--(-6.585,0.9549)--(-6.426,0.9898)--(-6.267,0.9999)--(-6.109,0.9848)--(-5.950,0.9450)--(-5.791,0.8815)--(-5.633,0.7958)--(-5.474,0.6901)--(-5.315,0.5671)--(-5.157,0.4298)--(-4.998,0.2817)--(-4.839,0.1266)--(-4.681,-0.03173)--(-4.522,-0.1893)--(-4.363,-0.3420)--(-4.205,-0.4862)--(-4.046,-0.6182)--(-3.887,-0.7346)--(-3.729,-0.8326)--(-3.570,-0.9096)--(-3.411,-0.9638)--(-3.253,-0.9938)--(-3.094,-0.9989)--(-2.935,-0.9788)--(-2.777,-0.9342)--(-2.618,-0.8660)--(-2.459,-0.7761)--(-2.301,-0.6668)--(-2.142,-0.5406)--(-1.983,-0.4009)--(-1.825,-0.2511)--(-1.666,-0.09506)--(-1.507,0.06342)--(-1.349,0.2203)--(-1.190,0.3717)--(-1.031,0.5137)--(-0.8727,0.6428)--(-0.7140,0.7558)--(-0.5553,0.8497)--(-0.3967,0.9224)--(-0.2380,0.9718)--(-0.07933,0.9969)--(0.07933,0.9969)--(0.2380,0.9718)--(0.3967,0.9224)--(0.5553,0.8497)--(0.7140,0.7558)--(0.8727,0.6428)--(1.031,0.5137)--(1.190,0.3717)--(1.349,0.2203)--(1.507,0.06342)--(1.666,-0.09506)--(1.825,-0.2511)--(1.983,-0.4009)--(2.142,-0.5406)--(2.301,-0.6668)--(2.459,-0.7761)--(2.618,-0.8660)--(2.777,-0.9342)--(2.935,-0.9788)--(3.094,-0.9989)--(3.253,-0.9938)--(3.411,-0.9638)--(3.570,-0.9096)--(3.729,-0.8326)--(3.887,-0.7346)--(4.046,-0.6182)--(4.205,-0.4862)--(4.363,-0.3420)--(4.522,-0.1893)--(4.681,-0.03173)--(4.839,0.1266)--(4.998,0.2817)--(5.157,0.4298)--(5.315,0.5671)--(5.474,0.6901)--(5.633,0.7958)--(5.791,0.8815)--(5.950,0.9450)--(6.109,0.9848)--(6.267,0.9999)--(6.426,0.9898)--(6.585,0.9549)--(6.743,0.8960)--(6.902,0.8146)--(7.061,0.7127)--(7.219,0.5929)--(7.378,0.4582)--(7.537,0.3120)--(7.695,0.1580)--(7.854,0);
-\draw (-7.853981634,-0.4207143333) node {$ -\frac{5}{2} \, \pi $};
+\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
 \draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.283185307,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.712388980,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.853981634,-0.4207143333) node {$ \frac{5}{2} \, \pi $};
+\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
 \draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks b/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
index 8d281f24f..ab72b83d5 100644
--- a/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
+++ b/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
@@ -92,8 +92,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.212388985,0) -- (5.212388985,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-5.2124,0) -- (5.2124,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -145,21 +145,21 @@
 \draw [] (4.71,-1.00) -- (4.71,0);
 
 \draw [color=blue] (-4.712,1.000)--(-4.617,0.9955)--(-4.522,0.9819)--(-4.427,0.9595)--(-4.332,0.9284)--(-4.236,0.8888)--(-4.141,0.8413)--(-4.046,0.7861)--(-3.951,0.7237)--(-3.856,0.6549)--(-3.760,0.5801)--(-3.665,0.5000)--(-3.570,0.4154)--(-3.475,0.3271)--(-3.380,0.2358)--(-3.284,0.1423)--(-3.189,0.04758)--(-3.094,-0.04758)--(-2.999,-0.1423)--(-2.904,-0.2358)--(-2.808,-0.3271)--(-2.713,-0.4154)--(-2.618,-0.5000)--(-2.523,-0.5801)--(-2.428,-0.6549)--(-2.332,-0.7237)--(-2.237,-0.7861)--(-2.142,-0.8413)--(-2.047,-0.8888)--(-1.952,-0.9284)--(-1.856,-0.9595)--(-1.761,-0.9819)--(-1.666,-0.9955)--(-1.571,-1.000)--(-1.476,-0.9955)--(-1.380,-0.9819)--(-1.285,-0.9595)--(-1.190,-0.9284)--(-1.095,-0.8888)--(-0.9996,-0.8413)--(-0.9044,-0.7861)--(-0.8092,-0.7237)--(-0.7140,-0.6549)--(-0.6188,-0.5801)--(-0.5236,-0.5000)--(-0.4284,-0.4154)--(-0.3332,-0.3271)--(-0.2380,-0.2358)--(-0.1428,-0.1423)--(-0.04760,-0.04758)--(0.04760,0.04758)--(0.1428,0.1423)--(0.2380,0.2358)--(0.3332,0.3271)--(0.4284,0.4154)--(0.5236,0.5000)--(0.6188,0.5801)--(0.7140,0.6549)--(0.8092,0.7237)--(0.9044,0.7861)--(0.9996,0.8413)--(1.095,0.8888)--(1.190,0.9284)--(1.285,0.9595)--(1.380,0.9819)--(1.476,0.9955)--(1.571,1.000)--(1.666,0.9955)--(1.761,0.9819)--(1.856,0.9595)--(1.952,0.9284)--(2.047,0.8888)--(2.142,0.8413)--(2.237,0.7861)--(2.332,0.7237)--(2.428,0.6549)--(2.523,0.5801)--(2.618,0.5000)--(2.713,0.4154)--(2.808,0.3271)--(2.904,0.2358)--(2.999,0.1423)--(3.094,0.04758)--(3.189,-0.04758)--(3.284,-0.1423)--(3.380,-0.2358)--(3.475,-0.3271)--(3.570,-0.4154)--(3.665,-0.5000)--(3.760,-0.5801)--(3.856,-0.6549)--(3.951,-0.7237)--(4.046,-0.7861)--(4.141,-0.8413)--(4.236,-0.8888)--(4.332,-0.9284)--(4.427,-0.9595)--(4.522,-0.9819)--(4.617,-0.9955)--(4.712,-1.000);
-\draw (-4.712388980,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks b/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
index 2f94eb547..649f91548 100644
--- a/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
+++ b/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
@@ -83,20 +83,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
-\draw (1.523347667,2.000000000) node {$(x,y)$};
-\draw [,->,>=latex] (0,0) -- (1.000000000,2.000000000);
-\draw (0.6845247985,0.4139141375) node {$\theta$};
+\draw (1.5233,2.0000) node {$(x,y)$};
+\draw [,->,>=latex] (0,0) -- (1.0000,2.0000);
+\draw (0.68452,0.41391) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00559)--(0.500,0.0112)--(0.500,0.0168)--(0.500,0.0224)--(0.499,0.0279)--(0.499,0.0335)--(0.498,0.0391)--(0.498,0.0447)--(0.497,0.0502)--(0.497,0.0558)--(0.496,0.0614)--(0.496,0.0669)--(0.495,0.0724)--(0.494,0.0780)--(0.493,0.0835)--(0.492,0.0890)--(0.491,0.0945)--(0.490,0.100)--(0.489,0.105)--(0.488,0.111)--(0.486,0.116)--(0.485,0.122)--(0.484,0.127)--(0.482,0.133)--(0.481,0.138)--(0.479,0.143)--(0.477,0.149)--(0.476,0.154)--(0.474,0.159)--(0.472,0.165)--(0.470,0.170)--(0.468,0.175)--(0.466,0.180)--(0.464,0.186)--(0.462,0.191)--(0.460,0.196)--(0.458,0.201)--(0.456,0.206)--(0.453,0.211)--(0.451,0.216)--(0.448,0.221)--(0.446,0.226)--(0.443,0.231)--(0.441,0.236)--(0.438,0.241)--(0.435,0.246)--(0.432,0.251)--(0.430,0.256)--(0.427,0.260)--(0.424,0.265)--(0.421,0.270)--(0.418,0.275)--(0.415,0.279)--(0.412,0.284)--(0.408,0.289)--(0.405,0.293)--(0.402,0.298)--(0.398,0.302)--(0.395,0.306)--(0.392,0.311)--(0.388,0.315)--(0.385,0.320)--(0.381,0.324)--(0.377,0.328)--(0.374,0.332)--(0.370,0.336)--(0.366,0.341)--(0.362,0.345)--(0.358,0.349)--(0.354,0.353)--(0.350,0.357)--(0.346,0.360)--(0.342,0.364)--(0.338,0.368)--(0.334,0.372)--(0.330,0.376)--(0.326,0.379)--(0.322,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.304,0.397)--(0.300,0.400)--(0.295,0.404)--(0.291,0.407)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416)--(0.272,0.419)--(0.267,0.422)--(0.263,0.425)--(0.258,0.428)--(0.253,0.431)--(0.248,0.434)--(0.243,0.437)--(0.238,0.439)--(0.234,0.442)--(0.229,0.445)--(0.224,0.447);
-\draw (0.2337087285,1.168018886) node {$r$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (0.23371,1.1680) node {$r$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks b/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks
index 9e95ea0bc..18f70b22a 100644
--- a/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks
+++ b/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.662277660) -- (0,3.662277660);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
 %DEFAULT
 
 \draw [color=red] (-3.000,3.162)--(-2.939,3.105)--(-2.879,3.048)--(-2.818,2.990)--(-2.758,2.933)--(-2.697,2.876)--(-2.636,2.820)--(-2.576,2.763)--(-2.515,2.707)--(-2.455,2.650)--(-2.394,2.594)--(-2.333,2.539)--(-2.273,2.483)--(-2.212,2.428)--(-2.152,2.373)--(-2.091,2.318)--(-2.030,2.263)--(-1.970,2.209)--(-1.909,2.155)--(-1.848,2.102)--(-1.788,2.049)--(-1.727,1.996)--(-1.667,1.944)--(-1.606,1.892)--(-1.545,1.841)--(-1.485,1.790)--(-1.424,1.740)--(-1.364,1.691)--(-1.303,1.643)--(-1.242,1.595)--(-1.182,1.548)--(-1.121,1.502)--(-1.061,1.458)--(-1.000,1.414)--(-0.9394,1.372)--(-0.8788,1.331)--(-0.8182,1.292)--(-0.7576,1.255)--(-0.6970,1.219)--(-0.6364,1.185)--(-0.5758,1.154)--(-0.5152,1.125)--(-0.4545,1.098)--(-0.3939,1.075)--(-0.3333,1.054)--(-0.2727,1.037)--(-0.2121,1.022)--(-0.1515,1.011)--(-0.09091,1.004)--(-0.03030,1.000)--(0.03030,1.000)--(0.09091,1.004)--(0.1515,1.011)--(0.2121,1.022)--(0.2727,1.037)--(0.3333,1.054)--(0.3939,1.075)--(0.4545,1.098)--(0.5152,1.125)--(0.5758,1.154)--(0.6364,1.185)--(0.6970,1.219)--(0.7576,1.255)--(0.8182,1.292)--(0.8788,1.331)--(0.9394,1.372)--(1.000,1.414)--(1.061,1.458)--(1.121,1.502)--(1.182,1.548)--(1.242,1.595)--(1.303,1.643)--(1.364,1.691)--(1.424,1.740)--(1.485,1.790)--(1.545,1.841)--(1.606,1.892)--(1.667,1.944)--(1.727,1.996)--(1.788,2.049)--(1.848,2.102)--(1.909,2.155)--(1.970,2.209)--(2.030,2.263)--(2.091,2.318)--(2.152,2.373)--(2.212,2.428)--(2.273,2.483)--(2.333,2.539)--(2.394,2.594)--(2.455,2.650)--(2.515,2.707)--(2.576,2.763)--(2.636,2.820)--(2.697,2.876)--(2.758,2.933)--(2.818,2.990)--(2.879,3.048)--(2.939,3.105)--(3.000,3.162);
@@ -104,33 +104,33 @@
 \draw [color=red] (-3.000,-3.162)--(-2.939,-3.105)--(-2.879,-3.048)--(-2.818,-2.990)--(-2.758,-2.933)--(-2.697,-2.876)--(-2.636,-2.820)--(-2.576,-2.763)--(-2.515,-2.707)--(-2.455,-2.650)--(-2.394,-2.594)--(-2.333,-2.539)--(-2.273,-2.483)--(-2.212,-2.428)--(-2.152,-2.373)--(-2.091,-2.318)--(-2.030,-2.263)--(-1.970,-2.209)--(-1.909,-2.155)--(-1.848,-2.102)--(-1.788,-2.049)--(-1.727,-1.996)--(-1.667,-1.944)--(-1.606,-1.892)--(-1.545,-1.841)--(-1.485,-1.790)--(-1.424,-1.740)--(-1.364,-1.691)--(-1.303,-1.643)--(-1.242,-1.595)--(-1.182,-1.548)--(-1.121,-1.502)--(-1.061,-1.458)--(-1.000,-1.414)--(-0.9394,-1.372)--(-0.8788,-1.331)--(-0.8182,-1.292)--(-0.7576,-1.255)--(-0.6970,-1.219)--(-0.6364,-1.185)--(-0.5758,-1.154)--(-0.5152,-1.125)--(-0.4545,-1.098)--(-0.3939,-1.075)--(-0.3333,-1.054)--(-0.2727,-1.037)--(-0.2121,-1.022)--(-0.1515,-1.011)--(-0.09091,-1.004)--(-0.03030,-1.000)--(0.03030,-1.000)--(0.09091,-1.004)--(0.1515,-1.011)--(0.2121,-1.022)--(0.2727,-1.037)--(0.3333,-1.054)--(0.3939,-1.075)--(0.4545,-1.098)--(0.5152,-1.125)--(0.5758,-1.154)--(0.6364,-1.185)--(0.6970,-1.219)--(0.7576,-1.255)--(0.8182,-1.292)--(0.8788,-1.331)--(0.9394,-1.372)--(1.000,-1.414)--(1.061,-1.458)--(1.121,-1.502)--(1.182,-1.548)--(1.242,-1.595)--(1.303,-1.643)--(1.364,-1.691)--(1.424,-1.740)--(1.485,-1.790)--(1.545,-1.841)--(1.606,-1.892)--(1.667,-1.944)--(1.727,-1.996)--(1.788,-2.049)--(1.848,-2.102)--(1.909,-2.155)--(1.970,-2.209)--(2.030,-2.263)--(2.091,-2.318)--(2.152,-2.373)--(2.212,-2.428)--(2.273,-2.483)--(2.333,-2.539)--(2.394,-2.594)--(2.455,-2.650)--(2.515,-2.707)--(2.576,-2.763)--(2.636,-2.820)--(2.697,-2.876)--(2.758,-2.933)--(2.818,-2.990)--(2.879,-3.048)--(2.939,-3.105)--(3.000,-3.162);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.2106906781,0.8045813219) node {$R$};
-\draw []  (0,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.1931375115,-0.8045813219) node {$S$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.21069,0.80458) node {$R$};
+\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.19314,-0.80458) node {$S$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks b/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks
index 2ba636293..e54d5c080 100644
--- a/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks
+++ b/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks
@@ -72,18 +72,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.448683298);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4487);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.03030,0.03029)--(0.06061,0.06049)--(0.09091,0.09054)--(0.1212,0.1203)--(0.1515,0.1498)--(0.1818,0.1789)--(0.2121,0.2075)--(0.2424,0.2356)--(0.2727,0.2631)--(0.3030,0.2900)--(0.3333,0.3162)--(0.3636,0.3417)--(0.3939,0.3665)--(0.4242,0.3905)--(0.4545,0.4138)--(0.4848,0.4363)--(0.5152,0.4580)--(0.5455,0.4789)--(0.5758,0.4990)--(0.6061,0.5183)--(0.6364,0.5369)--(0.6667,0.5547)--(0.6970,0.5718)--(0.7273,0.5882)--(0.7576,0.6039)--(0.7879,0.6189)--(0.8182,0.6332)--(0.8485,0.6470)--(0.8788,0.6601)--(0.9091,0.6727)--(0.9394,0.6847)--(0.9697,0.6961)--(1.000,0.7071)--(1.030,0.7176)--(1.061,0.7276)--(1.091,0.7372)--(1.121,0.7463)--(1.152,0.7550)--(1.182,0.7634)--(1.212,0.7714)--(1.242,0.7790)--(1.273,0.7863)--(1.303,0.7933)--(1.333,0.8000)--(1.364,0.8064)--(1.394,0.8125)--(1.424,0.8184)--(1.455,0.8240)--(1.485,0.8294)--(1.515,0.8346)--(1.545,0.8396)--(1.576,0.8443)--(1.606,0.8489)--(1.636,0.8533)--(1.667,0.8575)--(1.697,0.8615)--(1.727,0.8654)--(1.758,0.8692)--(1.788,0.8728)--(1.818,0.8762)--(1.848,0.8795)--(1.879,0.8827)--(1.909,0.8858)--(1.939,0.8888)--(1.970,0.8917)--(2.000,0.8944)--(2.030,0.8971)--(2.061,0.8997)--(2.091,0.9021)--(2.121,0.9045)--(2.152,0.9068)--(2.182,0.9091)--(2.212,0.9112)--(2.242,0.9133)--(2.273,0.9153)--(2.303,0.9173)--(2.333,0.9191)--(2.364,0.9210)--(2.394,0.9227)--(2.424,0.9244)--(2.455,0.9261)--(2.485,0.9277)--(2.515,0.9292)--(2.545,0.9307)--(2.576,0.9322)--(2.606,0.9336)--(2.636,0.9350)--(2.667,0.9363)--(2.697,0.9376)--(2.727,0.9389)--(2.758,0.9401)--(2.788,0.9413)--(2.818,0.9424)--(2.848,0.9435)--(2.879,0.9446)--(2.909,0.9457)--(2.939,0.9467)--(2.970,0.9477)--(3.000,0.9487);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks b/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks
index 7b135deb6..92685399c 100644
--- a/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks
+++ b/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (6.783185307,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
+\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
 %DEFAULT
-\draw []  (-1.570796327,0) node [rotate=0] {$\bullet$};
-\draw (-1.570796327,-0.2785761667) node {$a$};
-\draw []  (6.283185307,0) node [rotate=0] {$\bullet$};
-\draw (6.283185307,-0.3267360000) node {$b$};
+\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
+\draw (-1.5708,-0.27858) node {$a$};
+\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
+\draw (6.2832,-0.32674) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/auto/pictures_tex/Fig_KKRooHseDzC.pstricks b/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
index 597246ba4..62e8d054f 100644
--- a/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
+++ b/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.374994332);
+\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.3750);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -104,12 +104,12 @@
 \draw [color=red,style=dashed] (6.00,0) -- (6.00,2.82);
 \draw [color=red,style=dashed] (6.00,2.82) -- (5.00,2.82);
 \draw [color=red,style=dashed] (5.00,2.82) -- (5.00,0);
-\draw []  (5.000000000,2.819444444) node [rotate=0] {$\bullet$};
-\draw (5.441978850,3.392374269) node {$f(x)$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.2785761667) node {$x$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3455708333) node {$x+\Delta x$};
+\draw []  (5.0000,2.8194) node [rotate=0] {$\bullet$};
+\draw (5.4420,3.3924) node {$f(x)$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.27858) node {$x$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.34557) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_KScolorD.pstricks b/auto/pictures_tex/Fig_KScolorD.pstricks
index c23ebf509..091738bd0 100644
--- a/auto/pictures_tex/Fig_KScolorD.pstricks
+++ b/auto/pictures_tex/Fig_KScolorD.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.696851470,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.694039839);
+\draw [,->,>=latex] (-1.6969,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.6940);
 %DEFAULT
 
 \draw [color=blue] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
@@ -72,8 +72,8 @@
 \draw [color=black] plot [smooth,tension=1] coordinates {(0,1.00)(-0.119,1.19)(-0.159,0.784)(-0.354,1.15)(-0.311,0.737)(-0.575,1.05)(-0.452,0.660)(-0.773,0.918)(-0.574,0.557)(-0.940,0.746)(-0.673,0.433)(-1.07,0.544)(-0.745,0.290)(-1.16,0.322)(-0.788,0.137)(-1.20,0.0869)(-1.00,0)};
 
 \draw [color=black] (0,-1.00)--(0.0159,-1.00)--(0.0317,-1.00)--(0.0476,-0.999)--(0.0634,-0.998)--(0.0792,-0.997)--(0.0951,-0.995)--(0.111,-0.994)--(0.127,-0.992)--(0.142,-0.990)--(0.158,-0.987)--(0.174,-0.985)--(0.189,-0.982)--(0.205,-0.979)--(0.220,-0.975)--(0.236,-0.972)--(0.251,-0.968)--(0.266,-0.964)--(0.282,-0.959)--(0.297,-0.955)--(0.312,-0.950)--(0.327,-0.945)--(0.342,-0.940)--(0.357,-0.934)--(0.372,-0.928)--(0.386,-0.922)--(0.401,-0.916)--(0.415,-0.910)--(0.430,-0.903)--(0.444,-0.896)--(0.458,-0.889)--(0.472,-0.881)--(0.486,-0.874)--(0.500,-0.866)--(0.514,-0.858)--(0.527,-0.850)--(0.541,-0.841)--(0.554,-0.833)--(0.567,-0.824)--(0.580,-0.815)--(0.593,-0.805)--(0.606,-0.796)--(0.618,-0.786)--(0.631,-0.776)--(0.643,-0.766)--(0.655,-0.756)--(0.667,-0.745)--(0.679,-0.735)--(0.690,-0.724)--(0.701,-0.713)--(0.713,-0.701)--(0.724,-0.690)--(0.735,-0.679)--(0.745,-0.667)--(0.756,-0.655)--(0.766,-0.643)--(0.776,-0.631)--(0.786,-0.618)--(0.796,-0.606)--(0.805,-0.593)--(0.815,-0.580)--(0.824,-0.567)--(0.833,-0.554)--(0.841,-0.541)--(0.850,-0.527)--(0.858,-0.514)--(0.866,-0.500)--(0.874,-0.486)--(0.881,-0.472)--(0.889,-0.458)--(0.896,-0.444)--(0.903,-0.430)--(0.910,-0.415)--(0.916,-0.401)--(0.922,-0.386)--(0.928,-0.372)--(0.934,-0.357)--(0.940,-0.342)--(0.945,-0.327)--(0.950,-0.312)--(0.955,-0.297)--(0.959,-0.282)--(0.964,-0.266)--(0.968,-0.251)--(0.972,-0.236)--(0.975,-0.220)--(0.979,-0.205)--(0.982,-0.189)--(0.985,-0.174)--(0.987,-0.158)--(0.990,-0.142)--(0.992,-0.127)--(0.994,-0.111)--(0.995,-0.0951)--(0.997,-0.0792)--(0.998,-0.0634)--(0.999,-0.0476)--(1.00,-0.0317)--(1.00,-0.0159)--(1.00,0);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0,-1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LEJNDxI.pstricks b/auto/pictures_tex/Fig_LEJNDxI.pstricks
index 9a91fb79c..57274a81d 100644
--- a/auto/pictures_tex/Fig_LEJNDxI.pstricks
+++ b/auto/pictures_tex/Fig_LEJNDxI.pstricks
@@ -71,15 +71,15 @@
 \draw [style=dotted] (2.00,0) -- (3.00,0);
 \draw [] (3.00,0) -- (4.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.2000000000) node {};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2000000000) node {};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.2000000000) node {};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.2000000000) node {};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.2000000000) node {};
+\draw (0,0.20000) node {};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.20000) node {};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.20000) node {};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.20000) node {};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.20000) node {};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_LesSpheres.pstricks b/auto/pictures_tex/Fig_LesSpheres.pstricks
index cb22e70e5..3858e3937 100644
--- a/auto/pictures_tex/Fig_LesSpheres.pstricks
+++ b/auto/pictures_tex/Fig_LesSpheres.pstricks
@@ -41,20 +41,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 \draw [] (0,1.00) -- (-1.00,0);
 \draw [] (-1.00,0) -- (0,-1.00);
 \draw [] (0,-1.00) -- (1.00,0);
 \draw [] (1.00,0) -- (0,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -93,18 +93,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -143,20 +143,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 \draw [] (1.00,1.00) -- (-1.00,1.00);
 \draw [] (-1.00,1.00) -- (-1.00,-1.00);
 \draw [] (-1.00,-1.00) -- (1.00,-1.00);
 \draw [] (1.00,-1.00) -- (1.00,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LesSubFigures.pstricks b/auto/pictures_tex/Fig_LesSubFigures.pstricks
index 657447a76..29f58eaae 100644
--- a/auto/pictures_tex/Fig_LesSubFigures.pstricks
+++ b/auto/pictures_tex/Fig_LesSubFigures.pstricks
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.800000000,2.975000000) node [rotate=0] {$\bullet$};
-\draw (3.078729394,2.519944177) node {$Q_{0}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
+\draw (3.0787,2.5199) node {$Q_{0}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.397500000,2.886861314) node [rotate=0] {$\bullet$};
-\draw (2.695272866,2.436021235) node {$Q_{1}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
+\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.995000000,2.763157895) node [rotate=0] {$\bullet$};
-\draw (2.322383202,2.321545179) node {$Q_{2}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
+\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.778894979);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
 %DEFAULT
 \draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.592500000,2.576923077) node [rotate=0] {$\bullet$};
-\draw (1.966388364,2.157179026) node {$Q_{3}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
+\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks b/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks
index 442c58b99..27a5ac04b 100644
--- a/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks
+++ b/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.800000000,2.975000000) node [rotate=0] {$\bullet$};
-\draw (3.078729394,2.519944177) node {$Q_{0}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
+\draw (3.0787,2.5199) node {$Q_{0}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.397500000,2.886861314) node [rotate=0] {$\bullet$};
-\draw (2.695272866,2.436021235) node {$Q_{1}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
+\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.995000000,2.763157895) node [rotate=0] {$\bullet$};
-\draw (2.322383202,2.321545179) node {$Q_{2}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
+\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.778894979);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
 %DEFAULT
 \draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.592500000,2.576923077) node [rotate=0] {$\bullet$};
-\draw (1.966388364,2.157179026) node {$Q_{3}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
+\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MCKyvdk.pstricks b/auto/pictures_tex/Fig_MCKyvdk.pstricks
index 48b0eefa7..8e7561c6c 100644
--- a/auto/pictures_tex/Fig_MCKyvdk.pstricks
+++ b/auto/pictures_tex/Fig_MCKyvdk.pstricks
@@ -95,14 +95,14 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (-0.2782965229,3.266129356) node {\( A\)};
-\draw (3.000000000,3.324708000) node {\( B\)};
-\draw (3.284911356,-0.2661293562) node {\( C\)};
-\draw (-0.3561643333,0) node {\( D\)};
-\draw (1.012377249,4.016129356) node {\( E\)};
-\draw (4.641743106,3.750000000) node {\( F\)};
-\draw (4.642528439,0.7500000000) node {\( G\)};
-\draw (0.9910859161,1.016129356) node {\( H\)};
+\draw (-0.27830,3.2661) node {\( A\)};
+\draw (3.0000,3.3247) node {\( B\)};
+\draw (3.2849,-0.26613) node {\( C\)};
+\draw (-0.35616,0) node {\( D\)};
+\draw (1.0124,4.0161) node {\( E\)};
+\draw (4.6417,3.7500) node {\( F\)};
+\draw (4.6425,0.75000) node {\( G\)};
+\draw (0.99109,1.0161) node {\( H\)};
 \draw [] (0,3.00) -- (1.30,3.75);
 \draw [] (3.00,3.00) -- (4.30,3.75);
 \draw [] (3.00,0) -- (4.30,0.750);
diff --git a/auto/pictures_tex/Fig_MNICGhR.pstricks b/auto/pictures_tex/Fig_MNICGhR.pstricks
index d8fec2b34..ee87719a4 100644
--- a/auto/pictures_tex/Fig_MNICGhR.pstricks
+++ b/auto/pictures_tex/Fig_MNICGhR.pstricks
@@ -87,15 +87,15 @@
 \draw [style=dotted] (2.00,0) -- (3.00,0);
 \draw [] (3.00,0) -- (4.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3059510000) node {$\alpha_1$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.3059510000) node {$\alpha_2$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.3059510000) node {$\alpha_3$};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.3215388333) node {$\alpha_{l-1}$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.3083078333) node {$\alpha_l$};
+\draw (0,0.30595) node {$\alpha_1$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.30595) node {$\alpha_2$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.30595) node {$\alpha_3$};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.32154) node {$\alpha_{l-1}$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.30831) node {$\alpha_l$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_MaxVraissLp.pstricks b/auto/pictures_tex/Fig_MaxVraissLp.pstricks
index 85887b764..4d9809e18 100644
--- a/auto/pictures_tex/Fig_MaxVraissLp.pstricks
+++ b/auto/pictures_tex/Fig_MaxVraissLp.pstricks
@@ -87,24 +87,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.168279320);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
 %DEFAULT
-\draw []  (3.000000000,2.668279320) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,2.6683) node [rotate=0] {$\bullet$};
 
 \draw [color=blue] (0,0)--(0.101,0.00115)--(0.202,0.00858)--(0.303,0.0269)--(0.404,0.0593)--(0.505,0.108)--(0.606,0.172)--(0.707,0.254)--(0.808,0.351)--(0.909,0.463)--(1.01,0.587)--(1.11,0.722)--(1.21,0.865)--(1.31,1.01)--(1.41,1.17)--(1.52,1.32)--(1.62,1.47)--(1.72,1.63)--(1.82,1.77)--(1.92,1.91)--(2.02,2.04)--(2.12,2.16)--(2.22,2.27)--(2.32,2.36)--(2.42,2.45)--(2.53,2.52)--(2.63,2.58)--(2.73,2.62)--(2.83,2.65)--(2.93,2.67)--(3.03,2.67)--(3.13,2.66)--(3.23,2.64)--(3.33,2.60)--(3.43,2.56)--(3.54,2.50)--(3.64,2.44)--(3.74,2.37)--(3.84,2.29)--(3.94,2.20)--(4.04,2.11)--(4.14,2.02)--(4.24,1.92)--(4.34,1.82)--(4.44,1.72)--(4.55,1.62)--(4.65,1.52)--(4.75,1.42)--(4.85,1.32)--(4.95,1.22)--(5.05,1.12)--(5.15,1.03)--(5.25,0.945)--(5.35,0.861)--(5.45,0.781)--(5.56,0.705)--(5.66,0.633)--(5.76,0.567)--(5.86,0.504)--(5.96,0.446)--(6.06,0.393)--(6.16,0.345)--(6.26,0.300)--(6.36,0.260)--(6.46,0.224)--(6.57,0.191)--(6.67,0.163)--(6.77,0.137)--(6.87,0.115)--(6.97,0.0953)--(7.07,0.0785)--(7.17,0.0641)--(7.27,0.0518)--(7.37,0.0415)--(7.47,0.0328)--(7.58,0.0257)--(7.68,0.0198)--(7.78,0.0151)--(7.88,0.0113)--(7.98,0.00837)--(8.08,0.00607)--(8.18,0.00432)--(8.28,0.00300)--(8.38,0.00204)--(8.48,0.00134)--(8.59,0)--(8.69,0)--(8.79,0)--(8.89,0)--(8.99,0)--(9.09,0)--(9.19,0)--(9.29,0)--(9.39,0)--(9.50,0)--(9.60,0)--(9.70,0)--(9.80,0)--(9.90,0)--(10.0,0);
 \draw [style=dotted] (3.00,2.67) -- (3.00,0);
-\draw (10.50000000,-0.4140621667) node {$p$};
-\draw (10.50000000,-0.4140621667) node {$p$};
-\draw (3.000000000,-0.4207143333) node {$ \frac{3}{10} $};
+\draw (10.500,-0.41406) node {$p$};
+\draw (10.500,-0.41406) node {$p$};
+\draw (3.0000,-0.42071) node {$ \frac{3}{10} $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.000000000,-0.4207143333) node {$ \frac{3}{5} $};
+\draw (6.0000,-0.42071) node {$ \frac{3}{5} $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (9.000000000,-0.4207143333) node {$ \frac{9}{10} $};
+\draw (9.0000,-0.42071) node {$ \frac{9}{10} $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.6579310000,3.168279320) node {$L(p)$};
-\draw (-0.6579310000,3.168279320) node {$L(p)$};
-\draw (-0.3108333333,2.000000000) node {$ \frac{1}{5} $};
+\draw (-0.65793,3.1683) node {$L(p)$};
+\draw (-0.65793,3.1683) node {$L(p)$};
+\draw (-0.31083,2.0000) node {$ \frac{1}{5} $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MethodeChemin.pstricks b/auto/pictures_tex/Fig_MethodeChemin.pstricks
index b8239b7b4..8934d3069 100644
--- a/auto/pictures_tex/Fig_MethodeChemin.pstricks
+++ b/auto/pictures_tex/Fig_MethodeChemin.pstricks
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-2.000000000) -- (0,2.000000000);
+\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
 %DEFAULT
 \draw [color=red,style=dashed] (-1.50,1.50) -- (1.50,-1.50);
 \draw [color=blue,style=dashed] (-1.50,-0.750) -- (1.50,0.750);
-\draw (-1.500000000,1.941614833) node {$y=-x$};
-\draw (2.325053201,1.144587034) node {$y=x/2$};
+\draw (-1.5000,1.9416) node {$y=-x$};
+\draw (2.3251,1.1446) node {$y=x/2$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MethodeNewton.pstricks b/auto/pictures_tex/Fig_MethodeNewton.pstricks
index 23469e25c..c62cf7e8b 100644
--- a/auto/pictures_tex/Fig_MethodeNewton.pstricks
+++ b/auto/pictures_tex/Fig_MethodeNewton.pstricks
@@ -95,30 +95,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-1.287500000) -- (0,4.400000002);
+\draw [,->,>=latex] (-2.0000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-1.2875) -- (0,4.4000);
 %DEFAULT
 
 \draw [color=blue] (-1.500,3.900)--(-1.424,3.713)--(-1.348,3.529)--(-1.273,3.349)--(-1.197,3.173)--(-1.121,3.001)--(-1.045,2.833)--(-0.9697,2.668)--(-0.8939,2.507)--(-0.8182,2.350)--(-0.7424,2.197)--(-0.6667,2.048)--(-0.5909,1.903)--(-0.5152,1.761)--(-0.4394,1.623)--(-0.3636,1.490)--(-0.2879,1.359)--(-0.2121,1.233)--(-0.1364,1.111)--(-0.06061,0.9921)--(0.01515,0.8773)--(0.09091,0.7664)--(0.1667,0.6593)--(0.2424,0.5560)--(0.3182,0.4565)--(0.3939,0.3608)--(0.4697,0.2690)--(0.5455,0.1810)--(0.6212,0.09682)--(0.6970,0.01647)--(0.7727,-0.06006)--(0.8485,-0.1328)--(0.9242,-0.2016)--(1.000,-0.2667)--(1.076,-0.3279)--(1.152,-0.3853)--(1.227,-0.4388)--(1.303,-0.4886)--(1.379,-0.5345)--(1.455,-0.5766)--(1.530,-0.6148)--(1.606,-0.6493)--(1.682,-0.6799)--(1.758,-0.7067)--(1.833,-0.7296)--(1.909,-0.7488)--(1.985,-0.7641)--(2.061,-0.7755)--(2.136,-0.7832)--(2.212,-0.7870)--(2.288,-0.7870)--(2.364,-0.7832)--(2.439,-0.7755)--(2.515,-0.7641)--(2.591,-0.7488)--(2.667,-0.7296)--(2.742,-0.7067)--(2.818,-0.6799)--(2.894,-0.6493)--(2.970,-0.6148)--(3.045,-0.5766)--(3.121,-0.5345)--(3.197,-0.4886)--(3.273,-0.4388)--(3.348,-0.3853)--(3.424,-0.3279)--(3.500,-0.2667)--(3.576,-0.2016)--(3.652,-0.1328)--(3.727,-0.06006)--(3.803,0.01647)--(3.879,0.09682)--(3.955,0.1810)--(4.030,0.2690)--(4.106,0.3608)--(4.182,0.4565)--(4.258,0.5560)--(4.333,0.6593)--(4.409,0.7664)--(4.485,0.8773)--(4.561,0.9921)--(4.636,1.111)--(4.712,1.233)--(4.788,1.359)--(4.864,1.490)--(4.939,1.623)--(5.015,1.761)--(5.091,1.903)--(5.167,2.048)--(5.242,2.197)--(5.318,2.350)--(5.394,2.507)--(5.470,2.668)--(5.545,2.833)--(5.621,3.001)--(5.697,3.173)--(5.773,3.349)--(5.849,3.529)--(5.924,3.713)--(6.000,3.900);
 \draw [color=red,style=dotted] (-0.900,0) -- (-0.900,2.52);
 \draw [color=green,style=dashed] (-1.20,3.15) -- (0.600,-0.630);
-\draw []  (-0.9000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.9000000000,-0.4059510000) node {$x_n$};
-\draw []  (0.3000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.3000000000,-0.4191818333) node {$x_{n+1}$};
-\draw []  (-0.9000000000,2.520000000) node [rotate=0] {$\bullet$};
-\draw (-0.5041004656,2.846194201) node {$y_n$};
-\draw []  (0.7129573851,0) node [rotate=0] {$\bullet$};
-\draw (0.7129573851,0.4059510000) node {$r_0$};
-\draw []  (3.787042615,0) node [rotate=0] {$\bullet$};
-\draw (3.787042615,0.4059510000) node {$r_1$};
-\draw []  (2.250000000,-0.7875000000) node [rotate=0] {$\bullet$};
-\draw (2.250000000,-1.212208000) node {$S$};
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (-0.90000,0) node [rotate=0] {$\bullet$};
+\draw (-0.90000,-0.40595) node {$x_n$};
+\draw []  (0.30000,0) node [rotate=0] {$\bullet$};
+\draw (0.30000,-0.41918) node {$x_{n+1}$};
+\draw []  (-0.90000,2.5200) node [rotate=0] {$\bullet$};
+\draw (-0.50410,2.8462) node {$y_n$};
+\draw []  (0.71296,0) node [rotate=0] {$\bullet$};
+\draw (0.71296,0.40595) node {$r_0$};
+\draw []  (3.7870,0) node [rotate=0] {$\bullet$};
+\draw (3.7870,0.40595) node {$r_1$};
+\draw []  (2.2500,-0.78750) node [rotate=0] {$\bullet$};
+\draw (2.2500,-1.2122) node {$S$};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 2 $};
+\draw (6.0000,-0.31492) node {$ 2 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MomentForce.pstricks b/auto/pictures_tex/Fig_MomentForce.pstricks
index d2d50996a..6e2f5e73e 100644
--- a/auto/pictures_tex/Fig_MomentForce.pstricks
+++ b/auto/pictures_tex/Fig_MomentForce.pstricks
@@ -82,13 +82,13 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.4247080000) node {$O$};
-\draw [,->,>=latex] (0,0) -- (-1.000000000,-1.000000000);
-\draw (-1.000000000,-0.4419603333) node {$\overline{ R }$};
-\draw [,->,>=latex] (-1.000000000,-1.000000000) -- (-3.000000000,-1.500000000);
-\draw (-3.000000000,-1.041960333) node {$\overline{ F }$};
+\draw (0,0.42471) node {$O$};
+\draw [,->,>=latex] (0,0) -- (-1.0000,-1.0000);
+\draw (-1.0000,-0.44196) node {$\overline{ R }$};
+\draw [,->,>=latex] (-1.0000,-1.0000) -- (-3.0000,-1.5000);
+\draw (-3.0000,-1.0420) node {$\overline{ F }$};
 \draw [color=blue,style=dotted] (0,0) -- (0.176,-0.706);
-\draw (0.4742668775,-0.1534444890) node {$d$};
+\draw (0.47427,-0.15344) node {$d$};
 \draw [color=brown,style=dashed] (-1.00,-1.00) -- (0.467,-0.633);
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_MoulinEau.pstricks b/auto/pictures_tex/Fig_MoulinEau.pstricks
index ed42ed16a..a61cb53e6 100644
--- a/auto/pictures_tex/Fig_MoulinEau.pstricks
+++ b/auto/pictures_tex/Fig_MoulinEau.pstricks
@@ -36,15 +36,15 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -76,17 +76,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (-1.29,0.468);
-\draw [color=red,->,>=latex] (-0.6427876097,1.233955557) -- (-1.229611699,1.726359433);
-\draw [color=green,->,>=latex] (-0.6427876097,1.233955557) -- (-1.055963521,0.7415516804);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=red,->,>=latex] (-0.64279,1.2340) -- (-1.2296,1.7264);
+\draw [color=green,->,>=latex] (-0.64279,1.2340) -- (-1.0560,0.74155);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_NEtAchr.pstricks b/auto/pictures_tex/Fig_NEtAchr.pstricks
index 463979272..eb02886ee 100644
--- a/auto/pictures_tex/Fig_NEtAchr.pstricks
+++ b/auto/pictures_tex/Fig_NEtAchr.pstricks
@@ -70,18 +70,18 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=brown] plot [smooth,tension=1] coordinates {(-1.50,-0.500)(0.500,-0.300)(2.00,1.00)(3.50,1.50)(5.00,2.70)(5.80,2.70)};
-\draw []  (-1.500000000,-0.5000000000) node [rotate=0] {$\bullet$};
-\draw (-1.500000000,-0.1789703333) node {\( +\)};
-\draw []  (0.5000000000,-0.3000000000) node [rotate=0] {$\bullet$};
-\draw (0.5000000000,0.02102966667) node {\( +\)};
-\draw []  (2.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,1.321029667) node {\( +\)};
-\draw []  (3.500000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw (3.500000000,1.821029667) node {\( +\)};
-\draw []  (5.000000000,2.700000000) node [rotate=0] {$\bullet$};
-\draw (5.000000000,3.021029667) node {\( +\)};
-\draw []  (5.800000000,2.700000000) node [rotate=0] {$\bullet$};
-\draw (5.800000000,3.021029667) node {\( +\)};
+\draw []  (-1.5000,-0.50000) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.17897) node {\( +\)};
+\draw []  (0.50000,-0.30000) node [rotate=0] {$\bullet$};
+\draw (0.50000,0.021030) node {\( +\)};
+\draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,1.3210) node {\( +\)};
+\draw []  (3.5000,1.5000) node [rotate=0] {$\bullet$};
+\draw (3.5000,1.8210) node {\( +\)};
+\draw []  (5.0000,2.7000) node [rotate=0] {$\bullet$};
+\draw (5.0000,3.0210) node {\( +\)};
+\draw []  (5.8000,2.7000) node [rotate=0] {$\bullet$};
+\draw (5.8000,3.0210) node {\( +\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_NWDooOObSHB.pstricks b/auto/pictures_tex/Fig_NWDooOObSHB.pstricks
index 55206359f..0769eba3a 100644
--- a/auto/pictures_tex/Fig_NWDooOObSHB.pstricks
+++ b/auto/pictures_tex/Fig_NWDooOObSHB.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.300000000,0) -- (7.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.781250000);
+\draw [,->,>=latex] (-8.3000,0) -- (7.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.7812);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -102,12 +102,12 @@
 \draw [color=blue] (-2.600,0)--(-2.508,0)--(-2.416,0)--(-2.324,0)--(-2.232,0)--(-2.140,0)--(-2.048,0)--(-1.957,0)--(-1.865,0)--(-1.773,0)--(-1.681,0)--(-1.589,0)--(-1.497,0)--(-1.405,0)--(-1.313,0)--(-1.221,0)--(-1.129,0)--(-1.037,0)--(-0.9455,0)--(-0.8535,0)--(-0.7616,0)--(-0.6697,0)--(-0.5778,0)--(-0.4859,0)--(-0.3939,0)--(-0.3020,0)--(-0.2101,0)--(-0.1182,0)--(-0.02626,0)--(0.06566,0)--(0.1576,0)--(0.2495,0)--(0.3414,0)--(0.4333,0)--(0.5253,0)--(0.6172,0)--(0.7091,0)--(0.8010,0)--(0.8929,0)--(0.9848,0)--(1.077,0)--(1.169,0)--(1.261,0)--(1.353,0)--(1.444,0)--(1.536,0)--(1.628,0)--(1.720,0)--(1.812,0)--(1.904,0)--(1.996,0)--(2.088,0)--(2.180,0)--(2.272,0)--(2.364,0)--(2.456,0)--(2.547,0)--(2.639,0)--(2.731,0)--(2.823,0)--(2.915,0)--(3.007,0)--(3.099,0)--(3.191,0)--(3.283,0)--(3.375,0)--(3.467,0)--(3.559,0)--(3.651,0)--(3.742,0)--(3.834,0)--(3.926,0)--(4.018,0)--(4.110,0)--(4.202,0)--(4.294,0)--(4.386,0)--(4.478,0)--(4.570,0)--(4.662,0)--(4.754,0)--(4.845,0)--(4.937,0)--(5.029,0)--(5.121,0)--(5.213,0)--(5.305,0)--(5.397,0)--(5.489,0)--(5.581,0)--(5.673,0)--(5.765,0)--(5.857,0)--(5.948,0)--(6.040,0)--(6.132,0)--(6.224,0)--(6.316,0)--(6.408,0)--(6.500,0);
 \draw [] (-2.60,0) -- (-2.60,2.12);
 \draw [] (6.50,6.28) -- (6.50,0);
-\draw []  (-7.800000000,0) node [rotate=0] {$\bullet$};
-\draw (-7.800000000,-0.2785761667) node {\( a\)};
-\draw []  (-2.600000000,0) node [rotate=0] {$\bullet$};
-\draw (-2.600000000,-0.3267360000) node {\( b\)};
-\draw []  (6.500000000,0) node [rotate=0] {$\bullet$};
-\draw (6.500000000,-0.2785761667) node {\( c\)};
+\draw []  (-7.8000,0) node [rotate=0] {$\bullet$};
+\draw (-7.8000,-0.27858) node {\( a\)};
+\draw []  (-2.6000,0) node [rotate=0] {$\bullet$};
+\draw (-2.6000,-0.32674) node {\( b\)};
+\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.27858) node {\( c\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_NiveauHyperbole.pstricks b/auto/pictures_tex/Fig_NiveauHyperbole.pstricks
index d20738e11..342639491 100644
--- a/auto/pictures_tex/Fig_NiveauHyperbole.pstricks
+++ b/auto/pictures_tex/Fig_NiveauHyperbole.pstricks
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.020,0.2020)--(1.040,0.2871)--(1.061,0.3534)--(1.081,0.4101)--(1.101,0.4607)--(1.121,0.5071)--(1.141,0.5503)--(1.162,0.5911)--(1.182,0.6298)--(1.202,0.6670)--(1.222,0.7027)--(1.242,0.7373)--(1.263,0.7709)--(1.283,0.8035)--(1.303,0.8354)--(1.323,0.8666)--(1.343,0.8971)--(1.364,0.9271)--(1.384,0.9566)--(1.404,0.9856)--(1.424,1.014)--(1.444,1.042)--(1.465,1.070)--(1.485,1.098)--(1.505,1.125)--(1.525,1.152)--(1.545,1.178)--(1.566,1.205)--(1.586,1.231)--(1.606,1.257)--(1.626,1.282)--(1.646,1.308)--(1.667,1.333)--(1.687,1.359)--(1.707,1.383)--(1.727,1.408)--(1.747,1.433)--(1.768,1.458)--(1.788,1.482)--(1.808,1.506)--(1.828,1.531)--(1.848,1.555)--(1.869,1.579)--(1.889,1.602)--(1.909,1.626)--(1.929,1.650)--(1.949,1.673)--(1.970,1.697)--(1.990,1.720)--(2.010,1.744)--(2.030,1.767)--(2.051,1.790)--(2.071,1.813)--(2.091,1.836)--(2.111,1.859)--(2.131,1.882)--(2.152,1.905)--(2.172,1.928)--(2.192,1.951)--(2.212,1.973)--(2.232,1.996)--(2.253,2.018)--(2.273,2.041)--(2.293,2.063)--(2.313,2.086)--(2.333,2.108)--(2.354,2.131)--(2.374,2.153)--(2.394,2.175)--(2.414,2.197)--(2.434,2.219)--(2.455,2.242)--(2.475,2.264)--(2.495,2.286)--(2.515,2.308)--(2.535,2.330)--(2.556,2.352)--(2.576,2.374)--(2.596,2.396)--(2.616,2.418)--(2.636,2.439)--(2.657,2.461)--(2.677,2.483)--(2.697,2.505)--(2.717,2.526)--(2.737,2.548)--(2.758,2.570)--(2.778,2.592)--(2.798,2.613)--(2.818,2.635)--(2.838,2.656)--(2.859,2.678)--(2.879,2.700)--(2.899,2.721)--(2.919,2.743)--(2.939,2.764)--(2.960,2.786)--(2.980,2.807)--(3.000,2.828);
@@ -108,33 +108,33 @@
 \draw [color=blue] (-3.000,-2.828)--(-2.980,-2.807)--(-2.960,-2.786)--(-2.939,-2.764)--(-2.919,-2.743)--(-2.899,-2.721)--(-2.879,-2.700)--(-2.859,-2.678)--(-2.838,-2.656)--(-2.818,-2.635)--(-2.798,-2.613)--(-2.778,-2.592)--(-2.758,-2.570)--(-2.737,-2.548)--(-2.717,-2.526)--(-2.697,-2.505)--(-2.677,-2.483)--(-2.657,-2.461)--(-2.636,-2.439)--(-2.616,-2.418)--(-2.596,-2.396)--(-2.576,-2.374)--(-2.556,-2.352)--(-2.535,-2.330)--(-2.515,-2.308)--(-2.495,-2.286)--(-2.475,-2.264)--(-2.455,-2.242)--(-2.434,-2.219)--(-2.414,-2.197)--(-2.394,-2.175)--(-2.374,-2.153)--(-2.354,-2.131)--(-2.333,-2.108)--(-2.313,-2.086)--(-2.293,-2.063)--(-2.273,-2.041)--(-2.253,-2.018)--(-2.232,-1.996)--(-2.212,-1.973)--(-2.192,-1.951)--(-2.172,-1.928)--(-2.152,-1.905)--(-2.131,-1.882)--(-2.111,-1.859)--(-2.091,-1.836)--(-2.071,-1.813)--(-2.051,-1.790)--(-2.030,-1.767)--(-2.010,-1.744)--(-1.990,-1.720)--(-1.970,-1.697)--(-1.949,-1.673)--(-1.929,-1.650)--(-1.909,-1.626)--(-1.889,-1.602)--(-1.869,-1.579)--(-1.848,-1.555)--(-1.828,-1.531)--(-1.808,-1.506)--(-1.788,-1.482)--(-1.768,-1.458)--(-1.747,-1.433)--(-1.727,-1.408)--(-1.707,-1.383)--(-1.687,-1.359)--(-1.667,-1.333)--(-1.646,-1.308)--(-1.626,-1.282)--(-1.606,-1.257)--(-1.586,-1.231)--(-1.566,-1.205)--(-1.545,-1.178)--(-1.525,-1.152)--(-1.505,-1.125)--(-1.485,-1.098)--(-1.465,-1.070)--(-1.444,-1.042)--(-1.424,-1.014)--(-1.404,-0.9856)--(-1.384,-0.9566)--(-1.364,-0.9271)--(-1.343,-0.8971)--(-1.323,-0.8666)--(-1.303,-0.8354)--(-1.283,-0.8035)--(-1.263,-0.7709)--(-1.242,-0.7373)--(-1.222,-0.7027)--(-1.202,-0.6670)--(-1.182,-0.6298)--(-1.162,-0.5911)--(-1.141,-0.5503)--(-1.121,-0.5071)--(-1.101,-0.4607)--(-1.081,-0.4101)--(-1.061,-0.3534)--(-1.040,-0.2871)--(-1.020,-0.2020)--(-1.000,0);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.7867744886,0.1954186781) node {$P$};
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.7850131552,0.2309048448) node {$Q$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (0.78677,0.19542) node {$P$};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-0.78501,0.23090) node {$Q$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks b/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
index 152de5dbe..292a4a5f5 100644
--- a/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
+++ b/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
@@ -116,38 +116,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.900000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-2.813086867) -- (0,3.699326205);
+\draw [,->,>=latex] (-4.9000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-2.8131) -- (0,3.6993);
 %DEFAULT
 
 \draw [color=blue] (-4.4000,-2.3131)--(-4.3051,-1.7786)--(-4.2101,-1.3073)--(-4.1152,-0.89319)--(-4.0202,-0.53075)--(-3.9253,-0.21496)--(-3.8303,0.058716)--(-3.7354,0.29443)--(-3.6404,0.49595)--(-3.5455,0.66669)--(-3.4505,0.80977)--(-3.3556,0.92802)--(-3.2606,1.0240)--(-3.1657,1.1001)--(-3.0707,1.1584)--(-2.9758,1.2009)--(-2.8808,1.2293)--(-2.7859,1.2452)--(-2.6909,1.2501)--(-2.5960,1.2452)--(-2.5010,1.2319)--(-2.4061,1.2112)--(-2.3111,1.1841)--(-2.2162,1.1515)--(-2.1212,1.1143)--(-2.0263,1.0731)--(-1.9313,1.0287)--(-1.8364,0.98162)--(-1.7414,0.93253)--(-1.6465,0.88190)--(-1.5515,0.83018)--(-1.4566,0.77782)--(-1.3616,0.72520)--(-1.2667,0.67266)--(-1.1717,0.62052)--(-1.0768,0.56908)--(-0.98182,0.51859)--(-0.88687,0.46930)--(-0.79192,0.42143)--(-0.69697,0.37518)--(-0.60202,0.33071)--(-0.50707,0.28821)--(-0.41212,0.24782)--(-0.31717,0.20966)--(-0.22222,0.17388)--(-0.12727,0.14058)--(-0.032323,0.10985)--(0.062626,0.081805)--(0.15758,0.056515)--(0.25253,0.034060)--(0.34747,0.014511)--(0.44242,-0.0020696)--(0.53737,-0.015623)--(0.63232,-0.026098)--(0.72727,-0.033446)--(0.82222,-0.037624)--(0.91717,-0.038593)--(1.0121,-0.036315)--(1.1071,-0.030760)--(1.2020,-0.021896)--(1.2970,-0.0096977)--(1.3919,0.0058604)--(1.4869,0.024800)--(1.5818,0.047142)--(1.6768,0.072905)--(1.7717,0.10211)--(1.8667,0.13476)--(1.9616,0.17088)--(2.0566,0.21048)--(2.1515,0.25357)--(2.2465,0.30016)--(2.3414,0.35026)--(2.4364,0.40388)--(2.5313,0.46103)--(2.6263,0.52171)--(2.7212,0.58593)--(2.8162,0.65370)--(2.9111,0.72503)--(3.0061,0.79991)--(3.1010,0.87835)--(3.1960,0.96035)--(3.2909,1.0459)--(3.3859,1.1351)--(3.4808,1.2278)--(3.5758,1.3241)--(3.6707,1.4240)--(3.7657,1.5275)--(3.8606,1.6345)--(3.9556,1.7451)--(4.0505,1.8594)--(4.1455,1.9772)--(4.2404,2.0986)--(4.3354,2.2236)--(4.4303,2.3522)--(4.5253,2.4844)--(4.6202,2.6202)--(4.7151,2.7596)--(4.8101,2.9026)--(4.9051,3.0491)--(5.0000,3.1993);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.5244091667,-2.000000000) node {$ -20 $};
+\draw (-0.52441,-2.0000) node {$ -20 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.5244091667,-1.000000000) node {$ -10 $};
+\draw (-0.52441,-1.0000) node {$ -10 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.3824996667,1.000000000) node {$ 10 $};
+\draw (-0.38250,1.0000) node {$ 10 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.3824996667,2.000000000) node {$ 20 $};
+\draw (-0.38250,2.0000) node {$ 20 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.3824996667,3.000000000) node {$ 30 $};
+\draw (-0.38250,3.0000) node {$ 30 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Osculateur.pstricks b/auto/pictures_tex/Fig_Osculateur.pstricks
index ba1e0cdc4..3136c04e9 100644
--- a/auto/pictures_tex/Fig_Osculateur.pstricks
+++ b/auto/pictures_tex/Fig_Osculateur.pstricks
@@ -65,18 +65,18 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (-19.37824843,2.000000000) node [rotate=0] {$\bullet$};
+\draw []  (-19.378,2.0000) node [rotate=0] {$\bullet$};
 
 \draw [] (-19.068,1.6899)--(-19.069,1.7096)--(-19.071,1.7292)--(-19.074,1.7486)--(-19.078,1.7678)--(-19.084,1.7867)--(-19.090,1.8052)--(-19.098,1.8232)--(-19.107,1.8407)--(-19.117,1.8576)--(-19.129,1.8738)--(-19.141,1.8892)--(-19.154,1.9039)--(-19.168,1.9177)--(-19.183,1.9306)--(-19.198,1.9425)--(-19.215,1.9534)--(-19.232,1.9632)--(-19.249,1.9720)--(-19.268,1.9796)--(-19.286,1.9860)--(-19.305,1.9913)--(-19.324,1.9953)--(-19.344,1.9981)--(-19.363,1.9996)--(-19.383,2.0000)--(-19.403,1.9990)--(-19.422,1.9968)--(-19.442,1.9934)--(-19.461,1.9888)--(-19.480,1.9829)--(-19.498,1.9759)--(-19.516,1.9677)--(-19.533,1.9585)--(-19.550,1.9481)--(-19.566,1.9367)--(-19.581,1.9243)--(-19.596,1.9109)--(-19.609,1.8967)--(-19.622,1.8816)--(-19.634,1.8658)--(-19.644,1.8492)--(-19.654,1.8320)--(-19.662,1.8142)--(-19.670,1.7960)--(-19.676,1.7773)--(-19.681,1.7582)--(-19.684,1.7389)--(-19.687,1.7194)--(-19.688,1.6998)--(-19.688,1.6801)--(-19.687,1.6604)--(-19.684,1.6409)--(-19.681,1.6216)--(-19.676,1.6026)--(-19.670,1.5839)--(-19.662,1.5656)--(-19.654,1.5478)--(-19.644,1.5306)--(-19.634,1.5141)--(-19.622,1.4982)--(-19.609,1.4832)--(-19.596,1.4689)--(-19.581,1.4556)--(-19.566,1.4432)--(-19.550,1.4317)--(-19.533,1.4214)--(-19.516,1.4121)--(-19.498,1.4039)--(-19.480,1.3969)--(-19.461,1.3910)--(-19.442,1.3864)--(-19.422,1.3830)--(-19.403,1.3808)--(-19.383,1.3799)--(-19.363,1.3802)--(-19.344,1.3817)--(-19.324,1.3845)--(-19.305,1.3886)--(-19.286,1.3938)--(-19.268,1.4002)--(-19.249,1.4078)--(-19.232,1.4166)--(-19.215,1.4264)--(-19.198,1.4373)--(-19.183,1.4492)--(-19.168,1.4621)--(-19.154,1.4759)--(-19.141,1.4906)--(-19.129,1.5061)--(-19.117,1.5223)--(-19.107,1.5392)--(-19.098,1.5566)--(-19.090,1.5747)--(-19.084,1.5932)--(-19.078,1.6120)--(-19.074,1.6312)--(-19.071,1.6507)--(-19.069,1.6702)--(-19.068,1.6899);
-\draw []  (-19.37824843,2.000000000) node [rotate=0] {$\bullet$};
-\draw []  (-19.10672978,1.902113033) node [rotate=0] {$\bullet$};
+\draw []  (-19.378,2.0000) node [rotate=0] {$\bullet$};
+\draw []  (-19.107,1.9021) node [rotate=0] {$\bullet$};
 
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+\draw []  (-19.107,1.9021) node [rotate=0] {$\bullet$};
+\draw []  (-15.297,-1.9021) node [rotate=0] {$\bullet$};
 
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+\draw []  (-15.297,-1.9021) node [rotate=0] {$\bullet$};
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 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_PHTVjfk.pstricks b/auto/pictures_tex/Fig_PHTVjfk.pstricks
index 5b9ad53f5..833669663 100644
--- a/auto/pictures_tex/Fig_PHTVjfk.pstricks
+++ b/auto/pictures_tex/Fig_PHTVjfk.pstricks
@@ -79,28 +79,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-2.231453278) -- (0,2.231453278);
+\draw [,->,>=latex] (-2.2315,0) -- (2.2315,0);
+\draw [,->,>=latex] (0,-2.2315) -- (0,2.2315);
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diff --git a/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks b/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks
index 04928061f..2cc978bad 100644
--- a/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks
+++ b/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks
@@ -72,8 +72,8 @@
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-\draw []  (2.100000000,1.400000000) node [rotate=0] {$\bullet$};
-\draw (2.353454873,1.108881941) node {\( P\)};
+\draw []  (2.1000,1.4000) node [rotate=0] {$\bullet$};
+\draw (2.3535,1.1089) node {\( P\)};
 \draw [] (0,0) -- (3.00,2.00);
 \draw [style=dashed] (2.10,1.40) -- (0.100,4.40);
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_PONXooXYjEot.pstricks b/auto/pictures_tex/Fig_PONXooXYjEot.pstricks
index 94718c6f6..3adcf2189 100644
--- a/auto/pictures_tex/Fig_PONXooXYjEot.pstricks
+++ b/auto/pictures_tex/Fig_PONXooXYjEot.pstricks
@@ -79,22 +79,22 @@
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+\draw [,->,>=latex] (-2.4998,0) -- (2.4998,0);
+\draw [,->,>=latex] (0,-1.2068) -- (0,1.2068);
 %DEFAULT
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 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks b/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks
index d5e62e6d6..ab49695de 100644
--- a/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks
+++ b/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks
@@ -143,8 +143,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.997787144,0) -- (5.997787144,0);
-\draw [,->,>=latex] (0,-4.000000000) -- (0,4.000000000);
+\draw [,->,>=latex] (-5.9978,0) -- (5.9978,0);
+\draw [,->,>=latex] (0,-4.0000) -- (0,4.0000);
 %DEFAULT
 
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@@ -162,45 +162,45 @@
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 \draw [style=dashed] (3.30,-3.50) -- (3.30,3.50);
 \draw [style=dashed] (5.50,-3.50) -- (5.50,3.50);
-\draw (-5.497787144,-0.4207143333) node {$ -\frac{5}{2} \, \pi $};
+\draw (-5.4978,-0.42071) node {$ -\frac{5}{2} \, \pi $};
 \draw [] (-5.50,-0.100) -- (-5.50,0.100);
-\draw (-4.398229715,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-4.3982,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-4.40,-0.100) -- (-4.40,0.100);
-\draw (-3.298672286,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-3.2987,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.199114858,-0.3210296667) node {$ -\pi $};
+\draw (-2.1991,-0.32103) node {$ -\pi $};
 \draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.099557429,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.0996,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.099557429,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.0996,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.199114858,-0.2785761667) node {$ \pi $};
+\draw (2.1991,-0.27858) node {$ \pi $};
 \draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.298672286,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (3.2987,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (4.398229715,-0.3149246667) node {$ 2 \, \pi $};
+\draw (4.3982,-0.31492) node {$ 2 \, \pi $};
 \draw [] (4.40,-0.100) -- (4.40,0.100);
-\draw (5.497787144,-0.4207143333) node {$ \frac{5}{2} \, \pi $};
+\draw (5.4978,-0.42071) node {$ \frac{5}{2} \, \pi $};
 \draw [] (5.50,-0.100) -- (5.50,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks b/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks
index 4c3f272e9..e0da8fb27 100644
--- a/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks
+++ b/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.000000000,0);
-\draw (3.308599701,-0.2907082010) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.000000000,2.000000000);
-\draw (2.000000000,2.426736000) node {$b$};
-\draw []  (5.000000000,2.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (0,0) -- (3.0000,0);
+\draw (3.3086,-0.29071) node {$a$};
+\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw (2.0000,2.4267) node {$b$};
+\draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
 \draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
 \draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
 \draw [style=dashed] (2.00,2.00) -- (2.00,0);
-\draw (2.305148833,1.000000000) node {$h$};
-\draw (0.8061547663,0.3180777162) node {$\theta$};
+\draw (2.3051,1.0000) node {$h$};
+\draw (0.80616,0.31808) node {$\theta$};
 
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-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ParamTangente.pstricks b/auto/pictures_tex/Fig_ParamTangente.pstricks
index 20a074298..7a28085b0 100644
--- a/auto/pictures_tex/Fig_ParamTangente.pstricks
+++ b/auto/pictures_tex/Fig_ParamTangente.pstricks
@@ -87,35 +87,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.042356081,0) -- (3.556125852,0);
-\draw [,->,>=latex] (0,-3.875000000) -- (0,3.945386379);
+\draw [,->,>=latex] (-1.0424,0) -- (3.5561,0);
+\draw [,->,>=latex] (0,-3.8750) -- (0,3.9454);
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-\draw [color=red,->,>=latex] (2.716201962,2.504933430) -- (3.056125852,3.445386379);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw [color=red,->,>=latex] (2.5078,-1.9715) -- (2.1174,-1.0508);
+\draw [color=red,->,>=latex] (1.7548,-0.67544) -- (1.1001,0.080499);
+\draw [color=red,->,>=latex] (0.45465,-0.011748) -- (-0.54236,0.065536);
+\draw [color=red,->,>=latex] (1.0298,0.13650) -- (1.9590,0.50601);
+\draw [color=red,->,>=latex] (2.0918,1.1442) -- (2.6122,1.9981);
+\draw [color=red,->,>=latex] (2.7162,2.5049) -- (3.0561,3.4454);
+\draw (-1.0000,-0.32983) node {$ -1 $};
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
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-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
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-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_PartieEntiere.pstricks b/auto/pictures_tex/Fig_PartieEntiere.pstricks
index 418078f57..5820d4ebf 100644
--- a/auto/pictures_tex/Fig_PartieEntiere.pstricks
+++ b/auto/pictures_tex/Fig_PartieEntiere.pstricks
@@ -83,31 +83,31 @@
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+\draw [,->,>=latex] (-2.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,3.5000);
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-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Polirettangolo.pstricks b/auto/pictures_tex/Fig_Polirettangolo.pstricks
index de9a6ea44..8d5164762 100644
--- a/auto/pictures_tex/Fig_Polirettangolo.pstricks
+++ b/auto/pictures_tex/Fig_Polirettangolo.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -94,33 +94,57 @@
 \draw [style=dotted] (1.50,2.00) -- (1.50,1.00);
 \draw [style=dotted] (1.50,1.00) -- (0,1.00);
 \draw [style=dotted] (0,1.00) -- (0,2.00);
+
+% declaring the keys in tikz
+\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
+         hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
+% setting the default values
+\tikzset{hatchspread=3pt,
+         hatchthickness=0.4pt}
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.500,2.50) -- (2.00,2.50) -- (2.00,2.50) -- (2.00,2.00) -- (2.00,2.00) -- (0.500,2.00) -- (0.500,2.00) -- (0.500,2.50) --  cycle;
 \draw [style=dotted] (0.500,2.50) -- (2.00,2.50);
 \draw [style=dotted] (2.00,2.50) -- (2.00,2.00);
 \draw [style=dotted] (2.00,2.00) -- (0.500,2.00);
 \draw [style=dotted] (0.500,2.00) -- (0.500,2.50);
+
+% declaring the keys in tikz
+\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
+         hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
+% setting the default values
+\tikzset{hatchspread=3pt,
+         hatchthickness=0.4pt}
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,1.00) -- (3.00,1.00) -- (3.00,1.00) -- (3.00,0) -- (3.00,0) -- (2.00,0) -- (2.00,0) -- (2.00,1.00) --  cycle;
 \draw [style=dotted] (2.00,1.00) -- (3.00,1.00);
 \draw [style=dotted] (3.00,1.00) -- (3.00,0);
 \draw [style=dotted] (3.00,0) -- (2.00,0);
 \draw [style=dotted] (2.00,0) -- (2.00,1.00);
+
+% declaring the keys in tikz
+\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
+         hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
+% setting the default values
+\tikzset{hatchspread=3pt,
+         hatchthickness=0.4pt}
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,3.50) -- (3.50,3.50) -- (3.50,3.50) -- (3.50,1.50) -- (3.50,1.50) -- (2.00,1.50) -- (2.00,1.50) -- (2.00,3.50) --  cycle;
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 \draw [] (1.00,-0.100) -- (1.00,0.100);
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 \draw [] (2.00,-0.100) -- (2.00,0.100);
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 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 8 $};
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 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
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 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
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 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
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 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
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 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ProjPoly.pstricks b/auto/pictures_tex/Fig_ProjPoly.pstricks
index 0d22b5a65..2b8360aeb 100644
--- a/auto/pictures_tex/Fig_ProjPoly.pstricks
+++ b/auto/pictures_tex/Fig_ProjPoly.pstricks
@@ -111,56 +111,56 @@
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-\draw (-2.500000000,-0.3298256667) node {$ -5 $};
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 \draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -4 $};
+\draw (-2.0000,-0.32983) node {$ -4 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.500000000,-0.3298256667) node {$ -3 $};
+\draw (-1.5000,-0.32983) node {$ -3 $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -2 $};
+\draw (-1.0000,-0.32983) node {$ -2 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.5000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.50000,-0.32983) node {$ -1 $};
 \draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.5000000000,-0.3149246667) node {$ 1 $};
+\draw (0.50000,-0.31492) node {$ 1 $};
 \draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 2 $};
+\draw (1.0000,-0.31492) node {$ 2 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.500000000,-0.3149246667) node {$ 3 $};
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 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 4 $};
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 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.500000000,-0.3149246667) node {$ 5 $};
+\draw (2.5000,-0.31492) node {$ 5 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 6 $};
+\draw (3.0000,-0.31492) node {$ 6 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -6 $};
+\draw (-0.43316,-3.0000) node {$ -6 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.500000000) node {$ -5 $};
+\draw (-0.43316,-2.5000) node {$ -5 $};
 \draw [] (-0.100,-2.50) -- (0.100,-2.50);
-\draw (-0.4331593333,-2.000000000) node {$ -4 $};
+\draw (-0.43316,-2.0000) node {$ -4 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.500000000) node {$ -3 $};
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 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.4331593333,-1.000000000) node {$ -2 $};
+\draw (-0.43316,-1.0000) node {$ -2 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.4331593333,-0.5000000000) node {$ -1 $};
+\draw (-0.43316,-0.50000) node {$ -1 $};
 \draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.2912498333,0.5000000000) node {$ 1 $};
+\draw (-0.29125,0.50000) node {$ 1 $};
 \draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,1.500000000) node {$ 3 $};
+\draw (-0.29125,1.5000) node {$ 3 $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,2.500000000) node {$ 5 $};
+\draw (-0.29125,2.5000) node {$ 5 $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QCb.pstricks b/auto/pictures_tex/Fig_QCb.pstricks
index 5947cc800..389c36fef 100644
--- a/auto/pictures_tex/Fig_QCb.pstricks
+++ b/auto/pictures_tex/Fig_QCb.pstricks
@@ -71,15 +71,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw (-0.7996721667,1.000000000) node {\( xy\)};
-\draw (1.567623333,1.000000000) node {\( \sin(xy)\)};
-\draw (1.200327833,-1.000000000) node {\( xy\)};
-\draw (-0.7996721667,-1.000000000) node {\( xy\)};
+\draw (-0.79967,1.0000) node {\( xy\)};
+\draw (1.5676,1.0000) node {\( \sin(xy)\)};
+\draw (1.2003,-1.0000) node {\( xy\)};
+\draw (-0.79967,-1.0000) node {\( xy\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks b/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
index aee2e97e2..25a73757d 100644
--- a/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
+++ b/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
@@ -80,23 +80,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-4.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.000000000,-0.3247080000) node {\( A\)};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3247080000) node {\( B\)};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.32471) node {\( A\)};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.32471) node {\( B\)};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0.2856973562,-0.2661293562) node {\( O\)};
-\draw []  (0,2.828427125) node [rotate=0] {$\bullet$};
-\draw (0.2359663562,3.094556481) node {\( I\)};
+\draw (0.28570,-0.26613) node {\( O\)};
+\draw []  (0,2.8284) node [rotate=0] {$\bullet$};
+\draw (0.23597,3.0946) node {\( I\)};
 
 \draw [] (2.000,0)--(1.994,0.1903)--(1.976,0.3798)--(1.946,0.5678)--(1.904,0.7534)--(1.850,0.9361)--(1.785,1.115)--(1.709,1.289)--(1.622,1.459)--(1.524,1.622)--(1.416,1.779)--(1.298,1.928)--(1.171,2.070)--(1.036,2.204)--(0.8917,2.328)--(0.7402,2.444)--(0.5817,2.549)--(0.4168,2.644)--(0.2462,2.729)--(0.07066,2.802)--(-0.1092,2.865)--(-0.2927,2.915)--(-0.4791,2.954)--(-0.6675,2.982)--(-0.8573,2.997)--(-1.048,3.000)--(-1.238,2.991)--(-1.427,2.969)--(-1.614,2.936)--(-1.799,2.892)--(-1.981,2.835)--(-2.159,2.767)--(-2.332,2.688)--(-2.500,2.598)--(-2.662,2.498)--(-2.817,2.387)--(-2.965,2.267)--(-3.104,2.138)--(-3.236,2.000)--(-3.358,1.854)--(-3.471,1.701)--(-3.574,1.541)--(-3.667,1.375)--(-3.748,1.203)--(-3.819,1.026)--(-3.878,0.8452)--(-3.926,0.6609)--(-3.962,0.4740)--(-3.986,0.2852)--(-3.999,0.09518)--(-3.999,-0.09518)--(-3.986,-0.2852)--(-3.962,-0.4740)--(-3.926,-0.6609)--(-3.878,-0.8452)--(-3.819,-1.026)--(-3.748,-1.203)--(-3.667,-1.375)--(-3.574,-1.541)--(-3.471,-1.701)--(-3.358,-1.854)--(-3.236,-2.000)--(-3.104,-2.138)--(-2.965,-2.267)--(-2.817,-2.387)--(-2.662,-2.498)--(-2.500,-2.598)--(-2.332,-2.688)--(-2.159,-2.767)--(-1.981,-2.835)--(-1.799,-2.892)--(-1.614,-2.936)--(-1.427,-2.969)--(-1.238,-2.991)--(-1.048,-3.000)--(-0.8573,-2.997)--(-0.6675,-2.982)--(-0.4791,-2.954)--(-0.2927,-2.915)--(-0.1092,-2.865)--(0.07066,-2.802)--(0.2462,-2.729)--(0.4168,-2.644)--(0.5817,-2.549)--(0.7402,-2.444)--(0.8917,-2.328)--(1.036,-2.204)--(1.171,-2.070)--(1.298,-1.928)--(1.416,-1.779)--(1.524,-1.622)--(1.622,-1.459)--(1.709,-1.289)--(1.785,-1.115)--(1.850,-0.9361)--(1.904,-0.7534)--(1.946,-0.5678)--(1.976,-0.3798)--(1.994,-0.1903)--(2.000,0);
 
 \draw [] (4.000,0)--(3.994,0.1903)--(3.976,0.3798)--(3.946,0.5678)--(3.904,0.7534)--(3.850,0.9361)--(3.785,1.115)--(3.709,1.289)--(3.622,1.459)--(3.524,1.622)--(3.416,1.779)--(3.298,1.928)--(3.171,2.070)--(3.036,2.204)--(2.892,2.328)--(2.740,2.444)--(2.582,2.549)--(2.417,2.644)--(2.246,2.729)--(2.071,2.802)--(1.891,2.865)--(1.707,2.915)--(1.521,2.954)--(1.333,2.982)--(1.143,2.997)--(0.9524,3.000)--(0.7623,2.991)--(0.5731,2.969)--(0.3856,2.936)--(0.2006,2.892)--(0.01880,2.835)--(-0.1590,2.767)--(-0.3322,2.688)--(-0.5000,2.598)--(-0.6618,2.498)--(-0.8168,2.387)--(-0.9646,2.267)--(-1.104,2.138)--(-1.236,2.000)--(-1.358,1.854)--(-1.471,1.701)--(-1.574,1.541)--(-1.667,1.375)--(-1.748,1.203)--(-1.819,1.026)--(-1.878,0.8452)--(-1.926,0.6609)--(-1.962,0.4740)--(-1.986,0.2852)--(-1.998,0.09518)--(-1.998,-0.09518)--(-1.986,-0.2852)--(-1.962,-0.4740)--(-1.926,-0.6609)--(-1.878,-0.8452)--(-1.819,-1.026)--(-1.748,-1.203)--(-1.667,-1.375)--(-1.574,-1.541)--(-1.471,-1.701)--(-1.358,-1.854)--(-1.236,-2.000)--(-1.104,-2.138)--(-0.9646,-2.267)--(-0.8168,-2.387)--(-0.6618,-2.498)--(-0.5000,-2.598)--(-0.3322,-2.688)--(-0.1590,-2.767)--(0.01880,-2.835)--(0.2006,-2.892)--(0.3856,-2.936)--(0.5731,-2.969)--(0.7623,-2.991)--(0.9524,-3.000)--(1.143,-2.997)--(1.333,-2.982)--(1.521,-2.954)--(1.707,-2.915)--(1.891,-2.865)--(2.071,-2.802)--(2.246,-2.729)--(2.417,-2.644)--(2.582,-2.549)--(2.740,-2.444)--(2.892,-2.328)--(3.036,-2.204)--(3.171,-2.070)--(3.298,-1.928)--(3.416,-1.779)--(3.524,-1.622)--(3.622,-1.459)--(3.709,-1.289)--(3.785,-1.115)--(3.850,-0.9361)--(3.904,-0.7534)--(3.946,-0.5678)--(3.976,-0.3798)--(3.994,-0.1903)--(4.000,0);
-\draw []  (-0.7000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.9856975229,1.301615523) node {\( Q\)};
+\draw []  (-0.70000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.98570,1.3016) node {\( Q\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks b/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks
index b53a39525..21bc4eded 100644
--- a/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks
+++ b/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (1.000,2.925)--(1.061,2.844)--(1.121,2.765)--(1.182,2.687)--(1.242,2.611)--(1.303,2.537)--(1.364,2.464)--(1.424,2.393)--(1.485,2.323)--(1.545,2.256)--(1.606,2.189)--(1.667,2.125)--(1.727,2.062)--(1.788,2.001)--(1.848,1.942)--(1.909,1.884)--(1.970,1.827)--(2.030,1.773)--(2.091,1.720)--(2.152,1.669)--(2.212,1.619)--(2.273,1.571)--(2.333,1.525)--(2.394,1.480)--(2.455,1.437)--(2.515,1.396)--(2.576,1.356)--(2.636,1.318)--(2.697,1.282)--(2.758,1.247)--(2.818,1.214)--(2.879,1.183)--(2.939,1.153)--(3.000,1.125)--(3.061,1.099)--(3.121,1.074)--(3.182,1.051)--(3.242,1.029)--(3.303,1.009)--(3.364,0.9911)--(3.424,0.9746)--(3.485,0.9597)--(3.545,0.9465)--(3.606,0.9349)--(3.667,0.9250)--(3.727,0.9167)--(3.788,0.9101)--(3.848,0.9052)--(3.909,0.9019)--(3.970,0.9002)--(4.030,0.9002)--(4.091,0.9019)--(4.151,0.9052)--(4.212,0.9101)--(4.273,0.9167)--(4.333,0.9250)--(4.394,0.9349)--(4.455,0.9465)--(4.515,0.9597)--(4.576,0.9746)--(4.636,0.9911)--(4.697,1.009)--(4.758,1.029)--(4.818,1.051)--(4.879,1.074)--(4.939,1.099)--(5.000,1.125)--(5.061,1.153)--(5.121,1.183)--(5.182,1.214)--(5.242,1.247)--(5.303,1.282)--(5.364,1.318)--(5.424,1.356)--(5.485,1.396)--(5.545,1.437)--(5.606,1.480)--(5.667,1.525)--(5.727,1.571)--(5.788,1.619)--(5.849,1.669)--(5.909,1.720)--(5.970,1.773)--(6.030,1.827)--(6.091,1.884)--(6.151,1.942)--(6.212,2.001)--(6.273,2.062)--(6.333,2.125)--(6.394,2.189)--(6.455,2.256)--(6.515,2.323)--(6.576,2.393)--(6.636,2.464)--(6.697,2.537)--(6.758,2.611)--(6.818,2.687)--(6.879,2.765)--(6.939,2.844)--(7.000,2.925);
@@ -53,10 +53,10 @@
 \draw [color=blue] (2.174,0)--(2.211,0)--(2.248,0)--(2.285,0)--(2.322,0)--(2.359,0)--(2.396,0)--(2.432,0)--(2.469,0)--(2.506,0)--(2.543,0)--(2.580,0)--(2.617,0)--(2.654,0)--(2.691,0)--(2.728,0)--(2.764,0)--(2.801,0)--(2.838,0)--(2.875,0)--(2.912,0)--(2.949,0)--(2.986,0)--(3.023,0)--(3.059,0)--(3.096,0)--(3.133,0)--(3.170,0)--(3.207,0)--(3.244,0)--(3.281,0)--(3.318,0)--(3.355,0)--(3.391,0)--(3.428,0)--(3.465,0)--(3.502,0)--(3.539,0)--(3.576,0)--(3.613,0)--(3.650,0)--(3.686,0)--(3.723,0)--(3.760,0)--(3.797,0)--(3.834,0)--(3.871,0)--(3.908,0)--(3.945,0)--(3.982,0)--(4.018,0)--(4.055,0)--(4.092,0)--(4.129,0)--(4.166,0)--(4.203,0)--(4.240,0)--(4.277,0)--(4.314,0)--(4.350,0)--(4.387,0)--(4.424,0)--(4.461,0)--(4.498,0)--(4.535,0)--(4.572,0)--(4.609,0)--(4.645,0)--(4.682,0)--(4.719,0)--(4.756,0)--(4.793,0)--(4.830,0)--(4.867,0)--(4.904,0)--(4.941,0)--(4.977,0)--(5.014,0)--(5.051,0)--(5.088,0)--(5.125,0)--(5.162,0)--(5.199,0)--(5.236,0)--(5.272,0)--(5.309,0)--(5.346,0)--(5.383,0)--(5.420,0)--(5.457,0)--(5.494,0)--(5.531,0)--(5.568,0)--(5.604,0)--(5.641,0)--(5.678,0)--(5.715,0)--(5.752,0)--(5.789,0)--(5.826,0);
 \draw [] (2.17,0) -- (2.17,1.65);
 \draw [] (5.83,1.65) -- (5.83,0);
-\draw []  (2.174258142,0) node [rotate=0] {$\bullet$};
-\draw (2.174258142,-0.3785761667) node {$a$};
-\draw []  (5.825741858,0) node [rotate=0] {$\bullet$};
-\draw (5.825741858,-0.4267360000) node {$b$};
+\draw []  (2.1743,0) node [rotate=0] {$\bullet$};
+\draw (2.1743,-0.37858) node {$a$};
+\draw []  (5.8257,0) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (1.000,2.925)--(1.061,2.844)--(1.121,2.765)--(1.182,2.687)--(1.242,2.611)--(1.303,2.537)--(1.364,2.464)--(1.424,2.393)--(1.485,2.323)--(1.545,2.256)--(1.606,2.189)--(1.667,2.125)--(1.727,2.062)--(1.788,2.001)--(1.848,1.942)--(1.909,1.884)--(1.970,1.827)--(2.030,1.773)--(2.091,1.720)--(2.152,1.669)--(2.212,1.619)--(2.273,1.571)--(2.333,1.525)--(2.394,1.480)--(2.455,1.437)--(2.515,1.396)--(2.576,1.356)--(2.636,1.318)--(2.697,1.282)--(2.758,1.247)--(2.818,1.214)--(2.879,1.183)--(2.939,1.153)--(3.000,1.125)--(3.061,1.099)--(3.121,1.074)--(3.182,1.051)--(3.242,1.029)--(3.303,1.009)--(3.364,0.9911)--(3.424,0.9746)--(3.485,0.9597)--(3.545,0.9465)--(3.606,0.9349)--(3.667,0.9250)--(3.727,0.9167)--(3.788,0.9101)--(3.848,0.9052)--(3.909,0.9019)--(3.970,0.9002)--(4.030,0.9002)--(4.091,0.9019)--(4.151,0.9052)--(4.212,0.9101)--(4.273,0.9167)--(4.333,0.9250)--(4.394,0.9349)--(4.455,0.9465)--(4.515,0.9597)--(4.576,0.9746)--(4.636,0.9911)--(4.697,1.009)--(4.758,1.029)--(4.818,1.051)--(4.879,1.074)--(4.939,1.099)--(5.000,1.125)--(5.061,1.153)--(5.121,1.183)--(5.182,1.214)--(5.242,1.247)--(5.303,1.282)--(5.364,1.318)--(5.424,1.356)--(5.485,1.396)--(5.545,1.437)--(5.606,1.480)--(5.667,1.525)--(5.727,1.571)--(5.788,1.619)--(5.849,1.669)--(5.909,1.720)--(5.970,1.773)--(6.030,1.827)--(6.091,1.884)--(6.151,1.942)--(6.212,2.001)--(6.273,2.062)--(6.333,2.125)--(6.394,2.189)--(6.455,2.256)--(6.515,2.323)--(6.576,2.393)--(6.636,2.464)--(6.697,2.537)--(6.758,2.611)--(6.818,2.687)--(6.879,2.765)--(6.939,2.844)--(7.000,2.925);
@@ -107,10 +107,10 @@
 \draw [color=blue] (2.174,0)--(2.211,0)--(2.248,0)--(2.285,0)--(2.322,0)--(2.359,0)--(2.396,0)--(2.432,0)--(2.469,0)--(2.506,0)--(2.543,0)--(2.580,0)--(2.617,0)--(2.654,0)--(2.691,0)--(2.728,0)--(2.764,0)--(2.801,0)--(2.838,0)--(2.875,0)--(2.912,0)--(2.949,0)--(2.986,0)--(3.023,0)--(3.059,0)--(3.096,0)--(3.133,0)--(3.170,0)--(3.207,0)--(3.244,0)--(3.281,0)--(3.318,0)--(3.355,0)--(3.391,0)--(3.428,0)--(3.465,0)--(3.502,0)--(3.539,0)--(3.576,0)--(3.613,0)--(3.650,0)--(3.686,0)--(3.723,0)--(3.760,0)--(3.797,0)--(3.834,0)--(3.871,0)--(3.908,0)--(3.945,0)--(3.982,0)--(4.018,0)--(4.055,0)--(4.092,0)--(4.129,0)--(4.166,0)--(4.203,0)--(4.240,0)--(4.277,0)--(4.314,0)--(4.350,0)--(4.387,0)--(4.424,0)--(4.461,0)--(4.498,0)--(4.535,0)--(4.572,0)--(4.609,0)--(4.645,0)--(4.682,0)--(4.719,0)--(4.756,0)--(4.793,0)--(4.830,0)--(4.867,0)--(4.904,0)--(4.941,0)--(4.977,0)--(5.014,0)--(5.051,0)--(5.088,0)--(5.125,0)--(5.162,0)--(5.199,0)--(5.236,0)--(5.272,0)--(5.309,0)--(5.346,0)--(5.383,0)--(5.420,0)--(5.457,0)--(5.494,0)--(5.531,0)--(5.568,0)--(5.604,0)--(5.641,0)--(5.678,0)--(5.715,0)--(5.752,0)--(5.789,0)--(5.826,0);
 \draw [] (2.17,0) -- (2.17,1.65);
 \draw [] (5.83,1.65) -- (5.83,0);
-\draw []  (2.174258142,0) node [rotate=0] {$\bullet$};
-\draw (2.174258142,-0.3785761667) node {$a$};
-\draw []  (5.825741858,0) node [rotate=0] {$\bullet$};
-\draw (5.825741858,-0.4267360000) node {$b$};
+\draw []  (2.1743,0) node [rotate=0] {$\bullet$};
+\draw (2.1743,-0.37858) node {$a$};
+\draw []  (5.8257,0) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,8 +149,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (1.000,2.925)--(1.061,2.844)--(1.121,2.765)--(1.182,2.687)--(1.242,2.611)--(1.303,2.537)--(1.364,2.464)--(1.424,2.393)--(1.485,2.323)--(1.545,2.256)--(1.606,2.189)--(1.667,2.125)--(1.727,2.062)--(1.788,2.001)--(1.848,1.942)--(1.909,1.884)--(1.970,1.827)--(2.030,1.773)--(2.091,1.720)--(2.152,1.669)--(2.212,1.619)--(2.273,1.571)--(2.333,1.525)--(2.394,1.480)--(2.455,1.437)--(2.515,1.396)--(2.576,1.356)--(2.636,1.318)--(2.697,1.282)--(2.758,1.247)--(2.818,1.214)--(2.879,1.183)--(2.939,1.153)--(3.000,1.125)--(3.061,1.099)--(3.121,1.074)--(3.182,1.051)--(3.242,1.029)--(3.303,1.009)--(3.364,0.9911)--(3.424,0.9746)--(3.485,0.9597)--(3.545,0.9465)--(3.606,0.9349)--(3.667,0.9250)--(3.727,0.9167)--(3.788,0.9101)--(3.848,0.9052)--(3.909,0.9019)--(3.970,0.9002)--(4.030,0.9002)--(4.091,0.9019)--(4.151,0.9052)--(4.212,0.9101)--(4.273,0.9167)--(4.333,0.9250)--(4.394,0.9349)--(4.455,0.9465)--(4.515,0.9597)--(4.576,0.9746)--(4.636,0.9911)--(4.697,1.009)--(4.758,1.029)--(4.818,1.051)--(4.879,1.074)--(4.939,1.099)--(5.000,1.125)--(5.061,1.153)--(5.121,1.183)--(5.182,1.214)--(5.242,1.247)--(5.303,1.282)--(5.364,1.318)--(5.424,1.356)--(5.485,1.396)--(5.545,1.437)--(5.606,1.480)--(5.667,1.525)--(5.727,1.571)--(5.788,1.619)--(5.849,1.669)--(5.909,1.720)--(5.970,1.773)--(6.030,1.827)--(6.091,1.884)--(6.151,1.942)--(6.212,2.001)--(6.273,2.062)--(6.333,2.125)--(6.394,2.189)--(6.455,2.256)--(6.515,2.323)--(6.576,2.393)--(6.636,2.464)--(6.697,2.537)--(6.758,2.611)--(6.818,2.687)--(6.879,2.765)--(6.939,2.844)--(7.000,2.925);
@@ -161,10 +161,10 @@
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 \draw [] (2.17,1.65) -- (2.17,1.65);
 \draw [] (5.83,1.65) -- (5.83,1.65);
-\draw []  (2.174258142,0) node [rotate=0] {$\bullet$};
-\draw (2.174258142,-0.3785761667) node {$a$};
-\draw []  (5.825741858,0) node [rotate=0] {$\bullet$};
-\draw (5.825741858,-0.4267360000) node {$b$};
+\draw []  (2.1743,0) node [rotate=0] {$\bullet$};
+\draw (2.1743,-0.37858) node {$a$};
+\draw []  (5.8257,0) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QQa.pstricks b/auto/pictures_tex/Fig_QQa.pstricks
index 54142f0c4..b9cb733d7 100644
--- a/auto/pictures_tex/Fig_QQa.pstricks
+++ b/auto/pictures_tex/Fig_QQa.pstricks
@@ -33,8 +33,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (0.5000000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (0.50000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
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-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
@@ -102,8 +102,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
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@@ -136,8 +136,8 @@
 %GRID
 %PSTRICKS CODE
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-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.802585093) -- (0,1.235192735);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.8026) -- (0,1.2352);
 %DEFAULT
 
 \draw [color=red] plot [smooth,tension=1] coordinates {(0.100,-2.30)(0.0409,-2.20)(0.191,-2.11)(0.0654,-1.99)(0.218,-1.91)(0.0952,-1.79)(0.250,-1.72)(0.131,-1.60)(0.289,-1.53)(0.175,-1.40)(0.336,-1.33)(0.228,-1.20)(0.391,-1.15)(0.291,-1.01)(0.457,-0.962)(0.365,-0.817)(0.534,-0.783)(0.453,-0.631)(0.624,-0.609)(0.554,-0.452)(0.726,-0.443)(0.668,-0.281)(0.840,-0.286)(0.794,-0.120)(0.966,-0.137)(0.933,0.0318)(1.10,0.00217)(1.08,0.173)(1.25,0.133)(1.24,0.304)(1.40,0.253)(1.40,0.425)(1.57,0.365)(1.58,0.537)(1.73,0.469)(1.75,0.640)(1.90,0.565)(1.93,0.735)(2.00,0.693)};
@@ -171,8 +171,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
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@@ -206,8 +206,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
diff --git a/auto/pictures_tex/Fig_QXyVaKD.pstricks b/auto/pictures_tex/Fig_QXyVaKD.pstricks
index 8f57cfe01..c8f85cd29 100644
--- a/auto/pictures_tex/Fig_QXyVaKD.pstricks
+++ b/auto/pictures_tex/Fig_QXyVaKD.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
 %DEFAULT
 
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@@ -80,8 +80,8 @@
 \draw [color=cyan] (0,0) -- (0,0);
 \draw [color=cyan] (1.00,1.00) -- (1.00,1.00);
 \draw [color=red] (-0.500,-0.500) -- (1.50,1.50);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.354646868,0.6631599656) node {$P$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3546,0.66316) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QuelCote.pstricks b/auto/pictures_tex/Fig_QuelCote.pstricks
index 4a2b8ccb2..314ffc1de 100644
--- a/auto/pictures_tex/Fig_QuelCote.pstricks
+++ b/auto/pictures_tex/Fig_QuelCote.pstricks
@@ -87,16 +87,16 @@
 %DEFAULT
 \draw [color=blue] (-3.000,-1.750)--(-2.980,-1.720)--(-2.961,-1.691)--(-2.941,-1.661)--(-2.922,-1.631)--(-2.903,-1.601)--(-2.883,-1.572)--(-2.864,-1.541)--(-2.845,-1.511)--(-2.826,-1.481)--(-2.808,-1.451)--(-2.789,-1.421)--(-2.770,-1.390)--(-2.752,-1.360)--(-2.734,-1.329)--(-2.715,-1.298)--(-2.697,-1.268)--(-2.679,-1.237)--(-2.661,-1.206)--(-2.644,-1.175)--(-2.626,-1.143)--(-2.608,-1.112)--(-2.591,-1.081)--(-2.574,-1.049)--(-2.557,-1.017)--(-2.540,-0.9856)--(-2.523,-0.9537)--(-2.506,-0.9216)--(-2.490,-0.8893)--(-2.474,-0.8570)--(-2.457,-0.8244)--(-2.441,-0.7918)--(-2.426,-0.7589)--(-2.410,-0.7259)--(-2.394,-0.6927)--(-2.379,-0.6594)--(-2.364,-0.6258)--(-2.349,-0.5921)--(-2.334,-0.5582)--(-2.320,-0.5240)--(-2.306,-0.4897)--(-2.292,-0.4551)--(-2.278,-0.4203)--(-2.264,-0.3853)--(-2.251,-0.3501)--(-2.238,-0.3145)--(-2.225,-0.2788)--(-2.212,-0.2427)--(-2.200,-0.2064)--(-2.187,-0.1697)--(-2.176,-0.1328)--(-2.164,-0.09554)--(-2.153,-0.05796)--(-2.142,-0.02003)--(-2.131,0.01826)--(-2.121,0.05692)--(-2.111,0.09597)--(-2.101,0.1354)--(-2.092,0.1753)--(-2.083,0.2156)--(-2.074,0.2564)--(-2.066,0.2976)--(-2.058,0.3394)--(-2.051,0.3817)--(-2.044,0.4245)--(-2.037,0.4679)--(-2.031,0.5119)--(-2.026,0.5565)--(-2.021,0.6018)--(-2.016,0.6478)--(-2.012,0.6944)--(-2.009,0.7419)--(-2.006,0.7901)--(-2.003,0.8392)--(-2.002,0.8892)--(-2.000,0.9401)--(-2.000,0.9919)--(-2.000,1.045)--(-2.001,1.099)--(-2.003,1.154)--(-2.005,1.210)--(-2.009,1.268)--(-2.013,1.327)--(-2.018,1.387)--(-2.024,1.449)--(-2.031,1.512)--(-2.038,1.578)--(-2.047,1.645)--(-2.057,1.713)--(-2.068,1.784)--(-2.081,1.857)--(-2.094,1.933)--(-2.109,2.011)--(-2.126,2.091)--(-2.143,2.174)--(-2.163,2.261)--(-2.184,2.350)--(-2.206,2.443)--(-2.231,2.540)--(-2.257,2.641);
 \draw [color=brown,style=dashed] (-2.45,-1.67) -- (-1.69,2.26);
-\draw [color=brown,->,>=latex] (-2.066666667,0.2944444444) -- (-1.686622854,2.258004146);
-\draw (-0.8272048535,2.258004146) node {$\gamma'(t)$};
-\draw [,->,>=latex] (-2.066666667,0.2944444444) -- (-3.316056762,1.856182063);
-\draw (-2.711760061,2.254701931) node {$\gamma''(t)$};
-\draw [color=green,->,>=latex] (-2.066666667,0.2944444444) -- (-0.1031069655,-0.08559936869);
-\draw (-0.1031069655,0.3968556313) node {$n(t)$};
-\draw [color=green,style=dashed,->,>=latex] (-2.066666667,0.2944444444) -- (-4.030226368,0.6744882576);
-\draw (-4.030226368,1.156943258) node {$-n(t)$};
-\draw []  (-2.066666667,0.2944444444) node [rotate=0] {$\bullet$};
-\draw (-1.712019799,-0.04239558991) node {$P$};
+\draw [color=brown,->,>=latex] (-2.0667,0.29444) -- (-1.6866,2.2580);
+\draw (-0.82720,2.2580) node {$\gamma'(t)$};
+\draw [,->,>=latex] (-2.0667,0.29444) -- (-3.3161,1.8562);
+\draw (-2.7118,2.2547) node {$\gamma''(t)$};
+\draw [color=green,->,>=latex] (-2.0667,0.29444) -- (-0.10311,-0.085599);
+\draw (-0.10311,0.39686) node {$n(t)$};
+\draw [color=green,style=dashed,->,>=latex] (-2.0667,0.29444) -- (-4.0302,0.67449);
+\draw (-4.0302,1.1569) node {$-n(t)$};
+\draw []  (-2.0667,0.29444) node [rotate=0] {$\bullet$};
+\draw (-1.7120,-0.042396) node {$P$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RGjjpwF.pstricks b/auto/pictures_tex/Fig_RGjjpwF.pstricks
index eb1ded532..055d4425c 100644
--- a/auto/pictures_tex/Fig_RGjjpwF.pstricks
+++ b/auto/pictures_tex/Fig_RGjjpwF.pstricks
@@ -75,15 +75,15 @@
 \draw [style=dotted] (2.00,0) -- (3.00,0);
 \draw [] (3.00,0) -- (4.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3149246667) node {$1$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2000000000) node {};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.2000000000) node {};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.2000000000) node {};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.2000000000) node {};
+\draw (0,0.31492) node {$1$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.20000) node {};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.20000) node {};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.20000) node {};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.20000) node {};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RLuqsrr.pstricks b/auto/pictures_tex/Fig_RLuqsrr.pstricks
index ae4cd3b5d..f50e8f6d6 100644
--- a/auto/pictures_tex/Fig_RLuqsrr.pstricks
+++ b/auto/pictures_tex/Fig_RLuqsrr.pstricks
@@ -87,24 +87,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185311,0);
-\draw [,->,>=latex] (0,-0.9138130496) -- (0,2.914169059);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-0.91381) -- (0,2.9142);
 %DEFAULT
 
 \draw [color=red] (0,0)--(0.06347,0.1346)--(0.1269,0.2832)--(0.1904,0.4433)--(0.2539,0.6124)--(0.3173,0.7876)--(0.3808,0.9663)--(0.4443,1.146)--(0.5077,1.322)--(0.5712,1.494)--(0.6347,1.658)--(0.6981,1.811)--(0.7616,1.951)--(0.8251,2.076)--(0.8885,2.184)--(0.9520,2.272)--(1.015,2.340)--(1.079,2.387)--(1.142,2.411)--(1.206,2.412)--(1.269,2.391)--(1.333,2.347)--(1.396,2.282)--(1.460,2.196)--(1.523,2.091)--(1.587,1.968)--(1.650,1.829)--(1.714,1.678)--(1.777,1.515)--(1.841,1.344)--(1.904,1.168)--(1.967,0.9888)--(2.031,0.8098)--(2.094,0.6340)--(2.158,0.4640)--(2.221,0.3026)--(2.285,0.1525)--(2.348,0.01599)--(2.412,-0.1047)--(2.475,-0.2076)--(2.539,-0.2910)--(2.602,-0.3537)--(2.666,-0.3946)--(2.729,-0.4131)--(2.793,-0.4088)--(2.856,-0.3819)--(2.919,-0.3327)--(2.983,-0.2621)--(3.046,-0.1712)--(3.110,-0.06141)--(3.173,0.06544)--(3.237,0.2073)--(3.300,0.3620)--(3.364,0.5269)--(3.427,0.6994)--(3.491,0.8767)--(3.554,1.056)--(3.618,1.235)--(3.681,1.409)--(3.745,1.577)--(3.808,1.736)--(3.871,1.883)--(3.935,2.016)--(3.998,2.132)--(4.062,2.230)--(4.125,2.309)--(4.189,2.366)--(4.252,2.401)--(4.316,2.414)--(4.379,2.404)--(4.443,2.372)--(4.506,2.317)--(4.570,2.241)--(4.633,2.145)--(4.697,2.031)--(4.760,1.900)--(4.823,1.755)--(4.887,1.598)--(4.950,1.431)--(5.014,1.257)--(5.077,1.078)--(5.141,0.8991)--(5.204,0.7214)--(5.268,0.5481)--(5.331,0.3821)--(5.395,0.2260)--(5.458,0.08240)--(5.522,-0.04645)--(5.585,-0.1585)--(5.648,-0.2518)--(5.712,-0.3250)--(5.775,-0.3769)--(5.839,-0.4067)--(5.902,-0.4138)--(5.966,-0.3982)--(6.029,-0.3600)--(6.093,-0.3000)--(6.156,-0.2191)--(6.220,-0.1185)--(6.283,0);
 
 \draw [color=blue] (0,0)--(0.06347,0.008045)--(0.1269,0.03205)--(0.1904,0.07163)--(0.2539,0.1262)--(0.3173,0.1947)--(0.3808,0.2763)--(0.4443,0.3694)--(0.5077,0.4728)--(0.5712,0.5846)--(0.6347,0.7031)--(0.6981,0.8264)--(0.7616,0.9524)--(0.8251,1.079)--(0.8885,1.205)--(0.9520,1.327)--(1.015,1.444)--(1.079,1.554)--(1.142,1.655)--(1.206,1.745)--(1.269,1.824)--(1.333,1.889)--(1.396,1.940)--(1.460,1.975)--(1.523,1.995)--(1.587,1.999)--(1.650,1.987)--(1.714,1.959)--(1.777,1.916)--(1.841,1.858)--(1.904,1.786)--(1.967,1.701)--(2.031,1.606)--(2.094,1.500)--(2.158,1.386)--(2.221,1.266)--(2.285,1.142)--(2.348,1.016)--(2.412,0.8892)--(2.475,0.7642)--(2.539,0.6431)--(2.602,0.5277)--(2.666,0.4199)--(2.729,0.3215)--(2.793,0.2340)--(2.856,0.1587)--(2.919,0.09707)--(2.983,0.04993)--(3.046,0.01807)--(3.110,0.002013)--(3.173,0.002013)--(3.237,0.01807)--(3.300,0.04993)--(3.364,0.09707)--(3.427,0.1587)--(3.491,0.2340)--(3.554,0.3215)--(3.618,0.4199)--(3.681,0.5277)--(3.745,0.6431)--(3.808,0.7642)--(3.871,0.8892)--(3.935,1.016)--(3.998,1.142)--(4.062,1.266)--(4.125,1.386)--(4.189,1.500)--(4.252,1.606)--(4.316,1.701)--(4.379,1.786)--(4.443,1.858)--(4.506,1.916)--(4.570,1.959)--(4.633,1.987)--(4.697,1.999)--(4.760,1.995)--(4.823,1.975)--(4.887,1.940)--(4.950,1.889)--(5.014,1.824)--(5.077,1.745)--(5.141,1.655)--(5.204,1.554)--(5.268,1.444)--(5.331,1.327)--(5.395,1.205)--(5.458,1.079)--(5.522,0.9524)--(5.585,0.8264)--(5.648,0.7031)--(5.712,0.5846)--(5.775,0.4728)--(5.839,0.3694)--(5.902,0.2763)--(5.966,0.1947)--(6.029,0.1262)--(6.093,0.07163)--(6.156,0.03205)--(6.220,0.008045)--(6.283,0);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks b/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks
index 7103486be..204852b2e 100644
--- a/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks
+++ b/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,0);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,3.000000000);
-\draw (1.896467667,1.500000000) node {$a$};
-\draw []  (0,3.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
+\draw (1.8965,1.5000) node {$a$};
+\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
 \draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.1430327019,3.634515621) node {$R$};
+\draw (0.14303,3.6345) node {$R$};
 \draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
 
 %OTHER STUFF
diff --git a/auto/pictures_tex/Fig_RQsQKTl.pstricks b/auto/pictures_tex/Fig_RQsQKTl.pstricks
index d688dbefc..adc854ce0 100644
--- a/auto/pictures_tex/Fig_RQsQKTl.pstricks
+++ b/auto/pictures_tex/Fig_RQsQKTl.pstricks
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3149246667) node {\( 1\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
+\draw (0,0.31492) node {\( 1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks b/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks
index 8c7d4a199..27241d0e2 100644
--- a/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks
+++ b/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks
@@ -65,8 +65,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -80,12 +80,12 @@
 \draw [color=blue,style=solid] (1.000,5.000)--(1.030,5.060)--(1.061,5.118)--(1.091,5.174)--(1.121,5.228)--(1.152,5.280)--(1.182,5.331)--(1.212,5.379)--(1.242,5.426)--(1.273,5.471)--(1.303,5.514)--(1.333,5.556)--(1.364,5.595)--(1.394,5.633)--(1.424,5.669)--(1.455,5.702)--(1.485,5.735)--(1.515,5.765)--(1.545,5.793)--(1.576,5.820)--(1.606,5.845)--(1.636,5.868)--(1.667,5.889)--(1.697,5.908)--(1.727,5.926)--(1.758,5.941)--(1.788,5.955)--(1.818,5.967)--(1.848,5.977)--(1.879,5.985)--(1.909,5.992)--(1.939,5.996)--(1.970,5.999)--(2.000,6.000)--(2.030,5.999)--(2.061,5.996)--(2.091,5.992)--(2.121,5.985)--(2.152,5.977)--(2.182,5.967)--(2.212,5.955)--(2.242,5.941)--(2.273,5.926)--(2.303,5.908)--(2.333,5.889)--(2.364,5.868)--(2.394,5.845)--(2.424,5.820)--(2.455,5.793)--(2.485,5.765)--(2.515,5.735)--(2.545,5.702)--(2.576,5.669)--(2.606,5.633)--(2.636,5.595)--(2.667,5.556)--(2.697,5.514)--(2.727,5.471)--(2.758,5.426)--(2.788,5.379)--(2.818,5.331)--(2.848,5.280)--(2.879,5.228)--(2.909,5.174)--(2.939,5.118)--(2.970,5.060)--(3.000,5.000)--(3.030,4.938)--(3.061,4.875)--(3.091,4.810)--(3.121,4.743)--(3.152,4.674)--(3.182,4.603)--(3.212,4.531)--(3.242,4.456)--(3.273,4.380)--(3.303,4.302)--(3.333,4.222)--(3.364,4.141)--(3.394,4.057)--(3.424,3.972)--(3.455,3.884)--(3.485,3.795)--(3.515,3.704)--(3.545,3.612)--(3.576,3.517)--(3.606,3.421)--(3.636,3.322)--(3.667,3.222)--(3.697,3.120)--(3.727,3.017)--(3.758,2.911)--(3.788,2.803)--(3.818,2.694)--(3.848,2.583)--(3.879,2.470)--(3.909,2.355)--(3.939,2.239)--(3.970,2.120)--(4.000,2.000);
 \draw [style=dashed] (1.00,5.00) -- (1.00,2.91);
 \draw [style=dashed] (4.00,1.04) -- (4.00,2.00);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3785761667) node {$a$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.4267360000) node {$b$};
-\draw (2.128725951,1.316179486) node {$g_1$};
-\draw (2.878345368,6.076194201) node {$g_2$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.37858) node {$a$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.42674) node {$b$};
+\draw (2.1287,1.3162) node {$g_1$};
+\draw (2.8783,6.0762) node {$g_2$};
 \draw [style=dotted] (1.00,2.91) -- (1.00,0);
 \draw [style=dotted] (4.00,1.04) -- (4.00,0);
 
@@ -154,8 +154,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -168,12 +168,12 @@
 \draw [style=dashed] (1.24,5.00) -- (6.00,5.00);
 \draw [style=dashed] (2.00,1.00) -- (5.11,1.00);
 \draw [color=blue] (5.111,1.000)--(5.102,1.040)--(5.094,1.081)--(5.086,1.121)--(5.078,1.162)--(5.071,1.202)--(5.064,1.242)--(5.057,1.283)--(5.051,1.323)--(5.045,1.364)--(5.039,1.404)--(5.034,1.444)--(5.029,1.485)--(5.025,1.525)--(5.021,1.566)--(5.017,1.606)--(5.014,1.646)--(5.011,1.687)--(5.008,1.727)--(5.006,1.768)--(5.004,1.808)--(5.003,1.848)--(5.001,1.889)--(5.001,1.929)--(5.000,1.970)--(5.000,2.010)--(5.000,2.051)--(5.001,2.091)--(5.002,2.131)--(5.003,2.172)--(5.005,2.212)--(5.007,2.253)--(5.010,2.293)--(5.012,2.333)--(5.016,2.374)--(5.019,2.414)--(5.023,2.455)--(5.027,2.495)--(5.032,2.535)--(5.037,2.576)--(5.042,2.616)--(5.048,2.657)--(5.054,2.697)--(5.060,2.737)--(5.067,2.778)--(5.074,2.818)--(5.082,2.859)--(5.090,2.899)--(5.098,2.939)--(5.107,2.980)--(5.116,3.020)--(5.125,3.061)--(5.135,3.101)--(5.145,3.141)--(5.155,3.182)--(5.166,3.222)--(5.177,3.263)--(5.189,3.303)--(5.201,3.343)--(5.213,3.384)--(5.225,3.424)--(5.238,3.465)--(5.252,3.505)--(5.265,3.545)--(5.279,3.586)--(5.294,3.626)--(5.309,3.667)--(5.324,3.707)--(5.339,3.747)--(5.355,3.788)--(5.371,3.828)--(5.388,3.869)--(5.405,3.909)--(5.422,3.949)--(5.440,3.990)--(5.458,4.030)--(5.476,4.071)--(5.495,4.111)--(5.514,4.151)--(5.534,4.192)--(5.554,4.232)--(5.574,4.273)--(5.594,4.313)--(5.615,4.354)--(5.637,4.394)--(5.658,4.434)--(5.680,4.475)--(5.703,4.515)--(5.726,4.556)--(5.749,4.596)--(5.772,4.636)--(5.796,4.677)--(5.820,4.717)--(5.845,4.758)--(5.870,4.798)--(5.895,4.838)--(5.921,4.879)--(5.947,4.919)--(5.973,4.960)--(6.000,5.000);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.3789780000,1.000000000) node {$c$};
-\draw []  (0,5.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.3949888333,5.000000000) node {$d$};
-\draw (2.448007760,2.730627410) node {$h_1$};
-\draw (5.595426611,3.000000000) node {$h_2$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.37898,1.0000) node {$c$};
+\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.39499,5.0000) node {$d$};
+\draw (2.4480,2.7306) node {$h_1$};
+\draw (5.5954,3.0000) node {$h_2$};
 \draw [style=dotted] (2.00,1.00) -- (0,1.00);
 \draw [style=dotted] (1.24,5.00) -- (0,5.00);
 
diff --git a/auto/pictures_tex/Fig_SBTooEasQsT.pstricks b/auto/pictures_tex/Fig_SBTooEasQsT.pstricks
index f972ddd12..a50ca2224 100644
--- a/auto/pictures_tex/Fig_SBTooEasQsT.pstricks
+++ b/auto/pictures_tex/Fig_SBTooEasQsT.pstricks
@@ -165,8 +165,8 @@
 \draw [color=gray,style=solid] (-4.00,3.50) -- (2.00,3.50);
 \draw [color=gray,style=solid] (-4.00,4.00) -- (2.00,4.00);
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-4.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=blue] (-4.0000,-0.0091578)--(-3.9394,-0.0097300)--(-3.8788,-0.010338)--(-3.8182,-0.010984)--(-3.7576,-0.011670)--(-3.6970,-0.012399)--(-3.6364,-0.013174)--(-3.5758,-0.013997)--(-3.5152,-0.014872)--(-3.4545,-0.015801)--(-3.3939,-0.016788)--(-3.3333,-0.017837)--(-3.2727,-0.018951)--(-3.2121,-0.020136)--(-3.1515,-0.021394)--(-3.0909,-0.022730)--(-3.0303,-0.024150)--(-2.9697,-0.025659)--(-2.9091,-0.027263)--(-2.8485,-0.028966)--(-2.7879,-0.030776)--(-2.7273,-0.032699)--(-2.6667,-0.034742)--(-2.6061,-0.036912)--(-2.5455,-0.039219)--(-2.4848,-0.041669)--(-2.4242,-0.044273)--(-2.3636,-0.047039)--(-2.3030,-0.049978)--(-2.2424,-0.053100)--(-2.1818,-0.056418)--(-2.1212,-0.059943)--(-2.0606,-0.063688)--(-2.0000,-0.067668)--(-1.9394,-0.071895)--(-1.8788,-0.076388)--(-1.8182,-0.081160)--(-1.7576,-0.086231)--(-1.6970,-0.091619)--(-1.6364,-0.097343)--(-1.5758,-0.10343)--(-1.5152,-0.10989)--(-1.4545,-0.11675)--(-1.3939,-0.12405)--(-1.3333,-0.13180)--(-1.2727,-0.14003)--(-1.2121,-0.14878)--(-1.1515,-0.15808)--(-1.0909,-0.16796)--(-1.0303,-0.17845)--(-0.96970,-0.18960)--(-0.90909,-0.20145)--(-0.84848,-0.21403)--(-0.78788,-0.22740)--(-0.72727,-0.24161)--(-0.66667,-0.25671)--(-0.60606,-0.27275)--(-0.54545,-0.28979)--(-0.48485,-0.30790)--(-0.42424,-0.32713)--(-0.36364,-0.34757)--(-0.30303,-0.36929)--(-0.24242,-0.39236)--(-0.18182,-0.41688)--(-0.12121,-0.44292)--(-0.060606,-0.47060)--(0,-0.50000)--(0.060606,-0.53124)--(0.12121,-0.56443)--(0.18182,-0.59970)--(0.24242,-0.63717)--(0.30303,-0.67698)--(0.36364,-0.71928)--(0.42424,-0.76422)--(0.48485,-0.81196)--(0.54545,-0.86270)--(0.60606,-0.91660)--(0.66667,-0.97387)--(0.72727,-1.0347)--(0.78788,-1.0994)--(0.84848,-1.1681)--(0.90909,-1.2410)--(0.96970,-1.3186)--(1.0303,-1.4010)--(1.0909,-1.4885)--(1.1515,-1.5815)--(1.2121,-1.6803)--(1.2727,-1.7853)--(1.3333,-1.8968)--(1.3939,-2.0154)--(1.4545,-2.1413)--(1.5152,-2.2751)--(1.5758,-2.4172)--(1.6364,-2.5682)--(1.6970,-2.7287)--(1.7576,-2.8992)--(1.8182,-3.0803)--(1.8788,-3.2728)--(1.9394,-3.4773)--(2.0000,-3.6945);
@@ -199,33 +199,33 @@
 
 \draw [color=blue] (-4.0000,0.0091578)--(-3.9394,0.0097300)--(-3.8788,0.010338)--(-3.8182,0.010984)--(-3.7576,0.011670)--(-3.6970,0.012399)--(-3.6364,0.013174)--(-3.5758,0.013997)--(-3.5152,0.014872)--(-3.4545,0.015801)--(-3.3939,0.016788)--(-3.3333,0.017837)--(-3.2727,0.018951)--(-3.2121,0.020136)--(-3.1515,0.021394)--(-3.0909,0.022730)--(-3.0303,0.024150)--(-2.9697,0.025659)--(-2.9091,0.027263)--(-2.8485,0.028966)--(-2.7879,0.030776)--(-2.7273,0.032699)--(-2.6667,0.034742)--(-2.6061,0.036912)--(-2.5455,0.039219)--(-2.4848,0.041669)--(-2.4242,0.044273)--(-2.3636,0.047039)--(-2.3030,0.049978)--(-2.2424,0.053100)--(-2.1818,0.056418)--(-2.1212,0.059943)--(-2.0606,0.063688)--(-2.0000,0.067668)--(-1.9394,0.071895)--(-1.8788,0.076388)--(-1.8182,0.081160)--(-1.7576,0.086231)--(-1.6970,0.091619)--(-1.6364,0.097343)--(-1.5758,0.10343)--(-1.5152,0.10989)--(-1.4545,0.11675)--(-1.3939,0.12405)--(-1.3333,0.13180)--(-1.2727,0.14003)--(-1.2121,0.14878)--(-1.1515,0.15808)--(-1.0909,0.16796)--(-1.0303,0.17845)--(-0.96970,0.18960)--(-0.90909,0.20145)--(-0.84848,0.21403)--(-0.78788,0.22740)--(-0.72727,0.24161)--(-0.66667,0.25671)--(-0.60606,0.27275)--(-0.54545,0.28979)--(-0.48485,0.30790)--(-0.42424,0.32713)--(-0.36364,0.34757)--(-0.30303,0.36929)--(-0.24242,0.39236)--(-0.18182,0.41688)--(-0.12121,0.44292)--(-0.060606,0.47060)--(0,0.50000)--(0.060606,0.53124)--(0.12121,0.56443)--(0.18182,0.59970)--(0.24242,0.63717)--(0.30303,0.67698)--(0.36364,0.71928)--(0.42424,0.76422)--(0.48485,0.81196)--(0.54545,0.86270)--(0.60606,0.91660)--(0.66667,0.97387)--(0.72727,1.0347)--(0.78788,1.0994)--(0.84848,1.1681)--(0.90909,1.2410)--(0.96970,1.3186)--(1.0303,1.4010)--(1.0909,1.4885)--(1.1515,1.5815)--(1.2121,1.6803)--(1.2727,1.7853)--(1.3333,1.8968)--(1.3939,2.0154)--(1.4545,2.1413)--(1.5152,2.2751)--(1.5758,2.4172)--(1.6364,2.5682)--(1.6970,2.7287)--(1.7576,2.8992)--(1.8182,3.0803)--(1.8788,3.2728)--(1.9394,3.4773)--(2.0000,3.6945);
 
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.5244091667,-4.000000000) node {$ -40 $};
+\draw (-0.52441,-4.0000) node {$ -40 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.5244091667,-3.000000000) node {$ -30 $};
+\draw (-0.52441,-3.0000) node {$ -30 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.5244091667,-2.000000000) node {$ -20 $};
+\draw (-0.52441,-2.0000) node {$ -20 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.5244091667,-1.000000000) node {$ -10 $};
+\draw (-0.52441,-1.0000) node {$ -10 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.3824996667,1.000000000) node {$ 10 $};
+\draw (-0.38250,1.0000) node {$ 10 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.3824996667,2.000000000) node {$ 20 $};
+\draw (-0.38250,2.0000) node {$ 20 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.3824996667,3.000000000) node {$ 30 $};
+\draw (-0.38250,3.0000) node {$ 30 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.3824996667,4.000000000) node {$ 40 $};
+\draw (-0.38250,4.0000) node {$ 40 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SFdgHdO.pstricks b/auto/pictures_tex/Fig_SFdgHdO.pstricks
index d0c79a559..aa92dc4e9 100644
--- a/auto/pictures_tex/Fig_SFdgHdO.pstricks
+++ b/auto/pictures_tex/Fig_SFdgHdO.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.538972437,0) -- (1.547391058,0);
-\draw [,->,>=latex] (0,-1.549697438) -- (0,1.549697438);
+\draw [,->,>=latex] (-1.5390,0) -- (1.5474,0);
+\draw [,->,>=latex] (0,-1.5497) -- (0,1.5497);
 %DEFAULT
 
 \draw [color=black] plot [smooth,tension=1] coordinates {(0.00100,1.00)(0.105,1.04)(0.189,0.931)(0.310,1.00)(0.370,0.875)(0.504,0.921)(0.537,0.784)(0.677,0.803)(0.682,0.661)(0.823,0.652)(0.800,0.513)(0.936,0.476)(0.886,0.344)(1.01,0.281)(0.936,0.162)(1.05,0.0740)(0.999,0.0447)};
@@ -78,12 +78,12 @@
 \draw [color=green] plot [smooth,tension=1] coordinates {(0.00100,-1.00)(0.105,-1.04)(0.189,-0.931)(0.310,-1.00)(0.370,-0.875)(0.504,-0.921)(0.537,-0.784)(0.677,-0.803)(0.682,-0.661)(0.823,-0.652)(0.800,-0.513)(0.936,-0.476)(0.886,-0.344)(1.01,-0.281)(0.936,-0.162)(1.05,-0.0740)(0.999,-0.0447)};
 
 \draw [color=black] plot [smooth,tension=1] coordinates {(-0.999,-0.0447)(-1.04,-0.152)(-0.922,-0.230)(-0.988,-0.356)(-0.857,-0.410)(-0.898,-0.545)(-0.759,-0.571)(-0.772,-0.712)(-0.630,-0.711)(-0.615,-0.851)(-0.476,-0.822)(-0.433,-0.957)(-0.303,-0.900)(-0.233,-1.02)(-0.118,-0.943)(-0.0252,-1.05)(-0.00100,-1.00)};
-\draw [color=red]  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw [color=red]  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (2.049499592,-0.3650902010) node {$K_H$};
-\draw (2.049499592,-0.3650902010) node {$K_H$};
+\draw [color=red]  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw (2.0495,-0.36509) node {$K_H$};
+\draw (2.0495,-0.36509) node {$K_H$};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks b/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks
index fd4d556f1..8f3b3168c 100644
--- a/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks
+++ b/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks
@@ -65,226 +65,226 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.000000000,-4.000000000) -- (-4.044194174,-4.044194174);
-\draw [,->,>=latex] (-4.000000000,-3.428571429) -- (-4.054711138,-3.475466689);
-\draw [,->,>=latex] (-4.000000000,-2.857142857) -- (-4.067352939,-2.905252100);
-\draw [,->,>=latex] (-4.000000000,-2.285714286) -- (-4.081815219,-2.332465840);
-\draw [,->,>=latex] (-4.000000000,-1.714285714) -- (-4.097064885,-1.755884951);
-\draw [,->,>=latex] (-4.000000000,-1.142857143) -- (-4.111119513,-1.174605575);
-\draw [,->,>=latex] (-4.000000000,-0.5714285714) -- (-4.121268813,-0.5887526876);
-\draw [,->,>=latex] (-4.000000000,0) -- (-4.125000000,0);
-\draw [,->,>=latex] (-4.000000000,0.5714285714) -- (-4.121268813,0.5887526876);
-\draw [,->,>=latex] (-4.000000000,1.142857143) -- (-4.111119513,1.174605575);
-\draw [,->,>=latex] (-4.000000000,1.714285714) -- (-4.097064885,1.755884951);
-\draw [,->,>=latex] (-4.000000000,2.285714286) -- (-4.081815219,2.332465840);
-\draw [,->,>=latex] (-4.000000000,2.857142857) -- (-4.067352939,2.905252100);
-\draw [,->,>=latex] (-4.000000000,3.428571429) -- (-4.054711138,3.475466689);
-\draw [,->,>=latex] (-4.000000000,4.000000000) -- (-4.044194174,4.044194174);
-\draw [,->,>=latex] (-3.428571429,-4.000000000) -- (-3.475466689,-4.054711138);
-\draw [,->,>=latex] (-3.428571429,-3.428571429) -- (-3.488724610,-3.488724610);
-\draw [,->,>=latex] (-3.428571429,-2.857142857) -- (-3.505708401,-2.921423668);
-\draw [,->,>=latex] (-3.428571429,-2.285714286) -- (-3.526577353,-2.351051568);
-\draw [,->,>=latex] (-3.428571429,-1.714285714) -- (-3.550312907,-1.775156454);
-\draw [,->,>=latex] (-3.428571429,-1.142857143) -- (-3.573838559,-1.191279520);
-\draw [,->,>=latex] (-3.428571429,-0.5714285714) -- (-3.591859612,-0.5986432687);
-\draw [,->,>=latex] (-3.428571429,0) -- (-3.598710317,0);
-\draw [,->,>=latex] (-3.428571429,0.5714285714) -- (-3.591859612,0.5986432687);
-\draw [,->,>=latex] (-3.428571429,1.142857143) -- (-3.573838559,1.191279520);
-\draw [,->,>=latex] (-3.428571429,1.714285714) -- (-3.550312907,1.775156454);
-\draw [,->,>=latex] (-3.428571429,2.285714286) -- (-3.526577353,2.351051568);
-\draw [,->,>=latex] (-3.428571429,2.857142857) -- (-3.505708401,2.921423668);
-\draw [,->,>=latex] (-3.428571429,3.428571429) -- (-3.488724610,3.488724610);
-\draw [,->,>=latex] (-3.428571429,4.000000000) -- (-3.475466689,4.054711138);
-\draw [,->,>=latex] (-2.857142857,-4.000000000) -- (-2.905252100,-4.067352939);
-\draw [,->,>=latex] (-2.857142857,-3.428571429) -- (-2.921423668,-3.505708401);
-\draw [,->,>=latex] (-2.857142857,-2.857142857) -- (-2.943763438,-2.943763438);
-\draw [,->,>=latex] (-2.857142857,-2.285714286) -- (-2.973797039,-2.379037631);
-\draw [,->,>=latex] (-2.857142857,-1.714285714) -- (-3.011617686,-1.806970611);
-\draw [,->,>=latex] (-2.857142857,-1.142857143) -- (-3.053243538,-1.221297415);
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diff --git a/auto/pictures_tex/Fig_STdyNTH.pstricks b/auto/pictures_tex/Fig_STdyNTH.pstricks
index d81aeb717..b283a16d4 100644
--- a/auto/pictures_tex/Fig_STdyNTH.pstricks
+++ b/auto/pictures_tex/Fig_STdyNTH.pstricks
@@ -75,15 +75,15 @@
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-\draw (2.000000000,0.2000000000) node {};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.2000000000) node {};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.3149246667) node {$1$};
+\draw (0,0.20000) node {};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.20000) node {};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.20000) node {};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.20000) node {};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.31492) node {$1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks b/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks
index 939537d7b..bb5921328 100644
--- a/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks
+++ b/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks
@@ -119,48 +119,48 @@
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-\draw [,->,>=latex] (0,-3.525102241) -- (0,3.566144740);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-3.5251) -- (0,3.5661);
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 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,1.500000000) node {$ 3 $};
+\draw (-0.29125,1.5000) node {$ 3 $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
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 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,2.500000000) node {$ 5 $};
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 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
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diff --git a/auto/pictures_tex/Fig_SenoTopologo.pstricks b/auto/pictures_tex/Fig_SenoTopologo.pstricks
index 64c749052..18e3d2a7e 100644
--- a/auto/pictures_tex/Fig_SenoTopologo.pstricks
+++ b/auto/pictures_tex/Fig_SenoTopologo.pstricks
@@ -63,8 +63,8 @@
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diff --git a/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks b/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks
index 80fc2e34a..cb4d2a63a 100644
--- a/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks
+++ b/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.080796327,0) -- (2.080796327,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-2.0808,0) -- (2.0808,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
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@@ -88,13 +88,13 @@
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-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SpiraleLimite.pstricks b/auto/pictures_tex/Fig_SpiraleLimite.pstricks
index cb8a83eff..3908c2a7d 100644
--- a/auto/pictures_tex/Fig_SpiraleLimite.pstricks
+++ b/auto/pictures_tex/Fig_SpiraleLimite.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.710694496);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.7107);
 %DEFAULT
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 \draw [] (3.00,-0.100) -- (3.00,0.100);
diff --git a/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks b/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks
index 6f6b36964..19a5c0484 100644
--- a/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks
+++ b/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks
@@ -69,64 +69,64 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.490292088,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,3.130986314);
+\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,3.1310);
 %DEFAULT
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.000000000,1.000000000) -- (0,1.000000000);
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.0000,1.0000) -- (0,1.0000);
 \draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw [color=green,->,>=latex] (1.000000000,1.000000000) -- (1.707106781,0.2928932188);
-\draw []  (2.141592654,1.000000000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (2.141592654,1.000000000) -- (3.141592654,1.000000000);
+\draw [color=green,->,>=latex] (1.0000,1.0000) -- (1.7071,0.29289);
+\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (2.1416,1.0000) -- (3.1416,1.0000);
 \draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw [color=green,->,>=latex] (2.141592654,1.000000000) -- (1.434485872,0.2928932188);
-\draw []  (4.712388980,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (4.712388980,0) -- (4.712388980,1.000000000);
+\draw [color=green,->,>=latex] (2.1416,1.0000) -- (1.4345,0.29289);
+\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (4.7124,0) -- (4.7124,1.0000);
 \draw [color=brown] (3.71,0) -- (5.71,0);
-\draw [color=green,->,>=latex] (4.712388980,0) -- (4.712388980,-1.000000000);
-\draw []  (7.283185307,1.000000000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (7.283185307,1.000000000) -- (6.283185307,1.000000000);
+\draw [color=green,->,>=latex] (4.7124,0) -- (4.7124,-1.0000);
+\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (7.2832,1.0000) -- (6.2832,1.0000);
 \draw [color=brown] (6.58,0.293) -- (7.99,1.71);
-\draw [color=green,->,>=latex] (7.283185307,1.000000000) -- (7.990292088,0.2928932188);
-\draw []  (1.492504945,1.707106781) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.492504945,1.707106781) -- (0.7853981634,1.000000000);
+\draw [color=green,->,>=latex] (7.2832,1.0000) -- (7.9903,0.29289);
+\draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.4925,1.7071) -- (0.78540,1.0000);
 \draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw [color=green,->,>=latex] (1.492504945,1.707106781) -- (2.416384477,1.324423349);
-\draw []  (1.649087709,1.707106781) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.649087709,1.707106781) -- (2.356194490,1.000000000);
+\draw [color=green,->,>=latex] (1.4925,1.7071) -- (2.4164,1.3244);
+\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.6491,1.7071) -- (2.3562,1.0000);
 \draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw [color=green,->,>=latex] (1.649087709,1.707106781) -- (0.7252081765,1.324423349);
-\draw []  (3.219884036,0.2928932188) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (3.219884036,0.2928932188) -- (3.926990817,1.000000000);
+\draw [color=green,->,>=latex] (1.6491,1.7071) -- (0.72521,1.3244);
+\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (3.2199,0.29289) -- (3.9270,1.0000);
 \draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw [color=green,->,>=latex] (3.219884036,0.2928932188) -- (2.837200603,-0.6309863137);
-\draw []  (6.204893925,0.2928932188) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (6.204893925,0.2928932188) -- (5.497787144,1.000000000);
+\draw [color=green,->,>=latex] (3.2199,0.29289) -- (2.8372,-0.63099);
+\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (6.2049,0.29289) -- (5.4978,1.0000);
 \draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
-\draw [color=green,->,>=latex] (6.204893925,0.2928932188) -- (6.587577357,-0.6309863137);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw [color=green,->,>=latex] (6.2049,0.29289) -- (6.5876,-0.63099);
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -189,47 +189,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.490292088,0);
-\draw [,->,>=latex] (0,-0.5897902136) -- (0,3.130986314);
+\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
+\draw [,->,>=latex] (0,-0.58979) -- (0,3.1310);
 %DEFAULT
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw []  (2.141592654,1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw []  (4.712388980,0) node [rotate=0] {$\bullet$};
+\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
 \draw [color=brown] (3.71,0) -- (5.71,0);
-\draw []  (7.283185307,1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=brown] (6.58,0.293) -- (7.99,1.71);
-\draw []  (1.492504945,1.707106781) node [rotate=0] {$\bullet$};
+\draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
 \draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw []  (1.649087709,1.707106781) node [rotate=0] {$\bullet$};
+\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
 \draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw []  (3.219884036,0.2928932188) node [rotate=0] {$\bullet$};
+\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
 \draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw []  (6.204893925,0.2928932188) node [rotate=0] {$\bullet$};
+\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
 \draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
 \draw [color=blue,style=dashed] (1.000,1.000)--(1.061,1.063)--(1.119,1.127)--(1.172,1.189)--(1.222,1.251)--(1.267,1.312)--(1.309,1.372)--(1.347,1.430)--(1.382,1.486)--(1.412,1.541)--(1.440,1.593)--(1.464,1.643)--(1.485,1.690)--(1.504,1.735)--(1.519,1.776)--(1.532,1.815)--(1.543,1.850)--(1.551,1.881)--(1.558,1.910)--(1.563,1.934)--(1.566,1.955)--(1.569,1.972)--(1.570,1.985)--(1.571,1.994)--(1.571,1.999)--(1.571,2.000)--(1.571,1.997)--(1.571,1.990)--(1.572,1.979)--(1.574,1.964)--(1.577,1.945)--(1.581,1.922)--(1.587,1.896)--(1.594,1.866)--(1.604,1.833)--(1.616,1.796)--(1.630,1.756)--(1.647,1.713)--(1.666,1.667)--(1.689,1.618)--(1.715,1.567)--(1.744,1.514)--(1.777,1.458)--(1.813,1.401)--(1.853,1.342)--(1.896,1.282)--(1.944,1.220)--(1.995,1.158)--(2.051,1.095)--(2.110,1.032)--(2.174,0.9683)--(2.241,0.9049)--(2.313,0.8420)--(2.388,0.7797)--(2.468,0.7183)--(2.551,0.6580)--(2.638,0.5991)--(2.729,0.5418)--(2.823,0.4863)--(2.921,0.4329)--(3.022,0.3818)--(3.126,0.3332)--(3.233,0.2873)--(3.344,0.2443)--(3.456,0.2042)--(3.571,0.1674)--(3.689,0.1340)--(3.808,0.1040)--(3.929,0.07765)--(4.052,0.05500)--(4.176,0.03616)--(4.301,0.02120)--(4.427,0.01018)--(4.554,0.003145)--(4.681,0)--(4.808,0.001133)--(4.934,0.006162)--(5.061,0.01519)--(5.186,0.02819)--(5.311,0.04510)--(5.434,0.06585)--(5.556,0.09037)--(5.677,0.1185)--(5.795,0.1503)--(5.911,0.1854)--(6.025,0.2239)--(6.137,0.2654)--(6.245,0.3099)--(6.351,0.3572)--(6.454,0.4071)--(6.553,0.4594)--(6.649,0.5138)--(6.742,0.5702)--(6.831,0.6283)--(6.916,0.6880)--(6.997,0.7489)--(7.075,0.8107)--(7.148,0.8734)--(7.218,0.9366)--(7.283,1.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SuiteUnSurn.pstricks b/auto/pictures_tex/Fig_SuiteUnSurn.pstricks
index c39856e87..ad00f9bf4 100644
--- a/auto/pictures_tex/Fig_SuiteUnSurn.pstricks
+++ b/auto/pictures_tex/Fig_SuiteUnSurn.pstricks
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
-\draw []  (1.000000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (1.000000000,3.414924667) node {$1$};
-\draw []  (2.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,1.982455000) node {$1/2$};
-\draw []  (3.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.000000000,1.482455000) node {$1/3$};
-\draw []  (4.000000000,0.7500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,1.232455000) node {$1/4$};
-\draw []  (5.000000000,0.6000000000) node [rotate=0] {$\bullet$};
-\draw (5.000000000,1.082455000) node {$1/5$};
-\draw []  (6.000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (6.000000000,0.9824550000) node {$1/6$};
-\draw []  (7.000000000,0.4285714286) node [rotate=0] {$\bullet$};
-\draw (7.000000000,0.9110264286) node {$1/7$};
-\draw []  (8.000000000,0.3750000000) node [rotate=0] {$\bullet$};
-\draw (8.000000000,0.8574550000) node {$1/8$};
-\draw []  (9.000000000,0.3333333333) node [rotate=0] {$\bullet$};
-\draw (9.000000000,0.8157883333) node {$1/9$};
-\draw []  (10.00000000,0.3000000000) node [rotate=0] {$\bullet$};
-\draw (10.00000000,0.7824550000) node {$1/10$};
+\draw []  (1.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,3.4149) node {$1$};
+\draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (2.0000,1.9825) node {$1/2$};
+\draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,1.4825) node {$1/3$};
+\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2325) node {$1/4$};
+\draw []  (5.0000,0.60000) node [rotate=0] {$\bullet$};
+\draw (5.0000,1.0825) node {$1/5$};
+\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.98246) node {$1/6$};
+\draw []  (7.0000,0.42857) node [rotate=0] {$\bullet$};
+\draw (7.0000,0.91103) node {$1/7$};
+\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.85746) node {$1/8$};
+\draw []  (9.0000,0.33333) node [rotate=0] {$\bullet$};
+\draw (9.0000,0.81579) node {$1/9$};
+\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.78246) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfaceCercle.pstricks b/auto/pictures_tex/Fig_SurfaceCercle.pstricks
index 2a553dd3c..79caff591 100644
--- a/auto/pictures_tex/Fig_SurfaceCercle.pstricks
+++ b/auto/pictures_tex/Fig_SurfaceCercle.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.499897967) -- (0,2.499897967);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks b/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks
index 0bed20805..a3b7cc8a7 100644
--- a/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks
+++ b/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.499883862);
+\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -56,10 +56,10 @@
 \draw [color=blue,style=solid] (1.000,1.635)--(1.051,1.562)--(1.101,1.497)--(1.152,1.439)--(1.202,1.390)--(1.253,1.348)--(1.303,1.316)--(1.354,1.293)--(1.404,1.279)--(1.455,1.273)--(1.505,1.276)--(1.556,1.288)--(1.606,1.307)--(1.657,1.334)--(1.707,1.367)--(1.758,1.406)--(1.808,1.450)--(1.859,1.498)--(1.909,1.549)--(1.960,1.603)--(2.010,1.658)--(2.061,1.713)--(2.111,1.768)--(2.162,1.821)--(2.212,1.873)--(2.263,1.921)--(2.313,1.965)--(2.364,2.006)--(2.414,2.041)--(2.465,2.071)--(2.515,2.096)--(2.566,2.116)--(2.616,2.129)--(2.667,2.137)--(2.717,2.139)--(2.768,2.136)--(2.818,2.128)--(2.869,2.116)--(2.919,2.100)--(2.970,2.081)--(3.020,2.059)--(3.071,2.035)--(3.121,2.010)--(3.172,1.985)--(3.222,1.960)--(3.273,1.937)--(3.323,1.915)--(3.374,1.896)--(3.424,1.881)--(3.475,1.870)--(3.525,1.863)--(3.576,1.861)--(3.626,1.864)--(3.677,1.873)--(3.727,1.888)--(3.778,1.908)--(3.828,1.934)--(3.879,1.965)--(3.929,2.002)--(3.980,2.043)--(4.030,2.088)--(4.081,2.137)--(4.131,2.189)--(4.182,2.242)--(4.232,2.297)--(4.283,2.353)--(4.333,2.408)--(4.384,2.461)--(4.434,2.512)--(4.485,2.559)--(4.535,2.602)--(4.586,2.640)--(4.636,2.672)--(4.687,2.697)--(4.737,2.715)--(4.788,2.725)--(4.838,2.727)--(4.889,2.719)--(4.939,2.703)--(4.990,2.678)--(5.040,2.644)--(5.091,2.602)--(5.141,2.550)--(5.192,2.491)--(5.242,2.424)--(5.293,2.351)--(5.343,2.271)--(5.394,2.186)--(5.444,2.097)--(5.495,2.005)--(5.545,1.911)--(5.596,1.816)--(5.646,1.722)--(5.697,1.629)--(5.747,1.538)--(5.798,1.452)--(5.849,1.370)--(5.899,1.294)--(5.950,1.225)--(6.000,1.165);
 \draw [color=magenta,style=dashed] (1.00,1.63) -- (1.00,4.84);
 \draw [color=magenta,style=dashed] (6.00,3.72) -- (6.00,1.16);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.2785761667) node {$a$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3267360000) node {$b$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.27858) node {$a$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.32674) node {$b$};
 \draw [style=dotted] (1.00,0) -- (1.00,4.84);
 \draw [style=dotted] (6.00,0) -- (6.00,3.72);
 
@@ -109,8 +109,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.499883862,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -124,10 +124,10 @@
 \draw [color=blue] (1.635,1.000)--(1.562,1.051)--(1.497,1.101)--(1.439,1.152)--(1.390,1.202)--(1.348,1.253)--(1.316,1.303)--(1.293,1.354)--(1.279,1.404)--(1.273,1.455)--(1.276,1.505)--(1.288,1.556)--(1.307,1.606)--(1.334,1.657)--(1.367,1.707)--(1.406,1.758)--(1.450,1.808)--(1.498,1.859)--(1.549,1.909)--(1.603,1.960)--(1.658,2.010)--(1.713,2.061)--(1.768,2.111)--(1.821,2.162)--(1.873,2.212)--(1.921,2.263)--(1.965,2.313)--(2.006,2.364)--(2.041,2.414)--(2.071,2.465)--(2.096,2.515)--(2.116,2.566)--(2.129,2.616)--(2.137,2.667)--(2.139,2.717)--(2.136,2.768)--(2.128,2.818)--(2.116,2.869)--(2.100,2.919)--(2.081,2.970)--(2.059,3.020)--(2.035,3.071)--(2.010,3.121)--(1.985,3.172)--(1.960,3.222)--(1.937,3.273)--(1.915,3.323)--(1.896,3.374)--(1.881,3.424)--(1.870,3.475)--(1.863,3.525)--(1.861,3.576)--(1.864,3.626)--(1.873,3.677)--(1.888,3.727)--(1.908,3.778)--(1.934,3.828)--(1.965,3.879)--(2.002,3.929)--(2.043,3.980)--(2.088,4.030)--(2.137,4.081)--(2.189,4.131)--(2.242,4.182)--(2.297,4.232)--(2.353,4.283)--(2.408,4.333)--(2.461,4.384)--(2.512,4.434)--(2.559,4.485)--(2.602,4.535)--(2.640,4.586)--(2.672,4.636)--(2.697,4.687)--(2.715,4.737)--(2.725,4.788)--(2.727,4.838)--(2.719,4.889)--(2.703,4.939)--(2.678,4.990)--(2.644,5.040)--(2.602,5.091)--(2.550,5.141)--(2.491,5.192)--(2.424,5.242)--(2.351,5.293)--(2.271,5.343)--(2.186,5.394)--(2.097,5.444)--(2.005,5.495)--(1.911,5.545)--(1.816,5.596)--(1.722,5.646)--(1.629,5.697)--(1.538,5.747)--(1.452,5.798)--(1.370,5.849)--(1.294,5.899)--(1.225,5.950)--(1.165,6.000);
 \draw [color=magenta,style=dashed] (1.16,6.00) -- (3.72,6.00);
 \draw [color=magenta,style=dashed] (4.84,1.00) -- (1.63,1.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2789780000,1.000000000) node {$c$};
-\draw []  (0,6.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2949888333,6.000000000) node {$d$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.27898,1.0000) node {$c$};
+\draw []  (0,6.0000) node [rotate=0] {$\bullet$};
+\draw (-0.29499,6.0000) node {$d$};
 \draw [style=dotted] (0,1.00) -- (4.84,1.00);
 \draw [style=dotted] (0,6.00) -- (1.16,6.00);
 \draw [color=blue,style=solid] (4.841,1.000)--(4.868,1.051)--(4.892,1.101)--(4.913,1.152)--(4.933,1.202)--(4.950,1.253)--(4.964,1.303)--(4.977,1.354)--(4.986,1.404)--(4.993,1.455)--(4.998,1.505)--(5.000,1.556)--(4.999,1.606)--(4.996,1.657)--(4.991,1.707)--(4.983,1.758)--(4.972,1.808)--(4.959,1.859)--(4.943,1.909)--(4.925,1.960)--(4.905,2.010)--(4.882,2.061)--(4.858,2.111)--(4.831,2.162)--(4.801,2.212)--(4.770,2.263)--(4.737,2.313)--(4.702,2.364)--(4.665,2.414)--(4.626,2.465)--(4.586,2.515)--(4.545,2.566)--(4.502,2.616)--(4.457,2.667)--(4.412,2.717)--(4.365,2.768)--(4.318,2.818)--(4.270,2.869)--(4.221,2.919)--(4.171,2.970)--(4.121,3.020)--(4.071,3.071)--(4.020,3.121)--(3.970,3.172)--(3.919,3.222)--(3.869,3.273)--(3.819,3.323)--(3.770,3.374)--(3.721,3.424)--(3.673,3.475)--(3.626,3.525)--(3.579,3.576)--(3.534,3.626)--(3.490,3.677)--(3.447,3.727)--(3.406,3.778)--(3.366,3.828)--(3.328,3.879)--(3.291,3.929)--(3.257,3.980)--(3.224,4.030)--(3.193,4.081)--(3.164,4.131)--(3.137,4.182)--(3.113,4.232)--(3.091,4.283)--(3.071,4.333)--(3.053,4.384)--(3.038,4.434)--(3.026,4.485)--(3.016,4.535)--(3.008,4.586)--(3.003,4.636)--(3.000,4.687)--(3.000,4.737)--(3.003,4.788)--(3.008,4.838)--(3.016,4.889)--(3.026,4.939)--(3.038,4.990)--(3.053,5.040)--(3.071,5.091)--(3.091,5.141)--(3.113,5.192)--(3.137,5.242)--(3.164,5.293)--(3.193,5.343)--(3.223,5.394)--(3.256,5.444)--(3.291,5.495)--(3.327,5.545)--(3.366,5.596)--(3.405,5.646)--(3.447,5.697)--(3.490,5.747)--(3.534,5.798)--(3.579,5.849)--(3.625,5.899)--(3.672,5.950)--(3.721,6.000);
diff --git a/auto/pictures_tex/Fig_TGdUoZR.pstricks b/auto/pictures_tex/Fig_TGdUoZR.pstricks
index 8a44a7c03..feb371f98 100644
--- a/auto/pictures_tex/Fig_TGdUoZR.pstricks
+++ b/auto/pictures_tex/Fig_TGdUoZR.pstricks
@@ -66,37 +66,37 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (0,1.000000000) node {};
+\draw (0,1.0000) node {};
 \draw [] (-0.250,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,1.25);
 \draw [] (0.250,1.25) -- (-0.250,1.25);
 \draw [] (-0.250,1.25) -- (-0.250,0.750);
-\draw (0.5000000000,1.000000000) node {};
+\draw (0.50000,1.0000) node {};
 \draw [] (0.250,0.750) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.750,1.25);
 \draw [] (0.750,1.25) -- (0.250,1.25);
 \draw [] (0.250,1.25) -- (0.250,0.750);
-\draw (1.000000000,1.000000000) node {};
+\draw (1.0000,1.0000) node {};
 \draw [] (0.750,0.750) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (1.25,1.25);
 \draw [] (1.25,1.25) -- (0.750,1.25);
 \draw [] (0.750,1.25) -- (0.750,0.750);
-\draw (1.500000000,1.000000000) node {};
+\draw (1.5000,1.0000) node {};
 \draw [] (1.25,0.750) -- (1.75,0.750);
 \draw [] (1.75,0.750) -- (1.75,1.25);
 \draw [] (1.75,1.25) -- (1.25,1.25);
 \draw [] (1.25,1.25) -- (1.25,0.750);
-\draw (0,0.5000000000) node {};
+\draw (0,0.50000) node {};
 \draw [] (-0.250,0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (-0.250,0.750);
 \draw [] (-0.250,0.750) -- (-0.250,0.250);
-\draw (0.5000000000,0.5000000000) node {};
+\draw (0.50000,0.50000) node {};
 \draw [] (0.250,0.250) -- (0.750,0.250);
 \draw [] (0.750,0.250) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,0.250);
-\draw (1.000000000,0.5000000000) node {};
+\draw (1.0000,0.50000) node {};
 \draw [] (0.750,0.250) -- (1.25,0.250);
 \draw [] (1.25,0.250) -- (1.25,0.750);
 \draw [] (1.25,0.750) -- (0.750,0.750);
diff --git a/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks b/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks
index 37c82159c..067a5b0f8 100644
--- a/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks
+++ b/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (-2.10,0.700) -- (2.10,0.700);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3824550000) node {\( \pi(e_1)\)};
-\draw []  (0.7000000000,0.7000000000) node [rotate=0] {$\bullet$};
-\draw (0.7000000000,1.082455000) node {\( \pi(e_2)\)};
+\draw (0,-0.38245) node {\( \pi(e_1)\)};
+\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
+\draw (0.70000,1.0825) node {\( \pi(e_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks b/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
index 92ab9d915..4c1dc4011 100644
--- a/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
+++ b/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
@@ -68,55 +68,55 @@
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
 
 
-\draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
-\draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
-\draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
-\draw [,->,>=latex] (1.600000000,0) -- (2.225000000,0);
-\draw [,->,>=latex] (1.600000000,0.3333333333) -- (2.225000000,0.3333333333);
-\draw [,->,>=latex] (1.600000000,0.6666666667) -- (2.225000000,0.6666666667);
-\draw [,->,>=latex] (1.600000000,1.000000000) -- (2.225000000,1.000000000);
-\draw [,->,>=latex] (2.333333333,-1.000000000) -- (2.761904762,-1.000000000);
-\draw [,->,>=latex] (2.333333333,-0.6666666667) -- (2.761904762,-0.6666666667);
-\draw [,->,>=latex] (2.333333333,-0.3333333333) -- (2.761904762,-0.3333333333);
-\draw [,->,>=latex] (2.333333333,0) -- (2.761904762,0);
-\draw [,->,>=latex] (2.333333333,0.3333333333) -- (2.761904762,0.3333333333);
-\draw [,->,>=latex] (2.333333333,0.6666666667) -- (2.761904762,0.6666666667);
-\draw [,->,>=latex] (2.333333333,1.000000000) -- (2.761904762,1.000000000);
-\draw [,->,>=latex] (3.066666667,-1.000000000) -- (3.392753623,-1.000000000);
-\draw [,->,>=latex] (3.066666667,-0.6666666667) -- (3.392753623,-0.6666666667);
-\draw [,->,>=latex] (3.066666667,-0.3333333333) -- (3.392753623,-0.3333333333);
-\draw [,->,>=latex] (3.066666667,0) -- (3.392753623,0);
-\draw [,->,>=latex] (3.066666667,0.3333333333) -- (3.392753623,0.3333333333);
-\draw [,->,>=latex] (3.066666667,0.6666666667) -- (3.392753623,0.6666666667);
-\draw [,->,>=latex] (3.066666667,1.000000000) -- (3.392753623,1.000000000);
-\draw [,->,>=latex] (3.800000000,-1.000000000) -- (4.063157895,-1.000000000);
-\draw [,->,>=latex] (3.800000000,-0.6666666667) -- (4.063157895,-0.6666666667);
-\draw [,->,>=latex] (3.800000000,-0.3333333333) -- (4.063157895,-0.3333333333);
-\draw [,->,>=latex] (3.800000000,0) -- (4.063157895,0);
-\draw [,->,>=latex] (3.800000000,0.3333333333) -- (4.063157895,0.3333333333);
-\draw [,->,>=latex] (3.800000000,0.6666666667) -- (4.063157895,0.6666666667);
-\draw [,->,>=latex] (3.800000000,1.000000000) -- (4.063157895,1.000000000);
-\draw [,->,>=latex] (4.533333333,-1.000000000) -- (4.753921569,-1.000000000);
-\draw [,->,>=latex] (4.533333333,-0.6666666667) -- (4.753921569,-0.6666666667);
-\draw [,->,>=latex] (4.533333333,-0.3333333333) -- (4.753921569,-0.3333333333);
-\draw [,->,>=latex] (4.533333333,0) -- (4.753921569,0);
-\draw [,->,>=latex] (4.533333333,0.3333333333) -- (4.753921569,0.3333333333);
-\draw [,->,>=latex] (4.533333333,0.6666666667) -- (4.753921569,0.6666666667);
-\draw [,->,>=latex] (4.533333333,1.000000000) -- (4.753921569,1.000000000);
-\draw [,->,>=latex] (5.266666667,-1.000000000) -- (5.456540084,-1.000000000);
-\draw [,->,>=latex] (5.266666667,-0.6666666667) -- (5.456540084,-0.6666666667);
-\draw [,->,>=latex] (5.266666667,-0.3333333333) -- (5.456540084,-0.3333333333);
-\draw [,->,>=latex] (5.266666667,0) -- (5.456540084,0);
-\draw [,->,>=latex] (5.266666667,0.3333333333) -- (5.456540084,0.3333333333);
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+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks b/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
index 92b9e2f6a..feed9e843 100644
--- a/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
+++ b/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
@@ -88,28 +88,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.6113094300) -- (0,2.527798437);
+\draw [,->,>=latex] (-2.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.61131) -- (0,2.5278);
 %DEFAULT
 
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-\draw (2.000000000,-0.3149246667) node {$ 2 $};
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-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
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 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks b/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
index 42f5d021d..66e9b1259 100644
--- a/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
+++ b/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
@@ -108,34 +108,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
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-\draw (-7.853981634,-0.4207143333) node {$ -\frac{5}{2} \, \pi $};
+\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
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-\draw (-6.283185307,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.712388980,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
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 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.853981634,-0.4207143333) node {$ \frac{5}{2} \, \pi $};
+\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
 \draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangentSegment.pstricks b/auto/pictures_tex/Fig_TangentSegment.pstricks
index 49f2ac546..7ff8d9539 100644
--- a/auto/pictures_tex/Fig_TangentSegment.pstricks
+++ b/auto/pictures_tex/Fig_TangentSegment.pstricks
@@ -103,44 +103,44 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-3.068914101) -- (0,2.500000000);
+\draw [,->,>=latex] (-3.7038,0) -- (7.7832,0);
+\draw [,->,>=latex] (0,-3.0689) -- (0,2.5000);
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-\draw []  (-1.570796327,-1.414213562) node [rotate=0] {$\bullet$};
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-\draw []  (3.141592654,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
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 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
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-\draw (2.000000000,-0.3149246667) node {$ 2 $};
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-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteDetail.pstricks b/auto/pictures_tex/Fig_TangenteDetail.pstricks
index 86f02dcca..bfa327892 100644
--- a/auto/pictures_tex/Fig_TangenteDetail.pstricks
+++ b/auto/pictures_tex/Fig_TangenteDetail.pstricks
@@ -111,8 +111,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.105147059);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
 %DEFAULT
 \draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
 \draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
@@ -122,24 +122,24 @@
 \draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
 
 \draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.700000000,3.235294118) node [rotate=0] {$\bullet$};
-\draw (1.341429089,3.568136140) node {$P$};
-\draw []  (1.700000000,0) node [rotate=0] {$\bullet$};
-\draw (1.700000000,-0.2785761667) node {$a$};
-\draw []  (0,3.235294118) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,3.235294118) node {$f(a)$};
-\draw []  (4.000000000,4.250000000) node [rotate=0] {$\bullet$};
-\draw (3.800437273,4.705055823) node {$Q$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.2785761667) node {$x$};
-\draw []  (0,4.250000000) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,4.250000000) node {$f(x)$};
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (1.700000000,3.035294118);
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (4.000000000,3.035294118);
-\draw (2.850000000,2.714264451) node {$x-a$};
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,4.250000000);
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,3.235294118);
-\draw (5.325596167,3.742647059) node {$f(x)-f(a)$};
+\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw (1.3414,3.5681) node {$P$};
+\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.27858) node {$a$};
+\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
+\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (3.8004,4.7051) node {$Q$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.27858) node {$x$};
+\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.45521,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
+\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
+\draw (2.8500,2.7143) node {$x-a$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
+\draw (5.3256,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteDetailOM.pstricks b/auto/pictures_tex/Fig_TangenteDetailOM.pstricks
index 0869fa71e..4771ca36f 100644
--- a/auto/pictures_tex/Fig_TangenteDetailOM.pstricks
+++ b/auto/pictures_tex/Fig_TangenteDetailOM.pstricks
@@ -111,8 +111,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.105147059);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
 %DEFAULT
 \draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
 \draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
@@ -122,24 +122,24 @@
 \draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
 
 \draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.700000000,3.235294118) node [rotate=0] {$\bullet$};
-\draw (1.341429089,3.568136140) node {$P$};
-\draw []  (1.700000000,0) node [rotate=0] {$\bullet$};
-\draw (1.700000000,-0.2785761667) node {$a$};
-\draw []  (0,3.235294118) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,3.235294118) node {$f(a)$};
-\draw []  (4.000000000,4.250000000) node [rotate=0] {$\bullet$};
-\draw (3.800437273,4.705055823) node {$Q$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.2785761667) node {$x$};
-\draw []  (0,4.250000000) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,4.250000000) node {$f(x)$};
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (1.700000000,3.035294118);
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (4.000000000,3.035294118);
-\draw (2.850000000,2.714264451) node {$x-a$};
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,4.250000000);
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,3.235294118);
-\draw (5.325596167,3.742647059) node {$f(x)-f(a)$};
+\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw (1.3414,3.5681) node {$P$};
+\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.27858) node {$a$};
+\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
+\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (3.8004,4.7051) node {$Q$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.27858) node {$x$};
+\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.45521,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
+\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
+\draw (2.8500,2.7143) node {$x-a$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
+\draw (5.3256,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteQuestion.pstricks b/auto/pictures_tex/Fig_TangenteQuestion.pstricks
index 45df39124..ad59a76cd 100644
--- a/auto/pictures_tex/Fig_TangenteQuestion.pstricks
+++ b/auto/pictures_tex/Fig_TangenteQuestion.pstricks
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.580000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
 %DEFAULT
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks b/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks
index defcf2ea8..411ca3541 100644
--- a/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks
+++ b/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.580000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
 %DEFAULT
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ToreRevolution.pstricks b/auto/pictures_tex/Fig_ToreRevolution.pstricks
index b06e73eb4..044f09832 100644
--- a/auto/pictures_tex/Fig_ToreRevolution.pstricks
+++ b/auto/pictures_tex/Fig_ToreRevolution.pstricks
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,0);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,3.000000000);
-\draw (1.896467667,1.500000000) node {$a$};
-\draw []  (0,3.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
+\draw (1.8965,1.5000) node {$a$};
+\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
 \draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.1430327019,3.634515621) node {$R$};
+\draw (0.14303,3.6345) node {$R$};
 \draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
 
 %OTHER STUFF
diff --git a/auto/pictures_tex/Fig_Trajs.pstricks b/auto/pictures_tex/Fig_Trajs.pstricks
index 5a8041098..d6f17fd84 100644
--- a/auto/pictures_tex/Fig_Trajs.pstricks
+++ b/auto/pictures_tex/Fig_Trajs.pstricks
@@ -75,28 +75,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.642639664,0) -- (2.723244275,0);
-\draw [,->,>=latex] (0,-1.914172501) -- (0,2.735975350);
+\draw [,->,>=latex] (-1.6426,0) -- (2.7232,0);
+\draw [,->,>=latex] (0,-1.9142) -- (0,2.7360);
 %DEFAULT
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-\draw [color=brown]  (1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
 \draw [color=green] (0.540,0.841)--(0.557,0.830)--(0.574,0.819)--(0.590,0.807)--(0.606,0.795)--(0.622,0.783)--(0.638,0.770)--(0.654,0.757)--(0.669,0.744)--(0.684,0.730)--(0.698,0.716)--(0.712,0.702)--(0.727,0.687)--(0.740,0.672)--(0.754,0.657)--(0.767,0.642)--(0.780,0.626)--(0.792,0.610)--(0.804,0.594)--(0.816,0.578)--(0.828,0.561)--(0.839,0.544)--(0.850,0.527)--(0.860,0.510)--(0.870,0.493)--(0.880,0.475)--(0.889,0.457)--(0.898,0.439)--(0.907,0.421)--(0.915,0.402)--(0.923,0.384)--(0.931,0.365)--(0.938,0.346)--(0.945,0.327)--(0.951,0.308)--(0.957,0.289)--(0.963,0.269)--(0.968,0.250)--(0.973,0.230)--(0.978,0.211)--(0.982,0.191)--(0.985,0.171)--(0.989,0.151)--(0.991,0.131)--(0.994,0.111)--(0.996,0.0908)--(0.997,0.0706)--(0.999,0.0505)--(1.00,0.0303)--(1.00,0.0101)--(1.00,-0.0101)--(1.00,-0.0303)--(0.999,-0.0505)--(0.997,-0.0706)--(0.996,-0.0908)--(0.994,-0.111)--(0.991,-0.131)--(0.989,-0.151)--(0.985,-0.171)--(0.982,-0.191)--(0.978,-0.211)--(0.973,-0.230)--(0.968,-0.250)--(0.963,-0.269)--(0.957,-0.289)--(0.951,-0.308)--(0.945,-0.327)--(0.938,-0.346)--(0.931,-0.365)--(0.923,-0.384)--(0.915,-0.402)--(0.907,-0.421)--(0.898,-0.439)--(0.889,-0.457)--(0.880,-0.475)--(0.870,-0.493)--(0.860,-0.510)--(0.850,-0.527)--(0.839,-0.544)--(0.828,-0.561)--(0.816,-0.578)--(0.804,-0.594)--(0.792,-0.610)--(0.780,-0.626)--(0.767,-0.642)--(0.754,-0.657)--(0.740,-0.672)--(0.727,-0.687)--(0.712,-0.702)--(0.698,-0.716)--(0.684,-0.730)--(0.669,-0.744)--(0.654,-0.757)--(0.638,-0.770)--(0.622,-0.783)--(0.606,-0.795)--(0.590,-0.807)--(0.574,-0.819)--(0.557,-0.830)--(0.540,-0.841);
-\draw [color=brown]  (1.000000000,0) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (1.0000,0) node [rotate=0] {$\bullet$};
 \draw [color=blue] (-0.3012,1.382)--(-0.2732,1.388)--(-0.2451,1.393)--(-0.2169,1.397)--(-0.1886,1.402)--(-0.1603,1.405)--(-0.1319,1.408)--(-0.1034,1.410)--(-0.07490,1.412)--(-0.04635,1.413)--(-0.01779,1.414)--(0.01078,1.414)--(0.03934,1.414)--(0.06789,1.413)--(0.09641,1.411)--(0.1249,1.409)--(0.1533,1.406)--(0.1817,1.402)--(0.2100,1.399)--(0.2382,1.394)--(0.2663,1.389)--(0.2943,1.383)--(0.3222,1.377)--(0.3499,1.370)--(0.3776,1.363)--(0.4050,1.355)--(0.4323,1.347)--(0.4594,1.338)--(0.4863,1.328)--(0.5131,1.318)--(0.5396,1.307)--(0.5659,1.296)--(0.5919,1.284)--(0.6178,1.272)--(0.6433,1.259)--(0.6686,1.246)--(0.6937,1.232)--(0.7184,1.218)--(0.7429,1.203)--(0.7671,1.188)--(0.7909,1.172)--(0.8144,1.156)--(0.8376,1.139)--(0.8605,1.122)--(0.8829,1.105)--(0.9051,1.087)--(0.9268,1.068)--(0.9482,1.049)--(0.9692,1.030)--(0.9898,1.010)--(1.010,0.9898)--(1.030,0.9692)--(1.049,0.9482)--(1.068,0.9268)--(1.087,0.9051)--(1.105,0.8829)--(1.122,0.8605)--(1.139,0.8376)--(1.156,0.8144)--(1.172,0.7909)--(1.188,0.7671)--(1.203,0.7429)--(1.218,0.7184)--(1.232,0.6937)--(1.246,0.6686)--(1.259,0.6433)--(1.272,0.6178)--(1.284,0.5919)--(1.296,0.5659)--(1.307,0.5396)--(1.318,0.5131)--(1.328,0.4863)--(1.338,0.4594)--(1.347,0.4323)--(1.355,0.4050)--(1.363,0.3776)--(1.370,0.3499)--(1.377,0.3222)--(1.383,0.2943)--(1.389,0.2663)--(1.394,0.2382)--(1.399,0.2100)--(1.402,0.1817)--(1.406,0.1533)--(1.409,0.1249)--(1.411,0.09641)--(1.413,0.06789)--(1.414,0.03934)--(1.414,0.01078)--(1.414,-0.01779)--(1.413,-0.04635)--(1.412,-0.07490)--(1.410,-0.1034)--(1.408,-0.1319)--(1.405,-0.1603)--(1.402,-0.1886)--(1.397,-0.2169)--(1.393,-0.2451)--(1.388,-0.2732)--(1.382,-0.3012);
-\draw [color=brown]  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=cyan] (-1.143,1.922)--(-1.104,1.945)--(-1.064,1.967)--(-1.024,1.988)--(-0.9838,2.008)--(-0.9430,2.027)--(-0.9018,2.046)--(-0.8603,2.064)--(-0.8185,2.081)--(-0.7763,2.097)--(-0.7337,2.112)--(-0.6909,2.127)--(-0.6478,2.140)--(-0.6045,2.153)--(-0.5608,2.165)--(-0.5170,2.175)--(-0.4729,2.185)--(-0.4287,2.195)--(-0.3843,2.203)--(-0.3397,2.210)--(-0.2950,2.217)--(-0.2502,2.222)--(-0.2052,2.227)--(-0.1602,2.230)--(-0.1151,2.233)--(-0.06998,2.235)--(-0.02482,2.236)--(0.02035,2.236)--(0.06552,2.235)--(0.1107,2.233)--(0.1557,2.231)--(0.2008,2.227)--(0.2457,2.223)--(0.2906,2.217)--(0.3353,2.211)--(0.3799,2.204)--(0.4243,2.195)--(0.4686,2.186)--(0.5127,2.177)--(0.5565,2.166)--(0.6002,2.154)--(0.6435,2.141)--(0.6867,2.128)--(0.7295,2.114)--(0.7721,2.099)--(0.8143,2.083)--(0.8562,2.066)--(0.8978,2.048)--(0.9389,2.029)--(0.9797,2.010)--(1.020,1.990)--(1.060,1.969)--(1.100,1.947)--(1.139,1.924)--(1.177,1.901)--(1.216,1.877)--(1.253,1.852)--(1.290,1.826)--(1.327,1.800)--(1.363,1.773)--(1.399,1.745)--(1.434,1.716)--(1.468,1.687)--(1.502,1.657)--(1.535,1.626)--(1.567,1.595)--(1.599,1.563)--(1.631,1.530)--(1.661,1.497)--(1.691,1.463)--(1.720,1.429)--(1.749,1.393)--(1.777,1.358)--(1.804,1.322)--(1.830,1.285)--(1.856,1.248)--(1.880,1.210)--(1.904,1.172)--(1.928,1.133)--(1.950,1.094)--(1.972,1.054)--(1.993,1.014)--(2.013,0.9738)--(2.032,0.9329)--(2.051,0.8917)--(2.068,0.8501)--(2.085,0.8081)--(2.101,0.7658)--(2.116,0.7233)--(2.130,0.6804)--(2.143,0.6372)--(2.156,0.5938)--(2.167,0.5501)--(2.178,0.5062)--(2.188,0.4621)--(2.197,0.4178)--(2.205,0.3734)--(2.212,0.3287)--(2.218,0.2840)--(2.223,0.2391);
-\draw [color=brown]  (1.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw [color=brown]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TriangleRectangle.pstricks b/auto/pictures_tex/Fig_TriangleRectangle.pstricks
index 606ad3d81..1807898e1 100644
--- a/auto/pictures_tex/Fig_TriangleRectangle.pstricks
+++ b/auto/pictures_tex/Fig_TriangleRectangle.pstricks
@@ -93,20 +93,20 @@
 \draw [] (2.00,3.46) -- (4.00,0);
 \draw [] (4.00,0) -- (0,0);
 \draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.251784057,0.3649246667) node {$60$};
+\draw (3.2518,0.36492) node {$60$};
 
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-\draw (2.311909189,2.234016645) node {$30$};
+\draw (2.3119,2.2340) node {$30$};
 
 \draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
-\draw []  (2.000000000,3.464101615) node [rotate=0] {$\bullet$};
-\draw (2.000000000,3.888809615) node {$A$};
+\draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
+\draw (2.0000,3.8888) node {$A$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.4475840000,0) node {$B$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.443490000,0) node {$C$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.4247080000) node {$H$};
+\draw (-0.44758,0) node {$B$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.4435,0) node {$C$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.42471) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_TriangleUV.pstricks b/auto/pictures_tex/Fig_TriangleUV.pstricks
index f48a25681..69dd3cc1b 100644
--- a/auto/pictures_tex/Fig_TriangleUV.pstricks
+++ b/auto/pictures_tex/Fig_TriangleUV.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -91,14 +91,14 @@
          hatchthickness=0.4pt}
 \fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,0) -- (0,3.00) -- (0,3.00) -- (3.00,0) -- (3.00,0) -- (0,0) --  cycle;
 
-\draw [color=green,->,>=latex] (0,0) -- (1.000000000,0);
-\draw (1.000000000,-0.2059510000) node {\( e_u\)};
-\draw [color=red,->,>=latex] (0,0) -- (0,1.000000000);
-\draw (-0.2670763333,1.000000000) node {\( e_v\)};
-\draw [color=green,->,>=latex] (1.712132034,1.712132034) -- (2.419238816,2.419238816);
-\draw (2.246758804,2.568525660) node {\( \nu\)};
-\draw [color=red,->,>=latex] (1.712132034,1.712132034) -- (1.005025253,2.419238816);
-\draw (0.8023187417,2.614657494) node {\( T\)};
+\draw [color=green,->,>=latex] (0,0) -- (1.0000,0);
+\draw (1.0000,-0.20595) node {\( e_u\)};
+\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
+\draw (-0.26708,1.0000) node {\( e_v\)};
+\draw [color=green,->,>=latex] (1.7121,1.7121) -- (2.4192,2.4192);
+\draw (2.2468,2.5685) node {\( \nu\)};
+\draw [color=red,->,>=latex] (1.7121,1.7121) -- (1.0050,2.4192);
+\draw (0.80232,2.6147) node {\( T\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks b/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks
index 54a390d85..039dac099 100644
--- a/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks
+++ b/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks
@@ -60,8 +60,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (0.5000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,0.5000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (0.50000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,0.50000);
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
 
diff --git a/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks b/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks
index a8cef4db8..55b304951 100644
--- a/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks
+++ b/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks
@@ -116,51 +116,51 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (14.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (14.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
-\draw []  (0,4.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,2.400000000) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,1.440000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0.8640000000) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,0.5184000000) node [rotate=0] {$\bullet$};
-\draw []  (5.000000000,0.3110400000) node [rotate=0] {$\bullet$};
-\draw []  (6.000000000,0.1866240000) node [rotate=0] {$\bullet$};
-\draw []  (7.000000000,0.1119744000) node [rotate=0] {$\bullet$};
-\draw []  (8.000000000,0.06718464000) node [rotate=0] {$\bullet$};
-\draw []  (9.000000000,0.04031078400) node [rotate=0] {$\bullet$};
-\draw []  (10.00000000,0.02418647040) node [rotate=0] {$\bullet$};
-\draw []  (11.00000000,0.01451188224) node [rotate=0] {$\bullet$};
-\draw []  (12.00000000,0.008707129344) node [rotate=0] {$\bullet$};
-\draw []  (13.00000000,0.005224277606) node [rotate=0] {$\bullet$};
-\draw []  (14.00000000,0.003134566564) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,2.4000) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,1.4400) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.86400) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.51840) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0.31104) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0.18662) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0.11197) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0.067185) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0.040311) node [rotate=0] {$\bullet$};
+\draw []  (10.000,0.024186) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0.014512) node [rotate=0] {$\bullet$};
+\draw []  (12.000,0.0087071) node [rotate=0] {$\bullet$};
+\draw []  (13.000,0.0052243) node [rotate=0] {$\bullet$};
+\draw []  (14.000,0.0031346) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.00000000,-0.3149246667) node {$ 11 $};
+\draw (11.000,-0.31492) node {$ 11 $};
 \draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.00000000,-0.3149246667) node {$ 12 $};
+\draw (12.000,-0.31492) node {$ 12 $};
 \draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.00000000,-0.3149246667) node {$ 13 $};
+\draw (13.000,-0.31492) node {$ 13 $};
 \draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (14.00000000,-0.3149246667) node {$ 14 $};
+\draw (14.000,-0.31492) node {$ 14 $};
 \draw [] (14.0,-0.100) -- (14.0,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks b/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks
index 2635881ef..8310c82c8 100644
--- a/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks
+++ b/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks
@@ -142,47 +142,47 @@
 \draw [color=gray,style=solid] (-3.00,4.00) -- (3.00,4.00);
 \draw [color=gray,style=solid] (-3.00,5.00) -- (3.00,5.00);
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-5.500000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-5.5000) -- (0,5.5000);
 %DEFAULT
-\draw (-3.418346945,4.913809108) node {\( y=cosh(x)\)};
+\draw (-3.4183,4.9138) node {\( y=cosh(x)\)};
 
 \draw [color=blue] (-2.275,4.914)--(-2.229,4.698)--(-2.183,4.492)--(-2.137,4.295)--(-2.091,4.108)--(-2.045,3.929)--(-1.999,3.758)--(-1.953,3.596)--(-1.907,3.441)--(-1.861,3.293)--(-1.815,3.152)--(-1.769,3.018)--(-1.723,2.891)--(-1.677,2.769)--(-1.631,2.653)--(-1.585,2.543)--(-1.539,2.438)--(-1.493,2.339)--(-1.448,2.244)--(-1.402,2.154)--(-1.356,2.068)--(-1.310,1.987)--(-1.264,1.911)--(-1.218,1.838)--(-1.172,1.769)--(-1.126,1.704)--(-1.080,1.642)--(-1.034,1.584)--(-0.9880,1.529)--(-0.9420,1.478)--(-0.8961,1.429)--(-0.8501,1.384)--(-0.8042,1.341)--(-0.7582,1.301)--(-0.7123,1.265)--(-0.6663,1.230)--(-0.6204,1.199)--(-0.5744,1.170)--(-0.5285,1.143)--(-0.4825,1.119)--(-0.4366,1.097)--(-0.3906,1.077)--(-0.3446,1.060)--(-0.2987,1.045)--(-0.2527,1.032)--(-0.2068,1.021)--(-0.1608,1.013)--(-0.1149,1.007)--(-0.06893,1.002)--(-0.02298,1.000)--(0.02298,1.000)--(0.06893,1.002)--(0.1149,1.007)--(0.1608,1.013)--(0.2068,1.021)--(0.2527,1.032)--(0.2987,1.045)--(0.3446,1.060)--(0.3906,1.077)--(0.4366,1.097)--(0.4825,1.119)--(0.5285,1.143)--(0.5744,1.170)--(0.6204,1.199)--(0.6663,1.230)--(0.7123,1.265)--(0.7582,1.301)--(0.8042,1.341)--(0.8501,1.384)--(0.8961,1.429)--(0.9420,1.478)--(0.9880,1.529)--(1.034,1.584)--(1.080,1.642)--(1.126,1.704)--(1.172,1.769)--(1.218,1.838)--(1.264,1.911)--(1.310,1.987)--(1.356,2.068)--(1.402,2.154)--(1.448,2.244)--(1.493,2.339)--(1.539,2.438)--(1.585,2.543)--(1.631,2.653)--(1.677,2.769)--(1.723,2.891)--(1.769,3.018)--(1.815,3.152)--(1.861,3.293)--(1.907,3.441)--(1.953,3.596)--(1.999,3.758)--(2.045,3.929)--(2.091,4.108)--(2.137,4.295)--(2.183,4.492)--(2.229,4.698)--(2.275,4.914);
-\draw (-3.423323612,-4.810979105) node {\( y=sinh(x)\)};
+\draw (-3.4233,-4.8110) node {\( y=sinh(x)\)};
 
 \draw [color=blue] (-2.275,-4.811)--(-2.229,-4.590)--(-2.183,-4.379)--(-2.137,-4.177)--(-2.091,-3.984)--(-2.045,-3.800)--(-1.999,-3.623)--(-1.953,-3.454)--(-1.907,-3.292)--(-1.861,-3.138)--(-1.815,-2.990)--(-1.769,-2.848)--(-1.723,-2.712)--(-1.677,-2.582)--(-1.631,-2.458)--(-1.585,-2.338)--(-1.539,-2.224)--(-1.493,-2.114)--(-1.448,-2.009)--(-1.402,-1.908)--(-1.356,-1.811)--(-1.310,-1.718)--(-1.264,-1.628)--(-1.218,-1.542)--(-1.172,-1.459)--(-1.126,-1.379)--(-1.080,-1.302)--(-1.034,-1.228)--(-0.9880,-1.157)--(-0.9420,-1.088)--(-0.8961,-1.021)--(-0.8501,-0.9563)--(-0.8042,-0.8937)--(-0.7582,-0.8330)--(-0.7123,-0.7740)--(-0.6663,-0.7167)--(-0.6204,-0.6609)--(-0.5744,-0.6065)--(-0.5285,-0.5534)--(-0.4825,-0.5014)--(-0.4366,-0.4506)--(-0.3906,-0.4006)--(-0.3446,-0.3515)--(-0.2987,-0.3032)--(-0.2527,-0.2554)--(-0.2068,-0.2083)--(-0.1608,-0.1615)--(-0.1149,-0.1151)--(-0.06893,-0.06898)--(-0.02298,-0.02298)--(0.02298,0.02298)--(0.06893,0.06898)--(0.1149,0.1151)--(0.1608,0.1615)--(0.2068,0.2083)--(0.2527,0.2554)--(0.2987,0.3032)--(0.3446,0.3515)--(0.3906,0.4006)--(0.4366,0.4506)--(0.4825,0.5014)--(0.5285,0.5534)--(0.5744,0.6065)--(0.6204,0.6609)--(0.6663,0.7167)--(0.7123,0.7740)--(0.7582,0.8330)--(0.8042,0.8937)--(0.8501,0.9563)--(0.8961,1.021)--(0.9420,1.088)--(0.9880,1.157)--(1.034,1.228)--(1.080,1.302)--(1.126,1.379)--(1.172,1.459)--(1.218,1.542)--(1.264,1.628)--(1.310,1.718)--(1.356,1.811)--(1.402,1.908)--(1.448,2.009)--(1.493,2.114)--(1.539,2.224)--(1.585,2.338)--(1.631,2.458)--(1.677,2.582)--(1.723,2.712)--(1.769,2.848)--(1.815,2.990)--(1.861,3.138)--(1.907,3.292)--(1.953,3.454)--(1.999,3.623)--(2.045,3.800)--(2.091,3.984)--(2.137,4.177)--(2.183,4.379)--(2.229,4.590)--(2.275,4.811);
 
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
 \draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks b/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks
index 819ccd255..a266f3275 100644
--- a/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks
+++ b/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks
@@ -108,36 +108,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-2.070796327) -- (0,2.070796327);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-2.0708) -- (0,2.0708);
 %DEFAULT
 
 \draw [color=blue] (-5.000,-1.373)--(-4.899,-1.369)--(-4.798,-1.365)--(-4.697,-1.361)--(-4.596,-1.357)--(-4.495,-1.352)--(-4.394,-1.347)--(-4.293,-1.342)--(-4.192,-1.337)--(-4.091,-1.331)--(-3.990,-1.325)--(-3.889,-1.319)--(-3.788,-1.313)--(-3.687,-1.306)--(-3.586,-1.299)--(-3.485,-1.291)--(-3.384,-1.283)--(-3.283,-1.275)--(-3.182,-1.266)--(-3.081,-1.257)--(-2.980,-1.247)--(-2.879,-1.236)--(-2.778,-1.225)--(-2.677,-1.213)--(-2.576,-1.200)--(-2.475,-1.187)--(-2.374,-1.172)--(-2.273,-1.156)--(-2.172,-1.139)--(-2.071,-1.121)--(-1.970,-1.101)--(-1.869,-1.079)--(-1.768,-1.056)--(-1.667,-1.030)--(-1.566,-1.002)--(-1.465,-0.9717)--(-1.364,-0.9380)--(-1.263,-0.9010)--(-1.162,-0.8600)--(-1.061,-0.8148)--(-0.9596,-0.7648)--(-0.8586,-0.7095)--(-0.7576,-0.6483)--(-0.6566,-0.5810)--(-0.5556,-0.5071)--(-0.4545,-0.4266)--(-0.3535,-0.3398)--(-0.2525,-0.2474)--(-0.1515,-0.1504)--(-0.05051,-0.05046)--(0.05051,0.05046)--(0.1515,0.1504)--(0.2525,0.2474)--(0.3535,0.3398)--(0.4545,0.4266)--(0.5556,0.5071)--(0.6566,0.5810)--(0.7576,0.6483)--(0.8586,0.7095)--(0.9596,0.7648)--(1.061,0.8148)--(1.162,0.8600)--(1.263,0.9010)--(1.364,0.9380)--(1.465,0.9717)--(1.566,1.002)--(1.667,1.030)--(1.768,1.056)--(1.869,1.079)--(1.970,1.101)--(2.071,1.121)--(2.172,1.139)--(2.273,1.156)--(2.374,1.172)--(2.475,1.187)--(2.576,1.200)--(2.677,1.213)--(2.778,1.225)--(2.879,1.236)--(2.980,1.247)--(3.081,1.257)--(3.182,1.266)--(3.283,1.275)--(3.384,1.283)--(3.485,1.291)--(3.586,1.299)--(3.687,1.306)--(3.788,1.313)--(3.889,1.319)--(3.990,1.325)--(4.091,1.331)--(4.192,1.337)--(4.293,1.342)--(4.394,1.347)--(4.495,1.352)--(4.596,1.357)--(4.697,1.361)--(4.798,1.365)--(4.899,1.369)--(5.000,1.373);
 \draw [color=red,style=dashed] (-5.00,1.57) -- (5.00,1.57);
 \draw [color=red,style=dashed] (-5.00,-1.57) -- (5.00,-1.57);
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.5937365000,-1.570796327) node {$ -\frac{1}{2} \, \pi $};
+\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.4518270000,1.570796327) node {$ \frac{1}{2} \, \pi $};
+\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
 \draw [] (-0.100,1.57) -- (0.100,1.57);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks b/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks
index 800f0865f..b8a022bef 100644
--- a/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks
+++ b/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks
@@ -96,46 +96,46 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,4.5000);
 %DEFAULT
 \fill [color=lightgray] (2.83,1.00) -- (2.80,1.07) -- (2.78,1.14) -- (2.75,1.21) -- (2.72,1.27) -- (2.68,1.34) -- (2.65,1.41) -- (2.62,1.47) -- (2.58,1.53) -- (2.54,1.60) -- (2.50,1.66) -- (2.46,1.72) -- (2.41,1.78) -- (2.37,1.84) -- (2.32,1.90) -- (2.28,1.95) -- (2.23,2.01) -- (2.18,2.06) -- (2.13,2.12) -- (2.07,2.17) -- (2.02,2.22) -- (1.96,2.27) -- (1.91,2.31) -- (1.85,2.36) -- (1.79,2.41) -- (1.73,2.45) -- (1.67,2.49) -- (1.61,2.53) -- (1.55,2.57) -- (1.48,2.61) -- (1.42,2.64) -- (1.35,2.68) -- (1.29,2.71) -- (1.22,2.74) -- (1.15,2.77) -- (1.08,2.80) -- (1.01,2.82) -- (0.944,2.85) -- (0.873,2.87) -- (0.803,2.89) -- (0.731,2.91) -- (0.659,2.93) -- (0.587,2.94) -- (0.515,2.96) -- (0.442,2.97) -- (0.368,2.98) -- (0.295,2.99) -- (0.221,2.99) -- (0.148,3.00) -- (0.0740,3.00) -- (0,3.00) -- (2.00,1.00) -- (2.00,1.00) -- (2.83,1.00) --  cycle;
 
 \draw [] (3.000,0)--(2.994,0.1903)--(2.976,0.3798)--(2.946,0.5678)--(2.904,0.7534)--(2.850,0.9361)--(2.785,1.115)--(2.709,1.289)--(2.622,1.459)--(2.524,1.622)--(2.416,1.779)--(2.298,1.928)--(2.171,2.070)--(2.036,2.204)--(1.892,2.328)--(1.740,2.444)--(1.582,2.549)--(1.417,2.644)--(1.246,2.729)--(1.071,2.802)--(0.8908,2.865)--(0.7073,2.915)--(0.5209,2.954)--(0.3325,2.982)--(0.1427,2.997)--(-0.04760,3.000)--(-0.2377,2.991)--(-0.4269,2.969)--(-0.6144,2.936)--(-0.7994,2.892)--(-0.9812,2.835)--(-1.159,2.767)--(-1.332,2.688)--(-1.500,2.598)--(-1.662,2.498)--(-1.817,2.387)--(-1.965,2.267)--(-2.104,2.138)--(-2.236,2.000)--(-2.358,1.854)--(-2.471,1.701)--(-2.574,1.541)--(-2.667,1.375)--(-2.748,1.203)--(-2.819,1.026)--(-2.878,0.8452)--(-2.926,0.6609)--(-2.962,0.4740)--(-2.986,0.2852)--(-2.999,0.09518)--(-2.999,-0.09518)--(-2.986,-0.2852)--(-2.962,-0.4740)--(-2.926,-0.6609)--(-2.878,-0.8452)--(-2.819,-1.026)--(-2.748,-1.203)--(-2.667,-1.375)--(-2.574,-1.541)--(-2.471,-1.701)--(-2.358,-1.854)--(-2.236,-2.000)--(-2.104,-2.138)--(-1.965,-2.267)--(-1.817,-2.387)--(-1.662,-2.498)--(-1.500,-2.598)--(-1.332,-2.688)--(-1.159,-2.767)--(-0.9812,-2.835)--(-0.7994,-2.892)--(-0.6144,-2.936)--(-0.4269,-2.969)--(-0.2377,-2.991)--(-0.04760,-3.000)--(0.1427,-2.997)--(0.3325,-2.982)--(0.5209,-2.954)--(0.7073,-2.915)--(0.8908,-2.865)--(1.071,-2.802)--(1.246,-2.729)--(1.417,-2.644)--(1.582,-2.549)--(1.740,-2.444)--(1.892,-2.328)--(2.036,-2.204)--(2.171,-2.070)--(2.298,-1.928)--(2.416,-1.779)--(2.524,-1.622)--(2.622,-1.459)--(2.709,-1.289)--(2.785,-1.115)--(2.850,-0.9361)--(2.904,-0.7534)--(2.946,-0.5678)--(2.976,-0.3798)--(2.994,-0.1903)--(3.000,0);
-\draw []  (2.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.721703477,0.7338706438) node {\( A\)};
-\draw []  (2.828427125,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.117432481,0.7338706438) node {\( B\)};
+\draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.7217,0.73387) node {\( A\)};
+\draw []  (2.8284,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.1174,0.73387) node {\( B\)};
 \draw [] (-1.00,1.00) -- (4.00,1.00);
 
 \draw [color=blue] (-1.000,4.000)--(-0.9495,3.949)--(-0.8990,3.899)--(-0.8485,3.848)--(-0.7980,3.798)--(-0.7475,3.747)--(-0.6970,3.697)--(-0.6465,3.646)--(-0.5960,3.596)--(-0.5455,3.545)--(-0.4949,3.495)--(-0.4444,3.444)--(-0.3939,3.394)--(-0.3434,3.343)--(-0.2929,3.293)--(-0.2424,3.242)--(-0.1919,3.192)--(-0.1414,3.141)--(-0.09091,3.091)--(-0.04040,3.040)--(0.01010,2.990)--(0.06061,2.939)--(0.1111,2.889)--(0.1616,2.838)--(0.2121,2.788)--(0.2626,2.737)--(0.3131,2.687)--(0.3636,2.636)--(0.4141,2.586)--(0.4646,2.535)--(0.5152,2.485)--(0.5657,2.434)--(0.6162,2.384)--(0.6667,2.333)--(0.7172,2.283)--(0.7677,2.232)--(0.8182,2.182)--(0.8687,2.131)--(0.9192,2.081)--(0.9697,2.030)--(1.020,1.980)--(1.071,1.929)--(1.121,1.879)--(1.172,1.828)--(1.222,1.778)--(1.273,1.727)--(1.323,1.677)--(1.374,1.626)--(1.424,1.576)--(1.475,1.525)--(1.525,1.475)--(1.576,1.424)--(1.626,1.374)--(1.677,1.323)--(1.727,1.273)--(1.778,1.222)--(1.828,1.172)--(1.879,1.121)--(1.929,1.071)--(1.980,1.020)--(2.030,0.9697)--(2.081,0.9192)--(2.131,0.8687)--(2.182,0.8182)--(2.232,0.7677)--(2.283,0.7172)--(2.333,0.6667)--(2.384,0.6162)--(2.434,0.5657)--(2.485,0.5152)--(2.535,0.4646)--(2.586,0.4141)--(2.636,0.3636)--(2.687,0.3131)--(2.737,0.2626)--(2.788,0.2121)--(2.838,0.1616)--(2.889,0.1111)--(2.939,0.06061)--(2.990,0.01010)--(3.040,-0.04040)--(3.091,-0.09091)--(3.141,-0.1414)--(3.192,-0.1919)--(3.242,-0.2424)--(3.293,-0.2929)--(3.343,-0.3434)--(3.394,-0.3939)--(3.444,-0.4444)--(3.495,-0.4949)--(3.545,-0.5455)--(3.596,-0.5960)--(3.646,-0.6465)--(3.697,-0.6970)--(3.747,-0.7475)--(3.798,-0.7980)--(3.848,-0.8485)--(3.899,-0.8990)--(3.949,-0.9495)--(4.000,-1.000);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UnSurxInt.pstricks b/auto/pictures_tex/Fig_UnSurxInt.pstricks
index 8736d3043..702ef7c25 100644
--- a/auto/pictures_tex/Fig_UnSurxInt.pstricks
+++ b/auto/pictures_tex/Fig_UnSurxInt.pstricks
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.833333333) -- (0,3.833333333);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.8333) -- (0,3.8333);
 %DEFAULT
 
 \draw [color=blue] (-3.000,-0.3333)--(-2.973,-0.3364)--(-2.945,-0.3395)--(-2.918,-0.3427)--(-2.891,-0.3459)--(-2.864,-0.3492)--(-2.836,-0.3526)--(-2.809,-0.3560)--(-2.782,-0.3595)--(-2.755,-0.3630)--(-2.727,-0.3667)--(-2.700,-0.3704)--(-2.673,-0.3741)--(-2.645,-0.3780)--(-2.618,-0.3819)--(-2.591,-0.3860)--(-2.564,-0.3901)--(-2.536,-0.3943)--(-2.509,-0.3986)--(-2.482,-0.4029)--(-2.455,-0.4074)--(-2.427,-0.4120)--(-2.400,-0.4167)--(-2.373,-0.4215)--(-2.345,-0.4264)--(-2.318,-0.4314)--(-2.291,-0.4365)--(-2.264,-0.4418)--(-2.236,-0.4472)--(-2.209,-0.4527)--(-2.182,-0.4583)--(-2.155,-0.4641)--(-2.127,-0.4701)--(-2.100,-0.4762)--(-2.073,-0.4825)--(-2.045,-0.4889)--(-2.018,-0.4955)--(-1.991,-0.5023)--(-1.964,-0.5093)--(-1.936,-0.5164)--(-1.909,-0.5238)--(-1.882,-0.5314)--(-1.855,-0.5392)--(-1.827,-0.5473)--(-1.800,-0.5556)--(-1.773,-0.5641)--(-1.745,-0.5729)--(-1.718,-0.5820)--(-1.691,-0.5914)--(-1.664,-0.6011)--(-1.636,-0.6111)--(-1.609,-0.6215)--(-1.582,-0.6322)--(-1.555,-0.6433)--(-1.527,-0.6548)--(-1.500,-0.6667)--(-1.473,-0.6790)--(-1.445,-0.6918)--(-1.418,-0.7051)--(-1.391,-0.7190)--(-1.364,-0.7333)--(-1.336,-0.7483)--(-1.309,-0.7639)--(-1.282,-0.7801)--(-1.255,-0.7971)--(-1.227,-0.8148)--(-1.200,-0.8333)--(-1.173,-0.8527)--(-1.145,-0.8730)--(-1.118,-0.8943)--(-1.091,-0.9167)--(-1.064,-0.9402)--(-1.036,-0.9649)--(-1.009,-0.9910)--(-0.9818,-1.019)--(-0.9545,-1.048)--(-0.9273,-1.078)--(-0.9000,-1.111)--(-0.8727,-1.146)--(-0.8455,-1.183)--(-0.8182,-1.222)--(-0.7909,-1.264)--(-0.7636,-1.310)--(-0.7364,-1.358)--(-0.7091,-1.410)--(-0.6818,-1.467)--(-0.6545,-1.528)--(-0.6273,-1.594)--(-0.6000,-1.667)--(-0.5727,-1.746)--(-0.5455,-1.833)--(-0.5182,-1.930)--(-0.4909,-2.037)--(-0.4636,-2.157)--(-0.4364,-2.292)--(-0.4091,-2.444)--(-0.3818,-2.619)--(-0.3545,-2.821)--(-0.3273,-3.056)--(-0.3000,-3.333);
@@ -118,29 +118,29 @@
 \draw [color=blue] (1.000,0)--(1.010,0)--(1.020,0)--(1.030,0)--(1.040,0)--(1.051,0)--(1.061,0)--(1.071,0)--(1.081,0)--(1.091,0)--(1.101,0)--(1.111,0)--(1.121,0)--(1.131,0)--(1.141,0)--(1.152,0)--(1.162,0)--(1.172,0)--(1.182,0)--(1.192,0)--(1.202,0)--(1.212,0)--(1.222,0)--(1.232,0)--(1.242,0)--(1.253,0)--(1.263,0)--(1.273,0)--(1.283,0)--(1.293,0)--(1.303,0)--(1.313,0)--(1.323,0)--(1.333,0)--(1.343,0)--(1.354,0)--(1.364,0)--(1.374,0)--(1.384,0)--(1.394,0)--(1.404,0)--(1.414,0)--(1.424,0)--(1.434,0)--(1.444,0)--(1.455,0)--(1.465,0)--(1.475,0)--(1.485,0)--(1.495,0)--(1.505,0)--(1.515,0)--(1.525,0)--(1.535,0)--(1.545,0)--(1.556,0)--(1.566,0)--(1.576,0)--(1.586,0)--(1.596,0)--(1.606,0)--(1.616,0)--(1.626,0)--(1.636,0)--(1.646,0)--(1.657,0)--(1.667,0)--(1.677,0)--(1.687,0)--(1.697,0)--(1.707,0)--(1.717,0)--(1.727,0)--(1.737,0)--(1.747,0)--(1.758,0)--(1.768,0)--(1.778,0)--(1.788,0)--(1.798,0)--(1.808,0)--(1.818,0)--(1.828,0)--(1.838,0)--(1.848,0)--(1.859,0)--(1.869,0)--(1.879,0)--(1.889,0)--(1.899,0)--(1.909,0)--(1.919,0)--(1.929,0)--(1.939,0)--(1.949,0)--(1.960,0)--(1.970,0)--(1.980,0)--(1.990,0)--(2.000,0);
 \draw [color=brown,style=solid] (1.00,0) -- (1.00,1.00);
 \draw [color=brown,style=solid] (2.00,0.500) -- (2.00,0);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks b/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks
index 550c4f950..34b8ef2f9 100644
--- a/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks
+++ b/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks
@@ -66,14 +66,14 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.246812693,0.6811368787) -- (1.245451007,0.6910437354);
-\draw [,->,>=latex] (0.3575977601,1.218773069) -- (0.3480435445,1.215820618);
-\draw [,->,>=latex] (-0.2777141985,0.5083524306) -- (-0.2808616227,0.4988606594);
-\draw [,->,>=latex] (-0.7003261911,0.2054813022) -- (-0.7094196719,0.2013209407);
-\draw [,->,>=latex] (-1.246812693,-0.6811368787) -- (-1.245451007,-0.6910437354);
-\draw [,->,>=latex] (-0.3575977601,-1.218773069) -- (-0.3480435445,-1.215820618);
-\draw [,->,>=latex] (0.2777141985,-0.5083524306) -- (0.2808616227,-0.4988606594);
-\draw [,->,>=latex] (0.7003261912,-0.2054813022) -- (0.7094196719,-0.2013209407);
+\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
+\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
+\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
+\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
+\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
+\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
+\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
+\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks b/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks
index 22d34bd01..d8e38e460 100644
--- a/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks
+++ b/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -88,11 +88,11 @@
 \draw [color=red,style=solid] (1.00,2.00) -- (1.00,0);
 
 \draw [color=red] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks b/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks
index e5ae1d139..ba9700c1b 100644
--- a/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks
+++ b/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.650000000,0) -- (3.650000000,0);
-\draw [,->,>=latex] (0,-2.650000000) -- (0,2.650000000);
+\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
+\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
 %DEFAULT
 \draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
 \draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
@@ -102,21 +102,21 @@
 \draw [color=blue] (2.00,0) -- (1.00,-1.00);
 \draw [color=blue] (1.00,-1.00) -- (0,0);
 \draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks b/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks
index d7858707b..ff1ca43f9 100644
--- a/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks
+++ b/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks
@@ -79,13 +79,13 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3247080000) node {\( O\)};
-\draw []  (1.050000000,1.818653348) node [rotate=0] {$\bullet$};
-\draw (1.297584000,2.116566429) node {\( B\)};
-\draw []  (1.818653348,1.050000000) node [rotate=0] {$\bullet$};
-\draw (2.128733595,1.274708000) node {\( A\)};
-\draw []  (2.100000000,0) node [rotate=0] {$\bullet$};
-\draw (2.335966356,-0.2661293562) node {\( I\)};
+\draw (0,-0.32471) node {\( O\)};
+\draw []  (1.0500,1.8187) node [rotate=0] {$\bullet$};
+\draw (1.2976,2.1166) node {\( B\)};
+\draw []  (1.8187,1.0500) node [rotate=0] {$\bullet$};
+\draw (2.1287,1.2747) node {\( A\)};
+\draw []  (2.1000,0) node [rotate=0] {$\bullet$};
+\draw (2.3360,-0.26613) node {\( I\)};
 
 \draw [] (2.100,0)--(2.096,0.1332)--(2.083,0.2658)--(2.062,0.3974)--(2.033,0.5274)--(1.995,0.6553)--(1.950,0.7805)--(1.896,0.9026)--(1.835,1.021)--(1.767,1.135)--(1.691,1.245)--(1.609,1.350)--(1.520,1.449)--(1.425,1.543)--(1.324,1.630)--(1.218,1.711)--(1.107,1.784)--(0.9918,1.851)--(0.8724,1.910)--(0.7495,1.962)--(0.6235,2.005)--(0.4951,2.041)--(0.3647,2.068)--(0.2328,2.087)--(0.09992,2.098)--(-0.03332,2.100)--(-0.1664,2.093)--(-0.2989,2.079)--(-0.4301,2.055)--(-0.5596,2.024)--(-0.6868,1.984)--(-0.8113,1.937)--(-0.9325,1.882)--(-1.050,1.819)--(-1.163,1.748)--(-1.272,1.671)--(-1.375,1.587)--(-1.473,1.497)--(-1.565,1.400)--(-1.651,1.298)--(-1.730,1.191)--(-1.802,1.079)--(-1.867,0.9623)--(-1.924,0.8420)--(-1.973,0.7182)--(-2.015,0.5916)--(-2.048,0.4627)--(-2.074,0.3318)--(-2.090,0.1996)--(-2.099,0.06663)--(-2.099,-0.06663)--(-2.090,-0.1996)--(-2.074,-0.3318)--(-2.048,-0.4627)--(-2.015,-0.5916)--(-1.973,-0.7182)--(-1.924,-0.8420)--(-1.867,-0.9623)--(-1.802,-1.079)--(-1.730,-1.191)--(-1.651,-1.298)--(-1.565,-1.400)--(-1.473,-1.497)--(-1.375,-1.587)--(-1.272,-1.671)--(-1.163,-1.748)--(-1.050,-1.819)--(-0.9325,-1.882)--(-0.8113,-1.937)--(-0.6868,-1.984)--(-0.5596,-2.024)--(-0.4301,-2.055)--(-0.2989,-2.079)--(-0.1664,-2.093)--(-0.03332,-2.100)--(0.09992,-2.098)--(0.2328,-2.087)--(0.3647,-2.068)--(0.4951,-2.041)--(0.6235,-2.005)--(0.7495,-1.962)--(0.8724,-1.910)--(0.9918,-1.851)--(1.107,-1.784)--(1.218,-1.711)--(1.324,-1.630)--(1.425,-1.543)--(1.520,-1.449)--(1.609,-1.350)--(1.691,-1.245)--(1.767,-1.135)--(1.835,-1.021)--(1.896,-0.9026)--(1.950,-0.7805)--(1.995,-0.6553)--(2.033,-0.5274)--(2.062,-0.3974)--(2.083,-0.2658)--(2.096,-0.1332)--(2.100,0);
 
diff --git a/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks b/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
index fd2268f2a..9099168b7 100644
--- a/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
+++ b/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
@@ -119,40 +119,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
+\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
 %DEFAULT
 
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-\draw (-3.298672286,-0.4207143333) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.199114858,-0.3210296667) node {$-\mathit{\pi}$};
+\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
 \draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.099557429,-0.4207143333) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.099557429,-0.4207143333) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.199114858,-0.2785761667) node {$\mathit{\pi}$};
+\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
 \draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.298672286,-0.4207143333) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks b/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
index bd5d82aaf..ae1ef50a0 100644
--- a/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
+++ b/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
@@ -100,8 +100,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.212388980,0) -- (9.924777961,0);
-\draw [,->,>=latex] (0,-3.445045352) -- (0,3.317012042);
+\draw [,->,>=latex] (-5.2124,0) -- (9.9248,0);
+\draw [,->,>=latex] (0,-3.4450) -- (0,3.3170);
 %DEFAULT
 
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@@ -109,25 +109,25 @@
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 \draw [color=blue,style=dashed] (-1.072,-2.945)--(-0.9794,-2.436)--(-0.8868,-1.974)--(-0.7943,-1.558)--(-0.7017,-1.185)--(-0.6092,-0.8526)--(-0.5167,-0.5589)--(-0.4241,-0.3015)--(-0.3316,-0.07832)--(-0.2390,0.1128)--(-0.1465,0.2738)--(-0.05393,0.4068)--(0.03861,0.5136)--(0.1312,0.5962)--(0.2237,0.6564)--(0.3162,0.6960)--(0.4088,0.7166)--(0.5013,0.7200)--(0.5939,0.7079)--(0.6864,0.6817)--(0.7790,0.6430)--(0.8715,0.5932)--(0.9641,0.5339)--(1.057,0.4663)--(1.149,0.3917)--(1.242,0.3115)--(1.334,0.2269)--(1.427,0.1389)--(1.519,0.04875)--(1.612,-0.04255)--(1.704,-0.1340)--(1.797,-0.2246)--(1.889,-0.3135)--(1.982,-0.3998)--(2.075,-0.4828)--(2.167,-0.5616)--(2.260,-0.6357)--(2.352,-0.7043)--(2.445,-0.7669)--(2.537,-0.8229)--(2.630,-0.8719)--(2.722,-0.9134)--(2.815,-0.9470)--(2.907,-0.9725)--(3.000,-0.9894)--(3.093,-0.9977)--(3.185,-0.9971)--(3.278,-0.9875)--(3.370,-0.9689)--(3.463,-0.9411)--(3.555,-0.9044)--(3.648,-0.8586)--(3.740,-0.8041)--(3.833,-0.7409)--(3.925,-0.6693)--(4.018,-0.5897)--(4.111,-0.5022)--(4.203,-0.4074)--(4.296,-0.3058)--(4.388,-0.1977)--(4.481,-0.08370)--(4.573,0.03551)--(4.666,0.1593)--(4.758,0.2870)--(4.851,0.4178)--(4.943,0.5509)--(5.036,0.6855)--(5.129,0.8206)--(5.221,0.9553)--(5.314,1.089)--(5.406,1.219)--(5.499,1.346)--(5.591,1.469)--(5.684,1.585)--(5.776,1.694)--(5.869,1.794)--(5.961,1.884)--(6.054,1.963)--(6.146,2.028)--(6.239,2.080)--(6.332,2.115)--(6.424,2.132)--(6.517,2.130)--(6.609,2.107)--(6.702,2.061)--(6.794,1.990)--(6.887,1.893)--(6.979,1.766)--(7.072,1.609)--(7.164,1.420)--(7.257,1.195)--(7.350,0.9335)--(7.442,0.6328)--(7.535,0.2905)--(7.627,-0.09545)--(7.720,-0.5276)--(7.812,-1.008)--(7.905,-1.540)--(7.997,-2.125)--(8.090,-2.767);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (9.424777961,-0.3149246667) node {$ 3 \, \pi $};
+\draw (9.4248,-0.31492) node {$ 3 \, \pi $};
 \draw [] (9.42,-0.100) -- (9.42,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_WUYooCISzeB.pstricks b/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
index bf8eb2063..b138a1fb5 100644
--- a/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
+++ b/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
@@ -61,28 +61,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.9918454937) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.99185) -- (0,4.0276);
 %DEFAULT
 
 \draw [color=red] (-0.8400,2.700)--(-0.8013,2.668)--(-0.7626,2.636)--(-0.7239,2.603)--(-0.6852,2.571)--(-0.6466,2.539)--(-0.6079,2.507)--(-0.5692,2.474)--(-0.5305,2.442)--(-0.4918,2.410)--(-0.4531,2.378)--(-0.4144,2.345)--(-0.3757,2.313)--(-0.3370,2.281)--(-0.2984,2.249)--(-0.2597,2.216)--(-0.2210,2.184)--(-0.1823,2.152)--(-0.1436,2.120)--(-0.1049,2.087)--(-0.06622,2.055)--(-0.02753,2.023)--(0.01116,1.991)--(0.04985,1.958)--(0.08854,1.926)--(0.1272,1.894)--(0.1659,1.862)--(0.2046,1.829)--(0.2433,1.797)--(0.2820,1.765)--(0.3207,1.733)--(0.3594,1.701)--(0.3980,1.668)--(0.4367,1.636)--(0.4754,1.604)--(0.5141,1.572)--(0.5528,1.539)--(0.5915,1.507)--(0.6302,1.475)--(0.6689,1.443)--(0.7076,1.410)--(0.7463,1.378)--(0.7849,1.346)--(0.8236,1.314)--(0.8623,1.281)--(0.9010,1.249)--(0.9397,1.217)--(0.9784,1.185)--(1.017,1.152)--(1.056,1.120)--(1.094,1.088)--(1.133,1.056)--(1.172,1.023)--(1.211,0.9912)--(1.249,0.9590)--(1.288,0.9268)--(1.327,0.8945)--(1.365,0.8623)--(1.404,0.8300)--(1.443,0.7978)--(1.481,0.7655)--(1.520,0.7333)--(1.559,0.7011)--(1.597,0.6688)--(1.636,0.6366)--(1.675,0.6043)--(1.713,0.5721)--(1.752,0.5399)--(1.791,0.5076)--(1.830,0.4754)--(1.868,0.4431)--(1.907,0.4109)--(1.946,0.3787)--(1.984,0.3464)--(2.023,0.3142)--(2.062,0.2819)--(2.100,0.2497)--(2.139,0.2175)--(2.178,0.1852)--(2.216,0.1530)--(2.255,0.1207)--(2.294,0.08849)--(2.332,0.05625)--(2.371,0.02401)--(2.410,-0.008233)--(2.449,-0.04047)--(2.487,-0.07271)--(2.526,-0.1050)--(2.565,-0.1372)--(2.603,-0.1694)--(2.642,-0.2017)--(2.681,-0.2339)--(2.719,-0.2662)--(2.758,-0.2984)--(2.797,-0.3306)--(2.835,-0.3629)--(2.874,-0.3951)--(2.913,-0.4274)--(2.952,-0.4596)--(2.990,-0.4918);
 
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-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -141,28 +141,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
 %DEFAULT
 
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 \draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -221,28 +221,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.9831816482) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.98318) -- (0,4.0276);
 %DEFAULT
 
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 \draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -301,28 +301,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
 %DEFAULT
 
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-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks b/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks
index ed308a35a..766d698d6 100644
--- a/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks
+++ b/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks
@@ -144,8 +144,8 @@
 \draw [color=gray,style=solid] (-5.00,2.00) -- (5.00,2.00);
 \draw [color=gray,style=solid] (-5.00,3.00) -- (5.00,3.00);
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,3.5000);
 %DEFAULT
 
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@@ -153,46 +153,46 @@
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 \draw [color=blue] (1.000,3.000)--(1.040,2.960)--(1.081,2.919)--(1.121,2.879)--(1.162,2.838)--(1.202,2.798)--(1.242,2.758)--(1.283,2.717)--(1.323,2.677)--(1.364,2.636)--(1.404,2.596)--(1.444,2.556)--(1.485,2.515)--(1.525,2.475)--(1.566,2.434)--(1.606,2.394)--(1.646,2.354)--(1.687,2.313)--(1.727,2.273)--(1.768,2.232)--(1.808,2.192)--(1.848,2.152)--(1.889,2.111)--(1.929,2.071)--(1.970,2.030)--(2.010,1.990)--(2.051,1.949)--(2.091,1.909)--(2.131,1.869)--(2.172,1.828)--(2.212,1.788)--(2.253,1.747)--(2.293,1.707)--(2.333,1.667)--(2.374,1.626)--(2.414,1.586)--(2.455,1.545)--(2.495,1.505)--(2.535,1.465)--(2.576,1.424)--(2.616,1.384)--(2.657,1.343)--(2.697,1.303)--(2.737,1.263)--(2.778,1.222)--(2.818,1.182)--(2.859,1.141)--(2.899,1.101)--(2.939,1.061)--(2.980,1.020)--(3.020,0.9798)--(3.061,0.9394)--(3.101,0.8990)--(3.141,0.8586)--(3.182,0.8182)--(3.222,0.7778)--(3.263,0.7374)--(3.303,0.6970)--(3.343,0.6566)--(3.384,0.6162)--(3.424,0.5758)--(3.465,0.5354)--(3.505,0.4949)--(3.545,0.4545)--(3.586,0.4141)--(3.626,0.3737)--(3.667,0.3333)--(3.707,0.2929)--(3.747,0.2525)--(3.788,0.2121)--(3.828,0.1717)--(3.869,0.1313)--(3.909,0.09091)--(3.949,0.05051)--(3.990,0.01010)--(4.030,-0.03030)--(4.071,-0.07071)--(4.111,-0.1111)--(4.151,-0.1515)--(4.192,-0.1919)--(4.232,-0.2323)--(4.273,-0.2727)--(4.313,-0.3131)--(4.354,-0.3535)--(4.394,-0.3939)--(4.434,-0.4343)--(4.475,-0.4747)--(4.515,-0.5152)--(4.556,-0.5556)--(4.596,-0.5960)--(4.636,-0.6364)--(4.677,-0.6768)--(4.717,-0.7172)--(4.758,-0.7576)--(4.798,-0.7980)--(4.838,-0.8384)--(4.879,-0.8788)--(4.919,-0.9192)--(4.960,-0.9596)--(5.000,-1.000);
-\draw [color=blue]  (-1.000000000,-2.000000000) node [rotate=0] {$\bullet$};
-\draw [color=blue]  (1.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0.5000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.8451696896,0.7064958104) node {\( Z_1\)};
-\draw []  (3.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.345169690,1.293504190) node {\( Z_2\)};
+\draw [color=blue]  (-1.0000,-2.0000) node [rotate=0] {$\bullet$};
+\draw [color=blue]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.50000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.84517,0.70650) node {\( Z_1\)};
+\draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.3452,1.2935) node {\( Z_2\)};
 
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks b/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
index 78ec5fba6..824be3771 100644
--- a/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
+++ b/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
@@ -79,19 +79,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
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-\draw (2.538859511,3.253165678) node {$(R,h)$};
-\draw (0.7236718444,0.3359520608) node {$\alpha$};
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+\draw (2.5389,3.2532) node {$(R,h)$};
+\draw (0.72367,0.33595) node {$\alpha$};
 
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-\draw (2.000000000,-0.3257195000) node {$\mathit{R}$};
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 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.3027346667,3.000000000) node {$\mathit{h}$};
+\draw (-0.30273,3.0000) node {$\mathit{h}$};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks b/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks
index 2d02785ef..0ddf6c696 100644
--- a/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks
+++ b/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks
@@ -65,230 +65,230 @@
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 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_YWxOAkh.pstricks b/auto/pictures_tex/Fig_YWxOAkh.pstricks
index 8c87b74e9..78d8f53a2 100644
--- a/auto/pictures_tex/Fig_YWxOAkh.pstricks
+++ b/auto/pictures_tex/Fig_YWxOAkh.pstricks
@@ -71,8 +71,8 @@
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-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks b/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks
index 74061cebc..6b17f4219 100644
--- a/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks
+++ b/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks
@@ -104,42 +104,42 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (11.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.168279320);
+\draw [,->,>=latex] (-0.50000,0) -- (11.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,0.001377810000) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,0.01446700500) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0.09001692000) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,0.3675690900) node [rotate=0] {$\bullet$};
-\draw []  (5.000000000,1.029193452) node [rotate=0] {$\bullet$};
-\draw []  (6.000000000,2.001209490) node [rotate=0] {$\bullet$};
-\draw []  (7.000000000,2.668279320) node [rotate=0] {$\bullet$};
-\draw []  (8.000000000,2.334744405) node [rotate=0] {$\bullet$};
-\draw []  (9.000000000,1.210608210) node [rotate=0] {$\bullet$};
-\draw []  (10.00000000,0.2824752490) node [rotate=0] {$\bullet$};
-\draw []  (11.00000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (1.0000,0.0013778) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0.014467) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.090017) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.36757) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,1.0292) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,2.0012) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,2.6683) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,2.3347) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,1.2106) node [rotate=0] {$\bullet$};
+\draw []  (10.000,0.28248) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.00000000,-0.3149246667) node {$ 11 $};
+\draw (11.000,-0.31492) node {$ 11 $};
 \draw [] (11.0,-0.100) -- (11.0,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks b/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks
index 516b8372f..4dae1298f 100644
--- a/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks
+++ b/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks
@@ -69,47 +69,47 @@
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+\draw [,->,>=latex] (-1.2368,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7317,2.0120);
+\draw [,->,>=latex] (-1.6150,1.4940) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1405,1.5701);
+\draw [,->,>=latex] (-1.9100,1.0917) -- (-2.3047,1.3172);
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+\draw [,->,>=latex] (-2.1954,0.14221) -- (-2.6490,0.17159);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks b/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks
index f1ade295b..e2dc589d2 100644
--- a/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks
+++ b/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks
@@ -60,13 +60,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [] (-3.00,1.00) -- (0,1.00);
 \draw [] (3.00,2.00) -- (0.0700,2.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0,2.000000000) node [rotate=0] {$o$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0,2.0000) node [rotate=0] {$o$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks b/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks
index 676ebf81e..9a04c9614 100644
--- a/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks
+++ b/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks
@@ -49,8 +49,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,0.5000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,0.50000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -64,13 +64,13 @@
 \draw [] (2.00,0) -- (0,-2.00);
 \draw [] (0,-2.00) -- (0,-1.00);
 \draw [] (0,-1.00) -- (1.00,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -117,8 +117,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -132,17 +132,17 @@
 \draw [] (2.00,2.00) -- (1.00,1.00);
 \draw [] (1.00,1.00) -- (-1.00,1.00);
 \draw [] (-1.00,1.00) -- (-2.00,2.00);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_examsseptii.pstricks b/auto/pictures_tex/Fig_examsseptii.pstricks
index dea0c0c47..32328027d 100644
--- a/auto/pictures_tex/Fig_examsseptii.pstricks
+++ b/auto/pictures_tex/Fig_examsseptii.pstricks
@@ -107,38 +107,38 @@
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-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=blue] (0,-2.00) -- (0,2.00);
 \draw [color=blue] (1.00,-2.00) -- (1.00,2.00);
 \draw [color=blue] (4.00,-2.00) -- (4.00,2.00);
 \draw [color=blue] (9.00,-2.00) -- (9.00,2.00);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks b/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks
index 61797a55a..644298998 100644
--- a/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks
+++ b/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks
@@ -89,14 +89,14 @@
 \draw [style=dashed] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
 
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-\draw (0.6519509490,0.1894063750) node {$\theta$};
+\draw (0.65195,0.18941) node {$\theta$};
 
 \draw [] (0.492,-0.0868)--(0.493,-0.0807)--(0.494,-0.0746)--(0.495,-0.0685)--(0.496,-0.0624)--(0.497,-0.0563)--(0.497,-0.0502)--(0.498,-0.0440)--(0.499,-0.0379)--(0.499,-0.0317)--(0.499,-0.0256)--(0.500,-0.0194)--(0.500,-0.0132)--(0.500,-0.00705)--(0.500,0)--(0.500,0.00529)--(0.500,0.0115)--(0.500,0.0176)--(0.499,0.0238)--(0.499,0.0300)--(0.499,0.0361)--(0.498,0.0423)--(0.498,0.0484)--(0.497,0.0545)--(0.496,0.0607)--(0.496,0.0668)--(0.495,0.0729)--(0.494,0.0790)--(0.493,0.0851)--(0.492,0.0912)--(0.490,0.0972)--(0.489,0.103)--(0.488,0.109)--(0.487,0.115)--(0.485,0.121)--(0.484,0.127)--(0.482,0.133)--(0.480,0.139)--(0.478,0.145)--(0.477,0.151)--(0.475,0.157)--(0.473,0.163)--(0.471,0.169)--(0.469,0.174)--(0.466,0.180)--(0.464,0.186)--(0.462,0.192)--(0.459,0.197)--(0.457,0.203)--(0.454,0.209)--(0.452,0.214)--(0.449,0.220)--(0.446,0.225)--(0.444,0.231)--(0.441,0.236)--(0.438,0.242)--(0.435,0.247)--(0.432,0.252)--(0.429,0.258)--(0.425,0.263)--(0.422,0.268)--(0.419,0.273)--(0.415,0.278)--(0.412,0.284)--(0.408,0.289)--(0.405,0.294)--(0.401,0.299)--(0.397,0.303)--(0.394,0.308)--(0.390,0.313)--(0.386,0.318)--(0.382,0.323)--(0.378,0.327)--(0.374,0.332)--(0.370,0.337)--(0.366,0.341)--(0.361,0.346)--(0.357,0.350)--(0.353,0.354)--(0.348,0.359)--(0.344,0.363)--(0.339,0.367)--(0.335,0.371)--(0.330,0.376)--(0.325,0.380)--(0.321,0.384)--(0.316,0.388)--(0.311,0.391)--(0.306,0.395)--(0.301,0.399)--(0.296,0.403)--(0.291,0.406)--(0.286,0.410)--(0.281,0.413)--(0.276,0.417)--(0.271,0.420)--(0.266,0.423)--(0.261,0.427)--(0.255,0.430)--(0.250,0.433);
 \draw [] (0,0) -- (1.97,-0.347);
 \draw [] (0,0) -- (1.00,1.73);
-\draw (0.1002123789,1.140733404) node {$R$};
-\draw (2.429897999,-0.5535014753) node {$\theta_0$};
-\draw (1.314840167,2.145969095) node {$\theta_1$};
+\draw (0.10021,1.1407) node {$R$};
+\draw (2.4299,-0.55350) node {$\theta_0$};
+\draw (1.3148,2.1460) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ratrap.pstricks b/auto/pictures_tex/Fig_ratrap.pstricks
index 155eeb364..c0c32b68c 100644
--- a/auto/pictures_tex/Fig_ratrap.pstricks
+++ b/auto/pictures_tex/Fig_ratrap.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -86,14 +86,14 @@
 \draw [color=blue] (2.000,0)--(2.000,0.03173)--(1.999,0.06346)--(1.998,0.09516)--(1.996,0.1268)--(1.994,0.1585)--(1.991,0.1901)--(1.988,0.2217)--(1.984,0.2532)--(1.980,0.2846)--(1.975,0.3160)--(1.970,0.3473)--(1.964,0.3785)--(1.958,0.4096)--(1.951,0.4406)--(1.944,0.4715)--(1.936,0.5023)--(1.928,0.5330)--(1.919,0.5635)--(1.910,0.5938)--(1.900,0.6241)--(1.890,0.6541)--(1.879,0.6840)--(1.868,0.7138)--(1.857,0.7433)--(1.845,0.7727)--(1.832,0.8019)--(1.819,0.8308)--(1.806,0.8596)--(1.792,0.8881)--(1.778,0.9165)--(1.763,0.9445)--(1.748,0.9724)--(1.732,1.000)--(1.716,1.027)--(1.699,1.054)--(1.683,1.081)--(1.665,1.108)--(1.647,1.134)--(1.629,1.160)--(1.611,1.186)--(1.592,1.211)--(1.572,1.236)--(1.552,1.261)--(1.532,1.286)--(1.512,1.310)--(1.491,1.334)--(1.469,1.357)--(1.447,1.380)--(1.425,1.403)--(1.403,1.425)--(1.380,1.447)--(1.357,1.469)--(1.334,1.491)--(1.310,1.512)--(1.286,1.532)--(1.261,1.552)--(1.236,1.572)--(1.211,1.592)--(1.186,1.611)--(1.160,1.629)--(1.134,1.647)--(1.108,1.665)--(1.081,1.683)--(1.054,1.699)--(1.027,1.716)--(1.000,1.732)--(0.9724,1.748)--(0.9445,1.763)--(0.9165,1.778)--(0.8881,1.792)--(0.8596,1.806)--(0.8308,1.819)--(0.8019,1.832)--(0.7727,1.845)--(0.7433,1.857)--(0.7138,1.868)--(0.6840,1.879)--(0.6541,1.890)--(0.6241,1.900)--(0.5938,1.910)--(0.5635,1.919)--(0.5330,1.928)--(0.5023,1.936)--(0.4715,1.944)--(0.4406,1.951)--(0.4096,1.958)--(0.3785,1.964)--(0.3473,1.970)--(0.3160,1.975)--(0.2846,1.980)--(0.2532,1.984)--(0.2217,1.988)--(0.1901,1.991)--(0.1585,1.994)--(0.1268,1.996)--(0.09516,1.998)--(0.06346,1.999)--(0.03173,2.000)--(0,2.000);
 \draw [color=blue] (0,2.00) -- (0,1.00);
 \draw [color=blue] (1.00,0) -- (2.00,0);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.1785761667) node {$a$};
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.1964676667,1.000000000) node {$a$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.2267360000) node {$b$};
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.1783228333,2.000000000) node {$b$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.17858) node {$a$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.19647,1.0000) node {$a$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.22674) node {$b$};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.17832,2.0000) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_senotopologo.pstricks b/auto/pictures_tex/Fig_senotopologo.pstricks
index d3db436bf..180a3c929 100644
--- a/auto/pictures_tex/Fig_senotopologo.pstricks
+++ b/auto/pictures_tex/Fig_senotopologo.pstricks
@@ -95,33 +95,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.500000000,0) -- (8.500000000,0);
-\draw [,->,>=latex] (0,-2.202481950) -- (0,2.202481950);
+\draw [,->,>=latex] (-8.5000,0) -- (8.5000,0);
+\draw [,->,>=latex] (0,-2.2025) -- (0,2.2025);
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-\draw (-8.000000000,-0.3298256667) node {$ -4 $};
+\draw (-8.0000,-0.32983) node {$ -4 $};
 \draw [] (-8.00,-0.100) -- (-8.00,0.100);
-\draw (-6.000000000,-0.3298256667) node {$ -3 $};
+\draw (-6.0000,-0.32983) node {$ -3 $};
 \draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -2 $};
+\draw (-4.0000,-0.32983) node {$ -2 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 3 $};
+\draw (6.0000,-0.31492) node {$ 3 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 4 $};
+\draw (8.0000,-0.31492) node {$ 4 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall b/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
index 9e1c13cba..974934d32 100644
--- a/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
+++ b/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
@@ -49,20 +49,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.193147181);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
 %DEFAULT
 
 \draw [color=blue] (0.1353,-2.000)--(0.1542,-1.870)--(0.1730,-1.754)--(0.1918,-1.651)--(0.2107,-1.557)--(0.2295,-1.472)--(0.2483,-1.393)--(0.2672,-1.320)--(0.2860,-1.252)--(0.3049,-1.188)--(0.3237,-1.128)--(0.3425,-1.071)--(0.3614,-1.018)--(0.3802,-0.9671)--(0.3990,-0.9187)--(0.4179,-0.8726)--(0.4367,-0.8285)--(0.4555,-0.7863)--(0.4744,-0.7458)--(0.4932,-0.7068)--(0.5120,-0.6694)--(0.5309,-0.6332)--(0.5497,-0.5984)--(0.5685,-0.5647)--(0.5874,-0.5321)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6439,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3834)--(0.7004,-0.3561)--(0.7192,-0.3296)--(0.7381,-0.3037)--(0.7569,-0.2785)--(0.7757,-0.2540)--(0.7946,-0.2300)--(0.8134,-0.2065)--(0.8322,-0.1836)--(0.8511,-0.1613)--(0.8699,-0.1394)--(0.8887,-0.1180)--(0.9076,-0.09698)--(0.9264,-0.07644)--(0.9452,-0.05632)--(0.9641,-0.03659)--(0.9829,-0.01724)--(1.002,0.001744)--(1.021,0.02037)--(1.039,0.03866)--(1.058,0.05662)--(1.077,0.07426)--(1.096,0.09159)--(1.115,0.1086)--(1.134,0.1254)--(1.152,0.1419)--(1.171,0.1581)--(1.190,0.1740)--(1.209,0.1897)--(1.228,0.2052)--(1.247,0.2204)--(1.265,0.2354)--(1.284,0.2502)--(1.303,0.2648)--(1.322,0.2791)--(1.341,0.2932)--(1.360,0.3072)--(1.378,0.3210)--(1.397,0.3345)--(1.416,0.3479)--(1.435,0.3611)--(1.454,0.3742)--(1.473,0.3870)--(1.491,0.3998)--(1.510,0.4123)--(1.529,0.4247)--(1.548,0.4369)--(1.567,0.4490)--(1.586,0.4610)--(1.604,0.4728)--(1.623,0.4845)--(1.642,0.4960)--(1.661,0.5074)--(1.680,0.5187)--(1.699,0.5298)--(1.717,0.5409)--(1.736,0.5518)--(1.755,0.5626)--(1.774,0.5732)--(1.793,0.5838)--(1.812,0.5942)--(1.830,0.6046)--(1.849,0.6148)--(1.868,0.6250)--(1.887,0.6350)--(1.906,0.6449)--(1.925,0.6547)--(1.943,0.6645)--(1.962,0.6741)--(1.981,0.6837)--(2.000,0.6931);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -109,26 +109,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.193147181);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
 %DEFAULT
 
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-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -175,22 +175,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.364664717,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,1.598612289);
+\draw [,->,>=latex] (-1.3647,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,1.5986);
 %DEFAULT
 
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -233,20 +233,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,2.193147181);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,2.1931);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -289,20 +289,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -345,18 +345,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.548147075);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5481);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.020,0.1414)--(1.040,0.1990)--(1.061,0.2426)--(1.081,0.2788)--(1.101,0.3102)--(1.121,0.3382)--(1.141,0.3637)--(1.162,0.3871)--(1.182,0.4087)--(1.202,0.4290)--(1.222,0.4480)--(1.242,0.4659)--(1.263,0.4829)--(1.283,0.4991)--(1.303,0.5145)--(1.323,0.5292)--(1.343,0.5434)--(1.364,0.5569)--(1.384,0.5700)--(1.404,0.5825)--(1.424,0.5947)--(1.444,0.6064)--(1.465,0.6178)--(1.485,0.6287)--(1.505,0.6394)--(1.525,0.6497)--(1.545,0.6598)--(1.566,0.6696)--(1.586,0.6791)--(1.606,0.6883)--(1.626,0.6973)--(1.646,0.7061)--(1.667,0.7147)--(1.687,0.7231)--(1.707,0.7313)--(1.727,0.7393)--(1.747,0.7471)--(1.768,0.7548)--(1.788,0.7623)--(1.808,0.7696)--(1.828,0.7768)--(1.848,0.7838)--(1.869,0.7907)--(1.889,0.7975)--(1.909,0.8041)--(1.929,0.8107)--(1.949,0.8170)--(1.970,0.8233)--(1.990,0.8295)--(2.010,0.8356)--(2.030,0.8415)--(2.051,0.8474)--(2.071,0.8532)--(2.091,0.8588)--(2.111,0.8644)--(2.131,0.8699)--(2.152,0.8753)--(2.172,0.8806)--(2.192,0.8859)--(2.212,0.8910)--(2.232,0.8961)--(2.253,0.9011)--(2.273,0.9061)--(2.293,0.9109)--(2.313,0.9157)--(2.333,0.9205)--(2.354,0.9252)--(2.374,0.9298)--(2.394,0.9343)--(2.414,0.9388)--(2.434,0.9432)--(2.455,0.9476)--(2.475,0.9519)--(2.495,0.9562)--(2.515,0.9604)--(2.535,0.9645)--(2.556,0.9686)--(2.576,0.9727)--(2.596,0.9767)--(2.616,0.9807)--(2.636,0.9846)--(2.657,0.9884)--(2.677,0.9923)--(2.697,0.9961)--(2.717,0.9998)--(2.737,1.003)--(2.758,1.007)--(2.778,1.011)--(2.798,1.014)--(2.818,1.018)--(2.838,1.021)--(2.859,1.025)--(2.879,1.028)--(2.899,1.032)--(2.919,1.035)--(2.939,1.038)--(2.960,1.042)--(2.980,1.045)--(3.000,1.048);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -411,30 +411,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,1.886294361);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,1.8863);
 %DEFAULT
 
 \draw [color=blue] (0.1353,-4.000)--(0.1542,-3.739)--(0.1730,-3.509)--(0.1918,-3.302)--(0.2107,-3.115)--(0.2295,-2.944)--(0.2483,-2.786)--(0.2672,-2.640)--(0.2860,-2.503)--(0.3049,-2.376)--(0.3237,-2.256)--(0.3425,-2.143)--(0.3614,-2.036)--(0.3802,-1.934)--(0.3990,-1.837)--(0.4179,-1.745)--(0.4367,-1.657)--(0.4555,-1.573)--(0.4744,-1.492)--(0.4932,-1.414)--(0.5120,-1.339)--(0.5309,-1.266)--(0.5497,-1.197)--(0.5685,-1.129)--(0.5874,-1.064)--(0.6062,-1.001)--(0.6250,-0.9399)--(0.6439,-0.8805)--(0.6627,-0.8228)--(0.6815,-0.7668)--(0.7004,-0.7122)--(0.7192,-0.6592)--(0.7381,-0.6075)--(0.7569,-0.5571)--(0.7757,-0.5079)--(0.7946,-0.4599)--(0.8134,-0.4131)--(0.8322,-0.3673)--(0.8511,-0.3225)--(0.8699,-0.2788)--(0.8887,-0.2359)--(0.9076,-0.1940)--(0.9264,-0.1529)--(0.9452,-0.1126)--(0.9641,-0.07317)--(0.9829,-0.03448)--(1.002,0.003487)--(1.021,0.04074)--(1.039,0.07732)--(1.058,0.1132)--(1.077,0.1485)--(1.096,0.1832)--(1.115,0.2173)--(1.134,0.2508)--(1.152,0.2837)--(1.171,0.3162)--(1.190,0.3481)--(1.209,0.3795)--(1.228,0.4104)--(1.247,0.4408)--(1.265,0.4708)--(1.284,0.5004)--(1.303,0.5295)--(1.322,0.5582)--(1.341,0.5865)--(1.360,0.6144)--(1.378,0.6419)--(1.397,0.6691)--(1.416,0.6958)--(1.435,0.7223)--(1.454,0.7483)--(1.473,0.7741)--(1.491,0.7995)--(1.510,0.8246)--(1.529,0.8494)--(1.548,0.8739)--(1.567,0.8981)--(1.586,0.9220)--(1.604,0.9456)--(1.623,0.9689)--(1.642,0.9920)--(1.661,1.015)--(1.680,1.037)--(1.699,1.060)--(1.717,1.082)--(1.736,1.104)--(1.755,1.125)--(1.774,1.146)--(1.793,1.168)--(1.812,1.188)--(1.830,1.209)--(1.849,1.230)--(1.868,1.250)--(1.887,1.270)--(1.906,1.290)--(1.925,1.309)--(1.943,1.329)--(1.962,1.348)--(1.981,1.367)--(2.000,1.386);
 
 \draw [color=blue] (-2.000,1.386)--(-1.981,1.367)--(-1.962,1.348)--(-1.943,1.329)--(-1.925,1.309)--(-1.906,1.290)--(-1.887,1.270)--(-1.868,1.250)--(-1.849,1.230)--(-1.830,1.209)--(-1.812,1.188)--(-1.793,1.168)--(-1.774,1.146)--(-1.755,1.125)--(-1.736,1.104)--(-1.717,1.082)--(-1.699,1.060)--(-1.680,1.037)--(-1.661,1.015)--(-1.642,0.9920)--(-1.623,0.9689)--(-1.604,0.9456)--(-1.586,0.9220)--(-1.567,0.8981)--(-1.548,0.8739)--(-1.529,0.8494)--(-1.510,0.8246)--(-1.491,0.7995)--(-1.473,0.7741)--(-1.454,0.7483)--(-1.435,0.7223)--(-1.416,0.6958)--(-1.397,0.6691)--(-1.378,0.6419)--(-1.360,0.6144)--(-1.341,0.5865)--(-1.322,0.5582)--(-1.303,0.5295)--(-1.284,0.5004)--(-1.265,0.4708)--(-1.247,0.4408)--(-1.228,0.4104)--(-1.209,0.3795)--(-1.190,0.3481)--(-1.171,0.3162)--(-1.152,0.2837)--(-1.134,0.2508)--(-1.115,0.2173)--(-1.096,0.1832)--(-1.077,0.1485)--(-1.058,0.1132)--(-1.039,0.07732)--(-1.021,0.04074)--(-1.002,0.003487)--(-0.9829,-0.03448)--(-0.9641,-0.07317)--(-0.9452,-0.1126)--(-0.9264,-0.1529)--(-0.9076,-0.1940)--(-0.8887,-0.2359)--(-0.8699,-0.2788)--(-0.8511,-0.3225)--(-0.8322,-0.3673)--(-0.8134,-0.4131)--(-0.7946,-0.4599)--(-0.7757,-0.5079)--(-0.7569,-0.5571)--(-0.7381,-0.6075)--(-0.7192,-0.6592)--(-0.7004,-0.7122)--(-0.6815,-0.7668)--(-0.6627,-0.8228)--(-0.6439,-0.8805)--(-0.6250,-0.9399)--(-0.6062,-1.001)--(-0.5874,-1.064)--(-0.5685,-1.129)--(-0.5497,-1.197)--(-0.5309,-1.266)--(-0.5120,-1.339)--(-0.4932,-1.414)--(-0.4744,-1.492)--(-0.4555,-1.573)--(-0.4367,-1.657)--(-0.4179,-1.745)--(-0.3990,-1.837)--(-0.3802,-1.934)--(-0.3614,-2.036)--(-0.3425,-2.143)--(-0.3237,-2.256)--(-0.3049,-2.376)--(-0.2860,-2.503)--(-0.2672,-2.640)--(-0.2483,-2.786)--(-0.2295,-2.944)--(-0.2107,-3.115)--(-0.1918,-3.302)--(-0.1730,-3.509)--(-0.1542,-3.739)--(-0.1353,-4.000);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ADUGmRRB.pstricks.recall b/src_phystricks/Fig_ADUGmRRB.pstricks.recall
index 9435f083e..70dd143f3 100644
--- a/src_phystricks/Fig_ADUGmRRB.pstricks.recall
+++ b/src_phystricks/Fig_ADUGmRRB.pstricks.recall
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.2059510000) node {\( \alpha_i\)};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,0.2191818333) node {\( \alpha_{i+1}\)};
+\draw (0,0.20595) node {\( \alpha_i\)};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ADUGmRRC.pstricks.recall b/src_phystricks/Fig_ADUGmRRC.pstricks.recall
index 9f961ca53..8ffd11439 100644
--- a/src_phystricks/Fig_ADUGmRRC.pstricks.recall
+++ b/src_phystricks/Fig_ADUGmRRC.pstricks.recall
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.2059510000) node {\( \alpha_i\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2191818333) node {\( \alpha_{i+1}\)};
+\draw (0,0.20595) node {\( \alpha_i\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ALIzHFm.pstricks.recall b/src_phystricks/Fig_ALIzHFm.pstricks.recall
index 5d71b3b11..e5623dafb 100644
--- a/src_phystricks/Fig_ALIzHFm.pstricks.recall
+++ b/src_phystricks/Fig_ALIzHFm.pstricks.recall
@@ -75,10 +75,10 @@
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.5000000000,0.8660254038) node [rotate=0] {$\bullet$};
-\draw (0.7640381667,1.145181485) node {\( z_1\)};
-\draw []  (-0.5000000000,-0.8660254038) node [rotate=0] {$\bullet$};
-\draw (-0.7640381667,-1.145181485) node {\( z_2\)};
+\draw []  (0.50000,0.86602) node [rotate=0] {$\bullet$};
+\draw (0.76404,1.1452) node {\( z_1\)};
+\draw []  (-0.50000,-0.86602) node [rotate=0] {$\bullet$};
+\draw (-0.76404,-1.1452) node {\( z_2\)};
 \draw [] (2.67,-0.384) -- (-1.67,2.12);
 \draw [] (-2.67,0.384) -- (1.67,-2.12);
 \draw [style=dotted] (2.17,-1.25) -- (-2.17,1.25);
diff --git a/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall b/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall
index ac5b69a94..ff57c58d7 100644
--- a/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-1.0000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
 %DEFAULT
 
 \draw [color=gray,style=dashed] (-0.5000,0.5000)--(-0.4646,0.5354)--(-0.4293,0.5707)--(-0.3939,0.6061)--(-0.3586,0.6414)--(-0.3232,0.6768)--(-0.2879,0.7121)--(-0.2525,0.7475)--(-0.2172,0.7828)--(-0.1818,0.8182)--(-0.1465,0.8535)--(-0.1111,0.8889)--(-0.07576,0.9242)--(-0.04040,0.9596)--(-0.005050,0.9949)--(0.03030,1.030)--(0.06566,1.066)--(0.1010,1.101)--(0.1364,1.136)--(0.1717,1.172)--(0.2071,1.207)--(0.2424,1.242)--(0.2778,1.278)--(0.3131,1.313)--(0.3485,1.348)--(0.3838,1.384)--(0.4192,1.419)--(0.4545,1.455)--(0.4899,1.490)--(0.5253,1.525)--(0.5606,1.561)--(0.5960,1.596)--(0.6313,1.631)--(0.6667,1.667)--(0.7020,1.702)--(0.7374,1.737)--(0.7727,1.773)--(0.8081,1.808)--(0.8434,1.843)--(0.8788,1.879)--(0.9141,1.914)--(0.9495,1.949)--(0.9848,1.985)--(1.020,2.020)--(1.056,2.056)--(1.091,2.091)--(1.126,2.126)--(1.162,2.162)--(1.197,2.197)--(1.232,2.232)--(1.268,2.268)--(1.303,2.303)--(1.338,2.338)--(1.374,2.374)--(1.409,2.409)--(1.444,2.444)--(1.480,2.480)--(1.515,2.515)--(1.551,2.551)--(1.586,2.586)--(1.621,2.621)--(1.657,2.657)--(1.692,2.692)--(1.727,2.727)--(1.763,2.763)--(1.798,2.798)--(1.833,2.833)--(1.869,2.869)--(1.904,2.904)--(1.939,2.939)--(1.975,2.975)--(2.010,3.010)--(2.045,3.045)--(2.081,3.081)--(2.116,3.116)--(2.152,3.152)--(2.187,3.187)--(2.222,3.222)--(2.258,3.258)--(2.293,3.293)--(2.328,3.328)--(2.364,3.364)--(2.399,3.399)--(2.434,3.434)--(2.470,3.470)--(2.505,3.505)--(2.540,3.540)--(2.576,3.576)--(2.611,3.611)--(2.646,3.646)--(2.682,3.682)--(2.717,3.717)--(2.753,3.753)--(2.788,3.788)--(2.823,3.823)--(2.859,3.859)--(2.894,3.894)--(2.929,3.929)--(2.965,3.965)--(3.000,4.000);
@@ -110,30 +110,30 @@
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 \draw [] (1.00,2.00) -- (1.00,2.00);
 \draw [] (3.00,4.00) -- (3.00,6.00);
-\draw []  (1.000000000,2.000000000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,2.0000) node [rotate=0] {$\bullet$};
 
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-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_AdhIntFrSix.pstricks.recall b/src_phystricks/Fig_AdhIntFrSix.pstricks.recall
index a85c4e695..855693c3c 100644
--- a/src_phystricks/Fig_AdhIntFrSix.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFrSix.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.100000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.1000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
 \draw [color=blue] (4.00,0) -- (4.00,4.00);
 \draw [color=blue] (2.00,0) -- (2.00,4.00);
@@ -170,12 +170,12 @@
 \draw [color=blue] (0.0408,0) -- (0.0408,4.00);
 \draw [color=blue] (0.0404,0) -- (0.0404,4.00);
 \draw [] (0,0) -- (0,4.00);
-\draw []  (0,3.200000000) node [rotate=0] {$\bullet$};
+\draw []  (0,3.2000) node [rotate=0] {$\bullet$};
 
 \draw [] (0.600,3.20)--(0.599,3.24)--(0.595,3.28)--(0.589,3.31)--(0.581,3.35)--(0.570,3.39)--(0.557,3.42)--(0.542,3.46)--(0.524,3.49)--(0.505,3.52)--(0.483,3.56)--(0.460,3.59)--(0.434,3.61)--(0.407,3.64)--(0.378,3.67)--(0.348,3.69)--(0.316,3.71)--(0.283,3.73)--(0.249,3.75)--(0.214,3.76)--(0.178,3.77)--(0.141,3.78)--(0.104,3.79)--(0.0665,3.80)--(0.0285,3.80)--(-0.00952,3.80)--(-0.0476,3.80)--(-0.0854,3.79)--(-0.123,3.79)--(-0.160,3.78)--(-0.196,3.77)--(-0.232,3.75)--(-0.266,3.74)--(-0.300,3.72)--(-0.332,3.70)--(-0.363,3.68)--(-0.393,3.65)--(-0.421,3.63)--(-0.447,3.60)--(-0.472,3.57)--(-0.494,3.54)--(-0.515,3.51)--(-0.533,3.47)--(-0.550,3.44)--(-0.564,3.41)--(-0.576,3.37)--(-0.585,3.33)--(-0.592,3.29)--(-0.597,3.26)--(-0.600,3.22)--(-0.600,3.18)--(-0.597,3.14)--(-0.592,3.11)--(-0.585,3.07)--(-0.576,3.03)--(-0.564,2.99)--(-0.550,2.96)--(-0.533,2.93)--(-0.515,2.89)--(-0.494,2.86)--(-0.472,2.83)--(-0.447,2.80)--(-0.421,2.77)--(-0.393,2.75)--(-0.363,2.72)--(-0.332,2.70)--(-0.300,2.68)--(-0.266,2.66)--(-0.232,2.65)--(-0.196,2.63)--(-0.160,2.62)--(-0.123,2.61)--(-0.0854,2.61)--(-0.0476,2.60)--(-0.00952,2.60)--(0.0285,2.60)--(0.0665,2.60)--(0.104,2.61)--(0.141,2.62)--(0.178,2.63)--(0.214,2.64)--(0.249,2.65)--(0.283,2.67)--(0.316,2.69)--(0.348,2.71)--(0.378,2.73)--(0.407,2.76)--(0.434,2.79)--(0.460,2.81)--(0.483,2.84)--(0.505,2.88)--(0.524,2.91)--(0.542,2.94)--(0.557,2.98)--(0.570,3.01)--(0.581,3.05)--(0.589,3.09)--(0.595,3.12)--(0.599,3.16)--(0.600,3.20);
-\draw (4.000000000,-0.3149246667) node {$ 1 $};
+\draw (4.0000,-0.31492) node {$ 1 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,4.000000000) node {$ 1 $};
+\draw (-0.29125,4.0000) node {$ 1 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall b/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall
index e6c167ab0..0ccc5fe2d 100644
--- a/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (8.500000000,0);
-\draw [,->,>=latex] (0,-1.499999991) -- (0,3.500000000);
+\draw [,->,>=latex] (-4.5000,0) -- (8.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=gray,style=dashed] (-4.000,3.000)--(-3.960,3.000)--(-3.920,3.000)--(-3.879,3.000)--(-3.839,3.000)--(-3.799,3.000)--(-3.759,3.000)--(-3.719,3.000)--(-3.678,3.000)--(-3.638,3.000)--(-3.598,3.000)--(-3.558,3.000)--(-3.518,3.000)--(-3.477,3.000)--(-3.437,3.000)--(-3.397,3.000)--(-3.357,3.000)--(-3.317,3.000)--(-3.276,3.000)--(-3.236,3.000)--(-3.196,3.000)--(-3.156,3.000)--(-3.116,3.000)--(-3.075,3.000)--(-3.035,3.000)--(-2.995,3.000)--(-2.955,3.000)--(-2.915,3.000)--(-2.874,3.000)--(-2.834,3.000)--(-2.794,3.000)--(-2.754,3.000)--(-2.714,3.000)--(-2.673,3.000)--(-2.633,3.000)--(-2.593,3.000)--(-2.553,3.000)--(-2.513,3.000)--(-2.472,3.000)--(-2.432,3.000)--(-2.392,3.000)--(-2.352,3.000)--(-2.312,3.000)--(-2.271,3.000)--(-2.231,3.000)--(-2.191,3.000)--(-2.151,3.000)--(-2.111,3.000)--(-2.070,3.000)--(-2.030,3.000)--(-1.990,3.000)--(-1.950,3.000)--(-1.910,3.000)--(-1.869,3.000)--(-1.829,3.000)--(-1.789,3.000)--(-1.749,3.000)--(-1.708,3.000)--(-1.668,3.000)--(-1.628,3.000)--(-1.588,3.000)--(-1.548,3.000)--(-1.507,3.000)--(-1.467,3.000)--(-1.427,3.000)--(-1.387,3.000)--(-1.347,3.000)--(-1.306,3.000)--(-1.266,3.000)--(-1.226,3.000)--(-1.186,3.000)--(-1.146,3.000)--(-1.105,3.000)--(-1.065,3.000)--(-1.025,3.000)--(-0.9848,3.000)--(-0.9446,3.000)--(-0.9044,3.000)--(-0.8642,3.000)--(-0.8240,3.000)--(-0.7838,3.000)--(-0.7436,3.000)--(-0.7034,3.000)--(-0.6632,3.000)--(-0.6230,3.000)--(-0.5828,3.000)--(-0.5426,3.000)--(-0.5024,3.000)--(-0.4622,3.000)--(-0.4220,3.000)--(-0.3818,3.000)--(-0.3416,3.000)--(-0.3014,3.000)--(-0.2612,3.000)--(-0.2210,3.000)--(-0.1808,3.000)--(-0.1406,3.000)--(-0.1004,3.000)--(-0.06020,3.000)--(-0.02000,3.000);
@@ -129,22 +129,22 @@
 \draw [color=blue,style=solid] 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 \draw [color=blue,style=solid] (0.0200,-0.873) -- (0.0200,3.00);
 \draw [color=blue,style=solid] (4.00,3.00) -- (4.00,0.841);
-\draw []  (0,0.7000000000) node [rotate=0] {$\bullet$};
+\draw []  (0,0.70000) node [rotate=0] {$\bullet$};
 
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-\draw (-4.000000000,-0.3298256667) node {$ -1 $};
+\draw (-4.0000,-0.32983) node {$ -1 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 1 $};
+\draw (4.0000,-0.31492) node {$ 1 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 2 $};
+\draw (8.0000,-0.31492) node {$ 2 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_AireParabole.pstricks.recall b/src_phystricks/Fig_AireParabole.pstricks.recall
index 4edb09d9e..69294f452 100644
--- a/src_phystricks/Fig_AireParabole.pstricks.recall
+++ b/src_phystricks/Fig_AireParabole.pstricks.recall
@@ -91,33 +91,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
 %DEFAULT
 
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 \draw [] (2.00,-1.00) -- (2.00,3.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.2912498333,6.000000000) node {$ 6 $};
+\draw (-0.29125,6.0000) node {$ 6 $};
 \draw [] (-0.100,6.00) -- (0.100,6.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall b/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall
index 694e73559..a64d6dad8 100644
--- a/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall
+++ b/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall
@@ -66,59 +66,57 @@
 %PSTRICKS CODE
 %DEFAULT
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
-\draw [] (1.60,-1.20) -- (3.07,-1.20);
-\draw [] (3.07,-1.20) -- (3.07,1.20);
-\draw [] (3.07,1.20) -- (1.60,1.20);
-\draw [] (1.60,1.20) -- (1.60,-1.20);
-\draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
-\draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
-\draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
-\draw [,->,>=latex] (1.600000000,0) -- (2.225000000,0);
-\draw [,->,>=latex] (1.600000000,0.3333333333) -- (2.225000000,0.3333333333);
-\draw [,->,>=latex] (1.600000000,0.6666666667) -- (2.225000000,0.6666666667);
-\draw [,->,>=latex] (1.600000000,1.000000000) -- (2.225000000,1.000000000);
-\draw [,->,>=latex] (2.333333333,-1.000000000) -- (2.761904762,-1.000000000);
-\draw [,->,>=latex] (2.333333333,-0.6666666667) -- (2.761904762,-0.6666666667);
-\draw [,->,>=latex] (2.333333333,-0.3333333333) -- (2.761904762,-0.3333333333);
-\draw [,->,>=latex] (2.333333333,0) -- (2.761904762,0);
-\draw [,->,>=latex] (2.333333333,0.3333333333) -- (2.761904762,0.3333333333);
-\draw [,->,>=latex] (2.333333333,0.6666666667) -- (2.761904762,0.6666666667);
-\draw [,->,>=latex] (2.333333333,1.000000000) -- (2.761904762,1.000000000);
-\draw [,->,>=latex] (3.066666667,-1.000000000) -- (3.392753623,-1.000000000);
-\draw [,->,>=latex] (3.066666667,-0.6666666667) -- (3.392753623,-0.6666666667);
-\draw [,->,>=latex] (3.066666667,-0.3333333333) -- (3.392753623,-0.3333333333);
-\draw [,->,>=latex] (3.066666667,0) -- (3.392753623,0);
-\draw [,->,>=latex] (3.066666667,0.3333333333) -- (3.392753623,0.3333333333);
-\draw [,->,>=latex] (3.066666667,0.6666666667) -- (3.392753623,0.6666666667);
-\draw [,->,>=latex] (3.066666667,1.000000000) -- (3.392753623,1.000000000);
-\draw [,->,>=latex] (3.800000000,-1.000000000) -- (4.063157895,-1.000000000);
-\draw [,->,>=latex] (3.800000000,-0.6666666667) -- (4.063157895,-0.6666666667);
-\draw [,->,>=latex] (3.800000000,-0.3333333333) -- (4.063157895,-0.3333333333);
-\draw [,->,>=latex] (3.800000000,0) -- (4.063157895,0);
-\draw [,->,>=latex] (3.800000000,0.3333333333) -- (4.063157895,0.3333333333);
-\draw [,->,>=latex] (3.800000000,0.6666666667) -- (4.063157895,0.6666666667);
-\draw [,->,>=latex] (3.800000000,1.000000000) -- (4.063157895,1.000000000);
-\draw [,->,>=latex] (4.533333333,-1.000000000) -- (4.753921569,-1.000000000);
-\draw [,->,>=latex] (4.533333333,-0.6666666667) -- (4.753921569,-0.6666666667);
-\draw [,->,>=latex] (4.533333333,-0.3333333333) -- (4.753921569,-0.3333333333);
-\draw [,->,>=latex] (4.533333333,0) -- (4.753921569,0);
-\draw [,->,>=latex] (4.533333333,0.3333333333) -- (4.753921569,0.3333333333);
-\draw [,->,>=latex] (4.533333333,0.6666666667) -- (4.753921569,0.6666666667);
-\draw [,->,>=latex] (4.533333333,1.000000000) -- (4.753921569,1.000000000);
-\draw [,->,>=latex] (5.266666667,-1.000000000) -- (5.456540084,-1.000000000);
-\draw [,->,>=latex] (5.266666667,-0.6666666667) -- (5.456540084,-0.6666666667);
-\draw [,->,>=latex] (5.266666667,-0.3333333333) -- (5.456540084,-0.3333333333);
-\draw [,->,>=latex] (5.266666667,0) -- (5.456540084,0);
-\draw [,->,>=latex] (5.266666667,0.3333333333) -- (5.456540084,0.3333333333);
-\draw [,->,>=latex] (5.266666667,0.6666666667) -- (5.456540084,0.6666666667);
-\draw [,->,>=latex] (5.266666667,1.000000000) -- (5.456540084,1.000000000);
-\draw [,->,>=latex] (6.000000000,-1.000000000) -- (6.166666667,-1.000000000);
-\draw [,->,>=latex] (6.000000000,-0.6666666667) -- (6.166666667,-0.6666666667);
-\draw [,->,>=latex] (6.000000000,-0.3333333333) -- (6.166666667,-0.3333333333);
-\draw [,->,>=latex] (6.000000000,0) -- (6.166666667,0);
-\draw [,->,>=latex] (6.000000000,0.3333333333) -- (6.166666667,0.3333333333);
-\draw [,->,>=latex] (6.000000000,0.6666666667) -- (6.166666667,0.6666666667);
-\draw [,->,>=latex] (6.000000000,1.000000000) -- (6.166666667,1.000000000);
+
+
+\draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
+\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
+\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
+\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
+\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
+\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
+\draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
+\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
+\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
+\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
+\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
+\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
+\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
+\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
+\draw [,->,>=latex] (3.0667,-0.33333) -- (3.3928,-0.33333);
+\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
+\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
+\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
+\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
+\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
+\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
+\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
+\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
+\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
+\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
+\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
+\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
+\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
+\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
+\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
+\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
+\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
+\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
+\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
+\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
+\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
+\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
+\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall b/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall
index b0e1c2e24..a72e93213 100644
--- a/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall
+++ b/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.000000000,0);
-\draw (3.308599701,-0.2907082010) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.000000000,2.000000000);
-\draw (2.000000000,2.426736000) node {$b$};
-\draw []  (5.000000000,2.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (0,0) -- (3.0000,0);
+\draw (3.3086,-0.29071) node {$a$};
+\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw (2.0000,2.4267) node {$b$};
+\draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
 \draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
 \draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
 \draw [style=dashed] (2.00,2.00) -- (2.00,0);
-\draw (2.305148833,1.000000000) node {$h$};
-\draw (0.8091529067,0.3191815970) node {$\theta$};
+\draw (2.3051,1.0000) node {$h$};
+\draw (0.80915,0.31918) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall b/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall
index b84460edb..3f5418461 100644
--- a/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall
+++ b/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.249988663);
+\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (5.00,2.64) -- (5.00,0);
 
-\draw [color=brown] (-0.5000,3.500)--(-0.4242,3.575)--(-0.3485,3.649)--(-0.2727,3.722)--(-0.1970,3.793)--(-0.1212,3.863)--(-0.04545,3.932)--(0.03030,3.999)--(0.1061,4.065)--(0.1818,4.130)--(0.2576,4.194)--(0.3333,4.256)--(0.4091,4.317)--(0.4848,4.377)--(0.5606,4.436)--(0.6364,4.493)--(0.7121,4.549)--(0.7879,4.604)--(0.8636,4.657)--(0.9394,4.709)--(1.015,4.760)--(1.091,4.810)--(1.167,4.858)--(1.242,4.905)--(1.318,4.951)--(1.394,4.995)--(1.470,5.039)--(1.545,5.081)--(1.621,5.121)--(1.697,5.161)--(1.773,5.199)--(1.848,5.236)--(1.924,5.271)--(2.000,5.306)--(2.076,5.339)--(2.152,5.370)--(2.227,5.401)--(2.303,5.430)--(2.379,5.458)--(2.455,5.485)--(2.530,5.510)--(2.606,5.534)--(2.682,5.557)--(2.758,5.578)--(2.833,5.599)--(2.909,5.618)--(2.985,5.635)--(3.061,5.652)--(3.136,5.667)--(3.212,5.681)--(3.288,5.694)--(3.364,5.705)--(3.439,5.715)--(3.515,5.724)--(3.591,5.731)--(3.667,5.738)--(3.742,5.743)--(3.818,5.746)--(3.894,5.749)--(3.970,5.750)--(4.045,5.750)--(4.121,5.748)--(4.197,5.746)--(4.273,5.742)--(4.349,5.737)--(4.424,5.730)--(4.500,5.722)--(4.576,5.713)--(4.651,5.703)--(4.727,5.691)--(4.803,5.678)--(4.879,5.664)--(4.955,5.649)--(5.030,5.632)--(5.106,5.614)--(5.182,5.595)--(5.258,5.574)--(5.333,5.552)--(5.409,5.529)--(5.485,5.505)--(5.561,5.479)--(5.636,5.452)--(5.712,5.424)--(5.788,5.395)--(5.864,5.364)--(5.939,5.332)--(6.015,5.299)--(6.091,5.264)--(6.167,5.228)--(6.242,5.191)--(6.318,5.153)--(6.394,5.113)--(6.470,5.072)--(6.545,5.030)--(6.621,4.987)--(6.697,4.942)--(6.773,4.896)--(6.849,4.848)--(6.924,4.800)--(7.000,4.750);
+\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3485,0.6490)--(-0.2727,0.7215)--(-0.1970,0.7928)--(-0.1212,0.8628)--(-0.04545,0.9316)--(0.03030,0.9991)--(0.1061,1.065)--(0.1818,1.130)--(0.2576,1.194)--(0.3333,1.256)--(0.4091,1.317)--(0.4848,1.377)--(0.5606,1.436)--(0.6364,1.493)--(0.7121,1.549)--(0.7879,1.604)--(0.8636,1.657)--(0.9394,1.709)--(1.015,1.760)--(1.091,1.810)--(1.167,1.858)--(1.242,1.905)--(1.318,1.951)--(1.394,1.995)--(1.470,2.039)--(1.545,2.081)--(1.621,2.121)--(1.697,2.161)--(1.773,2.199)--(1.848,2.236)--(1.924,2.271)--(2.000,2.306)--(2.076,2.339)--(2.152,2.370)--(2.227,2.401)--(2.303,2.430)--(2.379,2.458)--(2.455,2.485)--(2.530,2.510)--(2.606,2.534)--(2.682,2.557)--(2.758,2.578)--(2.833,2.599)--(2.909,2.618)--(2.985,2.635)--(3.061,2.652)--(3.136,2.667)--(3.212,2.681)--(3.288,2.694)--(3.364,2.705)--(3.439,2.715)--(3.515,2.724)--(3.591,2.731)--(3.667,2.738)--(3.742,2.743)--(3.818,2.746)--(3.894,2.749)--(3.970,2.750)--(4.045,2.750)--(4.121,2.748)--(4.197,2.746)--(4.273,2.742)--(4.349,2.737)--(4.424,2.730)--(4.500,2.722)--(4.576,2.713)--(4.651,2.703)--(4.727,2.691)--(4.803,2.678)--(4.879,2.664)--(4.955,2.649)--(5.030,2.632)--(5.106,2.614)--(5.182,2.595)--(5.258,2.574)--(5.333,2.552)--(5.409,2.529)--(5.485,2.505)--(5.561,2.479)--(5.636,2.452)--(5.712,2.424)--(5.788,2.395)--(5.864,2.364)--(5.939,2.332)--(6.015,2.299)--(6.091,2.264)--(6.167,2.228)--(6.242,2.191)--(6.318,2.153)--(6.394,2.113)--(6.470,2.072)--(6.545,2.030)--(6.621,1.987)--(6.697,1.942)--(6.773,1.896)--(6.849,1.848)--(6.924,1.800)--(7.000,1.750);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,5.64) -- (6.00,5.64) -- (5.00,5.64) -- (5.00,5.64) -- (5.00,0) --  cycle;
-\draw [color=red] (5.00,0) -- (6.00,0);
-\draw [color=red] (6.00,0) -- (6.00,5.64);
-\draw [color=red] (6.00,5.64) -- (5.00,5.64);
-\draw [color=red] (5.00,5.64) -- (5.00,0);
-\draw []  (5.000000000,5.638888889) node [rotate=0] {$\bullet$};
-\draw (5.441978850,6.211818713) node {$f(x)$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.2785761667) node {$x$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3455708333) node {$x+\Delta x$};
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
+\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
+\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
+\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
+\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
+\draw (5.4420,3.2118) node {$f(x)$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.27858) node {$x$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.34557) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Bateau.pstricks.recall b/src_phystricks/Fig_Bateau.pstricks.recall
index d6ad0aca5..117a93c45 100644
--- a/src_phystricks/Fig_Bateau.pstricks.recall
+++ b/src_phystricks/Fig_Bateau.pstricks.recall
@@ -98,28 +98,28 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (-1.00,0) -- (5.00,0);
-\draw []  (0,2.500000000) node [rotate=0] {$\bullet$};
-\draw (0,2.924708000) node {$A$};
-\draw [,->,>=latex] (2.000000000,2.500000000) -- (0,2.500000000);
-\draw [,->,>=latex] (2.000000000,2.500000000) -- (4.000000000,2.500000000);
-\draw (2.000000000,2.725719500) node {$\unit{4}{\kilo\meter}$};
-\draw [,->,>=latex] (4.200000000,2.250000000) -- (4.200000000,4.500000000);
-\draw [,->,>=latex] (4.200000000,2.250000000) -- (4.200000000,0);
-\draw (4.671424833,2.250000000) node {$\unit{9}{\kilo\meter}$};
-\draw [,->,>=latex] (0,1.250000000) -- (0,2.500000000);
-\draw [,->,>=latex] (0,1.250000000) -- (0,0);
-\draw (-0.4714248333,1.250000000) node {$\unit{3}{\kilo\meter}$};
-\draw []  (1.428571429,0) node [rotate=0] {$\bullet$};
-\draw (1.735248463,0.3368400344) node {$I$};
+\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
+\draw (0,2.9247) node {$A$};
+\draw [,->,>=latex] (2.0000,2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (2.0000,2.5000) -- (4.0000,2.5000);
+\draw (2.0000,2.7257) node {$\unit{4}{\kilo\meter}$};
+\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,4.5000);
+\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,0);
+\draw (4.6714,2.2500) node {$\unit{9}{\kilo\meter}$};
+\draw [,->,>=latex] (0,1.2500) -- (0,2.5000);
+\draw [,->,>=latex] (0,1.2500) -- (0,0);
+\draw (-0.47143,1.2500) node {$\unit{3}{\kilo\meter}$};
+\draw []  (1.4286,0) node [rotate=0] {$\bullet$};
+\draw (1.7352,0.33684) node {$I$};
 \draw [color=brown,style=dashed] (4.00,4.50) -- (4.00,-4.50);
 \draw [color=blue,style=dashed] (0,2.50) -- (4.00,-4.50);
-\draw []  (4.000000000,4.500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,4.924708000) node {$B$};
-\draw []  (4.000000000,-4.500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-4.940774333) node {$B'$};
-\draw [,->,>=latex] (0.7142857143,-0.2000000000) -- (0,-0.2000000000);
-\draw [,->,>=latex] (0.7142857143,-0.2000000000) -- (1.428571429,-0.2000000000);
-\draw (0.7142857143,-0.4257195000) node {$x\kilo\meter$};
+\draw []  (4.0000,4.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,4.9247) node {$B$};
+\draw []  (4.0000,-4.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-4.9408) node {$B'$};
+\draw [,->,>=latex] (0.71429,-0.20000) -- (0,-0.20000);
+\draw [,->,>=latex] (0.71429,-0.20000) -- (1.4286,-0.20000);
+\draw (0.71429,-0.42572) node {$x\kilo\meter$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CELooGVvzMc.pstricks.recall b/src_phystricks/Fig_CELooGVvzMc.pstricks.recall
index a9d6c9369..3ceb434c7 100644
--- a/src_phystricks/Fig_CELooGVvzMc.pstricks.recall
+++ b/src_phystricks/Fig_CELooGVvzMc.pstricks.recall
@@ -80,33 +80,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.790000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.790000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.7900,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.7900);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.02323,0)--(0.04646,0.002159)--(0.06970,0.004858)--(0.09293,0.008636)--(0.1162,0.01349)--(0.1394,0.01943)--(0.1626,0.02645)--(0.1859,0.03454)--(0.2091,0.04372)--(0.2323,0.05397)--(0.2556,0.06531)--(0.2788,0.07772)--(0.3020,0.09122)--(0.3253,0.1058)--(0.3485,0.1214)--(0.3717,0.1382)--(0.3950,0.1560)--(0.4182,0.1749)--(0.4414,0.1948)--(0.4646,0.2159)--(0.4879,0.2380)--(0.5111,0.2612)--(0.5343,0.2855)--(0.5576,0.3109)--(0.5808,0.3373)--(0.6040,0.3649)--(0.6273,0.3935)--(0.6505,0.4232)--(0.6737,0.4539)--(0.6970,0.4858)--(0.7202,0.5187)--(0.7434,0.5527)--(0.7667,0.5878)--(0.7899,0.6239)--(0.8131,0.6612)--(0.8364,0.6995)--(0.8596,0.7389)--(0.8828,0.7794)--(0.9061,0.8209)--(0.9293,0.8636)--(0.9525,0.9073)--(0.9758,0.9521)--(0.9990,0.9980)--(1.022,1.045)--(1.045,1.093)--(1.069,1.142)--(1.092,1.192)--(1.115,1.244)--(1.138,1.296)--(1.162,1.349)--(1.185,1.404)--(1.208,1.459)--(1.231,1.516)--(1.255,1.574)--(1.278,1.633)--(1.301,1.693)--(1.324,1.754)--(1.347,1.816)--(1.371,1.879)--(1.394,1.943)--(1.417,2.008)--(1.440,2.075)--(1.464,2.142)--(1.487,2.211)--(1.510,2.280)--(1.533,2.351)--(1.557,2.423)--(1.580,2.496)--(1.603,2.570)--(1.626,2.645)--(1.649,2.721)--(1.673,2.798)--(1.696,2.876)--(1.719,2.956)--(1.742,3.036)--(1.766,3.118)--(1.789,3.200)--(1.812,3.284)--(1.835,3.369)--(1.859,3.454)--(1.882,3.541)--(1.905,3.629)--(1.928,3.718)--(1.952,3.808)--(1.975,3.900)--(1.998,3.992)--(2.021,4.085)--(2.044,4.180)--(2.068,4.275)--(2.091,4.372)--(2.114,4.470)--(2.137,4.568)--(2.161,4.668)--(2.184,4.769)--(2.207,4.871)--(2.230,4.974)--(2.254,5.078)--(2.277,5.184)--(2.300,5.290);
 
 \draw [color=blue] (0,0)--(0.05343,0.2312)--(0.1069,0.3269)--(0.1603,0.4004)--(0.2137,0.4623)--(0.2672,0.5169)--(0.3206,0.5662)--(0.3740,0.6116)--(0.4275,0.6538)--(0.4809,0.6935)--(0.5343,0.7310)--(0.5878,0.7667)--(0.6412,0.8008)--(0.6946,0.8335)--(0.7481,0.8649)--(0.8015,0.8953)--(0.8549,0.9246)--(0.9084,0.9531)--(0.9618,0.9807)--(1.015,1.008)--(1.069,1.034)--(1.122,1.059)--(1.176,1.084)--(1.229,1.109)--(1.282,1.132)--(1.336,1.156)--(1.389,1.179)--(1.443,1.201)--(1.496,1.223)--(1.550,1.245)--(1.603,1.266)--(1.656,1.287)--(1.710,1.308)--(1.763,1.328)--(1.817,1.348)--(1.870,1.368)--(1.924,1.387)--(1.977,1.406)--(2.031,1.425)--(2.084,1.444)--(2.137,1.462)--(2.191,1.480)--(2.244,1.498)--(2.298,1.516)--(2.351,1.533)--(2.405,1.551)--(2.458,1.568)--(2.511,1.585)--(2.565,1.602)--(2.618,1.618)--(2.672,1.635)--(2.725,1.651)--(2.779,1.667)--(2.832,1.683)--(2.885,1.699)--(2.939,1.714)--(2.992,1.730)--(3.046,1.745)--(3.099,1.760)--(3.153,1.776)--(3.206,1.791)--(3.259,1.805)--(3.313,1.820)--(3.366,1.835)--(3.420,1.849)--(3.473,1.864)--(3.527,1.878)--(3.580,1.892)--(3.634,1.906)--(3.687,1.920)--(3.740,1.934)--(3.794,1.948)--(3.847,1.961)--(3.901,1.975)--(3.954,1.988)--(4.008,2.002)--(4.061,2.015)--(4.114,2.028)--(4.168,2.042)--(4.221,2.055)--(4.275,2.068)--(4.328,2.080)--(4.382,2.093)--(4.435,2.106)--(4.488,2.119)--(4.542,2.131)--(4.595,2.144)--(4.649,2.156)--(4.702,2.168)--(4.756,2.181)--(4.809,2.193)--(4.863,2.205)--(4.916,2.217)--(4.969,2.229)--(5.023,2.241)--(5.076,2.253)--(5.130,2.265)--(5.183,2.277)--(5.237,2.288)--(5.290,2.300);
 \draw [style=dashed] (0,0) -- (3.79,3.79);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CMMAooQegASg.pstricks.recall b/src_phystricks/Fig_CMMAooQegASg.pstricks.recall
index 014c6bf78..6b8a9b57d 100644
--- a/src_phystricks/Fig_CMMAooQegASg.pstricks.recall
+++ b/src_phystricks/Fig_CMMAooQegASg.pstricks.recall
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.499897967) -- (0,2.499897967);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
 %DEFAULT
 
 \draw [color=blue] (-2.000,0)--(-1.960,0.4000)--(-1.919,0.5628)--(-1.879,0.6857)--(-1.838,0.7876)--(-1.798,0.8759)--(-1.758,0.9544)--(-1.717,1.025)--(-1.677,1.090)--(-1.636,1.150)--(-1.596,1.205)--(-1.556,1.257)--(-1.515,1.305)--(-1.475,1.351)--(-1.434,1.394)--(-1.394,1.434)--(-1.354,1.472)--(-1.313,1.509)--(-1.273,1.543)--(-1.232,1.575)--(-1.192,1.606)--(-1.152,1.635)--(-1.111,1.663)--(-1.071,1.689)--(-1.030,1.714)--(-0.9899,1.738)--(-0.9495,1.760)--(-0.9091,1.781)--(-0.8687,1.801)--(-0.8283,1.820)--(-0.7879,1.838)--(-0.7475,1.855)--(-0.7071,1.871)--(-0.6667,1.886)--(-0.6263,1.899)--(-0.5859,1.912)--(-0.5455,1.924)--(-0.5051,1.935)--(-0.4646,1.945)--(-0.4242,1.954)--(-0.3838,1.963)--(-0.3434,1.970)--(-0.3030,1.977)--(-0.2626,1.983)--(-0.2222,1.988)--(-0.1818,1.992)--(-0.1414,1.995)--(-0.1010,1.997)--(-0.06061,1.999)--(-0.02020,2.000)--(0.02020,2.000)--(0.06061,1.999)--(0.1010,1.997)--(0.1414,1.995)--(0.1818,1.992)--(0.2222,1.988)--(0.2626,1.983)--(0.3030,1.977)--(0.3434,1.970)--(0.3838,1.963)--(0.4242,1.954)--(0.4646,1.945)--(0.5051,1.935)--(0.5455,1.924)--(0.5859,1.912)--(0.6263,1.899)--(0.6667,1.886)--(0.7071,1.871)--(0.7475,1.855)--(0.7879,1.838)--(0.8283,1.820)--(0.8687,1.801)--(0.9091,1.781)--(0.9495,1.760)--(0.9899,1.738)--(1.030,1.714)--(1.071,1.689)--(1.111,1.663)--(1.152,1.635)--(1.192,1.606)--(1.232,1.575)--(1.273,1.543)--(1.313,1.509)--(1.354,1.472)--(1.394,1.434)--(1.434,1.394)--(1.475,1.351)--(1.515,1.305)--(1.556,1.257)--(1.596,1.205)--(1.636,1.150)--(1.677,1.090)--(1.717,1.025)--(1.758,0.9544)--(1.798,0.8759)--(1.838,0.7876)--(1.879,0.6857)--(1.919,0.5628)--(1.960,0.4000)--(2.000,0);
diff --git a/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall b/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall
index 574d6a7fe..90633a3ea 100644
--- a/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall
+++ b/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.020,0.2020)--(1.040,0.2871)--(1.061,0.3534)--(1.081,0.4101)--(1.101,0.4607)--(1.121,0.5071)--(1.141,0.5503)--(1.162,0.5911)--(1.182,0.6298)--(1.202,0.6670)--(1.222,0.7027)--(1.242,0.7373)--(1.263,0.7709)--(1.283,0.8035)--(1.303,0.8354)--(1.323,0.8666)--(1.343,0.8971)--(1.364,0.9271)--(1.384,0.9566)--(1.404,0.9856)--(1.424,1.014)--(1.444,1.042)--(1.465,1.070)--(1.485,1.098)--(1.505,1.125)--(1.525,1.152)--(1.545,1.178)--(1.566,1.205)--(1.586,1.231)--(1.606,1.257)--(1.626,1.282)--(1.646,1.308)--(1.667,1.333)--(1.687,1.359)--(1.707,1.383)--(1.727,1.408)--(1.747,1.433)--(1.768,1.458)--(1.788,1.482)--(1.808,1.506)--(1.828,1.531)--(1.848,1.555)--(1.869,1.579)--(1.889,1.602)--(1.909,1.626)--(1.929,1.650)--(1.949,1.673)--(1.970,1.697)--(1.990,1.720)--(2.010,1.744)--(2.030,1.767)--(2.051,1.790)--(2.071,1.813)--(2.091,1.836)--(2.111,1.859)--(2.131,1.882)--(2.152,1.905)--(2.172,1.928)--(2.192,1.951)--(2.212,1.973)--(2.232,1.996)--(2.253,2.018)--(2.273,2.041)--(2.293,2.063)--(2.313,2.086)--(2.333,2.108)--(2.354,2.131)--(2.374,2.153)--(2.394,2.175)--(2.414,2.197)--(2.434,2.219)--(2.455,2.242)--(2.475,2.264)--(2.495,2.286)--(2.515,2.308)--(2.535,2.330)--(2.556,2.352)--(2.576,2.374)--(2.596,2.396)--(2.616,2.418)--(2.636,2.439)--(2.657,2.461)--(2.677,2.483)--(2.697,2.505)--(2.717,2.526)--(2.737,2.548)--(2.758,2.570)--(2.778,2.592)--(2.798,2.613)--(2.818,2.635)--(2.838,2.656)--(2.859,2.678)--(2.879,2.700)--(2.899,2.721)--(2.919,2.743)--(2.939,2.764)--(2.960,2.786)--(2.980,2.807)--(3.000,2.828);
@@ -108,33 +108,33 @@
 \draw [color=blue] (-3.000,-2.828)--(-2.980,-2.807)--(-2.960,-2.786)--(-2.939,-2.764)--(-2.919,-2.743)--(-2.899,-2.721)--(-2.879,-2.700)--(-2.859,-2.678)--(-2.838,-2.656)--(-2.818,-2.635)--(-2.798,-2.613)--(-2.778,-2.592)--(-2.758,-2.570)--(-2.737,-2.548)--(-2.717,-2.526)--(-2.697,-2.505)--(-2.677,-2.483)--(-2.657,-2.461)--(-2.636,-2.439)--(-2.616,-2.418)--(-2.596,-2.396)--(-2.576,-2.374)--(-2.556,-2.352)--(-2.535,-2.330)--(-2.515,-2.308)--(-2.495,-2.286)--(-2.475,-2.264)--(-2.455,-2.242)--(-2.434,-2.219)--(-2.414,-2.197)--(-2.394,-2.175)--(-2.374,-2.153)--(-2.354,-2.131)--(-2.333,-2.108)--(-2.313,-2.086)--(-2.293,-2.063)--(-2.273,-2.041)--(-2.253,-2.018)--(-2.232,-1.996)--(-2.212,-1.973)--(-2.192,-1.951)--(-2.172,-1.928)--(-2.152,-1.905)--(-2.131,-1.882)--(-2.111,-1.859)--(-2.091,-1.836)--(-2.071,-1.813)--(-2.051,-1.790)--(-2.030,-1.767)--(-2.010,-1.744)--(-1.990,-1.720)--(-1.970,-1.697)--(-1.949,-1.673)--(-1.929,-1.650)--(-1.909,-1.626)--(-1.889,-1.602)--(-1.869,-1.579)--(-1.848,-1.555)--(-1.828,-1.531)--(-1.808,-1.506)--(-1.788,-1.482)--(-1.768,-1.458)--(-1.747,-1.433)--(-1.727,-1.408)--(-1.707,-1.383)--(-1.687,-1.359)--(-1.667,-1.333)--(-1.646,-1.308)--(-1.626,-1.282)--(-1.606,-1.257)--(-1.586,-1.231)--(-1.566,-1.205)--(-1.545,-1.178)--(-1.525,-1.152)--(-1.505,-1.125)--(-1.485,-1.098)--(-1.465,-1.070)--(-1.444,-1.042)--(-1.424,-1.014)--(-1.404,-0.9856)--(-1.384,-0.9566)--(-1.364,-0.9271)--(-1.343,-0.8971)--(-1.323,-0.8666)--(-1.303,-0.8354)--(-1.283,-0.8035)--(-1.263,-0.7709)--(-1.242,-0.7373)--(-1.222,-0.7027)--(-1.202,-0.6670)--(-1.182,-0.6298)--(-1.162,-0.5911)--(-1.141,-0.5503)--(-1.121,-0.5071)--(-1.101,-0.4607)--(-1.081,-0.4101)--(-1.061,-0.3534)--(-1.040,-0.2871)--(-1.020,-0.2020)--(-1.000,0);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.7867744886,0.1954186781) node {$P$};
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.7850131552,0.2309048448) node {$Q$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (0.78677,0.19542) node {$P$};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-0.78501,0.23090) node {$Q$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CSCvi.pstricks.recall b/src_phystricks/Fig_CSCvi.pstricks.recall
index 10e26577f..8a0a3f7b3 100644
--- a/src_phystricks/Fig_CSCvi.pstricks.recall
+++ b/src_phystricks/Fig_CSCvi.pstricks.recall
@@ -61,29 +61,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (2.070796327,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.433154876);
+\draw [,->,>=latex] (-2.0708,0) -- (2.0708,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4332);
 %DEFAULT
 
 \draw [color=blue] (-1.171,4.933)--(-1.143,4.605)--(-1.115,4.316)--(-1.088,4.059)--(-1.060,3.830)--(-1.032,3.624)--(-1.005,3.438)--(-0.9769,3.268)--(-0.9493,3.114)--(-0.9216,2.972)--(-0.8939,2.841)--(-0.8662,2.720)--(-0.8385,2.608)--(-0.8108,2.504)--(-0.7831,2.406)--(-0.7554,2.315)--(-0.7277,2.230)--(-0.7000,2.150)--(-0.6723,2.074)--(-0.6446,2.003)--(-0.6169,1.935)--(-0.5892,1.871)--(-0.5616,1.811)--(-0.5339,1.753)--(-0.5062,1.698)--(-0.4785,1.645)--(-0.4508,1.595)--(-0.4231,1.547)--(-0.3954,1.501)--(-0.3677,1.457)--(-0.3400,1.414)--(-0.3123,1.374)--(-0.2846,1.334)--(-0.2569,1.297)--(-0.2292,1.260)--(-0.2015,1.225)--(-0.1739,1.191)--(-0.1462,1.158)--(-0.1185,1.126)--(-0.09077,1.095)--(-0.06308,1.065)--(-0.03539,1.036)--(-0.007696,1.008)--(0.02000,0.9802)--(0.04769,0.9534)--(0.07538,0.9273)--(0.1031,0.9019)--(0.1308,0.8771)--(0.1585,0.8529)--(0.1862,0.8292)--(0.2138,0.8061)--(0.2415,0.7835)--(0.2692,0.7614)--(0.2969,0.7398)--(0.3246,0.7186)--(0.3523,0.6978)--(0.3800,0.6774)--(0.4077,0.6574)--(0.4354,0.6377)--(0.4631,0.6184)--(0.4908,0.5994)--(0.5185,0.5808)--(0.5462,0.5624)--(0.5739,0.5443)--(0.6015,0.5265)--(0.6292,0.5089)--(0.6569,0.4916)--(0.6846,0.4746)--(0.7123,0.4577)--(0.7400,0.4411)--(0.7677,0.4246)--(0.7954,0.4084)--(0.8231,0.3923)--(0.8508,0.3764)--(0.8785,0.3607)--(0.9062,0.3451)--(0.9339,0.3297)--(0.9616,0.3144)--(0.9893,0.2993)--(1.017,0.2842)--(1.045,0.2693)--(1.072,0.2545)--(1.100,0.2398)--(1.128,0.2252)--(1.155,0.2107)--(1.183,0.1963)--(1.211,0.1820)--(1.238,0.1677)--(1.266,0.1535)--(1.294,0.1394)--(1.322,0.1253)--(1.349,0.1112)--(1.377,0.09723)--(1.405,0.08327)--(1.432,0.06934)--(1.460,0.05544)--(1.488,0.04156)--(1.515,0.02770)--(1.543,0.01385)--(1.571,0);
 \draw [color=lightgray,style=dashed] (-1.57,0) -- (-1.57,4.93);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -142,23 +142,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.421060994,0);
-\draw [,->,>=latex] (0,-5.043736533) -- (0,0.8002358951);
+\draw [,->,>=latex] (-0.50000,0) -- (2.4211,0);
+\draw [,->,>=latex] (0,-5.0437) -- (0,0.80024);
 %DEFAULT
 \draw [color=blue] (1.921,-4.544)--(1.910,-4.190)--(1.898,-3.876)--(1.886,-3.595)--(1.872,-3.341)--(1.858,-3.111)--(1.844,-2.901)--(1.829,-2.709)--(1.813,-2.531)--(1.797,-2.367)--(1.779,-2.214)--(1.762,-2.072)--(1.744,-1.939)--(1.725,-1.815)--(1.705,-1.698)--(1.686,-1.587)--(1.665,-1.483)--(1.644,-1.385)--(1.623,-1.292)--(1.601,-1.204)--(1.579,-1.120)--(1.556,-1.040)--(1.533,-0.9641)--(1.509,-0.8919)--(1.485,-0.8231)--(1.460,-0.7575)--(1.436,-0.6949)--(1.411,-0.6352)--(1.385,-0.5781)--(1.359,-0.5237)--(1.333,-0.4717)--(1.307,-0.4221)--(1.281,-0.3747)--(1.254,-0.3295)--(1.227,-0.2864)--(1.200,-0.2452)--(1.173,-0.2060)--(1.146,-0.1686)--(1.118,-0.1331)--(1.091,-0.09928)--(1.063,-0.06715)--(1.035,-0.03666)--(1.008,-0.007756)--(0.9800,0.01960)--(0.9523,0.04545)--(0.9247,0.06984)--(0.8971,0.09280)--(0.8696,0.1144)--(0.8422,0.1346)--(0.8149,0.1535)--(0.7878,0.1711)--(0.7608,0.1874)--(0.7340,0.2025)--(0.7074,0.2164)--(0.6811,0.2292)--(0.6549,0.2408)--(0.6291,0.2513)--(0.6035,0.2607)--(0.5782,0.2690)--(0.5533,0.2763)--(0.5287,0.2825)--(0.5044,0.2878)--(0.4806,0.2921)--(0.4571,0.2955)--(0.4341,0.2980)--(0.4115,0.2995)--(0.3893,0.3002)--(0.3676,0.3001)--(0.3464,0.2991)--(0.3257,0.2974)--(0.3055,0.2949)--(0.2859,0.2916)--(0.2668,0.2877)--(0.2482,0.2830)--(0.2302,0.2776)--(0.2129,0.2717)--(0.1961,0.2650)--(0.1799,0.2578)--(0.1644,0.2501)--(0.1495,0.2417)--(0.1353,0.2329)--(0.1217,0.2236)--(0.1088,0.2137)--(0.09657,0.2035)--(0.08504,0.1928)--(0.07422,0.1817)--(0.06411,0.1703)--(0.05471,0.1585)--(0.04604,0.1464)--(0.03810,0.1340)--(0.03090,0.1214)--(0.02444,0.1085)--(0.01873,0.09541)--(0.01377,0.08212)--(0.009571,0.06868)--(0.006129,0.05510)--(0.003449,0.04142)--(0.001533,0.02766)--(0,0.01384)--(0,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
 \draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CURGooXvruWV.pstricks.recall b/src_phystricks/Fig_CURGooXvruWV.pstricks.recall
index 21cc5cf13..8d44d1bc7 100644
--- a/src_phystricks/Fig_CURGooXvruWV.pstricks.recall
+++ b/src_phystricks/Fig_CURGooXvruWV.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -90,13 +90,13 @@
 \draw [color=green] (0,2.000)--(0.02020,2.000)--(0.04040,2.000)--(0.06061,2.000)--(0.08081,2.000)--(0.1010,2.000)--(0.1212,2.000)--(0.1414,2.000)--(0.1616,2.000)--(0.1818,2.000)--(0.2020,2.000)--(0.2222,2.000)--(0.2424,2.000)--(0.2626,2.000)--(0.2828,2.000)--(0.3030,2.000)--(0.3232,2.000)--(0.3434,2.000)--(0.3636,2.000)--(0.3838,2.000)--(0.4040,2.000)--(0.4242,2.000)--(0.4444,2.000)--(0.4646,2.000)--(0.4848,2.000)--(0.5051,2.000)--(0.5253,2.000)--(0.5455,2.000)--(0.5657,2.000)--(0.5859,2.000)--(0.6061,2.000)--(0.6263,2.000)--(0.6465,2.000)--(0.6667,2.000)--(0.6869,2.000)--(0.7071,2.000)--(0.7273,2.000)--(0.7475,2.000)--(0.7677,2.000)--(0.7879,2.000)--(0.8081,2.000)--(0.8283,2.000)--(0.8485,2.000)--(0.8687,2.000)--(0.8889,2.000)--(0.9091,2.000)--(0.9293,2.000)--(0.9495,2.000)--(0.9697,2.000)--(0.9899,2.000)--(1.010,2.000)--(1.030,2.000)--(1.051,2.000)--(1.071,2.000)--(1.091,2.000)--(1.111,2.000)--(1.131,2.000)--(1.152,2.000)--(1.172,2.000)--(1.192,2.000)--(1.212,2.000)--(1.232,2.000)--(1.253,2.000)--(1.273,2.000)--(1.293,2.000)--(1.313,2.000)--(1.333,2.000)--(1.354,2.000)--(1.374,2.000)--(1.394,2.000)--(1.414,2.000)--(1.434,2.000)--(1.455,2.000)--(1.475,2.000)--(1.495,2.000)--(1.515,2.000)--(1.535,2.000)--(1.556,2.000)--(1.576,2.000)--(1.596,2.000)--(1.616,2.000)--(1.636,2.000)--(1.657,2.000)--(1.677,2.000)--(1.697,2.000)--(1.717,2.000)--(1.737,2.000)--(1.758,2.000)--(1.778,2.000)--(1.798,2.000)--(1.818,2.000)--(1.838,2.000)--(1.859,2.000)--(1.879,2.000)--(1.899,2.000)--(1.919,2.000)--(1.939,2.000)--(1.960,2.000)--(1.980,2.000)--(2.000,2.000);
 
 \draw [color=green] (0,0)--(0.02020,0.02020)--(0.04040,0.04040)--(0.06061,0.06061)--(0.08081,0.08081)--(0.1010,0.1010)--(0.1212,0.1212)--(0.1414,0.1414)--(0.1616,0.1616)--(0.1818,0.1818)--(0.2020,0.2020)--(0.2222,0.2222)--(0.2424,0.2424)--(0.2626,0.2626)--(0.2828,0.2828)--(0.3030,0.3030)--(0.3232,0.3232)--(0.3434,0.3434)--(0.3636,0.3636)--(0.3838,0.3838)--(0.4040,0.4040)--(0.4242,0.4242)--(0.4444,0.4444)--(0.4646,0.4646)--(0.4848,0.4848)--(0.5051,0.5051)--(0.5253,0.5253)--(0.5455,0.5455)--(0.5657,0.5657)--(0.5859,0.5859)--(0.6061,0.6061)--(0.6263,0.6263)--(0.6465,0.6465)--(0.6667,0.6667)--(0.6869,0.6869)--(0.7071,0.7071)--(0.7273,0.7273)--(0.7475,0.7475)--(0.7677,0.7677)--(0.7879,0.7879)--(0.8081,0.8081)--(0.8283,0.8283)--(0.8485,0.8485)--(0.8687,0.8687)--(0.8889,0.8889)--(0.9091,0.9091)--(0.9293,0.9293)--(0.9495,0.9495)--(0.9697,0.9697)--(0.9899,0.9899)--(1.010,1.010)--(1.030,1.030)--(1.051,1.051)--(1.071,1.071)--(1.091,1.091)--(1.111,1.111)--(1.131,1.131)--(1.152,1.152)--(1.172,1.172)--(1.192,1.192)--(1.212,1.212)--(1.232,1.232)--(1.253,1.253)--(1.273,1.273)--(1.293,1.293)--(1.313,1.313)--(1.333,1.333)--(1.354,1.354)--(1.374,1.374)--(1.394,1.394)--(1.414,1.414)--(1.434,1.434)--(1.455,1.455)--(1.475,1.475)--(1.495,1.495)--(1.515,1.515)--(1.535,1.535)--(1.556,1.556)--(1.576,1.576)--(1.596,1.596)--(1.616,1.616)--(1.636,1.636)--(1.657,1.657)--(1.677,1.677)--(1.697,1.697)--(1.717,1.717)--(1.737,1.737)--(1.758,1.758)--(1.778,1.778)--(1.798,1.798)--(1.818,1.818)--(1.838,1.838)--(1.859,1.859)--(1.879,1.879)--(1.899,1.899)--(1.919,1.919)--(1.939,1.939)--(1.960,1.960)--(1.980,1.980)--(2.000,2.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall b/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
index 4c1ce07d4..b289ededc 100644
--- a/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
+++ b/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
@@ -68,14 +68,14 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (1.000000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,0.5000000000);
+\draw [,->,>=latex] (-3.5000,0) -- (1.0000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,0.50000);
 %DEFAULT
-\draw [,->,>=latex] (-3.000000000,-1.000000000) -- (-3.000000000,0);
+\draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0);
 \draw [] (0,0) -- (-3.00,-1.00);
-\draw []  (-3.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (-3.828301160,-1.204882387) node {\( -x+\lambda i\)};
-\draw (0.5312334950,-0.7091140396) node {\( \arg(z)\)};
+\draw []  (-3.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (-3.8283,-1.2049) node {\( -x+\lambda i\)};
+\draw (0.53123,-0.70911) node {\( \arg(z)\)};
 
 \draw [] (-0.474,-0.158)--(-0.470,-0.172)--(-0.465,-0.185)--(-0.459,-0.198)--(-0.453,-0.211)--(-0.447,-0.224)--(-0.441,-0.236)--(-0.434,-0.249)--(-0.426,-0.261)--(-0.419,-0.273)--(-0.411,-0.285)--(-0.403,-0.297)--(-0.394,-0.308)--(-0.385,-0.319)--(-0.376,-0.330)--(-0.366,-0.340)--(-0.356,-0.351)--(-0.346,-0.361)--(-0.336,-0.370)--(-0.325,-0.380)--(-0.314,-0.389)--(-0.303,-0.398)--(-0.292,-0.406)--(-0.280,-0.414)--(-0.268,-0.422)--(-0.256,-0.430)--(-0.243,-0.437)--(-0.231,-0.444)--(-0.218,-0.450)--(-0.205,-0.456)--(-0.192,-0.462)--(-0.179,-0.467)--(-0.166,-0.472)--(-0.152,-0.476)--(-0.138,-0.480)--(-0.125,-0.484)--(-0.111,-0.488)--(-0.0970,-0.491)--(-0.0830,-0.493)--(-0.0689,-0.495)--(-0.0547,-0.497)--(-0.0406,-0.498)--(-0.0264,-0.499)--(-0.0121,-0.500)--(0.00211,-0.500)--(0.0163,-0.500)--(0.0306,-0.499)--(0.0448,-0.498)--(0.0589,-0.497)--(0.0731,-0.495)--(0.0871,-0.492)--(0.101,-0.490)--(0.115,-0.487)--(0.129,-0.483)--(0.143,-0.479)--(0.156,-0.475)--(0.170,-0.470)--(0.183,-0.465)--(0.196,-0.460)--(0.209,-0.454)--(0.222,-0.448)--(0.235,-0.442)--(0.247,-0.435)--(0.259,-0.427)--(0.271,-0.420)--(0.283,-0.412)--(0.295,-0.404)--(0.306,-0.395)--(0.317,-0.386)--(0.328,-0.377)--(0.339,-0.368)--(0.349,-0.358)--(0.359,-0.348)--(0.369,-0.337)--(0.379,-0.327)--(0.388,-0.316)--(0.396,-0.305)--(0.405,-0.293)--(0.413,-0.282)--(0.421,-0.270)--(0.429,-0.258)--(0.436,-0.245)--(0.443,-0.233)--(0.449,-0.220)--(0.455,-0.207)--(0.461,-0.194)--(0.466,-0.181)--(0.471,-0.168)--(0.476,-0.154)--(0.480,-0.141)--(0.484,-0.127)--(0.487,-0.113)--(0.490,-0.0990)--(0.493,-0.0850)--(0.495,-0.0710)--(0.497,-0.0568)--(0.498,-0.0427)--(0.499,-0.0285)--(0.500,-0.0142)--(0.500,0);
 
diff --git a/src_phystricks/Fig_CbCartTui.pstricks.recall b/src_phystricks/Fig_CbCartTui.pstricks.recall
index 398a015fa..ad4c00516 100644
--- a/src_phystricks/Fig_CbCartTui.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTui.pstricks.recall
@@ -103,60 +103,60 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.140000002,0) -- (4.140000002,0);
-\draw [,->,>=latex] (0,-3.972000001) -- (0,4.028000000);
+\draw [,->,>=latex] (-4.1400,0) -- (4.1400,0);
+\draw [,->,>=latex] (0,-3.9720) -- (0,4.0280);
 %DEFAULT
 \draw [color=blue] (-3.640,-3.472)--(-3.609,-3.440)--(-3.579,-3.407)--(-3.548,-3.375)--(-3.518,-3.343)--(-3.488,-3.310)--(-3.457,-3.278)--(-3.427,-3.245)--(-3.396,-3.213)--(-3.366,-3.180)--(-3.336,-3.148)--(-3.306,-3.115)--(-3.275,-3.083)--(-3.245,-3.050)--(-3.215,-3.018)--(-3.185,-2.985)--(-3.155,-2.953)--(-3.125,-2.920)--(-3.095,-2.887)--(-3.065,-2.855)--(-3.035,-2.822)--(-3.005,-2.789)--(-2.975,-2.756)--(-2.945,-2.723)--(-2.915,-2.691)--(-2.886,-2.658)--(-2.856,-2.625)--(-2.826,-2.592)--(-2.797,-2.559)--(-2.767,-2.526)--(-2.738,-2.493)--(-2.709,-2.459)--(-2.679,-2.426)--(-2.650,-2.393)--(-2.621,-2.360)--(-2.592,-2.326)--(-2.563,-2.293)--(-2.534,-2.259)--(-2.505,-2.226)--(-2.476,-2.192)--(-2.447,-2.158)--(-2.419,-2.124)--(-2.390,-2.090)--(-2.362,-2.056)--(-2.333,-2.022)--(-2.305,-1.988)--(-2.277,-1.954)--(-2.249,-1.919)--(-2.221,-1.885)--(-2.193,-1.850)--(-2.166,-1.815)--(-2.138,-1.780)--(-2.111,-1.745)--(-2.084,-1.709)--(-2.057,-1.674)--(-2.030,-1.638)--(-2.003,-1.602)--(-1.977,-1.566)--(-1.951,-1.529)--(-1.925,-1.492)--(-1.899,-1.455)--(-1.873,-1.418)--(-1.848,-1.380)--(-1.823,-1.342)--(-1.798,-1.304)--(-1.774,-1.265)--(-1.750,-1.225)--(-1.726,-1.185)--(-1.703,-1.144)--(-1.680,-1.103)--(-1.658,-1.061)--(-1.636,-1.018)--(-1.614,-0.9745)--(-1.593,-0.9298)--(-1.573,-0.8840)--(-1.554,-0.8371)--(-1.535,-0.7887)--(-1.517,-0.7389)--(-1.499,-0.6873)--(-1.483,-0.6338)--(-1.468,-0.5781)--(-1.454,-0.5199)--(-1.441,-0.4588)--(-1.429,-0.3943)--(-1.420,-0.3261)--(-1.411,-0.2534)--(-1.405,-0.1754)--(-1.401,-0.09136)--(-1.400,0)--(-1.402,0.1001)--(-1.406,0.2106)--(-1.415,0.3340)--(-1.428,0.4729)--(-1.447,0.6314)--(-1.472,0.8143)--(-1.504,1.029)--(-1.545,1.283)--(-1.598,1.591)--(-1.665,1.971)--(-1.750,2.450);
-\draw [,->,>=latex] (-3.505833333,-3.329618056) -- (-3.501040033,-3.324516656);
-\draw [,->,>=latex] (-2.333333333,-2.022222222) -- (-2.328870359,-2.016829462);
-\draw [,->,>=latex] (-1.750000000,-1.225000000) -- (-1.746398530,-1.218997550);
-\draw [,->,>=latex] (-1.490000000,0.9385714286) -- (-1.491054412,0.9454915598);
+\draw [,->,>=latex] (-3.5058,-3.3296) -- (-3.5010,-3.3245);
+\draw [,->,>=latex] (-2.3333,-2.0222) -- (-2.3289,-2.0168);
+\draw [,->,>=latex] (-1.7500,-1.2250) -- (-1.7464,-1.2190);
+\draw [,->,>=latex] (-1.4900,0.93857) -- (-1.4911,0.94549);
 \draw [color=blue] (1.750,3.150)--(1.740,3.098)--(1.729,3.047)--(1.719,2.998)--(1.710,2.951)--(1.700,2.905)--(1.691,2.860)--(1.682,2.817)--(1.674,2.775)--(1.665,2.735)--(1.657,2.695)--(1.649,2.657)--(1.641,2.620)--(1.633,2.584)--(1.626,2.549)--(1.619,2.515)--(1.612,2.482)--(1.605,2.450)--(1.598,2.418)--(1.592,2.388)--(1.585,2.359)--(1.579,2.330)--(1.573,2.302)--(1.567,2.275)--(1.562,2.249)--(1.556,2.223)--(1.551,2.198)--(1.545,2.174)--(1.540,2.150)--(1.535,2.128)--(1.530,2.105)--(1.526,2.083)--(1.521,2.062)--(1.517,2.042)--(1.512,2.022)--(1.508,2.002)--(1.504,1.983)--(1.500,1.965)--(1.496,1.946)--(1.492,1.929)--(1.489,1.912)--(1.485,1.895)--(1.481,1.879)--(1.478,1.863)--(1.475,1.848)--(1.472,1.833)--(1.468,1.818)--(1.465,1.804)--(1.463,1.790)--(1.460,1.776)--(1.457,1.763)--(1.454,1.750)--(1.452,1.737)--(1.449,1.725)--(1.447,1.713)--(1.444,1.702)--(1.442,1.690)--(1.440,1.679)--(1.438,1.668)--(1.436,1.658)--(1.434,1.648)--(1.432,1.638)--(1.430,1.628)--(1.428,1.618)--(1.427,1.609)--(1.425,1.600)--(1.423,1.591)--(1.422,1.583)--(1.420,1.574)--(1.419,1.566)--(1.418,1.558)--(1.416,1.551)--(1.415,1.543)--(1.414,1.536)--(1.413,1.529)--(1.412,1.522)--(1.411,1.515)--(1.410,1.508)--(1.409,1.502)--(1.408,1.495)--(1.407,1.489)--(1.406,1.483)--(1.406,1.478)--(1.405,1.472)--(1.404,1.466)--(1.404,1.461)--(1.403,1.456)--(1.403,1.451)--(1.402,1.446)--(1.402,1.441)--(1.402,1.436)--(1.401,1.432)--(1.401,1.428)--(1.401,1.423)--(1.400,1.419)--(1.400,1.415)--(1.400,1.411)--(1.400,1.407)--(1.400,1.404)--(1.400,1.400);
-\draw [,->,>=latex] (1.586666667,2.364444444) -- (1.585193677,2.357601178);
-\draw [,->,>=latex] (1.435000000,1.653750000) -- (1.433669847,1.646877541);
+\draw [,->,>=latex] (1.5867,2.3644) -- (1.5852,2.3576);
+\draw [,->,>=latex] (1.4350,1.6537) -- (1.4337,1.6469);
 \draw [color=blue] (1.400,1.400)--(1.401,1.375)--(1.404,1.356)--(1.409,1.342)--(1.416,1.332)--(1.424,1.326)--(1.433,1.323)--(1.444,1.323)--(1.455,1.326)--(1.468,1.331)--(1.481,1.338)--(1.496,1.347)--(1.511,1.357)--(1.527,1.369)--(1.543,1.382)--(1.560,1.396)--(1.578,1.411)--(1.596,1.427)--(1.614,1.444)--(1.633,1.461)--(1.653,1.480)--(1.673,1.499)--(1.693,1.518)--(1.713,1.539)--(1.734,1.559)--(1.755,1.580)--(1.777,1.602)--(1.798,1.624)--(1.820,1.646)--(1.843,1.669)--(1.865,1.692)--(1.888,1.715)--(1.910,1.738)--(1.933,1.762)--(1.957,1.786)--(1.980,1.810)--(2.003,1.834)--(2.027,1.859)--(2.051,1.884)--(2.075,1.909)--(2.099,1.934)--(2.123,1.959)--(2.147,1.984)--(2.172,2.010)--(2.196,2.035)--(2.221,2.061)--(2.246,2.087)--(2.271,2.113)--(2.296,2.139)--(2.321,2.165)--(2.346,2.191)--(2.371,2.217)--(2.396,2.243)--(2.422,2.270)--(2.447,2.296)--(2.473,2.323)--(2.498,2.350)--(2.524,2.376)--(2.550,2.403)--(2.576,2.430)--(2.601,2.457)--(2.627,2.484)--(2.653,2.511)--(2.679,2.538)--(2.705,2.565)--(2.731,2.592)--(2.758,2.619)--(2.784,2.646)--(2.810,2.673)--(2.836,2.700)--(2.863,2.728)--(2.889,2.755)--(2.915,2.782)--(2.942,2.810)--(2.968,2.837)--(2.995,2.864)--(3.021,2.892)--(3.048,2.919)--(3.075,2.947)--(3.101,2.974)--(3.128,3.002)--(3.155,3.029)--(3.181,3.057)--(3.208,3.084)--(3.235,3.112)--(3.262,3.140)--(3.289,3.167)--(3.316,3.195)--(3.343,3.223)--(3.369,3.250)--(3.396,3.278)--(3.423,3.306)--(3.450,3.333)--(3.477,3.361)--(3.504,3.389)--(3.532,3.417)--(3.559,3.445)--(3.586,3.472)--(3.613,3.500)--(3.640,3.528);
-\draw [,->,>=latex] (1.423333333,1.326111111) -- (1.429556155,1.322905415);
-\draw [,->,>=latex] (1.750000000,1.575000000) -- (1.754949747,1.579949747);
-\draw [,->,>=latex] (2.333333333,2.177777778) -- (2.338181056,2.182827489);
-\draw (-3.500000000,-0.3298256667) node {$ -5 $};
+\draw [,->,>=latex] (1.4233,1.3261) -- (1.4296,1.3229);
+\draw [,->,>=latex] (1.7500,1.5750) -- (1.7549,1.5800);
+\draw [,->,>=latex] (2.3333,2.1778) -- (2.3382,2.1828);
+\draw (-3.5000,-0.32983) node {$ -5 $};
 \draw [] (-3.50,-0.100) -- (-3.50,0.100);
-\draw (-2.800000000,-0.3298256667) node {$ -4 $};
+\draw (-2.8000,-0.32983) node {$ -4 $};
 \draw [] (-2.80,-0.100) -- (-2.80,0.100);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CbCartTuiii.pstricks.recall b/src_phystricks/Fig_CbCartTuiii.pstricks.recall
index 95ddf09e7..8b557fe13 100644
--- a/src_phystricks/Fig_CbCartTuiii.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTuiii.pstricks.recall
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.499748256,0) -- (2.499748256,0);
-\draw [,->,>=latex] (0,-2.497734679) -- (0,2.497734679);
+\draw [,->,>=latex] (-2.4997,0) -- (2.4997,0);
+\draw [,->,>=latex] (0,-2.4977) -- (0,2.4977);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.253,0.379)--(0.502,0.743)--(0.743,1.08)--(0.972,1.38)--(1.19,1.63)--(1.38,1.82)--(1.55,1.94)--(1.70,2.00)--(1.82,1.98)--(1.91,1.89)--(1.97,1.73)--(2.00,1.51)--(1.99,1.24)--(1.96,0.916)--(1.89,0.563)--(1.79,0.190)--(1.67,-0.190)--(1.51,-0.563)--(1.33,-0.916)--(1.13,-1.24)--(0.916,-1.51)--(0.684,-1.73)--(0.441,-1.89)--(0.190,-1.98)--(-0.0635,-2.00)--(-0.316,-1.94)--(-0.563,-1.82)--(-0.802,-1.63)--(-1.03,-1.38)--(-1.24,-1.08)--(-1.43,-0.743)--(-1.59,-0.379)--(-1.73,0)--(-1.84,0.379)--(-1.93,0.743)--(-1.98,1.08)--(-2.00,1.38)--(-1.99,1.63)--(-1.94,1.82)--(-1.87,1.94)--(-1.76,2.00)--(-1.63,1.98)--(-1.47,1.89)--(-1.29,1.73)--(-1.08,1.51)--(-0.860,1.24)--(-0.624,0.916)--(-0.379,0.563)--(-0.127,0.190)--(0.127,-0.190)--(0.379,-0.563)--(0.624,-0.916)--(0.860,-1.24)--(1.08,-1.51)--(1.29,-1.73)--(1.47,-1.89)--(1.63,-1.98)--(1.76,-2.00)--(1.87,-1.94)--(1.94,-1.82)--(1.99,-1.63)--(2.00,-1.38)--(1.98,-1.08)--(1.93,-0.743)--(1.84,-0.379)--(1.73,0)--(1.59,0.379)--(1.43,0.743)--(1.24,1.08)--(1.03,1.38)--(0.802,1.63)--(0.563,1.82)--(0.316,1.94)--(0.0635,2.00)--(-0.190,1.98)--(-0.441,1.89)--(-0.684,1.73)--(-0.916,1.51)--(-1.13,1.24)--(-1.33,0.916)--(-1.51,0.563)--(-1.67,0.190)--(-1.79,-0.190)--(-1.89,-0.563)--(-1.96,-0.916)--(-1.99,-1.24)--(-2.00,-1.51)--(-1.97,-1.73)--(-1.91,-1.89)--(-1.82,-1.98)--(-1.70,-2.00)--(-1.55,-1.94)--(-1.38,-1.82)--(-1.19,-1.63)--(-0.972,-1.38)--(-0.743,-1.08)--(-0.502,-0.743)--(-0.253,-0.379)--(0,0);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CercleImplicite.pstricks.recall b/src_phystricks/Fig_CercleImplicite.pstricks.recall
index de144d561..65090a7cf 100644
--- a/src_phystricks/Fig_CercleImplicite.pstricks.recall
+++ b/src_phystricks/Fig_CercleImplicite.pstricks.recall
@@ -79,19 +79,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 
 \draw [] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
-\draw []  (1.414213562,1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.768860430,1.751053597) node {$P$};
-\draw []  (1.414213562,-1.414213562) node [rotate=0] {$\bullet$};
-\draw (1.815841763,-1.767119930) node {\( P'\)};
-\draw []  (-2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-2.356408201,0.3723262010) node {\( Q\)};
-\draw []  (1.414213562,0) node [rotate=0] {$\bullet$};
-\draw (1.097777861,-0.2907082010) node {\( x\)};
+\draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.7689,1.7511) node {$P$};
+\draw []  (1.4142,-1.4142) node [rotate=0] {$\bullet$};
+\draw (1.8158,-1.7671) node {\( P'\)};
+\draw []  (-2.0000,0) node [rotate=0] {$\bullet$};
+\draw (-2.3564,0.37233) node {\( Q\)};
+\draw []  (1.4142,0) node [rotate=0] {$\bullet$};
+\draw (1.0978,-0.29071) node {\( x\)};
 \draw [color=red,style=dotted] (1.41,1.41) -- (1.41,-1.41);
 
 %OTHER STUFF
diff --git a/src_phystricks/Fig_ChampGraviation.pstricks.recall b/src_phystricks/Fig_ChampGraviation.pstricks.recall
index f49509765..518a4dcc8 100644
--- a/src_phystricks/Fig_ChampGraviation.pstricks.recall
+++ b/src_phystricks/Fig_ChampGraviation.pstricks.recall
@@ -65,226 +65,226 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.000000000,-4.000000000) -- (-4.044194174,-4.044194174);
-\draw [,->,>=latex] (-4.000000000,-3.428571429) -- (-4.054711138,-3.475466689);
-\draw [,->,>=latex] (-4.000000000,-2.857142857) -- (-4.067352939,-2.905252100);
-\draw [,->,>=latex] (-4.000000000,-2.285714286) -- (-4.081815219,-2.332465840);
-\draw [,->,>=latex] (-4.000000000,-1.714285714) -- (-4.097064885,-1.755884951);
-\draw [,->,>=latex] (-4.000000000,-1.142857143) -- (-4.111119513,-1.174605575);
-\draw [,->,>=latex] (-4.000000000,-0.5714285714) -- (-4.121268813,-0.5887526876);
-\draw [,->,>=latex] (-4.000000000,0) -- (-4.125000000,0);
-\draw [,->,>=latex] (-4.000000000,0.5714285714) -- (-4.121268813,0.5887526876);
-\draw [,->,>=latex] (-4.000000000,1.142857143) -- (-4.111119513,1.174605575);
-\draw [,->,>=latex] (-4.000000000,1.714285714) -- (-4.097064885,1.755884951);
-\draw [,->,>=latex] (-4.000000000,2.285714286) -- (-4.081815219,2.332465840);
-\draw [,->,>=latex] (-4.000000000,2.857142857) -- (-4.067352939,2.905252100);
-\draw [,->,>=latex] (-4.000000000,3.428571429) -- (-4.054711138,3.475466689);
-\draw [,->,>=latex] (-4.000000000,4.000000000) -- (-4.044194174,4.044194174);
-\draw [,->,>=latex] (-3.428571429,-4.000000000) -- (-3.475466689,-4.054711138);
-\draw [,->,>=latex] (-3.428571429,-3.428571429) -- (-3.488724610,-3.488724610);
-\draw [,->,>=latex] (-3.428571429,-2.857142857) -- (-3.505708401,-2.921423668);
-\draw [,->,>=latex] (-3.428571429,-2.285714286) -- (-3.526577353,-2.351051568);
-\draw [,->,>=latex] (-3.428571429,-1.714285714) -- (-3.550312907,-1.775156454);
-\draw [,->,>=latex] (-3.428571429,-1.142857143) -- (-3.573838559,-1.191279520);
-\draw [,->,>=latex] (-3.428571429,-0.5714285714) -- (-3.591859612,-0.5986432687);
-\draw [,->,>=latex] (-3.428571429,0) -- (-3.598710317,0);
-\draw [,->,>=latex] (-3.428571429,0.5714285714) -- (-3.591859612,0.5986432687);
-\draw [,->,>=latex] (-3.428571429,1.142857143) -- (-3.573838559,1.191279520);
-\draw [,->,>=latex] (-3.428571429,1.714285714) -- (-3.550312907,1.775156454);
-\draw [,->,>=latex] (-3.428571429,2.285714286) -- (-3.526577353,2.351051568);
-\draw [,->,>=latex] (-3.428571429,2.857142857) -- (-3.505708401,2.921423668);
-\draw [,->,>=latex] (-3.428571429,3.428571429) -- (-3.488724610,3.488724610);
-\draw [,->,>=latex] (-3.428571429,4.000000000) -- (-3.475466689,4.054711138);
-\draw [,->,>=latex] (-2.857142857,-4.000000000) -- (-2.905252100,-4.067352939);
-\draw [,->,>=latex] (-2.857142857,-3.428571429) -- (-2.921423668,-3.505708401);
-\draw [,->,>=latex] (-2.857142857,-2.857142857) -- (-2.943763438,-2.943763438);
-\draw [,->,>=latex] (-2.857142857,-2.285714286) -- (-2.973797039,-2.379037631);
-\draw [,->,>=latex] (-2.857142857,-1.714285714) -- (-3.011617686,-1.806970611);
-\draw [,->,>=latex] (-2.857142857,-1.142857143) -- (-3.053243538,-1.221297415);
-\draw [,->,>=latex] (-2.857142857,-0.5714285714) -- (-3.088145036,-0.6176290071);
-\draw [,->,>=latex] (-2.857142857,0) -- (-3.102142857,0);
-\draw [,->,>=latex] (-2.857142857,0.5714285714) -- (-3.088145036,0.6176290071);
-\draw [,->,>=latex] (-2.857142857,1.142857143) -- (-3.053243538,1.221297415);
-\draw [,->,>=latex] (-2.857142857,1.714285714) -- (-3.011617686,1.806970611);
-\draw [,->,>=latex] (-2.857142857,2.285714286) -- (-2.973797039,2.379037631);
-\draw [,->,>=latex] (-2.857142857,2.857142857) -- (-2.943763438,2.943763438);
-\draw [,->,>=latex] (-2.857142857,3.428571429) -- (-2.921423668,3.505708401);
-\draw [,->,>=latex] (-2.857142857,4.000000000) -- (-2.905252100,4.067352939);
-\draw [,->,>=latex] (-2.285714286,-4.000000000) -- (-2.332465840,-4.081815219);
-\draw [,->,>=latex] (-2.285714286,-3.428571429) -- (-2.351051568,-3.526577353);
-\draw [,->,>=latex] (-2.285714286,-2.857142857) -- (-2.379037631,-2.973797039);
-\draw [,->,>=latex] (-2.285714286,-2.285714286) -- (-2.421058943,-2.421058943);
-\draw [,->,>=latex] (-2.285714286,-1.714285714) -- (-2.481714286,-1.861285714);
-\draw [,->,>=latex] (-2.285714286,-1.142857143) -- (-2.559632613,-1.279816306);
-\draw [,->,>=latex] (-2.285714286,-0.5714285714) -- (-2.635250922,-0.6588127304);
-\draw [,->,>=latex] (-2.285714286,0) -- (-2.668526786,0);
-\draw [,->,>=latex] (-2.285714286,0.5714285714) -- (-2.635250922,0.6588127304);
-\draw [,->,>=latex] (-2.285714286,1.142857143) -- (-2.559632613,1.279816306);
-\draw [,->,>=latex] (-2.285714286,1.714285714) -- (-2.481714286,1.861285714);
-\draw [,->,>=latex] (-2.285714286,2.285714286) -- (-2.421058943,2.421058943);
-\draw [,->,>=latex] (-2.285714286,2.857142857) -- (-2.379037631,2.973797039);
-\draw [,->,>=latex] (-2.285714286,3.428571429) -- (-2.351051568,3.526577353);
-\draw [,->,>=latex] (-2.285714286,4.000000000) -- (-2.332465840,4.081815219);
-\draw [,->,>=latex] (-1.714285714,-4.000000000) -- (-1.755884951,-4.097064885);
-\draw [,->,>=latex] (-1.714285714,-3.428571429) -- (-1.775156454,-3.550312907);
-\draw [,->,>=latex] (-1.714285714,-2.857142857) -- (-1.806970611,-3.011617686);
-\draw [,->,>=latex] (-1.714285714,-2.285714286) -- (-1.861285714,-2.481714286);
-\draw [,->,>=latex] (-1.714285714,-1.714285714) -- (-1.954898438,-1.954898438);
-\draw [,->,>=latex] (-1.714285714,-1.142857143) -- (-2.106309411,-1.404206274);
-\draw [,->,>=latex] (-1.714285714,-0.5714285714) -- (-2.295354234,-0.7651180781);
-\draw [,->,>=latex] (-1.714285714,0) -- (-2.394841270,0);
-\draw [,->,>=latex] (-1.714285714,0.5714285714) -- (-2.295354234,0.7651180781);
-\draw [,->,>=latex] (-1.714285714,1.142857143) -- (-2.106309411,1.404206274);
-\draw [,->,>=latex] (-1.714285714,1.714285714) -- (-1.954898438,1.954898438);
-\draw [,->,>=latex] (-1.714285714,2.285714286) -- (-1.861285714,2.481714286);
-\draw [,->,>=latex] (-1.714285714,2.857142857) -- (-1.806970611,3.011617686);
-\draw [,->,>=latex] (-1.714285714,3.428571429) -- (-1.775156454,3.550312907);
-\draw [,->,>=latex] (-1.714285714,4.000000000) -- (-1.755884951,4.097064885);
-\draw [,->,>=latex] (-1.142857143,-4.000000000) -- (-1.174605575,-4.111119513);
-\draw [,->,>=latex] (-1.142857143,-3.428571429) -- (-1.191279520,-3.573838559);
-\draw [,->,>=latex] (-1.142857143,-2.857142857) -- (-1.221297415,-3.053243538);
-\draw [,->,>=latex] (-1.142857143,-2.285714286) -- (-1.279816306,-2.559632613);
-\draw [,->,>=latex] (-1.142857143,-1.714285714) -- (-1.404206274,-2.106309411);
-\draw [,->,>=latex] (-1.142857143,-1.142857143) -- (-1.684235772,-1.684235772);
-\draw [,->,>=latex] (-1.142857143,-0.5714285714) -- (-2.238530452,-1.119265226);
-\draw [,->,>=latex] (-1.142857143,0) -- (-2.674107143,0);
-\draw [,->,>=latex] (-1.142857143,0.5714285714) -- (-2.238530452,1.119265226);
-\draw [,->,>=latex] (-1.142857143,1.142857143) -- (-1.684235772,1.684235772);
-\draw [,->,>=latex] (-1.142857143,1.714285714) -- (-1.404206274,2.106309411);
-\draw [,->,>=latex] (-1.142857143,2.285714286) -- (-1.279816306,2.559632613);
-\draw [,->,>=latex] (-1.142857143,2.857142857) -- (-1.221297415,3.053243538);
-\draw [,->,>=latex] (-1.142857143,3.428571429) -- (-1.191279520,3.573838559);
-\draw [,->,>=latex] (-1.142857143,4.000000000) -- (-1.174605575,4.111119513);
-\draw [,->,>=latex] (-0.5714285714,-4.000000000) -- (-0.5887526876,-4.121268813);
-\draw [,->,>=latex] (-0.5714285714,-3.428571429) -- (-0.5986432687,-3.591859612);
-\draw [,->,>=latex] (-0.5714285714,-2.857142857) -- (-0.6176290071,-3.088145036);
-\draw [,->,>=latex] (-0.5714285714,-2.285714286) -- (-0.6588127304,-2.635250922);
-\draw [,->,>=latex] (-0.5714285714,-1.714285714) -- (-0.7651180781,-2.295354234);
-\draw [,->,>=latex] (-0.5714285714,-1.142857143) -- (-1.119265226,-2.238530452);
-\draw [,->,>=latex] (-0.5714285714,-0.5714285714) -- (-2.736943089,-2.736943089);
-\draw [,->,>=latex] (-0.5714285714,0.5714285714) -- (-2.736943089,2.736943089);
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+\draw [,->,>=latex] (4.0000,-2.8571) -- (4.0674,-2.9053);
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+\draw [,->,>=latex] (4.0000,-1.7143) -- (4.0971,-1.7559);
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+\draw [,->,>=latex] (4.0000,-0.57143) -- (4.1213,-0.58875);
+\draw [,->,>=latex] (4.0000,0) -- (4.1250,0);
+\draw [,->,>=latex] (4.0000,0.57143) -- (4.1213,0.58875);
+\draw [,->,>=latex] (4.0000,1.1429) -- (4.1111,1.1746);
+\draw [,->,>=latex] (4.0000,1.7143) -- (4.0971,1.7559);
+\draw [,->,>=latex] (4.0000,2.2857) -- (4.0818,2.3325);
+\draw [,->,>=latex] (4.0000,2.8571) -- (4.0674,2.9053);
+\draw [,->,>=latex] (4.0000,3.4286) -- (4.0547,3.4755);
+\draw [,->,>=latex] (4.0000,4.0000) -- (4.0442,4.0442);
 
 \draw [color=blue] (0.700,0)--(0.699,0.0444)--(0.694,0.0886)--(0.687,0.132)--(0.678,0.176)--(0.665,0.218)--(0.650,0.260)--(0.632,0.301)--(0.612,0.340)--(0.589,0.378)--(0.564,0.415)--(0.536,0.450)--(0.507,0.483)--(0.475,0.514)--(0.441,0.543)--(0.406,0.570)--(0.369,0.595)--(0.331,0.617)--(0.291,0.637)--(0.250,0.654)--(0.208,0.668)--(0.165,0.680)--(0.122,0.689)--(0.0776,0.696)--(0.0333,0.699)--(-0.0111,0.700)--(-0.0555,0.698)--(-0.0996,0.693)--(-0.143,0.685)--(-0.187,0.675)--(-0.229,0.661)--(-0.270,0.646)--(-0.311,0.627)--(-0.350,0.606)--(-0.388,0.583)--(-0.424,0.557)--(-0.458,0.529)--(-0.491,0.499)--(-0.522,0.467)--(-0.550,0.433)--(-0.577,0.397)--(-0.601,0.360)--(-0.622,0.321)--(-0.641,0.281)--(-0.658,0.239)--(-0.672,0.197)--(-0.683,0.154)--(-0.691,0.111)--(-0.697,0.0665)--(-0.700,0.0222)--(-0.700,-0.0222)--(-0.697,-0.0665)--(-0.691,-0.111)--(-0.683,-0.154)--(-0.672,-0.197)--(-0.658,-0.239)--(-0.641,-0.281)--(-0.622,-0.321)--(-0.601,-0.360)--(-0.577,-0.397)--(-0.550,-0.433)--(-0.522,-0.467)--(-0.491,-0.499)--(-0.458,-0.529)--(-0.424,-0.557)--(-0.388,-0.583)--(-0.350,-0.606)--(-0.311,-0.627)--(-0.270,-0.646)--(-0.229,-0.661)--(-0.187,-0.675)--(-0.143,-0.685)--(-0.0996,-0.693)--(-0.0555,-0.698)--(-0.0111,-0.700)--(0.0333,-0.699)--(0.0776,-0.696)--(0.122,-0.689)--(0.165,-0.680)--(0.208,-0.668)--(0.250,-0.654)--(0.291,-0.637)--(0.331,-0.617)--(0.369,-0.595)--(0.406,-0.570)--(0.441,-0.543)--(0.475,-0.514)--(0.507,-0.483)--(0.536,-0.450)--(0.564,-0.415)--(0.589,-0.378)--(0.612,-0.340)--(0.632,-0.301)--(0.650,-0.260)--(0.665,-0.218)--(0.678,-0.176)--(0.687,-0.132)--(0.694,-0.0886)--(0.699,-0.0444)--(0.700,0);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ChiSquared.pstricks.recall b/src_phystricks/Fig_ChiSquared.pstricks.recall
index 758660882..77a4f4ecd 100644
--- a/src_phystricks/Fig_ChiSquared.pstricks.recall
+++ b/src_phystricks/Fig_ChiSquared.pstricks.recall
@@ -95,26 +95,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (15.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.381905481);
+\draw [,->,>=latex] (-0.50000,0) -- (15.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.3819);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.15152,0)--(0.30303,0.0064874)--(0.45455,0.028225)--(0.60606,0.076663)--(0.75758,0.16085)--(0.90909,0.28665)--(1.0606,0.45638)--(1.2121,0.66911)--(1.3636,0.92109)--(1.5152,1.2065)--(1.6667,1.5181)--(1.8182,1.8478)--(1.9697,2.1872)--(2.1212,2.5283)--(2.2727,2.8634)--(2.4242,3.1856)--(2.5758,3.4891)--(2.7273,3.7688)--(2.8788,4.0209)--(3.0303,4.2426)--(3.1818,4.4318)--(3.3333,4.5877)--(3.4848,4.7099)--(3.6364,4.7989)--(3.7879,4.8558)--(3.9394,4.8819)--(4.0909,4.8792)--(4.2424,4.8498)--(4.3939,4.7960)--(4.5455,4.7203)--(4.6970,4.6252)--(4.8485,4.5132)--(5.0000,4.3867)--(5.1515,4.2481)--(5.3030,4.0997)--(5.4545,3.9435)--(5.6061,3.7816)--(5.7576,3.6158)--(5.9091,3.4477)--(6.0606,3.2787)--(6.2121,3.1103)--(6.3636,2.9435)--(6.5152,2.7793)--(6.6667,2.6186)--(6.8182,2.4621)--(6.9697,2.3104)--(7.1212,2.1639)--(7.2727,2.0231)--(7.4242,1.8881)--(7.5758,1.7592)--(7.7273,1.6365)--(7.8788,1.5200)--(8.0303,1.4097)--(8.1818,1.3056)--(8.3333,1.2075)--(8.4848,1.1153)--(8.6364,1.0288)--(8.7879,0.94785)--(8.9394,0.87223)--(9.0909,0.80173)--(9.2424,0.73610)--(9.3939,0.67512)--(9.5455,0.61855)--(9.6970,0.56615)--(9.8485,0.51768)--(10.000,0.47292)--(10.152,0.43162)--(10.303,0.39359)--(10.455,0.35859)--(10.606,0.32643)--(10.758,0.29691)--(10.909,0.26985)--(11.061,0.24507)--(11.212,0.22239)--(11.364,0.20167)--(11.515,0.18274)--(11.667,0.16548)--(11.818,0.14975)--(11.970,0.13542)--(12.121,0.12239)--(12.273,0.11054)--(12.424,0.099777)--(12.576,0.090008)--(12.727,0.081149)--(12.879,0.073121)--(13.030,0.065850)--(13.182,0.059271)--(13.333,0.053320)--(13.485,0.047942)--(13.636,0.043085)--(13.788,0.038701)--(13.939,0.034746)--(14.091,0.031180)--(14.242,0.027968)--(14.394,0.025075)--(14.545,0.022471)--(14.697,0.020129)--(14.848,0.018024)--(15.000,0.016132);
-\draw (2.500000000,-0.3149246667) node {$ 5 $};
+\draw (2.5000,-0.31492) node {$ 5 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 10 $};
+\draw (5.0000,-0.31492) node {$ 10 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (7.500000000,-0.3149246667) node {$ 15 $};
+\draw (7.5000,-0.31492) node {$ 15 $};
 \draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 20 $};
+\draw (10.000,-0.31492) node {$ 20 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (12.50000000,-0.3149246667) node {$ 25 $};
+\draw (12.500,-0.31492) node {$ 25 $};
 \draw [] (12.5,-0.100) -- (12.5,0.100);
-\draw (15.00000000,-0.3149246667) node {$ 30 $};
+\draw (15.000,-0.31492) node {$ 30 $};
 \draw [] (15.0,-0.100) -- (15.0,0.100);
-\draw (-0.3816666667,2.500000000) node {$ \frac{1}{20} $};
+\draw (-0.38167,2.5000) node {$ \frac{1}{20} $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.3816666667,5.000000000) node {$ \frac{1}{10} $};
+\draw (-0.38167,5.0000) node {$ \frac{1}{10} $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall b/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
index ff609a685..8e28a13e5 100644
--- a/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
+++ b/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (0,0) -- (9.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.384154943);
+\draw [,->,>=latex] (0,0) -- (9.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.3842);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.045226,0)--(0.090452,0)--(0.13568,0.0021725)--(0.18090,0.0063676)--(0.22613,0.014417)--(0.27136,0.027725)--(0.31658,0.047635)--(0.36181,0.075363)--(0.40704,0.11195)--(0.45226,0.15824)--(0.49749,0.21486)--(0.54271,0.28221)--(0.58794,0.36049)--(0.63317,0.44967)--(0.67839,0.54955)--(0.72362,0.65977)--(0.76884,0.77977)--(0.81407,0.90892)--(0.85930,1.0464)--(0.90452,1.1915)--(0.94975,1.3431)--(0.99498,1.5003)--(1.0402,1.6621)--(1.0854,1.8275)--(1.1307,1.9955)--(1.1759,2.1649)--(1.2211,2.3349)--(1.2663,2.5044)--(1.3116,2.6726)--(1.3568,2.8385)--(1.4020,3.0014)--(1.4472,3.1604)--(1.4925,3.3148)--(1.5377,3.4640)--(1.5829,3.6075)--(1.6281,3.7446)--(1.6734,3.8749)--(1.7186,3.9981)--(1.7638,4.1138)--(1.8090,4.2217)--(1.8543,4.3217)--(1.8995,4.4134)--(1.9447,4.4969)--(1.9900,4.5721)--(2.0352,4.6390)--(2.0804,4.6975)--(2.1256,4.7478)--(2.1709,4.7899)--(2.2161,4.8241)--(2.2613,4.8503)--(2.3065,4.8690)--(2.3518,4.8802)--(2.3970,4.8842)--(2.4422,4.8812)--(2.4874,4.8715)--(2.5327,4.8555)--(2.5779,4.8333)--(2.6231,4.8053)--(2.6683,4.7718)--(2.7136,4.7330)--(2.7588,4.6894)--(2.8040,4.6412)--(2.8492,4.5887)--(2.8945,4.5322)--(2.9397,4.4721)--(2.9849,4.4086)--(3.0302,4.3419)--(3.0754,4.2725)--(3.1206,4.2006)--(3.1658,4.1264)--(3.2111,4.0502)--(3.2563,3.9722)--(3.3015,3.8928)--(3.3467,3.8121)--(3.3920,3.7303)--(3.4372,3.6477)--(3.4824,3.5644)--(3.5276,3.4807)--(3.5729,3.3968)--(3.6181,3.3127)--(3.6633,3.2287)--(3.7085,3.1449)--(3.7538,3.0614)--(3.7990,2.9785)--(3.8442,2.8961)--(3.8894,2.8145)--(3.9347,2.7337)--(3.9799,2.6538)--(4.0251,2.5749)--(4.0704,2.4971)--(4.1156,2.4204)--(4.1608,2.3450)--(4.2060,2.2708)--(4.2513,2.1980)--(4.2965,2.1266)--(4.3417,2.0565)--(4.3869,1.9879)--(4.4322,1.9208)--(4.4774,1.8552)--(4.5226,1.7910)--(4.5678,1.7284)--(4.6131,1.6674)--(4.6583,1.6079)--(4.7035,1.5499)--(4.7487,1.4934)--(4.7940,1.4385)--(4.8392,1.3851)--(4.8844,1.3333)--(4.9296,1.2829)--(4.9749,1.2340)--(5.0201,1.1866)--(5.0653,1.1406)--(5.1105,1.0961)--(5.1558,1.0530)--(5.2010,1.0113)--(5.2462,0.97088)--(5.2915,0.93184)--(5.3367,0.89411)--(5.3819,0.85766)--(5.4271,0.82246)--(5.4724,0.78849)--(5.5176,0.75571)--(5.5628,0.72411)--(5.6080,0.69364)--(5.6533,0.66428)--(5.6985,0.63600)--(5.7437,0.60877)--(5.7889,0.58256)--(5.8342,0.55735)--(5.8794,0.53310)--(5.9246,0.50978)--(5.9698,0.48737)--(6.0151,0.46583)--(6.0603,0.44515)--(6.1055,0.42529)--(6.1508,0.40623)--(6.1960,0.38794)--(6.2412,0.37039)--(6.2864,0.35357)--(6.3317,0.33743)--(6.3769,0.32197)--(6.4221,0.30715)--(6.4673,0.29296)--(6.5126,0.27937)--(6.5578,0.26636)--(6.6030,0.25390)--(6.6482,0.24199)--(6.6935,0.23059)--(6.7387,0.21968)--(6.7839,0.20926)--(6.8291,0.19929)--(6.8744,0.18977)--(6.9196,0.18067)--(6.9648,0.17197)--(7.0100,0.16367)--(7.0553,0.15574)--(7.1005,0.14817)--(7.1457,0.14095)--(7.1910,0.13406)--(7.2362,0.12748)--(7.2814,0.12121)--(7.3266,0.11523)--(7.3719,0.10952)--(7.4171,0.10409)--(7.4623,0.098904)--(7.5075,0.093967)--(7.5528,0.089264)--(7.5980,0.084783)--(7.6432,0.080516)--(7.6884,0.076453)--(7.7337,0.072586)--(7.7789,0.068904)--(7.8241,0.065400)--(7.8693,0.062066)--(7.9146,0.058895)--(7.9598,0.055878)--(8.0050,0.053008)--(8.0502,0.050280)--(8.0955,0.047686)--(8.1407,0.045220)--(8.1859,0.042877)--(8.2312,0.040650)--(8.2764,0.038534)--(8.3216,0.036523)--(8.3668,0.034614)--(8.4121,0.032801)--(8.4573,0.031078)--(8.5025,0.029443)--(8.5477,0.027891)--(8.5930,0.026418)--(8.6382,0.025020)--(8.6834,0.023693)--(8.7286,0.022434)--(8.7739,0.021240)--(8.8191,0.020107)--(8.8643,0.019033)--(8.9095,0.018014)--(8.9548,0.017047)--(9.0000,0.016132);
@@ -124,21 +124,21 @@
 \draw [color=blue] (4.8000,0)--(4.8424,0)--(4.8848,0)--(4.9273,0)--(4.9697,0)--(5.0121,0)--(5.0545,0)--(5.0970,0)--(5.1394,0)--(5.1818,0)--(5.2242,0)--(5.2667,0)--(5.3091,0)--(5.3515,0)--(5.3939,0)--(5.4364,0)--(5.4788,0)--(5.5212,0)--(5.5636,0)--(5.6061,0)--(5.6485,0)--(5.6909,0)--(5.7333,0)--(5.7758,0)--(5.8182,0)--(5.8606,0)--(5.9030,0)--(5.9455,0)--(5.9879,0)--(6.0303,0)--(6.0727,0)--(6.1152,0)--(6.1576,0)--(6.2000,0)--(6.2424,0)--(6.2849,0)--(6.3273,0)--(6.3697,0)--(6.4121,0)--(6.4545,0)--(6.4970,0)--(6.5394,0)--(6.5818,0)--(6.6242,0)--(6.6667,0)--(6.7091,0)--(6.7515,0)--(6.7939,0)--(6.8364,0)--(6.8788,0)--(6.9212,0)--(6.9636,0)--(7.0061,0)--(7.0485,0)--(7.0909,0)--(7.1333,0)--(7.1758,0)--(7.2182,0)--(7.2606,0)--(7.3030,0)--(7.3455,0)--(7.3879,0)--(7.4303,0)--(7.4727,0)--(7.5152,0)--(7.5576,0)--(7.6000,0)--(7.6424,0)--(7.6848,0)--(7.7273,0)--(7.7697,0)--(7.8121,0)--(7.8545,0)--(7.8970,0)--(7.9394,0)--(7.9818,0)--(8.0242,0)--(8.0667,0)--(8.1091,0)--(8.1515,0)--(8.1939,0)--(8.2364,0)--(8.2788,0)--(8.3212,0)--(8.3636,0)--(8.4061,0)--(8.4485,0)--(8.4909,0)--(8.5333,0)--(8.5758,0)--(8.6182,0)--(8.6606,0)--(8.7030,0)--(8.7455,0)--(8.7879,0)--(8.8303,0)--(8.8727,0)--(8.9151,0)--(8.9576,0)--(9.0000,0);
 \draw [] (4.80,0) -- (4.80,1.43);
 \draw [] (9.00,0.0161) -- (9.00,0);
-\draw (1.500000000,-0.3149246667) node {$ 5 $};
+\draw (1.5000,-0.31492) node {$ 5 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 10 $};
+\draw (3.0000,-0.31492) node {$ 10 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.500000000,-0.3149246667) node {$ 15 $};
+\draw (4.5000,-0.31492) node {$ 15 $};
 \draw [] (4.50,-0.100) -- (4.50,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 20 $};
+\draw (6.0000,-0.31492) node {$ 20 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.500000000,-0.3149246667) node {$ 25 $};
+\draw (7.5000,-0.31492) node {$ 25 $};
 \draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 30 $};
+\draw (9.0000,-0.31492) node {$ 30 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.3816666667,2.500000000) node {$ \frac{1}{20} $};
+\draw (-0.38167,2.5000) node {$ \frac{1}{20} $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.3816666667,5.000000000) node {$ \frac{1}{10} $};
+\draw (-0.38167,5.0000) node {$ \frac{1}{10} $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ChoixInfini.pstricks.recall b/src_phystricks/Fig_ChoixInfini.pstricks.recall
index 07e01a9e1..40c07f5d0 100644
--- a/src_phystricks/Fig_ChoixInfini.pstricks.recall
+++ b/src_phystricks/Fig_ChoixInfini.pstricks.recall
@@ -57,24 +57,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
 %DEFAULT
 \draw [color=blue] (-3.00,1.00) -- (3.00,1.00);
-\draw [color=blue]  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw [color=blue]  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -121,22 +121,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [color=blue] (-2.00,0) -- (2.00,2.00);
-\draw [color=blue]  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw [color=blue]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CoinPasVar.pstricks.recall b/src_phystricks/Fig_CoinPasVar.pstricks.recall
index 92a1bf5a8..471cee114 100644
--- a/src_phystricks/Fig_CoinPasVar.pstricks.recall
+++ b/src_phystricks/Fig_CoinPasVar.pstricks.recall
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [color=blue] (-2.00,0) -- (0,2.00);
 \draw [color=blue] (2.00,0) -- (0,2.00);
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (0.2372415115,2.195418678) node {\( N\)};
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.215780345,1.210337511) node {\( t_1\)};
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.215780345,1.210337511) node {\( t_2\)};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (0.23724,2.1954) node {\( N\)};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.2158,1.2103) node {\( t_1\)};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.2158,1.2103) node {\( t_2\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ContourGreen.pstricks.recall b/src_phystricks/Fig_ContourGreen.pstricks.recall
index 8fe9b877d..6990892da 100644
--- a/src_phystricks/Fig_ContourGreen.pstricks.recall
+++ b/src_phystricks/Fig_ContourGreen.pstricks.recall
@@ -66,14 +66,14 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.246812693,0.6811368787) -- (1.245451007,0.6910437354);
-\draw [,->,>=latex] (0.3575977601,1.218773069) -- (0.3480435445,1.215820618);
-\draw [,->,>=latex] (-0.2777141985,0.5083524306) -- (-0.2808616227,0.4988606594);
-\draw [,->,>=latex] (-0.7003261911,0.2054813022) -- (-0.7094196719,0.2013209407);
-\draw [,->,>=latex] (-1.246812693,-0.6811368787) -- (-1.245451007,-0.6910437354);
-\draw [,->,>=latex] (-0.3575977601,-1.218773069) -- (-0.3480435445,-1.215820618);
-\draw [,->,>=latex] (0.2777141985,-0.5083524306) -- (0.2808616227,-0.4988606594);
-\draw [,->,>=latex] (0.7003261912,-0.2054813022) -- (0.7094196719,-0.2013209407);
+\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
+\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
+\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
+\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
+\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
+\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
+\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
+\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ContourSqL.pstricks.recall b/src_phystricks/Fig_ContourSqL.pstricks.recall
index 99087b680..766a75511 100644
--- a/src_phystricks/Fig_ContourSqL.pstricks.recall
+++ b/src_phystricks/Fig_ContourSqL.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -79,14 +79,14 @@
          hatchthickness=0.4pt}
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 \draw [color=blue] (0,0)--(0.0303,0)--(0.0606,0.00122)--(0.0909,0.00275)--(0.121,0.00490)--(0.152,0.00765)--(0.182,0.0110)--(0.212,0.0150)--(0.242,0.0196)--(0.273,0.0248)--(0.303,0.0306)--(0.333,0.0370)--(0.364,0.0441)--(0.394,0.0517)--(0.424,0.0600)--(0.455,0.0689)--(0.485,0.0784)--(0.515,0.0885)--(0.545,0.0992)--(0.576,0.110)--(0.606,0.122)--(0.636,0.135)--(0.667,0.148)--(0.697,0.162)--(0.727,0.176)--(0.758,0.191)--(0.788,0.207)--(0.818,0.223)--(0.849,0.240)--(0.879,0.257)--(0.909,0.275)--(0.939,0.294)--(0.970,0.313)--(1.00,0.333)--(1.03,0.354)--(1.06,0.375)--(1.09,0.397)--(1.12,0.419)--(1.15,0.442)--(1.18,0.466)--(1.21,0.490)--(1.24,0.515)--(1.27,0.540)--(1.30,0.566)--(1.33,0.593)--(1.36,0.620)--(1.39,0.648)--(1.42,0.676)--(1.45,0.705)--(1.48,0.735)--(1.52,0.765)--(1.55,0.796)--(1.58,0.828)--(1.61,0.860)--(1.64,0.893)--(1.67,0.926)--(1.70,0.960)--(1.73,0.995)--(1.76,1.03)--(1.79,1.07)--(1.82,1.10)--(1.85,1.14)--(1.88,1.18)--(1.91,1.21)--(1.94,1.25)--(1.97,1.29)--(2.00,1.33)--(2.03,1.37)--(2.06,1.42)--(2.09,1.46)--(2.12,1.50)--(2.15,1.54)--(2.18,1.59)--(2.21,1.63)--(2.24,1.68)--(2.27,1.72)--(2.30,1.77)--(2.33,1.81)--(2.36,1.86)--(2.39,1.91)--(2.42,1.96)--(2.45,2.01)--(2.48,2.06)--(2.52,2.11)--(2.55,2.16)--(2.58,2.21)--(2.61,2.26)--(2.64,2.32)--(2.67,2.37)--(2.70,2.42)--(2.73,2.48)--(2.76,2.53)--(2.79,2.59)--(2.82,2.65)--(2.85,2.70)--(2.88,2.76)--(2.91,2.82)--(2.94,2.88)--(2.97,2.94)--(3.00,3.00);
-\draw [,->,>=latex] (1.500000000,0.7500000000) -- (1.521213203,0.7712132034);
+\draw [,->,>=latex] (1.5000,0.75000) -- (1.5212,0.77121);
 \draw [] (0,0) -- (0,0);
 \draw [] (3.00,3.00) -- (3.00,3.00);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall b/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall
index ea9d6a1ea..e2b22fef2 100644
--- a/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall
+++ b/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall
@@ -65,31 +65,31 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red,->,>=latex] (2.000000000,0) -- (2.707106781,0.7071067812);
-\draw [color=green,->,>=latex] (2.000000000,0) -- (2.707106781,-0.7071067812);
-\draw [color=red,->,>=latex] (2.341413528,1.827257052) -- (1.861553169,2.704601930);
-\draw [color=green,->,>=latex] (2.341413528,1.827257052) -- (3.218758407,2.307117411);
-\draw [color=red,->,>=latex] (0.6118985586,2.401762329) -- (-0.3215114110,2.042950640);
-\draw [color=green,->,>=latex] (0.6118985586,2.401762329) -- (0.2530868690,3.335172299);
-\draw [color=red,->,>=latex] (-0.5954791539,0.9010655917) -- (-0.9467052076,-0.03522509989);
-\draw [color=green,->,>=latex] (-0.5954791539,0.9010655917) -- (-1.531769845,1.252291645);
-\draw [color=red,->,>=latex] (-2.176644828,-0.2044632369) -- (-2.772636306,-1.007453996);
-\draw [color=green,->,>=latex] (-2.176644828,-0.2044632369) -- (-2.979635587,0.3915282409);
-\draw [color=red,->,>=latex] (-2.172784878,-2.066762532) -- (-1.508100860,-2.813887125);
-\draw [color=green,->,>=latex] (-2.172784878,-2.066762532) -- (-2.919909470,-2.731446550);
-\draw [color=red,->,>=latex] (-0.3430145542,-2.269729649) -- (0.5125413216,-1.752019167);
-\draw [color=green,->,>=latex] (-0.3430145542,-2.269729649) -- (0.1746959280,-3.125285525);
-\draw [color=red,->,>=latex] (0.7768218803,-0.6530736828) -- (1.631807400,-0.1344218137);
-\draw [color=green,->,>=latex] (0.7768218803,-0.6530736828) -- (1.295473749,-1.508059202);
+\draw [color=red,->,>=latex] (2.0000,0) -- (2.7071,0.70711);
+\draw [color=green,->,>=latex] (2.0000,0) -- (2.7071,-0.70711);
+\draw [color=red,->,>=latex] (2.3414,1.8273) -- (1.8616,2.7046);
+\draw [color=green,->,>=latex] (2.3414,1.8273) -- (3.2188,2.3071);
+\draw [color=red,->,>=latex] (0.61190,2.4018) -- (-0.32151,2.0429);
+\draw [color=green,->,>=latex] (0.61190,2.4018) -- (0.25309,3.3352);
+\draw [color=red,->,>=latex] (-0.59548,0.90107) -- (-0.94670,-0.035225);
+\draw [color=green,->,>=latex] (-0.59548,0.90107) -- (-1.5318,1.2523);
+\draw [color=red,->,>=latex] (-2.1766,-0.20446) -- (-2.7726,-1.0075);
+\draw [color=green,->,>=latex] (-2.1766,-0.20446) -- (-2.9796,0.39153);
+\draw [color=red,->,>=latex] (-2.1728,-2.0668) -- (-1.5081,-2.8139);
+\draw [color=green,->,>=latex] (-2.1728,-2.0668) -- (-2.9199,-2.7314);
+\draw [color=red,->,>=latex] (-0.34301,-2.2697) -- (0.51254,-1.7520);
+\draw [color=green,->,>=latex] (-0.34301,-2.2697) -- (0.17470,-3.1253);
+\draw [color=red,->,>=latex] (0.77682,-0.65307) -- (1.6318,-0.13442);
+\draw [color=green,->,>=latex] (0.77682,-0.65307) -- (1.2955,-1.5081);
 \draw [color=blue] (2.000,0)--(2.122,0.1349)--(2.233,0.2850)--(2.329,0.4488)--(2.407,0.6244)--(2.463,0.8091)--(2.497,0.9998)--(2.507,1.193)--(2.490,1.386)--(2.448,1.573)--(2.379,1.752)--(2.286,1.919)--(2.170,2.069)--(2.033,2.201)--(1.878,2.312)--(1.708,2.399)--(1.527,2.461)--(1.338,2.497)--(1.145,2.507)--(0.9517,2.491)--(0.7622,2.451)--(0.5796,2.389)--(0.4067,2.306)--(0.2461,2.207)--(0.09969,2.093)--(-0.03123,1.968)--(-0.1460,1.836)--(-0.2445,1.701)--(-0.3275,1.565)--(-0.3961,1.433)--(-0.4520,1.306)--(-0.4973,1.187)--(-0.5348,1.079)--(-0.5670,0.9820)--(-0.5969,0.8972)--(-0.6275,0.8245)--(-0.6615,0.7634)--(-0.7016,0.7128)--(-0.7499,0.6709)--(-0.8082,0.6356)--(-0.8779,0.6044)--(-0.9597,0.5746)--(-1.054,0.5432)--(-1.159,0.5073)--(-1.275,0.4642)--(-1.400,0.4111)--(-1.532,0.3459)--(-1.667,0.2667)--(-1.803,0.1721)--(-1.936,0.06144)--(-2.062,-0.06547)--(-2.179,-0.2081)--(-2.283,-0.3653)--(-2.370,-0.5353)--(-2.438,-0.7158)--(-2.483,-0.9039)--(-2.505,-1.096)--(-2.502,-1.290)--(-2.472,-1.480)--(-2.417,-1.664)--(-2.336,-1.837)--(-2.231,-1.996)--(-2.104,-2.138)--(-1.958,-2.260)--(-1.795,-2.359)--(-1.619,-2.433)--(-1.433,-2.482)--(-1.242,-2.505)--(-1.048,-2.502)--(-0.8563,-2.474)--(-0.6698,-2.423)--(-0.4917,-2.350)--(-0.3247,-2.259)--(-0.1710,-2.151)--(-0.03224,-2.031)--(0.09064,-1.903)--(0.1973,-1.769)--(0.2879,-1.633)--(0.3635,-1.498)--(0.4255,-1.368)--(0.4758,-1.245)--(0.5169,-1.132)--(0.5513,-1.029)--(0.5821,-0.9381)--(0.6120,-0.8594)--(0.6439,-0.7926)--(0.6806,-0.7369)--(0.7246,-0.6909)--(0.7777,-0.6525)--(0.8416,-0.6196)--(0.9173,-0.5895)--(1.005,-0.5593)--(1.105,-0.5260)--(1.216,-0.4868)--(1.337,-0.4391)--(1.465,-0.3802)--(1.599,-0.3082)--(1.735,-0.2214)--(1.870,-0.1188)--(2.000,0);
-\draw [,->,>=latex] (2.000000000,0) -- (2.014142136,0.01414213562);
-\draw [,->,>=latex] (2.341413528,1.827257052) -- (2.331816321,1.844803950);
-\draw [,->,>=latex] (0.6118985586,2.401762329) -- (0.5932303592,2.394586095);
-\draw [,->,>=latex] (-0.5954791539,0.9010655917) -- (-0.6025036750,0.8823397779);
-\draw [,->,>=latex] (-2.176644828,-0.2044632369) -- (-2.188564658,-0.2205230521);
-\draw [,->,>=latex] (-2.172784878,-2.066762532) -- (-2.159491197,-2.081705024);
-\draw [,->,>=latex] (-0.3430145542,-2.269729649) -- (-0.3259034367,-2.259375439);
-\draw [,->,>=latex] (0.7768218803,-0.6530736828) -- (0.7939215907,-0.6427006454);
+\draw [,->,>=latex] (2.0000,0) -- (2.0141,0.014142);
+\draw [,->,>=latex] (2.3414,1.8273) -- (2.3318,1.8448);
+\draw [,->,>=latex] (0.61190,2.4018) -- (0.59323,2.3946);
+\draw [,->,>=latex] (-0.59548,0.90107) -- (-0.60250,0.88234);
+\draw [,->,>=latex] (-2.1766,-0.20446) -- (-2.1886,-0.22052);
+\draw [,->,>=latex] (-2.1728,-2.0668) -- (-2.1595,-2.0817);
+\draw [,->,>=latex] (-0.34301,-2.2697) -- (-0.32590,-2.2594);
+\draw [,->,>=latex] (0.77682,-0.65307) -- (0.79392,-0.64270);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CoordPolaires.pstricks.recall b/src_phystricks/Fig_CoordPolaires.pstricks.recall
index 37c5a05be..827399ee2 100644
--- a/src_phystricks/Fig_CoordPolaires.pstricks.recall
+++ b/src_phystricks/Fig_CoordPolaires.pstricks.recall
@@ -83,20 +83,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
-\draw (1.523347667,2.000000000) node {$(x,y)$};
-\draw [,->,>=latex] (0,0) -- (1.000000000,2.000000000);
-\draw (0.6845247985,0.4139141375) node {$\theta$};
+\draw (1.5233,2.0000) node {$(x,y)$};
+\draw [,->,>=latex] (0,0) -- (1.0000,2.0000);
+\draw (0.68452,0.41391) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00559)--(0.500,0.0112)--(0.500,0.0168)--(0.500,0.0224)--(0.499,0.0279)--(0.499,0.0335)--(0.498,0.0391)--(0.498,0.0447)--(0.497,0.0502)--(0.497,0.0558)--(0.496,0.0614)--(0.496,0.0669)--(0.495,0.0724)--(0.494,0.0780)--(0.493,0.0835)--(0.492,0.0890)--(0.491,0.0945)--(0.490,0.100)--(0.489,0.105)--(0.488,0.111)--(0.486,0.116)--(0.485,0.122)--(0.484,0.127)--(0.482,0.133)--(0.481,0.138)--(0.479,0.143)--(0.477,0.149)--(0.476,0.154)--(0.474,0.159)--(0.472,0.165)--(0.470,0.170)--(0.468,0.175)--(0.466,0.180)--(0.464,0.186)--(0.462,0.191)--(0.460,0.196)--(0.458,0.201)--(0.456,0.206)--(0.453,0.211)--(0.451,0.216)--(0.448,0.221)--(0.446,0.226)--(0.443,0.231)--(0.441,0.236)--(0.438,0.241)--(0.435,0.246)--(0.432,0.251)--(0.430,0.256)--(0.427,0.260)--(0.424,0.265)--(0.421,0.270)--(0.418,0.275)--(0.415,0.279)--(0.412,0.284)--(0.408,0.289)--(0.405,0.293)--(0.402,0.298)--(0.398,0.302)--(0.395,0.306)--(0.392,0.311)--(0.388,0.315)--(0.385,0.320)--(0.381,0.324)--(0.377,0.328)--(0.374,0.332)--(0.370,0.336)--(0.366,0.341)--(0.362,0.345)--(0.358,0.349)--(0.354,0.353)--(0.350,0.357)--(0.346,0.360)--(0.342,0.364)--(0.338,0.368)--(0.334,0.372)--(0.330,0.376)--(0.326,0.379)--(0.322,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.304,0.397)--(0.300,0.400)--(0.295,0.404)--(0.291,0.407)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416)--(0.272,0.419)--(0.267,0.422)--(0.263,0.425)--(0.258,0.428)--(0.253,0.431)--(0.248,0.434)--(0.243,0.437)--(0.238,0.439)--(0.234,0.442)--(0.229,0.445)--(0.224,0.447);
-\draw (0.2337087285,1.168018886) node {$r$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (0.23371,1.1680) node {$r$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CornetGlace.pstricks.recall b/src_phystricks/Fig_CornetGlace.pstricks.recall
index 38fbfdeac..a8a07a6d4 100644
--- a/src_phystricks/Fig_CornetGlace.pstricks.recall
+++ b/src_phystricks/Fig_CornetGlace.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.700000000,0) -- (1.700000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.700000000);
+\draw [,->,>=latex] (-1.7000,0) -- (1.7000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.7000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -87,11 +87,11 @@
 \draw [color=blue] (-1.200,1.200)--(-1.188,1.188)--(-1.176,1.176)--(-1.164,1.164)--(-1.152,1.152)--(-1.139,1.139)--(-1.127,1.127)--(-1.115,1.115)--(-1.103,1.103)--(-1.091,1.091)--(-1.079,1.079)--(-1.067,1.067)--(-1.055,1.055)--(-1.042,1.042)--(-1.030,1.030)--(-1.018,1.018)--(-1.006,1.006)--(-0.9939,0.9939)--(-0.9818,0.9818)--(-0.9697,0.9697)--(-0.9576,0.9576)--(-0.9455,0.9455)--(-0.9333,0.9333)--(-0.9212,0.9212)--(-0.9091,0.9091)--(-0.8970,0.8970)--(-0.8848,0.8848)--(-0.8727,0.8727)--(-0.8606,0.8606)--(-0.8485,0.8485)--(-0.8364,0.8364)--(-0.8242,0.8242)--(-0.8121,0.8121)--(-0.8000,0.8000)--(-0.7879,0.7879)--(-0.7758,0.7758)--(-0.7636,0.7636)--(-0.7515,0.7515)--(-0.7394,0.7394)--(-0.7273,0.7273)--(-0.7151,0.7151)--(-0.7030,0.7030)--(-0.6909,0.6909)--(-0.6788,0.6788)--(-0.6667,0.6667)--(-0.6545,0.6545)--(-0.6424,0.6424)--(-0.6303,0.6303)--(-0.6182,0.6182)--(-0.6061,0.6061)--(-0.5939,0.5939)--(-0.5818,0.5818)--(-0.5697,0.5697)--(-0.5576,0.5576)--(-0.5455,0.5455)--(-0.5333,0.5333)--(-0.5212,0.5212)--(-0.5091,0.5091)--(-0.4970,0.4970)--(-0.4848,0.4848)--(-0.4727,0.4727)--(-0.4606,0.4606)--(-0.4485,0.4485)--(-0.4364,0.4364)--(-0.4242,0.4242)--(-0.4121,0.4121)--(-0.4000,0.4000)--(-0.3879,0.3879)--(-0.3758,0.3758)--(-0.3636,0.3636)--(-0.3515,0.3515)--(-0.3394,0.3394)--(-0.3273,0.3273)--(-0.3152,0.3152)--(-0.3030,0.3030)--(-0.2909,0.2909)--(-0.2788,0.2788)--(-0.2667,0.2667)--(-0.2545,0.2545)--(-0.2424,0.2424)--(-0.2303,0.2303)--(-0.2182,0.2182)--(-0.2061,0.2061)--(-0.1939,0.1939)--(-0.1818,0.1818)--(-0.1697,0.1697)--(-0.1576,0.1576)--(-0.1455,0.1455)--(-0.1333,0.1333)--(-0.1212,0.1212)--(-0.1091,0.1091)--(-0.09697,0.09697)--(-0.08485,0.08485)--(-0.07273,0.07273)--(-0.06061,0.06061)--(-0.04848,0.04848)--(-0.03636,0.03636)--(-0.02424,0.02424)--(-0.01212,0.01212)--(0,0);
 
 \draw [color=blue] (0,0)--(0.01212,0.01212)--(0.02424,0.02424)--(0.03636,0.03636)--(0.04848,0.04848)--(0.06061,0.06061)--(0.07273,0.07273)--(0.08485,0.08485)--(0.09697,0.09697)--(0.1091,0.1091)--(0.1212,0.1212)--(0.1333,0.1333)--(0.1455,0.1455)--(0.1576,0.1576)--(0.1697,0.1697)--(0.1818,0.1818)--(0.1939,0.1939)--(0.2061,0.2061)--(0.2182,0.2182)--(0.2303,0.2303)--(0.2424,0.2424)--(0.2545,0.2545)--(0.2667,0.2667)--(0.2788,0.2788)--(0.2909,0.2909)--(0.3030,0.3030)--(0.3152,0.3152)--(0.3273,0.3273)--(0.3394,0.3394)--(0.3515,0.3515)--(0.3636,0.3636)--(0.3758,0.3758)--(0.3879,0.3879)--(0.4000,0.4000)--(0.4121,0.4121)--(0.4242,0.4242)--(0.4364,0.4364)--(0.4485,0.4485)--(0.4606,0.4606)--(0.4727,0.4727)--(0.4848,0.4848)--(0.4970,0.4970)--(0.5091,0.5091)--(0.5212,0.5212)--(0.5333,0.5333)--(0.5455,0.5455)--(0.5576,0.5576)--(0.5697,0.5697)--(0.5818,0.5818)--(0.5939,0.5939)--(0.6061,0.6061)--(0.6182,0.6182)--(0.6303,0.6303)--(0.6424,0.6424)--(0.6545,0.6545)--(0.6667,0.6667)--(0.6788,0.6788)--(0.6909,0.6909)--(0.7030,0.7030)--(0.7151,0.7151)--(0.7273,0.7273)--(0.7394,0.7394)--(0.7515,0.7515)--(0.7636,0.7636)--(0.7758,0.7758)--(0.7879,0.7879)--(0.8000,0.8000)--(0.8121,0.8121)--(0.8242,0.8242)--(0.8364,0.8364)--(0.8485,0.8485)--(0.8606,0.8606)--(0.8727,0.8727)--(0.8848,0.8848)--(0.8970,0.8970)--(0.9091,0.9091)--(0.9212,0.9212)--(0.9333,0.9333)--(0.9455,0.9455)--(0.9576,0.9576)--(0.9697,0.9697)--(0.9818,0.9818)--(0.9939,0.9939)--(1.006,1.006)--(1.018,1.018)--(1.030,1.030)--(1.042,1.042)--(1.055,1.055)--(1.067,1.067)--(1.079,1.079)--(1.091,1.091)--(1.103,1.103)--(1.115,1.115)--(1.127,1.127)--(1.139,1.139)--(1.152,1.152)--(1.164,1.164)--(1.176,1.176)--(1.188,1.188)--(1.200,1.200);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CourbeRectifiable.pstricks.recall b/src_phystricks/Fig_CourbeRectifiable.pstricks.recall
index dbfe50a3c..c87a9cf28 100644
--- a/src_phystricks/Fig_CourbeRectifiable.pstricks.recall
+++ b/src_phystricks/Fig_CourbeRectifiable.pstricks.recall
@@ -89,16 +89,16 @@
 \draw [color=red] (-13.6,1.40) -- (-12.3,0);
 \draw [color=red] (-12.3,0) -- (-10.2,-1.40);
 \draw [color=red] (-10.2,-1.40) -- (-7.56,0);
-\draw []  (-14.00000000,0) node [rotate=0] {$\bullet$};
-\draw (-14.59160333,0) node {$\gamma(t_{0})$};
-\draw []  (-13.56477390,1.400000000) node [rotate=0] {$\bullet$};
-\draw (-13.56477390,1.782455000) node {$\gamma(t_{1})$};
-\draw []  (-12.28615587,0) node [rotate=0] {$\bullet$};
-\draw (-12.83675930,-0.3037767567) node {$\gamma(t_{2})$};
-\draw []  (-10.24364416,-1.400000000) node [rotate=0] {$\bullet$};
-\draw (-10.24364416,-1.782455000) node {$\gamma(t_{3})$};
-\draw []  (-7.564232282,0) node [rotate=0] {$\bullet$};
-\draw (-8.075495931,0.3427092006) node {$\gamma(t_{4})$};
+\draw []  (-14.000,0) node [rotate=0] {$\bullet$};
+\draw (-14.592,0) node {$\gamma(t_{0})$};
+\draw []  (-13.565,1.4000) node [rotate=0] {$\bullet$};
+\draw (-13.565,1.7825) node {$\gamma(t_{1})$};
+\draw []  (-12.286,0) node [rotate=0] {$\bullet$};
+\draw (-12.837,-0.30378) node {$\gamma(t_{2})$};
+\draw []  (-10.244,-1.4000) node [rotate=0] {$\bullet$};
+\draw (-10.244,-1.7825) node {$\gamma(t_{3})$};
+\draw []  (-7.5642,0) node [rotate=0] {$\bullet$};
+\draw (-8.0755,0.34271) node {$\gamma(t_{4})$};
 \draw [color=blue] (-14.000,0)--(-13.999,0.088794)--(-13.997,0.17723)--(-13.994,0.26495)--(-13.989,0.35161)--(-13.982,0.43685)--(-13.974,0.52033)--(-13.965,0.60171)--(-13.954,0.68068)--(-13.942,0.75690)--(-13.929,0.83007)--(-13.914,0.89990)--(-13.897,0.96611)--(-13.879,1.0284)--(-13.860,1.0866)--(-13.840,1.1404)--(-13.818,1.1896)--(-13.794,1.2340)--(-13.769,1.2735)--(-13.743,1.3078)--(-13.715,1.3369)--(-13.686,1.3605)--(-13.656,1.3787)--(-13.624,1.3914)--(-13.591,1.3984)--(-13.556,1.3998)--(-13.520,1.3956)--(-13.483,1.3857)--(-13.444,1.3703)--(-13.404,1.3494)--(-13.362,1.3230)--(-13.319,1.2913)--(-13.275,1.2544)--(-13.229,1.2124)--(-13.182,1.1656)--(-13.134,1.1141)--(-13.085,1.0581)--(-13.034,0.99777)--(-12.981,0.93348)--(-12.928,0.86542)--(-12.873,0.79388)--(-12.816,0.71915)--(-12.759,0.64152)--(-12.700,0.56130)--(-12.640,0.47883)--(-12.578,0.39443)--(-12.516,0.30843)--(-12.452,0.22120)--(-12.386,0.13308)--(-12.320,0.044419)--(-12.252,-0.044419)--(-12.183,-0.13308)--(-12.113,-0.22120)--(-12.041,-0.30843)--(-11.968,-0.39443)--(-11.895,-0.47883)--(-11.819,-0.56130)--(-11.743,-0.64152)--(-11.665,-0.71915)--(-11.587,-0.79388)--(-11.507,-0.86542)--(-11.425,-0.93348)--(-11.343,-0.99777)--(-11.260,-1.0581)--(-11.175,-1.1141)--(-11.089,-1.1656)--(-11.002,-1.2124)--(-10.914,-1.2544)--(-10.825,-1.2913)--(-10.735,-1.3230)--(-10.644,-1.3494)--(-10.551,-1.3703)--(-10.458,-1.3857)--(-10.363,-1.3956)--(-10.268,-1.3998)--(-10.171,-1.3984)--(-10.073,-1.3914)--(-9.9746,-1.3787)--(-9.8749,-1.3605)--(-9.7742,-1.3369)--(-9.6724,-1.3078)--(-9.5697,-1.2735)--(-9.4660,-1.2340)--(-9.3613,-1.1896)--(-9.2557,-1.1404)--(-9.1491,-1.0866)--(-9.0416,-1.0284)--(-8.9332,-0.96611)--(-8.8239,-0.89990)--(-8.7136,-0.83007)--(-8.6025,-0.75690)--(-8.4905,-0.68068)--(-8.3776,-0.60171)--(-8.2639,-0.52033)--(-8.1493,-0.43685)--(-8.0339,-0.35161)--(-7.9177,-0.26495)--(-7.8007,-0.17723)--(-7.6828,-0.088794)--(-7.5642,0);
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CouroneExam.pstricks.recall b/src_phystricks/Fig_CouroneExam.pstricks.recall
index e90d72afd..33009d868 100644
--- a/src_phystricks/Fig_CouroneExam.pstricks.recall
+++ b/src_phystricks/Fig_CouroneExam.pstricks.recall
@@ -71,21 +71,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \fill [color=lightgray] (0,1.00) -- (0.0159,1.00) -- (0.0317,1.00) -- (0.0476,0.999) -- (0.0634,0.998) -- (0.0792,0.997) -- (0.0951,0.995) -- (0.111,0.994) -- (0.127,0.992) -- (0.142,0.990) -- (0.158,0.987) -- (0.174,0.985) -- (0.189,0.982) -- (0.205,0.979) -- (0.220,0.975) -- (0.236,0.972) -- (0.251,0.968) -- (0.266,0.964) -- (0.282,0.959) -- (0.297,0.955) -- (0.312,0.950) -- (0.327,0.945) -- (0.342,0.940) -- (0.357,0.934) -- (0.372,0.928) -- (0.386,0.922) -- (0.401,0.916) -- (0.415,0.910) -- (0.430,0.903) -- (0.444,0.896) -- (0.458,0.889) -- (0.472,0.881) -- (0.486,0.874) -- (0.500,0.866) -- (0.514,0.858) -- (0.527,0.850) -- (0.541,0.841) -- (0.554,0.833) -- (0.567,0.824) -- (0.580,0.815) -- (0.593,0.805) -- (0.606,0.796) -- (0.618,0.786) -- (0.631,0.776) -- (0.643,0.766) -- (0.655,0.756) -- (0.667,0.745) -- (0.679,0.735) -- (0.690,0.724) -- (0.701,0.713) -- (0.713,0.701) -- (0.724,0.690) -- (0.735,0.679) -- (0.745,0.667) -- (0.756,0.655) -- (0.766,0.643) -- (0.776,0.631) -- (0.786,0.618) -- (0.796,0.606) -- (0.805,0.593) -- (0.815,0.580) -- (0.824,0.567) -- (0.833,0.554) -- (0.841,0.541) -- (0.850,0.527) -- (0.858,0.514) -- (0.866,0.500) -- (0.874,0.486) -- (0.881,0.472) -- (0.889,0.458) -- (0.896,0.444) -- (0.903,0.430) -- (0.910,0.415) -- (0.916,0.401) -- (0.922,0.386) -- (0.928,0.372) -- (0.934,0.357) -- (0.940,0.342) -- (0.945,0.327) -- (0.950,0.312) -- (0.955,0.297) -- (0.959,0.282) -- (0.964,0.266) -- (0.968,0.251) -- (0.972,0.236) -- (0.975,0.220) -- (0.979,0.205) -- (0.982,0.189) -- (0.985,0.174) -- (0.987,0.158) -- (0.990,0.142) -- (0.992,0.127) -- (0.994,0.111) -- (0.995,0.0951) -- (0.997,0.0792) -- (0.998,0.0634) -- (0.999,0.0476) -- (1.00,0.0317) -- (1.00,0.0159) -- (1.00,0) -- (1.00,0) -- (2.00,0) -- (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (0,1.00) --  cycle;
 \draw [color=blue] (1.00,0)--(1.00,0.0159)--(1.00,0.0317)--(0.999,0.0476)--(0.998,0.0634)--(0.997,0.0792)--(0.995,0.0951)--(0.994,0.111)--(0.992,0.127)--(0.990,0.142)--(0.987,0.158)--(0.985,0.174)--(0.982,0.189)--(0.979,0.205)--(0.975,0.220)--(0.972,0.236)--(0.968,0.251)--(0.964,0.266)--(0.959,0.282)--(0.955,0.297)--(0.950,0.312)--(0.945,0.327)--(0.940,0.342)--(0.934,0.357)--(0.928,0.372)--(0.922,0.386)--(0.916,0.401)--(0.910,0.415)--(0.903,0.430)--(0.896,0.444)--(0.889,0.458)--(0.881,0.472)--(0.874,0.486)--(0.866,0.500)--(0.858,0.514)--(0.850,0.527)--(0.841,0.541)--(0.833,0.554)--(0.824,0.567)--(0.815,0.580)--(0.805,0.593)--(0.796,0.606)--(0.786,0.618)--(0.776,0.631)--(0.766,0.643)--(0.756,0.655)--(0.745,0.667)--(0.735,0.679)--(0.724,0.690)--(0.713,0.701)--(0.701,0.713)--(0.690,0.724)--(0.679,0.735)--(0.667,0.745)--(0.655,0.756)--(0.643,0.766)--(0.631,0.776)--(0.618,0.786)--(0.606,0.796)--(0.593,0.805)--(0.580,0.815)--(0.567,0.824)--(0.554,0.833)--(0.541,0.841)--(0.527,0.850)--(0.514,0.858)--(0.500,0.866)--(0.486,0.874)--(0.472,0.881)--(0.458,0.889)--(0.444,0.896)--(0.430,0.903)--(0.415,0.910)--(0.401,0.916)--(0.386,0.922)--(0.372,0.928)--(0.357,0.934)--(0.342,0.940)--(0.327,0.945)--(0.312,0.950)--(0.297,0.955)--(0.282,0.959)--(0.266,0.964)--(0.251,0.968)--(0.236,0.972)--(0.220,0.975)--(0.205,0.979)--(0.189,0.982)--(0.174,0.985)--(0.158,0.987)--(0.142,0.990)--(0.127,0.992)--(0.111,0.994)--(0.0951,0.995)--(0.0792,0.997)--(0.0634,0.998)--(0.0476,0.999)--(0.0317,1.00)--(0.0159,1.00)--(0,1.00);
 \draw [color=blue] (2.000,0)--(2.000,0.03173)--(1.999,0.06346)--(1.998,0.09516)--(1.996,0.1268)--(1.994,0.1585)--(1.991,0.1901)--(1.988,0.2217)--(1.984,0.2532)--(1.980,0.2846)--(1.975,0.3160)--(1.970,0.3473)--(1.964,0.3785)--(1.958,0.4096)--(1.951,0.4406)--(1.944,0.4715)--(1.936,0.5023)--(1.928,0.5330)--(1.919,0.5635)--(1.910,0.5938)--(1.900,0.6241)--(1.890,0.6541)--(1.879,0.6840)--(1.868,0.7138)--(1.857,0.7433)--(1.845,0.7727)--(1.832,0.8019)--(1.819,0.8308)--(1.806,0.8596)--(1.792,0.8881)--(1.778,0.9165)--(1.763,0.9445)--(1.748,0.9724)--(1.732,1.000)--(1.716,1.027)--(1.699,1.054)--(1.683,1.081)--(1.665,1.108)--(1.647,1.134)--(1.629,1.160)--(1.611,1.186)--(1.592,1.211)--(1.572,1.236)--(1.552,1.261)--(1.532,1.286)--(1.512,1.310)--(1.491,1.334)--(1.469,1.357)--(1.447,1.380)--(1.425,1.403)--(1.403,1.425)--(1.380,1.447)--(1.357,1.469)--(1.334,1.491)--(1.310,1.512)--(1.286,1.532)--(1.261,1.552)--(1.236,1.572)--(1.211,1.592)--(1.186,1.611)--(1.160,1.629)--(1.134,1.647)--(1.108,1.665)--(1.081,1.683)--(1.054,1.699)--(1.027,1.716)--(1.000,1.732)--(0.9724,1.748)--(0.9445,1.763)--(0.9165,1.778)--(0.8881,1.792)--(0.8596,1.806)--(0.8308,1.819)--(0.8019,1.832)--(0.7727,1.845)--(0.7433,1.857)--(0.7138,1.868)--(0.6840,1.879)--(0.6541,1.890)--(0.6241,1.900)--(0.5938,1.910)--(0.5635,1.919)--(0.5330,1.928)--(0.5023,1.936)--(0.4715,1.944)--(0.4406,1.951)--(0.4096,1.958)--(0.3785,1.964)--(0.3473,1.970)--(0.3160,1.975)--(0.2846,1.980)--(0.2532,1.984)--(0.2217,1.988)--(0.1901,1.991)--(0.1585,1.994)--(0.1268,1.996)--(0.09516,1.998)--(0.06346,1.999)--(0.03173,2.000)--(0,2.000);
 \draw [color=blue] (0,2.00) -- (0,1.00);
 \draw [color=blue] (1.00,0) -- (2.00,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall b/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall
index 2ab5032ab..401de7a4e 100644
--- a/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall
+++ b/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall
@@ -46,10 +46,10 @@
 
 \draw [color=brown,style=dashed] (2.000,0)--(2.000,0.02116)--(2.000,0.04231)--(1.999,0.06346)--(1.998,0.08460)--(1.997,0.1057)--(1.996,0.1268)--(1.995,0.1480)--(1.993,0.1690)--(1.991,0.1901)--(1.989,0.2112)--(1.986,0.2322)--(1.984,0.2532)--(1.981,0.2742)--(1.978,0.2951)--(1.975,0.3160)--(1.971,0.3369)--(1.968,0.3577)--(1.964,0.3785)--(1.960,0.3993)--(1.955,0.4200)--(1.951,0.4406)--(1.946,0.4612)--(1.941,0.4818)--(1.936,0.5023)--(1.930,0.5227)--(1.925,0.5431)--(1.919,0.5635)--(1.913,0.5837)--(1.907,0.6039)--(1.900,0.6241)--(1.893,0.6441)--(1.887,0.6641)--(1.879,0.6840)--(1.872,0.7039)--(1.865,0.7236)--(1.857,0.7433)--(1.849,0.7629)--(1.841,0.7824)--(1.832,0.8019)--(1.824,0.8212)--(1.815,0.8404)--(1.806,0.8596)--(1.797,0.8786)--(1.787,0.8976)--(1.778,0.9165)--(1.768,0.9352)--(1.758,0.9538)--(1.748,0.9724)--(1.737,0.9908)--(1.727,1.009)--(1.716,1.027)--(1.705,1.045)--(1.694,1.063)--(1.683,1.081)--(1.671,1.099)--(1.659,1.117)--(1.647,1.134)--(1.635,1.151)--(1.623,1.169)--(1.611,1.186)--(1.598,1.203)--(1.585,1.220)--(1.572,1.236)--(1.559,1.253)--(1.546,1.269)--(1.532,1.286)--(1.518,1.302)--(1.505,1.318)--(1.491,1.334)--(1.476,1.349)--(1.462,1.365)--(1.447,1.380)--(1.433,1.395)--(1.418,1.410)--(1.403,1.425)--(1.388,1.440)--(1.372,1.455)--(1.357,1.469)--(1.341,1.483)--(1.326,1.498)--(1.310,1.512)--(1.294,1.525)--(1.277,1.539)--(1.261,1.552)--(1.245,1.566)--(1.228,1.579)--(1.211,1.592)--(1.194,1.604)--(1.177,1.617)--(1.160,1.629)--(1.143,1.641)--(1.125,1.653)--(1.108,1.665)--(1.090,1.677)--(1.072,1.688)--(1.054,1.699)--(1.036,1.711)--(1.018,1.721)--(1.000,1.732);
 \draw [color=brown,style=dashed] (1.30,0.750) -- (3.03,1.75);
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (2.598076211,1.500000000);
-\draw (2.286902711,1.865758621) node {$e_{r}$};
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (1.232050808,1.866025404);
-\draw (1.607516675,2.186465271) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
+\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
+\draw (1.6075,2.1865) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -91,10 +91,10 @@
 
 \draw [color=brown,style=dashed] (-1.477,0.2605)--(-1.480,0.2448)--(-1.482,0.2292)--(-1.485,0.2135)--(-1.487,0.1978)--(-1.489,0.1820)--(-1.491,0.1663)--(-1.492,0.1505)--(-1.494,0.1347)--(-1.495,0.1189)--(-1.496,0.1031)--(-1.497,0.08722)--(-1.498,0.07137)--(-1.499,0.05552)--(-1.499,0.03966)--(-1.500,0.02380)--(-1.500,0.007933)--(-1.500,-0.007933)--(-1.500,-0.02380)--(-1.499,-0.03966)--(-1.499,-0.05552)--(-1.498,-0.07137)--(-1.497,-0.08722)--(-1.496,-0.1031)--(-1.495,-0.1189)--(-1.494,-0.1347)--(-1.492,-0.1505)--(-1.491,-0.1663)--(-1.489,-0.1820)--(-1.487,-0.1978)--(-1.485,-0.2135)--(-1.482,-0.2292)--(-1.480,-0.2448)--(-1.477,-0.2605)--(-1.474,-0.2761)--(-1.471,-0.2917)--(-1.468,-0.3072)--(-1.465,-0.3227)--(-1.461,-0.3382)--(-1.458,-0.3536)--(-1.454,-0.3690)--(-1.450,-0.3844)--(-1.446,-0.3997)--(-1.441,-0.4150)--(-1.437,-0.4302)--(-1.432,-0.4454)--(-1.428,-0.4605)--(-1.423,-0.4756)--(-1.417,-0.4906)--(-1.412,-0.5056)--(-1.407,-0.5205)--(-1.401,-0.5353)--(-1.395,-0.5501)--(-1.390,-0.5648)--(-1.384,-0.5795)--(-1.377,-0.5941)--(-1.371,-0.6087)--(-1.364,-0.6231)--(-1.358,-0.6375)--(-1.351,-0.6518)--(-1.344,-0.6661)--(-1.337,-0.6803)--(-1.330,-0.6944)--(-1.322,-0.7084)--(-1.315,-0.7224)--(-1.307,-0.7362)--(-1.299,-0.7500)--(-1.291,-0.7637)--(-1.283,-0.7773)--(-1.275,-0.7908)--(-1.266,-0.8043)--(-1.258,-0.8176)--(-1.249,-0.8309)--(-1.240,-0.8440)--(-1.231,-0.8571)--(-1.222,-0.8701)--(-1.213,-0.8830)--(-1.203,-0.8957)--(-1.194,-0.9084)--(-1.184,-0.9210)--(-1.174,-0.9335)--(-1.164,-0.9458)--(-1.154,-0.9581)--(-1.144,-0.9702)--(-1.134,-0.9823)--(-1.123,-0.9942)--(-1.113,-1.006)--(-1.102,-1.018)--(-1.091,-1.029)--(-1.080,-1.041)--(-1.069,-1.052)--(-1.058,-1.063)--(-1.047,-1.075)--(-1.035,-1.086)--(-1.024,-1.096)--(-1.012,-1.107)--(-1.000,-1.118)--(-0.9883,-1.128)--(-0.9763,-1.139)--(-0.9642,-1.149);
 \draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-2.349231552,-0.8550503583);
-\draw (-2.085452009,-1.242909145) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-1.067518788,-1.452722836);
-\draw (-0.6920529202,-1.132282968) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
+\draw (-2.0855,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
+\draw (-0.69205,-1.1323) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CycloideA.pstricks.recall b/src_phystricks/Fig_CycloideA.pstricks.recall
index 9f4df5aac..22616691c 100644
--- a/src_phystricks/Fig_CycloideA.pstricks.recall
+++ b/src_phystricks/Fig_CycloideA.pstricks.recall
@@ -115,39 +115,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (13.06637062,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.499496542);
+\draw [,->,>=latex] (-0.50000,0) -- (13.066,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.4995);
 %DEFAULT
 \draw [color=blue] (0,0)--(0,0.0080452)--(0.0027181,0.032051)--(0.0091366,0.071632)--(0.021535,0.12615)--(0.041757,0.19473)--(0.071519,0.27627)--(0.11238,0.36945)--(0.16574,0.47277)--(0.23277,0.58459)--(0.31443,0.70308)--(0.41146,0.82635)--(0.52433,0.95242)--(0.65327,1.0793)--(0.79826,1.2048)--(0.95899,1.3271)--(1.1349,1.4441)--(1.3253,1.5539)--(1.5290,1.6549)--(1.7450,1.7453)--(1.9716,1.8237)--(2.2074,1.8888)--(2.4505,1.9397)--(2.6992,1.9754)--(2.9513,1.9955)--(3.2051,1.9995)--(3.4583,1.9874)--(3.7089,1.9595)--(3.9551,1.9161)--(4.1947,1.8580)--(4.4261,1.7861)--(4.6476,1.7015)--(4.8576,1.6056)--(5.0548,1.5000)--(5.2381,1.3863)--(5.4065,1.2665)--(5.5594,1.1423)--(5.6964,1.0159)--(5.8173,0.88916)--(5.9222,0.76424)--(6.0115,0.64311)--(6.0857,0.52773)--(6.1458,0.41994)--(6.1927,0.32149)--(6.2278,0.23396)--(6.2526,0.15875)--(6.2687,0.097073)--(6.2779,0.049929)--(6.2820,0.018071)--(6.2831,0.0020133)--(6.2832,0.0020133)--(6.2843,0.018071)--(6.2885,0.049929)--(6.2977,0.097073)--(6.3137,0.15875)--(6.3385,0.23396)--(6.3737,0.32149)--(6.4206,0.41994)--(6.4807,0.52773)--(6.5549,0.64311)--(6.6442,0.76424)--(6.7491,0.88916)--(6.8700,1.0159)--(7.0070,1.1423)--(7.1599,1.2665)--(7.3283,1.3863)--(7.5116,1.5000)--(7.7087,1.6056)--(7.9188,1.7015)--(8.1402,1.7861)--(8.3716,1.8580)--(8.6113,1.9161)--(8.8575,1.9595)--(9.1081,1.9874)--(9.3613,1.9995)--(9.6150,1.9955)--(9.8672,1.9754)--(10.116,1.9397)--(10.359,1.8888)--(10.595,1.8237)--(10.821,1.7453)--(11.037,1.6549)--(11.241,1.5539)--(11.431,1.4441)--(11.607,1.3271)--(11.768,1.2048)--(11.913,1.0793)--(12.042,0.95242)--(12.155,0.82635)--(12.252,0.70308)--(12.334,0.58459)--(12.401,0.47277)--(12.454,0.36945)--(12.495,0.27627)--(12.525,0.19473)--(12.545,0.12615)--(12.557,0.071632)--(12.564,0.032051)--(12.566,0.0080452)--(12.566,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.00000000,-0.3149246667) node {$ 11 $};
+\draw (11.000,-0.31492) node {$ 11 $};
 \draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.00000000,-0.3149246667) node {$ 12 $};
+\draw (12.000,-0.31492) node {$ 12 $};
 \draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.00000000,-0.3149246667) node {$ 13 $};
+\draw (13.000,-0.31492) node {$ 13 $};
 \draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall b/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall
index 03ab6c0ac..bb8a4f9fd 100644
--- a/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall
+++ b/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall
@@ -65,31 +65,31 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red,->,>=latex] (2.000000000,0) -- (2.707106781,0.7071067812);
-\draw [color=green,->,>=latex] (2.000000000,0) -- (2.707106781,-0.7071067812);
-\draw [color=red,->,>=latex] (2.341413528,1.827257052) -- (1.861553169,2.704601930);
-\draw [color=green,->,>=latex] (2.341413528,1.827257052) -- (3.218758407,2.307117411);
-\draw [color=red,->,>=latex] (0.6118985586,2.401762329) -- (-0.3215114110,2.042950640);
-\draw [color=green,->,>=latex] (0.6118985586,2.401762329) -- (0.2530868690,3.335172299);
-\draw [color=red,->,>=latex] (-0.5954791539,0.9010655917) -- (-0.9467052076,-0.03522509989);
-\draw [color=green,->,>=latex] (-0.5954791539,0.9010655917) -- (-1.531769845,1.252291645);
-\draw [color=red,->,>=latex] (-2.176644828,-0.2044632369) -- (-2.772636306,-1.007453996);
-\draw [color=green,->,>=latex] (-2.176644828,-0.2044632369) -- (-2.979635587,0.3915282409);
-\draw [color=red,->,>=latex] (-2.172784878,-2.066762532) -- (-1.508100860,-2.813887125);
-\draw [color=green,->,>=latex] (-2.172784878,-2.066762532) -- (-2.919909470,-2.731446550);
-\draw [color=red,->,>=latex] (-0.3430145542,-2.269729649) -- (0.5125413216,-1.752019167);
-\draw [color=green,->,>=latex] (-0.3430145542,-2.269729649) -- (0.1746959280,-3.125285525);
-\draw [color=red,->,>=latex] (0.7768218803,-0.6530736828) -- (1.631807400,-0.1344218137);
-\draw [color=green,->,>=latex] (0.7768218803,-0.6530736828) -- (1.295473749,-1.508059202);
+\draw [color=red,->,>=latex] (2.0000,0) -- (2.7071,0.70711);
+\draw [color=green,->,>=latex] (2.0000,0) -- (2.7071,-0.70711);
+\draw [color=red,->,>=latex] (2.3414,1.8273) -- (1.8616,2.7046);
+\draw [color=green,->,>=latex] (2.3414,1.8273) -- (3.2188,2.3071);
+\draw [color=red,->,>=latex] (0.61190,2.4018) -- (-0.32151,2.0429);
+\draw [color=green,->,>=latex] (0.61190,2.4018) -- (0.25309,3.3352);
+\draw [color=red,->,>=latex] (-0.59548,0.90107) -- (-0.94670,-0.035225);
+\draw [color=green,->,>=latex] (-0.59548,0.90107) -- (-1.5318,1.2523);
+\draw [color=red,->,>=latex] (-2.1766,-0.20446) -- (-2.7726,-1.0075);
+\draw [color=green,->,>=latex] (-2.1766,-0.20446) -- (-2.9796,0.39153);
+\draw [color=red,->,>=latex] (-2.1728,-2.0668) -- (-1.5081,-2.8139);
+\draw [color=green,->,>=latex] (-2.1728,-2.0668) -- (-2.9199,-2.7314);
+\draw [color=red,->,>=latex] (-0.34301,-2.2697) -- (0.51254,-1.7520);
+\draw [color=green,->,>=latex] (-0.34301,-2.2697) -- (0.17470,-3.1253);
+\draw [color=red,->,>=latex] (0.77682,-0.65307) -- (1.6318,-0.13442);
+\draw [color=green,->,>=latex] (0.77682,-0.65307) -- (1.2955,-1.5081);
 \draw [color=blue] (2.000,0)--(2.122,0.1349)--(2.233,0.2850)--(2.329,0.4488)--(2.407,0.6244)--(2.463,0.8091)--(2.497,0.9998)--(2.507,1.193)--(2.490,1.386)--(2.448,1.573)--(2.379,1.752)--(2.286,1.919)--(2.170,2.069)--(2.033,2.201)--(1.878,2.312)--(1.708,2.399)--(1.527,2.461)--(1.338,2.497)--(1.145,2.507)--(0.9517,2.491)--(0.7622,2.451)--(0.5796,2.389)--(0.4067,2.306)--(0.2461,2.207)--(0.09969,2.093)--(-0.03123,1.968)--(-0.1460,1.836)--(-0.2445,1.701)--(-0.3275,1.565)--(-0.3961,1.433)--(-0.4520,1.306)--(-0.4973,1.187)--(-0.5348,1.079)--(-0.5670,0.9820)--(-0.5969,0.8972)--(-0.6275,0.8245)--(-0.6615,0.7634)--(-0.7016,0.7128)--(-0.7499,0.6709)--(-0.8082,0.6356)--(-0.8779,0.6044)--(-0.9597,0.5746)--(-1.054,0.5432)--(-1.159,0.5073)--(-1.275,0.4642)--(-1.400,0.4111)--(-1.532,0.3459)--(-1.667,0.2667)--(-1.803,0.1721)--(-1.936,0.06144)--(-2.062,-0.06547)--(-2.179,-0.2081)--(-2.283,-0.3653)--(-2.370,-0.5353)--(-2.438,-0.7158)--(-2.483,-0.9039)--(-2.505,-1.096)--(-2.502,-1.290)--(-2.472,-1.480)--(-2.417,-1.664)--(-2.336,-1.837)--(-2.231,-1.996)--(-2.104,-2.138)--(-1.958,-2.260)--(-1.795,-2.359)--(-1.619,-2.433)--(-1.433,-2.482)--(-1.242,-2.505)--(-1.048,-2.502)--(-0.8563,-2.474)--(-0.6698,-2.423)--(-0.4917,-2.350)--(-0.3247,-2.259)--(-0.1710,-2.151)--(-0.03224,-2.031)--(0.09064,-1.903)--(0.1973,-1.769)--(0.2879,-1.633)--(0.3635,-1.498)--(0.4255,-1.368)--(0.4758,-1.245)--(0.5169,-1.132)--(0.5513,-1.029)--(0.5821,-0.9381)--(0.6120,-0.8594)--(0.6439,-0.7926)--(0.6806,-0.7369)--(0.7246,-0.6909)--(0.7777,-0.6525)--(0.8416,-0.6196)--(0.9173,-0.5895)--(1.005,-0.5593)--(1.105,-0.5260)--(1.216,-0.4868)--(1.337,-0.4391)--(1.465,-0.3802)--(1.599,-0.3082)--(1.735,-0.2214)--(1.870,-0.1188)--(2.000,0);
-\draw [,->,>=latex] (2.000000000,0) -- (2.014142136,0.01414213562);
-\draw [,->,>=latex] (2.341413528,1.827257052) -- (2.331816321,1.844803950);
-\draw [,->,>=latex] (0.6118985586,2.401762329) -- (0.5932303592,2.394586095);
-\draw [,->,>=latex] (-0.5954791539,0.9010655917) -- (-0.6025036750,0.8823397779);
-\draw [,->,>=latex] (-2.176644828,-0.2044632369) -- (-2.188564658,-0.2205230521);
-\draw [,->,>=latex] (-2.172784878,-2.066762532) -- (-2.159491197,-2.081705024);
-\draw [,->,>=latex] (-0.3430145542,-2.269729649) -- (-0.3259034367,-2.259375439);
-\draw [,->,>=latex] (0.7768218803,-0.6530736828) -- (0.7939215907,-0.6427006454);
+\draw [,->,>=latex] (2.0000,0) -- (2.0141,0.014142);
+\draw [,->,>=latex] (2.3414,1.8273) -- (2.3318,1.8448);
+\draw [,->,>=latex] (0.61190,2.4018) -- (0.59323,2.3946);
+\draw [,->,>=latex] (-0.59548,0.90107) -- (-0.60250,0.88234);
+\draw [,->,>=latex] (-2.1766,-0.20446) -- (-2.1886,-0.22052);
+\draw [,->,>=latex] (-2.1728,-2.0668) -- (-2.1595,-2.0817);
+\draw [,->,>=latex] (-0.34301,-2.2697) -- (-0.32590,-2.2594);
+\draw [,->,>=latex] (0.77682,-0.65307) -- (0.79392,-0.64270);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall b/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall
index 421e7c1d6..28b9aef27 100644
--- a/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall
+++ b/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall
@@ -60,8 +60,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.250000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.2500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.0126,0.0101)--(0.0253,0.0202)--(0.0379,0.0303)--(0.0505,0.0404)--(0.0631,0.0505)--(0.0758,0.0606)--(0.0884,0.0707)--(0.101,0.0808)--(0.114,0.0909)--(0.126,0.101)--(0.139,0.111)--(0.152,0.121)--(0.164,0.131)--(0.177,0.141)--(0.189,0.152)--(0.202,0.162)--(0.215,0.172)--(0.227,0.182)--(0.240,0.192)--(0.253,0.202)--(0.265,0.212)--(0.278,0.222)--(0.290,0.232)--(0.303,0.242)--(0.316,0.253)--(0.328,0.263)--(0.341,0.273)--(0.354,0.283)--(0.366,0.293)--(0.379,0.303)--(0.391,0.313)--(0.404,0.323)--(0.417,0.333)--(0.429,0.343)--(0.442,0.354)--(0.455,0.364)--(0.467,0.374)--(0.480,0.384)--(0.492,0.394)--(0.505,0.404)--(0.518,0.414)--(0.530,0.424)--(0.543,0.434)--(0.556,0.444)--(0.568,0.455)--(0.581,0.465)--(0.593,0.475)--(0.606,0.485)--(0.619,0.495)--(0.631,0.505)--(0.644,0.515)--(0.657,0.525)--(0.669,0.535)--(0.682,0.545)--(0.694,0.556)--(0.707,0.566)--(0.720,0.576)--(0.732,0.586)--(0.745,0.596)--(0.758,0.606)--(0.770,0.616)--(0.783,0.626)--(0.795,0.636)--(0.808,0.646)--(0.821,0.657)--(0.833,0.667)--(0.846,0.677)--(0.859,0.687)--(0.871,0.697)--(0.884,0.707)--(0.896,0.717)--(0.909,0.727)--(0.922,0.737)--(0.934,0.747)--(0.947,0.758)--(0.960,0.768)--(0.972,0.778)--(0.985,0.788)--(0.997,0.798)--(1.01,0.808)--(1.02,0.818)--(1.04,0.828)--(1.05,0.838)--(1.06,0.849)--(1.07,0.859)--(1.09,0.869)--(1.10,0.879)--(1.11,0.889)--(1.12,0.899)--(1.14,0.909)--(1.15,0.919)--(1.16,0.929)--(1.17,0.939)--(1.19,0.950)--(1.20,0.960)--(1.21,0.970)--(1.22,0.980)--(1.24,0.990)--(1.25,1.00);
diff --git a/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall b/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
index fe50e66aa..aca707323 100644
--- a/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
+++ b/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
@@ -53,28 +53,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.723619130);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7236);
 %DEFAULT
 
 \draw [color=blue] (0.01000,3.224)--(0.05030,2.093)--(0.09061,1.681)--(0.1309,1.423)--(0.1712,1.235)--(0.2115,1.087)--(0.2518,0.9653)--(0.2921,0.8614)--(0.3324,0.7709)--(0.3727,0.6908)--(0.4130,0.6190)--(0.4533,0.5538)--(0.4936,0.4942)--(0.5339,0.4392)--(0.5742,0.3883)--(0.6145,0.3408)--(0.6548,0.2963)--(0.6952,0.2545)--(0.7355,0.2151)--(0.7758,0.1777)--(0.8161,0.1423)--(0.8564,0.1085)--(0.8967,0.07635)--(0.9370,0.04557)--(0.9773,0.01609)--(1.018,0.01220)--(1.058,0.03939)--(1.098,0.06556)--(1.138,0.09079)--(1.179,0.1151)--(1.219,0.1387)--(1.259,0.1614)--(1.300,0.1835)--(1.340,0.2049)--(1.380,0.2256)--(1.421,0.2458)--(1.461,0.2653)--(1.501,0.2844)--(1.542,0.3029)--(1.582,0.3210)--(1.622,0.3386)--(1.662,0.3558)--(1.703,0.3726)--(1.743,0.3889)--(1.783,0.4049)--(1.824,0.4206)--(1.864,0.4359)--(1.904,0.4509)--(1.945,0.4655)--(1.985,0.4799)--(2.025,0.4939)--(2.065,0.5077)--(2.106,0.5213)--(2.146,0.5345)--(2.186,0.5476)--(2.227,0.5604)--(2.267,0.5729)--(2.307,0.5852)--(2.348,0.5974)--(2.388,0.6093)--(2.428,0.6210)--(2.468,0.6325)--(2.509,0.6439)--(2.549,0.6550)--(2.589,0.6660)--(2.630,0.6768)--(2.670,0.6875)--(2.710,0.6979)--(2.751,0.7083)--(2.791,0.7185)--(2.831,0.7285)--(2.872,0.7384)--(2.912,0.7481)--(2.952,0.7578)--(2.992,0.7673)--(3.033,0.7766)--(3.073,0.7859)--(3.113,0.7950)--(3.154,0.8040)--(3.194,0.8129)--(3.234,0.8217)--(3.275,0.8303)--(3.315,0.8389)--(3.355,0.8474)--(3.395,0.8557)--(3.436,0.8640)--(3.476,0.8721)--(3.516,0.8802)--(3.557,0.8882)--(3.597,0.8961)--(3.637,0.9039)--(3.678,0.9116)--(3.718,0.9192)--(3.758,0.9268)--(3.798,0.9342)--(3.839,0.9416)--(3.879,0.9489)--(3.919,0.9562)--(3.960,0.9633)--(4.000,0.9704);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
@@ -141,32 +141,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.723619130) -- (0,1.470406053);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.7236) -- (0,1.4704);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
 %OTHER STUFF
 %END PSPICTURE
@@ -229,32 +229,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.023619131) -- (0,2.170406053);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.0236) -- (0,2.1704);
 %DEFAULT
 
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
 %OTHER STUFF
 %END PSPICTURE
@@ -301,20 +301,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.324187016);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.3242);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.030,0.1209)--(1.061,0.1698)--(1.091,0.2065)--(1.121,0.2368)--(1.152,0.2629)--(1.182,0.2861)--(1.212,0.3070)--(1.242,0.3261)--(1.273,0.3438)--(1.303,0.3601)--(1.333,0.3755)--(1.364,0.3898)--(1.394,0.4034)--(1.424,0.4163)--(1.455,0.4285)--(1.485,0.4401)--(1.515,0.4512)--(1.545,0.4618)--(1.576,0.4720)--(1.606,0.4818)--(1.636,0.4912)--(1.667,0.5003)--(1.697,0.5090)--(1.727,0.5175)--(1.758,0.5257)--(1.788,0.5336)--(1.818,0.5412)--(1.848,0.5487)--(1.879,0.5559)--(1.909,0.5629)--(1.939,0.5697)--(1.970,0.5763)--(2.000,0.5828)--(2.030,0.5891)--(2.061,0.5952)--(2.091,0.6012)--(2.121,0.6070)--(2.152,0.6127)--(2.182,0.6183)--(2.212,0.6237)--(2.242,0.6291)--(2.273,0.6343)--(2.303,0.6394)--(2.333,0.6443)--(2.364,0.6492)--(2.394,0.6540)--(2.424,0.6587)--(2.455,0.6633)--(2.485,0.6678)--(2.515,0.6723)--(2.545,0.6766)--(2.576,0.6809)--(2.606,0.6851)--(2.636,0.6892)--(2.667,0.6933)--(2.697,0.6972)--(2.727,0.7012)--(2.758,0.7050)--(2.788,0.7088)--(2.818,0.7125)--(2.848,0.7162)--(2.879,0.7198)--(2.909,0.7234)--(2.939,0.7269)--(2.970,0.7303)--(3.000,0.7337)--(3.030,0.7371)--(3.061,0.7403)--(3.091,0.7436)--(3.121,0.7468)--(3.152,0.7500)--(3.182,0.7531)--(3.212,0.7562)--(3.242,0.7592)--(3.273,0.7622)--(3.303,0.7652)--(3.333,0.7681)--(3.364,0.7710)--(3.394,0.7738)--(3.424,0.7766)--(3.455,0.7794)--(3.485,0.7821)--(3.515,0.7848)--(3.545,0.7875)--(3.576,0.7902)--(3.606,0.7928)--(3.636,0.7953)--(3.667,0.7979)--(3.697,0.8004)--(3.727,0.8029)--(3.758,0.8054)--(3.788,0.8078)--(3.818,0.8102)--(3.848,0.8126)--(3.879,0.8150)--(3.909,0.8173)--(3.939,0.8196)--(3.970,0.8219)--(4.000,0.8242);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -381,34 +381,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.490000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-3.723619130) -- (0,1.552854178);
+\draw [,->,>=latex] (-1.4900,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-3.7236) -- (0,1.5529);
 %DEFAULT
 
 \draw [color=blue] (-0.9900,-3.224)--(-0.9446,-2.026)--(-0.8993,-1.607)--(-0.8539,-1.347)--(-0.8086,-1.157)--(-0.7632,-1.008)--(-0.7179,-0.8858)--(-0.6725,-0.7814)--(-0.6272,-0.6906)--(-0.5818,-0.6103)--(-0.5365,-0.5382)--(-0.4911,-0.4729)--(-0.4458,-0.4131)--(-0.4004,-0.3581)--(-0.3550,-0.3070)--(-0.3097,-0.2594)--(-0.2643,-0.2149)--(-0.2190,-0.1730)--(-0.1736,-0.1335)--(-0.1283,-0.09610)--(-0.08293,-0.06060)--(-0.03758,-0.02681)--(0.007778,0.005423)--(0.05313,0.03624)--(0.09848,0.06575)--(0.1438,0.09407)--(0.1892,0.1213)--(0.2345,0.1475)--(0.2799,0.1727)--(0.3253,0.1971)--(0.3706,0.2207)--(0.4160,0.2435)--(0.4613,0.2655)--(0.5067,0.2869)--(0.5520,0.3077)--(0.5974,0.3279)--(0.6427,0.3475)--(0.6881,0.3665)--(0.7334,0.3851)--(0.7788,0.4032)--(0.8241,0.4208)--(0.8695,0.4380)--(0.9148,0.4547)--(0.9602,0.4711)--(1.006,0.4871)--(1.051,0.5028)--(1.096,0.5181)--(1.142,0.5331)--(1.187,0.5478)--(1.232,0.5621)--(1.278,0.5762)--(1.323,0.5900)--(1.368,0.6035)--(1.414,0.6168)--(1.459,0.6299)--(1.504,0.6426)--(1.550,0.6552)--(1.595,0.6675)--(1.641,0.6797)--(1.686,0.6916)--(1.731,0.7033)--(1.777,0.7149)--(1.822,0.7262)--(1.867,0.7374)--(1.913,0.7483)--(1.958,0.7592)--(2.003,0.7698)--(2.049,0.7803)--(2.094,0.7906)--(2.139,0.8008)--(2.185,0.8109)--(2.230,0.8208)--(2.275,0.8305)--(2.321,0.8401)--(2.366,0.8496)--(2.412,0.8590)--(2.457,0.8683)--(2.502,0.8774)--(2.548,0.8864)--(2.593,0.8953)--(2.638,0.9041)--(2.684,0.9127)--(2.729,0.9213)--(2.774,0.9298)--(2.820,0.9381)--(2.865,0.9464)--(2.910,0.9546)--(2.956,0.9626)--(3.001,0.9706)--(3.046,0.9785)--(3.092,0.9863)--(3.137,0.9940)--(3.183,1.002)--(3.228,1.009)--(3.273,1.017)--(3.319,1.024)--(3.364,1.031)--(3.409,1.039)--(3.455,1.046)--(3.500,1.053);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
 %OTHER STUFF
 %END PSPICTURE
@@ -471,38 +471,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.250000000,0) -- (2.250000000,0);
-\draw [,->,>=latex] (0,-3.772532402) -- (0,1.782807025);
+\draw [,->,>=latex] (-2.2500,0) -- (2.2500,0);
+\draw [,->,>=latex] (0,-3.7725) -- (0,1.7828);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
 %OTHER STUFF
 %END PSPICTURE
@@ -565,36 +565,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.600000000,0) -- (2.600000000,0);
-\draw [,->,>=latex] (0,-3.546698315) -- (0,1.269028602);
+\draw [,->,>=latex] (-2.6000,0) -- (2.6000,0);
+\draw [,->,>=latex] (0,-3.5467) -- (0,1.2690);
 %DEFAULT
 
 \draw [color=blue] (-2.100,0.7690)--(-2.079,0.7620)--(-2.058,0.7548)--(-2.037,0.7476)--(-2.016,0.7403)--(-1.994,0.7329)--(-1.973,0.7255)--(-1.952,0.7179)--(-1.931,0.7103)--(-1.910,0.7026)--(-1.889,0.6948)--(-1.868,0.6870)--(-1.847,0.6790)--(-1.825,0.6709)--(-1.804,0.6628)--(-1.783,0.6546)--(-1.762,0.6462)--(-1.741,0.6378)--(-1.720,0.6292)--(-1.699,0.6206)--(-1.678,0.6118)--(-1.656,0.6030)--(-1.635,0.5940)--(-1.614,0.5849)--(-1.593,0.5756)--(-1.572,0.5663)--(-1.551,0.5568)--(-1.530,0.5472)--(-1.509,0.5375)--(-1.487,0.5276)--(-1.466,0.5176)--(-1.445,0.5075)--(-1.424,0.4972)--(-1.403,0.4867)--(-1.382,0.4761)--(-1.361,0.4653)--(-1.340,0.4544)--(-1.319,0.4432)--(-1.297,0.4319)--(-1.276,0.4204)--(-1.255,0.4088)--(-1.234,0.3969)--(-1.213,0.3848)--(-1.192,0.3725)--(-1.171,0.3600)--(-1.150,0.3472)--(-1.128,0.3343)--(-1.107,0.3210)--(-1.086,0.3075)--(-1.065,0.2938)--(-1.044,0.2798)--(-1.023,0.2655)--(-1.002,0.2509)--(-0.9806,0.2359)--(-0.9595,0.2207)--(-0.9383,0.2051)--(-0.9172,0.1892)--(-0.8961,0.1729)--(-0.8750,0.1562)--(-0.8539,0.1391)--(-0.8327,0.1215)--(-0.8116,0.1036)--(-0.7905,0.08510)--(-0.7694,0.06615)--(-0.7483,0.04666)--(-0.7271,0.02662)--(-0.7060,0.005983)--(-0.6849,-0.01528)--(-0.6638,-0.03721)--(-0.6426,-0.05984)--(-0.6215,-0.08323)--(-0.6004,-0.1074)--(-0.5793,-0.1325)--(-0.5582,-0.1585)--(-0.5370,-0.1855)--(-0.5159,-0.2136)--(-0.4948,-0.2429)--(-0.4737,-0.2734)--(-0.4526,-0.3053)--(-0.4314,-0.3388)--(-0.4103,-0.3739)--(-0.3892,-0.4109)--(-0.3681,-0.4500)--(-0.3470,-0.4913)--(-0.3258,-0.5353)--(-0.3047,-0.5822)--(-0.2836,-0.6325)--(-0.2625,-0.6867)--(-0.2413,-0.7454)--(-0.2202,-0.8095)--(-0.1991,-0.8801)--(-0.1780,-0.9586)--(-0.1569,-1.047)--(-0.1357,-1.148)--(-0.1146,-1.267)--(-0.09350,-1.409)--(-0.07238,-1.588)--(-0.05126,-1.830)--(-0.03013,-2.202)--(-0.009013,-3.047);
 
 \draw [color=blue] (0.009013,-3.047)--(0.03013,-2.202)--(0.05126,-1.830)--(0.07238,-1.588)--(0.09350,-1.409)--(0.1146,-1.267)--(0.1357,-1.148)--(0.1569,-1.047)--(0.1780,-0.9586)--(0.1991,-0.8801)--(0.2202,-0.8095)--(0.2413,-0.7454)--(0.2625,-0.6867)--(0.2836,-0.6325)--(0.3047,-0.5822)--(0.3258,-0.5353)--(0.3470,-0.4913)--(0.3681,-0.4500)--(0.3892,-0.4109)--(0.4103,-0.3739)--(0.4314,-0.3388)--(0.4526,-0.3053)--(0.4737,-0.2734)--(0.4948,-0.2429)--(0.5159,-0.2136)--(0.5370,-0.1855)--(0.5582,-0.1585)--(0.5793,-0.1325)--(0.6004,-0.1074)--(0.6215,-0.08323)--(0.6426,-0.05984)--(0.6638,-0.03721)--(0.6849,-0.01528)--(0.7060,0.005983)--(0.7271,0.02662)--(0.7483,0.04666)--(0.7694,0.06615)--(0.7905,0.08510)--(0.8116,0.1036)--(0.8327,0.1215)--(0.8539,0.1391)--(0.8750,0.1562)--(0.8961,0.1729)--(0.9172,0.1892)--(0.9383,0.2051)--(0.9595,0.2207)--(0.9806,0.2359)--(1.002,0.2509)--(1.023,0.2655)--(1.044,0.2798)--(1.065,0.2938)--(1.086,0.3075)--(1.107,0.3210)--(1.128,0.3343)--(1.150,0.3472)--(1.171,0.3600)--(1.192,0.3725)--(1.213,0.3848)--(1.234,0.3969)--(1.255,0.4088)--(1.276,0.4204)--(1.297,0.4319)--(1.319,0.4432)--(1.340,0.4544)--(1.361,0.4653)--(1.382,0.4761)--(1.403,0.4867)--(1.424,0.4972)--(1.445,0.5075)--(1.466,0.5176)--(1.487,0.5276)--(1.509,0.5375)--(1.530,0.5472)--(1.551,0.5568)--(1.572,0.5663)--(1.593,0.5756)--(1.614,0.5849)--(1.635,0.5940)--(1.656,0.6030)--(1.678,0.6118)--(1.699,0.6206)--(1.720,0.6292)--(1.741,0.6378)--(1.762,0.6462)--(1.783,0.6546)--(1.804,0.6628)--(1.825,0.6709)--(1.847,0.6790)--(1.868,0.6870)--(1.889,0.6948)--(1.910,0.7026)--(1.931,0.7103)--(1.952,0.7179)--(1.973,0.7255)--(1.994,0.7329)--(2.016,0.7403)--(2.037,0.7476)--(2.058,0.7548)--(2.079,0.7620)--(2.100,0.7690);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DerivTangenteOM.pstricks.recall b/src_phystricks/Fig_DerivTangenteOM.pstricks.recall
index 8bfad4bc6..a0c5b4ece 100644
--- a/src_phystricks/Fig_DerivTangenteOM.pstricks.recall
+++ b/src_phystricks/Fig_DerivTangenteOM.pstricks.recall
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.875000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,7.851851852);
+\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
 %DEFAULT
 \draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
 \draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
@@ -98,22 +98,22 @@
 \draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
 
 \draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
-\draw []  (3.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,-0.2785761667) node {$a$};
-\draw []  (0,1.500000000) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,1.500000000) node {$f(a)$};
-\draw []  (6.500000000,6.085648148) node [rotate=0] {$\bullet$};
-\draw []  (6.500000000,0) node [rotate=0] {$\bullet$};
-\draw (6.500000000,-0.2785761667) node {$x$};
-\draw []  (0,6.085648148) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,6.085648148) node {$f(x)$};
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (3.000000000,1.300000000);
-\draw [,->,>=latex] (4.750000000,1.300000000) -- (6.500000000,1.300000000);
-\draw (4.750000000,0.9789703333) node {$x-a$};
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,6.085648148);
-\draw [,->,>=latex] (6.700000000,3.792824074) -- (6.700000000,1.500000000);
-\draw (7.825596167,3.792824074) node {$f(x)-f(a)$};
+\draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.27858) node {$a$};
+\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
+\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.27858) node {$x$};
+\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.45521,6.0856) node {$f(x)$};
+\draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
+\draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
+\draw (4.7500,0.97897) node {$x-a$};
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
+\draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
+\draw (7.8256,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DessinLim.pstricks.recall b/src_phystricks/Fig_DessinLim.pstricks.recall
index 1b95a76d9..59d0386bb 100644
--- a/src_phystricks/Fig_DessinLim.pstricks.recall
+++ b/src_phystricks/Fig_DessinLim.pstricks.recall
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.800000001,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.800000001);
+\draw [,->,>=latex] (-0.50000,0) -- (2.8000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.8000);
 %DEFAULT
 
 \draw [color=blue] (2.300,0)--(2.300,0.03649)--(2.299,0.07297)--(2.297,0.1094)--(2.295,0.1459)--(2.293,0.1823)--(2.290,0.2186)--(2.286,0.2549)--(2.281,0.2912)--(2.277,0.3273)--(2.271,0.3634)--(2.265,0.3994)--(2.258,0.4353)--(2.251,0.4711)--(2.243,0.5067)--(2.235,0.5422)--(2.226,0.5776)--(2.217,0.6129)--(2.207,0.6480)--(2.196,0.6829)--(2.185,0.7177)--(2.173,0.7523)--(2.161,0.7866)--(2.149,0.8208)--(2.135,0.8548)--(2.121,0.8886)--(2.107,0.9221)--(2.092,0.9555)--(2.077,0.9885)--(2.061,1.021)--(2.044,1.054)--(2.027,1.086)--(2.010,1.118)--(1.992,1.150)--(1.973,1.181)--(1.954,1.213)--(1.935,1.243)--(1.915,1.274)--(1.894,1.304)--(1.874,1.334)--(1.852,1.364)--(1.830,1.393)--(1.808,1.422)--(1.785,1.450)--(1.762,1.478)--(1.738,1.506)--(1.714,1.534)--(1.690,1.561)--(1.665,1.587)--(1.639,1.613)--(1.613,1.639)--(1.587,1.665)--(1.561,1.690)--(1.534,1.714)--(1.506,1.738)--(1.478,1.762)--(1.450,1.785)--(1.422,1.808)--(1.393,1.830)--(1.364,1.852)--(1.334,1.874)--(1.304,1.894)--(1.274,1.915)--(1.243,1.935)--(1.213,1.954)--(1.181,1.973)--(1.150,1.992)--(1.118,2.010)--(1.086,2.027)--(1.054,2.044)--(1.021,2.061)--(0.9885,2.077)--(0.9555,2.092)--(0.9221,2.107)--(0.8886,2.121)--(0.8548,2.135)--(0.8208,2.149)--(0.7866,2.161)--(0.7523,2.173)--(0.7177,2.185)--(0.6829,2.196)--(0.6480,2.207)--(0.6129,2.217)--(0.5776,2.226)--(0.5422,2.235)--(0.5067,2.243)--(0.4711,2.251)--(0.4353,2.258)--(0.3994,2.265)--(0.3634,2.271)--(0.3273,2.277)--(0.2912,2.281)--(0.2549,2.286)--(0.2186,2.290)--(0.1823,2.293)--(0.1459,2.295)--(0.1094,2.297)--(0.07297,2.299)--(0.03649,2.300)--(0,2.300);
@@ -96,16 +96,16 @@
 \draw [style=dashed] (0,1.63) -- (1.63,1.63);
 \draw [style=dashed] (1.63,0) -- (1.63,1.63);
 \draw [] (2.30,2.30) -- (2.30,0);
-\draw []  (0,1.626345597) node [rotate=0] {$\bullet$};
-\draw (-0.5715993333,1.626345597) node {\( \sin(x)\)};
-\draw []  (1.626345597,0) node [rotate=0] {$\bullet$};
-\draw (1.626345597,-0.2824550000) node {\( \cos(x)\)};
-\draw []  (2.300000000,0) node [rotate=0] {$\bullet$};
-\draw (2.486875167,-0.2113105404) node {\( A\)};
-\draw []  (2.300000000,2.300000000) node [rotate=0] {$\bullet$};
-\draw (2.531995833,2.300000000) node {\( T\)};
-\draw []  (1.626345597,1.626345597) node [rotate=0] {$\bullet$};
-\draw (2.068860430,1.626345597) node {\( P\)};
+\draw []  (0,1.6263) node [rotate=0] {$\bullet$};
+\draw (-0.57160,1.6263) node {\( \sin(x)\)};
+\draw []  (1.6263,0) node [rotate=0] {$\bullet$};
+\draw (1.6263,-0.28245) node {\( \cos(x)\)};
+\draw []  (2.3000,0) node [rotate=0] {$\bullet$};
+\draw (2.4869,-0.21131) node {\( A\)};
+\draw []  (2.3000,2.3000) node [rotate=0] {$\bullet$};
+\draw (2.5320,2.3000) node {\( T\)};
+\draw []  (1.6263,1.6263) node [rotate=0] {$\bullet$};
+\draw (2.0689,1.6263) node {\( P\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DeuxCercles.pstricks.recall b/src_phystricks/Fig_DeuxCercles.pstricks.recall
index 0433aa39d..b1136c0f4 100644
--- a/src_phystricks/Fig_DeuxCercles.pstricks.recall
+++ b/src_phystricks/Fig_DeuxCercles.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -83,9 +83,9 @@
 \draw [] (2.00,0)--(2.00,0.0634)--(1.99,0.127)--(1.98,0.189)--(1.97,0.251)--(1.95,0.312)--(1.93,0.372)--(1.90,0.430)--(1.87,0.486)--(1.84,0.541)--(1.81,0.593)--(1.77,0.643)--(1.72,0.690)--(1.68,0.735)--(1.63,0.776)--(1.58,0.815)--(1.53,0.850)--(1.47,0.881)--(1.42,0.910)--(1.36,0.934)--(1.30,0.955)--(1.24,0.972)--(1.17,0.985)--(1.11,0.994)--(1.05,0.999)--(0.984,1.00)--(0.921,0.997)--(0.858,0.990)--(0.795,0.979)--(0.734,0.964)--(0.673,0.945)--(0.614,0.922)--(0.556,0.896)--(0.500,0.866)--(0.446,0.833)--(0.394,0.796)--(0.345,0.756)--(0.299,0.713)--(0.255,0.667)--(0.214,0.618)--(0.176,0.567)--(0.142,0.514)--(0.111,0.458)--(0.0839,0.401)--(0.0603,0.342)--(0.0405,0.282)--(0.0246,0.220)--(0.0126,0.158)--(0.00453,0.0951)--(0,0.0317)--(0,-0.0317)--(0.00453,-0.0951)--(0.0126,-0.158)--(0.0246,-0.220)--(0.0405,-0.282)--(0.0603,-0.342)--(0.0839,-0.401)--(0.111,-0.458)--(0.142,-0.514)--(0.176,-0.567)--(0.214,-0.618)--(0.255,-0.667)--(0.299,-0.713)--(0.345,-0.756)--(0.394,-0.796)--(0.446,-0.833)--(0.500,-0.866)--(0.556,-0.896)--(0.614,-0.922)--(0.673,-0.945)--(0.734,-0.964)--(0.795,-0.979)--(0.858,-0.990)--(0.921,-0.997)--(0.984,-1.00)--(1.05,-0.999)--(1.11,-0.994)--(1.17,-0.985)--(1.24,-0.972)--(1.30,-0.955)--(1.36,-0.934)--(1.42,-0.910)--(1.47,-0.881)--(1.53,-0.850)--(1.58,-0.815)--(1.63,-0.776)--(1.68,-0.735)--(1.72,-0.690)--(1.77,-0.643)--(1.81,-0.593)--(1.84,-0.541)--(1.87,-0.486)--(1.90,-0.430)--(1.93,-0.372)--(1.95,-0.312)--(1.97,-0.251)--(1.98,-0.189)--(1.99,-0.127)--(2.00,-0.0634)--(2.00,0);
 \draw [color=red] (0,0)--(0.0159,0)--(0.0317,0)--(0.0476,0.00113)--(0.0634,0.00201)--(0.0792,0.00315)--(0.0951,0.00453)--(0.111,0.00616)--(0.127,0.00805)--(0.142,0.0102)--(0.158,0.0126)--(0.174,0.0152)--(0.189,0.0181)--(0.205,0.0212)--(0.220,0.0246)--(0.236,0.0282)--(0.251,0.0321)--(0.266,0.0362)--(0.282,0.0405)--(0.297,0.0451)--(0.312,0.0499)--(0.327,0.0550)--(0.342,0.0603)--(0.357,0.0658)--(0.372,0.0716)--(0.386,0.0776)--(0.401,0.0839)--(0.415,0.0904)--(0.430,0.0971)--(0.444,0.104)--(0.458,0.111)--(0.472,0.119)--(0.486,0.126)--(0.500,0.134)--(0.514,0.142)--(0.527,0.150)--(0.541,0.159)--(0.554,0.167)--(0.567,0.176)--(0.580,0.185)--(0.593,0.195)--(0.606,0.204)--(0.618,0.214)--(0.631,0.224)--(0.643,0.234)--(0.655,0.244)--(0.667,0.255)--(0.679,0.265)--(0.690,0.276)--(0.701,0.287)--(0.713,0.299)--(0.724,0.310)--(0.735,0.322)--(0.745,0.333)--(0.756,0.345)--(0.766,0.357)--(0.776,0.369)--(0.786,0.382)--(0.796,0.394)--(0.805,0.407)--(0.815,0.420)--(0.824,0.433)--(0.833,0.446)--(0.841,0.459)--(0.850,0.473)--(0.858,0.486)--(0.866,0.500)--(0.874,0.514)--(0.881,0.528)--(0.889,0.542)--(0.896,0.556)--(0.903,0.570)--(0.910,0.585)--(0.916,0.599)--(0.922,0.614)--(0.928,0.628)--(0.934,0.643)--(0.940,0.658)--(0.945,0.673)--(0.950,0.688)--(0.955,0.703)--(0.959,0.718)--(0.964,0.734)--(0.968,0.749)--(0.972,0.764)--(0.975,0.780)--(0.979,0.795)--(0.982,0.811)--(0.985,0.826)--(0.987,0.842)--(0.990,0.858)--(0.992,0.873)--(0.994,0.889)--(0.995,0.905)--(0.997,0.921)--(0.998,0.937)--(0.999,0.952)--(1.00,0.968)--(1.00,0.984)--(1.00,1.00);
 \draw [color=red] (0,0)--(0.00113,-0.0476)--(0.00453,-0.0951)--(0.0102,-0.142)--(0.0181,-0.189)--(0.0282,-0.236)--(0.0405,-0.282)--(0.0550,-0.327)--(0.0716,-0.372)--(0.0904,-0.415)--(0.111,-0.458)--(0.134,-0.500)--(0.159,-0.541)--(0.185,-0.580)--(0.214,-0.618)--(0.244,-0.655)--(0.276,-0.690)--(0.310,-0.724)--(0.345,-0.756)--(0.382,-0.786)--(0.420,-0.815)--(0.459,-0.841)--(0.500,-0.866)--(0.542,-0.889)--(0.585,-0.910)--(0.628,-0.928)--(0.673,-0.945)--(0.718,-0.959)--(0.764,-0.972)--(0.811,-0.982)--(0.858,-0.990)--(0.905,-0.995)--(0.952,-0.999)--(1.00,-1.00)--(1.05,-0.999)--(1.10,-0.995)--(1.14,-0.990)--(1.19,-0.982)--(1.24,-0.972)--(1.28,-0.959)--(1.33,-0.945)--(1.37,-0.928)--(1.42,-0.910)--(1.46,-0.889)--(1.50,-0.866)--(1.54,-0.841)--(1.58,-0.815)--(1.62,-0.786)--(1.65,-0.756)--(1.69,-0.724)--(1.72,-0.690)--(1.76,-0.655)--(1.79,-0.618)--(1.81,-0.580)--(1.84,-0.541)--(1.87,-0.500)--(1.89,-0.458)--(1.91,-0.415)--(1.93,-0.372)--(1.94,-0.327)--(1.96,-0.282)--(1.97,-0.236)--(1.98,-0.189)--(1.99,-0.142)--(2.00,-0.0951)--(2.00,-0.0476)--(2.00,0)--(2.00,0.0476)--(2.00,0.0951)--(1.99,0.142)--(1.98,0.189)--(1.97,0.236)--(1.96,0.282)--(1.94,0.327)--(1.93,0.372)--(1.91,0.415)--(1.89,0.458)--(1.87,0.500)--(1.84,0.541)--(1.81,0.580)--(1.79,0.618)--(1.76,0.655)--(1.72,0.690)--(1.69,0.724)--(1.65,0.756)--(1.62,0.786)--(1.58,0.815)--(1.54,0.841)--(1.50,0.866)--(1.46,0.889)--(1.42,0.910)--(1.37,0.928)--(1.33,0.945)--(1.28,0.959)--(1.24,0.972)--(1.19,0.982)--(1.14,0.990)--(1.10,0.995)--(1.05,0.999)--(1.00,1.00);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Differentielle.pstricks.recall b/src_phystricks/Fig_Differentielle.pstricks.recall
index c47af80f4..00d72fbeb 100644
--- a/src_phystricks/Fig_Differentielle.pstricks.recall
+++ b/src_phystricks/Fig_Differentielle.pstricks.recall
@@ -91,8 +91,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.700000003,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.700000003);
+\draw [,->,>=latex] (-0.50000,0) -- (4.7000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.7000);
 %DEFAULT
 \draw [style=dotted] (2.00,2.00) -- (4.00,2.00);
 \draw [style=dotted] (4.00,2.00) -- (4.00,4.00);
@@ -100,25 +100,25 @@
 \draw [color=red] (1.000,1.000)--(1.032,1.032)--(1.065,1.065)--(1.097,1.097)--(1.129,1.129)--(1.162,1.162)--(1.194,1.194)--(1.226,1.226)--(1.259,1.259)--(1.291,1.291)--(1.323,1.323)--(1.356,1.356)--(1.388,1.388)--(1.420,1.420)--(1.453,1.453)--(1.485,1.485)--(1.517,1.517)--(1.549,1.549)--(1.582,1.582)--(1.614,1.614)--(1.646,1.646)--(1.679,1.679)--(1.711,1.711)--(1.743,1.743)--(1.776,1.776)--(1.808,1.808)--(1.840,1.840)--(1.873,1.873)--(1.905,1.905)--(1.937,1.937)--(1.970,1.970)--(2.002,2.002)--(2.034,2.034)--(2.067,2.067)--(2.099,2.099)--(2.131,2.131)--(2.164,2.164)--(2.196,2.196)--(2.228,2.228)--(2.261,2.261)--(2.293,2.293)--(2.325,2.325)--(2.358,2.358)--(2.390,2.390)--(2.422,2.422)--(2.455,2.455)--(2.487,2.487)--(2.519,2.519)--(2.552,2.552)--(2.584,2.584)--(2.616,2.616)--(2.648,2.648)--(2.681,2.681)--(2.713,2.713)--(2.745,2.745)--(2.778,2.778)--(2.810,2.810)--(2.842,2.842)--(2.875,2.875)--(2.907,2.907)--(2.939,2.939)--(2.972,2.972)--(3.004,3.004)--(3.036,3.036)--(3.069,3.069)--(3.101,3.101)--(3.133,3.133)--(3.166,3.166)--(3.198,3.198)--(3.230,3.230)--(3.263,3.263)--(3.295,3.295)--(3.327,3.327)--(3.360,3.360)--(3.392,3.392)--(3.424,3.424)--(3.457,3.457)--(3.489,3.489)--(3.521,3.521)--(3.554,3.554)--(3.586,3.586)--(3.618,3.618)--(3.651,3.651)--(3.683,3.683)--(3.715,3.715)--(3.747,3.747)--(3.780,3.780)--(3.812,3.812)--(3.844,3.844)--(3.877,3.877)--(3.909,3.909)--(3.941,3.941)--(3.974,3.974)--(4.006,4.006)--(4.038,4.038)--(4.071,4.071)--(4.103,4.103)--(4.135,4.135)--(4.168,4.168)--(4.200,4.200);
 
 \draw [color=blue] (1.000,0.6137)--(1.032,0.6773)--(1.065,0.7390)--(1.097,0.7988)--(1.129,0.8569)--(1.162,0.9133)--(1.194,0.9682)--(1.226,1.022)--(1.259,1.074)--(1.291,1.124)--(1.323,1.174)--(1.356,1.222)--(1.388,1.269)--(1.420,1.315)--(1.453,1.360)--(1.485,1.404)--(1.517,1.447)--(1.549,1.490)--(1.582,1.531)--(1.614,1.571)--(1.646,1.611)--(1.679,1.650)--(1.711,1.688)--(1.743,1.725)--(1.776,1.762)--(1.808,1.798)--(1.840,1.834)--(1.873,1.868)--(1.905,1.903)--(1.937,1.936)--(1.970,1.969)--(2.002,2.002)--(2.034,2.034)--(2.067,2.066)--(2.099,2.097)--(2.131,2.127)--(2.164,2.157)--(2.196,2.187)--(2.228,2.216)--(2.261,2.245)--(2.293,2.273)--(2.325,2.301)--(2.358,2.329)--(2.390,2.356)--(2.422,2.383)--(2.455,2.410)--(2.487,2.436)--(2.519,2.462)--(2.552,2.487)--(2.584,2.512)--(2.616,2.537)--(2.648,2.562)--(2.681,2.586)--(2.713,2.610)--(2.745,2.634)--(2.778,2.657)--(2.810,2.680)--(2.842,2.703)--(2.875,2.726)--(2.907,2.748)--(2.939,2.770)--(2.972,2.792)--(3.004,2.814)--(3.036,2.835)--(3.069,2.856)--(3.101,2.877)--(3.133,2.898)--(3.166,2.918)--(3.198,2.939)--(3.230,2.959)--(3.263,2.979)--(3.295,2.998)--(3.327,3.018)--(3.360,3.037)--(3.392,3.056)--(3.424,3.075)--(3.457,3.094)--(3.489,3.113)--(3.521,3.131)--(3.554,3.150)--(3.586,3.168)--(3.618,3.186)--(3.651,3.203)--(3.683,3.221)--(3.715,3.239)--(3.747,3.256)--(3.780,3.273)--(3.812,3.290)--(3.844,3.307)--(3.877,3.324)--(3.909,3.340)--(3.941,3.357)--(3.974,3.373)--(4.006,3.389)--(4.038,3.405)--(4.071,3.421)--(4.103,3.437)--(4.135,3.453)--(4.168,3.468)--(4.200,3.484);
-\draw [,->,>=latex] (4.300000000,3.693147181) -- (4.300000000,3.386294361);
-\draw [,->,>=latex] (4.300000000,3.693147181) -- (4.300000000,4.000000000);
-\draw (5.121135500,3.693147181) node {$\epsilon(h)$};
-\draw [,->,>=latex] (5.500000000,3.000000000) -- (5.500000000,2.000000000);
-\draw [,->,>=latex] (5.500000000,3.000000000) -- (5.500000000,4.000000000);
-\draw (6.379054000,3.000000000) node {$T(h)$};
-\draw [,->,>=latex] (3.000000000,1.500000000) -- (2.000000000,1.500000000);
-\draw [,->,>=latex] (3.000000000,1.500000000) -- (4.000000000,1.500000000);
-\draw (3.000000000,1.073264000) node {$h$};
-\draw []  (2.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (1.299076276,2.536008391) node {$f(a)$};
-\draw []  (4.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,3.386294361) node [rotate=0] {$\bullet$};
-\draw (4.850181414,2.708864614) node {$f(x)$};
-\draw []  (4.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3785761667) node {$a$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.3785761667) node {$x$};
+\draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,3.3863);
+\draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,4.0000);
+\draw (5.1211,3.6931) node {$\epsilon(h)$};
+\draw [,->,>=latex] (5.5000,3.0000) -- (5.5000,2.0000);
+\draw [,->,>=latex] (5.5000,3.0000) -- (5.5000,4.0000);
+\draw (6.3791,3.0000) node {$T(h)$};
+\draw [,->,>=latex] (3.0000,1.5000) -- (2.0000,1.5000);
+\draw [,->,>=latex] (3.0000,1.5000) -- (4.0000,1.5000);
+\draw (3.0000,1.0733) node {$h$};
+\draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (1.2991,2.5360) node {$f(a)$};
+\draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,3.3863) node [rotate=0] {$\bullet$};
+\draw (4.8502,2.7089) node {$f(x)$};
+\draw []  (4.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.37858) node {$a$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.37858) node {$x$};
 \draw [style=dotted] (2.00,2.00) -- (2.00,0);
 \draw [style=dotted] (4.00,2.00) -- (4.00,0);
 
diff --git a/src_phystricks/Fig_DistanceEnsemble.pstricks.recall b/src_phystricks/Fig_DistanceEnsemble.pstricks.recall
index 581792965..fb287b1fd 100644
--- a/src_phystricks/Fig_DistanceEnsemble.pstricks.recall
+++ b/src_phystricks/Fig_DistanceEnsemble.pstricks.recall
@@ -77,18 +77,18 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (1.763220763,1.751053597) node {$A$};
+\draw (1.7632,1.7511) node {$A$};
 \draw [] (-3.98,-2.30) -- (-1.73,-1.00);
 \draw [style=dotted] (-3.98,-2.30) -- (-0.347,1.97);
 \draw [style=dotted] (-3.98,-2.30) -- (0,-1.40);
 
 \draw [] (2.00,0)--(2.00,0.127)--(1.98,0.253)--(1.96,0.379)--(1.94,0.502)--(1.90,0.624)--(1.86,0.743)--(1.81,0.860)--(1.75,0.972)--(1.68,1.08)--(1.61,1.19)--(1.53,1.29)--(1.45,1.38)--(1.36,1.47)--(1.26,1.55)--(1.16,1.63)--(1.05,1.70)--(0.945,1.76)--(0.831,1.82)--(0.714,1.87)--(0.594,1.91)--(0.472,1.94)--(0.347,1.97)--(0.222,1.99)--(0.0952,2.00)--(-0.0317,2.00)--(-0.158,1.99)--(-0.285,1.98)--(-0.410,1.96)--(-0.533,1.93)--(-0.654,1.89)--(-0.773,1.84)--(-0.888,1.79)--(-1.00,1.73)--(-1.11,1.67)--(-1.21,1.59)--(-1.31,1.51)--(-1.40,1.43)--(-1.49,1.33)--(-1.57,1.24)--(-1.65,1.13)--(-1.72,1.03)--(-1.78,0.916)--(-1.83,0.802)--(-1.88,0.684)--(-1.92,0.563)--(-1.95,0.441)--(-1.97,0.316)--(-1.99,0.190)--(-2.00,0.0635)--(-2.00,-0.0635)--(-1.99,-0.190)--(-1.97,-0.316)--(-1.95,-0.441)--(-1.92,-0.563)--(-1.88,-0.684)--(-1.83,-0.802)--(-1.78,-0.916)--(-1.72,-1.03)--(-1.65,-1.13)--(-1.57,-1.24)--(-1.49,-1.33)--(-1.40,-1.43)--(-1.31,-1.51)--(-1.21,-1.59)--(-1.11,-1.67)--(-1.00,-1.73)--(-0.888,-1.79)--(-0.773,-1.84)--(-0.654,-1.89)--(-0.533,-1.93)--(-0.410,-1.96)--(-0.285,-1.98)--(-0.158,-1.99)--(-0.0317,-2.00)--(0.0952,-2.00)--(0.222,-1.99)--(0.347,-1.97)--(0.472,-1.94)--(0.594,-1.91)--(0.714,-1.87)--(0.831,-1.82)--(0.945,-1.76)--(1.05,-1.70)--(1.16,-1.63)--(1.26,-1.55)--(1.36,-1.47)--(1.45,-1.38)--(1.53,-1.29)--(1.61,-1.19)--(1.68,-1.08)--(1.75,-0.972)--(1.81,-0.860)--(1.86,-0.743)--(1.90,-0.624)--(1.94,-0.502)--(1.96,-0.379)--(1.98,-0.253)--(2.00,-0.127)--(2.00,0);
-\draw []  (-1.732050808,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.732050808,-1.614062167) node {$p$};
-\draw []  (0,-1.400000000) node [rotate=0] {$\bullet$};
-\draw []  (-0.3472963553,1.969615506) node [rotate=0] {$\bullet$};
-\draw []  (-3.983716857,-2.300000000) node [rotate=0] {$\bullet$};
-\draw (-4.388020524,-2.300000000) node {$x$};
+\draw []  (-1.7320,-1.0000) node [rotate=0] {$\bullet$};
+\draw (-1.7320,-1.6141) node {$p$};
+\draw []  (0,-1.4000) node [rotate=0] {$\bullet$};
+\draw []  (-0.34729,1.9696) node [rotate=0] {$\bullet$};
+\draw []  (-3.9837,-2.3000) node [rotate=0] {$\bullet$};
+\draw (-4.3880,-2.3000) node {$x$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DivergenceDeux.pstricks.recall b/src_phystricks/Fig_DivergenceDeux.pstricks.recall
index cc1aa1379..1b692ca44 100644
--- a/src_phystricks/Fig_DivergenceDeux.pstricks.recall
+++ b/src_phystricks/Fig_DivergenceDeux.pstricks.recall
@@ -65,230 +65,230 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.000000000,-4.000000000) -- (-4.235702260,-3.764297740);
-\draw [,->,>=latex] (-4.000000000,-3.428571429) -- (-4.216930458,-3.175485894);
-\draw [,->,>=latex] (-4.000000000,-2.857142857) -- (-4.193746065,-2.585898367);
-\draw [,->,>=latex] (-4.000000000,-2.285714286) -- (-4.165379646,-1.996299905);
-\draw [,->,>=latex] (-4.000000000,-1.714285714) -- (-4.131306433,-1.407904038);
-\draw [,->,>=latex] (-4.000000000,-1.142857143) -- (-4.091573709,-0.8223491603);
-\draw [,->,>=latex] (-4.000000000,-0.5714285714) -- (-4.047140452,-0.2414454069);
-\draw [,->,>=latex] (-4.000000000,0) -- (-4.000000000,0.3333333333);
-\draw [,->,>=latex] (-4.000000000,0.5714285714) -- (-3.952859548,0.9014117360);
-\draw [,->,>=latex] (-4.000000000,1.142857143) -- (-3.908426291,1.463365125);
-\draw [,->,>=latex] (-4.000000000,1.714285714) -- (-3.868693567,2.020667391);
-\draw [,->,>=latex] (-4.000000000,2.285714286) -- (-3.834620354,2.575128666);
-\draw [,->,>=latex] (-4.000000000,2.857142857) -- (-3.806253935,3.128387348);
-\draw [,->,>=latex] (-4.000000000,3.428571429) -- (-3.783069542,3.681656963);
-\draw [,->,>=latex] (-4.000000000,4.000000000) -- (-3.764297740,4.235702260);
-\draw [,->,>=latex] (-3.428571429,-4.000000000) -- (-3.681656963,-3.783069542);
-\draw [,->,>=latex] (-3.428571429,-3.428571429) -- (-3.664273689,-3.192869168);
-\draw [,->,>=latex] (-3.428571429,-2.857142857) -- (-3.641966228,-2.601069097);
-\draw [,->,>=latex] (-3.428571429,-2.285714286) -- (-3.613471494,-2.008364188);
-\draw [,->,>=latex] (-3.428571429,-1.714285714) -- (-3.577642627,-1.416143317);
-\draw [,->,>=latex] (-3.428571429,-1.142857143) -- (-3.533980684,-0.8266293768);
-\draw [,->,>=latex] (-3.428571429,-0.5714285714) -- (-3.483371091,-0.2426305968);
-\draw [,->,>=latex] (-3.428571429,0) -- (-3.428571429,0.3333333333);
-\draw [,->,>=latex] (-3.428571429,0.5714285714) -- (-3.373771766,0.9002265460);
-\draw [,->,>=latex] (-3.428571429,1.142857143) -- (-3.323162173,1.459084909);
-\draw [,->,>=latex] (-3.428571429,1.714285714) -- (-3.279500230,2.012428111);
-\draw [,->,>=latex] (-3.428571429,2.285714286) -- (-3.243671363,2.563064384);
-\draw [,->,>=latex] (-3.428571429,2.857142857) -- (-3.215176629,3.113216617);
-\draw [,->,>=latex] (-3.428571429,3.428571429) -- (-3.192869168,3.664273689);
-\draw [,->,>=latex] (-3.428571429,4.000000000) -- (-3.175485894,4.216930458);
-\draw [,->,>=latex] (-2.857142857,-4.000000000) -- (-3.128387348,-3.806253935);
-\draw [,->,>=latex] (-2.857142857,-3.428571429) -- (-3.113216617,-3.215176629);
-\draw [,->,>=latex] (-2.857142857,-2.857142857) -- (-3.092845118,-2.621440597);
-\draw [,->,>=latex] (-2.857142857,-2.285714286) -- (-3.065374540,-2.025424683);
-\draw [,->,>=latex] (-2.857142857,-1.714285714) -- (-3.028641442,-1.428454739);
-\draw [,->,>=latex] (-2.857142857,-1.142857143) -- (-2.980939749,-0.8333649126);
-\draw [,->,>=latex] (-2.857142857,-0.5714285714) -- (-2.922514902,-0.2445683462);
-\draw [,->,>=latex] (-2.857142857,0) -- (-2.857142857,0.3333333333);
-\draw [,->,>=latex] (-2.857142857,0.5714285714) -- (-2.791770812,0.8982887967);
-\draw [,->,>=latex] (-2.857142857,1.142857143) -- (-2.733345965,1.452349373);
-\draw [,->,>=latex] (-2.857142857,1.714285714) -- (-2.685644272,2.000116690);
-\draw [,->,>=latex] (-2.857142857,2.285714286) -- (-2.648911175,2.546003889);
-\draw [,->,>=latex] (-2.857142857,2.857142857) -- (-2.621440597,3.092845118);
-\draw [,->,>=latex] (-2.857142857,3.428571429) -- (-2.601069097,3.641966228);
-\draw [,->,>=latex] (-2.857142857,4.000000000) -- (-2.585898367,4.193746065);
-\draw [,->,>=latex] (-2.285714286,-4.000000000) -- (-2.575128666,-3.834620354);
-\draw [,->,>=latex] (-2.285714286,-3.428571429) -- (-2.563064384,-3.243671363);
-\draw [,->,>=latex] (-2.285714286,-2.857142857) -- (-2.546003889,-2.648911175);
-\draw [,->,>=latex] (-2.285714286,-2.285714286) -- (-2.521416546,-2.050012025);
-\draw [,->,>=latex] (-2.285714286,-1.714285714) -- (-2.485714286,-1.447619048);
-\draw [,->,>=latex] (-2.285714286,-1.142857143) -- (-2.434785484,-0.8447147459);
-\draw [,->,>=latex] (-2.285714286,-0.5714285714) -- (-2.366559494,-0.2480477380);
-\draw [,->,>=latex] (-2.285714286,0) -- (-2.285714286,0.3333333333);
-\draw [,->,>=latex] (-2.285714286,0.5714285714) -- (-2.204869077,0.8948094048);
-\draw [,->,>=latex] (-2.285714286,1.142857143) -- (-2.136643087,1.440999540);
-\draw [,->,>=latex] (-2.285714286,1.714285714) -- (-2.085714286,1.980952381);
-\draw [,->,>=latex] (-2.285714286,2.285714286) -- (-2.050012025,2.521416546);
-\draw [,->,>=latex] (-2.285714286,2.857142857) -- (-2.025424683,3.065374540);
-\draw [,->,>=latex] (-2.285714286,3.428571429) -- (-2.008364188,3.613471494);
-\draw [,->,>=latex] (-2.285714286,4.000000000) -- (-1.996299905,4.165379646);
-\draw [,->,>=latex] (-1.714285714,-4.000000000) -- (-2.020667391,-3.868693567);
-\draw [,->,>=latex] (-1.714285714,-3.428571429) -- (-2.012428111,-3.279500230);
-\draw [,->,>=latex] (-1.714285714,-2.857142857) -- (-2.000116690,-2.685644272);
-\draw [,->,>=latex] (-1.714285714,-2.285714286) -- (-1.980952381,-2.085714286);
-\draw [,->,>=latex] (-1.714285714,-1.714285714) -- (-1.949987975,-1.478583454);
-\draw [,->,>=latex] (-1.714285714,-1.142857143) -- (-1.899185780,-0.8655070447);
-\draw [,->,>=latex] (-1.714285714,-0.5714285714) -- (-1.819694970,-0.2552008054);
-\draw [,->,>=latex] (-1.714285714,0) -- (-1.714285714,0.3333333333);
-\draw [,->,>=latex] (-1.714285714,0.5714285714) -- (-1.608876459,0.8876563374);
-\draw [,->,>=latex] (-1.714285714,1.142857143) -- (-1.529385649,1.420207241);
-\draw [,->,>=latex] (-1.714285714,1.714285714) -- (-1.478583454,1.949987975);
-\draw [,->,>=latex] (-1.714285714,2.285714286) -- (-1.447619048,2.485714286);
-\draw [,->,>=latex] (-1.714285714,2.857142857) -- (-1.428454739,3.028641442);
-\draw [,->,>=latex] (-1.714285714,3.428571429) -- (-1.416143317,3.577642627);
-\draw [,->,>=latex] (-1.714285714,4.000000000) -- (-1.407904038,4.131306433);
-\draw [,->,>=latex] (-1.142857143,-4.000000000) -- (-1.463365125,-3.908426291);
-\draw [,->,>=latex] (-1.142857143,-3.428571429) -- (-1.459084909,-3.323162173);
-\draw [,->,>=latex] (-1.142857143,-2.857142857) -- (-1.452349373,-2.733345965);
-\draw [,->,>=latex] (-1.142857143,-2.285714286) -- (-1.440999540,-2.136643087);
-\draw [,->,>=latex] (-1.142857143,-1.714285714) -- (-1.420207241,-1.529385649);
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-\draw [,->,>=latex] (-1.142857143,1.714285714) -- (-0.8655070447,1.899185780);
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-\draw [,->,>=latex] (-0.5714285714,-4.000000000) -- (-0.9014117360,-3.952859548);
-\draw [,->,>=latex] (-0.5714285714,-3.428571429) -- (-0.9002265460,-3.373771766);
-\draw [,->,>=latex] (-0.5714285714,-2.857142857) -- (-0.8982887967,-2.791770812);
-\draw [,->,>=latex] (-0.5714285714,-2.285714286) -- (-0.8948094048,-2.204869077);
-\draw [,->,>=latex] (-0.5714285714,-1.714285714) -- (-0.8876563374,-1.608876459);
-\draw [,->,>=latex] (-0.5714285714,-1.142857143) -- (-0.8695709684,-0.9937859444);
-\draw [,->,>=latex] (-0.5714285714,-0.5714285714) -- (-0.8071308318,-0.3357263110);
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-\draw [,->,>=latex] (-0.5714285714,1.142857143) -- (-0.2732861744,1.291928341);
-\draw [,->,>=latex] (-0.5714285714,1.714285714) -- (-0.2552008054,1.819694970);
-\draw [,->,>=latex] (-0.5714285714,2.285714286) -- (-0.2480477380,2.366559494);
-\draw [,->,>=latex] (-0.5714285714,2.857142857) -- (-0.2445683462,2.922514902);
-\draw [,->,>=latex] (-0.5714285714,3.428571429) -- (-0.2426305968,3.483371091);
-\draw [,->,>=latex] (-0.5714285714,4.000000000) -- (-0.2414454069,4.047140452);
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-\draw [,->,>=latex] (0,-2.285714286) -- (-0.3333333333,-2.285714286);
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diff --git a/src_phystricks/Fig_DivergenceTrois.pstricks.recall b/src_phystricks/Fig_DivergenceTrois.pstricks.recall
index 762748ea8..8a25fb77c 100644
--- a/src_phystricks/Fig_DivergenceTrois.pstricks.recall
+++ b/src_phystricks/Fig_DivergenceTrois.pstricks.recall
@@ -69,47 +69,47 @@
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+\draw [,->,>=latex] (-0.18683,0.98239) -- (-0.37366,1.9648);
+\draw [,->,>=latex] (-0.42407,0.90563) -- (-0.84813,1.8113);
+\draw [,->,>=latex] (-0.63494,0.77256) -- (-1.2699,1.5451);
+\draw [,->,>=latex] (-0.80634,0.59146) -- (-1.6127,1.1829);
+\draw [,->,>=latex] (-0.92760,0.37358) -- (-1.8552,0.74716);
+\draw [,->,>=latex] (-0.99119,0.13247) -- (-1.9824,0.26495);
+\draw [,->,>=latex] (2.1999,0.019198) -- (2.6544,0.023165);
+\draw [,->,>=latex] (2.1836,0.26853) -- (2.6347,0.32401);
+\draw [,->,>=latex] (2.1390,0.51439) -- (2.5810,0.62067);
+\draw [,->,>=latex] (2.0669,0.75362) -- (2.4939,0.90933);
+\draw [,->,>=latex] (1.9681,0.98313) -- (2.3747,1.1863);
+\draw [,->,>=latex] (1.8439,1.2000) -- (2.2249,1.4479);
+\draw [,->,>=latex] (1.6960,1.4013) -- (2.0464,1.6908);
+\draw [,->,>=latex] (1.5261,1.5846) -- (1.8415,1.9120);
+\draw [,->,>=latex] (1.3366,1.7474) -- (1.6128,2.1084);
+\draw [,->,>=latex] (1.1299,1.8877) -- (1.3633,2.2777);
+\draw [,->,>=latex] (0.90852,2.0036) -- (1.0962,2.4176);
+\draw [,->,>=latex] (0.67546,2.0937) -- (0.81502,2.5263);
+\draw [,->,>=latex] (0.43369,2.1568) -- (0.52330,2.6025);
+\draw [,->,>=latex] (0.18633,2.1921) -- (0.22483,2.6450);
+\draw [,->,>=latex] (-0.063439,2.1991) -- (-0.076546,2.6534);
+\draw [,->,>=latex] (-0.31239,2.1777) -- (-0.37693,2.6276);
+\draw [,->,>=latex] (-0.55731,2.1282) -- (-0.67245,2.5680);
+\draw [,->,>=latex] (-0.79504,2.0513) -- (-0.95931,2.4751);
+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2338,2.3504);
+\draw [,->,>=latex] (-1.2368,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7317,2.0120);
+\draw [,->,>=latex] (-1.6150,1.4940) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1405,1.5701);
+\draw [,->,>=latex] (-1.9100,1.0917) -- (-2.3047,1.3172);
+\draw [,->,>=latex] (-2.0215,0.86804) -- (-2.4392,1.0474);
+\draw [,->,>=latex] (-2.1069,0.63322) -- (-2.5422,0.76405);
+\draw [,->,>=latex] (-2.1651,0.39022) -- (-2.6125,0.47085);
+\draw [,->,>=latex] (-2.1954,0.14221) -- (-2.6490,0.17159);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DivergenceUn.pstricks.recall b/src_phystricks/Fig_DivergenceUn.pstricks.recall
index d20b21eaf..8f1c5afca 100644
--- a/src_phystricks/Fig_DivergenceUn.pstricks.recall
+++ b/src_phystricks/Fig_DivergenceUn.pstricks.recall
@@ -66,59 +66,57 @@
 %PSTRICKS CODE
 %DEFAULT
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
-\draw [] (1.60,-1.20) -- (3.07,-1.20);
-\draw [] (3.07,-1.20) -- (3.07,1.20);
-\draw [] (3.07,1.20) -- (1.60,1.20);
-\draw [] (1.60,1.20) -- (1.60,-1.20);
-\draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
-\draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
-\draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
-\draw [,->,>=latex] (1.600000000,0) -- (2.225000000,0);
-\draw [,->,>=latex] (1.600000000,0.3333333333) -- (2.225000000,0.3333333333);
-\draw [,->,>=latex] (1.600000000,0.6666666667) -- (2.225000000,0.6666666667);
-\draw [,->,>=latex] (1.600000000,1.000000000) -- (2.225000000,1.000000000);
-\draw [,->,>=latex] (2.333333333,-1.000000000) -- (2.761904762,-1.000000000);
-\draw [,->,>=latex] (2.333333333,-0.6666666667) -- (2.761904762,-0.6666666667);
-\draw [,->,>=latex] (2.333333333,-0.3333333333) -- (2.761904762,-0.3333333333);
-\draw [,->,>=latex] (2.333333333,0) -- (2.761904762,0);
-\draw [,->,>=latex] (2.333333333,0.3333333333) -- (2.761904762,0.3333333333);
-\draw [,->,>=latex] (2.333333333,0.6666666667) -- (2.761904762,0.6666666667);
-\draw [,->,>=latex] (2.333333333,1.000000000) -- (2.761904762,1.000000000);
-\draw [,->,>=latex] (3.066666667,-1.000000000) -- (3.392753623,-1.000000000);
-\draw [,->,>=latex] (3.066666667,-0.6666666667) -- (3.392753623,-0.6666666667);
-\draw [,->,>=latex] (3.066666667,-0.3333333333) -- (3.392753623,-0.3333333333);
-\draw [,->,>=latex] (3.066666667,0) -- (3.392753623,0);
-\draw [,->,>=latex] (3.066666667,0.3333333333) -- (3.392753623,0.3333333333);
-\draw [,->,>=latex] (3.066666667,0.6666666667) -- (3.392753623,0.6666666667);
-\draw [,->,>=latex] (3.066666667,1.000000000) -- (3.392753623,1.000000000);
-\draw [,->,>=latex] (3.800000000,-1.000000000) -- (4.063157895,-1.000000000);
-\draw [,->,>=latex] (3.800000000,-0.6666666667) -- (4.063157895,-0.6666666667);
-\draw [,->,>=latex] (3.800000000,-0.3333333333) -- (4.063157895,-0.3333333333);
-\draw [,->,>=latex] (3.800000000,0) -- (4.063157895,0);
-\draw [,->,>=latex] (3.800000000,0.3333333333) -- (4.063157895,0.3333333333);
-\draw [,->,>=latex] (3.800000000,0.6666666667) -- (4.063157895,0.6666666667);
-\draw [,->,>=latex] (3.800000000,1.000000000) -- (4.063157895,1.000000000);
-\draw [,->,>=latex] (4.533333333,-1.000000000) -- (4.753921569,-1.000000000);
-\draw [,->,>=latex] (4.533333333,-0.6666666667) -- (4.753921569,-0.6666666667);
-\draw [,->,>=latex] (4.533333333,-0.3333333333) -- (4.753921569,-0.3333333333);
-\draw [,->,>=latex] (4.533333333,0) -- (4.753921569,0);
-\draw [,->,>=latex] (4.533333333,0.3333333333) -- (4.753921569,0.3333333333);
-\draw [,->,>=latex] (4.533333333,0.6666666667) -- (4.753921569,0.6666666667);
-\draw [,->,>=latex] (4.533333333,1.000000000) -- (4.753921569,1.000000000);
-\draw [,->,>=latex] (5.266666667,-1.000000000) -- (5.456540084,-1.000000000);
-\draw [,->,>=latex] (5.266666667,-0.6666666667) -- (5.456540084,-0.6666666667);
-\draw [,->,>=latex] (5.266666667,-0.3333333333) -- (5.456540084,-0.3333333333);
-\draw [,->,>=latex] (5.266666667,0) -- (5.456540084,0);
-\draw [,->,>=latex] (5.266666667,0.3333333333) -- (5.456540084,0.3333333333);
-\draw [,->,>=latex] (5.266666667,0.6666666667) -- (5.456540084,0.6666666667);
-\draw [,->,>=latex] (5.266666667,1.000000000) -- (5.456540084,1.000000000);
-\draw [,->,>=latex] (6.000000000,-1.000000000) -- (6.166666667,-1.000000000);
-\draw [,->,>=latex] (6.000000000,-0.6666666667) -- (6.166666667,-0.6666666667);
-\draw [,->,>=latex] (6.000000000,-0.3333333333) -- (6.166666667,-0.3333333333);
-\draw [,->,>=latex] (6.000000000,0) -- (6.166666667,0);
-\draw [,->,>=latex] (6.000000000,0.3333333333) -- (6.166666667,0.3333333333);
-\draw [,->,>=latex] (6.000000000,0.6666666667) -- (6.166666667,0.6666666667);
-\draw [,->,>=latex] (6.000000000,1.000000000) -- (6.166666667,1.000000000);
+
+
+\draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
+\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
+\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
+\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
+\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
+\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
+\draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
+\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
+\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
+\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
+\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
+\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
+\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
+\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
+\draw [,->,>=latex] (3.0667,-0.33333) -- (3.3928,-0.33333);
+\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
+\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
+\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
+\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
+\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
+\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
+\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
+\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
+\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
+\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
+\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
+\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
+\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
+\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
+\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
+\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
+\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
+\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
+\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
+\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
+\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
+\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
+\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall b/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall
index 7abd61204..40c268f53 100644
--- a/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall
+++ b/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall
@@ -44,12 +44,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3149246667) node {\(1\)};
+\draw (0,0.31492) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -89,12 +89,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.3149246667) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.31492) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -134,12 +134,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.3149246667) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.31492) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -179,12 +179,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw (3.232671190,-0.2436539771) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw (3.2327,-0.24365) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -224,12 +224,12 @@
 \draw [] (2.00,0) -- (3.00,-0.500);
 \draw [] (2.00,0) -- (3.00,0.500);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw []  (3.000000000,-0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw []  (3.000000000,0.5000000000) node [rotate=0] {$o$};
-\draw (3.232671190,0.7563460229) node {\(1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
+\draw (3.2327,0.75635) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall b/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall
index 5b742a721..569d4db5b 100644
--- a/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall
+++ b/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2644441667) node {\( 2\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.26444) node {\( 2\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall b/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall
index b6c9314d7..ad2371f2a 100644
--- a/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall
+++ b/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall
@@ -95,27 +95,27 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,0) -- (8.00,2.00);
-\draw []  (1.600000000,0.4000000000) node [rotate=0] {$\bullet$};
-\draw (1.744974792,0.1273953333) node {\( a\)};
-\draw []  (4.800000000,1.200000000) node [rotate=0] {$\bullet$};
-\draw (4.926829958,0.8792355000) node {\( b\)};
-\draw []  (6.400000000,1.600000000) node [rotate=0] {$\bullet$};
-\draw (6.527485125,1.327395333) node {\( c\)};
-\draw []  (4.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (4.152810792,0.7273953333) node {\( x\)};
+\draw []  (1.6000,0.40000) node [rotate=0] {$\bullet$};
+\draw (1.7450,0.12740) node {\( a\)};
+\draw []  (4.8000,1.2000) node [rotate=0] {$\bullet$};
+\draw (4.9268,0.87924) node {\( b\)};
+\draw []  (6.4000,1.6000) node [rotate=0] {$\bullet$};
+\draw (6.5275,1.3274) node {\( c\)};
+\draw []  (4.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (4.1528,0.72740) node {\( x\)};
 \draw [] (1.60,0.400) -- (2.60,3.40);
 \draw [] (4.80,1.20) -- (2.60,3.40);
-\draw []  (2.200000000,2.200000000) node [rotate=0] {$\bullet$};
-\draw (1.918443007,2.377307720) node {\( p\)};
-\draw []  (4.066666667,1.933333333) node [rotate=0] {$\bullet$};
-\draw (4.335311801,2.133267258) node {\( q\)};
+\draw []  (2.2000,2.2000) node [rotate=0] {$\bullet$};
+\draw (1.9184,2.3773) node {\( p\)};
+\draw []  (4.0667,1.9333) node [rotate=0] {$\bullet$};
+\draw (4.3353,2.1333) node {\( q\)};
 \draw [] (6.40,1.60) -- (0.940,2.38);
-\draw []  (2.600000000,3.400000000) node [rotate=0] {$\bullet$};
-\draw (2.881727355,3.637447733) node {\( m\)};
+\draw []  (2.6000,3.4000) node [rotate=0] {$\bullet$};
+\draw (2.8817,3.6374) node {\( m\)};
 \draw [] (4.07,1.93) -- (1.60,0.400);
 \draw [] (2.20,2.20) -- (4.80,1.20);
-\draw []  (3.618181818,1.654545455) node [rotate=0] {$\bullet$};
-\draw (3.326753965,1.492795736) node {\( n\)};
+\draw []  (3.6182,1.6545) node [rotate=0] {$\bullet$};
+\draw (3.3268,1.4928) node {\( n\)};
 \draw [] (2.60,3.40) -- (4.00,1.00);
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall b/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall
index 3a2443411..61e7d4c9d 100644
--- a/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall
+++ b/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall
@@ -83,15 +83,15 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.3247080000) node {\( A\)};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.289005356,0.2661293562) node {\( B\)};
-\draw []  (12.00000000,-3.000000000) node [rotate=0] {$\bullet$};
-\draw (12.28491136,-3.266129356) node {\( C\)};
-\draw []  (4.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.690526810,-1.266129356) node {\( K\)};
-\draw []  (1.333333333,0) node [rotate=0] {$\bullet$};
-\draw (1.333333333,0.3247080000) node {\( L\)};
+\draw (0,0.32471) node {\( A\)};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.2890,0.26613) node {\( B\)};
+\draw []  (12.000,-3.0000) node [rotate=0] {$\bullet$};
+\draw (12.285,-3.2661) node {\( C\)};
+\draw []  (4.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (3.6905,-1.2661) node {\( K\)};
+\draw []  (1.3333,0) node [rotate=0] {$\bullet$};
+\draw (1.3333,0.32471) node {\( L\)};
 \draw [] (0,0) -- (4.00,0);
 \draw [] (0,0) -- (12.0,-3.00);
 \draw [style=dashed] (12.0,-3.00) -- (4.00,0);
diff --git a/src_phystricks/Fig_EJRsWXw.pstricks.recall b/src_phystricks/Fig_EJRsWXw.pstricks.recall
index 810f409ea..c26000343 100644
--- a/src_phystricks/Fig_EJRsWXw.pstricks.recall
+++ b/src_phystricks/Fig_EJRsWXw.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -88,10 +88,10 @@
 \draw [color=green] (-1.00,1.00) -- (-2.00,2.00);
 \draw [color=green] (-1.00,1.00) -- (1.00,3.00);
 \draw [color=red] (-2.00,2.00) -- (1.00,3.00);
-\draw (1.500000000,-0.2785761667) node {\( x\)};
-\draw (1.500000000,-0.2785761667) node {\( x\)};
-\draw (0.2659030000,3.500000000) node {\( t\)};
-\draw (0.2659030000,3.500000000) node {\( t\)};
+\draw (1.5000,-0.27858) node {\( x\)};
+\draw (1.5000,-0.27858) node {\( x\)};
+\draw (0.26590,3.5000) node {\( t\)};
+\draw (0.26590,3.5000) node {\( t\)};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall b/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall
index 2d0b6e358..31c8bb72e 100644
--- a/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall
+++ b/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall
@@ -77,35 +77,35 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (2.086833187,1.759539037) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.410240275,2.935347274);
-\draw []  (1.735039751,1.464349294) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.3472874529,0.3652246914);
-\draw []  (2.657517092,2.437667613) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.247852509,3.098210238);
-\draw []  (0.9681152512,1.020091151) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.2150970834,0.5534262785);
-\draw []  (2.853656194,2.793571599) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (3.155827251,3.175606475);
-\draw []  (0.5835590482,0.8887339218) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.1548566875,0.6574931816);
-\draw []  (3.013566327,3.308802930) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.999711929,3.287338346);
-\draw []  (0.04838713566,0.8207221648) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (0.07193141522,0.8306387974);
-\draw []  (2.896085542,4.007090670) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.915111108,1.606969024) -- (2.671786357,3.458304371);
-\draw []  (-0.6188917196,1.057674546) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.915111108,1.606969024) -- (-0.03949357359,1.183270382);
+\draw []  (2.0868,1.7595) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.4102,2.9353);
+\draw []  (1.7350,1.4643) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.34725,0.36521);
+\draw []  (2.6575,2.4377) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.2478,3.0982);
+\draw []  (0.96811,1.0201) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.21509,0.55343);
+\draw []  (2.8537,2.7936) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.1558,3.1756);
+\draw []  (0.58356,0.88873) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.15485,0.65749);
+\draw []  (3.0136,3.3088) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.9997,3.2873);
+\draw []  (0.048375,0.82073) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.071918,0.83064);
+\draw []  (2.8961,4.0071) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.6718,3.4583);
+\draw []  (-0.61890,1.0577) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (-0.039497,1.1833);
 \draw [] (-3.00,0) -- (3.00,0);
 \draw [] (0,0) -- (3.83,3.21);
-\draw (0.7745541682,0.2495862383) node {\( \alpha\)};
+\draw (0.77455,0.24959) node {\( \alpha\)};
 
 \draw [] (0.500,0)--(0.500,0.00353)--(0.500,0.00705)--(0.500,0.0106)--(0.500,0.0141)--(0.500,0.0176)--(0.500,0.0211)--(0.499,0.0247)--(0.499,0.0282)--(0.499,0.0317)--(0.499,0.0352)--(0.499,0.0387)--(0.498,0.0423)--(0.498,0.0458)--(0.498,0.0493)--(0.497,0.0528)--(0.497,0.0563)--(0.496,0.0598)--(0.496,0.0633)--(0.496,0.0668)--(0.495,0.0703)--(0.495,0.0738)--(0.494,0.0773)--(0.493,0.0807)--(0.493,0.0842)--(0.492,0.0877)--(0.492,0.0912)--(0.491,0.0946)--(0.490,0.0981)--(0.490,0.102)--(0.489,0.105)--(0.488,0.108)--(0.487,0.112)--(0.487,0.115)--(0.486,0.119)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.132)--(0.481,0.136)--(0.480,0.139)--(0.479,0.143)--(0.478,0.146)--(0.477,0.149)--(0.476,0.153)--(0.475,0.156)--(0.474,0.159)--(0.473,0.163)--(0.472,0.166)--(0.470,0.169)--(0.469,0.173)--(0.468,0.176)--(0.467,0.179)--(0.465,0.183)--(0.464,0.186)--(0.463,0.189)--(0.462,0.192)--(0.460,0.196)--(0.459,0.199)--(0.457,0.202)--(0.456,0.205)--(0.454,0.209)--(0.453,0.212)--(0.451,0.215)--(0.450,0.218)--(0.448,0.221)--(0.447,0.224)--(0.445,0.228)--(0.444,0.231)--(0.442,0.234)--(0.440,0.237)--(0.439,0.240)--(0.437,0.243)--(0.435,0.246)--(0.433,0.249)--(0.432,0.252)--(0.430,0.255)--(0.428,0.258)--(0.426,0.261)--(0.424,0.264)--(0.423,0.267)--(0.421,0.270)--(0.419,0.273)--(0.417,0.276)--(0.415,0.279)--(0.413,0.282)--(0.411,0.285)--(0.409,0.288)--(0.407,0.291)--(0.405,0.294)--(0.403,0.296)--(0.401,0.299)--(0.398,0.302)--(0.396,0.305)--(0.394,0.308)--(0.392,0.310)--(0.390,0.313)--(0.388,0.316)--(0.385,0.319)--(0.383,0.321);
-\draw []  (1.915111108,1.606969024) node [rotate=0] {$\bullet$};
-\draw (2.186183463,1.329052136) node {\( P\)};
+\draw []  (1.9151,1.6070) node [rotate=0] {$\bullet$};
+\draw (2.1862,1.3291) node {\( P\)};
 \draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
-\draw (4.139119438,3.522532237) node {\( \ell_p\)};
+\draw (4.1391,3.5225) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ExPolygone.pstricks.recall b/src_phystricks/Fig_ExPolygone.pstricks.recall
index 9de4be9a0..c729554c1 100644
--- a/src_phystricks/Fig_ExPolygone.pstricks.recall
+++ b/src_phystricks/Fig_ExPolygone.pstricks.recall
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.650000000,0) -- (3.650000000,0);
-\draw [,->,>=latex] (0,-2.650000000) -- (0,2.650000000);
+\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
+\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
 %DEFAULT
 \draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
 \draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
@@ -102,21 +102,21 @@
 \draw [color=blue] (2.00,0) -- (1.00,-1.00);
 \draw [color=blue] (1.00,-1.00) -- (0,0);
 \draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExSinLarge.pstricks.recall b/src_phystricks/Fig_ExSinLarge.pstricks.recall
index 8cdc23e1e..306d962f8 100644
--- a/src_phystricks/Fig_ExSinLarge.pstricks.recall
+++ b/src_phystricks/Fig_ExSinLarge.pstricks.recall
@@ -75,25 +75,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.641592654,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499874128);
+\draw [,->,>=latex] (-0.50000,0) -- (3.6416,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4999);
 %DEFAULT
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 \draw [] (3.14,1.00) -- (3.14,2.00);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExampleIntegration.pstricks.recall b/src_phystricks/Fig_ExampleIntegration.pstricks.recall
index f5c2ea7a1..63012a8e2 100644
--- a/src_phystricks/Fig_ExampleIntegration.pstricks.recall
+++ b/src_phystricks/Fig_ExampleIntegration.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -98,17 +98,17 @@
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-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall b/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall
index d674e887c..8e4adfdc0 100644
--- a/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall
+++ b/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall
@@ -103,8 +103,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,5.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -136,41 +136,41 @@
 \draw [color=blue,style=solid] (-3.000,0)--(-2.909,0.4264)--(-2.818,0.6030)--(-2.727,0.7385)--(-2.636,0.8528)--(-2.545,0.9535)--(-2.455,1.044)--(-2.364,1.128)--(-2.273,1.206)--(-2.182,1.279)--(-2.091,1.348)--(-2.000,1.414)--(-1.909,1.477)--(-1.818,1.537)--(-1.727,1.595)--(-1.636,1.651)--(-1.545,1.706)--(-1.455,1.758)--(-1.364,1.809)--(-1.273,1.859)--(-1.182,1.907)--(-1.091,1.954)--(-1.000,2.000)--(-0.9091,2.045)--(-0.8182,2.089)--(-0.7273,2.132)--(-0.6364,2.174)--(-0.5455,2.216)--(-0.4545,2.256)--(-0.3636,2.296)--(-0.2727,2.336)--(-0.1818,2.374)--(-0.09091,2.412)--(0,2.449)--(0.09091,2.486)--(0.1818,2.523)--(0.2727,2.558)--(0.3636,2.594)--(0.4545,2.629)--(0.5455,2.663)--(0.6364,2.697)--(0.7273,2.730)--(0.8182,2.763)--(0.9091,2.796)--(1.000,2.828)--(1.091,2.860)--(1.182,2.892)--(1.273,2.923)--(1.364,2.954)--(1.455,2.985)--(1.545,3.015)--(1.636,3.045)--(1.727,3.075)--(1.818,3.104)--(1.909,3.133)--(2.000,3.162)--(2.091,3.191)--(2.182,3.219)--(2.273,3.247)--(2.364,3.275)--(2.455,3.303)--(2.545,3.330)--(2.636,3.357)--(2.727,3.384)--(2.818,3.411)--(2.909,3.438)--(3.000,3.464)--(3.091,3.490)--(3.182,3.516)--(3.273,3.542)--(3.364,3.568)--(3.455,3.593)--(3.545,3.618)--(3.636,3.643)--(3.727,3.668)--(3.818,3.693)--(3.909,3.717)--(4.000,3.742)--(4.091,3.766)--(4.182,3.790)--(4.273,3.814)--(4.364,3.838)--(4.455,3.861)--(4.545,3.885)--(4.636,3.908)--(4.727,3.931)--(4.818,3.954)--(4.909,3.977)--(5.000,4.000)--(5.091,4.023)--(5.182,4.045)--(5.273,4.068)--(5.364,4.090)--(5.455,4.112)--(5.545,4.134)--(5.636,4.156)--(5.727,4.178)--(5.818,4.200)--(5.909,4.221)--(6.000,4.243);
 
 \draw [color=blue,style=solid] (-3.000,0)--(-2.970,-0.2462)--(-2.939,-0.3482)--(-2.909,-0.4264)--(-2.879,-0.4924)--(-2.848,-0.5505)--(-2.818,-0.6030)--(-2.788,-0.6513)--(-2.758,-0.6963)--(-2.727,-0.7385)--(-2.697,-0.7785)--(-2.667,-0.8165)--(-2.636,-0.8528)--(-2.606,-0.8876)--(-2.576,-0.9211)--(-2.545,-0.9535)--(-2.515,-0.9847)--(-2.485,-1.015)--(-2.455,-1.044)--(-2.424,-1.073)--(-2.394,-1.101)--(-2.364,-1.128)--(-2.333,-1.155)--(-2.303,-1.181)--(-2.273,-1.206)--(-2.242,-1.231)--(-2.212,-1.255)--(-2.182,-1.279)--(-2.152,-1.303)--(-2.121,-1.326)--(-2.091,-1.348)--(-2.061,-1.371)--(-2.030,-1.393)--(-2.000,-1.414)--(-1.970,-1.435)--(-1.939,-1.456)--(-1.909,-1.477)--(-1.879,-1.497)--(-1.848,-1.518)--(-1.818,-1.537)--(-1.788,-1.557)--(-1.758,-1.576)--(-1.727,-1.595)--(-1.697,-1.614)--(-1.667,-1.633)--(-1.636,-1.651)--(-1.606,-1.670)--(-1.576,-1.688)--(-1.545,-1.706)--(-1.515,-1.723)--(-1.485,-1.741)--(-1.455,-1.758)--(-1.424,-1.775)--(-1.394,-1.792)--(-1.364,-1.809)--(-1.333,-1.826)--(-1.303,-1.842)--(-1.273,-1.859)--(-1.242,-1.875)--(-1.212,-1.891)--(-1.182,-1.907)--(-1.152,-1.923)--(-1.121,-1.938)--(-1.091,-1.954)--(-1.061,-1.969)--(-1.030,-1.985)--(-1.000,-2.000)--(-0.9697,-2.015)--(-0.9394,-2.030)--(-0.9091,-2.045)--(-0.8788,-2.060)--(-0.8485,-2.074)--(-0.8182,-2.089)--(-0.7879,-2.103)--(-0.7576,-2.118)--(-0.7273,-2.132)--(-0.6970,-2.146)--(-0.6667,-2.160)--(-0.6364,-2.174)--(-0.6061,-2.188)--(-0.5758,-2.202)--(-0.5455,-2.216)--(-0.5152,-2.229)--(-0.4848,-2.243)--(-0.4545,-2.256)--(-0.4242,-2.270)--(-0.3939,-2.283)--(-0.3636,-2.296)--(-0.3333,-2.309)--(-0.3030,-2.322)--(-0.2727,-2.336)--(-0.2424,-2.348)--(-0.2121,-2.361)--(-0.1818,-2.374)--(-0.1515,-2.387)--(-0.1212,-2.400)--(-0.09091,-2.412)--(-0.06061,-2.425)--(-0.03030,-2.437)--(0,-2.449);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExempleArcParam.pstricks.recall b/src_phystricks/Fig_ExempleArcParam.pstricks.recall
index 0ff051d15..3f874e1c9 100644
--- a/src_phystricks/Fig_ExempleArcParam.pstricks.recall
+++ b/src_phystricks/Fig_ExempleArcParam.pstricks.recall
@@ -65,29 +65,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.899823779,0) -- (1.899823779,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.898229715);
+\draw [,->,>=latex] (-1.8998,0) -- (1.8998,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.8982);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.08879,0.04443)--(0.1772,0.08885)--(0.2650,0.1333)--(0.3516,0.1777)--(0.4368,0.2221)--(0.5203,0.2666)--(0.6017,0.3110)--(0.6807,0.3554)--(0.7569,0.3998)--(0.8301,0.4443)--(0.8999,0.4887)--(0.9661,0.5331)--(1.028,0.5775)--(1.087,0.6220)--(1.140,0.6664)--(1.190,0.7108)--(1.234,0.7552)--(1.273,0.7997)--(1.308,0.8441)--(1.337,0.8885)--(1.361,0.9330)--(1.379,0.9774)--(1.391,1.022)--(1.398,1.066)--(1.400,1.111)--(1.396,1.155)--(1.386,1.200)--(1.370,1.244)--(1.349,1.288)--(1.323,1.333)--(1.291,1.377)--(1.254,1.422)--(1.212,1.466)--(1.166,1.510)--(1.114,1.555)--(1.058,1.599)--(0.9978,1.644)--(0.9335,1.688)--(0.8654,1.733)--(0.7939,1.777)--(0.7191,1.821)--(0.6415,1.866)--(0.5613,1.910)--(0.4788,1.955)--(0.3944,1.999)--(0.3084,2.044)--(0.2212,2.088)--(0.1331,2.132)--(0.04442,2.177)--(-0.04442,2.221)--(-0.1331,2.266)--(-0.2212,2.310)--(-0.3084,2.355)--(-0.3944,2.399)--(-0.4788,2.443)--(-0.5613,2.488)--(-0.6415,2.532)--(-0.7191,2.577)--(-0.7939,2.621)--(-0.8654,2.666)--(-0.9335,2.710)--(-0.9978,2.754)--(-1.058,2.799)--(-1.114,2.843)--(-1.166,2.888)--(-1.212,2.932)--(-1.254,2.977)--(-1.291,3.021)--(-1.323,3.065)--(-1.349,3.110)--(-1.370,3.154)--(-1.386,3.199)--(-1.396,3.243)--(-1.400,3.288)--(-1.398,3.332)--(-1.391,3.376)--(-1.379,3.421)--(-1.361,3.465)--(-1.337,3.510)--(-1.308,3.554)--(-1.273,3.599)--(-1.234,3.643)--(-1.190,3.687)--(-1.140,3.732)--(-1.087,3.776)--(-1.028,3.821)--(-0.9661,3.865)--(-0.8999,3.910)--(-0.8301,3.954)--(-0.7569,3.998)--(-0.6807,4.043)--(-0.6017,4.087)--(-0.5203,4.132)--(-0.4368,4.176)--(-0.3516,4.221)--(-0.2650,4.265)--(-0.1772,4.309)--(-0.08879,4.354)--(0,4.398);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.2912498333,4.200000000) node {$ 6 $};
+\draw (-0.29125,4.2000) node {$ 6 $};
 \draw [] (-0.100,4.20) -- (0.100,4.20);
 %OTHER STUFF
 %END PSPICTURE
@@ -126,17 +126,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.999811192,0) -- (1.999811192,0);
-\draw [,->,>=latex] (0,-1.999811192) -- (0,1.999811192);
+\draw [,->,>=latex] (-1.9998,0) -- (1.9998,0);
+\draw [,->,>=latex] (0,-1.9998) -- (0,1.9998);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.190,-0.0951)--(0.377,-0.190)--(0.557,-0.284)--(0.729,-0.377)--(0.889,-0.468)--(1.04,-0.557)--(1.16,-0.645)--(1.27,-0.729)--(1.36,-0.811)--(1.43,-0.889)--(1.48,-0.964)--(1.50,-1.04)--(1.50,-1.10)--(1.47,-1.16)--(1.42,-1.22)--(1.34,-1.27)--(1.25,-1.32)--(1.13,-1.36)--(1.00,-1.40)--(0.851,-1.43)--(0.687,-1.46)--(0.513,-1.48)--(0.330,-1.49)--(0.143,-1.50)--(-0.0476,-1.50)--(-0.237,-1.50)--(-0.423,-1.48)--(-0.601,-1.47)--(-0.771,-1.45)--(-0.927,-1.42)--(-1.07,-1.38)--(-1.19,-1.34)--(-1.30,-1.30)--(-1.38,-1.25)--(-1.45,-1.19)--(-1.48,-1.13)--(-1.50,-1.07)--(-1.49,-1.00)--(-1.46,-0.927)--(-1.40,-0.851)--(-1.32,-0.771)--(-1.22,-0.687)--(-1.10,-0.601)--(-0.964,-0.513)--(-0.811,-0.423)--(-0.645,-0.330)--(-0.468,-0.237)--(-0.284,-0.143)--(-0.0951,-0.0476)--(0.0951,0.0476)--(0.284,0.143)--(0.468,0.237)--(0.645,0.330)--(0.811,0.423)--(0.964,0.513)--(1.10,0.601)--(1.22,0.687)--(1.32,0.771)--(1.40,0.851)--(1.46,0.927)--(1.49,1.00)--(1.50,1.07)--(1.48,1.13)--(1.45,1.19)--(1.38,1.25)--(1.30,1.30)--(1.19,1.34)--(1.07,1.38)--(0.927,1.42)--(0.771,1.45)--(0.601,1.47)--(0.423,1.48)--(0.237,1.50)--(0.0476,1.50)--(-0.143,1.50)--(-0.330,1.49)--(-0.513,1.48)--(-0.687,1.46)--(-0.851,1.43)--(-1.00,1.40)--(-1.13,1.36)--(-1.25,1.32)--(-1.34,1.27)--(-1.42,1.22)--(-1.47,1.16)--(-1.50,1.10)--(-1.50,1.04)--(-1.48,0.964)--(-1.43,0.889)--(-1.36,0.811)--(-1.27,0.729)--(-1.16,0.645)--(-1.04,0.557)--(-0.889,0.468)--(-0.729,0.377)--(-0.557,0.284)--(-0.377,0.190)--(-0.190,0.0951)--(0,0);
-\draw (-1.500000000,-0.3298256667) node {$ -1 $};
+\draw (-1.5000,-0.32983) node {$ -1 $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (1.500000000,-0.3149246667) node {$ 1 $};
+\draw (1.5000,-0.31492) node {$ 1 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (-0.4331593333,-1.500000000) node {$ -1 $};
+\draw (-0.43316,-1.5000) node {$ -1 $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.2912498333,1.500000000) node {$ 1 $};
+\draw (-0.29125,1.5000) node {$ 1 $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExempleNonRang.pstricks.recall b/src_phystricks/Fig_ExempleNonRang.pstricks.recall
index 446d77124..89185f4ea 100644
--- a/src_phystricks/Fig_ExempleNonRang.pstricks.recall
+++ b/src_phystricks/Fig_ExempleNonRang.pstricks.recall
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.328427125) -- (0,3.328427125);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.3284) -- (0,3.3284);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.04040,0.002871)--(0.08081,0.008121)--(0.1212,0.01492)--(0.1616,0.02297)--(0.2020,0.03210)--(0.2424,0.04220)--(0.2828,0.05318)--(0.3232,0.06497)--(0.3636,0.07753)--(0.4040,0.09080)--(0.4444,0.1048)--(0.4848,0.1194)--(0.5253,0.1346)--(0.5657,0.1504)--(0.6061,0.1668)--(0.6465,0.1838)--(0.6869,0.2013)--(0.7273,0.2193)--(0.7677,0.2378)--(0.8081,0.2568)--(0.8485,0.2763)--(0.8889,0.2963)--(0.9293,0.3167)--(0.9697,0.3376)--(1.010,0.3589)--(1.051,0.3807)--(1.091,0.4028)--(1.131,0.4254)--(1.172,0.4484)--(1.212,0.4718)--(1.253,0.4956)--(1.293,0.5198)--(1.333,0.5443)--(1.374,0.5693)--(1.414,0.5946)--(1.455,0.6202)--(1.495,0.6462)--(1.535,0.6726)--(1.576,0.6993)--(1.616,0.7264)--(1.657,0.7538)--(1.697,0.7816)--(1.737,0.8096)--(1.778,0.8381)--(1.818,0.8668)--(1.859,0.8958)--(1.899,0.9252)--(1.939,0.9549)--(1.980,0.9849)--(2.020,1.015)--(2.061,1.046)--(2.101,1.077)--(2.141,1.108)--(2.182,1.139)--(2.222,1.171)--(2.263,1.203)--(2.303,1.236)--(2.343,1.268)--(2.384,1.301)--(2.424,1.335)--(2.465,1.368)--(2.505,1.402)--(2.545,1.436)--(2.586,1.470)--(2.626,1.505)--(2.667,1.540)--(2.707,1.575)--(2.747,1.610)--(2.788,1.646)--(2.828,1.682)--(2.869,1.718)--(2.909,1.754)--(2.949,1.791)--(2.990,1.828)--(3.030,1.865)--(3.071,1.902)--(3.111,1.940)--(3.152,1.978)--(3.192,2.016)--(3.232,2.055)--(3.273,2.093)--(3.313,2.132)--(3.354,2.171)--(3.394,2.211)--(3.434,2.250)--(3.475,2.290)--(3.515,2.330)--(3.556,2.370)--(3.596,2.411)--(3.636,2.452)--(3.677,2.493)--(3.717,2.534)--(3.758,2.575)--(3.798,2.617)--(3.838,2.659)--(3.879,2.701)--(3.919,2.743)--(3.960,2.786)--(4.000,2.828);
diff --git a/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
index f7088ee5d..637a3a657 100644
--- a/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
+++ b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
@@ -65,30 +65,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.200000000) -- (0,1.199647580);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.2000) -- (0,1.1996);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -151,28 +151,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -239,34 +239,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796328,0) -- (2.856194490,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.499748271);
+\draw [,->,>=latex] (-2.0708,0) -- (2.8562,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4997);
 %DEFAULT
 
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-\draw (-2.000000000,-0.3298256667) node {$ -4 $};
+\draw (-2.0000,-0.32983) node {$ -4 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.500000000,-0.3298256667) node {$ -3 $};
+\draw (-1.5000,-0.32983) node {$ -3 $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -2 $};
+\draw (-1.0000,-0.32983) node {$ -2 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.5000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.50000,-0.32983) node {$ -1 $};
 \draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.5000000000,-0.3149246667) node {$ 1 $};
+\draw (0.50000,-0.31492) node {$ 1 $};
 \draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 2 $};
+\draw (1.0000,-0.31492) node {$ 2 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.500000000,-0.3149246667) node {$ 3 $};
+\draw (1.5000,-0.31492) node {$ 3 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 4 $};
+\draw (2.0000,-0.31492) node {$ 4 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.500000000,-0.3149246667) node {$ 5 $};
+\draw (2.5000,-0.31492) node {$ 5 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (-0.4331593333,-0.5000000000) node {$ -1 $};
+\draw (-0.43316,-0.50000) node {$ -1 $};
 \draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.2912498333,0.5000000000) node {$ 1 $};
+\draw (-0.29125,0.50000) node {$ 1 $};
 \draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -329,30 +329,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
 %DEFAULT
 
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-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -415,31 +415,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.199998598) -- (0,1.199904439);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.2000) -- (0,1.1999);
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+\draw (2.8000,-0.31492) node {$ 4 $};
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-\draw (3.500000000,-0.3149246667) node {$ 5 $};
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 %OTHER STUFF
 %END PSPICTURE
@@ -502,30 +502,30 @@
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 %AXES
-\draw [,->,>=latex] (-1.599557429,0) -- (4.898229717,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.200000000);
+\draw [,->,>=latex] (-1.5996,0) -- (4.8982,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
 %DEFAULT
 
 \draw [color=blue] (-1.100,0)--(-1.077,0.1247)--(-1.055,0.1763)--(-1.033,0.2158)--(-1.011,0.2491)--(-0.9885,0.2782)--(-0.9663,0.3045)--(-0.9441,0.3286)--(-0.9219,0.3508)--(-0.8996,0.3715)--(-0.8774,0.3910)--(-0.8552,0.4094)--(-0.8330,0.4268)--(-0.8108,0.4432)--(-0.7886,0.4589)--(-0.7664,0.4738)--(-0.7441,0.4881)--(-0.7219,0.5017)--(-0.6997,0.5147)--(-0.6775,0.5271)--(-0.6553,0.5390)--(-0.6331,0.5504)--(-0.6109,0.5612)--(-0.5887,0.5716)--(-0.5664,0.5815)--(-0.5442,0.5910)--(-0.5220,0.6000)--(-0.4998,0.6085)--(-0.4776,0.6167)--(-0.4554,0.6244)--(-0.4332,0.6318)--(-0.4109,0.6387)--(-0.3887,0.6453)--(-0.3665,0.6514)--(-0.3443,0.6572)--(-0.3221,0.6626)--(-0.2999,0.6676)--(-0.2777,0.6723)--(-0.2555,0.6766)--(-0.2332,0.6805)--(-0.2110,0.6840)--(-0.1888,0.6872)--(-0.1666,0.6901)--(-0.1444,0.6925)--(-0.1222,0.6947)--(-0.09996,0.6964)--(-0.07775,0.6978)--(-0.05553,0.6989)--(-0.03332,0.6996)--(-0.01111,0.7000)--(0.01111,0.7000)--(0.03332,0.6996)--(0.05553,0.6989)--(0.07775,0.6978)--(0.09996,0.6964)--(0.1222,0.6947)--(0.1444,0.6925)--(0.1666,0.6901)--(0.1888,0.6872)--(0.2110,0.6840)--(0.2332,0.6805)--(0.2555,0.6766)--(0.2777,0.6723)--(0.2999,0.6676)--(0.3221,0.6626)--(0.3443,0.6572)--(0.3665,0.6514)--(0.3887,0.6453)--(0.4109,0.6387)--(0.4332,0.6318)--(0.4554,0.6244)--(0.4776,0.6167)--(0.4998,0.6085)--(0.5220,0.6000)--(0.5442,0.5910)--(0.5664,0.5815)--(0.5887,0.5716)--(0.6109,0.5612)--(0.6331,0.5504)--(0.6553,0.5390)--(0.6775,0.5271)--(0.6997,0.5147)--(0.7219,0.5017)--(0.7441,0.4881)--(0.7664,0.4738)--(0.7886,0.4589)--(0.8108,0.4432)--(0.8330,0.4268)--(0.8552,0.4094)--(0.8774,0.3910)--(0.8996,0.3715)--(0.9219,0.3508)--(0.9441,0.3286)--(0.9663,0.3045)--(0.9885,0.2782)--(1.011,0.2491)--(1.033,0.2158)--(1.055,0.1763)--(1.077,0.1247)--(1.100,0);
 
 \draw [color=blue] (3.299,0)--(3.310,0.08817)--(3.321,0.1247)--(3.332,0.1527)--(3.343,0.1763)--(3.354,0.1971)--(3.365,0.2158)--(3.376,0.2330)--(3.388,0.2491)--(3.399,0.2641)--(3.410,0.2782)--(3.421,0.2917)--(3.432,0.3045)--(3.443,0.3168)--(3.454,0.3286)--(3.465,0.3399)--(3.476,0.3508)--(3.487,0.3613)--(3.499,0.3715)--(3.510,0.3814)--(3.521,0.3910)--(3.532,0.4003)--(3.543,0.4094)--(3.554,0.4182)--(3.565,0.4268)--(3.576,0.4351)--(3.587,0.4432)--(3.599,0.4512)--(3.610,0.4589)--(3.621,0.4665)--(3.632,0.4738)--(3.643,0.4811)--(3.654,0.4881)--(3.665,0.4950)--(3.676,0.5017)--(3.687,0.5083)--(3.699,0.5147)--(3.710,0.5210)--(3.721,0.5271)--(3.732,0.5331)--(3.743,0.5390)--(3.754,0.5447)--(3.765,0.5504)--(3.776,0.5559)--(3.787,0.5612)--(3.798,0.5665)--(3.810,0.5716)--(3.821,0.5766)--(3.832,0.5815)--(3.843,0.5863)--(3.854,0.5910)--(3.865,0.5955)--(3.876,0.6000)--(3.887,0.6043)--(3.898,0.6085)--(3.910,0.6127)--(3.921,0.6167)--(3.932,0.6206)--(3.943,0.6244)--(3.954,0.6282)--(3.965,0.6318)--(3.976,0.6353)--(3.987,0.6387)--(3.998,0.6420)--(4.010,0.6453)--(4.021,0.6484)--(4.032,0.6514)--(4.043,0.6544)--(4.054,0.6572)--(4.065,0.6600)--(4.076,0.6626)--(4.087,0.6652)--(4.098,0.6676)--(4.109,0.6700)--(4.121,0.6723)--(4.132,0.6745)--(4.143,0.6766)--(4.154,0.6786)--(4.165,0.6805)--(4.176,0.6823)--(4.187,0.6840)--(4.198,0.6857)--(4.209,0.6872)--(4.221,0.6887)--(4.232,0.6901)--(4.243,0.6913)--(4.254,0.6925)--(4.265,0.6936)--(4.276,0.6947)--(4.287,0.6956)--(4.298,0.6964)--(4.309,0.6972)--(4.320,0.6978)--(4.332,0.6984)--(4.343,0.6989)--(4.354,0.6993)--(4.365,0.6996)--(4.376,0.6998)--(4.387,0.7000)--(4.398,0.7000);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (4.200000000,-0.3149246667) node {$ 6 $};
+\draw (4.2000,-0.31492) node {$ 6 $};
 \draw [] (4.20,-0.100) -- (4.20,0.100);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
@@ -588,30 +588,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.699114859,0) -- (3.798672286,0);
-\draw [,->,>=latex] (0,-1.199647580) -- (0,1.200000000);
+\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
+\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
 %DEFAULT
 
 \draw [color=blue] (-2.199,0.7000)--(-2.144,0.6650)--(-2.088,0.5637)--(-2.033,0.4060)--(-1.977,0.2078)--(-1.921,-0.01111)--(-1.866,-0.2289)--(-1.810,-0.4239)--(-1.755,-0.5766)--(-1.699,-0.6716)--(-1.644,-0.6996)--(-1.588,-0.6578)--(-1.533,-0.5502)--(-1.477,-0.3877)--(-1.422,-0.1865)--(-1.366,0.03331)--(-1.311,0.2498)--(-1.255,0.4414)--(-1.200,0.5889)--(-1.144,0.6776)--(-1.088,0.6986)--(-1.033,0.6499)--(-0.9774,0.5362)--(-0.9219,0.3691)--(-0.8663,0.1650)--(-0.8108,-0.05547)--(-0.7552,-0.2704)--(-0.6997,-0.4584)--(-0.6442,-0.6006)--(-0.5887,-0.6828)--(-0.5331,-0.6968)--(-0.4776,-0.6413)--(-0.4221,-0.5217)--(-0.3665,-0.3500)--(-0.3110,-0.1434)--(-0.2555,0.07759)--(-0.1999,0.2908)--(-0.1444,0.4750)--(-0.08885,0.6117)--(-0.03332,0.6873)--(0.02221,0.6944)--(0.07775,0.6320)--(0.1333,0.5066)--(0.1888,0.3306)--(0.2443,0.1216)--(0.2999,-0.09962)--(0.3554,-0.3108)--(0.4109,-0.4910)--(0.4665,-0.6222)--(0.5220,-0.6912)--(0.5775,-0.6912)--(0.6331,-0.6222)--(0.6886,-0.4910)--(0.7441,-0.3108)--(0.7997,-0.09962)--(0.8552,0.1216)--(0.9107,0.3306)--(0.9663,0.5066)--(1.022,0.6320)--(1.077,0.6944)--(1.133,0.6873)--(1.188,0.6117)--(1.244,0.4750)--(1.299,0.2908)--(1.355,0.07759)--(1.411,-0.1434)--(1.466,-0.3500)--(1.522,-0.5217)--(1.577,-0.6413)--(1.633,-0.6968)--(1.688,-0.6828)--(1.744,-0.6006)--(1.799,-0.4584)--(1.855,-0.2704)--(1.910,-0.05547)--(1.966,0.1650)--(2.021,0.3691)--(2.077,0.5362)--(2.132,0.6499)--(2.188,0.6986)--(2.244,0.6776)--(2.299,0.5889)--(2.355,0.4414)--(2.410,0.2498)--(2.466,0.03331)--(2.521,-0.1865)--(2.577,-0.3877)--(2.632,-0.5502)--(2.688,-0.6578)--(2.743,-0.6996)--(2.799,-0.6716)--(2.854,-0.5766)--(2.910,-0.4239)--(2.965,-0.2289)--(3.021,-0.01111)--(3.077,0.2078)--(3.132,0.4060)--(3.188,0.5637)--(3.243,0.6650)--(3.299,0.7000);
-\draw (-2.100000000,-0.3298256667) node {$ -3 $};
+\draw (-2.1000,-0.32983) node {$ -3 $};
 \draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.400000000,-0.3298256667) node {$ -2 $};
+\draw (-1.4000,-0.32983) node {$ -2 $};
 \draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.7000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.70000,-0.32983) node {$ -1 $};
 \draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.7000000000,-0.3149246667) node {$ 1 $};
+\draw (0.70000,-0.31492) node {$ 1 $};
 \draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.400000000,-0.3149246667) node {$ 2 $};
+\draw (1.4000,-0.31492) node {$ 2 $};
 \draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.100000000,-0.3149246667) node {$ 3 $};
+\draw (2.1000,-0.31492) node {$ 3 $};
 \draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.800000000,-0.3149246667) node {$ 4 $};
+\draw (2.8000,-0.31492) node {$ 4 $};
 \draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.500000000,-0.3149246667) node {$ 5 $};
+\draw (3.5000,-0.31492) node {$ 5 $};
 \draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExoMagnetique.pstricks.recall b/src_phystricks/Fig_ExoMagnetique.pstricks.recall
index fe440b24b..6466d25a8 100644
--- a/src_phystricks/Fig_ExoMagnetique.pstricks.recall
+++ b/src_phystricks/Fig_ExoMagnetique.pstricks.recall
@@ -78,12 +78,12 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [style=dashed] (0,-2.00) -- (0,2.00);
-\draw [color=red,->,>=latex] (0,0) -- (0,1.000000000);
-\draw (0.3945450000,1.000000000) node {$I$};
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-2.000000000,1.000000000);
-\draw (-1.763589810,0.7318426438) node {$d$};
-\draw []  (-2.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-2.706459690,1.323876356) node {$(r,\theta,z)$};
+\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
+\draw (0.39455,1.0000) node {$I$};
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-2.0000,1.0000);
+\draw (-1.7636,0.73184) node {$d$};
+\draw []  (-2.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-2.7065,1.3239) node {$(r,\theta,z)$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ExoPolaire.pstricks.recall b/src_phystricks/Fig_ExoPolaire.pstricks.recall
index 853914bfa..6980c9bb4 100644
--- a/src_phystricks/Fig_ExoPolaire.pstricks.recall
+++ b/src_phystricks/Fig_ExoPolaire.pstricks.recall
@@ -75,15 +75,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.232050808,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.2321,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
 %DEFAULT
-\draw (2.389616308,1.000000000) node {$(\sqrt{3},1)$};
-\draw (0.6579799038,0.8865436211) node {$l$};
-\draw (1.012732295,0.2561455226) node {$\theta$};
+\draw (2.3896,1.0000) node {$(\sqrt{3},1)$};
+\draw (0.65798,0.88654) node {$l$};
+\draw (1.0127,0.25615) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw [,->,>=latex] (0,0) -- (1.732050808,1.000000000);
+\draw [,->,>=latex] (0,0) -- (1.7320,1.0000);
 \draw [] (1.00,-0.100) -- (1.00,0.100);
 \draw [] (2.00,-0.100) -- (2.00,0.100);
 \draw [] (-0.100,1.00) -- (0.100,1.00);
diff --git a/src_phystricks/Fig_ExoProjection.pstricks.recall b/src_phystricks/Fig_ExoProjection.pstricks.recall
index 2c166c030..7d7fed300 100644
--- a/src_phystricks/Fig_ExoProjection.pstricks.recall
+++ b/src_phystricks/Fig_ExoProjection.pstricks.recall
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.600000000,0) -- (3.300000000,0);
-\draw [,->,>=latex] (0,-1.550000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-2.6000,0) -- (3.3000,0);
+\draw [,->,>=latex] (0,-1.5500) -- (0,3.5000);
 %DEFAULT
 \draw [style=dashed] (2.80,1.40) -- (-2.10,-1.05);
-\draw []  (2.400000000,1.200000000) node [rotate=0] {$\bullet$};
-\draw (3.246325678,0.9435588219) node {$\pr_w(A)$};
-\draw [color=red,->,>=latex] (1.500000000,3.000000000) -- (2.400000000,1.200000000);
-\draw [color=blue,->,>=latex] (0,0) -- (1.400000000,0.7000000000);
-\draw (1.071598050,1.008389500) node {$w$};
-\draw []  (-1.000000000,-0.5000000000) node [rotate=0] {$\bullet$};
-\draw (-1.390882667,-0.7824550000) node {$P(\lambda)$};
-\draw [color=cyan,->,>=latex] (1.500000000,3.000000000) -- (-1.000000000,-0.5000000000);
-\draw []  (1.500000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (1.500000000,3.424708000) node {$A$};
+\draw []  (2.4000,1.2000) node [rotate=0] {$\bullet$};
+\draw (3.2463,0.94356) node {$\pr_w(A)$};
+\draw [color=red,->,>=latex] (1.5000,3.0000) -- (2.4000,1.2000);
+\draw [color=blue,->,>=latex] (0,0) -- (1.4000,0.70000);
+\draw (1.0716,1.0084) node {$w$};
+\draw []  (-1.0000,-0.50000) node [rotate=0] {$\bullet$};
+\draw (-1.3909,-0.78246) node {$P(\lambda)$};
+\draw [color=cyan,->,>=latex] (1.5000,3.0000) -- (-1.0000,-0.50000);
+\draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
+\draw (1.5000,3.4247) node {$A$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExoVarj.pstricks.recall b/src_phystricks/Fig_ExoVarj.pstricks.recall
index 5fc4b4037..9f183c7db 100644
--- a/src_phystricks/Fig_ExoVarj.pstricks.recall
+++ b/src_phystricks/Fig_ExoVarj.pstricks.recall
@@ -71,16 +71,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-1.999875734) -- (0,1.999875734);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-1.9999) -- (0,1.9999);
 %DEFAULT
 
 \draw [color=blue] (-3.00,1.46)--(-2.94,1.47)--(-2.88,1.48)--(-2.82,1.49)--(-2.76,1.49)--(-2.70,1.50)--(-2.64,1.50)--(-2.58,1.50)--(-2.52,1.50)--(-2.45,1.49)--(-2.39,1.49)--(-2.33,1.48)--(-2.27,1.47)--(-2.21,1.46)--(-2.15,1.45)--(-2.09,1.44)--(-2.03,1.42)--(-1.97,1.41)--(-1.91,1.39)--(-1.85,1.37)--(-1.79,1.34)--(-1.73,1.32)--(-1.67,1.29)--(-1.61,1.27)--(-1.55,1.24)--(-1.48,1.21)--(-1.42,1.18)--(-1.36,1.14)--(-1.30,1.11)--(-1.24,1.07)--(-1.18,1.03)--(-1.12,0.990)--(-1.06,0.947)--(-1.00,0.904)--(-0.939,0.858)--(-0.879,0.812)--(-0.818,0.763)--(-0.758,0.713)--(-0.697,0.662)--(-0.636,0.609)--(-0.576,0.556)--(-0.515,0.501)--(-0.455,0.444)--(-0.394,0.387)--(-0.333,0.329)--(-0.273,0.271)--(-0.212,0.211)--(-0.152,0.151)--(-0.0909,0.0908)--(-0.0303,0.0303)--(0.0303,0.0303)--(0.0909,0.0908)--(0.152,0.151)--(0.212,0.211)--(0.273,0.271)--(0.333,0.329)--(0.394,0.387)--(0.455,0.444)--(0.515,0.501)--(0.576,0.556)--(0.636,0.609)--(0.697,0.662)--(0.758,0.713)--(0.818,0.763)--(0.879,0.812)--(0.939,0.858)--(1.00,0.904)--(1.06,0.947)--(1.12,0.990)--(1.18,1.03)--(1.24,1.07)--(1.30,1.11)--(1.36,1.14)--(1.42,1.18)--(1.48,1.21)--(1.55,1.24)--(1.61,1.27)--(1.67,1.29)--(1.73,1.32)--(1.79,1.34)--(1.85,1.37)--(1.91,1.39)--(1.97,1.41)--(2.03,1.42)--(2.09,1.44)--(2.15,1.45)--(2.21,1.46)--(2.27,1.47)--(2.33,1.48)--(2.39,1.49)--(2.45,1.49)--(2.52,1.50)--(2.58,1.50)--(2.64,1.50)--(2.70,1.50)--(2.76,1.49)--(2.82,1.49)--(2.88,1.48)--(2.94,1.47)--(3.00,1.46);
 
 \draw [color=red] (-3.00,-1.46)--(-2.94,-1.47)--(-2.88,-1.48)--(-2.82,-1.49)--(-2.76,-1.49)--(-2.70,-1.50)--(-2.64,-1.50)--(-2.58,-1.50)--(-2.52,-1.50)--(-2.45,-1.49)--(-2.39,-1.49)--(-2.33,-1.48)--(-2.27,-1.47)--(-2.21,-1.46)--(-2.15,-1.45)--(-2.09,-1.44)--(-2.03,-1.42)--(-1.97,-1.41)--(-1.91,-1.39)--(-1.85,-1.37)--(-1.79,-1.34)--(-1.73,-1.32)--(-1.67,-1.29)--(-1.61,-1.27)--(-1.55,-1.24)--(-1.48,-1.21)--(-1.42,-1.18)--(-1.36,-1.14)--(-1.30,-1.11)--(-1.24,-1.07)--(-1.18,-1.03)--(-1.12,-0.990)--(-1.06,-0.947)--(-1.00,-0.904)--(-0.939,-0.858)--(-0.879,-0.812)--(-0.818,-0.763)--(-0.758,-0.713)--(-0.697,-0.662)--(-0.636,-0.609)--(-0.576,-0.556)--(-0.515,-0.501)--(-0.455,-0.444)--(-0.394,-0.387)--(-0.333,-0.329)--(-0.273,-0.271)--(-0.212,-0.211)--(-0.152,-0.151)--(-0.0909,-0.0908)--(-0.0303,-0.0303)--(0.0303,-0.0303)--(0.0909,-0.0908)--(0.152,-0.151)--(0.212,-0.211)--(0.273,-0.271)--(0.333,-0.329)--(0.394,-0.387)--(0.455,-0.444)--(0.515,-0.501)--(0.576,-0.556)--(0.636,-0.609)--(0.697,-0.662)--(0.758,-0.713)--(0.818,-0.763)--(0.879,-0.812)--(0.939,-0.858)--(1.00,-0.904)--(1.06,-0.947)--(1.12,-0.990)--(1.18,-1.03)--(1.24,-1.07)--(1.30,-1.11)--(1.36,-1.14)--(1.42,-1.18)--(1.48,-1.21)--(1.55,-1.24)--(1.61,-1.27)--(1.67,-1.29)--(1.73,-1.32)--(1.79,-1.34)--(1.85,-1.37)--(1.91,-1.39)--(1.97,-1.41)--(2.03,-1.42)--(2.09,-1.44)--(2.15,-1.45)--(2.21,-1.46)--(2.27,-1.47)--(2.33,-1.48)--(2.39,-1.49)--(2.45,-1.49)--(2.52,-1.50)--(2.58,-1.50)--(2.64,-1.50)--(2.70,-1.50)--(2.76,-1.49)--(2.82,-1.49)--(2.88,-1.48)--(2.94,-1.47)--(3.00,-1.46);
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExoXLVL.pstricks.recall b/src_phystricks/Fig_ExoXLVL.pstricks.recall
index 5007c3cbe..9dd1f83e3 100644
--- a/src_phystricks/Fig_ExoXLVL.pstricks.recall
+++ b/src_phystricks/Fig_ExoXLVL.pstricks.recall
@@ -79,29 +79,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
-\draw [color=blue] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue] (-2.50,2.50) -- (-2.50,0.100);
+\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
+\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
+\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
+\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
 \draw [color=red] (0,2.50) -- (2.50,2.50);
 \draw [color=red] (2.50,2.50) -- (2.50,0);
 \draw [color=red] (2.50,0) -- (0,0);
 \draw [color=red] (0,0) -- (0,2.50);
-\draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
-\draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
-\draw [color=cyan] (-2.50,0) -- (0,0);
-\draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green] (0.100,-0.100) -- (0.100,-2.50);
-\draw (-1.099672167,1.300000000) node {\( xy\)};
-\draw (1.673347000,1.250000000) node {\( x-y\)};
-\draw (-0.9705055000,-1.250000000) node {\( x^2y\)};
-\draw (1.723347000,-1.300000000) node {\( x+y\)};
+\draw [color=cyan,style=dashed] (-0.100,-2.50) -- (-2.50,-2.50);
+\draw [color=cyan,style=dashed] (-2.50,-2.50) -- (-2.50,-0.100);
+\draw [color=cyan,style=dashed] (-2.50,-0.100) -- (-0.100,-0.100);
+\draw [color=cyan,style=dashed] (-0.100,-0.100) -- (-0.100,-2.50);
+\draw [color=green] (0,-2.50) -- (2.50,-2.50);
+\draw [color=green] (2.50,-2.50) -- (2.50,0);
+\draw [color=green] (2.50,0) -- (0,0);
+\draw [color=green] (0,0) -- (0,-2.50);
+\draw (-1.0997,1.3000) node {\( xy\)};
+\draw (1.6733,1.2500) node {\( x-y\)};
+\draw (-1.0205,-1.3000) node {\( x^2y\)};
+\draw (1.6733,-1.2500) node {\( x+y\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall b/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall
index 042e0423e..aacb17a58 100644
--- a/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall
+++ b/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall
@@ -99,7 +99,7 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (1.456249167,-1.184375000) node {$
+\draw (1.4562,-1.1844) node {$
         \begin{pmatrix}
         \phantom{
         \begin{matrix}
@@ -108,31 +108,31 @@
         }
         \end{pmatrix}$
         };
-\draw (0.2912498333,-0.2368750000) node {*};
-\draw (0.8737495000,-0.2368750000) node {*};
-\draw (1.456249167,-0.2368750000) node {*};
-\draw (2.038748833,-0.2368750000) node {*};
-\draw (2.621248500,-0.2368750000) node {*};
-\draw (0.2912498333,-0.7106250000) node {0};
-\draw (0.8737495000,-0.7106250000) node {*};
-\draw (1.456249167,-0.7106250000) node {*};
-\draw (2.038748833,-0.7106250000) node {*};
-\draw (2.621248500,-0.7106250000) node {*};
-\draw (0.2912498333,-1.184375000) node {0};
-\draw (0.8737495000,-1.184375000) node {0};
-\draw (1.456249167,-1.184375000) node {*};
-\draw (2.038748833,-1.184375000) node {*};
-\draw (2.621248500,-1.184375000) node {*};
-\draw (0.2912498333,-1.658125000) node {0};
-\draw (0.8737495000,-1.658125000) node {0};
-\draw (1.456249167,-1.658125000) node {*};
-\draw (2.038748833,-1.658125000) node {*};
-\draw (2.621248500,-1.658125000) node {*};
-\draw (0.2912498333,-2.131875000) node {0};
-\draw (0.8737495000,-2.131875000) node {0};
-\draw (1.456249167,-2.131875000) node {*};
-\draw (2.038748833,-2.131875000) node {*};
-\draw (2.621248500,-2.131875000) node {*};
+\draw (0.29125,-0.23688) node {*};
+\draw (0.87375,-0.23688) node {*};
+\draw (1.4562,-0.23688) node {*};
+\draw (2.0387,-0.23688) node {*};
+\draw (2.6213,-0.23688) node {*};
+\draw (0.29125,-0.71062) node {0};
+\draw (0.87375,-0.71062) node {*};
+\draw (1.4562,-0.71062) node {*};
+\draw (2.0387,-0.71062) node {*};
+\draw (2.6213,-0.71062) node {*};
+\draw (0.29125,-1.1844) node {0};
+\draw (0.87375,-1.1844) node {0};
+\draw (1.4562,-1.1844) node {*};
+\draw (2.0387,-1.1844) node {*};
+\draw (2.6213,-1.1844) node {*};
+\draw (0.29125,-1.6581) node {0};
+\draw (0.87375,-1.6581) node {0};
+\draw (1.4562,-1.6581) node {*};
+\draw (2.0387,-1.6581) node {*};
+\draw (2.6213,-1.6581) node {*};
+\draw (0.29125,-2.1319) node {0};
+\draw (0.87375,-2.1319) node {0};
+\draw (1.4562,-2.1319) node {*};
+\draw (2.0387,-2.1319) node {*};
+\draw (2.6213,-2.1319) node {*};
 \draw [color=red] (0.100,-0.0500) -- (1.06,-0.0500);
 \draw [color=red] (1.06,-0.0500) -- (1.06,-0.898);
 \draw [color=red] (1.06,-0.898) -- (0.100,-0.898);
@@ -141,8 +141,8 @@
 \draw [color=blue] (2.81,-0.997) -- (2.81,-2.32);
 \draw [color=blue] (2.81,-2.32) -- (1.27,-2.32);
 \draw [color=blue] (1.27,-2.32) -- (1.27,-0.997);
-\draw (1.212378848,0.2264242621) node {\( \Delta_k(A_2)\)};
-\draw (2.038748833,-2.601205000) node {\( \Omega_{k+1}(A_2)\)};
+\draw (1.2124,0.22642) node {\( \Delta_k(A_2)\)};
+\draw (2.0387,-2.6012) node {\( \Omega_{k+1}(A_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_FGRooDhFkch.pstricks.recall b/src_phystricks/Fig_FGRooDhFkch.pstricks.recall
index 7b0b588b1..f4993c771 100644
--- a/src_phystricks/Fig_FGRooDhFkch.pstricks.recall
+++ b/src_phystricks/Fig_FGRooDhFkch.pstricks.recall
@@ -62,6 +62,10 @@
 \immediate\write\writeOfphystricks{totalheightof2f9abdf237016df2db0c349fe50e30dbad8bcb6c:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ 1 $}}%
 \immediate\write\writeOfphystricks{widthof2f9abdf237016df2db0c349fe50e30dbad8bcb6c:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$ -\pi $}}%
+\immediate\write\writeOfphystricks{totalheightofae0a42324905a92dfd1fa03440abc5b447c8e6b8:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$ -\pi $}}%
+\immediate\write\writeOfphystricks{widthofae0a42324905a92dfd1fa03440abc5b447c8e6b8:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\totalheightof{$ -\frac{1}{2} \, \pi $}}%
 \immediate\write\writeOfphystricks{totalheightof8244d948eb981a434cc8bc83b6ec2b87e2ede7b5:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ -\frac{1}{2} \, \pi $}}%
@@ -70,6 +74,10 @@
 \immediate\write\writeOfphystricks{totalheightof85c8661aa83890a29f3568c5ed92d1300142f127:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ \frac{1}{2} \, \pi $}}%
 \immediate\write\writeOfphystricks{widthof85c8661aa83890a29f3568c5ed92d1300142f127:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$ \pi $}}%
+\immediate\write\writeOfphystricks{totalheightof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$ \pi $}}%
+\immediate\write\writeOfphystricks{widthof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
 %CLOSE_WRITE_AND_LABEL
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
@@ -78,31 +86,39 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-3.00,-1.57) -- (-3.00,1.57);
-\draw [color=gray,style=solid] (0,-1.57) -- (0,1.57);
-\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,1.57);
-\draw [color=gray,style=dotted] (-1.50,-1.57) -- (-1.50,1.57);
-\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,1.57);
+\draw [color=gray,style=solid] (-3.00,-3.14) -- (-3.00,3.14);
+\draw [color=gray,style=solid] (0,-3.14) -- (0,3.14);
+\draw [color=gray,style=solid] (3.00,-3.14) -- (3.00,3.14);
+\draw [color=gray,style=dotted] (-1.50,-3.14) -- (-1.50,3.14);
+\draw [color=gray,style=dotted] (1.50,-3.14) -- (1.50,3.14);
+\draw [color=gray,style=dotted] (-3.00,-2.36) -- (3.00,-2.36);
 \draw [color=gray,style=dotted] (-3.00,-0.785) -- (3.00,-0.785);
 \draw [color=gray,style=dotted] (-3.00,0.785) -- (3.00,0.785);
+\draw [color=gray,style=dotted] (-3.00,2.36) -- (3.00,2.36);
+\draw [color=gray,style=solid] (-3.00,-3.14) -- (3.00,-3.14);
 \draw [color=gray,style=solid] (-3.00,-1.57) -- (3.00,-1.57);
 \draw [color=gray,style=solid] (-3.00,0) -- (3.00,0);
 \draw [color=gray,style=solid] (-3.00,1.57) -- (3.00,1.57);
+\draw [color=gray,style=solid] (-3.00,3.14) -- (3.00,3.14);
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-2.070796327) -- (0,2.070796327);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.6416) -- (0,3.6416);
 %DEFAULT
 
 \draw [color=blue] (-3.000,-1.571)--(-2.939,-1.369)--(-2.879,-1.286)--(-2.818,-1.221)--(-2.758,-1.166)--(-2.697,-1.117)--(-2.636,-1.073)--(-2.576,-1.033)--(-2.515,-0.9943)--(-2.455,-0.9582)--(-2.394,-0.9239)--(-2.333,-0.8911)--(-2.273,-0.8596)--(-2.212,-0.8292)--(-2.152,-0.7997)--(-2.091,-0.7712)--(-2.030,-0.7434)--(-1.970,-0.7163)--(-1.909,-0.6898)--(-1.848,-0.6639)--(-1.788,-0.6385)--(-1.727,-0.6135)--(-1.667,-0.5890)--(-1.606,-0.5649)--(-1.545,-0.5412)--(-1.485,-0.5178)--(-1.424,-0.4947)--(-1.364,-0.4719)--(-1.303,-0.4493)--(-1.242,-0.4270)--(-1.182,-0.4049)--(-1.121,-0.3830)--(-1.061,-0.3613)--(-1.000,-0.3398)--(-0.9394,-0.3185)--(-0.8788,-0.2973)--(-0.8182,-0.2762)--(-0.7576,-0.2553)--(-0.6970,-0.2345)--(-0.6364,-0.2137)--(-0.5758,-0.1931)--(-0.5152,-0.1726)--(-0.4545,-0.1521)--(-0.3939,-0.1317)--(-0.3333,-0.1113)--(-0.2727,-0.09103)--(-0.2121,-0.07077)--(-0.1515,-0.05053)--(-0.09091,-0.03031)--(-0.03030,-0.01010)--(0.03030,0.01010)--(0.09091,0.03031)--(0.1515,0.05053)--(0.2121,0.07077)--(0.2727,0.09103)--(0.3333,0.1113)--(0.3939,0.1317)--(0.4545,0.1521)--(0.5152,0.1726)--(0.5758,0.1931)--(0.6364,0.2137)--(0.6970,0.2345)--(0.7576,0.2553)--(0.8182,0.2762)--(0.8788,0.2973)--(0.9394,0.3185)--(1.000,0.3398)--(1.061,0.3613)--(1.121,0.3830)--(1.182,0.4049)--(1.242,0.4270)--(1.303,0.4493)--(1.364,0.4719)--(1.424,0.4947)--(1.485,0.5178)--(1.545,0.5412)--(1.606,0.5649)--(1.667,0.5890)--(1.727,0.6135)--(1.788,0.6385)--(1.848,0.6639)--(1.909,0.6898)--(1.970,0.7163)--(2.030,0.7434)--(2.091,0.7712)--(2.152,0.7997)--(2.212,0.8292)--(2.273,0.8596)--(2.333,0.8911)--(2.394,0.9239)--(2.455,0.9582)--(2.515,0.9943)--(2.576,1.033)--(2.636,1.073)--(2.697,1.117)--(2.758,1.166)--(2.818,1.221)--(2.879,1.286)--(2.939,1.369)--(3.000,1.571);
 
-\draw (-3.000000000,-0.3298256667) node {$ -1 $};
+\draw (-3.0000,-0.32983) node {$ -1 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.5937365000,-1.570796327) node {$ -\frac{1}{2} \, \pi $};
+\draw (-0.45249,-3.1416) node {$ -\pi $};
+\draw [] (-0.100,-3.14) -- (0.100,-3.14);
+\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.4518270000,1.570796327) node {$ \frac{1}{2} \, \pi $};
+\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
 \draw [] (-0.100,1.57) -- (0.100,1.57);
+\draw (-0.31058,3.1416) node {$ \pi $};
+\draw [] (-0.100,3.14) -- (0.100,3.14);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_FGWjJBX.pstricks.recall b/src_phystricks/Fig_FGWjJBX.pstricks.recall
index 1af925b5d..78a04f112 100644
--- a/src_phystricks/Fig_FGWjJBX.pstricks.recall
+++ b/src_phystricks/Fig_FGWjJBX.pstricks.recall
@@ -88,16 +88,16 @@
 \draw [] (3.00,0) -- (4.00,0.500);
 \draw [] (3.00,0) -- (4.00,-0.500);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.3059510000) node {\( \alpha_1\)};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,0.3059510000) node {\( \alpha_2\)};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0) node [rotate=0] {$\bullet$};
-\draw (3.000000000,0.3215388333) node {\( \alpha_{l-2}\)};
-\draw []  (4.000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.8215388333) node {\( \alpha_{l-1}\)};
-\draw []  (4.000000000,-0.5000000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.1916921667) node {\( \alpha_l\)};
+\draw (0,0.30595) node {\( \alpha_1\)};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.30595) node {\( \alpha_2\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
+\draw (3.0000,0.32154) node {\( \alpha_{l-2}\)};
+\draw []  (4.0000,0.50000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.82154) node {\( \alpha_{l-1}\)};
+\draw []  (4.0000,-0.50000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.19169) node {\( \alpha_l\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall b/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall
index 33fb116ff..1957bbd5f 100644
--- a/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall
+++ b/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall
@@ -75,12 +75,12 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=blue] (3.000,0)--(2.994,0.06342)--(2.976,0.1266)--(2.946,0.1893)--(2.904,0.2511)--(2.850,0.3120)--(2.785,0.3717)--(2.709,0.4298)--(2.622,0.4862)--(2.524,0.5406)--(2.416,0.5929)--(2.298,0.6428)--(2.171,0.6901)--(2.036,0.7346)--(1.892,0.7761)--(1.740,0.8146)--(1.582,0.8497)--(1.417,0.8815)--(1.246,0.9096)--(1.071,0.9342)--(0.8908,0.9549)--(0.7073,0.9718)--(0.5209,0.9848)--(0.3325,0.9938)--(0.1427,0.9989)--(-0.04760,0.9999)--(-0.2377,0.9969)--(-0.4269,0.9898)--(-0.6144,0.9788)--(-0.7994,0.9638)--(-0.9812,0.9450)--(-1.159,0.9224)--(-1.332,0.8960)--(-1.500,0.8660)--(-1.662,0.8326)--(-1.817,0.7958)--(-1.965,0.7558)--(-2.104,0.7127)--(-2.236,0.6668)--(-2.358,0.6182)--(-2.471,0.5671)--(-2.574,0.5137)--(-2.667,0.4582)--(-2.748,0.4009)--(-2.819,0.3420)--(-2.878,0.2817)--(-2.926,0.2203)--(-2.962,0.1580)--(-2.986,0.09506)--(-2.999,0.03173)--(-2.999,-0.03173)--(-2.986,-0.09506)--(-2.962,-0.1580)--(-2.926,-0.2203)--(-2.878,-0.2817)--(-2.819,-0.3420)--(-2.748,-0.4009)--(-2.667,-0.4582)--(-2.574,-0.5137)--(-2.471,-0.5671)--(-2.358,-0.6182)--(-2.236,-0.6668)--(-2.104,-0.7127)--(-1.965,-0.7558)--(-1.817,-0.7958)--(-1.662,-0.8326)--(-1.500,-0.8660)--(-1.332,-0.8960)--(-1.159,-0.9224)--(-0.9812,-0.9450)--(-0.7994,-0.9638)--(-0.6144,-0.9788)--(-0.4269,-0.9898)--(-0.2377,-0.9969)--(-0.04760,-0.9999)--(0.1427,-0.9989)--(0.3325,-0.9938)--(0.5209,-0.9848)--(0.7073,-0.9718)--(0.8908,-0.9549)--(1.071,-0.9342)--(1.246,-0.9096)--(1.417,-0.8815)--(1.582,-0.8497)--(1.740,-0.8146)--(1.892,-0.7761)--(2.036,-0.7346)--(2.171,-0.6901)--(2.298,-0.6428)--(2.416,-0.5929)--(2.524,-0.5406)--(2.622,-0.4862)--(2.709,-0.4298)--(2.785,-0.3717)--(2.850,-0.3120)--(2.904,-0.2511)--(2.946,-0.1893)--(2.976,-0.1266)--(2.994,-0.06342)--(3.000,0);
-\draw []  (2.121320344,0.7071067812) node [rotate=0] {$\bullet$};
-\draw (2.011554297,0.9856083399) node {\( x\)};
-\draw [,->,>=latex] (2.121320344,0.7071067812) -- (2.437548110,1.655790079);
-\draw (1.761496450,1.901490632) node {\( \nabla q(x)\)};
-\draw [,->,>=latex] (2.121320344,0.7071067812) -- (2.148632243,-0.2925201793);
-\draw (2.589736468,-0.1623497994) node {\( Ax\)};
+\draw []  (2.1213,0.70711) node [rotate=0] {$\bullet$};
+\draw (2.0116,0.98561) node {\( x\)};
+\draw [,->,>=latex] (2.1213,0.70711) -- (2.4375,1.6558);
+\draw (1.7615,1.9015) node {\( \nabla q(x)\)};
+\draw [,->,>=latex] (2.1213,0.70711) -- (2.1486,-0.29252);
+\draw (2.5897,-0.16235) node {\( Ax\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_FWJuNhU.pstricks.recall b/src_phystricks/Fig_FWJuNhU.pstricks.recall
index ef8cb70ee..c5cad1966 100644
--- a/src_phystricks/Fig_FWJuNhU.pstricks.recall
+++ b/src_phystricks/Fig_FWJuNhU.pstricks.recall
@@ -79,29 +79,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
-\draw [color=blue] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red] (0,2.50) -- (2.50,2.50);
-\draw [color=red] (2.50,2.50) -- (2.50,0);
-\draw [color=red] (2.50,0) -- (0,0);
-\draw [color=red] (0,0) -- (0,2.50);
+\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
+\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
+\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
+\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
+\draw [color=red,style=dashed] (0,2.50) -- (2.50,2.50);
+\draw [color=red,style=dashed] (2.50,2.50) -- (2.50,0);
+\draw [color=red,style=dashed] (2.50,0) -- (0,0);
+\draw [color=red,style=dashed] (0,0) -- (0,2.50);
 \draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
 \draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
 \draw [color=cyan] (-2.50,0) -- (0,0);
 \draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green] (0.100,-0.100) -- (0.100,-2.50);
-\draw (-1.099672167,1.300000000) node {\( xy\)};
-\draw (1.673347000,1.250000000) node {\( x-y\)};
-\draw (-0.9705055000,-1.250000000) node {\( x^2y\)};
-\draw (1.723347000,-1.300000000) node {\( x+y\)};
+\draw [color=green,style=dashed] (0.100,-2.50) -- (2.50,-2.50);
+\draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
+\draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
+\draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
+\draw (-1.0997,1.3000) node {\( xy\)};
+\draw (1.6733,1.2500) node {\( x-y\)};
+\draw (-0.97051,-1.2500) node {\( x^2y\)};
+\draw (1.7233,-1.3000) node {\( x+y\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall b/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
index 85c01337c..5fcac2505 100644
--- a/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
+++ b/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
@@ -119,40 +119,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
+\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
 %DEFAULT
 
 \draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
 
 \draw [color=red] (-3.500,3.555)--(-3.429,3.500)--(-3.359,3.406)--(-3.288,3.277)--(-3.217,3.114)--(-3.146,2.921)--(-3.076,2.702)--(-3.005,2.459)--(-2.934,2.198)--(-2.864,1.921)--(-2.793,1.632)--(-2.722,1.337)--(-2.652,1.038)--(-2.581,0.7401)--(-2.510,0.4468)--(-2.439,0.1618)--(-2.369,-0.1114)--(-2.298,-0.3696)--(-2.227,-0.6099)--(-2.157,-0.8297)--(-2.086,-1.027)--(-2.015,-1.199)--(-1.944,-1.346)--(-1.874,-1.466)--(-1.803,-1.558)--(-1.732,-1.622)--(-1.662,-1.658)--(-1.591,-1.667)--(-1.520,-1.650)--(-1.449,-1.608)--(-1.379,-1.542)--(-1.308,-1.456)--(-1.237,-1.350)--(-1.167,-1.228)--(-1.096,-1.092)--(-1.025,-0.9453)--(-0.9545,-0.7902)--(-0.8838,-0.6299)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.002540)--(-0.5303,0.1441)--(-0.4596,0.2739)--(-0.3889,0.3896)--(-0.3182,0.4892)--(-0.2475,0.5710)--(-0.1768,0.6336)--(-0.1061,0.6760)--(-0.03535,0.6973)--(0.03535,0.6973)--(0.1061,0.6760)--(0.1768,0.6336)--(0.2475,0.5710)--(0.3182,0.4892)--(0.3889,0.3896)--(0.4596,0.2739)--(0.5303,0.1441)--(0.6010,0.002540)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6299)--(0.9545,-0.7902)--(1.025,-0.9453)--(1.096,-1.092)--(1.167,-1.228)--(1.237,-1.350)--(1.308,-1.456)--(1.379,-1.542)--(1.449,-1.608)--(1.520,-1.650)--(1.591,-1.667)--(1.662,-1.658)--(1.732,-1.622)--(1.803,-1.558)--(1.874,-1.466)--(1.944,-1.346)--(2.015,-1.199)--(2.086,-1.027)--(2.157,-0.8297)--(2.227,-0.6099)--(2.298,-0.3696)--(2.369,-0.1114)--(2.439,0.1618)--(2.510,0.4468)--(2.581,0.7401)--(2.652,1.038)--(2.722,1.337)--(2.793,1.632)--(2.864,1.921)--(2.934,2.198)--(3.005,2.459)--(3.076,2.702)--(3.146,2.921)--(3.217,3.114)--(3.288,3.277)--(3.359,3.406)--(3.429,3.500)--(3.500,3.555);
-\draw (-3.298672286,-0.4207143333) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.199114858,-0.3210296667) node {$-\mathit{\pi}$};
+\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
 \draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.099557429,-0.4207143333) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.099557429,-0.4207143333) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.199114858,-0.2785761667) node {$\mathit{\pi}$};
+\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
 \draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.298672286,-0.4207143333) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall b/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall
index 81ec9b593..8c1bda278 100644
--- a/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall
+++ b/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall
@@ -91,30 +91,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.400000000,0) -- (1.400000000,0);
-\draw [,->,>=latex] (0,-2.525000000) -- (0,4.550000000);
+\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
+\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
 %DEFAULT
 
 \draw [color=blue] (-0.9000,-2.025)--(-0.8818,-1.905)--(-0.8636,-1.789)--(-0.8455,-1.679)--(-0.8273,-1.573)--(-0.8091,-1.471)--(-0.7909,-1.374)--(-0.7727,-1.282)--(-0.7545,-1.193)--(-0.7364,-1.109)--(-0.7182,-1.029)--(-0.7000,-0.9528)--(-0.6818,-0.8804)--(-0.6636,-0.8119)--(-0.6455,-0.7470)--(-0.6273,-0.6856)--(-0.6091,-0.6277)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5364,-0.4286)--(-0.5182,-0.3865)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4455,-0.2455)--(-0.4273,-0.2167)--(-0.4091,-0.1902)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1238)--(-0.3364,-0.1057)--(-0.3182,-0.08948)--(-0.3000,-0.07500)--(-0.2818,-0.06217)--(-0.2636,-0.05090)--(-0.2455,-0.04108)--(-0.2273,-0.03261)--(-0.2091,-0.02539)--(-0.1909,-0.01933)--(-0.1727,-0.01431)--(-0.1545,-0.01025)--(-0.1364,-0.007044)--(-0.1182,-0.004585)--(-0.1000,-0.002778)--(-0.08182,-0.001521)--(-0.06364,0)--(-0.04545,0)--(-0.02727,0)--(-0.009091,0)--(0.009091,0)--(0.02727,0)--(0.04545,0)--(0.06364,0)--(0.08182,0.001521)--(0.1000,0.002778)--(0.1182,0.004585)--(0.1364,0.007044)--(0.1545,0.01025)--(0.1727,0.01431)--(0.1909,0.01933)--(0.2091,0.02539)--(0.2273,0.03261)--(0.2455,0.04108)--(0.2636,0.05090)--(0.2818,0.06217)--(0.3000,0.07500)--(0.3182,0.08948)--(0.3364,0.1057)--(0.3545,0.1238)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4091,0.1902)--(0.4273,0.2167)--(0.4455,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5182,0.3865)--(0.5364,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6091,0.6277)--(0.6273,0.6856)--(0.6455,0.7470)--(0.6636,0.8119)--(0.6818,0.8804)--(0.7000,0.9528)--(0.7182,1.029)--(0.7364,1.109)--(0.7545,1.193)--(0.7727,1.282)--(0.7909,1.374)--(0.8091,1.471)--(0.8273,1.573)--(0.8455,1.679)--(0.8636,1.789)--(0.8818,1.905)--(0.9000,2.025);
 
 \draw [color=red] (-0.9000,4.050)--(-0.8818,3.888)--(-0.8636,3.729)--(-0.8455,3.574)--(-0.8273,3.422)--(-0.8091,3.273)--(-0.7909,3.128)--(-0.7727,2.986)--(-0.7545,2.847)--(-0.7364,2.711)--(-0.7182,2.579)--(-0.7000,2.450)--(-0.6818,2.324)--(-0.6636,2.202)--(-0.6455,2.083)--(-0.6273,1.967)--(-0.6091,1.855)--(-0.5909,1.746)--(-0.5727,1.640)--(-0.5545,1.538)--(-0.5364,1.438)--(-0.5182,1.343)--(-0.5000,1.250)--(-0.4818,1.161)--(-0.4636,1.075)--(-0.4455,0.9921)--(-0.4273,0.9128)--(-0.4091,0.8368)--(-0.3909,0.7641)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3364,0.5657)--(-0.3182,0.5062)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2455,0.3012)--(-0.2273,0.2583)--(-0.2091,0.2186)--(-0.1909,0.1822)--(-0.1727,0.1492)--(-0.1545,0.1194)--(-0.1364,0.09298)--(-0.1182,0.06983)--(-0.1000,0.05000)--(-0.08182,0.03347)--(-0.06364,0.02025)--(-0.04545,0.01033)--(-0.02727,0.003719)--(-0.009091,0)--(0.009091,0)--(0.02727,0.003719)--(0.04545,0.01033)--(0.06364,0.02025)--(0.08182,0.03347)--(0.1000,0.05000)--(0.1182,0.06983)--(0.1364,0.09298)--(0.1545,0.1194)--(0.1727,0.1492)--(0.1909,0.1822)--(0.2091,0.2186)--(0.2273,0.2583)--(0.2455,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3182,0.5062)--(0.3364,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7641)--(0.4091,0.8368)--(0.4273,0.9128)--(0.4455,0.9921)--(0.4636,1.075)--(0.4818,1.161)--(0.5000,1.250)--(0.5182,1.343)--(0.5364,1.438)--(0.5545,1.538)--(0.5727,1.640)--(0.5909,1.746)--(0.6091,1.855)--(0.6273,1.967)--(0.6455,2.083)--(0.6636,2.202)--(0.6818,2.324)--(0.7000,2.450)--(0.7182,2.579)--(0.7364,2.711)--(0.7545,2.847)--(0.7727,2.986)--(0.7909,3.128)--(0.8091,3.273)--(0.8273,3.422)--(0.8455,3.574)--(0.8636,3.729)--(0.8818,3.888)--(0.9000,4.050);
-\draw (-1.200000000,-0.3298256667) node {$ -2 $};
+\draw (-1.2000,-0.32983) node {$ -2 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (-0.6000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.60000,-0.32983) node {$ -1 $};
 \draw [] (-0.600,-0.100) -- (-0.600,0.100);
-\draw (0.6000000000,-0.3149246667) node {$ 1 $};
+\draw (0.60000,-0.31492) node {$ 1 $};
 \draw [] (0.600,-0.100) -- (0.600,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 2 $};
+\draw (1.2000,-0.31492) node {$ 2 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (-0.4331593333,-2.400000000) node {$ -4 $};
+\draw (-0.43316,-2.4000) node {$ -4 $};
 \draw [] (-0.100,-2.40) -- (0.100,-2.40);
-\draw (-0.4331593333,-1.200000000) node {$ -2 $};
+\draw (-0.43316,-1.2000) node {$ -2 $};
 \draw [] (-0.100,-1.20) -- (0.100,-1.20);
-\draw (-0.2912498333,1.200000000) node {$ 2 $};
+\draw (-0.29125,1.2000) node {$ 2 $};
 \draw [] (-0.100,1.20) -- (0.100,1.20);
-\draw (-0.2912498333,2.400000000) node {$ 4 $};
+\draw (-0.29125,2.4000) node {$ 4 $};
 \draw [] (-0.100,2.40) -- (0.100,2.40);
-\draw (-0.2912498333,3.600000000) node {$ 6 $};
+\draw (-0.29125,3.6000) node {$ 6 $};
 \draw [] (-0.100,3.60) -- (0.100,3.60);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall b/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall
index 2a7e1e559..115bccffb 100644
--- a/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall
+++ b/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall
@@ -70,22 +70,22 @@
 \draw [color=gray,style=solid] (0,0) -- (3.00,0);
 \draw [color=gray,style=solid] (0,3.00) -- (3.00,3.00);
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.0152,0.0152)--(0.0303,0.0303)--(0.0455,0.0455)--(0.0606,0.0606)--(0.0758,0.0758)--(0.0909,0.0909)--(0.106,0.106)--(0.121,0.121)--(0.136,0.136)--(0.152,0.152)--(0.167,0.167)--(0.182,0.182)--(0.197,0.197)--(0.212,0.212)--(0.227,0.227)--(0.242,0.242)--(0.258,0.258)--(0.273,0.273)--(0.288,0.288)--(0.303,0.303)--(0.318,0.318)--(0.333,0.333)--(0.348,0.348)--(0.364,0.364)--(0.379,0.379)--(0.394,0.394)--(0.409,0.409)--(0.424,0.424)--(0.439,0.439)--(0.455,0.455)--(0.470,0.470)--(0.485,0.485)--(0.500,0.500)--(0.515,0.515)--(0.530,0.530)--(0.545,0.545)--(0.561,0.561)--(0.576,0.576)--(0.591,0.591)--(0.606,0.606)--(0.621,0.621)--(0.636,0.636)--(0.651,0.651)--(0.667,0.667)--(0.682,0.682)--(0.697,0.697)--(0.712,0.712)--(0.727,0.727)--(0.742,0.742)--(0.758,0.758)--(0.773,0.773)--(0.788,0.788)--(0.803,0.803)--(0.818,0.818)--(0.833,0.833)--(0.849,0.849)--(0.864,0.864)--(0.879,0.879)--(0.894,0.894)--(0.909,0.909)--(0.924,0.924)--(0.939,0.939)--(0.955,0.955)--(0.970,0.970)--(0.985,0.985)--(1.00,1.00)--(1.02,1.02)--(1.03,1.03)--(1.05,1.05)--(1.06,1.06)--(1.08,1.08)--(1.09,1.09)--(1.11,1.11)--(1.12,1.12)--(1.14,1.14)--(1.15,1.15)--(1.17,1.17)--(1.18,1.18)--(1.20,1.20)--(1.21,1.21)--(1.23,1.23)--(1.24,1.24)--(1.26,1.26)--(1.27,1.27)--(1.29,1.29)--(1.30,1.30)--(1.32,1.32)--(1.33,1.33)--(1.35,1.35)--(1.36,1.36)--(1.38,1.38)--(1.39,1.39)--(1.41,1.41)--(1.42,1.42)--(1.44,1.44)--(1.45,1.45)--(1.47,1.47)--(1.48,1.48)--(1.50,1.50);
 
 \draw [color=blue] (1.50,3.00)--(1.52,2.98)--(1.53,2.97)--(1.55,2.95)--(1.56,2.94)--(1.58,2.92)--(1.59,2.91)--(1.61,2.89)--(1.62,2.88)--(1.64,2.86)--(1.65,2.85)--(1.67,2.83)--(1.68,2.82)--(1.70,2.80)--(1.71,2.79)--(1.73,2.77)--(1.74,2.76)--(1.76,2.74)--(1.77,2.73)--(1.79,2.71)--(1.80,2.70)--(1.82,2.68)--(1.83,2.67)--(1.85,2.65)--(1.86,2.64)--(1.88,2.62)--(1.89,2.61)--(1.91,2.59)--(1.92,2.58)--(1.94,2.56)--(1.95,2.55)--(1.97,2.53)--(1.98,2.52)--(2.00,2.50)--(2.02,2.48)--(2.03,2.47)--(2.05,2.45)--(2.06,2.44)--(2.08,2.42)--(2.09,2.41)--(2.11,2.39)--(2.12,2.38)--(2.14,2.36)--(2.15,2.35)--(2.17,2.33)--(2.18,2.32)--(2.20,2.30)--(2.21,2.29)--(2.23,2.27)--(2.24,2.26)--(2.26,2.24)--(2.27,2.23)--(2.29,2.21)--(2.30,2.20)--(2.32,2.18)--(2.33,2.17)--(2.35,2.15)--(2.36,2.14)--(2.38,2.12)--(2.39,2.11)--(2.41,2.09)--(2.42,2.08)--(2.44,2.06)--(2.45,2.05)--(2.47,2.03)--(2.48,2.02)--(2.50,2.00)--(2.52,1.98)--(2.53,1.97)--(2.55,1.95)--(2.56,1.94)--(2.58,1.92)--(2.59,1.91)--(2.61,1.89)--(2.62,1.88)--(2.64,1.86)--(2.65,1.85)--(2.67,1.83)--(2.68,1.82)--(2.70,1.80)--(2.71,1.79)--(2.73,1.77)--(2.74,1.76)--(2.76,1.74)--(2.77,1.73)--(2.79,1.71)--(2.80,1.70)--(2.82,1.68)--(2.83,1.67)--(2.85,1.65)--(2.86,1.64)--(2.88,1.62)--(2.89,1.61)--(2.91,1.59)--(2.92,1.58)--(2.94,1.56)--(2.95,1.55)--(2.97,1.53)--(2.98,1.52)--(3.00,1.50);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw []  (1.500000000,1.500000000) node [rotate=0] {$o$};
-\draw []  (1.500000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,1.500000000) node [rotate=0] {$\bullet$};
+\draw []  (1.5000,1.5000) node [rotate=0] {$o$};
+\draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
 \draw [style=dashed] (1.50,1.50) -- (1.50,3.00);
 
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall b/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall
index 545b53b85..83e1fe1b2 100644
--- a/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall
+++ b/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall
@@ -83,6 +83,10 @@
 \immediate\write\writeOfphystricks{totalheightof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
 \setlength{\lengthOfforphystricks}{\widthof{$ \pi $}}%
 \immediate\write\writeOfphystricks{widthof4fa392bd2560f31afbf4405a31edbffd44ae5e14:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$ \frac{3}{2} \, \pi $}}%
+\immediate\write\writeOfphystricks{totalheightofe3fecee9a68c3a0c0d73e0eaa33c6afc47ee8bde:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$ \frac{3}{2} \, \pi $}}%
+\immediate\write\writeOfphystricks{widthofe3fecee9a68c3a0c0d73e0eaa33c6afc47ee8bde:\the\lengthOfforphystricks-}
 %CLOSE_WRITE_AND_LABEL
 \immediate\closeout\writeOfphystricks%
 %BEFORE PSPICTURE
@@ -91,49 +95,53 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-1.00,-1.57) -- (-1.00,3.14);
-\draw [color=gray,style=solid] (0,-1.57) -- (0,3.14);
-\draw [color=gray,style=solid] (1.00,-1.57) -- (1.00,3.14);
-\draw [color=gray,style=solid] (2.00,-1.57) -- (2.00,3.14);
-\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,3.14);
-\draw [color=gray,style=solid] (4.00,-1.57) -- (4.00,3.14);
-\draw [color=gray,style=dotted] (-0.500,-1.57) -- (-0.500,3.14);
-\draw [color=gray,style=dotted] (0.500,-1.57) -- (0.500,3.14);
-\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,3.14);
-\draw [color=gray,style=dotted] (2.50,-1.57) -- (2.50,3.14);
-\draw [color=gray,style=dotted] (3.50,-1.57) -- (3.50,3.14);
+\draw [color=gray,style=solid] (-1.00,-1.57) -- (-1.00,4.71);
+\draw [color=gray,style=solid] (0,-1.57) -- (0,4.71);
+\draw [color=gray,style=solid] (1.00,-1.57) -- (1.00,4.71);
+\draw [color=gray,style=solid] (2.00,-1.57) -- (2.00,4.71);
+\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,4.71);
+\draw [color=gray,style=solid] (4.00,-1.57) -- (4.00,4.71);
+\draw [color=gray,style=dotted] (-0.500,-1.57) -- (-0.500,4.71);
+\draw [color=gray,style=dotted] (0.500,-1.57) -- (0.500,4.71);
+\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,4.71);
+\draw [color=gray,style=dotted] (2.50,-1.57) -- (2.50,4.71);
+\draw [color=gray,style=dotted] (3.50,-1.57) -- (3.50,4.71);
 \draw [color=gray,style=dotted] (-1.00,-0.785) -- (4.00,-0.785);
 \draw [color=gray,style=dotted] (-1.00,0.785) -- (4.00,0.785);
 \draw [color=gray,style=dotted] (-1.00,2.36) -- (4.00,2.36);
+\draw [color=gray,style=dotted] (-1.00,3.93) -- (4.00,3.93);
 \draw [color=gray,style=solid] (-1.00,-1.57) -- (4.00,-1.57);
 \draw [color=gray,style=solid] (-1.00,0) -- (4.00,0);
 \draw [color=gray,style=solid] (-1.00,1.57) -- (4.00,1.57);
 \draw [color=gray,style=solid] (-1.00,3.14) -- (4.00,3.14);
+\draw [color=gray,style=solid] (-1.00,4.71) -- (4.00,4.71);
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-2.070796327) -- (0,3.641592654);
+\draw [,->,>=latex] (-1.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-2.0708) -- (0,5.2124);
 %DEFAULT
 
 \draw [color=blue] (0,1.000)--(0.03173,0.9995)--(0.06347,0.9980)--(0.09520,0.9955)--(0.1269,0.9920)--(0.1587,0.9874)--(0.1904,0.9819)--(0.2221,0.9754)--(0.2539,0.9679)--(0.2856,0.9595)--(0.3173,0.9501)--(0.3491,0.9397)--(0.3808,0.9284)--(0.4125,0.9161)--(0.4443,0.9029)--(0.4760,0.8888)--(0.5077,0.8738)--(0.5395,0.8580)--(0.5712,0.8413)--(0.6029,0.8237)--(0.6347,0.8053)--(0.6664,0.7861)--(0.6981,0.7660)--(0.7299,0.7453)--(0.7616,0.7237)--(0.7933,0.7015)--(0.8251,0.6785)--(0.8568,0.6549)--(0.8885,0.6306)--(0.9203,0.6056)--(0.9520,0.5801)--(0.9837,0.5539)--(1.015,0.5272)--(1.047,0.5000)--(1.079,0.4723)--(1.111,0.4441)--(1.142,0.4154)--(1.174,0.3863)--(1.206,0.3569)--(1.238,0.3271)--(1.269,0.2969)--(1.301,0.2665)--(1.333,0.2358)--(1.365,0.2048)--(1.396,0.1736)--(1.428,0.1423)--(1.460,0.1108)--(1.491,0.07925)--(1.523,0.04758)--(1.555,0.01587)--(1.587,-0.01587)--(1.618,-0.04758)--(1.650,-0.07925)--(1.682,-0.1108)--(1.714,-0.1423)--(1.745,-0.1736)--(1.777,-0.2048)--(1.809,-0.2358)--(1.841,-0.2665)--(1.872,-0.2969)--(1.904,-0.3271)--(1.936,-0.3569)--(1.967,-0.3863)--(1.999,-0.4154)--(2.031,-0.4441)--(2.063,-0.4723)--(2.094,-0.5000)--(2.126,-0.5272)--(2.158,-0.5539)--(2.190,-0.5801)--(2.221,-0.6056)--(2.253,-0.6306)--(2.285,-0.6549)--(2.317,-0.6785)--(2.348,-0.7015)--(2.380,-0.7237)--(2.412,-0.7453)--(2.443,-0.7660)--(2.475,-0.7861)--(2.507,-0.8053)--(2.539,-0.8237)--(2.570,-0.8413)--(2.602,-0.8580)--(2.634,-0.8738)--(2.666,-0.8888)--(2.697,-0.9029)--(2.729,-0.9161)--(2.761,-0.9284)--(2.793,-0.9397)--(2.824,-0.9501)--(2.856,-0.9595)--(2.888,-0.9679)--(2.919,-0.9754)--(2.951,-0.9819)--(2.983,-0.9874)--(3.015,-0.9920)--(3.046,-0.9955)--(3.078,-0.9980)--(3.110,-0.9995)--(3.142,-1.000);
 
 \draw [color=blue] (-1.000,3.142)--(-0.9798,2.940)--(-0.9596,2.856)--(-0.9394,2.792)--(-0.9192,2.737)--(-0.8990,2.688)--(-0.8788,2.644)--(-0.8586,2.603)--(-0.8384,2.565)--(-0.8182,2.529)--(-0.7980,2.495)--(-0.7778,2.462)--(-0.7576,2.430)--(-0.7374,2.400)--(-0.7172,2.371)--(-0.6970,2.342)--(-0.6768,2.314)--(-0.6566,2.287)--(-0.6364,2.261)--(-0.6162,2.235)--(-0.5960,2.209)--(-0.5758,2.184)--(-0.5556,2.160)--(-0.5354,2.136)--(-0.5152,2.112)--(-0.4949,2.089)--(-0.4747,2.065)--(-0.4545,2.043)--(-0.4343,2.020)--(-0.4141,1.998)--(-0.3939,1.976)--(-0.3737,1.954)--(-0.3535,1.932)--(-0.3333,1.911)--(-0.3131,1.889)--(-0.2929,1.868)--(-0.2727,1.847)--(-0.2525,1.826)--(-0.2323,1.805)--(-0.2121,1.785)--(-0.1919,1.764)--(-0.1717,1.743)--(-0.1515,1.723)--(-0.1313,1.702)--(-0.1111,1.682)--(-0.09091,1.662)--(-0.07071,1.642)--(-0.05051,1.621)--(-0.03030,1.601)--(-0.01010,1.581)--(0.01010,1.561)--(0.03030,1.540)--(0.05051,1.520)--(0.07071,1.500)--(0.09091,1.480)--(0.1111,1.459)--(0.1313,1.439)--(0.1515,1.419)--(0.1717,1.398)--(0.1919,1.378)--(0.2121,1.357)--(0.2323,1.336)--(0.2525,1.316)--(0.2727,1.295)--(0.2929,1.274)--(0.3131,1.252)--(0.3333,1.231)--(0.3535,1.209)--(0.3737,1.188)--(0.3939,1.166)--(0.4141,1.144)--(0.4343,1.121)--(0.4545,1.099)--(0.4747,1.076)--(0.4949,1.053)--(0.5152,1.030)--(0.5354,1.006)--(0.5556,0.9818)--(0.5758,0.9573)--(0.5960,0.9323)--(0.6162,0.9069)--(0.6364,0.8810)--(0.6566,0.8545)--(0.6768,0.8274)--(0.6970,0.7996)--(0.7172,0.7711)--(0.7374,0.7416)--(0.7576,0.7112)--(0.7778,0.6797)--(0.7980,0.6469)--(0.8182,0.6126)--(0.8384,0.5765)--(0.8586,0.5383)--(0.8788,0.4975)--(0.8990,0.4533)--(0.9192,0.4048)--(0.9394,0.3499)--(0.9596,0.2852)--(0.9798,0.2013)--(1.000,0);
 
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.5937365000,-1.570796327) node {$ -\frac{1}{2} \, \pi $};
+\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.4518270000,1.570796327) node {$ \frac{1}{2} \, \pi $};
+\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
 \draw [] (-0.100,1.57) -- (0.100,1.57);
-\draw (-0.3105776667,3.141592654) node {$ \pi $};
+\draw (-0.31058,3.1416) node {$ \pi $};
 \draw [] (-0.100,3.14) -- (0.100,3.14);
+\draw (-0.45183,4.7124) node {$ \frac{3}{2} \, \pi $};
+\draw [] (-0.100,4.71) -- (0.100,4.71);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall b/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall
index 4dc7f5db6..b6547c9ce 100644
--- a/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall
+++ b/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall
@@ -83,19 +83,19 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.4240027746,-0.1283777105) node {\(A\)};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.281992727,-0.3611696314) node {\(B\)};
-\draw []  (6.000000000,4.000000000) node [rotate=0] {$\bullet$};
-\draw (6.260698653,4.369932668) node {\(C\)};
+\draw (-0.42400,-0.12838) node {\(A\)};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.2820,-0.36117) node {\(B\)};
+\draw []  (6.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (6.2607,4.3699) node {\(C\)};
 \draw [] (0,0) -- (5.00,0);
 \draw [] (5.00,0) -- (6.00,4.00);
 \draw [] (6.00,4.00) -- (0,0);
 \draw [] (0,0) -- (5.60,2.40);
-\draw []  (4.480000000,1.920000000) node [rotate=0] {$\bullet$};
-\draw (4.725314693,1.611462994) node {\( N\)};
-\draw []  (5.600000000,2.400000000) node [rotate=0] {$\bullet$};
-\draw (5.936543333,2.226784875) node {\( P\)};
+\draw []  (4.4800,1.9200) node [rotate=0] {$\bullet$};
+\draw (4.7253,1.6115) node {\( N\)};
+\draw []  (5.6000,2.4000) node [rotate=0] {$\bullet$};
+\draw (5.9365,2.2268) node {\( P\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall b/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall
index 201617687..feb15aa45 100644
--- a/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall
+++ b/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall
@@ -46,10 +46,10 @@
 
 \draw [color=brown,style=dashed] (2.000,0)--(2.000,0.02116)--(2.000,0.04231)--(1.999,0.06346)--(1.998,0.08460)--(1.997,0.1057)--(1.996,0.1268)--(1.995,0.1480)--(1.993,0.1690)--(1.991,0.1901)--(1.989,0.2112)--(1.986,0.2322)--(1.984,0.2532)--(1.981,0.2742)--(1.978,0.2951)--(1.975,0.3160)--(1.971,0.3369)--(1.968,0.3577)--(1.964,0.3785)--(1.960,0.3993)--(1.955,0.4200)--(1.951,0.4406)--(1.946,0.4612)--(1.941,0.4818)--(1.936,0.5023)--(1.930,0.5227)--(1.925,0.5431)--(1.919,0.5635)--(1.913,0.5837)--(1.907,0.6039)--(1.900,0.6241)--(1.893,0.6441)--(1.887,0.6641)--(1.879,0.6840)--(1.872,0.7039)--(1.865,0.7236)--(1.857,0.7433)--(1.849,0.7629)--(1.841,0.7824)--(1.832,0.8019)--(1.824,0.8212)--(1.815,0.8404)--(1.806,0.8596)--(1.797,0.8786)--(1.787,0.8976)--(1.778,0.9165)--(1.768,0.9352)--(1.758,0.9538)--(1.748,0.9724)--(1.737,0.9908)--(1.727,1.009)--(1.716,1.027)--(1.705,1.045)--(1.694,1.063)--(1.683,1.081)--(1.671,1.099)--(1.659,1.117)--(1.647,1.134)--(1.635,1.151)--(1.623,1.169)--(1.611,1.186)--(1.598,1.203)--(1.585,1.220)--(1.572,1.236)--(1.559,1.253)--(1.546,1.269)--(1.532,1.286)--(1.518,1.302)--(1.505,1.318)--(1.491,1.334)--(1.476,1.349)--(1.462,1.365)--(1.447,1.380)--(1.433,1.395)--(1.418,1.410)--(1.403,1.425)--(1.388,1.440)--(1.372,1.455)--(1.357,1.469)--(1.341,1.483)--(1.326,1.498)--(1.310,1.512)--(1.294,1.525)--(1.277,1.539)--(1.261,1.552)--(1.245,1.566)--(1.228,1.579)--(1.211,1.592)--(1.194,1.604)--(1.177,1.617)--(1.160,1.629)--(1.143,1.641)--(1.125,1.653)--(1.108,1.665)--(1.090,1.677)--(1.072,1.688)--(1.054,1.699)--(1.036,1.711)--(1.018,1.721)--(1.000,1.732);
 \draw [color=brown,style=dashed] (1.30,0.750) -- (3.03,1.75);
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (2.598076211,1.500000000);
-\draw (2.286902711,1.865758621) node {$e_{r}$};
-\draw [color=red,->,>=latex] (1.732050808,1.000000000) -- (1.232050808,1.866025404);
-\draw (1.607516675,2.186465271) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
+\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
+\draw (1.6075,2.1865) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -91,10 +91,10 @@
 
 \draw [color=brown,style=dashed] (-1.477,0.2605)--(-1.480,0.2448)--(-1.482,0.2292)--(-1.485,0.2135)--(-1.487,0.1978)--(-1.489,0.1820)--(-1.491,0.1663)--(-1.492,0.1505)--(-1.494,0.1347)--(-1.495,0.1189)--(-1.496,0.1031)--(-1.497,0.08722)--(-1.498,0.07137)--(-1.499,0.05552)--(-1.499,0.03966)--(-1.500,0.02380)--(-1.500,0.007933)--(-1.500,-0.007933)--(-1.500,-0.02380)--(-1.499,-0.03966)--(-1.499,-0.05552)--(-1.498,-0.07137)--(-1.497,-0.08722)--(-1.496,-0.1031)--(-1.495,-0.1189)--(-1.494,-0.1347)--(-1.492,-0.1505)--(-1.491,-0.1663)--(-1.489,-0.1820)--(-1.487,-0.1978)--(-1.485,-0.2135)--(-1.482,-0.2292)--(-1.480,-0.2448)--(-1.477,-0.2605)--(-1.474,-0.2761)--(-1.471,-0.2917)--(-1.468,-0.3072)--(-1.465,-0.3227)--(-1.461,-0.3382)--(-1.458,-0.3536)--(-1.454,-0.3690)--(-1.450,-0.3844)--(-1.446,-0.3997)--(-1.441,-0.4150)--(-1.437,-0.4302)--(-1.432,-0.4454)--(-1.428,-0.4605)--(-1.423,-0.4756)--(-1.417,-0.4906)--(-1.412,-0.5056)--(-1.407,-0.5205)--(-1.401,-0.5353)--(-1.395,-0.5501)--(-1.390,-0.5648)--(-1.384,-0.5795)--(-1.377,-0.5941)--(-1.371,-0.6087)--(-1.364,-0.6231)--(-1.358,-0.6375)--(-1.351,-0.6518)--(-1.344,-0.6661)--(-1.337,-0.6803)--(-1.330,-0.6944)--(-1.322,-0.7084)--(-1.315,-0.7224)--(-1.307,-0.7362)--(-1.299,-0.7500)--(-1.291,-0.7637)--(-1.283,-0.7773)--(-1.275,-0.7908)--(-1.266,-0.8043)--(-1.258,-0.8176)--(-1.249,-0.8309)--(-1.240,-0.8440)--(-1.231,-0.8571)--(-1.222,-0.8701)--(-1.213,-0.8830)--(-1.203,-0.8957)--(-1.194,-0.9084)--(-1.184,-0.9210)--(-1.174,-0.9335)--(-1.164,-0.9458)--(-1.154,-0.9581)--(-1.144,-0.9702)--(-1.134,-0.9823)--(-1.123,-0.9942)--(-1.113,-1.006)--(-1.102,-1.018)--(-1.091,-1.029)--(-1.080,-1.041)--(-1.069,-1.052)--(-1.058,-1.063)--(-1.047,-1.075)--(-1.035,-1.086)--(-1.024,-1.096)--(-1.012,-1.107)--(-1.000,-1.118)--(-0.9883,-1.128)--(-0.9763,-1.139)--(-0.9642,-1.149);
 \draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-2.349231552,-0.8550503583);
-\draw (-2.085452009,-1.242909145) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.409538931,-0.5130302150) -- (-1.067518788,-1.452722836);
-\draw (-0.6920529202,-1.132282968) node {$e_{\theta}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
+\draw (-2.0855,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
+\draw (-0.69205,-1.1323) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_HNxitLj.pstricks.recall b/src_phystricks/Fig_HNxitLj.pstricks.recall
index f80923d7c..ea0ee8ecf 100644
--- a/src_phystricks/Fig_HNxitLj.pstricks.recall
+++ b/src_phystricks/Fig_HNxitLj.pstricks.recall
@@ -75,26 +75,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (-1.000000000,0) node [rotate=0] {$\diamondsuit$};
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\diamondsuit$};
-\draw []  (-1.000000000,1.000000000) node [rotate=0] {$\diamondsuit$};
-\draw []  (-1.000000000,-1.000000000) node [rotate=0] {$\diamondsuit$};
-\draw (1.500000000,-0.3824895000) node {\( \sA^*_{\sH}\)};
-\draw (1.500000000,-0.3824895000) node {\( \sA^*_{\sH}\)};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (-1.0000,0) node [rotate=0] {$\diamondsuit$};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
+\draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
+\draw []  (-1.0000,-1.0000) node [rotate=0] {$\diamondsuit$};
+\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
+\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_HasseAGdfdy.pstricks.recall b/src_phystricks/Fig_HasseAGdfdy.pstricks.recall
index 969e6fbb9..95b7c5107 100644
--- a/src_phystricks/Fig_HasseAGdfdy.pstricks.recall
+++ b/src_phystricks/Fig_HasseAGdfdy.pstricks.recall
@@ -87,17 +87,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.2785761667) node {\( \alpha\)};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.3622220000) node {\( \beta\)};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.3140621667) node {\( \gamma\)};
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (0,2.278576167) node {\( a\)};
-\draw []  (2.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,2.326736000) node {\( b\)};
-\draw []  (4.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,2.278576167) node {\( c\)};
+\draw (0,-0.27858) node {\( \alpha\)};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.36222) node {\( \beta\)};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.31406) node {\( \gamma\)};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (0,2.2786) node {\( a\)};
+\draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,2.3267) node {\( b\)};
+\draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,2.2786) node {\( c\)};
 \draw [] (0,0) -- (0,2.00);
 \draw [] (2.00,0) -- (2.00,2.00);
 \draw [] (4.00,0) -- (4.00,2.00);
diff --git a/src_phystricks/Fig_IWuPxFc.pstricks.recall b/src_phystricks/Fig_IWuPxFc.pstricks.recall
index 5ee6a5c4e..a40e3d90c 100644
--- a/src_phystricks/Fig_IWuPxFc.pstricks.recall
+++ b/src_phystricks/Fig_IWuPxFc.pstricks.recall
@@ -95,39 +95,39 @@
 %GRID
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+\draw [,->,>=latex] (0,-3.1457) -- (0,3.1457);
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 \draw [] plot [smooth,tension=1] coordinates {(5.00,1.73)(4.93,1.71)(4.93,1.71)(4.87,1.69)(4.80,1.67)(4.73,1.65)(4.70,1.64)(4.66,1.63)(4.60,1.61)(4.53,1.59)(4.49,1.58)(4.46,1.57)(4.40,1.55)(4.33,1.53)(4.28,1.51)(4.26,1.50)(4.19,1.48)(4.13,1.46)(4.08,1.44)(4.06,1.44)(3.99,1.41)(3.93,1.39)(3.89,1.38)(3.86,1.36)(3.79,1.34)(3.72,1.31)(3.71,1.31)(3.66,1.29)(3.59,1.26)(3.54,1.24)(3.52,1.23)(3.46,1.21)(3.39,1.18)(3.38,1.17)(3.32,1.15)(3.26,1.12)(3.23,1.11)(3.19,1.09)(3.12,1.06)(3.08,1.04)(3.05,1.03)(2.99,0.993)(2.95,0.973)(2.92,0.958)(2.85,0.923)(2.82,0.906)(2.79,0.886)(2.72,0.847)(2.70,0.839)(2.65,0.806)(2.60,0.772)(2.58,0.764)(2.52,0.718)(2.50,0.705)(2.45,0.670)(2.41,0.638)(2.38,0.618)(2.33,0.570)(2.32,0.561)(2.25,0.503)(2.25,0.498)(2.19,0.436)(2.18,0.425)(2.14,0.369)(2.11,0.336)(2.09,0.302)(2.06,0.235)(2.05,0.215)(2.03,0.168)(2.01,0.101)(2.00,0.0336)(2.00,-0.0336)(2.01,-0.101)(2.03,-0.168)(2.05,-0.215)(2.06,-0.235)(2.09,-0.302)(2.11,-0.336)(2.14,-0.369)(2.18,-0.425)(2.19,-0.436)(2.25,-0.498)(2.25,-0.503)(2.32,-0.561)(2.33,-0.570)(2.38,-0.618)(2.41,-0.638)(2.45,-0.670)(2.50,-0.705)(2.52,-0.718)(2.58,-0.764)(2.60,-0.772)(2.65,-0.806)(2.70,-0.839)(2.72,-0.847)(2.79,-0.886)(2.82,-0.906)(2.85,-0.923)(2.92,-0.958)(2.95,-0.973)(2.99,-0.993)(3.05,-1.03)(3.08,-1.04)(3.12,-1.06)(3.19,-1.09)(3.23,-1.11)(3.26,-1.12)(3.32,-1.15)(3.38,-1.17)(3.39,-1.18)(3.46,-1.21)(3.52,-1.23)(3.54,-1.24)(3.59,-1.26)(3.66,-1.29)(3.71,-1.31)(3.72,-1.31)(3.79,-1.34)(3.86,-1.36)(3.89,-1.38)(3.93,-1.39)(3.99,-1.41)(4.06,-1.44)(4.08,-1.44)(4.13,-1.46)(4.19,-1.48)(4.26,-1.50)(4.28,-1.51)(4.33,-1.53)(4.40,-1.55)(4.46,-1.57)(4.49,-1.58)(4.53,-1.59)(4.60,-1.61)(4.66,-1.63)(4.70,-1.64)(4.73,-1.65)(4.80,-1.67)(4.87,-1.69)(4.93,-1.71)(4.93,-1.71)(5.00,-1.73)};
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IYAvSvI.pstricks.recall b/src_phystricks/Fig_IYAvSvI.pstricks.recall
index fd1a6616f..9bf4d113f 100644
--- a/src_phystricks/Fig_IYAvSvI.pstricks.recall
+++ b/src_phystricks/Fig_IYAvSvI.pstricks.recall
@@ -78,22 +78,22 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (0.09124983333,0.5000000000) node {1};
+\draw (0.091250,0.50000) node {1};
 \draw [] (-0.250,0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (-0.250,0.750);
 \draw [] (-0.250,0.750) -- (-0.250,0.250);
-\draw (0.5912498333,0.5000000000) node {3};
+\draw (0.59125,0.50000) node {3};
 \draw [] (0.250,0.250) -- (0.750,0.250);
 \draw [] (0.750,0.250) -- (0.750,0.750);
 \draw [] (0.750,0.750) -- (0.250,0.750);
 \draw [] (0.250,0.750) -- (0.250,0.250);
-\draw (0.09124983333,0) node {2};
+\draw (0.091250,0) node {2};
 \draw [] (-0.250,-0.250) -- (0.250,-0.250);
 \draw [] (0.250,-0.250) -- (0.250,0.250);
 \draw [] (0.250,0.250) -- (-0.250,0.250);
 \draw [] (-0.250,0.250) -- (-0.250,-0.250);
-\draw (0.5912498333,0) node {4};
+\draw (0.59125,0) node {4};
 \draw [] (0.250,-0.250) -- (0.750,-0.250);
 \draw [] (0.750,-0.250) -- (0.750,0.250);
 \draw [] (0.750,0.250) -- (0.250,0.250);
diff --git a/src_phystricks/Fig_IntBoutCercle.pstricks.recall b/src_phystricks/Fig_IntBoutCercle.pstricks.recall
index a01592ed6..01492c84e 100644
--- a/src_phystricks/Fig_IntBoutCercle.pstricks.recall
+++ b/src_phystricks/Fig_IntBoutCercle.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
 %DEFAULT
 
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@@ -80,8 +80,8 @@
 \draw [] (0,0) -- (0,0);
 \draw [] (1.00,1.00) -- (1.00,1.00);
 \draw [color=red] (-0.500,-0.500) -- (1.50,1.50);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.354646868,0.6631599656) node {$P$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3546,0.66316) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IntRectangle.pstricks.recall b/src_phystricks/Fig_IntRectangle.pstricks.recall
index a3d7d1732..1ac58ccc5 100644
--- a/src_phystricks/Fig_IntRectangle.pstricks.recall
+++ b/src_phystricks/Fig_IntRectangle.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
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 \draw [color=red,style=solid] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
@@ -81,11 +81,11 @@
 \draw [color=red,style=solid] (1.00,2.00) -- (1.00,0);
 
 \draw [color=red] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IntTriangle.pstricks.recall b/src_phystricks/Fig_IntTriangle.pstricks.recall
index 5dc2cfc1d..5831092a4 100644
--- a/src_phystricks/Fig_IntTriangle.pstricks.recall
+++ b/src_phystricks/Fig_IntTriangle.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -90,13 +90,13 @@
 \draw [color=green] (0,2.000)--(0.02020,2.000)--(0.04040,2.000)--(0.06061,2.000)--(0.08081,2.000)--(0.1010,2.000)--(0.1212,2.000)--(0.1414,2.000)--(0.1616,2.000)--(0.1818,2.000)--(0.2020,2.000)--(0.2222,2.000)--(0.2424,2.000)--(0.2626,2.000)--(0.2828,2.000)--(0.3030,2.000)--(0.3232,2.000)--(0.3434,2.000)--(0.3636,2.000)--(0.3838,2.000)--(0.4040,2.000)--(0.4242,2.000)--(0.4444,2.000)--(0.4646,2.000)--(0.4848,2.000)--(0.5051,2.000)--(0.5253,2.000)--(0.5455,2.000)--(0.5657,2.000)--(0.5859,2.000)--(0.6061,2.000)--(0.6263,2.000)--(0.6465,2.000)--(0.6667,2.000)--(0.6869,2.000)--(0.7071,2.000)--(0.7273,2.000)--(0.7475,2.000)--(0.7677,2.000)--(0.7879,2.000)--(0.8081,2.000)--(0.8283,2.000)--(0.8485,2.000)--(0.8687,2.000)--(0.8889,2.000)--(0.9091,2.000)--(0.9293,2.000)--(0.9495,2.000)--(0.9697,2.000)--(0.9899,2.000)--(1.010,2.000)--(1.030,2.000)--(1.051,2.000)--(1.071,2.000)--(1.091,2.000)--(1.111,2.000)--(1.131,2.000)--(1.152,2.000)--(1.172,2.000)--(1.192,2.000)--(1.212,2.000)--(1.232,2.000)--(1.253,2.000)--(1.273,2.000)--(1.293,2.000)--(1.313,2.000)--(1.333,2.000)--(1.354,2.000)--(1.374,2.000)--(1.394,2.000)--(1.414,2.000)--(1.434,2.000)--(1.455,2.000)--(1.475,2.000)--(1.495,2.000)--(1.515,2.000)--(1.535,2.000)--(1.556,2.000)--(1.576,2.000)--(1.596,2.000)--(1.616,2.000)--(1.636,2.000)--(1.657,2.000)--(1.677,2.000)--(1.697,2.000)--(1.717,2.000)--(1.737,2.000)--(1.758,2.000)--(1.778,2.000)--(1.798,2.000)--(1.818,2.000)--(1.838,2.000)--(1.859,2.000)--(1.879,2.000)--(1.899,2.000)--(1.919,2.000)--(1.939,2.000)--(1.960,2.000)--(1.980,2.000)--(2.000,2.000);
 
 \draw [color=green] (0,0)--(0.02020,0.02020)--(0.04040,0.04040)--(0.06061,0.06061)--(0.08081,0.08081)--(0.1010,0.1010)--(0.1212,0.1212)--(0.1414,0.1414)--(0.1616,0.1616)--(0.1818,0.1818)--(0.2020,0.2020)--(0.2222,0.2222)--(0.2424,0.2424)--(0.2626,0.2626)--(0.2828,0.2828)--(0.3030,0.3030)--(0.3232,0.3232)--(0.3434,0.3434)--(0.3636,0.3636)--(0.3838,0.3838)--(0.4040,0.4040)--(0.4242,0.4242)--(0.4444,0.4444)--(0.4646,0.4646)--(0.4848,0.4848)--(0.5051,0.5051)--(0.5253,0.5253)--(0.5455,0.5455)--(0.5657,0.5657)--(0.5859,0.5859)--(0.6061,0.6061)--(0.6263,0.6263)--(0.6465,0.6465)--(0.6667,0.6667)--(0.6869,0.6869)--(0.7071,0.7071)--(0.7273,0.7273)--(0.7475,0.7475)--(0.7677,0.7677)--(0.7879,0.7879)--(0.8081,0.8081)--(0.8283,0.8283)--(0.8485,0.8485)--(0.8687,0.8687)--(0.8889,0.8889)--(0.9091,0.9091)--(0.9293,0.9293)--(0.9495,0.9495)--(0.9697,0.9697)--(0.9899,0.9899)--(1.010,1.010)--(1.030,1.030)--(1.051,1.051)--(1.071,1.071)--(1.091,1.091)--(1.111,1.111)--(1.131,1.131)--(1.152,1.152)--(1.172,1.172)--(1.192,1.192)--(1.212,1.212)--(1.232,1.232)--(1.253,1.253)--(1.273,1.273)--(1.293,1.293)--(1.313,1.313)--(1.333,1.333)--(1.354,1.354)--(1.374,1.374)--(1.394,1.394)--(1.414,1.414)--(1.434,1.434)--(1.455,1.455)--(1.475,1.475)--(1.495,1.495)--(1.515,1.515)--(1.535,1.535)--(1.556,1.556)--(1.576,1.576)--(1.596,1.596)--(1.616,1.616)--(1.636,1.636)--(1.657,1.657)--(1.677,1.677)--(1.697,1.697)--(1.717,1.717)--(1.737,1.737)--(1.758,1.758)--(1.778,1.778)--(1.798,1.798)--(1.818,1.818)--(1.838,1.838)--(1.859,1.859)--(1.879,1.879)--(1.899,1.899)--(1.919,1.919)--(1.939,1.939)--(1.960,1.960)--(1.980,1.980)--(2.000,2.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IntegraleSimple.pstricks.recall b/src_phystricks/Fig_IntegraleSimple.pstricks.recall
index 5f46e6eb9..50af4c229 100644
--- a/src_phystricks/Fig_IntegraleSimple.pstricks.recall
+++ b/src_phystricks/Fig_IntegraleSimple.pstricks.recall
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796328,0) -- (6.783185311,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
+\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
 %DEFAULT
-\draw []  (-1.570796327,0) node [rotate=0] {$\bullet$};
-\draw (-1.570796327,-0.2785761667) node {$a$};
-\draw []  (6.283185307,0) node [rotate=0] {$\bullet$};
-\draw (6.283185307,-0.3267360000) node {$b$};
+\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
+\draw (-1.5708,-0.27858) node {$a$};
+\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
+\draw (6.2832,-0.32674) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/src_phystricks/Fig_JJAooWpimYW.pstricks.recall b/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
index 00789f209..9f0dd8da9 100644
--- a/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
+++ b/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
@@ -108,34 +108,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
-\draw [,->,>=latex] (0,-1.498867339) -- (0,1.499874128);
+\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
+\draw [,->,>=latex] (0,-1.4989) -- (0,1.4999);
 %DEFAULT
 
 \draw [color=blue] (-7.854,0)--(-7.695,0.1580)--(-7.537,0.3120)--(-7.378,0.4582)--(-7.219,0.5929)--(-7.061,0.7127)--(-6.902,0.8146)--(-6.743,0.8960)--(-6.585,0.9549)--(-6.426,0.9898)--(-6.267,0.9999)--(-6.109,0.9848)--(-5.950,0.9450)--(-5.791,0.8815)--(-5.633,0.7958)--(-5.474,0.6901)--(-5.315,0.5671)--(-5.157,0.4298)--(-4.998,0.2817)--(-4.839,0.1266)--(-4.681,-0.03173)--(-4.522,-0.1893)--(-4.363,-0.3420)--(-4.205,-0.4862)--(-4.046,-0.6182)--(-3.887,-0.7346)--(-3.729,-0.8326)--(-3.570,-0.9096)--(-3.411,-0.9638)--(-3.253,-0.9938)--(-3.094,-0.9989)--(-2.935,-0.9788)--(-2.777,-0.9342)--(-2.618,-0.8660)--(-2.459,-0.7761)--(-2.301,-0.6668)--(-2.142,-0.5406)--(-1.983,-0.4009)--(-1.825,-0.2511)--(-1.666,-0.09506)--(-1.507,0.06342)--(-1.349,0.2203)--(-1.190,0.3717)--(-1.031,0.5137)--(-0.8727,0.6428)--(-0.7140,0.7558)--(-0.5553,0.8497)--(-0.3967,0.9224)--(-0.2380,0.9718)--(-0.07933,0.9969)--(0.07933,0.9969)--(0.2380,0.9718)--(0.3967,0.9224)--(0.5553,0.8497)--(0.7140,0.7558)--(0.8727,0.6428)--(1.031,0.5137)--(1.190,0.3717)--(1.349,0.2203)--(1.507,0.06342)--(1.666,-0.09506)--(1.825,-0.2511)--(1.983,-0.4009)--(2.142,-0.5406)--(2.301,-0.6668)--(2.459,-0.7761)--(2.618,-0.8660)--(2.777,-0.9342)--(2.935,-0.9788)--(3.094,-0.9989)--(3.253,-0.9938)--(3.411,-0.9638)--(3.570,-0.9096)--(3.729,-0.8326)--(3.887,-0.7346)--(4.046,-0.6182)--(4.205,-0.4862)--(4.363,-0.3420)--(4.522,-0.1893)--(4.681,-0.03173)--(4.839,0.1266)--(4.998,0.2817)--(5.157,0.4298)--(5.315,0.5671)--(5.474,0.6901)--(5.633,0.7958)--(5.791,0.8815)--(5.950,0.9450)--(6.109,0.9848)--(6.267,0.9999)--(6.426,0.9898)--(6.585,0.9549)--(6.743,0.8960)--(6.902,0.8146)--(7.061,0.7127)--(7.219,0.5929)--(7.378,0.4582)--(7.537,0.3120)--(7.695,0.1580)--(7.854,0);
-\draw (-7.853981634,-0.4207143333) node {$ -\frac{5}{2} \, \pi $};
+\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
 \draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.283185307,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.712388980,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.853981634,-0.4207143333) node {$ \frac{5}{2} \, \pi $};
+\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
 \draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall b/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
index 8d281f24f..ab72b83d5 100644
--- a/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
+++ b/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
@@ -92,8 +92,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.212388985,0) -- (5.212388985,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-5.2124,0) -- (5.2124,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -145,21 +145,21 @@
 \draw [] (4.71,-1.00) -- (4.71,0);
 
 \draw [color=blue] (-4.712,1.000)--(-4.617,0.9955)--(-4.522,0.9819)--(-4.427,0.9595)--(-4.332,0.9284)--(-4.236,0.8888)--(-4.141,0.8413)--(-4.046,0.7861)--(-3.951,0.7237)--(-3.856,0.6549)--(-3.760,0.5801)--(-3.665,0.5000)--(-3.570,0.4154)--(-3.475,0.3271)--(-3.380,0.2358)--(-3.284,0.1423)--(-3.189,0.04758)--(-3.094,-0.04758)--(-2.999,-0.1423)--(-2.904,-0.2358)--(-2.808,-0.3271)--(-2.713,-0.4154)--(-2.618,-0.5000)--(-2.523,-0.5801)--(-2.428,-0.6549)--(-2.332,-0.7237)--(-2.237,-0.7861)--(-2.142,-0.8413)--(-2.047,-0.8888)--(-1.952,-0.9284)--(-1.856,-0.9595)--(-1.761,-0.9819)--(-1.666,-0.9955)--(-1.571,-1.000)--(-1.476,-0.9955)--(-1.380,-0.9819)--(-1.285,-0.9595)--(-1.190,-0.9284)--(-1.095,-0.8888)--(-0.9996,-0.8413)--(-0.9044,-0.7861)--(-0.8092,-0.7237)--(-0.7140,-0.6549)--(-0.6188,-0.5801)--(-0.5236,-0.5000)--(-0.4284,-0.4154)--(-0.3332,-0.3271)--(-0.2380,-0.2358)--(-0.1428,-0.1423)--(-0.04760,-0.04758)--(0.04760,0.04758)--(0.1428,0.1423)--(0.2380,0.2358)--(0.3332,0.3271)--(0.4284,0.4154)--(0.5236,0.5000)--(0.6188,0.5801)--(0.7140,0.6549)--(0.8092,0.7237)--(0.9044,0.7861)--(0.9996,0.8413)--(1.095,0.8888)--(1.190,0.9284)--(1.285,0.9595)--(1.380,0.9819)--(1.476,0.9955)--(1.571,1.000)--(1.666,0.9955)--(1.761,0.9819)--(1.856,0.9595)--(1.952,0.9284)--(2.047,0.8888)--(2.142,0.8413)--(2.237,0.7861)--(2.332,0.7237)--(2.428,0.6549)--(2.523,0.5801)--(2.618,0.5000)--(2.713,0.4154)--(2.808,0.3271)--(2.904,0.2358)--(2.999,0.1423)--(3.094,0.04758)--(3.189,-0.04758)--(3.284,-0.1423)--(3.380,-0.2358)--(3.475,-0.3271)--(3.570,-0.4154)--(3.665,-0.5000)--(3.760,-0.5801)--(3.856,-0.6549)--(3.951,-0.7237)--(4.046,-0.7861)--(4.141,-0.8413)--(4.236,-0.8888)--(4.332,-0.9284)--(4.427,-0.9595)--(4.522,-0.9819)--(4.617,-0.9955)--(4.712,-1.000);
-\draw (-4.712388980,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall b/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall
index 2f94eb547..649f91548 100644
--- a/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall
+++ b/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall
@@ -83,20 +83,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
-\draw (1.523347667,2.000000000) node {$(x,y)$};
-\draw [,->,>=latex] (0,0) -- (1.000000000,2.000000000);
-\draw (0.6845247985,0.4139141375) node {$\theta$};
+\draw (1.5233,2.0000) node {$(x,y)$};
+\draw [,->,>=latex] (0,0) -- (1.0000,2.0000);
+\draw (0.68452,0.41391) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00559)--(0.500,0.0112)--(0.500,0.0168)--(0.500,0.0224)--(0.499,0.0279)--(0.499,0.0335)--(0.498,0.0391)--(0.498,0.0447)--(0.497,0.0502)--(0.497,0.0558)--(0.496,0.0614)--(0.496,0.0669)--(0.495,0.0724)--(0.494,0.0780)--(0.493,0.0835)--(0.492,0.0890)--(0.491,0.0945)--(0.490,0.100)--(0.489,0.105)--(0.488,0.111)--(0.486,0.116)--(0.485,0.122)--(0.484,0.127)--(0.482,0.133)--(0.481,0.138)--(0.479,0.143)--(0.477,0.149)--(0.476,0.154)--(0.474,0.159)--(0.472,0.165)--(0.470,0.170)--(0.468,0.175)--(0.466,0.180)--(0.464,0.186)--(0.462,0.191)--(0.460,0.196)--(0.458,0.201)--(0.456,0.206)--(0.453,0.211)--(0.451,0.216)--(0.448,0.221)--(0.446,0.226)--(0.443,0.231)--(0.441,0.236)--(0.438,0.241)--(0.435,0.246)--(0.432,0.251)--(0.430,0.256)--(0.427,0.260)--(0.424,0.265)--(0.421,0.270)--(0.418,0.275)--(0.415,0.279)--(0.412,0.284)--(0.408,0.289)--(0.405,0.293)--(0.402,0.298)--(0.398,0.302)--(0.395,0.306)--(0.392,0.311)--(0.388,0.315)--(0.385,0.320)--(0.381,0.324)--(0.377,0.328)--(0.374,0.332)--(0.370,0.336)--(0.366,0.341)--(0.362,0.345)--(0.358,0.349)--(0.354,0.353)--(0.350,0.357)--(0.346,0.360)--(0.342,0.364)--(0.338,0.368)--(0.334,0.372)--(0.330,0.376)--(0.326,0.379)--(0.322,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.304,0.397)--(0.300,0.400)--(0.295,0.404)--(0.291,0.407)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416)--(0.272,0.419)--(0.267,0.422)--(0.263,0.425)--(0.258,0.428)--(0.253,0.431)--(0.248,0.434)--(0.243,0.437)--(0.238,0.439)--(0.234,0.442)--(0.229,0.445)--(0.224,0.447);
-\draw (0.2337087285,1.168018886) node {$r$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (0.23371,1.1680) node {$r$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall b/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall
index 9e95ea0bc..18f70b22a 100644
--- a/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall
+++ b/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.662277660) -- (0,3.662277660);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
 %DEFAULT
 
 \draw [color=red] (-3.000,3.162)--(-2.939,3.105)--(-2.879,3.048)--(-2.818,2.990)--(-2.758,2.933)--(-2.697,2.876)--(-2.636,2.820)--(-2.576,2.763)--(-2.515,2.707)--(-2.455,2.650)--(-2.394,2.594)--(-2.333,2.539)--(-2.273,2.483)--(-2.212,2.428)--(-2.152,2.373)--(-2.091,2.318)--(-2.030,2.263)--(-1.970,2.209)--(-1.909,2.155)--(-1.848,2.102)--(-1.788,2.049)--(-1.727,1.996)--(-1.667,1.944)--(-1.606,1.892)--(-1.545,1.841)--(-1.485,1.790)--(-1.424,1.740)--(-1.364,1.691)--(-1.303,1.643)--(-1.242,1.595)--(-1.182,1.548)--(-1.121,1.502)--(-1.061,1.458)--(-1.000,1.414)--(-0.9394,1.372)--(-0.8788,1.331)--(-0.8182,1.292)--(-0.7576,1.255)--(-0.6970,1.219)--(-0.6364,1.185)--(-0.5758,1.154)--(-0.5152,1.125)--(-0.4545,1.098)--(-0.3939,1.075)--(-0.3333,1.054)--(-0.2727,1.037)--(-0.2121,1.022)--(-0.1515,1.011)--(-0.09091,1.004)--(-0.03030,1.000)--(0.03030,1.000)--(0.09091,1.004)--(0.1515,1.011)--(0.2121,1.022)--(0.2727,1.037)--(0.3333,1.054)--(0.3939,1.075)--(0.4545,1.098)--(0.5152,1.125)--(0.5758,1.154)--(0.6364,1.185)--(0.6970,1.219)--(0.7576,1.255)--(0.8182,1.292)--(0.8788,1.331)--(0.9394,1.372)--(1.000,1.414)--(1.061,1.458)--(1.121,1.502)--(1.182,1.548)--(1.242,1.595)--(1.303,1.643)--(1.364,1.691)--(1.424,1.740)--(1.485,1.790)--(1.545,1.841)--(1.606,1.892)--(1.667,1.944)--(1.727,1.996)--(1.788,2.049)--(1.848,2.102)--(1.909,2.155)--(1.970,2.209)--(2.030,2.263)--(2.091,2.318)--(2.152,2.373)--(2.212,2.428)--(2.273,2.483)--(2.333,2.539)--(2.394,2.594)--(2.455,2.650)--(2.515,2.707)--(2.576,2.763)--(2.636,2.820)--(2.697,2.876)--(2.758,2.933)--(2.818,2.990)--(2.879,3.048)--(2.939,3.105)--(3.000,3.162);
@@ -104,33 +104,33 @@
 \draw [color=red] (-3.000,-3.162)--(-2.939,-3.105)--(-2.879,-3.048)--(-2.818,-2.990)--(-2.758,-2.933)--(-2.697,-2.876)--(-2.636,-2.820)--(-2.576,-2.763)--(-2.515,-2.707)--(-2.455,-2.650)--(-2.394,-2.594)--(-2.333,-2.539)--(-2.273,-2.483)--(-2.212,-2.428)--(-2.152,-2.373)--(-2.091,-2.318)--(-2.030,-2.263)--(-1.970,-2.209)--(-1.909,-2.155)--(-1.848,-2.102)--(-1.788,-2.049)--(-1.727,-1.996)--(-1.667,-1.944)--(-1.606,-1.892)--(-1.545,-1.841)--(-1.485,-1.790)--(-1.424,-1.740)--(-1.364,-1.691)--(-1.303,-1.643)--(-1.242,-1.595)--(-1.182,-1.548)--(-1.121,-1.502)--(-1.061,-1.458)--(-1.000,-1.414)--(-0.9394,-1.372)--(-0.8788,-1.331)--(-0.8182,-1.292)--(-0.7576,-1.255)--(-0.6970,-1.219)--(-0.6364,-1.185)--(-0.5758,-1.154)--(-0.5152,-1.125)--(-0.4545,-1.098)--(-0.3939,-1.075)--(-0.3333,-1.054)--(-0.2727,-1.037)--(-0.2121,-1.022)--(-0.1515,-1.011)--(-0.09091,-1.004)--(-0.03030,-1.000)--(0.03030,-1.000)--(0.09091,-1.004)--(0.1515,-1.011)--(0.2121,-1.022)--(0.2727,-1.037)--(0.3333,-1.054)--(0.3939,-1.075)--(0.4545,-1.098)--(0.5152,-1.125)--(0.5758,-1.154)--(0.6364,-1.185)--(0.6970,-1.219)--(0.7576,-1.255)--(0.8182,-1.292)--(0.8788,-1.331)--(0.9394,-1.372)--(1.000,-1.414)--(1.061,-1.458)--(1.121,-1.502)--(1.182,-1.548)--(1.242,-1.595)--(1.303,-1.643)--(1.364,-1.691)--(1.424,-1.740)--(1.485,-1.790)--(1.545,-1.841)--(1.606,-1.892)--(1.667,-1.944)--(1.727,-1.996)--(1.788,-2.049)--(1.848,-2.102)--(1.909,-2.155)--(1.970,-2.209)--(2.030,-2.263)--(2.091,-2.318)--(2.152,-2.373)--(2.212,-2.428)--(2.273,-2.483)--(2.333,-2.539)--(2.394,-2.594)--(2.455,-2.650)--(2.515,-2.707)--(2.576,-2.763)--(2.636,-2.820)--(2.697,-2.876)--(2.758,-2.933)--(2.818,-2.990)--(2.879,-3.048)--(2.939,-3.105)--(3.000,-3.162);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.2106906781,0.8045813219) node {$R$};
-\draw []  (0,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.1931375115,-0.8045813219) node {$S$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.21069,0.80458) node {$R$};
+\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.19314,-0.80458) node {$S$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall b/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall
index 2ba636293..e54d5c080 100644
--- a/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall
+++ b/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall
@@ -72,18 +72,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.448683298);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.4487);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.03030,0.03029)--(0.06061,0.06049)--(0.09091,0.09054)--(0.1212,0.1203)--(0.1515,0.1498)--(0.1818,0.1789)--(0.2121,0.2075)--(0.2424,0.2356)--(0.2727,0.2631)--(0.3030,0.2900)--(0.3333,0.3162)--(0.3636,0.3417)--(0.3939,0.3665)--(0.4242,0.3905)--(0.4545,0.4138)--(0.4848,0.4363)--(0.5152,0.4580)--(0.5455,0.4789)--(0.5758,0.4990)--(0.6061,0.5183)--(0.6364,0.5369)--(0.6667,0.5547)--(0.6970,0.5718)--(0.7273,0.5882)--(0.7576,0.6039)--(0.7879,0.6189)--(0.8182,0.6332)--(0.8485,0.6470)--(0.8788,0.6601)--(0.9091,0.6727)--(0.9394,0.6847)--(0.9697,0.6961)--(1.000,0.7071)--(1.030,0.7176)--(1.061,0.7276)--(1.091,0.7372)--(1.121,0.7463)--(1.152,0.7550)--(1.182,0.7634)--(1.212,0.7714)--(1.242,0.7790)--(1.273,0.7863)--(1.303,0.7933)--(1.333,0.8000)--(1.364,0.8064)--(1.394,0.8125)--(1.424,0.8184)--(1.455,0.8240)--(1.485,0.8294)--(1.515,0.8346)--(1.545,0.8396)--(1.576,0.8443)--(1.606,0.8489)--(1.636,0.8533)--(1.667,0.8575)--(1.697,0.8615)--(1.727,0.8654)--(1.758,0.8692)--(1.788,0.8728)--(1.818,0.8762)--(1.848,0.8795)--(1.879,0.8827)--(1.909,0.8858)--(1.939,0.8888)--(1.970,0.8917)--(2.000,0.8944)--(2.030,0.8971)--(2.061,0.8997)--(2.091,0.9021)--(2.121,0.9045)--(2.152,0.9068)--(2.182,0.9091)--(2.212,0.9112)--(2.242,0.9133)--(2.273,0.9153)--(2.303,0.9173)--(2.333,0.9191)--(2.364,0.9210)--(2.394,0.9227)--(2.424,0.9244)--(2.455,0.9261)--(2.485,0.9277)--(2.515,0.9292)--(2.545,0.9307)--(2.576,0.9322)--(2.606,0.9336)--(2.636,0.9350)--(2.667,0.9363)--(2.697,0.9376)--(2.727,0.9389)--(2.758,0.9401)--(2.788,0.9413)--(2.818,0.9424)--(2.848,0.9435)--(2.879,0.9446)--(2.909,0.9457)--(2.939,0.9467)--(2.970,0.9477)--(3.000,0.9487);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall b/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall
index 7b135deb6..92685399c 100644
--- a/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall
+++ b/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.070796327,0) -- (6.783185307,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.499496542);
+\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
 %DEFAULT
-\draw []  (-1.570796327,0) node [rotate=0] {$\bullet$};
-\draw (-1.570796327,-0.2785761667) node {$a$};
-\draw []  (6.283185307,0) node [rotate=0] {$\bullet$};
-\draw (6.283185307,-0.3267360000) node {$b$};
+\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
+\draw (-1.5708,-0.27858) node {$a$};
+\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
+\draw (6.2832,-0.32674) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
diff --git a/src_phystricks/Fig_KKRooHseDzC.pstricks.recall b/src_phystricks/Fig_KKRooHseDzC.pstricks.recall
index f12ee2ecf..62e8d054f 100644
--- a/src_phystricks/Fig_KKRooHseDzC.pstricks.recall
+++ b/src_phystricks/Fig_KKRooHseDzC.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.374994332);
+\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.3750);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -100,16 +100,16 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.82) -- (6.00,2.82) -- (5.00,2.82) -- (5.00,2.82) -- (5.00,0) --  cycle;
-\draw [color=red] (5.00,0) -- (6.00,0);
-\draw [color=red] (6.00,0) -- (6.00,2.82);
-\draw [color=red] (6.00,2.82) -- (5.00,2.82);
-\draw [color=red] (5.00,2.82) -- (5.00,0);
-\draw []  (5.000000000,2.819444444) node [rotate=0] {$\bullet$};
-\draw (5.441978850,3.392374269) node {$f(x)$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.2785761667) node {$x$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3455708333) node {$x+\Delta x$};
+\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.82);
+\draw [color=red,style=dashed] (6.00,2.82) -- (5.00,2.82);
+\draw [color=red,style=dashed] (5.00,2.82) -- (5.00,0);
+\draw []  (5.0000,2.8194) node [rotate=0] {$\bullet$};
+\draw (5.4420,3.3924) node {$f(x)$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.27858) node {$x$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.34557) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_KScolorD.pstricks.recall b/src_phystricks/Fig_KScolorD.pstricks.recall
index c23ebf509..091738bd0 100644
--- a/src_phystricks/Fig_KScolorD.pstricks.recall
+++ b/src_phystricks/Fig_KScolorD.pstricks.recall
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.696851470,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.694039839);
+\draw [,->,>=latex] (-1.6969,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.6940);
 %DEFAULT
 
 \draw [color=blue] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
@@ -72,8 +72,8 @@
 \draw [color=black] plot [smooth,tension=1] coordinates {(0,1.00)(-0.119,1.19)(-0.159,0.784)(-0.354,1.15)(-0.311,0.737)(-0.575,1.05)(-0.452,0.660)(-0.773,0.918)(-0.574,0.557)(-0.940,0.746)(-0.673,0.433)(-1.07,0.544)(-0.745,0.290)(-1.16,0.322)(-0.788,0.137)(-1.20,0.0869)(-1.00,0)};
 
 \draw [color=black] (0,-1.00)--(0.0159,-1.00)--(0.0317,-1.00)--(0.0476,-0.999)--(0.0634,-0.998)--(0.0792,-0.997)--(0.0951,-0.995)--(0.111,-0.994)--(0.127,-0.992)--(0.142,-0.990)--(0.158,-0.987)--(0.174,-0.985)--(0.189,-0.982)--(0.205,-0.979)--(0.220,-0.975)--(0.236,-0.972)--(0.251,-0.968)--(0.266,-0.964)--(0.282,-0.959)--(0.297,-0.955)--(0.312,-0.950)--(0.327,-0.945)--(0.342,-0.940)--(0.357,-0.934)--(0.372,-0.928)--(0.386,-0.922)--(0.401,-0.916)--(0.415,-0.910)--(0.430,-0.903)--(0.444,-0.896)--(0.458,-0.889)--(0.472,-0.881)--(0.486,-0.874)--(0.500,-0.866)--(0.514,-0.858)--(0.527,-0.850)--(0.541,-0.841)--(0.554,-0.833)--(0.567,-0.824)--(0.580,-0.815)--(0.593,-0.805)--(0.606,-0.796)--(0.618,-0.786)--(0.631,-0.776)--(0.643,-0.766)--(0.655,-0.756)--(0.667,-0.745)--(0.679,-0.735)--(0.690,-0.724)--(0.701,-0.713)--(0.713,-0.701)--(0.724,-0.690)--(0.735,-0.679)--(0.745,-0.667)--(0.756,-0.655)--(0.766,-0.643)--(0.776,-0.631)--(0.786,-0.618)--(0.796,-0.606)--(0.805,-0.593)--(0.815,-0.580)--(0.824,-0.567)--(0.833,-0.554)--(0.841,-0.541)--(0.850,-0.527)--(0.858,-0.514)--(0.866,-0.500)--(0.874,-0.486)--(0.881,-0.472)--(0.889,-0.458)--(0.896,-0.444)--(0.903,-0.430)--(0.910,-0.415)--(0.916,-0.401)--(0.922,-0.386)--(0.928,-0.372)--(0.934,-0.357)--(0.940,-0.342)--(0.945,-0.327)--(0.950,-0.312)--(0.955,-0.297)--(0.959,-0.282)--(0.964,-0.266)--(0.968,-0.251)--(0.972,-0.236)--(0.975,-0.220)--(0.979,-0.205)--(0.982,-0.189)--(0.985,-0.174)--(0.987,-0.158)--(0.990,-0.142)--(0.992,-0.127)--(0.994,-0.111)--(0.995,-0.0951)--(0.997,-0.0792)--(0.998,-0.0634)--(0.999,-0.0476)--(1.00,-0.0317)--(1.00,-0.0159)--(1.00,0);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0,-1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LesSpheres.pstricks.recall b/src_phystricks/Fig_LesSpheres.pstricks.recall
index cb22e70e5..3858e3937 100644
--- a/src_phystricks/Fig_LesSpheres.pstricks.recall
+++ b/src_phystricks/Fig_LesSpheres.pstricks.recall
@@ -41,20 +41,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 \draw [] (0,1.00) -- (-1.00,0);
 \draw [] (-1.00,0) -- (0,-1.00);
 \draw [] (0,-1.00) -- (1.00,0);
 \draw [] (1.00,0) -- (0,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -93,18 +93,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -143,20 +143,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 \draw [] (1.00,1.00) -- (-1.00,1.00);
 \draw [] (-1.00,1.00) -- (-1.00,-1.00);
 \draw [] (-1.00,-1.00) -- (1.00,-1.00);
 \draw [] (1.00,-1.00) -- (1.00,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LesSubFigures.pstricks.recall b/src_phystricks/Fig_LesSubFigures.pstricks.recall
index 657447a76..29f58eaae 100644
--- a/src_phystricks/Fig_LesSubFigures.pstricks.recall
+++ b/src_phystricks/Fig_LesSubFigures.pstricks.recall
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.800000000,2.975000000) node [rotate=0] {$\bullet$};
-\draw (3.078729394,2.519944177) node {$Q_{0}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
+\draw (3.0787,2.5199) node {$Q_{0}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.397500000,2.886861314) node [rotate=0] {$\bullet$};
-\draw (2.695272866,2.436021235) node {$Q_{1}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
+\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.995000000,2.763157895) node [rotate=0] {$\bullet$};
-\draw (2.322383202,2.321545179) node {$Q_{2}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
+\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.778894979);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
 %DEFAULT
 \draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.592500000,2.576923077) node [rotate=0] {$\bullet$};
-\draw (1.966388364,2.157179026) node {$Q_{3}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
+\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall b/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall
index 442c58b99..27a5ac04b 100644
--- a/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall
+++ b/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.800000000,2.975000000) node [rotate=0] {$\bullet$};
-\draw (3.078729394,2.519944177) node {$Q_{0}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
+\draw (3.0787,2.5199) node {$Q_{0}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.397500000,2.886861314) node [rotate=0] {$\bullet$};
-\draw (2.695272866,2.436021235) node {$Q_{1}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
+\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.772967580);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
 %DEFAULT
 \draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.995000000,2.763157895) node [rotate=0] {$\bullet$};
-\draw (2.322383202,2.321545179) node {$Q_{2}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
+\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.778894979);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
 %DEFAULT
 \draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
 \draw [color=red] (0.219,1.26) -- (2.16,3.27);
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.592500000,2.576923077) node [rotate=0] {$\bullet$};
-\draw (1.966388364,2.157179026) node {$Q_{3}$};
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
+\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MCKyvdk.pstricks.recall b/src_phystricks/Fig_MCKyvdk.pstricks.recall
index 48b0eefa7..8e7561c6c 100644
--- a/src_phystricks/Fig_MCKyvdk.pstricks.recall
+++ b/src_phystricks/Fig_MCKyvdk.pstricks.recall
@@ -95,14 +95,14 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (-0.2782965229,3.266129356) node {\( A\)};
-\draw (3.000000000,3.324708000) node {\( B\)};
-\draw (3.284911356,-0.2661293562) node {\( C\)};
-\draw (-0.3561643333,0) node {\( D\)};
-\draw (1.012377249,4.016129356) node {\( E\)};
-\draw (4.641743106,3.750000000) node {\( F\)};
-\draw (4.642528439,0.7500000000) node {\( G\)};
-\draw (0.9910859161,1.016129356) node {\( H\)};
+\draw (-0.27830,3.2661) node {\( A\)};
+\draw (3.0000,3.3247) node {\( B\)};
+\draw (3.2849,-0.26613) node {\( C\)};
+\draw (-0.35616,0) node {\( D\)};
+\draw (1.0124,4.0161) node {\( E\)};
+\draw (4.6417,3.7500) node {\( F\)};
+\draw (4.6425,0.75000) node {\( G\)};
+\draw (0.99109,1.0161) node {\( H\)};
 \draw [] (0,3.00) -- (1.30,3.75);
 \draw [] (3.00,3.00) -- (4.30,3.75);
 \draw [] (3.00,0) -- (4.30,0.750);
diff --git a/src_phystricks/Fig_MNICGhR.pstricks.recall b/src_phystricks/Fig_MNICGhR.pstricks.recall
index d8fec2b34..ee87719a4 100644
--- a/src_phystricks/Fig_MNICGhR.pstricks.recall
+++ b/src_phystricks/Fig_MNICGhR.pstricks.recall
@@ -87,15 +87,15 @@
 \draw [style=dotted] (2.00,0) -- (3.00,0);
 \draw [] (3.00,0) -- (4.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3059510000) node {$\alpha_1$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.3059510000) node {$\alpha_2$};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.3059510000) node {$\alpha_3$};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.3215388333) node {$\alpha_{l-1}$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.3083078333) node {$\alpha_l$};
+\draw (0,0.30595) node {$\alpha_1$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.30595) node {$\alpha_2$};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.30595) node {$\alpha_3$};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.32154) node {$\alpha_{l-1}$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.30831) node {$\alpha_l$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_MaxVraissLp.pstricks.recall b/src_phystricks/Fig_MaxVraissLp.pstricks.recall
index 85887b764..4d9809e18 100644
--- a/src_phystricks/Fig_MaxVraissLp.pstricks.recall
+++ b/src_phystricks/Fig_MaxVraissLp.pstricks.recall
@@ -87,24 +87,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.168279320);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
 %DEFAULT
-\draw []  (3.000000000,2.668279320) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,2.6683) node [rotate=0] {$\bullet$};
 
 \draw [color=blue] (0,0)--(0.101,0.00115)--(0.202,0.00858)--(0.303,0.0269)--(0.404,0.0593)--(0.505,0.108)--(0.606,0.172)--(0.707,0.254)--(0.808,0.351)--(0.909,0.463)--(1.01,0.587)--(1.11,0.722)--(1.21,0.865)--(1.31,1.01)--(1.41,1.17)--(1.52,1.32)--(1.62,1.47)--(1.72,1.63)--(1.82,1.77)--(1.92,1.91)--(2.02,2.04)--(2.12,2.16)--(2.22,2.27)--(2.32,2.36)--(2.42,2.45)--(2.53,2.52)--(2.63,2.58)--(2.73,2.62)--(2.83,2.65)--(2.93,2.67)--(3.03,2.67)--(3.13,2.66)--(3.23,2.64)--(3.33,2.60)--(3.43,2.56)--(3.54,2.50)--(3.64,2.44)--(3.74,2.37)--(3.84,2.29)--(3.94,2.20)--(4.04,2.11)--(4.14,2.02)--(4.24,1.92)--(4.34,1.82)--(4.44,1.72)--(4.55,1.62)--(4.65,1.52)--(4.75,1.42)--(4.85,1.32)--(4.95,1.22)--(5.05,1.12)--(5.15,1.03)--(5.25,0.945)--(5.35,0.861)--(5.45,0.781)--(5.56,0.705)--(5.66,0.633)--(5.76,0.567)--(5.86,0.504)--(5.96,0.446)--(6.06,0.393)--(6.16,0.345)--(6.26,0.300)--(6.36,0.260)--(6.46,0.224)--(6.57,0.191)--(6.67,0.163)--(6.77,0.137)--(6.87,0.115)--(6.97,0.0953)--(7.07,0.0785)--(7.17,0.0641)--(7.27,0.0518)--(7.37,0.0415)--(7.47,0.0328)--(7.58,0.0257)--(7.68,0.0198)--(7.78,0.0151)--(7.88,0.0113)--(7.98,0.00837)--(8.08,0.00607)--(8.18,0.00432)--(8.28,0.00300)--(8.38,0.00204)--(8.48,0.00134)--(8.59,0)--(8.69,0)--(8.79,0)--(8.89,0)--(8.99,0)--(9.09,0)--(9.19,0)--(9.29,0)--(9.39,0)--(9.50,0)--(9.60,0)--(9.70,0)--(9.80,0)--(9.90,0)--(10.0,0);
 \draw [style=dotted] (3.00,2.67) -- (3.00,0);
-\draw (10.50000000,-0.4140621667) node {$p$};
-\draw (10.50000000,-0.4140621667) node {$p$};
-\draw (3.000000000,-0.4207143333) node {$ \frac{3}{10} $};
+\draw (10.500,-0.41406) node {$p$};
+\draw (10.500,-0.41406) node {$p$};
+\draw (3.0000,-0.42071) node {$ \frac{3}{10} $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.000000000,-0.4207143333) node {$ \frac{3}{5} $};
+\draw (6.0000,-0.42071) node {$ \frac{3}{5} $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (9.000000000,-0.4207143333) node {$ \frac{9}{10} $};
+\draw (9.0000,-0.42071) node {$ \frac{9}{10} $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.6579310000,3.168279320) node {$L(p)$};
-\draw (-0.6579310000,3.168279320) node {$L(p)$};
-\draw (-0.3108333333,2.000000000) node {$ \frac{1}{5} $};
+\draw (-0.65793,3.1683) node {$L(p)$};
+\draw (-0.65793,3.1683) node {$L(p)$};
+\draw (-0.31083,2.0000) node {$ \frac{1}{5} $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MethodeChemin.pstricks.recall b/src_phystricks/Fig_MethodeChemin.pstricks.recall
index b8239b7b4..8934d3069 100644
--- a/src_phystricks/Fig_MethodeChemin.pstricks.recall
+++ b/src_phystricks/Fig_MethodeChemin.pstricks.recall
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-2.000000000) -- (0,2.000000000);
+\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
 %DEFAULT
 \draw [color=red,style=dashed] (-1.50,1.50) -- (1.50,-1.50);
 \draw [color=blue,style=dashed] (-1.50,-0.750) -- (1.50,0.750);
-\draw (-1.500000000,1.941614833) node {$y=-x$};
-\draw (2.325053201,1.144587034) node {$y=x/2$};
+\draw (-1.5000,1.9416) node {$y=-x$};
+\draw (2.3251,1.1446) node {$y=x/2$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MethodeNewton.pstricks.recall b/src_phystricks/Fig_MethodeNewton.pstricks.recall
index 23469e25c..c62cf7e8b 100644
--- a/src_phystricks/Fig_MethodeNewton.pstricks.recall
+++ b/src_phystricks/Fig_MethodeNewton.pstricks.recall
@@ -95,30 +95,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-1.287500000) -- (0,4.400000002);
+\draw [,->,>=latex] (-2.0000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-1.2875) -- (0,4.4000);
 %DEFAULT
 
 \draw [color=blue] (-1.500,3.900)--(-1.424,3.713)--(-1.348,3.529)--(-1.273,3.349)--(-1.197,3.173)--(-1.121,3.001)--(-1.045,2.833)--(-0.9697,2.668)--(-0.8939,2.507)--(-0.8182,2.350)--(-0.7424,2.197)--(-0.6667,2.048)--(-0.5909,1.903)--(-0.5152,1.761)--(-0.4394,1.623)--(-0.3636,1.490)--(-0.2879,1.359)--(-0.2121,1.233)--(-0.1364,1.111)--(-0.06061,0.9921)--(0.01515,0.8773)--(0.09091,0.7664)--(0.1667,0.6593)--(0.2424,0.5560)--(0.3182,0.4565)--(0.3939,0.3608)--(0.4697,0.2690)--(0.5455,0.1810)--(0.6212,0.09682)--(0.6970,0.01647)--(0.7727,-0.06006)--(0.8485,-0.1328)--(0.9242,-0.2016)--(1.000,-0.2667)--(1.076,-0.3279)--(1.152,-0.3853)--(1.227,-0.4388)--(1.303,-0.4886)--(1.379,-0.5345)--(1.455,-0.5766)--(1.530,-0.6148)--(1.606,-0.6493)--(1.682,-0.6799)--(1.758,-0.7067)--(1.833,-0.7296)--(1.909,-0.7488)--(1.985,-0.7641)--(2.061,-0.7755)--(2.136,-0.7832)--(2.212,-0.7870)--(2.288,-0.7870)--(2.364,-0.7832)--(2.439,-0.7755)--(2.515,-0.7641)--(2.591,-0.7488)--(2.667,-0.7296)--(2.742,-0.7067)--(2.818,-0.6799)--(2.894,-0.6493)--(2.970,-0.6148)--(3.045,-0.5766)--(3.121,-0.5345)--(3.197,-0.4886)--(3.273,-0.4388)--(3.348,-0.3853)--(3.424,-0.3279)--(3.500,-0.2667)--(3.576,-0.2016)--(3.652,-0.1328)--(3.727,-0.06006)--(3.803,0.01647)--(3.879,0.09682)--(3.955,0.1810)--(4.030,0.2690)--(4.106,0.3608)--(4.182,0.4565)--(4.258,0.5560)--(4.333,0.6593)--(4.409,0.7664)--(4.485,0.8773)--(4.561,0.9921)--(4.636,1.111)--(4.712,1.233)--(4.788,1.359)--(4.864,1.490)--(4.939,1.623)--(5.015,1.761)--(5.091,1.903)--(5.167,2.048)--(5.242,2.197)--(5.318,2.350)--(5.394,2.507)--(5.470,2.668)--(5.545,2.833)--(5.621,3.001)--(5.697,3.173)--(5.773,3.349)--(5.849,3.529)--(5.924,3.713)--(6.000,3.900);
 \draw [color=red,style=dotted] (-0.900,0) -- (-0.900,2.52);
 \draw [color=green,style=dashed] (-1.20,3.15) -- (0.600,-0.630);
-\draw []  (-0.9000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.9000000000,-0.4059510000) node {$x_n$};
-\draw []  (0.3000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.3000000000,-0.4191818333) node {$x_{n+1}$};
-\draw []  (-0.9000000000,2.520000000) node [rotate=0] {$\bullet$};
-\draw (-0.5041004656,2.846194201) node {$y_n$};
-\draw []  (0.7129573851,0) node [rotate=0] {$\bullet$};
-\draw (0.7129573851,0.4059510000) node {$r_0$};
-\draw []  (3.787042615,0) node [rotate=0] {$\bullet$};
-\draw (3.787042615,0.4059510000) node {$r_1$};
-\draw []  (2.250000000,-0.7875000000) node [rotate=0] {$\bullet$};
-\draw (2.250000000,-1.212208000) node {$S$};
-\draw (3.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (-0.90000,0) node [rotate=0] {$\bullet$};
+\draw (-0.90000,-0.40595) node {$x_n$};
+\draw []  (0.30000,0) node [rotate=0] {$\bullet$};
+\draw (0.30000,-0.41918) node {$x_{n+1}$};
+\draw []  (-0.90000,2.5200) node [rotate=0] {$\bullet$};
+\draw (-0.50410,2.8462) node {$y_n$};
+\draw []  (0.71296,0) node [rotate=0] {$\bullet$};
+\draw (0.71296,0.40595) node {$r_0$};
+\draw []  (3.7870,0) node [rotate=0] {$\bullet$};
+\draw (3.7870,0.40595) node {$r_1$};
+\draw []  (2.2500,-0.78750) node [rotate=0] {$\bullet$};
+\draw (2.2500,-1.2122) node {$S$};
+\draw (3.0000,-0.31492) node {$ 1 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 2 $};
+\draw (6.0000,-0.31492) node {$ 2 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.2912498333,3.000000000) node {$ 1 $};
+\draw (-0.29125,3.0000) node {$ 1 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MomentForce.pstricks.recall b/src_phystricks/Fig_MomentForce.pstricks.recall
index d2d50996a..6e2f5e73e 100644
--- a/src_phystricks/Fig_MomentForce.pstricks.recall
+++ b/src_phystricks/Fig_MomentForce.pstricks.recall
@@ -82,13 +82,13 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.4247080000) node {$O$};
-\draw [,->,>=latex] (0,0) -- (-1.000000000,-1.000000000);
-\draw (-1.000000000,-0.4419603333) node {$\overline{ R }$};
-\draw [,->,>=latex] (-1.000000000,-1.000000000) -- (-3.000000000,-1.500000000);
-\draw (-3.000000000,-1.041960333) node {$\overline{ F }$};
+\draw (0,0.42471) node {$O$};
+\draw [,->,>=latex] (0,0) -- (-1.0000,-1.0000);
+\draw (-1.0000,-0.44196) node {$\overline{ R }$};
+\draw [,->,>=latex] (-1.0000,-1.0000) -- (-3.0000,-1.5000);
+\draw (-3.0000,-1.0420) node {$\overline{ F }$};
 \draw [color=blue,style=dotted] (0,0) -- (0.176,-0.706);
-\draw (0.4742668775,-0.1534444890) node {$d$};
+\draw (0.47427,-0.15344) node {$d$};
 \draw [color=brown,style=dashed] (-1.00,-1.00) -- (0.467,-0.633);
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_MoulinEau.pstricks.recall b/src_phystricks/Fig_MoulinEau.pstricks.recall
index ed42ed16a..a61cb53e6 100644
--- a/src_phystricks/Fig_MoulinEau.pstricks.recall
+++ b/src_phystricks/Fig_MoulinEau.pstricks.recall
@@ -36,15 +36,15 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -76,17 +76,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [] (0,2.00) -- (-1.29,0.468);
-\draw [color=red,->,>=latex] (-0.6427876097,1.233955557) -- (-1.229611699,1.726359433);
-\draw [color=green,->,>=latex] (-0.6427876097,1.233955557) -- (-1.055963521,0.7415516804);
-\draw [color=blue,->,>=latex] (-2.000000000,0) -- (-3.000000000,0);
-\draw [color=blue,->,>=latex] (-2.000000000,1.000000000) -- (-3.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (-2.000000000,2.000000000) -- (-3.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.000000000,0);
-\draw [color=blue,->,>=latex] (0,1.000000000) -- (-1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (0,2.000000000) -- (-1.000000000,2.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,0) -- (1.000000000,0);
-\draw [color=blue,->,>=latex] (2.000000000,1.000000000) -- (1.000000000,1.000000000);
-\draw [color=blue,->,>=latex] (2.000000000,2.000000000) -- (1.000000000,2.000000000);
+\draw [color=red,->,>=latex] (-0.64279,1.2340) -- (-1.2296,1.7264);
+\draw [color=green,->,>=latex] (-0.64279,1.2340) -- (-1.0560,0.74155);
+\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
+\draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
+\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
+\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_NEtAchr.pstricks.recall b/src_phystricks/Fig_NEtAchr.pstricks.recall
index 463979272..eb02886ee 100644
--- a/src_phystricks/Fig_NEtAchr.pstricks.recall
+++ b/src_phystricks/Fig_NEtAchr.pstricks.recall
@@ -70,18 +70,18 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=brown] plot [smooth,tension=1] coordinates {(-1.50,-0.500)(0.500,-0.300)(2.00,1.00)(3.50,1.50)(5.00,2.70)(5.80,2.70)};
-\draw []  (-1.500000000,-0.5000000000) node [rotate=0] {$\bullet$};
-\draw (-1.500000000,-0.1789703333) node {\( +\)};
-\draw []  (0.5000000000,-0.3000000000) node [rotate=0] {$\bullet$};
-\draw (0.5000000000,0.02102966667) node {\( +\)};
-\draw []  (2.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,1.321029667) node {\( +\)};
-\draw []  (3.500000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw (3.500000000,1.821029667) node {\( +\)};
-\draw []  (5.000000000,2.700000000) node [rotate=0] {$\bullet$};
-\draw (5.000000000,3.021029667) node {\( +\)};
-\draw []  (5.800000000,2.700000000) node [rotate=0] {$\bullet$};
-\draw (5.800000000,3.021029667) node {\( +\)};
+\draw []  (-1.5000,-0.50000) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.17897) node {\( +\)};
+\draw []  (0.50000,-0.30000) node [rotate=0] {$\bullet$};
+\draw (0.50000,0.021030) node {\( +\)};
+\draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,1.3210) node {\( +\)};
+\draw []  (3.5000,1.5000) node [rotate=0] {$\bullet$};
+\draw (3.5000,1.8210) node {\( +\)};
+\draw []  (5.0000,2.7000) node [rotate=0] {$\bullet$};
+\draw (5.0000,3.0210) node {\( +\)};
+\draw []  (5.8000,2.7000) node [rotate=0] {$\bullet$};
+\draw (5.8000,3.0210) node {\( +\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall b/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall
index 039e10817..fc28bdea2 100644
--- a/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall
+++ b/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.249988663);
+\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -100,16 +100,16 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
-\draw [color=red] (5.00,0) -- (6.00,0);
-\draw [color=red] (6.00,0) -- (6.00,2.64);
-\draw [color=red] (6.00,2.64) -- (5.00,2.64);
-\draw [color=red] (5.00,2.64) -- (5.00,0);
-\draw []  (5.000000000,2.638888889) node [rotate=0] {$\bullet$};
-\draw (5.441978850,3.211818713) node {$f(x)$};
-\draw []  (5.000000000,0) node [rotate=0] {$\bullet$};
-\draw (5.000000000,-0.2785761667) node {$x$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3455708333) node {$x+\Delta x$};
+\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
+\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
+\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
+\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
+\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
+\draw (5.4420,3.2118) node {$f(x)$};
+\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.27858) node {$x$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.34557) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_NWDooOObSHB.pstricks.recall b/src_phystricks/Fig_NWDooOObSHB.pstricks.recall
index 55206359f..0769eba3a 100644
--- a/src_phystricks/Fig_NWDooOObSHB.pstricks.recall
+++ b/src_phystricks/Fig_NWDooOObSHB.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.300000000,0) -- (7.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.781250000);
+\draw [,->,>=latex] (-8.3000,0) -- (7.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.7812);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -102,12 +102,12 @@
 \draw [color=blue] (-2.600,0)--(-2.508,0)--(-2.416,0)--(-2.324,0)--(-2.232,0)--(-2.140,0)--(-2.048,0)--(-1.957,0)--(-1.865,0)--(-1.773,0)--(-1.681,0)--(-1.589,0)--(-1.497,0)--(-1.405,0)--(-1.313,0)--(-1.221,0)--(-1.129,0)--(-1.037,0)--(-0.9455,0)--(-0.8535,0)--(-0.7616,0)--(-0.6697,0)--(-0.5778,0)--(-0.4859,0)--(-0.3939,0)--(-0.3020,0)--(-0.2101,0)--(-0.1182,0)--(-0.02626,0)--(0.06566,0)--(0.1576,0)--(0.2495,0)--(0.3414,0)--(0.4333,0)--(0.5253,0)--(0.6172,0)--(0.7091,0)--(0.8010,0)--(0.8929,0)--(0.9848,0)--(1.077,0)--(1.169,0)--(1.261,0)--(1.353,0)--(1.444,0)--(1.536,0)--(1.628,0)--(1.720,0)--(1.812,0)--(1.904,0)--(1.996,0)--(2.088,0)--(2.180,0)--(2.272,0)--(2.364,0)--(2.456,0)--(2.547,0)--(2.639,0)--(2.731,0)--(2.823,0)--(2.915,0)--(3.007,0)--(3.099,0)--(3.191,0)--(3.283,0)--(3.375,0)--(3.467,0)--(3.559,0)--(3.651,0)--(3.742,0)--(3.834,0)--(3.926,0)--(4.018,0)--(4.110,0)--(4.202,0)--(4.294,0)--(4.386,0)--(4.478,0)--(4.570,0)--(4.662,0)--(4.754,0)--(4.845,0)--(4.937,0)--(5.029,0)--(5.121,0)--(5.213,0)--(5.305,0)--(5.397,0)--(5.489,0)--(5.581,0)--(5.673,0)--(5.765,0)--(5.857,0)--(5.948,0)--(6.040,0)--(6.132,0)--(6.224,0)--(6.316,0)--(6.408,0)--(6.500,0);
 \draw [] (-2.60,0) -- (-2.60,2.12);
 \draw [] (6.50,6.28) -- (6.50,0);
-\draw []  (-7.800000000,0) node [rotate=0] {$\bullet$};
-\draw (-7.800000000,-0.2785761667) node {\( a\)};
-\draw []  (-2.600000000,0) node [rotate=0] {$\bullet$};
-\draw (-2.600000000,-0.3267360000) node {\( b\)};
-\draw []  (6.500000000,0) node [rotate=0] {$\bullet$};
-\draw (6.500000000,-0.2785761667) node {\( c\)};
+\draw []  (-7.8000,0) node [rotate=0] {$\bullet$};
+\draw (-7.8000,-0.27858) node {\( a\)};
+\draw []  (-2.6000,0) node [rotate=0] {$\bullet$};
+\draw (-2.6000,-0.32674) node {\( b\)};
+\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.27858) node {\( c\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_NiveauHyperbole.pstricks.recall b/src_phystricks/Fig_NiveauHyperbole.pstricks.recall
index d20738e11..342639491 100644
--- a/src_phystricks/Fig_NiveauHyperbole.pstricks.recall
+++ b/src_phystricks/Fig_NiveauHyperbole.pstricks.recall
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (1.000,0)--(1.020,0.2020)--(1.040,0.2871)--(1.061,0.3534)--(1.081,0.4101)--(1.101,0.4607)--(1.121,0.5071)--(1.141,0.5503)--(1.162,0.5911)--(1.182,0.6298)--(1.202,0.6670)--(1.222,0.7027)--(1.242,0.7373)--(1.263,0.7709)--(1.283,0.8035)--(1.303,0.8354)--(1.323,0.8666)--(1.343,0.8971)--(1.364,0.9271)--(1.384,0.9566)--(1.404,0.9856)--(1.424,1.014)--(1.444,1.042)--(1.465,1.070)--(1.485,1.098)--(1.505,1.125)--(1.525,1.152)--(1.545,1.178)--(1.566,1.205)--(1.586,1.231)--(1.606,1.257)--(1.626,1.282)--(1.646,1.308)--(1.667,1.333)--(1.687,1.359)--(1.707,1.383)--(1.727,1.408)--(1.747,1.433)--(1.768,1.458)--(1.788,1.482)--(1.808,1.506)--(1.828,1.531)--(1.848,1.555)--(1.869,1.579)--(1.889,1.602)--(1.909,1.626)--(1.929,1.650)--(1.949,1.673)--(1.970,1.697)--(1.990,1.720)--(2.010,1.744)--(2.030,1.767)--(2.051,1.790)--(2.071,1.813)--(2.091,1.836)--(2.111,1.859)--(2.131,1.882)--(2.152,1.905)--(2.172,1.928)--(2.192,1.951)--(2.212,1.973)--(2.232,1.996)--(2.253,2.018)--(2.273,2.041)--(2.293,2.063)--(2.313,2.086)--(2.333,2.108)--(2.354,2.131)--(2.374,2.153)--(2.394,2.175)--(2.414,2.197)--(2.434,2.219)--(2.455,2.242)--(2.475,2.264)--(2.495,2.286)--(2.515,2.308)--(2.535,2.330)--(2.556,2.352)--(2.576,2.374)--(2.596,2.396)--(2.616,2.418)--(2.636,2.439)--(2.657,2.461)--(2.677,2.483)--(2.697,2.505)--(2.717,2.526)--(2.737,2.548)--(2.758,2.570)--(2.778,2.592)--(2.798,2.613)--(2.818,2.635)--(2.838,2.656)--(2.859,2.678)--(2.879,2.700)--(2.899,2.721)--(2.919,2.743)--(2.939,2.764)--(2.960,2.786)--(2.980,2.807)--(3.000,2.828);
@@ -108,33 +108,33 @@
 \draw [color=blue] (-3.000,-2.828)--(-2.980,-2.807)--(-2.960,-2.786)--(-2.939,-2.764)--(-2.919,-2.743)--(-2.899,-2.721)--(-2.879,-2.700)--(-2.859,-2.678)--(-2.838,-2.656)--(-2.818,-2.635)--(-2.798,-2.613)--(-2.778,-2.592)--(-2.758,-2.570)--(-2.737,-2.548)--(-2.717,-2.526)--(-2.697,-2.505)--(-2.677,-2.483)--(-2.657,-2.461)--(-2.636,-2.439)--(-2.616,-2.418)--(-2.596,-2.396)--(-2.576,-2.374)--(-2.556,-2.352)--(-2.535,-2.330)--(-2.515,-2.308)--(-2.495,-2.286)--(-2.475,-2.264)--(-2.455,-2.242)--(-2.434,-2.219)--(-2.414,-2.197)--(-2.394,-2.175)--(-2.374,-2.153)--(-2.354,-2.131)--(-2.333,-2.108)--(-2.313,-2.086)--(-2.293,-2.063)--(-2.273,-2.041)--(-2.253,-2.018)--(-2.232,-1.996)--(-2.212,-1.973)--(-2.192,-1.951)--(-2.172,-1.928)--(-2.152,-1.905)--(-2.131,-1.882)--(-2.111,-1.859)--(-2.091,-1.836)--(-2.071,-1.813)--(-2.051,-1.790)--(-2.030,-1.767)--(-2.010,-1.744)--(-1.990,-1.720)--(-1.970,-1.697)--(-1.949,-1.673)--(-1.929,-1.650)--(-1.909,-1.626)--(-1.889,-1.602)--(-1.869,-1.579)--(-1.848,-1.555)--(-1.828,-1.531)--(-1.808,-1.506)--(-1.788,-1.482)--(-1.768,-1.458)--(-1.747,-1.433)--(-1.727,-1.408)--(-1.707,-1.383)--(-1.687,-1.359)--(-1.667,-1.333)--(-1.646,-1.308)--(-1.626,-1.282)--(-1.606,-1.257)--(-1.586,-1.231)--(-1.566,-1.205)--(-1.545,-1.178)--(-1.525,-1.152)--(-1.505,-1.125)--(-1.485,-1.098)--(-1.465,-1.070)--(-1.444,-1.042)--(-1.424,-1.014)--(-1.404,-0.9856)--(-1.384,-0.9566)--(-1.364,-0.9271)--(-1.343,-0.8971)--(-1.323,-0.8666)--(-1.303,-0.8354)--(-1.283,-0.8035)--(-1.263,-0.7709)--(-1.242,-0.7373)--(-1.222,-0.7027)--(-1.202,-0.6670)--(-1.182,-0.6298)--(-1.162,-0.5911)--(-1.141,-0.5503)--(-1.121,-0.5071)--(-1.101,-0.4607)--(-1.081,-0.4101)--(-1.061,-0.3534)--(-1.040,-0.2871)--(-1.020,-0.2020)--(-1.000,0);
 \draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
 \draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (0.7867744886,0.1954186781) node {$P$};
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-0.7850131552,0.2309048448) node {$Q$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (0.78677,0.19542) node {$P$};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-0.78501,0.23090) node {$Q$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall b/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
index 152de5dbe..292a4a5f5 100644
--- a/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
+++ b/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
@@ -116,38 +116,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.900000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-2.813086867) -- (0,3.699326205);
+\draw [,->,>=latex] (-4.9000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-2.8131) -- (0,3.6993);
 %DEFAULT
 
 \draw [color=blue] (-4.4000,-2.3131)--(-4.3051,-1.7786)--(-4.2101,-1.3073)--(-4.1152,-0.89319)--(-4.0202,-0.53075)--(-3.9253,-0.21496)--(-3.8303,0.058716)--(-3.7354,0.29443)--(-3.6404,0.49595)--(-3.5455,0.66669)--(-3.4505,0.80977)--(-3.3556,0.92802)--(-3.2606,1.0240)--(-3.1657,1.1001)--(-3.0707,1.1584)--(-2.9758,1.2009)--(-2.8808,1.2293)--(-2.7859,1.2452)--(-2.6909,1.2501)--(-2.5960,1.2452)--(-2.5010,1.2319)--(-2.4061,1.2112)--(-2.3111,1.1841)--(-2.2162,1.1515)--(-2.1212,1.1143)--(-2.0263,1.0731)--(-1.9313,1.0287)--(-1.8364,0.98162)--(-1.7414,0.93253)--(-1.6465,0.88190)--(-1.5515,0.83018)--(-1.4566,0.77782)--(-1.3616,0.72520)--(-1.2667,0.67266)--(-1.1717,0.62052)--(-1.0768,0.56908)--(-0.98182,0.51859)--(-0.88687,0.46930)--(-0.79192,0.42143)--(-0.69697,0.37518)--(-0.60202,0.33071)--(-0.50707,0.28821)--(-0.41212,0.24782)--(-0.31717,0.20966)--(-0.22222,0.17388)--(-0.12727,0.14058)--(-0.032323,0.10985)--(0.062626,0.081805)--(0.15758,0.056515)--(0.25253,0.034060)--(0.34747,0.014511)--(0.44242,-0.0020696)--(0.53737,-0.015623)--(0.63232,-0.026098)--(0.72727,-0.033446)--(0.82222,-0.037624)--(0.91717,-0.038593)--(1.0121,-0.036315)--(1.1071,-0.030760)--(1.2020,-0.021896)--(1.2970,-0.0096977)--(1.3919,0.0058604)--(1.4869,0.024800)--(1.5818,0.047142)--(1.6768,0.072905)--(1.7717,0.10211)--(1.8667,0.13476)--(1.9616,0.17088)--(2.0566,0.21048)--(2.1515,0.25357)--(2.2465,0.30016)--(2.3414,0.35026)--(2.4364,0.40388)--(2.5313,0.46103)--(2.6263,0.52171)--(2.7212,0.58593)--(2.8162,0.65370)--(2.9111,0.72503)--(3.0061,0.79991)--(3.1010,0.87835)--(3.1960,0.96035)--(3.2909,1.0459)--(3.3859,1.1351)--(3.4808,1.2278)--(3.5758,1.3241)--(3.6707,1.4240)--(3.7657,1.5275)--(3.8606,1.6345)--(3.9556,1.7451)--(4.0505,1.8594)--(4.1455,1.9772)--(4.2404,2.0986)--(4.3354,2.2236)--(4.4303,2.3522)--(4.5253,2.4844)--(4.6202,2.6202)--(4.7151,2.7596)--(4.8101,2.9026)--(4.9051,3.0491)--(5.0000,3.1993);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.5244091667,-2.000000000) node {$ -20 $};
+\draw (-0.52441,-2.0000) node {$ -20 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.5244091667,-1.000000000) node {$ -10 $};
+\draw (-0.52441,-1.0000) node {$ -10 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.3824996667,1.000000000) node {$ 10 $};
+\draw (-0.38250,1.0000) node {$ 10 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.3824996667,2.000000000) node {$ 20 $};
+\draw (-0.38250,2.0000) node {$ 20 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.3824996667,3.000000000) node {$ 30 $};
+\draw (-0.38250,3.0000) node {$ 30 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Osculateur.pstricks.recall b/src_phystricks/Fig_Osculateur.pstricks.recall
index ba1e0cdc4..3136c04e9 100644
--- a/src_phystricks/Fig_Osculateur.pstricks.recall
+++ b/src_phystricks/Fig_Osculateur.pstricks.recall
@@ -65,18 +65,18 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (-19.37824843,2.000000000) node [rotate=0] {$\bullet$};
+\draw []  (-19.378,2.0000) node [rotate=0] {$\bullet$};
 
 \draw [] (-19.068,1.6899)--(-19.069,1.7096)--(-19.071,1.7292)--(-19.074,1.7486)--(-19.078,1.7678)--(-19.084,1.7867)--(-19.090,1.8052)--(-19.098,1.8232)--(-19.107,1.8407)--(-19.117,1.8576)--(-19.129,1.8738)--(-19.141,1.8892)--(-19.154,1.9039)--(-19.168,1.9177)--(-19.183,1.9306)--(-19.198,1.9425)--(-19.215,1.9534)--(-19.232,1.9632)--(-19.249,1.9720)--(-19.268,1.9796)--(-19.286,1.9860)--(-19.305,1.9913)--(-19.324,1.9953)--(-19.344,1.9981)--(-19.363,1.9996)--(-19.383,2.0000)--(-19.403,1.9990)--(-19.422,1.9968)--(-19.442,1.9934)--(-19.461,1.9888)--(-19.480,1.9829)--(-19.498,1.9759)--(-19.516,1.9677)--(-19.533,1.9585)--(-19.550,1.9481)--(-19.566,1.9367)--(-19.581,1.9243)--(-19.596,1.9109)--(-19.609,1.8967)--(-19.622,1.8816)--(-19.634,1.8658)--(-19.644,1.8492)--(-19.654,1.8320)--(-19.662,1.8142)--(-19.670,1.7960)--(-19.676,1.7773)--(-19.681,1.7582)--(-19.684,1.7389)--(-19.687,1.7194)--(-19.688,1.6998)--(-19.688,1.6801)--(-19.687,1.6604)--(-19.684,1.6409)--(-19.681,1.6216)--(-19.676,1.6026)--(-19.670,1.5839)--(-19.662,1.5656)--(-19.654,1.5478)--(-19.644,1.5306)--(-19.634,1.5141)--(-19.622,1.4982)--(-19.609,1.4832)--(-19.596,1.4689)--(-19.581,1.4556)--(-19.566,1.4432)--(-19.550,1.4317)--(-19.533,1.4214)--(-19.516,1.4121)--(-19.498,1.4039)--(-19.480,1.3969)--(-19.461,1.3910)--(-19.442,1.3864)--(-19.422,1.3830)--(-19.403,1.3808)--(-19.383,1.3799)--(-19.363,1.3802)--(-19.344,1.3817)--(-19.324,1.3845)--(-19.305,1.3886)--(-19.286,1.3938)--(-19.268,1.4002)--(-19.249,1.4078)--(-19.232,1.4166)--(-19.215,1.4264)--(-19.198,1.4373)--(-19.183,1.4492)--(-19.168,1.4621)--(-19.154,1.4759)--(-19.141,1.4906)--(-19.129,1.5061)--(-19.117,1.5223)--(-19.107,1.5392)--(-19.098,1.5566)--(-19.090,1.5747)--(-19.084,1.5932)--(-19.078,1.6120)--(-19.074,1.6312)--(-19.071,1.6507)--(-19.069,1.6702)--(-19.068,1.6899);
-\draw []  (-19.37824843,2.000000000) node [rotate=0] {$\bullet$};
-\draw []  (-19.10672978,1.902113033) node [rotate=0] {$\bullet$};
+\draw []  (-19.378,2.0000) node [rotate=0] {$\bullet$};
+\draw []  (-19.107,1.9021) node [rotate=0] {$\bullet$};
 
 \draw [] (-18.675,1.1024)--(-18.677,1.1631)--(-18.683,1.2235)--(-18.693,1.2835)--(-18.706,1.3427)--(-18.723,1.4010)--(-18.744,1.4580)--(-18.768,1.5137)--(-18.796,1.5676)--(-18.827,1.6197)--(-18.862,1.6697)--(-18.899,1.7175)--(-18.940,1.7627)--(-18.983,1.8053)--(-19.029,1.8451)--(-19.077,1.8818)--(-19.128,1.9155)--(-19.180,1.9458)--(-19.235,1.9728)--(-19.291,1.9963)--(-19.348,2.0161)--(-19.407,2.0323)--(-19.466,2.0447)--(-19.526,2.0534)--(-19.587,2.0582)--(-19.647,2.0592)--(-19.708,2.0563)--(-19.768,2.0495)--(-19.828,2.0390)--(-19.887,2.0247)--(-19.945,2.0066)--(-20.002,1.9850)--(-20.057,1.9598)--(-20.111,1.9311)--(-20.162,1.8991)--(-20.212,1.8638)--(-20.259,1.8256)--(-20.303,1.7844)--(-20.345,1.7404)--(-20.384,1.6939)--(-20.420,1.6450)--(-20.453,1.5939)--(-20.483,1.5409)--(-20.509,1.4860)--(-20.531,1.4297)--(-20.550,1.3720)--(-20.565,1.3132)--(-20.577,1.2536)--(-20.585,1.1934)--(-20.589,1.1328)--(-20.589,1.0721)--(-20.585,1.0115)--(-20.577,0.95123)--(-20.565,0.89161)--(-20.550,0.83284)--(-20.531,0.77515)--(-20.509,0.71878)--(-20.483,0.66396)--(-20.453,0.61090)--(-20.420,0.55982)--(-20.384,0.51093)--(-20.345,0.46441)--(-20.303,0.42047)--(-20.259,0.37927)--(-20.212,0.34099)--(-20.162,0.30577)--(-20.111,0.27375)--(-20.057,0.24508)--(-20.002,0.21985)--(-19.945,0.19819)--(-19.887,0.18016)--(-19.828,0.16584)--(-19.768,0.15530)--(-19.708,0.14857)--(-19.647,0.14568)--(-19.587,0.14664)--(-19.526,0.15145)--(-19.466,0.16010)--(-19.407,0.17253)--(-19.348,0.18871)--(-19.291,0.20857)--(-19.235,0.23203)--(-19.180,0.25899)--(-19.128,0.28935)--(-19.077,0.32298)--(-19.029,0.35975)--(-18.983,0.39952)--(-18.940,0.44211)--(-18.899,0.48736)--(-18.862,0.53509)--(-18.827,0.58510)--(-18.796,0.63719)--(-18.768,0.69116)--(-18.744,0.74679)--(-18.723,0.80384)--(-18.706,0.86210)--(-18.693,0.92133)--(-18.683,0.98129)--(-18.677,1.0417)--(-18.675,1.1024);
-\draw []  (-19.10672978,1.902113033) node [rotate=0] {$\bullet$};
-\draw []  (-15.29684375,-1.902113033) node [rotate=0] {$\bullet$};
+\draw []  (-19.107,1.9021) node [rotate=0] {$\bullet$};
+\draw []  (-15.297,-1.9021) node [rotate=0] {$\bullet$};
 
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-\draw []  (-15.29684375,-1.902113033) node [rotate=0] {$\bullet$};
+\draw []  (-15.297,-1.9021) node [rotate=0] {$\bullet$};
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 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_PHTVjfk.pstricks.recall b/src_phystricks/Fig_PHTVjfk.pstricks.recall
index 5b9ad53f5..833669663 100644
--- a/src_phystricks/Fig_PHTVjfk.pstricks.recall
+++ b/src_phystricks/Fig_PHTVjfk.pstricks.recall
@@ -79,28 +79,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.231453278,0) -- (2.231453278,0);
-\draw [,->,>=latex] (0,-2.231453278) -- (0,2.231453278);
+\draw [,->,>=latex] (-2.2315,0) -- (2.2315,0);
+\draw [,->,>=latex] (0,-2.2315) -- (0,2.2315);
 %DEFAULT
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+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall b/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall
index 04928061f..2cc978bad 100644
--- a/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall
+++ b/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall
@@ -72,8 +72,8 @@
 \draw [color=red] (2.903,2.900)--(2.899,3.014)--(2.888,3.128)--(2.870,3.241)--(2.845,3.353)--(2.813,3.463)--(2.774,3.570)--(2.728,3.675)--(2.675,3.776)--(2.617,3.875)--(2.552,3.969)--(2.481,4.059)--(2.405,4.144)--(2.323,4.224)--(2.237,4.299)--(2.146,4.368)--(2.050,4.432)--(1.951,4.489)--(1.849,4.540)--(1.743,4.584)--(1.635,4.621)--(1.525,4.652)--(1.413,4.675)--(1.300,4.692)--(1.186,4.701)--(1.071,4.703)--(0.9571,4.697)--(0.8434,4.684)--(0.7308,4.665)--(0.6196,4.638)--(0.5104,4.604)--(0.4035,4.563)--(0.2994,4.515)--(0.1986,4.461)--(0.1014,4.401)--(0.008222,4.335)--(-0.08057,4.262)--(-0.1646,4.185)--(-0.2435,4.102)--(-0.3171,4.014)--(-0.3849,3.922)--(-0.4468,3.826)--(-0.5024,3.726)--(-0.5515,3.623)--(-0.5941,3.517)--(-0.6298,3.408)--(-0.6585,3.297)--(-0.6801,3.185)--(-0.6946,3.071)--(-0.7019,2.957)--(-0.7019,2.843)--(-0.6946,2.729)--(-0.6801,2.615)--(-0.6585,2.503)--(-0.6298,2.392)--(-0.5941,2.283)--(-0.5515,2.177)--(-0.5024,2.074)--(-0.4468,1.974)--(-0.3849,1.878)--(-0.3171,1.786)--(-0.2435,1.698)--(-0.1646,1.615)--(-0.08057,1.538)--(0.008222,1.465)--(0.1014,1.399)--(0.1986,1.339)--(0.2994,1.285)--(0.4035,1.237)--(0.5104,1.196)--(0.6196,1.162)--(0.7308,1.135)--(0.8434,1.116)--(0.9571,1.103)--(1.071,1.097)--(1.186,1.099)--(1.300,1.108)--(1.413,1.125)--(1.525,1.148)--(1.635,1.179)--(1.743,1.216)--(1.849,1.260)--(1.951,1.311)--(2.050,1.368)--(2.146,1.432)--(2.237,1.501)--(2.323,1.576)--(2.405,1.656)--(2.481,1.741)--(2.552,1.831)--(2.617,1.925)--(2.675,2.023)--(2.728,2.125)--(2.774,2.230)--(2.813,2.337)--(2.845,2.447)--(2.870,2.559)--(2.888,2.672)--(2.899,2.786)--(2.903,2.900);
 
 \draw [color=red] (3.224,3.500)--(3.219,3.660)--(3.204,3.820)--(3.178,3.978)--(3.143,4.134)--(3.098,4.288)--(3.043,4.438)--(2.979,4.585)--(2.905,4.727)--(2.823,4.865)--(2.732,4.996)--(2.633,5.122)--(2.527,5.242)--(2.412,5.354)--(2.291,5.459)--(2.164,5.556)--(2.031,5.645)--(1.892,5.725)--(1.748,5.796)--(1.601,5.858)--(1.449,5.910)--(1.295,5.953)--(1.138,5.986)--(0.9797,6.008)--(0.8201,6.021)--(0.6600,6.024)--(0.5000,6.016)--(0.3408,5.998)--(0.1831,5.970)--(0.02745,5.933)--(-0.1255,5.885)--(-0.2751,5.828)--(-0.4208,5.761)--(-0.5619,5.686)--(-0.6980,5.601)--(-0.8285,5.508)--(-0.9528,5.407)--(-1.070,5.299)--(-1.181,5.183)--(-1.284,5.060)--(-1.379,4.931)--(-1.465,4.796)--(-1.543,4.656)--(-1.612,4.512)--(-1.672,4.363)--(-1.722,4.211)--(-1.762,4.056)--(-1.792,3.899)--(-1.812,3.740)--(-1.823,3.580)--(-1.823,3.420)--(-1.812,3.260)--(-1.792,3.101)--(-1.762,2.944)--(-1.722,2.789)--(-1.672,2.637)--(-1.612,2.488)--(-1.543,2.343)--(-1.465,2.204)--(-1.379,2.069)--(-1.284,1.940)--(-1.181,1.817)--(-1.070,1.701)--(-0.9528,1.593)--(-0.8285,1.492)--(-0.6980,1.399)--(-0.5619,1.314)--(-0.4208,1.239)--(-0.2751,1.172)--(-0.1255,1.115)--(0.02745,1.067)--(0.1831,1.030)--(0.3408,1.002)--(0.5000,0.9841)--(0.6600,0.9764)--(0.8201,0.9790)--(0.9797,0.9917)--(1.138,1.014)--(1.295,1.047)--(1.449,1.090)--(1.601,1.142)--(1.748,1.204)--(1.892,1.275)--(2.031,1.355)--(2.164,1.444)--(2.291,1.541)--(2.412,1.646)--(2.527,1.758)--(2.633,1.878)--(2.732,2.004)--(2.823,2.135)--(2.905,2.273)--(2.979,2.415)--(3.043,2.562)--(3.098,2.712)--(3.143,2.866)--(3.178,3.022)--(3.204,3.180)--(3.219,3.340)--(3.224,3.500);
-\draw []  (2.100000000,1.400000000) node [rotate=0] {$\bullet$};
-\draw (2.353454873,1.108881941) node {\( P\)};
+\draw []  (2.1000,1.4000) node [rotate=0] {$\bullet$};
+\draw (2.3535,1.1089) node {\( P\)};
 \draw [] (0,0) -- (3.00,2.00);
 \draw [style=dashed] (2.10,1.40) -- (0.100,4.40);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_PONXooXYjEot.pstricks.recall b/src_phystricks/Fig_PONXooXYjEot.pstricks.recall
index 94718c6f6..3adcf2189 100644
--- a/src_phystricks/Fig_PONXooXYjEot.pstricks.recall
+++ b/src_phystricks/Fig_PONXooXYjEot.pstricks.recall
@@ -79,22 +79,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.499811191,0) -- (2.499811191,0);
-\draw [,->,>=latex] (0,-1.206842173) -- (0,1.206842173);
+\draw [,->,>=latex] (-2.4998,0) -- (2.4998,0);
+\draw [,->,>=latex] (0,-1.2068) -- (0,1.2068);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.2559,-0.2479)--(0.3673,-0.3447)--(0.4563,-0.4148)--(0.5341,-0.4702)--(0.6049,-0.5158)--(0.6709,-0.5540)--(0.7333,-0.5861)--(0.7927,-0.6133)--(0.8498,-0.6362)--(0.9049,-0.6552)--(0.9581,-0.6709)--(1.010,-0.6834)--(1.060,-0.6931)--(1.109,-0.7001)--(1.156,-0.7046)--(1.202,-0.7068)--(1.247,-0.7068)--(1.291,-0.7048)--(1.333,-0.7007)--(1.374,-0.6948)--(1.414,-0.6871)--(1.453,-0.6777)--(1.491,-0.6666)--(1.527,-0.6540)--(1.562,-0.6399)--(1.596,-0.6244)--(1.629,-0.6076)--(1.660,-0.5895)--(1.691,-0.5701)--(1.719,-0.5496)--(1.747,-0.5280)--(1.773,-0.5054)--(1.798,-0.4817)--(1.821,-0.4571)--(1.843,-0.4317)--(1.864,-0.4055)--(1.883,-0.3785)--(1.901,-0.3508)--(1.917,-0.3224)--(1.932,-0.2935)--(1.946,-0.2640)--(1.958,-0.2341)--(1.968,-0.2037)--(1.977,-0.1730)--(1.985,-0.1420)--(1.991,-0.1107)--(1.995,-0.07919)--(1.998,-0.04757)--(2.000,-0.01587)--(2.000,0.01587)--(1.998,0.04757)--(1.995,0.07919)--(1.991,0.1107)--(1.985,0.1420)--(1.977,0.1730)--(1.968,0.2037)--(1.958,0.2341)--(1.946,0.2640)--(1.932,0.2935)--(1.917,0.3224)--(1.901,0.3508)--(1.883,0.3785)--(1.864,0.4055)--(1.843,0.4317)--(1.821,0.4571)--(1.798,0.4817)--(1.773,0.5054)--(1.747,0.5280)--(1.719,0.5496)--(1.691,0.5701)--(1.660,0.5895)--(1.629,0.6076)--(1.596,0.6244)--(1.562,0.6399)--(1.527,0.6540)--(1.491,0.6666)--(1.453,0.6777)--(1.414,0.6871)--(1.374,0.6948)--(1.333,0.7007)--(1.291,0.7048)--(1.247,0.7068)--(1.202,0.7068)--(1.156,0.7046)--(1.109,0.7001)--(1.060,0.6931)--(1.010,0.6834)--(0.9581,0.6709)--(0.9049,0.6552)--(0.8498,0.6362)--(0.7927,0.6133)--(0.7333,0.5861)--(0.6709,0.5540)--(0.6049,0.5158)--(0.5341,0.4702)--(0.4563,0.4148)--(0.3673,0.3447)--(0.2559,0.2479)--(0,0);
 \draw [color=blue] (0,0)--(-0.2559,0.2479)--(-0.3673,0.3447)--(-0.4563,0.4148)--(-0.5341,0.4702)--(-0.6049,0.5158)--(-0.6709,0.5540)--(-0.7333,0.5861)--(-0.7927,0.6133)--(-0.8498,0.6362)--(-0.9049,0.6552)--(-0.9581,0.6709)--(-1.010,0.6834)--(-1.060,0.6931)--(-1.109,0.7001)--(-1.156,0.7046)--(-1.202,0.7068)--(-1.247,0.7068)--(-1.291,0.7048)--(-1.333,0.7007)--(-1.374,0.6948)--(-1.414,0.6871)--(-1.453,0.6777)--(-1.491,0.6666)--(-1.527,0.6540)--(-1.562,0.6399)--(-1.596,0.6244)--(-1.629,0.6076)--(-1.660,0.5895)--(-1.691,0.5701)--(-1.719,0.5496)--(-1.747,0.5280)--(-1.773,0.5054)--(-1.798,0.4817)--(-1.821,0.4571)--(-1.843,0.4317)--(-1.864,0.4055)--(-1.883,0.3785)--(-1.901,0.3508)--(-1.917,0.3224)--(-1.932,0.2935)--(-1.946,0.2640)--(-1.958,0.2341)--(-1.968,0.2037)--(-1.977,0.1730)--(-1.985,0.1420)--(-1.991,0.1107)--(-1.995,0.07919)--(-1.998,0.04757)--(-2.000,0.01587)--(-2.000,-0.01587)--(-1.998,-0.04757)--(-1.995,-0.07919)--(-1.991,-0.1107)--(-1.985,-0.1420)--(-1.977,-0.1730)--(-1.968,-0.2037)--(-1.958,-0.2341)--(-1.946,-0.2640)--(-1.932,-0.2935)--(-1.917,-0.3224)--(-1.901,-0.3508)--(-1.883,-0.3785)--(-1.864,-0.4055)--(-1.843,-0.4317)--(-1.821,-0.4571)--(-1.798,-0.4817)--(-1.773,-0.5054)--(-1.747,-0.5280)--(-1.719,-0.5496)--(-1.691,-0.5701)--(-1.660,-0.5895)--(-1.629,-0.6076)--(-1.596,-0.6244)--(-1.562,-0.6399)--(-1.527,-0.6540)--(-1.491,-0.6666)--(-1.453,-0.6777)--(-1.414,-0.6871)--(-1.374,-0.6948)--(-1.333,-0.7007)--(-1.291,-0.7048)--(-1.247,-0.7068)--(-1.202,-0.7068)--(-1.156,-0.7046)--(-1.109,-0.7001)--(-1.060,-0.6931)--(-1.010,-0.6834)--(-0.9581,-0.6709)--(-0.9049,-0.6552)--(-0.8498,-0.6362)--(-0.7927,-0.6133)--(-0.7333,-0.5861)--(-0.6709,-0.5540)--(-0.6049,-0.5158)--(-0.5341,-0.4702)--(-0.4563,-0.4148)--(-0.3673,-0.3447)--(-0.2559,-0.2479)--(0,0);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall b/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall
index d5e62e6d6..ab49695de 100644
--- a/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall
+++ b/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall
@@ -143,8 +143,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.997787144,0) -- (5.997787144,0);
-\draw [,->,>=latex] (0,-4.000000000) -- (0,4.000000000);
+\draw [,->,>=latex] (-5.9978,0) -- (5.9978,0);
+\draw [,->,>=latex] (0,-4.0000) -- (0,4.0000);
 %DEFAULT
 
 \draw [color=blue] (-5.356,-3.414)--(-5.337,-2.997)--(-5.318,-2.666)--(-5.299,-2.398)--(-5.280,-2.176)--(-5.261,-1.989)--(-5.242,-1.828)--(-5.223,-1.689)--(-5.204,-1.567)--(-5.185,-1.459)--(-5.166,-1.362)--(-5.146,-1.276)--(-5.127,-1.197)--(-5.108,-1.126)--(-5.089,-1.060)--(-5.070,-0.9998)--(-5.051,-0.9440)--(-5.032,-0.8921)--(-5.013,-0.8438)--(-4.994,-0.7985)--(-4.975,-0.7559)--(-4.956,-0.7158)--(-4.937,-0.6778)--(-4.918,-0.6418)--(-4.899,-0.6076)--(-4.880,-0.5749)--(-4.860,-0.5437)--(-4.841,-0.5137)--(-4.822,-0.4849)--(-4.803,-0.4572)--(-4.784,-0.4305)--(-4.765,-0.4047)--(-4.746,-0.3796)--(-4.727,-0.3553)--(-4.708,-0.3316)--(-4.689,-0.3086)--(-4.670,-0.2861)--(-4.651,-0.2640)--(-4.632,-0.2425)--(-4.613,-0.2213)--(-4.594,-0.2005)--(-4.574,-0.1800)--(-4.555,-0.1599)--(-4.536,-0.1399)--(-4.517,-0.1202)--(-4.498,-0.1006)--(-4.479,-0.08125)--(-4.460,-0.06199)--(-4.441,-0.04281)--(-4.422,-0.02370)--(-4.403,-0.004624)--(-4.384,0.01445)--(-4.365,0.03354)--(-4.346,0.05268)--(-4.327,0.07190)--(-4.307,0.09122)--(-4.288,0.1107)--(-4.269,0.1303)--(-4.250,0.1502)--(-4.231,0.1702)--(-4.212,0.1906)--(-4.193,0.2112)--(-4.174,0.2322)--(-4.155,0.2535)--(-4.136,0.2753)--(-4.117,0.2976)--(-4.098,0.3204)--(-4.079,0.3437)--(-4.060,0.3677)--(-4.041,0.3924)--(-4.021,0.4179)--(-4.002,0.4442)--(-3.983,0.4714)--(-3.964,0.4996)--(-3.945,0.5290)--(-3.926,0.5596)--(-3.907,0.5915)--(-3.888,0.6250)--(-3.869,0.6601)--(-3.850,0.6971)--(-3.831,0.7361)--(-3.812,0.7775)--(-3.793,0.8214)--(-3.774,0.8683)--(-3.755,0.9184)--(-3.736,0.9722)--(-3.716,1.030)--(-3.697,1.093)--(-3.678,1.162)--(-3.659,1.237)--(-3.640,1.319)--(-3.621,1.411)--(-3.602,1.513)--(-3.583,1.628)--(-3.564,1.758)--(-3.545,1.908)--(-3.526,2.082)--(-3.507,2.286)--(-3.488,2.530)--(-3.469,2.827);
@@ -162,45 +162,45 @@
 \draw [style=dashed] (1.10,-3.50) -- (1.10,3.50);
 \draw [style=dashed] (3.30,-3.50) -- (3.30,3.50);
 \draw [style=dashed] (5.50,-3.50) -- (5.50,3.50);
-\draw (-5.497787144,-0.4207143333) node {$ -\frac{5}{2} \, \pi $};
+\draw (-5.4978,-0.42071) node {$ -\frac{5}{2} \, \pi $};
 \draw [] (-5.50,-0.100) -- (-5.50,0.100);
-\draw (-4.398229715,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-4.3982,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-4.40,-0.100) -- (-4.40,0.100);
-\draw (-3.298672286,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-3.2987,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.199114858,-0.3210296667) node {$ -\pi $};
+\draw (-2.1991,-0.32103) node {$ -\pi $};
 \draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.099557429,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.0996,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.099557429,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.0996,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.199114858,-0.2785761667) node {$ \pi $};
+\draw (2.1991,-0.27858) node {$ \pi $};
 \draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.298672286,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (3.2987,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (4.398229715,-0.3149246667) node {$ 2 \, \pi $};
+\draw (4.3982,-0.31492) node {$ 2 \, \pi $};
 \draw [] (4.40,-0.100) -- (4.40,0.100);
-\draw (5.497787144,-0.4207143333) node {$ \frac{5}{2} \, \pi $};
+\draw (5.4978,-0.42071) node {$ \frac{5}{2} \, \pi $};
 \draw [] (5.50,-0.100) -- (5.50,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -5 $};
+\draw (-0.43316,-3.5000) node {$ -5 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-2.800000000) node {$ -4 $};
+\draw (-0.43316,-2.8000) node {$ -4 $};
 \draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall b/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall
index 4c3f272e9..e0da8fb27 100644
--- a/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall
+++ b/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.000000000,0);
-\draw (3.308599701,-0.2907082010) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.000000000,2.000000000);
-\draw (2.000000000,2.426736000) node {$b$};
-\draw []  (5.000000000,2.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (0,0) -- (3.0000,0);
+\draw (3.3086,-0.29071) node {$a$};
+\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw (2.0000,2.4267) node {$b$};
+\draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
 \draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
 \draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
 \draw [style=dashed] (2.00,2.00) -- (2.00,0);
-\draw (2.305148833,1.000000000) node {$h$};
-\draw (0.8061547663,0.3180777162) node {$\theta$};
+\draw (2.3051,1.0000) node {$h$};
+\draw (0.80616,0.31808) node {$\theta$};
 
 \draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ParamTangente.pstricks.recall b/src_phystricks/Fig_ParamTangente.pstricks.recall
index 20a074298..7a28085b0 100644
--- a/src_phystricks/Fig_ParamTangente.pstricks.recall
+++ b/src_phystricks/Fig_ParamTangente.pstricks.recall
@@ -87,35 +87,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.042356081,0) -- (3.556125852,0);
-\draw [,->,>=latex] (0,-3.875000000) -- (0,3.945386379);
+\draw [,->,>=latex] (-1.0424,0) -- (3.5561,0);
+\draw [,->,>=latex] (0,-3.8750) -- (0,3.9454);
 %DEFAULT
 \draw [color=blue] (3.000,-3.375)--(2.939,-3.175)--(2.879,-2.982)--(2.818,-2.798)--(2.758,-2.621)--(2.697,-2.452)--(2.636,-2.290)--(2.576,-2.136)--(2.515,-1.989)--(2.455,-1.849)--(2.394,-1.715)--(2.333,-1.588)--(2.273,-1.467)--(2.212,-1.353)--(2.152,-1.245)--(2.091,-1.143)--(2.030,-1.046)--(1.970,-0.9552)--(1.909,-0.8697)--(1.848,-0.7895)--(1.788,-0.7144)--(1.727,-0.6442)--(1.667,-0.5787)--(1.606,-0.5178)--(1.545,-0.4614)--(1.485,-0.4092)--(1.424,-0.3611)--(1.364,-0.3170)--(1.303,-0.2766)--(1.242,-0.2397)--(1.182,-0.2063)--(1.121,-0.1762)--(1.061,-0.1491)--(1.000,-0.1250)--(0.9394,-0.1036)--(0.8788,-0.08483)--(0.8182,-0.06846)--(0.7576,-0.05435)--(0.6970,-0.04232)--(0.6364,-0.03221)--(0.5758,-0.02386)--(0.5152,-0.01709)--(0.4545,-0.01174)--(0.3939,-0.007642)--(0.3333,-0.004630)--(0.2727,-0.002536)--(0.2121,-0.001193)--(0.1515,0)--(0.09091,0)--(0.03030,0)--(0.03030,0)--(0.09091,0)--(0.1515,0)--(0.2121,0.001193)--(0.2727,0.002536)--(0.3333,0.004630)--(0.3939,0.007642)--(0.4545,0.01174)--(0.5152,0.01709)--(0.5758,0.02386)--(0.6364,0.03221)--(0.6970,0.04232)--(0.7576,0.05435)--(0.8182,0.06846)--(0.8788,0.08483)--(0.9394,0.1036)--(1.000,0.1250)--(1.061,0.1491)--(1.121,0.1762)--(1.182,0.2063)--(1.242,0.2397)--(1.303,0.2766)--(1.364,0.3170)--(1.424,0.3611)--(1.485,0.4092)--(1.545,0.4614)--(1.606,0.5178)--(1.667,0.5787)--(1.727,0.6442)--(1.788,0.7144)--(1.848,0.7895)--(1.909,0.8697)--(1.970,0.9552)--(2.030,1.046)--(2.091,1.143)--(2.152,1.245)--(2.212,1.353)--(2.273,1.467)--(2.333,1.588)--(2.394,1.715)--(2.455,1.849)--(2.515,1.989)--(2.576,2.136)--(2.636,2.290)--(2.697,2.452)--(2.758,2.621)--(2.818,2.798)--(2.879,2.982)--(2.939,3.175)--(3.000,3.375);
-\draw [color=red,->,>=latex] (2.507812500,-1.971492827) -- (2.117442095,-1.050834939);
-\draw [color=red,->,>=latex] (1.754791260,-0.6754394161) -- (1.100148636,0.08049909656);
-\draw [color=red,->,>=latex] (0.4546530247,-0.01174763020) -- (-0.5423560809,0.06553654240);
-\draw [color=red,->,>=latex] (1.029768197,0.1364986758) -- (1.958993278,0.5060128831);
-\draw [color=red,->,>=latex] (2.091846278,1.144193070) -- (2.612237915,1.998120786);
-\draw [color=red,->,>=latex] (2.716201962,2.504933430) -- (3.056125852,3.445386379);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw [color=red,->,>=latex] (2.5078,-1.9715) -- (2.1174,-1.0508);
+\draw [color=red,->,>=latex] (1.7548,-0.67544) -- (1.1001,0.080499);
+\draw [color=red,->,>=latex] (0.45465,-0.011748) -- (-0.54236,0.065536);
+\draw [color=red,->,>=latex] (1.0298,0.13650) -- (1.9590,0.50601);
+\draw [color=red,->,>=latex] (2.0918,1.1442) -- (2.6122,1.9981);
+\draw [color=red,->,>=latex] (2.7162,2.5049) -- (3.0561,3.4454);
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_PartieEntiere.pstricks.recall b/src_phystricks/Fig_PartieEntiere.pstricks.recall
index 418078f57..5820d4ebf 100644
--- a/src_phystricks/Fig_PartieEntiere.pstricks.recall
+++ b/src_phystricks/Fig_PartieEntiere.pstricks.recall
@@ -83,31 +83,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,3.5000);
 %DEFAULT
 
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-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Polirettangolo.pstricks.recall b/src_phystricks/Fig_Polirettangolo.pstricks.recall
index f41ef73c0..8d5164762 100644
--- a/src_phystricks/Fig_Polirettangolo.pstricks.recall
+++ b/src_phystricks/Fig_Polirettangolo.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -130,21 +130,21 @@
 \draw [style=dotted] (3.50,3.50) -- (3.50,1.50);
 \draw [style=dotted] (3.50,1.50) -- (2.00,1.50);
 \draw [style=dotted] (2.00,1.50) -- (2.00,3.50);
-\draw (1.000000000,-0.3149246667) node {$ 2 $};
+\draw (1.0000,-0.31492) node {$ 2 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 4 $};
+\draw (2.0000,-0.31492) node {$ 4 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 6 $};
+\draw (3.0000,-0.31492) node {$ 6 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 8 $};
+\draw (4.0000,-0.31492) node {$ 8 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ProjPoly.pstricks.recall b/src_phystricks/Fig_ProjPoly.pstricks.recall
index 0d22b5a65..2b8360aeb 100644
--- a/src_phystricks/Fig_ProjPoly.pstricks.recall
+++ b/src_phystricks/Fig_ProjPoly.pstricks.recall
@@ -111,56 +111,56 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.811685490,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-3.000000000) -- (0,3.000000000);
+\draw [,->,>=latex] (-2.8117,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
 %DEFAULT
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 \draw [] plot [smooth,tension=1] coordinates {(2.50,2.18)(2.49,2.16)(2.47,2.14)(2.45,2.13)(2.43,2.11)(2.42,2.10)(2.40,2.08)(2.39,2.06)(2.37,2.04)(2.36,2.03)(2.33,2.01)(2.32,2.00)(2.30,1.97)(2.29,1.96)(2.27,1.94)(2.26,1.93)(2.23,1.90)(2.23,1.90)(2.20,1.87)(2.19,1.86)(2.16,1.83)(2.16,1.83)(2.13,1.80)(2.13,1.80)(2.10,1.76)(2.10,1.76)(2.07,1.73)(2.06,1.73)(2.03,1.69)(2.03,1.69)(2.00,1.66)(2.00,1.65)(1.97,1.63)(1.96,1.62)(1.94,1.59)(1.93,1.58)(1.91,1.56)(1.90,1.55)(1.88,1.53)(1.86,1.51)(1.84,1.49)(1.83,1.48)(1.81,1.46)(1.80,1.44)(1.78,1.43)(1.76,1.40)(1.75,1.39)(1.73,1.37)(1.72,1.36)(1.69,1.33)(1.69,1.33)(1.66,1.30)(1.66,1.29)(1.63,1.26)(1.63,1.26)(1.60,1.22)(1.59,1.22)(1.57,1.19)(1.56,1.19)(1.54,1.16)(1.53,1.15)(1.51,1.12)(1.49,1.11)(1.48,1.09)(1.46,1.07)(1.45,1.06)(1.43,1.03)(1.42,1.02)(1.39,0.996)(1.39,0.990)(1.36,0.958)(1.36,0.956)(1.33,0.923)(1.33,0.919)(1.30,0.889)(1.29,0.879)(1.27,0.856)(1.26,0.839)(1.24,0.822)(1.22,0.798)(1.22,0.789)(1.19,0.757)(1.19,0.755)(1.16,0.721)(1.16,0.715)(1.14,0.688)(1.12,0.672)(1.11,0.654)(1.09,0.628)(1.09,0.621)(1.06,0.587)(1.06,0.582)(1.04,0.554)(1.02,0.534)(1.01,0.520)(0.991,0.487)(0.990,0.485)(0.969,0.453)(0.956,0.432)(0.949,0.419)(0.929,0.386)(0.923,0.374)(0.911,0.352)(0.893,0.319)(0.889,0.310)(0.877,0.285)(0.863,0.252)(0.856,0.233)(0.850,0.218)(0.838,0.185)(0.829,0.151)(0.822,0.121)(0.821,0.117)(0.815,0.0839)(0.811,0.0503)(0.809,0.0168)(0.809,-0.0168)(0.811,-0.0503)(0.815,-0.0839)(0.821,-0.117)(0.822,-0.121)(0.829,-0.151)(0.838,-0.185)(0.850,-0.218)(0.856,-0.233)(0.863,-0.252)(0.877,-0.285)(0.889,-0.310)(0.893,-0.319)(0.911,-0.352)(0.923,-0.374)(0.929,-0.386)(0.949,-0.419)(0.956,-0.432)(0.969,-0.453)(0.990,-0.485)(0.991,-0.487)(1.01,-0.520)(1.02,-0.534)(1.04,-0.554)(1.06,-0.582)(1.06,-0.587)(1.09,-0.621)(1.09,-0.628)(1.11,-0.654)(1.12,-0.672)(1.14,-0.688)(1.16,-0.715)(1.16,-0.721)(1.19,-0.755)(1.19,-0.757)(1.22,-0.789)(1.22,-0.798)(1.24,-0.822)(1.26,-0.839)(1.27,-0.856)(1.29,-0.879)(1.30,-0.889)(1.33,-0.919)(1.33,-0.923)(1.36,-0.956)(1.36,-0.958)(1.39,-0.990)(1.39,-0.996)(1.42,-1.02)(1.43,-1.03)(1.45,-1.06)(1.46,-1.07)(1.48,-1.09)(1.49,-1.11)(1.51,-1.12)(1.53,-1.15)(1.54,-1.16)(1.56,-1.19)(1.57,-1.19)(1.59,-1.22)(1.60,-1.22)(1.63,-1.26)(1.63,-1.26)(1.66,-1.29)(1.66,-1.30)(1.69,-1.33)(1.69,-1.33)(1.72,-1.36)(1.73,-1.37)(1.75,-1.39)(1.76,-1.40)(1.78,-1.43)(1.80,-1.44)(1.81,-1.46)(1.83,-1.48)(1.84,-1.49)(1.86,-1.51)(1.88,-1.53)(1.90,-1.55)(1.91,-1.56)(1.93,-1.58)(1.94,-1.59)(1.96,-1.62)(1.97,-1.63)(2.00,-1.65)(2.00,-1.66)(2.03,-1.69)(2.03,-1.69)(2.06,-1.73)(2.07,-1.73)(2.10,-1.76)(2.10,-1.76)(2.13,-1.80)(2.13,-1.80)(2.16,-1.83)(2.16,-1.83)(2.19,-1.86)(2.20,-1.87)(2.23,-1.90)(2.23,-1.90)(2.26,-1.93)(2.27,-1.94)(2.29,-1.96)(2.30,-1.97)(2.32,-2.00)(2.33,-2.01)(2.36,-2.03)(2.37,-2.04)(2.39,-2.06)(2.40,-2.08)(2.42,-2.10)(2.43,-2.11)(2.45,-2.13)(2.47,-2.14)(2.49,-2.16)(2.50,-2.18)};
-\draw (-2.500000000,-0.3298256667) node {$ -5 $};
+\draw (-2.5000,-0.32983) node {$ -5 $};
 \draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -4 $};
+\draw (-2.0000,-0.32983) node {$ -4 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.500000000,-0.3298256667) node {$ -3 $};
+\draw (-1.5000,-0.32983) node {$ -3 $};
 \draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -2 $};
+\draw (-1.0000,-0.32983) node {$ -2 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.5000000000,-0.3298256667) node {$ -1 $};
+\draw (-0.50000,-0.32983) node {$ -1 $};
 \draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.5000000000,-0.3149246667) node {$ 1 $};
+\draw (0.50000,-0.31492) node {$ 1 $};
 \draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 2 $};
+\draw (1.0000,-0.31492) node {$ 2 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.500000000,-0.3149246667) node {$ 3 $};
+\draw (1.5000,-0.31492) node {$ 3 $};
 \draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 4 $};
+\draw (2.0000,-0.31492) node {$ 4 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.500000000,-0.3149246667) node {$ 5 $};
+\draw (2.5000,-0.31492) node {$ 5 $};
 \draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 6 $};
+\draw (3.0000,-0.31492) node {$ 6 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -6 $};
+\draw (-0.43316,-3.0000) node {$ -6 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.500000000) node {$ -5 $};
+\draw (-0.43316,-2.5000) node {$ -5 $};
 \draw [] (-0.100,-2.50) -- (0.100,-2.50);
-\draw (-0.4331593333,-2.000000000) node {$ -4 $};
+\draw (-0.43316,-2.0000) node {$ -4 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.500000000) node {$ -3 $};
+\draw (-0.43316,-1.5000) node {$ -3 $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.4331593333,-1.000000000) node {$ -2 $};
+\draw (-0.43316,-1.0000) node {$ -2 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.4331593333,-0.5000000000) node {$ -1 $};
+\draw (-0.43316,-0.50000) node {$ -1 $};
 \draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.2912498333,0.5000000000) node {$ 1 $};
+\draw (-0.29125,0.50000) node {$ 1 $};
 \draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,1.500000000) node {$ 3 $};
+\draw (-0.29125,1.5000) node {$ 3 $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,2.500000000) node {$ 5 $};
+\draw (-0.29125,2.5000) node {$ 5 $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QCb.pstricks.recall b/src_phystricks/Fig_QCb.pstricks.recall
index 5947cc800..389c36fef 100644
--- a/src_phystricks/Fig_QCb.pstricks.recall
+++ b/src_phystricks/Fig_QCb.pstricks.recall
@@ -71,15 +71,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw (-0.7996721667,1.000000000) node {\( xy\)};
-\draw (1.567623333,1.000000000) node {\( \sin(xy)\)};
-\draw (1.200327833,-1.000000000) node {\( xy\)};
-\draw (-0.7996721667,-1.000000000) node {\( xy\)};
+\draw (-0.79967,1.0000) node {\( xy\)};
+\draw (1.5676,1.0000) node {\( \sin(xy)\)};
+\draw (1.2003,-1.0000) node {\( xy\)};
+\draw (-0.79967,-1.0000) node {\( xy\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall b/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall
index aee2e97e2..25a73757d 100644
--- a/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall
+++ b/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall
@@ -80,23 +80,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-4.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
 %DEFAULT
-\draw []  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (-1.000000000,-0.3247080000) node {\( A\)};
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3247080000) node {\( B\)};
+\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.32471) node {\( A\)};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.32471) node {\( B\)};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0.2856973562,-0.2661293562) node {\( O\)};
-\draw []  (0,2.828427125) node [rotate=0] {$\bullet$};
-\draw (0.2359663562,3.094556481) node {\( I\)};
+\draw (0.28570,-0.26613) node {\( O\)};
+\draw []  (0,2.8284) node [rotate=0] {$\bullet$};
+\draw (0.23597,3.0946) node {\( I\)};
 
 \draw [] (2.000,0)--(1.994,0.1903)--(1.976,0.3798)--(1.946,0.5678)--(1.904,0.7534)--(1.850,0.9361)--(1.785,1.115)--(1.709,1.289)--(1.622,1.459)--(1.524,1.622)--(1.416,1.779)--(1.298,1.928)--(1.171,2.070)--(1.036,2.204)--(0.8917,2.328)--(0.7402,2.444)--(0.5817,2.549)--(0.4168,2.644)--(0.2462,2.729)--(0.07066,2.802)--(-0.1092,2.865)--(-0.2927,2.915)--(-0.4791,2.954)--(-0.6675,2.982)--(-0.8573,2.997)--(-1.048,3.000)--(-1.238,2.991)--(-1.427,2.969)--(-1.614,2.936)--(-1.799,2.892)--(-1.981,2.835)--(-2.159,2.767)--(-2.332,2.688)--(-2.500,2.598)--(-2.662,2.498)--(-2.817,2.387)--(-2.965,2.267)--(-3.104,2.138)--(-3.236,2.000)--(-3.358,1.854)--(-3.471,1.701)--(-3.574,1.541)--(-3.667,1.375)--(-3.748,1.203)--(-3.819,1.026)--(-3.878,0.8452)--(-3.926,0.6609)--(-3.962,0.4740)--(-3.986,0.2852)--(-3.999,0.09518)--(-3.999,-0.09518)--(-3.986,-0.2852)--(-3.962,-0.4740)--(-3.926,-0.6609)--(-3.878,-0.8452)--(-3.819,-1.026)--(-3.748,-1.203)--(-3.667,-1.375)--(-3.574,-1.541)--(-3.471,-1.701)--(-3.358,-1.854)--(-3.236,-2.000)--(-3.104,-2.138)--(-2.965,-2.267)--(-2.817,-2.387)--(-2.662,-2.498)--(-2.500,-2.598)--(-2.332,-2.688)--(-2.159,-2.767)--(-1.981,-2.835)--(-1.799,-2.892)--(-1.614,-2.936)--(-1.427,-2.969)--(-1.238,-2.991)--(-1.048,-3.000)--(-0.8573,-2.997)--(-0.6675,-2.982)--(-0.4791,-2.954)--(-0.2927,-2.915)--(-0.1092,-2.865)--(0.07066,-2.802)--(0.2462,-2.729)--(0.4168,-2.644)--(0.5817,-2.549)--(0.7402,-2.444)--(0.8917,-2.328)--(1.036,-2.204)--(1.171,-2.070)--(1.298,-1.928)--(1.416,-1.779)--(1.524,-1.622)--(1.622,-1.459)--(1.709,-1.289)--(1.785,-1.115)--(1.850,-0.9361)--(1.904,-0.7534)--(1.946,-0.5678)--(1.976,-0.3798)--(1.994,-0.1903)--(2.000,0);
 
 \draw [] (4.000,0)--(3.994,0.1903)--(3.976,0.3798)--(3.946,0.5678)--(3.904,0.7534)--(3.850,0.9361)--(3.785,1.115)--(3.709,1.289)--(3.622,1.459)--(3.524,1.622)--(3.416,1.779)--(3.298,1.928)--(3.171,2.070)--(3.036,2.204)--(2.892,2.328)--(2.740,2.444)--(2.582,2.549)--(2.417,2.644)--(2.246,2.729)--(2.071,2.802)--(1.891,2.865)--(1.707,2.915)--(1.521,2.954)--(1.333,2.982)--(1.143,2.997)--(0.9524,3.000)--(0.7623,2.991)--(0.5731,2.969)--(0.3856,2.936)--(0.2006,2.892)--(0.01880,2.835)--(-0.1590,2.767)--(-0.3322,2.688)--(-0.5000,2.598)--(-0.6618,2.498)--(-0.8168,2.387)--(-0.9646,2.267)--(-1.104,2.138)--(-1.236,2.000)--(-1.358,1.854)--(-1.471,1.701)--(-1.574,1.541)--(-1.667,1.375)--(-1.748,1.203)--(-1.819,1.026)--(-1.878,0.8452)--(-1.926,0.6609)--(-1.962,0.4740)--(-1.986,0.2852)--(-1.998,0.09518)--(-1.998,-0.09518)--(-1.986,-0.2852)--(-1.962,-0.4740)--(-1.926,-0.6609)--(-1.878,-0.8452)--(-1.819,-1.026)--(-1.748,-1.203)--(-1.667,-1.375)--(-1.574,-1.541)--(-1.471,-1.701)--(-1.358,-1.854)--(-1.236,-2.000)--(-1.104,-2.138)--(-0.9646,-2.267)--(-0.8168,-2.387)--(-0.6618,-2.498)--(-0.5000,-2.598)--(-0.3322,-2.688)--(-0.1590,-2.767)--(0.01880,-2.835)--(0.2006,-2.892)--(0.3856,-2.936)--(0.5731,-2.969)--(0.7623,-2.991)--(0.9524,-3.000)--(1.143,-2.997)--(1.333,-2.982)--(1.521,-2.954)--(1.707,-2.915)--(1.891,-2.865)--(2.071,-2.802)--(2.246,-2.729)--(2.417,-2.644)--(2.582,-2.549)--(2.740,-2.444)--(2.892,-2.328)--(3.036,-2.204)--(3.171,-2.070)--(3.298,-1.928)--(3.416,-1.779)--(3.524,-1.622)--(3.622,-1.459)--(3.709,-1.289)--(3.785,-1.115)--(3.850,-0.9361)--(3.904,-0.7534)--(3.946,-0.5678)--(3.976,-0.3798)--(3.994,-0.1903)--(4.000,0);
-\draw []  (-0.7000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.9856975229,1.301615523) node {\( Q\)};
+\draw []  (-0.70000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.98570,1.3016) node {\( Q\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall b/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall
index b53a39525..21bc4eded 100644
--- a/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall
+++ b/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (1.000,2.925)--(1.061,2.844)--(1.121,2.765)--(1.182,2.687)--(1.242,2.611)--(1.303,2.537)--(1.364,2.464)--(1.424,2.393)--(1.485,2.323)--(1.545,2.256)--(1.606,2.189)--(1.667,2.125)--(1.727,2.062)--(1.788,2.001)--(1.848,1.942)--(1.909,1.884)--(1.970,1.827)--(2.030,1.773)--(2.091,1.720)--(2.152,1.669)--(2.212,1.619)--(2.273,1.571)--(2.333,1.525)--(2.394,1.480)--(2.455,1.437)--(2.515,1.396)--(2.576,1.356)--(2.636,1.318)--(2.697,1.282)--(2.758,1.247)--(2.818,1.214)--(2.879,1.183)--(2.939,1.153)--(3.000,1.125)--(3.061,1.099)--(3.121,1.074)--(3.182,1.051)--(3.242,1.029)--(3.303,1.009)--(3.364,0.9911)--(3.424,0.9746)--(3.485,0.9597)--(3.545,0.9465)--(3.606,0.9349)--(3.667,0.9250)--(3.727,0.9167)--(3.788,0.9101)--(3.848,0.9052)--(3.909,0.9019)--(3.970,0.9002)--(4.030,0.9002)--(4.091,0.9019)--(4.151,0.9052)--(4.212,0.9101)--(4.273,0.9167)--(4.333,0.9250)--(4.394,0.9349)--(4.455,0.9465)--(4.515,0.9597)--(4.576,0.9746)--(4.636,0.9911)--(4.697,1.009)--(4.758,1.029)--(4.818,1.051)--(4.879,1.074)--(4.939,1.099)--(5.000,1.125)--(5.061,1.153)--(5.121,1.183)--(5.182,1.214)--(5.242,1.247)--(5.303,1.282)--(5.364,1.318)--(5.424,1.356)--(5.485,1.396)--(5.545,1.437)--(5.606,1.480)--(5.667,1.525)--(5.727,1.571)--(5.788,1.619)--(5.849,1.669)--(5.909,1.720)--(5.970,1.773)--(6.030,1.827)--(6.091,1.884)--(6.151,1.942)--(6.212,2.001)--(6.273,2.062)--(6.333,2.125)--(6.394,2.189)--(6.455,2.256)--(6.515,2.323)--(6.576,2.393)--(6.636,2.464)--(6.697,2.537)--(6.758,2.611)--(6.818,2.687)--(6.879,2.765)--(6.939,2.844)--(7.000,2.925);
@@ -53,10 +53,10 @@
 \draw [color=blue] (2.174,0)--(2.211,0)--(2.248,0)--(2.285,0)--(2.322,0)--(2.359,0)--(2.396,0)--(2.432,0)--(2.469,0)--(2.506,0)--(2.543,0)--(2.580,0)--(2.617,0)--(2.654,0)--(2.691,0)--(2.728,0)--(2.764,0)--(2.801,0)--(2.838,0)--(2.875,0)--(2.912,0)--(2.949,0)--(2.986,0)--(3.023,0)--(3.059,0)--(3.096,0)--(3.133,0)--(3.170,0)--(3.207,0)--(3.244,0)--(3.281,0)--(3.318,0)--(3.355,0)--(3.391,0)--(3.428,0)--(3.465,0)--(3.502,0)--(3.539,0)--(3.576,0)--(3.613,0)--(3.650,0)--(3.686,0)--(3.723,0)--(3.760,0)--(3.797,0)--(3.834,0)--(3.871,0)--(3.908,0)--(3.945,0)--(3.982,0)--(4.018,0)--(4.055,0)--(4.092,0)--(4.129,0)--(4.166,0)--(4.203,0)--(4.240,0)--(4.277,0)--(4.314,0)--(4.350,0)--(4.387,0)--(4.424,0)--(4.461,0)--(4.498,0)--(4.535,0)--(4.572,0)--(4.609,0)--(4.645,0)--(4.682,0)--(4.719,0)--(4.756,0)--(4.793,0)--(4.830,0)--(4.867,0)--(4.904,0)--(4.941,0)--(4.977,0)--(5.014,0)--(5.051,0)--(5.088,0)--(5.125,0)--(5.162,0)--(5.199,0)--(5.236,0)--(5.272,0)--(5.309,0)--(5.346,0)--(5.383,0)--(5.420,0)--(5.457,0)--(5.494,0)--(5.531,0)--(5.568,0)--(5.604,0)--(5.641,0)--(5.678,0)--(5.715,0)--(5.752,0)--(5.789,0)--(5.826,0);
 \draw [] (2.17,0) -- (2.17,1.65);
 \draw [] (5.83,1.65) -- (5.83,0);
-\draw []  (2.174258142,0) node [rotate=0] {$\bullet$};
-\draw (2.174258142,-0.3785761667) node {$a$};
-\draw []  (5.825741858,0) node [rotate=0] {$\bullet$};
-\draw (5.825741858,-0.4267360000) node {$b$};
+\draw []  (2.1743,0) node [rotate=0] {$\bullet$};
+\draw (2.1743,-0.37858) node {$a$};
+\draw []  (5.8257,0) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,8 +95,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (1.000,2.925)--(1.061,2.844)--(1.121,2.765)--(1.182,2.687)--(1.242,2.611)--(1.303,2.537)--(1.364,2.464)--(1.424,2.393)--(1.485,2.323)--(1.545,2.256)--(1.606,2.189)--(1.667,2.125)--(1.727,2.062)--(1.788,2.001)--(1.848,1.942)--(1.909,1.884)--(1.970,1.827)--(2.030,1.773)--(2.091,1.720)--(2.152,1.669)--(2.212,1.619)--(2.273,1.571)--(2.333,1.525)--(2.394,1.480)--(2.455,1.437)--(2.515,1.396)--(2.576,1.356)--(2.636,1.318)--(2.697,1.282)--(2.758,1.247)--(2.818,1.214)--(2.879,1.183)--(2.939,1.153)--(3.000,1.125)--(3.061,1.099)--(3.121,1.074)--(3.182,1.051)--(3.242,1.029)--(3.303,1.009)--(3.364,0.9911)--(3.424,0.9746)--(3.485,0.9597)--(3.545,0.9465)--(3.606,0.9349)--(3.667,0.9250)--(3.727,0.9167)--(3.788,0.9101)--(3.848,0.9052)--(3.909,0.9019)--(3.970,0.9002)--(4.030,0.9002)--(4.091,0.9019)--(4.151,0.9052)--(4.212,0.9101)--(4.273,0.9167)--(4.333,0.9250)--(4.394,0.9349)--(4.455,0.9465)--(4.515,0.9597)--(4.576,0.9746)--(4.636,0.9911)--(4.697,1.009)--(4.758,1.029)--(4.818,1.051)--(4.879,1.074)--(4.939,1.099)--(5.000,1.125)--(5.061,1.153)--(5.121,1.183)--(5.182,1.214)--(5.242,1.247)--(5.303,1.282)--(5.364,1.318)--(5.424,1.356)--(5.485,1.396)--(5.545,1.437)--(5.606,1.480)--(5.667,1.525)--(5.727,1.571)--(5.788,1.619)--(5.849,1.669)--(5.909,1.720)--(5.970,1.773)--(6.030,1.827)--(6.091,1.884)--(6.151,1.942)--(6.212,2.001)--(6.273,2.062)--(6.333,2.125)--(6.394,2.189)--(6.455,2.256)--(6.515,2.323)--(6.576,2.393)--(6.636,2.464)--(6.697,2.537)--(6.758,2.611)--(6.818,2.687)--(6.879,2.765)--(6.939,2.844)--(7.000,2.925);
@@ -107,10 +107,10 @@
 \draw [color=blue] (2.174,0)--(2.211,0)--(2.248,0)--(2.285,0)--(2.322,0)--(2.359,0)--(2.396,0)--(2.432,0)--(2.469,0)--(2.506,0)--(2.543,0)--(2.580,0)--(2.617,0)--(2.654,0)--(2.691,0)--(2.728,0)--(2.764,0)--(2.801,0)--(2.838,0)--(2.875,0)--(2.912,0)--(2.949,0)--(2.986,0)--(3.023,0)--(3.059,0)--(3.096,0)--(3.133,0)--(3.170,0)--(3.207,0)--(3.244,0)--(3.281,0)--(3.318,0)--(3.355,0)--(3.391,0)--(3.428,0)--(3.465,0)--(3.502,0)--(3.539,0)--(3.576,0)--(3.613,0)--(3.650,0)--(3.686,0)--(3.723,0)--(3.760,0)--(3.797,0)--(3.834,0)--(3.871,0)--(3.908,0)--(3.945,0)--(3.982,0)--(4.018,0)--(4.055,0)--(4.092,0)--(4.129,0)--(4.166,0)--(4.203,0)--(4.240,0)--(4.277,0)--(4.314,0)--(4.350,0)--(4.387,0)--(4.424,0)--(4.461,0)--(4.498,0)--(4.535,0)--(4.572,0)--(4.609,0)--(4.645,0)--(4.682,0)--(4.719,0)--(4.756,0)--(4.793,0)--(4.830,0)--(4.867,0)--(4.904,0)--(4.941,0)--(4.977,0)--(5.014,0)--(5.051,0)--(5.088,0)--(5.125,0)--(5.162,0)--(5.199,0)--(5.236,0)--(5.272,0)--(5.309,0)--(5.346,0)--(5.383,0)--(5.420,0)--(5.457,0)--(5.494,0)--(5.531,0)--(5.568,0)--(5.604,0)--(5.641,0)--(5.678,0)--(5.715,0)--(5.752,0)--(5.789,0)--(5.826,0);
 \draw [] (2.17,0) -- (2.17,1.65);
 \draw [] (5.83,1.65) -- (5.83,0);
-\draw []  (2.174258142,0) node [rotate=0] {$\bullet$};
-\draw (2.174258142,-0.3785761667) node {$a$};
-\draw []  (5.825741858,0) node [rotate=0] {$\bullet$};
-\draw (5.825741858,-0.4267360000) node {$b$};
+\draw []  (2.1743,0) node [rotate=0] {$\bullet$};
+\draw (2.1743,-0.37858) node {$a$};
+\draw []  (5.8257,0) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,8 +149,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (7.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.425000000);
+\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
 %DEFAULT
 
 \draw [color=blue] (1.000,2.925)--(1.061,2.844)--(1.121,2.765)--(1.182,2.687)--(1.242,2.611)--(1.303,2.537)--(1.364,2.464)--(1.424,2.393)--(1.485,2.323)--(1.545,2.256)--(1.606,2.189)--(1.667,2.125)--(1.727,2.062)--(1.788,2.001)--(1.848,1.942)--(1.909,1.884)--(1.970,1.827)--(2.030,1.773)--(2.091,1.720)--(2.152,1.669)--(2.212,1.619)--(2.273,1.571)--(2.333,1.525)--(2.394,1.480)--(2.455,1.437)--(2.515,1.396)--(2.576,1.356)--(2.636,1.318)--(2.697,1.282)--(2.758,1.247)--(2.818,1.214)--(2.879,1.183)--(2.939,1.153)--(3.000,1.125)--(3.061,1.099)--(3.121,1.074)--(3.182,1.051)--(3.242,1.029)--(3.303,1.009)--(3.364,0.9911)--(3.424,0.9746)--(3.485,0.9597)--(3.545,0.9465)--(3.606,0.9349)--(3.667,0.9250)--(3.727,0.9167)--(3.788,0.9101)--(3.848,0.9052)--(3.909,0.9019)--(3.970,0.9002)--(4.030,0.9002)--(4.091,0.9019)--(4.151,0.9052)--(4.212,0.9101)--(4.273,0.9167)--(4.333,0.9250)--(4.394,0.9349)--(4.455,0.9465)--(4.515,0.9597)--(4.576,0.9746)--(4.636,0.9911)--(4.697,1.009)--(4.758,1.029)--(4.818,1.051)--(4.879,1.074)--(4.939,1.099)--(5.000,1.125)--(5.061,1.153)--(5.121,1.183)--(5.182,1.214)--(5.242,1.247)--(5.303,1.282)--(5.364,1.318)--(5.424,1.356)--(5.485,1.396)--(5.545,1.437)--(5.606,1.480)--(5.667,1.525)--(5.727,1.571)--(5.788,1.619)--(5.849,1.669)--(5.909,1.720)--(5.970,1.773)--(6.030,1.827)--(6.091,1.884)--(6.151,1.942)--(6.212,2.001)--(6.273,2.062)--(6.333,2.125)--(6.394,2.189)--(6.455,2.256)--(6.515,2.323)--(6.576,2.393)--(6.636,2.464)--(6.697,2.537)--(6.758,2.611)--(6.818,2.687)--(6.879,2.765)--(6.939,2.844)--(7.000,2.925);
@@ -161,10 +161,10 @@
 \draw [color=blue] (2.174,1.650)--(2.211,1.680)--(2.248,1.709)--(2.285,1.738)--(2.322,1.766)--(2.359,1.794)--(2.396,1.821)--(2.432,1.847)--(2.469,1.873)--(2.506,1.898)--(2.543,1.922)--(2.580,1.946)--(2.617,1.970)--(2.654,1.992)--(2.691,2.014)--(2.728,2.036)--(2.764,2.056)--(2.801,2.077)--(2.838,2.096)--(2.875,2.115)--(2.912,2.134)--(2.949,2.151)--(2.986,2.169)--(3.023,2.185)--(3.059,2.201)--(3.096,2.216)--(3.133,2.231)--(3.170,2.245)--(3.207,2.259)--(3.244,2.271)--(3.281,2.284)--(3.318,2.295)--(3.355,2.306)--(3.391,2.317)--(3.428,2.326)--(3.465,2.336)--(3.502,2.344)--(3.539,2.352)--(3.576,2.360)--(3.613,2.366)--(3.650,2.372)--(3.686,2.378)--(3.723,2.383)--(3.760,2.387)--(3.797,2.391)--(3.834,2.394)--(3.871,2.396)--(3.908,2.398)--(3.945,2.399)--(3.982,2.400)--(4.018,2.400)--(4.055,2.399)--(4.092,2.398)--(4.129,2.396)--(4.166,2.394)--(4.203,2.391)--(4.240,2.387)--(4.277,2.383)--(4.314,2.378)--(4.350,2.372)--(4.387,2.366)--(4.424,2.360)--(4.461,2.352)--(4.498,2.344)--(4.535,2.336)--(4.572,2.326)--(4.609,2.317)--(4.645,2.306)--(4.682,2.295)--(4.719,2.284)--(4.756,2.271)--(4.793,2.259)--(4.830,2.245)--(4.867,2.231)--(4.904,2.216)--(4.941,2.201)--(4.977,2.185)--(5.014,2.169)--(5.051,2.151)--(5.088,2.134)--(5.125,2.115)--(5.162,2.096)--(5.199,2.077)--(5.236,2.056)--(5.272,2.036)--(5.309,2.014)--(5.346,1.992)--(5.383,1.970)--(5.420,1.946)--(5.457,1.922)--(5.494,1.898)--(5.531,1.873)--(5.568,1.847)--(5.604,1.821)--(5.641,1.794)--(5.678,1.766)--(5.715,1.738)--(5.752,1.709)--(5.789,1.680)--(5.826,1.650);
 \draw [] (2.17,1.65) -- (2.17,1.65);
 \draw [] (5.83,1.65) -- (5.83,1.65);
-\draw []  (2.174258142,0) node [rotate=0] {$\bullet$};
-\draw (2.174258142,-0.3785761667) node {$a$};
-\draw []  (5.825741858,0) node [rotate=0] {$\bullet$};
-\draw (5.825741858,-0.4267360000) node {$b$};
+\draw []  (2.1743,0) node [rotate=0] {$\bullet$};
+\draw (2.1743,-0.37858) node {$a$};
+\draw []  (5.8257,0) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.42674) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QQa.pstricks.recall b/src_phystricks/Fig_QQa.pstricks.recall
index 54142f0c4..b9cb733d7 100644
--- a/src_phystricks/Fig_QQa.pstricks.recall
+++ b/src_phystricks/Fig_QQa.pstricks.recall
@@ -33,8 +33,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (0.5000000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (0.50000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
@@ -102,8 +102,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,-2.00)(-1.98,-1.88)(-1.81,-1.91)(-1.84,-1.74)(-1.66,-1.76)(-1.69,-1.59)(-1.52,-1.62)(-1.55,-1.45)(-1.38,-1.48)(-1.41,-1.31)(-1.24,-1.34)(-1.26,-1.16)(-1.09,-1.19)(-1.12,-1.02)(-0.951,-1.05)(-0.978,-0.879)(-0.808,-0.907)(-0.835,-0.736)(-0.665,-0.764)(-0.692,-0.593)(-0.522,-0.621)(-0.549,-0.451)(-0.379,-0.478)(-0.407,-0.308)(-0.236,-0.335)(-0.264,-0.165)(-0.0934,-0.192)(-0.121,-0.0219)(0.0495,-0.0495)(0.0219,0.121)(0.192,0.0934)(0.165,0.264)(0.335,0.236)(0.308,0.407)(0.478,0.379)(0.451,0.549)(0.621,0.522)(0.593,0.692)(0.764,0.665)(0.736,0.835)(0.907,0.808)(0.879,0.978)(1.05,0.951)(1.02,1.12)(1.19,1.09)(1.16,1.26)(1.34,1.24)(1.31,1.41)(1.48,1.38)(1.45,1.55)(1.62,1.52)(1.59,1.69)(1.76,1.66)(1.74,1.84)(1.91,1.81)(1.88,1.98)(2.00,2.00)};
 
@@ -136,8 +136,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.802585093) -- (0,1.235192735);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.8026) -- (0,1.2352);
 %DEFAULT
 
 \draw [color=red] plot [smooth,tension=1] coordinates {(0.100,-2.30)(0.0409,-2.20)(0.191,-2.11)(0.0654,-1.99)(0.218,-1.91)(0.0952,-1.79)(0.250,-1.72)(0.131,-1.60)(0.289,-1.53)(0.175,-1.40)(0.336,-1.33)(0.228,-1.20)(0.391,-1.15)(0.291,-1.01)(0.457,-0.962)(0.365,-0.817)(0.534,-0.783)(0.453,-0.631)(0.624,-0.609)(0.554,-0.452)(0.726,-0.443)(0.668,-0.281)(0.840,-0.286)(0.794,-0.120)(0.966,-0.137)(0.933,0.0318)(1.10,0.00217)(1.08,0.173)(1.25,0.133)(1.24,0.304)(1.40,0.253)(1.40,0.425)(1.57,0.365)(1.58,0.537)(1.73,0.469)(1.75,0.640)(1.90,0.565)(1.93,0.735)(2.00,0.693)};
@@ -171,8 +171,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
@@ -206,8 +206,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
 \draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
diff --git a/src_phystricks/Fig_QXyVaKD.pstricks.recall b/src_phystricks/Fig_QXyVaKD.pstricks.recall
index 8f57cfe01..c8f85cd29 100644
--- a/src_phystricks/Fig_QXyVaKD.pstricks.recall
+++ b/src_phystricks/Fig_QXyVaKD.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-1.000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=red] (0,0)--(0.0159,0)--(0.0317,0)--(0.0476,0.00113)--(0.0634,0.00201)--(0.0792,0.00315)--(0.0951,0.00453)--(0.111,0.00616)--(0.127,0.00805)--(0.142,0.0102)--(0.158,0.0126)--(0.174,0.0152)--(0.189,0.0181)--(0.205,0.0212)--(0.220,0.0246)--(0.236,0.0282)--(0.251,0.0321)--(0.266,0.0362)--(0.282,0.0405)--(0.297,0.0451)--(0.312,0.0499)--(0.327,0.0550)--(0.342,0.0603)--(0.357,0.0658)--(0.372,0.0716)--(0.386,0.0776)--(0.401,0.0839)--(0.415,0.0904)--(0.430,0.0971)--(0.444,0.104)--(0.458,0.111)--(0.472,0.119)--(0.486,0.126)--(0.500,0.134)--(0.514,0.142)--(0.527,0.150)--(0.541,0.159)--(0.554,0.167)--(0.567,0.176)--(0.580,0.185)--(0.593,0.195)--(0.606,0.204)--(0.618,0.214)--(0.631,0.224)--(0.643,0.234)--(0.655,0.244)--(0.667,0.255)--(0.679,0.265)--(0.690,0.276)--(0.701,0.287)--(0.713,0.299)--(0.724,0.310)--(0.735,0.322)--(0.745,0.333)--(0.756,0.345)--(0.766,0.357)--(0.776,0.369)--(0.786,0.382)--(0.796,0.394)--(0.805,0.407)--(0.815,0.420)--(0.824,0.433)--(0.833,0.446)--(0.841,0.459)--(0.850,0.473)--(0.858,0.486)--(0.866,0.500)--(0.874,0.514)--(0.881,0.528)--(0.889,0.542)--(0.896,0.556)--(0.903,0.570)--(0.910,0.585)--(0.916,0.599)--(0.922,0.614)--(0.928,0.628)--(0.934,0.643)--(0.940,0.658)--(0.945,0.673)--(0.950,0.688)--(0.955,0.703)--(0.959,0.718)--(0.964,0.734)--(0.968,0.749)--(0.972,0.764)--(0.975,0.780)--(0.979,0.795)--(0.982,0.811)--(0.985,0.826)--(0.987,0.842)--(0.990,0.858)--(0.992,0.873)--(0.994,0.889)--(0.995,0.905)--(0.997,0.921)--(0.998,0.937)--(0.999,0.952)--(1.00,0.968)--(1.00,0.984)--(1.00,1.00);
@@ -80,8 +80,8 @@
 \draw [color=cyan] (0,0) -- (0,0);
 \draw [color=cyan] (1.00,1.00) -- (1.00,1.00);
 \draw [color=red] (-0.500,-0.500) -- (1.50,1.50);
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.354646868,0.6631599656) node {$P$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.3546,0.66316) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QuelCote.pstricks.recall b/src_phystricks/Fig_QuelCote.pstricks.recall
index 4a2b8ccb2..314ffc1de 100644
--- a/src_phystricks/Fig_QuelCote.pstricks.recall
+++ b/src_phystricks/Fig_QuelCote.pstricks.recall
@@ -87,16 +87,16 @@
 %DEFAULT
 \draw [color=blue] (-3.000,-1.750)--(-2.980,-1.720)--(-2.961,-1.691)--(-2.941,-1.661)--(-2.922,-1.631)--(-2.903,-1.601)--(-2.883,-1.572)--(-2.864,-1.541)--(-2.845,-1.511)--(-2.826,-1.481)--(-2.808,-1.451)--(-2.789,-1.421)--(-2.770,-1.390)--(-2.752,-1.360)--(-2.734,-1.329)--(-2.715,-1.298)--(-2.697,-1.268)--(-2.679,-1.237)--(-2.661,-1.206)--(-2.644,-1.175)--(-2.626,-1.143)--(-2.608,-1.112)--(-2.591,-1.081)--(-2.574,-1.049)--(-2.557,-1.017)--(-2.540,-0.9856)--(-2.523,-0.9537)--(-2.506,-0.9216)--(-2.490,-0.8893)--(-2.474,-0.8570)--(-2.457,-0.8244)--(-2.441,-0.7918)--(-2.426,-0.7589)--(-2.410,-0.7259)--(-2.394,-0.6927)--(-2.379,-0.6594)--(-2.364,-0.6258)--(-2.349,-0.5921)--(-2.334,-0.5582)--(-2.320,-0.5240)--(-2.306,-0.4897)--(-2.292,-0.4551)--(-2.278,-0.4203)--(-2.264,-0.3853)--(-2.251,-0.3501)--(-2.238,-0.3145)--(-2.225,-0.2788)--(-2.212,-0.2427)--(-2.200,-0.2064)--(-2.187,-0.1697)--(-2.176,-0.1328)--(-2.164,-0.09554)--(-2.153,-0.05796)--(-2.142,-0.02003)--(-2.131,0.01826)--(-2.121,0.05692)--(-2.111,0.09597)--(-2.101,0.1354)--(-2.092,0.1753)--(-2.083,0.2156)--(-2.074,0.2564)--(-2.066,0.2976)--(-2.058,0.3394)--(-2.051,0.3817)--(-2.044,0.4245)--(-2.037,0.4679)--(-2.031,0.5119)--(-2.026,0.5565)--(-2.021,0.6018)--(-2.016,0.6478)--(-2.012,0.6944)--(-2.009,0.7419)--(-2.006,0.7901)--(-2.003,0.8392)--(-2.002,0.8892)--(-2.000,0.9401)--(-2.000,0.9919)--(-2.000,1.045)--(-2.001,1.099)--(-2.003,1.154)--(-2.005,1.210)--(-2.009,1.268)--(-2.013,1.327)--(-2.018,1.387)--(-2.024,1.449)--(-2.031,1.512)--(-2.038,1.578)--(-2.047,1.645)--(-2.057,1.713)--(-2.068,1.784)--(-2.081,1.857)--(-2.094,1.933)--(-2.109,2.011)--(-2.126,2.091)--(-2.143,2.174)--(-2.163,2.261)--(-2.184,2.350)--(-2.206,2.443)--(-2.231,2.540)--(-2.257,2.641);
 \draw [color=brown,style=dashed] (-2.45,-1.67) -- (-1.69,2.26);
-\draw [color=brown,->,>=latex] (-2.066666667,0.2944444444) -- (-1.686622854,2.258004146);
-\draw (-0.8272048535,2.258004146) node {$\gamma'(t)$};
-\draw [,->,>=latex] (-2.066666667,0.2944444444) -- (-3.316056762,1.856182063);
-\draw (-2.711760061,2.254701931) node {$\gamma''(t)$};
-\draw [color=green,->,>=latex] (-2.066666667,0.2944444444) -- (-0.1031069655,-0.08559936869);
-\draw (-0.1031069655,0.3968556313) node {$n(t)$};
-\draw [color=green,style=dashed,->,>=latex] (-2.066666667,0.2944444444) -- (-4.030226368,0.6744882576);
-\draw (-4.030226368,1.156943258) node {$-n(t)$};
-\draw []  (-2.066666667,0.2944444444) node [rotate=0] {$\bullet$};
-\draw (-1.712019799,-0.04239558991) node {$P$};
+\draw [color=brown,->,>=latex] (-2.0667,0.29444) -- (-1.6866,2.2580);
+\draw (-0.82720,2.2580) node {$\gamma'(t)$};
+\draw [,->,>=latex] (-2.0667,0.29444) -- (-3.3161,1.8562);
+\draw (-2.7118,2.2547) node {$\gamma''(t)$};
+\draw [color=green,->,>=latex] (-2.0667,0.29444) -- (-0.10311,-0.085599);
+\draw (-0.10311,0.39686) node {$n(t)$};
+\draw [color=green,style=dashed,->,>=latex] (-2.0667,0.29444) -- (-4.0302,0.67449);
+\draw (-4.0302,1.1569) node {$-n(t)$};
+\draw []  (-2.0667,0.29444) node [rotate=0] {$\bullet$};
+\draw (-1.7120,-0.042396) node {$P$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_RGjjpwF.pstricks.recall b/src_phystricks/Fig_RGjjpwF.pstricks.recall
index eb1ded532..055d4425c 100644
--- a/src_phystricks/Fig_RGjjpwF.pstricks.recall
+++ b/src_phystricks/Fig_RGjjpwF.pstricks.recall
@@ -75,15 +75,15 @@
 \draw [style=dotted] (2.00,0) -- (3.00,0);
 \draw [] (3.00,0) -- (4.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3149246667) node {$1$};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2000000000) node {};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.2000000000) node {};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.2000000000) node {};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.2000000000) node {};
+\draw (0,0.31492) node {$1$};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.20000) node {};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.20000) node {};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.20000) node {};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.20000) node {};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_RLuqsrr.pstricks.recall b/src_phystricks/Fig_RLuqsrr.pstricks.recall
index ae4cd3b5d..f50e8f6d6 100644
--- a/src_phystricks/Fig_RLuqsrr.pstricks.recall
+++ b/src_phystricks/Fig_RLuqsrr.pstricks.recall
@@ -87,24 +87,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.783185311,0);
-\draw [,->,>=latex] (0,-0.9138130496) -- (0,2.914169059);
+\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
+\draw [,->,>=latex] (0,-0.91381) -- (0,2.9142);
 %DEFAULT
 
 \draw [color=red] (0,0)--(0.06347,0.1346)--(0.1269,0.2832)--(0.1904,0.4433)--(0.2539,0.6124)--(0.3173,0.7876)--(0.3808,0.9663)--(0.4443,1.146)--(0.5077,1.322)--(0.5712,1.494)--(0.6347,1.658)--(0.6981,1.811)--(0.7616,1.951)--(0.8251,2.076)--(0.8885,2.184)--(0.9520,2.272)--(1.015,2.340)--(1.079,2.387)--(1.142,2.411)--(1.206,2.412)--(1.269,2.391)--(1.333,2.347)--(1.396,2.282)--(1.460,2.196)--(1.523,2.091)--(1.587,1.968)--(1.650,1.829)--(1.714,1.678)--(1.777,1.515)--(1.841,1.344)--(1.904,1.168)--(1.967,0.9888)--(2.031,0.8098)--(2.094,0.6340)--(2.158,0.4640)--(2.221,0.3026)--(2.285,0.1525)--(2.348,0.01599)--(2.412,-0.1047)--(2.475,-0.2076)--(2.539,-0.2910)--(2.602,-0.3537)--(2.666,-0.3946)--(2.729,-0.4131)--(2.793,-0.4088)--(2.856,-0.3819)--(2.919,-0.3327)--(2.983,-0.2621)--(3.046,-0.1712)--(3.110,-0.06141)--(3.173,0.06544)--(3.237,0.2073)--(3.300,0.3620)--(3.364,0.5269)--(3.427,0.6994)--(3.491,0.8767)--(3.554,1.056)--(3.618,1.235)--(3.681,1.409)--(3.745,1.577)--(3.808,1.736)--(3.871,1.883)--(3.935,2.016)--(3.998,2.132)--(4.062,2.230)--(4.125,2.309)--(4.189,2.366)--(4.252,2.401)--(4.316,2.414)--(4.379,2.404)--(4.443,2.372)--(4.506,2.317)--(4.570,2.241)--(4.633,2.145)--(4.697,2.031)--(4.760,1.900)--(4.823,1.755)--(4.887,1.598)--(4.950,1.431)--(5.014,1.257)--(5.077,1.078)--(5.141,0.8991)--(5.204,0.7214)--(5.268,0.5481)--(5.331,0.3821)--(5.395,0.2260)--(5.458,0.08240)--(5.522,-0.04645)--(5.585,-0.1585)--(5.648,-0.2518)--(5.712,-0.3250)--(5.775,-0.3769)--(5.839,-0.4067)--(5.902,-0.4138)--(5.966,-0.3982)--(6.029,-0.3600)--(6.093,-0.3000)--(6.156,-0.2191)--(6.220,-0.1185)--(6.283,0);
 
 \draw [color=blue] (0,0)--(0.06347,0.008045)--(0.1269,0.03205)--(0.1904,0.07163)--(0.2539,0.1262)--(0.3173,0.1947)--(0.3808,0.2763)--(0.4443,0.3694)--(0.5077,0.4728)--(0.5712,0.5846)--(0.6347,0.7031)--(0.6981,0.8264)--(0.7616,0.9524)--(0.8251,1.079)--(0.8885,1.205)--(0.9520,1.327)--(1.015,1.444)--(1.079,1.554)--(1.142,1.655)--(1.206,1.745)--(1.269,1.824)--(1.333,1.889)--(1.396,1.940)--(1.460,1.975)--(1.523,1.995)--(1.587,1.999)--(1.650,1.987)--(1.714,1.959)--(1.777,1.916)--(1.841,1.858)--(1.904,1.786)--(1.967,1.701)--(2.031,1.606)--(2.094,1.500)--(2.158,1.386)--(2.221,1.266)--(2.285,1.142)--(2.348,1.016)--(2.412,0.8892)--(2.475,0.7642)--(2.539,0.6431)--(2.602,0.5277)--(2.666,0.4199)--(2.729,0.3215)--(2.793,0.2340)--(2.856,0.1587)--(2.919,0.09707)--(2.983,0.04993)--(3.046,0.01807)--(3.110,0.002013)--(3.173,0.002013)--(3.237,0.01807)--(3.300,0.04993)--(3.364,0.09707)--(3.427,0.1587)--(3.491,0.2340)--(3.554,0.3215)--(3.618,0.4199)--(3.681,0.5277)--(3.745,0.6431)--(3.808,0.7642)--(3.871,0.8892)--(3.935,1.016)--(3.998,1.142)--(4.062,1.266)--(4.125,1.386)--(4.189,1.500)--(4.252,1.606)--(4.316,1.701)--(4.379,1.786)--(4.443,1.858)--(4.506,1.916)--(4.570,1.959)--(4.633,1.987)--(4.697,1.999)--(4.760,1.995)--(4.823,1.975)--(4.887,1.940)--(4.950,1.889)--(5.014,1.824)--(5.077,1.745)--(5.141,1.655)--(5.204,1.554)--(5.268,1.444)--(5.331,1.327)--(5.395,1.205)--(5.458,1.079)--(5.522,0.9524)--(5.585,0.8264)--(5.648,0.7031)--(5.712,0.5846)--(5.775,0.4728)--(5.839,0.3694)--(5.902,0.2763)--(5.966,0.1947)--(6.029,0.1262)--(6.093,0.07163)--(6.156,0.03205)--(6.220,0.008045)--(6.283,0);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall b/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall
index 7103486be..204852b2e 100644
--- a/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall
+++ b/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,0);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,3.000000000);
-\draw (1.896467667,1.500000000) node {$a$};
-\draw []  (0,3.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
+\draw (1.8965,1.5000) node {$a$};
+\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
 \draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.1430327019,3.634515621) node {$R$};
+\draw (0.14303,3.6345) node {$R$};
 \draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
 
 %OTHER STUFF
diff --git a/src_phystricks/Fig_RQsQKTl.pstricks.recall b/src_phystricks/Fig_RQsQKTl.pstricks.recall
index d688dbefc..adc854ce0 100644
--- a/src_phystricks/Fig_RQsQKTl.pstricks.recall
+++ b/src_phystricks/Fig_RQsQKTl.pstricks.recall
@@ -68,8 +68,8 @@
 %DEFAULT
 \draw [] (0,0) -- (1.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.3149246667) node {\( 1\)};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
+\draw (0,0.31492) node {\( 1\)};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall b/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall
index 8c7d4a199..27241d0e2 100644
--- a/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall
+++ b/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall
@@ -65,8 +65,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -80,12 +80,12 @@
 \draw [color=blue,style=solid] (1.000,5.000)--(1.030,5.060)--(1.061,5.118)--(1.091,5.174)--(1.121,5.228)--(1.152,5.280)--(1.182,5.331)--(1.212,5.379)--(1.242,5.426)--(1.273,5.471)--(1.303,5.514)--(1.333,5.556)--(1.364,5.595)--(1.394,5.633)--(1.424,5.669)--(1.455,5.702)--(1.485,5.735)--(1.515,5.765)--(1.545,5.793)--(1.576,5.820)--(1.606,5.845)--(1.636,5.868)--(1.667,5.889)--(1.697,5.908)--(1.727,5.926)--(1.758,5.941)--(1.788,5.955)--(1.818,5.967)--(1.848,5.977)--(1.879,5.985)--(1.909,5.992)--(1.939,5.996)--(1.970,5.999)--(2.000,6.000)--(2.030,5.999)--(2.061,5.996)--(2.091,5.992)--(2.121,5.985)--(2.152,5.977)--(2.182,5.967)--(2.212,5.955)--(2.242,5.941)--(2.273,5.926)--(2.303,5.908)--(2.333,5.889)--(2.364,5.868)--(2.394,5.845)--(2.424,5.820)--(2.455,5.793)--(2.485,5.765)--(2.515,5.735)--(2.545,5.702)--(2.576,5.669)--(2.606,5.633)--(2.636,5.595)--(2.667,5.556)--(2.697,5.514)--(2.727,5.471)--(2.758,5.426)--(2.788,5.379)--(2.818,5.331)--(2.848,5.280)--(2.879,5.228)--(2.909,5.174)--(2.939,5.118)--(2.970,5.060)--(3.000,5.000)--(3.030,4.938)--(3.061,4.875)--(3.091,4.810)--(3.121,4.743)--(3.152,4.674)--(3.182,4.603)--(3.212,4.531)--(3.242,4.456)--(3.273,4.380)--(3.303,4.302)--(3.333,4.222)--(3.364,4.141)--(3.394,4.057)--(3.424,3.972)--(3.455,3.884)--(3.485,3.795)--(3.515,3.704)--(3.545,3.612)--(3.576,3.517)--(3.606,3.421)--(3.636,3.322)--(3.667,3.222)--(3.697,3.120)--(3.727,3.017)--(3.758,2.911)--(3.788,2.803)--(3.818,2.694)--(3.848,2.583)--(3.879,2.470)--(3.909,2.355)--(3.939,2.239)--(3.970,2.120)--(4.000,2.000);
 \draw [style=dashed] (1.00,5.00) -- (1.00,2.91);
 \draw [style=dashed] (4.00,1.04) -- (4.00,2.00);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3785761667) node {$a$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.4267360000) node {$b$};
-\draw (2.128725951,1.316179486) node {$g_1$};
-\draw (2.878345368,6.076194201) node {$g_2$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.37858) node {$a$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.42674) node {$b$};
+\draw (2.1287,1.3162) node {$g_1$};
+\draw (2.8783,6.0762) node {$g_2$};
 \draw [style=dotted] (1.00,2.91) -- (1.00,0);
 \draw [style=dotted] (4.00,1.04) -- (4.00,0);
 
@@ -154,8 +154,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -168,12 +168,12 @@
 \draw [style=dashed] (1.24,5.00) -- (6.00,5.00);
 \draw [style=dashed] (2.00,1.00) -- (5.11,1.00);
 \draw [color=blue] (5.111,1.000)--(5.102,1.040)--(5.094,1.081)--(5.086,1.121)--(5.078,1.162)--(5.071,1.202)--(5.064,1.242)--(5.057,1.283)--(5.051,1.323)--(5.045,1.364)--(5.039,1.404)--(5.034,1.444)--(5.029,1.485)--(5.025,1.525)--(5.021,1.566)--(5.017,1.606)--(5.014,1.646)--(5.011,1.687)--(5.008,1.727)--(5.006,1.768)--(5.004,1.808)--(5.003,1.848)--(5.001,1.889)--(5.001,1.929)--(5.000,1.970)--(5.000,2.010)--(5.000,2.051)--(5.001,2.091)--(5.002,2.131)--(5.003,2.172)--(5.005,2.212)--(5.007,2.253)--(5.010,2.293)--(5.012,2.333)--(5.016,2.374)--(5.019,2.414)--(5.023,2.455)--(5.027,2.495)--(5.032,2.535)--(5.037,2.576)--(5.042,2.616)--(5.048,2.657)--(5.054,2.697)--(5.060,2.737)--(5.067,2.778)--(5.074,2.818)--(5.082,2.859)--(5.090,2.899)--(5.098,2.939)--(5.107,2.980)--(5.116,3.020)--(5.125,3.061)--(5.135,3.101)--(5.145,3.141)--(5.155,3.182)--(5.166,3.222)--(5.177,3.263)--(5.189,3.303)--(5.201,3.343)--(5.213,3.384)--(5.225,3.424)--(5.238,3.465)--(5.252,3.505)--(5.265,3.545)--(5.279,3.586)--(5.294,3.626)--(5.309,3.667)--(5.324,3.707)--(5.339,3.747)--(5.355,3.788)--(5.371,3.828)--(5.388,3.869)--(5.405,3.909)--(5.422,3.949)--(5.440,3.990)--(5.458,4.030)--(5.476,4.071)--(5.495,4.111)--(5.514,4.151)--(5.534,4.192)--(5.554,4.232)--(5.574,4.273)--(5.594,4.313)--(5.615,4.354)--(5.637,4.394)--(5.658,4.434)--(5.680,4.475)--(5.703,4.515)--(5.726,4.556)--(5.749,4.596)--(5.772,4.636)--(5.796,4.677)--(5.820,4.717)--(5.845,4.758)--(5.870,4.798)--(5.895,4.838)--(5.921,4.879)--(5.947,4.919)--(5.973,4.960)--(6.000,5.000);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.3789780000,1.000000000) node {$c$};
-\draw []  (0,5.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.3949888333,5.000000000) node {$d$};
-\draw (2.448007760,2.730627410) node {$h_1$};
-\draw (5.595426611,3.000000000) node {$h_2$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.37898,1.0000) node {$c$};
+\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.39499,5.0000) node {$d$};
+\draw (2.4480,2.7306) node {$h_1$};
+\draw (5.5954,3.0000) node {$h_2$};
 \draw [style=dotted] (2.00,1.00) -- (0,1.00);
 \draw [style=dotted] (1.24,5.00) -- (0,5.00);
 
diff --git a/src_phystricks/Fig_SBTooEasQsT.pstricks.recall b/src_phystricks/Fig_SBTooEasQsT.pstricks.recall
index f972ddd12..a50ca2224 100644
--- a/src_phystricks/Fig_SBTooEasQsT.pstricks.recall
+++ b/src_phystricks/Fig_SBTooEasQsT.pstricks.recall
@@ -165,8 +165,8 @@
 \draw [color=gray,style=solid] (-4.00,3.50) -- (2.00,3.50);
 \draw [color=gray,style=solid] (-4.00,4.00) -- (2.00,4.00);
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-4.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=blue] (-4.0000,-0.0091578)--(-3.9394,-0.0097300)--(-3.8788,-0.010338)--(-3.8182,-0.010984)--(-3.7576,-0.011670)--(-3.6970,-0.012399)--(-3.6364,-0.013174)--(-3.5758,-0.013997)--(-3.5152,-0.014872)--(-3.4545,-0.015801)--(-3.3939,-0.016788)--(-3.3333,-0.017837)--(-3.2727,-0.018951)--(-3.2121,-0.020136)--(-3.1515,-0.021394)--(-3.0909,-0.022730)--(-3.0303,-0.024150)--(-2.9697,-0.025659)--(-2.9091,-0.027263)--(-2.8485,-0.028966)--(-2.7879,-0.030776)--(-2.7273,-0.032699)--(-2.6667,-0.034742)--(-2.6061,-0.036912)--(-2.5455,-0.039219)--(-2.4848,-0.041669)--(-2.4242,-0.044273)--(-2.3636,-0.047039)--(-2.3030,-0.049978)--(-2.2424,-0.053100)--(-2.1818,-0.056418)--(-2.1212,-0.059943)--(-2.0606,-0.063688)--(-2.0000,-0.067668)--(-1.9394,-0.071895)--(-1.8788,-0.076388)--(-1.8182,-0.081160)--(-1.7576,-0.086231)--(-1.6970,-0.091619)--(-1.6364,-0.097343)--(-1.5758,-0.10343)--(-1.5152,-0.10989)--(-1.4545,-0.11675)--(-1.3939,-0.12405)--(-1.3333,-0.13180)--(-1.2727,-0.14003)--(-1.2121,-0.14878)--(-1.1515,-0.15808)--(-1.0909,-0.16796)--(-1.0303,-0.17845)--(-0.96970,-0.18960)--(-0.90909,-0.20145)--(-0.84848,-0.21403)--(-0.78788,-0.22740)--(-0.72727,-0.24161)--(-0.66667,-0.25671)--(-0.60606,-0.27275)--(-0.54545,-0.28979)--(-0.48485,-0.30790)--(-0.42424,-0.32713)--(-0.36364,-0.34757)--(-0.30303,-0.36929)--(-0.24242,-0.39236)--(-0.18182,-0.41688)--(-0.12121,-0.44292)--(-0.060606,-0.47060)--(0,-0.50000)--(0.060606,-0.53124)--(0.12121,-0.56443)--(0.18182,-0.59970)--(0.24242,-0.63717)--(0.30303,-0.67698)--(0.36364,-0.71928)--(0.42424,-0.76422)--(0.48485,-0.81196)--(0.54545,-0.86270)--(0.60606,-0.91660)--(0.66667,-0.97387)--(0.72727,-1.0347)--(0.78788,-1.0994)--(0.84848,-1.1681)--(0.90909,-1.2410)--(0.96970,-1.3186)--(1.0303,-1.4010)--(1.0909,-1.4885)--(1.1515,-1.5815)--(1.2121,-1.6803)--(1.2727,-1.7853)--(1.3333,-1.8968)--(1.3939,-2.0154)--(1.4545,-2.1413)--(1.5152,-2.2751)--(1.5758,-2.4172)--(1.6364,-2.5682)--(1.6970,-2.7287)--(1.7576,-2.8992)--(1.8182,-3.0803)--(1.8788,-3.2728)--(1.9394,-3.4773)--(2.0000,-3.6945);
@@ -199,33 +199,33 @@
 
 \draw [color=blue] (-4.0000,0.0091578)--(-3.9394,0.0097300)--(-3.8788,0.010338)--(-3.8182,0.010984)--(-3.7576,0.011670)--(-3.6970,0.012399)--(-3.6364,0.013174)--(-3.5758,0.013997)--(-3.5152,0.014872)--(-3.4545,0.015801)--(-3.3939,0.016788)--(-3.3333,0.017837)--(-3.2727,0.018951)--(-3.2121,0.020136)--(-3.1515,0.021394)--(-3.0909,0.022730)--(-3.0303,0.024150)--(-2.9697,0.025659)--(-2.9091,0.027263)--(-2.8485,0.028966)--(-2.7879,0.030776)--(-2.7273,0.032699)--(-2.6667,0.034742)--(-2.6061,0.036912)--(-2.5455,0.039219)--(-2.4848,0.041669)--(-2.4242,0.044273)--(-2.3636,0.047039)--(-2.3030,0.049978)--(-2.2424,0.053100)--(-2.1818,0.056418)--(-2.1212,0.059943)--(-2.0606,0.063688)--(-2.0000,0.067668)--(-1.9394,0.071895)--(-1.8788,0.076388)--(-1.8182,0.081160)--(-1.7576,0.086231)--(-1.6970,0.091619)--(-1.6364,0.097343)--(-1.5758,0.10343)--(-1.5152,0.10989)--(-1.4545,0.11675)--(-1.3939,0.12405)--(-1.3333,0.13180)--(-1.2727,0.14003)--(-1.2121,0.14878)--(-1.1515,0.15808)--(-1.0909,0.16796)--(-1.0303,0.17845)--(-0.96970,0.18960)--(-0.90909,0.20145)--(-0.84848,0.21403)--(-0.78788,0.22740)--(-0.72727,0.24161)--(-0.66667,0.25671)--(-0.60606,0.27275)--(-0.54545,0.28979)--(-0.48485,0.30790)--(-0.42424,0.32713)--(-0.36364,0.34757)--(-0.30303,0.36929)--(-0.24242,0.39236)--(-0.18182,0.41688)--(-0.12121,0.44292)--(-0.060606,0.47060)--(0,0.50000)--(0.060606,0.53124)--(0.12121,0.56443)--(0.18182,0.59970)--(0.24242,0.63717)--(0.30303,0.67698)--(0.36364,0.71928)--(0.42424,0.76422)--(0.48485,0.81196)--(0.54545,0.86270)--(0.60606,0.91660)--(0.66667,0.97387)--(0.72727,1.0347)--(0.78788,1.0994)--(0.84848,1.1681)--(0.90909,1.2410)--(0.96970,1.3186)--(1.0303,1.4010)--(1.0909,1.4885)--(1.1515,1.5815)--(1.2121,1.6803)--(1.2727,1.7853)--(1.3333,1.8968)--(1.3939,2.0154)--(1.4545,2.1413)--(1.5152,2.2751)--(1.5758,2.4172)--(1.6364,2.5682)--(1.6970,2.7287)--(1.7576,2.8992)--(1.8182,3.0803)--(1.8788,3.2728)--(1.9394,3.4773)--(2.0000,3.6945);
 
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.5244091667,-4.000000000) node {$ -40 $};
+\draw (-0.52441,-4.0000) node {$ -40 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.5244091667,-3.000000000) node {$ -30 $};
+\draw (-0.52441,-3.0000) node {$ -30 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.5244091667,-2.000000000) node {$ -20 $};
+\draw (-0.52441,-2.0000) node {$ -20 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.5244091667,-1.000000000) node {$ -10 $};
+\draw (-0.52441,-1.0000) node {$ -10 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.3824996667,1.000000000) node {$ 10 $};
+\draw (-0.38250,1.0000) node {$ 10 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.3824996667,2.000000000) node {$ 20 $};
+\draw (-0.38250,2.0000) node {$ 20 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.3824996667,3.000000000) node {$ 30 $};
+\draw (-0.38250,3.0000) node {$ 30 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.3824996667,4.000000000) node {$ 40 $};
+\draw (-0.38250,4.0000) node {$ 40 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SFdgHdO.pstricks.recall b/src_phystricks/Fig_SFdgHdO.pstricks.recall
index d0c79a559..aa92dc4e9 100644
--- a/src_phystricks/Fig_SFdgHdO.pstricks.recall
+++ b/src_phystricks/Fig_SFdgHdO.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.538972437,0) -- (1.547391058,0);
-\draw [,->,>=latex] (0,-1.549697438) -- (0,1.549697438);
+\draw [,->,>=latex] (-1.5390,0) -- (1.5474,0);
+\draw [,->,>=latex] (0,-1.5497) -- (0,1.5497);
 %DEFAULT
 
 \draw [color=black] plot [smooth,tension=1] coordinates {(0.00100,1.00)(0.105,1.04)(0.189,0.931)(0.310,1.00)(0.370,0.875)(0.504,0.921)(0.537,0.784)(0.677,0.803)(0.682,0.661)(0.823,0.652)(0.800,0.513)(0.936,0.476)(0.886,0.344)(1.01,0.281)(0.936,0.162)(1.05,0.0740)(0.999,0.0447)};
@@ -78,12 +78,12 @@
 \draw [color=green] plot [smooth,tension=1] coordinates {(0.00100,-1.00)(0.105,-1.04)(0.189,-0.931)(0.310,-1.00)(0.370,-0.875)(0.504,-0.921)(0.537,-0.784)(0.677,-0.803)(0.682,-0.661)(0.823,-0.652)(0.800,-0.513)(0.936,-0.476)(0.886,-0.344)(1.01,-0.281)(0.936,-0.162)(1.05,-0.0740)(0.999,-0.0447)};
 
 \draw [color=black] plot [smooth,tension=1] coordinates {(-0.999,-0.0447)(-1.04,-0.152)(-0.922,-0.230)(-0.988,-0.356)(-0.857,-0.410)(-0.898,-0.545)(-0.759,-0.571)(-0.772,-0.712)(-0.630,-0.711)(-0.615,-0.851)(-0.476,-0.822)(-0.433,-0.957)(-0.303,-0.900)(-0.233,-1.02)(-0.118,-0.943)(-0.0252,-1.05)(-0.00100,-1.00)};
-\draw [color=red]  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw [color=red]  (-1.000000000,0) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,-1.000000000) node [rotate=0] {$\bullet$};
-\draw (2.049499592,-0.3650902010) node {$K_H$};
-\draw (2.049499592,-0.3650902010) node {$K_H$};
+\draw [color=red]  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw (2.0495,-0.36509) node {$K_H$};
+\draw (2.0495,-0.36509) node {$K_H$};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall b/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall
index fd4d556f1..8f3b3168c 100644
--- a/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall
+++ b/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall
@@ -65,226 +65,226 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.000000000,-4.000000000) -- (-4.044194174,-4.044194174);
-\draw [,->,>=latex] (-4.000000000,-3.428571429) -- (-4.054711138,-3.475466689);
-\draw [,->,>=latex] (-4.000000000,-2.857142857) -- (-4.067352939,-2.905252100);
-\draw [,->,>=latex] (-4.000000000,-2.285714286) -- (-4.081815219,-2.332465840);
-\draw [,->,>=latex] (-4.000000000,-1.714285714) -- (-4.097064885,-1.755884951);
-\draw [,->,>=latex] (-4.000000000,-1.142857143) -- (-4.111119513,-1.174605575);
-\draw [,->,>=latex] (-4.000000000,-0.5714285714) -- (-4.121268813,-0.5887526876);
-\draw [,->,>=latex] (-4.000000000,0) -- (-4.125000000,0);
-\draw [,->,>=latex] (-4.000000000,0.5714285714) -- (-4.121268813,0.5887526876);
-\draw [,->,>=latex] (-4.000000000,1.142857143) -- (-4.111119513,1.174605575);
-\draw [,->,>=latex] (-4.000000000,1.714285714) -- (-4.097064885,1.755884951);
-\draw [,->,>=latex] (-4.000000000,2.285714286) -- (-4.081815219,2.332465840);
-\draw [,->,>=latex] (-4.000000000,2.857142857) -- (-4.067352939,2.905252100);
-\draw [,->,>=latex] (-4.000000000,3.428571429) -- (-4.054711138,3.475466689);
-\draw [,->,>=latex] (-4.000000000,4.000000000) -- (-4.044194174,4.044194174);
-\draw [,->,>=latex] (-3.428571429,-4.000000000) -- (-3.475466689,-4.054711138);
-\draw [,->,>=latex] (-3.428571429,-3.428571429) -- (-3.488724610,-3.488724610);
-\draw [,->,>=latex] (-3.428571429,-2.857142857) -- (-3.505708401,-2.921423668);
-\draw [,->,>=latex] (-3.428571429,-2.285714286) -- (-3.526577353,-2.351051568);
-\draw [,->,>=latex] (-3.428571429,-1.714285714) -- (-3.550312907,-1.775156454);
-\draw [,->,>=latex] (-3.428571429,-1.142857143) -- (-3.573838559,-1.191279520);
-\draw [,->,>=latex] (-3.428571429,-0.5714285714) -- (-3.591859612,-0.5986432687);
-\draw [,->,>=latex] (-3.428571429,0) -- (-3.598710317,0);
-\draw [,->,>=latex] (-3.428571429,0.5714285714) -- (-3.591859612,0.5986432687);
-\draw [,->,>=latex] (-3.428571429,1.142857143) -- (-3.573838559,1.191279520);
-\draw [,->,>=latex] (-3.428571429,1.714285714) -- (-3.550312907,1.775156454);
-\draw [,->,>=latex] (-3.428571429,2.285714286) -- (-3.526577353,2.351051568);
-\draw [,->,>=latex] (-3.428571429,2.857142857) -- (-3.505708401,2.921423668);
-\draw [,->,>=latex] (-3.428571429,3.428571429) -- (-3.488724610,3.488724610);
-\draw [,->,>=latex] (-3.428571429,4.000000000) -- (-3.475466689,4.054711138);
-\draw [,->,>=latex] (-2.857142857,-4.000000000) -- (-2.905252100,-4.067352939);
-\draw [,->,>=latex] (-2.857142857,-3.428571429) -- (-2.921423668,-3.505708401);
-\draw [,->,>=latex] (-2.857142857,-2.857142857) -- (-2.943763438,-2.943763438);
-\draw [,->,>=latex] (-2.857142857,-2.285714286) -- (-2.973797039,-2.379037631);
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+\draw [,->,>=latex] (4.0000,4.0000) -- (4.0442,4.0442);
 
 \draw [color=blue] (0.700,0)--(0.699,0.0444)--(0.694,0.0886)--(0.687,0.132)--(0.678,0.176)--(0.665,0.218)--(0.650,0.260)--(0.632,0.301)--(0.612,0.340)--(0.589,0.378)--(0.564,0.415)--(0.536,0.450)--(0.507,0.483)--(0.475,0.514)--(0.441,0.543)--(0.406,0.570)--(0.369,0.595)--(0.331,0.617)--(0.291,0.637)--(0.250,0.654)--(0.208,0.668)--(0.165,0.680)--(0.122,0.689)--(0.0776,0.696)--(0.0333,0.699)--(-0.0111,0.700)--(-0.0555,0.698)--(-0.0996,0.693)--(-0.143,0.685)--(-0.187,0.675)--(-0.229,0.661)--(-0.270,0.646)--(-0.311,0.627)--(-0.350,0.606)--(-0.388,0.583)--(-0.424,0.557)--(-0.458,0.529)--(-0.491,0.499)--(-0.522,0.467)--(-0.550,0.433)--(-0.577,0.397)--(-0.601,0.360)--(-0.622,0.321)--(-0.641,0.281)--(-0.658,0.239)--(-0.672,0.197)--(-0.683,0.154)--(-0.691,0.111)--(-0.697,0.0665)--(-0.700,0.0222)--(-0.700,-0.0222)--(-0.697,-0.0665)--(-0.691,-0.111)--(-0.683,-0.154)--(-0.672,-0.197)--(-0.658,-0.239)--(-0.641,-0.281)--(-0.622,-0.321)--(-0.601,-0.360)--(-0.577,-0.397)--(-0.550,-0.433)--(-0.522,-0.467)--(-0.491,-0.499)--(-0.458,-0.529)--(-0.424,-0.557)--(-0.388,-0.583)--(-0.350,-0.606)--(-0.311,-0.627)--(-0.270,-0.646)--(-0.229,-0.661)--(-0.187,-0.675)--(-0.143,-0.685)--(-0.0996,-0.693)--(-0.0555,-0.698)--(-0.0111,-0.700)--(0.0333,-0.699)--(0.0776,-0.696)--(0.122,-0.689)--(0.165,-0.680)--(0.208,-0.668)--(0.250,-0.654)--(0.291,-0.637)--(0.331,-0.617)--(0.369,-0.595)--(0.406,-0.570)--(0.441,-0.543)--(0.475,-0.514)--(0.507,-0.483)--(0.536,-0.450)--(0.564,-0.415)--(0.589,-0.378)--(0.612,-0.340)--(0.632,-0.301)--(0.650,-0.260)--(0.665,-0.218)--(0.678,-0.176)--(0.687,-0.132)--(0.694,-0.0886)--(0.699,-0.0444)--(0.700,0);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_STdyNTH.pstricks.recall b/src_phystricks/Fig_STdyNTH.pstricks.recall
index d81aeb717..b283a16d4 100644
--- a/src_phystricks/Fig_STdyNTH.pstricks.recall
+++ b/src_phystricks/Fig_STdyNTH.pstricks.recall
@@ -75,15 +75,15 @@
 \draw [style=dotted] (2.00,0) -- (3.00,0);
 \draw [] (3.00,0) -- (4.00,0);
 \draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.2000000000) node {};
-\draw []  (1.000000000,0) node [rotate=0] {$o$};
-\draw (1.000000000,0.2000000000) node {};
-\draw []  (2.000000000,0) node [rotate=0] {$o$};
-\draw (2.000000000,0.2000000000) node {};
-\draw []  (3.000000000,0) node [rotate=0] {$o$};
-\draw (3.000000000,0.2000000000) node {};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,0.3149246667) node {$1$};
+\draw (0,0.20000) node {};
+\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw (1.0000,0.20000) node {};
+\draw []  (2.0000,0) node [rotate=0] {$o$};
+\draw (2.0000,0.20000) node {};
+\draw []  (3.0000,0) node [rotate=0] {$o$};
+\draw (3.0000,0.20000) node {};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.31492) node {$1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall b/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall
index 939537d7b..bb5921328 100644
--- a/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall
+++ b/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall
@@ -119,48 +119,48 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-3.525102241) -- (0,3.566144740);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-3.5251) -- (0,3.5661);
 %DEFAULT
 
 \draw [color=red] (-5.000,-3.025)--(-4.899,-2.874)--(-4.798,-2.730)--(-4.697,-2.594)--(-4.596,-2.463)--(-4.495,-2.340)--(-4.394,-2.222)--(-4.293,-2.109)--(-4.192,-2.003)--(-4.091,-1.901)--(-3.990,-1.804)--(-3.889,-1.712)--(-3.788,-1.624)--(-3.687,-1.540)--(-3.586,-1.460)--(-3.485,-1.384)--(-3.384,-1.311)--(-3.283,-1.242)--(-3.182,-1.176)--(-3.081,-1.113)--(-2.980,-1.053)--(-2.879,-0.9953)--(-2.778,-0.9403)--(-2.677,-0.8876)--(-2.576,-0.8373)--(-2.475,-0.7891)--(-2.374,-0.7429)--(-2.273,-0.6986)--(-2.172,-0.6561)--(-2.071,-0.6152)--(-1.970,-0.5760)--(-1.869,-0.5382)--(-1.768,-0.5017)--(-1.667,-0.4666)--(-1.566,-0.4326)--(-1.465,-0.3998)--(-1.364,-0.3679)--(-1.263,-0.3370)--(-1.162,-0.3070)--(-1.061,-0.2778)--(-0.9596,-0.2492)--(-0.8586,-0.2213)--(-0.7576,-0.1940)--(-0.6566,-0.1671)--(-0.5556,-0.1407)--(-0.4545,-0.1146)--(-0.3535,-0.08885)--(-0.2525,-0.06330)--(-0.1515,-0.03792)--(-0.05051,-0.01263)--(0.05051,0.01263)--(0.1515,0.03792)--(0.2525,0.06330)--(0.3535,0.08885)--(0.4545,0.1146)--(0.5556,0.1407)--(0.6566,0.1671)--(0.7576,0.1940)--(0.8586,0.2213)--(0.9596,0.2492)--(1.061,0.2778)--(1.162,0.3070)--(1.263,0.3370)--(1.364,0.3679)--(1.465,0.3998)--(1.566,0.4326)--(1.667,0.4666)--(1.768,0.5017)--(1.869,0.5382)--(1.970,0.5760)--(2.071,0.6152)--(2.172,0.6561)--(2.273,0.6986)--(2.374,0.7429)--(2.475,0.7891)--(2.576,0.8373)--(2.677,0.8876)--(2.778,0.9403)--(2.879,0.9953)--(2.980,1.053)--(3.081,1.113)--(3.182,1.176)--(3.283,1.242)--(3.384,1.311)--(3.485,1.384)--(3.586,1.460)--(3.687,1.540)--(3.788,1.624)--(3.889,1.712)--(3.990,1.804)--(4.091,1.901)--(4.192,2.003)--(4.293,2.109)--(4.394,2.222)--(4.495,2.340)--(4.596,2.463)--(4.697,2.594)--(4.798,2.730)--(4.899,2.874)--(5.000,3.025);
 
 \draw [color=blue] (-5.000,3.066)--(-4.899,2.917)--(-4.798,2.776)--(-4.697,2.641)--(-4.596,2.514)--(-4.495,2.392)--(-4.394,2.277)--(-4.293,2.168)--(-4.192,2.064)--(-4.091,1.965)--(-3.990,1.872)--(-3.889,1.783)--(-3.788,1.699)--(-3.687,1.619)--(-3.586,1.543)--(-3.485,1.472)--(-3.384,1.404)--(-3.283,1.339)--(-3.182,1.278)--(-3.081,1.220)--(-2.980,1.166)--(-2.879,1.114)--(-2.778,1.065)--(-2.677,1.019)--(-2.576,0.9752)--(-2.475,0.9342)--(-2.374,0.8955)--(-2.273,0.8591)--(-2.172,0.8249)--(-2.071,0.7928)--(-1.970,0.7627)--(-1.869,0.7346)--(-1.768,0.7083)--(-1.667,0.6839)--(-1.566,0.6612)--(-1.465,0.6402)--(-1.364,0.6208)--(-1.263,0.6030)--(-1.162,0.5867)--(-1.061,0.5720)--(-0.9596,0.5587)--(-0.8586,0.5468)--(-0.7576,0.5363)--(-0.6566,0.5272)--(-0.5556,0.5194)--(-0.4545,0.5130)--(-0.3535,0.5078)--(-0.2525,0.5040)--(-0.1515,0.5014)--(-0.05051,0.5002)--(0.05051,0.5002)--(0.1515,0.5014)--(0.2525,0.5040)--(0.3535,0.5078)--(0.4545,0.5130)--(0.5556,0.5194)--(0.6566,0.5272)--(0.7576,0.5363)--(0.8586,0.5468)--(0.9596,0.5587)--(1.061,0.5720)--(1.162,0.5867)--(1.263,0.6030)--(1.364,0.6208)--(1.465,0.6402)--(1.566,0.6612)--(1.667,0.6839)--(1.768,0.7083)--(1.869,0.7346)--(1.970,0.7627)--(2.071,0.7928)--(2.172,0.8249)--(2.273,0.8591)--(2.374,0.8955)--(2.475,0.9342)--(2.576,0.9752)--(2.677,1.019)--(2.778,1.065)--(2.879,1.114)--(2.980,1.166)--(3.081,1.220)--(3.182,1.278)--(3.283,1.339)--(3.384,1.404)--(3.485,1.472)--(3.586,1.543)--(3.687,1.619)--(3.788,1.699)--(3.889,1.783)--(3.990,1.872)--(4.091,1.965)--(4.192,2.064)--(4.293,2.168)--(4.394,2.277)--(4.495,2.392)--(4.596,2.514)--(4.697,2.641)--(4.798,2.776)--(4.899,2.917)--(5.000,3.066);
-\draw (-4.000000000,-0.3298256667) node {$ -2 $};
+\draw (-4.0000,-0.32983) node {$ -2 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.500000000) node {$ -7 $};
+\draw (-0.43316,-3.5000) node {$ -7 $};
 \draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.4331593333,-3.000000000) node {$ -6 $};
+\draw (-0.43316,-3.0000) node {$ -6 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.500000000) node {$ -5 $};
+\draw (-0.43316,-2.5000) node {$ -5 $};
 \draw [] (-0.100,-2.50) -- (0.100,-2.50);
-\draw (-0.4331593333,-2.000000000) node {$ -4 $};
+\draw (-0.43316,-2.0000) node {$ -4 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.500000000) node {$ -3 $};
+\draw (-0.43316,-1.5000) node {$ -3 $};
 \draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.4331593333,-1.000000000) node {$ -2 $};
+\draw (-0.43316,-1.0000) node {$ -2 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.4331593333,-0.5000000000) node {$ -1 $};
+\draw (-0.43316,-0.50000) node {$ -1 $};
 \draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.2912498333,0.5000000000) node {$ 1 $};
+\draw (-0.29125,0.50000) node {$ 1 $};
 \draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,1.500000000) node {$ 3 $};
+\draw (-0.29125,1.5000) node {$ 3 $};
 \draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,2.500000000) node {$ 5 $};
+\draw (-0.29125,2.5000) node {$ 5 $};
 \draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,3.500000000) node {$ 7 $};
+\draw (-0.29125,3.5000) node {$ 7 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SenoTopologo.pstricks.recall b/src_phystricks/Fig_SenoTopologo.pstricks.recall
index 64c749052..18e3d2a7e 100644
--- a/src_phystricks/Fig_SenoTopologo.pstricks.recall
+++ b/src_phystricks/Fig_SenoTopologo.pstricks.recall
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.000000000,0) -- (3.000000000,0);
-\draw [,->,>=latex] (0,-1.586160393) -- (0,2.773243568);
+\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
+\draw [,->,>=latex] (0,-1.5862) -- (0,2.7732);
 %DEFAULT
 
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diff --git a/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall b/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall
index 80fc2e34a..cb4d2a63a 100644
--- a/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall
+++ b/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.080796327,0) -- (2.080796327,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-2.0808,0) -- (2.0808,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
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@@ -88,13 +88,13 @@
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-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SpiraleLimite.pstricks.recall b/src_phystricks/Fig_SpiraleLimite.pstricks.recall
index cb8a83eff..3908c2a7d 100644
--- a/src_phystricks/Fig_SpiraleLimite.pstricks.recall
+++ b/src_phystricks/Fig_SpiraleLimite.pstricks.recall
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,1.710694496);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,1.7107);
 %DEFAULT
 \draw [color=blue] (0,0)--(0.0214,0.0214)--(0.0429,0.0429)--(0.0643,0.0643)--(0.0857,0.0857)--(0.107,0.107)--(0.129,0.129)--(0.150,0.150)--(0.171,0.171)--(0.193,0.193)--(0.214,0.214)--(0.236,0.236)--(0.257,0.257)--(0.279,0.278)--(0.300,0.300)--(0.322,0.321)--(0.344,0.342)--(0.365,0.363)--(0.387,0.385)--(0.409,0.406)--(0.430,0.427)--(0.452,0.448)--(0.474,0.469)--(0.496,0.490)--(0.518,0.511)--(0.540,0.531)--(0.562,0.552)--(0.584,0.573)--(0.607,0.593)--(0.629,0.614)--(0.652,0.634)--(0.674,0.654)--(0.697,0.674)--(0.720,0.694)--(0.743,0.714)--(0.766,0.733)--(0.790,0.753)--(0.813,0.772)--(0.837,0.791)--(0.861,0.810)--(0.885,0.828)--(0.909,0.847)--(0.934,0.865)--(0.958,0.883)--(0.983,0.900)--(1.01,0.918)--(1.03,0.935)--(1.06,0.952)--(1.09,0.968)--(1.11,0.984)--(1.14,1.00)--(1.17,1.02)--(1.19,1.03)--(1.22,1.04)--(1.25,1.06)--(1.28,1.07)--(1.30,1.09)--(1.33,1.10)--(1.36,1.11)--(1.39,1.12)--(1.42,1.13)--(1.45,1.14)--(1.48,1.15)--(1.51,1.16)--(1.55,1.17)--(1.58,1.18)--(1.61,1.19)--(1.64,1.19)--(1.68,1.20)--(1.71,1.20)--(1.74,1.21)--(1.78,1.21)--(1.82,1.21)--(1.85,1.21)--(1.89,1.21)--(1.92,1.21)--(1.96,1.21)--(2.00,1.20)--(2.04,1.19)--(2.08,1.19)--(2.12,1.18)--(2.16,1.17)--(2.20,1.15)--(2.24,1.14)--(2.28,1.12)--(2.33,1.10)--(2.37,1.08)--(2.42,1.06)--(2.46,1.03)--(2.51,0.997)--(2.55,0.962)--(2.60,0.922)--(2.65,0.876)--(2.70,0.824)--(2.74,0.764)--(2.79,0.694)--(2.84,0.611)--(2.90,0.507)--(2.95,0.364)--(3.00,0);
 \draw [] (3.00,-0.100) -- (3.00,0.100);
diff --git a/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall b/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall
index 6f6b36964..19a5c0484 100644
--- a/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall
+++ b/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall
@@ -69,64 +69,64 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.490292088,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,3.130986314);
+\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,3.1310);
 %DEFAULT
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.000000000,1.000000000) -- (0,1.000000000);
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.0000,1.0000) -- (0,1.0000);
 \draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw [color=green,->,>=latex] (1.000000000,1.000000000) -- (1.707106781,0.2928932188);
-\draw []  (2.141592654,1.000000000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (2.141592654,1.000000000) -- (3.141592654,1.000000000);
+\draw [color=green,->,>=latex] (1.0000,1.0000) -- (1.7071,0.29289);
+\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (2.1416,1.0000) -- (3.1416,1.0000);
 \draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw [color=green,->,>=latex] (2.141592654,1.000000000) -- (1.434485872,0.2928932188);
-\draw []  (4.712388980,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (4.712388980,0) -- (4.712388980,1.000000000);
+\draw [color=green,->,>=latex] (2.1416,1.0000) -- (1.4345,0.29289);
+\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (4.7124,0) -- (4.7124,1.0000);
 \draw [color=brown] (3.71,0) -- (5.71,0);
-\draw [color=green,->,>=latex] (4.712388980,0) -- (4.712388980,-1.000000000);
-\draw []  (7.283185307,1.000000000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (7.283185307,1.000000000) -- (6.283185307,1.000000000);
+\draw [color=green,->,>=latex] (4.7124,0) -- (4.7124,-1.0000);
+\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (7.2832,1.0000) -- (6.2832,1.0000);
 \draw [color=brown] (6.58,0.293) -- (7.99,1.71);
-\draw [color=green,->,>=latex] (7.283185307,1.000000000) -- (7.990292088,0.2928932188);
-\draw []  (1.492504945,1.707106781) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.492504945,1.707106781) -- (0.7853981634,1.000000000);
+\draw [color=green,->,>=latex] (7.2832,1.0000) -- (7.9903,0.29289);
+\draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.4925,1.7071) -- (0.78540,1.0000);
 \draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw [color=green,->,>=latex] (1.492504945,1.707106781) -- (2.416384477,1.324423349);
-\draw []  (1.649087709,1.707106781) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.649087709,1.707106781) -- (2.356194490,1.000000000);
+\draw [color=green,->,>=latex] (1.4925,1.7071) -- (2.4164,1.3244);
+\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.6491,1.7071) -- (2.3562,1.0000);
 \draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw [color=green,->,>=latex] (1.649087709,1.707106781) -- (0.7252081765,1.324423349);
-\draw []  (3.219884036,0.2928932188) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (3.219884036,0.2928932188) -- (3.926990817,1.000000000);
+\draw [color=green,->,>=latex] (1.6491,1.7071) -- (0.72521,1.3244);
+\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (3.2199,0.29289) -- (3.9270,1.0000);
 \draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw [color=green,->,>=latex] (3.219884036,0.2928932188) -- (2.837200603,-0.6309863137);
-\draw []  (6.204893925,0.2928932188) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (6.204893925,0.2928932188) -- (5.497787144,1.000000000);
+\draw [color=green,->,>=latex] (3.2199,0.29289) -- (2.8372,-0.63099);
+\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (6.2049,0.29289) -- (5.4978,1.0000);
 \draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
-\draw [color=green,->,>=latex] (6.204893925,0.2928932188) -- (6.587577357,-0.6309863137);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw [color=green,->,>=latex] (6.2049,0.29289) -- (6.5876,-0.63099);
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -189,47 +189,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (8.490292088,0);
-\draw [,->,>=latex] (0,-0.5897902136) -- (0,3.130986314);
+\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
+\draw [,->,>=latex] (0,-0.58979) -- (0,3.1310);
 %DEFAULT
-\draw []  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw []  (2.141592654,1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw []  (4.712388980,0) node [rotate=0] {$\bullet$};
+\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
 \draw [color=brown] (3.71,0) -- (5.71,0);
-\draw []  (7.283185307,1.000000000) node [rotate=0] {$\bullet$};
+\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=brown] (6.58,0.293) -- (7.99,1.71);
-\draw []  (1.492504945,1.707106781) node [rotate=0] {$\bullet$};
+\draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
 \draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw []  (1.649087709,1.707106781) node [rotate=0] {$\bullet$};
+\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
 \draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw []  (3.219884036,0.2928932188) node [rotate=0] {$\bullet$};
+\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
 \draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw []  (6.204893925,0.2928932188) node [rotate=0] {$\bullet$};
+\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
 \draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
 \draw [color=blue,style=dashed] (1.000,1.000)--(1.061,1.063)--(1.119,1.127)--(1.172,1.189)--(1.222,1.251)--(1.267,1.312)--(1.309,1.372)--(1.347,1.430)--(1.382,1.486)--(1.412,1.541)--(1.440,1.593)--(1.464,1.643)--(1.485,1.690)--(1.504,1.735)--(1.519,1.776)--(1.532,1.815)--(1.543,1.850)--(1.551,1.881)--(1.558,1.910)--(1.563,1.934)--(1.566,1.955)--(1.569,1.972)--(1.570,1.985)--(1.571,1.994)--(1.571,1.999)--(1.571,2.000)--(1.571,1.997)--(1.571,1.990)--(1.572,1.979)--(1.574,1.964)--(1.577,1.945)--(1.581,1.922)--(1.587,1.896)--(1.594,1.866)--(1.604,1.833)--(1.616,1.796)--(1.630,1.756)--(1.647,1.713)--(1.666,1.667)--(1.689,1.618)--(1.715,1.567)--(1.744,1.514)--(1.777,1.458)--(1.813,1.401)--(1.853,1.342)--(1.896,1.282)--(1.944,1.220)--(1.995,1.158)--(2.051,1.095)--(2.110,1.032)--(2.174,0.9683)--(2.241,0.9049)--(2.313,0.8420)--(2.388,0.7797)--(2.468,0.7183)--(2.551,0.6580)--(2.638,0.5991)--(2.729,0.5418)--(2.823,0.4863)--(2.921,0.4329)--(3.022,0.3818)--(3.126,0.3332)--(3.233,0.2873)--(3.344,0.2443)--(3.456,0.2042)--(3.571,0.1674)--(3.689,0.1340)--(3.808,0.1040)--(3.929,0.07765)--(4.052,0.05500)--(4.176,0.03616)--(4.301,0.02120)--(4.427,0.01018)--(4.554,0.003145)--(4.681,0)--(4.808,0.001133)--(4.934,0.006162)--(5.061,0.01519)--(5.186,0.02819)--(5.311,0.04510)--(5.434,0.06585)--(5.556,0.09037)--(5.677,0.1185)--(5.795,0.1503)--(5.911,0.1854)--(6.025,0.2239)--(6.137,0.2654)--(6.245,0.3099)--(6.351,0.3572)--(6.454,0.4071)--(6.553,0.4594)--(6.649,0.5138)--(6.742,0.5702)--(6.831,0.6283)--(6.916,0.6880)--(6.997,0.7489)--(7.075,0.8107)--(7.148,0.8734)--(7.218,0.9366)--(7.283,1.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SuiteUnSurn.pstricks.recall b/src_phystricks/Fig_SuiteUnSurn.pstricks.recall
index c39856e87..ad00f9bf4 100644
--- a/src_phystricks/Fig_SuiteUnSurn.pstricks.recall
+++ b/src_phystricks/Fig_SuiteUnSurn.pstricks.recall
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (10.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
-\draw []  (1.000000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (1.000000000,3.414924667) node {$1$};
-\draw []  (2.000000000,1.500000000) node [rotate=0] {$\bullet$};
-\draw (2.000000000,1.982455000) node {$1/2$};
-\draw []  (3.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.000000000,1.482455000) node {$1/3$};
-\draw []  (4.000000000,0.7500000000) node [rotate=0] {$\bullet$};
-\draw (4.000000000,1.232455000) node {$1/4$};
-\draw []  (5.000000000,0.6000000000) node [rotate=0] {$\bullet$};
-\draw (5.000000000,1.082455000) node {$1/5$};
-\draw []  (6.000000000,0.5000000000) node [rotate=0] {$\bullet$};
-\draw (6.000000000,0.9824550000) node {$1/6$};
-\draw []  (7.000000000,0.4285714286) node [rotate=0] {$\bullet$};
-\draw (7.000000000,0.9110264286) node {$1/7$};
-\draw []  (8.000000000,0.3750000000) node [rotate=0] {$\bullet$};
-\draw (8.000000000,0.8574550000) node {$1/8$};
-\draw []  (9.000000000,0.3333333333) node [rotate=0] {$\bullet$};
-\draw (9.000000000,0.8157883333) node {$1/9$};
-\draw []  (10.00000000,0.3000000000) node [rotate=0] {$\bullet$};
-\draw (10.00000000,0.7824550000) node {$1/10$};
+\draw []  (1.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,3.4149) node {$1$};
+\draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (2.0000,1.9825) node {$1/2$};
+\draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,1.4825) node {$1/3$};
+\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2325) node {$1/4$};
+\draw []  (5.0000,0.60000) node [rotate=0] {$\bullet$};
+\draw (5.0000,1.0825) node {$1/5$};
+\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.98246) node {$1/6$};
+\draw []  (7.0000,0.42857) node [rotate=0] {$\bullet$};
+\draw (7.0000,0.91103) node {$1/7$};
+\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.85746) node {$1/8$};
+\draw []  (9.0000,0.33333) node [rotate=0] {$\bullet$};
+\draw (9.0000,0.81579) node {$1/9$};
+\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.78246) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SurfaceCercle.pstricks.recall b/src_phystricks/Fig_SurfaceCercle.pstricks.recall
index 2a553dd3c..79caff591 100644
--- a/src_phystricks/Fig_SurfaceCercle.pstricks.recall
+++ b/src_phystricks/Fig_SurfaceCercle.pstricks.recall
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.499897967) -- (0,2.499897967);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
 %DEFAULT
 
 \draw [color=blue] (-2.000,0)--(-1.960,0.4000)--(-1.919,0.5628)--(-1.879,0.6857)--(-1.838,0.7876)--(-1.798,0.8759)--(-1.758,0.9544)--(-1.717,1.025)--(-1.677,1.090)--(-1.636,1.150)--(-1.596,1.205)--(-1.556,1.257)--(-1.515,1.305)--(-1.475,1.351)--(-1.434,1.394)--(-1.394,1.434)--(-1.354,1.472)--(-1.313,1.509)--(-1.273,1.543)--(-1.232,1.575)--(-1.192,1.606)--(-1.152,1.635)--(-1.111,1.663)--(-1.071,1.689)--(-1.030,1.714)--(-0.9899,1.738)--(-0.9495,1.760)--(-0.9091,1.781)--(-0.8687,1.801)--(-0.8283,1.820)--(-0.7879,1.838)--(-0.7475,1.855)--(-0.7071,1.871)--(-0.6667,1.886)--(-0.6263,1.899)--(-0.5859,1.912)--(-0.5455,1.924)--(-0.5051,1.935)--(-0.4646,1.945)--(-0.4242,1.954)--(-0.3838,1.963)--(-0.3434,1.970)--(-0.3030,1.977)--(-0.2626,1.983)--(-0.2222,1.988)--(-0.1818,1.992)--(-0.1414,1.995)--(-0.1010,1.997)--(-0.06061,1.999)--(-0.02020,2.000)--(0.02020,2.000)--(0.06061,1.999)--(0.1010,1.997)--(0.1414,1.995)--(0.1818,1.992)--(0.2222,1.988)--(0.2626,1.983)--(0.3030,1.977)--(0.3434,1.970)--(0.3838,1.963)--(0.4242,1.954)--(0.4646,1.945)--(0.5051,1.935)--(0.5455,1.924)--(0.5859,1.912)--(0.6263,1.899)--(0.6667,1.886)--(0.7071,1.871)--(0.7475,1.855)--(0.7879,1.838)--(0.8283,1.820)--(0.8687,1.801)--(0.9091,1.781)--(0.9495,1.760)--(0.9899,1.738)--(1.030,1.714)--(1.071,1.689)--(1.111,1.663)--(1.152,1.635)--(1.192,1.606)--(1.232,1.575)--(1.273,1.543)--(1.313,1.509)--(1.354,1.472)--(1.394,1.434)--(1.434,1.394)--(1.475,1.351)--(1.515,1.305)--(1.556,1.257)--(1.596,1.205)--(1.636,1.150)--(1.677,1.090)--(1.717,1.025)--(1.758,0.9544)--(1.798,0.8759)--(1.838,0.7876)--(1.879,0.6857)--(1.919,0.5628)--(1.960,0.4000)--(2.000,0);
diff --git a/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall b/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall
index 0bed20805..a3b7cc8a7 100644
--- a/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall
+++ b/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (6.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.499883862);
+\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -56,10 +56,10 @@
 \draw [color=blue,style=solid] (1.000,1.635)--(1.051,1.562)--(1.101,1.497)--(1.152,1.439)--(1.202,1.390)--(1.253,1.348)--(1.303,1.316)--(1.354,1.293)--(1.404,1.279)--(1.455,1.273)--(1.505,1.276)--(1.556,1.288)--(1.606,1.307)--(1.657,1.334)--(1.707,1.367)--(1.758,1.406)--(1.808,1.450)--(1.859,1.498)--(1.909,1.549)--(1.960,1.603)--(2.010,1.658)--(2.061,1.713)--(2.111,1.768)--(2.162,1.821)--(2.212,1.873)--(2.263,1.921)--(2.313,1.965)--(2.364,2.006)--(2.414,2.041)--(2.465,2.071)--(2.515,2.096)--(2.566,2.116)--(2.616,2.129)--(2.667,2.137)--(2.717,2.139)--(2.768,2.136)--(2.818,2.128)--(2.869,2.116)--(2.919,2.100)--(2.970,2.081)--(3.020,2.059)--(3.071,2.035)--(3.121,2.010)--(3.172,1.985)--(3.222,1.960)--(3.273,1.937)--(3.323,1.915)--(3.374,1.896)--(3.424,1.881)--(3.475,1.870)--(3.525,1.863)--(3.576,1.861)--(3.626,1.864)--(3.677,1.873)--(3.727,1.888)--(3.778,1.908)--(3.828,1.934)--(3.879,1.965)--(3.929,2.002)--(3.980,2.043)--(4.030,2.088)--(4.081,2.137)--(4.131,2.189)--(4.182,2.242)--(4.232,2.297)--(4.283,2.353)--(4.333,2.408)--(4.384,2.461)--(4.434,2.512)--(4.485,2.559)--(4.535,2.602)--(4.586,2.640)--(4.636,2.672)--(4.687,2.697)--(4.737,2.715)--(4.788,2.725)--(4.838,2.727)--(4.889,2.719)--(4.939,2.703)--(4.990,2.678)--(5.040,2.644)--(5.091,2.602)--(5.141,2.550)--(5.192,2.491)--(5.242,2.424)--(5.293,2.351)--(5.343,2.271)--(5.394,2.186)--(5.444,2.097)--(5.495,2.005)--(5.545,1.911)--(5.596,1.816)--(5.646,1.722)--(5.697,1.629)--(5.747,1.538)--(5.798,1.452)--(5.849,1.370)--(5.899,1.294)--(5.950,1.225)--(6.000,1.165);
 \draw [color=magenta,style=dashed] (1.00,1.63) -- (1.00,4.84);
 \draw [color=magenta,style=dashed] (6.00,3.72) -- (6.00,1.16);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.2785761667) node {$a$};
-\draw []  (6.000000000,0) node [rotate=0] {$\bullet$};
-\draw (6.000000000,-0.3267360000) node {$b$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.27858) node {$a$};
+\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.32674) node {$b$};
 \draw [style=dotted] (1.00,0) -- (1.00,4.84);
 \draw [style=dotted] (6.00,0) -- (6.00,3.72);
 
@@ -109,8 +109,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.499883862,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,6.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -124,10 +124,10 @@
 \draw [color=blue] (1.635,1.000)--(1.562,1.051)--(1.497,1.101)--(1.439,1.152)--(1.390,1.202)--(1.348,1.253)--(1.316,1.303)--(1.293,1.354)--(1.279,1.404)--(1.273,1.455)--(1.276,1.505)--(1.288,1.556)--(1.307,1.606)--(1.334,1.657)--(1.367,1.707)--(1.406,1.758)--(1.450,1.808)--(1.498,1.859)--(1.549,1.909)--(1.603,1.960)--(1.658,2.010)--(1.713,2.061)--(1.768,2.111)--(1.821,2.162)--(1.873,2.212)--(1.921,2.263)--(1.965,2.313)--(2.006,2.364)--(2.041,2.414)--(2.071,2.465)--(2.096,2.515)--(2.116,2.566)--(2.129,2.616)--(2.137,2.667)--(2.139,2.717)--(2.136,2.768)--(2.128,2.818)--(2.116,2.869)--(2.100,2.919)--(2.081,2.970)--(2.059,3.020)--(2.035,3.071)--(2.010,3.121)--(1.985,3.172)--(1.960,3.222)--(1.937,3.273)--(1.915,3.323)--(1.896,3.374)--(1.881,3.424)--(1.870,3.475)--(1.863,3.525)--(1.861,3.576)--(1.864,3.626)--(1.873,3.677)--(1.888,3.727)--(1.908,3.778)--(1.934,3.828)--(1.965,3.879)--(2.002,3.929)--(2.043,3.980)--(2.088,4.030)--(2.137,4.081)--(2.189,4.131)--(2.242,4.182)--(2.297,4.232)--(2.353,4.283)--(2.408,4.333)--(2.461,4.384)--(2.512,4.434)--(2.559,4.485)--(2.602,4.535)--(2.640,4.586)--(2.672,4.636)--(2.697,4.687)--(2.715,4.737)--(2.725,4.788)--(2.727,4.838)--(2.719,4.889)--(2.703,4.939)--(2.678,4.990)--(2.644,5.040)--(2.602,5.091)--(2.550,5.141)--(2.491,5.192)--(2.424,5.242)--(2.351,5.293)--(2.271,5.343)--(2.186,5.394)--(2.097,5.444)--(2.005,5.495)--(1.911,5.545)--(1.816,5.596)--(1.722,5.646)--(1.629,5.697)--(1.538,5.747)--(1.452,5.798)--(1.370,5.849)--(1.294,5.899)--(1.225,5.950)--(1.165,6.000);
 \draw [color=magenta,style=dashed] (1.16,6.00) -- (3.72,6.00);
 \draw [color=magenta,style=dashed] (4.84,1.00) -- (1.63,1.00);
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2789780000,1.000000000) node {$c$};
-\draw []  (0,6.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.2949888333,6.000000000) node {$d$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.27898,1.0000) node {$c$};
+\draw []  (0,6.0000) node [rotate=0] {$\bullet$};
+\draw (-0.29499,6.0000) node {$d$};
 \draw [style=dotted] (0,1.00) -- (4.84,1.00);
 \draw [style=dotted] (0,6.00) -- (1.16,6.00);
 \draw [color=blue,style=solid] (4.841,1.000)--(4.868,1.051)--(4.892,1.101)--(4.913,1.152)--(4.933,1.202)--(4.950,1.253)--(4.964,1.303)--(4.977,1.354)--(4.986,1.404)--(4.993,1.455)--(4.998,1.505)--(5.000,1.556)--(4.999,1.606)--(4.996,1.657)--(4.991,1.707)--(4.983,1.758)--(4.972,1.808)--(4.959,1.859)--(4.943,1.909)--(4.925,1.960)--(4.905,2.010)--(4.882,2.061)--(4.858,2.111)--(4.831,2.162)--(4.801,2.212)--(4.770,2.263)--(4.737,2.313)--(4.702,2.364)--(4.665,2.414)--(4.626,2.465)--(4.586,2.515)--(4.545,2.566)--(4.502,2.616)--(4.457,2.667)--(4.412,2.717)--(4.365,2.768)--(4.318,2.818)--(4.270,2.869)--(4.221,2.919)--(4.171,2.970)--(4.121,3.020)--(4.071,3.071)--(4.020,3.121)--(3.970,3.172)--(3.919,3.222)--(3.869,3.273)--(3.819,3.323)--(3.770,3.374)--(3.721,3.424)--(3.673,3.475)--(3.626,3.525)--(3.579,3.576)--(3.534,3.626)--(3.490,3.677)--(3.447,3.727)--(3.406,3.778)--(3.366,3.828)--(3.328,3.879)--(3.291,3.929)--(3.257,3.980)--(3.224,4.030)--(3.193,4.081)--(3.164,4.131)--(3.137,4.182)--(3.113,4.232)--(3.091,4.283)--(3.071,4.333)--(3.053,4.384)--(3.038,4.434)--(3.026,4.485)--(3.016,4.535)--(3.008,4.586)--(3.003,4.636)--(3.000,4.687)--(3.000,4.737)--(3.003,4.788)--(3.008,4.838)--(3.016,4.889)--(3.026,4.939)--(3.038,4.990)--(3.053,5.040)--(3.071,5.091)--(3.091,5.141)--(3.113,5.192)--(3.137,5.242)--(3.164,5.293)--(3.193,5.343)--(3.223,5.394)--(3.256,5.444)--(3.291,5.495)--(3.327,5.545)--(3.366,5.596)--(3.405,5.646)--(3.447,5.697)--(3.490,5.747)--(3.534,5.798)--(3.579,5.849)--(3.625,5.899)--(3.672,5.950)--(3.721,6.000);
diff --git a/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall b/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall
index 37c82159c..067a5b0f8 100644
--- a/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall
+++ b/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall
@@ -72,9 +72,9 @@
 %DEFAULT
 \draw [] (-2.10,0.700) -- (2.10,0.700);
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3824550000) node {\( \pi(e_1)\)};
-\draw []  (0.7000000000,0.7000000000) node [rotate=0] {$\bullet$};
-\draw (0.7000000000,1.082455000) node {\( \pi(e_2)\)};
+\draw (0,-0.38245) node {\( \pi(e_1)\)};
+\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
+\draw (0.70000,1.0825) node {\( \pi(e_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall b/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall
index b6179f2bd..4c1dc4011 100644
--- a/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall
+++ b/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall
@@ -66,59 +66,57 @@
 %PSTRICKS CODE
 %DEFAULT
 \fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
-\draw [] (1.60,-1.20) -- (3.07,-1.20);
-\draw [] (3.07,-1.20) -- (3.07,1.20);
-\draw [] (3.07,1.20) -- (1.60,1.20);
-\draw [] (1.60,1.20) -- (1.60,-1.20);
-\draw [,->,>=latex] (1.600000000,-1.000000000) -- (2.225000000,-1.000000000);
-\draw [,->,>=latex] (1.600000000,-0.6666666667) -- (2.225000000,-0.6666666667);
-\draw [,->,>=latex] (1.600000000,-0.3333333333) -- (2.225000000,-0.3333333333);
-\draw [,->,>=latex] (1.600000000,0) -- (2.225000000,0);
-\draw [,->,>=latex] (1.600000000,0.3333333333) -- (2.225000000,0.3333333333);
-\draw [,->,>=latex] (1.600000000,0.6666666667) -- (2.225000000,0.6666666667);
-\draw [,->,>=latex] (1.600000000,1.000000000) -- (2.225000000,1.000000000);
-\draw [,->,>=latex] (2.333333333,-1.000000000) -- (2.761904762,-1.000000000);
-\draw [,->,>=latex] (2.333333333,-0.6666666667) -- (2.761904762,-0.6666666667);
-\draw [,->,>=latex] (2.333333333,-0.3333333333) -- (2.761904762,-0.3333333333);
-\draw [,->,>=latex] (2.333333333,0) -- (2.761904762,0);
-\draw [,->,>=latex] (2.333333333,0.3333333333) -- (2.761904762,0.3333333333);
-\draw [,->,>=latex] (2.333333333,0.6666666667) -- (2.761904762,0.6666666667);
-\draw [,->,>=latex] (2.333333333,1.000000000) -- (2.761904762,1.000000000);
-\draw [,->,>=latex] (3.066666667,-1.000000000) -- (3.392753623,-1.000000000);
-\draw [,->,>=latex] (3.066666667,-0.6666666667) -- (3.392753623,-0.6666666667);
-\draw [,->,>=latex] (3.066666667,-0.3333333333) -- (3.392753623,-0.3333333333);
-\draw [,->,>=latex] (3.066666667,0) -- (3.392753623,0);
-\draw [,->,>=latex] (3.066666667,0.3333333333) -- (3.392753623,0.3333333333);
-\draw [,->,>=latex] (3.066666667,0.6666666667) -- (3.392753623,0.6666666667);
-\draw [,->,>=latex] (3.066666667,1.000000000) -- (3.392753623,1.000000000);
-\draw [,->,>=latex] (3.800000000,-1.000000000) -- (4.063157895,-1.000000000);
-\draw [,->,>=latex] (3.800000000,-0.6666666667) -- (4.063157895,-0.6666666667);
-\draw [,->,>=latex] (3.800000000,-0.3333333333) -- (4.063157895,-0.3333333333);
-\draw [,->,>=latex] (3.800000000,0) -- (4.063157895,0);
-\draw [,->,>=latex] (3.800000000,0.3333333333) -- (4.063157895,0.3333333333);
-\draw [,->,>=latex] (3.800000000,0.6666666667) -- (4.063157895,0.6666666667);
-\draw [,->,>=latex] (3.800000000,1.000000000) -- (4.063157895,1.000000000);
-\draw [,->,>=latex] (4.533333333,-1.000000000) -- (4.753921569,-1.000000000);
-\draw [,->,>=latex] (4.533333333,-0.6666666667) -- (4.753921569,-0.6666666667);
-\draw [,->,>=latex] (4.533333333,-0.3333333333) -- (4.753921569,-0.3333333333);
-\draw [,->,>=latex] (4.533333333,0) -- (4.753921569,0);
-\draw [,->,>=latex] (4.533333333,0.3333333333) -- (4.753921569,0.3333333333);
-\draw [,->,>=latex] (4.533333333,0.6666666667) -- (4.753921569,0.6666666667);
-\draw [,->,>=latex] (4.533333333,1.000000000) -- (4.753921569,1.000000000);
-\draw [,->,>=latex] (5.266666667,-1.000000000) -- (5.456540084,-1.000000000);
-\draw [,->,>=latex] (5.266666667,-0.6666666667) -- (5.456540084,-0.6666666667);
-\draw [,->,>=latex] (5.266666667,-0.3333333333) -- (5.456540084,-0.3333333333);
-\draw [,->,>=latex] (5.266666667,0) -- (5.456540084,0);
-\draw [,->,>=latex] (5.266666667,0.3333333333) -- (5.456540084,0.3333333333);
-\draw [,->,>=latex] (5.266666667,0.6666666667) -- (5.456540084,0.6666666667);
-\draw [,->,>=latex] (5.266666667,1.000000000) -- (5.456540084,1.000000000);
-\draw [,->,>=latex] (6.000000000,-1.000000000) -- (6.166666667,-1.000000000);
-\draw [,->,>=latex] (6.000000000,-0.6666666667) -- (6.166666667,-0.6666666667);
-\draw [,->,>=latex] (6.000000000,-0.3333333333) -- (6.166666667,-0.3333333333);
-\draw [,->,>=latex] (6.000000000,0) -- (6.166666667,0);
-\draw [,->,>=latex] (6.000000000,0.3333333333) -- (6.166666667,0.3333333333);
-\draw [,->,>=latex] (6.000000000,0.6666666667) -- (6.166666667,0.6666666667);
-\draw [,->,>=latex] (6.000000000,1.000000000) -- (6.166666667,1.000000000);
+
+
+\draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
+\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
+\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
+\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
+\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
+\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
+\draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
+\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
+\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
+\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
+\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
+\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
+\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
+\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
+\draw [,->,>=latex] (3.0667,-0.33333) -- (3.3928,-0.33333);
+\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
+\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
+\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
+\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
+\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
+\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
+\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
+\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
+\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
+\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
+\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
+\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
+\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
+\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
+\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
+\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
+\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
+\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
+\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
+\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
+\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
+\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
+\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall b/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
index 92b9e2f6a..feed9e843 100644
--- a/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
+++ b/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
@@ -88,28 +88,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.6113094300) -- (0,2.527798437);
+\draw [,->,>=latex] (-2.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.61131) -- (0,2.5278);
 %DEFAULT
 
 \draw [color=blue] (-2.000,1.127)--(-1.929,1.000)--(-1.859,0.8813)--(-1.788,0.7700)--(-1.717,0.6663)--(-1.646,0.5699)--(-1.576,0.4807)--(-1.505,0.3985)--(-1.434,0.3231)--(-1.364,0.2544)--(-1.293,0.1922)--(-1.222,0.1363)--(-1.152,0.08657)--(-1.081,0.04284)--(-1.010,0.004948)--(-0.9394,-0.02729)--(-0.8687,-0.05404)--(-0.7980,-0.07546)--(-0.7273,-0.09173)--(-0.6566,-0.1030)--(-0.5859,-0.1095)--(-0.5152,-0.1113)--(-0.4444,-0.1087)--(-0.3737,-0.1017)--(-0.3030,-0.09057)--(-0.2323,-0.07548)--(-0.1616,-0.05658)--(-0.09091,-0.03405)--(-0.02020,-0.008044)--(0.05051,0.02126)--(0.1212,0.05370)--(0.1919,0.08911)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5455,0.3047)--(0.6162,0.3544)--(0.6869,0.4058)--(0.7576,0.4589)--(0.8283,0.5134)--(0.8990,0.5692)--(0.9697,0.6261)--(1.040,0.6840)--(1.111,0.7426)--(1.182,0.8018)--(1.253,0.8615)--(1.323,0.9215)--(1.394,0.9815)--(1.465,1.042)--(1.535,1.101)--(1.606,1.161)--(1.677,1.219)--(1.747,1.277)--(1.818,1.335)--(1.889,1.391)--(1.960,1.445)--(2.030,1.499)--(2.101,1.551)--(2.172,1.601)--(2.242,1.649)--(2.313,1.695)--(2.384,1.739)--(2.455,1.780)--(2.525,1.819)--(2.596,1.855)--(2.667,1.888)--(2.737,1.918)--(2.808,1.945)--(2.879,1.968)--(2.949,1.988)--(3.020,2.004)--(3.091,2.016)--(3.162,2.024)--(3.232,2.028)--(3.303,2.027)--(3.374,2.022)--(3.444,2.011)--(3.515,1.996)--(3.586,1.976)--(3.657,1.951)--(3.727,1.920)--(3.798,1.883)--(3.869,1.841)--(3.939,1.792)--(4.010,1.738)--(4.081,1.677)--(4.151,1.610)--(4.222,1.536)--(4.293,1.455)--(4.364,1.368)--(4.434,1.273)--(4.505,1.171)--(4.576,1.062)--(4.646,0.9444)--(4.717,0.8194)--(4.788,0.6865)--(4.859,0.5454)--(4.929,0.3960)--(5.000,0.2381);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall b/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
index 42f5d021d..66e9b1259 100644
--- a/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
+++ b/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
@@ -108,34 +108,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.353981636,0) -- (8.353981636,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
 \draw [color=blue] (-7.854,-1.000)--(-7.695,-0.9874)--(-7.537,-0.9501)--(-7.378,-0.8888)--(-7.219,-0.8053)--(-7.061,-0.7015)--(-6.902,-0.5801)--(-6.743,-0.4441)--(-6.585,-0.2969)--(-6.426,-0.1423)--(-6.267,0.01587)--(-6.109,0.1736)--(-5.950,0.3271)--(-5.791,0.4723)--(-5.633,0.6056)--(-5.474,0.7237)--(-5.315,0.8237)--(-5.157,0.9029)--(-4.998,0.9595)--(-4.839,0.9920)--(-4.681,0.9995)--(-4.522,0.9819)--(-4.363,0.9397)--(-4.205,0.8738)--(-4.046,0.7861)--(-3.887,0.6785)--(-3.729,0.5539)--(-3.570,0.4154)--(-3.411,0.2665)--(-3.253,0.1108)--(-3.094,-0.04758)--(-2.935,-0.2048)--(-2.777,-0.3569)--(-2.618,-0.5000)--(-2.459,-0.6306)--(-2.301,-0.7453)--(-2.142,-0.8413)--(-1.983,-0.9161)--(-1.825,-0.9679)--(-1.666,-0.9955)--(-1.507,-0.9980)--(-1.349,-0.9754)--(-1.190,-0.9284)--(-1.031,-0.8580)--(-0.8727,-0.7660)--(-0.7140,-0.6549)--(-0.5553,-0.5272)--(-0.3967,-0.3863)--(-0.2380,-0.2358)--(-0.07933,-0.07925)--(0.07933,0.07925)--(0.2380,0.2358)--(0.3967,0.3863)--(0.5553,0.5272)--(0.7140,0.6549)--(0.8727,0.7660)--(1.031,0.8580)--(1.190,0.9284)--(1.349,0.9754)--(1.507,0.9980)--(1.666,0.9955)--(1.825,0.9679)--(1.983,0.9161)--(2.142,0.8413)--(2.301,0.7453)--(2.459,0.6306)--(2.618,0.5000)--(2.777,0.3569)--(2.935,0.2048)--(3.094,0.04758)--(3.253,-0.1108)--(3.411,-0.2665)--(3.570,-0.4154)--(3.729,-0.5539)--(3.887,-0.6785)--(4.046,-0.7861)--(4.205,-0.8738)--(4.363,-0.9397)--(4.522,-0.9819)--(4.681,-0.9995)--(4.839,-0.9920)--(4.998,-0.9595)--(5.157,-0.9029)--(5.315,-0.8237)--(5.474,-0.7237)--(5.633,-0.6056)--(5.791,-0.4723)--(5.950,-0.3271)--(6.109,-0.1736)--(6.267,-0.01587)--(6.426,0.1423)--(6.585,0.2969)--(6.743,0.4441)--(6.902,0.5801)--(7.061,0.7015)--(7.219,0.8053)--(7.378,0.8888)--(7.537,0.9501)--(7.695,0.9874)--(7.854,1.000);
-\draw (-7.853981634,-0.4207143333) node {$ -\frac{5}{2} \, \pi $};
+\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
 \draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.283185307,-0.3298256667) node {$ -2 \, \pi $};
+\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
 \draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.712388980,-0.4207143333) node {$ -\frac{3}{2} \, \pi $};
+\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
 \draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.570796327,-0.4207143333) node {$ -\frac{1}{2} \, \pi $};
+\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.570796327,-0.4207143333) node {$ \frac{1}{2} \, \pi $};
+\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
 \draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.712388980,-0.4207143333) node {$ \frac{3}{2} \, \pi $};
+\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
 \draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.853981634,-0.4207143333) node {$ \frac{5}{2} \, \pi $};
+\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
 \draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangentSegment.pstricks.recall b/src_phystricks/Fig_TangentSegment.pstricks.recall
index 49f2ac546..7ff8d9539 100644
--- a/src_phystricks/Fig_TangentSegment.pstricks.recall
+++ b/src_phystricks/Fig_TangentSegment.pstricks.recall
@@ -103,44 +103,44 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.703789489,0) -- (7.783185311,0);
-\draw [,->,>=latex] (0,-3.068914101) -- (0,2.500000000);
+\draw [,->,>=latex] (-3.7038,0) -- (7.7832,0);
+\draw [,->,>=latex] (0,-3.0689) -- (0,2.5000);
 %DEFAULT
 
 \draw [color=blue] (-3.142,-2.000)--(-3.036,-1.997)--(-2.931,-1.989)--(-2.826,-1.975)--(-2.720,-1.956)--(-2.615,-1.931)--(-2.510,-1.901)--(-2.404,-1.866)--(-2.299,-1.825)--(-2.194,-1.780)--(-2.089,-1.729)--(-1.983,-1.674)--(-1.878,-1.614)--(-1.773,-1.550)--(-1.667,-1.481)--(-1.562,-1.408)--(-1.457,-1.331)--(-1.351,-1.251)--(-1.246,-1.167)--(-1.141,-1.080)--(-1.036,-0.9899)--(-0.9303,-0.8971)--(-0.8250,-0.8018)--(-0.7197,-0.7042)--(-0.6144,-0.6048)--(-0.5091,-0.5036)--(-0.4038,-0.4010)--(-0.2985,-0.2974)--(-0.1932,-0.1929)--(-0.08787,-0.08784)--(0.01743,0.01743)--(0.1227,0.1227)--(0.2280,0.2275)--(0.3333,0.3318)--(0.4386,0.4351)--(0.5439,0.5373)--(0.6492,0.6379)--(0.7545,0.7368)--(0.8598,0.8336)--(0.9651,0.9281)--(1.070,1.020)--(1.176,1.109)--(1.281,1.195)--(1.386,1.278)--(1.492,1.357)--(1.597,1.433)--(1.702,1.504)--(1.808,1.571)--(1.913,1.634)--(2.018,1.693)--(2.123,1.746)--(2.229,1.795)--(2.334,1.839)--(2.439,1.878)--(2.545,1.912)--(2.650,1.940)--(2.755,1.963)--(2.861,1.980)--(2.966,1.992)--(3.071,1.999)--(3.176,2.000)--(3.282,1.995)--(3.387,1.985)--(3.492,1.969)--(3.598,1.948)--(3.703,1.922)--(3.808,1.890)--(3.914,1.853)--(4.019,1.811)--(4.124,1.763)--(4.229,1.711)--(4.335,1.655)--(4.440,1.593)--(4.545,1.527)--(4.651,1.457)--(4.756,1.383)--(4.861,1.305)--(4.967,1.224)--(5.072,1.139)--(5.177,1.050)--(5.282,0.9595)--(5.388,0.8658)--(5.493,0.7697)--(5.598,0.6715)--(5.704,0.5714)--(5.809,0.4698)--(5.914,0.3668)--(6.020,0.2628)--(6.125,0.1581)--(6.230,0.05300)--(6.335,-0.05229)--(6.441,-0.1574)--(6.546,-0.2621)--(6.651,-0.3661)--(6.757,-0.4691)--(6.862,-0.5708)--(6.967,-0.6708)--(7.073,-0.7691)--(7.178,-0.8652)--(7.283,-0.9589);
 \draw [] (-3.20,-2.57) -- (0.0622,-0.259);
-\draw []  (-1.570796327,-1.414213562) node [rotate=0] {$\bullet$};
+\draw []  (-1.5708,-1.4142) node [rotate=0] {$\bullet$};
 \draw [] (1.14,2.00) -- (5.14,2.00);
-\draw []  (3.141592654,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw []  (3.1416,2.0000) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangenteDetail.pstricks.recall b/src_phystricks/Fig_TangenteDetail.pstricks.recall
index 86f02dcca..bfa327892 100644
--- a/src_phystricks/Fig_TangenteDetail.pstricks.recall
+++ b/src_phystricks/Fig_TangenteDetail.pstricks.recall
@@ -111,8 +111,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.105147059);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
 %DEFAULT
 \draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
 \draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
@@ -122,24 +122,24 @@
 \draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
 
 \draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.700000000,3.235294118) node [rotate=0] {$\bullet$};
-\draw (1.341429089,3.568136140) node {$P$};
-\draw []  (1.700000000,0) node [rotate=0] {$\bullet$};
-\draw (1.700000000,-0.2785761667) node {$a$};
-\draw []  (0,3.235294118) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,3.235294118) node {$f(a)$};
-\draw []  (4.000000000,4.250000000) node [rotate=0] {$\bullet$};
-\draw (3.800437273,4.705055823) node {$Q$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.2785761667) node {$x$};
-\draw []  (0,4.250000000) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,4.250000000) node {$f(x)$};
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (1.700000000,3.035294118);
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (4.000000000,3.035294118);
-\draw (2.850000000,2.714264451) node {$x-a$};
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,4.250000000);
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,3.235294118);
-\draw (5.325596167,3.742647059) node {$f(x)-f(a)$};
+\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw (1.3414,3.5681) node {$P$};
+\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.27858) node {$a$};
+\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
+\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (3.8004,4.7051) node {$Q$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.27858) node {$x$};
+\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.45521,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
+\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
+\draw (2.8500,2.7143) node {$x-a$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
+\draw (5.3256,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangenteDetailOM.pstricks.recall b/src_phystricks/Fig_TangenteDetailOM.pstricks.recall
index 0869fa71e..4771ca36f 100644
--- a/src_phystricks/Fig_TangenteDetailOM.pstricks.recall
+++ b/src_phystricks/Fig_TangenteDetailOM.pstricks.recall
@@ -111,8 +111,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,5.105147059);
+\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
 %DEFAULT
 \draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
 \draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
@@ -122,24 +122,24 @@
 \draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
 
 \draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.700000000,3.235294118) node [rotate=0] {$\bullet$};
-\draw (1.341429089,3.568136140) node {$P$};
-\draw []  (1.700000000,0) node [rotate=0] {$\bullet$};
-\draw (1.700000000,-0.2785761667) node {$a$};
-\draw []  (0,3.235294118) node [rotate=0] {$\bullet$};
-\draw (-0.4473703333,3.235294118) node {$f(a)$};
-\draw []  (4.000000000,4.250000000) node [rotate=0] {$\bullet$};
-\draw (3.800437273,4.705055823) node {$Q$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.000000000,-0.2785761667) node {$x$};
-\draw []  (0,4.250000000) node [rotate=0] {$\bullet$};
-\draw (-0.4552066667,4.250000000) node {$f(x)$};
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (1.700000000,3.035294118);
-\draw [,->,>=latex] (2.850000000,3.035294118) -- (4.000000000,3.035294118);
-\draw (2.850000000,2.714264451) node {$x-a$};
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,4.250000000);
-\draw [,->,>=latex] (4.200000000,3.742647059) -- (4.200000000,3.235294118);
-\draw (5.325596167,3.742647059) node {$f(x)-f(a)$};
+\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw (1.3414,3.5681) node {$P$};
+\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.27858) node {$a$};
+\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
+\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (3.8004,4.7051) node {$Q$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.27858) node {$x$};
+\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.45521,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
+\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
+\draw (2.8500,2.7143) node {$x-a$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
+\draw (5.3256,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangenteQuestion.pstricks.recall b/src_phystricks/Fig_TangenteQuestion.pstricks.recall
index 45df39124..ad59a76cd 100644
--- a/src_phystricks/Fig_TangenteQuestion.pstricks.recall
+++ b/src_phystricks/Fig_TangenteQuestion.pstricks.recall
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.580000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
 %DEFAULT
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall b/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall
index defcf2ea8..411ca3541 100644
--- a/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall
+++ b/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.580000000);
+\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
 %DEFAULT
 
 \draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.190000000,2.264705882) node [rotate=0] {$\bullet$};
-\draw (0.8314290885,2.597547904) node {$P$};
+\draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
+\draw (0.83143,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ToreRevolution.pstricks.recall b/src_phystricks/Fig_ToreRevolution.pstricks.recall
index b06e73eb4..044f09832 100644
--- a/src_phystricks/Fig_ToreRevolution.pstricks.recall
+++ b/src_phystricks/Fig_ToreRevolution.pstricks.recall
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (2.000000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
 
 \draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,0);
-\draw [,->,>=latex] (1.500000000,1.500000000) -- (1.500000000,3.000000000);
-\draw (1.896467667,1.500000000) node {$a$};
-\draw []  (0,3.000000000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
+\draw (1.8965,1.5000) node {$a$};
+\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
 \draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.1430327019,3.634515621) node {$R$};
+\draw (0.14303,3.6345) node {$R$};
 \draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
 
 %OTHER STUFF
diff --git a/src_phystricks/Fig_Trajs.pstricks.recall b/src_phystricks/Fig_Trajs.pstricks.recall
index 5a8041098..d6f17fd84 100644
--- a/src_phystricks/Fig_Trajs.pstricks.recall
+++ b/src_phystricks/Fig_Trajs.pstricks.recall
@@ -75,28 +75,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.642639664,0) -- (2.723244275,0);
-\draw [,->,>=latex] (0,-1.914172501) -- (0,2.735975350);
+\draw [,->,>=latex] (-1.6426,0) -- (2.7232,0);
+\draw [,->,>=latex] (0,-1.9142) -- (0,2.7360);
 %DEFAULT
 \draw [color=red] (1.382,0.3012)--(1.388,0.2732)--(1.393,0.2451)--(1.397,0.2169)--(1.402,0.1886)--(1.405,0.1603)--(1.408,0.1319)--(1.410,0.1034)--(1.412,0.07490)--(1.413,0.04635)--(1.414,0.01779)--(1.414,-0.01078)--(1.414,-0.03934)--(1.413,-0.06789)--(1.411,-0.09641)--(1.409,-0.1249)--(1.406,-0.1533)--(1.402,-0.1817)--(1.399,-0.2100)--(1.394,-0.2382)--(1.389,-0.2663)--(1.383,-0.2943)--(1.377,-0.3222)--(1.370,-0.3499)--(1.363,-0.3776)--(1.355,-0.4050)--(1.347,-0.4323)--(1.338,-0.4594)--(1.328,-0.4863)--(1.318,-0.5131)--(1.307,-0.5396)--(1.296,-0.5659)--(1.284,-0.5919)--(1.272,-0.6178)--(1.259,-0.6433)--(1.246,-0.6686)--(1.232,-0.6937)--(1.218,-0.7184)--(1.203,-0.7429)--(1.188,-0.7671)--(1.172,-0.7909)--(1.156,-0.8144)--(1.139,-0.8376)--(1.122,-0.8605)--(1.105,-0.8829)--(1.087,-0.9051)--(1.068,-0.9268)--(1.049,-0.9482)--(1.030,-0.9692)--(1.010,-0.9898)--(0.9898,-1.010)--(0.9692,-1.030)--(0.9482,-1.049)--(0.9268,-1.068)--(0.9051,-1.087)--(0.8829,-1.105)--(0.8605,-1.122)--(0.8376,-1.139)--(0.8144,-1.156)--(0.7909,-1.172)--(0.7671,-1.188)--(0.7429,-1.203)--(0.7184,-1.218)--(0.6937,-1.232)--(0.6686,-1.246)--(0.6433,-1.259)--(0.6178,-1.272)--(0.5919,-1.284)--(0.5659,-1.296)--(0.5396,-1.307)--(0.5131,-1.318)--(0.4863,-1.328)--(0.4594,-1.338)--(0.4323,-1.347)--(0.4050,-1.355)--(0.3776,-1.363)--(0.3499,-1.370)--(0.3222,-1.377)--(0.2943,-1.383)--(0.2663,-1.389)--(0.2382,-1.394)--(0.2100,-1.399)--(0.1817,-1.402)--(0.1533,-1.406)--(0.1249,-1.409)--(0.09641,-1.411)--(0.06789,-1.413)--(0.03934,-1.414)--(0.01078,-1.414)--(-0.01779,-1.414)--(-0.04635,-1.413)--(-0.07490,-1.412)--(-0.1034,-1.410)--(-0.1319,-1.408)--(-0.1603,-1.405)--(-0.1886,-1.402)--(-0.2169,-1.397)--(-0.2451,-1.393)--(-0.2732,-1.388)--(-0.3012,-1.382);
-\draw [color=brown]  (1.000000000,-1.000000000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
 \draw [color=green] (0.540,0.841)--(0.557,0.830)--(0.574,0.819)--(0.590,0.807)--(0.606,0.795)--(0.622,0.783)--(0.638,0.770)--(0.654,0.757)--(0.669,0.744)--(0.684,0.730)--(0.698,0.716)--(0.712,0.702)--(0.727,0.687)--(0.740,0.672)--(0.754,0.657)--(0.767,0.642)--(0.780,0.626)--(0.792,0.610)--(0.804,0.594)--(0.816,0.578)--(0.828,0.561)--(0.839,0.544)--(0.850,0.527)--(0.860,0.510)--(0.870,0.493)--(0.880,0.475)--(0.889,0.457)--(0.898,0.439)--(0.907,0.421)--(0.915,0.402)--(0.923,0.384)--(0.931,0.365)--(0.938,0.346)--(0.945,0.327)--(0.951,0.308)--(0.957,0.289)--(0.963,0.269)--(0.968,0.250)--(0.973,0.230)--(0.978,0.211)--(0.982,0.191)--(0.985,0.171)--(0.989,0.151)--(0.991,0.131)--(0.994,0.111)--(0.996,0.0908)--(0.997,0.0706)--(0.999,0.0505)--(1.00,0.0303)--(1.00,0.0101)--(1.00,-0.0101)--(1.00,-0.0303)--(0.999,-0.0505)--(0.997,-0.0706)--(0.996,-0.0908)--(0.994,-0.111)--(0.991,-0.131)--(0.989,-0.151)--(0.985,-0.171)--(0.982,-0.191)--(0.978,-0.211)--(0.973,-0.230)--(0.968,-0.250)--(0.963,-0.269)--(0.957,-0.289)--(0.951,-0.308)--(0.945,-0.327)--(0.938,-0.346)--(0.931,-0.365)--(0.923,-0.384)--(0.915,-0.402)--(0.907,-0.421)--(0.898,-0.439)--(0.889,-0.457)--(0.880,-0.475)--(0.870,-0.493)--(0.860,-0.510)--(0.850,-0.527)--(0.839,-0.544)--(0.828,-0.561)--(0.816,-0.578)--(0.804,-0.594)--(0.792,-0.610)--(0.780,-0.626)--(0.767,-0.642)--(0.754,-0.657)--(0.740,-0.672)--(0.727,-0.687)--(0.712,-0.702)--(0.698,-0.716)--(0.684,-0.730)--(0.669,-0.744)--(0.654,-0.757)--(0.638,-0.770)--(0.622,-0.783)--(0.606,-0.795)--(0.590,-0.807)--(0.574,-0.819)--(0.557,-0.830)--(0.540,-0.841);
-\draw [color=brown]  (1.000000000,0) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (1.0000,0) node [rotate=0] {$\bullet$};
 \draw [color=blue] (-0.3012,1.382)--(-0.2732,1.388)--(-0.2451,1.393)--(-0.2169,1.397)--(-0.1886,1.402)--(-0.1603,1.405)--(-0.1319,1.408)--(-0.1034,1.410)--(-0.07490,1.412)--(-0.04635,1.413)--(-0.01779,1.414)--(0.01078,1.414)--(0.03934,1.414)--(0.06789,1.413)--(0.09641,1.411)--(0.1249,1.409)--(0.1533,1.406)--(0.1817,1.402)--(0.2100,1.399)--(0.2382,1.394)--(0.2663,1.389)--(0.2943,1.383)--(0.3222,1.377)--(0.3499,1.370)--(0.3776,1.363)--(0.4050,1.355)--(0.4323,1.347)--(0.4594,1.338)--(0.4863,1.328)--(0.5131,1.318)--(0.5396,1.307)--(0.5659,1.296)--(0.5919,1.284)--(0.6178,1.272)--(0.6433,1.259)--(0.6686,1.246)--(0.6937,1.232)--(0.7184,1.218)--(0.7429,1.203)--(0.7671,1.188)--(0.7909,1.172)--(0.8144,1.156)--(0.8376,1.139)--(0.8605,1.122)--(0.8829,1.105)--(0.9051,1.087)--(0.9268,1.068)--(0.9482,1.049)--(0.9692,1.030)--(0.9898,1.010)--(1.010,0.9898)--(1.030,0.9692)--(1.049,0.9482)--(1.068,0.9268)--(1.087,0.9051)--(1.105,0.8829)--(1.122,0.8605)--(1.139,0.8376)--(1.156,0.8144)--(1.172,0.7909)--(1.188,0.7671)--(1.203,0.7429)--(1.218,0.7184)--(1.232,0.6937)--(1.246,0.6686)--(1.259,0.6433)--(1.272,0.6178)--(1.284,0.5919)--(1.296,0.5659)--(1.307,0.5396)--(1.318,0.5131)--(1.328,0.4863)--(1.338,0.4594)--(1.347,0.4323)--(1.355,0.4050)--(1.363,0.3776)--(1.370,0.3499)--(1.377,0.3222)--(1.383,0.2943)--(1.389,0.2663)--(1.394,0.2382)--(1.399,0.2100)--(1.402,0.1817)--(1.406,0.1533)--(1.409,0.1249)--(1.411,0.09641)--(1.413,0.06789)--(1.414,0.03934)--(1.414,0.01078)--(1.414,-0.01779)--(1.413,-0.04635)--(1.412,-0.07490)--(1.410,-0.1034)--(1.408,-0.1319)--(1.405,-0.1603)--(1.402,-0.1886)--(1.397,-0.2169)--(1.393,-0.2451)--(1.388,-0.2732)--(1.382,-0.3012);
-\draw [color=brown]  (1.000000000,1.000000000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw [color=cyan] (-1.143,1.922)--(-1.104,1.945)--(-1.064,1.967)--(-1.024,1.988)--(-0.9838,2.008)--(-0.9430,2.027)--(-0.9018,2.046)--(-0.8603,2.064)--(-0.8185,2.081)--(-0.7763,2.097)--(-0.7337,2.112)--(-0.6909,2.127)--(-0.6478,2.140)--(-0.6045,2.153)--(-0.5608,2.165)--(-0.5170,2.175)--(-0.4729,2.185)--(-0.4287,2.195)--(-0.3843,2.203)--(-0.3397,2.210)--(-0.2950,2.217)--(-0.2502,2.222)--(-0.2052,2.227)--(-0.1602,2.230)--(-0.1151,2.233)--(-0.06998,2.235)--(-0.02482,2.236)--(0.02035,2.236)--(0.06552,2.235)--(0.1107,2.233)--(0.1557,2.231)--(0.2008,2.227)--(0.2457,2.223)--(0.2906,2.217)--(0.3353,2.211)--(0.3799,2.204)--(0.4243,2.195)--(0.4686,2.186)--(0.5127,2.177)--(0.5565,2.166)--(0.6002,2.154)--(0.6435,2.141)--(0.6867,2.128)--(0.7295,2.114)--(0.7721,2.099)--(0.8143,2.083)--(0.8562,2.066)--(0.8978,2.048)--(0.9389,2.029)--(0.9797,2.010)--(1.020,1.990)--(1.060,1.969)--(1.100,1.947)--(1.139,1.924)--(1.177,1.901)--(1.216,1.877)--(1.253,1.852)--(1.290,1.826)--(1.327,1.800)--(1.363,1.773)--(1.399,1.745)--(1.434,1.716)--(1.468,1.687)--(1.502,1.657)--(1.535,1.626)--(1.567,1.595)--(1.599,1.563)--(1.631,1.530)--(1.661,1.497)--(1.691,1.463)--(1.720,1.429)--(1.749,1.393)--(1.777,1.358)--(1.804,1.322)--(1.830,1.285)--(1.856,1.248)--(1.880,1.210)--(1.904,1.172)--(1.928,1.133)--(1.950,1.094)--(1.972,1.054)--(1.993,1.014)--(2.013,0.9738)--(2.032,0.9329)--(2.051,0.8917)--(2.068,0.8501)--(2.085,0.8081)--(2.101,0.7658)--(2.116,0.7233)--(2.130,0.6804)--(2.143,0.6372)--(2.156,0.5938)--(2.167,0.5501)--(2.178,0.5062)--(2.188,0.4621)--(2.197,0.4178)--(2.205,0.3734)--(2.212,0.3287)--(2.218,0.2840)--(2.223,0.2391);
-\draw [color=brown]  (1.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw [color=brown]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TriangleRectangle.pstricks.recall b/src_phystricks/Fig_TriangleRectangle.pstricks.recall
index 606ad3d81..1807898e1 100644
--- a/src_phystricks/Fig_TriangleRectangle.pstricks.recall
+++ b/src_phystricks/Fig_TriangleRectangle.pstricks.recall
@@ -93,20 +93,20 @@
 \draw [] (2.00,3.46) -- (4.00,0);
 \draw [] (4.00,0) -- (0,0);
 \draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.251784057,0.3649246667) node {$60$};
+\draw (3.2518,0.36492) node {$60$};
 
 \draw [color=red] (3.75,0.433)--(3.75,0.430)--(3.74,0.428)--(3.74,0.425)--(3.73,0.422)--(3.73,0.419)--(3.72,0.416)--(3.72,0.413)--(3.71,0.410)--(3.71,0.407)--(3.71,0.404)--(3.70,0.401)--(3.70,0.398)--(3.69,0.395)--(3.69,0.391)--(3.68,0.388)--(3.68,0.385)--(3.68,0.381)--(3.67,0.378)--(3.67,0.374)--(3.66,0.371)--(3.66,0.367)--(3.66,0.364)--(3.65,0.360)--(3.65,0.356)--(3.65,0.353)--(3.64,0.349)--(3.64,0.345)--(3.63,0.341)--(3.63,0.337)--(3.63,0.333)--(3.62,0.329)--(3.62,0.325)--(3.62,0.321)--(3.61,0.317)--(3.61,0.313)--(3.61,0.309)--(3.60,0.305)--(3.60,0.301)--(3.60,0.296)--(3.59,0.292)--(3.59,0.288)--(3.59,0.284)--(3.59,0.279)--(3.58,0.275)--(3.58,0.270)--(3.58,0.266)--(3.57,0.261)--(3.57,0.257)--(3.57,0.252)--(3.57,0.248)--(3.56,0.243)--(3.56,0.238)--(3.56,0.234)--(3.56,0.229)--(3.55,0.224)--(3.55,0.220)--(3.55,0.215)--(3.55,0.210)--(3.54,0.205)--(3.54,0.200)--(3.54,0.196)--(3.54,0.191)--(3.54,0.186)--(3.53,0.181)--(3.53,0.176)--(3.53,0.171)--(3.53,0.166)--(3.53,0.161)--(3.52,0.156)--(3.52,0.151)--(3.52,0.146)--(3.52,0.141)--(3.52,0.136)--(3.52,0.131)--(3.52,0.126)--(3.51,0.120)--(3.51,0.115)--(3.51,0.110)--(3.51,0.105)--(3.51,0.0998)--(3.51,0.0946)--(3.51,0.0894)--(3.51,0.0842)--(3.51,0.0790)--(3.51,0.0738)--(3.50,0.0685)--(3.50,0.0633)--(3.50,0.0580)--(3.50,0.0528)--(3.50,0.0475)--(3.50,0.0423)--(3.50,0.0370)--(3.50,0.0317)--(3.50,0.0264)--(3.50,0.0211)--(3.50,0.0159)--(3.50,0.0106)--(3.50,0.00529)--(3.50,0);
-\draw (2.311909189,2.234016645) node {$30$};
+\draw (2.3119,2.2340) node {$30$};
 
 \draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
-\draw []  (2.000000000,3.464101615) node [rotate=0] {$\bullet$};
-\draw (2.000000000,3.888809615) node {$A$};
+\draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
+\draw (2.0000,3.8888) node {$A$};
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.4475840000,0) node {$B$};
-\draw []  (4.000000000,0) node [rotate=0] {$\bullet$};
-\draw (4.443490000,0) node {$C$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.4247080000) node {$H$};
+\draw (-0.44758,0) node {$B$};
+\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
+\draw (4.4435,0) node {$C$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.42471) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_TriangleUV.pstricks.recall b/src_phystricks/Fig_TriangleUV.pstricks.recall
index 1a1899f26..69dd3cc1b 100644
--- a/src_phystricks/Fig_TriangleUV.pstricks.recall
+++ b/src_phystricks/Fig_TriangleUV.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -90,17 +90,15 @@
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
 \fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,0) -- (0,3.00) -- (0,3.00) -- (3.00,0) -- (3.00,0) -- (0,0) --  cycle;
-\draw [] (0,0) -- (0,3.00);
-\draw [] (0,3.00) -- (3.00,0);
-\draw [] (3.00,0) -- (0,0);
-\draw [color=green,->,>=latex] (0,0) -- (1.000000000,0);
-\draw (1.000000000,-0.2059510000) node {\( e_u\)};
-\draw [color=red,->,>=latex] (0,0) -- (0,1.000000000);
-\draw (-0.2670763333,1.000000000) node {\( e_v\)};
-\draw [color=green,->,>=latex] (1.712132034,1.712132034) -- (2.419238816,2.419238816);
-\draw (2.246758804,2.568525660) node {\( \nu\)};
-\draw [color=red,->,>=latex] (1.712132034,1.712132034) -- (1.005025253,2.419238816);
-\draw (0.8023187417,2.614657494) node {\( T\)};
+
+\draw [color=green,->,>=latex] (0,0) -- (1.0000,0);
+\draw (1.0000,-0.20595) node {\( e_u\)};
+\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
+\draw (-0.26708,1.0000) node {\( e_v\)};
+\draw [color=green,->,>=latex] (1.7121,1.7121) -- (2.4192,2.4192);
+\draw (2.2468,2.5685) node {\( \nu\)};
+\draw [color=red,->,>=latex] (1.7121,1.7121) -- (1.0050,2.4192);
+\draw (0.80232,2.6147) node {\( T\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall b/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall
index a8cef4db8..55b304951 100644
--- a/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall
+++ b/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall
@@ -116,51 +116,51 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (14.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (14.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
 %DEFAULT
-\draw []  (0,4.000000000) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,2.400000000) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,1.440000000) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0.8640000000) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,0.5184000000) node [rotate=0] {$\bullet$};
-\draw []  (5.000000000,0.3110400000) node [rotate=0] {$\bullet$};
-\draw []  (6.000000000,0.1866240000) node [rotate=0] {$\bullet$};
-\draw []  (7.000000000,0.1119744000) node [rotate=0] {$\bullet$};
-\draw []  (8.000000000,0.06718464000) node [rotate=0] {$\bullet$};
-\draw []  (9.000000000,0.04031078400) node [rotate=0] {$\bullet$};
-\draw []  (10.00000000,0.02418647040) node [rotate=0] {$\bullet$};
-\draw []  (11.00000000,0.01451188224) node [rotate=0] {$\bullet$};
-\draw []  (12.00000000,0.008707129344) node [rotate=0] {$\bullet$};
-\draw []  (13.00000000,0.005224277606) node [rotate=0] {$\bullet$};
-\draw []  (14.00000000,0.003134566564) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,2.4000) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,1.4400) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.86400) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.51840) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0.31104) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0.18662) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0.11197) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0.067185) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0.040311) node [rotate=0] {$\bullet$};
+\draw []  (10.000,0.024186) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0.014512) node [rotate=0] {$\bullet$};
+\draw []  (12.000,0.0087071) node [rotate=0] {$\bullet$};
+\draw []  (13.000,0.0052243) node [rotate=0] {$\bullet$};
+\draw []  (14.000,0.0031346) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.00000000,-0.3149246667) node {$ 11 $};
+\draw (11.000,-0.31492) node {$ 11 $};
 \draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.00000000,-0.3149246667) node {$ 12 $};
+\draw (12.000,-0.31492) node {$ 12 $};
 \draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.00000000,-0.3149246667) node {$ 13 $};
+\draw (13.000,-0.31492) node {$ 13 $};
 \draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (14.00000000,-0.3149246667) node {$ 14 $};
+\draw (14.000,-0.31492) node {$ 14 $};
 \draw [] (14.0,-0.100) -- (14.0,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall b/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall
index 2635881ef..8310c82c8 100644
--- a/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall
+++ b/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall
@@ -142,47 +142,47 @@
 \draw [color=gray,style=solid] (-3.00,4.00) -- (3.00,4.00);
 \draw [color=gray,style=solid] (-3.00,5.00) -- (3.00,5.00);
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-5.500000000) -- (0,5.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-5.5000) -- (0,5.5000);
 %DEFAULT
-\draw (-3.418346945,4.913809108) node {\( y=cosh(x)\)};
+\draw (-3.4183,4.9138) node {\( y=cosh(x)\)};
 
 \draw [color=blue] (-2.275,4.914)--(-2.229,4.698)--(-2.183,4.492)--(-2.137,4.295)--(-2.091,4.108)--(-2.045,3.929)--(-1.999,3.758)--(-1.953,3.596)--(-1.907,3.441)--(-1.861,3.293)--(-1.815,3.152)--(-1.769,3.018)--(-1.723,2.891)--(-1.677,2.769)--(-1.631,2.653)--(-1.585,2.543)--(-1.539,2.438)--(-1.493,2.339)--(-1.448,2.244)--(-1.402,2.154)--(-1.356,2.068)--(-1.310,1.987)--(-1.264,1.911)--(-1.218,1.838)--(-1.172,1.769)--(-1.126,1.704)--(-1.080,1.642)--(-1.034,1.584)--(-0.9880,1.529)--(-0.9420,1.478)--(-0.8961,1.429)--(-0.8501,1.384)--(-0.8042,1.341)--(-0.7582,1.301)--(-0.7123,1.265)--(-0.6663,1.230)--(-0.6204,1.199)--(-0.5744,1.170)--(-0.5285,1.143)--(-0.4825,1.119)--(-0.4366,1.097)--(-0.3906,1.077)--(-0.3446,1.060)--(-0.2987,1.045)--(-0.2527,1.032)--(-0.2068,1.021)--(-0.1608,1.013)--(-0.1149,1.007)--(-0.06893,1.002)--(-0.02298,1.000)--(0.02298,1.000)--(0.06893,1.002)--(0.1149,1.007)--(0.1608,1.013)--(0.2068,1.021)--(0.2527,1.032)--(0.2987,1.045)--(0.3446,1.060)--(0.3906,1.077)--(0.4366,1.097)--(0.4825,1.119)--(0.5285,1.143)--(0.5744,1.170)--(0.6204,1.199)--(0.6663,1.230)--(0.7123,1.265)--(0.7582,1.301)--(0.8042,1.341)--(0.8501,1.384)--(0.8961,1.429)--(0.9420,1.478)--(0.9880,1.529)--(1.034,1.584)--(1.080,1.642)--(1.126,1.704)--(1.172,1.769)--(1.218,1.838)--(1.264,1.911)--(1.310,1.987)--(1.356,2.068)--(1.402,2.154)--(1.448,2.244)--(1.493,2.339)--(1.539,2.438)--(1.585,2.543)--(1.631,2.653)--(1.677,2.769)--(1.723,2.891)--(1.769,3.018)--(1.815,3.152)--(1.861,3.293)--(1.907,3.441)--(1.953,3.596)--(1.999,3.758)--(2.045,3.929)--(2.091,4.108)--(2.137,4.295)--(2.183,4.492)--(2.229,4.698)--(2.275,4.914);
-\draw (-3.423323612,-4.810979105) node {\( y=sinh(x)\)};
+\draw (-3.4233,-4.8110) node {\( y=sinh(x)\)};
 
 \draw [color=blue] (-2.275,-4.811)--(-2.229,-4.590)--(-2.183,-4.379)--(-2.137,-4.177)--(-2.091,-3.984)--(-2.045,-3.800)--(-1.999,-3.623)--(-1.953,-3.454)--(-1.907,-3.292)--(-1.861,-3.138)--(-1.815,-2.990)--(-1.769,-2.848)--(-1.723,-2.712)--(-1.677,-2.582)--(-1.631,-2.458)--(-1.585,-2.338)--(-1.539,-2.224)--(-1.493,-2.114)--(-1.448,-2.009)--(-1.402,-1.908)--(-1.356,-1.811)--(-1.310,-1.718)--(-1.264,-1.628)--(-1.218,-1.542)--(-1.172,-1.459)--(-1.126,-1.379)--(-1.080,-1.302)--(-1.034,-1.228)--(-0.9880,-1.157)--(-0.9420,-1.088)--(-0.8961,-1.021)--(-0.8501,-0.9563)--(-0.8042,-0.8937)--(-0.7582,-0.8330)--(-0.7123,-0.7740)--(-0.6663,-0.7167)--(-0.6204,-0.6609)--(-0.5744,-0.6065)--(-0.5285,-0.5534)--(-0.4825,-0.5014)--(-0.4366,-0.4506)--(-0.3906,-0.4006)--(-0.3446,-0.3515)--(-0.2987,-0.3032)--(-0.2527,-0.2554)--(-0.2068,-0.2083)--(-0.1608,-0.1615)--(-0.1149,-0.1151)--(-0.06893,-0.06898)--(-0.02298,-0.02298)--(0.02298,0.02298)--(0.06893,0.06898)--(0.1149,0.1151)--(0.1608,0.1615)--(0.2068,0.2083)--(0.2527,0.2554)--(0.2987,0.3032)--(0.3446,0.3515)--(0.3906,0.4006)--(0.4366,0.4506)--(0.4825,0.5014)--(0.5285,0.5534)--(0.5744,0.6065)--(0.6204,0.6609)--(0.6663,0.7167)--(0.7123,0.7740)--(0.7582,0.8330)--(0.8042,0.8937)--(0.8501,0.9563)--(0.8961,1.021)--(0.9420,1.088)--(0.9880,1.157)--(1.034,1.228)--(1.080,1.302)--(1.126,1.379)--(1.172,1.459)--(1.218,1.542)--(1.264,1.628)--(1.310,1.718)--(1.356,1.811)--(1.402,1.908)--(1.448,2.009)--(1.493,2.114)--(1.539,2.224)--(1.585,2.338)--(1.631,2.458)--(1.677,2.582)--(1.723,2.712)--(1.769,2.848)--(1.815,2.990)--(1.861,3.138)--(1.907,3.292)--(1.953,3.454)--(1.999,3.623)--(2.045,3.800)--(2.091,3.984)--(2.137,4.177)--(2.183,4.379)--(2.229,4.590)--(2.275,4.811);
 
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-5.000000000) node {$ -5 $};
+\draw (-0.43316,-5.0000) node {$ -5 $};
 \draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.2912498333,5.000000000) node {$ 5 $};
+\draw (-0.29125,5.0000) node {$ 5 $};
 \draw [] (-0.100,5.00) -- (0.100,5.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall b/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall
index 819ccd255..a266f3275 100644
--- a/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall
+++ b/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall
@@ -108,36 +108,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-2.070796327) -- (0,2.070796327);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-2.0708) -- (0,2.0708);
 %DEFAULT
 
 \draw [color=blue] (-5.000,-1.373)--(-4.899,-1.369)--(-4.798,-1.365)--(-4.697,-1.361)--(-4.596,-1.357)--(-4.495,-1.352)--(-4.394,-1.347)--(-4.293,-1.342)--(-4.192,-1.337)--(-4.091,-1.331)--(-3.990,-1.325)--(-3.889,-1.319)--(-3.788,-1.313)--(-3.687,-1.306)--(-3.586,-1.299)--(-3.485,-1.291)--(-3.384,-1.283)--(-3.283,-1.275)--(-3.182,-1.266)--(-3.081,-1.257)--(-2.980,-1.247)--(-2.879,-1.236)--(-2.778,-1.225)--(-2.677,-1.213)--(-2.576,-1.200)--(-2.475,-1.187)--(-2.374,-1.172)--(-2.273,-1.156)--(-2.172,-1.139)--(-2.071,-1.121)--(-1.970,-1.101)--(-1.869,-1.079)--(-1.768,-1.056)--(-1.667,-1.030)--(-1.566,-1.002)--(-1.465,-0.9717)--(-1.364,-0.9380)--(-1.263,-0.9010)--(-1.162,-0.8600)--(-1.061,-0.8148)--(-0.9596,-0.7648)--(-0.8586,-0.7095)--(-0.7576,-0.6483)--(-0.6566,-0.5810)--(-0.5556,-0.5071)--(-0.4545,-0.4266)--(-0.3535,-0.3398)--(-0.2525,-0.2474)--(-0.1515,-0.1504)--(-0.05051,-0.05046)--(0.05051,0.05046)--(0.1515,0.1504)--(0.2525,0.2474)--(0.3535,0.3398)--(0.4545,0.4266)--(0.5556,0.5071)--(0.6566,0.5810)--(0.7576,0.6483)--(0.8586,0.7095)--(0.9596,0.7648)--(1.061,0.8148)--(1.162,0.8600)--(1.263,0.9010)--(1.364,0.9380)--(1.465,0.9717)--(1.566,1.002)--(1.667,1.030)--(1.768,1.056)--(1.869,1.079)--(1.970,1.101)--(2.071,1.121)--(2.172,1.139)--(2.273,1.156)--(2.374,1.172)--(2.475,1.187)--(2.576,1.200)--(2.677,1.213)--(2.778,1.225)--(2.879,1.236)--(2.980,1.247)--(3.081,1.257)--(3.182,1.266)--(3.283,1.275)--(3.384,1.283)--(3.485,1.291)--(3.586,1.299)--(3.687,1.306)--(3.788,1.313)--(3.889,1.319)--(3.990,1.325)--(4.091,1.331)--(4.192,1.337)--(4.293,1.342)--(4.394,1.347)--(4.495,1.352)--(4.596,1.357)--(4.697,1.361)--(4.798,1.365)--(4.899,1.369)--(5.000,1.373);
 \draw [color=red,style=dashed] (-5.00,1.57) -- (5.00,1.57);
 \draw [color=red,style=dashed] (-5.00,-1.57) -- (5.00,-1.57);
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.5937365000,-1.570796327) node {$ -\frac{1}{2} \, \pi $};
+\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
 \draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.4518270000,1.570796327) node {$ \frac{1}{2} \, \pi $};
+\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
 \draw [] (-0.100,1.57) -- (0.100,1.57);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall b/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall
index 800f0865f..b8a022bef 100644
--- a/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall
+++ b/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall
@@ -96,46 +96,46 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-3.500000000) -- (0,4.500000000);
+\draw [,->,>=latex] (-3.5000,0) -- (4.5000,0);
+\draw [,->,>=latex] (0,-3.5000) -- (0,4.5000);
 %DEFAULT
 \fill [color=lightgray] (2.83,1.00) -- (2.80,1.07) -- (2.78,1.14) -- (2.75,1.21) -- (2.72,1.27) -- (2.68,1.34) -- (2.65,1.41) -- (2.62,1.47) -- (2.58,1.53) -- (2.54,1.60) -- (2.50,1.66) -- (2.46,1.72) -- (2.41,1.78) -- (2.37,1.84) -- (2.32,1.90) -- (2.28,1.95) -- (2.23,2.01) -- (2.18,2.06) -- (2.13,2.12) -- (2.07,2.17) -- (2.02,2.22) -- (1.96,2.27) -- (1.91,2.31) -- (1.85,2.36) -- (1.79,2.41) -- (1.73,2.45) -- (1.67,2.49) -- (1.61,2.53) -- (1.55,2.57) -- (1.48,2.61) -- (1.42,2.64) -- (1.35,2.68) -- (1.29,2.71) -- (1.22,2.74) -- (1.15,2.77) -- (1.08,2.80) -- (1.01,2.82) -- (0.944,2.85) -- (0.873,2.87) -- (0.803,2.89) -- (0.731,2.91) -- (0.659,2.93) -- (0.587,2.94) -- (0.515,2.96) -- (0.442,2.97) -- (0.368,2.98) -- (0.295,2.99) -- (0.221,2.99) -- (0.148,3.00) -- (0.0740,3.00) -- (0,3.00) -- (2.00,1.00) -- (2.00,1.00) -- (2.83,1.00) --  cycle;
 
 \draw [] (3.000,0)--(2.994,0.1903)--(2.976,0.3798)--(2.946,0.5678)--(2.904,0.7534)--(2.850,0.9361)--(2.785,1.115)--(2.709,1.289)--(2.622,1.459)--(2.524,1.622)--(2.416,1.779)--(2.298,1.928)--(2.171,2.070)--(2.036,2.204)--(1.892,2.328)--(1.740,2.444)--(1.582,2.549)--(1.417,2.644)--(1.246,2.729)--(1.071,2.802)--(0.8908,2.865)--(0.7073,2.915)--(0.5209,2.954)--(0.3325,2.982)--(0.1427,2.997)--(-0.04760,3.000)--(-0.2377,2.991)--(-0.4269,2.969)--(-0.6144,2.936)--(-0.7994,2.892)--(-0.9812,2.835)--(-1.159,2.767)--(-1.332,2.688)--(-1.500,2.598)--(-1.662,2.498)--(-1.817,2.387)--(-1.965,2.267)--(-2.104,2.138)--(-2.236,2.000)--(-2.358,1.854)--(-2.471,1.701)--(-2.574,1.541)--(-2.667,1.375)--(-2.748,1.203)--(-2.819,1.026)--(-2.878,0.8452)--(-2.926,0.6609)--(-2.962,0.4740)--(-2.986,0.2852)--(-2.999,0.09518)--(-2.999,-0.09518)--(-2.986,-0.2852)--(-2.962,-0.4740)--(-2.926,-0.6609)--(-2.878,-0.8452)--(-2.819,-1.026)--(-2.748,-1.203)--(-2.667,-1.375)--(-2.574,-1.541)--(-2.471,-1.701)--(-2.358,-1.854)--(-2.236,-2.000)--(-2.104,-2.138)--(-1.965,-2.267)--(-1.817,-2.387)--(-1.662,-2.498)--(-1.500,-2.598)--(-1.332,-2.688)--(-1.159,-2.767)--(-0.9812,-2.835)--(-0.7994,-2.892)--(-0.6144,-2.936)--(-0.4269,-2.969)--(-0.2377,-2.991)--(-0.04760,-3.000)--(0.1427,-2.997)--(0.3325,-2.982)--(0.5209,-2.954)--(0.7073,-2.915)--(0.8908,-2.865)--(1.071,-2.802)--(1.246,-2.729)--(1.417,-2.644)--(1.582,-2.549)--(1.740,-2.444)--(1.892,-2.328)--(2.036,-2.204)--(2.171,-2.070)--(2.298,-1.928)--(2.416,-1.779)--(2.524,-1.622)--(2.622,-1.459)--(2.709,-1.289)--(2.785,-1.115)--(2.850,-0.9361)--(2.904,-0.7534)--(2.946,-0.5678)--(2.976,-0.3798)--(2.994,-0.1903)--(3.000,0);
-\draw []  (2.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (1.721703477,0.7338706438) node {\( A\)};
-\draw []  (2.828427125,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.117432481,0.7338706438) node {\( B\)};
+\draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (1.7217,0.73387) node {\( A\)};
+\draw []  (2.8284,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.1174,0.73387) node {\( B\)};
 \draw [] (-1.00,1.00) -- (4.00,1.00);
 
 \draw [color=blue] (-1.000,4.000)--(-0.9495,3.949)--(-0.8990,3.899)--(-0.8485,3.848)--(-0.7980,3.798)--(-0.7475,3.747)--(-0.6970,3.697)--(-0.6465,3.646)--(-0.5960,3.596)--(-0.5455,3.545)--(-0.4949,3.495)--(-0.4444,3.444)--(-0.3939,3.394)--(-0.3434,3.343)--(-0.2929,3.293)--(-0.2424,3.242)--(-0.1919,3.192)--(-0.1414,3.141)--(-0.09091,3.091)--(-0.04040,3.040)--(0.01010,2.990)--(0.06061,2.939)--(0.1111,2.889)--(0.1616,2.838)--(0.2121,2.788)--(0.2626,2.737)--(0.3131,2.687)--(0.3636,2.636)--(0.4141,2.586)--(0.4646,2.535)--(0.5152,2.485)--(0.5657,2.434)--(0.6162,2.384)--(0.6667,2.333)--(0.7172,2.283)--(0.7677,2.232)--(0.8182,2.182)--(0.8687,2.131)--(0.9192,2.081)--(0.9697,2.030)--(1.020,1.980)--(1.071,1.929)--(1.121,1.879)--(1.172,1.828)--(1.222,1.778)--(1.273,1.727)--(1.323,1.677)--(1.374,1.626)--(1.424,1.576)--(1.475,1.525)--(1.525,1.475)--(1.576,1.424)--(1.626,1.374)--(1.677,1.323)--(1.727,1.273)--(1.778,1.222)--(1.828,1.172)--(1.879,1.121)--(1.929,1.071)--(1.980,1.020)--(2.030,0.9697)--(2.081,0.9192)--(2.131,0.8687)--(2.182,0.8182)--(2.232,0.7677)--(2.283,0.7172)--(2.333,0.6667)--(2.384,0.6162)--(2.434,0.5657)--(2.485,0.5152)--(2.535,0.4646)--(2.586,0.4141)--(2.636,0.3636)--(2.687,0.3131)--(2.737,0.2626)--(2.788,0.2121)--(2.838,0.1616)--(2.889,0.1111)--(2.939,0.06061)--(2.990,0.01010)--(3.040,-0.04040)--(3.091,-0.09091)--(3.141,-0.1414)--(3.192,-0.1919)--(3.242,-0.2424)--(3.293,-0.2929)--(3.343,-0.3434)--(3.394,-0.3939)--(3.444,-0.4444)--(3.495,-0.4949)--(3.545,-0.5455)--(3.596,-0.5960)--(3.646,-0.6465)--(3.697,-0.6970)--(3.747,-0.7475)--(3.798,-0.7980)--(3.848,-0.8485)--(3.899,-0.8990)--(3.949,-0.9495)--(4.000,-1.000);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 4 $};
+\draw (-0.29125,4.0000) node {$ 4 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UnSurxInt.pstricks.recall b/src_phystricks/Fig_UnSurxInt.pstricks.recall
index 8736d3043..702ef7c25 100644
--- a/src_phystricks/Fig_UnSurxInt.pstricks.recall
+++ b/src_phystricks/Fig_UnSurxInt.pstricks.recall
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.500000000,0) -- (3.500000000,0);
-\draw [,->,>=latex] (0,-3.833333333) -- (0,3.833333333);
+\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
+\draw [,->,>=latex] (0,-3.8333) -- (0,3.8333);
 %DEFAULT
 
 \draw [color=blue] (-3.000,-0.3333)--(-2.973,-0.3364)--(-2.945,-0.3395)--(-2.918,-0.3427)--(-2.891,-0.3459)--(-2.864,-0.3492)--(-2.836,-0.3526)--(-2.809,-0.3560)--(-2.782,-0.3595)--(-2.755,-0.3630)--(-2.727,-0.3667)--(-2.700,-0.3704)--(-2.673,-0.3741)--(-2.645,-0.3780)--(-2.618,-0.3819)--(-2.591,-0.3860)--(-2.564,-0.3901)--(-2.536,-0.3943)--(-2.509,-0.3986)--(-2.482,-0.4029)--(-2.455,-0.4074)--(-2.427,-0.4120)--(-2.400,-0.4167)--(-2.373,-0.4215)--(-2.345,-0.4264)--(-2.318,-0.4314)--(-2.291,-0.4365)--(-2.264,-0.4418)--(-2.236,-0.4472)--(-2.209,-0.4527)--(-2.182,-0.4583)--(-2.155,-0.4641)--(-2.127,-0.4701)--(-2.100,-0.4762)--(-2.073,-0.4825)--(-2.045,-0.4889)--(-2.018,-0.4955)--(-1.991,-0.5023)--(-1.964,-0.5093)--(-1.936,-0.5164)--(-1.909,-0.5238)--(-1.882,-0.5314)--(-1.855,-0.5392)--(-1.827,-0.5473)--(-1.800,-0.5556)--(-1.773,-0.5641)--(-1.745,-0.5729)--(-1.718,-0.5820)--(-1.691,-0.5914)--(-1.664,-0.6011)--(-1.636,-0.6111)--(-1.609,-0.6215)--(-1.582,-0.6322)--(-1.555,-0.6433)--(-1.527,-0.6548)--(-1.500,-0.6667)--(-1.473,-0.6790)--(-1.445,-0.6918)--(-1.418,-0.7051)--(-1.391,-0.7190)--(-1.364,-0.7333)--(-1.336,-0.7483)--(-1.309,-0.7639)--(-1.282,-0.7801)--(-1.255,-0.7971)--(-1.227,-0.8148)--(-1.200,-0.8333)--(-1.173,-0.8527)--(-1.145,-0.8730)--(-1.118,-0.8943)--(-1.091,-0.9167)--(-1.064,-0.9402)--(-1.036,-0.9649)--(-1.009,-0.9910)--(-0.9818,-1.019)--(-0.9545,-1.048)--(-0.9273,-1.078)--(-0.9000,-1.111)--(-0.8727,-1.146)--(-0.8455,-1.183)--(-0.8182,-1.222)--(-0.7909,-1.264)--(-0.7636,-1.310)--(-0.7364,-1.358)--(-0.7091,-1.410)--(-0.6818,-1.467)--(-0.6545,-1.528)--(-0.6273,-1.594)--(-0.6000,-1.667)--(-0.5727,-1.746)--(-0.5455,-1.833)--(-0.5182,-1.930)--(-0.4909,-2.037)--(-0.4636,-2.157)--(-0.4364,-2.292)--(-0.4091,-2.444)--(-0.3818,-2.619)--(-0.3545,-2.821)--(-0.3273,-3.056)--(-0.3000,-3.333);
@@ -118,29 +118,29 @@
 \draw [color=blue] (1.000,0)--(1.010,0)--(1.020,0)--(1.030,0)--(1.040,0)--(1.051,0)--(1.061,0)--(1.071,0)--(1.081,0)--(1.091,0)--(1.101,0)--(1.111,0)--(1.121,0)--(1.131,0)--(1.141,0)--(1.152,0)--(1.162,0)--(1.172,0)--(1.182,0)--(1.192,0)--(1.202,0)--(1.212,0)--(1.222,0)--(1.232,0)--(1.242,0)--(1.253,0)--(1.263,0)--(1.273,0)--(1.283,0)--(1.293,0)--(1.303,0)--(1.313,0)--(1.323,0)--(1.333,0)--(1.343,0)--(1.354,0)--(1.364,0)--(1.374,0)--(1.384,0)--(1.394,0)--(1.404,0)--(1.414,0)--(1.424,0)--(1.434,0)--(1.444,0)--(1.455,0)--(1.465,0)--(1.475,0)--(1.485,0)--(1.495,0)--(1.505,0)--(1.515,0)--(1.525,0)--(1.535,0)--(1.545,0)--(1.556,0)--(1.566,0)--(1.576,0)--(1.586,0)--(1.596,0)--(1.606,0)--(1.616,0)--(1.626,0)--(1.636,0)--(1.646,0)--(1.657,0)--(1.667,0)--(1.677,0)--(1.687,0)--(1.697,0)--(1.707,0)--(1.717,0)--(1.727,0)--(1.737,0)--(1.747,0)--(1.758,0)--(1.768,0)--(1.778,0)--(1.788,0)--(1.798,0)--(1.808,0)--(1.818,0)--(1.828,0)--(1.838,0)--(1.848,0)--(1.859,0)--(1.869,0)--(1.879,0)--(1.889,0)--(1.899,0)--(1.909,0)--(1.919,0)--(1.929,0)--(1.939,0)--(1.949,0)--(1.960,0)--(1.970,0)--(1.980,0)--(1.990,0)--(2.000,0);
 \draw [color=brown,style=solid] (1.00,0) -- (1.00,1.00);
 \draw [color=brown,style=solid] (2.00,0.500) -- (2.00,0);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall b/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall
index 550c4f950..34b8ef2f9 100644
--- a/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall
+++ b/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall
@@ -66,14 +66,14 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.246812693,0.6811368787) -- (1.245451007,0.6910437354);
-\draw [,->,>=latex] (0.3575977601,1.218773069) -- (0.3480435445,1.215820618);
-\draw [,->,>=latex] (-0.2777141985,0.5083524306) -- (-0.2808616227,0.4988606594);
-\draw [,->,>=latex] (-0.7003261911,0.2054813022) -- (-0.7094196719,0.2013209407);
-\draw [,->,>=latex] (-1.246812693,-0.6811368787) -- (-1.245451007,-0.6910437354);
-\draw [,->,>=latex] (-0.3575977601,-1.218773069) -- (-0.3480435445,-1.215820618);
-\draw [,->,>=latex] (0.2777141985,-0.5083524306) -- (0.2808616227,-0.4988606594);
-\draw [,->,>=latex] (0.7003261912,-0.2054813022) -- (0.7094196719,-0.2013209407);
+\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
+\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
+\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
+\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
+\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
+\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
+\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
+\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall b/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall
index 22d34bd01..d8e38e460 100644
--- a/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall
+++ b/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -88,11 +88,11 @@
 \draw [color=red,style=solid] (1.00,2.00) -- (1.00,0);
 
 \draw [color=red] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall b/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall
index e5ae1d139..ba9700c1b 100644
--- a/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall
+++ b/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall
@@ -83,8 +83,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.650000000,0) -- (3.650000000,0);
-\draw [,->,>=latex] (0,-2.650000000) -- (0,2.650000000);
+\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
+\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
 %DEFAULT
 \draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
 \draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
@@ -102,21 +102,21 @@
 \draw [color=blue] (2.00,0) -- (1.00,-1.00);
 \draw [color=blue] (1.00,-1.00) -- (0,0);
 \draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall b/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall
index d7858707b..ff1ca43f9 100644
--- a/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall
+++ b/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall
@@ -79,13 +79,13 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.3247080000) node {\( O\)};
-\draw []  (1.050000000,1.818653348) node [rotate=0] {$\bullet$};
-\draw (1.297584000,2.116566429) node {\( B\)};
-\draw []  (1.818653348,1.050000000) node [rotate=0] {$\bullet$};
-\draw (2.128733595,1.274708000) node {\( A\)};
-\draw []  (2.100000000,0) node [rotate=0] {$\bullet$};
-\draw (2.335966356,-0.2661293562) node {\( I\)};
+\draw (0,-0.32471) node {\( O\)};
+\draw []  (1.0500,1.8187) node [rotate=0] {$\bullet$};
+\draw (1.2976,2.1166) node {\( B\)};
+\draw []  (1.8187,1.0500) node [rotate=0] {$\bullet$};
+\draw (2.1287,1.2747) node {\( A\)};
+\draw []  (2.1000,0) node [rotate=0] {$\bullet$};
+\draw (2.3360,-0.26613) node {\( I\)};
 
 \draw [] (2.100,0)--(2.096,0.1332)--(2.083,0.2658)--(2.062,0.3974)--(2.033,0.5274)--(1.995,0.6553)--(1.950,0.7805)--(1.896,0.9026)--(1.835,1.021)--(1.767,1.135)--(1.691,1.245)--(1.609,1.350)--(1.520,1.449)--(1.425,1.543)--(1.324,1.630)--(1.218,1.711)--(1.107,1.784)--(0.9918,1.851)--(0.8724,1.910)--(0.7495,1.962)--(0.6235,2.005)--(0.4951,2.041)--(0.3647,2.068)--(0.2328,2.087)--(0.09992,2.098)--(-0.03332,2.100)--(-0.1664,2.093)--(-0.2989,2.079)--(-0.4301,2.055)--(-0.5596,2.024)--(-0.6868,1.984)--(-0.8113,1.937)--(-0.9325,1.882)--(-1.050,1.819)--(-1.163,1.748)--(-1.272,1.671)--(-1.375,1.587)--(-1.473,1.497)--(-1.565,1.400)--(-1.651,1.298)--(-1.730,1.191)--(-1.802,1.079)--(-1.867,0.9623)--(-1.924,0.8420)--(-1.973,0.7182)--(-2.015,0.5916)--(-2.048,0.4627)--(-2.074,0.3318)--(-2.090,0.1996)--(-2.099,0.06663)--(-2.099,-0.06663)--(-2.090,-0.1996)--(-2.074,-0.3318)--(-2.048,-0.4627)--(-2.015,-0.5916)--(-1.973,-0.7182)--(-1.924,-0.8420)--(-1.867,-0.9623)--(-1.802,-1.079)--(-1.730,-1.191)--(-1.651,-1.298)--(-1.565,-1.400)--(-1.473,-1.497)--(-1.375,-1.587)--(-1.272,-1.671)--(-1.163,-1.748)--(-1.050,-1.819)--(-0.9325,-1.882)--(-0.8113,-1.937)--(-0.6868,-1.984)--(-0.5596,-2.024)--(-0.4301,-2.055)--(-0.2989,-2.079)--(-0.1664,-2.093)--(-0.03332,-2.100)--(0.09992,-2.098)--(0.2328,-2.087)--(0.3647,-2.068)--(0.4951,-2.041)--(0.6235,-2.005)--(0.7495,-1.962)--(0.8724,-1.910)--(0.9918,-1.851)--(1.107,-1.784)--(1.218,-1.711)--(1.324,-1.630)--(1.425,-1.543)--(1.520,-1.449)--(1.609,-1.350)--(1.691,-1.245)--(1.767,-1.135)--(1.835,-1.021)--(1.896,-0.9026)--(1.950,-0.7805)--(1.995,-0.6553)--(2.033,-0.5274)--(2.062,-0.3974)--(2.083,-0.2658)--(2.096,-0.1332)--(2.100,0);
 
diff --git a/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall b/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
index fd2268f2a..9099168b7 100644
--- a/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
+++ b/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
@@ -119,40 +119,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.000000000,0) -- (4.000000000,0);
-\draw [,->,>=latex] (0,-2.799525264) -- (0,4.054798491);
+\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
+\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
 %DEFAULT
 
 \draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
 
 \draw [color=red] (-3.500,3.555)--(-3.429,3.500)--(-3.359,3.406)--(-3.288,3.277)--(-3.217,3.114)--(-3.146,2.921)--(-3.076,2.702)--(-3.005,2.459)--(-2.934,2.198)--(-2.864,1.921)--(-2.793,1.632)--(-2.722,1.337)--(-2.652,1.038)--(-2.581,0.7401)--(-2.510,0.4468)--(-2.439,0.1618)--(-2.369,-0.1114)--(-2.298,-0.3696)--(-2.227,-0.6099)--(-2.157,-0.8297)--(-2.086,-1.027)--(-2.015,-1.199)--(-1.944,-1.346)--(-1.874,-1.466)--(-1.803,-1.558)--(-1.732,-1.622)--(-1.662,-1.658)--(-1.591,-1.667)--(-1.520,-1.650)--(-1.449,-1.608)--(-1.379,-1.542)--(-1.308,-1.456)--(-1.237,-1.350)--(-1.167,-1.228)--(-1.096,-1.092)--(-1.025,-0.9453)--(-0.9545,-0.7902)--(-0.8838,-0.6299)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.002540)--(-0.5303,0.1441)--(-0.4596,0.2739)--(-0.3889,0.3896)--(-0.3182,0.4892)--(-0.2475,0.5710)--(-0.1768,0.6336)--(-0.1061,0.6760)--(-0.03535,0.6973)--(0.03535,0.6973)--(0.1061,0.6760)--(0.1768,0.6336)--(0.2475,0.5710)--(0.3182,0.4892)--(0.3889,0.3896)--(0.4596,0.2739)--(0.5303,0.1441)--(0.6010,0.002540)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6299)--(0.9545,-0.7902)--(1.025,-0.9453)--(1.096,-1.092)--(1.167,-1.228)--(1.237,-1.350)--(1.308,-1.456)--(1.379,-1.542)--(1.449,-1.608)--(1.520,-1.650)--(1.591,-1.667)--(1.662,-1.658)--(1.732,-1.622)--(1.803,-1.558)--(1.874,-1.466)--(1.944,-1.346)--(2.015,-1.199)--(2.086,-1.027)--(2.157,-0.8297)--(2.227,-0.6099)--(2.298,-0.3696)--(2.369,-0.1114)--(2.439,0.1618)--(2.510,0.4468)--(2.581,0.7401)--(2.652,1.038)--(2.722,1.337)--(2.793,1.632)--(2.864,1.921)--(2.934,2.198)--(3.005,2.459)--(3.076,2.702)--(3.146,2.921)--(3.217,3.114)--(3.288,3.277)--(3.359,3.406)--(3.429,3.500)--(3.500,3.555);
-\draw (-3.298672286,-0.4207143333) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.199114858,-0.3210296667) node {$-\mathit{\pi}$};
+\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
 \draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.099557429,-0.4207143333) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.099557429,-0.4207143333) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
 \draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.199114858,-0.2785761667) node {$\mathit{\pi}$};
+\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
 \draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.298672286,-0.4207143333) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
 \draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.4331593333,-2.100000000) node {$ -3 $};
+\draw (-0.43316,-2.1000) node {$ -3 $};
 \draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.4331593333,-1.400000000) node {$ -2 $};
+\draw (-0.43316,-1.4000) node {$ -2 $};
 \draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.4331593333,-0.7000000000) node {$ -1 $};
+\draw (-0.43316,-0.70000) node {$ -1 $};
 \draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.2912498333,0.7000000000) node {$ 1 $};
+\draw (-0.29125,0.70000) node {$ 1 $};
 \draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.2912498333,1.400000000) node {$ 2 $};
+\draw (-0.29125,1.4000) node {$ 2 $};
 \draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.2912498333,2.100000000) node {$ 3 $};
+\draw (-0.29125,2.1000) node {$ 3 $};
 \draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.2912498333,2.800000000) node {$ 4 $};
+\draw (-0.29125,2.8000) node {$ 4 $};
 \draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.2912498333,3.500000000) node {$ 5 $};
+\draw (-0.29125,3.5000) node {$ 5 $};
 \draw [] (-0.100,3.50) -- (0.100,3.50);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall b/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
index bd5d82aaf..ae1ef50a0 100644
--- a/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
+++ b/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
@@ -100,8 +100,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.212388980,0) -- (9.924777961,0);
-\draw [,->,>=latex] (0,-3.445045352) -- (0,3.317012042);
+\draw [,->,>=latex] (-5.2124,0) -- (9.9248,0);
+\draw [,->,>=latex] (0,-3.4450) -- (0,3.3170);
 %DEFAULT
 
 \draw [color=black] (-4.712,0)--(-4.570,-0.1423)--(-4.427,-0.2817)--(-4.284,-0.4154)--(-4.141,-0.5406)--(-3.998,-0.6549)--(-3.856,-0.7558)--(-3.713,-0.8413)--(-3.570,-0.9096)--(-3.427,-0.9595)--(-3.284,-0.9898)--(-3.142,-1.000)--(-2.999,-0.9898)--(-2.856,-0.9595)--(-2.713,-0.9096)--(-2.570,-0.8413)--(-2.428,-0.7558)--(-2.285,-0.6549)--(-2.142,-0.5406)--(-1.999,-0.4154)--(-1.856,-0.2817)--(-1.714,-0.1423)--(-1.571,0)--(-1.428,0.1423)--(-1.285,0.2817)--(-1.142,0.4154)--(-0.9996,0.5406)--(-0.8568,0.6549)--(-0.7140,0.7558)--(-0.5712,0.8413)--(-0.4284,0.9096)--(-0.2856,0.9595)--(-0.1428,0.9898)--(0,1.000)--(0.1428,0.9898)--(0.2856,0.9595)--(0.4284,0.9096)--(0.5712,0.8413)--(0.7140,0.7558)--(0.8568,0.6549)--(0.9996,0.5406)--(1.142,0.4154)--(1.285,0.2817)--(1.428,0.1423)--(1.571,0)--(1.714,-0.1423)--(1.856,-0.2817)--(1.999,-0.4154)--(2.142,-0.5406)--(2.285,-0.6549)--(2.428,-0.7558)--(2.570,-0.8413)--(2.713,-0.9096)--(2.856,-0.9595)--(2.999,-0.9898)--(3.142,-1.000)--(3.284,-0.9898)--(3.427,-0.9595)--(3.570,-0.9096)--(3.713,-0.8413)--(3.856,-0.7558)--(3.998,-0.6549)--(4.141,-0.5406)--(4.284,-0.4154)--(4.427,-0.2817)--(4.570,-0.1423)--(4.712,0)--(4.855,0.1423)--(4.998,0.2817)--(5.141,0.4154)--(5.284,0.5406)--(5.426,0.6549)--(5.569,0.7558)--(5.712,0.8413)--(5.855,0.9096)--(5.998,0.9595)--(6.140,0.9898)--(6.283,1.000)--(6.426,0.9898)--(6.569,0.9595)--(6.712,0.9096)--(6.854,0.8413)--(6.997,0.7558)--(7.140,0.6549)--(7.283,0.5406)--(7.426,0.4154)--(7.568,0.2817)--(7.711,0.1423)--(7.854,0)--(7.997,-0.1423)--(8.140,-0.2817)--(8.282,-0.4154)--(8.425,-0.5406)--(8.568,-0.6549)--(8.711,-0.7558)--(8.854,-0.8413)--(8.996,-0.9096)--(9.139,-0.9595)--(9.282,-0.9898)--(9.425,-1.000);
@@ -109,25 +109,25 @@
 \draw [color=red,style=dashed] (-3.863,2.817)--(-3.785,2.389)--(-3.707,1.999)--(-3.629,1.644)--(-3.552,1.323)--(-3.474,1.034)--(-3.396,0.7753)--(-3.318,0.5458)--(-3.240,0.3435)--(-3.162,0.1670)--(-3.085,0.01472)--(-3.007,-0.1148)--(-2.929,-0.2229)--(-2.851,-0.3112)--(-2.773,-0.3809)--(-2.695,-0.4333)--(-2.618,-0.4698)--(-2.540,-0.4916)--(-2.462,-0.4998)--(-2.384,-0.4958)--(-2.306,-0.4807)--(-2.228,-0.4554)--(-2.151,-0.4212)--(-2.073,-0.3791)--(-1.995,-0.3299)--(-1.917,-0.2748)--(-1.839,-0.2146)--(-1.761,-0.1502)--(-1.684,-0.08246)--(-1.606,-0.01219)--(-1.528,0.05984)--(-1.450,0.1329)--(-1.372,0.2062)--(-1.294,0.2792)--(-1.217,0.3513)--(-1.139,0.4217)--(-1.061,0.4900)--(-0.9830,0.5557)--(-0.9052,0.6183)--(-0.8274,0.6772)--(-0.7495,0.7322)--(-0.6717,0.7829)--(-0.5939,0.8288)--(-0.5160,0.8698)--(-0.4382,0.9055)--(-0.3604,0.9358)--(-0.2825,0.9604)--(-0.2047,0.9791)--(-0.1269,0.9920)--(-0.04903,0.9988)--(0.02881,0.9996)--(0.1066,0.9943)--(0.1845,0.9830)--(0.2623,0.9658)--(0.3401,0.9427)--(0.4180,0.9139)--(0.4958,0.8796)--(0.5736,0.8400)--(0.6515,0.7953)--(0.7293,0.7458)--(0.8072,0.6919)--(0.8850,0.6340)--(0.9628,0.5723)--(1.041,0.5074)--(1.118,0.4397)--(1.196,0.3697)--(1.274,0.2981)--(1.352,0.2253)--(1.430,0.1519)--(1.508,0.07875)--(1.586,0.006392)--(1.663,-0.06441)--(1.741,-0.1329)--(1.819,-0.1982)--(1.897,-0.2596)--(1.975,-0.3161)--(2.053,-0.3669)--(2.130,-0.4110)--(2.208,-0.4474)--(2.286,-0.4750)--(2.364,-0.4929)--(2.442,-0.4999)--(2.520,-0.4950)--(2.597,-0.4768)--(2.675,-0.4443)--(2.753,-0.3961)--(2.831,-0.3310)--(2.909,-0.2477)--(2.987,-0.1449)--(3.064,-0.02106)--(3.142,0.1252)--(3.220,0.2952)--(3.298,0.4906)--(3.376,0.7129)--(3.454,0.9637)--(3.531,1.245)--(3.609,1.557)--(3.687,1.903)--(3.765,2.284)--(3.843,2.702);
 
 \draw [color=blue,style=dashed] (-1.072,-2.945)--(-0.9794,-2.436)--(-0.8868,-1.974)--(-0.7943,-1.558)--(-0.7017,-1.185)--(-0.6092,-0.8526)--(-0.5167,-0.5589)--(-0.4241,-0.3015)--(-0.3316,-0.07832)--(-0.2390,0.1128)--(-0.1465,0.2738)--(-0.05393,0.4068)--(0.03861,0.5136)--(0.1312,0.5962)--(0.2237,0.6564)--(0.3162,0.6960)--(0.4088,0.7166)--(0.5013,0.7200)--(0.5939,0.7079)--(0.6864,0.6817)--(0.7790,0.6430)--(0.8715,0.5932)--(0.9641,0.5339)--(1.057,0.4663)--(1.149,0.3917)--(1.242,0.3115)--(1.334,0.2269)--(1.427,0.1389)--(1.519,0.04875)--(1.612,-0.04255)--(1.704,-0.1340)--(1.797,-0.2246)--(1.889,-0.3135)--(1.982,-0.3998)--(2.075,-0.4828)--(2.167,-0.5616)--(2.260,-0.6357)--(2.352,-0.7043)--(2.445,-0.7669)--(2.537,-0.8229)--(2.630,-0.8719)--(2.722,-0.9134)--(2.815,-0.9470)--(2.907,-0.9725)--(3.000,-0.9894)--(3.093,-0.9977)--(3.185,-0.9971)--(3.278,-0.9875)--(3.370,-0.9689)--(3.463,-0.9411)--(3.555,-0.9044)--(3.648,-0.8586)--(3.740,-0.8041)--(3.833,-0.7409)--(3.925,-0.6693)--(4.018,-0.5897)--(4.111,-0.5022)--(4.203,-0.4074)--(4.296,-0.3058)--(4.388,-0.1977)--(4.481,-0.08370)--(4.573,0.03551)--(4.666,0.1593)--(4.758,0.2870)--(4.851,0.4178)--(4.943,0.5509)--(5.036,0.6855)--(5.129,0.8206)--(5.221,0.9553)--(5.314,1.089)--(5.406,1.219)--(5.499,1.346)--(5.591,1.469)--(5.684,1.585)--(5.776,1.694)--(5.869,1.794)--(5.961,1.884)--(6.054,1.963)--(6.146,2.028)--(6.239,2.080)--(6.332,2.115)--(6.424,2.132)--(6.517,2.130)--(6.609,2.107)--(6.702,2.061)--(6.794,1.990)--(6.887,1.893)--(6.979,1.766)--(7.072,1.609)--(7.164,1.420)--(7.257,1.195)--(7.350,0.9335)--(7.442,0.6328)--(7.535,0.2905)--(7.627,-0.09545)--(7.720,-0.5276)--(7.812,-1.008)--(7.905,-1.540)--(7.997,-2.125)--(8.090,-2.767);
-\draw (-3.141592654,-0.3210296667) node {$ -\pi $};
+\draw (-3.1416,-0.32103) node {$ -\pi $};
 \draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (3.141592654,-0.2785761667) node {$ \pi $};
+\draw (3.1416,-0.27858) node {$ \pi $};
 \draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (6.283185307,-0.3149246667) node {$ 2 \, \pi $};
+\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
 \draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (9.424777961,-0.3149246667) node {$ 3 \, \pi $};
+\draw (9.4248,-0.31492) node {$ 3 \, \pi $};
 \draw [] (9.42,-0.100) -- (9.42,0.100);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_WUYooCISzeB.pstricks.recall b/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
index bf8eb2063..b138a1fb5 100644
--- a/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
+++ b/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
@@ -61,28 +61,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.9918454937) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.99185) -- (0,4.0276);
 %DEFAULT
 
 \draw [color=red] (-0.8400,2.700)--(-0.8013,2.668)--(-0.7626,2.636)--(-0.7239,2.603)--(-0.6852,2.571)--(-0.6466,2.539)--(-0.6079,2.507)--(-0.5692,2.474)--(-0.5305,2.442)--(-0.4918,2.410)--(-0.4531,2.378)--(-0.4144,2.345)--(-0.3757,2.313)--(-0.3370,2.281)--(-0.2984,2.249)--(-0.2597,2.216)--(-0.2210,2.184)--(-0.1823,2.152)--(-0.1436,2.120)--(-0.1049,2.087)--(-0.06622,2.055)--(-0.02753,2.023)--(0.01116,1.991)--(0.04985,1.958)--(0.08854,1.926)--(0.1272,1.894)--(0.1659,1.862)--(0.2046,1.829)--(0.2433,1.797)--(0.2820,1.765)--(0.3207,1.733)--(0.3594,1.701)--(0.3980,1.668)--(0.4367,1.636)--(0.4754,1.604)--(0.5141,1.572)--(0.5528,1.539)--(0.5915,1.507)--(0.6302,1.475)--(0.6689,1.443)--(0.7076,1.410)--(0.7463,1.378)--(0.7849,1.346)--(0.8236,1.314)--(0.8623,1.281)--(0.9010,1.249)--(0.9397,1.217)--(0.9784,1.185)--(1.017,1.152)--(1.056,1.120)--(1.094,1.088)--(1.133,1.056)--(1.172,1.023)--(1.211,0.9912)--(1.249,0.9590)--(1.288,0.9268)--(1.327,0.8945)--(1.365,0.8623)--(1.404,0.8300)--(1.443,0.7978)--(1.481,0.7655)--(1.520,0.7333)--(1.559,0.7011)--(1.597,0.6688)--(1.636,0.6366)--(1.675,0.6043)--(1.713,0.5721)--(1.752,0.5399)--(1.791,0.5076)--(1.830,0.4754)--(1.868,0.4431)--(1.907,0.4109)--(1.946,0.3787)--(1.984,0.3464)--(2.023,0.3142)--(2.062,0.2819)--(2.100,0.2497)--(2.139,0.2175)--(2.178,0.1852)--(2.216,0.1530)--(2.255,0.1207)--(2.294,0.08849)--(2.332,0.05625)--(2.371,0.02401)--(2.410,-0.008233)--(2.449,-0.04047)--(2.487,-0.07271)--(2.526,-0.1050)--(2.565,-0.1372)--(2.603,-0.1694)--(2.642,-0.2017)--(2.681,-0.2339)--(2.719,-0.2662)--(2.758,-0.2984)--(2.797,-0.3306)--(2.835,-0.3629)--(2.874,-0.3951)--(2.913,-0.4274)--(2.952,-0.4596)--(2.990,-0.4918);
 
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-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -141,28 +141,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
 %DEFAULT
 
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 \draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -221,28 +221,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.9831816482) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.98318) -- (0,4.0276);
 %DEFAULT
 
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 \draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -301,28 +301,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.340000000,0) -- (4.100000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,4.027599983);
+\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
 %DEFAULT
 
 \draw [color=red] (-0.8400,3.499)--(-0.8086,3.401)--(-0.7773,3.307)--(-0.7459,3.218)--(-0.7145,3.133)--(-0.6831,3.051)--(-0.6517,2.974)--(-0.6204,2.900)--(-0.5890,2.830)--(-0.5576,2.763)--(-0.5263,2.699)--(-0.4949,2.639)--(-0.4635,2.581)--(-0.4321,2.527)--(-0.4008,2.474)--(-0.3694,2.425)--(-0.3380,2.378)--(-0.3066,2.333)--(-0.2753,2.291)--(-0.2439,2.251)--(-0.2125,2.212)--(-0.1811,2.176)--(-0.1498,2.142)--(-0.1184,2.109)--(-0.08701,2.078)--(-0.05563,2.049)--(-0.02426,2.021)--(0.007117,1.994)--(0.03849,1.969)--(0.06987,1.945)--(0.1012,1.922)--(0.1326,1.901)--(0.1640,1.880)--(0.1954,1.861)--(0.2267,1.843)--(0.2581,1.825)--(0.2895,1.809)--(0.3209,1.793)--(0.3522,1.778)--(0.3836,1.764)--(0.4150,1.751)--(0.4464,1.738)--(0.4777,1.727)--(0.5091,1.716)--(0.5405,1.705)--(0.5719,1.696)--(0.6032,1.687)--(0.6346,1.678)--(0.6660,1.671)--(0.6974,1.664)--(0.7287,1.658)--(0.7601,1.652)--(0.7915,1.647)--(0.8229,1.643)--(0.8542,1.640)--(0.8856,1.638)--(0.9170,1.636)--(0.9484,1.635)--(0.9797,1.635)--(1.011,1.637)--(1.042,1.639)--(1.074,1.642)--(1.105,1.646)--(1.137,1.652)--(1.168,1.658)--(1.199,1.666)--(1.231,1.676)--(1.262,1.687)--(1.293,1.699)--(1.325,1.713)--(1.356,1.729)--(1.388,1.746)--(1.419,1.765)--(1.450,1.786)--(1.482,1.810)--(1.513,1.835)--(1.544,1.863)--(1.576,1.893)--(1.607,1.925)--(1.639,1.961)--(1.670,1.998)--(1.701,2.039)--(1.733,2.083)--(1.764,2.130)--(1.795,2.180)--(1.827,2.234)--(1.858,2.291)--(1.890,2.351)--(1.921,2.416)--(1.952,2.485)--(1.984,2.557)--(2.015,2.634)--(2.046,2.716)--(2.078,2.802)--(2.109,2.893)--(2.141,2.989)--(2.172,3.090)--(2.203,3.197)--(2.235,3.309)--(2.266,3.427);
 
 \draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.200000000,-0.3298256667) node {$ -1 $};
+\draw (-1.2000,-0.32983) node {$ -1 $};
 \draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.200000000,-0.3149246667) node {$ 1 $};
+\draw (1.2000,-0.31492) node {$ 1 $};
 \draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.400000000,-0.3149246667) node {$ 2 $};
+\draw (2.4000,-0.31492) node {$ 2 $};
 \draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.600000000,-0.3149246667) node {$ 3 $};
+\draw (3.6000,-0.31492) node {$ 3 $};
 \draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 2 $};
+\draw (-0.29125,1.0000) node {$ 2 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 4 $};
+\draw (-0.29125,2.0000) node {$ 4 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 6 $};
+\draw (-0.29125,3.0000) node {$ 6 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.2912498333,4.000000000) node {$ 8 $};
+\draw (-0.29125,4.0000) node {$ 8 $};
 \draw [] (-0.100,4.00) -- (0.100,4.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall b/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall
index ed308a35a..766d698d6 100644
--- a/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall
+++ b/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall
@@ -144,8 +144,8 @@
 \draw [color=gray,style=solid] (-5.00,2.00) -- (5.00,2.00);
 \draw [color=gray,style=solid] (-5.00,3.00) -- (5.00,3.00);
 %AXES
-\draw [,->,>=latex] (-5.500000000,0) -- (5.500000000,0);
-\draw [,->,>=latex] (0,-4.500000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
+\draw [,->,>=latex] (0,-4.5000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (-5.000,-4.000)--(-4.960,-3.960)--(-4.919,-3.919)--(-4.879,-3.879)--(-4.838,-3.838)--(-4.798,-3.798)--(-4.758,-3.758)--(-4.717,-3.717)--(-4.677,-3.677)--(-4.636,-3.636)--(-4.596,-3.596)--(-4.556,-3.556)--(-4.515,-3.515)--(-4.475,-3.475)--(-4.434,-3.434)--(-4.394,-3.394)--(-4.354,-3.354)--(-4.313,-3.313)--(-4.273,-3.273)--(-4.232,-3.232)--(-4.192,-3.192)--(-4.151,-3.152)--(-4.111,-3.111)--(-4.071,-3.071)--(-4.030,-3.030)--(-3.990,-2.990)--(-3.949,-2.949)--(-3.909,-2.909)--(-3.869,-2.869)--(-3.828,-2.828)--(-3.788,-2.788)--(-3.747,-2.747)--(-3.707,-2.707)--(-3.667,-2.667)--(-3.626,-2.626)--(-3.586,-2.586)--(-3.545,-2.545)--(-3.505,-2.505)--(-3.465,-2.465)--(-3.424,-2.424)--(-3.384,-2.384)--(-3.343,-2.343)--(-3.303,-2.303)--(-3.263,-2.263)--(-3.222,-2.222)--(-3.182,-2.182)--(-3.141,-2.141)--(-3.101,-2.101)--(-3.061,-2.061)--(-3.020,-2.020)--(-2.980,-1.980)--(-2.939,-1.939)--(-2.899,-1.899)--(-2.859,-1.859)--(-2.818,-1.818)--(-2.778,-1.778)--(-2.737,-1.737)--(-2.697,-1.697)--(-2.657,-1.657)--(-2.616,-1.616)--(-2.576,-1.576)--(-2.535,-1.535)--(-2.495,-1.495)--(-2.455,-1.455)--(-2.414,-1.414)--(-2.374,-1.374)--(-2.333,-1.333)--(-2.293,-1.293)--(-2.253,-1.253)--(-2.212,-1.212)--(-2.172,-1.172)--(-2.131,-1.131)--(-2.091,-1.091)--(-2.051,-1.051)--(-2.010,-1.010)--(-1.970,-0.9697)--(-1.929,-0.9293)--(-1.889,-0.8889)--(-1.848,-0.8485)--(-1.808,-0.8081)--(-1.768,-0.7677)--(-1.727,-0.7273)--(-1.687,-0.6869)--(-1.646,-0.6465)--(-1.606,-0.6061)--(-1.566,-0.5657)--(-1.525,-0.5253)--(-1.485,-0.4848)--(-1.444,-0.4444)--(-1.404,-0.4040)--(-1.364,-0.3636)--(-1.323,-0.3232)--(-1.283,-0.2828)--(-1.242,-0.2424)--(-1.202,-0.2020)--(-1.162,-0.1616)--(-1.121,-0.1212)--(-1.081,-0.08081)--(-1.040,-0.04040)--(-1.000,0);
@@ -153,46 +153,46 @@
 \draw [color=blue] (-1.000,-2.000)--(-0.9798,-1.960)--(-0.9596,-1.919)--(-0.9394,-1.879)--(-0.9192,-1.838)--(-0.8990,-1.798)--(-0.8788,-1.758)--(-0.8586,-1.717)--(-0.8384,-1.677)--(-0.8182,-1.636)--(-0.7980,-1.596)--(-0.7778,-1.556)--(-0.7576,-1.515)--(-0.7374,-1.475)--(-0.7172,-1.434)--(-0.6970,-1.394)--(-0.6768,-1.354)--(-0.6566,-1.313)--(-0.6364,-1.273)--(-0.6162,-1.232)--(-0.5960,-1.192)--(-0.5758,-1.152)--(-0.5556,-1.111)--(-0.5354,-1.071)--(-0.5152,-1.030)--(-0.4949,-0.9899)--(-0.4747,-0.9495)--(-0.4545,-0.9091)--(-0.4343,-0.8687)--(-0.4141,-0.8283)--(-0.3939,-0.7879)--(-0.3737,-0.7475)--(-0.3535,-0.7071)--(-0.3333,-0.6667)--(-0.3131,-0.6263)--(-0.2929,-0.5859)--(-0.2727,-0.5455)--(-0.2525,-0.5051)--(-0.2323,-0.4646)--(-0.2121,-0.4242)--(-0.1919,-0.3838)--(-0.1717,-0.3434)--(-0.1515,-0.3030)--(-0.1313,-0.2626)--(-0.1111,-0.2222)--(-0.09091,-0.1818)--(-0.07071,-0.1414)--(-0.05051,-0.1010)--(-0.03030,-0.06061)--(-0.01010,-0.02020)--(0.01010,0.02020)--(0.03030,0.06061)--(0.05051,0.1010)--(0.07071,0.1414)--(0.09091,0.1818)--(0.1111,0.2222)--(0.1313,0.2626)--(0.1515,0.3030)--(0.1717,0.3434)--(0.1919,0.3838)--(0.2121,0.4242)--(0.2323,0.4646)--(0.2525,0.5051)--(0.2727,0.5455)--(0.2929,0.5859)--(0.3131,0.6263)--(0.3333,0.6667)--(0.3535,0.7071)--(0.3737,0.7475)--(0.3939,0.7879)--(0.4141,0.8283)--(0.4343,0.8687)--(0.4545,0.9091)--(0.4747,0.9495)--(0.4949,0.9899)--(0.5152,1.030)--(0.5354,1.071)--(0.5556,1.111)--(0.5758,1.152)--(0.5960,1.192)--(0.6162,1.232)--(0.6364,1.273)--(0.6566,1.313)--(0.6768,1.354)--(0.6970,1.394)--(0.7172,1.434)--(0.7374,1.475)--(0.7576,1.515)--(0.7778,1.556)--(0.7980,1.596)--(0.8182,1.636)--(0.8384,1.677)--(0.8586,1.717)--(0.8788,1.758)--(0.8990,1.798)--(0.9192,1.838)--(0.9394,1.879)--(0.9596,1.919)--(0.9798,1.960)--(1.000,2.000);
 
 \draw [color=blue] (1.000,3.000)--(1.040,2.960)--(1.081,2.919)--(1.121,2.879)--(1.162,2.838)--(1.202,2.798)--(1.242,2.758)--(1.283,2.717)--(1.323,2.677)--(1.364,2.636)--(1.404,2.596)--(1.444,2.556)--(1.485,2.515)--(1.525,2.475)--(1.566,2.434)--(1.606,2.394)--(1.646,2.354)--(1.687,2.313)--(1.727,2.273)--(1.768,2.232)--(1.808,2.192)--(1.848,2.152)--(1.889,2.111)--(1.929,2.071)--(1.970,2.030)--(2.010,1.990)--(2.051,1.949)--(2.091,1.909)--(2.131,1.869)--(2.172,1.828)--(2.212,1.788)--(2.253,1.747)--(2.293,1.707)--(2.333,1.667)--(2.374,1.626)--(2.414,1.586)--(2.455,1.545)--(2.495,1.505)--(2.535,1.465)--(2.576,1.424)--(2.616,1.384)--(2.657,1.343)--(2.697,1.303)--(2.737,1.263)--(2.778,1.222)--(2.818,1.182)--(2.859,1.141)--(2.899,1.101)--(2.939,1.061)--(2.980,1.020)--(3.020,0.9798)--(3.061,0.9394)--(3.101,0.8990)--(3.141,0.8586)--(3.182,0.8182)--(3.222,0.7778)--(3.263,0.7374)--(3.303,0.6970)--(3.343,0.6566)--(3.384,0.6162)--(3.424,0.5758)--(3.465,0.5354)--(3.505,0.4949)--(3.545,0.4545)--(3.586,0.4141)--(3.626,0.3737)--(3.667,0.3333)--(3.707,0.2929)--(3.747,0.2525)--(3.788,0.2121)--(3.828,0.1717)--(3.869,0.1313)--(3.909,0.09091)--(3.949,0.05051)--(3.990,0.01010)--(4.030,-0.03030)--(4.071,-0.07071)--(4.111,-0.1111)--(4.151,-0.1515)--(4.192,-0.1919)--(4.232,-0.2323)--(4.273,-0.2727)--(4.313,-0.3131)--(4.354,-0.3535)--(4.394,-0.3939)--(4.434,-0.4343)--(4.475,-0.4747)--(4.515,-0.5152)--(4.556,-0.5556)--(4.596,-0.5960)--(4.636,-0.6364)--(4.677,-0.6768)--(4.717,-0.7172)--(4.758,-0.7576)--(4.798,-0.7980)--(4.838,-0.8384)--(4.879,-0.8788)--(4.919,-0.9192)--(4.960,-0.9596)--(5.000,-1.000);
-\draw [color=blue]  (-1.000000000,-2.000000000) node [rotate=0] {$\bullet$};
-\draw [color=blue]  (1.000000000,2.000000000) node [rotate=0] {$\bullet$};
-\draw []  (0.5000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (0.8451696896,0.7064958104) node {\( Z_1\)};
-\draw []  (3.000000000,1.000000000) node [rotate=0] {$\bullet$};
-\draw (3.345169690,1.293504190) node {\( Z_2\)};
+\draw [color=blue]  (-1.0000,-2.0000) node [rotate=0] {$\bullet$};
+\draw [color=blue]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.50000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.84517,0.70650) node {\( Z_1\)};
+\draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (3.3452,1.2935) node {\( Z_2\)};
 
-\draw (-5.000000000,-0.3298256667) node {$ -5 $};
+\draw (-5.0000,-0.32983) node {$ -5 $};
 \draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -4 $};
+\draw (-4.0000,-0.32983) node {$ -4 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.000000000,-0.3298256667) node {$ -3 $};
+\draw (-3.0000,-0.32983) node {$ -3 $};
 \draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.4331593333,-4.000000000) node {$ -4 $};
+\draw (-0.43316,-4.0000) node {$ -4 $};
 \draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.4331593333,-3.000000000) node {$ -3 $};
+\draw (-0.43316,-3.0000) node {$ -3 $};
 \draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.2912498333,3.000000000) node {$ 3 $};
+\draw (-0.29125,3.0000) node {$ 3 $};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall b/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall
index 78ec5fba6..824be3771 100644
--- a/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall
+++ b/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall
@@ -79,19 +79,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
 %DEFAULT
 
 \draw [color=blue] (0,0)--(0.02020,0.03030)--(0.04040,0.06061)--(0.06061,0.09091)--(0.08081,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5152)--(0.3636,0.5455)--(0.3838,0.5758)--(0.4040,0.6061)--(0.4242,0.6364)--(0.4444,0.6667)--(0.4646,0.6970)--(0.4848,0.7273)--(0.5051,0.7576)--(0.5253,0.7879)--(0.5455,0.8182)--(0.5657,0.8485)--(0.5859,0.8788)--(0.6061,0.9091)--(0.6263,0.9394)--(0.6465,0.9697)--(0.6667,1.000)--(0.6869,1.030)--(0.7071,1.061)--(0.7273,1.091)--(0.7475,1.121)--(0.7677,1.152)--(0.7879,1.182)--(0.8081,1.212)--(0.8283,1.242)--(0.8485,1.273)--(0.8687,1.303)--(0.8889,1.333)--(0.9091,1.364)--(0.9293,1.394)--(0.9495,1.424)--(0.9697,1.455)--(0.9899,1.485)--(1.010,1.515)--(1.030,1.545)--(1.051,1.576)--(1.071,1.606)--(1.091,1.636)--(1.111,1.667)--(1.131,1.697)--(1.152,1.727)--(1.172,1.758)--(1.192,1.788)--(1.212,1.818)--(1.232,1.848)--(1.253,1.879)--(1.273,1.909)--(1.293,1.939)--(1.313,1.970)--(1.333,2.000)--(1.354,2.030)--(1.374,2.061)--(1.394,2.091)--(1.414,2.121)--(1.434,2.152)--(1.455,2.182)--(1.475,2.212)--(1.495,2.242)--(1.515,2.273)--(1.535,2.303)--(1.556,2.333)--(1.576,2.364)--(1.596,2.394)--(1.616,2.424)--(1.636,2.455)--(1.657,2.485)--(1.677,2.515)--(1.697,2.545)--(1.717,2.576)--(1.737,2.606)--(1.758,2.636)--(1.778,2.667)--(1.798,2.697)--(1.818,2.727)--(1.838,2.758)--(1.859,2.788)--(1.879,2.818)--(1.899,2.848)--(1.919,2.879)--(1.939,2.909)--(1.960,2.939)--(1.980,2.970)--(2.000,3.000);
-\draw []  (2.000000000,3.000000000) node [rotate=0] {$\bullet$};
-\draw (2.538859511,3.253165678) node {$(R,h)$};
-\draw (0.7236718444,0.3359520608) node {$\alpha$};
+\draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw (2.5389,3.2532) node {$(R,h)$};
+\draw (0.72367,0.33595) node {$\alpha$};
 
 \draw [color=red] (0.500,0)--(0.500,0.00496)--(0.500,0.00993)--(0.500,0.0149)--(0.500,0.0198)--(0.499,0.0248)--(0.499,0.0298)--(0.499,0.0347)--(0.498,0.0397)--(0.498,0.0446)--(0.498,0.0496)--(0.497,0.0545)--(0.496,0.0594)--(0.496,0.0643)--(0.495,0.0693)--(0.494,0.0742)--(0.494,0.0791)--(0.493,0.0840)--(0.492,0.0889)--(0.491,0.0938)--(0.490,0.0986)--(0.489,0.103)--(0.488,0.108)--(0.487,0.113)--(0.486,0.118)--(0.485,0.123)--(0.483,0.128)--(0.482,0.132)--(0.481,0.137)--(0.479,0.142)--(0.478,0.147)--(0.477,0.151)--(0.475,0.156)--(0.473,0.161)--(0.472,0.166)--(0.470,0.170)--(0.468,0.175)--(0.467,0.180)--(0.465,0.184)--(0.463,0.189)--(0.461,0.193)--(0.459,0.198)--(0.457,0.202)--(0.455,0.207)--(0.453,0.212)--(0.451,0.216)--(0.449,0.220)--(0.447,0.225)--(0.444,0.229)--(0.442,0.234)--(0.440,0.238)--(0.437,0.242)--(0.435,0.247)--(0.432,0.251)--(0.430,0.255)--(0.427,0.260)--(0.425,0.264)--(0.422,0.268)--(0.419,0.272)--(0.417,0.276)--(0.414,0.281)--(0.411,0.285)--(0.408,0.289)--(0.405,0.293)--(0.402,0.297)--(0.399,0.301)--(0.396,0.305)--(0.393,0.309)--(0.390,0.312)--(0.387,0.316)--(0.384,0.320)--(0.381,0.324)--(0.378,0.328)--(0.374,0.331)--(0.371,0.335)--(0.368,0.339)--(0.364,0.342)--(0.361,0.346)--(0.357,0.350)--(0.354,0.353)--(0.350,0.357)--(0.347,0.360)--(0.343,0.364)--(0.340,0.367)--(0.336,0.370)--(0.332,0.374)--(0.329,0.377)--(0.325,0.380)--(0.321,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.306,0.396)--(0.302,0.399)--(0.298,0.402)--(0.294,0.405)--(0.290,0.408)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416);
-\draw (2.000000000,-0.3257195000) node {$\mathit{R}$};
+\draw (2.0000,-0.32572) node {$\mathit{R}$};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.3027346667,3.000000000) node {$\mathit{h}$};
+\draw (-0.30273,3.0000) node {$\mathit{h}$};
 \draw [] (-0.100,3.00) -- (0.100,3.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall b/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall
index 2d02785ef..0ddf6c696 100644
--- a/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall
+++ b/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall
@@ -65,230 +65,230 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.000000000,-4.000000000) -- (-4.235702260,-3.764297740);
-\draw [,->,>=latex] (-4.000000000,-3.428571429) -- (-4.216930458,-3.175485894);
-\draw [,->,>=latex] (-4.000000000,-2.857142857) -- (-4.193746065,-2.585898367);
-\draw [,->,>=latex] (-4.000000000,-2.285714286) -- (-4.165379646,-1.996299905);
-\draw [,->,>=latex] (-4.000000000,-1.714285714) -- (-4.131306433,-1.407904038);
-\draw [,->,>=latex] (-4.000000000,-1.142857143) -- (-4.091573709,-0.8223491603);
-\draw [,->,>=latex] (-4.000000000,-0.5714285714) -- (-4.047140452,-0.2414454069);
-\draw [,->,>=latex] (-4.000000000,0) -- (-4.000000000,0.3333333333);
-\draw [,->,>=latex] (-4.000000000,0.5714285714) -- (-3.952859548,0.9014117360);
-\draw [,->,>=latex] (-4.000000000,1.142857143) -- (-3.908426291,1.463365125);
-\draw [,->,>=latex] (-4.000000000,1.714285714) -- (-3.868693567,2.020667391);
-\draw [,->,>=latex] (-4.000000000,2.285714286) -- (-3.834620354,2.575128666);
-\draw [,->,>=latex] (-4.000000000,2.857142857) -- (-3.806253935,3.128387348);
-\draw [,->,>=latex] (-4.000000000,3.428571429) -- (-3.783069542,3.681656963);
-\draw [,->,>=latex] (-4.000000000,4.000000000) -- (-3.764297740,4.235702260);
-\draw [,->,>=latex] (-3.428571429,-4.000000000) -- (-3.681656963,-3.783069542);
-\draw [,->,>=latex] (-3.428571429,-3.428571429) -- (-3.664273689,-3.192869168);
-\draw [,->,>=latex] (-3.428571429,-2.857142857) -- (-3.641966228,-2.601069097);
-\draw [,->,>=latex] (-3.428571429,-2.285714286) -- (-3.613471494,-2.008364188);
-\draw [,->,>=latex] (-3.428571429,-1.714285714) -- (-3.577642627,-1.416143317);
-\draw [,->,>=latex] (-3.428571429,-1.142857143) -- (-3.533980684,-0.8266293768);
-\draw [,->,>=latex] (-3.428571429,-0.5714285714) -- (-3.483371091,-0.2426305968);
-\draw [,->,>=latex] (-3.428571429,0) -- (-3.428571429,0.3333333333);
-\draw [,->,>=latex] (-3.428571429,0.5714285714) -- (-3.373771766,0.9002265460);
-\draw [,->,>=latex] (-3.428571429,1.142857143) -- (-3.323162173,1.459084909);
-\draw [,->,>=latex] (-3.428571429,1.714285714) -- (-3.279500230,2.012428111);
-\draw [,->,>=latex] (-3.428571429,2.285714286) -- (-3.243671363,2.563064384);
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-\draw [,->,>=latex] (-3.428571429,3.428571429) -- (-3.192869168,3.664273689);
-\draw [,->,>=latex] (-3.428571429,4.000000000) -- (-3.175485894,4.216930458);
-\draw [,->,>=latex] (-2.857142857,-4.000000000) -- (-3.128387348,-3.806253935);
-\draw [,->,>=latex] (-2.857142857,-3.428571429) -- (-3.113216617,-3.215176629);
-\draw [,->,>=latex] (-2.857142857,-2.857142857) -- (-3.092845118,-2.621440597);
-\draw [,->,>=latex] (-2.857142857,-2.285714286) -- (-3.065374540,-2.025424683);
-\draw [,->,>=latex] (-2.857142857,-1.714285714) -- (-3.028641442,-1.428454739);
-\draw [,->,>=latex] (-2.857142857,-1.142857143) -- (-2.980939749,-0.8333649126);
-\draw [,->,>=latex] (-2.857142857,-0.5714285714) -- (-2.922514902,-0.2445683462);
-\draw [,->,>=latex] (-2.857142857,0) -- (-2.857142857,0.3333333333);
-\draw [,->,>=latex] (-2.857142857,0.5714285714) -- (-2.791770812,0.8982887967);
-\draw [,->,>=latex] (-2.857142857,1.142857143) -- (-2.733345965,1.452349373);
-\draw [,->,>=latex] (-2.857142857,1.714285714) -- (-2.685644272,2.000116690);
-\draw [,->,>=latex] (-2.857142857,2.285714286) -- (-2.648911175,2.546003889);
-\draw [,->,>=latex] (-2.857142857,2.857142857) -- (-2.621440597,3.092845118);
-\draw [,->,>=latex] (-2.857142857,3.428571429) -- (-2.601069097,3.641966228);
-\draw [,->,>=latex] (-2.857142857,4.000000000) -- (-2.585898367,4.193746065);
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-\draw [,->,>=latex] (-2.285714286,-3.428571429) -- (-2.563064384,-3.243671363);
-\draw [,->,>=latex] (-2.285714286,-2.857142857) -- (-2.546003889,-2.648911175);
-\draw [,->,>=latex] (-2.285714286,-2.285714286) -- (-2.521416546,-2.050012025);
-\draw [,->,>=latex] (-2.285714286,-1.714285714) -- (-2.485714286,-1.447619048);
-\draw [,->,>=latex] (-2.285714286,-1.142857143) -- (-2.434785484,-0.8447147459);
-\draw [,->,>=latex] (-2.285714286,-0.5714285714) -- (-2.366559494,-0.2480477380);
-\draw [,->,>=latex] (-2.285714286,0) -- (-2.285714286,0.3333333333);
-\draw [,->,>=latex] (-2.285714286,0.5714285714) -- (-2.204869077,0.8948094048);
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-\draw [,->,>=latex] (-2.285714286,2.285714286) -- (-2.050012025,2.521416546);
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-\draw [,->,>=latex] (-1.714285714,-3.428571429) -- (-2.012428111,-3.279500230);
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-\draw [,->,>=latex] (-1.714285714,1.714285714) -- (-1.478583454,1.949987975);
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-\draw [,->,>=latex] (-1.714285714,4.000000000) -- (-1.407904038,4.131306433);
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-\draw [,->,>=latex] (-1.142857143,1.714285714) -- (-0.8655070447,1.899185780);
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-\draw [,->,>=latex] (-0.5714285714,-0.5714285714) -- (-0.8071308318,-0.3357263110);
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-\draw [,->,>=latex] (-0.5714285714,0.5714285714) -- (-0.3357263110,0.8071308318);
-\draw [,->,>=latex] (-0.5714285714,1.142857143) -- (-0.2732861744,1.291928341);
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+\draw [,->,>=latex] (3.4286,1.1429) -- (3.5340,0.82663);
+\draw [,->,>=latex] (3.4286,1.7143) -- (3.5776,1.4161);
+\draw [,->,>=latex] (3.4286,2.2857) -- (3.6135,2.0084);
+\draw [,->,>=latex] (3.4286,2.8571) -- (3.6420,2.6011);
+\draw [,->,>=latex] (3.4286,3.4286) -- (3.6643,3.1929);
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+\draw [,->,>=latex] (4.0000,-4.0000) -- (3.7643,-4.2357);
+\draw [,->,>=latex] (4.0000,-3.4286) -- (3.7831,-3.6817);
+\draw [,->,>=latex] (4.0000,-2.8571) -- (3.8063,-3.1284);
+\draw [,->,>=latex] (4.0000,-2.2857) -- (3.8346,-2.5751);
+\draw [,->,>=latex] (4.0000,-1.7143) -- (3.8687,-2.0207);
+\draw [,->,>=latex] (4.0000,-1.1429) -- (3.9084,-1.4634);
+\draw [,->,>=latex] (4.0000,-0.57143) -- (3.9529,-0.90141);
+\draw [,->,>=latex] (4.0000,0) -- (4.0000,-0.33333);
+\draw [,->,>=latex] (4.0000,0.57143) -- (4.0471,0.24145);
+\draw [,->,>=latex] (4.0000,1.1429) -- (4.0916,0.82235);
+\draw [,->,>=latex] (4.0000,1.7143) -- (4.1313,1.4079);
+\draw [,->,>=latex] (4.0000,2.2857) -- (4.1654,1.9963);
+\draw [,->,>=latex] (4.0000,2.8571) -- (4.1937,2.5859);
+\draw [,->,>=latex] (4.0000,3.4286) -- (4.2169,3.1755);
+\draw [,->,>=latex] (4.0000,4.0000) -- (4.2357,3.7643);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_YWxOAkh.pstricks.recall b/src_phystricks/Fig_YWxOAkh.pstricks.recall
index 8c87b74e9..78d8f53a2 100644
--- a/src_phystricks/Fig_YWxOAkh.pstricks.recall
+++ b/src_phystricks/Fig_YWxOAkh.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-1.499999910) -- (0,1.499998029);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
 %DEFAULT
 
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-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall b/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall
index 74061cebc..6b17f4219 100644
--- a/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall
+++ b/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall
@@ -104,42 +104,42 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (11.50000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,3.168279320);
+\draw [,->,>=latex] (-0.50000,0) -- (11.500,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
 %DEFAULT
 \draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw []  (1.000000000,0.001377810000) node [rotate=0] {$\bullet$};
-\draw []  (2.000000000,0.01446700500) node [rotate=0] {$\bullet$};
-\draw []  (3.000000000,0.09001692000) node [rotate=0] {$\bullet$};
-\draw []  (4.000000000,0.3675690900) node [rotate=0] {$\bullet$};
-\draw []  (5.000000000,1.029193452) node [rotate=0] {$\bullet$};
-\draw []  (6.000000000,2.001209490) node [rotate=0] {$\bullet$};
-\draw []  (7.000000000,2.668279320) node [rotate=0] {$\bullet$};
-\draw []  (8.000000000,2.334744405) node [rotate=0] {$\bullet$};
-\draw []  (9.000000000,1.210608210) node [rotate=0] {$\bullet$};
-\draw []  (10.00000000,0.2824752490) node [rotate=0] {$\bullet$};
-\draw []  (11.00000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw []  (1.0000,0.0013778) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0.014467) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.090017) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.36757) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,1.0292) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,2.0012) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,2.6683) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,2.3347) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,1.2106) node [rotate=0] {$\bullet$};
+\draw []  (10.000,0.28248) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.00000000,-0.3149246667) node {$ 10 $};
+\draw (10.000,-0.31492) node {$ 10 $};
 \draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.00000000,-0.3149246667) node {$ 11 $};
+\draw (11.000,-0.31492) node {$ 11 $};
 \draw [] (11.0,-0.100) -- (11.0,0.100);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall b/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall
index 516b8372f..4dae1298f 100644
--- a/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall
+++ b/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall
@@ -69,47 +69,47 @@
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+\draw [,->,>=latex] (2.1999,0.019198) -- (2.6544,0.023165);
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+\draw [,->,>=latex] (2.0669,0.75362) -- (2.4939,0.90933);
+\draw [,->,>=latex] (1.9681,0.98313) -- (2.3747,1.1863);
+\draw [,->,>=latex] (1.8439,1.2000) -- (2.2249,1.4479);
+\draw [,->,>=latex] (1.6960,1.4013) -- (2.0464,1.6908);
+\draw [,->,>=latex] (1.5261,1.5846) -- (1.8415,1.9120);
+\draw [,->,>=latex] (1.3366,1.7474) -- (1.6128,2.1084);
+\draw [,->,>=latex] (1.1299,1.8877) -- (1.3633,2.2777);
+\draw [,->,>=latex] (0.90852,2.0036) -- (1.0962,2.4176);
+\draw [,->,>=latex] (0.67546,2.0937) -- (0.81502,2.5263);
+\draw [,->,>=latex] (0.43369,2.1568) -- (0.52330,2.6025);
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+\draw [,->,>=latex] (-0.063439,2.1991) -- (-0.076546,2.6534);
+\draw [,->,>=latex] (-0.31239,2.1777) -- (-0.37693,2.6276);
+\draw [,->,>=latex] (-0.55731,2.1282) -- (-0.67245,2.5680);
+\draw [,->,>=latex] (-0.79504,2.0513) -- (-0.95931,2.4751);
+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2338,2.3504);
+\draw [,->,>=latex] (-1.2368,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7317,2.0120);
+\draw [,->,>=latex] (-1.6150,1.4940) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1405,1.5701);
+\draw [,->,>=latex] (-1.9100,1.0917) -- (-2.3047,1.3172);
+\draw [,->,>=latex] (-2.0215,0.86804) -- (-2.4392,1.0474);
+\draw [,->,>=latex] (-2.1069,0.63322) -- (-2.5422,0.76405);
+\draw [,->,>=latex] (-2.1651,0.39022) -- (-2.6125,0.47085);
+\draw [,->,>=latex] (-2.1954,0.14221) -- (-2.6490,0.17159);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall b/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall
index 676ebf81e..9a04c9614 100644
--- a/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall
+++ b/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall
@@ -49,8 +49,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,0.5000000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,0.50000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -64,13 +64,13 @@
 \draw [] (2.00,0) -- (0,-2.00);
 \draw [] (0,-2.00) -- (0,-1.00);
 \draw [] (0,-1.00) -- (1.00,0);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
 %OTHER STUFF
 %END PSPICTURE
@@ -117,8 +117,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.500000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -132,17 +132,17 @@
 \draw [] (2.00,2.00) -- (1.00,1.00);
 \draw [] (1.00,1.00) -- (-1.00,1.00);
 \draw [] (-1.00,1.00) -- (-2.00,2.00);
-\draw (-2.000000000,-0.3298256667) node {$ -2 $};
+\draw (-2.0000,-0.32983) node {$ -2 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.000000000,-0.3298256667) node {$ -1 $};
+\draw (-1.0000,-0.32983) node {$ -1 $};
 \draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_examsseptii.pstricks.recall b/src_phystricks/Fig_examsseptii.pstricks.recall
index dea0c0c47..32328027d 100644
--- a/src_phystricks/Fig_examsseptii.pstricks.recall
+++ b/src_phystricks/Fig_examsseptii.pstricks.recall
@@ -107,38 +107,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (9.500000000,0);
-\draw [,->,>=latex] (0,-2.500000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
+\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
 %DEFAULT
 \draw [color=blue] (0,-2.00) -- (0,2.00);
 \draw [color=blue] (1.00,-2.00) -- (1.00,2.00);
 \draw [color=blue] (4.00,-2.00) -- (4.00,2.00);
 \draw [color=blue] (9.00,-2.00) -- (9.00,2.00);
-\draw (1.000000000,-0.3149246667) node {$ 1 $};
+\draw (1.0000,-0.31492) node {$ 1 $};
 \draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 2 $};
+\draw (2.0000,-0.31492) node {$ 2 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.000000000,-0.3149246667) node {$ 3 $};
+\draw (3.0000,-0.31492) node {$ 3 $};
 \draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 4 $};
+\draw (4.0000,-0.31492) node {$ 4 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.000000000,-0.3149246667) node {$ 5 $};
+\draw (5.0000,-0.31492) node {$ 5 $};
 \draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 6 $};
+\draw (6.0000,-0.31492) node {$ 6 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.000000000,-0.3149246667) node {$ 7 $};
+\draw (7.0000,-0.31492) node {$ 7 $};
 \draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 8 $};
+\draw (8.0000,-0.31492) node {$ 8 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.000000000,-0.3149246667) node {$ 9 $};
+\draw (9.0000,-0.31492) node {$ 9 $};
 \draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -2 $};
+\draw (-0.43316,-2.0000) node {$ -2 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.4331593333,-1.000000000) node {$ -1 $};
+\draw (-0.43316,-1.0000) node {$ -1 $};
 \draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.2912498333,1.000000000) node {$ 1 $};
+\draw (-0.29125,1.0000) node {$ 1 $};
 \draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.2912498333,2.000000000) node {$ 2 $};
+\draw (-0.29125,2.0000) node {$ 2 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall b/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall
index 61797a55a..644298998 100644
--- a/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall
+++ b/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall
@@ -89,14 +89,14 @@
 \draw [style=dashed] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
 
 \draw [color=blue,style=dashed] (1.970,-0.3473)--(1.974,-0.3230)--(1.978,-0.2986)--(1.981,-0.2742)--(1.984,-0.2497)--(1.987,-0.2252)--(1.990,-0.2006)--(1.992,-0.1761)--(1.994,-0.1515)--(1.996,-0.1268)--(1.997,-0.1022)--(1.998,-0.07755)--(1.999,-0.05288)--(2.000,-0.02821)--(2.000,-0.003526)--(2.000,0.02116)--(1.999,0.04583)--(1.999,0.07050)--(1.998,0.09516)--(1.996,0.1198)--(1.995,0.1444)--(1.993,0.1690)--(1.991,0.1936)--(1.988,0.2182)--(1.985,0.2427)--(1.982,0.2672)--(1.979,0.2916)--(1.975,0.3160)--(1.971,0.3404)--(1.966,0.3646)--(1.962,0.3889)--(1.957,0.4131)--(1.952,0.4372)--(1.946,0.4612)--(1.940,0.4852)--(1.934,0.5091)--(1.928,0.5330)--(1.921,0.5567)--(1.914,0.5804)--(1.907,0.6039)--(1.899,0.6274)--(1.891,0.6508)--(1.883,0.6741)--(1.875,0.6973)--(1.866,0.7204)--(1.857,0.7433)--(1.847,0.7662)--(1.838,0.7889)--(1.828,0.8115)--(1.818,0.8340)--(1.807,0.8564)--(1.797,0.8786)--(1.786,0.9007)--(1.774,0.9227)--(1.763,0.9445)--(1.751,0.9662)--(1.739,0.9878)--(1.727,1.009)--(1.714,1.030)--(1.701,1.051)--(1.688,1.072)--(1.675,1.093)--(1.661,1.114)--(1.647,1.134)--(1.633,1.154)--(1.619,1.174)--(1.604,1.194)--(1.589,1.214)--(1.574,1.234)--(1.559,1.253)--(1.543,1.272)--(1.528,1.291)--(1.512,1.310)--(1.495,1.328)--(1.479,1.347)--(1.462,1.365)--(1.445,1.383)--(1.428,1.400)--(1.410,1.418)--(1.393,1.435)--(1.375,1.452)--(1.357,1.469)--(1.339,1.486)--(1.320,1.502)--(1.302,1.518)--(1.283,1.534)--(1.264,1.550)--(1.245,1.566)--(1.225,1.581)--(1.206,1.596)--(1.186,1.611)--(1.166,1.625)--(1.146,1.639)--(1.125,1.653)--(1.105,1.667)--(1.084,1.681)--(1.063,1.694)--(1.042,1.707)--(1.021,1.720)--(1.000,1.732);
-\draw (0.6519509490,0.1894063750) node {$\theta$};
+\draw (0.65195,0.18941) node {$\theta$};
 
 \draw [] (0.492,-0.0868)--(0.493,-0.0807)--(0.494,-0.0746)--(0.495,-0.0685)--(0.496,-0.0624)--(0.497,-0.0563)--(0.497,-0.0502)--(0.498,-0.0440)--(0.499,-0.0379)--(0.499,-0.0317)--(0.499,-0.0256)--(0.500,-0.0194)--(0.500,-0.0132)--(0.500,-0.00705)--(0.500,0)--(0.500,0.00529)--(0.500,0.0115)--(0.500,0.0176)--(0.499,0.0238)--(0.499,0.0300)--(0.499,0.0361)--(0.498,0.0423)--(0.498,0.0484)--(0.497,0.0545)--(0.496,0.0607)--(0.496,0.0668)--(0.495,0.0729)--(0.494,0.0790)--(0.493,0.0851)--(0.492,0.0912)--(0.490,0.0972)--(0.489,0.103)--(0.488,0.109)--(0.487,0.115)--(0.485,0.121)--(0.484,0.127)--(0.482,0.133)--(0.480,0.139)--(0.478,0.145)--(0.477,0.151)--(0.475,0.157)--(0.473,0.163)--(0.471,0.169)--(0.469,0.174)--(0.466,0.180)--(0.464,0.186)--(0.462,0.192)--(0.459,0.197)--(0.457,0.203)--(0.454,0.209)--(0.452,0.214)--(0.449,0.220)--(0.446,0.225)--(0.444,0.231)--(0.441,0.236)--(0.438,0.242)--(0.435,0.247)--(0.432,0.252)--(0.429,0.258)--(0.425,0.263)--(0.422,0.268)--(0.419,0.273)--(0.415,0.278)--(0.412,0.284)--(0.408,0.289)--(0.405,0.294)--(0.401,0.299)--(0.397,0.303)--(0.394,0.308)--(0.390,0.313)--(0.386,0.318)--(0.382,0.323)--(0.378,0.327)--(0.374,0.332)--(0.370,0.337)--(0.366,0.341)--(0.361,0.346)--(0.357,0.350)--(0.353,0.354)--(0.348,0.359)--(0.344,0.363)--(0.339,0.367)--(0.335,0.371)--(0.330,0.376)--(0.325,0.380)--(0.321,0.384)--(0.316,0.388)--(0.311,0.391)--(0.306,0.395)--(0.301,0.399)--(0.296,0.403)--(0.291,0.406)--(0.286,0.410)--(0.281,0.413)--(0.276,0.417)--(0.271,0.420)--(0.266,0.423)--(0.261,0.427)--(0.255,0.430)--(0.250,0.433);
 \draw [] (0,0) -- (1.97,-0.347);
 \draw [] (0,0) -- (1.00,1.73);
-\draw (0.1002123789,1.140733404) node {$R$};
-\draw (2.429897999,-0.5535014753) node {$\theta_0$};
-\draw (1.314840167,2.145969095) node {$\theta_1$};
+\draw (0.10021,1.1407) node {$R$};
+\draw (2.4299,-0.55350) node {$\theta_0$};
+\draw (1.3148,2.1460) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ratrap.pstricks.recall b/src_phystricks/Fig_ratrap.pstricks.recall
index 155eeb364..c0c32b68c 100644
--- a/src_phystricks/Fig_ratrap.pstricks.recall
+++ b/src_phystricks/Fig_ratrap.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000000000,0) -- (2.500000000,0);
-\draw [,->,>=latex] (0,-0.5000000000) -- (0,2.500000000);
+\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
+\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -86,14 +86,14 @@
 \draw [color=blue] (2.000,0)--(2.000,0.03173)--(1.999,0.06346)--(1.998,0.09516)--(1.996,0.1268)--(1.994,0.1585)--(1.991,0.1901)--(1.988,0.2217)--(1.984,0.2532)--(1.980,0.2846)--(1.975,0.3160)--(1.970,0.3473)--(1.964,0.3785)--(1.958,0.4096)--(1.951,0.4406)--(1.944,0.4715)--(1.936,0.5023)--(1.928,0.5330)--(1.919,0.5635)--(1.910,0.5938)--(1.900,0.6241)--(1.890,0.6541)--(1.879,0.6840)--(1.868,0.7138)--(1.857,0.7433)--(1.845,0.7727)--(1.832,0.8019)--(1.819,0.8308)--(1.806,0.8596)--(1.792,0.8881)--(1.778,0.9165)--(1.763,0.9445)--(1.748,0.9724)--(1.732,1.000)--(1.716,1.027)--(1.699,1.054)--(1.683,1.081)--(1.665,1.108)--(1.647,1.134)--(1.629,1.160)--(1.611,1.186)--(1.592,1.211)--(1.572,1.236)--(1.552,1.261)--(1.532,1.286)--(1.512,1.310)--(1.491,1.334)--(1.469,1.357)--(1.447,1.380)--(1.425,1.403)--(1.403,1.425)--(1.380,1.447)--(1.357,1.469)--(1.334,1.491)--(1.310,1.512)--(1.286,1.532)--(1.261,1.552)--(1.236,1.572)--(1.211,1.592)--(1.186,1.611)--(1.160,1.629)--(1.134,1.647)--(1.108,1.665)--(1.081,1.683)--(1.054,1.699)--(1.027,1.716)--(1.000,1.732)--(0.9724,1.748)--(0.9445,1.763)--(0.9165,1.778)--(0.8881,1.792)--(0.8596,1.806)--(0.8308,1.819)--(0.8019,1.832)--(0.7727,1.845)--(0.7433,1.857)--(0.7138,1.868)--(0.6840,1.879)--(0.6541,1.890)--(0.6241,1.900)--(0.5938,1.910)--(0.5635,1.919)--(0.5330,1.928)--(0.5023,1.936)--(0.4715,1.944)--(0.4406,1.951)--(0.4096,1.958)--(0.3785,1.964)--(0.3473,1.970)--(0.3160,1.975)--(0.2846,1.980)--(0.2532,1.984)--(0.2217,1.988)--(0.1901,1.991)--(0.1585,1.994)--(0.1268,1.996)--(0.09516,1.998)--(0.06346,1.999)--(0.03173,2.000)--(0,2.000);
 \draw [color=blue] (0,2.00) -- (0,1.00);
 \draw [color=blue] (1.00,0) -- (2.00,0);
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw (1.000000000,-0.1785761667) node {$a$};
-\draw []  (0,1.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.1964676667,1.000000000) node {$a$};
-\draw []  (2.000000000,0) node [rotate=0] {$\bullet$};
-\draw (2.000000000,-0.2267360000) node {$b$};
-\draw []  (0,2.000000000) node [rotate=0] {$\bullet$};
-\draw (-0.1783228333,2.000000000) node {$b$};
+\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.17858) node {$a$};
+\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.19647,1.0000) node {$a$};
+\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.22674) node {$b$};
+\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.17832,2.0000) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_senotopologo.pstricks.recall b/src_phystricks/Fig_senotopologo.pstricks.recall
index d3db436bf..180a3c929 100644
--- a/src_phystricks/Fig_senotopologo.pstricks.recall
+++ b/src_phystricks/Fig_senotopologo.pstricks.recall
@@ -95,33 +95,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.500000000,0) -- (8.500000000,0);
-\draw [,->,>=latex] (0,-2.202481950) -- (0,2.202481950);
+\draw [,->,>=latex] (-8.5000,0) -- (8.5000,0);
+\draw [,->,>=latex] (0,-2.2025) -- (0,2.2025);
 %DEFAULT
 
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.6142)--(6.511,0.6134)--(6.519,0.6127)--(6.527,0.6119)--(6.535,0.6112)--(6.543,0.6104)--(6.551,0.6097)--(6.559,0.6089)--(6.567,0.6082)--(6.575,0.6075)--(6.583,0.6067)--(6.591,0.6060)--(6.599,0.6053)--(6.607,0.6045)--(6.615,0.6038)--(6.623,0.6031)--(6.631,0.6024)--(6.639,0.6016)--(6.647,0.6009)--(6.655,0.6002)--(6.663,0.5995)--(6.671,0.5988)--(6.679,0.5981)--(6.687,0.5974)--(6.695,0.5966)--(6.703,0.5959)--(6.711,0.5952)--(6.719,0.5945)--(6.727,0.5938)--(6.735,0.5931)--(6.743,0.5924)--(6.751,0.5917)--(6.759,0.5910)--(6.767,0.5903)--(6.775,0.5896)--(6.783,0.5889)--(6.791,0.5882)--(6.799,0.5876)--(6.807,0.5869)--(6.815,0.5862)--(6.823,0.5855)--(6.831,0.5848)--(6.839,0.5841)--(6.847,0.5835)--(6.855,0.5828)--(6.863,0.5821)--(6.871,0.5814)--(6.879,0.5807)--(6.887,0.5801)--(6.895,0.5794)--(6.903,0.5787)--(6.911,0.5781)--(6.919,0.5774)--(6.927,0.5767)--(6.935,0.5761)--(6.943,0.5754)--(6.951,0.5748)--(6.959,0.5741)--(6.967,0.5734)--(6.975,0.5728)--(6.984,0.5721)--(6.992,0.5715)--(7.000,0.5708)--(7.008,0.5702)--(7.016,0.5695)--(7.023,0.5689)--(7.031,0.5682)--(7.039,0.5676)--(7.048,0.5670)--(7.056,0.5663)--(7.064,0.5657)--(7.072,0.5650)--(7.080,0.5644)--(7.088,0.5638)--(7.096,0.5631)--(7.104,0.5625)--(7.112,0.5619)--(7.120,0.5612)--(7.128,0.5606)--(7.136,0.5600)--(7.144,0.5594)--(7.152,0.5587)--(7.160,0.5581)--(7.168,0.5575)--(7.176,0.5569)--(7.184,0.5563)--(7.192,0.5556)--(7.200,0.5550)--(7.208,0.5544)--(7.216,0.5538)--(7.224,0.5532)--(7.232,0.5526)--(7.240,0.5520)--(7.248,0.5514)--(7.256,0.5508)--(7.264,0.5502)--(7.272,0.5496)--(7.280,0.5490)--(7.288,0.5484)--(7.296,0.5478)--(7.304,0.5472)--(7.312,0.5466)--(7.320,0.5460)--(7.328,0.5454)--(7.336,0.5448)--(7.344,0.5442)--(7.352,0.5436)--(7.360,0.5430)--(7.368,0.5424)--(7.376,0.5418)--(7.384,0.5413)--(7.392,0.5407)--(7.400,0.5401)--(7.408,0.5395)--(7.416,0.5389)--(7.424,0.5383)--(7.432,0.5378)--(7.440,0.5372)--(7.448,0.5366)--(7.456,0.5360)--(7.464,0.5355)--(7.472,0.5349)--(7.480,0.5343)--(7.488,0.5338)--(7.496,0.5332)--(7.504,0.5326)--(7.512,0.5321)--(7.520,0.5315)--(7.528,0.5309)--(7.536,0.5304)--(7.544,0.5298)--(7.552,0.5292)--(7.560,0.5287)--(7.568,0.5281)--(7.576,0.5276)--(7.584,0.5270)--(7.592,0.5265)--(7.600,0.5259)--(7.608,0.5254)--(7.616,0.5248)--(7.624,0.5243)--(7.632,0.5237)--(7.640,0.5232)--(7.648,0.5226)--(7.656,0.5221)--(7.664,0.5215)--(7.672,0.5210)--(7.680,0.5204)--(7.688,0.5199)--(7.696,0.5194)--(7.704,0.5188)--(7.712,0.5183)--(7.720,0.5178)--(7.728,0.5172)--(7.736,0.5167)--(7.744,0.5162)--(7.752,0.5156)--(7.760,0.5151)--(7.768,0.5146)--(7.776,0.5140)--(7.784,0.5135)--(7.792,0.5130)--(7.800,0.5125)--(7.808,0.5119)--(7.816,0.5114)--(7.824,0.5109)--(7.832,0.5104)--(7.840,0.5098)--(7.848,0.5093)--(7.856,0.5088)--(7.864,0.5083)--(7.872,0.5078)--(7.880,0.5073)--(7.888,0.5068)--(7.896,0.5062)--(7.904,0.5057)--(7.912,0.5052)--(7.920,0.5047)--(7.928,0.5042)--(7.936,0.5037)--(7.944,0.5032)--(7.952,0.5027)--(7.960,0.5022)--(7.968,0.5017)--(7.976,0.5012)--(7.984,0.5007)--(7.992,0.5002)--(8.000,0.4997);
-\draw (-8.000000000,-0.3298256667) node {$ -4 $};
+\draw (-8.0000,-0.32983) node {$ -4 $};
 \draw [] (-8.00,-0.100) -- (-8.00,0.100);
-\draw (-6.000000000,-0.3298256667) node {$ -3 $};
+\draw (-6.0000,-0.32983) node {$ -3 $};
 \draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.000000000,-0.3298256667) node {$ -2 $};
+\draw (-4.0000,-0.32983) node {$ -2 $};
 \draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.000000000,-0.3298256667) node {$ -1 $};
+\draw (-2.0000,-0.32983) node {$ -1 $};
 \draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.000000000,-0.3149246667) node {$ 1 $};
+\draw (2.0000,-0.31492) node {$ 1 $};
 \draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.000000000,-0.3149246667) node {$ 2 $};
+\draw (4.0000,-0.31492) node {$ 2 $};
 \draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.000000000,-0.3149246667) node {$ 3 $};
+\draw (6.0000,-0.31492) node {$ 3 $};
 \draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (8.000000000,-0.3149246667) node {$ 4 $};
+\draw (8.0000,-0.31492) node {$ 4 $};
 \draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.4331593333,-2.000000000) node {$ -1 $};
+\draw (-0.43316,-2.0000) node {$ -1 $};
 \draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.2912498333,2.000000000) node {$ 1 $};
+\draw (-0.29125,2.0000) node {$ 1 $};
 \draw [] (-0.100,2.00) -- (0.100,2.00);
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/phystricksBQXKooPqSEMN.py b/src_phystricks/phystricksBQXKooPqSEMN.py
index 9c6793935..95608a0bb 100644
--- a/src_phystricks/phystricksBQXKooPqSEMN.py
+++ b/src_phystricks/phystricksBQXKooPqSEMN.py
@@ -7,7 +7,7 @@ def BQXKooPqSEMN():                # ex SurfaceDerive  (February 2016)
 	x0 = 5
 	dx = 1
 	x=var('x')
-	f = phyFunction(-((x+0.5)/3)**2+4+x).graph(mx,Mx)
+	f = phyFunction(-((x+0.5)/3)**2+1+x).graph(mx,Mx)
 	f.parameters.color="brown"
 
 	P=f.get_point(x0)
diff --git a/src_phystricks/phystricksExoXLVL.py b/src_phystricks/phystricksExoXLVL.py
index 501c141b0..fcdc1fbc5 100644
--- a/src_phystricks/phystricksExoXLVL.py
+++ b/src_phystricks/phystricksExoXLVL.py
@@ -8,8 +8,8 @@ def ExoXLVL():
 
     C1=Rectangle( Point(-l,l),Point(-dist,dist) )
     C2=Rectangle( Point(0,0),Point(l,l) )
-    C3=Rectangle( Point(0,0),Point(-l,-l) )
-    C4=Rectangle( Point(dist,-dist),Point(l,-l) )
+    C3=Rectangle( Point(-dist,-dist),Point(-l,-l) )
+    C4=Rectangle( Point(0,0),Point(l,-l) )
 
     C1.edges_parameters.color="blue"
     C2.edges_parameters.color="red"
@@ -17,9 +17,9 @@ def ExoXLVL():
     C4.edges_parameters.color="green"
 
     C1.edges_parameters.style="dashed"
-    C2.edges_parameters.style=C1.edges_parameters.style
-    C2.edges_parameters.style=C1.edges_parameters.style
-    C4.edges_parameters.style=C1.edges_parameters.style
+    C3.edges_parameters.style=C1.edges_parameters.style
+    #C2.edges_parameters.style=C1.edges_parameters.style
+    #C4.edges_parameters.style=C1.edges_parameters.style
 
     a1=C1.center()
     a1.parameters.symbol=""
diff --git a/src_phystricks/phystricksPolirettangolo.py b/src_phystricks/phystricksPolirettangolo.py
index a53359343..42e66f282 100644
--- a/src_phystricks/phystricksPolirettangolo.py
+++ b/src_phystricks/phystricksPolirettangolo.py
@@ -23,6 +23,7 @@ def Polirettangolo():
         rect.edges_parameters.style="dotted"
 
     for rect in [R1,R2,R3]:
+        rect.hatched()
         rect.edges_parameters=R.edges_parameters.copy()
         rect.hatch_parameters=R.hatch_parameters.copy()
 

From 62b6cd2f7a64832afe5326dbb781de4b8602bd93 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 21 Jun 2017 06:21:27 +0200
Subject: [PATCH 27/64] (coding style) Replace some tabs by spaces for the
 picture 'TraceCycloide'

---
 src_phystricks/phystricksCommuns.py       | 57 +++++++++++------------
 src_phystricks/phystricksTraceCycloide.py | 16 +++----
 2 files changed, 34 insertions(+), 39 deletions(-)

diff --git a/src_phystricks/phystricksCommuns.py b/src_phystricks/phystricksCommuns.py
index 04265b470..f8e8bbbf3 100644
--- a/src_phystricks/phystricksCommuns.py
+++ b/src_phystricks/phystricksCommuns.py
@@ -1,35 +1,32 @@
 # -*- coding: utf8 -*-
 
-# Note : si tu modifie ce fichier, tu dois le copier à la main vers le répertoire de test.
-# ~/script/modules/phystricks/tests
-
 from phystricks import *
 def CorrectionParametrique(curve,LLms,name,dilatation=1):
-	fig = GenericFigure("SubfiguresCDU"+name,script_filename="Communs")
-
-	ssfig1 = fig.new_subfigure(u"Quelque points de repères","SS1"+name)
-	pspict1 = ssfig1.new_pspicture(name+"psp1")
-	ssfig2 = fig.new_subfigure(u"La courbe","SS2"+name)
-	pspict2 = ssfig2.new_pspicture(name+"psp2")
-
-	for llam in LLms :
-		P=curve(llam)
-		tangent=curve.get_tangent_segment(llam)
-		second=curve.get_second_derivative_vector(llam)
-		normal=curve.get_normal_vector(llam)
-		normal.parameters.color="green"
-		tangent.parameters.color="brown"
-
-		pspict1.DrawGraphs(P,second,tangent,normal)
-		pspict2.DrawGraphs(P,tangent)
-
-	curve.parameters.style="dashed"
-	pspict2.DrawGraphs(curve)
-	pspict1.DrawDefaultAxes()
-	pspict1.dilatation(dilatation)
-	pspict2.DrawDefaultAxes()
-	pspict2.dilatation(dilatation)
-
-	fig.conclude()
-	fig.write_the_file()
+    fig = GenericFigure("SubfiguresCDU"+name,script_filename="Communs")
+
+    ssfig1 = fig.new_subfigure(u"Quelque points de repères","SS1"+name)
+    pspict1 = ssfig1.new_pspicture(name+"psp1")
+    ssfig2 = fig.new_subfigure(u"La courbe","SS2"+name)
+    pspict2 = ssfig2.new_pspicture(name+"psp2")
+
+    for llam in LLms :
+        P=curve(llam)
+        tangent=curve.get_tangent_segment(llam)
+        second=curve.get_second_derivative_vector(llam)
+        normal=curve.get_normal_vector(llam)
+        normal.parameters.color="green"
+        tangent.parameters.color="brown"
+
+        pspict1.DrawGraphs(P,second,tangent,normal)
+        pspict2.DrawGraphs(P,tangent)
+
+    curve.parameters.style="dashed"
+    pspict2.DrawGraphs(curve)
+    pspict1.DrawDefaultAxes()
+    pspict1.dilatation(dilatation)
+    pspict2.DrawDefaultAxes()
+    pspict2.dilatation(dilatation)
+
+    fig.conclude()
+    fig.write_the_file()
 
diff --git a/src_phystricks/phystricksTraceCycloide.py b/src_phystricks/phystricksTraceCycloide.py
index d77917256..eeb944c3a 100644
--- a/src_phystricks/phystricksTraceCycloide.py
+++ b/src_phystricks/phystricksTraceCycloide.py
@@ -1,13 +1,11 @@
 from phystricks import *
 import phystricksCommuns as Communs
 def TraceCycloide():
+    x=var('x')
+    f1=phyFunction(x+cos(x))
+    f2=phyFunction(1+sin(x))
+    curve=ParametricCurve(f1,f2).graph(0,2*pi)
+    curve.parameters.color="blue"
+    LLms=[0,pi,3*pi/2,2*pi, pi/4,3*pi/4,5*pi/4,7*pi/4]
 
-	x=var('x')
-	f1=phyFunction(x+cos(x))
-	f2=phyFunction(1+sin(x))
-	curve=ParametricCurve(f1,f2).graph(0,2*pi)
-	curve.parameters.color="blue"
-	LLms=[0,pi,3*pi/2,2*pi, pi/4,3*pi/4,5*pi/4,7*pi/4]
-
-	Communs.CorrectionParametrique(curve,LLms,"TraceCycloide")
-
+    Communs.CorrectionParametrique(curve,LLms,"TraceCycloide")

From cbff8adc997fa31cc5636623ba6dd603e60e3848 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Wed, 21 Jun 2017 06:22:13 +0200
Subject: [PATCH 28/64] (pictures) Request two passes instead of one for the
 picture 'TraceCycloide'

---
 src_phystricks/figures_mazhe.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src_phystricks/figures_mazhe.py b/src_phystricks/figures_mazhe.py
index 93e25fa17..64354e80c 100755
--- a/src_phystricks/figures_mazhe.py
+++ b/src_phystricks/figures_mazhe.py
@@ -540,7 +540,7 @@ def append_picture(fun,number):
 append_picture(Cardioid,1)
 append_picture(ArcLongueurFinesse,1)
 append_picture(SenoTopologo,1)
-append_picture(TraceCycloide,1)
+append_picture(TraceCycloide,2)
 append_picture(Osculateur,1)
 append_picture(JGuKEjH,1)
 append_picture(ExerciceGraphesbis,2)

From e9be471aedc3d565ba135b88d9d64c2250248f13 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 02:59:49 +0200
Subject: [PATCH 29/64] (pictures) Update (once again) the '.recall' files for
 which we have a 'small point move'

---
 auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks   |  18 +-
 auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks |   6 +-
 .../Fig_ACUooQwcDMZ.pstricks.recall           | 210 ++++-----
 src_phystricks/Fig_ADUGmRRA.pstricks.recall   |  10 +-
 src_phystricks/Fig_ADUGmRRB.pstricks.recall   |  10 +-
 src_phystricks/Fig_ADUGmRRC.pstricks.recall   |  10 +-
 .../Fig_ASHYooUVHkak.pstricks.recall          |  28 +-
 .../Fig_AccumulationIsole.pstricks.recall     |  18 +-
 .../Fig_AireParabole.pstricks.recall          |  52 +--
 .../Fig_BNHLooLDxdPA.pstricks.recall          |  18 +-
 src_phystricks/Fig_Bateau.pstricks.recall     |  32 +-
 src_phystricks/Fig_BiaisOuPas.pstricks.recall |  46 +-
 .../Fig_CMMAooQegASg.pstricks.recall          |   8 +-
 .../Fig_CQIXooBEDnfK.pstricks.recall          |  72 +--
 src_phystricks/Fig_CSCii.pstricks.recall      |  28 +-
 src_phystricks/Fig_CSCiv.pstricks.recall      |  52 +--
 src_phystricks/Fig_CSCv.pstricks.recall       |  62 +--
 .../Fig_CWKJooppMsZXjw.pstricks.recall        |  14 +-
 src_phystricks/Fig_Cardioid.pstricks.recall   |   2 +-
 src_phystricks/Fig_CbCartTui.pstricks.recall  | 106 ++---
 .../Fig_ChampGraviation.pstricks.recall       | 442 +++++++++---------
 .../Fig_CheminFresnel.pstricks.recall         |  22 +-
 .../Fig_ChoixInfini.pstricks.recall           |  66 +--
 src_phystricks/Fig_CoinPasVar.pstricks.recall |  16 +-
 .../Fig_ConeRevolution.pstricks.recall        |  20 +-
 .../Fig_ContourGreen.pstricks.recall          |  18 +-
 .../Fig_ContourTgNDivergence.pstricks.recall  |  50 +-
 .../Fig_CoordPolaires.pstricks.recall         |  24 +-
 .../Fig_CornetGlace.pstricks.recall           |  24 +-
 .../Fig_DDCTooYscVzA.pstricks.recall          |  50 +-
 .../Fig_DNHRooqGtffLkd.pstricks.recall        |  66 +--
 .../Fig_DNRRooJWRHgOCw.pstricks.recall        |  42 +-
 src_phystricks/Fig_DTIYKkP.pstricks.recall    |   6 +-
 .../Fig_DZVooQZLUtf.pstricks.recall           | 338 +++++++-------
 ...Fig_DefinitionCartesiennes.pstricks.recall |  78 ++--
 .../Fig_Differentielle.pstricks.recall        |  36 +-
 .../Fig_DistanceEuclide.pstricks.recall       |  40 +-
 .../Fig_DivergenceDeux.pstricks.recall        | 416 ++++++++---------
 .../Fig_DynkinNUtPJx.pstricks.recall          | 110 ++---
 .../Fig_DynkinpWjUbE.pstricks.recall          |   8 +-
 .../Fig_DynkinqlgIQl.pstricks.recall          |   8 +-
 .../Fig_DynkinrjbHIu.pstricks.recall          |   8 +-
 src_phystricks/Fig_EJRsWXw.pstricks.recall    |  26 +-
 src_phystricks/Fig_ExPolygone.pstricks.recall |  54 +--
 .../Fig_ExempleNonRang.pstricks.recall        |   8 +-
 .../Fig_ExerciceGraphesbis.pstricks.recall    | 326 ++++++-------
 src_phystricks/Fig_ExoCUd.pstricks.recall     |  48 +-
 .../Fig_ExoMagnetique.pstricks.recall         |  12 +-
 src_phystricks/Fig_ExoPolaire.pstricks.recall |  18 +-
 .../Fig_ExoProjection.pstricks.recall         |  18 +-
 .../Fig_ExoUnSurxPolaire.pstricks.recall      |  80 ++--
 src_phystricks/Fig_ExoXLVL.pstricks.recall    |  38 +-
 src_phystricks/Fig_FGWjJBX.pstricks.recall    |  32 +-
 .../Fig_FNBQooYgkAmS.pstricks.recall          |  14 +-
 src_phystricks/Fig_FWJuNhU.pstricks.recall    |  40 +-
 .../Fig_FXVooJYAfif.pstricks.recall           |  10 +-
 .../Fig_FonctionXtroisOM.pstricks.recall      |  44 +-
 src_phystricks/Fig_GBnUivi.pstricks.recall    | 106 ++---
 .../Fig_GYODoojTiGZSkJ.pstricks.recall        |  22 +-
 .../Fig_HLJooGDZnqF.pstricks.recall           | 148 +++---
 src_phystricks/Fig_HNxitLj.pstricks.recall    |  30 +-
 .../Fig_HasseAGdfdy.pstricks.recall           |  30 +-
 src_phystricks/Fig_IYAvSvI.pstricks.recall    |  40 +-
 .../Fig_IntRectangle.pstricks.recall          |  28 +-
 .../Fig_IntervalleUn.pstricks.recall          |  26 +-
 src_phystricks/Fig_IsomCarre.pstricks.recall  |  18 +-
 src_phystricks/Fig_JGuKEjH.pstricks.recall    |  14 +-
 .../Fig_JWINooSfKCeA.pstricks.recall          |  24 +-
 .../Fig_KGQXooZFNVnW.pstricks.recall          |  68 +--
 .../Fig_KKJAooubQzgBgP.pstricks.recall        |  22 +-
 src_phystricks/Fig_LAfWmaN.pstricks.recall    |  48 +-
 .../Fig_LBGooAdteCt.pstricks.recall           | 106 ++---
 .../Fig_LMHMooCscXNNdU.pstricks.recall        |  58 +--
 src_phystricks/Fig_Laurin.pstricks.recall     |  60 +--
 src_phystricks/Fig_LesSpheres.pstricks.recall |  78 ++--
 src_phystricks/Fig_MNICGhR.pstricks.recall    |  28 +-
 src_phystricks/Fig_Mantisse.pstricks.recall   |  42 +-
 .../Fig_MethodeChemin.pstricks.recall         |  10 +-
 .../Fig_MethodeNewton.pstricks.recall         |  44 +-
 .../Fig_MomentForce.pstricks.recall           |  16 +-
 src_phystricks/Fig_NEtAchr.pstricks.recall    |  10 +-
 .../Fig_NiveauHyperbole.pstricks.recall       |  72 +--
 .../Fig_NiveauHyperboleDeux.pstricks.recall   |  68 +--
 .../Fig_OQTEoodIwAPfZE.pstricks.recall        |  62 +--
 .../Fig_PLTWoocPNeiZir.pstricks.recall        |  12 +-
 .../Fig_PONXooXYjEot.pstricks.recall          |  32 +-
 .../Fig_ParallelogrammeOM.pstricks.recall     |  18 +-
 .../Fig_ParamTangente.pstricks.recall         |  58 +--
 .../Fig_PartieEntiere.pstricks.recall         |  48 +-
 .../Fig_Polirettangolo.pstricks.recall        |  76 +--
 .../Fig_ProjectionScalaire.pstricks.recall    |  22 +-
 src_phystricks/Fig_QCb.pstricks.recall        |  12 +-
 .../Fig_QMWKooRRulrgcH.pstricks.recall        |  28 +-
 src_phystricks/Fig_QPcdHwP.pstricks.recall    |  10 +-
 .../Fig_QSKDooujUbDCsu.pstricks.recall        |  10 +-
 src_phystricks/Fig_RGjjpwF.pstricks.recall    |  28 +-
 .../Fig_ROAOooPgUZIt.pstricks.recall          |  18 +-
 src_phystricks/Fig_RQsQKTl.pstricks.recall    |   8 +-
 src_phystricks/Fig_Refraction.pstricks.recall |  18 +-
 .../Fig_SBTooEasQsT.pstricks.recall           | 182 ++++----
 .../Fig_SQNPooPTrLRQ.pstricks.recall          | 442 +++++++++---------
 src_phystricks/Fig_STdyNTH.pstricks.recall    |  28 +-
 .../Fig_SYNKooZBuEWsWw.pstricks.recall        |  80 ++--
 .../Fig_SolsEqDiffSin.pstricks.recall         |  26 +-
 .../Fig_SuiteInverseAlterne.pstricks.recall   |  36 +-
 .../Fig_SuiteUnSurn.pstricks.recall           |  36 +-
 .../Fig_SurfaceCercle.pstricks.recall         |   8 +-
 .../Fig_TIMYoochXZZNGP.pstricks.recall        |  10 +-
 .../Fig_TVXooWoKkqV.pstricks.recall           |  42 +-
 .../Fig_TangenteQuestion.pstricks.recall      |   8 +-
 .../Fig_TangenteQuestionOM.pstricks.recall    |   8 +-
 .../Fig_ToreRevolution.pstricks.recall        |  18 +-
 src_phystricks/Fig_TracerUn.pstricks.recall   | 198 ++++----
 src_phystricks/Fig_Trajs.pstricks.recall      |  38 +-
 src_phystricks/Fig_TriangleUV.pstricks.recall |  18 +-
 .../Fig_UGCFooQoCihh.pstricks.recall          |  56 +--
 .../Fig_UIEHooSlbzIJ.pstricks.recall          |  86 ++--
 .../Fig_UMEBooVTMyfD.pstricks.recall          |  54 +--
 .../Fig_UNVooMsXxHa.pstricks.recall           | 142 +++---
 .../Fig_UQZooGFLNEq.pstricks.recall           |  58 +--
 src_phystricks/Fig_UneCellule.pstricks.recall |  94 ++--
 .../Fig_VBOIooRHhKOH.pstricks.recall          |  44 +-
 .../Fig_VDFMooHMmFZr.pstricks.recall          |  18 +-
 .../Fig_VNBGooSqMsGU.pstricks.recall          |  28 +-
 .../Fig_VWFLooPSrOqz.pstricks.recall          |  54 +--
 .../Fig_WUYooCISzeB.pstricks.recall           | 160 +++----
 .../Fig_XJMooCQTlNL.pstricks.recall           |  54 +--
 .../Fig_XOLBooGcrjiwoU.pstricks.recall        |  66 +--
 .../Fig_XTGooSFFtPu.pstricks.recall           | 156 +++----
 .../Fig_YHJYooTEXLLn.pstricks.recall          |  20 +-
 .../Fig_YQIDooBqpAdbIM.pstricks.recall        |  26 +-
 .../Fig_YQVHooYsGLHQ.pstricks.recall          | 416 ++++++++---------
 .../Fig_YYECooQlnKtD.pstricks.recall          |  66 +--
 .../Fig_ZTTooXtHkci.pstricks.recall           |  68 +--
 .../Fig_examsseptii.pstricks.recall           |  64 +--
 .../Fig_senotopologo.pstricks.recall          |  52 +--
 src_phystricks/Fig_trigoWedd.pstricks.recall  |  18 +-
 137 files changed, 4056 insertions(+), 4056 deletions(-)

diff --git a/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks b/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
index 6fbdb6220..37bbbdd11 100644
--- a/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
+++ b/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
@@ -86,17 +86,17 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [style=dashed] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
+\draw [style=dashed] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 
-\draw [color=blue,style=] (1.970,-0.3473)--(1.974,-0.3230)--(1.978,-0.2986)--(1.981,-0.2742)--(1.984,-0.2497)--(1.987,-0.2252)--(1.990,-0.2006)--(1.992,-0.1761)--(1.994,-0.1515)--(1.996,-0.1268)--(1.997,-0.1022)--(1.998,-0.07755)--(1.999,-0.05288)--(2.000,-0.02821)--(2.000,-0.003526)--(2.000,0.02116)--(1.999,0.04583)--(1.999,0.07050)--(1.998,0.09516)--(1.996,0.1198)--(1.995,0.1444)--(1.993,0.1690)--(1.991,0.1936)--(1.988,0.2182)--(1.985,0.2427)--(1.982,0.2672)--(1.979,0.2916)--(1.975,0.3160)--(1.971,0.3404)--(1.966,0.3646)--(1.962,0.3889)--(1.957,0.4131)--(1.952,0.4372)--(1.946,0.4612)--(1.940,0.4852)--(1.934,0.5091)--(1.928,0.5330)--(1.921,0.5567)--(1.914,0.5804)--(1.907,0.6039)--(1.899,0.6274)--(1.891,0.6508)--(1.883,0.6741)--(1.875,0.6973)--(1.866,0.7204)--(1.857,0.7433)--(1.847,0.7662)--(1.838,0.7889)--(1.828,0.8115)--(1.818,0.8340)--(1.807,0.8564)--(1.797,0.8786)--(1.786,0.9007)--(1.774,0.9227)--(1.763,0.9445)--(1.751,0.9662)--(1.739,0.9878)--(1.727,1.009)--(1.714,1.030)--(1.701,1.051)--(1.688,1.072)--(1.675,1.093)--(1.661,1.114)--(1.647,1.134)--(1.633,1.154)--(1.619,1.174)--(1.604,1.194)--(1.589,1.214)--(1.574,1.234)--(1.559,1.253)--(1.543,1.272)--(1.528,1.291)--(1.512,1.310)--(1.495,1.328)--(1.479,1.347)--(1.462,1.365)--(1.445,1.383)--(1.428,1.400)--(1.410,1.418)--(1.393,1.435)--(1.375,1.452)--(1.357,1.469)--(1.339,1.486)--(1.320,1.502)--(1.302,1.518)--(1.283,1.534)--(1.264,1.550)--(1.245,1.566)--(1.225,1.581)--(1.206,1.596)--(1.186,1.611)--(1.166,1.625)--(1.146,1.639)--(1.125,1.653)--(1.105,1.667)--(1.084,1.681)--(1.063,1.694)--(1.042,1.707)--(1.021,1.720)--(1.000,1.732);
-\draw (0.65195,0.18941) node {$\theta$};
+\draw [color=blue,style=] (1.9696,-0.3472)--(1.9737,-0.3229)--(1.9775,-0.2985)--(1.9811,-0.2741)--(1.9843,-0.2496)--(1.9872,-0.2251)--(1.9899,-0.2006)--(1.9922,-0.1760)--(1.9942,-0.1514)--(1.9959,-0.1268)--(1.9973,-0.1022)--(1.9984,-0.0775)--(1.9993,-0.0528)--(1.9998,-0.0282)--(1.9999,-0.0035)--(1.9998,0.0211)--(1.9994,0.0458)--(1.9987,0.0705)--(1.9977,0.0951)--(1.9964,0.1198)--(1.9947,0.1444)--(1.9928,0.1690)--(1.9906,0.1936)--(1.9880,0.2181)--(1.9852,0.2426)--(1.9820,0.2671)--(1.9786,0.2916)--(1.9748,0.3160)--(1.9708,0.3403)--(1.9664,0.3646)--(1.9618,0.3888)--(1.9568,0.4130)--(1.9516,0.4371)--(1.9460,0.4612)--(1.9402,0.4852)--(1.9341,0.5091)--(1.9276,0.5329)--(1.9209,0.5566)--(1.9139,0.5803)--(1.9066,0.6039)--(1.8990,0.6274)--(1.8911,0.6508)--(1.8829,0.6740)--(1.8745,0.6972)--(1.8657,0.7203)--(1.8567,0.7433)--(1.8474,0.7661)--(1.8378,0.7889)--(1.8279,0.8115)--(1.8177,0.8340)--(1.8073,0.8564)--(1.7966,0.8786)--(1.7856,0.9007)--(1.7744,0.9227)--(1.7629,0.9445)--(1.7511,0.9662)--(1.7390,0.9877)--(1.7267,1.0091)--(1.7141,1.0303)--(1.7013,1.0514)--(1.6882,1.0723)--(1.6748,1.0931)--(1.6612,1.1137)--(1.6473,1.1341)--(1.6332,1.1543)--(1.6188,1.1744)--(1.6042,1.1943)--(1.5893,1.2140)--(1.5742,1.2335)--(1.5589,1.2528)--(1.5433,1.2720)--(1.5275,1.2909)--(1.5114,1.3097)--(1.4952,1.3282)--(1.4787,1.3466)--(1.4619,1.3647)--(1.4450,1.3827)--(1.4278,1.4004)--(1.4104,1.4179)--(1.3928,1.4352)--(1.3750,1.4523)--(1.3570,1.4691)--(1.3387,1.4858)--(1.3203,1.5022)--(1.3017,1.5184)--(1.2828,1.5343)--(1.2638,1.5500)--(1.2446,1.5655)--(1.2252,1.5807)--(1.2056,1.5957)--(1.1858,1.6105)--(1.1658,1.6250)--(1.1457,1.6393)--(1.1253,1.6533)--(1.1049,1.6670)--(1.0842,1.6805)--(1.0634,1.6938)--(1.0424,1.7068)--(1.0212,1.7195)--(1.0000,1.7320);
+\draw (0.6519,0.1894) node {$\theta$};
 
-\draw [] (0.492,-0.0868)--(0.493,-0.0807)--(0.494,-0.0746)--(0.495,-0.0685)--(0.496,-0.0624)--(0.497,-0.0563)--(0.497,-0.0502)--(0.498,-0.0440)--(0.499,-0.0379)--(0.499,-0.0317)--(0.499,-0.0256)--(0.500,-0.0194)--(0.500,-0.0132)--(0.500,-0.00705)--(0.500,0)--(0.500,0.00529)--(0.500,0.0115)--(0.500,0.0176)--(0.499,0.0238)--(0.499,0.0300)--(0.499,0.0361)--(0.498,0.0423)--(0.498,0.0484)--(0.497,0.0545)--(0.496,0.0607)--(0.496,0.0668)--(0.495,0.0729)--(0.494,0.0790)--(0.493,0.0851)--(0.492,0.0912)--(0.490,0.0972)--(0.489,0.103)--(0.488,0.109)--(0.487,0.115)--(0.485,0.121)--(0.484,0.127)--(0.482,0.133)--(0.480,0.139)--(0.478,0.145)--(0.477,0.151)--(0.475,0.157)--(0.473,0.163)--(0.471,0.169)--(0.469,0.174)--(0.466,0.180)--(0.464,0.186)--(0.462,0.192)--(0.459,0.197)--(0.457,0.203)--(0.454,0.209)--(0.452,0.214)--(0.449,0.220)--(0.446,0.225)--(0.444,0.231)--(0.441,0.236)--(0.438,0.242)--(0.435,0.247)--(0.432,0.252)--(0.429,0.258)--(0.425,0.263)--(0.422,0.268)--(0.419,0.273)--(0.415,0.278)--(0.412,0.284)--(0.408,0.289)--(0.405,0.294)--(0.401,0.299)--(0.397,0.303)--(0.394,0.308)--(0.390,0.313)--(0.386,0.318)--(0.382,0.323)--(0.378,0.327)--(0.374,0.332)--(0.370,0.337)--(0.366,0.341)--(0.361,0.346)--(0.357,0.350)--(0.353,0.354)--(0.348,0.359)--(0.344,0.363)--(0.339,0.367)--(0.335,0.371)--(0.330,0.376)--(0.325,0.380)--(0.321,0.384)--(0.316,0.388)--(0.311,0.391)--(0.306,0.395)--(0.301,0.399)--(0.296,0.403)--(0.291,0.406)--(0.286,0.410)--(0.281,0.413)--(0.276,0.417)--(0.271,0.420)--(0.266,0.423)--(0.261,0.427)--(0.255,0.430)--(0.250,0.433);
-\draw [] (0,0) -- (1.97,-0.347);
-\draw [] (0,0) -- (1.00,1.73);
-\draw (0.10021,1.1407) node {$R$};
-\draw (2.4299,-0.55350) node {$\theta_0$};
-\draw (1.3148,2.1460) node {$\theta_1$};
+\draw [] (0.4924,-0.0868)--(0.4934,-0.0807)--(0.4943,-0.0746)--(0.4952,-0.0685)--(0.4960,-0.0624)--(0.4968,-0.0562)--(0.4974,-0.0501)--(0.4980,-0.0440)--(0.4985,-0.0378)--(0.4989,-0.0317)--(0.4993,-0.0255)--(0.4996,-0.0193)--(0.4998,-0.0132)--(0.4999,-0.0070)--(0.4999,0.0000)--(0.4999,0.0052)--(0.4998,0.0114)--(0.4996,0.0176)--(0.4994,0.0237)--(0.4991,0.0299)--(0.4986,0.0361)--(0.4982,0.0422)--(0.4976,0.0484)--(0.4970,0.0545)--(0.4963,0.0606)--(0.4955,0.0667)--(0.4946,0.0729)--(0.4937,0.0790)--(0.4927,0.0850)--(0.4916,0.0911)--(0.4904,0.0972)--(0.4892,0.1032)--(0.4879,0.1092)--(0.4865,0.1153)--(0.4850,0.1213)--(0.4835,0.1272)--(0.4819,0.1332)--(0.4802,0.1391)--(0.4784,0.1450)--(0.4766,0.1509)--(0.4747,0.1568)--(0.4727,0.1627)--(0.4707,0.1685)--(0.4686,0.1743)--(0.4664,0.1800)--(0.4641,0.1858)--(0.4618,0.1915)--(0.4594,0.1972)--(0.4569,0.2028)--(0.4544,0.2085)--(0.4518,0.2141)--(0.4491,0.2196)--(0.4464,0.2251)--(0.4436,0.2306)--(0.4407,0.2361)--(0.4377,0.2415)--(0.4347,0.2469)--(0.4316,0.2522)--(0.4285,0.2575)--(0.4253,0.2628)--(0.4220,0.2680)--(0.4187,0.2732)--(0.4153,0.2784)--(0.4118,0.2835)--(0.4083,0.2885)--(0.4047,0.2936)--(0.4010,0.2985)--(0.3973,0.3035)--(0.3935,0.3083)--(0.3897,0.3132)--(0.3858,0.3180)--(0.3818,0.3227)--(0.3778,0.3274)--(0.3738,0.3320)--(0.3696,0.3366)--(0.3654,0.3411)--(0.3612,0.3456)--(0.3569,0.3501)--(0.3526,0.3544)--(0.3482,0.3588)--(0.3437,0.3630)--(0.3392,0.3672)--(0.3346,0.3714)--(0.3300,0.3755)--(0.3254,0.3796)--(0.3207,0.3835)--(0.3159,0.3875)--(0.3111,0.3913)--(0.3063,0.3951)--(0.3014,0.3989)--(0.2964,0.4026)--(0.2914,0.4062)--(0.2864,0.4098)--(0.2813,0.4133)--(0.2762,0.4167)--(0.2710,0.4201)--(0.2658,0.4234)--(0.2606,0.4267)--(0.2553,0.4298)--(0.2500,0.4330);
+\draw [] (0.0000,0.0000) -- (1.9696,-0.3472);
+\draw [] (0.0000,0.0000) -- (1.0000,1.7320);
+\draw (0.1002,1.1407) node {$R$};
+\draw (2.4298,-0.5535) node {$\theta_0$};
+\draw (1.3148,2.1459) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
index 31c8bb72e..9a27504e5 100644
--- a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
+++ b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
@@ -80,7 +80,7 @@
 \draw []  (2.0868,1.7595) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.4102,2.9353);
 \draw []  (1.7350,1.4643) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.34725,0.36521);
+\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.34725,0.36520);
 \draw []  (2.6575,2.4377) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.2478,3.0982);
 \draw []  (0.96811,1.0201) node [rotate=0] {$\bullet$};
@@ -95,7 +95,7 @@
 \draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.071918,0.83064);
 \draw []  (2.8961,4.0071) node [rotate=0] {$\bullet$};
 \draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.6718,3.4583);
-\draw []  (-0.61890,1.0577) node [rotate=0] {$\bullet$};
+\draw []  (-0.61889,1.0577) node [rotate=0] {$\bullet$};
 \draw [color=blue,->,>=latex] (1.9151,1.6070) -- (-0.039497,1.1833);
 \draw [] (-3.00,0) -- (3.00,0);
 \draw [] (0,0) -- (3.83,3.21);
@@ -104,7 +104,7 @@
 \draw [] (0.500,0)--(0.500,0.00353)--(0.500,0.00705)--(0.500,0.0106)--(0.500,0.0141)--(0.500,0.0176)--(0.500,0.0211)--(0.499,0.0247)--(0.499,0.0282)--(0.499,0.0317)--(0.499,0.0352)--(0.499,0.0387)--(0.498,0.0423)--(0.498,0.0458)--(0.498,0.0493)--(0.497,0.0528)--(0.497,0.0563)--(0.496,0.0598)--(0.496,0.0633)--(0.496,0.0668)--(0.495,0.0703)--(0.495,0.0738)--(0.494,0.0773)--(0.493,0.0807)--(0.493,0.0842)--(0.492,0.0877)--(0.492,0.0912)--(0.491,0.0946)--(0.490,0.0981)--(0.490,0.102)--(0.489,0.105)--(0.488,0.108)--(0.487,0.112)--(0.487,0.115)--(0.486,0.119)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.132)--(0.481,0.136)--(0.480,0.139)--(0.479,0.143)--(0.478,0.146)--(0.477,0.149)--(0.476,0.153)--(0.475,0.156)--(0.474,0.159)--(0.473,0.163)--(0.472,0.166)--(0.470,0.169)--(0.469,0.173)--(0.468,0.176)--(0.467,0.179)--(0.465,0.183)--(0.464,0.186)--(0.463,0.189)--(0.462,0.192)--(0.460,0.196)--(0.459,0.199)--(0.457,0.202)--(0.456,0.205)--(0.454,0.209)--(0.453,0.212)--(0.451,0.215)--(0.450,0.218)--(0.448,0.221)--(0.447,0.224)--(0.445,0.228)--(0.444,0.231)--(0.442,0.234)--(0.440,0.237)--(0.439,0.240)--(0.437,0.243)--(0.435,0.246)--(0.433,0.249)--(0.432,0.252)--(0.430,0.255)--(0.428,0.258)--(0.426,0.261)--(0.424,0.264)--(0.423,0.267)--(0.421,0.270)--(0.419,0.273)--(0.417,0.276)--(0.415,0.279)--(0.413,0.282)--(0.411,0.285)--(0.409,0.288)--(0.407,0.291)--(0.405,0.294)--(0.403,0.296)--(0.401,0.299)--(0.398,0.302)--(0.396,0.305)--(0.394,0.308)--(0.392,0.310)--(0.390,0.313)--(0.388,0.316)--(0.385,0.319)--(0.383,0.321);
 \draw []  (1.9151,1.6070) node [rotate=0] {$\bullet$};
 \draw (2.1862,1.3291) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.38)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
 \draw (4.1391,3.5225) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall b/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
index 974934d32..ce0f78fb5 100644
--- a/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
+++ b/src_phystricks/Fig_ACUooQwcDMZ.pstricks.recall
@@ -49,21 +49,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.1931);
 %DEFAULT
 
-\draw [color=blue] (0.1353,-2.000)--(0.1542,-1.870)--(0.1730,-1.754)--(0.1918,-1.651)--(0.2107,-1.557)--(0.2295,-1.472)--(0.2483,-1.393)--(0.2672,-1.320)--(0.2860,-1.252)--(0.3049,-1.188)--(0.3237,-1.128)--(0.3425,-1.071)--(0.3614,-1.018)--(0.3802,-0.9671)--(0.3990,-0.9187)--(0.4179,-0.8726)--(0.4367,-0.8285)--(0.4555,-0.7863)--(0.4744,-0.7458)--(0.4932,-0.7068)--(0.5120,-0.6694)--(0.5309,-0.6332)--(0.5497,-0.5984)--(0.5685,-0.5647)--(0.5874,-0.5321)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6439,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3834)--(0.7004,-0.3561)--(0.7192,-0.3296)--(0.7381,-0.3037)--(0.7569,-0.2785)--(0.7757,-0.2540)--(0.7946,-0.2300)--(0.8134,-0.2065)--(0.8322,-0.1836)--(0.8511,-0.1613)--(0.8699,-0.1394)--(0.8887,-0.1180)--(0.9076,-0.09698)--(0.9264,-0.07644)--(0.9452,-0.05632)--(0.9641,-0.03659)--(0.9829,-0.01724)--(1.002,0.001744)--(1.021,0.02037)--(1.039,0.03866)--(1.058,0.05662)--(1.077,0.07426)--(1.096,0.09159)--(1.115,0.1086)--(1.134,0.1254)--(1.152,0.1419)--(1.171,0.1581)--(1.190,0.1740)--(1.209,0.1897)--(1.228,0.2052)--(1.247,0.2204)--(1.265,0.2354)--(1.284,0.2502)--(1.303,0.2648)--(1.322,0.2791)--(1.341,0.2932)--(1.360,0.3072)--(1.378,0.3210)--(1.397,0.3345)--(1.416,0.3479)--(1.435,0.3611)--(1.454,0.3742)--(1.473,0.3870)--(1.491,0.3998)--(1.510,0.4123)--(1.529,0.4247)--(1.548,0.4369)--(1.567,0.4490)--(1.586,0.4610)--(1.604,0.4728)--(1.623,0.4845)--(1.642,0.4960)--(1.661,0.5074)--(1.680,0.5187)--(1.699,0.5298)--(1.717,0.5409)--(1.736,0.5518)--(1.755,0.5626)--(1.774,0.5732)--(1.793,0.5838)--(1.812,0.5942)--(1.830,0.6046)--(1.849,0.6148)--(1.868,0.6250)--(1.887,0.6350)--(1.906,0.6449)--(1.925,0.6547)--(1.943,0.6645)--(1.962,0.6741)--(1.981,0.6837)--(2.000,0.6931);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.1353,-2.0000)--(0.1541,-1.8696)--(0.1730,-1.7544)--(0.1918,-1.6510)--(0.2106,-1.5574)--(0.2295,-1.4718)--(0.2483,-1.3929)--(0.2671,-1.3198)--(0.2860,-1.2517)--(0.3048,-1.1879)--(0.3236,-1.1279)--(0.3425,-1.0714)--(0.3613,-1.0178)--(0.3801,-0.9670)--(0.3990,-0.9187)--(0.4178,-0.8726)--(0.4366,-0.8285)--(0.4555,-0.7862)--(0.4743,-0.7457)--(0.4932,-0.7068)--(0.5120,-0.6693)--(0.5308,-0.6332)--(0.5497,-0.5983)--(0.5685,-0.5646)--(0.5873,-0.5320)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6438,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3833)--(0.7003,-0.3561)--(0.7192,-0.3295)--(0.7380,-0.3037)--(0.7568,-0.2785)--(0.7757,-0.2539)--(0.7945,-0.2299)--(0.8133,-0.2065)--(0.8322,-0.1836)--(0.8510,-0.1612)--(0.8699,-0.1393)--(0.8887,-0.1179)--(0.9075,-0.0969)--(0.9264,-0.0764)--(0.9452,-0.0563)--(0.9640,-0.0365)--(0.9829,-0.0172)--(1.0017,0.0017)--(1.0205,0.0203)--(1.0394,0.0386)--(1.0582,0.0566)--(1.0770,0.0742)--(1.0959,0.0915)--(1.1147,0.1086)--(1.1335,0.1253)--(1.1524,0.1418)--(1.1712,0.1580)--(1.1900,0.1740)--(1.2089,0.1897)--(1.2277,0.2051)--(1.2466,0.2204)--(1.2654,0.2354)--(1.2842,0.2501)--(1.3031,0.2647)--(1.3219,0.2791)--(1.3407,0.2932)--(1.3596,0.3071)--(1.3784,0.3209)--(1.3972,0.3345)--(1.4161,0.3479)--(1.4349,0.3611)--(1.4537,0.3741)--(1.4726,0.3870)--(1.4914,0.3997)--(1.5102,0.4123)--(1.5291,0.4246)--(1.5479,0.4369)--(1.5667,0.4490)--(1.5856,0.4609)--(1.6044,0.4727)--(1.6233,0.4844)--(1.6421,0.4959)--(1.6609,0.5074)--(1.6798,0.5186)--(1.6986,0.5298)--(1.7174,0.5408)--(1.7363,0.5517)--(1.7551,0.5625)--(1.7739,0.5732)--(1.7928,0.5837)--(1.8116,0.5942)--(1.8304,0.6045)--(1.8493,0.6148)--(1.8681,0.6249)--(1.8869,0.6349)--(1.9058,0.6449)--(1.9246,0.6547)--(1.9434,0.6644)--(1.9623,0.6741)--(1.9811,0.6836)--(2.0000,0.6931);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -109,27 +109,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.1931);
 %DEFAULT
 
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+\draw [color=blue] (0.1353,-2.0000)--(0.1541,-1.8696)--(0.1730,-1.7544)--(0.1918,-1.6510)--(0.2106,-1.5574)--(0.2295,-1.4718)--(0.2483,-1.3929)--(0.2671,-1.3198)--(0.2860,-1.2517)--(0.3048,-1.1879)--(0.3236,-1.1279)--(0.3425,-1.0714)--(0.3613,-1.0178)--(0.3801,-0.9670)--(0.3990,-0.9187)--(0.4178,-0.8726)--(0.4366,-0.8285)--(0.4555,-0.7862)--(0.4743,-0.7457)--(0.4932,-0.7068)--(0.5120,-0.6693)--(0.5308,-0.6332)--(0.5497,-0.5983)--(0.5685,-0.5646)--(0.5873,-0.5320)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6438,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3833)--(0.7003,-0.3561)--(0.7192,-0.3295)--(0.7380,-0.3037)--(0.7568,-0.2785)--(0.7757,-0.2539)--(0.7945,-0.2299)--(0.8133,-0.2065)--(0.8322,-0.1836)--(0.8510,-0.1612)--(0.8699,-0.1393)--(0.8887,-0.1179)--(0.9075,-0.0969)--(0.9264,-0.0764)--(0.9452,-0.0563)--(0.9640,-0.0365)--(0.9829,-0.0172)--(1.0017,0.0017)--(1.0205,0.0203)--(1.0394,0.0386)--(1.0582,0.0566)--(1.0770,0.0742)--(1.0959,0.0915)--(1.1147,0.1086)--(1.1335,0.1253)--(1.1524,0.1418)--(1.1712,0.1580)--(1.1900,0.1740)--(1.2089,0.1897)--(1.2277,0.2051)--(1.2466,0.2204)--(1.2654,0.2354)--(1.2842,0.2501)--(1.3031,0.2647)--(1.3219,0.2791)--(1.3407,0.2932)--(1.3596,0.3071)--(1.3784,0.3209)--(1.3972,0.3345)--(1.4161,0.3479)--(1.4349,0.3611)--(1.4537,0.3741)--(1.4726,0.3870)--(1.4914,0.3997)--(1.5102,0.4123)--(1.5291,0.4246)--(1.5479,0.4369)--(1.5667,0.4490)--(1.5856,0.4609)--(1.6044,0.4727)--(1.6233,0.4844)--(1.6421,0.4959)--(1.6609,0.5074)--(1.6798,0.5186)--(1.6986,0.5298)--(1.7174,0.5408)--(1.7363,0.5517)--(1.7551,0.5625)--(1.7739,0.5732)--(1.7928,0.5837)--(1.8116,0.5942)--(1.8304,0.6045)--(1.8493,0.6148)--(1.8681,0.6249)--(1.8869,0.6349)--(1.9058,0.6449)--(1.9246,0.6547)--(1.9434,0.6644)--(1.9623,0.6741)--(1.9811,0.6836)--(2.0000,0.6931);
 
-\draw [color=blue] (-2.000,0.6931)--(-1.981,0.6837)--(-1.962,0.6741)--(-1.943,0.6645)--(-1.925,0.6547)--(-1.906,0.6449)--(-1.887,0.6350)--(-1.868,0.6250)--(-1.849,0.6148)--(-1.830,0.6046)--(-1.812,0.5942)--(-1.793,0.5838)--(-1.774,0.5732)--(-1.755,0.5626)--(-1.736,0.5518)--(-1.717,0.5409)--(-1.699,0.5298)--(-1.680,0.5187)--(-1.661,0.5074)--(-1.642,0.4960)--(-1.623,0.4845)--(-1.604,0.4728)--(-1.586,0.4610)--(-1.567,0.4490)--(-1.548,0.4369)--(-1.529,0.4247)--(-1.510,0.4123)--(-1.491,0.3998)--(-1.473,0.3870)--(-1.454,0.3742)--(-1.435,0.3611)--(-1.416,0.3479)--(-1.397,0.3345)--(-1.378,0.3210)--(-1.360,0.3072)--(-1.341,0.2932)--(-1.322,0.2791)--(-1.303,0.2648)--(-1.284,0.2502)--(-1.265,0.2354)--(-1.247,0.2204)--(-1.228,0.2052)--(-1.209,0.1897)--(-1.190,0.1740)--(-1.171,0.1581)--(-1.152,0.1419)--(-1.134,0.1254)--(-1.115,0.1086)--(-1.096,0.09159)--(-1.077,0.07426)--(-1.058,0.05662)--(-1.039,0.03866)--(-1.021,0.02037)--(-1.002,0.001744)--(-0.9829,-0.01724)--(-0.9641,-0.03659)--(-0.9452,-0.05632)--(-0.9264,-0.07644)--(-0.9076,-0.09698)--(-0.8887,-0.1180)--(-0.8699,-0.1394)--(-0.8511,-0.1613)--(-0.8322,-0.1836)--(-0.8134,-0.2065)--(-0.7946,-0.2300)--(-0.7757,-0.2540)--(-0.7569,-0.2785)--(-0.7381,-0.3037)--(-0.7192,-0.3296)--(-0.7004,-0.3561)--(-0.6815,-0.3834)--(-0.6627,-0.4114)--(-0.6439,-0.4402)--(-0.6250,-0.4699)--(-0.6062,-0.5005)--(-0.5874,-0.5321)--(-0.5685,-0.5647)--(-0.5497,-0.5984)--(-0.5309,-0.6332)--(-0.5120,-0.6694)--(-0.4932,-0.7068)--(-0.4744,-0.7458)--(-0.4555,-0.7863)--(-0.4367,-0.8285)--(-0.4179,-0.8726)--(-0.3990,-0.9187)--(-0.3802,-0.9671)--(-0.3614,-1.018)--(-0.3425,-1.071)--(-0.3237,-1.128)--(-0.3049,-1.188)--(-0.2860,-1.252)--(-0.2672,-1.320)--(-0.2483,-1.393)--(-0.2295,-1.472)--(-0.2107,-1.557)--(-0.1918,-1.651)--(-0.1730,-1.754)--(-0.1542,-1.870)--(-0.1353,-2.000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-2.0000,0.6931)--(-1.9811,0.6836)--(-1.9623,0.6741)--(-1.9434,0.6644)--(-1.9246,0.6547)--(-1.9058,0.6449)--(-1.8869,0.6349)--(-1.8681,0.6249)--(-1.8493,0.6148)--(-1.8304,0.6045)--(-1.8116,0.5942)--(-1.7928,0.5837)--(-1.7739,0.5732)--(-1.7551,0.5625)--(-1.7363,0.5517)--(-1.7174,0.5408)--(-1.6986,0.5298)--(-1.6798,0.5186)--(-1.6609,0.5074)--(-1.6421,0.4959)--(-1.6233,0.4844)--(-1.6044,0.4727)--(-1.5856,0.4609)--(-1.5667,0.4490)--(-1.5479,0.4369)--(-1.5291,0.4246)--(-1.5102,0.4123)--(-1.4914,0.3997)--(-1.4726,0.3870)--(-1.4537,0.3741)--(-1.4349,0.3611)--(-1.4161,0.3479)--(-1.3972,0.3345)--(-1.3784,0.3209)--(-1.3596,0.3071)--(-1.3407,0.2932)--(-1.3219,0.2791)--(-1.3031,0.2647)--(-1.2842,0.2501)--(-1.2654,0.2354)--(-1.2466,0.2204)--(-1.2277,0.2051)--(-1.2089,0.1897)--(-1.1900,0.1740)--(-1.1712,0.1580)--(-1.1524,0.1418)--(-1.1335,0.1253)--(-1.1147,0.1086)--(-1.0959,0.0915)--(-1.0770,0.0742)--(-1.0582,0.0566)--(-1.0394,0.0386)--(-1.0205,0.0203)--(-1.0017,0.0017)--(-0.9829,-0.0172)--(-0.9640,-0.0365)--(-0.9452,-0.0563)--(-0.9264,-0.0764)--(-0.9075,-0.0969)--(-0.8887,-0.1179)--(-0.8699,-0.1393)--(-0.8510,-0.1612)--(-0.8322,-0.1836)--(-0.8133,-0.2065)--(-0.7945,-0.2299)--(-0.7757,-0.2539)--(-0.7568,-0.2785)--(-0.7380,-0.3037)--(-0.7192,-0.3295)--(-0.7003,-0.3561)--(-0.6815,-0.3833)--(-0.6627,-0.4114)--(-0.6438,-0.4402)--(-0.6250,-0.4699)--(-0.6062,-0.5005)--(-0.5873,-0.5320)--(-0.5685,-0.5646)--(-0.5497,-0.5983)--(-0.5308,-0.6332)--(-0.5120,-0.6693)--(-0.4932,-0.7068)--(-0.4743,-0.7457)--(-0.4555,-0.7862)--(-0.4366,-0.8285)--(-0.4178,-0.8726)--(-0.3990,-0.9187)--(-0.3801,-0.9670)--(-0.3613,-1.0178)--(-0.3425,-1.0714)--(-0.3236,-1.1279)--(-0.3048,-1.1879)--(-0.2860,-1.2517)--(-0.2671,-1.3198)--(-0.2483,-1.3929)--(-0.2295,-1.4718)--(-0.2106,-1.5574)--(-0.1918,-1.6510)--(-0.1730,-1.7544)--(-0.1541,-1.8696)--(-0.1353,-2.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -175,23 +175,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3647,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.5986);
+\draw [,->,>=latex] (-1.3646,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.5986);
 %DEFAULT
 
-\draw [color=blue] (-0.8647,-2.000)--(-0.8357,-1.806)--(-0.8068,-1.644)--(-0.7779,-1.504)--(-0.7489,-1.382)--(-0.7200,-1.273)--(-0.6910,-1.175)--(-0.6621,-1.085)--(-0.6332,-1.003)--(-0.6042,-0.9269)--(-0.5753,-0.8564)--(-0.5464,-0.7905)--(-0.5174,-0.7286)--(-0.4885,-0.6704)--(-0.4596,-0.6154)--(-0.4306,-0.5632)--(-0.4017,-0.5136)--(-0.3728,-0.4664)--(-0.3438,-0.4213)--(-0.3149,-0.3782)--(-0.2859,-0.3368)--(-0.2570,-0.2971)--(-0.2281,-0.2589)--(-0.1991,-0.2221)--(-0.1702,-0.1866)--(-0.1413,-0.1523)--(-0.1123,-0.1192)--(-0.08339,-0.08708)--(-0.05446,-0.05600)--(-0.02552,-0.02585)--(0.003415,0.003410)--(0.03235,0.03184)--(0.06129,0.05948)--(0.09022,0.08638)--(0.1192,0.1126)--(0.1481,0.1381)--(0.1770,0.1630)--(0.2060,0.1873)--(0.2349,0.2110)--(0.2638,0.2342)--(0.2928,0.2568)--(0.3217,0.2789)--(0.3506,0.3006)--(0.3796,0.3218)--(0.4085,0.3425)--(0.4375,0.3629)--(0.4664,0.3828)--(0.4953,0.4023)--(0.5243,0.4215)--(0.5532,0.4403)--(0.5821,0.4588)--(0.6111,0.4769)--(0.6400,0.4947)--(0.6689,0.5122)--(0.6979,0.5294)--(0.7268,0.5463)--(0.7558,0.5629)--(0.7847,0.5792)--(0.8136,0.5953)--(0.8426,0.6112)--(0.8715,0.6267)--(0.9004,0.6421)--(0.9294,0.6572)--(0.9583,0.6721)--(0.9872,0.6867)--(1.016,0.7012)--(1.045,0.7155)--(1.074,0.7295)--(1.103,0.7434)--(1.132,0.7570)--(1.161,0.7705)--(1.190,0.7838)--(1.219,0.7969)--(1.248,0.8099)--(1.277,0.8227)--(1.306,0.8353)--(1.334,0.8478)--(1.363,0.8601)--(1.392,0.8723)--(1.421,0.8843)--(1.450,0.8962)--(1.479,0.9079)--(1.508,0.9195)--(1.537,0.9310)--(1.566,0.9423)--(1.595,0.9535)--(1.624,0.9646)--(1.653,0.9756)--(1.682,0.9865)--(1.711,0.9972)--(1.740,1.008)--(1.769,1.018)--(1.797,1.029)--(1.826,1.039)--(1.855,1.049)--(1.884,1.059)--(1.913,1.069)--(1.942,1.079)--(1.971,1.089)--(2.000,1.099);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-0.8646,-2.0000)--(-0.8357,-1.8062)--(-0.8067,-1.6439)--(-0.7778,-1.5044)--(-0.7489,-1.3819)--(-0.7199,-1.2729)--(-0.6910,-1.1745)--(-0.6621,-1.0850)--(-0.6331,-1.0028)--(-0.6042,-0.9269)--(-0.5753,-0.8563)--(-0.5463,-0.7904)--(-0.5174,-0.7286)--(-0.4884,-0.6704)--(-0.4595,-0.6153)--(-0.4306,-0.5632)--(-0.4016,-0.5136)--(-0.3727,-0.4664)--(-0.3438,-0.4213)--(-0.3148,-0.3781)--(-0.2859,-0.3367)--(-0.2570,-0.2970)--(-0.2280,-0.2588)--(-0.1991,-0.2220)--(-0.1702,-0.1865)--(-0.1412,-0.1522)--(-0.1123,-0.1191)--(-0.0833,-0.0870)--(-0.0544,-0.0559)--(-0.0255,-0.0258)--(0.0034,0.0034)--(0.0323,0.0318)--(0.0612,0.0594)--(0.0902,0.0863)--(0.1191,0.1125)--(0.1480,0.1381)--(0.1770,0.1629)--(0.2059,0.1872)--(0.2349,0.2109)--(0.2638,0.2341)--(0.2927,0.2567)--(0.3217,0.2789)--(0.3506,0.3005)--(0.3795,0.3217)--(0.4085,0.3425)--(0.4374,0.3628)--(0.4663,0.3828)--(0.4953,0.4023)--(0.5242,0.4215)--(0.5531,0.4403)--(0.5821,0.4587)--(0.6110,0.4768)--(0.6400,0.4947)--(0.6689,0.5121)--(0.6978,0.5293)--(0.7268,0.5462)--(0.7557,0.5628)--(0.7846,0.5792)--(0.8136,0.5953)--(0.8425,0.6111)--(0.8714,0.6267)--(0.9004,0.6420)--(0.9293,0.6571)--(0.9583,0.6720)--(0.9872,0.6867)--(1.0161,0.7012)--(1.0451,0.7154)--(1.0740,0.7295)--(1.1029,0.7433)--(1.1319,0.7570)--(1.1608,0.7705)--(1.1897,0.7838)--(1.2187,0.7969)--(1.2476,0.8098)--(1.2765,0.8226)--(1.3055,0.8353)--(1.3344,0.8477)--(1.3634,0.8601)--(1.3923,0.8722)--(1.4212,0.8842)--(1.4502,0.8961)--(1.4791,0.9079)--(1.5080,0.9195)--(1.5370,0.9309)--(1.5659,0.9423)--(1.5948,0.9535)--(1.6238,0.9646)--(1.6527,0.9756)--(1.6817,0.9864)--(1.7106,0.9971)--(1.7395,1.0078)--(1.7685,1.0183)--(1.7974,1.0287)--(1.8263,1.0389)--(1.8553,1.0491)--(1.8842,1.0592)--(1.9131,1.0692)--(1.9421,1.0791)--(1.9710,1.0889)--(2.0000,1.0986);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -233,21 +233,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,2.1931);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.1931);
 %DEFAULT
 
-\draw [color=blue] (0.1353,-1.000)--(0.1542,-0.8697)--(0.1730,-0.7544)--(0.1918,-0.6511)--(0.2107,-0.5574)--(0.2295,-0.4718)--(0.2483,-0.3929)--(0.2672,-0.3198)--(0.2860,-0.2517)--(0.3049,-0.1879)--(0.3237,-0.1280)--(0.3425,-0.07142)--(0.3614,-0.01789)--(0.3802,0.03292)--(0.3990,0.08127)--(0.4179,0.1274)--(0.4367,0.1715)--(0.4555,0.2137)--(0.4744,0.2542)--(0.4932,0.2932)--(0.5120,0.3306)--(0.5309,0.3668)--(0.5497,0.4016)--(0.5685,0.4353)--(0.5874,0.4679)--(0.6062,0.4995)--(0.6250,0.5301)--(0.6439,0.5598)--(0.6627,0.5886)--(0.6815,0.6166)--(0.7004,0.6439)--(0.7192,0.6704)--(0.7381,0.6963)--(0.7569,0.7215)--(0.7757,0.7460)--(0.7946,0.7700)--(0.8134,0.7935)--(0.8322,0.8164)--(0.8511,0.8387)--(0.8699,0.8606)--(0.8887,0.8820)--(0.9076,0.9030)--(0.9264,0.9236)--(0.9452,0.9437)--(0.9641,0.9634)--(0.9829,0.9828)--(1.002,1.002)--(1.021,1.020)--(1.039,1.039)--(1.058,1.057)--(1.077,1.074)--(1.096,1.092)--(1.115,1.109)--(1.134,1.125)--(1.152,1.142)--(1.171,1.158)--(1.190,1.174)--(1.209,1.190)--(1.228,1.205)--(1.247,1.220)--(1.265,1.235)--(1.284,1.250)--(1.303,1.265)--(1.322,1.279)--(1.341,1.293)--(1.360,1.307)--(1.378,1.321)--(1.397,1.335)--(1.416,1.348)--(1.435,1.361)--(1.454,1.374)--(1.473,1.387)--(1.491,1.400)--(1.510,1.412)--(1.529,1.425)--(1.548,1.437)--(1.567,1.449)--(1.586,1.461)--(1.604,1.473)--(1.623,1.484)--(1.642,1.496)--(1.661,1.507)--(1.680,1.519)--(1.699,1.530)--(1.717,1.541)--(1.736,1.552)--(1.755,1.563)--(1.774,1.573)--(1.793,1.584)--(1.812,1.594)--(1.830,1.605)--(1.849,1.615)--(1.868,1.625)--(1.887,1.635)--(1.906,1.645)--(1.925,1.655)--(1.943,1.664)--(1.962,1.674)--(1.981,1.684)--(2.000,1.693);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.1353,-1.0000)--(0.1541,-0.8696)--(0.1730,-0.7544)--(0.1918,-0.6510)--(0.2106,-0.5574)--(0.2295,-0.4718)--(0.2483,-0.3929)--(0.2671,-0.3198)--(0.2860,-0.2517)--(0.3048,-0.1879)--(0.3236,-0.1279)--(0.3425,-0.0714)--(0.3613,-0.0178)--(0.3801,0.0329)--(0.3990,0.0812)--(0.4178,0.1273)--(0.4366,0.1714)--(0.4555,0.2137)--(0.4743,0.2542)--(0.4932,0.2931)--(0.5120,0.3306)--(0.5308,0.3667)--(0.5497,0.4016)--(0.5685,0.4353)--(0.5873,0.4679)--(0.6062,0.4994)--(0.6250,0.5300)--(0.6438,0.5597)--(0.6627,0.5885)--(0.6815,0.6166)--(0.7003,0.6438)--(0.7192,0.6704)--(0.7380,0.6962)--(0.7568,0.7214)--(0.7757,0.7460)--(0.7945,0.7700)--(0.8133,0.7934)--(0.8322,0.8163)--(0.8510,0.8387)--(0.8699,0.8606)--(0.8887,0.8820)--(0.9075,0.9030)--(0.9264,0.9235)--(0.9452,0.9436)--(0.9640,0.9634)--(0.9829,0.9827)--(1.0017,1.0017)--(1.0205,1.0203)--(1.0394,1.0386)--(1.0582,1.0566)--(1.0770,1.0742)--(1.0959,1.0915)--(1.1147,1.1086)--(1.1335,1.1253)--(1.1524,1.1418)--(1.1712,1.1580)--(1.1900,1.1740)--(1.2089,1.1897)--(1.2277,1.2051)--(1.2466,1.2204)--(1.2654,1.2354)--(1.2842,1.2501)--(1.3031,1.2647)--(1.3219,1.2791)--(1.3407,1.2932)--(1.3596,1.3071)--(1.3784,1.3209)--(1.3972,1.3345)--(1.4161,1.3479)--(1.4349,1.3611)--(1.4537,1.3741)--(1.4726,1.3870)--(1.4914,1.3997)--(1.5102,1.4123)--(1.5291,1.4246)--(1.5479,1.4369)--(1.5667,1.4490)--(1.5856,1.4609)--(1.6044,1.4727)--(1.6233,1.4844)--(1.6421,1.4959)--(1.6609,1.5074)--(1.6798,1.5186)--(1.6986,1.5298)--(1.7174,1.5408)--(1.7363,1.5517)--(1.7551,1.5625)--(1.7739,1.5732)--(1.7928,1.5837)--(1.8116,1.5942)--(1.8304,1.6045)--(1.8493,1.6148)--(1.8681,1.6249)--(1.8869,1.6349)--(1.9058,1.6449)--(1.9246,1.6547)--(1.9434,1.6644)--(1.9623,1.6741)--(1.9811,1.6836)--(2.0000,1.6931);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -289,21 +289,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (0.1353,2.000)--(0.1643,1.806)--(0.1932,1.644)--(0.2221,1.504)--(0.2511,1.382)--(0.2800,1.273)--(0.3090,1.175)--(0.3379,1.085)--(0.3668,1.003)--(0.3958,0.9269)--(0.4247,0.8564)--(0.4536,0.7905)--(0.4826,0.7286)--(0.5115,0.6704)--(0.5404,0.6154)--(0.5694,0.5632)--(0.5983,0.5136)--(0.6273,0.4664)--(0.6562,0.4213)--(0.6851,0.3782)--(0.7141,0.3368)--(0.7430,0.2971)--(0.7719,0.2589)--(0.8009,0.2221)--(0.8298,0.1866)--(0.8587,0.1523)--(0.8877,0.1192)--(0.9166,0.08708)--(0.9455,0.05600)--(0.9745,0.02585)--(1.003,0.003410)--(1.032,0.03184)--(1.061,0.05948)--(1.090,0.08638)--(1.119,0.1126)--(1.148,0.1381)--(1.177,0.1630)--(1.206,0.1873)--(1.235,0.2110)--(1.264,0.2342)--(1.293,0.2568)--(1.322,0.2789)--(1.351,0.3006)--(1.380,0.3218)--(1.409,0.3425)--(1.437,0.3629)--(1.466,0.3828)--(1.495,0.4023)--(1.524,0.4215)--(1.553,0.4403)--(1.582,0.4588)--(1.611,0.4769)--(1.640,0.4947)--(1.669,0.5122)--(1.698,0.5294)--(1.727,0.5463)--(1.756,0.5629)--(1.785,0.5792)--(1.814,0.5953)--(1.843,0.6112)--(1.871,0.6267)--(1.900,0.6421)--(1.929,0.6572)--(1.958,0.6721)--(1.987,0.6867)--(2.016,0.7012)--(2.045,0.7155)--(2.074,0.7295)--(2.103,0.7434)--(2.132,0.7570)--(2.161,0.7705)--(2.190,0.7838)--(2.219,0.7969)--(2.248,0.8099)--(2.277,0.8227)--(2.306,0.8353)--(2.334,0.8478)--(2.363,0.8601)--(2.392,0.8723)--(2.421,0.8843)--(2.450,0.8962)--(2.479,0.9079)--(2.508,0.9195)--(2.537,0.9310)--(2.566,0.9423)--(2.595,0.9535)--(2.624,0.9646)--(2.653,0.9756)--(2.682,0.9865)--(2.711,0.9972)--(2.740,1.008)--(2.769,1.018)--(2.797,1.029)--(2.826,1.039)--(2.855,1.049)--(2.884,1.059)--(2.913,1.069)--(2.942,1.079)--(2.971,1.089)--(3.000,1.099);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.1353,2.0000)--(0.1642,1.8062)--(0.1932,1.6439)--(0.2221,1.5044)--(0.2510,1.3819)--(0.2800,1.2729)--(0.3089,1.1745)--(0.3378,1.0850)--(0.3668,1.0028)--(0.3957,0.9269)--(0.4246,0.8563)--(0.4536,0.7904)--(0.4825,0.7286)--(0.5115,0.6704)--(0.5404,0.6153)--(0.5693,0.5632)--(0.5983,0.5136)--(0.6272,0.4664)--(0.6561,0.4213)--(0.6851,0.3781)--(0.7140,0.3367)--(0.7429,0.2970)--(0.7719,0.2588)--(0.8008,0.2220)--(0.8297,0.1865)--(0.8587,0.1522)--(0.8876,0.1191)--(0.9166,0.0870)--(0.9455,0.0559)--(0.9744,0.0258)--(1.0034,0.0034)--(1.0323,0.0318)--(1.0612,0.0594)--(1.0902,0.0863)--(1.1191,0.1125)--(1.1480,0.1381)--(1.1770,0.1629)--(1.2059,0.1872)--(1.2349,0.2109)--(1.2638,0.2341)--(1.2927,0.2567)--(1.3217,0.2789)--(1.3506,0.3005)--(1.3795,0.3217)--(1.4085,0.3425)--(1.4374,0.3628)--(1.4663,0.3828)--(1.4953,0.4023)--(1.5242,0.4215)--(1.5531,0.4403)--(1.5821,0.4587)--(1.6110,0.4768)--(1.6400,0.4947)--(1.6689,0.5121)--(1.6978,0.5293)--(1.7268,0.5462)--(1.7557,0.5628)--(1.7846,0.5792)--(1.8136,0.5953)--(1.8425,0.6111)--(1.8714,0.6267)--(1.9004,0.6420)--(1.9293,0.6571)--(1.9583,0.6720)--(1.9872,0.6867)--(2.0161,0.7012)--(2.0451,0.7154)--(2.0740,0.7295)--(2.1029,0.7433)--(2.1319,0.7570)--(2.1608,0.7705)--(2.1897,0.7838)--(2.2187,0.7969)--(2.2476,0.8098)--(2.2765,0.8226)--(2.3055,0.8353)--(2.3344,0.8477)--(2.3634,0.8601)--(2.3923,0.8722)--(2.4212,0.8842)--(2.4502,0.8961)--(2.4791,0.9079)--(2.5080,0.9195)--(2.5370,0.9309)--(2.5659,0.9423)--(2.5948,0.9535)--(2.6238,0.9646)--(2.6527,0.9756)--(2.6817,0.9864)--(2.7106,0.9971)--(2.7395,1.0078)--(2.7685,1.0183)--(2.7974,1.0287)--(2.8263,1.0389)--(2.8553,1.0491)--(2.8842,1.0592)--(2.9131,1.0692)--(2.9421,1.0791)--(2.9710,1.0889)--(3.0000,1.0986);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -345,19 +345,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5481);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5481);
 %DEFAULT
 
-\draw [color=blue] (1.000,0)--(1.020,0.1414)--(1.040,0.1990)--(1.061,0.2426)--(1.081,0.2788)--(1.101,0.3102)--(1.121,0.3382)--(1.141,0.3637)--(1.162,0.3871)--(1.182,0.4087)--(1.202,0.4290)--(1.222,0.4480)--(1.242,0.4659)--(1.263,0.4829)--(1.283,0.4991)--(1.303,0.5145)--(1.323,0.5292)--(1.343,0.5434)--(1.364,0.5569)--(1.384,0.5700)--(1.404,0.5825)--(1.424,0.5947)--(1.444,0.6064)--(1.465,0.6178)--(1.485,0.6287)--(1.505,0.6394)--(1.525,0.6497)--(1.545,0.6598)--(1.566,0.6696)--(1.586,0.6791)--(1.606,0.6883)--(1.626,0.6973)--(1.646,0.7061)--(1.667,0.7147)--(1.687,0.7231)--(1.707,0.7313)--(1.727,0.7393)--(1.747,0.7471)--(1.768,0.7548)--(1.788,0.7623)--(1.808,0.7696)--(1.828,0.7768)--(1.848,0.7838)--(1.869,0.7907)--(1.889,0.7975)--(1.909,0.8041)--(1.929,0.8107)--(1.949,0.8170)--(1.970,0.8233)--(1.990,0.8295)--(2.010,0.8356)--(2.030,0.8415)--(2.051,0.8474)--(2.071,0.8532)--(2.091,0.8588)--(2.111,0.8644)--(2.131,0.8699)--(2.152,0.8753)--(2.172,0.8806)--(2.192,0.8859)--(2.212,0.8910)--(2.232,0.8961)--(2.253,0.9011)--(2.273,0.9061)--(2.293,0.9109)--(2.313,0.9157)--(2.333,0.9205)--(2.354,0.9252)--(2.374,0.9298)--(2.394,0.9343)--(2.414,0.9388)--(2.434,0.9432)--(2.455,0.9476)--(2.475,0.9519)--(2.495,0.9562)--(2.515,0.9604)--(2.535,0.9645)--(2.556,0.9686)--(2.576,0.9727)--(2.596,0.9767)--(2.616,0.9807)--(2.636,0.9846)--(2.657,0.9884)--(2.677,0.9923)--(2.697,0.9961)--(2.717,0.9998)--(2.737,1.003)--(2.758,1.007)--(2.778,1.011)--(2.798,1.014)--(2.818,1.018)--(2.838,1.021)--(2.859,1.025)--(2.879,1.028)--(2.899,1.032)--(2.919,1.035)--(2.939,1.038)--(2.960,1.042)--(2.980,1.045)--(3.000,1.048);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (1.0000,0.0000)--(1.0202,0.1414)--(1.0404,0.1990)--(1.0606,0.2425)--(1.0808,0.2787)--(1.1010,0.3102)--(1.1212,0.3382)--(1.1414,0.3636)--(1.1616,0.3870)--(1.1818,0.4087)--(1.2020,0.4289)--(1.2222,0.4479)--(1.2424,0.4659)--(1.2626,0.4829)--(1.2828,0.4990)--(1.3030,0.5144)--(1.3232,0.5292)--(1.3434,0.5433)--(1.3636,0.5569)--(1.3838,0.5699)--(1.4040,0.5825)--(1.4242,0.5946)--(1.4444,0.6064)--(1.4646,0.6177)--(1.4848,0.6287)--(1.5050,0.6393)--(1.5252,0.6497)--(1.5454,0.6597)--(1.5656,0.6695)--(1.5858,0.6790)--(1.6060,0.6883)--(1.6262,0.6973)--(1.6464,0.7061)--(1.6666,0.7147)--(1.6868,0.7231)--(1.7070,0.7312)--(1.7272,0.7392)--(1.7474,0.7471)--(1.7676,0.7547)--(1.7878,0.7622)--(1.8080,0.7695)--(1.8282,0.7767)--(1.8484,0.7838)--(1.8686,0.7907)--(1.8888,0.7974)--(1.9090,0.8041)--(1.9292,0.8106)--(1.9494,0.8170)--(1.9696,0.8233)--(1.9898,0.8295)--(2.0101,0.8355)--(2.0303,0.8415)--(2.0505,0.8473)--(2.0707,0.8531)--(2.0909,0.8588)--(2.1111,0.8644)--(2.1313,0.8699)--(2.1515,0.8753)--(2.1717,0.8806)--(2.1919,0.8858)--(2.2121,0.8910)--(2.2323,0.8961)--(2.2525,0.9011)--(2.2727,0.9060)--(2.2929,0.9109)--(2.3131,0.9157)--(2.3333,0.9204)--(2.3535,0.9251)--(2.3737,0.9297)--(2.3939,0.9343)--(2.4141,0.9387)--(2.4343,0.9432)--(2.4545,0.9475)--(2.4747,0.9519)--(2.4949,0.9561)--(2.5151,0.9603)--(2.5353,0.9645)--(2.5555,0.9686)--(2.5757,0.9726)--(2.5959,0.9767)--(2.6161,0.9806)--(2.6363,0.9845)--(2.6565,0.9884)--(2.6767,0.9922)--(2.6969,0.9960)--(2.7171,0.9997)--(2.7373,1.0034)--(2.7575,1.0071)--(2.7777,1.0107)--(2.7979,1.0143)--(2.8181,1.0178)--(2.8383,1.0213)--(2.8585,1.0248)--(2.8787,1.0282)--(2.8989,1.0316)--(2.9191,1.0350)--(2.9393,1.0383)--(2.9595,1.0416)--(2.9797,1.0449)--(3.0000,1.0481);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -411,31 +411,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,1.8863);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,1.8862);
 %DEFAULT
 
-\draw [color=blue] (0.1353,-4.000)--(0.1542,-3.739)--(0.1730,-3.509)--(0.1918,-3.302)--(0.2107,-3.115)--(0.2295,-2.944)--(0.2483,-2.786)--(0.2672,-2.640)--(0.2860,-2.503)--(0.3049,-2.376)--(0.3237,-2.256)--(0.3425,-2.143)--(0.3614,-2.036)--(0.3802,-1.934)--(0.3990,-1.837)--(0.4179,-1.745)--(0.4367,-1.657)--(0.4555,-1.573)--(0.4744,-1.492)--(0.4932,-1.414)--(0.5120,-1.339)--(0.5309,-1.266)--(0.5497,-1.197)--(0.5685,-1.129)--(0.5874,-1.064)--(0.6062,-1.001)--(0.6250,-0.9399)--(0.6439,-0.8805)--(0.6627,-0.8228)--(0.6815,-0.7668)--(0.7004,-0.7122)--(0.7192,-0.6592)--(0.7381,-0.6075)--(0.7569,-0.5571)--(0.7757,-0.5079)--(0.7946,-0.4599)--(0.8134,-0.4131)--(0.8322,-0.3673)--(0.8511,-0.3225)--(0.8699,-0.2788)--(0.8887,-0.2359)--(0.9076,-0.1940)--(0.9264,-0.1529)--(0.9452,-0.1126)--(0.9641,-0.07317)--(0.9829,-0.03448)--(1.002,0.003487)--(1.021,0.04074)--(1.039,0.07732)--(1.058,0.1132)--(1.077,0.1485)--(1.096,0.1832)--(1.115,0.2173)--(1.134,0.2508)--(1.152,0.2837)--(1.171,0.3162)--(1.190,0.3481)--(1.209,0.3795)--(1.228,0.4104)--(1.247,0.4408)--(1.265,0.4708)--(1.284,0.5004)--(1.303,0.5295)--(1.322,0.5582)--(1.341,0.5865)--(1.360,0.6144)--(1.378,0.6419)--(1.397,0.6691)--(1.416,0.6958)--(1.435,0.7223)--(1.454,0.7483)--(1.473,0.7741)--(1.491,0.7995)--(1.510,0.8246)--(1.529,0.8494)--(1.548,0.8739)--(1.567,0.8981)--(1.586,0.9220)--(1.604,0.9456)--(1.623,0.9689)--(1.642,0.9920)--(1.661,1.015)--(1.680,1.037)--(1.699,1.060)--(1.717,1.082)--(1.736,1.104)--(1.755,1.125)--(1.774,1.146)--(1.793,1.168)--(1.812,1.188)--(1.830,1.209)--(1.849,1.230)--(1.868,1.250)--(1.887,1.270)--(1.906,1.290)--(1.925,1.309)--(1.943,1.329)--(1.962,1.348)--(1.981,1.367)--(2.000,1.386);
+\draw [color=blue] (0.1353,-4.0000)--(0.1541,-3.7393)--(0.1730,-3.5088)--(0.1918,-3.3021)--(0.2106,-3.1148)--(0.2295,-2.9436)--(0.2483,-2.7858)--(0.2671,-2.6396)--(0.2860,-2.5034)--(0.3048,-2.3758)--(0.3236,-2.2559)--(0.3425,-2.1428)--(0.3613,-2.0357)--(0.3801,-1.9341)--(0.3990,-1.8374)--(0.4178,-1.7452)--(0.4366,-1.6570)--(0.4555,-1.5725)--(0.4743,-1.4915)--(0.4932,-1.4136)--(0.5120,-1.3387)--(0.5308,-1.2664)--(0.5497,-1.1967)--(0.5685,-1.1293)--(0.5873,-1.0641)--(0.6062,-1.0010)--(0.6250,-0.9398)--(0.6438,-0.8804)--(0.6627,-0.8228)--(0.6815,-0.7667)--(0.7003,-0.7122)--(0.7192,-0.6591)--(0.7380,-0.6074)--(0.7568,-0.5570)--(0.7757,-0.5079)--(0.7945,-0.4599)--(0.8133,-0.4130)--(0.8322,-0.3672)--(0.8510,-0.3225)--(0.8699,-0.2787)--(0.8887,-0.2359)--(0.9075,-0.1939)--(0.9264,-0.1528)--(0.9452,-0.1126)--(0.9640,-0.0731)--(0.9829,-0.0344)--(1.0017,0.0034)--(1.0205,0.0407)--(1.0394,0.0773)--(1.0582,0.1132)--(1.0770,0.1485)--(1.0959,0.1831)--(1.1147,0.2172)--(1.1335,0.2507)--(1.1524,0.2837)--(1.1712,0.3161)--(1.1900,0.3480)--(1.2089,0.3794)--(1.2277,0.4103)--(1.2466,0.4408)--(1.2654,0.4708)--(1.2842,0.5003)--(1.3031,0.5294)--(1.3219,0.5582)--(1.3407,0.5864)--(1.3596,0.6143)--(1.3784,0.6419)--(1.3972,0.6690)--(1.4161,0.6958)--(1.4349,0.7222)--(1.4537,0.7483)--(1.4726,0.7740)--(1.4914,0.7995)--(1.5102,0.8246)--(1.5291,0.8493)--(1.5479,0.8738)--(1.5667,0.8980)--(1.5856,0.9219)--(1.6044,0.9455)--(1.6233,0.9689)--(1.6421,0.9919)--(1.6609,1.0148)--(1.6798,1.0373)--(1.6986,1.0596)--(1.7174,1.0817)--(1.7363,1.1035)--(1.7551,1.1251)--(1.7739,1.1464)--(1.7928,1.1675)--(1.8116,1.1884)--(1.8304,1.2091)--(1.8493,1.2296)--(1.8681,1.2499)--(1.8869,1.2699)--(1.9058,1.2898)--(1.9246,1.3094)--(1.9434,1.3289)--(1.9623,1.3482)--(1.9811,1.3673)--(2.0000,1.3862);
 
-\draw [color=blue] (-2.000,1.386)--(-1.981,1.367)--(-1.962,1.348)--(-1.943,1.329)--(-1.925,1.309)--(-1.906,1.290)--(-1.887,1.270)--(-1.868,1.250)--(-1.849,1.230)--(-1.830,1.209)--(-1.812,1.188)--(-1.793,1.168)--(-1.774,1.146)--(-1.755,1.125)--(-1.736,1.104)--(-1.717,1.082)--(-1.699,1.060)--(-1.680,1.037)--(-1.661,1.015)--(-1.642,0.9920)--(-1.623,0.9689)--(-1.604,0.9456)--(-1.586,0.9220)--(-1.567,0.8981)--(-1.548,0.8739)--(-1.529,0.8494)--(-1.510,0.8246)--(-1.491,0.7995)--(-1.473,0.7741)--(-1.454,0.7483)--(-1.435,0.7223)--(-1.416,0.6958)--(-1.397,0.6691)--(-1.378,0.6419)--(-1.360,0.6144)--(-1.341,0.5865)--(-1.322,0.5582)--(-1.303,0.5295)--(-1.284,0.5004)--(-1.265,0.4708)--(-1.247,0.4408)--(-1.228,0.4104)--(-1.209,0.3795)--(-1.190,0.3481)--(-1.171,0.3162)--(-1.152,0.2837)--(-1.134,0.2508)--(-1.115,0.2173)--(-1.096,0.1832)--(-1.077,0.1485)--(-1.058,0.1132)--(-1.039,0.07732)--(-1.021,0.04074)--(-1.002,0.003487)--(-0.9829,-0.03448)--(-0.9641,-0.07317)--(-0.9452,-0.1126)--(-0.9264,-0.1529)--(-0.9076,-0.1940)--(-0.8887,-0.2359)--(-0.8699,-0.2788)--(-0.8511,-0.3225)--(-0.8322,-0.3673)--(-0.8134,-0.4131)--(-0.7946,-0.4599)--(-0.7757,-0.5079)--(-0.7569,-0.5571)--(-0.7381,-0.6075)--(-0.7192,-0.6592)--(-0.7004,-0.7122)--(-0.6815,-0.7668)--(-0.6627,-0.8228)--(-0.6439,-0.8805)--(-0.6250,-0.9399)--(-0.6062,-1.001)--(-0.5874,-1.064)--(-0.5685,-1.129)--(-0.5497,-1.197)--(-0.5309,-1.266)--(-0.5120,-1.339)--(-0.4932,-1.414)--(-0.4744,-1.492)--(-0.4555,-1.573)--(-0.4367,-1.657)--(-0.4179,-1.745)--(-0.3990,-1.837)--(-0.3802,-1.934)--(-0.3614,-2.036)--(-0.3425,-2.143)--(-0.3237,-2.256)--(-0.3049,-2.376)--(-0.2860,-2.503)--(-0.2672,-2.640)--(-0.2483,-2.786)--(-0.2295,-2.944)--(-0.2107,-3.115)--(-0.1918,-3.302)--(-0.1730,-3.509)--(-0.1542,-3.739)--(-0.1353,-4.000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-2.0000,1.3862)--(-1.9811,1.3673)--(-1.9623,1.3482)--(-1.9434,1.3289)--(-1.9246,1.3094)--(-1.9058,1.2898)--(-1.8869,1.2699)--(-1.8681,1.2499)--(-1.8493,1.2296)--(-1.8304,1.2091)--(-1.8116,1.1884)--(-1.7928,1.1675)--(-1.7739,1.1464)--(-1.7551,1.1251)--(-1.7363,1.1035)--(-1.7174,1.0817)--(-1.6986,1.0596)--(-1.6798,1.0373)--(-1.6609,1.0148)--(-1.6421,0.9919)--(-1.6233,0.9689)--(-1.6044,0.9455)--(-1.5856,0.9219)--(-1.5667,0.8980)--(-1.5479,0.8738)--(-1.5291,0.8493)--(-1.5102,0.8246)--(-1.4914,0.7995)--(-1.4726,0.7740)--(-1.4537,0.7483)--(-1.4349,0.7222)--(-1.4161,0.6958)--(-1.3972,0.6690)--(-1.3784,0.6419)--(-1.3596,0.6143)--(-1.3407,0.5864)--(-1.3219,0.5582)--(-1.3031,0.5294)--(-1.2842,0.5003)--(-1.2654,0.4708)--(-1.2466,0.4408)--(-1.2277,0.4103)--(-1.2089,0.3794)--(-1.1900,0.3480)--(-1.1712,0.3161)--(-1.1524,0.2837)--(-1.1335,0.2507)--(-1.1147,0.2172)--(-1.0959,0.1831)--(-1.0770,0.1485)--(-1.0582,0.1132)--(-1.0394,0.0773)--(-1.0205,0.0407)--(-1.0017,0.0034)--(-0.9829,-0.0344)--(-0.9640,-0.0731)--(-0.9452,-0.1126)--(-0.9264,-0.1528)--(-0.9075,-0.1939)--(-0.8887,-0.2359)--(-0.8699,-0.2787)--(-0.8510,-0.3225)--(-0.8322,-0.3672)--(-0.8133,-0.4130)--(-0.7945,-0.4599)--(-0.7757,-0.5079)--(-0.7568,-0.5570)--(-0.7380,-0.6074)--(-0.7192,-0.6591)--(-0.7003,-0.7122)--(-0.6815,-0.7667)--(-0.6627,-0.8228)--(-0.6438,-0.8804)--(-0.6250,-0.9398)--(-0.6062,-1.0010)--(-0.5873,-1.0641)--(-0.5685,-1.1293)--(-0.5497,-1.1967)--(-0.5308,-1.2664)--(-0.5120,-1.3387)--(-0.4932,-1.4136)--(-0.4743,-1.4915)--(-0.4555,-1.5725)--(-0.4366,-1.6570)--(-0.4178,-1.7452)--(-0.3990,-1.8374)--(-0.3801,-1.9341)--(-0.3613,-2.0357)--(-0.3425,-2.1428)--(-0.3236,-2.2559)--(-0.3048,-2.3758)--(-0.2860,-2.5034)--(-0.2671,-2.6396)--(-0.2483,-2.7858)--(-0.2295,-2.9436)--(-0.2106,-3.1148)--(-0.1918,-3.3021)--(-0.1730,-3.5088)--(-0.1541,-3.7393)--(-0.1353,-4.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ADUGmRRA.pstricks.recall b/src_phystricks/Fig_ADUGmRRA.pstricks.recall
index 3eafef08f..16ac4b910 100644
--- a/src_phystricks/Fig_ADUGmRRA.pstricks.recall
+++ b/src_phystricks/Fig_ADUGmRRA.pstricks.recall
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.20595) node {\( \alpha_i\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2059) node {\( \alpha_i\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2191) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ADUGmRRB.pstricks.recall b/src_phystricks/Fig_ADUGmRRB.pstricks.recall
index 70dd143f3..4932d0624 100644
--- a/src_phystricks/Fig_ADUGmRRB.pstricks.recall
+++ b/src_phystricks/Fig_ADUGmRRB.pstricks.recall
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.20595) node {\( \alpha_i\)};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.2059) node {\( \alpha_i\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.2191) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ADUGmRRC.pstricks.recall b/src_phystricks/Fig_ADUGmRRC.pstricks.recall
index 8ffd11439..e384890b1 100644
--- a/src_phystricks/Fig_ADUGmRRC.pstricks.recall
+++ b/src_phystricks/Fig_ADUGmRRC.pstricks.recall
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.20595) node {\( \alpha_i\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.2059) node {\( \alpha_i\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2191) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall b/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall
index 4e8788b1b..10f8b5696 100644
--- a/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall
+++ b/src_phystricks/Fig_ASHYooUVHkak.pstricks.recall
@@ -72,21 +72,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.9000);
+\draw [,->,>=latex] (-2.4000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.9000);
 %DEFAULT
-\draw [style=dashed] (-1.90,1.20) -- (3.00,1.20);
-\draw []  (0,1.2000) node [rotate=0] {$\bullet$};
-\draw (0.22944,1.4682) node {\( \delta\)};
-\draw [] (-0.300,1.40) -- (0.300,1.00);
-\draw [] (1.70,1.40) -- (2.30,1.00);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.33963) node {\( t_1\)};
-\draw []  (-1.5000,0) node [rotate=0] {$\bullet$};
-\draw (-1.5000,-0.33963) node {\( t_2\)};
-\draw [style=dotted] (2.00,0) -- (2.00,1.20);
-\draw [style=dotted] (-1.50,0) -- (-1.50,1.20);
-\draw [] (-1.80,1.40) -- (-1.20,1.00);
+\draw [style=dashed] (-1.9000,1.2000) -- (3.0000,1.2000);
+\draw []  (0.0000,1.2000) node [rotate=0] {$\bullet$};
+\draw (0.2294,1.4681) node {\( \delta\)};
+\draw [] (-0.3000,1.4000) -- (0.3000,1.0000);
+\draw [] (1.7000,1.4000) -- (2.3000,1.0000);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3396) node {\( t_1\)};
+\draw []  (-1.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.3396) node {\( t_2\)};
+\draw [style=dotted] (2.0000,0.0000) -- (2.0000,1.2000);
+\draw [style=dotted] (-1.5000,0.0000) -- (-1.5000,1.2000);
+\draw [] (-1.8000,1.4000) -- (-1.2000,1.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_AccumulationIsole.pstricks.recall b/src_phystricks/Fig_AccumulationIsole.pstricks.recall
index 3b674d22b..9277d4a0b 100644
--- a/src_phystricks/Fig_AccumulationIsole.pstricks.recall
+++ b/src_phystricks/Fig_AccumulationIsole.pstricks.recall
@@ -77,20 +77,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\fill [color=lightgray] (1.00,0) -- (0.998,0.0634) -- (0.992,0.127) -- (0.982,0.189) -- (0.968,0.251) -- (0.950,0.312) -- (0.928,0.372) -- (0.903,0.430) -- (0.874,0.486) -- (0.841,0.541) -- (0.805,0.593) -- (0.766,0.643) -- (0.724,0.690) -- (0.679,0.735) -- (0.631,0.776) -- (0.580,0.815) -- (0.527,0.850) -- (0.472,0.881) -- (0.415,0.910) -- (0.357,0.934) -- (0.297,0.955) -- (0.236,0.972) -- (0.174,0.985) -- (0.111,0.994) -- (0.0476,0.999) -- (-0.0159,1.00) -- (-0.0792,0.997) -- (-0.142,0.990) -- (-0.205,0.979) -- (-0.266,0.964) -- (-0.327,0.945) -- (-0.386,0.922) -- (-0.444,0.896) -- (-0.500,0.866) -- (-0.554,0.833) -- (-0.606,0.796) -- (-0.655,0.756) -- (-0.701,0.713) -- (-0.745,0.667) -- (-0.786,0.618) -- (-0.824,0.567) -- (-0.858,0.514) -- (-0.889,0.458) -- (-0.916,0.401) -- (-0.940,0.342) -- (-0.959,0.282) -- (-0.975,0.220) -- (-0.987,0.158) -- (-0.995,0.0951) -- (-1.00,0.0317) -- (-1.00,-0.0317) -- (-0.995,-0.0951) -- (-0.987,-0.158) -- (-0.975,-0.220) -- (-0.959,-0.282) -- (-0.940,-0.342) -- (-0.916,-0.401) -- (-0.889,-0.458) -- (-0.858,-0.514) -- (-0.824,-0.567) -- (-0.786,-0.618) -- (-0.745,-0.667) -- (-0.701,-0.713) -- (-0.655,-0.756) -- (-0.606,-0.796) -- (-0.554,-0.833) -- (-0.500,-0.866) -- (-0.444,-0.896) -- (-0.386,-0.922) -- (-0.327,-0.945) -- (-0.266,-0.964) -- (-0.205,-0.979) -- (-0.142,-0.990) -- (-0.0792,-0.997) -- (-0.0159,-1.00) -- (0.0476,-0.999) -- (0.111,-0.994) -- (0.174,-0.985) -- (0.236,-0.972) -- (0.297,-0.955) -- (0.357,-0.934) -- (0.415,-0.910) -- (0.472,-0.881) -- (0.527,-0.850) -- (0.580,-0.815) -- (0.631,-0.776) -- (0.679,-0.735) -- (0.724,-0.690) -- (0.766,-0.643) -- (0.805,-0.593) -- (0.841,-0.541) -- (0.874,-0.486) -- (0.903,-0.430) -- (0.928,-0.372) -- (0.950,-0.312) -- (0.968,-0.251) -- (0.982,-0.189) -- (0.992,-0.127) -- (0.998,-0.0634) -- (1.00,0) --  cycle;
-\draw [color=red] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
+\fill [color=lightgray] (1.0000,0.0000) -- (0.9979,0.0634) -- (0.9919,0.1265) -- (0.9819,0.1892) -- (0.9679,0.2511) -- (0.9500,0.3120) -- (0.9283,0.3716) -- (0.9029,0.4297) -- (0.8738,0.4861) -- (0.8412,0.5406) -- (0.8052,0.5929) -- (0.7660,0.6427) -- (0.7237,0.6900) -- (0.6785,0.7345) -- (0.6305,0.7761) -- (0.5800,0.8145) -- (0.5272,0.8497) -- (0.4722,0.8814) -- (0.4154,0.9096) -- (0.3568,0.9341) -- (0.2969,0.9549) -- (0.2357,0.9718) -- (0.1736,0.9848) -- (0.1108,0.9938) -- (0.0475,0.9988) -- (-0.0158,0.9998) -- (-0.0792,0.9968) -- (-0.1423,0.9898) -- (-0.2048,0.9788) -- (-0.2664,0.9638) -- (-0.3270,0.9450) -- (-0.3863,0.9223) -- (-0.4440,0.8959) -- (-0.5000,0.8660) -- (-0.5539,0.8325) -- (-0.6056,0.7957) -- (-0.6548,0.7557) -- (-0.7014,0.7126) -- (-0.7452,0.6667) -- (-0.7860,0.6181) -- (-0.8236,0.5670) -- (-0.8579,0.5136) -- (-0.8888,0.4582) -- (-0.9161,0.4009) -- (-0.9396,0.3420) -- (-0.9594,0.2817) -- (-0.9754,0.2203) -- (-0.9874,0.1580) -- (-0.9954,0.0950) -- (-0.9994,0.0317) -- (-0.9994,-0.0317) -- (-0.9954,-0.0950) -- (-0.9874,-0.1580) -- (-0.9754,-0.2203) -- (-0.9594,-0.2817) -- (-0.9396,-0.3420) -- (-0.9161,-0.4009) -- (-0.8888,-0.4582) -- (-0.8579,-0.5136) -- (-0.8236,-0.5670) -- (-0.7860,-0.6181) -- (-0.7452,-0.6667) -- (-0.7014,-0.7126) -- (-0.6548,-0.7557) -- (-0.6056,-0.7957) -- (-0.5539,-0.8325) -- (-0.5000,-0.8660) -- (-0.4440,-0.8959) -- (-0.3863,-0.9223) -- (-0.3270,-0.9450) -- (-0.2664,-0.9638) -- (-0.2048,-0.9788) -- (-0.1423,-0.9898) -- (-0.0792,-0.9968) -- (-0.0158,-0.9998) -- (0.0475,-0.9988) -- (0.1108,-0.9938) -- (0.1736,-0.9848) -- (0.2357,-0.9718) -- (0.2969,-0.9549) -- (0.3568,-0.9341) -- (0.4154,-0.9096) -- (0.4722,-0.8814) -- (0.5272,-0.8497) -- (0.5800,-0.8145) -- (0.6305,-0.7761) -- (0.6785,-0.7345) -- (0.7237,-0.6900) -- (0.7660,-0.6427) -- (0.8052,-0.5929) -- (0.8412,-0.5406) -- (0.8738,-0.4861) -- (0.9029,-0.4297) -- (0.9283,-0.3716) -- (0.9500,-0.3120) -- (0.9679,-0.2511) -- (0.9819,-0.1892) -- (0.9919,-0.1265) -- (0.9979,-0.0634) -- (1.0000,0.0000) --  cycle;
+\draw [color=red] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw (1.5425,1.0000) node {$P$};
-\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-1.5443,0) node {$Q$};
-\draw []  (0.50000,0.50000) node [rotate=0] {$\bullet$};
-\draw (0.16544,0.16316) node {$S$};
+\draw [color=red]  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5442,0.0000) node {$Q$};
+\draw []  (0.5000,0.5000) node [rotate=0] {$\bullet$};
+\draw (0.1654,0.1631) node {$S$};
 
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_AireParabole.pstricks.recall b/src_phystricks/Fig_AireParabole.pstricks.recall
index 69294f452..0d48af449 100644
--- a/src_phystricks/Fig_AireParabole.pstricks.recall
+++ b/src_phystricks/Fig_AireParabole.pstricks.recall
@@ -91,34 +91,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,6.5000);
 %DEFAULT
 
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-\draw (-0.29125,6.0000) node {$ 6 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw [color=blue] (-1.0000,2.0000)--(-0.9595,1.8804)--(-0.9191,1.7641)--(-0.8787,1.6510)--(-0.8383,1.5412)--(-0.7979,1.4347)--(-0.7575,1.3314)--(-0.7171,1.2315)--(-0.6767,1.1347)--(-0.6363,1.0413)--(-0.5959,0.9511)--(-0.5555,0.8641)--(-0.5151,0.7805)--(-0.4747,0.7001)--(-0.4343,0.6229)--(-0.3939,0.5491)--(-0.3535,0.4785)--(-0.3131,0.4111)--(-0.2727,0.3471)--(-0.2323,0.2862)--(-0.1919,0.2287)--(-0.1515,0.1744)--(-0.1111,0.1234)--(-0.0707,0.0757)--(-0.0303,0.0312)--(0.0101,-0.0099)--(0.0505,-0.0479)--(0.0909,-0.0826)--(0.1313,-0.1140)--(0.1717,-0.1422)--(0.2121,-0.1671)--(0.2525,-0.1887)--(0.2929,-0.2071)--(0.3333,-0.2222)--(0.3737,-0.2340)--(0.4141,-0.2426)--(0.4545,-0.2479)--(0.4949,-0.2499)--(0.5353,-0.2487)--(0.5757,-0.2442)--(0.6161,-0.2365)--(0.6565,-0.2254)--(0.6969,-0.2112)--(0.7373,-0.1936)--(0.7777,-0.1728)--(0.8181,-0.1487)--(0.8585,-0.1214)--(0.8989,-0.0908)--(0.9393,-0.0569)--(0.9797,-0.0197)--(1.0202,0.0206)--(1.0606,0.0642)--(1.1010,0.1112)--(1.1414,0.1614)--(1.1818,0.2148)--(1.2222,0.2716)--(1.2626,0.3315)--(1.3030,0.3948)--(1.3434,0.4613)--(1.3838,0.5311)--(1.4242,0.6042)--(1.4646,0.6805)--(1.5050,0.7601)--(1.5454,0.8429)--(1.5858,0.9290)--(1.6262,1.0184)--(1.6666,1.1111)--(1.7070,1.2070)--(1.7474,1.3061)--(1.7878,1.4086)--(1.8282,1.5143)--(1.8686,1.6233)--(1.9090,1.7355)--(1.9494,1.8510)--(1.9898,1.9697)--(2.0303,2.0918)--(2.0707,2.2171)--(2.1111,2.3456)--(2.1515,2.4775)--(2.1919,2.6125)--(2.2323,2.7509)--(2.2727,2.8925)--(2.3131,3.0374)--(2.3535,3.1855)--(2.3939,3.3370)--(2.4343,3.4916)--(2.4747,3.6496)--(2.5151,3.8108)--(2.5555,3.9753)--(2.5959,4.1430)--(2.6363,4.3140)--(2.6767,4.4883)--(2.7171,4.6658)--(2.7575,4.8466)--(2.7979,5.0307)--(2.8383,5.2180)--(2.8787,5.4086)--(2.9191,5.6024)--(2.9595,5.7996)--(3.0000,6.0000);
+\draw [] (2.0000,-1.0000) -- (2.0000,3.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall b/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall
index a72e93213..6c66b5674 100644
--- a/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall
+++ b/src_phystricks/Fig_BNHLooLDxdPA.pstricks.recall
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.0000,0);
-\draw (3.3086,-0.29071) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw [,->,>=latex] (0.0000,0.0000) -- (3.0000,0.0000);
+\draw (3.3085,-0.2907) node {$a$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (2.0000,2.0000);
 \draw (2.0000,2.4267) node {$b$};
 \draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
-\draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
-\draw [style=dashed] (2.00,2.00) -- (2.00,0);
+\draw [color=blue,style=dotted] (2.0000,2.0000) -- (5.0000,2.0000);
+\draw [color=blue,style=dotted] (3.0000,0.0000) -- (5.0000,2.0000);
+\draw [style=dashed] (2.0000,2.0000) -- (2.0000,0.0000);
 \draw (2.3051,1.0000) node {$h$};
-\draw (0.80915,0.31918) node {$\theta$};
+\draw (0.8091,0.3191) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw [] (0.5000,0.0000)--(0.4999,0.0039)--(0.4999,0.0079)--(0.4998,0.0118)--(0.4997,0.0158)--(0.4996,0.0198)--(0.4994,0.0237)--(0.4992,0.0277)--(0.4989,0.0317)--(0.4987,0.0356)--(0.4984,0.0396)--(0.4980,0.0435)--(0.4977,0.0475)--(0.4973,0.0514)--(0.4969,0.0554)--(0.4964,0.0593)--(0.4959,0.0632)--(0.4954,0.0672)--(0.4949,0.0711)--(0.4943,0.0750)--(0.4937,0.0790)--(0.4930,0.0829)--(0.4924,0.0868)--(0.4916,0.0907)--(0.4909,0.0946)--(0.4901,0.0985)--(0.4894,0.1024)--(0.4885,0.1062)--(0.4877,0.1101)--(0.4868,0.1140)--(0.4859,0.1178)--(0.4849,0.1217)--(0.4839,0.1255)--(0.4829,0.1294)--(0.4819,0.1332)--(0.4808,0.1370)--(0.4797,0.1408)--(0.4786,0.1446)--(0.4774,0.1484)--(0.4762,0.1522)--(0.4750,0.1560)--(0.4737,0.1597)--(0.4725,0.1635)--(0.4711,0.1672)--(0.4698,0.1710)--(0.4684,0.1747)--(0.4670,0.1784)--(0.4656,0.1821)--(0.4641,0.1858)--(0.4626,0.1895)--(0.4611,0.1931)--(0.4596,0.1968)--(0.4580,0.2004)--(0.4564,0.2040)--(0.4548,0.2077)--(0.4531,0.2113)--(0.4514,0.2148)--(0.4497,0.2184)--(0.4479,0.2220)--(0.4462,0.2255)--(0.4444,0.2291)--(0.4425,0.2326)--(0.4407,0.2361)--(0.4388,0.2396)--(0.4369,0.2430)--(0.4349,0.2465)--(0.4330,0.2500)--(0.4310,0.2534)--(0.4289,0.2568)--(0.4269,0.2602)--(0.4248,0.2636)--(0.4227,0.2669)--(0.4206,0.2703)--(0.4184,0.2736)--(0.4162,0.2769)--(0.4140,0.2802)--(0.4118,0.2835)--(0.4095,0.2867)--(0.4072,0.2900)--(0.4049,0.2932)--(0.4026,0.2964)--(0.4002,0.2996)--(0.3978,0.3028)--(0.3954,0.3059)--(0.3930,0.3090)--(0.3905,0.3121)--(0.3880,0.3152)--(0.3855,0.3183)--(0.3830,0.3213)--(0.3804,0.3244)--(0.3778,0.3274)--(0.3752,0.3304)--(0.3726,0.3333)--(0.3699,0.3363)--(0.3672,0.3392)--(0.3645,0.3421)--(0.3618,0.3450)--(0.3591,0.3478)--(0.3563,0.3507)--(0.3535,0.3535);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_Bateau.pstricks.recall b/src_phystricks/Fig_Bateau.pstricks.recall
index 117a93c45..0ad5f1e6d 100644
--- a/src_phystricks/Fig_Bateau.pstricks.recall
+++ b/src_phystricks/Fig_Bateau.pstricks.recall
@@ -97,29 +97,29 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-1.00,0) -- (5.00,0);
-\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
-\draw (0,2.9247) node {$A$};
-\draw [,->,>=latex] (2.0000,2.5000) -- (0,2.5000);
+\draw [] (-1.0000,0.0000) -- (5.0000,0.0000);
+\draw []  (0.0000,2.5000) node [rotate=0] {$\bullet$};
+\draw (0.0000,2.9247) node {$A$};
+\draw [,->,>=latex] (2.0000,2.5000) -- (0.0000,2.5000);
 \draw [,->,>=latex] (2.0000,2.5000) -- (4.0000,2.5000);
 \draw (2.0000,2.7257) node {$\unit{4}{\kilo\meter}$};
 \draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,4.5000);
-\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,0);
+\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,0.0000);
 \draw (4.6714,2.2500) node {$\unit{9}{\kilo\meter}$};
-\draw [,->,>=latex] (0,1.2500) -- (0,2.5000);
-\draw [,->,>=latex] (0,1.2500) -- (0,0);
-\draw (-0.47143,1.2500) node {$\unit{3}{\kilo\meter}$};
-\draw []  (1.4286,0) node [rotate=0] {$\bullet$};
-\draw (1.7352,0.33684) node {$I$};
-\draw [color=brown,style=dashed] (4.00,4.50) -- (4.00,-4.50);
-\draw [color=blue,style=dashed] (0,2.50) -- (4.00,-4.50);
+\draw [,->,>=latex] (0.0000,1.2500) -- (0.0000,2.5000);
+\draw [,->,>=latex] (0.0000,1.2500) -- (0.0000,0.0000);
+\draw (-0.4714,1.2500) node {$\unit{3}{\kilo\meter}$};
+\draw []  (1.4285,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.7352,0.3368) node {$I$};
+\draw [color=brown,style=dashed] (4.0000,4.5000) -- (4.0000,-4.5000);
+\draw [color=blue,style=dashed] (0.0000,2.5000) -- (4.0000,-4.5000);
 \draw []  (4.0000,4.5000) node [rotate=0] {$\bullet$};
 \draw (4.0000,4.9247) node {$B$};
 \draw []  (4.0000,-4.5000) node [rotate=0] {$\bullet$};
-\draw (4.0000,-4.9408) node {$B'$};
-\draw [,->,>=latex] (0.71429,-0.20000) -- (0,-0.20000);
-\draw [,->,>=latex] (0.71429,-0.20000) -- (1.4286,-0.20000);
-\draw (0.71429,-0.42572) node {$x\kilo\meter$};
+\draw (4.0000,-4.9407) node {$B'$};
+\draw [,->,>=latex] (0.7142,-0.2000) -- (0.0000,-0.2000);
+\draw [,->,>=latex] (0.7142,-0.2000) -- (1.4285,-0.2000);
+\draw (0.7142,-0.4257) node {$x\kilo\meter$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_BiaisOuPas.pstricks.recall b/src_phystricks/Fig_BiaisOuPas.pstricks.recall
index 5a720b6dc..b2ab27c07 100644
--- a/src_phystricks/Fig_BiaisOuPas.pstricks.recall
+++ b/src_phystricks/Fig_BiaisOuPas.pstricks.recall
@@ -91,32 +91,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.0000,0) -- (8.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4685);
+\draw [,->,>=latex] (-8.0000,0.0000) -- (8.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4684);
 %DEFAULT
 
-\draw [color=blue] (-7.500,0)--(-7.349,0)--(-7.197,0)--(-7.045,0)--(-6.894,0)--(-6.742,0)--(-6.591,0)--(-6.439,0.001311)--(-6.288,0.001784)--(-6.136,0.002412)--(-5.985,0.003235)--(-5.833,0.004309)--(-5.682,0.005696)--(-5.530,0.007475)--(-5.379,0.009739)--(-5.227,0.01259)--(-5.076,0.01617)--(-4.924,0.02060)--(-4.773,0.02606)--(-4.621,0.03273)--(-4.470,0.04080)--(-4.318,0.05048)--(-4.167,0.06201)--(-4.015,0.07562)--(-3.864,0.09153)--(-3.712,0.1100)--(-3.561,0.1312)--(-3.409,0.1553)--(-3.258,0.1826)--(-3.106,0.2130)--(-2.955,0.2468)--(-2.803,0.2837)--(-2.652,0.3238)--(-2.500,0.3669)--(-2.348,0.4127)--(-2.197,0.4607)--(-2.045,0.5107)--(-1.894,0.5618)--(-1.742,0.6136)--(-1.591,0.6652)--(-1.439,0.7160)--(-1.288,0.7649)--(-1.136,0.8112)--(-0.9848,0.8540)--(-0.8333,0.8925)--(-0.6818,0.9259)--(-0.5303,0.9535)--(-0.3788,0.9747)--(-0.2273,0.9891)--(-0.07576,0.9964)--(0.07576,0.9964)--(0.2273,0.9891)--(0.3788,0.9747)--(0.5303,0.9535)--(0.6818,0.9259)--(0.8333,0.8925)--(0.9848,0.8540)--(1.136,0.8112)--(1.288,0.7649)--(1.439,0.7160)--(1.591,0.6652)--(1.742,0.6136)--(1.894,0.5618)--(2.045,0.5107)--(2.197,0.4607)--(2.348,0.4127)--(2.500,0.3669)--(2.652,0.3238)--(2.803,0.2837)--(2.955,0.2468)--(3.106,0.2130)--(3.258,0.1826)--(3.409,0.1553)--(3.561,0.1312)--(3.712,0.1100)--(3.864,0.09153)--(4.015,0.07562)--(4.167,0.06201)--(4.318,0.05048)--(4.470,0.04080)--(4.621,0.03273)--(4.773,0.02606)--(4.924,0.02060)--(5.076,0.01617)--(5.227,0.01259)--(5.379,0.009739)--(5.530,0.007475)--(5.682,0.005696)--(5.833,0.004309)--(5.985,0.003235)--(6.136,0.002412)--(6.288,0.001784)--(6.439,0.001311)--(6.591,0)--(6.742,0)--(6.894,0)--(7.045,0)--(7.197,0)--(7.349,0)--(7.500,0);
+\draw [color=blue] (-7.5000,0.0000)--(-7.3484,0.0000)--(-7.1969,0.0000)--(-7.0454,0.0000)--(-6.8939,0.0000)--(-6.7424,0.0000)--(-6.5909,0.0000)--(-6.4393,0.0013)--(-6.2878,0.0017)--(-6.1363,0.0024)--(-5.9848,0.0032)--(-5.8333,0.0043)--(-5.6818,0.0056)--(-5.5303,0.0074)--(-5.3787,0.0097)--(-5.2272,0.0125)--(-5.0757,0.0161)--(-4.9242,0.0206)--(-4.7727,0.0260)--(-4.6212,0.0327)--(-4.4696,0.0407)--(-4.3181,0.0504)--(-4.1666,0.0620)--(-4.0151,0.0756)--(-3.8636,0.0915)--(-3.7121,0.1099)--(-3.5606,0.1311)--(-3.4090,0.1553)--(-3.2575,0.1825)--(-3.1060,0.2130)--(-2.9545,0.2467)--(-2.8030,0.2837)--(-2.6515,0.3238)--(-2.5000,0.3669)--(-2.3484,0.4126)--(-2.1969,0.4607)--(-2.0454,0.5106)--(-1.8939,0.5618)--(-1.7424,0.6135)--(-1.5909,0.6652)--(-1.4393,0.7159)--(-1.2878,0.7648)--(-1.1363,0.8111)--(-0.9848,0.8539)--(-0.8333,0.8924)--(-0.6818,0.9258)--(-0.5303,0.9534)--(-0.3787,0.9747)--(-0.2272,0.9891)--(-0.0757,0.9964)--(0.0757,0.9964)--(0.2272,0.9891)--(0.3787,0.9747)--(0.5303,0.9534)--(0.6818,0.9258)--(0.8333,0.8924)--(0.9848,0.8539)--(1.1363,0.8111)--(1.2878,0.7648)--(1.4393,0.7159)--(1.5909,0.6652)--(1.7424,0.6135)--(1.8939,0.5618)--(2.0454,0.5106)--(2.1969,0.4607)--(2.3484,0.4126)--(2.5000,0.3669)--(2.6515,0.3238)--(2.8030,0.2837)--(2.9545,0.2467)--(3.1060,0.2130)--(3.2575,0.1825)--(3.4090,0.1553)--(3.5606,0.1311)--(3.7121,0.1099)--(3.8636,0.0915)--(4.0151,0.0756)--(4.1666,0.0620)--(4.3181,0.0504)--(4.4696,0.0407)--(4.6212,0.0327)--(4.7727,0.0260)--(4.9242,0.0206)--(5.0757,0.0161)--(5.2272,0.0125)--(5.3787,0.0097)--(5.5303,0.0074)--(5.6818,0.0056)--(5.8333,0.0043)--(5.9848,0.0032)--(6.1363,0.0024)--(6.2878,0.0017)--(6.4393,0.0013)--(6.5909,0.0000)--(6.7424,0.0000)--(6.8939,0.0000)--(7.0454,0.0000)--(7.1969,0.0000)--(7.3484,0.0000)--(7.5000,0.0000);
 
-\draw [color=red] (-7.500,0)--(-7.349,0)--(-7.197,0)--(-7.045,0)--(-6.894,0)--(-6.742,0)--(-6.591,0)--(-6.439,0)--(-6.288,0)--(-6.136,0)--(-5.985,0)--(-5.833,0)--(-5.682,0)--(-5.530,0)--(-5.379,0)--(-5.227,0)--(-5.076,0)--(-4.924,0)--(-4.773,0)--(-4.621,0)--(-4.470,0)--(-4.318,0)--(-4.167,0)--(-4.015,0)--(-3.864,0)--(-3.712,0)--(-3.561,0)--(-3.409,0)--(-3.258,0)--(-3.106,0)--(-2.955,0)--(-2.803,0)--(-2.652,0)--(-2.500,0)--(-2.348,0)--(-2.197,0)--(-2.045,0)--(-1.894,0.002101)--(-1.742,0.01038)--(-1.591,0.04270)--(-1.439,0.1462)--(-1.288,0.4164)--(-1.136,0.9870)--(-0.9848,1.947)--(-0.8333,3.197)--(-0.6818,4.369)--(-0.5303,4.969)--(-0.3788,4.702)--(-0.2273,3.703)--(-0.07576,2.427)--(0.07576,1.324)--(0.2273,0.6012)--(0.3788,0.2271)--(0.5303,0.07141)--(0.6818,0.01869)--(0.8333,0.004069)--(0.9848,0)--(1.136,0)--(1.288,0)--(1.439,0)--(1.591,0)--(1.742,0)--(1.894,0)--(2.045,0)--(2.197,0)--(2.348,0)--(2.500,0)--(2.652,0)--(2.803,0)--(2.955,0)--(3.106,0)--(3.258,0)--(3.409,0)--(3.561,0)--(3.712,0)--(3.864,0)--(4.015,0)--(4.167,0)--(4.318,0)--(4.470,0)--(4.621,0)--(4.773,0)--(4.924,0)--(5.076,0)--(5.227,0)--(5.379,0)--(5.530,0)--(5.682,0)--(5.833,0)--(5.985,0)--(6.136,0)--(6.288,0)--(6.439,0)--(6.591,0)--(6.742,0)--(6.894,0)--(7.045,0)--(7.197,0)--(7.349,0)--(7.500,0);
-\draw [color=cyan,->,>=latex] (0,-0.50000) -- (-1.2500,-0.50000);
-\draw [color=cyan,->,>=latex] (0,-0.50000) -- (1.2500,-0.50000);
-\draw (0,-0.92471) node {\( I\)};
-\draw (-7.5000,-0.32983) node {$ -3 $};
-\draw [] (-7.50,-0.100) -- (-7.50,0.100);
-\draw (-5.0000,-0.32983) node {$ -2 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-2.5000,-0.32983) node {$ -1 $};
-\draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (2.5000,-0.31492) node {$ 1 $};
-\draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (5.0000,-0.31492) node {$ 2 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (7.5000,-0.31492) node {$ 3 $};
-\draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (-0.29125,2.5000) node {$ 1 $};
-\draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.29125,5.0000) node {$ 2 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=red] (-7.5000,0.0000)--(-7.3484,0.0000)--(-7.1969,0.0000)--(-7.0454,0.0000)--(-6.8939,0.0000)--(-6.7424,0.0000)--(-6.5909,0.0000)--(-6.4393,0.0000)--(-6.2878,0.0000)--(-6.1363,0.0000)--(-5.9848,0.0000)--(-5.8333,0.0000)--(-5.6818,0.0000)--(-5.5303,0.0000)--(-5.3787,0.0000)--(-5.2272,0.0000)--(-5.0757,0.0000)--(-4.9242,0.0000)--(-4.7727,0.0000)--(-4.6212,0.0000)--(-4.4696,0.0000)--(-4.3181,0.0000)--(-4.1666,0.0000)--(-4.0151,0.0000)--(-3.8636,0.0000)--(-3.7121,0.0000)--(-3.5606,0.0000)--(-3.4090,0.0000)--(-3.2575,0.0000)--(-3.1060,0.0000)--(-2.9545,0.0000)--(-2.8030,0.0000)--(-2.6515,0.0000)--(-2.5000,0.0000)--(-2.3484,0.0000)--(-2.1969,0.0000)--(-2.0454,0.0000)--(-1.8939,0.0021)--(-1.7424,0.0103)--(-1.5909,0.0427)--(-1.4393,0.1461)--(-1.2878,0.4163)--(-1.1363,0.9870)--(-0.9848,1.9473)--(-0.8333,3.1974)--(-0.6818,4.3691)--(-0.5303,4.9684)--(-0.3787,4.7021)--(-0.2272,3.7034)--(-0.0757,2.4275)--(0.0757,1.3241)--(0.2272,0.6011)--(0.3787,0.2271)--(0.5303,0.0714)--(0.6818,0.0186)--(0.8333,0.0040)--(0.9848,0.0000)--(1.1363,0.0000)--(1.2878,0.0000)--(1.4393,0.0000)--(1.5909,0.0000)--(1.7424,0.0000)--(1.8939,0.0000)--(2.0454,0.0000)--(2.1969,0.0000)--(2.3484,0.0000)--(2.5000,0.0000)--(2.6515,0.0000)--(2.8030,0.0000)--(2.9545,0.0000)--(3.1060,0.0000)--(3.2575,0.0000)--(3.4090,0.0000)--(3.5606,0.0000)--(3.7121,0.0000)--(3.8636,0.0000)--(4.0151,0.0000)--(4.1666,0.0000)--(4.3181,0.0000)--(4.4696,0.0000)--(4.6212,0.0000)--(4.7727,0.0000)--(4.9242,0.0000)--(5.0757,0.0000)--(5.2272,0.0000)--(5.3787,0.0000)--(5.5303,0.0000)--(5.6818,0.0000)--(5.8333,0.0000)--(5.9848,0.0000)--(6.1363,0.0000)--(6.2878,0.0000)--(6.4393,0.0000)--(6.5909,0.0000)--(6.7424,0.0000)--(6.8939,0.0000)--(7.0454,0.0000)--(7.1969,0.0000)--(7.3484,0.0000)--(7.5000,0.0000);
+\draw [color=cyan,->,>=latex] (0.0000,-0.5000) -- (-1.2500,-0.5000);
+\draw [color=cyan,->,>=latex] (0.0000,-0.5000) -- (1.2500,-0.5000);
+\draw (0.0000,-0.9247) node {\( I\)};
+\draw (-7.5000,-0.3298) node {$ -3 $};
+\draw [] (-7.5000,-0.1000) -- (-7.5000,0.1000);
+\draw (-5.0000,-0.3298) node {$ -2 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-2.5000,-0.3298) node {$ -1 $};
+\draw [] (-2.5000,-0.1000) -- (-2.5000,0.1000);
+\draw (2.5000,-0.3149) node {$ 1 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (5.0000,-0.3149) node {$ 2 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (7.5000,-0.3149) node {$ 3 $};
+\draw [] (7.5000,-0.1000) -- (7.5000,0.1000);
+\draw (-0.2912,2.5000) node {$ 1 $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.2912,5.0000) node {$ 2 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CMMAooQegASg.pstricks.recall b/src_phystricks/Fig_CMMAooQegASg.pstricks.recall
index 6b8a9b57d..ebf8af8aa 100644
--- a/src_phystricks/Fig_CMMAooQegASg.pstricks.recall
+++ b/src_phystricks/Fig_CMMAooQegASg.pstricks.recall
@@ -63,13 +63,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4998) -- (0.0000,2.4998);
 %DEFAULT
 
-\draw [color=blue] (-2.000,0)--(-1.960,0.4000)--(-1.919,0.5628)--(-1.879,0.6857)--(-1.838,0.7876)--(-1.798,0.8759)--(-1.758,0.9544)--(-1.717,1.025)--(-1.677,1.090)--(-1.636,1.150)--(-1.596,1.205)--(-1.556,1.257)--(-1.515,1.305)--(-1.475,1.351)--(-1.434,1.394)--(-1.394,1.434)--(-1.354,1.472)--(-1.313,1.509)--(-1.273,1.543)--(-1.232,1.575)--(-1.192,1.606)--(-1.152,1.635)--(-1.111,1.663)--(-1.071,1.689)--(-1.030,1.714)--(-0.9899,1.738)--(-0.9495,1.760)--(-0.9091,1.781)--(-0.8687,1.801)--(-0.8283,1.820)--(-0.7879,1.838)--(-0.7475,1.855)--(-0.7071,1.871)--(-0.6667,1.886)--(-0.6263,1.899)--(-0.5859,1.912)--(-0.5455,1.924)--(-0.5051,1.935)--(-0.4646,1.945)--(-0.4242,1.954)--(-0.3838,1.963)--(-0.3434,1.970)--(-0.3030,1.977)--(-0.2626,1.983)--(-0.2222,1.988)--(-0.1818,1.992)--(-0.1414,1.995)--(-0.1010,1.997)--(-0.06061,1.999)--(-0.02020,2.000)--(0.02020,2.000)--(0.06061,1.999)--(0.1010,1.997)--(0.1414,1.995)--(0.1818,1.992)--(0.2222,1.988)--(0.2626,1.983)--(0.3030,1.977)--(0.3434,1.970)--(0.3838,1.963)--(0.4242,1.954)--(0.4646,1.945)--(0.5051,1.935)--(0.5455,1.924)--(0.5859,1.912)--(0.6263,1.899)--(0.6667,1.886)--(0.7071,1.871)--(0.7475,1.855)--(0.7879,1.838)--(0.8283,1.820)--(0.8687,1.801)--(0.9091,1.781)--(0.9495,1.760)--(0.9899,1.738)--(1.030,1.714)--(1.071,1.689)--(1.111,1.663)--(1.152,1.635)--(1.192,1.606)--(1.232,1.575)--(1.273,1.543)--(1.313,1.509)--(1.354,1.472)--(1.394,1.434)--(1.434,1.394)--(1.475,1.351)--(1.515,1.305)--(1.556,1.257)--(1.596,1.205)--(1.636,1.150)--(1.677,1.090)--(1.717,1.025)--(1.758,0.9544)--(1.798,0.8759)--(1.838,0.7876)--(1.879,0.6857)--(1.919,0.5628)--(1.960,0.4000)--(2.000,0);
+\draw [color=blue] (-2.0000,0.0000)--(-1.9595,0.3999)--(-1.9191,0.5627)--(-1.8787,0.6856)--(-1.8383,0.7876)--(-1.7979,0.8759)--(-1.7575,0.9544)--(-1.7171,1.0253)--(-1.6767,1.0901)--(-1.6363,1.1499)--(-1.5959,1.2053)--(-1.5555,1.2570)--(-1.5151,1.3054)--(-1.4747,1.3509)--(-1.4343,1.3937)--(-1.3939,1.4342)--(-1.3535,1.4723)--(-1.3131,1.5085)--(-1.2727,1.5427)--(-1.2323,1.5752)--(-1.1919,1.6060)--(-1.1515,1.6352)--(-1.1111,1.6629)--(-1.0707,1.6892)--(-1.0303,1.7141)--(-0.9898,1.7378)--(-0.9494,1.7602)--(-0.9090,1.7814)--(-0.8686,1.8014)--(-0.8282,1.8204)--(-0.7878,1.8382)--(-0.7474,1.8550)--(-0.7070,1.8708)--(-0.6666,1.8856)--(-0.6262,1.8994)--(-0.5858,1.9122)--(-0.5454,1.9241)--(-0.5050,1.9351)--(-0.4646,1.9452)--(-0.4242,1.9544)--(-0.3838,1.9628)--(-0.3434,1.9702)--(-0.3030,1.9769)--(-0.2626,1.9826)--(-0.2222,1.9876)--(-0.1818,1.9917)--(-0.1414,1.9949)--(-0.1010,1.9974)--(-0.0606,1.9990)--(-0.0202,1.9998)--(0.0202,1.9998)--(0.0606,1.9990)--(0.1010,1.9974)--(0.1414,1.9949)--(0.1818,1.9917)--(0.2222,1.9876)--(0.2626,1.9826)--(0.3030,1.9769)--(0.3434,1.9702)--(0.3838,1.9628)--(0.4242,1.9544)--(0.4646,1.9452)--(0.5050,1.9351)--(0.5454,1.9241)--(0.5858,1.9122)--(0.6262,1.8994)--(0.6666,1.8856)--(0.7070,1.8708)--(0.7474,1.8550)--(0.7878,1.8382)--(0.8282,1.8204)--(0.8686,1.8014)--(0.9090,1.7814)--(0.9494,1.7602)--(0.9898,1.7378)--(1.0303,1.7141)--(1.0707,1.6892)--(1.1111,1.6629)--(1.1515,1.6352)--(1.1919,1.6060)--(1.2323,1.5752)--(1.2727,1.5427)--(1.3131,1.5085)--(1.3535,1.4723)--(1.3939,1.4342)--(1.4343,1.3937)--(1.4747,1.3509)--(1.5151,1.3054)--(1.5555,1.2570)--(1.5959,1.2053)--(1.6363,1.1499)--(1.6767,1.0901)--(1.7171,1.0253)--(1.7575,0.9544)--(1.7979,0.8759)--(1.8383,0.7876)--(1.8787,0.6856)--(1.9191,0.5627)--(1.9595,0.3999)--(2.0000,0.0000);
 
-\draw [color=red] (-2.000,0)--(-1.960,-0.4000)--(-1.919,-0.5628)--(-1.879,-0.6857)--(-1.838,-0.7876)--(-1.798,-0.8759)--(-1.758,-0.9544)--(-1.717,-1.025)--(-1.677,-1.090)--(-1.636,-1.150)--(-1.596,-1.205)--(-1.556,-1.257)--(-1.515,-1.305)--(-1.475,-1.351)--(-1.434,-1.394)--(-1.394,-1.434)--(-1.354,-1.472)--(-1.313,-1.509)--(-1.273,-1.543)--(-1.232,-1.575)--(-1.192,-1.606)--(-1.152,-1.635)--(-1.111,-1.663)--(-1.071,-1.689)--(-1.030,-1.714)--(-0.9899,-1.738)--(-0.9495,-1.760)--(-0.9091,-1.781)--(-0.8687,-1.801)--(-0.8283,-1.820)--(-0.7879,-1.838)--(-0.7475,-1.855)--(-0.7071,-1.871)--(-0.6667,-1.886)--(-0.6263,-1.899)--(-0.5859,-1.912)--(-0.5455,-1.924)--(-0.5051,-1.935)--(-0.4646,-1.945)--(-0.4242,-1.954)--(-0.3838,-1.963)--(-0.3434,-1.970)--(-0.3030,-1.977)--(-0.2626,-1.983)--(-0.2222,-1.988)--(-0.1818,-1.992)--(-0.1414,-1.995)--(-0.1010,-1.997)--(-0.06061,-1.999)--(-0.02020,-2.000)--(0.02020,-2.000)--(0.06061,-1.999)--(0.1010,-1.997)--(0.1414,-1.995)--(0.1818,-1.992)--(0.2222,-1.988)--(0.2626,-1.983)--(0.3030,-1.977)--(0.3434,-1.970)--(0.3838,-1.963)--(0.4242,-1.954)--(0.4646,-1.945)--(0.5051,-1.935)--(0.5455,-1.924)--(0.5859,-1.912)--(0.6263,-1.899)--(0.6667,-1.886)--(0.7071,-1.871)--(0.7475,-1.855)--(0.7879,-1.838)--(0.8283,-1.820)--(0.8687,-1.801)--(0.9091,-1.781)--(0.9495,-1.760)--(0.9899,-1.738)--(1.030,-1.714)--(1.071,-1.689)--(1.111,-1.663)--(1.152,-1.635)--(1.192,-1.606)--(1.232,-1.575)--(1.273,-1.543)--(1.313,-1.509)--(1.354,-1.472)--(1.394,-1.434)--(1.434,-1.394)--(1.475,-1.351)--(1.515,-1.305)--(1.556,-1.257)--(1.596,-1.205)--(1.636,-1.150)--(1.677,-1.090)--(1.717,-1.025)--(1.758,-0.9544)--(1.798,-0.8759)--(1.838,-0.7876)--(1.879,-0.6857)--(1.919,-0.5628)--(1.960,-0.4000)--(2.000,0);
+\draw [color=red] (-2.0000,0.0000)--(-1.9595,-0.3999)--(-1.9191,-0.5627)--(-1.8787,-0.6856)--(-1.8383,-0.7876)--(-1.7979,-0.8759)--(-1.7575,-0.9544)--(-1.7171,-1.0253)--(-1.6767,-1.0901)--(-1.6363,-1.1499)--(-1.5959,-1.2053)--(-1.5555,-1.2570)--(-1.5151,-1.3054)--(-1.4747,-1.3509)--(-1.4343,-1.3937)--(-1.3939,-1.4342)--(-1.3535,-1.4723)--(-1.3131,-1.5085)--(-1.2727,-1.5427)--(-1.2323,-1.5752)--(-1.1919,-1.6060)--(-1.1515,-1.6352)--(-1.1111,-1.6629)--(-1.0707,-1.6892)--(-1.0303,-1.7141)--(-0.9898,-1.7378)--(-0.9494,-1.7602)--(-0.9090,-1.7814)--(-0.8686,-1.8014)--(-0.8282,-1.8204)--(-0.7878,-1.8382)--(-0.7474,-1.8550)--(-0.7070,-1.8708)--(-0.6666,-1.8856)--(-0.6262,-1.8994)--(-0.5858,-1.9122)--(-0.5454,-1.9241)--(-0.5050,-1.9351)--(-0.4646,-1.9452)--(-0.4242,-1.9544)--(-0.3838,-1.9628)--(-0.3434,-1.9702)--(-0.3030,-1.9769)--(-0.2626,-1.9826)--(-0.2222,-1.9876)--(-0.1818,-1.9917)--(-0.1414,-1.9949)--(-0.1010,-1.9974)--(-0.0606,-1.9990)--(-0.0202,-1.9998)--(0.0202,-1.9998)--(0.0606,-1.9990)--(0.1010,-1.9974)--(0.1414,-1.9949)--(0.1818,-1.9917)--(0.2222,-1.9876)--(0.2626,-1.9826)--(0.3030,-1.9769)--(0.3434,-1.9702)--(0.3838,-1.9628)--(0.4242,-1.9544)--(0.4646,-1.9452)--(0.5050,-1.9351)--(0.5454,-1.9241)--(0.5858,-1.9122)--(0.6262,-1.8994)--(0.6666,-1.8856)--(0.7070,-1.8708)--(0.7474,-1.8550)--(0.7878,-1.8382)--(0.8282,-1.8204)--(0.8686,-1.8014)--(0.9090,-1.7814)--(0.9494,-1.7602)--(0.9898,-1.7378)--(1.0303,-1.7141)--(1.0707,-1.6892)--(1.1111,-1.6629)--(1.1515,-1.6352)--(1.1919,-1.6060)--(1.2323,-1.5752)--(1.2727,-1.5427)--(1.3131,-1.5085)--(1.3535,-1.4723)--(1.3939,-1.4342)--(1.4343,-1.3937)--(1.4747,-1.3509)--(1.5151,-1.3054)--(1.5555,-1.2570)--(1.5959,-1.2053)--(1.6363,-1.1499)--(1.6767,-1.0901)--(1.7171,-1.0253)--(1.7575,-0.9544)--(1.7979,-0.8759)--(1.8383,-0.7876)--(1.8787,-0.6856)--(1.9191,-0.5627)--(1.9595,-0.3999)--(2.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall b/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall
index 90633a3ea..36f0bf98c 100644
--- a/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall
+++ b/src_phystricks/Fig_CQIXooBEDnfK.pstricks.recall
@@ -95,47 +95,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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-\draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
-\draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (0.78677,0.19542) node {$P$};
-\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-0.78501,0.23090) node {$Q$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-3.0000,-2.8284)--(-2.9797,-2.8069)--(-2.9595,-2.7855)--(-2.9393,-2.7640)--(-2.9191,-2.7425)--(-2.8989,-2.7210)--(-2.8787,-2.6995)--(-2.8585,-2.6779)--(-2.8383,-2.6563)--(-2.8181,-2.6347)--(-2.7979,-2.6131)--(-2.7777,-2.5915)--(-2.7575,-2.5698)--(-2.7373,-2.5481)--(-2.7171,-2.5264)--(-2.6969,-2.5047)--(-2.6767,-2.4829)--(-2.6565,-2.4611)--(-2.6363,-2.4393)--(-2.6161,-2.4174)--(-2.5959,-2.3956)--(-2.5757,-2.3737)--(-2.5555,-2.3517)--(-2.5353,-2.3298)--(-2.5151,-2.3078)--(-2.4949,-2.2857)--(-2.4747,-2.2637)--(-2.4545,-2.2416)--(-2.4343,-2.2194)--(-2.4141,-2.1972)--(-2.3939,-2.1750)--(-2.3737,-2.1528)--(-2.3535,-2.1305)--(-2.3333,-2.1081)--(-2.3131,-2.0858)--(-2.2929,-2.0633)--(-2.2727,-2.0409)--(-2.2525,-2.0183)--(-2.2323,-1.9958)--(-2.2121,-1.9731)--(-2.1919,-1.9505)--(-2.1717,-1.9277)--(-2.1515,-1.9049)--(-2.1313,-1.8821)--(-2.1111,-1.8592)--(-2.0909,-1.8362)--(-2.0707,-1.8132)--(-2.0505,-1.7901)--(-2.0303,-1.7669)--(-2.0101,-1.7437)--(-1.9898,-1.7203)--(-1.9696,-1.6969)--(-1.9494,-1.6734)--(-1.9292,-1.6499)--(-1.9090,-1.6262)--(-1.8888,-1.6024)--(-1.8686,-1.5786)--(-1.8484,-1.5546)--(-1.8282,-1.5305)--(-1.8080,-1.5063)--(-1.7878,-1.4820)--(-1.7676,-1.4576)--(-1.7474,-1.4330)--(-1.7272,-1.4083)--(-1.7070,-1.3835)--(-1.6868,-1.3585)--(-1.6666,-1.3333)--(-1.6464,-1.3079)--(-1.6262,-1.2824)--(-1.6060,-1.2567)--(-1.5858,-1.2308)--(-1.5656,-1.2046)--(-1.5454,-1.1783)--(-1.5252,-1.1516)--(-1.5050,-1.1248)--(-1.4848,-1.0976)--(-1.4646,-1.0701)--(-1.4444,-1.0423)--(-1.4242,-1.0141)--(-1.4040,-0.9855)--(-1.3838,-0.9565)--(-1.3636,-0.9270)--(-1.3434,-0.8971)--(-1.3232,-0.8665)--(-1.3030,-0.8353)--(-1.2828,-0.8035)--(-1.2626,-0.7708)--(-1.2424,-0.7373)--(-1.2222,-0.7027)--(-1.2020,-0.6669)--(-1.1818,-0.6298)--(-1.1616,-0.5910)--(-1.1414,-0.5502)--(-1.1212,-0.5070)--(-1.1010,-0.4606)--(-1.0808,-0.4100)--(-1.0606,-0.3533)--(-1.0404,-0.2871)--(-1.0202,-0.2020)--(-1.0000,0.0000);
+\draw [color=brown] (-3.0000,3.0000) -- (3.0000,-3.0000);
+\draw [color=brown] (-3.0000,-3.0000) -- (3.0000,3.0000);
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.7867,0.1954) node {$P$};
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.7850,0.2309) node {$Q$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CSCii.pstricks.recall b/src_phystricks/Fig_CSCii.pstricks.recall
index fe4d62089..2997d784d 100644
--- a/src_phystricks/Fig_CSCii.pstricks.recall
+++ b/src_phystricks/Fig_CSCii.pstricks.recall
@@ -71,22 +71,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.2698,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.2698) -- (0,1.5000);
+\draw [,->,>=latex] (-1.2697,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2697) -- (0.0000,1.5000);
 %DEFAULT
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+\draw [color=lightgray] (1.0000,1.0000) -- (1.0000,0.0000);
+\draw [color=lightgray] (1.0000,1.0000) -- (0.0000,1.0000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CSCiv.pstricks.recall b/src_phystricks/Fig_CSCiv.pstricks.recall
index 42c742b85..14477e968 100644
--- a/src_phystricks/Fig_CSCiv.pstricks.recall
+++ b/src_phystricks/Fig_CSCiv.pstricks.recall
@@ -91,33 +91,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4935,0) -- (1.8801,0);
-\draw [,->,>=latex] (0,-5.4875) -- (0,2.0459);
+\draw [,->,>=latex] (-2.4935,0.0000) -- (1.8801,0.0000);
+\draw [,->,>=latex] (0.0000,-5.4874) -- (0.0000,2.0458);
 %DEFAULT
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+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CSCv.pstricks.recall b/src_phystricks/Fig_CSCv.pstricks.recall
index eafafce59..1065eab20 100644
--- a/src_phystricks/Fig_CSCv.pstricks.recall
+++ b/src_phystricks/Fig_CSCv.pstricks.recall
@@ -65,29 +65,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-2.4975) -- (0,1.6246);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4974) -- (0.0000,1.6245);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.06347,0.006032)--(0.1269,0.02401)--(0.1904,0.05356)--(0.2539,0.09410)--(0.3173,0.1448)--(0.3808,0.2046)--(0.4443,0.2724)--(0.5077,0.3466)--(0.5712,0.4258)--(0.6347,0.5083)--(0.6981,0.5924)--(0.7616,0.6762)--(0.8251,0.7578)--(0.8885,0.8354)--(0.9520,0.9071)--(1.015,0.9713)--(1.079,1.026)--(1.142,1.070)--(1.206,1.102)--(1.269,1.121)--(1.333,1.125)--(1.396,1.113)--(1.460,1.086)--(1.523,1.043)--(1.587,0.9836)--(1.650,0.9082)--(1.714,0.8172)--(1.777,0.7113)--(1.841,0.5915)--(1.904,0.4590)--(1.967,0.3151)--(2.031,0.1615)--(2.094,0)--(2.158,-0.1676)--(2.221,-0.3391)--(2.285,-0.5125)--(2.348,-0.6856)--(2.412,-0.8561)--(2.475,-1.022)--(2.539,-1.181)--(2.602,-1.330)--(2.666,-1.469)--(2.729,-1.595)--(2.793,-1.706)--(2.856,-1.801)--(2.919,-1.878)--(2.983,-1.938)--(3.046,-1.977)--(3.110,-1.997)--(3.173,-1.997)--(3.237,-1.977)--(3.300,-1.938)--(3.364,-1.878)--(3.427,-1.801)--(3.491,-1.706)--(3.554,-1.595)--(3.618,-1.469)--(3.681,-1.330)--(3.745,-1.181)--(3.808,-1.022)--(3.871,-0.8561)--(3.935,-0.6856)--(3.998,-0.5125)--(4.062,-0.3391)--(4.125,-0.1676)--(4.189,0)--(4.252,0.1615)--(4.316,0.3151)--(4.379,0.4590)--(4.443,0.5915)--(4.506,0.7113)--(4.570,0.8172)--(4.633,0.9082)--(4.697,0.9836)--(4.760,1.043)--(4.823,1.086)--(4.887,1.113)--(4.950,1.125)--(5.014,1.121)--(5.077,1.102)--(5.141,1.070)--(5.204,1.026)--(5.268,0.9713)--(5.331,0.9071)--(5.395,0.8354)--(5.458,0.7578)--(5.522,0.6762)--(5.585,0.5924)--(5.648,0.5083)--(5.712,0.4258)--(5.775,0.3466)--(5.839,0.2724)--(5.902,0.2046)--(5.966,0.1448)--(6.029,0.09410)--(6.093,0.05356)--(6.156,0.02401)--(6.220,0.006032)--(6.283,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0634,0.0060)--(0.1269,0.0240)--(0.1903,0.0535)--(0.2538,0.0940)--(0.3173,0.1448)--(0.3807,0.2046)--(0.4442,0.2723)--(0.5077,0.3466)--(0.5711,0.4258)--(0.6346,0.5083)--(0.6981,0.5923)--(0.7615,0.6761)--(0.8250,0.7577)--(0.8885,0.8353)--(0.9519,0.9071)--(1.0154,0.9712)--(1.0789,1.0261)--(1.1423,1.0702)--(1.2058,1.1021)--(1.2693,1.1205)--(1.3327,1.1245)--(1.3962,1.1133)--(1.4597,1.0862)--(1.5231,1.0430)--(1.5866,0.9836)--(1.6501,0.9081)--(1.7135,0.8171)--(1.7770,0.7113)--(1.8405,0.5915)--(1.9039,0.4589)--(1.9674,0.3151)--(2.0309,0.1615)--(2.0943,0.0000)--(2.1578,-0.1675)--(2.2213,-0.3391)--(2.2847,-0.5125)--(2.3482,-0.6856)--(2.4117,-0.8561)--(2.4751,-1.0218)--(2.5386,-1.1805)--(2.6021,-1.3302)--(2.6655,-1.4688)--(2.7290,-1.5946)--(2.7925,-1.7057)--(2.8559,-1.8007)--(2.9194,-1.8783)--(2.9829,-1.9375)--(3.0463,-1.9774)--(3.1098,-1.9974)--(3.1733,-1.9974)--(3.2367,-1.9774)--(3.3002,-1.9375)--(3.3637,-1.8783)--(3.4271,-1.8007)--(3.4906,-1.7057)--(3.5541,-1.5946)--(3.6175,-1.4688)--(3.6810,-1.3302)--(3.7445,-1.1805)--(3.8079,-1.0218)--(3.8714,-0.8561)--(3.9349,-0.6856)--(3.9983,-0.5125)--(4.0618,-0.3391)--(4.1253,-0.1675)--(4.1887,0.0000)--(4.2522,0.1615)--(4.3157,0.3151)--(4.3791,0.4589)--(4.4426,0.5915)--(4.5061,0.7113)--(4.5695,0.8171)--(4.6330,0.9081)--(4.6965,0.9836)--(4.7599,1.0430)--(4.8234,1.0862)--(4.8869,1.1133)--(4.9503,1.1245)--(5.0138,1.1205)--(5.0773,1.1021)--(5.1407,1.0702)--(5.2042,1.0261)--(5.2677,0.9712)--(5.3311,0.9071)--(5.3946,0.8353)--(5.4581,0.7577)--(5.5215,0.6761)--(5.5850,0.5923)--(5.6485,0.5083)--(5.7119,0.4258)--(5.7754,0.3466)--(5.8389,0.2723)--(5.9023,0.2046)--(5.9658,0.1448)--(6.0293,0.0940)--(6.0927,0.0535)--(6.1562,0.0240)--(6.2197,0.0060)--(6.2831,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -125,17 +125,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.65762,0) -- (1.0280,0);
-\draw [,->,>=latex] (0,-1.5975) -- (0,1.5975);
+\draw [,->,>=latex] (-0.6576,0.0000) -- (1.0280,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5974) -- (0.0000,1.5974);
 %DEFAULT
-\draw [color=blue] (0,0)--(0,0)--(0.002681,0)--(0.006020,0)--(0.01067,0)--(0.01661,0.001764)--(0.02381,0.003039)--(0.03224,0.004809)--(0.04185,0.007151)--(0.05259,0.01014)--(0.06442,0.01384)--(0.07727,0.01831)--(0.09108,0.02363)--(0.1058,0.02985)--(0.1213,0.03702)--(0.1376,0.04518)--(0.1545,0.05439)--(0.1720,0.06467)--(0.1900,0.07605)--(0.2084,0.08857)--(0.2270,0.1022)--(0.2459,0.1171)--(0.2649,0.1331)--(0.2840,0.1502)--(0.3029,0.1685)--(0.3217,0.1880)--(0.3401,0.2086)--(0.3582,0.2302)--(0.3759,0.2530)--(0.3929,0.2767)--(0.4094,0.3014)--(0.4250,0.3270)--(0.4399,0.3535)--(0.4538,0.3808)--(0.4667,0.4088)--(0.4786,0.4374)--(0.4894,0.4666)--(0.4989,0.4963)--(0.5072,0.5263)--(0.5141,0.5566)--(0.5198,0.5872)--(0.5240,0.6178)--(0.5267,0.6484)--(0.5280,0.6788)--(0.5279,0.7090)--(0.5262,0.7389)--(0.5230,0.7683)--(0.5183,0.7972)--(0.5121,0.8253)--(0.5044,0.8527)--(0.4952,0.8791)--(0.4846,0.9045)--(0.4726,0.9288)--(0.4593,0.9519)--(0.4446,0.9736)--(0.4287,0.9938)--(0.4116,1.013)--(0.3933,1.030)--(0.3741,1.045)--(0.3538,1.058)--(0.3327,1.070)--(0.3108,1.080)--(0.2883,1.087)--(0.2651,1.093)--(0.2415,1.096)--(0.2175,1.097)--(0.1933,1.096)--(0.1690,1.093)--(0.1446,1.087)--(0.1204,1.080)--(0.09640,1.069)--(0.07277,1.057)--(0.04963,1.042)--(0.02710,1.025)--(0.005317,1.005)--(-0.01561,0.9835)--(-0.03554,0.9596)--(-0.05437,0.9335)--(-0.07197,0.9053)--(-0.08824,0.8751)--(-0.1030,0.8429)--(-0.1163,0.8089)--(-0.1279,0.7730)--(-0.1377,0.7354)--(-0.1457,0.6962)--(-0.1517,0.6555)--(-0.1557,0.6135)--(-0.1576,0.5701)--(-0.1574,0.5256)--(-0.1549,0.4801)--(-0.1501,0.4337)--(-0.1430,0.3866)--(-0.1336,0.3389)--(-0.1217,0.2907)--(-0.1075,0.2421)--(-0.09083,0.1935)--(-0.07174,0.1447)--(-0.05022,0.09616)--(-0.02630,0.04787)--(0,0);
-\draw [color=blue] (0,0)--(-0.02630,-0.04787)--(-0.05022,-0.09616)--(-0.07174,-0.1447)--(-0.09083,-0.1935)--(-0.1075,-0.2421)--(-0.1217,-0.2907)--(-0.1336,-0.3389)--(-0.1430,-0.3866)--(-0.1501,-0.4337)--(-0.1549,-0.4801)--(-0.1574,-0.5256)--(-0.1576,-0.5701)--(-0.1557,-0.6135)--(-0.1517,-0.6555)--(-0.1457,-0.6962)--(-0.1377,-0.7354)--(-0.1279,-0.7730)--(-0.1163,-0.8089)--(-0.1030,-0.8429)--(-0.08824,-0.8751)--(-0.07197,-0.9053)--(-0.05437,-0.9335)--(-0.03554,-0.9596)--(-0.01561,-0.9835)--(0.005317,-1.005)--(0.02710,-1.025)--(0.04963,-1.042)--(0.07277,-1.057)--(0.09640,-1.069)--(0.1204,-1.080)--(0.1446,-1.087)--(0.1690,-1.093)--(0.1933,-1.096)--(0.2175,-1.097)--(0.2415,-1.096)--(0.2651,-1.093)--(0.2883,-1.087)--(0.3108,-1.080)--(0.3327,-1.070)--(0.3538,-1.058)--(0.3741,-1.045)--(0.3933,-1.030)--(0.4116,-1.013)--(0.4287,-0.9938)--(0.4446,-0.9736)--(0.4593,-0.9519)--(0.4726,-0.9288)--(0.4846,-0.9045)--(0.4952,-0.8791)--(0.5044,-0.8527)--(0.5121,-0.8253)--(0.5183,-0.7972)--(0.5230,-0.7683)--(0.5262,-0.7389)--(0.5279,-0.7090)--(0.5280,-0.6788)--(0.5267,-0.6484)--(0.5240,-0.6178)--(0.5198,-0.5872)--(0.5141,-0.5566)--(0.5072,-0.5263)--(0.4989,-0.4963)--(0.4894,-0.4666)--(0.4786,-0.4374)--(0.4667,-0.4088)--(0.4538,-0.3808)--(0.4399,-0.3535)--(0.4250,-0.3270)--(0.4094,-0.3014)--(0.3929,-0.2767)--(0.3759,-0.2530)--(0.3582,-0.2302)--(0.3401,-0.2086)--(0.3217,-0.1880)--(0.3029,-0.1685)--(0.2840,-0.1502)--(0.2649,-0.1331)--(0.2459,-0.1171)--(0.2270,-0.1022)--(0.2084,-0.08857)--(0.1900,-0.07605)--(0.1720,-0.06467)--(0.1545,-0.05439)--(0.1376,-0.04518)--(0.1213,-0.03702)--(0.1058,-0.02985)--(0.09108,-0.02363)--(0.07727,-0.01831)--(0.06442,-0.01384)--(0.05259,-0.01014)--(0.04185,-0.007151)--(0.03224,-0.004809)--(0.02381,-0.003039)--(0.01661,-0.001764)--(0.01067,0)--(0.006020,0)--(0.002681,0)--(0,0)--(0,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0000,0.0000)--(0.0026,0.0000)--(0.0060,0.0000)--(0.0106,0.0000)--(0.0166,0.0017)--(0.0238,0.0030)--(0.0322,0.0048)--(0.0418,0.0071)--(0.0525,0.0101)--(0.0644,0.0138)--(0.0772,0.0183)--(0.0910,0.0236)--(0.1057,0.0298)--(0.1213,0.0370)--(0.1375,0.0451)--(0.1544,0.0543)--(0.1719,0.0646)--(0.1899,0.0760)--(0.2083,0.0885)--(0.2270,0.1022)--(0.2459,0.1170)--(0.2649,0.1330)--(0.2839,0.1502)--(0.3028,0.1685)--(0.3216,0.1879)--(0.3401,0.2085)--(0.3582,0.2302)--(0.3758,0.2529)--(0.3929,0.2766)--(0.4093,0.3014)--(0.4250,0.3270)--(0.4398,0.3535)--(0.4538,0.3807)--(0.4667,0.4087)--(0.4786,0.4374)--(0.4893,0.4665)--(0.4988,0.4962)--(0.5071,0.5263)--(0.5141,0.5566)--(0.5197,0.5871)--(0.5239,0.6177)--(0.5267,0.6483)--(0.5280,0.6788)--(0.5278,0.7090)--(0.5261,0.7389)--(0.5229,0.7683)--(0.5182,0.7971)--(0.5120,0.8253)--(0.5043,0.8526)--(0.4952,0.8791)--(0.4846,0.9045)--(0.4726,0.9288)--(0.4592,0.9518)--(0.4446,0.9735)--(0.4286,0.9938)--(0.4115,1.0125)--(0.3933,1.0295)--(0.3740,1.0449)--(0.3538,1.0584)--(0.3327,1.0700)--(0.3108,1.0797)--(0.2882,1.0873)--(0.2651,1.0928)--(0.2415,1.0962)--(0.2175,1.0974)--(0.1933,1.0964)--(0.1689,1.0931)--(0.1446,1.0874)--(0.1203,1.0795)--(0.0964,1.0693)--(0.0727,1.0567)--(0.0496,1.0418)--(0.0271,1.0246)--(0.0053,1.0052)--(-0.0156,0.9835)--(-0.0355,0.9595)--(-0.0543,0.9335)--(-0.0719,0.9053)--(-0.0882,0.8751)--(-0.1030,0.8429)--(-0.1162,0.8088)--(-0.1278,0.7729)--(-0.1377,0.7354)--(-0.1456,0.6962)--(-0.1517,0.6555)--(-0.1557,0.6134)--(-0.1576,0.5701)--(-0.1573,0.5256)--(-0.1548,0.4801)--(-0.1501,0.4337)--(-0.1430,0.3866)--(-0.1335,0.3388)--(-0.1217,0.2906)--(-0.1075,0.2421)--(-0.0908,0.1934)--(-0.0717,0.1447)--(-0.0502,0.0961)--(-0.0263,0.0478)--(0.0000,0.0000);
+\draw [color=blue] (0.0000,0.0000)--(-0.0263,-0.0478)--(-0.0502,-0.0961)--(-0.0717,-0.1447)--(-0.0908,-0.1934)--(-0.1075,-0.2421)--(-0.1217,-0.2906)--(-0.1335,-0.3388)--(-0.1430,-0.3866)--(-0.1501,-0.4337)--(-0.1548,-0.4801)--(-0.1573,-0.5256)--(-0.1576,-0.5701)--(-0.1557,-0.6134)--(-0.1517,-0.6555)--(-0.1456,-0.6962)--(-0.1377,-0.7354)--(-0.1278,-0.7729)--(-0.1162,-0.8088)--(-0.1030,-0.8429)--(-0.0882,-0.8751)--(-0.0719,-0.9053)--(-0.0543,-0.9335)--(-0.0355,-0.9595)--(-0.0156,-0.9835)--(0.0053,-1.0052)--(0.0271,-1.0246)--(0.0496,-1.0418)--(0.0727,-1.0567)--(0.0964,-1.0693)--(0.1203,-1.0795)--(0.1446,-1.0874)--(0.1689,-1.0931)--(0.1933,-1.0964)--(0.2175,-1.0974)--(0.2415,-1.0962)--(0.2651,-1.0928)--(0.2882,-1.0873)--(0.3108,-1.0797)--(0.3327,-1.0700)--(0.3538,-1.0584)--(0.3740,-1.0449)--(0.3933,-1.0295)--(0.4115,-1.0125)--(0.4286,-0.9938)--(0.4446,-0.9735)--(0.4592,-0.9518)--(0.4726,-0.9288)--(0.4846,-0.9045)--(0.4952,-0.8791)--(0.5043,-0.8526)--(0.5120,-0.8253)--(0.5182,-0.7971)--(0.5229,-0.7683)--(0.5261,-0.7389)--(0.5278,-0.7090)--(0.5280,-0.6788)--(0.5267,-0.6483)--(0.5239,-0.6177)--(0.5197,-0.5871)--(0.5141,-0.5566)--(0.5071,-0.5263)--(0.4988,-0.4962)--(0.4893,-0.4665)--(0.4786,-0.4374)--(0.4667,-0.4087)--(0.4538,-0.3807)--(0.4398,-0.3535)--(0.4250,-0.3270)--(0.4093,-0.3014)--(0.3929,-0.2766)--(0.3758,-0.2529)--(0.3582,-0.2302)--(0.3401,-0.2085)--(0.3216,-0.1879)--(0.3028,-0.1685)--(0.2839,-0.1502)--(0.2649,-0.1330)--(0.2459,-0.1170)--(0.2270,-0.1022)--(0.2083,-0.0885)--(0.1899,-0.0760)--(0.1719,-0.0646)--(0.1544,-0.0543)--(0.1375,-0.0451)--(0.1213,-0.0370)--(0.1057,-0.0298)--(0.0910,-0.0236)--(0.0772,-0.0183)--(0.0644,-0.0138)--(0.0525,-0.0101)--(0.0418,-0.0071)--(0.0322,-0.0048)--(0.0238,-0.0030)--(0.0166,-0.0017)--(0.0106,0.0000)--(0.0060,0.0000)--(0.0026,0.0000)--(0.0000,0.0000)--(0.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall b/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
index b289ededc..53a4f6010 100644
--- a/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
+++ b/src_phystricks/Fig_CWKJooppMsZXjw.pstricks.recall
@@ -68,16 +68,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (1.0000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,0.50000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (1.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,0.5000);
 %DEFAULT
-\draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0);
-\draw [] (0,0) -- (-3.00,-1.00);
+\draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (-3.0000,-1.0000);
 \draw []  (-3.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw (-3.8283,-1.2049) node {\( -x+\lambda i\)};
-\draw (0.53123,-0.70911) node {\( \arg(z)\)};
+\draw (-3.8283,-1.2048) node {\( -x+\lambda i\)};
+\draw (0.5312,-0.7091) node {\( \arg(z)\)};
 
-\draw [] (-0.474,-0.158)--(-0.470,-0.172)--(-0.465,-0.185)--(-0.459,-0.198)--(-0.453,-0.211)--(-0.447,-0.224)--(-0.441,-0.236)--(-0.434,-0.249)--(-0.426,-0.261)--(-0.419,-0.273)--(-0.411,-0.285)--(-0.403,-0.297)--(-0.394,-0.308)--(-0.385,-0.319)--(-0.376,-0.330)--(-0.366,-0.340)--(-0.356,-0.351)--(-0.346,-0.361)--(-0.336,-0.370)--(-0.325,-0.380)--(-0.314,-0.389)--(-0.303,-0.398)--(-0.292,-0.406)--(-0.280,-0.414)--(-0.268,-0.422)--(-0.256,-0.430)--(-0.243,-0.437)--(-0.231,-0.444)--(-0.218,-0.450)--(-0.205,-0.456)--(-0.192,-0.462)--(-0.179,-0.467)--(-0.166,-0.472)--(-0.152,-0.476)--(-0.138,-0.480)--(-0.125,-0.484)--(-0.111,-0.488)--(-0.0970,-0.491)--(-0.0830,-0.493)--(-0.0689,-0.495)--(-0.0547,-0.497)--(-0.0406,-0.498)--(-0.0264,-0.499)--(-0.0121,-0.500)--(0.00211,-0.500)--(0.0163,-0.500)--(0.0306,-0.499)--(0.0448,-0.498)--(0.0589,-0.497)--(0.0731,-0.495)--(0.0871,-0.492)--(0.101,-0.490)--(0.115,-0.487)--(0.129,-0.483)--(0.143,-0.479)--(0.156,-0.475)--(0.170,-0.470)--(0.183,-0.465)--(0.196,-0.460)--(0.209,-0.454)--(0.222,-0.448)--(0.235,-0.442)--(0.247,-0.435)--(0.259,-0.427)--(0.271,-0.420)--(0.283,-0.412)--(0.295,-0.404)--(0.306,-0.395)--(0.317,-0.386)--(0.328,-0.377)--(0.339,-0.368)--(0.349,-0.358)--(0.359,-0.348)--(0.369,-0.337)--(0.379,-0.327)--(0.388,-0.316)--(0.396,-0.305)--(0.405,-0.293)--(0.413,-0.282)--(0.421,-0.270)--(0.429,-0.258)--(0.436,-0.245)--(0.443,-0.233)--(0.449,-0.220)--(0.455,-0.207)--(0.461,-0.194)--(0.466,-0.181)--(0.471,-0.168)--(0.476,-0.154)--(0.480,-0.141)--(0.484,-0.127)--(0.487,-0.113)--(0.490,-0.0990)--(0.493,-0.0850)--(0.495,-0.0710)--(0.497,-0.0568)--(0.498,-0.0427)--(0.499,-0.0285)--(0.500,-0.0142)--(0.500,0);
+\draw [] (-0.4743,-0.1581)--(-0.4696,-0.1715)--(-0.4645,-0.1848)--(-0.4591,-0.1980)--(-0.4532,-0.2110)--(-0.4470,-0.2238)--(-0.4405,-0.2364)--(-0.4336,-0.2489)--(-0.4263,-0.2611)--(-0.4187,-0.2732)--(-0.4108,-0.2850)--(-0.4025,-0.2966)--(-0.3939,-0.3079)--(-0.3849,-0.3190)--(-0.3757,-0.3298)--(-0.3661,-0.3404)--(-0.3563,-0.3507)--(-0.3462,-0.3607)--(-0.3357,-0.3704)--(-0.3251,-0.3798)--(-0.3141,-0.3889)--(-0.3029,-0.3977)--(-0.2914,-0.4062)--(-0.2798,-0.4143)--(-0.2678,-0.4221)--(-0.2557,-0.4296)--(-0.2434,-0.4367)--(-0.2308,-0.4434)--(-0.2181,-0.4498)--(-0.2052,-0.4559)--(-0.1921,-0.4615)--(-0.1789,-0.4668)--(-0.1656,-0.4717)--(-0.1520,-0.4763)--(-0.1384,-0.4804)--(-0.1247,-0.4841)--(-0.1108,-0.4875)--(-0.0969,-0.4905)--(-0.0829,-0.4930)--(-0.0688,-0.4952)--(-0.0547,-0.4969)--(-0.0405,-0.4983)--(-0.0263,-0.4993)--(-0.0121,-0.4998)--(0.0021,-0.4999)--(0.0163,-0.4997)--(0.0305,-0.4990)--(0.0447,-0.4979)--(0.0589,-0.4965)--(0.0730,-0.4946)--(0.0871,-0.4923)--(0.1010,-0.4896)--(0.1150,-0.4865)--(0.1288,-0.4831)--(0.1425,-0.4792)--(0.1561,-0.4750)--(0.1695,-0.4703)--(0.1829,-0.4653)--(0.1960,-0.4599)--(0.2091,-0.4541)--(0.2219,-0.4480)--(0.2346,-0.4415)--(0.2471,-0.4346)--(0.2593,-0.4274)--(0.2714,-0.4198)--(0.2832,-0.4119)--(0.2949,-0.4037)--(0.3062,-0.3952)--(0.3174,-0.3863)--(0.3282,-0.3771)--(0.3389,-0.3676)--(0.3492,-0.3578)--(0.3592,-0.3477)--(0.3690,-0.3373)--(0.3785,-0.3267)--(0.3876,-0.3157)--(0.3964,-0.3046)--(0.4050,-0.2932)--(0.4131,-0.2815)--(0.4210,-0.2696)--(0.4285,-0.2575)--(0.4357,-0.2452)--(0.4425,-0.2327)--(0.4489,-0.2200)--(0.4550,-0.2071)--(0.4607,-0.1941)--(0.4661,-0.1809)--(0.4710,-0.1675)--(0.4756,-0.1541)--(0.4798,-0.1404)--(0.4836,-0.1267)--(0.4870,-0.1129)--(0.4900,-0.0990)--(0.4927,-0.0850)--(0.4949,-0.0709)--(0.4967,-0.0568)--(0.4981,-0.0426)--(0.4991,-0.0284)--(0.4997,-0.0142)--(0.5000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Cardioid.pstricks.recall b/src_phystricks/Fig_Cardioid.pstricks.recall
index aa20450b4..dc102ab60 100644
--- a/src_phystricks/Fig_Cardioid.pstricks.recall
+++ b/src_phystricks/Fig_Cardioid.pstricks.recall
@@ -65,7 +65,7 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red] (3.000,0)--(2.991,0.1901)--(2.964,0.3783)--(2.919,0.5626)--(2.857,0.7414)--(2.779,0.9127)--(2.685,1.075)--(2.577,1.227)--(2.456,1.367)--(2.323,1.493)--(2.181,1.606)--(2.029,1.703)--(1.871,1.784)--(1.708,1.850)--(1.542,1.898)--(1.375,1.931)--(1.208,1.947)--(1.043,1.947)--(0.8820,1.931)--(0.7264,1.901)--(0.5776,1.858)--(0.4370,1.801)--(0.3057,1.734)--(0.1847,1.656)--(0.07477,1.570)--(-0.02342,1.476)--(-0.1095,1.377)--(-0.1831,1.273)--(-0.2443,1.168)--(-0.2932,1.061)--(-0.3301,0.9539)--(-0.3556,0.8490)--(-0.3703,0.7472)--(-0.3750,0.6495)--(-0.3706,0.5571)--(-0.3583,0.4708)--(-0.3390,0.3913)--(-0.3141,0.3191)--(-0.2848,0.2548)--(-0.2523,0.1984)--(-0.2178,0.1500)--(-0.1828,0.1094)--(-0.1482,0.07641)--(-0.1153,0.05045)--(-0.08501,0.03094)--(-0.05830,0.01712)--(-0.03595,0.008120)--(-0.01860,0.002977)--(-0.006761,0)--(0,0)--(0,0)--(-0.006761,0)--(-0.01860,-0.002977)--(-0.03595,-0.008120)--(-0.05830,-0.01712)--(-0.08501,-0.03094)--(-0.1153,-0.05045)--(-0.1482,-0.07641)--(-0.1828,-0.1094)--(-0.2178,-0.1500)--(-0.2523,-0.1984)--(-0.2848,-0.2548)--(-0.3141,-0.3191)--(-0.3390,-0.3913)--(-0.3583,-0.4708)--(-0.3706,-0.5571)--(-0.3750,-0.6495)--(-0.3703,-0.7472)--(-0.3556,-0.8490)--(-0.3301,-0.9539)--(-0.2932,-1.061)--(-0.2443,-1.168)--(-0.1831,-1.273)--(-0.1095,-1.377)--(-0.02342,-1.476)--(0.07477,-1.570)--(0.1847,-1.656)--(0.3057,-1.734)--(0.4370,-1.801)--(0.5776,-1.858)--(0.7264,-1.901)--(0.8820,-1.931)--(1.043,-1.947)--(1.208,-1.947)--(1.375,-1.931)--(1.542,-1.898)--(1.708,-1.850)--(1.871,-1.784)--(2.029,-1.703)--(2.181,-1.606)--(2.323,-1.493)--(2.456,-1.367)--(2.577,-1.227)--(2.685,-1.075)--(2.779,-0.9127)--(2.857,-0.7414)--(2.919,-0.5626)--(2.964,-0.3783)--(2.991,-0.1901)--(3.000,0);
+\draw [color=red] (3.0000,0.0000)--(2.9909,0.1900)--(2.9638,0.3782)--(2.9191,0.5626)--(2.8573,0.7413)--(2.7790,0.9127)--(2.6853,1.0750)--(2.5773,1.2268)--(2.4561,1.3665)--(2.3234,1.4931)--(2.1805,1.6055)--(2.0293,1.7027)--(1.8712,1.7842)--(1.7083,1.8495)--(1.5422,1.8983)--(1.3747,1.9306)--(1.2077,1.9465)--(1.0429,1.9466)--(0.8819,1.9312)--(0.7263,1.9012)--(0.5776,1.8576)--(0.4370,1.8013)--(0.3057,1.7337)--(0.1846,1.6559)--(0.0747,1.5695)--(-0.0234,1.4760)--(-0.1094,1.3767)--(-0.1830,1.2734)--(-0.2442,1.1675)--(-0.2931,1.0605)--(-0.3301,0.9538)--(-0.3556,0.8490)--(-0.3703,0.7471)--(-0.3750,0.6495)--(-0.3706,0.5570)--(-0.3582,0.4707)--(-0.3390,0.3912)--(-0.3141,0.3191)--(-0.2847,0.2547)--(-0.2522,0.1983)--(-0.2178,0.1499)--(-0.1827,0.1094)--(-0.1482,0.0764)--(-0.1152,0.0504)--(-0.0850,0.0309)--(-0.0582,0.0171)--(-0.0359,0.0081)--(-0.0186,0.0029)--(-0.0067,0.0000)--(0.0000,0.0000)--(0.0000,0.0000)--(-0.0067,0.0000)--(-0.0186,-0.0029)--(-0.0359,-0.0081)--(-0.0582,-0.0171)--(-0.0850,-0.0309)--(-0.1152,-0.0504)--(-0.1482,-0.0764)--(-0.1827,-0.1094)--(-0.2178,-0.1499)--(-0.2522,-0.1983)--(-0.2847,-0.2547)--(-0.3141,-0.3191)--(-0.3390,-0.3912)--(-0.3582,-0.4707)--(-0.3706,-0.5570)--(-0.3750,-0.6495)--(-0.3703,-0.7471)--(-0.3556,-0.8490)--(-0.3301,-0.9538)--(-0.2931,-1.0605)--(-0.2442,-1.1675)--(-0.1830,-1.2734)--(-0.1094,-1.3767)--(-0.0234,-1.4760)--(0.0747,-1.5695)--(0.1846,-1.6559)--(0.3057,-1.7337)--(0.4370,-1.8013)--(0.5776,-1.8576)--(0.7263,-1.9012)--(0.8819,-1.9312)--(1.0429,-1.9466)--(1.2077,-1.9465)--(1.3747,-1.9306)--(1.5422,-1.8983)--(1.7083,-1.8495)--(1.8712,-1.7842)--(2.0293,-1.7027)--(2.1805,-1.6055)--(2.3234,-1.4931)--(2.4561,-1.3665)--(2.5773,-1.2268)--(2.6853,-1.0750)--(2.7790,-0.9127)--(2.8573,-0.7413)--(2.9191,-0.5626)--(2.9638,-0.3782)--(2.9909,-0.1900)--(3.0000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CbCartTui.pstricks.recall b/src_phystricks/Fig_CbCartTui.pstricks.recall
index ad4c00516..6771f4c1f 100644
--- a/src_phystricks/Fig_CbCartTui.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTui.pstricks.recall
@@ -103,61 +103,61 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.1400,0) -- (4.1400,0);
-\draw [,->,>=latex] (0,-3.9720) -- (0,4.0280);
+\draw [,->,>=latex] (-4.1400,0.0000) -- (4.1400,0.0000);
+\draw [,->,>=latex] (0.0000,-3.9720) -- (0.0000,4.0280);
 %DEFAULT
-\draw [color=blue] (-3.640,-3.472)--(-3.609,-3.440)--(-3.579,-3.407)--(-3.548,-3.375)--(-3.518,-3.343)--(-3.488,-3.310)--(-3.457,-3.278)--(-3.427,-3.245)--(-3.396,-3.213)--(-3.366,-3.180)--(-3.336,-3.148)--(-3.306,-3.115)--(-3.275,-3.083)--(-3.245,-3.050)--(-3.215,-3.018)--(-3.185,-2.985)--(-3.155,-2.953)--(-3.125,-2.920)--(-3.095,-2.887)--(-3.065,-2.855)--(-3.035,-2.822)--(-3.005,-2.789)--(-2.975,-2.756)--(-2.945,-2.723)--(-2.915,-2.691)--(-2.886,-2.658)--(-2.856,-2.625)--(-2.826,-2.592)--(-2.797,-2.559)--(-2.767,-2.526)--(-2.738,-2.493)--(-2.709,-2.459)--(-2.679,-2.426)--(-2.650,-2.393)--(-2.621,-2.360)--(-2.592,-2.326)--(-2.563,-2.293)--(-2.534,-2.259)--(-2.505,-2.226)--(-2.476,-2.192)--(-2.447,-2.158)--(-2.419,-2.124)--(-2.390,-2.090)--(-2.362,-2.056)--(-2.333,-2.022)--(-2.305,-1.988)--(-2.277,-1.954)--(-2.249,-1.919)--(-2.221,-1.885)--(-2.193,-1.850)--(-2.166,-1.815)--(-2.138,-1.780)--(-2.111,-1.745)--(-2.084,-1.709)--(-2.057,-1.674)--(-2.030,-1.638)--(-2.003,-1.602)--(-1.977,-1.566)--(-1.951,-1.529)--(-1.925,-1.492)--(-1.899,-1.455)--(-1.873,-1.418)--(-1.848,-1.380)--(-1.823,-1.342)--(-1.798,-1.304)--(-1.774,-1.265)--(-1.750,-1.225)--(-1.726,-1.185)--(-1.703,-1.144)--(-1.680,-1.103)--(-1.658,-1.061)--(-1.636,-1.018)--(-1.614,-0.9745)--(-1.593,-0.9298)--(-1.573,-0.8840)--(-1.554,-0.8371)--(-1.535,-0.7887)--(-1.517,-0.7389)--(-1.499,-0.6873)--(-1.483,-0.6338)--(-1.468,-0.5781)--(-1.454,-0.5199)--(-1.441,-0.4588)--(-1.429,-0.3943)--(-1.420,-0.3261)--(-1.411,-0.2534)--(-1.405,-0.1754)--(-1.401,-0.09136)--(-1.400,0)--(-1.402,0.1001)--(-1.406,0.2106)--(-1.415,0.3340)--(-1.428,0.4729)--(-1.447,0.6314)--(-1.472,0.8143)--(-1.504,1.029)--(-1.545,1.283)--(-1.598,1.591)--(-1.665,1.971)--(-1.750,2.450);
+\draw [color=blue] (-3.6400,-3.4720)--(-3.6094,-3.4396)--(-3.5789,-3.4073)--(-3.5484,-3.3749)--(-3.5180,-3.3425)--(-3.4875,-3.3101)--(-3.4571,-3.2777)--(-3.4267,-3.2453)--(-3.3964,-3.2128)--(-3.3661,-3.1804)--(-3.3358,-3.1479)--(-3.3055,-3.1154)--(-3.2753,-3.0829)--(-3.2451,-3.0503)--(-3.2149,-3.0177)--(-3.1848,-2.9851)--(-3.1547,-2.9525)--(-3.1246,-2.9199)--(-3.0946,-2.8872)--(-3.0646,-2.8545)--(-3.0347,-2.8218)--(-3.0048,-2.7890)--(-2.9750,-2.7562)--(-2.9451,-2.7234)--(-2.9154,-2.6905)--(-2.8857,-2.6576)--(-2.8560,-2.6247)--(-2.8264,-2.5917)--(-2.7968,-2.5587)--(-2.7673,-2.5256)--(-2.7379,-2.4925)--(-2.7085,-2.4593)--(-2.6792,-2.4261)--(-2.6500,-2.3928)--(-2.6208,-2.3595)--(-2.5916,-2.3261)--(-2.5626,-2.2926)--(-2.5336,-2.2591)--(-2.5047,-2.2255)--(-2.4759,-2.1918)--(-2.4472,-2.1581)--(-2.4186,-2.1242)--(-2.3901,-2.0903)--(-2.3616,-2.0563)--(-2.3333,-2.0222)--(-2.3051,-1.9879)--(-2.2769,-1.9536)--(-2.2489,-1.9191)--(-2.2211,-1.8845)--(-2.1933,-1.8498)--(-2.1657,-1.8149)--(-2.1382,-1.7799)--(-2.1109,-1.7447)--(-2.0838,-1.7093)--(-2.0568,-1.6737)--(-2.0300,-1.6380)--(-2.0033,-1.6019)--(-1.9769,-1.5657)--(-1.9506,-1.5292)--(-1.9246,-1.4924)--(-1.8989,-1.4553)--(-1.8733,-1.4179)--(-1.8481,-1.3802)--(-1.8231,-1.3420)--(-1.7984,-1.3035)--(-1.7740,-1.2645)--(-1.7500,-1.2250)--(-1.7263,-1.1849)--(-1.7030,-1.1443)--(-1.6801,-1.1029)--(-1.6577,-1.0609)--(-1.6357,-1.0181)--(-1.6143,-0.9744)--(-1.5934,-0.9297)--(-1.5732,-0.8840)--(-1.5536,-0.8370)--(-1.5347,-0.7887)--(-1.5166,-0.7388)--(-1.4994,-0.6873)--(-1.4831,-0.6338)--(-1.4678,-0.5781)--(-1.4537,-0.5198)--(-1.4409,-0.4587)--(-1.4294,-0.3943)--(-1.4195,-0.3260)--(-1.4114,-0.2533)--(-1.4053,-0.1754)--(-1.4013,-0.0913)--(-1.4000,0.0000)--(-1.4015,0.1000)--(-1.4063,0.2106)--(-1.4150,0.3339)--(-1.4282,0.4729)--(-1.4467,0.6314)--(-1.4715,0.8143)--(-1.5039,1.0285)--(-1.5454,1.2831)--(-1.5982,1.5910)--(-1.6651,1.9709)--(-1.7500,2.4500);
 \draw [,->,>=latex] (-3.5058,-3.3296) -- (-3.5010,-3.3245);
-\draw [,->,>=latex] (-2.3333,-2.0222) -- (-2.3289,-2.0168);
-\draw [,->,>=latex] (-1.7500,-1.2250) -- (-1.7464,-1.2190);
-\draw [,->,>=latex] (-1.4900,0.93857) -- (-1.4911,0.94549);
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-\draw [,->,>=latex] (1.4233,1.3261) -- (1.4296,1.3229);
-\draw [,->,>=latex] (1.7500,1.5750) -- (1.7549,1.5800);
-\draw [,->,>=latex] (2.3333,2.1778) -- (2.3382,2.1828);
-\draw (-3.5000,-0.32983) node {$ -5 $};
-\draw [] (-3.50,-0.100) -- (-3.50,0.100);
-\draw (-2.8000,-0.32983) node {$ -4 $};
-\draw [] (-2.80,-0.100) -- (-2.80,0.100);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [,->,>=latex] (-2.3333,-2.0222) -- (-2.3288,-2.0168);
+\draw [,->,>=latex] (-1.7500,-1.2250) -- (-1.7463,-1.2189);
+\draw [,->,>=latex] (-1.4900,0.9385) -- (-1.4910,0.9454);
+\draw [color=blue] (1.7500,3.1500)--(1.7395,3.0978)--(1.7293,3.0472)--(1.7194,2.9983)--(1.7097,2.9508)--(1.7003,2.9049)--(1.6912,2.8603)--(1.6822,2.8171)--(1.6736,2.7752)--(1.6651,2.7345)--(1.6569,2.6951)--(1.6488,2.6568)--(1.6410,2.6197)--(1.6334,2.5836)--(1.6260,2.5486)--(1.6188,2.5146)--(1.6117,2.4816)--(1.6049,2.4495)--(1.5982,2.4183)--(1.5917,2.3880)--(1.5854,2.3586)--(1.5792,2.3299)--(1.5732,2.3021)--(1.5673,2.2750)--(1.5616,2.2487)--(1.5561,2.2231)--(1.5507,2.1982)--(1.5454,2.1740)--(1.5403,2.1504)--(1.5353,2.1275)--(1.5304,2.1051)--(1.5257,2.0834)--(1.5211,2.0622)--(1.5166,2.0416)--(1.5123,2.0216)--(1.5080,2.0020)--(1.5039,1.9830)--(1.4999,1.9645)--(1.4960,1.9464)--(1.4922,1.9288)--(1.4885,1.9117)--(1.4849,1.8950)--(1.4814,1.8788)--(1.4780,1.8630)--(1.4747,1.8475)--(1.4715,1.8325)--(1.4684,1.8178)--(1.4654,1.8035)--(1.4625,1.7896)--(1.4597,1.7760)--(1.4569,1.7628)--(1.4543,1.7499)--(1.4517,1.7374)--(1.4492,1.7251)--(1.4467,1.7132)--(1.4444,1.7015)--(1.4421,1.6902)--(1.4399,1.6791)--(1.4378,1.6683)--(1.4358,1.6578)--(1.4338,1.6476)--(1.4319,1.6376)--(1.4300,1.6279)--(1.4282,1.6184)--(1.4265,1.6091)--(1.4249,1.6001)--(1.4233,1.5913)--(1.4218,1.5827)--(1.4203,1.5744)--(1.4189,1.5662)--(1.4175,1.5583)--(1.4163,1.5505)--(1.4150,1.5430)--(1.4138,1.5357)--(1.4127,1.5285)--(1.4117,1.5215)--(1.4106,1.5147)--(1.4097,1.5081)--(1.4088,1.5017)--(1.4079,1.4954)--(1.4071,1.4893)--(1.4063,1.4833)--(1.4056,1.4775)--(1.4049,1.4719)--(1.4043,1.4664)--(1.4037,1.4610)--(1.4032,1.4558)--(1.4027,1.4508)--(1.4022,1.4458)--(1.4018,1.4410)--(1.4015,1.4364)--(1.4011,1.4319)--(1.4009,1.4275)--(1.4006,1.4232)--(1.4004,1.4190)--(1.4002,1.4150)--(1.4001,1.4110)--(1.4000,1.4072)--(1.4000,1.4035)--(1.4000,1.4000);
+\draw [,->,>=latex] (1.5866,2.3644) -- (1.5851,2.3576);
+\draw [,->,>=latex] (1.4350,1.6537) -- (1.4336,1.6468);
+\draw [color=blue] (1.4000,1.4000)--(1.4010,1.3749)--(1.4042,1.3558)--(1.4091,1.3416)--(1.4157,1.3318)--(1.4237,1.3258)--(1.4331,1.3231)--(1.4436,1.3233)--(1.4552,1.3260)--(1.4678,1.3309)--(1.4813,1.3379)--(1.4957,1.3466)--(1.5108,1.3568)--(1.5266,1.3685)--(1.5430,1.3815)--(1.5600,1.3956)--(1.5776,1.4107)--(1.5957,1.4268)--(1.6143,1.4437)--(1.6333,1.4613)--(1.6528,1.4797)--(1.6726,1.4988)--(1.6928,1.5184)--(1.7133,1.5385)--(1.7341,1.5592)--(1.7553,1.5803)--(1.7767,1.6018)--(1.7984,1.6237)--(1.8203,1.6460)--(1.8425,1.6686)--(1.8649,1.6915)--(1.8875,1.7147)--(1.9103,1.7381)--(1.9333,1.7619)--(1.9565,1.7858)--(1.9798,1.8100)--(2.0033,1.8343)--(2.0270,1.8589)--(2.0508,1.8836)--(2.0747,1.9085)--(2.0988,1.9335)--(2.1230,1.9587)--(2.1474,1.9841)--(2.1718,2.0095)--(2.1964,2.0351)--(2.2211,2.0608)--(2.2458,2.0866)--(2.2707,2.1125)--(2.2957,2.1385)--(2.3207,2.1646)--(2.3459,2.1908)--(2.3711,2.2171)--(2.3964,2.2435)--(2.4218,2.2699)--(2.4472,2.2964)--(2.4727,2.3229)--(2.4983,2.3495)--(2.5240,2.3762)--(2.5497,2.4030)--(2.5755,2.4298)--(2.6013,2.4566)--(2.6272,2.4835)--(2.6532,2.5105)--(2.6792,2.5375)--(2.7053,2.5645)--(2.7314,2.5916)--(2.7575,2.6187)--(2.7837,2.6458)--(2.8100,2.6730)--(2.8363,2.7003)--(2.8626,2.7275)--(2.8890,2.7548)--(2.9154,2.7821)--(2.9418,2.8095)--(2.9683,2.8369)--(2.9948,2.8643)--(3.0214,2.8917)--(3.0480,2.9191)--(3.0746,2.9466)--(3.1013,2.9741)--(3.1280,3.0017)--(3.1547,3.0292)--(3.1814,3.0568)--(3.2082,3.0844)--(3.2350,3.1120)--(3.2618,3.1396)--(3.2887,3.1672)--(3.3156,3.1949)--(3.3425,3.2226)--(3.3694,3.2503)--(3.3964,3.2780)--(3.4234,3.3057)--(3.4504,3.3334)--(3.4774,3.3612)--(3.5044,3.3889)--(3.5315,3.4167)--(3.5586,3.4445)--(3.5857,3.4723)--(3.6128,3.5001)--(3.6400,3.5280);
+\draw [,->,>=latex] (1.4233,1.3261) -- (1.4295,1.3229);
+\draw [,->,>=latex] (1.7500,1.5750) -- (1.7549,1.5799);
+\draw [,->,>=latex] (2.3333,2.1777) -- (2.3381,2.1828);
+\draw (-3.5000,-0.3298) node {$ -5 $};
+\draw [] (-3.5000,-0.1000) -- (-3.5000,0.1000);
+\draw (-2.8000,-0.3298) node {$ -4 $};
+\draw [] (-2.8000,-0.1000) -- (-2.8000,0.1000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ChampGraviation.pstricks.recall b/src_phystricks/Fig_ChampGraviation.pstricks.recall
index 518a4dcc8..6adcbc888 100644
--- a/src_phystricks/Fig_ChampGraviation.pstricks.recall
+++ b/src_phystricks/Fig_ChampGraviation.pstricks.recall
@@ -65,228 +65,228 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.0000,-4.0000) -- (-4.0442,-4.0442);
-\draw [,->,>=latex] (-4.0000,-3.4286) -- (-4.0547,-3.4755);
-\draw [,->,>=latex] (-4.0000,-2.8571) -- (-4.0674,-2.9053);
-\draw [,->,>=latex] (-4.0000,-2.2857) -- (-4.0818,-2.3325);
-\draw [,->,>=latex] (-4.0000,-1.7143) -- (-4.0971,-1.7559);
-\draw [,->,>=latex] (-4.0000,-1.1429) -- (-4.1111,-1.1746);
-\draw [,->,>=latex] (-4.0000,-0.57143) -- (-4.1213,-0.58875);
-\draw [,->,>=latex] (-4.0000,0) -- (-4.1250,0);
-\draw [,->,>=latex] (-4.0000,0.57143) -- (-4.1213,0.58875);
-\draw [,->,>=latex] (-4.0000,1.1429) -- (-4.1111,1.1746);
-\draw [,->,>=latex] (-4.0000,1.7143) -- (-4.0971,1.7559);
-\draw [,->,>=latex] (-4.0000,2.2857) -- (-4.0818,2.3325);
-\draw [,->,>=latex] (-4.0000,2.8571) -- (-4.0674,2.9053);
-\draw [,->,>=latex] (-4.0000,3.4286) -- (-4.0547,3.4755);
-\draw [,->,>=latex] (-4.0000,4.0000) -- (-4.0442,4.0442);
-\draw [,->,>=latex] (-3.4286,-4.0000) -- (-3.4755,-4.0547);
-\draw [,->,>=latex] (-3.4286,-3.4286) -- (-3.4887,-3.4887);
-\draw [,->,>=latex] (-3.4286,-2.8571) -- (-3.5057,-2.9214);
-\draw [,->,>=latex] (-3.4286,-2.2857) -- (-3.5266,-2.3511);
-\draw [,->,>=latex] (-3.4286,-1.7143) -- (-3.5503,-1.7752);
-\draw [,->,>=latex] (-3.4286,-1.1429) -- (-3.5738,-1.1913);
-\draw [,->,>=latex] (-3.4286,-0.57143) -- (-3.5919,-0.59864);
-\draw [,->,>=latex] (-3.4286,0) -- (-3.5987,0);
-\draw [,->,>=latex] (-3.4286,0.57143) -- (-3.5919,0.59864);
-\draw [,->,>=latex] (-3.4286,1.1429) -- (-3.5738,1.1913);
-\draw [,->,>=latex] (-3.4286,1.7143) -- (-3.5503,1.7752);
-\draw [,->,>=latex] (-3.4286,2.2857) -- (-3.5266,2.3511);
-\draw [,->,>=latex] (-3.4286,2.8571) -- (-3.5057,2.9214);
-\draw [,->,>=latex] (-3.4286,3.4286) -- (-3.4887,3.4887);
-\draw [,->,>=latex] (-3.4286,4.0000) -- (-3.4755,4.0547);
-\draw [,->,>=latex] (-2.8571,-4.0000) -- (-2.9053,-4.0674);
-\draw [,->,>=latex] (-2.8571,-3.4286) -- (-2.9214,-3.5057);
-\draw [,->,>=latex] (-2.8571,-2.8571) -- (-2.9438,-2.9438);
-\draw [,->,>=latex] (-2.8571,-2.2857) -- (-2.9738,-2.3790);
-\draw [,->,>=latex] (-2.8571,-1.7143) -- (-3.0116,-1.8070);
-\draw [,->,>=latex] (-2.8571,-1.1429) -- (-3.0532,-1.2213);
-\draw [,->,>=latex] (-2.8571,-0.57143) -- (-3.0881,-0.61763);
-\draw [,->,>=latex] (-2.8571,0) -- (-3.1021,0);
-\draw [,->,>=latex] (-2.8571,0.57143) -- (-3.0881,0.61763);
-\draw [,->,>=latex] (-2.8571,1.1429) -- (-3.0532,1.2213);
-\draw [,->,>=latex] (-2.8571,1.7143) -- (-3.0116,1.8070);
-\draw [,->,>=latex] (-2.8571,2.2857) -- (-2.9738,2.3790);
-\draw [,->,>=latex] (-2.8571,2.8571) -- (-2.9438,2.9438);
-\draw [,->,>=latex] (-2.8571,3.4286) -- (-2.9214,3.5057);
-\draw [,->,>=latex] (-2.8571,4.0000) -- (-2.9053,4.0674);
-\draw [,->,>=latex] (-2.2857,-4.0000) -- (-2.3325,-4.0818);
-\draw [,->,>=latex] (-2.2857,-3.4286) -- (-2.3511,-3.5266);
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CheminFresnel.pstricks.recall b/src_phystricks/Fig_CheminFresnel.pstricks.recall
index 5589fe1bc..67f8f4c4b 100644
--- a/src_phystricks/Fig_CheminFresnel.pstricks.recall
+++ b/src_phystricks/Fig_CheminFresnel.pstricks.recall
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.9142);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.9142);
 %DEFAULT
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-\draw (1.0000,-0.21406) node {\( \gamma_1\)};
-\draw (2.1138,0.91770) node {\( \gamma_2\)};
-\draw (0.46274,0.89188) node {\( \gamma_3\)};
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+\draw [,->,>=latex] (1.0000,0.0000) -- (1.0100,0.0000);
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+\draw [,->,>=latex] (1.8477,0.7653) -- (1.8439,0.7746);
+\draw [color=blue] (0.0000,0.0000)--(0.0142,0.0142)--(0.0285,0.0285)--(0.0428,0.0428)--(0.0571,0.0571)--(0.0714,0.0714)--(0.0857,0.0857)--(0.0999,0.0999)--(0.1142,0.1142)--(0.1285,0.1285)--(0.1428,0.1428)--(0.1571,0.1571)--(0.1714,0.1714)--(0.1857,0.1857)--(0.1999,0.1999)--(0.2142,0.2142)--(0.2285,0.2285)--(0.2428,0.2428)--(0.2571,0.2571)--(0.2714,0.2714)--(0.2856,0.2856)--(0.2999,0.2999)--(0.3142,0.3142)--(0.3285,0.3285)--(0.3428,0.3428)--(0.3571,0.3571)--(0.3714,0.3714)--(0.3856,0.3856)--(0.3999,0.3999)--(0.4142,0.4142)--(0.4285,0.4285)--(0.4428,0.4428)--(0.4571,0.4571)--(0.4714,0.4714)--(0.4856,0.4856)--(0.4999,0.4999)--(0.5142,0.5142)--(0.5285,0.5285)--(0.5428,0.5428)--(0.5571,0.5571)--(0.5713,0.5713)--(0.5856,0.5856)--(0.5999,0.5999)--(0.6142,0.6142)--(0.6285,0.6285)--(0.6428,0.6428)--(0.6571,0.6571)--(0.6713,0.6713)--(0.6856,0.6856)--(0.6999,0.6999)--(0.7142,0.7142)--(0.7285,0.7285)--(0.7428,0.7428)--(0.7571,0.7571)--(0.7713,0.7713)--(0.7856,0.7856)--(0.7999,0.7999)--(0.8142,0.8142)--(0.8285,0.8285)--(0.8428,0.8428)--(0.8570,0.8570)--(0.8713,0.8713)--(0.8856,0.8856)--(0.8999,0.8999)--(0.9142,0.9142)--(0.9285,0.9285)--(0.9428,0.9428)--(0.9570,0.9570)--(0.9713,0.9713)--(0.9856,0.9856)--(0.9999,0.9999)--(1.0142,1.0142)--(1.0285,1.0285)--(1.0428,1.0428)--(1.0570,1.0570)--(1.0713,1.0713)--(1.0856,1.0856)--(1.0999,1.0999)--(1.1142,1.1142)--(1.1285,1.1285)--(1.1427,1.1427)--(1.1570,1.1570)--(1.1713,1.1713)--(1.1856,1.1856)--(1.1999,1.1999)--(1.2142,1.2142)--(1.2285,1.2285)--(1.2427,1.2427)--(1.2570,1.2570)--(1.2713,1.2713)--(1.2856,1.2856)--(1.2999,1.2999)--(1.3142,1.3142)--(1.3285,1.3285)--(1.3427,1.3427)--(1.3570,1.3570)--(1.3713,1.3713)--(1.3856,1.3856)--(1.3999,1.3999)--(1.4142,1.4142);
+\draw [,->,>=latex] (0.7071,0.7071) -- (0.7141,0.7141);
+\draw (1.0000,-0.2140) node {\( \gamma_1\)};
+\draw (2.1137,0.9176) node {\( \gamma_2\)};
+\draw (0.4627,0.8918) node {\( \gamma_3\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ChoixInfini.pstricks.recall b/src_phystricks/Fig_ChoixInfini.pstricks.recall
index 40c07f5d0..c75de4bed 100644
--- a/src_phystricks/Fig_ChoixInfini.pstricks.recall
+++ b/src_phystricks/Fig_ChoixInfini.pstricks.recall
@@ -57,25 +57,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw [color=blue] (-3.00,1.00) -- (3.00,1.00);
-\draw [color=blue]  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-3.0000,1.0000) -- (3.0000,1.0000);
+\draw [color=blue]  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -121,23 +121,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue] (-2.00,0) -- (2.00,2.00);
+\draw [color=blue] (-2.0000,0.0000) -- (2.0000,2.0000);
 \draw [color=blue]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CoinPasVar.pstricks.recall b/src_phystricks/Fig_CoinPasVar.pstricks.recall
index 471cee114..83860cb3d 100644
--- a/src_phystricks/Fig_CoinPasVar.pstricks.recall
+++ b/src_phystricks/Fig_CoinPasVar.pstricks.recall
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue] (-2.00,0) -- (0,2.00);
-\draw [color=blue] (2.00,0) -- (0,2.00);
-\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
-\draw (0.23724,2.1954) node {\( N\)};
+\draw [color=blue] (-2.0000,0.0000) -- (0.0000,2.0000);
+\draw [color=blue] (2.0000,0.0000) -- (0.0000,2.0000);
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (0.2372,2.1954) node {\( N\)};
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.2158,1.2103) node {\( t_1\)};
+\draw (-1.2157,1.2103) node {\( t_1\)};
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.2158,1.2103) node {\( t_2\)};
+\draw (1.2157,1.2103) node {\( t_2\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ConeRevolution.pstricks.recall b/src_phystricks/Fig_ConeRevolution.pstricks.recall
index 1c9a11ba0..8f2baeb65 100644
--- a/src_phystricks/Fig_ConeRevolution.pstricks.recall
+++ b/src_phystricks/Fig_ConeRevolution.pstricks.recall
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.02020,0.03030)--(0.04040,0.06061)--(0.06061,0.09091)--(0.08081,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5152)--(0.3636,0.5455)--(0.3838,0.5758)--(0.4040,0.6061)--(0.4242,0.6364)--(0.4444,0.6667)--(0.4646,0.6970)--(0.4848,0.7273)--(0.5051,0.7576)--(0.5253,0.7879)--(0.5455,0.8182)--(0.5657,0.8485)--(0.5859,0.8788)--(0.6061,0.9091)--(0.6263,0.9394)--(0.6465,0.9697)--(0.6667,1.000)--(0.6869,1.030)--(0.7071,1.061)--(0.7273,1.091)--(0.7475,1.121)--(0.7677,1.152)--(0.7879,1.182)--(0.8081,1.212)--(0.8283,1.242)--(0.8485,1.273)--(0.8687,1.303)--(0.8889,1.333)--(0.9091,1.364)--(0.9293,1.394)--(0.9495,1.424)--(0.9697,1.455)--(0.9899,1.485)--(1.010,1.515)--(1.030,1.545)--(1.051,1.576)--(1.071,1.606)--(1.091,1.636)--(1.111,1.667)--(1.131,1.697)--(1.152,1.727)--(1.172,1.758)--(1.192,1.788)--(1.212,1.818)--(1.232,1.848)--(1.253,1.879)--(1.273,1.909)--(1.293,1.939)--(1.313,1.970)--(1.333,2.000)--(1.354,2.030)--(1.374,2.061)--(1.394,2.091)--(1.414,2.121)--(1.434,2.152)--(1.455,2.182)--(1.475,2.212)--(1.495,2.242)--(1.515,2.273)--(1.535,2.303)--(1.556,2.333)--(1.576,2.364)--(1.596,2.394)--(1.616,2.424)--(1.636,2.455)--(1.657,2.485)--(1.677,2.515)--(1.697,2.545)--(1.717,2.576)--(1.737,2.606)--(1.758,2.636)--(1.778,2.667)--(1.798,2.697)--(1.818,2.727)--(1.838,2.758)--(1.859,2.788)--(1.879,2.818)--(1.899,2.848)--(1.919,2.879)--(1.939,2.909)--(1.960,2.939)--(1.980,2.970)--(2.000,3.000);
+\draw [color=blue] (0.0000,0.0000)--(0.0202,0.0303)--(0.0404,0.0606)--(0.0606,0.0909)--(0.0808,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5151)--(0.3636,0.5454)--(0.3838,0.5757)--(0.4040,0.6060)--(0.4242,0.6363)--(0.4444,0.6666)--(0.4646,0.6969)--(0.4848,0.7272)--(0.5050,0.7575)--(0.5252,0.7878)--(0.5454,0.8181)--(0.5656,0.8484)--(0.5858,0.8787)--(0.6060,0.9090)--(0.6262,0.9393)--(0.6464,0.9696)--(0.6666,1.0000)--(0.6868,1.0303)--(0.7070,1.0606)--(0.7272,1.0909)--(0.7474,1.1212)--(0.7676,1.1515)--(0.7878,1.1818)--(0.8080,1.2121)--(0.8282,1.2424)--(0.8484,1.2727)--(0.8686,1.3030)--(0.8888,1.3333)--(0.9090,1.3636)--(0.9292,1.3939)--(0.9494,1.4242)--(0.9696,1.4545)--(0.9898,1.4848)--(1.0101,1.5151)--(1.0303,1.5454)--(1.0505,1.5757)--(1.0707,1.6060)--(1.0909,1.6363)--(1.1111,1.6666)--(1.1313,1.6969)--(1.1515,1.7272)--(1.1717,1.7575)--(1.1919,1.7878)--(1.2121,1.8181)--(1.2323,1.8484)--(1.2525,1.8787)--(1.2727,1.9090)--(1.2929,1.9393)--(1.3131,1.9696)--(1.3333,2.0000)--(1.3535,2.0303)--(1.3737,2.0606)--(1.3939,2.0909)--(1.4141,2.1212)--(1.4343,2.1515)--(1.4545,2.1818)--(1.4747,2.2121)--(1.4949,2.2424)--(1.5151,2.2727)--(1.5353,2.3030)--(1.5555,2.3333)--(1.5757,2.3636)--(1.5959,2.3939)--(1.6161,2.4242)--(1.6363,2.4545)--(1.6565,2.4848)--(1.6767,2.5151)--(1.6969,2.5454)--(1.7171,2.5757)--(1.7373,2.6060)--(1.7575,2.6363)--(1.7777,2.6666)--(1.7979,2.6969)--(1.8181,2.7272)--(1.8383,2.7575)--(1.8585,2.7878)--(1.8787,2.8181)--(1.8989,2.8484)--(1.9191,2.8787)--(1.9393,2.9090)--(1.9595,2.9393)--(1.9797,2.9696)--(2.0000,3.0000);
 \draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
-\draw (2.5389,3.2532) node {$(R,h)$};
-\draw (0.72367,0.33595) node {$\alpha$};
+\draw (2.5388,3.2531) node {$(R,h)$};
+\draw (0.7236,0.3359) node {$\alpha$};
 
-\draw [color=red] (0.500,0)--(0.500,0.00496)--(0.500,0.00993)--(0.500,0.0149)--(0.500,0.0198)--(0.499,0.0248)--(0.499,0.0298)--(0.499,0.0347)--(0.498,0.0397)--(0.498,0.0446)--(0.498,0.0496)--(0.497,0.0545)--(0.496,0.0594)--(0.496,0.0643)--(0.495,0.0693)--(0.494,0.0742)--(0.494,0.0791)--(0.493,0.0840)--(0.492,0.0889)--(0.491,0.0938)--(0.490,0.0986)--(0.489,0.103)--(0.488,0.108)--(0.487,0.113)--(0.486,0.118)--(0.485,0.123)--(0.483,0.128)--(0.482,0.132)--(0.481,0.137)--(0.479,0.142)--(0.478,0.147)--(0.477,0.151)--(0.475,0.156)--(0.473,0.161)--(0.472,0.166)--(0.470,0.170)--(0.468,0.175)--(0.467,0.180)--(0.465,0.184)--(0.463,0.189)--(0.461,0.193)--(0.459,0.198)--(0.457,0.202)--(0.455,0.207)--(0.453,0.212)--(0.451,0.216)--(0.449,0.220)--(0.447,0.225)--(0.444,0.229)--(0.442,0.234)--(0.440,0.238)--(0.437,0.242)--(0.435,0.247)--(0.432,0.251)--(0.430,0.255)--(0.427,0.260)--(0.425,0.264)--(0.422,0.268)--(0.419,0.272)--(0.417,0.276)--(0.414,0.281)--(0.411,0.285)--(0.408,0.289)--(0.405,0.293)--(0.402,0.297)--(0.399,0.301)--(0.396,0.305)--(0.393,0.309)--(0.390,0.312)--(0.387,0.316)--(0.384,0.320)--(0.381,0.324)--(0.378,0.328)--(0.374,0.331)--(0.371,0.335)--(0.368,0.339)--(0.364,0.342)--(0.361,0.346)--(0.357,0.350)--(0.354,0.353)--(0.350,0.357)--(0.347,0.360)--(0.343,0.364)--(0.340,0.367)--(0.336,0.370)--(0.332,0.374)--(0.329,0.377)--(0.325,0.380)--(0.321,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.306,0.396)--(0.302,0.399)--(0.298,0.402)--(0.294,0.405)--(0.290,0.408)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416);
-\draw (2.0000,-0.32572) node {$\mathit{R}$};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.30273,3.0000) node {$\mathit{h}$};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (0.5000,0.0000)--(0.4999,0.0049)--(0.4999,0.0099)--(0.4997,0.0148)--(0.4996,0.0198)--(0.4993,0.0248)--(0.4991,0.0297)--(0.4987,0.0347)--(0.4984,0.0396)--(0.4980,0.0446)--(0.4975,0.0495)--(0.4970,0.0544)--(0.4964,0.0594)--(0.4958,0.0643)--(0.4951,0.0692)--(0.4944,0.0741)--(0.4937,0.0790)--(0.4928,0.0839)--(0.4920,0.0888)--(0.4911,0.0937)--(0.4901,0.0986)--(0.4891,0.1034)--(0.4881,0.1083)--(0.4870,0.1131)--(0.4858,0.1180)--(0.4846,0.1228)--(0.4834,0.1276)--(0.4821,0.1324)--(0.4808,0.1371)--(0.4794,0.1419)--(0.4779,0.1467)--(0.4765,0.1514)--(0.4749,0.1561)--(0.4734,0.1608)--(0.4717,0.1655)--(0.4701,0.1702)--(0.4684,0.1749)--(0.4666,0.1795)--(0.4648,0.1841)--(0.4629,0.1887)--(0.4610,0.1933)--(0.4591,0.1979)--(0.4571,0.2024)--(0.4551,0.2070)--(0.4530,0.2115)--(0.4509,0.2160)--(0.4487,0.2204)--(0.4465,0.2249)--(0.4443,0.2293)--(0.4420,0.2337)--(0.4396,0.2381)--(0.4372,0.2424)--(0.4348,0.2467)--(0.4323,0.2511)--(0.4298,0.2553)--(0.4273,0.2596)--(0.4247,0.2638)--(0.4220,0.2680)--(0.4193,0.2722)--(0.4166,0.2763)--(0.4138,0.2805)--(0.4110,0.2846)--(0.4082,0.2886)--(0.4053,0.2927)--(0.4024,0.2967)--(0.3994,0.3007)--(0.3964,0.3046)--(0.3934,0.3085)--(0.3903,0.3124)--(0.3872,0.3163)--(0.3840,0.3201)--(0.3808,0.3239)--(0.3776,0.3277)--(0.3743,0.3314)--(0.3710,0.3351)--(0.3676,0.3388)--(0.3643,0.3424)--(0.3609,0.3460)--(0.3574,0.3496)--(0.3539,0.3531)--(0.3504,0.3566)--(0.3468,0.3601)--(0.3432,0.3635)--(0.3396,0.3669)--(0.3360,0.3702)--(0.3323,0.3735)--(0.3285,0.3768)--(0.3248,0.3801)--(0.3210,0.3833)--(0.3172,0.3864)--(0.3133,0.3896)--(0.3094,0.3927)--(0.3055,0.3957)--(0.3016,0.3987)--(0.2976,0.4017)--(0.2936,0.4046)--(0.2896,0.4075)--(0.2855,0.4104)--(0.2814,0.4132)--(0.2773,0.4160);
+\draw (2.0000,-0.3257) node {$\mathit{R}$};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.3027,3.0000) node {$\mathit{h}$};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ContourGreen.pstricks.recall b/src_phystricks/Fig_ContourGreen.pstricks.recall
index 6990892da..355cff092 100644
--- a/src_phystricks/Fig_ContourGreen.pstricks.recall
+++ b/src_phystricks/Fig_ContourGreen.pstricks.recall
@@ -65,15 +65,15 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
-\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
-\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
-\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
-\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
-\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
-\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
-\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
+\draw [color=blue] (1.0000,0.0000)--(1.0611,0.0674)--(1.1165,0.1424)--(1.1644,0.2244)--(1.2032,0.3122)--(1.2317,0.4045)--(1.2486,0.4999)--(1.2533,0.5965)--(1.2451,0.6927)--(1.2238,0.7865)--(1.1897,0.8759)--(1.1432,0.9592)--(1.0851,1.0347)--(1.0166,1.1007)--(0.9391,1.1559)--(0.8541,1.1994)--(0.7634,1.2303)--(0.6688,1.2483)--(0.5723,1.2533)--(0.4758,1.2455)--(0.3811,1.2256)--(0.2897,1.1944)--(0.2033,1.1532)--(0.1230,1.1033)--(0.0498,1.0463)--(-0.0156,0.9840)--(-0.0729,0.9181)--(-0.1222,0.8503)--(-0.1637,0.7825)--(-0.1980,0.7162)--(-0.2259,0.6529)--(-0.2486,0.5936)--(-0.2673,0.5394)--(-0.2834,0.4910)--(-0.2984,0.4486)--(-0.3137,0.4122)--(-0.3307,0.3817)--(-0.3507,0.3563)--(-0.3749,0.3354)--(-0.4041,0.3177)--(-0.4389,0.3022)--(-0.4798,0.2872)--(-0.5268,0.2715)--(-0.5796,0.2536)--(-0.6376,0.2320)--(-0.7001,0.2055)--(-0.7658,0.1729)--(-0.8333,0.1333)--(-0.9012,0.0860)--(-0.9678,0.0307)--(-1.0311,-0.0327)--(-1.0896,-0.1040)--(-1.1414,-0.1826)--(-1.1850,-0.2676)--(-1.2188,-0.3578)--(-1.2417,-0.4519)--(-1.2525,-0.5481)--(-1.2508,-0.6448)--(-1.2361,-0.7400)--(-1.2083,-0.8319)--(-1.1680,-0.9185)--(-1.1156,-0.9980)--(-1.0521,-1.0689)--(-0.9789,-1.1297)--(-0.8974,-1.1792)--(-0.8093,-1.2165)--(-0.7165,-1.2410)--(-0.6207,-1.2524)--(-0.5240,-1.2510)--(-0.4281,-1.2370)--(-0.3349,-1.2113)--(-0.2458,-1.1750)--(-0.1623,-1.1292)--(-0.0855,-1.0756)--(-0.0161,-1.0157)--(0.0453,-0.9513)--(0.0986,-0.8843)--(0.1439,-0.8163)--(0.1817,-0.7491)--(0.2127,-0.6841)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2756,-0.5145)--(0.2910,-0.4690)--(0.3059,-0.4296)--(0.3219,-0.3962)--(0.3403,-0.3684)--(0.3622,-0.3454)--(0.3888,-0.3262)--(0.4207,-0.3098)--(0.4586,-0.2947)--(0.5025,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1900)--(0.7994,-0.1540)--(0.8673,-0.1106)--(0.9348,-0.0594)--(1.0000,0.0000);
+\draw [,->,>=latex] (1.2468,0.6811) -- (1.2454,0.6910);
+\draw [,->,>=latex] (0.3575,1.2187) -- (0.3480,1.2158);
+\draw [,->,>=latex] (-0.2777,0.5083) -- (-0.2808,0.4988);
+\draw [,->,>=latex] (-0.7003,0.2054) -- (-0.7094,0.2013);
+\draw [,->,>=latex] (-1.2468,-0.6811) -- (-1.2454,-0.6910);
+\draw [,->,>=latex] (-0.3575,-1.2187) -- (-0.3480,-1.2158);
+\draw [,->,>=latex] (0.2777,-0.5083) -- (0.2808,-0.4988);
+\draw [,->,>=latex] (0.7003,-0.2054) -- (0.7094,-0.2013);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall b/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall
index e2b22fef2..7c6762cc9 100644
--- a/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall
+++ b/src_phystricks/Fig_ContourTgNDivergence.pstricks.recall
@@ -65,31 +65,31 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red,->,>=latex] (2.0000,0) -- (2.7071,0.70711);
-\draw [color=green,->,>=latex] (2.0000,0) -- (2.7071,-0.70711);
-\draw [color=red,->,>=latex] (2.3414,1.8273) -- (1.8616,2.7046);
-\draw [color=green,->,>=latex] (2.3414,1.8273) -- (3.2188,2.3071);
-\draw [color=red,->,>=latex] (0.61190,2.4018) -- (-0.32151,2.0429);
-\draw [color=green,->,>=latex] (0.61190,2.4018) -- (0.25309,3.3352);
-\draw [color=red,->,>=latex] (-0.59548,0.90107) -- (-0.94670,-0.035225);
-\draw [color=green,->,>=latex] (-0.59548,0.90107) -- (-1.5318,1.2523);
-\draw [color=red,->,>=latex] (-2.1766,-0.20446) -- (-2.7726,-1.0075);
-\draw [color=green,->,>=latex] (-2.1766,-0.20446) -- (-2.9796,0.39153);
-\draw [color=red,->,>=latex] (-2.1728,-2.0668) -- (-1.5081,-2.8139);
-\draw [color=green,->,>=latex] (-2.1728,-2.0668) -- (-2.9199,-2.7314);
-\draw [color=red,->,>=latex] (-0.34301,-2.2697) -- (0.51254,-1.7520);
-\draw [color=green,->,>=latex] (-0.34301,-2.2697) -- (0.17470,-3.1253);
-\draw [color=red,->,>=latex] (0.77682,-0.65307) -- (1.6318,-0.13442);
-\draw [color=green,->,>=latex] (0.77682,-0.65307) -- (1.2955,-1.5081);
-\draw [color=blue] (2.000,0)--(2.122,0.1349)--(2.233,0.2850)--(2.329,0.4488)--(2.407,0.6244)--(2.463,0.8091)--(2.497,0.9998)--(2.507,1.193)--(2.490,1.386)--(2.448,1.573)--(2.379,1.752)--(2.286,1.919)--(2.170,2.069)--(2.033,2.201)--(1.878,2.312)--(1.708,2.399)--(1.527,2.461)--(1.338,2.497)--(1.145,2.507)--(0.9517,2.491)--(0.7622,2.451)--(0.5796,2.389)--(0.4067,2.306)--(0.2461,2.207)--(0.09969,2.093)--(-0.03123,1.968)--(-0.1460,1.836)--(-0.2445,1.701)--(-0.3275,1.565)--(-0.3961,1.433)--(-0.4520,1.306)--(-0.4973,1.187)--(-0.5348,1.079)--(-0.5670,0.9820)--(-0.5969,0.8972)--(-0.6275,0.8245)--(-0.6615,0.7634)--(-0.7016,0.7128)--(-0.7499,0.6709)--(-0.8082,0.6356)--(-0.8779,0.6044)--(-0.9597,0.5746)--(-1.054,0.5432)--(-1.159,0.5073)--(-1.275,0.4642)--(-1.400,0.4111)--(-1.532,0.3459)--(-1.667,0.2667)--(-1.803,0.1721)--(-1.936,0.06144)--(-2.062,-0.06547)--(-2.179,-0.2081)--(-2.283,-0.3653)--(-2.370,-0.5353)--(-2.438,-0.7158)--(-2.483,-0.9039)--(-2.505,-1.096)--(-2.502,-1.290)--(-2.472,-1.480)--(-2.417,-1.664)--(-2.336,-1.837)--(-2.231,-1.996)--(-2.104,-2.138)--(-1.958,-2.260)--(-1.795,-2.359)--(-1.619,-2.433)--(-1.433,-2.482)--(-1.242,-2.505)--(-1.048,-2.502)--(-0.8563,-2.474)--(-0.6698,-2.423)--(-0.4917,-2.350)--(-0.3247,-2.259)--(-0.1710,-2.151)--(-0.03224,-2.031)--(0.09064,-1.903)--(0.1973,-1.769)--(0.2879,-1.633)--(0.3635,-1.498)--(0.4255,-1.368)--(0.4758,-1.245)--(0.5169,-1.132)--(0.5513,-1.029)--(0.5821,-0.9381)--(0.6120,-0.8594)--(0.6439,-0.7926)--(0.6806,-0.7369)--(0.7246,-0.6909)--(0.7777,-0.6525)--(0.8416,-0.6196)--(0.9173,-0.5895)--(1.005,-0.5593)--(1.105,-0.5260)--(1.216,-0.4868)--(1.337,-0.4391)--(1.465,-0.3802)--(1.599,-0.3082)--(1.735,-0.2214)--(1.870,-0.1188)--(2.000,0);
-\draw [,->,>=latex] (2.0000,0) -- (2.0141,0.014142);
-\draw [,->,>=latex] (2.3414,1.8273) -- (2.3318,1.8448);
-\draw [,->,>=latex] (0.61190,2.4018) -- (0.59323,2.3946);
-\draw [,->,>=latex] (-0.59548,0.90107) -- (-0.60250,0.88234);
-\draw [,->,>=latex] (-2.1766,-0.20446) -- (-2.1886,-0.22052);
-\draw [,->,>=latex] (-2.1728,-2.0668) -- (-2.1595,-2.0817);
-\draw [,->,>=latex] (-0.34301,-2.2697) -- (-0.32590,-2.2594);
-\draw [,->,>=latex] (0.77682,-0.65307) -- (0.79392,-0.64270);
+\draw [color=red,->,>=latex] (2.0000,0.0000) -- (2.7071,0.7071);
+\draw [color=green,->,>=latex] (2.0000,0.0000) -- (2.7071,-0.7071);
+\draw [color=red,->,>=latex] (2.3414,1.8272) -- (1.8615,2.7046);
+\draw [color=green,->,>=latex] (2.3414,1.8272) -- (3.2187,2.3071);
+\draw [color=red,->,>=latex] (0.6118,2.4017) -- (-0.3215,2.0429);
+\draw [color=green,->,>=latex] (0.6118,2.4017) -- (0.2530,3.3351);
+\draw [color=red,->,>=latex] (-0.5954,0.9010) -- (-0.9467,-0.0352);
+\draw [color=green,->,>=latex] (-0.5954,0.9010) -- (-1.5317,1.2522);
+\draw [color=red,->,>=latex] (-2.1766,-0.2044) -- (-2.7726,-1.0074);
+\draw [color=green,->,>=latex] (-2.1766,-0.2044) -- (-2.9796,0.3915);
+\draw [color=red,->,>=latex] (-2.1727,-2.0667) -- (-1.5081,-2.8138);
+\draw [color=green,->,>=latex] (-2.1727,-2.0667) -- (-2.9199,-2.7314);
+\draw [color=red,->,>=latex] (-0.3430,-2.2697) -- (0.5125,-1.7520);
+\draw [color=green,->,>=latex] (-0.3430,-2.2697) -- (0.1746,-3.1252);
+\draw [color=red,->,>=latex] (0.7768,-0.6530) -- (1.6318,-0.1344);
+\draw [color=green,->,>=latex] (0.7768,-0.6530) -- (1.2954,-1.5080);
+\draw [color=blue] (2.0000,0.0000)--(2.1223,0.1348)--(2.2330,0.2849)--(2.3288,0.4488)--(2.4065,0.6244)--(2.4634,0.8090)--(2.4973,0.9998)--(2.5066,1.1931)--(2.4902,1.3855)--(2.4477,1.5730)--(2.3794,1.7519)--(2.2864,1.9185)--(2.1703,2.0694)--(2.0333,2.2014)--(1.8782,2.3119)--(1.7082,2.3989)--(1.5268,2.4607)--(1.3377,2.4967)--(1.1447,2.5067)--(0.9517,2.4911)--(0.7622,2.4512)--(0.5795,2.3889)--(0.4066,2.3064)--(0.2460,2.2066)--(0.0996,2.0926)--(-0.0312,1.9680)--(-0.1459,1.8362)--(-0.2445,1.7007)--(-0.3275,1.5651)--(-0.3960,1.4325)--(-0.4519,1.3058)--(-0.4973,1.1873)--(-0.5347,1.0789)--(-0.5669,0.9820)--(-0.5969,0.8972)--(-0.6275,0.8245)--(-0.6615,0.7634)--(-0.7015,0.7127)--(-0.7498,0.6708)--(-0.8082,0.6355)--(-0.8779,0.6044)--(-0.9596,0.5745)--(-1.0536,0.5431)--(-1.1592,0.5073)--(-1.2753,0.4641)--(-1.4002,0.4111)--(-1.5316,0.3459)--(-1.6667,0.2667)--(-1.8025,0.1721)--(-1.9356,0.0614)--(-2.0623,-0.0654)--(-2.1793,-0.2081)--(-2.2829,-0.3653)--(-2.3700,-0.5353)--(-2.4377,-0.7157)--(-2.4834,-0.9038)--(-2.5051,-1.0963)--(-2.5016,-1.2897)--(-2.4722,-1.4801)--(-2.4167,-1.6638)--(-2.3360,-1.8370)--(-2.2312,-1.9961)--(-2.1043,-2.1379)--(-1.9579,-2.2595)--(-1.7949,-2.3585)--(-1.6187,-2.4330)--(-1.4330,-2.4820)--(-1.2415,-2.5049)--(-1.0480,-2.5020)--(-0.8563,-2.4741)--(-0.6698,-2.4227)--(-0.4917,-2.3500)--(-0.3247,-2.2585)--(-0.1710,-2.1512)--(-0.0322,-2.0314)--(0.0906,-1.9027)--(0.1972,-1.7687)--(0.2879,-1.6327)--(0.3634,-1.4983)--(0.4254,-1.3683)--(0.4758,-1.2454)--(0.5168,-1.1318)--(0.5513,-1.0290)--(0.5820,-0.9381)--(0.6119,-0.8593)--(0.6439,-0.7925)--(0.6806,-0.7369)--(0.7245,-0.6908)--(0.7776,-0.6525)--(0.8415,-0.6196)--(0.9172,-0.5894)--(1.0051,-0.5592)--(1.1050,-0.5260)--(1.2160,-0.4868)--(1.3368,-0.4390)--(1.4652,-0.3801)--(1.5989,-0.3081)--(1.7347,-0.2213)--(1.8696,-0.1188)--(2.0000,0.0000);
+\draw [,->,>=latex] (2.0000,0.0000) -- (2.0141,0.0141);
+\draw [,->,>=latex] (2.3414,1.8272) -- (2.3318,1.8448);
+\draw [,->,>=latex] (0.6118,2.4017) -- (0.5932,2.3945);
+\draw [,->,>=latex] (-0.5954,0.9010) -- (-0.6025,0.8823);
+\draw [,->,>=latex] (-2.1766,-0.2044) -- (-2.1885,-0.2205);
+\draw [,->,>=latex] (-2.1727,-2.0667) -- (-2.1594,-2.0817);
+\draw [,->,>=latex] (-0.3430,-2.2697) -- (-0.3259,-2.2593);
+\draw [,->,>=latex] (0.7768,-0.6530) -- (0.7939,-0.6427);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CoordPolaires.pstricks.recall b/src_phystricks/Fig_CoordPolaires.pstricks.recall
index 827399ee2..e35320430 100644
--- a/src_phystricks/Fig_CoordPolaires.pstricks.recall
+++ b/src_phystricks/Fig_CoordPolaires.pstricks.recall
@@ -83,21 +83,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 \draw (1.5233,2.0000) node {$(x,y)$};
-\draw [,->,>=latex] (0,0) -- (1.0000,2.0000);
-\draw (0.68452,0.41391) node {$\theta$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (1.0000,2.0000);
+\draw (0.6845,0.4139) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00559)--(0.500,0.0112)--(0.500,0.0168)--(0.500,0.0224)--(0.499,0.0279)--(0.499,0.0335)--(0.498,0.0391)--(0.498,0.0447)--(0.497,0.0502)--(0.497,0.0558)--(0.496,0.0614)--(0.496,0.0669)--(0.495,0.0724)--(0.494,0.0780)--(0.493,0.0835)--(0.492,0.0890)--(0.491,0.0945)--(0.490,0.100)--(0.489,0.105)--(0.488,0.111)--(0.486,0.116)--(0.485,0.122)--(0.484,0.127)--(0.482,0.133)--(0.481,0.138)--(0.479,0.143)--(0.477,0.149)--(0.476,0.154)--(0.474,0.159)--(0.472,0.165)--(0.470,0.170)--(0.468,0.175)--(0.466,0.180)--(0.464,0.186)--(0.462,0.191)--(0.460,0.196)--(0.458,0.201)--(0.456,0.206)--(0.453,0.211)--(0.451,0.216)--(0.448,0.221)--(0.446,0.226)--(0.443,0.231)--(0.441,0.236)--(0.438,0.241)--(0.435,0.246)--(0.432,0.251)--(0.430,0.256)--(0.427,0.260)--(0.424,0.265)--(0.421,0.270)--(0.418,0.275)--(0.415,0.279)--(0.412,0.284)--(0.408,0.289)--(0.405,0.293)--(0.402,0.298)--(0.398,0.302)--(0.395,0.306)--(0.392,0.311)--(0.388,0.315)--(0.385,0.320)--(0.381,0.324)--(0.377,0.328)--(0.374,0.332)--(0.370,0.336)--(0.366,0.341)--(0.362,0.345)--(0.358,0.349)--(0.354,0.353)--(0.350,0.357)--(0.346,0.360)--(0.342,0.364)--(0.338,0.368)--(0.334,0.372)--(0.330,0.376)--(0.326,0.379)--(0.322,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.304,0.397)--(0.300,0.400)--(0.295,0.404)--(0.291,0.407)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416)--(0.272,0.419)--(0.267,0.422)--(0.263,0.425)--(0.258,0.428)--(0.253,0.431)--(0.248,0.434)--(0.243,0.437)--(0.238,0.439)--(0.234,0.442)--(0.229,0.445)--(0.224,0.447);
-\draw (0.23371,1.1680) node {$r$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [] (0.5000,0.0000)--(0.4999,0.0055)--(0.4998,0.0111)--(0.4997,0.0167)--(0.4994,0.0223)--(0.4992,0.0279)--(0.4988,0.0335)--(0.4984,0.0391)--(0.4980,0.0446)--(0.4974,0.0502)--(0.4968,0.0558)--(0.4962,0.0613)--(0.4955,0.0668)--(0.4947,0.0724)--(0.4938,0.0779)--(0.4929,0.0834)--(0.4920,0.0889)--(0.4909,0.0944)--(0.4899,0.0999)--(0.4887,0.1054)--(0.4875,0.1109)--(0.4862,0.1163)--(0.4849,0.1217)--(0.4835,0.1271)--(0.4820,0.1325)--(0.4805,0.1379)--(0.4790,0.1433)--(0.4773,0.1486)--(0.4756,0.1540)--(0.4739,0.1593)--(0.4721,0.1646)--(0.4702,0.1698)--(0.4683,0.1751)--(0.4663,0.1803)--(0.4642,0.1855)--(0.4621,0.1907)--(0.4600,0.1959)--(0.4578,0.2010)--(0.4555,0.2061)--(0.4531,0.2112)--(0.4508,0.2162)--(0.4483,0.2213)--(0.4458,0.2263)--(0.4432,0.2312)--(0.4406,0.2362)--(0.4380,0.2411)--(0.4352,0.2460)--(0.4325,0.2508)--(0.4296,0.2556)--(0.4267,0.2604)--(0.4238,0.2652)--(0.4208,0.2699)--(0.4178,0.2746)--(0.4147,0.2793)--(0.4115,0.2839)--(0.4083,0.2885)--(0.4051,0.2930)--(0.4018,0.2975)--(0.3984,0.3020)--(0.3950,0.3064)--(0.3916,0.3108)--(0.3880,0.3152)--(0.3845,0.3195)--(0.3809,0.3238)--(0.3773,0.3280)--(0.3736,0.3322)--(0.3698,0.3364)--(0.3660,0.3405)--(0.3622,0.3446)--(0.3583,0.3486)--(0.3544,0.3526)--(0.3504,0.3565)--(0.3464,0.3604)--(0.3424,0.3643)--(0.3383,0.3681)--(0.3341,0.3719)--(0.3300,0.3756)--(0.3257,0.3792)--(0.3215,0.3829)--(0.3172,0.3864)--(0.3128,0.3899)--(0.3085,0.3934)--(0.3040,0.3969)--(0.2996,0.4002)--(0.2951,0.4036)--(0.2906,0.4068)--(0.2860,0.4101)--(0.2814,0.4132)--(0.2767,0.4163)--(0.2721,0.4194)--(0.2674,0.4224)--(0.2626,0.4254)--(0.2578,0.4283)--(0.2530,0.4312)--(0.2482,0.4340)--(0.2433,0.4367)--(0.2384,0.4394)--(0.2335,0.4421)--(0.2285,0.4446)--(0.2236,0.4472);
+\draw (0.2337,1.1680) node {$r$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CornetGlace.pstricks.recall b/src_phystricks/Fig_CornetGlace.pstricks.recall
index a8a07a6d4..532fd1e8c 100644
--- a/src_phystricks/Fig_CornetGlace.pstricks.recall
+++ b/src_phystricks/Fig_CornetGlace.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.7000,0) -- (1.7000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.7000);
+\draw [,->,>=latex] (-1.7000,0.0000) -- (1.7000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.7000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -80,19 +80,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-0.707,0.707) -- (-0.700,0.700) -- (-0.693,0.693) -- (-0.686,0.686) -- (-0.679,0.679) -- (-0.671,0.671) -- (-0.664,0.664) -- (-0.657,0.657) -- (-0.650,0.650) -- (-0.643,0.643) -- (-0.636,0.636) -- (-0.629,0.629) -- (-0.621,0.621) -- (-0.614,0.614) -- (-0.607,0.607) -- (-0.600,0.600) -- (-0.593,0.593) -- (-0.586,0.586) -- (-0.579,0.579) -- (-0.571,0.571) -- (-0.564,0.564) -- (-0.557,0.557) -- (-0.550,0.550) -- (-0.543,0.543) -- (-0.536,0.536) -- (-0.529,0.529) -- (-0.521,0.521) -- (-0.514,0.514) -- (-0.507,0.507) -- (-0.500,0.500) -- (-0.493,0.493) -- (-0.486,0.486) -- (-0.479,0.479) -- (-0.471,0.471) -- (-0.464,0.464) -- (-0.457,0.457) -- (-0.450,0.450) -- (-0.443,0.443) -- (-0.436,0.436) -- (-0.429,0.429) -- (-0.421,0.421) -- (-0.414,0.414) -- (-0.407,0.407) -- (-0.400,0.400) -- (-0.393,0.393) -- (-0.386,0.386) -- (-0.379,0.379) -- (-0.371,0.371) -- (-0.364,0.364) -- (-0.357,0.357) -- 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+\draw [color=blue] (-1.2000,1.2000)--(-1.1878,1.1878)--(-1.1757,1.1757)--(-1.1636,1.1636)--(-1.1515,1.1515)--(-1.1393,1.1393)--(-1.1272,1.1272)--(-1.1151,1.1151)--(-1.1030,1.1030)--(-1.0909,1.0909)--(-1.0787,1.0787)--(-1.0666,1.0666)--(-1.0545,1.0545)--(-1.0424,1.0424)--(-1.0303,1.0303)--(-1.0181,1.0181)--(-1.0060,1.0060)--(-0.9939,0.9939)--(-0.9818,0.9818)--(-0.9696,0.9696)--(-0.9575,0.9575)--(-0.9454,0.9454)--(-0.9333,0.9333)--(-0.9212,0.9212)--(-0.9090,0.9090)--(-0.8969,0.8969)--(-0.8848,0.8848)--(-0.8727,0.8727)--(-0.8606,0.8606)--(-0.8484,0.8484)--(-0.8363,0.8363)--(-0.8242,0.8242)--(-0.8121,0.8121)--(-0.8000,0.8000)--(-0.7878,0.7878)--(-0.7757,0.7757)--(-0.7636,0.7636)--(-0.7515,0.7515)--(-0.7393,0.7393)--(-0.7272,0.7272)--(-0.7151,0.7151)--(-0.7030,0.7030)--(-0.6909,0.6909)--(-0.6787,0.6787)--(-0.6666,0.6666)--(-0.6545,0.6545)--(-0.6424,0.6424)--(-0.6303,0.6303)--(-0.6181,0.6181)--(-0.6060,0.6060)--(-0.5939,0.5939)--(-0.5818,0.5818)--(-0.5696,0.5696)--(-0.5575,0.5575)--(-0.5454,0.5454)--(-0.5333,0.5333)--(-0.5212,0.5212)--(-0.5090,0.5090)--(-0.4969,0.4969)--(-0.4848,0.4848)--(-0.4727,0.4727)--(-0.4606,0.4606)--(-0.4484,0.4484)--(-0.4363,0.4363)--(-0.4242,0.4242)--(-0.4121,0.4121)--(-0.4000,0.4000)--(-0.3878,0.3878)--(-0.3757,0.3757)--(-0.3636,0.3636)--(-0.3515,0.3515)--(-0.3393,0.3393)--(-0.3272,0.3272)--(-0.3151,0.3151)--(-0.3030,0.3030)--(-0.2909,0.2909)--(-0.2787,0.2787)--(-0.2666,0.2666)--(-0.2545,0.2545)--(-0.2424,0.2424)--(-0.2303,0.2303)--(-0.2181,0.2181)--(-0.2060,0.2060)--(-0.1939,0.1939)--(-0.1818,0.1818)--(-0.1696,0.1696)--(-0.1575,0.1575)--(-0.1454,0.1454)--(-0.1333,0.1333)--(-0.1212,0.1212)--(-0.1090,0.1090)--(-0.0969,0.0969)--(-0.0848,0.0848)--(-0.0727,0.0727)--(-0.0606,0.0606)--(-0.0484,0.0484)--(-0.0363,0.0363)--(-0.0242,0.0242)--(-0.0121,0.0121)--(0.0000,0.0000);
 
-\draw [color=blue] (0,0)--(0.01212,0.01212)--(0.02424,0.02424)--(0.03636,0.03636)--(0.04848,0.04848)--(0.06061,0.06061)--(0.07273,0.07273)--(0.08485,0.08485)--(0.09697,0.09697)--(0.1091,0.1091)--(0.1212,0.1212)--(0.1333,0.1333)--(0.1455,0.1455)--(0.1576,0.1576)--(0.1697,0.1697)--(0.1818,0.1818)--(0.1939,0.1939)--(0.2061,0.2061)--(0.2182,0.2182)--(0.2303,0.2303)--(0.2424,0.2424)--(0.2545,0.2545)--(0.2667,0.2667)--(0.2788,0.2788)--(0.2909,0.2909)--(0.3030,0.3030)--(0.3152,0.3152)--(0.3273,0.3273)--(0.3394,0.3394)--(0.3515,0.3515)--(0.3636,0.3636)--(0.3758,0.3758)--(0.3879,0.3879)--(0.4000,0.4000)--(0.4121,0.4121)--(0.4242,0.4242)--(0.4364,0.4364)--(0.4485,0.4485)--(0.4606,0.4606)--(0.4727,0.4727)--(0.4848,0.4848)--(0.4970,0.4970)--(0.5091,0.5091)--(0.5212,0.5212)--(0.5333,0.5333)--(0.5455,0.5455)--(0.5576,0.5576)--(0.5697,0.5697)--(0.5818,0.5818)--(0.5939,0.5939)--(0.6061,0.6061)--(0.6182,0.6182)--(0.6303,0.6303)--(0.6424,0.6424)--(0.6545,0.6545)--(0.6667,0.6667)--(0.6788,0.6788)--(0.6909,0.6909)--(0.7030,0.7030)--(0.7151,0.7151)--(0.7273,0.7273)--(0.7394,0.7394)--(0.7515,0.7515)--(0.7636,0.7636)--(0.7758,0.7758)--(0.7879,0.7879)--(0.8000,0.8000)--(0.8121,0.8121)--(0.8242,0.8242)--(0.8364,0.8364)--(0.8485,0.8485)--(0.8606,0.8606)--(0.8727,0.8727)--(0.8848,0.8848)--(0.8970,0.8970)--(0.9091,0.9091)--(0.9212,0.9212)--(0.9333,0.9333)--(0.9455,0.9455)--(0.9576,0.9576)--(0.9697,0.9697)--(0.9818,0.9818)--(0.9939,0.9939)--(1.006,1.006)--(1.018,1.018)--(1.030,1.030)--(1.042,1.042)--(1.055,1.055)--(1.067,1.067)--(1.079,1.079)--(1.091,1.091)--(1.103,1.103)--(1.115,1.115)--(1.127,1.127)--(1.139,1.139)--(1.152,1.152)--(1.164,1.164)--(1.176,1.176)--(1.188,1.188)--(1.200,1.200);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0121,0.0121)--(0.0242,0.0242)--(0.0363,0.0363)--(0.0484,0.0484)--(0.0606,0.0606)--(0.0727,0.0727)--(0.0848,0.0848)--(0.0969,0.0969)--(0.1090,0.1090)--(0.1212,0.1212)--(0.1333,0.1333)--(0.1454,0.1454)--(0.1575,0.1575)--(0.1696,0.1696)--(0.1818,0.1818)--(0.1939,0.1939)--(0.2060,0.2060)--(0.2181,0.2181)--(0.2303,0.2303)--(0.2424,0.2424)--(0.2545,0.2545)--(0.2666,0.2666)--(0.2787,0.2787)--(0.2909,0.2909)--(0.3030,0.3030)--(0.3151,0.3151)--(0.3272,0.3272)--(0.3393,0.3393)--(0.3515,0.3515)--(0.3636,0.3636)--(0.3757,0.3757)--(0.3878,0.3878)--(0.4000,0.4000)--(0.4121,0.4121)--(0.4242,0.4242)--(0.4363,0.4363)--(0.4484,0.4484)--(0.4606,0.4606)--(0.4727,0.4727)--(0.4848,0.4848)--(0.4969,0.4969)--(0.5090,0.5090)--(0.5212,0.5212)--(0.5333,0.5333)--(0.5454,0.5454)--(0.5575,0.5575)--(0.5696,0.5696)--(0.5818,0.5818)--(0.5939,0.5939)--(0.6060,0.6060)--(0.6181,0.6181)--(0.6303,0.6303)--(0.6424,0.6424)--(0.6545,0.6545)--(0.6666,0.6666)--(0.6787,0.6787)--(0.6909,0.6909)--(0.7030,0.7030)--(0.7151,0.7151)--(0.7272,0.7272)--(0.7393,0.7393)--(0.7515,0.7515)--(0.7636,0.7636)--(0.7757,0.7757)--(0.7878,0.7878)--(0.8000,0.8000)--(0.8121,0.8121)--(0.8242,0.8242)--(0.8363,0.8363)--(0.8484,0.8484)--(0.8606,0.8606)--(0.8727,0.8727)--(0.8848,0.8848)--(0.8969,0.8969)--(0.9090,0.9090)--(0.9212,0.9212)--(0.9333,0.9333)--(0.9454,0.9454)--(0.9575,0.9575)--(0.9696,0.9696)--(0.9818,0.9818)--(0.9939,0.9939)--(1.0060,1.0060)--(1.0181,1.0181)--(1.0303,1.0303)--(1.0424,1.0424)--(1.0545,1.0545)--(1.0666,1.0666)--(1.0787,1.0787)--(1.0909,1.0909)--(1.1030,1.1030)--(1.1151,1.1151)--(1.1272,1.1272)--(1.1393,1.1393)--(1.1515,1.1515)--(1.1636,1.1636)--(1.1757,1.1757)--(1.1878,1.1878)--(1.2000,1.2000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall b/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall
index bb8a4f9fd..190c40404 100644
--- a/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall
+++ b/src_phystricks/Fig_DDCTooYscVzA.pstricks.recall
@@ -65,31 +65,31 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red,->,>=latex] (2.0000,0) -- (2.7071,0.70711);
-\draw [color=green,->,>=latex] (2.0000,0) -- (2.7071,-0.70711);
-\draw [color=red,->,>=latex] (2.3414,1.8273) -- (1.8616,2.7046);
-\draw [color=green,->,>=latex] (2.3414,1.8273) -- (3.2188,2.3071);
-\draw [color=red,->,>=latex] (0.61190,2.4018) -- (-0.32151,2.0429);
-\draw [color=green,->,>=latex] (0.61190,2.4018) -- (0.25309,3.3352);
-\draw [color=red,->,>=latex] (-0.59548,0.90107) -- (-0.94670,-0.035225);
-\draw [color=green,->,>=latex] (-0.59548,0.90107) -- (-1.5318,1.2523);
-\draw [color=red,->,>=latex] (-2.1766,-0.20446) -- (-2.7726,-1.0075);
-\draw [color=green,->,>=latex] (-2.1766,-0.20446) -- (-2.9796,0.39153);
-\draw [color=red,->,>=latex] (-2.1728,-2.0668) -- (-1.5081,-2.8139);
-\draw [color=green,->,>=latex] (-2.1728,-2.0668) -- (-2.9199,-2.7314);
-\draw [color=red,->,>=latex] (-0.34301,-2.2697) -- (0.51254,-1.7520);
-\draw [color=green,->,>=latex] (-0.34301,-2.2697) -- (0.17470,-3.1253);
-\draw [color=red,->,>=latex] (0.77682,-0.65307) -- (1.6318,-0.13442);
-\draw [color=green,->,>=latex] (0.77682,-0.65307) -- (1.2955,-1.5081);
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-\draw [,->,>=latex] (2.0000,0) -- (2.0141,0.014142);
-\draw [,->,>=latex] (2.3414,1.8273) -- (2.3318,1.8448);
-\draw [,->,>=latex] (0.61190,2.4018) -- (0.59323,2.3946);
-\draw [,->,>=latex] (-0.59548,0.90107) -- (-0.60250,0.88234);
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-\draw [,->,>=latex] (-0.34301,-2.2697) -- (-0.32590,-2.2594);
-\draw [,->,>=latex] (0.77682,-0.65307) -- (0.79392,-0.64270);
+\draw [color=red,->,>=latex] (2.0000,0.0000) -- (2.7071,0.7071);
+\draw [color=green,->,>=latex] (2.0000,0.0000) -- (2.7071,-0.7071);
+\draw [color=red,->,>=latex] (2.3414,1.8272) -- (1.8615,2.7046);
+\draw [color=green,->,>=latex] (2.3414,1.8272) -- (3.2187,2.3071);
+\draw [color=red,->,>=latex] (0.6118,2.4017) -- (-0.3215,2.0429);
+\draw [color=green,->,>=latex] (0.6118,2.4017) -- (0.2530,3.3351);
+\draw [color=red,->,>=latex] (-0.5954,0.9010) -- (-0.9467,-0.0352);
+\draw [color=green,->,>=latex] (-0.5954,0.9010) -- (-1.5317,1.2522);
+\draw [color=red,->,>=latex] (-2.1766,-0.2044) -- (-2.7726,-1.0074);
+\draw [color=green,->,>=latex] (-2.1766,-0.2044) -- (-2.9796,0.3915);
+\draw [color=red,->,>=latex] (-2.1727,-2.0667) -- (-1.5081,-2.8138);
+\draw [color=green,->,>=latex] (-2.1727,-2.0667) -- (-2.9199,-2.7314);
+\draw [color=red,->,>=latex] (-0.3430,-2.2697) -- (0.5125,-1.7520);
+\draw [color=green,->,>=latex] (-0.3430,-2.2697) -- (0.1746,-3.1252);
+\draw [color=red,->,>=latex] (0.7768,-0.6530) -- (1.6318,-0.1344);
+\draw [color=green,->,>=latex] (0.7768,-0.6530) -- (1.2954,-1.5080);
+\draw [color=blue] (2.0000,0.0000)--(2.1223,0.1348)--(2.2330,0.2849)--(2.3288,0.4488)--(2.4065,0.6244)--(2.4634,0.8090)--(2.4973,0.9998)--(2.5066,1.1931)--(2.4902,1.3855)--(2.4477,1.5730)--(2.3794,1.7519)--(2.2864,1.9185)--(2.1703,2.0694)--(2.0333,2.2014)--(1.8782,2.3119)--(1.7082,2.3989)--(1.5268,2.4607)--(1.3377,2.4967)--(1.1447,2.5067)--(0.9517,2.4911)--(0.7622,2.4512)--(0.5795,2.3889)--(0.4066,2.3064)--(0.2460,2.2066)--(0.0996,2.0926)--(-0.0312,1.9680)--(-0.1459,1.8362)--(-0.2445,1.7007)--(-0.3275,1.5651)--(-0.3960,1.4325)--(-0.4519,1.3058)--(-0.4973,1.1873)--(-0.5347,1.0789)--(-0.5669,0.9820)--(-0.5969,0.8972)--(-0.6275,0.8245)--(-0.6615,0.7634)--(-0.7015,0.7127)--(-0.7498,0.6708)--(-0.8082,0.6355)--(-0.8779,0.6044)--(-0.9596,0.5745)--(-1.0536,0.5431)--(-1.1592,0.5073)--(-1.2753,0.4641)--(-1.4002,0.4111)--(-1.5316,0.3459)--(-1.6667,0.2667)--(-1.8025,0.1721)--(-1.9356,0.0614)--(-2.0623,-0.0654)--(-2.1793,-0.2081)--(-2.2829,-0.3653)--(-2.3700,-0.5353)--(-2.4377,-0.7157)--(-2.4834,-0.9038)--(-2.5051,-1.0963)--(-2.5016,-1.2897)--(-2.4722,-1.4801)--(-2.4167,-1.6638)--(-2.3360,-1.8370)--(-2.2312,-1.9961)--(-2.1043,-2.1379)--(-1.9579,-2.2595)--(-1.7949,-2.3585)--(-1.6187,-2.4330)--(-1.4330,-2.4820)--(-1.2415,-2.5049)--(-1.0480,-2.5020)--(-0.8563,-2.4741)--(-0.6698,-2.4227)--(-0.4917,-2.3500)--(-0.3247,-2.2585)--(-0.1710,-2.1512)--(-0.0322,-2.0314)--(0.0906,-1.9027)--(0.1972,-1.7687)--(0.2879,-1.6327)--(0.3634,-1.4983)--(0.4254,-1.3683)--(0.4758,-1.2454)--(0.5168,-1.1318)--(0.5513,-1.0290)--(0.5820,-0.9381)--(0.6119,-0.8593)--(0.6439,-0.7925)--(0.6806,-0.7369)--(0.7245,-0.6908)--(0.7776,-0.6525)--(0.8415,-0.6196)--(0.9172,-0.5894)--(1.0051,-0.5592)--(1.1050,-0.5260)--(1.2160,-0.4868)--(1.3368,-0.4390)--(1.4652,-0.3801)--(1.5989,-0.3081)--(1.7347,-0.2213)--(1.8696,-0.1188)--(2.0000,0.0000);
+\draw [,->,>=latex] (2.0000,0.0000) -- (2.0141,0.0141);
+\draw [,->,>=latex] (2.3414,1.8272) -- (2.3318,1.8448);
+\draw [,->,>=latex] (0.6118,2.4017) -- (0.5932,2.3945);
+\draw [,->,>=latex] (-0.5954,0.9010) -- (-0.6025,0.8823);
+\draw [,->,>=latex] (-2.1766,-0.2044) -- (-2.1885,-0.2205);
+\draw [,->,>=latex] (-2.1727,-2.0667) -- (-2.1594,-2.0817);
+\draw [,->,>=latex] (-0.3430,-2.2697) -- (-0.3259,-2.2593);
+\draw [,->,>=latex] (0.7768,-0.6530) -- (0.7939,-0.6427);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall b/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall
index 7432ae77f..2e7cbdaeb 100644
--- a/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall
+++ b/src_phystricks/Fig_DNHRooqGtffLkd.pstricks.recall
@@ -107,45 +107,45 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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+\draw [] (3.0000,0.0000)--(2.9959,0.1268)--(2.9839,0.2531)--(2.9638,0.3785)--(2.9358,0.5022)--(2.9001,0.6240)--(2.8567,0.7433)--(2.8058,0.8595)--(2.7476,0.9723)--(2.6825,1.0812)--(2.6105,1.1858)--(2.5320,1.2855)--(2.4474,1.3801)--(2.3570,1.4691)--(2.2611,1.5522)--(2.1601,1.6291)--(2.0544,1.6994)--(1.9445,1.7629)--(1.8308,1.8192)--(1.7137,1.8682)--(1.5938,1.9098)--(1.4715,1.9436)--(1.3472,1.9696)--(1.2216,1.9876)--(1.0951,1.9977)--(0.9682,1.9997)--(0.8415,1.9937)--(0.7153,1.9796)--(0.5903,1.9576)--(0.4670,1.9276)--(0.3458,1.8900)--(0.2273,1.8447)--(0.1118,1.7919)--(0.0000,1.7320)--(-0.1078,1.6651)--(-0.2112,1.5915)--(-0.3097,1.5114)--(-0.4029,1.4253)--(-0.4905,1.3335)--(-0.5721,1.2363)--(-0.6473,1.1341)--(-0.7159,1.0273)--(-0.7776,0.9164)--(-0.8322,0.8018)--(-0.8793,0.6840)--(-0.9189,0.5634)--(-0.9508,0.4406)--(-0.9748,0.3160)--(-0.9909,0.1901)--(-0.9989,0.0634)--(-0.9989,-0.0634)--(-0.9909,-0.1901)--(-0.9748,-0.3160)--(-0.9508,-0.4406)--(-0.9189,-0.5634)--(-0.8793,-0.6840)--(-0.8322,-0.8018)--(-0.7776,-0.9164)--(-0.7159,-1.0273)--(-0.6473,-1.1341)--(-0.5721,-1.2363)--(-0.4905,-1.3335)--(-0.4029,-1.4253)--(-0.3097,-1.5114)--(-0.2112,-1.5915)--(-0.1078,-1.6651)--(0.0000,-1.7320)--(0.1118,-1.7919)--(0.2273,-1.8447)--(0.3458,-1.8900)--(0.4670,-1.9276)--(0.5903,-1.9576)--(0.7153,-1.9796)--(0.8415,-1.9937)--(0.9682,-1.9997)--(1.0951,-1.9977)--(1.2216,-1.9876)--(1.3472,-1.9696)--(1.4715,-1.9436)--(1.5938,-1.9098)--(1.7137,-1.8682)--(1.8308,-1.8192)--(1.9445,-1.7629)--(2.0544,-1.6994)--(2.1601,-1.6291)--(2.2611,-1.5522)--(2.3570,-1.4691)--(2.4474,-1.3801)--(2.5320,-1.2855)--(2.6105,-1.1858)--(2.6825,-1.0812)--(2.7476,-0.9723)--(2.8058,-0.8595)--(2.8567,-0.7433)--(2.9001,-0.6240)--(2.9358,-0.5022)--(2.9638,-0.3785)--(2.9839,-0.2531)--(2.9959,-0.1268)--(3.0000,0.0000);
 
-\draw [] (5.000,0)--(4.994,0.1903)--(4.976,0.3798)--(4.946,0.5678)--(4.904,0.7534)--(4.850,0.9361)--(4.785,1.115)--(4.709,1.289)--(4.622,1.459)--(4.524,1.622)--(4.416,1.779)--(4.298,1.928)--(4.171,2.070)--(4.036,2.204)--(3.892,2.328)--(3.740,2.444)--(3.582,2.549)--(3.417,2.644)--(3.246,2.729)--(3.071,2.802)--(2.891,2.865)--(2.707,2.915)--(2.521,2.954)--(2.333,2.982)--(2.143,2.997)--(1.952,3.000)--(1.762,2.991)--(1.573,2.969)--(1.386,2.936)--(1.201,2.892)--(1.019,2.835)--(0.8410,2.767)--(0.6678,2.688)--(0.5000,2.598)--(0.3382,2.498)--(0.1832,2.387)--(0.03542,2.267)--(-0.1044,2.138)--(-0.2358,2.000)--(-0.3582,1.854)--(-0.4710,1.701)--(-0.5740,1.541)--(-0.6665,1.375)--(-0.7483,1.203)--(-0.8191,1.026)--(-0.8785,0.8452)--(-0.9263,0.6609)--(-0.9623,0.4740)--(-0.9864,0.2852)--(-0.9985,0.09518)--(-0.9985,-0.09518)--(-0.9864,-0.2852)--(-0.9623,-0.4740)--(-0.9263,-0.6609)--(-0.8785,-0.8452)--(-0.8191,-1.026)--(-0.7483,-1.203)--(-0.6665,-1.375)--(-0.5740,-1.541)--(-0.4710,-1.701)--(-0.3582,-1.854)--(-0.2358,-2.000)--(-0.1044,-2.138)--(0.03542,-2.267)--(0.1832,-2.387)--(0.3382,-2.498)--(0.5000,-2.598)--(0.6678,-2.688)--(0.8410,-2.767)--(1.019,-2.835)--(1.201,-2.892)--(1.386,-2.936)--(1.573,-2.969)--(1.762,-2.991)--(1.952,-3.000)--(2.143,-2.997)--(2.333,-2.982)--(2.521,-2.954)--(2.707,-2.915)--(2.891,-2.865)--(3.071,-2.802)--(3.246,-2.729)--(3.417,-2.644)--(3.582,-2.549)--(3.740,-2.444)--(3.892,-2.328)--(4.036,-2.204)--(4.171,-2.070)--(4.298,-1.928)--(4.416,-1.779)--(4.524,-1.622)--(4.622,-1.459)--(4.709,-1.289)--(4.785,-1.115)--(4.850,-0.9361)--(4.904,-0.7534)--(4.946,-0.5678)--(4.976,-0.3798)--(4.994,-0.1903)--(5.000,0);
-\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-1.3270,0.29553) node {\( \lambda_1\)};
+\draw [] (5.0000,0.0000)--(4.9939,0.1902)--(4.9758,0.3797)--(4.9457,0.5677)--(4.9038,0.7534)--(4.8502,0.9361)--(4.7851,1.1149)--(4.7087,1.2893)--(4.6215,1.4585)--(4.5237,1.6219)--(4.4158,1.7787)--(4.2981,1.9283)--(4.1712,2.0702)--(4.0355,2.2037)--(3.8916,2.3284)--(3.7401,2.4437)--(3.5816,2.5491)--(3.4168,2.6443)--(3.2462,2.7288)--(3.0706,2.8024)--(2.8907,2.8647)--(2.7072,2.9154)--(2.5209,2.9544)--(2.3325,2.9815)--(2.1427,2.9966)--(1.9524,2.9996)--(1.7622,2.9905)--(1.5730,2.9694)--(1.3855,2.9364)--(1.2005,2.8915)--(1.0187,2.8350)--(0.8409,2.7670)--(0.6678,2.6879)--(0.4999,2.5980)--(0.3382,2.4977)--(0.1831,2.3872)--(0.0354,2.2672)--(-0.1044,2.1380)--(-0.2357,2.0003)--(-0.3581,1.8544)--(-0.4710,1.7011)--(-0.5739,1.5410)--(-0.6665,1.3746)--(-0.7483,1.2027)--(-0.8190,1.0260)--(-0.8784,0.8451)--(-0.9262,0.6609)--(-0.9623,0.4740)--(-0.9864,0.2851)--(-0.9984,0.0951)--(-0.9984,-0.0951)--(-0.9864,-0.2851)--(-0.9623,-0.4740)--(-0.9262,-0.6609)--(-0.8784,-0.8451)--(-0.8190,-1.0260)--(-0.7483,-1.2027)--(-0.6665,-1.3746)--(-0.5739,-1.5410)--(-0.4710,-1.7011)--(-0.3581,-1.8544)--(-0.2357,-2.0003)--(-0.1044,-2.1380)--(0.0354,-2.2672)--(0.1831,-2.3872)--(0.3382,-2.4977)--(0.5000,-2.5980)--(0.6678,-2.6879)--(0.8409,-2.7670)--(1.0187,-2.8350)--(1.2005,-2.8915)--(1.3855,-2.9364)--(1.5730,-2.9694)--(1.7622,-2.9905)--(1.9524,-2.9996)--(2.1427,-2.9966)--(2.3325,-2.9815)--(2.5209,-2.9544)--(2.7072,-2.9154)--(2.8907,-2.8647)--(3.0706,-2.8024)--(3.2462,-2.7288)--(3.4168,-2.6443)--(3.5816,-2.5491)--(3.7401,-2.4437)--(3.8916,-2.3284)--(4.0355,-2.2037)--(4.1712,-2.0702)--(4.2981,-1.9283)--(4.4158,-1.7787)--(4.5237,-1.6219)--(4.6215,-1.4585)--(4.7087,-1.2893)--(4.7851,-1.1149)--(4.8502,-0.9361)--(4.9038,-0.7534)--(4.9457,-0.5677)--(4.9758,-0.3797)--(4.9939,-0.1902)--(5.0000,0.0000);
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.3270,0.2955) node {\( \lambda_1\)};
 \draw []  (2.0000,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.6730,1.1187) node {\( \lambda_2\)};
+\draw (1.6729,1.1186) node {\( \lambda_2\)};
 \draw []  (2.0000,-1.4142) node [rotate=0] {$\bullet$};
-\draw (1.6730,-1.1187) node {\( \lambda_3\)};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (1.6729,-1.1186) node {\( \lambda_3\)};
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall b/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall
index 86b9be42e..83dad0aaf 100644
--- a/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall
+++ b/src_phystricks/Fig_DNRRooJWRHgOCw.pstricks.recall
@@ -95,33 +95,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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+\draw [] (6.0000,0.0000)--(5.9959,0.1268)--(5.9839,0.2531)--(5.9638,0.3785)--(5.9358,0.5022)--(5.9001,0.6240)--(5.8567,0.7433)--(5.8058,0.8595)--(5.7476,0.9723)--(5.6825,1.0812)--(5.6105,1.1858)--(5.5320,1.2855)--(5.4474,1.3801)--(5.3570,1.4691)--(5.2611,1.5522)--(5.1601,1.6291)--(5.0544,1.6994)--(4.9445,1.7629)--(4.8308,1.8192)--(4.7137,1.8682)--(4.5938,1.9098)--(4.4715,1.9436)--(4.3472,1.9696)--(4.2216,1.9876)--(4.0951,1.9977)--(3.9682,1.9997)--(3.8415,1.9937)--(3.7153,1.9796)--(3.5903,1.9576)--(3.4670,1.9276)--(3.3458,1.8900)--(3.2273,1.8447)--(3.1118,1.7919)--(3.0000,1.7320)--(2.8921,1.6651)--(2.7887,1.5915)--(2.6902,1.5114)--(2.5970,1.4253)--(2.5094,1.3335)--(2.4278,1.2363)--(2.3526,1.1341)--(2.2840,1.0273)--(2.2223,0.9164)--(2.1677,0.8018)--(2.1206,0.6840)--(2.0810,0.5634)--(2.0491,0.4406)--(2.0251,0.3160)--(2.0090,0.1901)--(2.0010,0.0634)--(2.0010,-0.0634)--(2.0090,-0.1901)--(2.0251,-0.3160)--(2.0491,-0.4406)--(2.0810,-0.5634)--(2.1206,-0.6840)--(2.1677,-0.8018)--(2.2223,-0.9164)--(2.2840,-1.0273)--(2.3526,-1.1341)--(2.4278,-1.2363)--(2.5094,-1.3335)--(2.5970,-1.4253)--(2.6902,-1.5114)--(2.7887,-1.5915)--(2.8921,-1.6651)--(3.0000,-1.7320)--(3.1118,-1.7919)--(3.2273,-1.8447)--(3.3458,-1.8900)--(3.4670,-1.9276)--(3.5903,-1.9576)--(3.7153,-1.9796)--(3.8415,-1.9937)--(3.9682,-1.9997)--(4.0951,-1.9977)--(4.2216,-1.9876)--(4.3472,-1.9696)--(4.4715,-1.9436)--(4.5938,-1.9098)--(4.7137,-1.8682)--(4.8308,-1.8192)--(4.9445,-1.7629)--(5.0544,-1.6994)--(5.1601,-1.6291)--(5.2611,-1.5522)--(5.3570,-1.4691)--(5.4474,-1.3801)--(5.5320,-1.2855)--(5.6105,-1.1858)--(5.6825,-1.0812)--(5.7476,-0.9723)--(5.8058,-0.8595)--(5.8567,-0.7433)--(5.9001,-0.6240)--(5.9358,-0.5022)--(5.9638,-0.3785)--(5.9839,-0.2531)--(5.9959,-0.1268)--(6.0000,0.0000);
 
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+\draw [] (3.0000,0.0000)--(2.9979,0.0634)--(2.9919,0.1265)--(2.9819,0.1892)--(2.9679,0.2511)--(2.9500,0.3120)--(2.9283,0.3716)--(2.9029,0.4297)--(2.8738,0.4861)--(2.8412,0.5406)--(2.8052,0.5929)--(2.7660,0.6427)--(2.7237,0.6900)--(2.6785,0.7345)--(2.6305,0.7761)--(2.5800,0.8145)--(2.5272,0.8497)--(2.4722,0.8814)--(2.4154,0.9096)--(2.3568,0.9341)--(2.2969,0.9549)--(2.2357,0.9718)--(2.1736,0.9848)--(2.1108,0.9938)--(2.0475,0.9988)--(1.9841,0.9998)--(1.9207,0.9968)--(1.8576,0.9898)--(1.7951,0.9788)--(1.7335,0.9638)--(1.6729,0.9450)--(1.6136,0.9223)--(1.5559,0.8959)--(1.5000,0.8660)--(1.4460,0.8325)--(1.3943,0.7957)--(1.3451,0.7557)--(1.2985,0.7126)--(1.2547,0.6667)--(1.2139,0.6181)--(1.1763,0.5670)--(1.1420,0.5136)--(1.1111,0.4582)--(1.0838,0.4009)--(1.0603,0.3420)--(1.0405,0.2817)--(1.0245,0.2203)--(1.0125,0.1580)--(1.0045,0.0950)--(1.0005,0.0317)--(1.0005,-0.0317)--(1.0045,-0.0950)--(1.0125,-0.1580)--(1.0245,-0.2203)--(1.0405,-0.2817)--(1.0603,-0.3420)--(1.0838,-0.4009)--(1.1111,-0.4582)--(1.1420,-0.5136)--(1.1763,-0.5670)--(1.2139,-0.6181)--(1.2547,-0.6667)--(1.2985,-0.7126)--(1.3451,-0.7557)--(1.3943,-0.7957)--(1.4460,-0.8325)--(1.5000,-0.8660)--(1.5559,-0.8959)--(1.6136,-0.9223)--(1.6729,-0.9450)--(1.7335,-0.9638)--(1.7951,-0.9788)--(1.8576,-0.9898)--(1.9207,-0.9968)--(1.9841,-0.9998)--(2.0475,-0.9988)--(2.1108,-0.9938)--(2.1736,-0.9848)--(2.2357,-0.9718)--(2.2969,-0.9549)--(2.3568,-0.9341)--(2.4154,-0.9096)--(2.4722,-0.8814)--(2.5272,-0.8497)--(2.5800,-0.8145)--(2.6305,-0.7761)--(2.6785,-0.7345)--(2.7237,-0.6900)--(2.7660,-0.6427)--(2.8052,-0.5929)--(2.8412,-0.5406)--(2.8738,-0.4861)--(2.9029,-0.4297)--(2.9283,-0.3716)--(2.9500,-0.3120)--(2.9679,-0.2511)--(2.9819,-0.1892)--(2.9919,-0.1265)--(2.9979,-0.0634)--(3.0000,0.0000);
 
-\draw [] (8.000,0)--(7.996,0.1268)--(7.984,0.2532)--(7.964,0.3785)--(7.936,0.5023)--(7.900,0.6241)--(7.857,0.7433)--(7.806,0.8596)--(7.748,0.9724)--(7.682,1.081)--(7.611,1.186)--(7.532,1.286)--(7.447,1.380)--(7.357,1.469)--(7.261,1.552)--(7.160,1.629)--(7.054,1.699)--(6.945,1.763)--(6.831,1.819)--(6.714,1.868)--(6.594,1.910)--(6.471,1.944)--(6.347,1.970)--(6.222,1.988)--(6.095,1.998)--(5.968,2.000)--(5.841,1.994)--(5.715,1.980)--(5.590,1.958)--(5.467,1.928)--(5.346,1.890)--(5.227,1.845)--(5.112,1.792)--(5.000,1.732)--(4.892,1.665)--(4.789,1.592)--(4.690,1.512)--(4.597,1.425)--(4.509,1.334)--(4.428,1.236)--(4.353,1.134)--(4.284,1.027)--(4.222,0.9165)--(4.168,0.8019)--(4.121,0.6840)--(4.081,0.5635)--(4.049,0.4406)--(4.025,0.3160)--(4.009,0.1901)--(4.001,0.06346)--(4.001,-0.06346)--(4.009,-0.1901)--(4.025,-0.3160)--(4.049,-0.4406)--(4.081,-0.5635)--(4.121,-0.6840)--(4.168,-0.8019)--(4.222,-0.9165)--(4.284,-1.027)--(4.353,-1.134)--(4.428,-1.236)--(4.509,-1.334)--(4.597,-1.425)--(4.690,-1.512)--(4.789,-1.592)--(4.892,-1.665)--(5.000,-1.732)--(5.112,-1.792)--(5.227,-1.845)--(5.346,-1.890)--(5.467,-1.928)--(5.590,-1.958)--(5.715,-1.980)--(5.841,-1.994)--(5.968,-2.000)--(6.095,-1.998)--(6.222,-1.988)--(6.347,-1.970)--(6.471,-1.944)--(6.594,-1.910)--(6.714,-1.868)--(6.831,-1.819)--(6.945,-1.763)--(7.054,-1.699)--(7.160,-1.629)--(7.261,-1.552)--(7.357,-1.469)--(7.447,-1.380)--(7.532,-1.286)--(7.611,-1.186)--(7.682,-1.081)--(7.748,-0.9724)--(7.806,-0.8596)--(7.857,-0.7433)--(7.900,-0.6241)--(7.936,-0.5023)--(7.964,-0.3785)--(7.984,-0.2532)--(7.996,-0.1268)--(8.000,0);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.6730,0.29553) node {\( \lambda_1\)};
+\draw [] (8.0000,0.0000)--(7.9959,0.1268)--(7.9839,0.2531)--(7.9638,0.3785)--(7.9358,0.5022)--(7.9001,0.6240)--(7.8567,0.7433)--(7.8058,0.8595)--(7.7476,0.9723)--(7.6825,1.0812)--(7.6105,1.1858)--(7.5320,1.2855)--(7.4474,1.3801)--(7.3570,1.4691)--(7.2611,1.5522)--(7.1601,1.6291)--(7.0544,1.6994)--(6.9445,1.7629)--(6.8308,1.8192)--(6.7137,1.8682)--(6.5938,1.9098)--(6.4715,1.9436)--(6.3472,1.9696)--(6.2216,1.9876)--(6.0951,1.9977)--(5.9682,1.9997)--(5.8415,1.9937)--(5.7153,1.9796)--(5.5903,1.9576)--(5.4670,1.9276)--(5.3458,1.8900)--(5.2273,1.8447)--(5.1118,1.7919)--(5.0000,1.7320)--(4.8921,1.6651)--(4.7887,1.5915)--(4.6902,1.5114)--(4.5970,1.4253)--(4.5094,1.3335)--(4.4278,1.2363)--(4.3526,1.1341)--(4.2840,1.0273)--(4.2223,0.9164)--(4.1677,0.8018)--(4.1206,0.6840)--(4.0810,0.5634)--(4.0491,0.4406)--(4.0251,0.3160)--(4.0090,0.1901)--(4.0010,0.0634)--(4.0010,-0.0634)--(4.0090,-0.1901)--(4.0251,-0.3160)--(4.0491,-0.4406)--(4.0810,-0.5634)--(4.1206,-0.6840)--(4.1677,-0.8018)--(4.2223,-0.9164)--(4.2840,-1.0273)--(4.3526,-1.1341)--(4.4278,-1.2363)--(4.5094,-1.3335)--(4.5970,-1.4253)--(4.6902,-1.5114)--(4.7887,-1.5915)--(4.8921,-1.6651)--(5.0000,-1.7320)--(5.1118,-1.7919)--(5.2273,-1.8447)--(5.3458,-1.8900)--(5.4670,-1.9276)--(5.5903,-1.9576)--(5.7153,-1.9796)--(5.8415,-1.9937)--(5.9682,-1.9997)--(6.0951,-1.9977)--(6.2216,-1.9876)--(6.3472,-1.9696)--(6.4715,-1.9436)--(6.5938,-1.9098)--(6.7137,-1.8682)--(6.8308,-1.8192)--(6.9445,-1.7629)--(7.0544,-1.6994)--(7.1601,-1.6291)--(7.2611,-1.5522)--(7.3570,-1.4691)--(7.4474,-1.3801)--(7.5320,-1.2855)--(7.6105,-1.1858)--(7.6825,-1.0812)--(7.7476,-0.9723)--(7.8058,-0.8595)--(7.8567,-0.7433)--(7.9001,-0.6240)--(7.9358,-0.5022)--(7.9638,-0.3785)--(7.9839,-0.2531)--(7.9959,-0.1268)--(8.0000,0.0000);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.6729,0.2955) node {\( \lambda_1\)};
 \draw []  (5.0000,1.7320) node [rotate=0] {$\bullet$};
-\draw (5.0000,2.0862) node {\( \lambda_2\)};
+\draw (5.0000,2.0861) node {\( \lambda_2\)};
 \draw []  (5.0000,-1.7320) node [rotate=0] {$\bullet$};
-\draw (5.0000,-2.0862) node {\( \lambda_3\)};
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.0000,-0.31492) node {$ 2 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.0000,-0.31492) node {$ 3 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (8.0000,-0.31492) node {$ 4 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -1 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (5.0000,-2.0861) node {\( \lambda_3\)};
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 2 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 3 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 4 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DTIYKkP.pstricks.recall b/src_phystricks/Fig_DTIYKkP.pstricks.recall
index 2166b0fcf..78fbacbb2 100644
--- a/src_phystricks/Fig_DTIYKkP.pstricks.recall
+++ b/src_phystricks/Fig_DTIYKkP.pstricks.recall
@@ -82,13 +82,13 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [color=blue] (-5.000,0)--(-4.970,0.1206)--(-4.939,0.2400)--(-4.909,0.3581)--(-4.879,0.4751)--(-4.849,0.5908)--(-4.818,0.7052)--(-4.788,0.8185)--(-4.758,0.9305)--(-4.727,1.041)--(-4.697,1.151)--(-4.667,1.259)--(-4.636,1.366)--(-4.606,1.472)--(-4.576,1.577)--(-4.545,1.680)--(-4.515,1.783)--(-4.485,1.884)--(-4.455,1.983)--(-4.424,2.082)--(-4.394,2.179)--(-4.364,2.275)--(-4.333,2.370)--(-4.303,2.464)--(-4.273,2.556)--(-4.242,2.648)--(-4.212,2.738)--(-4.182,2.826)--(-4.151,2.914)--(-4.121,3.000)--(-4.091,3.085)--(-4.061,3.169)--(-4.030,3.252)--(-4.000,3.333)--(-3.970,3.414)--(-3.939,3.492)--(-3.909,3.570)--(-3.879,3.647)--(-3.848,3.722)--(-3.818,3.796)--(-3.788,3.869)--(-3.758,3.941)--(-3.727,4.011)--(-3.697,4.080)--(-3.667,4.148)--(-3.636,4.215)--(-3.606,4.280)--(-3.576,4.345)--(-3.545,4.408)--(-3.515,4.470)--(-3.485,4.530)--(-3.455,4.590)--(-3.424,4.648)--(-3.394,4.705)--(-3.364,4.760)--(-3.333,4.815)--(-3.303,4.868)--(-3.273,4.920)--(-3.242,4.971)--(-3.212,5.021)--(-3.182,5.069)--(-3.152,5.116)--(-3.121,5.162)--(-3.091,5.207)--(-3.061,5.250)--(-3.030,5.292)--(-3.000,5.333)--(-2.970,5.373)--(-2.939,5.412)--(-2.909,5.449)--(-2.879,5.485)--(-2.848,5.520)--(-2.818,5.554)--(-2.788,5.586)--(-2.758,5.617)--(-2.727,5.647)--(-2.697,5.676)--(-2.667,5.704)--(-2.636,5.730)--(-2.606,5.755)--(-2.576,5.779)--(-2.545,5.802)--(-2.515,5.823)--(-2.485,5.843)--(-2.455,5.862)--(-2.424,5.880)--(-2.394,5.897)--(-2.364,5.912)--(-2.333,5.926)--(-2.303,5.939)--(-2.273,5.950)--(-2.242,5.961)--(-2.212,5.970)--(-2.182,5.978)--(-2.152,5.985)--(-2.121,5.990)--(-2.091,5.995)--(-2.061,5.998)--(-2.030,5.999)--(-2.000,6.000);
+\draw [color=blue] (-5.0000,0.0000)--(-4.9696,0.1205)--(-4.9393,0.2399)--(-4.9090,0.3581)--(-4.8787,0.4750)--(-4.8484,0.5907)--(-4.8181,0.7052)--(-4.7878,0.8184)--(-4.7575,0.9305)--(-4.7272,1.0413)--(-4.6969,1.1509)--(-4.6666,1.2592)--(-4.6363,1.3663)--(-4.6060,1.4722)--(-4.5757,1.5769)--(-4.5454,1.6804)--(-4.5151,1.7826)--(-4.4848,1.8836)--(-4.4545,1.9834)--(-4.4242,2.0820)--(-4.3939,2.1793)--(-4.3636,2.2754)--(-4.3333,2.3703)--(-4.3030,2.4640)--(-4.2727,2.5564)--(-4.2424,2.6476)--(-4.2121,2.7376)--(-4.1818,2.8264)--(-4.1515,2.9139)--(-4.1212,3.0003)--(-4.0909,3.0853)--(-4.0606,3.1692)--(-4.0303,3.2519)--(-4.0000,3.3333)--(-3.9696,3.4135)--(-3.9393,3.4925)--(-3.9090,3.5702)--(-3.8787,3.6467)--(-3.8484,3.7220)--(-3.8181,3.7961)--(-3.7878,3.8689)--(-3.7575,3.9406)--(-3.7272,4.0110)--(-3.6969,4.0801)--(-3.6666,4.1481)--(-3.6363,4.2148)--(-3.6060,4.2803)--(-3.5757,4.3446)--(-3.5454,4.4077)--(-3.5151,4.4695)--(-3.4848,4.5301)--(-3.4545,4.5895)--(-3.4242,4.6476)--(-3.3939,4.7046)--(-3.3636,4.7603)--(-3.3333,4.8148)--(-3.3030,4.8680)--(-3.2727,4.9201)--(-3.2424,4.9709)--(-3.2121,5.0205)--(-3.1818,5.0688)--(-3.1515,5.1160)--(-3.1212,5.1619)--(-3.0909,5.2066)--(-3.0606,5.2500)--(-3.0303,5.2923)--(-3.0000,5.3333)--(-2.9696,5.3731)--(-2.9393,5.4116)--(-2.9090,5.4490)--(-2.8787,5.4851)--(-2.8484,5.5200)--(-2.8181,5.5537)--(-2.7878,5.5861)--(-2.7575,5.6173)--(-2.7272,5.6473)--(-2.6969,5.6761)--(-2.6666,5.7037)--(-2.6363,5.7300)--(-2.6060,5.7551)--(-2.5757,5.7790)--(-2.5454,5.8016)--(-2.5151,5.8230)--(-2.4848,5.8432)--(-2.4545,5.8622)--(-2.4242,5.8800)--(-2.3939,5.8965)--(-2.3636,5.9118)--(-2.3333,5.9259)--(-2.3030,5.9387)--(-2.2727,5.9504)--(-2.2424,5.9608)--(-2.2121,5.9700)--(-2.1818,5.9779)--(-2.1515,5.9846)--(-2.1212,5.9902)--(-2.0909,5.9944)--(-2.0606,5.9975)--(-2.0303,5.9993)--(-2.0000,6.0000);
 \draw [color=brown]  (-4.0000,3.3333) node [rotate=0] {$\bullet$};
-\draw (-4.7275,3.6358) node {$o=[\mtu]$};
+\draw (-4.7274,3.6357) node {$o=[\mtu]$};
 \draw (-1.1978,6.0000) node {$[\SO(2)]$};
 \draw [color=cyan,->,>=latex] (-4.5000,1.8333) -- (-2.5000,2.3333);
 \draw (-1.4416,1.9799) node {$[ e^{sE(w)} e^{xq_0}]$};
-\draw (-5.1029,2.1187) node {$[ e^{xq_0}]$};
+\draw (-5.1029,2.1186) node {$[ e^{xq_0}]$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall b/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
index aca707323..d2188f5a2 100644
--- a/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
+++ b/src_phystricks/Fig_DZVooQZLUtf.pstricks.recall
@@ -53,29 +53,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7236);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7236);
 %DEFAULT
 
-\draw [color=blue] (0.01000,3.224)--(0.05030,2.093)--(0.09061,1.681)--(0.1309,1.423)--(0.1712,1.235)--(0.2115,1.087)--(0.2518,0.9653)--(0.2921,0.8614)--(0.3324,0.7709)--(0.3727,0.6908)--(0.4130,0.6190)--(0.4533,0.5538)--(0.4936,0.4942)--(0.5339,0.4392)--(0.5742,0.3883)--(0.6145,0.3408)--(0.6548,0.2963)--(0.6952,0.2545)--(0.7355,0.2151)--(0.7758,0.1777)--(0.8161,0.1423)--(0.8564,0.1085)--(0.8967,0.07635)--(0.9370,0.04557)--(0.9773,0.01609)--(1.018,0.01220)--(1.058,0.03939)--(1.098,0.06556)--(1.138,0.09079)--(1.179,0.1151)--(1.219,0.1387)--(1.259,0.1614)--(1.300,0.1835)--(1.340,0.2049)--(1.380,0.2256)--(1.421,0.2458)--(1.461,0.2653)--(1.501,0.2844)--(1.542,0.3029)--(1.582,0.3210)--(1.622,0.3386)--(1.662,0.3558)--(1.703,0.3726)--(1.743,0.3889)--(1.783,0.4049)--(1.824,0.4206)--(1.864,0.4359)--(1.904,0.4509)--(1.945,0.4655)--(1.985,0.4799)--(2.025,0.4939)--(2.065,0.5077)--(2.106,0.5213)--(2.146,0.5345)--(2.186,0.5476)--(2.227,0.5604)--(2.267,0.5729)--(2.307,0.5852)--(2.348,0.5974)--(2.388,0.6093)--(2.428,0.6210)--(2.468,0.6325)--(2.509,0.6439)--(2.549,0.6550)--(2.589,0.6660)--(2.630,0.6768)--(2.670,0.6875)--(2.710,0.6979)--(2.751,0.7083)--(2.791,0.7185)--(2.831,0.7285)--(2.872,0.7384)--(2.912,0.7481)--(2.952,0.7578)--(2.992,0.7673)--(3.033,0.7766)--(3.073,0.7859)--(3.113,0.7950)--(3.154,0.8040)--(3.194,0.8129)--(3.234,0.8217)--(3.275,0.8303)--(3.315,0.8389)--(3.355,0.8474)--(3.395,0.8557)--(3.436,0.8640)--(3.476,0.8721)--(3.516,0.8802)--(3.557,0.8882)--(3.597,0.8961)--(3.637,0.9039)--(3.678,0.9116)--(3.718,0.9192)--(3.758,0.9268)--(3.798,0.9342)--(3.839,0.9416)--(3.879,0.9489)--(3.919,0.9562)--(3.960,0.9633)--(4.000,0.9704);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=blue] (0.0100,3.2236)--(0.0503,2.0927)--(0.0906,1.6808)--(0.1309,1.4232)--(0.1712,1.2353)--(0.2115,1.0874)--(0.2518,0.9653)--(0.2921,0.8614)--(0.3324,0.7709)--(0.3727,0.6908)--(0.4130,0.6189)--(0.4533,0.5537)--(0.4936,0.4941)--(0.5339,0.4392)--(0.5742,0.3882)--(0.6145,0.3408)--(0.6548,0.2963)--(0.6951,0.2545)--(0.7354,0.2150)--(0.7757,0.1777)--(0.8160,0.1422)--(0.8563,0.1085)--(0.8966,0.0763)--(0.9369,0.0455)--(0.9772,0.0160)--(1.0175,0.0121)--(1.0578,0.0393)--(1.0981,0.0655)--(1.1384,0.0907)--(1.1787,0.1151)--(1.2190,0.1386)--(1.2593,0.1614)--(1.2996,0.1834)--(1.3400,0.2048)--(1.3803,0.2256)--(1.4206,0.2457)--(1.4609,0.2653)--(1.5012,0.2843)--(1.5415,0.3029)--(1.5818,0.3210)--(1.6221,0.3386)--(1.6624,0.3557)--(1.7027,0.3725)--(1.7430,0.3889)--(1.7833,0.4049)--(1.8236,0.4205)--(1.8639,0.4358)--(1.9042,0.4508)--(1.9445,0.4655)--(1.9848,0.4798)--(2.0251,0.4939)--(2.0654,0.5077)--(2.1057,0.5212)--(2.1460,0.5345)--(2.1863,0.5475)--(2.2266,0.5603)--(2.2669,0.5729)--(2.3072,0.5852)--(2.3475,0.5973)--(2.3878,0.6092)--(2.4281,0.6209)--(2.4684,0.6325)--(2.5087,0.6438)--(2.5490,0.6550)--(2.5893,0.6659)--(2.6296,0.6768)--(2.6700,0.6874)--(2.7103,0.6979)--(2.7506,0.7082)--(2.7909,0.7184)--(2.8312,0.7284)--(2.8715,0.7383)--(2.9118,0.7481)--(2.9521,0.7577)--(2.9924,0.7672)--(3.0327,0.7766)--(3.0730,0.7858)--(3.1133,0.7949)--(3.1536,0.8039)--(3.1939,0.8128)--(3.2342,0.8216)--(3.2745,0.8303)--(3.3148,0.8388)--(3.3551,0.8473)--(3.3954,0.8557)--(3.4357,0.8639)--(3.4760,0.8721)--(3.5163,0.8801)--(3.5566,0.8881)--(3.5969,0.8960)--(3.6372,0.9038)--(3.6775,0.9115)--(3.7178,0.9192)--(3.7581,0.9267)--(3.7984,0.9342)--(3.8387,0.9416)--(3.8790,0.9489)--(3.9193,0.9561)--(3.9596,0.9633)--(4.0000,0.9704);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -141,33 +141,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.7236) -- (0,1.4704);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7236) -- (0.0000,1.4704);
 %DEFAULT
 
-\draw [color=blue] (0.01000,-3.224)--(0.05030,-2.093)--(0.09061,-1.681)--(0.1309,-1.423)--(0.1712,-1.235)--(0.2115,-1.087)--(0.2518,-0.9653)--(0.2921,-0.8614)--(0.3324,-0.7709)--(0.3727,-0.6908)--(0.4130,-0.6190)--(0.4533,-0.5538)--(0.4936,-0.4942)--(0.5339,-0.4392)--(0.5742,-0.3883)--(0.6145,-0.3408)--(0.6548,-0.2963)--(0.6952,-0.2545)--(0.7355,-0.2151)--(0.7758,-0.1777)--(0.8161,-0.1423)--(0.8564,-0.1085)--(0.8967,-0.07635)--(0.9370,-0.04557)--(0.9773,-0.01609)--(1.018,0.01220)--(1.058,0.03939)--(1.098,0.06556)--(1.138,0.09079)--(1.179,0.1151)--(1.219,0.1387)--(1.259,0.1614)--(1.300,0.1835)--(1.340,0.2049)--(1.380,0.2256)--(1.421,0.2458)--(1.461,0.2653)--(1.501,0.2844)--(1.542,0.3029)--(1.582,0.3210)--(1.622,0.3386)--(1.662,0.3558)--(1.703,0.3726)--(1.743,0.3889)--(1.783,0.4049)--(1.824,0.4206)--(1.864,0.4359)--(1.904,0.4509)--(1.945,0.4655)--(1.985,0.4799)--(2.025,0.4939)--(2.065,0.5077)--(2.106,0.5213)--(2.146,0.5345)--(2.186,0.5476)--(2.227,0.5604)--(2.267,0.5729)--(2.307,0.5852)--(2.348,0.5974)--(2.388,0.6093)--(2.428,0.6210)--(2.468,0.6325)--(2.509,0.6439)--(2.549,0.6550)--(2.589,0.6660)--(2.630,0.6768)--(2.670,0.6875)--(2.710,0.6979)--(2.751,0.7083)--(2.791,0.7185)--(2.831,0.7285)--(2.872,0.7384)--(2.912,0.7481)--(2.952,0.7578)--(2.992,0.7673)--(3.033,0.7766)--(3.073,0.7859)--(3.113,0.7950)--(3.154,0.8040)--(3.194,0.8129)--(3.234,0.8217)--(3.275,0.8303)--(3.315,0.8389)--(3.355,0.8474)--(3.395,0.8557)--(3.436,0.8640)--(3.476,0.8721)--(3.516,0.8802)--(3.557,0.8882)--(3.597,0.8961)--(3.637,0.9039)--(3.678,0.9116)--(3.718,0.9192)--(3.758,0.9268)--(3.798,0.9342)--(3.839,0.9416)--(3.879,0.9489)--(3.919,0.9562)--(3.960,0.9633)--(4.000,0.9704);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
+\draw [color=blue] (0.0100,-3.2236)--(0.0503,-2.0927)--(0.0906,-1.6808)--(0.1309,-1.4232)--(0.1712,-1.2353)--(0.2115,-1.0874)--(0.2518,-0.9653)--(0.2921,-0.8614)--(0.3324,-0.7709)--(0.3727,-0.6908)--(0.4130,-0.6189)--(0.4533,-0.5537)--(0.4936,-0.4941)--(0.5339,-0.4392)--(0.5742,-0.3882)--(0.6145,-0.3408)--(0.6548,-0.2963)--(0.6951,-0.2545)--(0.7354,-0.2150)--(0.7757,-0.1777)--(0.8160,-0.1422)--(0.8563,-0.1085)--(0.8966,-0.0763)--(0.9369,-0.0455)--(0.9772,-0.0160)--(1.0175,0.0121)--(1.0578,0.0393)--(1.0981,0.0655)--(1.1384,0.0907)--(1.1787,0.1151)--(1.2190,0.1386)--(1.2593,0.1614)--(1.2996,0.1834)--(1.3400,0.2048)--(1.3803,0.2256)--(1.4206,0.2457)--(1.4609,0.2653)--(1.5012,0.2843)--(1.5415,0.3029)--(1.5818,0.3210)--(1.6221,0.3386)--(1.6624,0.3557)--(1.7027,0.3725)--(1.7430,0.3889)--(1.7833,0.4049)--(1.8236,0.4205)--(1.8639,0.4358)--(1.9042,0.4508)--(1.9445,0.4655)--(1.9848,0.4798)--(2.0251,0.4939)--(2.0654,0.5077)--(2.1057,0.5212)--(2.1460,0.5345)--(2.1863,0.5475)--(2.2266,0.5603)--(2.2669,0.5729)--(2.3072,0.5852)--(2.3475,0.5973)--(2.3878,0.6092)--(2.4281,0.6209)--(2.4684,0.6325)--(2.5087,0.6438)--(2.5490,0.6550)--(2.5893,0.6659)--(2.6296,0.6768)--(2.6700,0.6874)--(2.7103,0.6979)--(2.7506,0.7082)--(2.7909,0.7184)--(2.8312,0.7284)--(2.8715,0.7383)--(2.9118,0.7481)--(2.9521,0.7577)--(2.9924,0.7672)--(3.0327,0.7766)--(3.0730,0.7858)--(3.1133,0.7949)--(3.1536,0.8039)--(3.1939,0.8128)--(3.2342,0.8216)--(3.2745,0.8303)--(3.3148,0.8388)--(3.3551,0.8473)--(3.3954,0.8557)--(3.4357,0.8639)--(3.4760,0.8721)--(3.5163,0.8801)--(3.5566,0.8881)--(3.5969,0.8960)--(3.6372,0.9038)--(3.6775,0.9115)--(3.7178,0.9192)--(3.7581,0.9267)--(3.7984,0.9342)--(3.8387,0.9416)--(3.8790,0.9489)--(3.9193,0.9561)--(3.9596,0.9633)--(4.0000,0.9704);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -229,33 +229,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.0236) -- (0,2.1704);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0236) -- (0.0000,2.1704);
 %DEFAULT
 
-\draw [color=blue] (0.01000,-2.524)--(0.05030,-1.393)--(0.09061,-0.9809)--(0.1309,-0.7233)--(0.1712,-0.5354)--(0.2115,-0.3874)--(0.2518,-0.2653)--(0.2921,-0.1614)--(0.3324,-0.07094)--(0.3727,0.009164)--(0.4130,0.08104)--(0.4533,0.1462)--(0.4936,0.2058)--(0.5339,0.2608)--(0.5742,0.3117)--(0.6145,0.3592)--(0.6548,0.4037)--(0.6952,0.4455)--(0.7355,0.4849)--(0.7758,0.5223)--(0.8161,0.5577)--(0.8564,0.5915)--(0.8967,0.6236)--(0.9370,0.6544)--(0.9773,0.6839)--(1.018,0.7122)--(1.058,0.7394)--(1.098,0.7656)--(1.138,0.7908)--(1.179,0.8151)--(1.219,0.8387)--(1.259,0.8614)--(1.300,0.8835)--(1.340,0.9049)--(1.380,0.9256)--(1.421,0.9458)--(1.461,0.9653)--(1.501,0.9844)--(1.542,1.003)--(1.582,1.021)--(1.622,1.039)--(1.662,1.056)--(1.703,1.073)--(1.743,1.089)--(1.783,1.105)--(1.824,1.121)--(1.864,1.136)--(1.904,1.151)--(1.945,1.166)--(1.985,1.180)--(2.025,1.194)--(2.065,1.208)--(2.106,1.221)--(2.146,1.235)--(2.186,1.248)--(2.227,1.260)--(2.267,1.273)--(2.307,1.285)--(2.348,1.297)--(2.388,1.309)--(2.428,1.321)--(2.468,1.333)--(2.509,1.344)--(2.549,1.355)--(2.589,1.366)--(2.630,1.377)--(2.670,1.387)--(2.710,1.398)--(2.751,1.408)--(2.791,1.418)--(2.831,1.428)--(2.872,1.438)--(2.912,1.448)--(2.952,1.458)--(2.992,1.467)--(3.033,1.477)--(3.073,1.486)--(3.113,1.495)--(3.154,1.504)--(3.194,1.513)--(3.234,1.522)--(3.275,1.530)--(3.315,1.539)--(3.355,1.547)--(3.395,1.556)--(3.436,1.564)--(3.476,1.572)--(3.516,1.580)--(3.557,1.588)--(3.597,1.596)--(3.637,1.604)--(3.678,1.612)--(3.718,1.619)--(3.758,1.627)--(3.798,1.634)--(3.839,1.642)--(3.879,1.649)--(3.919,1.656)--(3.960,1.663)--(4.000,1.670);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
+\draw [color=blue] (0.0100,-2.5236)--(0.0503,-1.3927)--(0.0906,-0.9808)--(0.1309,-0.7232)--(0.1712,-0.5353)--(0.2115,-0.3874)--(0.2518,-0.2653)--(0.2921,-0.1614)--(0.3324,-0.0709)--(0.3727,0.0091)--(0.4130,0.0810)--(0.4533,0.1462)--(0.4936,0.2058)--(0.5339,0.2607)--(0.5742,0.3117)--(0.6145,0.3591)--(0.6548,0.4036)--(0.6951,0.4454)--(0.7354,0.4849)--(0.7757,0.5222)--(0.8160,0.5577)--(0.8563,0.5914)--(0.8966,0.6236)--(0.9369,0.6544)--(0.9772,0.6839)--(1.0175,0.7121)--(1.0578,0.7393)--(1.0981,0.7655)--(1.1384,0.7907)--(1.1787,0.8151)--(1.2190,0.8386)--(1.2593,0.8614)--(1.2996,0.8834)--(1.3400,0.9048)--(1.3803,0.9256)--(1.4206,0.9457)--(1.4609,0.9653)--(1.5012,0.9843)--(1.5415,1.0029)--(1.5818,1.0210)--(1.6221,1.0386)--(1.6624,1.0557)--(1.7027,1.0725)--(1.7430,1.0889)--(1.7833,1.1049)--(1.8236,1.1205)--(1.8639,1.1358)--(1.9042,1.1508)--(1.9445,1.1655)--(1.9848,1.1798)--(2.0251,1.1939)--(2.0654,1.2077)--(2.1057,1.2212)--(2.1460,1.2345)--(2.1863,1.2475)--(2.2266,1.2603)--(2.2669,1.2729)--(2.3072,1.2852)--(2.3475,1.2973)--(2.3878,1.3092)--(2.4281,1.3209)--(2.4684,1.3325)--(2.5087,1.3438)--(2.5490,1.3550)--(2.5893,1.3659)--(2.6296,1.3768)--(2.6700,1.3874)--(2.7103,1.3979)--(2.7506,1.4082)--(2.7909,1.4184)--(2.8312,1.4284)--(2.8715,1.4383)--(2.9118,1.4481)--(2.9521,1.4577)--(2.9924,1.4672)--(3.0327,1.4766)--(3.0730,1.4858)--(3.1133,1.4949)--(3.1536,1.5039)--(3.1939,1.5128)--(3.2342,1.5216)--(3.2745,1.5303)--(3.3148,1.5388)--(3.3551,1.5473)--(3.3954,1.5557)--(3.4357,1.5639)--(3.4760,1.5721)--(3.5163,1.5801)--(3.5566,1.5881)--(3.5969,1.5960)--(3.6372,1.6038)--(3.6775,1.6115)--(3.7178,1.6192)--(3.7581,1.6267)--(3.7984,1.6342)--(3.8387,1.6416)--(3.8790,1.6489)--(3.9193,1.6561)--(3.9596,1.6633)--(4.0000,1.6704);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -301,21 +301,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.3242);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.3241);
 %DEFAULT
 
-\draw [color=blue] (1.000,0)--(1.030,0.1209)--(1.061,0.1698)--(1.091,0.2065)--(1.121,0.2368)--(1.152,0.2629)--(1.182,0.2861)--(1.212,0.3070)--(1.242,0.3261)--(1.273,0.3438)--(1.303,0.3601)--(1.333,0.3755)--(1.364,0.3898)--(1.394,0.4034)--(1.424,0.4163)--(1.455,0.4285)--(1.485,0.4401)--(1.515,0.4512)--(1.545,0.4618)--(1.576,0.4720)--(1.606,0.4818)--(1.636,0.4912)--(1.667,0.5003)--(1.697,0.5090)--(1.727,0.5175)--(1.758,0.5257)--(1.788,0.5336)--(1.818,0.5412)--(1.848,0.5487)--(1.879,0.5559)--(1.909,0.5629)--(1.939,0.5697)--(1.970,0.5763)--(2.000,0.5828)--(2.030,0.5891)--(2.061,0.5952)--(2.091,0.6012)--(2.121,0.6070)--(2.152,0.6127)--(2.182,0.6183)--(2.212,0.6237)--(2.242,0.6291)--(2.273,0.6343)--(2.303,0.6394)--(2.333,0.6443)--(2.364,0.6492)--(2.394,0.6540)--(2.424,0.6587)--(2.455,0.6633)--(2.485,0.6678)--(2.515,0.6723)--(2.545,0.6766)--(2.576,0.6809)--(2.606,0.6851)--(2.636,0.6892)--(2.667,0.6933)--(2.697,0.6972)--(2.727,0.7012)--(2.758,0.7050)--(2.788,0.7088)--(2.818,0.7125)--(2.848,0.7162)--(2.879,0.7198)--(2.909,0.7234)--(2.939,0.7269)--(2.970,0.7303)--(3.000,0.7337)--(3.030,0.7371)--(3.061,0.7403)--(3.091,0.7436)--(3.121,0.7468)--(3.152,0.7500)--(3.182,0.7531)--(3.212,0.7562)--(3.242,0.7592)--(3.273,0.7622)--(3.303,0.7652)--(3.333,0.7681)--(3.364,0.7710)--(3.394,0.7738)--(3.424,0.7766)--(3.455,0.7794)--(3.485,0.7821)--(3.515,0.7848)--(3.545,0.7875)--(3.576,0.7902)--(3.606,0.7928)--(3.636,0.7953)--(3.667,0.7979)--(3.697,0.8004)--(3.727,0.8029)--(3.758,0.8054)--(3.788,0.8078)--(3.818,0.8102)--(3.848,0.8126)--(3.879,0.8150)--(3.909,0.8173)--(3.939,0.8196)--(3.970,0.8219)--(4.000,0.8242);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (1.0000,0.0000)--(1.0303,0.1209)--(1.0606,0.1697)--(1.0909,0.2064)--(1.1212,0.2367)--(1.1515,0.2629)--(1.1818,0.2861)--(1.2121,0.3070)--(1.2424,0.3261)--(1.2727,0.3437)--(1.3030,0.3601)--(1.3333,0.3754)--(1.3636,0.3898)--(1.3939,0.4034)--(1.4242,0.4162)--(1.4545,0.4284)--(1.4848,0.4401)--(1.5151,0.4512)--(1.5454,0.4618)--(1.5757,0.4720)--(1.6060,0.4818)--(1.6363,0.4912)--(1.6666,0.5003)--(1.6969,0.5090)--(1.7272,0.5175)--(1.7575,0.5256)--(1.7878,0.5335)--(1.8181,0.5412)--(1.8484,0.5486)--(1.8787,0.5558)--(1.9090,0.5628)--(1.9393,0.5697)--(1.9696,0.5763)--(2.0000,0.5827)--(2.0303,0.5890)--(2.0606,0.5952)--(2.0909,0.6011)--(2.1212,0.6070)--(2.1515,0.6127)--(2.1818,0.6182)--(2.2121,0.6237)--(2.2424,0.6290)--(2.2727,0.6342)--(2.3030,0.6393)--(2.3333,0.6443)--(2.3636,0.6492)--(2.3939,0.6540)--(2.4242,0.6587)--(2.4545,0.6633)--(2.4848,0.6678)--(2.5151,0.6722)--(2.5454,0.6766)--(2.5757,0.6808)--(2.6060,0.6850)--(2.6363,0.6892)--(2.6666,0.6932)--(2.6969,0.6972)--(2.7272,0.7011)--(2.7575,0.7050)--(2.7878,0.7087)--(2.8181,0.7125)--(2.8484,0.7161)--(2.8787,0.7197)--(2.9090,0.7233)--(2.9393,0.7268)--(2.9696,0.7303)--(3.0000,0.7337)--(3.0303,0.7370)--(3.0606,0.7403)--(3.0909,0.7436)--(3.1212,0.7468)--(3.1515,0.7499)--(3.1818,0.7530)--(3.2121,0.7561)--(3.2424,0.7592)--(3.2727,0.7622)--(3.3030,0.7651)--(3.3333,0.7680)--(3.3636,0.7709)--(3.3939,0.7738)--(3.4242,0.7766)--(3.4545,0.7793)--(3.4848,0.7821)--(3.5151,0.7848)--(3.5454,0.7875)--(3.5757,0.7901)--(3.6060,0.7927)--(3.6363,0.7953)--(3.6666,0.7979)--(3.6969,0.8004)--(3.7272,0.8029)--(3.7575,0.8053)--(3.7878,0.8078)--(3.8181,0.8102)--(3.8484,0.8126)--(3.8787,0.8149)--(3.9090,0.8173)--(3.9393,0.8196)--(3.9696,0.8219)--(4.0000,0.8241);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -381,35 +381,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.4900,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-3.7236) -- (0,1.5529);
+\draw [,->,>=latex] (-1.4900,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7236) -- (0.0000,1.5528);
 %DEFAULT
 
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-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
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-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
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+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -471,39 +471,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-3.7725) -- (0,1.7828);
+\draw [,->,>=latex] (-2.2500,0.0000) -- (2.2500,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7725) -- (0.0000,1.7828);
 %DEFAULT
 
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-\draw (0.70000,-0.31492) node {$ 1 $};
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-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
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-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
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-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
+\draw [color=blue] (0.0675,-3.2725)--(0.0845,-2.9585)--(0.1015,-2.7022)--(0.1185,-2.4857)--(0.1355,-2.2982)--(0.1525,-2.1328)--(0.1695,-1.9850)--(0.1865,-1.8513)--(0.2035,-1.7292)--(0.2205,-1.6169)--(0.2375,-1.5130)--(0.2545,-1.4163)--(0.2715,-1.3258)--(0.2885,-1.2408)--(0.3055,-1.1607)--(0.3225,-1.0849)--(0.3395,-1.0130)--(0.3564,-0.9446)--(0.3734,-0.8794)--(0.3904,-0.8171)--(0.4074,-0.7575)--(0.4244,-0.7003)--(0.4414,-0.6453)--(0.4584,-0.5924)--(0.4754,-0.5415)--(0.4924,-0.4923)--(0.5094,-0.4448)--(0.5264,-0.3989)--(0.5434,-0.3544)--(0.5604,-0.3113)--(0.5774,-0.2695)--(0.5944,-0.2289)--(0.6114,-0.1894)--(0.6283,-0.1510)--(0.6453,-0.1137)--(0.6623,-0.0773)--(0.6793,-0.0418)--(0.6963,-0.0072)--(0.7133,0.0264)--(0.7303,0.0594)--(0.7473,0.0916)--(0.7643,0.1231)--(0.7813,0.1539)--(0.7983,0.1840)--(0.8153,0.2135)--(0.8323,0.2424)--(0.8493,0.2706)--(0.8663,0.2984)--(0.8833,0.3256)--(0.9003,0.3523)--(0.9172,0.3784)--(0.9342,0.4041)--(0.9512,0.4294)--(0.9682,0.4542)--(0.9852,0.4785)--(1.0022,0.5025)--(1.0192,0.5260)--(1.0362,0.5492)--(1.0532,0.5719)--(1.0702,0.5943)--(1.0872,0.6164)--(1.1042,0.6381)--(1.1212,0.6595)--(1.1382,0.6805)--(1.1552,0.7013)--(1.1722,0.7217)--(1.1891,0.7419)--(1.2061,0.7618)--(1.2231,0.7813)--(1.2401,0.8007)--(1.2571,0.8197)--(1.2741,0.8385)--(1.2911,0.8571)--(1.3081,0.8754)--(1.3251,0.8934)--(1.3421,0.9113)--(1.3591,0.9289)--(1.3761,0.9463)--(1.3931,0.9635)--(1.4101,0.9804)--(1.4271,0.9972)--(1.4441,1.0138)--(1.4611,1.0302)--(1.4780,1.0464)--(1.4950,1.0624)--(1.5120,1.0782)--(1.5290,1.0938)--(1.5460,1.1093)--(1.5630,1.1246)--(1.5800,1.1397)--(1.5970,1.1547)--(1.6140,1.1695)--(1.6310,1.1842)--(1.6480,1.1987)--(1.6650,1.2131)--(1.6820,1.2273)--(1.6990,1.2414)--(1.7160,1.2553)--(1.7330,1.2691)--(1.7500,1.2828);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -565,37 +565,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6000,0) -- (2.6000,0);
-\draw [,->,>=latex] (0,-3.5467) -- (0,1.2690);
+\draw [,->,>=latex] (-2.6000,0.0000) -- (2.6000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5466) -- (0.0000,1.2690);
 %DEFAULT
 
-\draw [color=blue] (-2.100,0.7690)--(-2.079,0.7620)--(-2.058,0.7548)--(-2.037,0.7476)--(-2.016,0.7403)--(-1.994,0.7329)--(-1.973,0.7255)--(-1.952,0.7179)--(-1.931,0.7103)--(-1.910,0.7026)--(-1.889,0.6948)--(-1.868,0.6870)--(-1.847,0.6790)--(-1.825,0.6709)--(-1.804,0.6628)--(-1.783,0.6546)--(-1.762,0.6462)--(-1.741,0.6378)--(-1.720,0.6292)--(-1.699,0.6206)--(-1.678,0.6118)--(-1.656,0.6030)--(-1.635,0.5940)--(-1.614,0.5849)--(-1.593,0.5756)--(-1.572,0.5663)--(-1.551,0.5568)--(-1.530,0.5472)--(-1.509,0.5375)--(-1.487,0.5276)--(-1.466,0.5176)--(-1.445,0.5075)--(-1.424,0.4972)--(-1.403,0.4867)--(-1.382,0.4761)--(-1.361,0.4653)--(-1.340,0.4544)--(-1.319,0.4432)--(-1.297,0.4319)--(-1.276,0.4204)--(-1.255,0.4088)--(-1.234,0.3969)--(-1.213,0.3848)--(-1.192,0.3725)--(-1.171,0.3600)--(-1.150,0.3472)--(-1.128,0.3343)--(-1.107,0.3210)--(-1.086,0.3075)--(-1.065,0.2938)--(-1.044,0.2798)--(-1.023,0.2655)--(-1.002,0.2509)--(-0.9806,0.2359)--(-0.9595,0.2207)--(-0.9383,0.2051)--(-0.9172,0.1892)--(-0.8961,0.1729)--(-0.8750,0.1562)--(-0.8539,0.1391)--(-0.8327,0.1215)--(-0.8116,0.1036)--(-0.7905,0.08510)--(-0.7694,0.06615)--(-0.7483,0.04666)--(-0.7271,0.02662)--(-0.7060,0.005983)--(-0.6849,-0.01528)--(-0.6638,-0.03721)--(-0.6426,-0.05984)--(-0.6215,-0.08323)--(-0.6004,-0.1074)--(-0.5793,-0.1325)--(-0.5582,-0.1585)--(-0.5370,-0.1855)--(-0.5159,-0.2136)--(-0.4948,-0.2429)--(-0.4737,-0.2734)--(-0.4526,-0.3053)--(-0.4314,-0.3388)--(-0.4103,-0.3739)--(-0.3892,-0.4109)--(-0.3681,-0.4500)--(-0.3470,-0.4913)--(-0.3258,-0.5353)--(-0.3047,-0.5822)--(-0.2836,-0.6325)--(-0.2625,-0.6867)--(-0.2413,-0.7454)--(-0.2202,-0.8095)--(-0.1991,-0.8801)--(-0.1780,-0.9586)--(-0.1569,-1.047)--(-0.1357,-1.148)--(-0.1146,-1.267)--(-0.09350,-1.409)--(-0.07238,-1.588)--(-0.05126,-1.830)--(-0.03013,-2.202)--(-0.009013,-3.047);
+\draw [color=blue] (-2.1000,0.7690)--(-2.0788,0.7619)--(-2.0577,0.7548)--(-2.0366,0.7475)--(-2.0155,0.7402)--(-1.9943,0.7329)--(-1.9732,0.7254)--(-1.9521,0.7179)--(-1.9310,0.7103)--(-1.9099,0.7026)--(-1.8887,0.6948)--(-1.8676,0.6869)--(-1.8465,0.6789)--(-1.8254,0.6709)--(-1.8043,0.6627)--(-1.7831,0.6545)--(-1.7620,0.6462)--(-1.7409,0.6377)--(-1.7198,0.6292)--(-1.6986,0.6205)--(-1.6775,0.6118)--(-1.6564,0.6029)--(-1.6353,0.5939)--(-1.6142,0.5848)--(-1.5930,0.5756)--(-1.5719,0.5663)--(-1.5508,0.5568)--(-1.5297,0.5472)--(-1.5086,0.5375)--(-1.4874,0.5276)--(-1.4663,0.5176)--(-1.4452,0.5074)--(-1.4241,0.4971)--(-1.4030,0.4867)--(-1.3818,0.4760)--(-1.3607,0.4653)--(-1.3396,0.4543)--(-1.3185,0.4432)--(-1.2973,0.4319)--(-1.2762,0.4204)--(-1.2551,0.4087)--(-1.2340,0.3968)--(-1.2129,0.3847)--(-1.1917,0.3724)--(-1.1706,0.3599)--(-1.1495,0.3472)--(-1.1284,0.3342)--(-1.1073,0.3210)--(-1.0861,0.3075)--(-1.0650,0.2937)--(-1.0439,0.2797)--(-1.0228,0.2654)--(-1.0017,0.2508)--(-0.9805,0.2359)--(-0.9594,0.2207)--(-0.9383,0.2051)--(-0.9172,0.1891)--(-0.8960,0.1728)--(-0.8749,0.1561)--(-0.8538,0.1390)--(-0.8327,0.1215)--(-0.8116,0.1035)--(-0.7904,0.0851)--(-0.7693,0.0661)--(-0.7482,0.0466)--(-0.7271,0.0266)--(-0.7060,0.0059)--(-0.6848,-0.0152)--(-0.6637,-0.0372)--(-0.6426,-0.0598)--(-0.6215,-0.0832)--(-0.6004,-0.1074)--(-0.5792,-0.1325)--(-0.5581,-0.1585)--(-0.5370,-0.1855)--(-0.5159,-0.2135)--(-0.4947,-0.2428)--(-0.4736,-0.2733)--(-0.4525,-0.3053)--(-0.4314,-0.3387)--(-0.4103,-0.3739)--(-0.3891,-0.4109)--(-0.3680,-0.4499)--(-0.3469,-0.4913)--(-0.3258,-0.5352)--(-0.3047,-0.5822)--(-0.2835,-0.6324)--(-0.2624,-0.6866)--(-0.2413,-0.7453)--(-0.2202,-0.8095)--(-0.1991,-0.8800)--(-0.1779,-0.9585)--(-0.1568,-1.0470)--(-0.1357,-1.1482)--(-0.1146,-1.2666)--(-0.0934,-1.4092)--(-0.0723,-1.5884)--(-0.0512,-1.8299)--(-0.0301,-2.2017)--(-0.0090,-3.0466);
 
-\draw [color=blue] (0.009013,-3.047)--(0.03013,-2.202)--(0.05126,-1.830)--(0.07238,-1.588)--(0.09350,-1.409)--(0.1146,-1.267)--(0.1357,-1.148)--(0.1569,-1.047)--(0.1780,-0.9586)--(0.1991,-0.8801)--(0.2202,-0.8095)--(0.2413,-0.7454)--(0.2625,-0.6867)--(0.2836,-0.6325)--(0.3047,-0.5822)--(0.3258,-0.5353)--(0.3470,-0.4913)--(0.3681,-0.4500)--(0.3892,-0.4109)--(0.4103,-0.3739)--(0.4314,-0.3388)--(0.4526,-0.3053)--(0.4737,-0.2734)--(0.4948,-0.2429)--(0.5159,-0.2136)--(0.5370,-0.1855)--(0.5582,-0.1585)--(0.5793,-0.1325)--(0.6004,-0.1074)--(0.6215,-0.08323)--(0.6426,-0.05984)--(0.6638,-0.03721)--(0.6849,-0.01528)--(0.7060,0.005983)--(0.7271,0.02662)--(0.7483,0.04666)--(0.7694,0.06615)--(0.7905,0.08510)--(0.8116,0.1036)--(0.8327,0.1215)--(0.8539,0.1391)--(0.8750,0.1562)--(0.8961,0.1729)--(0.9172,0.1892)--(0.9383,0.2051)--(0.9595,0.2207)--(0.9806,0.2359)--(1.002,0.2509)--(1.023,0.2655)--(1.044,0.2798)--(1.065,0.2938)--(1.086,0.3075)--(1.107,0.3210)--(1.128,0.3343)--(1.150,0.3472)--(1.171,0.3600)--(1.192,0.3725)--(1.213,0.3848)--(1.234,0.3969)--(1.255,0.4088)--(1.276,0.4204)--(1.297,0.4319)--(1.319,0.4432)--(1.340,0.4544)--(1.361,0.4653)--(1.382,0.4761)--(1.403,0.4867)--(1.424,0.4972)--(1.445,0.5075)--(1.466,0.5176)--(1.487,0.5276)--(1.509,0.5375)--(1.530,0.5472)--(1.551,0.5568)--(1.572,0.5663)--(1.593,0.5756)--(1.614,0.5849)--(1.635,0.5940)--(1.656,0.6030)--(1.678,0.6118)--(1.699,0.6206)--(1.720,0.6292)--(1.741,0.6378)--(1.762,0.6462)--(1.783,0.6546)--(1.804,0.6628)--(1.825,0.6709)--(1.847,0.6790)--(1.868,0.6870)--(1.889,0.6948)--(1.910,0.7026)--(1.931,0.7103)--(1.952,0.7179)--(1.973,0.7255)--(1.994,0.7329)--(2.016,0.7403)--(2.037,0.7476)--(2.058,0.7548)--(2.079,0.7620)--(2.100,0.7690);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (0.0090,-3.0466)--(0.0301,-2.2017)--(0.0512,-1.8299)--(0.0723,-1.5884)--(0.0934,-1.4092)--(0.1146,-1.2666)--(0.1357,-1.1482)--(0.1568,-1.0470)--(0.1779,-0.9585)--(0.1991,-0.8800)--(0.2202,-0.8095)--(0.2413,-0.7453)--(0.2624,-0.6866)--(0.2835,-0.6324)--(0.3047,-0.5822)--(0.3258,-0.5352)--(0.3469,-0.4913)--(0.3680,-0.4499)--(0.3891,-0.4109)--(0.4103,-0.3739)--(0.4314,-0.3387)--(0.4525,-0.3053)--(0.4736,-0.2733)--(0.4947,-0.2428)--(0.5159,-0.2135)--(0.5370,-0.1855)--(0.5581,-0.1585)--(0.5792,-0.1325)--(0.6004,-0.1074)--(0.6215,-0.0832)--(0.6426,-0.0598)--(0.6637,-0.0372)--(0.6848,-0.0152)--(0.7060,0.0059)--(0.7271,0.0266)--(0.7482,0.0466)--(0.7693,0.0661)--(0.7904,0.0851)--(0.8116,0.1035)--(0.8327,0.1215)--(0.8538,0.1390)--(0.8749,0.1561)--(0.8960,0.1728)--(0.9172,0.1891)--(0.9383,0.2051)--(0.9594,0.2207)--(0.9805,0.2359)--(1.0017,0.2508)--(1.0228,0.2654)--(1.0439,0.2797)--(1.0650,0.2937)--(1.0861,0.3075)--(1.1073,0.3210)--(1.1284,0.3342)--(1.1495,0.3472)--(1.1706,0.3599)--(1.1917,0.3724)--(1.2129,0.3847)--(1.2340,0.3968)--(1.2551,0.4087)--(1.2762,0.4204)--(1.2973,0.4319)--(1.3185,0.4432)--(1.3396,0.4543)--(1.3607,0.4653)--(1.3818,0.4760)--(1.4030,0.4867)--(1.4241,0.4971)--(1.4452,0.5074)--(1.4663,0.5176)--(1.4874,0.5276)--(1.5086,0.5375)--(1.5297,0.5472)--(1.5508,0.5568)--(1.5719,0.5663)--(1.5930,0.5756)--(1.6142,0.5848)--(1.6353,0.5939)--(1.6564,0.6029)--(1.6775,0.6118)--(1.6986,0.6205)--(1.7198,0.6292)--(1.7409,0.6377)--(1.7620,0.6462)--(1.7831,0.6545)--(1.8043,0.6627)--(1.8254,0.6709)--(1.8465,0.6789)--(1.8676,0.6869)--(1.8887,0.6948)--(1.9099,0.7026)--(1.9310,0.7103)--(1.9521,0.7179)--(1.9732,0.7254)--(1.9943,0.7329)--(2.0155,0.7402)--(2.0366,0.7475)--(2.0577,0.7548)--(2.0788,0.7619)--(2.1000,0.7690);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall b/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall
index 0ed1188bd..75c750781 100644
--- a/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall
+++ b/src_phystricks/Fig_DefinitionCartesiennes.pstricks.recall
@@ -103,47 +103,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-2.0000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
-\draw [color=blue,style=dashed] (3.00,1.00) -- (3.00,0);
-\draw [color=blue,style=dashed] (3.00,1.00) -- (0,1.00);
-\draw (3.5004,1.2141) node {$(3,1)$};
-\draw [color=blue,->,>=latex] (0,0) -- (3.0000,1.0000);
-\draw [color=green,style=dashed] (-1.50,-2.50) -- (-1.50,0);
-\draw [color=green,style=dashed] (-1.50,-2.50) -- (0,-2.50);
-\draw (-2.5247,-2.7682) node {$(-1.5,-2.5)$};
-\draw [color=green,->,>=latex] (0,0) -- (-1.5000,-2.5000);
-\draw [color=brown,style=dashed] (-1.00,2.50) -- (-1.00,0);
-\draw [color=brown,style=dashed] (-1.00,2.50) -- (0,2.50);
+\draw [color=blue,style=dashed] (3.0000,1.0000) -- (3.0000,0.0000);
+\draw [color=blue,style=dashed] (3.0000,1.0000) -- (0.0000,1.0000);
+\draw (3.5003,1.2140) node {$(3,1)$};
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (3.0000,1.0000);
+\draw [color=green,style=dashed] (-1.5000,-2.5000) -- (-1.5000,0.0000);
+\draw [color=green,style=dashed] (-1.5000,-2.5000) -- (0.0000,-2.5000);
+\draw (-2.5246,-2.7682) node {$(-1.5,-2.5)$};
+\draw [color=green,->,>=latex] (0.0000,0.0000) -- (-1.5000,-2.5000);
+\draw [color=brown,style=dashed] (-1.0000,2.5000) -- (-1.0000,0.0000);
+\draw [color=brown,style=dashed] (-1.0000,2.5000) -- (0.0000,2.5000);
 \draw (-1.7265,2.7753) node {$(-1,2.5)$};
-\draw [color=brown,->,>=latex] (0,0) -- (-1.0000,2.5000);
-\draw [color=cyan,style=dashed] (1.50,-1.00) -- (1.50,0);
-\draw [color=cyan,style=dashed] (1.50,-1.00) -- (0,-1.00);
-\draw (2.2726,-1.2379) node {$(1.5,-1)$};
-\draw [color=cyan,->,>=latex] (0,0) -- (1.5000,-1.0000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=brown,->,>=latex] (0.0000,0.0000) -- (-1.0000,2.5000);
+\draw [color=cyan,style=dashed] (1.5000,-1.0000) -- (1.5000,0.0000);
+\draw [color=cyan,style=dashed] (1.5000,-1.0000) -- (0.0000,-1.0000);
+\draw (2.2725,-1.2379) node {$(1.5,-1)$};
+\draw [color=cyan,->,>=latex] (0.0000,0.0000) -- (1.5000,-1.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_Differentielle.pstricks.recall b/src_phystricks/Fig_Differentielle.pstricks.recall
index 00d72fbeb..311716e53 100644
--- a/src_phystricks/Fig_Differentielle.pstricks.recall
+++ b/src_phystricks/Fig_Differentielle.pstricks.recall
@@ -91,36 +91,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.7000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.7000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.7000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.7000);
 %DEFAULT
-\draw [style=dotted] (2.00,2.00) -- (4.00,2.00);
-\draw [style=dotted] (4.00,2.00) -- (4.00,4.00);
+\draw [style=dotted] (2.0000,2.0000) -- (4.0000,2.0000);
+\draw [style=dotted] (4.0000,2.0000) -- (4.0000,4.0000);
 
-\draw [color=red] (1.000,1.000)--(1.032,1.032)--(1.065,1.065)--(1.097,1.097)--(1.129,1.129)--(1.162,1.162)--(1.194,1.194)--(1.226,1.226)--(1.259,1.259)--(1.291,1.291)--(1.323,1.323)--(1.356,1.356)--(1.388,1.388)--(1.420,1.420)--(1.453,1.453)--(1.485,1.485)--(1.517,1.517)--(1.549,1.549)--(1.582,1.582)--(1.614,1.614)--(1.646,1.646)--(1.679,1.679)--(1.711,1.711)--(1.743,1.743)--(1.776,1.776)--(1.808,1.808)--(1.840,1.840)--(1.873,1.873)--(1.905,1.905)--(1.937,1.937)--(1.970,1.970)--(2.002,2.002)--(2.034,2.034)--(2.067,2.067)--(2.099,2.099)--(2.131,2.131)--(2.164,2.164)--(2.196,2.196)--(2.228,2.228)--(2.261,2.261)--(2.293,2.293)--(2.325,2.325)--(2.358,2.358)--(2.390,2.390)--(2.422,2.422)--(2.455,2.455)--(2.487,2.487)--(2.519,2.519)--(2.552,2.552)--(2.584,2.584)--(2.616,2.616)--(2.648,2.648)--(2.681,2.681)--(2.713,2.713)--(2.745,2.745)--(2.778,2.778)--(2.810,2.810)--(2.842,2.842)--(2.875,2.875)--(2.907,2.907)--(2.939,2.939)--(2.972,2.972)--(3.004,3.004)--(3.036,3.036)--(3.069,3.069)--(3.101,3.101)--(3.133,3.133)--(3.166,3.166)--(3.198,3.198)--(3.230,3.230)--(3.263,3.263)--(3.295,3.295)--(3.327,3.327)--(3.360,3.360)--(3.392,3.392)--(3.424,3.424)--(3.457,3.457)--(3.489,3.489)--(3.521,3.521)--(3.554,3.554)--(3.586,3.586)--(3.618,3.618)--(3.651,3.651)--(3.683,3.683)--(3.715,3.715)--(3.747,3.747)--(3.780,3.780)--(3.812,3.812)--(3.844,3.844)--(3.877,3.877)--(3.909,3.909)--(3.941,3.941)--(3.974,3.974)--(4.006,4.006)--(4.038,4.038)--(4.071,4.071)--(4.103,4.103)--(4.135,4.135)--(4.168,4.168)--(4.200,4.200);
+\draw [color=red] (1.0000,1.0000)--(1.0323,1.0323)--(1.0646,1.0646)--(1.0969,1.0969)--(1.1292,1.1292)--(1.1616,1.1616)--(1.1939,1.1939)--(1.2262,1.2262)--(1.2585,1.2585)--(1.2909,1.2909)--(1.3232,1.3232)--(1.3555,1.3555)--(1.3878,1.3878)--(1.4202,1.4202)--(1.4525,1.4525)--(1.4848,1.4848)--(1.5171,1.5171)--(1.5494,1.5494)--(1.5818,1.5818)--(1.6141,1.6141)--(1.6464,1.6464)--(1.6787,1.6787)--(1.7111,1.7111)--(1.7434,1.7434)--(1.7757,1.7757)--(1.8080,1.8080)--(1.8404,1.8404)--(1.8727,1.8727)--(1.9050,1.9050)--(1.9373,1.9373)--(1.9696,1.9696)--(2.0020,2.0020)--(2.0343,2.0343)--(2.0666,2.0666)--(2.0989,2.0989)--(2.1313,2.1313)--(2.1636,2.1636)--(2.1959,2.1959)--(2.2282,2.2282)--(2.2606,2.2606)--(2.2929,2.2929)--(2.3252,2.3252)--(2.3575,2.3575)--(2.3898,2.3898)--(2.4222,2.4222)--(2.4545,2.4545)--(2.4868,2.4868)--(2.5191,2.5191)--(2.5515,2.5515)--(2.5838,2.5838)--(2.6161,2.6161)--(2.6484,2.6484)--(2.6808,2.6808)--(2.7131,2.7131)--(2.7454,2.7454)--(2.7777,2.7777)--(2.8101,2.8101)--(2.8424,2.8424)--(2.8747,2.8747)--(2.9070,2.9070)--(2.9393,2.9393)--(2.9717,2.9717)--(3.0040,3.0040)--(3.0363,3.0363)--(3.0686,3.0686)--(3.1010,3.1010)--(3.1333,3.1333)--(3.1656,3.1656)--(3.1979,3.1979)--(3.2303,3.2303)--(3.2626,3.2626)--(3.2949,3.2949)--(3.3272,3.3272)--(3.3595,3.3595)--(3.3919,3.3919)--(3.4242,3.4242)--(3.4565,3.4565)--(3.4888,3.4888)--(3.5212,3.5212)--(3.5535,3.5535)--(3.5858,3.5858)--(3.6181,3.6181)--(3.6505,3.6505)--(3.6828,3.6828)--(3.7151,3.7151)--(3.7474,3.7474)--(3.7797,3.7797)--(3.8121,3.8121)--(3.8444,3.8444)--(3.8767,3.8767)--(3.9090,3.9090)--(3.9414,3.9414)--(3.9737,3.9737)--(4.0060,4.0060)--(4.0383,4.0383)--(4.0707,4.0707)--(4.1030,4.1030)--(4.1353,4.1353)--(4.1676,4.1676)--(4.2000,4.2000);
 
-\draw [color=blue] (1.000,0.6137)--(1.032,0.6773)--(1.065,0.7390)--(1.097,0.7988)--(1.129,0.8569)--(1.162,0.9133)--(1.194,0.9682)--(1.226,1.022)--(1.259,1.074)--(1.291,1.124)--(1.323,1.174)--(1.356,1.222)--(1.388,1.269)--(1.420,1.315)--(1.453,1.360)--(1.485,1.404)--(1.517,1.447)--(1.549,1.490)--(1.582,1.531)--(1.614,1.571)--(1.646,1.611)--(1.679,1.650)--(1.711,1.688)--(1.743,1.725)--(1.776,1.762)--(1.808,1.798)--(1.840,1.834)--(1.873,1.868)--(1.905,1.903)--(1.937,1.936)--(1.970,1.969)--(2.002,2.002)--(2.034,2.034)--(2.067,2.066)--(2.099,2.097)--(2.131,2.127)--(2.164,2.157)--(2.196,2.187)--(2.228,2.216)--(2.261,2.245)--(2.293,2.273)--(2.325,2.301)--(2.358,2.329)--(2.390,2.356)--(2.422,2.383)--(2.455,2.410)--(2.487,2.436)--(2.519,2.462)--(2.552,2.487)--(2.584,2.512)--(2.616,2.537)--(2.648,2.562)--(2.681,2.586)--(2.713,2.610)--(2.745,2.634)--(2.778,2.657)--(2.810,2.680)--(2.842,2.703)--(2.875,2.726)--(2.907,2.748)--(2.939,2.770)--(2.972,2.792)--(3.004,2.814)--(3.036,2.835)--(3.069,2.856)--(3.101,2.877)--(3.133,2.898)--(3.166,2.918)--(3.198,2.939)--(3.230,2.959)--(3.263,2.979)--(3.295,2.998)--(3.327,3.018)--(3.360,3.037)--(3.392,3.056)--(3.424,3.075)--(3.457,3.094)--(3.489,3.113)--(3.521,3.131)--(3.554,3.150)--(3.586,3.168)--(3.618,3.186)--(3.651,3.203)--(3.683,3.221)--(3.715,3.239)--(3.747,3.256)--(3.780,3.273)--(3.812,3.290)--(3.844,3.307)--(3.877,3.324)--(3.909,3.340)--(3.941,3.357)--(3.974,3.373)--(4.006,3.389)--(4.038,3.405)--(4.071,3.421)--(4.103,3.437)--(4.135,3.453)--(4.168,3.468)--(4.200,3.484);
-\draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,3.3863);
+\draw [color=blue] (1.0000,0.6137)--(1.0323,0.6773)--(1.0646,0.7389)--(1.0969,0.7988)--(1.1292,0.8568)--(1.1616,0.9133)--(1.1939,0.9682)--(1.2262,1.0216)--(1.2585,1.0736)--(1.2909,1.1243)--(1.3232,1.1738)--(1.3555,1.2221)--(1.3878,1.2692)--(1.4202,1.3153)--(1.4525,1.3603)--(1.4848,1.4043)--(1.5171,1.4474)--(1.5494,1.4895)--(1.5818,1.5308)--(1.6141,1.5713)--(1.6464,1.6109)--(1.6787,1.6498)--(1.7111,1.6879)--(1.7434,1.7254)--(1.7757,1.7621)--(1.8080,1.7982)--(1.8404,1.8336)--(1.8727,1.8684)--(1.9050,1.9027)--(1.9373,1.9363)--(1.9696,1.9694)--(2.0020,2.0020)--(2.0343,2.0340)--(2.0666,2.0655)--(2.0989,2.0966)--(2.1313,2.1271)--(2.1636,2.1572)--(2.1959,2.1869)--(2.2282,2.2161)--(2.2606,2.2449)--(2.2929,2.2733)--(2.3252,2.3013)--(2.3575,2.3289)--(2.3898,2.3562)--(2.4222,2.3830)--(2.4545,2.4095)--(2.4868,2.4357)--(2.5191,2.4615)--(2.5515,2.4870)--(2.5838,2.5122)--(2.6161,2.5371)--(2.6484,2.5616)--(2.6808,2.5859)--(2.7131,2.6099)--(2.7454,2.6335)--(2.7777,2.6570)--(2.8101,2.6801)--(2.8424,2.7030)--(2.8747,2.7256)--(2.9070,2.7479)--(2.9393,2.7701)--(2.9717,2.7919)--(3.0040,2.8136)--(3.0363,2.8350)--(3.0686,2.8562)--(3.1010,2.8771)--(3.1333,2.8979)--(3.1656,2.9184)--(3.1979,2.9387)--(3.2303,2.9588)--(3.2626,2.9787)--(3.2949,2.9984)--(3.3272,3.0180)--(3.3595,3.0373)--(3.3919,3.0564)--(3.4242,3.0754)--(3.4565,3.0942)--(3.4888,3.1128)--(3.5212,3.1313)--(3.5535,3.1495)--(3.5858,3.1677)--(3.6181,3.1856)--(3.6505,3.2034)--(3.6828,3.2210)--(3.7151,3.2385)--(3.7474,3.2558)--(3.7797,3.2730)--(3.8121,3.2900)--(3.8444,3.3069)--(3.8767,3.3237)--(3.9090,3.3403)--(3.9414,3.3567)--(3.9737,3.3731)--(4.0060,3.3893)--(4.0383,3.4053)--(4.0707,3.4213)--(4.1030,3.4371)--(4.1353,3.4528)--(4.1676,3.4684)--(4.2000,3.4838);
+\draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,3.3862);
 \draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,4.0000);
 \draw (5.1211,3.6931) node {$\epsilon(h)$};
 \draw [,->,>=latex] (5.5000,3.0000) -- (5.5000,2.0000);
 \draw [,->,>=latex] (5.5000,3.0000) -- (5.5000,4.0000);
-\draw (6.3791,3.0000) node {$T(h)$};
+\draw (6.3790,3.0000) node {$T(h)$};
 \draw [,->,>=latex] (3.0000,1.5000) -- (2.0000,1.5000);
 \draw [,->,>=latex] (3.0000,1.5000) -- (4.0000,1.5000);
-\draw (3.0000,1.0733) node {$h$};
+\draw (3.0000,1.0732) node {$h$};
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (1.2991,2.5360) node {$f(a)$};
+\draw (1.2990,2.5360) node {$f(a)$};
 \draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,3.3863) node [rotate=0] {$\bullet$};
-\draw (4.8502,2.7089) node {$f(x)$};
+\draw []  (4.0000,3.3862) node [rotate=0] {$\bullet$};
+\draw (4.8501,2.7088) node {$f(x)$};
 \draw []  (4.0000,4.0000) node [rotate=0] {$\bullet$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.37858) node {$a$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.37858) node {$x$};
-\draw [style=dotted] (2.00,2.00) -- (2.00,0);
-\draw [style=dotted] (4.00,2.00) -- (4.00,0);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3785) node {$a$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.3785) node {$x$};
+\draw [style=dotted] (2.0000,2.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (4.0000,2.0000) -- (4.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DistanceEuclide.pstricks.recall b/src_phystricks/Fig_DistanceEuclide.pstricks.recall
index de1bd21a0..c4015af51 100644
--- a/src_phystricks/Fig_DistanceEuclide.pstricks.recall
+++ b/src_phystricks/Fig_DistanceEuclide.pstricks.recall
@@ -91,32 +91,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
 \draw []  (1.0000,4.0000) node [rotate=0] {$\bullet$};
 \draw (1.0000,4.4901) node {$(A_x,A_y)$};
 \draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.9711,1.0000) node {$(B_x,B_y)$};
+\draw (3.9710,1.0000) node {$(B_x,B_y)$};
 \draw []  (3.0000,4.0000) node [rotate=0] {$\bullet$};
 \draw (3.3556,4.3368) node {$C$};
-\draw [] (1.00,4.00) -- (3.00,1.00);
-\draw [] (1.00,4.00) -- (3.00,4.00);
-\draw [] (3.00,1.00) -- (3.00,4.00);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [] (1.0000,4.0000) -- (3.0000,1.0000);
+\draw [] (1.0000,4.0000) -- (3.0000,4.0000);
+\draw [] (3.0000,1.0000) -- (3.0000,4.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DivergenceDeux.pstricks.recall b/src_phystricks/Fig_DivergenceDeux.pstricks.recall
index 1b692ca44..86dad7615 100644
--- a/src_phystricks/Fig_DivergenceDeux.pstricks.recall
+++ b/src_phystricks/Fig_DivergenceDeux.pstricks.recall
@@ -65,230 +65,230 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.0000,-4.0000) -- (-4.2357,-3.7643);
-\draw [,->,>=latex] (-4.0000,-3.4286) -- (-4.2169,-3.1755);
-\draw [,->,>=latex] (-4.0000,-2.8571) -- (-4.1937,-2.5859);
-\draw [,->,>=latex] (-4.0000,-2.2857) -- (-4.1654,-1.9963);
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diff --git a/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall b/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall
index 40c268f53..d26af81de 100644
--- a/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall
+++ b/src_phystricks/Fig_DynkinNUtPJx.pstricks.recall
@@ -39,17 +39,17 @@
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-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.31492) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3149) node {\(1\)};
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 \end{tikzpicture}
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@@ -84,17 +84,17 @@
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-\draw [] (0,0) -- (1.00,0);
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-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.31492) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
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-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.31492) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.3149) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -174,17 +174,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw (3.2327,-0.24365) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw (3.2326,-0.2436) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -219,17 +219,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw (3.2327,0.75635) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw (3.2326,0.7563) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall b/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall
index aad544245..10a89f5ff 100644
--- a/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall
+++ b/src_phystricks/Fig_DynkinpWjUbE.pstricks.recall
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.26444) node {\( 1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2644) node {\( 1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall b/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall
index 569d4db5b..9dd3d2fc7 100644
--- a/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall
+++ b/src_phystricks/Fig_DynkinqlgIQl.pstricks.recall
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.26444) node {\( 2\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2644) node {\( 2\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall b/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall
index de90bcf5a..61a64d0bf 100644
--- a/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall
+++ b/src_phystricks/Fig_DynkinrjbHIu.pstricks.recall
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.26444) node {\( 1\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2644) node {\( 1\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_EJRsWXw.pstricks.recall b/src_phystricks/Fig_EJRsWXw.pstricks.recall
index c26000343..552d43246 100644
--- a/src_phystricks/Fig_EJRsWXw.pstricks.recall
+++ b/src_phystricks/Fig_EJRsWXw.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,17 +81,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-1.00,1.00) -- (-2.00,2.00) -- (-2.00,2.00) -- (1.00,3.00) -- (1.00,3.00) -- (-1.00,1.00) --  cycle;
-\draw [] (-1.00,1.00) -- (-2.00,2.00);
-\draw [] (-2.00,2.00) -- (1.00,3.00);
-\draw [] (1.00,3.00) -- (-1.00,1.00);
-\draw [color=green] (-1.00,1.00) -- (-2.00,2.00);
-\draw [color=green] (-1.00,1.00) -- (1.00,3.00);
-\draw [color=red] (-2.00,2.00) -- (1.00,3.00);
-\draw (1.5000,-0.27858) node {\( x\)};
-\draw (1.5000,-0.27858) node {\( x\)};
-\draw (0.26590,3.5000) node {\( t\)};
-\draw (0.26590,3.5000) node {\( t\)};
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-1.0000,1.0000) -- (-2.0000,2.0000) -- (-2.0000,2.0000) -- (1.0000,3.0000) -- (1.0000,3.0000) -- (-1.0000,1.0000) --  cycle;
+\draw [] (-1.0000,1.0000) -- (-2.0000,2.0000);
+\draw [] (-2.0000,2.0000) -- (1.0000,3.0000);
+\draw [] (1.0000,3.0000) -- (-1.0000,1.0000);
+\draw [color=green] (-1.0000,1.0000) -- (-2.0000,2.0000);
+\draw [color=green] (-1.0000,1.0000) -- (1.0000,3.0000);
+\draw [color=red] (-2.0000,2.0000) -- (1.0000,3.0000);
+\draw (1.5000,-0.2785) node {\( x\)};
+\draw (1.5000,-0.2785) node {\( x\)};
+\draw (0.2659,3.5000) node {\( t\)};
+\draw (0.2659,3.5000) node {\( t\)};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExPolygone.pstricks.recall b/src_phystricks/Fig_ExPolygone.pstricks.recall
index c729554c1..fec2fe5fb 100644
--- a/src_phystricks/Fig_ExPolygone.pstricks.recall
+++ b/src_phystricks/Fig_ExPolygone.pstricks.recall
@@ -83,13 +83,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
-\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
+\draw [,->,>=latex] (-1.6500,0.0000) -- (3.6500,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6500) -- (0.0000,2.6500);
 %DEFAULT
-\draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
-\draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
-\draw [color=red] (2.15,-2.15) -- (-1.15,1.15);
-\draw [color=red] (-1.15,-1.15) -- (2.15,2.15);
+\draw [color=red] (-0.1500,2.1500) -- (3.1500,-1.1500);
+\draw [color=red] (3.1500,1.1500) -- (-0.1500,-2.1500);
+\draw [color=red] (2.1500,-2.1500) -- (-1.1500,1.1500);
+\draw [color=red] (-1.1500,-1.1500) -- (2.1500,2.1500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,27 +97,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,1.00) -- (2.00,0) -- (2.00,0) -- (1.00,-1.00) -- (1.00,-1.00) -- (0,0) -- (0,0) -- (1.00,1.00) --  cycle;
-\draw [color=blue] (1.00,1.00) -- (2.00,0);
-\draw [color=blue] (2.00,0) -- (1.00,-1.00);
-\draw [color=blue] (1.00,-1.00) -- (0,0);
-\draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,1.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (1.0000,-1.0000) -- (1.0000,-1.0000) -- (0.0000,0.0000) -- (0.0000,0.0000) -- (1.0000,1.0000) --  cycle;
+\draw [color=blue] (1.0000,1.0000) -- (2.0000,0.0000);
+\draw [color=blue] (2.0000,0.0000) -- (1.0000,-1.0000);
+\draw [color=blue] (1.0000,-1.0000) -- (0.0000,0.0000);
+\draw [color=blue] (0.0000,0.0000) -- (1.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExempleNonRang.pstricks.recall b/src_phystricks/Fig_ExempleNonRang.pstricks.recall
index 89185f4ea..b0f5de877 100644
--- a/src_phystricks/Fig_ExempleNonRang.pstricks.recall
+++ b/src_phystricks/Fig_ExempleNonRang.pstricks.recall
@@ -63,13 +63,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.3284) -- (0,3.3284);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.3284) -- (0.0000,3.3284);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.04040,0.002871)--(0.08081,0.008121)--(0.1212,0.01492)--(0.1616,0.02297)--(0.2020,0.03210)--(0.2424,0.04220)--(0.2828,0.05318)--(0.3232,0.06497)--(0.3636,0.07753)--(0.4040,0.09080)--(0.4444,0.1048)--(0.4848,0.1194)--(0.5253,0.1346)--(0.5657,0.1504)--(0.6061,0.1668)--(0.6465,0.1838)--(0.6869,0.2013)--(0.7273,0.2193)--(0.7677,0.2378)--(0.8081,0.2568)--(0.8485,0.2763)--(0.8889,0.2963)--(0.9293,0.3167)--(0.9697,0.3376)--(1.010,0.3589)--(1.051,0.3807)--(1.091,0.4028)--(1.131,0.4254)--(1.172,0.4484)--(1.212,0.4718)--(1.253,0.4956)--(1.293,0.5198)--(1.333,0.5443)--(1.374,0.5693)--(1.414,0.5946)--(1.455,0.6202)--(1.495,0.6462)--(1.535,0.6726)--(1.576,0.6993)--(1.616,0.7264)--(1.657,0.7538)--(1.697,0.7816)--(1.737,0.8096)--(1.778,0.8381)--(1.818,0.8668)--(1.859,0.8958)--(1.899,0.9252)--(1.939,0.9549)--(1.980,0.9849)--(2.020,1.015)--(2.061,1.046)--(2.101,1.077)--(2.141,1.108)--(2.182,1.139)--(2.222,1.171)--(2.263,1.203)--(2.303,1.236)--(2.343,1.268)--(2.384,1.301)--(2.424,1.335)--(2.465,1.368)--(2.505,1.402)--(2.545,1.436)--(2.586,1.470)--(2.626,1.505)--(2.667,1.540)--(2.707,1.575)--(2.747,1.610)--(2.788,1.646)--(2.828,1.682)--(2.869,1.718)--(2.909,1.754)--(2.949,1.791)--(2.990,1.828)--(3.030,1.865)--(3.071,1.902)--(3.111,1.940)--(3.152,1.978)--(3.192,2.016)--(3.232,2.055)--(3.273,2.093)--(3.313,2.132)--(3.354,2.171)--(3.394,2.211)--(3.434,2.250)--(3.475,2.290)--(3.515,2.330)--(3.556,2.370)--(3.596,2.411)--(3.636,2.452)--(3.677,2.493)--(3.717,2.534)--(3.758,2.575)--(3.798,2.617)--(3.838,2.659)--(3.879,2.701)--(3.919,2.743)--(3.960,2.786)--(4.000,2.828);
+\draw [color=blue] (0.0000,0.0000)--(0.0404,0.0028)--(0.0808,0.0081)--(0.1212,0.0149)--(0.1616,0.0229)--(0.2020,0.0321)--(0.2424,0.0422)--(0.2828,0.0531)--(0.3232,0.0649)--(0.3636,0.0775)--(0.4040,0.0908)--(0.4444,0.1047)--(0.4848,0.1193)--(0.5252,0.1345)--(0.5656,0.1504)--(0.6060,0.1668)--(0.6464,0.1837)--(0.6868,0.2012)--(0.7272,0.2192)--(0.7676,0.2378)--(0.8080,0.2568)--(0.8484,0.2763)--(0.8888,0.2962)--(0.9292,0.3167)--(0.9696,0.3376)--(1.0101,0.3589)--(1.0505,0.3806)--(1.0909,0.4028)--(1.1313,0.4254)--(1.1717,0.4484)--(1.2121,0.4718)--(1.2525,0.4956)--(1.2929,0.5197)--(1.3333,0.5443)--(1.3737,0.5692)--(1.4141,0.5945)--(1.4545,0.6202)--(1.4949,0.6462)--(1.5353,0.6726)--(1.5757,0.6993)--(1.6161,0.7264)--(1.6565,0.7538)--(1.6969,0.7815)--(1.7373,0.8096)--(1.7777,0.8380)--(1.8181,0.8667)--(1.8585,0.8958)--(1.8989,0.9252)--(1.9393,0.9548)--(1.9797,0.9848)--(2.0202,1.0151)--(2.0606,1.0457)--(2.1010,1.0767)--(2.1414,1.1079)--(2.1818,1.1394)--(2.2222,1.1712)--(2.2626,1.2033)--(2.3030,1.2356)--(2.3434,1.2683)--(2.3838,1.3012)--(2.4242,1.3345)--(2.4646,1.3680)--(2.5050,1.4017)--(2.5454,1.4358)--(2.5858,1.4701)--(2.6262,1.5047)--(2.6666,1.5396)--(2.7070,1.5747)--(2.7474,1.6101)--(2.7878,1.6457)--(2.8282,1.6816)--(2.8686,1.7178)--(2.9090,1.7542)--(2.9494,1.7909)--(2.9898,1.8278)--(3.0303,1.8650)--(3.0707,1.9024)--(3.1111,1.9401)--(3.1515,1.9780)--(3.1919,2.0161)--(3.2323,2.0545)--(3.2727,2.0932)--(3.3131,2.1321)--(3.3535,2.1712)--(3.3939,2.2106)--(3.4343,2.2501)--(3.4747,2.2900)--(3.5151,2.3300)--(3.5555,2.3703)--(3.5959,2.4108)--(3.6363,2.4516)--(3.6767,2.4926)--(3.7171,2.5338)--(3.7575,2.5752)--(3.7979,2.6168)--(3.8383,2.6587)--(3.8787,2.7008)--(3.9191,2.7431)--(3.9595,2.7856)--(4.0000,2.8284);
 
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+\draw [color=blue] (0.0000,0.0000)--(0.0404,-0.0028)--(0.0808,-0.0081)--(0.1212,-0.0149)--(0.1616,-0.0229)--(0.2020,-0.0321)--(0.2424,-0.0422)--(0.2828,-0.0531)--(0.3232,-0.0649)--(0.3636,-0.0775)--(0.4040,-0.0908)--(0.4444,-0.1047)--(0.4848,-0.1193)--(0.5252,-0.1345)--(0.5656,-0.1504)--(0.6060,-0.1668)--(0.6464,-0.1837)--(0.6868,-0.2012)--(0.7272,-0.2192)--(0.7676,-0.2378)--(0.8080,-0.2568)--(0.8484,-0.2763)--(0.8888,-0.2962)--(0.9292,-0.3167)--(0.9696,-0.3376)--(1.0101,-0.3589)--(1.0505,-0.3806)--(1.0909,-0.4028)--(1.1313,-0.4254)--(1.1717,-0.4484)--(1.2121,-0.4718)--(1.2525,-0.4956)--(1.2929,-0.5197)--(1.3333,-0.5443)--(1.3737,-0.5692)--(1.4141,-0.5945)--(1.4545,-0.6202)--(1.4949,-0.6462)--(1.5353,-0.6726)--(1.5757,-0.6993)--(1.6161,-0.7264)--(1.6565,-0.7538)--(1.6969,-0.7815)--(1.7373,-0.8096)--(1.7777,-0.8380)--(1.8181,-0.8667)--(1.8585,-0.8958)--(1.8989,-0.9252)--(1.9393,-0.9548)--(1.9797,-0.9848)--(2.0202,-1.0151)--(2.0606,-1.0457)--(2.1010,-1.0767)--(2.1414,-1.1079)--(2.1818,-1.1394)--(2.2222,-1.1712)--(2.2626,-1.2033)--(2.3030,-1.2356)--(2.3434,-1.2683)--(2.3838,-1.3012)--(2.4242,-1.3345)--(2.4646,-1.3680)--(2.5050,-1.4017)--(2.5454,-1.4358)--(2.5858,-1.4701)--(2.6262,-1.5047)--(2.6666,-1.5396)--(2.7070,-1.5747)--(2.7474,-1.6101)--(2.7878,-1.6457)--(2.8282,-1.6816)--(2.8686,-1.7178)--(2.9090,-1.7542)--(2.9494,-1.7909)--(2.9898,-1.8278)--(3.0303,-1.8650)--(3.0707,-1.9024)--(3.1111,-1.9401)--(3.1515,-1.9780)--(3.1919,-2.0161)--(3.2323,-2.0545)--(3.2727,-2.0932)--(3.3131,-2.1321)--(3.3535,-2.1712)--(3.3939,-2.2106)--(3.4343,-2.2501)--(3.4747,-2.2900)--(3.5151,-2.3300)--(3.5555,-2.3703)--(3.5959,-2.4108)--(3.6363,-2.4516)--(3.6767,-2.4926)--(3.7171,-2.5338)--(3.7575,-2.5752)--(3.7979,-2.6168)--(3.8383,-2.6587)--(3.8787,-2.7008)--(3.9191,-2.7431)--(3.9595,-2.7856)--(4.0000,-2.8284);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
index 637a3a657..8571a0820 100644
--- a/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
+++ b/src_phystricks/Fig_ExerciceGraphesbis.pstricks.recall
@@ -65,31 +65,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.2000) -- (0,1.1996);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2000) -- (0.0000,1.1996);
 %DEFAULT
 
-\draw [color=blue] (-2.199,-0.7000)--(-2.144,-0.6978)--(-2.088,-0.6912)--(-2.033,-0.6803)--(-1.977,-0.6650)--(-1.921,-0.6456)--(-1.866,-0.6222)--(-1.810,-0.5948)--(-1.755,-0.5637)--(-1.699,-0.5290)--(-1.644,-0.4910)--(-1.588,-0.4500)--(-1.533,-0.4060)--(-1.477,-0.3596)--(-1.422,-0.3108)--(-1.366,-0.2602)--(-1.311,-0.2078)--(-1.255,-0.1542)--(-1.200,-0.09962)--(-1.144,-0.04440)--(-1.088,0.01111)--(-1.033,0.06654)--(-0.9774,0.1216)--(-0.9219,0.1758)--(-0.8663,0.2289)--(-0.8108,0.2807)--(-0.7552,0.3306)--(-0.6997,0.3784)--(-0.6442,0.4239)--(-0.5887,0.4667)--(-0.5331,0.5066)--(-0.4776,0.5433)--(-0.4221,0.5766)--(-0.3665,0.6062)--(-0.3110,0.6320)--(-0.2555,0.6539)--(-0.1999,0.6716)--(-0.1444,0.6852)--(-0.08885,0.6944)--(-0.03332,0.6992)--(0.02221,0.6996)--(0.07775,0.6957)--(0.1333,0.6873)--(0.1888,0.6747)--(0.2443,0.6578)--(0.2999,0.6367)--(0.3554,0.6117)--(0.4109,0.5828)--(0.4665,0.5502)--(0.5220,0.5142)--(0.5775,0.4750)--(0.6331,0.4327)--(0.6886,0.3877)--(0.7441,0.3403)--(0.7997,0.2908)--(0.8552,0.2394)--(0.9107,0.1865)--(0.9663,0.1325)--(1.022,0.07759)--(1.077,0.02221)--(1.133,-0.03331)--(1.188,-0.08861)--(1.244,-0.1434)--(1.299,-0.1972)--(1.355,-0.2498)--(1.411,-0.3009)--(1.466,-0.3500)--(1.522,-0.3969)--(1.577,-0.4414)--(1.633,-0.4831)--(1.688,-0.5217)--(1.744,-0.5570)--(1.799,-0.5889)--(1.855,-0.6170)--(1.910,-0.6413)--(1.966,-0.6615)--(2.021,-0.6776)--(2.077,-0.6894)--(2.132,-0.6968)--(2.188,-0.6999)--(2.244,-0.6986)--(2.299,-0.6929)--(2.355,-0.6828)--(2.410,-0.6684)--(2.466,-0.6499)--(2.521,-0.6272)--(2.577,-0.6006)--(2.632,-0.5702)--(2.688,-0.5362)--(2.743,-0.4989)--(2.799,-0.4584)--(2.854,-0.4150)--(2.910,-0.3691)--(2.965,-0.3208)--(3.021,-0.2704)--(3.077,-0.2184)--(3.132,-0.1650)--(3.188,-0.1106)--(3.243,-0.05547)--(3.299,0);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-2.1991,-0.7000)--(-2.1435,-0.6977)--(-2.0880,-0.6912)--(-2.0325,-0.6802)--(-1.9769,-0.6650)--(-1.9214,-0.6456)--(-1.8659,-0.6221)--(-1.8103,-0.5948)--(-1.7548,-0.5636)--(-1.6993,-0.5290)--(-1.6437,-0.4910)--(-1.5882,-0.4499)--(-1.5327,-0.4060)--(-1.4771,-0.3595)--(-1.4216,-0.3108)--(-1.3661,-0.2601)--(-1.3105,-0.2078)--(-1.2550,-0.1542)--(-1.1995,-0.0996)--(-1.1439,-0.0443)--(-1.0884,0.0111)--(-1.0329,0.0665)--(-0.9773,0.1215)--(-0.9218,0.1758)--(-0.8663,0.2289)--(-0.8107,0.2806)--(-0.7552,0.3305)--(-0.6997,0.3784)--(-0.6441,0.4239)--(-0.5886,0.4667)--(-0.5331,0.5066)--(-0.4775,0.5433)--(-0.4220,0.5765)--(-0.3665,0.6062)--(-0.3109,0.6320)--(-0.2554,0.6539)--(-0.1999,0.6716)--(-0.1443,0.6851)--(-0.0888,0.6943)--(-0.0333,0.6992)--(0.0222,0.6996)--(0.0777,0.6956)--(0.1332,0.6873)--(0.1888,0.6746)--(0.2443,0.6577)--(0.2998,0.6367)--(0.3554,0.6116)--(0.4109,0.5827)--(0.4664,0.5502)--(0.5220,0.5142)--(0.5775,0.4749)--(0.6330,0.4327)--(0.6886,0.3877)--(0.7441,0.3403)--(0.7996,0.2907)--(0.8552,0.2394)--(0.9107,0.1865)--(0.9662,0.1324)--(1.0218,0.0775)--(1.0773,0.0222)--(1.1328,-0.0333)--(1.1884,-0.0886)--(1.2439,-0.1433)--(1.2994,-0.1972)--(1.3550,-0.2498)--(1.4105,-0.3008)--(1.4660,-0.3500)--(1.5216,-0.3969)--(1.5771,-0.4413)--(1.6326,-0.4830)--(1.6882,-0.5216)--(1.7437,-0.5570)--(1.7992,-0.5888)--(1.8548,-0.6170)--(1.9103,-0.6412)--(1.9658,-0.6615)--(2.0214,-0.6775)--(2.0769,-0.6893)--(2.1324,-0.6968)--(2.1880,-0.6999)--(2.2435,-0.6985)--(2.2990,-0.6928)--(2.3546,-0.6828)--(2.4101,-0.6684)--(2.4656,-0.6498)--(2.5212,-0.6271)--(2.5767,-0.6005)--(2.6322,-0.5702)--(2.6878,-0.5362)--(2.7433,-0.4988)--(2.7988,-0.4584)--(2.8544,-0.4150)--(2.9099,-0.3690)--(2.9654,-0.3207)--(3.0210,-0.2704)--(3.0765,-0.2184)--(3.1320,-0.1650)--(3.1876,-0.1106)--(3.2431,-0.0554)--(3.2986,0.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -151,29 +151,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0.7000)--(-2.144,0.6978)--(-2.088,0.6912)--(-2.033,0.6803)--(-1.977,0.6650)--(-1.921,0.6456)--(-1.866,0.6222)--(-1.810,0.5948)--(-1.755,0.5637)--(-1.699,0.5290)--(-1.644,0.4910)--(-1.588,0.4500)--(-1.533,0.4060)--(-1.477,0.3596)--(-1.422,0.3108)--(-1.366,0.2602)--(-1.311,0.2078)--(-1.255,0.1542)--(-1.200,0.09962)--(-1.144,0.04440)--(-1.088,0.01111)--(-1.033,0.06654)--(-0.9774,0.1216)--(-0.9219,0.1758)--(-0.8663,0.2289)--(-0.8108,0.2807)--(-0.7552,0.3306)--(-0.6997,0.3784)--(-0.6442,0.4239)--(-0.5887,0.4667)--(-0.5331,0.5066)--(-0.4776,0.5433)--(-0.4221,0.5766)--(-0.3665,0.6062)--(-0.3110,0.6320)--(-0.2555,0.6539)--(-0.1999,0.6716)--(-0.1444,0.6852)--(-0.08885,0.6944)--(-0.03332,0.6992)--(0.02221,0.6996)--(0.07775,0.6957)--(0.1333,0.6873)--(0.1888,0.6747)--(0.2443,0.6578)--(0.2999,0.6367)--(0.3554,0.6117)--(0.4109,0.5828)--(0.4665,0.5502)--(0.5220,0.5142)--(0.5775,0.4750)--(0.6331,0.4327)--(0.6886,0.3877)--(0.7441,0.3403)--(0.7997,0.2908)--(0.8552,0.2394)--(0.9107,0.1865)--(0.9663,0.1325)--(1.022,0.07759)--(1.077,0.02221)--(1.133,0.03331)--(1.188,0.08861)--(1.244,0.1434)--(1.299,0.1972)--(1.355,0.2498)--(1.411,0.3009)--(1.466,0.3500)--(1.522,0.3969)--(1.577,0.4414)--(1.633,0.4831)--(1.688,0.5217)--(1.744,0.5570)--(1.799,0.5889)--(1.855,0.6170)--(1.910,0.6413)--(1.966,0.6615)--(2.021,0.6776)--(2.077,0.6894)--(2.132,0.6968)--(2.188,0.6999)--(2.244,0.6986)--(2.299,0.6929)--(2.355,0.6828)--(2.410,0.6684)--(2.466,0.6499)--(2.521,0.6272)--(2.577,0.6006)--(2.632,0.5702)--(2.688,0.5362)--(2.743,0.4989)--(2.799,0.4584)--(2.854,0.4150)--(2.910,0.3691)--(2.965,0.3208)--(3.021,0.2704)--(3.077,0.2184)--(3.132,0.1650)--(3.188,0.1106)--(3.243,0.05547)--(3.299,0);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-2.1991,0.7000)--(-2.1435,0.6977)--(-2.0880,0.6912)--(-2.0325,0.6802)--(-1.9769,0.6650)--(-1.9214,0.6456)--(-1.8659,0.6221)--(-1.8103,0.5948)--(-1.7548,0.5636)--(-1.6993,0.5290)--(-1.6437,0.4910)--(-1.5882,0.4499)--(-1.5327,0.4060)--(-1.4771,0.3595)--(-1.4216,0.3108)--(-1.3661,0.2601)--(-1.3105,0.2078)--(-1.2550,0.1542)--(-1.1995,0.0996)--(-1.1439,0.0443)--(-1.0884,0.0111)--(-1.0329,0.0665)--(-0.9773,0.1215)--(-0.9218,0.1758)--(-0.8663,0.2289)--(-0.8107,0.2806)--(-0.7552,0.3305)--(-0.6997,0.3784)--(-0.6441,0.4239)--(-0.5886,0.4667)--(-0.5331,0.5066)--(-0.4775,0.5433)--(-0.4220,0.5765)--(-0.3665,0.6062)--(-0.3109,0.6320)--(-0.2554,0.6539)--(-0.1999,0.6716)--(-0.1443,0.6851)--(-0.0888,0.6943)--(-0.0333,0.6992)--(0.0222,0.6996)--(0.0777,0.6956)--(0.1332,0.6873)--(0.1888,0.6746)--(0.2443,0.6577)--(0.2998,0.6367)--(0.3554,0.6116)--(0.4109,0.5827)--(0.4664,0.5502)--(0.5220,0.5142)--(0.5775,0.4749)--(0.6330,0.4327)--(0.6886,0.3877)--(0.7441,0.3403)--(0.7996,0.2907)--(0.8552,0.2394)--(0.9107,0.1865)--(0.9662,0.1324)--(1.0218,0.0775)--(1.0773,0.0222)--(1.1328,0.0333)--(1.1884,0.0886)--(1.2439,0.1433)--(1.2994,0.1972)--(1.3550,0.2498)--(1.4105,0.3008)--(1.4660,0.3500)--(1.5216,0.3969)--(1.5771,0.4413)--(1.6326,0.4830)--(1.6882,0.5216)--(1.7437,0.5570)--(1.7992,0.5888)--(1.8548,0.6170)--(1.9103,0.6412)--(1.9658,0.6615)--(2.0214,0.6775)--(2.0769,0.6893)--(2.1324,0.6968)--(2.1880,0.6999)--(2.2435,0.6985)--(2.2990,0.6928)--(2.3546,0.6828)--(2.4101,0.6684)--(2.4656,0.6498)--(2.5212,0.6271)--(2.5767,0.6005)--(2.6322,0.5702)--(2.6878,0.5362)--(2.7433,0.4988)--(2.7988,0.4584)--(2.8544,0.4150)--(2.9099,0.3690)--(2.9654,0.3207)--(3.0210,0.2704)--(3.0765,0.2184)--(3.1320,0.1650)--(3.1876,0.1106)--(3.2431,0.0554)--(3.2986,0.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -239,35 +239,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (2.8562,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4997);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (2.8561,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4997);
 %DEFAULT
 
-\draw [color=blue] (-1.571,0)--(-1.531,0.001573)--(-1.491,0.006281)--(-1.452,0.01409)--(-1.412,0.02496)--(-1.372,0.03882)--(-1.333,0.05558)--(-1.293,0.07514)--(-1.253,0.09736)--(-1.214,0.1221)--(-1.174,0.1493)--(-1.134,0.1786)--(-1.095,0.2100)--(-1.055,0.2432)--(-1.015,0.2780)--(-0.9758,0.3142)--(-0.9361,0.3515)--(-0.8965,0.3898)--(-0.8568,0.4288)--(-0.8171,0.4683)--(-0.7775,0.5079)--(-0.7378,0.5475)--(-0.6981,0.5868)--(-0.6585,0.6256)--(-0.6188,0.6635)--(-0.5791,0.7005)--(-0.5395,0.7361)--(-0.4998,0.7703)--(-0.4601,0.8028)--(-0.4205,0.8334)--(-0.3808,0.8619)--(-0.3411,0.8881)--(-0.3015,0.9118)--(-0.2618,0.9330)--(-0.2221,0.9515)--(-0.1825,0.9671)--(-0.1428,0.9797)--(-0.1031,0.9894)--(-0.06347,0.9960)--(-0.02380,0.9994)--(0.01587,0.9997)--(0.05553,0.9969)--(0.09520,0.9910)--(0.1349,0.9819)--(0.1745,0.9698)--(0.2142,0.9548)--(0.2539,0.9369)--(0.2935,0.9163)--(0.3332,0.8930)--(0.3729,0.8673)--(0.4125,0.8393)--(0.4522,0.8091)--(0.4919,0.7770)--(0.5315,0.7431)--(0.5712,0.7077)--(0.6109,0.6710)--(0.6505,0.6332)--(0.6902,0.5946)--(0.7299,0.5554)--(0.7695,0.5159)--(0.8092,0.4762)--(0.8489,0.4367)--(0.8885,0.3976)--(0.9282,0.3591)--(0.9679,0.3216)--(1.008,0.2851)--(1.047,0.2500)--(1.087,0.2165)--(1.127,0.1847)--(1.166,0.1550)--(1.206,0.1274)--(1.246,0.1021)--(1.285,0.07937)--(1.325,0.05927)--(1.365,0.04195)--(1.404,0.02750)--(1.444,0.01603)--(1.484,0.007596)--(1.523,0.002264)--(1.563,0)--(1.603,0.001007)--(1.642,0.005089)--(1.682,0.01229)--(1.722,0.02255)--(1.761,0.03582)--(1.801,0.05200)--(1.841,0.07101)--(1.880,0.09271)--(1.920,0.1170)--(1.960,0.1437)--(1.999,0.1726)--(2.039,0.2035)--(2.079,0.2364)--(2.118,0.2709)--(2.158,0.3068)--(2.198,0.3440)--(2.237,0.3821)--(2.277,0.4210)--(2.317,0.4604)--(2.356,0.5000);
-\draw (-2.0000,-0.32983) node {$ -4 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.5000,-0.32983) node {$ -3 $};
-\draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.0000,-0.32983) node {$ -2 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.50000,-0.32983) node {$ -1 $};
-\draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.50000,-0.31492) node {$ 1 $};
-\draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.0000,-0.31492) node {$ 2 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.5000,-0.31492) node {$ 3 $};
-\draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.0000,-0.31492) node {$ 4 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.5000,-0.31492) node {$ 5 $};
-\draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (-0.43316,-0.50000) node {$ -1 $};
-\draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.29125,0.50000) node {$ 1 $};
-\draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-1.5707,0.0000)--(-1.5311,0.0015)--(-1.4914,0.0062)--(-1.4517,0.0140)--(-1.4121,0.0249)--(-1.3724,0.0388)--(-1.3327,0.0555)--(-1.2931,0.0751)--(-1.2534,0.0973)--(-1.2137,0.1221)--(-1.1741,0.1492)--(-1.1344,0.1786)--(-1.0947,0.2099)--(-1.0551,0.2431)--(-1.0154,0.2779)--(-0.9757,0.3141)--(-0.9361,0.3515)--(-0.8964,0.3898)--(-0.8567,0.4288)--(-0.8171,0.4682)--(-0.7774,0.5079)--(-0.7377,0.5475)--(-0.6981,0.5868)--(-0.6584,0.6255)--(-0.6187,0.6635)--(-0.5791,0.7004)--(-0.5394,0.7361)--(-0.4997,0.7703)--(-0.4601,0.8028)--(-0.4204,0.8333)--(-0.3807,0.8618)--(-0.3411,0.8880)--(-0.3014,0.9118)--(-0.2617,0.9330)--(-0.2221,0.9514)--(-0.1824,0.9670)--(-0.1427,0.9797)--(-0.1031,0.9894)--(-0.0634,0.9959)--(-0.0237,0.9994)--(0.0158,0.9997)--(0.0555,0.9969)--(0.0951,0.9909)--(0.1348,0.9819)--(0.1745,0.9698)--(0.2141,0.9548)--(0.2538,0.9369)--(0.2935,0.9162)--(0.3331,0.8930)--(0.3728,0.8672)--(0.4125,0.8392)--(0.4521,0.8090)--(0.4918,0.7769)--(0.5315,0.7430)--(0.5711,0.7077)--(0.6108,0.6710)--(0.6505,0.6332)--(0.6901,0.5946)--(0.7298,0.5554)--(0.7695,0.5158)--(0.8091,0.4762)--(0.8488,0.4367)--(0.8885,0.3975)--(0.9281,0.3591)--(0.9678,0.3215)--(1.0075,0.2851)--(1.0471,0.2500)--(1.0868,0.2164)--(1.1265,0.1847)--(1.1661,0.1549)--(1.2058,0.1273)--(1.2455,0.1021)--(1.2851,0.0793)--(1.3248,0.0592)--(1.3645,0.0419)--(1.4041,0.0274)--(1.4438,0.0160)--(1.4835,0.0075)--(1.5231,0.0022)--(1.5628,0.0000)--(1.6025,0.0010)--(1.6421,0.0050)--(1.6818,0.0122)--(1.7215,0.0225)--(1.7611,0.0358)--(1.8008,0.0520)--(1.8405,0.0710)--(1.8801,0.0927)--(1.9198,0.1169)--(1.9595,0.1436)--(1.9991,0.1725)--(2.0388,0.2035)--(2.0785,0.2363)--(2.1181,0.2708)--(2.1578,0.3068)--(2.1975,0.3439)--(2.2371,0.3821)--(2.2768,0.4209)--(2.3165,0.4603)--(2.3561,0.5000);
+\draw (-2.0000,-0.3298) node {$ -4 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.5000,-0.3298) node {$ -3 $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -2 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (-0.5000,-0.3298) node {$ -1 $};
+\draw [] (-0.5000,-0.1000) -- (-0.5000,0.1000);
+\draw (0.5000,-0.3149) node {$ 1 $};
+\draw [] (0.5000,-0.1000) -- (0.5000,0.1000);
+\draw (1.0000,-0.3149) node {$ 2 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (1.5000,-0.3149) node {$ 3 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (2.0000,-0.3149) node {$ 4 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (2.5000,-0.3149) node {$ 5 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (-0.4331,-0.5000) node {$ -1 $};
+\draw [] (-0.1000,-0.5000) -- (0.1000,-0.5000);
+\draw (-0.2912,0.5000) node {$ 1 $};
+\draw [] (-0.1000,0.5000) -- (0.1000,0.5000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -329,31 +329,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1996) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0)--(-2.144,0.05547)--(-2.088,0.1106)--(-2.033,0.1650)--(-1.977,0.2184)--(-1.921,0.2704)--(-1.866,0.3208)--(-1.810,0.3691)--(-1.755,0.4150)--(-1.699,0.4584)--(-1.644,0.4989)--(-1.588,0.5362)--(-1.533,0.5702)--(-1.477,0.6006)--(-1.422,0.6272)--(-1.366,0.6499)--(-1.311,0.6684)--(-1.255,0.6828)--(-1.200,0.6929)--(-1.144,0.6986)--(-1.088,0.6999)--(-1.033,0.6968)--(-0.9774,0.6894)--(-0.9219,0.6776)--(-0.8663,0.6615)--(-0.8108,0.6413)--(-0.7552,0.6170)--(-0.6997,0.5889)--(-0.6442,0.5570)--(-0.5887,0.5217)--(-0.5331,0.4831)--(-0.4776,0.4414)--(-0.4221,0.3969)--(-0.3665,0.3500)--(-0.3110,0.3009)--(-0.2555,0.2498)--(-0.1999,0.1972)--(-0.1444,0.1434)--(-0.08885,0.08861)--(-0.03332,0.03331)--(0.02221,-0.02221)--(0.07775,-0.07759)--(0.1333,-0.1325)--(0.1888,-0.1865)--(0.2443,-0.2394)--(0.2999,-0.2908)--(0.3554,-0.3403)--(0.4109,-0.3877)--(0.4665,-0.4327)--(0.5220,-0.4750)--(0.5775,-0.5142)--(0.6331,-0.5502)--(0.6886,-0.5828)--(0.7441,-0.6117)--(0.7997,-0.6367)--(0.8552,-0.6578)--(0.9107,-0.6747)--(0.9663,-0.6873)--(1.022,-0.6957)--(1.077,-0.6996)--(1.133,-0.6992)--(1.188,-0.6944)--(1.244,-0.6852)--(1.299,-0.6716)--(1.355,-0.6539)--(1.411,-0.6320)--(1.466,-0.6062)--(1.522,-0.5766)--(1.577,-0.5433)--(1.633,-0.5066)--(1.688,-0.4667)--(1.744,-0.4239)--(1.799,-0.3784)--(1.855,-0.3306)--(1.910,-0.2807)--(1.966,-0.2289)--(2.021,-0.1758)--(2.077,-0.1216)--(2.132,-0.06654)--(2.188,-0.01111)--(2.244,0.04440)--(2.299,0.09962)--(2.355,0.1542)--(2.410,0.2078)--(2.466,0.2602)--(2.521,0.3108)--(2.577,0.3596)--(2.632,0.4060)--(2.688,0.4500)--(2.743,0.4910)--(2.799,0.5290)--(2.854,0.5637)--(2.910,0.5948)--(2.965,0.6222)--(3.021,0.6456)--(3.077,0.6650)--(3.132,0.6803)--(3.188,0.6912)--(3.243,0.6978)--(3.299,0.7000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
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+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -415,32 +415,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.2000) -- (0,1.1999);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1999) -- (0.0000,1.1999);
 %DEFAULT
 
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+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -502,31 +502,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5996,0) -- (4.8982,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
+\draw [,->,>=latex] (-1.5995,0.0000) -- (4.8982,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-1.100,0)--(-1.077,0.1247)--(-1.055,0.1763)--(-1.033,0.2158)--(-1.011,0.2491)--(-0.9885,0.2782)--(-0.9663,0.3045)--(-0.9441,0.3286)--(-0.9219,0.3508)--(-0.8996,0.3715)--(-0.8774,0.3910)--(-0.8552,0.4094)--(-0.8330,0.4268)--(-0.8108,0.4432)--(-0.7886,0.4589)--(-0.7664,0.4738)--(-0.7441,0.4881)--(-0.7219,0.5017)--(-0.6997,0.5147)--(-0.6775,0.5271)--(-0.6553,0.5390)--(-0.6331,0.5504)--(-0.6109,0.5612)--(-0.5887,0.5716)--(-0.5664,0.5815)--(-0.5442,0.5910)--(-0.5220,0.6000)--(-0.4998,0.6085)--(-0.4776,0.6167)--(-0.4554,0.6244)--(-0.4332,0.6318)--(-0.4109,0.6387)--(-0.3887,0.6453)--(-0.3665,0.6514)--(-0.3443,0.6572)--(-0.3221,0.6626)--(-0.2999,0.6676)--(-0.2777,0.6723)--(-0.2555,0.6766)--(-0.2332,0.6805)--(-0.2110,0.6840)--(-0.1888,0.6872)--(-0.1666,0.6901)--(-0.1444,0.6925)--(-0.1222,0.6947)--(-0.09996,0.6964)--(-0.07775,0.6978)--(-0.05553,0.6989)--(-0.03332,0.6996)--(-0.01111,0.7000)--(0.01111,0.7000)--(0.03332,0.6996)--(0.05553,0.6989)--(0.07775,0.6978)--(0.09996,0.6964)--(0.1222,0.6947)--(0.1444,0.6925)--(0.1666,0.6901)--(0.1888,0.6872)--(0.2110,0.6840)--(0.2332,0.6805)--(0.2555,0.6766)--(0.2777,0.6723)--(0.2999,0.6676)--(0.3221,0.6626)--(0.3443,0.6572)--(0.3665,0.6514)--(0.3887,0.6453)--(0.4109,0.6387)--(0.4332,0.6318)--(0.4554,0.6244)--(0.4776,0.6167)--(0.4998,0.6085)--(0.5220,0.6000)--(0.5442,0.5910)--(0.5664,0.5815)--(0.5887,0.5716)--(0.6109,0.5612)--(0.6331,0.5504)--(0.6553,0.5390)--(0.6775,0.5271)--(0.6997,0.5147)--(0.7219,0.5017)--(0.7441,0.4881)--(0.7664,0.4738)--(0.7886,0.4589)--(0.8108,0.4432)--(0.8330,0.4268)--(0.8552,0.4094)--(0.8774,0.3910)--(0.8996,0.3715)--(0.9219,0.3508)--(0.9441,0.3286)--(0.9663,0.3045)--(0.9885,0.2782)--(1.011,0.2491)--(1.033,0.2158)--(1.055,0.1763)--(1.077,0.1247)--(1.100,0);
+\draw [color=blue] (-1.0995,0.0000)--(-1.0773,0.1246)--(-1.0551,0.1762)--(-1.0329,0.2158)--(-1.0107,0.2490)--(-0.9884,0.2782)--(-0.9662,0.3045)--(-0.9440,0.3285)--(-0.9218,0.3508)--(-0.8996,0.3715)--(-0.8774,0.3910)--(-0.8552,0.4093)--(-0.8329,0.4267)--(-0.8107,0.4432)--(-0.7885,0.4589)--(-0.7663,0.4738)--(-0.7441,0.4880)--(-0.7219,0.5016)--(-0.6997,0.5146)--(-0.6775,0.5271)--(-0.6552,0.5390)--(-0.6330,0.5503)--(-0.6108,0.5612)--(-0.5886,0.5715)--(-0.5664,0.5814)--(-0.5442,0.5909)--(-0.5220,0.5999)--(-0.4997,0.6085)--(-0.4775,0.6166)--(-0.4553,0.6244)--(-0.4331,0.6317)--(-0.4109,0.6387)--(-0.3887,0.6452)--(-0.3665,0.6514)--(-0.3443,0.6572)--(-0.3220,0.6625)--(-0.2998,0.6676)--(-0.2776,0.6722)--(-0.2554,0.6765)--(-0.2332,0.6804)--(-0.2110,0.6840)--(-0.1888,0.6872)--(-0.1665,0.6900)--(-0.1443,0.6925)--(-0.1221,0.6946)--(-0.0999,0.6964)--(-0.0777,0.6978)--(-0.0555,0.6988)--(-0.0333,0.6996)--(-0.0111,0.6999)--(0.0111,0.6999)--(0.0333,0.6996)--(0.0555,0.6988)--(0.0777,0.6978)--(0.0999,0.6964)--(0.1221,0.6946)--(0.1443,0.6925)--(0.1665,0.6900)--(0.1888,0.6872)--(0.2110,0.6840)--(0.2332,0.6804)--(0.2554,0.6765)--(0.2776,0.6722)--(0.2998,0.6676)--(0.3220,0.6625)--(0.3443,0.6572)--(0.3665,0.6514)--(0.3887,0.6452)--(0.4109,0.6387)--(0.4331,0.6317)--(0.4553,0.6244)--(0.4775,0.6166)--(0.4997,0.6085)--(0.5220,0.5999)--(0.5442,0.5909)--(0.5664,0.5814)--(0.5886,0.5715)--(0.6108,0.5612)--(0.6330,0.5503)--(0.6552,0.5390)--(0.6775,0.5271)--(0.6997,0.5146)--(0.7219,0.5016)--(0.7441,0.4880)--(0.7663,0.4738)--(0.7885,0.4589)--(0.8107,0.4432)--(0.8329,0.4267)--(0.8552,0.4093)--(0.8774,0.3910)--(0.8996,0.3715)--(0.9218,0.3508)--(0.9440,0.3285)--(0.9662,0.3045)--(0.9884,0.2782)--(1.0107,0.2490)--(1.0329,0.2158)--(1.0551,0.1762)--(1.0773,0.1246)--(1.0995,0.0000);
 
-\draw [color=blue] (3.299,0)--(3.310,0.08817)--(3.321,0.1247)--(3.332,0.1527)--(3.343,0.1763)--(3.354,0.1971)--(3.365,0.2158)--(3.376,0.2330)--(3.388,0.2491)--(3.399,0.2641)--(3.410,0.2782)--(3.421,0.2917)--(3.432,0.3045)--(3.443,0.3168)--(3.454,0.3286)--(3.465,0.3399)--(3.476,0.3508)--(3.487,0.3613)--(3.499,0.3715)--(3.510,0.3814)--(3.521,0.3910)--(3.532,0.4003)--(3.543,0.4094)--(3.554,0.4182)--(3.565,0.4268)--(3.576,0.4351)--(3.587,0.4432)--(3.599,0.4512)--(3.610,0.4589)--(3.621,0.4665)--(3.632,0.4738)--(3.643,0.4811)--(3.654,0.4881)--(3.665,0.4950)--(3.676,0.5017)--(3.687,0.5083)--(3.699,0.5147)--(3.710,0.5210)--(3.721,0.5271)--(3.732,0.5331)--(3.743,0.5390)--(3.754,0.5447)--(3.765,0.5504)--(3.776,0.5559)--(3.787,0.5612)--(3.798,0.5665)--(3.810,0.5716)--(3.821,0.5766)--(3.832,0.5815)--(3.843,0.5863)--(3.854,0.5910)--(3.865,0.5955)--(3.876,0.6000)--(3.887,0.6043)--(3.898,0.6085)--(3.910,0.6127)--(3.921,0.6167)--(3.932,0.6206)--(3.943,0.6244)--(3.954,0.6282)--(3.965,0.6318)--(3.976,0.6353)--(3.987,0.6387)--(3.998,0.6420)--(4.010,0.6453)--(4.021,0.6484)--(4.032,0.6514)--(4.043,0.6544)--(4.054,0.6572)--(4.065,0.6600)--(4.076,0.6626)--(4.087,0.6652)--(4.098,0.6676)--(4.109,0.6700)--(4.121,0.6723)--(4.132,0.6745)--(4.143,0.6766)--(4.154,0.6786)--(4.165,0.6805)--(4.176,0.6823)--(4.187,0.6840)--(4.198,0.6857)--(4.209,0.6872)--(4.221,0.6887)--(4.232,0.6901)--(4.243,0.6913)--(4.254,0.6925)--(4.265,0.6936)--(4.276,0.6947)--(4.287,0.6956)--(4.298,0.6964)--(4.309,0.6972)--(4.320,0.6978)--(4.332,0.6984)--(4.343,0.6989)--(4.354,0.6993)--(4.365,0.6996)--(4.376,0.6998)--(4.387,0.7000)--(4.398,0.7000);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (4.2000,-0.31492) node {$ 6 $};
-\draw [] (4.20,-0.100) -- (4.20,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (3.2986,0.0000)--(3.3097,0.0881)--(3.3208,0.1246)--(3.3319,0.1526)--(3.3430,0.1762)--(3.3542,0.1970)--(3.3653,0.2158)--(3.3764,0.2330)--(3.3875,0.2490)--(3.3986,0.2640)--(3.4097,0.2782)--(3.4208,0.2916)--(3.4319,0.3045)--(3.4430,0.3167)--(3.4541,0.3285)--(3.4652,0.3398)--(3.4763,0.3508)--(3.4874,0.3613)--(3.4985,0.3715)--(3.5096,0.3814)--(3.5208,0.3910)--(3.5319,0.4003)--(3.5430,0.4093)--(3.5541,0.4181)--(3.5652,0.4267)--(3.5763,0.4350)--(3.5874,0.4432)--(3.5985,0.4511)--(3.6096,0.4589)--(3.6207,0.4664)--(3.6318,0.4738)--(3.6429,0.4810)--(3.6540,0.4880)--(3.6651,0.4949)--(3.6762,0.5016)--(3.6874,0.5082)--(3.6985,0.5146)--(3.7096,0.5209)--(3.7207,0.5271)--(3.7318,0.5331)--(3.7429,0.5390)--(3.7540,0.5447)--(3.7651,0.5503)--(3.7762,0.5558)--(3.7873,0.5612)--(3.7984,0.5664)--(3.8095,0.5715)--(3.8206,0.5766)--(3.8317,0.5814)--(3.8428,0.5862)--(3.8540,0.5909)--(3.8651,0.5955)--(3.8762,0.5999)--(3.8873,0.6043)--(3.8984,0.6085)--(3.9095,0.6126)--(3.9206,0.6166)--(3.9317,0.6206)--(3.9428,0.6244)--(3.9539,0.6281)--(3.9650,0.6317)--(3.9761,0.6352)--(3.9872,0.6387)--(3.9983,0.6420)--(4.0094,0.6452)--(4.0206,0.6483)--(4.0317,0.6514)--(4.0428,0.6543)--(4.0539,0.6572)--(4.0650,0.6599)--(4.0761,0.6625)--(4.0872,0.6651)--(4.0983,0.6676)--(4.1094,0.6699)--(4.1205,0.6722)--(4.1316,0.6744)--(4.1427,0.6765)--(4.1538,0.6785)--(4.1649,0.6804)--(4.1760,0.6823)--(4.1872,0.6840)--(4.1983,0.6856)--(4.2094,0.6872)--(4.2205,0.6886)--(4.2316,0.6900)--(4.2427,0.6913)--(4.2538,0.6925)--(4.2649,0.6936)--(4.2760,0.6946)--(4.2871,0.6955)--(4.2982,0.6964)--(4.3093,0.6971)--(4.3204,0.6978)--(4.3315,0.6984)--(4.3426,0.6988)--(4.3538,0.6992)--(4.3649,0.6996)--(4.3760,0.6998)--(4.3871,0.6999)--(4.3982,0.7000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (4.2000,-0.3149) node {$ 6 $};
+\draw [] (4.2000,-0.1000) -- (4.2000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -588,31 +588,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1996) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0.7000)--(-2.144,0.6650)--(-2.088,0.5637)--(-2.033,0.4060)--(-1.977,0.2078)--(-1.921,-0.01111)--(-1.866,-0.2289)--(-1.810,-0.4239)--(-1.755,-0.5766)--(-1.699,-0.6716)--(-1.644,-0.6996)--(-1.588,-0.6578)--(-1.533,-0.5502)--(-1.477,-0.3877)--(-1.422,-0.1865)--(-1.366,0.03331)--(-1.311,0.2498)--(-1.255,0.4414)--(-1.200,0.5889)--(-1.144,0.6776)--(-1.088,0.6986)--(-1.033,0.6499)--(-0.9774,0.5362)--(-0.9219,0.3691)--(-0.8663,0.1650)--(-0.8108,-0.05547)--(-0.7552,-0.2704)--(-0.6997,-0.4584)--(-0.6442,-0.6006)--(-0.5887,-0.6828)--(-0.5331,-0.6968)--(-0.4776,-0.6413)--(-0.4221,-0.5217)--(-0.3665,-0.3500)--(-0.3110,-0.1434)--(-0.2555,0.07759)--(-0.1999,0.2908)--(-0.1444,0.4750)--(-0.08885,0.6117)--(-0.03332,0.6873)--(0.02221,0.6944)--(0.07775,0.6320)--(0.1333,0.5066)--(0.1888,0.3306)--(0.2443,0.1216)--(0.2999,-0.09962)--(0.3554,-0.3108)--(0.4109,-0.4910)--(0.4665,-0.6222)--(0.5220,-0.6912)--(0.5775,-0.6912)--(0.6331,-0.6222)--(0.6886,-0.4910)--(0.7441,-0.3108)--(0.7997,-0.09962)--(0.8552,0.1216)--(0.9107,0.3306)--(0.9663,0.5066)--(1.022,0.6320)--(1.077,0.6944)--(1.133,0.6873)--(1.188,0.6117)--(1.244,0.4750)--(1.299,0.2908)--(1.355,0.07759)--(1.411,-0.1434)--(1.466,-0.3500)--(1.522,-0.5217)--(1.577,-0.6413)--(1.633,-0.6968)--(1.688,-0.6828)--(1.744,-0.6006)--(1.799,-0.4584)--(1.855,-0.2704)--(1.910,-0.05547)--(1.966,0.1650)--(2.021,0.3691)--(2.077,0.5362)--(2.132,0.6499)--(2.188,0.6986)--(2.244,0.6776)--(2.299,0.5889)--(2.355,0.4414)--(2.410,0.2498)--(2.466,0.03331)--(2.521,-0.1865)--(2.577,-0.3877)--(2.632,-0.5502)--(2.688,-0.6578)--(2.743,-0.6996)--(2.799,-0.6716)--(2.854,-0.5766)--(2.910,-0.4239)--(2.965,-0.2289)--(3.021,-0.01111)--(3.077,0.2078)--(3.132,0.4060)--(3.188,0.5637)--(3.243,0.6650)--(3.299,0.7000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-2.1991,0.7000)--(-2.1435,0.6650)--(-2.0880,0.5636)--(-2.0325,0.4060)--(-1.9769,0.2078)--(-1.9214,-0.0111)--(-1.8659,-0.2289)--(-1.8103,-0.4239)--(-1.7548,-0.5765)--(-1.6993,-0.6716)--(-1.6437,-0.6996)--(-1.5882,-0.6577)--(-1.5327,-0.5502)--(-1.4771,-0.3877)--(-1.4216,-0.1865)--(-1.3661,0.0333)--(-1.3105,0.2498)--(-1.2550,0.4413)--(-1.1995,0.5888)--(-1.1439,0.6775)--(-1.0884,0.6985)--(-1.0329,0.6498)--(-0.9773,0.5362)--(-0.9218,0.3690)--(-0.8663,0.1650)--(-0.8107,-0.0554)--(-0.7552,-0.2704)--(-0.6997,-0.4584)--(-0.6441,-0.6005)--(-0.5886,-0.6828)--(-0.5331,-0.6968)--(-0.4775,-0.6412)--(-0.4220,-0.5216)--(-0.3665,-0.3500)--(-0.3109,-0.1433)--(-0.2554,0.0775)--(-0.1999,0.2907)--(-0.1443,0.4749)--(-0.0888,0.6116)--(-0.0333,0.6873)--(0.0222,0.6943)--(0.0777,0.6320)--(0.1332,0.5066)--(0.1888,0.3305)--(0.2443,0.1215)--(0.2998,-0.0996)--(0.3554,-0.3108)--(0.4109,-0.4910)--(0.4664,-0.6221)--(0.5220,-0.6912)--(0.5775,-0.6912)--(0.6330,-0.6221)--(0.6886,-0.4910)--(0.7441,-0.3108)--(0.7996,-0.0996)--(0.8552,0.1215)--(0.9107,0.3305)--(0.9662,0.5066)--(1.0218,0.6320)--(1.0773,0.6943)--(1.1328,0.6873)--(1.1884,0.6116)--(1.2439,0.4749)--(1.2994,0.2907)--(1.3550,0.0775)--(1.4105,-0.1433)--(1.4660,-0.3500)--(1.5216,-0.5216)--(1.5771,-0.6412)--(1.6326,-0.6968)--(1.6882,-0.6828)--(1.7437,-0.6005)--(1.7992,-0.4584)--(1.8548,-0.2704)--(1.9103,-0.0554)--(1.9658,0.1650)--(2.0214,0.3690)--(2.0769,0.5362)--(2.1324,0.6498)--(2.1880,0.6985)--(2.2435,0.6775)--(2.2990,0.5888)--(2.3546,0.4413)--(2.4101,0.2498)--(2.4656,0.0333)--(2.5212,-0.1865)--(2.5767,-0.3877)--(2.6322,-0.5502)--(2.6878,-0.6577)--(2.7433,-0.6996)--(2.7988,-0.6716)--(2.8544,-0.5765)--(2.9099,-0.4239)--(2.9654,-0.2289)--(3.0210,-0.0111)--(3.0765,0.2078)--(3.1320,0.4060)--(3.1876,0.5636)--(3.2431,0.6650)--(3.2986,0.7000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExoCUd.pstricks.recall b/src_phystricks/Fig_ExoCUd.pstricks.recall
index 271934b9a..3dac3bca8 100644
--- a/src_phystricks/Fig_ExoCUd.pstricks.recall
+++ b/src_phystricks/Fig_ExoCUd.pstricks.recall
@@ -75,34 +75,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-1.0000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.0000) -- (0.0000,4.5000);
 %DEFAULT
 
-\draw [color=red] (-1.000,4.000)--(-0.9798,3.920)--(-0.9596,3.840)--(-0.9394,3.761)--(-0.9192,3.683)--(-0.8990,3.606)--(-0.8788,3.530)--(-0.8586,3.454)--(-0.8384,3.380)--(-0.8182,3.306)--(-0.7980,3.233)--(-0.7778,3.160)--(-0.7576,3.089)--(-0.7374,3.018)--(-0.7172,2.949)--(-0.6970,2.880)--(-0.6768,2.812)--(-0.6566,2.744)--(-0.6364,2.678)--(-0.6162,2.612)--(-0.5960,2.547)--(-0.5758,2.483)--(-0.5556,2.420)--(-0.5354,2.357)--(-0.5152,2.296)--(-0.4949,2.235)--(-0.4747,2.175)--(-0.4545,2.116)--(-0.4343,2.057)--(-0.4141,2.000)--(-0.3939,1.943)--(-0.3737,1.887)--(-0.3535,1.832)--(-0.3333,1.778)--(-0.3131,1.724)--(-0.2929,1.672)--(-0.2727,1.620)--(-0.2525,1.569)--(-0.2323,1.519)--(-0.2121,1.469)--(-0.1919,1.421)--(-0.1717,1.373)--(-0.1515,1.326)--(-0.1313,1.280)--(-0.1111,1.235)--(-0.09091,1.190)--(-0.07071,1.146)--(-0.05051,1.104)--(-0.03030,1.062)--(-0.01010,1.020)--(0.01010,0.9799)--(0.03030,0.9403)--(0.05051,0.9015)--(0.07071,0.8636)--(0.09091,0.8264)--(0.1111,0.7901)--(0.1313,0.7546)--(0.1515,0.7199)--(0.1717,0.6861)--(0.1919,0.6530)--(0.2121,0.6208)--(0.2323,0.5893)--(0.2525,0.5587)--(0.2727,0.5289)--(0.2929,0.5000)--(0.3131,0.4718)--(0.3333,0.4444)--(0.3535,0.4179)--(0.3737,0.3922)--(0.3939,0.3673)--(0.4141,0.3432)--(0.4343,0.3200)--(0.4545,0.2975)--(0.4747,0.2759)--(0.4949,0.2551)--(0.5152,0.2351)--(0.5354,0.2159)--(0.5556,0.1975)--(0.5758,0.1800)--(0.5960,0.1632)--(0.6162,0.1473)--(0.6364,0.1322)--(0.6566,0.1179)--(0.6768,0.1045)--(0.6970,0.09183)--(0.7172,0.07999)--(0.7374,0.06897)--(0.7576,0.05877)--(0.7778,0.04938)--(0.7980,0.04081)--(0.8182,0.03306)--(0.8384,0.02612)--(0.8586,0.02000)--(0.8788,0.01469)--(0.8990,0.01020)--(0.9192,0.006530)--(0.9394,0.003673)--(0.9596,0.001632)--(0.9798,0)--(1.000,0);
+\draw [color=red] (-1.0000,4.0000)--(-0.9797,3.9196)--(-0.9595,3.8400)--(-0.9393,3.7612)--(-0.9191,3.6832)--(-0.8989,3.6061)--(-0.8787,3.5298)--(-0.8585,3.4543)--(-0.8383,3.3796)--(-0.8181,3.3057)--(-0.7979,3.2327)--(-0.7777,3.1604)--(-0.7575,3.0890)--(-0.7373,3.0184)--(-0.7171,2.9486)--(-0.6969,2.8797)--(-0.6767,2.8115)--(-0.6565,2.7442)--(-0.6363,2.6776)--(-0.6161,2.6119)--(-0.5959,2.5470)--(-0.5757,2.4830)--(-0.5555,2.4197)--(-0.5353,2.3573)--(-0.5151,2.2956)--(-0.4949,2.2348)--(-0.4747,2.1748)--(-0.4545,2.1157)--(-0.4343,2.0573)--(-0.4141,1.9997)--(-0.3939,1.9430)--(-0.3737,1.8871)--(-0.3535,1.8320)--(-0.3333,1.7777)--(-0.3131,1.7243)--(-0.2929,1.6716)--(-0.2727,1.6198)--(-0.2525,1.5688)--(-0.2323,1.5186)--(-0.2121,1.4692)--(-0.1919,1.4206)--(-0.1717,1.3729)--(-0.1515,1.3259)--(-0.1313,1.2798)--(-0.1111,1.2345)--(-0.0909,1.1900)--(-0.0707,1.1464)--(-0.0505,1.1035)--(-0.0303,1.0615)--(-0.0101,1.0203)--(0.0101,0.9799)--(0.0303,0.9403)--(0.0505,0.9015)--(0.0707,0.8635)--(0.0909,0.8264)--(0.1111,0.7901)--(0.1313,0.7546)--(0.1515,0.7199)--(0.1717,0.6860)--(0.1919,0.6529)--(0.2121,0.6207)--(0.2323,0.5893)--(0.2525,0.5587)--(0.2727,0.5289)--(0.2929,0.4999)--(0.3131,0.4717)--(0.3333,0.4444)--(0.3535,0.4179)--(0.3737,0.3922)--(0.3939,0.3673)--(0.4141,0.3432)--(0.4343,0.3199)--(0.4545,0.2975)--(0.4747,0.2758)--(0.4949,0.2550)--(0.5151,0.2350)--(0.5353,0.2158)--(0.5555,0.1975)--(0.5757,0.1799)--(0.5959,0.1632)--(0.6161,0.1473)--(0.6363,0.1322)--(0.6565,0.1179)--(0.6767,0.1044)--(0.6969,0.0918)--(0.7171,0.0799)--(0.7373,0.0689)--(0.7575,0.0587)--(0.7777,0.0493)--(0.7979,0.0408)--(0.8181,0.0330)--(0.8383,0.0261)--(0.8585,0.0199)--(0.8787,0.0146)--(0.8989,0.0102)--(0.9191,0.0065)--(0.9393,0.0036)--(0.9595,0.0016)--(0.9797,0.0000)--(1.0000,0.0000);
 
-\draw [color=blue] (1.000,0)--(1.020,0)--(1.040,0.001632)--(1.061,0.003673)--(1.081,0.006530)--(1.101,0.01020)--(1.121,0.01469)--(1.141,0.02000)--(1.162,0.02612)--(1.182,0.03306)--(1.202,0.04081)--(1.222,0.04938)--(1.242,0.05877)--(1.263,0.06897)--(1.283,0.07999)--(1.303,0.09183)--(1.323,0.1045)--(1.343,0.1179)--(1.364,0.1322)--(1.384,0.1473)--(1.404,0.1632)--(1.424,0.1800)--(1.444,0.1975)--(1.465,0.2159)--(1.485,0.2351)--(1.505,0.2551)--(1.525,0.2759)--(1.545,0.2975)--(1.566,0.3200)--(1.586,0.3432)--(1.606,0.3673)--(1.626,0.3922)--(1.646,0.4179)--(1.667,0.4444)--(1.687,0.4718)--(1.707,0.5000)--(1.727,0.5289)--(1.747,0.5587)--(1.768,0.5893)--(1.788,0.6208)--(1.808,0.6530)--(1.828,0.6861)--(1.848,0.7199)--(1.869,0.7546)--(1.889,0.7901)--(1.909,0.8264)--(1.929,0.8636)--(1.949,0.9015)--(1.970,0.9403)--(1.990,0.9799)--(2.010,1.020)--(2.030,1.062)--(2.051,1.104)--(2.071,1.146)--(2.091,1.190)--(2.111,1.235)--(2.131,1.280)--(2.152,1.326)--(2.172,1.373)--(2.192,1.421)--(2.212,1.469)--(2.232,1.519)--(2.253,1.569)--(2.273,1.620)--(2.293,1.672)--(2.313,1.724)--(2.333,1.778)--(2.354,1.832)--(2.374,1.887)--(2.394,1.943)--(2.414,2.000)--(2.434,2.057)--(2.455,2.116)--(2.475,2.175)--(2.495,2.235)--(2.515,2.296)--(2.535,2.357)--(2.556,2.420)--(2.576,2.483)--(2.596,2.547)--(2.616,2.612)--(2.636,2.678)--(2.657,2.744)--(2.677,2.812)--(2.697,2.880)--(2.717,2.949)--(2.737,3.018)--(2.758,3.089)--(2.778,3.160)--(2.798,3.233)--(2.818,3.306)--(2.838,3.380)--(2.859,3.454)--(2.879,3.530)--(2.899,3.606)--(2.919,3.683)--(2.939,3.761)--(2.960,3.840)--(2.980,3.920)--(3.000,4.000);
-\draw [color=gray,style=dashed] (1.00,-0.500) -- (1.00,4.00);
-\draw []  (0,2.5600) node [rotate=0] {$\bullet$};
-\draw (0.30816,2.8862) node {$y$};
+\draw [color=blue] (1.0000,0.0000)--(1.0202,0.0000)--(1.0404,0.0016)--(1.0606,0.0036)--(1.0808,0.0065)--(1.1010,0.0102)--(1.1212,0.0146)--(1.1414,0.0199)--(1.1616,0.0261)--(1.1818,0.0330)--(1.2020,0.0408)--(1.2222,0.0493)--(1.2424,0.0587)--(1.2626,0.0689)--(1.2828,0.0799)--(1.3030,0.0918)--(1.3232,0.1044)--(1.3434,0.1179)--(1.3636,0.1322)--(1.3838,0.1473)--(1.4040,0.1632)--(1.4242,0.1799)--(1.4444,0.1975)--(1.4646,0.2158)--(1.4848,0.2350)--(1.5050,0.2550)--(1.5252,0.2758)--(1.5454,0.2975)--(1.5656,0.3199)--(1.5858,0.3432)--(1.6060,0.3673)--(1.6262,0.3922)--(1.6464,0.4179)--(1.6666,0.4444)--(1.6868,0.4717)--(1.7070,0.4999)--(1.7272,0.5289)--(1.7474,0.5587)--(1.7676,0.5893)--(1.7878,0.6207)--(1.8080,0.6529)--(1.8282,0.6860)--(1.8484,0.7199)--(1.8686,0.7546)--(1.8888,0.7901)--(1.9090,0.8264)--(1.9292,0.8635)--(1.9494,0.9015)--(1.9696,0.9403)--(1.9898,0.9799)--(2.0101,1.0203)--(2.0303,1.0615)--(2.0505,1.1035)--(2.0707,1.1464)--(2.0909,1.1900)--(2.1111,1.2345)--(2.1313,1.2798)--(2.1515,1.3259)--(2.1717,1.3729)--(2.1919,1.4206)--(2.2121,1.4692)--(2.2323,1.5186)--(2.2525,1.5688)--(2.2727,1.6198)--(2.2929,1.6716)--(2.3131,1.7243)--(2.3333,1.7777)--(2.3535,1.8320)--(2.3737,1.8871)--(2.3939,1.9430)--(2.4141,1.9997)--(2.4343,2.0573)--(2.4545,2.1157)--(2.4747,2.1748)--(2.4949,2.2348)--(2.5151,2.2956)--(2.5353,2.3573)--(2.5555,2.4197)--(2.5757,2.4830)--(2.5959,2.5470)--(2.6161,2.6119)--(2.6363,2.6776)--(2.6565,2.7442)--(2.6767,2.8115)--(2.6969,2.8797)--(2.7171,2.9486)--(2.7373,3.0184)--(2.7575,3.0890)--(2.7777,3.1604)--(2.7979,3.2327)--(2.8181,3.3057)--(2.8383,3.3796)--(2.8585,3.4543)--(2.8787,3.5298)--(2.8989,3.6061)--(2.9191,3.6832)--(2.9393,3.7612)--(2.9595,3.8400)--(2.9797,3.9196)--(3.0000,4.0000);
+\draw [color=gray,style=dashed] (1.0000,-0.5000) -- (1.0000,4.0000);
+\draw []  (0.0000,2.5600) node [rotate=0] {$\bullet$};
+\draw (0.3081,2.8861) node {$y$};
 \draw []  (2.6000,2.5600) node [rotate=0] {$\bullet$};
-\draw []  (-0.60000,2.5600) node [rotate=0] {$\bullet$};
-\draw []  (2.6000,0) node [rotate=0] {$\bullet$};
-\draw (2.6000,-0.41918) node {$x_+$};
-\draw []  (-0.60000,0) node [rotate=0] {$\bullet$};
-\draw (-0.60000,-0.41918) node {$x_-$};
-\draw [style=dashed] (2.60,2.56) -- (2.60,0);
-\draw [style=dashed] (-0.600,2.56) -- (-0.600,0);
-\draw [style=dotted] (2.60,2.56) -- (-0.600,2.56);
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw []  (-0.6000,2.5600) node [rotate=0] {$\bullet$};
+\draw []  (2.6000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.6000,-0.4191) node {$x_+$};
+\draw []  (-0.6000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.6000,-0.4191) node {$x_-$};
+\draw [style=dashed] (2.6000,2.5600) -- (2.6000,0.0000);
+\draw [style=dashed] (-0.6000,2.5600) -- (-0.6000,0.0000);
+\draw [style=dotted] (2.6000,2.5600) -- (-0.6000,2.5600);
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExoMagnetique.pstricks.recall b/src_phystricks/Fig_ExoMagnetique.pstricks.recall
index 6466d25a8..ad8e03dc3 100644
--- a/src_phystricks/Fig_ExoMagnetique.pstricks.recall
+++ b/src_phystricks/Fig_ExoMagnetique.pstricks.recall
@@ -77,13 +77,13 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [style=dashed] (0,-2.00) -- (0,2.00);
-\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
-\draw (0.39455,1.0000) node {$I$};
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-2.0000,1.0000);
-\draw (-1.7636,0.73184) node {$d$};
+\draw [style=dashed] (0.0000,-2.0000) -- (0.0000,2.0000);
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (0.0000,1.0000);
+\draw (0.3945,1.0000) node {$I$};
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-2.0000,1.0000);
+\draw (-1.7635,0.7318) node {$d$};
 \draw []  (-2.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-2.7065,1.3239) node {$(r,\theta,z)$};
+\draw (-2.7064,1.3238) node {$(r,\theta,z)$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ExoPolaire.pstricks.recall b/src_phystricks/Fig_ExoPolaire.pstricks.recall
index 6980c9bb4..f17182b99 100644
--- a/src_phystricks/Fig_ExoPolaire.pstricks.recall
+++ b/src_phystricks/Fig_ExoPolaire.pstricks.recall
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.2321,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.2320,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
 \draw (2.3896,1.0000) node {$(\sqrt{3},1)$};
-\draw (0.65798,0.88654) node {$l$};
-\draw (1.0127,0.25615) node {$\theta$};
+\draw (0.6579,0.8865) node {$l$};
+\draw (1.0127,0.2561) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw [,->,>=latex] (0,0) -- (1.7320,1.0000);
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (0.5000,0.0000)--(0.4999,0.0026)--(0.4999,0.0052)--(0.4999,0.0079)--(0.4998,0.0105)--(0.4998,0.0132)--(0.4997,0.0158)--(0.4996,0.0185)--(0.4995,0.0211)--(0.4994,0.0237)--(0.4993,0.0264)--(0.4991,0.0290)--(0.4989,0.0317)--(0.4988,0.0343)--(0.4986,0.0369)--(0.4984,0.0396)--(0.4982,0.0422)--(0.4979,0.0448)--(0.4977,0.0475)--(0.4974,0.0501)--(0.4972,0.0527)--(0.4969,0.0554)--(0.4966,0.0580)--(0.4963,0.0606)--(0.4959,0.0632)--(0.4956,0.0659)--(0.4952,0.0685)--(0.4949,0.0711)--(0.4945,0.0737)--(0.4941,0.0763)--(0.4937,0.0790)--(0.4932,0.0816)--(0.4928,0.0842)--(0.4924,0.0868)--(0.4919,0.0894)--(0.4914,0.0920)--(0.4909,0.0946)--(0.4904,0.0972)--(0.4899,0.0998)--(0.4894,0.1024)--(0.4888,0.1049)--(0.4882,0.1075)--(0.4877,0.1101)--(0.4871,0.1127)--(0.4865,0.1153)--(0.4859,0.1178)--(0.4852,0.1204)--(0.4846,0.1230)--(0.4839,0.1255)--(0.4833,0.1281)--(0.4826,0.1306)--(0.4819,0.1332)--(0.4812,0.1357)--(0.4804,0.1383)--(0.4797,0.1408)--(0.4789,0.1434)--(0.4782,0.1459)--(0.4774,0.1484)--(0.4766,0.1509)--(0.4758,0.1535)--(0.4750,0.1560)--(0.4742,0.1585)--(0.4733,0.1610)--(0.4725,0.1635)--(0.4716,0.1660)--(0.4707,0.1685)--(0.4698,0.1710)--(0.4689,0.1734)--(0.4680,0.1759)--(0.4670,0.1784)--(0.4661,0.1809)--(0.4651,0.1833)--(0.4641,0.1858)--(0.4631,0.1882)--(0.4621,0.1907)--(0.4611,0.1931)--(0.4601,0.1956)--(0.4591,0.1980)--(0.4580,0.2004)--(0.4569,0.2028)--(0.4559,0.2052)--(0.4548,0.2077)--(0.4537,0.2101)--(0.4525,0.2125)--(0.4514,0.2148)--(0.4503,0.2172)--(0.4491,0.2196)--(0.4479,0.2220)--(0.4468,0.2243)--(0.4456,0.2267)--(0.4444,0.2291)--(0.4431,0.2314)--(0.4419,0.2338)--(0.4407,0.2361)--(0.4394,0.2384)--(0.4382,0.2407)--(0.4369,0.2430)--(0.4356,0.2454)--(0.4343,0.2477)--(0.4330,0.2500);
+\draw [,->,>=latex] (0.0000,0.0000) -- (1.7320,1.0000);
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExoProjection.pstricks.recall b/src_phystricks/Fig_ExoProjection.pstricks.recall
index 7d7fed300..fb7d5e3ae 100644
--- a/src_phystricks/Fig_ExoProjection.pstricks.recall
+++ b/src_phystricks/Fig_ExoProjection.pstricks.recall
@@ -79,18 +79,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6000,0) -- (3.3000,0);
-\draw [,->,>=latex] (0,-1.5500) -- (0,3.5000);
+\draw [,->,>=latex] (-2.6000,0.0000) -- (3.3000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5500) -- (0.0000,3.5000);
 %DEFAULT
-\draw [style=dashed] (2.80,1.40) -- (-2.10,-1.05);
+\draw [style=dashed] (2.8000,1.4000) -- (-2.1000,-1.0500);
 \draw []  (2.4000,1.2000) node [rotate=0] {$\bullet$};
-\draw (3.2463,0.94356) node {$\pr_w(A)$};
+\draw (3.2463,0.9435) node {$\pr_w(A)$};
 \draw [color=red,->,>=latex] (1.5000,3.0000) -- (2.4000,1.2000);
-\draw [color=blue,->,>=latex] (0,0) -- (1.4000,0.70000);
-\draw (1.0716,1.0084) node {$w$};
-\draw []  (-1.0000,-0.50000) node [rotate=0] {$\bullet$};
-\draw (-1.3909,-0.78246) node {$P(\lambda)$};
-\draw [color=cyan,->,>=latex] (1.5000,3.0000) -- (-1.0000,-0.50000);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (1.4000,0.7000);
+\draw (1.0715,1.0083) node {$w$};
+\draw []  (-1.0000,-0.5000) node [rotate=0] {$\bullet$};
+\draw (-1.3908,-0.7824) node {$P(\lambda)$};
+\draw [color=cyan,->,>=latex] (1.5000,3.0000) -- (-1.0000,-0.5000);
 \draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
 \draw (1.5000,3.4247) node {$A$};
 
diff --git a/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall b/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall
index c9969ffe2..917499bce 100644
--- a/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall
+++ b/src_phystricks/Fig_ExoUnSurxPolaire.pstricks.recall
@@ -103,49 +103,49 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,4.5000);
 %DEFAULT
 
-\draw [color=blue] (-5.000,-0.2000)--(-4.952,-0.2019)--(-4.904,-0.2039)--(-4.856,-0.2059)--(-4.808,-0.2080)--(-4.760,-0.2101)--(-4.712,-0.2122)--(-4.664,-0.2144)--(-4.616,-0.2166)--(-4.568,-0.2189)--(-4.520,-0.2212)--(-4.472,-0.2236)--(-4.424,-0.2260)--(-4.376,-0.2285)--(-4.328,-0.2310)--(-4.280,-0.2336)--(-4.232,-0.2363)--(-4.184,-0.2390)--(-4.136,-0.2418)--(-4.088,-0.2446)--(-4.040,-0.2475)--(-3.992,-0.2505)--(-3.944,-0.2535)--(-3.896,-0.2566)--(-3.848,-0.2598)--(-3.801,-0.2631)--(-3.753,-0.2665)--(-3.705,-0.2699)--(-3.657,-0.2735)--(-3.609,-0.2771)--(-3.561,-0.2808)--(-3.513,-0.2847)--(-3.465,-0.2886)--(-3.417,-0.2927)--(-3.369,-0.2969)--(-3.321,-0.3011)--(-3.273,-0.3056)--(-3.225,-0.3101)--(-3.177,-0.3148)--(-3.129,-0.3196)--(-3.081,-0.3246)--(-3.033,-0.3297)--(-2.985,-0.3350)--(-2.937,-0.3405)--(-2.889,-0.3462)--(-2.841,-0.3520)--(-2.793,-0.3580)--(-2.745,-0.3643)--(-2.697,-0.3708)--(-2.649,-0.3775)--(-2.601,-0.3845)--(-2.553,-0.3917)--(-2.505,-0.3992)--(-2.457,-0.4070)--(-2.409,-0.4151)--(-2.361,-0.4235)--(-2.313,-0.4323)--(-2.265,-0.4415)--(-2.217,-0.4510)--(-2.169,-0.4610)--(-2.121,-0.4714)--(-2.073,-0.4823)--(-2.025,-0.4938)--(-1.977,-0.5057)--(-1.929,-0.5183)--(-1.881,-0.5315)--(-1.833,-0.5455)--(-1.785,-0.5601)--(-1.737,-0.5756)--(-1.689,-0.5919)--(-1.641,-0.6092)--(-1.593,-0.6276)--(-1.545,-0.6471)--(-1.497,-0.6678)--(-1.449,-0.6899)--(-1.402,-0.7135)--(-1.354,-0.7388)--(-1.306,-0.7660)--(-1.258,-0.7952)--(-1.210,-0.8267)--(-1.162,-0.8609)--(-1.114,-0.8980)--(-1.066,-0.9384)--(-1.018,-0.9826)--(-0.9697,-1.031)--(-0.9217,-1.085)--(-0.8737,-1.145)--(-0.8258,-1.211)--(-0.7778,-1.286)--(-0.7298,-1.370)--(-0.6818,-1.467)--(-0.6338,-1.578)--(-0.5859,-1.707)--(-0.5379,-1.859)--(-0.4899,-2.041)--(-0.4419,-2.263)--(-0.3939,-2.538)--(-0.3460,-2.891)--(-0.2980,-3.356)--(-0.2500,-4.000);
+\draw [color=blue] (-5.0000,-0.2000)--(-4.9520,-0.2019)--(-4.9040,-0.2039)--(-4.8560,-0.2059)--(-4.8080,-0.2079)--(-4.7601,-0.2100)--(-4.7121,-0.2122)--(-4.6641,-0.2144)--(-4.6161,-0.2166)--(-4.5681,-0.2189)--(-4.5202,-0.2212)--(-4.4722,-0.2236)--(-4.4242,-0.2260)--(-4.3762,-0.2285)--(-4.3282,-0.2310)--(-4.2803,-0.2336)--(-4.2323,-0.2362)--(-4.1843,-0.2389)--(-4.1363,-0.2417)--(-4.0883,-0.2445)--(-4.0404,-0.2475)--(-3.9924,-0.2504)--(-3.9444,-0.2535)--(-3.8964,-0.2566)--(-3.8484,-0.2598)--(-3.8005,-0.2631)--(-3.7525,-0.2664)--(-3.7045,-0.2699)--(-3.6565,-0.2734)--(-3.6085,-0.2771)--(-3.5606,-0.2808)--(-3.5126,-0.2846)--(-3.4646,-0.2886)--(-3.4166,-0.2926)--(-3.3686,-0.2968)--(-3.3207,-0.3011)--(-3.2727,-0.3055)--(-3.2247,-0.3101)--(-3.1767,-0.3147)--(-3.1287,-0.3196)--(-3.0808,-0.3245)--(-3.0328,-0.3297)--(-2.9848,-0.3350)--(-2.9368,-0.3404)--(-2.8888,-0.3461)--(-2.8409,-0.3520)--(-2.7929,-0.3580)--(-2.7449,-0.3643)--(-2.6969,-0.3707)--(-2.6489,-0.3775)--(-2.6010,-0.3844)--(-2.5530,-0.3916)--(-2.5050,-0.3991)--(-2.4570,-0.4069)--(-2.4090,-0.4150)--(-2.3611,-0.4235)--(-2.3131,-0.4323)--(-2.2651,-0.4414)--(-2.2171,-0.4510)--(-2.1691,-0.4610)--(-2.1212,-0.4714)--(-2.0732,-0.4823)--(-2.0252,-0.4937)--(-1.9772,-0.5057)--(-1.9292,-0.5183)--(-1.8813,-0.5315)--(-1.8333,-0.5454)--(-1.7853,-0.5601)--(-1.7373,-0.5755)--(-1.6893,-0.5919)--(-1.6414,-0.6092)--(-1.5934,-0.6275)--(-1.5454,-0.6470)--(-1.4974,-0.6677)--(-1.4494,-0.6898)--(-1.4015,-0.7135)--(-1.3535,-0.7388)--(-1.3055,-0.7659)--(-1.2575,-0.7951)--(-1.2095,-0.8267)--(-1.1616,-0.8608)--(-1.1136,-0.8979)--(-1.0656,-0.9383)--(-1.0176,-0.9826)--(-0.9696,-1.0312)--(-0.9217,-1.0849)--(-0.8737,-1.1445)--(-0.8257,-1.2110)--(-0.7777,-1.2857)--(-0.7297,-1.3702)--(-0.6818,-1.4666)--(-0.6338,-1.5776)--(-0.5858,-1.7068)--(-0.5378,-1.8591)--(-0.4898,-2.0412)--(-0.4419,-2.2628)--(-0.3939,-2.5384)--(-0.3459,-2.8905)--(-0.2979,-3.3559)--(-0.2500,-4.0000);
 
-\draw [color=blue] (0.2500,4.000)--(0.2980,3.356)--(0.3460,2.891)--(0.3939,2.538)--(0.4419,2.263)--(0.4899,2.041)--(0.5379,1.859)--(0.5859,1.707)--(0.6338,1.578)--(0.6818,1.467)--(0.7298,1.370)--(0.7778,1.286)--(0.8258,1.211)--(0.8737,1.145)--(0.9217,1.085)--(0.9697,1.031)--(1.018,0.9826)--(1.066,0.9384)--(1.114,0.8980)--(1.162,0.8609)--(1.210,0.8267)--(1.258,0.7952)--(1.306,0.7660)--(1.354,0.7388)--(1.402,0.7135)--(1.449,0.6899)--(1.497,0.6678)--(1.545,0.6471)--(1.593,0.6276)--(1.641,0.6092)--(1.689,0.5919)--(1.737,0.5756)--(1.785,0.5601)--(1.833,0.5455)--(1.881,0.5315)--(1.929,0.5183)--(1.977,0.5057)--(2.025,0.4938)--(2.073,0.4823)--(2.121,0.4714)--(2.169,0.4610)--(2.217,0.4510)--(2.265,0.4415)--(2.313,0.4323)--(2.361,0.4235)--(2.409,0.4151)--(2.457,0.4070)--(2.505,0.3992)--(2.553,0.3917)--(2.601,0.3845)--(2.649,0.3775)--(2.697,0.3708)--(2.745,0.3643)--(2.793,0.3580)--(2.841,0.3520)--(2.889,0.3462)--(2.937,0.3405)--(2.985,0.3350)--(3.033,0.3297)--(3.081,0.3246)--(3.129,0.3196)--(3.177,0.3148)--(3.225,0.3101)--(3.273,0.3056)--(3.321,0.3011)--(3.369,0.2969)--(3.417,0.2927)--(3.465,0.2886)--(3.513,0.2847)--(3.561,0.2808)--(3.609,0.2771)--(3.657,0.2735)--(3.705,0.2699)--(3.753,0.2665)--(3.801,0.2631)--(3.848,0.2598)--(3.896,0.2566)--(3.944,0.2535)--(3.992,0.2505)--(4.040,0.2475)--(4.088,0.2446)--(4.136,0.2418)--(4.184,0.2390)--(4.232,0.2363)--(4.280,0.2336)--(4.328,0.2310)--(4.376,0.2285)--(4.424,0.2260)--(4.472,0.2236)--(4.520,0.2212)--(4.568,0.2189)--(4.616,0.2166)--(4.664,0.2144)--(4.712,0.2122)--(4.760,0.2101)--(4.808,0.2080)--(4.856,0.2059)--(4.904,0.2039)--(4.952,0.2019)--(5.000,0.2000);
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (0.2500,4.0000)--(0.2979,3.3559)--(0.3459,2.8905)--(0.3939,2.5384)--(0.4419,2.2628)--(0.4898,2.0412)--(0.5378,1.8591)--(0.5858,1.7068)--(0.6338,1.5776)--(0.6818,1.4666)--(0.7297,1.3702)--(0.7777,1.2857)--(0.8257,1.2110)--(0.8737,1.1445)--(0.9217,1.0849)--(0.9696,1.0312)--(1.0176,0.9826)--(1.0656,0.9383)--(1.1136,0.8979)--(1.1616,0.8608)--(1.2095,0.8267)--(1.2575,0.7951)--(1.3055,0.7659)--(1.3535,0.7388)--(1.4015,0.7135)--(1.4494,0.6898)--(1.4974,0.6677)--(1.5454,0.6470)--(1.5934,0.6275)--(1.6414,0.6092)--(1.6893,0.5919)--(1.7373,0.5755)--(1.7853,0.5601)--(1.8333,0.5454)--(1.8813,0.5315)--(1.9292,0.5183)--(1.9772,0.5057)--(2.0252,0.4937)--(2.0732,0.4823)--(2.1212,0.4714)--(2.1691,0.4610)--(2.2171,0.4510)--(2.2651,0.4414)--(2.3131,0.4323)--(2.3611,0.4235)--(2.4090,0.4150)--(2.4570,0.4069)--(2.5050,0.3991)--(2.5530,0.3916)--(2.6010,0.3844)--(2.6489,0.3775)--(2.6969,0.3707)--(2.7449,0.3643)--(2.7929,0.3580)--(2.8409,0.3520)--(2.8888,0.3461)--(2.9368,0.3404)--(2.9848,0.3350)--(3.0328,0.3297)--(3.0808,0.3245)--(3.1287,0.3196)--(3.1767,0.3147)--(3.2247,0.3101)--(3.2727,0.3055)--(3.3207,0.3011)--(3.3686,0.2968)--(3.4166,0.2926)--(3.4646,0.2886)--(3.5126,0.2846)--(3.5606,0.2808)--(3.6085,0.2771)--(3.6565,0.2734)--(3.7045,0.2699)--(3.7525,0.2664)--(3.8005,0.2631)--(3.8484,0.2598)--(3.8964,0.2566)--(3.9444,0.2535)--(3.9924,0.2504)--(4.0404,0.2475)--(4.0883,0.2445)--(4.1363,0.2417)--(4.1843,0.2389)--(4.2323,0.2362)--(4.2803,0.2336)--(4.3282,0.2310)--(4.3762,0.2285)--(4.4242,0.2260)--(4.4722,0.2236)--(4.5202,0.2212)--(4.5681,0.2189)--(4.6161,0.2166)--(4.6641,0.2144)--(4.7121,0.2122)--(4.7601,0.2100)--(4.8080,0.2079)--(4.8560,0.2059)--(4.9040,0.2039)--(4.9520,0.2019)--(5.0000,0.2000);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExoXLVL.pstricks.recall b/src_phystricks/Fig_ExoXLVL.pstricks.recall
index 9dd1f83e3..0ac411b31 100644
--- a/src_phystricks/Fig_ExoXLVL.pstricks.recall
+++ b/src_phystricks/Fig_ExoXLVL.pstricks.recall
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
-\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red] (0,2.50) -- (2.50,2.50);
-\draw [color=red] (2.50,2.50) -- (2.50,0);
-\draw [color=red] (2.50,0) -- (0,0);
-\draw [color=red] (0,0) -- (0,2.50);
-\draw [color=cyan,style=dashed] (-0.100,-2.50) -- (-2.50,-2.50);
-\draw [color=cyan,style=dashed] (-2.50,-2.50) -- (-2.50,-0.100);
-\draw [color=cyan,style=dashed] (-2.50,-0.100) -- (-0.100,-0.100);
-\draw [color=cyan,style=dashed] (-0.100,-0.100) -- (-0.100,-2.50);
-\draw [color=green] (0,-2.50) -- (2.50,-2.50);
-\draw [color=green] (2.50,-2.50) -- (2.50,0);
-\draw [color=green] (2.50,0) -- (0,0);
-\draw [color=green] (0,0) -- (0,-2.50);
-\draw (-1.0997,1.3000) node {\( xy\)};
+\draw [color=blue,style=dashed] (-2.5000,0.1000) -- (-0.1000,0.1000);
+\draw [color=blue,style=dashed] (-0.1000,0.1000) -- (-0.1000,2.5000);
+\draw [color=blue,style=dashed] (-0.1000,2.5000) -- (-2.5000,2.5000);
+\draw [color=blue,style=dashed] (-2.5000,2.5000) -- (-2.5000,0.1000);
+\draw [color=red] (0.0000,2.5000) -- (2.5000,2.5000);
+\draw [color=red] (2.5000,2.5000) -- (2.5000,0.0000);
+\draw [color=red] (2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=red] (0.0000,0.0000) -- (0.0000,2.5000);
+\draw [color=cyan,style=dashed] (-0.1000,-2.5000) -- (-2.5000,-2.5000);
+\draw [color=cyan,style=dashed] (-2.5000,-2.5000) -- (-2.5000,-0.1000);
+\draw [color=cyan,style=dashed] (-2.5000,-0.1000) -- (-0.1000,-0.1000);
+\draw [color=cyan,style=dashed] (-0.1000,-0.1000) -- (-0.1000,-2.5000);
+\draw [color=green] (0.0000,-2.5000) -- (2.5000,-2.5000);
+\draw [color=green] (2.5000,-2.5000) -- (2.5000,0.0000);
+\draw [color=green] (2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=green] (0.0000,0.0000) -- (0.0000,-2.5000);
+\draw (-1.0996,1.3000) node {\( xy\)};
 \draw (1.6733,1.2500) node {\( x-y\)};
 \draw (-1.0205,-1.3000) node {\( x^2y\)};
 \draw (1.6733,-1.2500) node {\( x+y\)};
diff --git a/src_phystricks/Fig_FGWjJBX.pstricks.recall b/src_phystricks/Fig_FGWjJBX.pstricks.recall
index 78a04f112..a3aed9a03 100644
--- a/src_phystricks/Fig_FGWjJBX.pstricks.recall
+++ b/src_phystricks/Fig_FGWjJBX.pstricks.recall
@@ -82,22 +82,22 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [style=dotted] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0.500);
-\draw [] (3.00,0) -- (4.00,-0.500);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.30595) node {\( \alpha_1\)};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,0.30595) node {\( \alpha_2\)};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,0.32154) node {\( \alpha_{l-2}\)};
-\draw []  (4.0000,0.50000) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.82154) node {\( \alpha_{l-1}\)};
-\draw []  (4.0000,-0.50000) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.19169) node {\( \alpha_l\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [style=dotted] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.5000);
+\draw [] (3.0000,0.0000) -- (4.0000,-0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.3059) node {\( \alpha_1\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.3059) node {\( \alpha_2\)};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,0.3215) node {\( \alpha_{l-2}\)};
+\draw []  (4.0000,0.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.8215) node {\( \alpha_{l-1}\)};
+\draw []  (4.0000,-0.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.1916) node {\( \alpha_l\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall b/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall
index 1957bbd5f..bab9a7af9 100644
--- a/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall
+++ b/src_phystricks/Fig_FNBQooYgkAmS.pstricks.recall
@@ -74,13 +74,13 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (3.000,0)--(2.994,0.06342)--(2.976,0.1266)--(2.946,0.1893)--(2.904,0.2511)--(2.850,0.3120)--(2.785,0.3717)--(2.709,0.4298)--(2.622,0.4862)--(2.524,0.5406)--(2.416,0.5929)--(2.298,0.6428)--(2.171,0.6901)--(2.036,0.7346)--(1.892,0.7761)--(1.740,0.8146)--(1.582,0.8497)--(1.417,0.8815)--(1.246,0.9096)--(1.071,0.9342)--(0.8908,0.9549)--(0.7073,0.9718)--(0.5209,0.9848)--(0.3325,0.9938)--(0.1427,0.9989)--(-0.04760,0.9999)--(-0.2377,0.9969)--(-0.4269,0.9898)--(-0.6144,0.9788)--(-0.7994,0.9638)--(-0.9812,0.9450)--(-1.159,0.9224)--(-1.332,0.8960)--(-1.500,0.8660)--(-1.662,0.8326)--(-1.817,0.7958)--(-1.965,0.7558)--(-2.104,0.7127)--(-2.236,0.6668)--(-2.358,0.6182)--(-2.471,0.5671)--(-2.574,0.5137)--(-2.667,0.4582)--(-2.748,0.4009)--(-2.819,0.3420)--(-2.878,0.2817)--(-2.926,0.2203)--(-2.962,0.1580)--(-2.986,0.09506)--(-2.999,0.03173)--(-2.999,-0.03173)--(-2.986,-0.09506)--(-2.962,-0.1580)--(-2.926,-0.2203)--(-2.878,-0.2817)--(-2.819,-0.3420)--(-2.748,-0.4009)--(-2.667,-0.4582)--(-2.574,-0.5137)--(-2.471,-0.5671)--(-2.358,-0.6182)--(-2.236,-0.6668)--(-2.104,-0.7127)--(-1.965,-0.7558)--(-1.817,-0.7958)--(-1.662,-0.8326)--(-1.500,-0.8660)--(-1.332,-0.8960)--(-1.159,-0.9224)--(-0.9812,-0.9450)--(-0.7994,-0.9638)--(-0.6144,-0.9788)--(-0.4269,-0.9898)--(-0.2377,-0.9969)--(-0.04760,-0.9999)--(0.1427,-0.9989)--(0.3325,-0.9938)--(0.5209,-0.9848)--(0.7073,-0.9718)--(0.8908,-0.9549)--(1.071,-0.9342)--(1.246,-0.9096)--(1.417,-0.8815)--(1.582,-0.8497)--(1.740,-0.8146)--(1.892,-0.7761)--(2.036,-0.7346)--(2.171,-0.6901)--(2.298,-0.6428)--(2.416,-0.5929)--(2.524,-0.5406)--(2.622,-0.4862)--(2.709,-0.4298)--(2.785,-0.3717)--(2.850,-0.3120)--(2.904,-0.2511)--(2.946,-0.1893)--(2.976,-0.1266)--(2.994,-0.06342)--(3.000,0);
-\draw []  (2.1213,0.70711) node [rotate=0] {$\bullet$};
-\draw (2.0116,0.98561) node {\( x\)};
-\draw [,->,>=latex] (2.1213,0.70711) -- (2.4375,1.6558);
-\draw (1.7615,1.9015) node {\( \nabla q(x)\)};
-\draw [,->,>=latex] (2.1213,0.70711) -- (2.1486,-0.29252);
-\draw (2.5897,-0.16235) node {\( Ax\)};
+\draw [color=blue] (3.0000,0.0000)--(2.9939,0.0634)--(2.9758,0.1265)--(2.9457,0.1892)--(2.9038,0.2511)--(2.8502,0.3120)--(2.7851,0.3716)--(2.7087,0.4297)--(2.6215,0.4861)--(2.5237,0.5406)--(2.4158,0.5929)--(2.2981,0.6427)--(2.1712,0.6900)--(2.0355,0.7345)--(1.8916,0.7761)--(1.7401,0.8145)--(1.5816,0.8497)--(1.4168,0.8814)--(1.2462,0.9096)--(1.0706,0.9341)--(0.8907,0.9549)--(0.7072,0.9718)--(0.5209,0.9848)--(0.3325,0.9938)--(0.1427,0.9988)--(-0.0475,0.9998)--(-0.2377,0.9968)--(-0.4269,0.9898)--(-0.6144,0.9788)--(-0.7994,0.9638)--(-0.9812,0.9450)--(-1.1590,0.9223)--(-1.3321,0.8959)--(-1.5000,0.8660)--(-1.6617,0.8325)--(-1.8168,0.7957)--(-1.9645,0.7557)--(-2.1044,0.7126)--(-2.2357,0.6667)--(-2.3581,0.6181)--(-2.4710,0.5670)--(-2.5739,0.5136)--(-2.6665,0.4582)--(-2.7483,0.4009)--(-2.8190,0.3420)--(-2.8784,0.2817)--(-2.9262,0.2203)--(-2.9623,0.1580)--(-2.9864,0.0950)--(-2.9984,0.0317)--(-2.9984,-0.0317)--(-2.9864,-0.0950)--(-2.9623,-0.1580)--(-2.9262,-0.2203)--(-2.8784,-0.2817)--(-2.8190,-0.3420)--(-2.7483,-0.4009)--(-2.6665,-0.4582)--(-2.5739,-0.5136)--(-2.4710,-0.5670)--(-2.3581,-0.6181)--(-2.2357,-0.6667)--(-2.1044,-0.7126)--(-1.9645,-0.7557)--(-1.8168,-0.7957)--(-1.6617,-0.8325)--(-1.5000,-0.8660)--(-1.3321,-0.8959)--(-1.1590,-0.9223)--(-0.9812,-0.9450)--(-0.7994,-0.9638)--(-0.6144,-0.9788)--(-0.4269,-0.9898)--(-0.2377,-0.9968)--(-0.0475,-0.9998)--(0.1427,-0.9988)--(0.3325,-0.9938)--(0.5209,-0.9848)--(0.7072,-0.9718)--(0.8907,-0.9549)--(1.0706,-0.9341)--(1.2462,-0.9096)--(1.4168,-0.8814)--(1.5816,-0.8497)--(1.7401,-0.8145)--(1.8916,-0.7761)--(2.0355,-0.7345)--(2.1712,-0.6900)--(2.2981,-0.6427)--(2.4158,-0.5929)--(2.5237,-0.5406)--(2.6215,-0.4861)--(2.7087,-0.4297)--(2.7851,-0.3716)--(2.8502,-0.3120)--(2.9038,-0.2511)--(2.9457,-0.1892)--(2.9758,-0.1265)--(2.9939,-0.0634)--(3.0000,0.0000);
+\draw []  (2.1213,0.7071) node [rotate=0] {$\bullet$};
+\draw (2.0115,0.9856) node {\( x\)};
+\draw [,->,>=latex] (2.1213,0.7071) -- (2.4375,1.6557);
+\draw (1.7614,1.9014) node {\( \nabla q(x)\)};
+\draw [,->,>=latex] (2.1213,0.7071) -- (2.1486,-0.2925);
+\draw (2.5897,-0.1623) node {\( Ax\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_FWJuNhU.pstricks.recall b/src_phystricks/Fig_FWJuNhU.pstricks.recall
index c5cad1966..2dae2d4c7 100644
--- a/src_phystricks/Fig_FWJuNhU.pstricks.recall
+++ b/src_phystricks/Fig_FWJuNhU.pstricks.recall
@@ -79,28 +79,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
-\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red,style=dashed] (0,2.50) -- (2.50,2.50);
-\draw [color=red,style=dashed] (2.50,2.50) -- (2.50,0);
-\draw [color=red,style=dashed] (2.50,0) -- (0,0);
-\draw [color=red,style=dashed] (0,0) -- (0,2.50);
-\draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
-\draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
-\draw [color=cyan] (-2.50,0) -- (0,0);
-\draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green,style=dashed] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
-\draw (-1.0997,1.3000) node {\( xy\)};
+\draw [color=blue,style=dashed] (-2.5000,0.1000) -- (-0.1000,0.1000);
+\draw [color=blue,style=dashed] (-0.1000,0.1000) -- (-0.1000,2.5000);
+\draw [color=blue,style=dashed] (-0.1000,2.5000) -- (-2.5000,2.5000);
+\draw [color=blue,style=dashed] (-2.5000,2.5000) -- (-2.5000,0.1000);
+\draw [color=red,style=dashed] (0.0000,2.5000) -- (2.5000,2.5000);
+\draw [color=red,style=dashed] (2.5000,2.5000) -- (2.5000,0.0000);
+\draw [color=red,style=dashed] (2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=red,style=dashed] (0.0000,0.0000) -- (0.0000,2.5000);
+\draw [color=cyan] (0.0000,-2.5000) -- (-2.5000,-2.5000);
+\draw [color=cyan] (-2.5000,-2.5000) -- (-2.5000,0.0000);
+\draw [color=cyan] (-2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=cyan] (0.0000,0.0000) -- (0.0000,-2.5000);
+\draw [color=green,style=dashed] (0.1000,-2.5000) -- (2.5000,-2.5000);
+\draw [color=green,style=dashed] (2.5000,-2.5000) -- (2.5000,-0.1000);
+\draw [color=green,style=dashed] (2.5000,-0.1000) -- (0.1000,-0.1000);
+\draw [color=green,style=dashed] (0.1000,-0.1000) -- (0.1000,-2.5000);
+\draw (-1.0996,1.3000) node {\( xy\)};
 \draw (1.6733,1.2500) node {\( x-y\)};
-\draw (-0.97051,-1.2500) node {\( x^2y\)};
+\draw (-0.9705,-1.2500) node {\( x^2y\)};
 \draw (1.7233,-1.3000) node {\( x+y\)};
 
 %OTHER STUFF
diff --git a/src_phystricks/Fig_FXVooJYAfif.pstricks.recall b/src_phystricks/Fig_FXVooJYAfif.pstricks.recall
index e272810db..b4d41699b 100644
--- a/src_phystricks/Fig_FXVooJYAfif.pstricks.recall
+++ b/src_phystricks/Fig_FXVooJYAfif.pstricks.recall
@@ -60,13 +60,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.500000000,0) -- (4.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [color=blue] (-4.000,-0.8440)--(-3.919,-0.8410)--(-3.838,-0.8377)--(-3.758,-0.8344)--(-3.677,-0.8309)--(-3.596,-0.8273)--(-3.515,-0.8236)--(-3.434,-0.8196)--(-3.354,-0.8155)--(-3.273,-0.8112)--(-3.192,-0.8067)--(-3.111,-0.8020)--(-3.030,-0.7971)--(-2.949,-0.7919)--(-2.869,-0.7865)--(-2.788,-0.7807)--(-2.707,-0.7747)--(-2.626,-0.7684)--(-2.545,-0.7617)--(-2.465,-0.7546)--(-2.384,-0.7471)--(-2.303,-0.7392)--(-2.222,-0.7308)--(-2.141,-0.7219)--(-2.061,-0.7124)--(-1.980,-0.7022)--(-1.899,-0.6914)--(-1.818,-0.6799)--(-1.737,-0.6675)--(-1.657,-0.6543)--(-1.576,-0.6400)--(-1.495,-0.6247)--(-1.414,-0.6082)--(-1.333,-0.5903)--(-1.253,-0.5711)--(-1.172,-0.5502)--(-1.091,-0.5277)--(-1.010,-0.5032)--(-0.9293,-0.4767)--(-0.8485,-0.4479)--(-0.7677,-0.4168)--(-0.6869,-0.3832)--(-0.6061,-0.3469)--(-0.5253,-0.3079)--(-0.4444,-0.2663)--(-0.3636,-0.2220)--(-0.2828,-0.1755)--(-0.2020,-0.1269)--(-0.1212,-0.07679)--(-0.04040,-0.02571)--(0.04040,0.02571)--(0.1212,0.07679)--(0.2020,0.1269)--(0.2828,0.1755)--(0.3636,0.2220)--(0.4444,0.2663)--(0.5253,0.3079)--(0.6061,0.3469)--(0.6869,0.3832)--(0.7677,0.4168)--(0.8485,0.4479)--(0.9293,0.4767)--(1.010,0.5032)--(1.091,0.5277)--(1.172,0.5502)--(1.253,0.5711)--(1.333,0.5903)--(1.414,0.6082)--(1.495,0.6247)--(1.576,0.6400)--(1.657,0.6543)--(1.737,0.6675)--(1.818,0.6799)--(1.899,0.6914)--(1.980,0.7022)--(2.061,0.7124)--(2.141,0.7219)--(2.222,0.7308)--(2.303,0.7392)--(2.384,0.7471)--(2.465,0.7546)--(2.545,0.7617)--(2.626,0.7684)--(2.707,0.7747)--(2.788,0.7807)--(2.869,0.7865)--(2.949,0.7919)--(3.030,0.7971)--(3.111,0.8020)--(3.192,0.8067)--(3.273,0.8112)--(3.354,0.8155)--(3.434,0.8196)--(3.515,0.8236)--(3.596,0.8273)--(3.677,0.8309)--(3.758,0.8344)--(3.838,0.8377)--(3.919,0.8410)--(4.000,0.8440);
-\draw [color=red] (-4.00,1.00) -- (4.00,1.00);
-\draw [color=red] (-4.00,-1.00) -- (4.00,-1.00);
+\draw [color=blue] (-4.0000,-0.8440)--(-3.9191,-0.8409)--(-3.8383,-0.8377)--(-3.7575,-0.8344)--(-3.6767,-0.8309)--(-3.5959,-0.8273)--(-3.5151,-0.8235)--(-3.4343,-0.8196)--(-3.3535,-0.8155)--(-3.2727,-0.8112)--(-3.1919,-0.8067)--(-3.1111,-0.8020)--(-3.0303,-0.7970)--(-2.9494,-0.7919)--(-2.8686,-0.7864)--(-2.7878,-0.7807)--(-2.7070,-0.7747)--(-2.6262,-0.7683)--(-2.5454,-0.7616)--(-2.4646,-0.7546)--(-2.3838,-0.7471)--(-2.3030,-0.7392)--(-2.2222,-0.7308)--(-2.1414,-0.7218)--(-2.0606,-0.7123)--(-1.9797,-0.7022)--(-1.8989,-0.6914)--(-1.8181,-0.6798)--(-1.7373,-0.6675)--(-1.6565,-0.6542)--(-1.5757,-0.6400)--(-1.4949,-0.6246)--(-1.4141,-0.6081)--(-1.3333,-0.5903)--(-1.2525,-0.5710)--(-1.1717,-0.5502)--(-1.0909,-0.5276)--(-1.0101,-0.5031)--(-0.9292,-0.4766)--(-0.8484,-0.4479)--(-0.7676,-0.4168)--(-0.6868,-0.3831)--(-0.6060,-0.3468)--(-0.5252,-0.3078)--(-0.4444,-0.2662)--(-0.3636,-0.2220)--(-0.2828,-0.1754)--(-0.2020,-0.1269)--(-0.1212,-0.0767)--(-0.0404,-0.0257)--(0.0404,0.0257)--(0.1212,0.0767)--(0.2020,0.1269)--(0.2828,0.1754)--(0.3636,0.2220)--(0.4444,0.2662)--(0.5252,0.3078)--(0.6060,0.3468)--(0.6868,0.3831)--(0.7676,0.4168)--(0.8484,0.4479)--(0.9292,0.4766)--(1.0101,0.5031)--(1.0909,0.5276)--(1.1717,0.5502)--(1.2525,0.5710)--(1.3333,0.5903)--(1.4141,0.6081)--(1.4949,0.6246)--(1.5757,0.6400)--(1.6565,0.6542)--(1.7373,0.6675)--(1.8181,0.6798)--(1.8989,0.6914)--(1.9797,0.7022)--(2.0606,0.7123)--(2.1414,0.7218)--(2.2222,0.7308)--(2.3030,0.7392)--(2.3838,0.7471)--(2.4646,0.7546)--(2.5454,0.7616)--(2.6262,0.7683)--(2.7070,0.7747)--(2.7878,0.7807)--(2.8686,0.7864)--(2.9494,0.7919)--(3.0303,0.7970)--(3.1111,0.8020)--(3.1919,0.8067)--(3.2727,0.8112)--(3.3535,0.8155)--(3.4343,0.8196)--(3.5151,0.8235)--(3.5959,0.8273)--(3.6767,0.8309)--(3.7575,0.8344)--(3.8383,0.8377)--(3.9191,0.8409)--(4.0000,0.8440);
+\draw [color=red] (-4.0000,1.0000) -- (4.0000,1.0000);
+\draw [color=red] (-4.0000,-1.0000) -- (4.0000,-1.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall b/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall
index 8c1bda278..0211c51c8 100644
--- a/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall
+++ b/src_phystricks/Fig_FonctionXtroisOM.pstricks.recall
@@ -91,31 +91,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
-\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
+\draw [,->,>=latex] (-1.4000,0.0000) -- (1.4000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5250) -- (0.0000,4.5500);
 %DEFAULT
 
-\draw [color=blue] (-0.9000,-2.025)--(-0.8818,-1.905)--(-0.8636,-1.789)--(-0.8455,-1.679)--(-0.8273,-1.573)--(-0.8091,-1.471)--(-0.7909,-1.374)--(-0.7727,-1.282)--(-0.7545,-1.193)--(-0.7364,-1.109)--(-0.7182,-1.029)--(-0.7000,-0.9528)--(-0.6818,-0.8804)--(-0.6636,-0.8119)--(-0.6455,-0.7470)--(-0.6273,-0.6856)--(-0.6091,-0.6277)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5364,-0.4286)--(-0.5182,-0.3865)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4455,-0.2455)--(-0.4273,-0.2167)--(-0.4091,-0.1902)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1238)--(-0.3364,-0.1057)--(-0.3182,-0.08948)--(-0.3000,-0.07500)--(-0.2818,-0.06217)--(-0.2636,-0.05090)--(-0.2455,-0.04108)--(-0.2273,-0.03261)--(-0.2091,-0.02539)--(-0.1909,-0.01933)--(-0.1727,-0.01431)--(-0.1545,-0.01025)--(-0.1364,-0.007044)--(-0.1182,-0.004585)--(-0.1000,-0.002778)--(-0.08182,-0.001521)--(-0.06364,0)--(-0.04545,0)--(-0.02727,0)--(-0.009091,0)--(0.009091,0)--(0.02727,0)--(0.04545,0)--(0.06364,0)--(0.08182,0.001521)--(0.1000,0.002778)--(0.1182,0.004585)--(0.1364,0.007044)--(0.1545,0.01025)--(0.1727,0.01431)--(0.1909,0.01933)--(0.2091,0.02539)--(0.2273,0.03261)--(0.2455,0.04108)--(0.2636,0.05090)--(0.2818,0.06217)--(0.3000,0.07500)--(0.3182,0.08948)--(0.3364,0.1057)--(0.3545,0.1238)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4091,0.1902)--(0.4273,0.2167)--(0.4455,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5182,0.3865)--(0.5364,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6091,0.6277)--(0.6273,0.6856)--(0.6455,0.7470)--(0.6636,0.8119)--(0.6818,0.8804)--(0.7000,0.9528)--(0.7182,1.029)--(0.7364,1.109)--(0.7545,1.193)--(0.7727,1.282)--(0.7909,1.374)--(0.8091,1.471)--(0.8273,1.573)--(0.8455,1.679)--(0.8636,1.789)--(0.8818,1.905)--(0.9000,2.025);
+\draw [color=blue] (-0.9000,-2.0250)--(-0.8818,-1.9047)--(-0.8636,-1.7893)--(-0.8454,-1.6786)--(-0.8272,-1.5726)--(-0.8090,-1.4712)--(-0.7909,-1.3742)--(-0.7727,-1.2816)--(-0.7545,-1.1933)--(-0.7363,-1.1091)--(-0.7181,-1.0289)--(-0.7000,-0.9527)--(-0.6818,-0.8804)--(-0.6636,-0.8118)--(-0.6454,-0.7469)--(-0.6272,-0.6855)--(-0.6090,-0.6276)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5363,-0.4286)--(-0.5181,-0.3864)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4454,-0.2455)--(-0.4272,-0.2166)--(-0.4090,-0.1901)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1237)--(-0.3363,-0.1057)--(-0.3181,-0.0894)--(-0.3000,-0.0750)--(-0.2818,-0.0621)--(-0.2636,-0.0508)--(-0.2454,-0.0410)--(-0.2272,-0.0326)--(-0.2090,-0.0253)--(-0.1909,-0.0193)--(-0.1727,-0.0143)--(-0.1545,-0.0102)--(-0.1363,-0.0070)--(-0.1181,-0.0045)--(-0.0999,-0.0027)--(-0.0818,-0.0015)--(-0.0636,0.0000)--(-0.0454,0.0000)--(-0.0272,0.0000)--(-0.0090,0.0000)--(0.0090,0.0000)--(0.0272,0.0000)--(0.0454,0.0000)--(0.0636,0.0000)--(0.0818,0.0015)--(0.1000,0.0027)--(0.1181,0.0045)--(0.1363,0.0070)--(0.1545,0.0102)--(0.1727,0.0143)--(0.1909,0.0193)--(0.2090,0.0253)--(0.2272,0.0326)--(0.2454,0.0410)--(0.2636,0.0508)--(0.2818,0.0621)--(0.3000,0.0750)--(0.3181,0.0894)--(0.3363,0.1057)--(0.3545,0.1237)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4090,0.1901)--(0.4272,0.2166)--(0.4454,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5181,0.3864)--(0.5363,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6090,0.6276)--(0.6272,0.6855)--(0.6454,0.7469)--(0.6636,0.8118)--(0.6818,0.8804)--(0.7000,0.9527)--(0.7181,1.0289)--(0.7363,1.1091)--(0.7545,1.1933)--(0.7727,1.2816)--(0.7909,1.3742)--(0.8090,1.4712)--(0.8272,1.5726)--(0.8454,1.6786)--(0.8636,1.7893)--(0.8818,1.9047)--(0.9000,2.0250);
 
-\draw [color=red] (-0.9000,4.050)--(-0.8818,3.888)--(-0.8636,3.729)--(-0.8455,3.574)--(-0.8273,3.422)--(-0.8091,3.273)--(-0.7909,3.128)--(-0.7727,2.986)--(-0.7545,2.847)--(-0.7364,2.711)--(-0.7182,2.579)--(-0.7000,2.450)--(-0.6818,2.324)--(-0.6636,2.202)--(-0.6455,2.083)--(-0.6273,1.967)--(-0.6091,1.855)--(-0.5909,1.746)--(-0.5727,1.640)--(-0.5545,1.538)--(-0.5364,1.438)--(-0.5182,1.343)--(-0.5000,1.250)--(-0.4818,1.161)--(-0.4636,1.075)--(-0.4455,0.9921)--(-0.4273,0.9128)--(-0.4091,0.8368)--(-0.3909,0.7641)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3364,0.5657)--(-0.3182,0.5062)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2455,0.3012)--(-0.2273,0.2583)--(-0.2091,0.2186)--(-0.1909,0.1822)--(-0.1727,0.1492)--(-0.1545,0.1194)--(-0.1364,0.09298)--(-0.1182,0.06983)--(-0.1000,0.05000)--(-0.08182,0.03347)--(-0.06364,0.02025)--(-0.04545,0.01033)--(-0.02727,0.003719)--(-0.009091,0)--(0.009091,0)--(0.02727,0.003719)--(0.04545,0.01033)--(0.06364,0.02025)--(0.08182,0.03347)--(0.1000,0.05000)--(0.1182,0.06983)--(0.1364,0.09298)--(0.1545,0.1194)--(0.1727,0.1492)--(0.1909,0.1822)--(0.2091,0.2186)--(0.2273,0.2583)--(0.2455,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3182,0.5062)--(0.3364,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7641)--(0.4091,0.8368)--(0.4273,0.9128)--(0.4455,0.9921)--(0.4636,1.075)--(0.4818,1.161)--(0.5000,1.250)--(0.5182,1.343)--(0.5364,1.438)--(0.5545,1.538)--(0.5727,1.640)--(0.5909,1.746)--(0.6091,1.855)--(0.6273,1.967)--(0.6455,2.083)--(0.6636,2.202)--(0.6818,2.324)--(0.7000,2.450)--(0.7182,2.579)--(0.7364,2.711)--(0.7545,2.847)--(0.7727,2.986)--(0.7909,3.128)--(0.8091,3.273)--(0.8273,3.422)--(0.8455,3.574)--(0.8636,3.729)--(0.8818,3.888)--(0.9000,4.050);
-\draw (-1.2000,-0.32983) node {$ -2 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (-0.60000,-0.32983) node {$ -1 $};
-\draw [] (-0.600,-0.100) -- (-0.600,0.100);
-\draw (0.60000,-0.31492) node {$ 1 $};
-\draw [] (0.600,-0.100) -- (0.600,0.100);
-\draw (1.2000,-0.31492) node {$ 2 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (-0.43316,-2.4000) node {$ -4 $};
-\draw [] (-0.100,-2.40) -- (0.100,-2.40);
-\draw (-0.43316,-1.2000) node {$ -2 $};
-\draw [] (-0.100,-1.20) -- (0.100,-1.20);
-\draw (-0.29125,1.2000) node {$ 2 $};
-\draw [] (-0.100,1.20) -- (0.100,1.20);
-\draw (-0.29125,2.4000) node {$ 4 $};
-\draw [] (-0.100,2.40) -- (0.100,2.40);
-\draw (-0.29125,3.6000) node {$ 6 $};
-\draw [] (-0.100,3.60) -- (0.100,3.60);
+\draw [color=red] (-0.9000,4.0500)--(-0.8818,3.8880)--(-0.8636,3.7293)--(-0.8454,3.5739)--(-0.8272,3.4219)--(-0.8090,3.2731)--(-0.7909,3.1276)--(-0.7727,2.9855)--(-0.7545,2.8466)--(-0.7363,2.7111)--(-0.7181,2.5789)--(-0.7000,2.4500)--(-0.6818,2.3243)--(-0.6636,2.2020)--(-0.6454,2.0830)--(-0.6272,1.9673)--(-0.6090,1.8549)--(-0.5909,1.7458)--(-0.5727,1.6400)--(-0.5545,1.5376)--(-0.5363,1.4384)--(-0.5181,1.3425)--(-0.5000,1.2500)--(-0.4818,1.1607)--(-0.4636,1.0747)--(-0.4454,0.9921)--(-0.4272,0.9128)--(-0.4090,0.8367)--(-0.3909,0.7640)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3363,0.5657)--(-0.3181,0.5061)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2454,0.3012)--(-0.2272,0.2582)--(-0.2090,0.2185)--(-0.1909,0.1822)--(-0.1727,0.1491)--(-0.1545,0.1194)--(-0.1363,0.0929)--(-0.1181,0.0698)--(-0.0999,0.0499)--(-0.0818,0.0334)--(-0.0636,0.0202)--(-0.0454,0.0103)--(-0.0272,0.0037)--(-0.0090,0.0000)--(0.0090,0.0000)--(0.0272,0.0037)--(0.0454,0.0103)--(0.0636,0.0202)--(0.0818,0.0334)--(0.1000,0.0500)--(0.1181,0.0698)--(0.1363,0.0929)--(0.1545,0.1194)--(0.1727,0.1491)--(0.1909,0.1822)--(0.2090,0.2185)--(0.2272,0.2582)--(0.2454,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3181,0.5061)--(0.3363,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7640)--(0.4090,0.8367)--(0.4272,0.9128)--(0.4454,0.9921)--(0.4636,1.0747)--(0.4818,1.1607)--(0.5000,1.2500)--(0.5181,1.3425)--(0.5363,1.4384)--(0.5545,1.5376)--(0.5727,1.6400)--(0.5909,1.7458)--(0.6090,1.8549)--(0.6272,1.9673)--(0.6454,2.0830)--(0.6636,2.2020)--(0.6818,2.3243)--(0.7000,2.4500)--(0.7181,2.5789)--(0.7363,2.7111)--(0.7545,2.8466)--(0.7727,2.9855)--(0.7909,3.1276)--(0.8090,3.2731)--(0.8272,3.4219)--(0.8454,3.5739)--(0.8636,3.7293)--(0.8818,3.8880)--(0.9000,4.0500);
+\draw (-1.2000,-0.3298) node {$ -2 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (-0.6000,-0.3298) node {$ -1 $};
+\draw [] (-0.6000,-0.1000) -- (-0.6000,0.1000);
+\draw (0.6000,-0.3149) node {$ 1 $};
+\draw [] (0.6000,-0.1000) -- (0.6000,0.1000);
+\draw (1.2000,-0.3149) node {$ 2 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (-0.4331,-2.4000) node {$ -4 $};
+\draw [] (-0.1000,-2.4000) -- (0.1000,-2.4000);
+\draw (-0.4331,-1.2000) node {$ -2 $};
+\draw [] (-0.1000,-1.2000) -- (0.1000,-1.2000);
+\draw (-0.2912,1.2000) node {$ 2 $};
+\draw [] (-0.1000,1.2000) -- (0.1000,1.2000);
+\draw (-0.2912,2.4000) node {$ 4 $};
+\draw [] (-0.1000,2.4000) -- (0.1000,2.4000);
+\draw (-0.2912,3.6000) node {$ 6 $};
+\draw [] (-0.1000,3.6000) -- (0.1000,3.6000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_GBnUivi.pstricks.recall b/src_phystricks/Fig_GBnUivi.pstricks.recall
index 447475bac..eb4ed3200 100644
--- a/src_phystricks/Fig_GBnUivi.pstricks.recall
+++ b/src_phystricks/Fig_GBnUivi.pstricks.recall
@@ -102,61 +102,61 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (0.091250,1.0000) node {1};
-\draw [] (-0.250,0.750) -- (0.250,0.750);
-\draw [] (0.250,0.750) -- (0.250,1.25);
-\draw [] (0.250,1.25) -- (-0.250,1.25);
-\draw [] (-0.250,1.25) -- (-0.250,0.750);
-\draw (0.59125,1.0000) node {2};
-\draw [] (0.250,0.750) -- (0.750,0.750);
-\draw [] (0.750,0.750) -- (0.750,1.25);
-\draw [] (0.750,1.25) -- (0.250,1.25);
-\draw [] (0.250,1.25) -- (0.250,0.750);
+\draw (0.0912,1.0000) node {1};
+\draw [] (-0.2500,0.7500) -- (0.2500,0.7500);
+\draw [] (0.2500,0.7500) -- (0.2500,1.2500);
+\draw [] (0.2500,1.2500) -- (-0.2500,1.2500);
+\draw [] (-0.2500,1.2500) -- (-0.2500,0.7500);
+\draw (0.5912,1.0000) node {2};
+\draw [] (0.2500,0.7500) -- (0.7500,0.7500);
+\draw [] (0.7500,0.7500) -- (0.7500,1.2500);
+\draw [] (0.7500,1.2500) -- (0.2500,1.2500);
+\draw [] (0.2500,1.2500) -- (0.2500,0.7500);
 \draw (1.0912,1.0000) node {3};
-\draw [] (0.750,0.750) -- (1.25,0.750);
-\draw [] (1.25,0.750) -- (1.25,1.25);
-\draw [] (1.25,1.25) -- (0.750,1.25);
-\draw [] (0.750,1.25) -- (0.750,0.750);
+\draw [] (0.7500,0.7500) -- (1.2500,0.7500);
+\draw [] (1.2500,0.7500) -- (1.2500,1.2500);
+\draw [] (1.2500,1.2500) -- (0.7500,1.2500);
+\draw [] (0.7500,1.2500) -- (0.7500,0.7500);
 \draw (1.5912,1.0000) node {4};
-\draw [] (1.25,0.750) -- (1.75,0.750);
-\draw [] (1.75,0.750) -- (1.75,1.25);
-\draw [] (1.75,1.25) -- (1.25,1.25);
-\draw [] (1.25,1.25) -- (1.25,0.750);
-\draw (2.0913,1.0000) node {7};
-\draw [] (1.75,0.750) -- (2.25,0.750);
-\draw [] (2.25,0.750) -- (2.25,1.25);
-\draw [] (2.25,1.25) -- (1.75,1.25);
-\draw [] (1.75,1.25) -- (1.75,0.750);
-\draw (2.5913,1.0000) node {8};
-\draw [] (2.25,0.750) -- (2.75,0.750);
-\draw [] (2.75,0.750) -- (2.75,1.25);
-\draw [] (2.75,1.25) -- (2.25,1.25);
-\draw [] (2.25,1.25) -- (2.25,0.750);
-\draw (0.091250,0.50000) node {3};
-\draw [] (-0.250,0.250) -- (0.250,0.250);
-\draw [] (0.250,0.250) -- (0.250,0.750);
-\draw [] (0.250,0.750) -- (-0.250,0.750);
-\draw [] (-0.250,0.750) -- (-0.250,0.250);
-\draw (0.59125,0.50000) node {5};
-\draw [] (0.250,0.250) -- (0.750,0.250);
-\draw [] (0.750,0.250) -- (0.750,0.750);
-\draw [] (0.750,0.750) -- (0.250,0.750);
-\draw [] (0.250,0.750) -- (0.250,0.250);
-\draw (1.0912,0.50000) node {6};
-\draw [] (0.750,0.250) -- (1.25,0.250);
-\draw [] (1.25,0.250) -- (1.25,0.750);
-\draw [] (1.25,0.750) -- (0.750,0.750);
-\draw [] (0.750,0.750) -- (0.750,0.250);
-\draw (1.5912,0.50000) node {9};
-\draw [] (1.25,0.250) -- (1.75,0.250);
-\draw [] (1.75,0.250) -- (1.75,0.750);
-\draw [] (1.75,0.750) -- (1.25,0.750);
-\draw [] (1.25,0.750) -- (1.25,0.250);
-\draw (0.18250,0) node {10};
-\draw [] (-0.250,-0.250) -- (0.250,-0.250);
-\draw [] (0.250,-0.250) -- (0.250,0.250);
-\draw [] (0.250,0.250) -- (-0.250,0.250);
-\draw [] (-0.250,0.250) -- (-0.250,-0.250);
+\draw [] (1.2500,0.7500) -- (1.7500,0.7500);
+\draw [] (1.7500,0.7500) -- (1.7500,1.2500);
+\draw [] (1.7500,1.2500) -- (1.2500,1.2500);
+\draw [] (1.2500,1.2500) -- (1.2500,0.7500);
+\draw (2.0912,1.0000) node {7};
+\draw [] (1.7500,0.7500) -- (2.2500,0.7500);
+\draw [] (2.2500,0.7500) -- (2.2500,1.2500);
+\draw [] (2.2500,1.2500) -- (1.7500,1.2500);
+\draw [] (1.7500,1.2500) -- (1.7500,0.7500);
+\draw (2.5912,1.0000) node {8};
+\draw [] (2.2500,0.7500) -- (2.7500,0.7500);
+\draw [] (2.7500,0.7500) -- (2.7500,1.2500);
+\draw [] (2.7500,1.2500) -- (2.2500,1.2500);
+\draw [] (2.2500,1.2500) -- (2.2500,0.7500);
+\draw (0.0912,0.5000) node {3};
+\draw [] (-0.2500,0.2500) -- (0.2500,0.2500);
+\draw [] (0.2500,0.2500) -- (0.2500,0.7500);
+\draw [] (0.2500,0.7500) -- (-0.2500,0.7500);
+\draw [] (-0.2500,0.7500) -- (-0.2500,0.2500);
+\draw (0.5912,0.5000) node {5};
+\draw [] (0.2500,0.2500) -- (0.7500,0.2500);
+\draw [] (0.7500,0.2500) -- (0.7500,0.7500);
+\draw [] (0.7500,0.7500) -- (0.2500,0.7500);
+\draw [] (0.2500,0.7500) -- (0.2500,0.2500);
+\draw (1.0912,0.5000) node {6};
+\draw [] (0.7500,0.2500) -- (1.2500,0.2500);
+\draw [] (1.2500,0.2500) -- (1.2500,0.7500);
+\draw [] (1.2500,0.7500) -- (0.7500,0.7500);
+\draw [] (0.7500,0.7500) -- (0.7500,0.2500);
+\draw (1.5912,0.5000) node {9};
+\draw [] (1.2500,0.2500) -- (1.7500,0.2500);
+\draw [] (1.7500,0.2500) -- (1.7500,0.7500);
+\draw [] (1.7500,0.7500) -- (1.2500,0.7500);
+\draw [] (1.2500,0.7500) -- (1.2500,0.2500);
+\draw (0.1824,0.0000) node {10};
+\draw [] (-0.2500,-0.2500) -- (0.2500,-0.2500);
+\draw [] (0.2500,-0.2500) -- (0.2500,0.2500);
+\draw [] (0.2500,0.2500) -- (-0.2500,0.2500);
+\draw [] (-0.2500,0.2500) -- (-0.2500,-0.2500);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall b/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall
index b6547c9ce..44e397bc3 100644
--- a/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall
+++ b/src_phystricks/Fig_GYODoojTiGZSkJ.pstricks.recall
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.42400,-0.12838) node {\(A\)};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.2820,-0.36117) node {\(B\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.4240,-0.1283) node {\(A\)};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.2819,-0.3611) node {\(B\)};
 \draw []  (6.0000,4.0000) node [rotate=0] {$\bullet$};
-\draw (6.2607,4.3699) node {\(C\)};
-\draw [] (0,0) -- (5.00,0);
-\draw [] (5.00,0) -- (6.00,4.00);
-\draw [] (6.00,4.00) -- (0,0);
-\draw [] (0,0) -- (5.60,2.40);
+\draw (6.2606,4.3699) node {\(C\)};
+\draw [] (0.0000,0.0000) -- (5.0000,0.0000);
+\draw [] (5.0000,0.0000) -- (6.0000,4.0000);
+\draw [] (6.0000,4.0000) -- (0.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (5.6000,2.4000);
 \draw []  (4.4800,1.9200) node [rotate=0] {$\bullet$};
-\draw (4.7253,1.6115) node {\( N\)};
+\draw (4.7253,1.6114) node {\( N\)};
 \draw []  (5.6000,2.4000) node [rotate=0] {$\bullet$};
-\draw (5.9365,2.2268) node {\( P\)};
+\draw (5.9365,2.2267) node {\( P\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall b/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall
index d8605c418..5a89561a2 100644
--- a/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall
+++ b/src_phystricks/Fig_HLJooGDZnqF.pstricks.recall
@@ -103,87 +103,87 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-5.00,-3.00) -- (-5.00,6.00);
-\draw [color=gray,style=solid] (-4.00,-3.00) -- (-4.00,6.00);
-\draw [color=gray,style=solid] (-3.00,-3.00) -- (-3.00,6.00);
-\draw [color=gray,style=solid] (-2.00,-3.00) -- (-2.00,6.00);
-\draw [color=gray,style=solid] (-1.00,-3.00) -- (-1.00,6.00);
-\draw [color=gray,style=solid] (0,-3.00) -- (0,6.00);
-\draw [color=gray,style=solid] (1.00,-3.00) -- (1.00,6.00);
-\draw [color=gray,style=solid] (2.00,-3.00) -- (2.00,6.00);
-\draw [color=gray,style=dotted] (-4.50,-3.00) -- (-4.50,6.00);
-\draw [color=gray,style=dotted] (-3.50,-3.00) -- (-3.50,6.00);
-\draw [color=gray,style=dotted] (-2.50,-3.00) -- (-2.50,6.00);
-\draw [color=gray,style=dotted] (-1.50,-3.00) -- (-1.50,6.00);
-\draw [color=gray,style=dotted] (-0.500,-3.00) -- (-0.500,6.00);
-\draw [color=gray,style=dotted] (0.500,-3.00) -- (0.500,6.00);
-\draw [color=gray,style=dotted] (1.50,-3.00) -- (1.50,6.00);
-\draw [color=gray,style=dotted] (-5.00,-2.50) -- (2.00,-2.50);
-\draw [color=gray,style=dotted] (-5.00,-1.50) -- (2.00,-1.50);
-\draw [color=gray,style=dotted] (-5.00,-0.500) -- (2.00,-0.500);
-\draw [color=gray,style=dotted] (-5.00,0.500) -- (2.00,0.500);
-\draw [color=gray,style=dotted] (-5.00,1.50) -- (2.00,1.50);
-\draw [color=gray,style=dotted] (-5.00,2.50) -- (2.00,2.50);
-\draw [color=gray,style=dotted] (-5.00,3.50) -- (2.00,3.50);
-\draw [color=gray,style=dotted] (-5.00,4.50) -- (2.00,4.50);
-\draw [color=gray,style=dotted] (-5.00,5.50) -- (2.00,5.50);
-\draw [color=gray,style=solid] (-5.00,-3.00) -- (2.00,-3.00);
-\draw [color=gray,style=solid] (-5.00,-2.00) -- (2.00,-2.00);
-\draw [color=gray,style=solid] (-5.00,-1.00) -- (2.00,-1.00);
-\draw [color=gray,style=solid] (-5.00,0) -- (2.00,0);
-\draw [color=gray,style=solid] (-5.00,1.00) -- (2.00,1.00);
-\draw [color=gray,style=solid] (-5.00,2.00) -- (2.00,2.00);
-\draw [color=gray,style=solid] (-5.00,3.00) -- (2.00,3.00);
-\draw [color=gray,style=solid] (-5.00,4.00) -- (2.00,4.00);
-\draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
-\draw [color=gray,style=solid] (-5.00,6.00) -- (2.00,6.00);
+\draw [color=gray,style=solid] (-5.0000,-3.0000) -- (-5.0000,6.0000);
+\draw [color=gray,style=solid] (-4.0000,-3.0000) -- (-4.0000,6.0000);
+\draw [color=gray,style=solid] (-3.0000,-3.0000) -- (-3.0000,6.0000);
+\draw [color=gray,style=solid] (-2.0000,-3.0000) -- (-2.0000,6.0000);
+\draw [color=gray,style=solid] (-1.0000,-3.0000) -- (-1.0000,6.0000);
+\draw [color=gray,style=solid] (0.0000,-3.0000) -- (0.0000,6.0000);
+\draw [color=gray,style=solid] (1.0000,-3.0000) -- (1.0000,6.0000);
+\draw [color=gray,style=solid] (2.0000,-3.0000) -- (2.0000,6.0000);
+\draw [color=gray,style=dotted] (-4.5000,-3.0000) -- (-4.5000,6.0000);
+\draw [color=gray,style=dotted] (-3.5000,-3.0000) -- (-3.5000,6.0000);
+\draw [color=gray,style=dotted] (-2.5000,-3.0000) -- (-2.5000,6.0000);
+\draw [color=gray,style=dotted] (-1.5000,-3.0000) -- (-1.5000,6.0000);
+\draw [color=gray,style=dotted] (-0.5000,-3.0000) -- (-0.5000,6.0000);
+\draw [color=gray,style=dotted] (0.5000,-3.0000) -- (0.5000,6.0000);
+\draw [color=gray,style=dotted] (1.5000,-3.0000) -- (1.5000,6.0000);
+\draw [color=gray,style=dotted] (-5.0000,-2.5000) -- (2.0000,-2.5000);
+\draw [color=gray,style=dotted] (-5.0000,-1.5000) -- (2.0000,-1.5000);
+\draw [color=gray,style=dotted] (-5.0000,-0.5000) -- (2.0000,-0.5000);
+\draw [color=gray,style=dotted] (-5.0000,0.5000) -- (2.0000,0.5000);
+\draw [color=gray,style=dotted] (-5.0000,1.5000) -- (2.0000,1.5000);
+\draw [color=gray,style=dotted] (-5.0000,2.5000) -- (2.0000,2.5000);
+\draw [color=gray,style=dotted] (-5.0000,3.5000) -- (2.0000,3.5000);
+\draw [color=gray,style=dotted] (-5.0000,4.5000) -- (2.0000,4.5000);
+\draw [color=gray,style=dotted] (-5.0000,5.5000) -- (2.0000,5.5000);
+\draw [color=gray,style=solid] (-5.0000,-3.0000) -- (2.0000,-3.0000);
+\draw [color=gray,style=solid] (-5.0000,-2.0000) -- (2.0000,-2.0000);
+\draw [color=gray,style=solid] (-5.0000,-1.0000) -- (2.0000,-1.0000);
+\draw [color=gray,style=solid] (-5.0000,0.0000) -- (2.0000,0.0000);
+\draw [color=gray,style=solid] (-5.0000,1.0000) -- (2.0000,1.0000);
+\draw [color=gray,style=solid] (-5.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=gray,style=solid] (-5.0000,3.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (-5.0000,4.0000) -- (2.0000,4.0000);
+\draw [color=gray,style=solid] (-5.0000,5.0000) -- (2.0000,5.0000);
+\draw [color=gray,style=solid] (-5.0000,6.0000) -- (2.0000,6.0000);
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,6.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,6.5000);
 %DEFAULT
 
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+\draw [color=blue] (-5.0000,-2.5000)--(-4.9494,-2.4747)--(-4.8989,-2.4494)--(-4.8484,-2.4242)--(-4.7979,-2.3989)--(-4.7474,-2.3737)--(-4.6969,-2.3484)--(-4.6464,-2.3232)--(-4.5959,-2.2979)--(-4.5454,-2.2727)--(-4.4949,-2.2474)--(-4.4444,-2.2222)--(-4.3939,-2.1969)--(-4.3434,-2.1717)--(-4.2929,-2.1464)--(-4.2424,-2.1212)--(-4.1919,-2.0959)--(-4.1414,-2.0707)--(-4.0909,-2.0454)--(-4.0404,-2.0202)--(-3.9898,-1.9949)--(-3.9393,-1.9696)--(-3.8888,-1.9444)--(-3.8383,-1.9191)--(-3.7878,-1.8939)--(-3.7373,-1.8686)--(-3.6868,-1.8434)--(-3.6363,-1.8181)--(-3.5858,-1.7929)--(-3.5353,-1.7676)--(-3.4848,-1.7424)--(-3.4343,-1.7171)--(-3.3838,-1.6919)--(-3.3333,-1.6666)--(-3.2828,-1.6414)--(-3.2323,-1.6161)--(-3.1818,-1.5909)--(-3.1313,-1.5656)--(-3.0808,-1.5404)--(-3.0303,-1.5151)--(-2.9797,-1.4898)--(-2.9292,-1.4646)--(-2.8787,-1.4393)--(-2.8282,-1.4141)--(-2.7777,-1.3888)--(-2.7272,-1.3636)--(-2.6767,-1.3383)--(-2.6262,-1.3131)--(-2.5757,-1.2878)--(-2.5252,-1.2626)--(-2.4747,-1.2373)--(-2.4242,-1.2121)--(-2.3737,-1.1868)--(-2.3232,-1.1616)--(-2.2727,-1.1363)--(-2.2222,-1.1111)--(-2.1717,-1.0858)--(-2.1212,-1.0606)--(-2.0707,-1.0353)--(-2.0202,-1.0101)--(-1.9696,-0.9848)--(-1.9191,-0.9595)--(-1.8686,-0.9343)--(-1.8181,-0.9090)--(-1.7676,-0.8838)--(-1.7171,-0.8585)--(-1.6666,-0.8333)--(-1.6161,-0.8080)--(-1.5656,-0.7828)--(-1.5151,-0.7575)--(-1.4646,-0.7323)--(-1.4141,-0.7070)--(-1.3636,-0.6818)--(-1.3131,-0.6565)--(-1.2626,-0.6313)--(-1.2121,-0.6060)--(-1.1616,-0.5808)--(-1.1111,-0.5555)--(-1.0606,-0.5303)--(-1.0101,-0.5050)--(-0.9595,-0.4797)--(-0.9090,-0.4545)--(-0.8585,-0.4292)--(-0.8080,-0.4040)--(-0.7575,-0.3787)--(-0.7070,-0.3535)--(-0.6565,-0.3282)--(-0.6060,-0.3030)--(-0.5555,-0.2777)--(-0.5050,-0.2525)--(-0.4545,-0.2272)--(-0.4040,-0.2020)--(-0.3535,-0.1767)--(-0.3030,-0.1515)--(-0.2525,-0.1262)--(-0.2020,-0.1010)--(-0.1515,-0.0757)--(-0.1010,-0.0505)--(-0.0505,-0.0252)--(0.0000,0.0000);
 
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-\draw [color=blue] (1.000,3.000)--(1.010,3.030)--(1.020,3.061)--(1.030,3.091)--(1.040,3.121)--(1.051,3.152)--(1.061,3.182)--(1.071,3.212)--(1.081,3.242)--(1.091,3.273)--(1.101,3.303)--(1.111,3.333)--(1.121,3.364)--(1.131,3.394)--(1.141,3.424)--(1.152,3.455)--(1.162,3.485)--(1.172,3.515)--(1.182,3.545)--(1.192,3.576)--(1.202,3.606)--(1.212,3.636)--(1.222,3.667)--(1.232,3.697)--(1.242,3.727)--(1.253,3.758)--(1.263,3.788)--(1.273,3.818)--(1.283,3.848)--(1.293,3.879)--(1.303,3.909)--(1.313,3.939)--(1.323,3.970)--(1.333,4.000)--(1.343,4.030)--(1.354,4.061)--(1.364,4.091)--(1.374,4.121)--(1.384,4.151)--(1.394,4.182)--(1.404,4.212)--(1.414,4.242)--(1.424,4.273)--(1.434,4.303)--(1.444,4.333)--(1.455,4.364)--(1.465,4.394)--(1.475,4.424)--(1.485,4.455)--(1.495,4.485)--(1.505,4.515)--(1.515,4.545)--(1.525,4.576)--(1.535,4.606)--(1.545,4.636)--(1.556,4.667)--(1.566,4.697)--(1.576,4.727)--(1.586,4.758)--(1.596,4.788)--(1.606,4.818)--(1.616,4.849)--(1.626,4.879)--(1.636,4.909)--(1.646,4.939)--(1.657,4.970)--(1.667,5.000)--(1.677,5.030)--(1.687,5.061)--(1.697,5.091)--(1.707,5.121)--(1.717,5.151)--(1.727,5.182)--(1.737,5.212)--(1.747,5.242)--(1.758,5.273)--(1.768,5.303)--(1.778,5.333)--(1.788,5.364)--(1.798,5.394)--(1.808,5.424)--(1.818,5.455)--(1.828,5.485)--(1.838,5.515)--(1.848,5.545)--(1.859,5.576)--(1.869,5.606)--(1.879,5.636)--(1.889,5.667)--(1.899,5.697)--(1.909,5.727)--(1.919,5.758)--(1.929,5.788)--(1.939,5.818)--(1.949,5.849)--(1.960,5.879)--(1.970,5.909)--(1.980,5.939)--(1.990,5.970)--(2.000,6.000);
-\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (0,0) node [rotate=0] {$o$};
+\draw [color=blue] (1.0000,3.0000)--(1.0101,3.0303)--(1.0202,3.0606)--(1.0303,3.0909)--(1.0404,3.1212)--(1.0505,3.1515)--(1.0606,3.1818)--(1.0707,3.2121)--(1.0808,3.2424)--(1.0909,3.2727)--(1.1010,3.3030)--(1.1111,3.3333)--(1.1212,3.3636)--(1.1313,3.3939)--(1.1414,3.4242)--(1.1515,3.4545)--(1.1616,3.4848)--(1.1717,3.5151)--(1.1818,3.5454)--(1.1919,3.5757)--(1.2020,3.6060)--(1.2121,3.6363)--(1.2222,3.6666)--(1.2323,3.6969)--(1.2424,3.7272)--(1.2525,3.7575)--(1.2626,3.7878)--(1.2727,3.8181)--(1.2828,3.8484)--(1.2929,3.8787)--(1.3030,3.9090)--(1.3131,3.9393)--(1.3232,3.9696)--(1.3333,4.0000)--(1.3434,4.0303)--(1.3535,4.0606)--(1.3636,4.0909)--(1.3737,4.1212)--(1.3838,4.1515)--(1.3939,4.1818)--(1.4040,4.2121)--(1.4141,4.2424)--(1.4242,4.2727)--(1.4343,4.3030)--(1.4444,4.3333)--(1.4545,4.3636)--(1.4646,4.3939)--(1.4747,4.4242)--(1.4848,4.4545)--(1.4949,4.4848)--(1.5050,4.5151)--(1.5151,4.5454)--(1.5252,4.5757)--(1.5353,4.6060)--(1.5454,4.6363)--(1.5555,4.6666)--(1.5656,4.6969)--(1.5757,4.7272)--(1.5858,4.7575)--(1.5959,4.7878)--(1.6060,4.8181)--(1.6161,4.8484)--(1.6262,4.8787)--(1.6363,4.9090)--(1.6464,4.9393)--(1.6565,4.9696)--(1.6666,5.0000)--(1.6767,5.0303)--(1.6868,5.0606)--(1.6969,5.0909)--(1.7070,5.1212)--(1.7171,5.1515)--(1.7272,5.1818)--(1.7373,5.2121)--(1.7474,5.2424)--(1.7575,5.2727)--(1.7676,5.3030)--(1.7777,5.3333)--(1.7878,5.3636)--(1.7979,5.3939)--(1.8080,5.4242)--(1.8181,5.4545)--(1.8282,5.4848)--(1.8383,5.5151)--(1.8484,5.5454)--(1.8585,5.5757)--(1.8686,5.6060)--(1.8787,5.6363)--(1.8888,5.6666)--(1.8989,5.6969)--(1.9090,5.7272)--(1.9191,5.7575)--(1.9292,5.7878)--(1.9393,5.8181)--(1.9494,5.8484)--(1.9595,5.8787)--(1.9696,5.9090)--(1.9797,5.9393)--(1.9898,5.9696)--(2.0000,6.0000);
+\draw []  (0.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
 \draw []  (1.0000,3.0000) node [rotate=0] {$o$};
 
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.29125,6.0000) node {$ 6 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_HNxitLj.pstricks.recall b/src_phystricks/Fig_HNxitLj.pstricks.recall
index ea0ee8ecf..9c98b55bc 100644
--- a/src_phystricks/Fig_HNxitLj.pstricks.recall
+++ b/src_phystricks/Fig_HNxitLj.pstricks.recall
@@ -75,27 +75,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw []  (-1.0000,0) node [rotate=0] {$\diamondsuit$};
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\diamondsuit$};
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
 \draw []  (-1.0000,-1.0000) node [rotate=0] {$\diamondsuit$};
-\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
-\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw (1.5000,-0.3824) node {\( \sA^*_{\sH}\)};
+\draw (1.5000,-0.3824) node {\( \sA^*_{\sH}\)};
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_HasseAGdfdy.pstricks.recall b/src_phystricks/Fig_HasseAGdfdy.pstricks.recall
index 95b7c5107..2da7f01c3 100644
--- a/src_phystricks/Fig_HasseAGdfdy.pstricks.recall
+++ b/src_phystricks/Fig_HasseAGdfdy.pstricks.recall
@@ -86,24 +86,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.27858) node {\( \alpha\)};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.36222) node {\( \beta\)};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.31406) node {\( \gamma\)};
-\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
-\draw (0,2.2786) node {\( a\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.2785) node {\( \alpha\)};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3622) node {\( \beta\)};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.3140) node {\( \gamma\)};
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,2.2785) node {\( a\)};
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
 \draw (2.0000,2.3267) node {\( b\)};
 \draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (4.0000,2.2786) node {\( c\)};
-\draw [] (0,0) -- (0,2.00);
-\draw [] (2.00,0) -- (2.00,2.00);
-\draw [] (4.00,0) -- (4.00,2.00);
-\draw [] (0,2.00) -- (2.00,0);
-\draw [] (2.00,2.00) -- (4.00,0);
-\draw [] (0,0) -- (4.00,2.00);
+\draw (4.0000,2.2785) node {\( c\)};
+\draw [] (0.0000,0.0000) -- (0.0000,2.0000);
+\draw [] (2.0000,0.0000) -- (2.0000,2.0000);
+\draw [] (4.0000,0.0000) -- (4.0000,2.0000);
+\draw [] (0.0000,2.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,2.0000) -- (4.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (4.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IYAvSvI.pstricks.recall b/src_phystricks/Fig_IYAvSvI.pstricks.recall
index 9bf4d113f..23d56cc1b 100644
--- a/src_phystricks/Fig_IYAvSvI.pstricks.recall
+++ b/src_phystricks/Fig_IYAvSvI.pstricks.recall
@@ -78,26 +78,26 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (0.091250,0.50000) node {1};
-\draw [] (-0.250,0.250) -- (0.250,0.250);
-\draw [] (0.250,0.250) -- (0.250,0.750);
-\draw [] (0.250,0.750) -- (-0.250,0.750);
-\draw [] (-0.250,0.750) -- (-0.250,0.250);
-\draw (0.59125,0.50000) node {3};
-\draw [] (0.250,0.250) -- (0.750,0.250);
-\draw [] (0.750,0.250) -- (0.750,0.750);
-\draw [] (0.750,0.750) -- (0.250,0.750);
-\draw [] (0.250,0.750) -- (0.250,0.250);
-\draw (0.091250,0) node {2};
-\draw [] (-0.250,-0.250) -- (0.250,-0.250);
-\draw [] (0.250,-0.250) -- (0.250,0.250);
-\draw [] (0.250,0.250) -- (-0.250,0.250);
-\draw [] (-0.250,0.250) -- (-0.250,-0.250);
-\draw (0.59125,0) node {4};
-\draw [] (0.250,-0.250) -- (0.750,-0.250);
-\draw [] (0.750,-0.250) -- (0.750,0.250);
-\draw [] (0.750,0.250) -- (0.250,0.250);
-\draw [] (0.250,0.250) -- (0.250,-0.250);
+\draw (0.0912,0.5000) node {1};
+\draw [] (-0.2500,0.2500) -- (0.2500,0.2500);
+\draw [] (0.2500,0.2500) -- (0.2500,0.7500);
+\draw [] (0.2500,0.7500) -- (-0.2500,0.7500);
+\draw [] (-0.2500,0.7500) -- (-0.2500,0.2500);
+\draw (0.5912,0.5000) node {3};
+\draw [] (0.2500,0.2500) -- (0.7500,0.2500);
+\draw [] (0.7500,0.2500) -- (0.7500,0.7500);
+\draw [] (0.7500,0.7500) -- (0.2500,0.7500);
+\draw [] (0.2500,0.7500) -- (0.2500,0.2500);
+\draw (0.0912,0.0000) node {2};
+\draw [] (-0.2500,-0.2500) -- (0.2500,-0.2500);
+\draw [] (0.2500,-0.2500) -- (0.2500,0.2500);
+\draw [] (0.2500,0.2500) -- (-0.2500,0.2500);
+\draw [] (-0.2500,0.2500) -- (-0.2500,-0.2500);
+\draw (0.5912,0.0000) node {4};
+\draw [] (0.2500,-0.2500) -- (0.7500,-0.2500);
+\draw [] (0.7500,-0.2500) -- (0.7500,0.2500);
+\draw [] (0.7500,0.2500) -- (0.2500,0.2500);
+\draw [] (0.2500,0.2500) -- (0.2500,-0.2500);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IntRectangle.pstricks.recall b/src_phystricks/Fig_IntRectangle.pstricks.recall
index 1ac58ccc5..21ba95b9f 100644
--- a/src_phystricks/Fig_IntRectangle.pstricks.recall
+++ b/src_phystricks/Fig_IntRectangle.pstricks.recall
@@ -71,22 +71,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\fill [color=green] (0,2.00) -- (0.0101,2.00) -- (0.0202,2.00) -- (0.0303,2.00) -- (0.0404,2.00) -- (0.0505,2.00) -- (0.0606,2.00) -- (0.0707,2.00) -- (0.0808,2.00) -- (0.0909,2.00) -- (0.101,2.00) -- (0.111,2.00) -- (0.121,2.00) -- (0.131,2.00) -- (0.141,2.00) -- (0.152,2.00) -- (0.162,2.00) -- (0.172,2.00) -- (0.182,2.00) -- (0.192,2.00) -- (0.202,2.00) -- (0.212,2.00) -- (0.222,2.00) -- (0.232,2.00) -- (0.242,2.00) -- (0.253,2.00) -- (0.263,2.00) -- (0.273,2.00) -- (0.283,2.00) -- (0.293,2.00) -- (0.303,2.00) -- (0.313,2.00) -- (0.323,2.00) -- (0.333,2.00) -- (0.343,2.00) -- (0.354,2.00) -- (0.364,2.00) -- (0.374,2.00) -- (0.384,2.00) -- (0.394,2.00) -- (0.404,2.00) -- (0.414,2.00) -- (0.424,2.00) -- (0.434,2.00) -- (0.444,2.00) -- (0.455,2.00) -- (0.465,2.00) -- (0.475,2.00) -- (0.485,2.00) -- (0.495,2.00) -- (0.505,2.00) -- (0.515,2.00) -- (0.525,2.00) -- (0.535,2.00) -- (0.545,2.00) -- (0.556,2.00) -- (0.566,2.00) -- (0.576,2.00) -- (0.586,2.00) -- (0.596,2.00) -- (0.606,2.00) -- (0.616,2.00) -- (0.626,2.00) -- (0.636,2.00) -- (0.646,2.00) -- (0.657,2.00) -- (0.667,2.00) -- (0.677,2.00) -- (0.687,2.00) -- (0.697,2.00) -- (0.707,2.00) -- (0.717,2.00) -- (0.727,2.00) -- (0.737,2.00) -- (0.747,2.00) -- (0.758,2.00) -- (0.768,2.00) -- (0.778,2.00) -- (0.788,2.00) -- (0.798,2.00) -- (0.808,2.00) -- (0.818,2.00) -- (0.828,2.00) -- (0.838,2.00) -- (0.849,2.00) -- (0.859,2.00) -- (0.869,2.00) -- (0.879,2.00) -- (0.889,2.00) -- (0.899,2.00) -- (0.909,2.00) -- (0.919,2.00) -- (0.929,2.00) -- (0.939,2.00) -- (0.950,2.00) -- (0.960,2.00) -- (0.970,2.00) -- (0.980,2.00) -- (0.990,2.00) -- (1.00,2.00) -- (1.00,2.00) -- (1.00,0) -- (1.00,0) -- (0.990,0) -- (0.980,0) -- (0.970,0) -- (0.960,0) -- (0.950,0) -- (0.939,0) -- (0.929,0) -- (0.919,0) -- (0.909,0) -- (0.899,0) -- (0.889,0) -- (0.879,0) -- (0.869,0) -- (0.859,0) -- (0.849,0) -- (0.838,0) -- (0.828,0) -- (0.818,0) -- (0.808,0) -- (0.798,0) -- (0.788,0) -- (0.778,0) -- (0.768,0) -- (0.758,0) -- (0.747,0) -- (0.737,0) -- (0.727,0) -- (0.717,0) -- (0.707,0) -- (0.697,0) -- (0.687,0) -- (0.677,0) -- (0.667,0) -- (0.657,0) -- (0.646,0) -- (0.636,0) -- (0.626,0) -- (0.616,0) -- (0.606,0) -- (0.596,0) -- (0.586,0) -- (0.576,0) -- (0.566,0) -- (0.556,0) -- (0.545,0) -- (0.535,0) -- (0.525,0) -- (0.515,0) -- (0.505,0) -- (0.495,0) -- (0.485,0) -- (0.475,0) -- (0.465,0) -- (0.455,0) -- (0.444,0) -- (0.434,0) -- (0.424,0) -- (0.414,0) -- (0.404,0) -- (0.394,0) -- (0.384,0) -- (0.374,0) -- (0.364,0) -- (0.354,0) -- (0.343,0) -- (0.333,0) -- (0.323,0) -- (0.313,0) -- (0.303,0) -- (0.293,0) -- (0.283,0) -- (0.273,0) -- (0.263,0) -- (0.253,0) -- (0.242,0) -- (0.232,0) -- (0.222,0) -- (0.212,0) -- (0.202,0) -- (0.192,0) -- (0.182,0) -- (0.172,0) -- (0.162,0) -- (0.152,0) -- (0.141,0) -- (0.131,0) -- (0.121,0) -- (0.111,0) -- (0.101,0) -- (0.0909,0) -- (0.0808,0) -- (0.0707,0) -- (0.0606,0) -- (0.0505,0) -- (0.0404,0) -- (0.0303,0) -- (0.0202,0) -- (0.0101,0) -- (0,0) -- (0,0) -- (0,2.00) --  cycle;
-\draw [color=red,style=solid] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
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+\draw [color=red,style=solid] (0.0000,2.0000)--(0.0101,2.0000)--(0.0202,2.0000)--(0.0303,2.0000)--(0.0404,2.0000)--(0.0505,2.0000)--(0.0606,2.0000)--(0.0707,2.0000)--(0.0808,2.0000)--(0.0909,2.0000)--(0.1010,2.0000)--(0.1111,2.0000)--(0.1212,2.0000)--(0.1313,2.0000)--(0.1414,2.0000)--(0.1515,2.0000)--(0.1616,2.0000)--(0.1717,2.0000)--(0.1818,2.0000)--(0.1919,2.0000)--(0.2020,2.0000)--(0.2121,2.0000)--(0.2222,2.0000)--(0.2323,2.0000)--(0.2424,2.0000)--(0.2525,2.0000)--(0.2626,2.0000)--(0.2727,2.0000)--(0.2828,2.0000)--(0.2929,2.0000)--(0.3030,2.0000)--(0.3131,2.0000)--(0.3232,2.0000)--(0.3333,2.0000)--(0.3434,2.0000)--(0.3535,2.0000)--(0.3636,2.0000)--(0.3737,2.0000)--(0.3838,2.0000)--(0.3939,2.0000)--(0.4040,2.0000)--(0.4141,2.0000)--(0.4242,2.0000)--(0.4343,2.0000)--(0.4444,2.0000)--(0.4545,2.0000)--(0.4646,2.0000)--(0.4747,2.0000)--(0.4848,2.0000)--(0.4949,2.0000)--(0.5050,2.0000)--(0.5151,2.0000)--(0.5252,2.0000)--(0.5353,2.0000)--(0.5454,2.0000)--(0.5555,2.0000)--(0.5656,2.0000)--(0.5757,2.0000)--(0.5858,2.0000)--(0.5959,2.0000)--(0.6060,2.0000)--(0.6161,2.0000)--(0.6262,2.0000)--(0.6363,2.0000)--(0.6464,2.0000)--(0.6565,2.0000)--(0.6666,2.0000)--(0.6767,2.0000)--(0.6868,2.0000)--(0.6969,2.0000)--(0.7070,2.0000)--(0.7171,2.0000)--(0.7272,2.0000)--(0.7373,2.0000)--(0.7474,2.0000)--(0.7575,2.0000)--(0.7676,2.0000)--(0.7777,2.0000)--(0.7878,2.0000)--(0.7979,2.0000)--(0.8080,2.0000)--(0.8181,2.0000)--(0.8282,2.0000)--(0.8383,2.0000)--(0.8484,2.0000)--(0.8585,2.0000)--(0.8686,2.0000)--(0.8787,2.0000)--(0.8888,2.0000)--(0.8989,2.0000)--(0.9090,2.0000)--(0.9191,2.0000)--(0.9292,2.0000)--(0.9393,2.0000)--(0.9494,2.0000)--(0.9595,2.0000)--(0.9696,2.0000)--(0.9797,2.0000)--(0.9898,2.0000)--(1.0000,2.0000);
+\draw [color=blue] (0.0000,0.0000)--(0.0101,0.0000)--(0.0202,0.0000)--(0.0303,0.0000)--(0.0404,0.0000)--(0.0505,0.0000)--(0.0606,0.0000)--(0.0707,0.0000)--(0.0808,0.0000)--(0.0909,0.0000)--(0.1010,0.0000)--(0.1111,0.0000)--(0.1212,0.0000)--(0.1313,0.0000)--(0.1414,0.0000)--(0.1515,0.0000)--(0.1616,0.0000)--(0.1717,0.0000)--(0.1818,0.0000)--(0.1919,0.0000)--(0.2020,0.0000)--(0.2121,0.0000)--(0.2222,0.0000)--(0.2323,0.0000)--(0.2424,0.0000)--(0.2525,0.0000)--(0.2626,0.0000)--(0.2727,0.0000)--(0.2828,0.0000)--(0.2929,0.0000)--(0.3030,0.0000)--(0.3131,0.0000)--(0.3232,0.0000)--(0.3333,0.0000)--(0.3434,0.0000)--(0.3535,0.0000)--(0.3636,0.0000)--(0.3737,0.0000)--(0.3838,0.0000)--(0.3939,0.0000)--(0.4040,0.0000)--(0.4141,0.0000)--(0.4242,0.0000)--(0.4343,0.0000)--(0.4444,0.0000)--(0.4545,0.0000)--(0.4646,0.0000)--(0.4747,0.0000)--(0.4848,0.0000)--(0.4949,0.0000)--(0.5050,0.0000)--(0.5151,0.0000)--(0.5252,0.0000)--(0.5353,0.0000)--(0.5454,0.0000)--(0.5555,0.0000)--(0.5656,0.0000)--(0.5757,0.0000)--(0.5858,0.0000)--(0.5959,0.0000)--(0.6060,0.0000)--(0.6161,0.0000)--(0.6262,0.0000)--(0.6363,0.0000)--(0.6464,0.0000)--(0.6565,0.0000)--(0.6666,0.0000)--(0.6767,0.0000)--(0.6868,0.0000)--(0.6969,0.0000)--(0.7070,0.0000)--(0.7171,0.0000)--(0.7272,0.0000)--(0.7373,0.0000)--(0.7474,0.0000)--(0.7575,0.0000)--(0.7676,0.0000)--(0.7777,0.0000)--(0.7878,0.0000)--(0.7979,0.0000)--(0.8080,0.0000)--(0.8181,0.0000)--(0.8282,0.0000)--(0.8383,0.0000)--(0.8484,0.0000)--(0.8585,0.0000)--(0.8686,0.0000)--(0.8787,0.0000)--(0.8888,0.0000)--(0.8989,0.0000)--(0.9090,0.0000)--(0.9191,0.0000)--(0.9292,0.0000)--(0.9393,0.0000)--(0.9494,0.0000)--(0.9595,0.0000)--(0.9696,0.0000)--(0.9797,0.0000)--(0.9898,0.0000)--(1.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (0.0000,2.0000);
+\draw [color=red,style=solid] (1.0000,2.0000) -- (1.0000,0.0000);
 
-\draw [color=red] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=red] (0.0000,2.0000)--(0.0101,2.0000)--(0.0202,2.0000)--(0.0303,2.0000)--(0.0404,2.0000)--(0.0505,2.0000)--(0.0606,2.0000)--(0.0707,2.0000)--(0.0808,2.0000)--(0.0909,2.0000)--(0.1010,2.0000)--(0.1111,2.0000)--(0.1212,2.0000)--(0.1313,2.0000)--(0.1414,2.0000)--(0.1515,2.0000)--(0.1616,2.0000)--(0.1717,2.0000)--(0.1818,2.0000)--(0.1919,2.0000)--(0.2020,2.0000)--(0.2121,2.0000)--(0.2222,2.0000)--(0.2323,2.0000)--(0.2424,2.0000)--(0.2525,2.0000)--(0.2626,2.0000)--(0.2727,2.0000)--(0.2828,2.0000)--(0.2929,2.0000)--(0.3030,2.0000)--(0.3131,2.0000)--(0.3232,2.0000)--(0.3333,2.0000)--(0.3434,2.0000)--(0.3535,2.0000)--(0.3636,2.0000)--(0.3737,2.0000)--(0.3838,2.0000)--(0.3939,2.0000)--(0.4040,2.0000)--(0.4141,2.0000)--(0.4242,2.0000)--(0.4343,2.0000)--(0.4444,2.0000)--(0.4545,2.0000)--(0.4646,2.0000)--(0.4747,2.0000)--(0.4848,2.0000)--(0.4949,2.0000)--(0.5050,2.0000)--(0.5151,2.0000)--(0.5252,2.0000)--(0.5353,2.0000)--(0.5454,2.0000)--(0.5555,2.0000)--(0.5656,2.0000)--(0.5757,2.0000)--(0.5858,2.0000)--(0.5959,2.0000)--(0.6060,2.0000)--(0.6161,2.0000)--(0.6262,2.0000)--(0.6363,2.0000)--(0.6464,2.0000)--(0.6565,2.0000)--(0.6666,2.0000)--(0.6767,2.0000)--(0.6868,2.0000)--(0.6969,2.0000)--(0.7070,2.0000)--(0.7171,2.0000)--(0.7272,2.0000)--(0.7373,2.0000)--(0.7474,2.0000)--(0.7575,2.0000)--(0.7676,2.0000)--(0.7777,2.0000)--(0.7878,2.0000)--(0.7979,2.0000)--(0.8080,2.0000)--(0.8181,2.0000)--(0.8282,2.0000)--(0.8383,2.0000)--(0.8484,2.0000)--(0.8585,2.0000)--(0.8686,2.0000)--(0.8787,2.0000)--(0.8888,2.0000)--(0.8989,2.0000)--(0.9090,2.0000)--(0.9191,2.0000)--(0.9292,2.0000)--(0.9393,2.0000)--(0.9494,2.0000)--(0.9595,2.0000)--(0.9696,2.0000)--(0.9797,2.0000)--(0.9898,2.0000)--(1.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_IntervalleUn.pstricks.recall b/src_phystricks/Fig_IntervalleUn.pstricks.recall
index 27457d70b..e9032c9e1 100644
--- a/src_phystricks/Fig_IntervalleUn.pstricks.recall
+++ b/src_phystricks/Fig_IntervalleUn.pstricks.recall
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-1.50,0) -- (4.50,0);
-\draw []  (0.90000,0) node [rotate=0] {$\bullet$};
-\draw (0.90000,-0.37858) node {$a$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.41492) node {$0$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,-0.41492) node {$1$};
-\draw [,->,>=latex] (0.45000,0.30000) -- (0,0.30000);
-\draw [,->,>=latex] (0.45000,0.30000) -- (0.90000,0.30000);
-\draw (0.45000,0.67858) node {$a$};
-\draw [,->,>=latex] (1.9500,0.30000) -- (0.90000,0.30000);
-\draw [,->,>=latex] (1.9500,0.30000) -- (3.0000,0.30000);
-\draw (1.9500,0.72983) node {$1-a$};
+\draw [] (-1.5000,0.0000) -- (4.5000,0.0000);
+\draw []  (0.9000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.9000,-0.3785) node {$a$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.4149) node {$0$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.4149) node {$1$};
+\draw [,->,>=latex] (0.4500,0.3000) -- (0.0000,0.3000);
+\draw [,->,>=latex] (0.4500,0.3000) -- (0.9000,0.3000);
+\draw (0.4500,0.6785) node {$a$};
+\draw [,->,>=latex] (1.9500,0.3000) -- (0.9000,0.3000);
+\draw [,->,>=latex] (1.9500,0.3000) -- (3.0000,0.3000);
+\draw (1.9500,0.7298) node {$1-a$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IsomCarre.pstricks.recall b/src_phystricks/Fig_IsomCarre.pstricks.recall
index 52c75fa0f..6d9487b98 100644
--- a/src_phystricks/Fig_IsomCarre.pstricks.recall
+++ b/src_phystricks/Fig_IsomCarre.pstricks.recall
@@ -85,20 +85,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (-1.00,-1.00) -- (1.00,-1.00);
-\draw [color=blue] (1.00,-1.00) -- (1.00,1.00);
-\draw [color=blue] (1.00,1.00) -- (-1.00,1.00);
-\draw [color=blue] (-1.00,1.00) -- (-1.00,-1.00);
-\draw [] (0,-1.50) -- (0,1.50);
+\draw [color=blue] (-1.0000,-1.0000) -- (1.0000,-1.0000);
+\draw [color=blue] (1.0000,-1.0000) -- (1.0000,1.0000);
+\draw [color=blue] (1.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue] (-1.0000,1.0000) -- (-1.0000,-1.0000);
+\draw [] (0.0000,-1.5000) -- (0.0000,1.5000);
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.2076,1.1954) node {\( A\)};
+\draw (-1.2075,1.1954) node {\( A\)};
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.2183,1.1954) node {\( B\)};
+\draw (1.2182,1.1954) node {\( B\)};
 \draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
 \draw (1.2142,-1.1954) node {\( C\)};
 \draw []  (-1.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.2269,-1.1954) node {\( D\)};
-\draw (0.15626,1.6493) node {\( s\)};
+\draw (-1.2268,-1.1954) node {\( D\)};
+\draw (0.1562,1.6492) node {\( s\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_JGuKEjH.pstricks.recall b/src_phystricks/Fig_JGuKEjH.pstricks.recall
index f879ed6f9..3f97fb0e7 100644
--- a/src_phystricks/Fig_JGuKEjH.pstricks.recall
+++ b/src_phystricks/Fig_JGuKEjH.pstricks.recall
@@ -63,21 +63,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.500000000,0) -- (1.500000000,0);
-\draw [,->,>=latex] (0,-1.500000000) -- (0,1.500000000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
          hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,0) -- (-0.643,0.766) -- (-0.643,0.766) -- (-0.669,0.743) -- (-0.695,0.719) -- (-0.719,0.695) -- (-0.743,0.669) -- (-0.766,0.643) -- (-0.788,0.616) -- (-0.809,0.588) -- (-0.829,0.559) -- (-0.848,0.530) -- (-0.866,0.500) -- (-0.883,0.470) -- (-0.899,0.438) -- (-0.914,0.407) -- (-0.927,0.374) -- (-0.940,0.342) -- (-0.951,0.309) -- (-0.961,0.276) -- (-0.970,0.242) -- (-0.978,0.208) -- (-0.985,0.174) -- (-0.990,0.139) -- (-0.995,0.105) -- (-0.998,0.0698) -- (-0.999,0.0349) -- (-1.00,0) -- (-0.999,-0.0349) -- (-0.998,-0.0698) -- (-0.995,-0.105) -- (-0.990,-0.139) -- (-0.985,-0.174) -- (-0.978,-0.208) -- (-0.970,-0.242) -- (-0.961,-0.276) -- (-0.951,-0.309) -- (-0.940,-0.342) -- (-0.927,-0.374) -- (-0.914,-0.407) -- (-0.899,-0.438) -- (-0.883,-0.469) -- (-0.866,-0.500) -- (-0.848,-0.530) -- (-0.829,-0.559) -- (-0.809,-0.588) -- (-0.788,-0.616) -- (-0.766,-0.643) -- (-0.743,-0.669) -- (-0.719,-0.695) -- (-0.695,-0.719) -- (-0.669,-0.743) -- (1.00,0) -- (-0.643,-0.766) --  cycle;
-\draw []  (1.000000000,0) node [rotate=0] {$\bullet$};
-\draw [color=lightgray] (1.00,0) -- (-0.643,0.766);
-\draw [color=lightgray] (1.00,0) -- (-0.643,-0.766);
+\fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,0.0000) -- (-0.6427,0.7660) -- (-0.6427,0.7660) -- (-0.6691,0.7431) -- (-0.6946,0.7193) -- (-0.7193,0.6946) -- (-0.7431,0.6691) -- (-0.7660,0.6427) -- (-0.7880,0.6156) -- (-0.8090,0.5877) -- (-0.8290,0.5591) -- (-0.8480,0.5299) -- (-0.8660,0.5000) -- (-0.8829,0.4694) -- (-0.8987,0.4383) -- (-0.9135,0.4067) -- (-0.9271,0.3746) -- (-0.9396,0.3420) -- (-0.9510,0.3090) -- (-0.9612,0.2756) -- (-0.9702,0.2419) -- (-0.9781,0.2079) -- (-0.9848,0.1736) -- (-0.9902,0.1391) -- (-0.9945,0.1045) -- (-0.9975,0.0697) -- (-0.9993,0.0348) -- (-1.0000,0.0000) -- (-0.9993,-0.0348) -- (-0.9975,-0.0697) -- (-0.9945,-0.1045) -- (-0.9902,-0.1391) -- (-0.9848,-0.1736) -- (-0.9781,-0.2079) -- (-0.9702,-0.2419) -- (-0.9612,-0.2756) -- (-0.9510,-0.3090) -- (-0.9396,-0.3420) -- (-0.9271,-0.3746) -- (-0.9135,-0.4067) -- (-0.8987,-0.4383) -- (-0.8829,-0.4694) -- (-0.8660,-0.5000) -- (-0.8480,-0.5299) -- (-0.8290,-0.5591) -- (-0.8090,-0.5877) -- (-0.7880,-0.6156) -- (-0.7660,-0.6427) -- (-0.7431,-0.6691) -- (-0.7193,-0.6946) -- (-0.6946,-0.7193) -- (-0.6691,-0.7431) -- (1.0000,0.0000) -- (-0.6427,-0.7660) --  cycle;
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=lightgray] (1.0000,0.0000) -- (-0.6427,0.7660);
+\draw [color=lightgray] (1.0000,0.0000) -- (-0.6427,-0.7660);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall b/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall
index 649f91548..66eb5ab83 100644
--- a/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall
+++ b/src_phystricks/Fig_JWINooSfKCeA.pstricks.recall
@@ -83,21 +83,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 \draw (1.5233,2.0000) node {$(x,y)$};
-\draw [,->,>=latex] (0,0) -- (1.0000,2.0000);
-\draw (0.68452,0.41391) node {$\theta$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (1.0000,2.0000);
+\draw (0.6845,0.4139) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00559)--(0.500,0.0112)--(0.500,0.0168)--(0.500,0.0224)--(0.499,0.0279)--(0.499,0.0335)--(0.498,0.0391)--(0.498,0.0447)--(0.497,0.0502)--(0.497,0.0558)--(0.496,0.0614)--(0.496,0.0669)--(0.495,0.0724)--(0.494,0.0780)--(0.493,0.0835)--(0.492,0.0890)--(0.491,0.0945)--(0.490,0.100)--(0.489,0.105)--(0.488,0.111)--(0.486,0.116)--(0.485,0.122)--(0.484,0.127)--(0.482,0.133)--(0.481,0.138)--(0.479,0.143)--(0.477,0.149)--(0.476,0.154)--(0.474,0.159)--(0.472,0.165)--(0.470,0.170)--(0.468,0.175)--(0.466,0.180)--(0.464,0.186)--(0.462,0.191)--(0.460,0.196)--(0.458,0.201)--(0.456,0.206)--(0.453,0.211)--(0.451,0.216)--(0.448,0.221)--(0.446,0.226)--(0.443,0.231)--(0.441,0.236)--(0.438,0.241)--(0.435,0.246)--(0.432,0.251)--(0.430,0.256)--(0.427,0.260)--(0.424,0.265)--(0.421,0.270)--(0.418,0.275)--(0.415,0.279)--(0.412,0.284)--(0.408,0.289)--(0.405,0.293)--(0.402,0.298)--(0.398,0.302)--(0.395,0.306)--(0.392,0.311)--(0.388,0.315)--(0.385,0.320)--(0.381,0.324)--(0.377,0.328)--(0.374,0.332)--(0.370,0.336)--(0.366,0.341)--(0.362,0.345)--(0.358,0.349)--(0.354,0.353)--(0.350,0.357)--(0.346,0.360)--(0.342,0.364)--(0.338,0.368)--(0.334,0.372)--(0.330,0.376)--(0.326,0.379)--(0.322,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.304,0.397)--(0.300,0.400)--(0.295,0.404)--(0.291,0.407)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416)--(0.272,0.419)--(0.267,0.422)--(0.263,0.425)--(0.258,0.428)--(0.253,0.431)--(0.248,0.434)--(0.243,0.437)--(0.238,0.439)--(0.234,0.442)--(0.229,0.445)--(0.224,0.447);
-\draw (0.23371,1.1680) node {$r$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [] (0.5000,0.0000)--(0.4999,0.0055)--(0.4998,0.0111)--(0.4997,0.0167)--(0.4994,0.0223)--(0.4992,0.0279)--(0.4988,0.0335)--(0.4984,0.0391)--(0.4980,0.0446)--(0.4974,0.0502)--(0.4968,0.0558)--(0.4962,0.0613)--(0.4955,0.0668)--(0.4947,0.0724)--(0.4938,0.0779)--(0.4929,0.0834)--(0.4920,0.0889)--(0.4909,0.0944)--(0.4899,0.0999)--(0.4887,0.1054)--(0.4875,0.1109)--(0.4862,0.1163)--(0.4849,0.1217)--(0.4835,0.1271)--(0.4820,0.1325)--(0.4805,0.1379)--(0.4790,0.1433)--(0.4773,0.1486)--(0.4756,0.1540)--(0.4739,0.1593)--(0.4721,0.1646)--(0.4702,0.1698)--(0.4683,0.1751)--(0.4663,0.1803)--(0.4642,0.1855)--(0.4621,0.1907)--(0.4600,0.1959)--(0.4578,0.2010)--(0.4555,0.2061)--(0.4531,0.2112)--(0.4508,0.2162)--(0.4483,0.2213)--(0.4458,0.2263)--(0.4432,0.2312)--(0.4406,0.2362)--(0.4380,0.2411)--(0.4352,0.2460)--(0.4325,0.2508)--(0.4296,0.2556)--(0.4267,0.2604)--(0.4238,0.2652)--(0.4208,0.2699)--(0.4178,0.2746)--(0.4147,0.2793)--(0.4115,0.2839)--(0.4083,0.2885)--(0.4051,0.2930)--(0.4018,0.2975)--(0.3984,0.3020)--(0.3950,0.3064)--(0.3916,0.3108)--(0.3880,0.3152)--(0.3845,0.3195)--(0.3809,0.3238)--(0.3773,0.3280)--(0.3736,0.3322)--(0.3698,0.3364)--(0.3660,0.3405)--(0.3622,0.3446)--(0.3583,0.3486)--(0.3544,0.3526)--(0.3504,0.3565)--(0.3464,0.3604)--(0.3424,0.3643)--(0.3383,0.3681)--(0.3341,0.3719)--(0.3300,0.3756)--(0.3257,0.3792)--(0.3215,0.3829)--(0.3172,0.3864)--(0.3128,0.3899)--(0.3085,0.3934)--(0.3040,0.3969)--(0.2996,0.4002)--(0.2951,0.4036)--(0.2906,0.4068)--(0.2860,0.4101)--(0.2814,0.4132)--(0.2767,0.4163)--(0.2721,0.4194)--(0.2674,0.4224)--(0.2626,0.4254)--(0.2578,0.4283)--(0.2530,0.4312)--(0.2482,0.4340)--(0.2433,0.4367)--(0.2384,0.4394)--(0.2335,0.4421)--(0.2285,0.4446)--(0.2236,0.4472);
+\draw (0.2337,1.1680) node {$r$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall b/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall
index 18f70b22a..abc3a7474 100644
--- a/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall
+++ b/src_phystricks/Fig_KGQXooZFNVnW.pstricks.recall
@@ -95,43 +95,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.6622) -- (0.0000,3.6622);
 %DEFAULT
 
-\draw [color=red] (-3.000,3.162)--(-2.939,3.105)--(-2.879,3.048)--(-2.818,2.990)--(-2.758,2.933)--(-2.697,2.876)--(-2.636,2.820)--(-2.576,2.763)--(-2.515,2.707)--(-2.455,2.650)--(-2.394,2.594)--(-2.333,2.539)--(-2.273,2.483)--(-2.212,2.428)--(-2.152,2.373)--(-2.091,2.318)--(-2.030,2.263)--(-1.970,2.209)--(-1.909,2.155)--(-1.848,2.102)--(-1.788,2.049)--(-1.727,1.996)--(-1.667,1.944)--(-1.606,1.892)--(-1.545,1.841)--(-1.485,1.790)--(-1.424,1.740)--(-1.364,1.691)--(-1.303,1.643)--(-1.242,1.595)--(-1.182,1.548)--(-1.121,1.502)--(-1.061,1.458)--(-1.000,1.414)--(-0.9394,1.372)--(-0.8788,1.331)--(-0.8182,1.292)--(-0.7576,1.255)--(-0.6970,1.219)--(-0.6364,1.185)--(-0.5758,1.154)--(-0.5152,1.125)--(-0.4545,1.098)--(-0.3939,1.075)--(-0.3333,1.054)--(-0.2727,1.037)--(-0.2121,1.022)--(-0.1515,1.011)--(-0.09091,1.004)--(-0.03030,1.000)--(0.03030,1.000)--(0.09091,1.004)--(0.1515,1.011)--(0.2121,1.022)--(0.2727,1.037)--(0.3333,1.054)--(0.3939,1.075)--(0.4545,1.098)--(0.5152,1.125)--(0.5758,1.154)--(0.6364,1.185)--(0.6970,1.219)--(0.7576,1.255)--(0.8182,1.292)--(0.8788,1.331)--(0.9394,1.372)--(1.000,1.414)--(1.061,1.458)--(1.121,1.502)--(1.182,1.548)--(1.242,1.595)--(1.303,1.643)--(1.364,1.691)--(1.424,1.740)--(1.485,1.790)--(1.545,1.841)--(1.606,1.892)--(1.667,1.944)--(1.727,1.996)--(1.788,2.049)--(1.848,2.102)--(1.909,2.155)--(1.970,2.209)--(2.030,2.263)--(2.091,2.318)--(2.152,2.373)--(2.212,2.428)--(2.273,2.483)--(2.333,2.539)--(2.394,2.594)--(2.455,2.650)--(2.515,2.707)--(2.576,2.763)--(2.636,2.820)--(2.697,2.876)--(2.758,2.933)--(2.818,2.990)--(2.879,3.048)--(2.939,3.105)--(3.000,3.162);
+\draw [color=red] (-3.0000,3.1622)--(-2.9393,3.1048)--(-2.8787,3.0475)--(-2.8181,2.9903)--(-2.7575,2.9332)--(-2.6969,2.8763)--(-2.6363,2.8196)--(-2.5757,2.7630)--(-2.5151,2.7066)--(-2.4545,2.6504)--(-2.3939,2.5944)--(-2.3333,2.5385)--(-2.2727,2.4830)--(-2.2121,2.4276)--(-2.1515,2.3725)--(-2.0909,2.3177)--(-2.0303,2.2632)--(-1.9696,2.2090)--(-1.9090,2.1551)--(-1.8484,2.1016)--(-1.7878,2.0485)--(-1.7272,1.9958)--(-1.6666,1.9436)--(-1.6060,1.8919)--(-1.5454,1.8407)--(-1.4848,1.7901)--(-1.4242,1.7402)--(-1.3636,1.6910)--(-1.3030,1.6425)--(-1.2424,1.5948)--(-1.1818,1.5481)--(-1.1212,1.5023)--(-1.0606,1.4576)--(-1.0000,1.4142)--(-0.9393,1.3720)--(-0.8787,1.3312)--(-0.8181,1.2920)--(-0.7575,1.2545)--(-0.6969,1.2189)--(-0.6363,1.1853)--(-0.5757,1.1539)--(-0.5151,1.1248)--(-0.4545,1.0984)--(-0.3939,1.0747)--(-0.3333,1.0540)--(-0.2727,1.0365)--(-0.2121,1.0222)--(-0.1515,1.0114)--(-0.0909,1.0041)--(-0.0303,1.0004)--(0.0303,1.0004)--(0.0909,1.0041)--(0.1515,1.0114)--(0.2121,1.0222)--(0.2727,1.0365)--(0.3333,1.0540)--(0.3939,1.0747)--(0.4545,1.0984)--(0.5151,1.1248)--(0.5757,1.1539)--(0.6363,1.1853)--(0.6969,1.2189)--(0.7575,1.2545)--(0.8181,1.2920)--(0.8787,1.3312)--(0.9393,1.3720)--(1.0000,1.4142)--(1.0606,1.4576)--(1.1212,1.5023)--(1.1818,1.5481)--(1.2424,1.5948)--(1.3030,1.6425)--(1.3636,1.6910)--(1.4242,1.7402)--(1.4848,1.7901)--(1.5454,1.8407)--(1.6060,1.8919)--(1.6666,1.9436)--(1.7272,1.9958)--(1.7878,2.0485)--(1.8484,2.1016)--(1.9090,2.1551)--(1.9696,2.2090)--(2.0303,2.2632)--(2.0909,2.3177)--(2.1515,2.3725)--(2.2121,2.4276)--(2.2727,2.4830)--(2.3333,2.5385)--(2.3939,2.5944)--(2.4545,2.6504)--(2.5151,2.7066)--(2.5757,2.7630)--(2.6363,2.8196)--(2.6969,2.8763)--(2.7575,2.9332)--(2.8181,2.9903)--(2.8787,3.0475)--(2.9393,3.1048)--(3.0000,3.1622);
 
-\draw [color=red] (-3.000,-3.162)--(-2.939,-3.105)--(-2.879,-3.048)--(-2.818,-2.990)--(-2.758,-2.933)--(-2.697,-2.876)--(-2.636,-2.820)--(-2.576,-2.763)--(-2.515,-2.707)--(-2.455,-2.650)--(-2.394,-2.594)--(-2.333,-2.539)--(-2.273,-2.483)--(-2.212,-2.428)--(-2.152,-2.373)--(-2.091,-2.318)--(-2.030,-2.263)--(-1.970,-2.209)--(-1.909,-2.155)--(-1.848,-2.102)--(-1.788,-2.049)--(-1.727,-1.996)--(-1.667,-1.944)--(-1.606,-1.892)--(-1.545,-1.841)--(-1.485,-1.790)--(-1.424,-1.740)--(-1.364,-1.691)--(-1.303,-1.643)--(-1.242,-1.595)--(-1.182,-1.548)--(-1.121,-1.502)--(-1.061,-1.458)--(-1.000,-1.414)--(-0.9394,-1.372)--(-0.8788,-1.331)--(-0.8182,-1.292)--(-0.7576,-1.255)--(-0.6970,-1.219)--(-0.6364,-1.185)--(-0.5758,-1.154)--(-0.5152,-1.125)--(-0.4545,-1.098)--(-0.3939,-1.075)--(-0.3333,-1.054)--(-0.2727,-1.037)--(-0.2121,-1.022)--(-0.1515,-1.011)--(-0.09091,-1.004)--(-0.03030,-1.000)--(0.03030,-1.000)--(0.09091,-1.004)--(0.1515,-1.011)--(0.2121,-1.022)--(0.2727,-1.037)--(0.3333,-1.054)--(0.3939,-1.075)--(0.4545,-1.098)--(0.5152,-1.125)--(0.5758,-1.154)--(0.6364,-1.185)--(0.6970,-1.219)--(0.7576,-1.255)--(0.8182,-1.292)--(0.8788,-1.331)--(0.9394,-1.372)--(1.000,-1.414)--(1.061,-1.458)--(1.121,-1.502)--(1.182,-1.548)--(1.242,-1.595)--(1.303,-1.643)--(1.364,-1.691)--(1.424,-1.740)--(1.485,-1.790)--(1.545,-1.841)--(1.606,-1.892)--(1.667,-1.944)--(1.727,-1.996)--(1.788,-2.049)--(1.848,-2.102)--(1.909,-2.155)--(1.970,-2.209)--(2.030,-2.263)--(2.091,-2.318)--(2.152,-2.373)--(2.212,-2.428)--(2.273,-2.483)--(2.333,-2.539)--(2.394,-2.594)--(2.455,-2.650)--(2.515,-2.707)--(2.576,-2.763)--(2.636,-2.820)--(2.697,-2.876)--(2.758,-2.933)--(2.818,-2.990)--(2.879,-3.048)--(2.939,-3.105)--(3.000,-3.162);
-\draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
-\draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw (0.21069,0.80458) node {$R$};
-\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
-\draw (0.19314,-0.80458) node {$S$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (-3.0000,-3.1622)--(-2.9393,-3.1048)--(-2.8787,-3.0475)--(-2.8181,-2.9903)--(-2.7575,-2.9332)--(-2.6969,-2.8763)--(-2.6363,-2.8196)--(-2.5757,-2.7630)--(-2.5151,-2.7066)--(-2.4545,-2.6504)--(-2.3939,-2.5944)--(-2.3333,-2.5385)--(-2.2727,-2.4830)--(-2.2121,-2.4276)--(-2.1515,-2.3725)--(-2.0909,-2.3177)--(-2.0303,-2.2632)--(-1.9696,-2.2090)--(-1.9090,-2.1551)--(-1.8484,-2.1016)--(-1.7878,-2.0485)--(-1.7272,-1.9958)--(-1.6666,-1.9436)--(-1.6060,-1.8919)--(-1.5454,-1.8407)--(-1.4848,-1.7901)--(-1.4242,-1.7402)--(-1.3636,-1.6910)--(-1.3030,-1.6425)--(-1.2424,-1.5948)--(-1.1818,-1.5481)--(-1.1212,-1.5023)--(-1.0606,-1.4576)--(-1.0000,-1.4142)--(-0.9393,-1.3720)--(-0.8787,-1.3312)--(-0.8181,-1.2920)--(-0.7575,-1.2545)--(-0.6969,-1.2189)--(-0.6363,-1.1853)--(-0.5757,-1.1539)--(-0.5151,-1.1248)--(-0.4545,-1.0984)--(-0.3939,-1.0747)--(-0.3333,-1.0540)--(-0.2727,-1.0365)--(-0.2121,-1.0222)--(-0.1515,-1.0114)--(-0.0909,-1.0041)--(-0.0303,-1.0004)--(0.0303,-1.0004)--(0.0909,-1.0041)--(0.1515,-1.0114)--(0.2121,-1.0222)--(0.2727,-1.0365)--(0.3333,-1.0540)--(0.3939,-1.0747)--(0.4545,-1.0984)--(0.5151,-1.1248)--(0.5757,-1.1539)--(0.6363,-1.1853)--(0.6969,-1.2189)--(0.7575,-1.2545)--(0.8181,-1.2920)--(0.8787,-1.3312)--(0.9393,-1.3720)--(1.0000,-1.4142)--(1.0606,-1.4576)--(1.1212,-1.5023)--(1.1818,-1.5481)--(1.2424,-1.5948)--(1.3030,-1.6425)--(1.3636,-1.6910)--(1.4242,-1.7402)--(1.4848,-1.7901)--(1.5454,-1.8407)--(1.6060,-1.8919)--(1.6666,-1.9436)--(1.7272,-1.9958)--(1.7878,-2.0485)--(1.8484,-2.1016)--(1.9090,-2.1551)--(1.9696,-2.2090)--(2.0303,-2.2632)--(2.0909,-2.3177)--(2.1515,-2.3725)--(2.2121,-2.4276)--(2.2727,-2.4830)--(2.3333,-2.5385)--(2.3939,-2.5944)--(2.4545,-2.6504)--(2.5151,-2.7066)--(2.5757,-2.7630)--(2.6363,-2.8196)--(2.6969,-2.8763)--(2.7575,-2.9332)--(2.8181,-2.9903)--(2.8787,-3.0475)--(2.9393,-3.1048)--(3.0000,-3.1622);
+\draw [color=brown] (-3.0000,3.0000) -- (3.0000,-3.0000);
+\draw [color=brown] (-3.0000,-3.0000) -- (3.0000,3.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.2106,0.8045) node {$R$};
+\draw []  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.1931,-0.8045) node {$S$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall b/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall
index e54d5c080..9723d869d 100644
--- a/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall
+++ b/src_phystricks/Fig_KKJAooubQzgBgP.pstricks.recall
@@ -72,19 +72,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4487);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4486);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.03030,0.03029)--(0.06061,0.06049)--(0.09091,0.09054)--(0.1212,0.1203)--(0.1515,0.1498)--(0.1818,0.1789)--(0.2121,0.2075)--(0.2424,0.2356)--(0.2727,0.2631)--(0.3030,0.2900)--(0.3333,0.3162)--(0.3636,0.3417)--(0.3939,0.3665)--(0.4242,0.3905)--(0.4545,0.4138)--(0.4848,0.4363)--(0.5152,0.4580)--(0.5455,0.4789)--(0.5758,0.4990)--(0.6061,0.5183)--(0.6364,0.5369)--(0.6667,0.5547)--(0.6970,0.5718)--(0.7273,0.5882)--(0.7576,0.6039)--(0.7879,0.6189)--(0.8182,0.6332)--(0.8485,0.6470)--(0.8788,0.6601)--(0.9091,0.6727)--(0.9394,0.6847)--(0.9697,0.6961)--(1.000,0.7071)--(1.030,0.7176)--(1.061,0.7276)--(1.091,0.7372)--(1.121,0.7463)--(1.152,0.7550)--(1.182,0.7634)--(1.212,0.7714)--(1.242,0.7790)--(1.273,0.7863)--(1.303,0.7933)--(1.333,0.8000)--(1.364,0.8064)--(1.394,0.8125)--(1.424,0.8184)--(1.455,0.8240)--(1.485,0.8294)--(1.515,0.8346)--(1.545,0.8396)--(1.576,0.8443)--(1.606,0.8489)--(1.636,0.8533)--(1.667,0.8575)--(1.697,0.8615)--(1.727,0.8654)--(1.758,0.8692)--(1.788,0.8728)--(1.818,0.8762)--(1.848,0.8795)--(1.879,0.8827)--(1.909,0.8858)--(1.939,0.8888)--(1.970,0.8917)--(2.000,0.8944)--(2.030,0.8971)--(2.061,0.8997)--(2.091,0.9021)--(2.121,0.9045)--(2.152,0.9068)--(2.182,0.9091)--(2.212,0.9112)--(2.242,0.9133)--(2.273,0.9153)--(2.303,0.9173)--(2.333,0.9191)--(2.364,0.9210)--(2.394,0.9227)--(2.424,0.9244)--(2.455,0.9261)--(2.485,0.9277)--(2.515,0.9292)--(2.545,0.9307)--(2.576,0.9322)--(2.606,0.9336)--(2.636,0.9350)--(2.667,0.9363)--(2.697,0.9376)--(2.727,0.9389)--(2.758,0.9401)--(2.788,0.9413)--(2.818,0.9424)--(2.848,0.9435)--(2.879,0.9446)--(2.909,0.9457)--(2.939,0.9467)--(2.970,0.9477)--(3.000,0.9487);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0303,0.0302)--(0.0606,0.0604)--(0.0909,0.0905)--(0.1212,0.1203)--(0.1515,0.1498)--(0.1818,0.1788)--(0.2121,0.2075)--(0.2424,0.2356)--(0.2727,0.2631)--(0.3030,0.2900)--(0.3333,0.3162)--(0.3636,0.3417)--(0.3939,0.3665)--(0.4242,0.3905)--(0.4545,0.4138)--(0.4848,0.4362)--(0.5151,0.4579)--(0.5454,0.4788)--(0.5757,0.4989)--(0.6060,0.5183)--(0.6363,0.5368)--(0.6666,0.5547)--(0.6969,0.5717)--(0.7272,0.5881)--(0.7575,0.6038)--(0.7878,0.6188)--(0.8181,0.6332)--(0.8484,0.6469)--(0.8787,0.6601)--(0.9090,0.6726)--(0.9393,0.6846)--(0.9696,0.6961)--(1.0000,0.7071)--(1.0303,0.7175)--(1.0606,0.7275)--(1.0909,0.7371)--(1.1212,0.7462)--(1.1515,0.7550)--(1.1818,0.7633)--(1.2121,0.7713)--(1.2424,0.7790)--(1.2727,0.7863)--(1.3030,0.7933)--(1.3333,0.8000)--(1.3636,0.8064)--(1.3939,0.8125)--(1.4242,0.8184)--(1.4545,0.8240)--(1.4848,0.8294)--(1.5151,0.8346)--(1.5454,0.8395)--(1.5757,0.8443)--(1.6060,0.8488)--(1.6363,0.8532)--(1.6666,0.8574)--(1.6969,0.8615)--(1.7272,0.8654)--(1.7575,0.8691)--(1.7878,0.8727)--(1.8181,0.8762)--(1.8484,0.8795)--(1.8787,0.8827)--(1.9090,0.8858)--(1.9393,0.8888)--(1.9696,0.8916)--(2.0000,0.8944)--(2.0303,0.8970)--(2.0606,0.8996)--(2.0909,0.9021)--(2.1212,0.9045)--(2.1515,0.9068)--(2.1818,0.9090)--(2.2121,0.9112)--(2.2424,0.9133)--(2.2727,0.9153)--(2.3030,0.9172)--(2.3333,0.9191)--(2.3636,0.9209)--(2.3939,0.9227)--(2.4242,0.9244)--(2.4545,0.9260)--(2.4848,0.9276)--(2.5151,0.9292)--(2.5454,0.9307)--(2.5757,0.9322)--(2.6060,0.9336)--(2.6363,0.9349)--(2.6666,0.9363)--(2.6969,0.9376)--(2.7272,0.9388)--(2.7575,0.9400)--(2.7878,0.9412)--(2.8181,0.9424)--(2.8484,0.9435)--(2.8787,0.9446)--(2.9090,0.9456)--(2.9393,0.9467)--(2.9696,0.9477)--(3.0000,0.9486);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_LAfWmaN.pstricks.recall b/src_phystricks/Fig_LAfWmaN.pstricks.recall
index 45c4fc2fa..10fefba50 100644
--- a/src_phystricks/Fig_LAfWmaN.pstricks.recall
+++ b/src_phystricks/Fig_LAfWmaN.pstricks.recall
@@ -91,33 +91,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.6200,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.1000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (5.6200,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.1000);
 %DEFAULT
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-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-1.0000,-1.5000)--(-0.9393,-1.5000)--(-0.8787,-1.5000)--(-0.8181,-1.5000)--(-0.7575,-1.5000)--(-0.6969,-1.5000)--(-0.6363,-1.5000)--(-0.5757,-1.5000)--(-0.5151,-1.5000)--(-0.4545,-1.5000)--(-0.3939,-1.5000)--(-0.3333,-1.5000)--(-0.2727,-1.5000)--(-0.2121,-1.5000)--(-0.1515,-1.5000)--(-0.0909,-1.5000)--(-0.0303,-1.5000)--(0.0303,-1.5000)--(0.0909,-1.5000)--(0.1515,-1.5000)--(0.2121,-1.5000)--(0.2727,-1.5000)--(0.3333,-1.5000)--(0.3939,-1.5000)--(0.4545,-1.5000)--(0.5151,-1.5000)--(0.5757,-1.5000)--(0.6363,-1.5000)--(0.6969,-1.5000)--(0.7575,-1.5000)--(0.8181,-1.5000)--(0.8787,-1.5000)--(0.9393,-1.5000)--(1.0000,-1.5000)--(1.0606,-1.5000)--(1.1212,-1.5000)--(1.1818,-1.5000)--(1.2424,-1.5000)--(1.3030,-1.5000)--(1.3636,-1.5000)--(1.4242,-1.5000)--(1.4848,-1.5000)--(1.5454,-1.5000)--(1.6060,-1.5000)--(1.6666,-1.5000)--(1.7272,-1.5000)--(1.7878,-1.5000)--(1.8484,-1.5000)--(1.9090,-1.5000)--(1.9696,-1.5000)--(2.0303,-1.5000)--(2.0909,-1.5000)--(2.1515,-1.5000)--(2.2121,-1.5000)--(2.2727,-1.5000)--(2.3333,-1.5000)--(2.3939,-1.5000)--(2.4545,-1.5000)--(2.5151,-1.5000)--(2.5757,-1.5000)--(2.6363,-1.5000)--(2.6969,-1.5000)--(2.7575,-1.5000)--(2.8181,-1.5000)--(2.8787,-1.5000)--(2.9393,-1.5000)--(3.0000,-1.5000)--(3.0606,-1.5000)--(3.1212,-1.5000)--(3.1818,-1.5000)--(3.2424,-1.5000)--(3.3030,-1.5000)--(3.3636,-1.5000)--(3.4242,-1.5000)--(3.4848,-1.5000)--(3.5454,-1.5000)--(3.6060,-1.5000)--(3.6666,-1.5000)--(3.7272,-1.5000)--(3.7878,-1.5000)--(3.8484,-1.5000)--(3.9090,-1.5000)--(3.9696,-1.5000)--(4.0303,-1.5000)--(4.0909,-1.5000)--(4.1515,-1.5000)--(4.2121,-1.5000)--(4.2727,-1.5000)--(4.3333,-1.5000)--(4.3939,-1.5000)--(4.4545,-1.5000)--(4.5151,-1.5000)--(4.5757,-1.5000)--(4.6363,-1.5000)--(4.6969,-1.5000)--(4.7575,-1.5000)--(4.8181,-1.5000)--(4.8787,-1.5000)--(4.9393,-1.5000)--(5.0000,-1.5000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_LBGooAdteCt.pstricks.recall b/src_phystricks/Fig_LBGooAdteCt.pstricks.recall
index 5bc36bb3b..1b9248e31 100644
--- a/src_phystricks/Fig_LBGooAdteCt.pstricks.recall
+++ b/src_phystricks/Fig_LBGooAdteCt.pstricks.recall
@@ -99,63 +99,63 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-5.00,0) -- (-5.00,5.00);
-\draw [color=gray,style=solid] (-4.00,0) -- (-4.00,5.00);
-\draw [color=gray,style=solid] (-3.00,0) -- (-3.00,5.00);
-\draw [color=gray,style=solid] (-2.00,0) -- (-2.00,5.00);
-\draw [color=gray,style=solid] (-1.00,0) -- (-1.00,5.00);
-\draw [color=gray,style=solid] (0,0) -- (0,5.00);
-\draw [color=gray,style=solid] (1.00,0) -- (1.00,5.00);
-\draw [color=gray,style=solid] (2.00,0) -- (2.00,5.00);
-\draw [color=gray,style=dotted] (-4.50,0) -- (-4.50,5.00);
-\draw [color=gray,style=dotted] (-3.50,0) -- (-3.50,5.00);
-\draw [color=gray,style=dotted] (-2.50,0) -- (-2.50,5.00);
-\draw [color=gray,style=dotted] (-1.50,0) -- (-1.50,5.00);
-\draw [color=gray,style=dotted] (-0.500,0) -- (-0.500,5.00);
-\draw [color=gray,style=dotted] (0.500,0) -- (0.500,5.00);
-\draw [color=gray,style=dotted] (1.50,0) -- (1.50,5.00);
-\draw [color=gray,style=dotted] (-5.00,0.500) -- (2.00,0.500);
-\draw [color=gray,style=dotted] (-5.00,1.50) -- (2.00,1.50);
-\draw [color=gray,style=dotted] (-5.00,2.50) -- (2.00,2.50);
-\draw [color=gray,style=dotted] (-5.00,3.50) -- (2.00,3.50);
-\draw [color=gray,style=dotted] (-5.00,4.50) -- (2.00,4.50);
-\draw [color=gray,style=solid] (-5.00,0) -- (2.00,0);
-\draw [color=gray,style=solid] (-5.00,1.00) -- (2.00,1.00);
-\draw [color=gray,style=solid] (-5.00,2.00) -- (2.00,2.00);
-\draw [color=gray,style=solid] (-5.00,3.00) -- (2.00,3.00);
-\draw [color=gray,style=solid] (-5.00,4.00) -- (2.00,4.00);
-\draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
+\draw [color=gray,style=solid] (-5.0000,0.0000) -- (-5.0000,5.0000);
+\draw [color=gray,style=solid] (-4.0000,0.0000) -- (-4.0000,5.0000);
+\draw [color=gray,style=solid] (-3.0000,0.0000) -- (-3.0000,5.0000);
+\draw [color=gray,style=solid] (-2.0000,0.0000) -- (-2.0000,5.0000);
+\draw [color=gray,style=solid] (-1.0000,0.0000) -- (-1.0000,5.0000);
+\draw [color=gray,style=solid] (0.0000,0.0000) -- (0.0000,5.0000);
+\draw [color=gray,style=solid] (1.0000,0.0000) -- (1.0000,5.0000);
+\draw [color=gray,style=solid] (2.0000,0.0000) -- (2.0000,5.0000);
+\draw [color=gray,style=dotted] (-4.5000,0.0000) -- (-4.5000,5.0000);
+\draw [color=gray,style=dotted] (-3.5000,0.0000) -- (-3.5000,5.0000);
+\draw [color=gray,style=dotted] (-2.5000,0.0000) -- (-2.5000,5.0000);
+\draw [color=gray,style=dotted] (-1.5000,0.0000) -- (-1.5000,5.0000);
+\draw [color=gray,style=dotted] (-0.5000,0.0000) -- (-0.5000,5.0000);
+\draw [color=gray,style=dotted] (0.5000,0.0000) -- (0.5000,5.0000);
+\draw [color=gray,style=dotted] (1.5000,0.0000) -- (1.5000,5.0000);
+\draw [color=gray,style=dotted] (-5.0000,0.5000) -- (2.0000,0.5000);
+\draw [color=gray,style=dotted] (-5.0000,1.5000) -- (2.0000,1.5000);
+\draw [color=gray,style=dotted] (-5.0000,2.5000) -- (2.0000,2.5000);
+\draw [color=gray,style=dotted] (-5.0000,3.5000) -- (2.0000,3.5000);
+\draw [color=gray,style=dotted] (-5.0000,4.5000) -- (2.0000,4.5000);
+\draw [color=gray,style=solid] (-5.0000,0.0000) -- (2.0000,0.0000);
+\draw [color=gray,style=solid] (-5.0000,1.0000) -- (2.0000,1.0000);
+\draw [color=gray,style=solid] (-5.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=gray,style=solid] (-5.0000,3.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (-5.0000,4.0000) -- (2.0000,4.0000);
+\draw [color=gray,style=solid] (-5.0000,5.0000) -- (2.0000,5.0000);
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
 
-\draw [color=blue] (-5.000,0.006738)--(-4.934,0.007200)--(-4.867,0.007694)--(-4.801,0.008221)--(-4.735,0.008785)--(-4.668,0.009388)--(-4.602,0.01003)--(-4.536,0.01072)--(-4.469,0.01145)--(-4.403,0.01224)--(-4.337,0.01308)--(-4.270,0.01398)--(-4.204,0.01493)--(-4.138,0.01596)--(-4.071,0.01705)--(-4.005,0.01822)--(-3.939,0.01947)--(-3.872,0.02081)--(-3.806,0.02223)--(-3.740,0.02376)--(-3.673,0.02539)--(-3.607,0.02713)--(-3.541,0.02899)--(-3.474,0.03098)--(-3.408,0.03310)--(-3.342,0.03537)--(-3.275,0.03780)--(-3.209,0.04039)--(-3.143,0.04316)--(-3.076,0.04612)--(-3.010,0.04928)--(-2.944,0.05266)--(-2.878,0.05628)--(-2.811,0.06014)--(-2.745,0.06426)--(-2.678,0.06867)--(-2.612,0.07337)--(-2.546,0.07841)--(-2.480,0.08378)--(-2.413,0.08953)--(-2.347,0.09567)--(-2.281,0.1022)--(-2.214,0.1092)--(-2.148,0.1167)--(-2.082,0.1247)--(-2.015,0.1333)--(-1.949,0.1424)--(-1.883,0.1522)--(-1.816,0.1626)--(-1.750,0.1738)--(-1.684,0.1857)--(-1.617,0.1984)--(-1.551,0.2121)--(-1.485,0.2266)--(-1.418,0.2421)--(-1.352,0.2587)--(-1.286,0.2765)--(-1.219,0.2954)--(-1.153,0.3157)--(-1.087,0.3374)--(-1.020,0.3605)--(-0.9540,0.3852)--(-0.8876,0.4116)--(-0.8213,0.4399)--(-0.7550,0.4700)--(-0.6886,0.5023)--(-0.6223,0.5367)--(-0.5560,0.5735)--(-0.4897,0.6128)--(-0.4233,0.6549)--(-0.3570,0.6998)--(-0.2907,0.7478)--(-0.2243,0.7990)--(-0.1580,0.8538)--(-0.09169,0.9124)--(-0.02536,0.9750)--(0.04097,1.042)--(0.1073,1.113)--(0.1736,1.190)--(0.2400,1.271)--(0.3063,1.358)--(0.3726,1.452)--(0.4389,1.551)--(0.5053,1.657)--(0.5716,1.771)--(0.6379,1.893)--(0.7043,2.022)--(0.7706,2.161)--(0.8369,2.309)--(0.9032,2.468)--(0.9696,2.637)--(1.036,2.818)--(1.102,3.011)--(1.169,3.217)--(1.235,3.438)--(1.301,3.674)--(1.368,3.926)--(1.434,4.195)--(1.500,4.483)--(1.567,4.790);
+\draw [color=blue] (-5.0000,0.0067)--(-4.9336,0.0072)--(-4.8673,0.0076)--(-4.8010,0.0082)--(-4.7346,0.0087)--(-4.6683,0.0093)--(-4.6020,0.0100)--(-4.5357,0.0107)--(-4.4693,0.0114)--(-4.4030,0.0122)--(-4.3367,0.0130)--(-4.2703,0.0139)--(-4.2040,0.0149)--(-4.1377,0.0159)--(-4.0714,0.0170)--(-4.0050,0.0182)--(-3.9387,0.0194)--(-3.8724,0.0208)--(-3.8060,0.0222)--(-3.7397,0.0237)--(-3.6734,0.0253)--(-3.6071,0.0271)--(-3.5407,0.0289)--(-3.4744,0.0309)--(-3.4081,0.0331)--(-3.3417,0.0353)--(-3.2754,0.0377)--(-3.2091,0.0403)--(-3.1428,0.0431)--(-3.0764,0.0461)--(-3.0101,0.0492)--(-2.9438,0.0526)--(-2.8774,0.0562)--(-2.8111,0.0601)--(-2.7448,0.0642)--(-2.6785,0.0686)--(-2.6121,0.0733)--(-2.5458,0.0784)--(-2.4795,0.0837)--(-2.4131,0.0895)--(-2.3468,0.0956)--(-2.2805,0.1022)--(-2.2142,0.1092)--(-2.1478,0.1167)--(-2.0815,0.1247)--(-2.0152,0.1332)--(-1.9488,0.1424)--(-1.8825,0.1522)--(-1.8162,0.1626)--(-1.7499,0.1737)--(-1.6835,0.1857)--(-1.6172,0.1984)--(-1.5509,0.2120)--(-1.4845,0.2265)--(-1.4182,0.2421)--(-1.3519,0.2587)--(-1.2856,0.2764)--(-1.2192,0.2954)--(-1.1529,0.3157)--(-1.0866,0.3373)--(-1.0202,0.3604)--(-0.9539,0.3852)--(-0.8876,0.4116)--(-0.8213,0.4398)--(-0.7549,0.4700)--(-0.6886,0.5022)--(-0.6223,0.5366)--(-0.5559,0.5735)--(-0.4896,0.6128)--(-0.4233,0.6548)--(-0.3570,0.6997)--(-0.2906,0.7477)--(-0.2243,0.7990)--(-0.1580,0.8538)--(-0.0916,0.9123)--(-0.0253,0.9749)--(0.0409,1.0418)--(0.1072,1.1132)--(0.1736,1.1896)--(0.2399,1.2711)--(0.3062,1.3583)--(0.3726,1.4515)--(0.4389,1.5510)--(0.5052,1.6574)--(0.5715,1.7710)--(0.6379,1.8925)--(0.7042,2.0223)--(0.7705,2.1610)--(0.8369,2.3092)--(0.9032,2.4675)--(0.9695,2.6368)--(1.0358,2.8176)--(1.1022,3.0108)--(1.1685,3.2173)--(1.2348,3.4379)--(1.3012,3.6737)--(1.3675,3.9256)--(1.4338,4.1948)--(1.5001,4.4825)--(1.5665,4.7899);
 
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall b/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall
index d1387e358..a2739b882 100644
--- a/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall
+++ b/src_phystricks/Fig_LMHMooCscXNNdU.pstricks.recall
@@ -100,38 +100,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (-5.000,1.857)--(-4.939,1.856)--(-4.879,1.855)--(-4.818,1.853)--(-4.758,1.852)--(-4.697,1.851)--(-4.636,1.849)--(-4.576,1.848)--(-4.515,1.847)--(-4.455,1.845)--(-4.394,1.844)--(-4.333,1.842)--(-4.273,1.841)--(-4.212,1.839)--(-4.151,1.837)--(-4.091,1.836)--(-4.030,1.834)--(-3.970,1.832)--(-3.909,1.831)--(-3.848,1.829)--(-3.788,1.827)--(-3.727,1.825)--(-3.667,1.824)--(-3.606,1.822)--(-3.545,1.820)--(-3.485,1.818)--(-3.424,1.816)--(-3.364,1.814)--(-3.303,1.811)--(-3.242,1.809)--(-3.182,1.807)--(-3.121,1.805)--(-3.061,1.802)--(-3.000,1.800)--(-2.939,1.798)--(-2.879,1.795)--(-2.818,1.792)--(-2.758,1.790)--(-2.697,1.787)--(-2.636,1.784)--(-2.576,1.781)--(-2.515,1.779)--(-2.455,1.776)--(-2.394,1.772)--(-2.333,1.769)--(-2.273,1.766)--(-2.212,1.763)--(-2.152,1.759)--(-2.091,1.756)--(-2.030,1.752)--(-1.970,1.748)--(-1.909,1.744)--(-1.848,1.740)--(-1.788,1.736)--(-1.727,1.732)--(-1.667,1.727)--(-1.606,1.723)--(-1.545,1.718)--(-1.485,1.713)--(-1.424,1.708)--(-1.364,1.703)--(-1.303,1.697)--(-1.242,1.692)--(-1.182,1.686)--(-1.121,1.680)--(-1.061,1.673)--(-1.000,1.667)--(-0.9394,1.660)--(-0.8788,1.653)--(-0.8182,1.645)--(-0.7576,1.637)--(-0.6970,1.629)--(-0.6364,1.621)--(-0.5758,1.612)--(-0.5152,1.602)--(-0.4545,1.593)--(-0.3939,1.582)--(-0.3333,1.571)--(-0.2727,1.560)--(-0.2121,1.548)--(-0.1515,1.535)--(-0.09091,1.522)--(-0.03030,1.507)--(0.03030,1.492)--(0.09091,1.476)--(0.1515,1.459)--(0.2121,1.441)--(0.2727,1.421)--(0.3333,1.400)--(0.3939,1.377)--(0.4545,1.353)--(0.5152,1.327)--(0.5758,1.298)--(0.6364,1.267)--(0.6970,1.233)--(0.7576,1.195)--(0.8182,1.154)--(0.8788,1.108)--(0.9394,1.057)--(1.000,1.000);
+\draw [color=blue] (-5.0000,1.8571)--(-4.9393,1.8558)--(-4.8787,1.8546)--(-4.8181,1.8533)--(-4.7575,1.8520)--(-4.6969,1.8506)--(-4.6363,1.8493)--(-4.5757,1.8479)--(-4.5151,1.8465)--(-4.4545,1.8450)--(-4.3939,1.8436)--(-4.3333,1.8421)--(-4.2727,1.8405)--(-4.2121,1.8390)--(-4.1515,1.8374)--(-4.0909,1.8358)--(-4.0303,1.8341)--(-3.9696,1.8324)--(-3.9090,1.8307)--(-3.8484,1.8290)--(-3.7878,1.8272)--(-3.7272,1.8253)--(-3.6666,1.8235)--(-3.6060,1.8216)--(-3.5454,1.8196)--(-3.4848,1.8176)--(-3.4242,1.8156)--(-3.3636,1.8135)--(-3.3030,1.8114)--(-3.2424,1.8092)--(-3.1818,1.8070)--(-3.1212,1.8047)--(-3.0606,1.8023)--(-3.0000,1.8000)--(-2.9393,1.7975)--(-2.8787,1.7950)--(-2.8181,1.7924)--(-2.7575,1.7898)--(-2.6969,1.7870)--(-2.6363,1.7843)--(-2.5757,1.7814)--(-2.5151,1.7785)--(-2.4545,1.7755)--(-2.3939,1.7724)--(-2.3333,1.7692)--(-2.2727,1.7659)--(-2.2121,1.7625)--(-2.1515,1.7591)--(-2.0909,1.7555)--(-2.0303,1.7518)--(-1.9696,1.7480)--(-1.9090,1.7441)--(-1.8484,1.7401)--(-1.7878,1.7360)--(-1.7272,1.7317)--(-1.6666,1.7272)--(-1.6060,1.7226)--(-1.5454,1.7179)--(-1.4848,1.7130)--(-1.4242,1.7079)--(-1.3636,1.7027)--(-1.3030,1.6972)--(-1.2424,1.6915)--(-1.1818,1.6857)--(-1.1212,1.6796)--(-1.0606,1.6732)--(-1.0000,1.6666)--(-0.9393,1.6597)--(-0.8787,1.6526)--(-0.8181,1.6451)--(-0.7575,1.6373)--(-0.6969,1.6292)--(-0.6363,1.6206)--(-0.5757,1.6117)--(-0.5151,1.6024)--(-0.4545,1.5925)--(-0.3939,1.5822)--(-0.3333,1.5714)--(-0.2727,1.5600)--(-0.2121,1.5479)--(-0.1515,1.5352)--(-0.0909,1.5217)--(-0.0303,1.5074)--(0.0303,1.4923)--(0.0909,1.4761)--(0.1515,1.4590)--(0.2121,1.4406)--(0.2727,1.4210)--(0.3333,1.4000)--(0.3939,1.3773)--(0.4545,1.3529)--(0.5151,1.3265)--(0.5757,1.2978)--(0.6363,1.2666)--(0.6969,1.2325)--(0.7575,1.1951)--(0.8181,1.1538)--(0.8787,1.1081)--(0.9393,1.0571)--(1.0000,1.0000);
 
-\draw [color=blue] (1.000,1.000)--(1.040,0.9612)--(1.081,0.9252)--(1.121,0.8919)--(1.162,0.8609)--(1.202,0.8319)--(1.242,0.8049)--(1.283,0.7795)--(1.323,0.7557)--(1.364,0.7333)--(1.404,0.7122)--(1.444,0.6923)--(1.485,0.6735)--(1.525,0.6556)--(1.566,0.6387)--(1.606,0.6226)--(1.646,0.6074)--(1.687,0.5928)--(1.727,0.5789)--(1.768,0.5657)--(1.808,0.5531)--(1.848,0.5410)--(1.889,0.5294)--(1.929,0.5183)--(1.970,0.5077)--(2.010,0.4975)--(2.051,0.4877)--(2.091,0.4783)--(2.131,0.4692)--(2.172,0.4605)--(2.212,0.4521)--(2.253,0.4439)--(2.293,0.4361)--(2.333,0.4286)--(2.374,0.4213)--(2.414,0.4142)--(2.455,0.4074)--(2.495,0.4008)--(2.535,0.3944)--(2.576,0.3882)--(2.616,0.3822)--(2.657,0.3764)--(2.697,0.3708)--(2.737,0.3653)--(2.778,0.3600)--(2.818,0.3548)--(2.859,0.3498)--(2.899,0.3449)--(2.939,0.3402)--(2.980,0.3356)--(3.020,0.3311)--(3.061,0.3267)--(3.101,0.3225)--(3.141,0.3183)--(3.182,0.3143)--(3.222,0.3103)--(3.263,0.3065)--(3.303,0.3028)--(3.343,0.2991)--(3.384,0.2955)--(3.424,0.2920)--(3.465,0.2886)--(3.505,0.2853)--(3.545,0.2821)--(3.586,0.2789)--(3.626,0.2758)--(3.667,0.2727)--(3.707,0.2698)--(3.747,0.2668)--(3.788,0.2640)--(3.828,0.2612)--(3.869,0.2585)--(3.909,0.2558)--(3.949,0.2532)--(3.990,0.2506)--(4.030,0.2481)--(4.071,0.2457)--(4.111,0.2432)--(4.151,0.2409)--(4.192,0.2386)--(4.232,0.2363)--(4.273,0.2340)--(4.313,0.2318)--(4.354,0.2297)--(4.394,0.2276)--(4.434,0.2255)--(4.475,0.2235)--(4.515,0.2215)--(4.556,0.2195)--(4.596,0.2176)--(4.636,0.2157)--(4.677,0.2138)--(4.717,0.2120)--(4.758,0.2102)--(4.798,0.2084)--(4.838,0.2067)--(4.879,0.2050)--(4.919,0.2033)--(4.960,0.2016)--(5.000,0.2000);
-\draw [style=dashed] (-5.00,2.00) -- (5.00,2.00);
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (1.0000,1.0000)--(1.0404,0.9611)--(1.0808,0.9252)--(1.1212,0.8918)--(1.1616,0.8608)--(1.2020,0.8319)--(1.2424,0.8048)--(1.2828,0.7795)--(1.3232,0.7557)--(1.3636,0.7333)--(1.4040,0.7122)--(1.4444,0.6923)--(1.4848,0.6734)--(1.5252,0.6556)--(1.5656,0.6387)--(1.6060,0.6226)--(1.6464,0.6073)--(1.6868,0.5928)--(1.7272,0.5789)--(1.7676,0.5657)--(1.8080,0.5530)--(1.8484,0.5409)--(1.8888,0.5294)--(1.9292,0.5183)--(1.9696,0.5076)--(2.0101,0.4974)--(2.0505,0.4876)--(2.0909,0.4782)--(2.1313,0.4691)--(2.1717,0.4604)--(2.2121,0.4520)--(2.2525,0.4439)--(2.2929,0.4361)--(2.3333,0.4285)--(2.3737,0.4212)--(2.4141,0.4142)--(2.4545,0.4074)--(2.4949,0.4008)--(2.5353,0.3944)--(2.5757,0.3882)--(2.6161,0.3822)--(2.6565,0.3764)--(2.6969,0.3707)--(2.7373,0.3653)--(2.7777,0.3600)--(2.8181,0.3548)--(2.8585,0.3498)--(2.8989,0.3449)--(2.9393,0.3402)--(2.9797,0.3355)--(3.0202,0.3311)--(3.0606,0.3267)--(3.1010,0.3224)--(3.1414,0.3183)--(3.1818,0.3142)--(3.2222,0.3103)--(3.2626,0.3065)--(3.3030,0.3027)--(3.3434,0.2990)--(3.3838,0.2955)--(3.4242,0.2920)--(3.4646,0.2886)--(3.5050,0.2853)--(3.5454,0.2820)--(3.5858,0.2788)--(3.6262,0.2757)--(3.6666,0.2727)--(3.7070,0.2697)--(3.7474,0.2668)--(3.7878,0.2640)--(3.8282,0.2612)--(3.8686,0.2584)--(3.9090,0.2558)--(3.9494,0.2531)--(3.9898,0.2506)--(4.0303,0.2481)--(4.0707,0.2456)--(4.1111,0.2432)--(4.1515,0.2408)--(4.1919,0.2385)--(4.2323,0.2362)--(4.2727,0.2340)--(4.3131,0.2318)--(4.3535,0.2296)--(4.3939,0.2275)--(4.4343,0.2255)--(4.4747,0.2234)--(4.5151,0.2214)--(4.5555,0.2195)--(4.5959,0.2175)--(4.6363,0.2156)--(4.6767,0.2138)--(4.7171,0.2119)--(4.7575,0.2101)--(4.7979,0.2084)--(4.8383,0.2066)--(4.8787,0.2049)--(4.9191,0.2032)--(4.9595,0.2016)--(5.0000,0.2000);
+\draw [style=dashed] (-5.0000,2.0000) -- (5.0000,2.0000);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_Laurin.pstricks.recall b/src_phystricks/Fig_Laurin.pstricks.recall
index bd70fac13..5e2d5a7bc 100644
--- a/src_phystricks/Fig_Laurin.pstricks.recall
+++ b/src_phystricks/Fig_Laurin.pstricks.recall
@@ -99,41 +99,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,7.8891);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,7.8890);
 %DEFAULT
 
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-\draw (-0.29125,7.0000) node {$ 7 $};
-\draw [] (-0.100,7.00) -- (0.100,7.00);
+\draw [color=red] (-2.0000,1.0000)--(-1.9595,0.9604)--(-1.9191,0.9224)--(-1.8787,0.8861)--(-1.8383,0.8514)--(-1.7979,0.8183)--(-1.7575,0.7869)--(-1.7171,0.7571)--(-1.6767,0.7290)--(-1.6363,0.7024)--(-1.5959,0.6775)--(-1.5555,0.6543)--(-1.5151,0.6326)--(-1.4747,0.6126)--(-1.4343,0.5943)--(-1.3939,0.5775)--(-1.3535,0.5624)--(-1.3131,0.5490)--(-1.2727,0.5371)--(-1.2323,0.5269)--(-1.1919,0.5184)--(-1.1515,0.5114)--(-1.1111,0.5061)--(-1.0707,0.5024)--(-1.0303,0.5004)--(-0.9898,0.5000)--(-0.9494,0.5012)--(-0.9090,0.5041)--(-0.8686,0.5086)--(-0.8282,0.5147)--(-0.7878,0.5224)--(-0.7474,0.5318)--(-0.7070,0.5429)--(-0.6666,0.5555)--(-0.6262,0.5698)--(-0.5858,0.5857)--(-0.5454,0.6033)--(-0.5050,0.6224)--(-0.4646,0.6433)--(-0.4242,0.6657)--(-0.3838,0.6898)--(-0.3434,0.7155)--(-0.3030,0.7428)--(-0.2626,0.7718)--(-0.2222,0.8024)--(-0.1818,0.8347)--(-0.1414,0.8685)--(-0.1010,0.9040)--(-0.0606,0.9412)--(-0.0202,0.9800)--(0.0202,1.0204)--(0.0606,1.0624)--(0.1010,1.1061)--(0.1414,1.1514)--(0.1818,1.1983)--(0.2222,1.2469)--(0.2626,1.2971)--(0.3030,1.3489)--(0.3434,1.4024)--(0.3838,1.4575)--(0.4242,1.5142)--(0.4646,1.5725)--(0.5050,1.6325)--(0.5454,1.6942)--(0.5858,1.7574)--(0.6262,1.8223)--(0.6666,1.8888)--(0.7070,1.9570)--(0.7474,2.0268)--(0.7878,2.0982)--(0.8282,2.1713)--(0.8686,2.2459)--(0.9090,2.3223)--(0.9494,2.4002)--(0.9898,2.4798)--(1.0303,2.5610)--(1.0707,2.6439)--(1.1111,2.7283)--(1.1515,2.8145)--(1.1919,2.9022)--(1.2323,2.9916)--(1.2727,3.0826)--(1.3131,3.1752)--(1.3535,3.2695)--(1.3939,3.3654)--(1.4343,3.4630)--(1.4747,3.5621)--(1.5151,3.6629)--(1.5555,3.7654)--(1.5959,3.8695)--(1.6363,3.9752)--(1.6767,4.0825)--(1.7171,4.1915)--(1.7575,4.3021)--(1.7979,4.4143)--(1.8383,4.5282)--(1.8787,4.6437)--(1.9191,4.7608)--(1.9595,4.8796)--(2.0000,5.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
+\draw (-0.2912,7.0000) node {$ 7 $};
+\draw [] (-0.1000,7.0000) -- (0.1000,7.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_LesSpheres.pstricks.recall b/src_phystricks/Fig_LesSpheres.pstricks.recall
index 3858e3937..cd673c46c 100644
--- a/src_phystricks/Fig_LesSpheres.pstricks.recall
+++ b/src_phystricks/Fig_LesSpheres.pstricks.recall
@@ -41,21 +41,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw [] (0,1.00) -- (-1.00,0);
-\draw [] (-1.00,0) -- (0,-1.00);
-\draw [] (0,-1.00) -- (1.00,0);
-\draw [] (1.00,0) -- (0,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (0.0000,1.0000) -- (-1.0000,0.0000);
+\draw [] (-1.0000,0.0000) -- (0.0000,-1.0000);
+\draw [] (0.0000,-1.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (0.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -93,19 +93,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -143,21 +143,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw [] (1.00,1.00) -- (-1.00,1.00);
-\draw [] (-1.00,1.00) -- (-1.00,-1.00);
-\draw [] (-1.00,-1.00) -- (1.00,-1.00);
-\draw [] (1.00,-1.00) -- (1.00,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (1.0000,1.0000) -- (-1.0000,1.0000);
+\draw [] (-1.0000,1.0000) -- (-1.0000,-1.0000);
+\draw [] (-1.0000,-1.0000) -- (1.0000,-1.0000);
+\draw [] (1.0000,-1.0000) -- (1.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_MNICGhR.pstricks.recall b/src_phystricks/Fig_MNICGhR.pstricks.recall
index ee87719a4..902511998 100644
--- a/src_phystricks/Fig_MNICGhR.pstricks.recall
+++ b/src_phystricks/Fig_MNICGhR.pstricks.recall
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.30595) node {$\alpha_1$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.30595) node {$\alpha_2$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.30595) node {$\alpha_3$};
-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.32154) node {$\alpha_{l-1}$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.30831) node {$\alpha_l$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3059) node {$\alpha_1$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.3059) node {$\alpha_2$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.3059) node {$\alpha_3$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$o$};
+\draw (3.0000,0.3215) node {$\alpha_{l-1}$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.3083) node {$\alpha_l$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_Mantisse.pstricks.recall b/src_phystricks/Fig_Mantisse.pstricks.recall
index 1c275f207..c33f42fe7 100644
--- a/src_phystricks/Fig_Mantisse.pstricks.recall
+++ b/src_phystricks/Fig_Mantisse.pstricks.recall
@@ -83,29 +83,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4990);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4989);
 %DEFAULT
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+\draw [color=blue] 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+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_MethodeChemin.pstricks.recall b/src_phystricks/Fig_MethodeChemin.pstricks.recall
index 8934d3069..51e3cf49d 100644
--- a/src_phystricks/Fig_MethodeChemin.pstricks.recall
+++ b/src_phystricks/Fig_MethodeChemin.pstricks.recall
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
+\draw [,->,>=latex] (-2.0000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0000) -- (0.0000,2.0000);
 %DEFAULT
-\draw [color=red,style=dashed] (-1.50,1.50) -- (1.50,-1.50);
-\draw [color=blue,style=dashed] (-1.50,-0.750) -- (1.50,0.750);
+\draw [color=red,style=dashed] (-1.5000,1.5000) -- (1.5000,-1.5000);
+\draw [color=blue,style=dashed] (-1.5000,-0.7500) -- (1.5000,0.7500);
 \draw (-1.5000,1.9416) node {$y=-x$};
-\draw (2.3251,1.1446) node {$y=x/2$};
+\draw (2.3250,1.1445) node {$y=x/2$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MethodeNewton.pstricks.recall b/src_phystricks/Fig_MethodeNewton.pstricks.recall
index c62cf7e8b..b9a349019 100644
--- a/src_phystricks/Fig_MethodeNewton.pstricks.recall
+++ b/src_phystricks/Fig_MethodeNewton.pstricks.recall
@@ -95,31 +95,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-1.2875) -- (0,4.4000);
+\draw [,->,>=latex] (-2.0000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2875) -- (0.0000,4.4000);
 %DEFAULT
 
-\draw [color=blue] (-1.500,3.900)--(-1.424,3.713)--(-1.348,3.529)--(-1.273,3.349)--(-1.197,3.173)--(-1.121,3.001)--(-1.045,2.833)--(-0.9697,2.668)--(-0.8939,2.507)--(-0.8182,2.350)--(-0.7424,2.197)--(-0.6667,2.048)--(-0.5909,1.903)--(-0.5152,1.761)--(-0.4394,1.623)--(-0.3636,1.490)--(-0.2879,1.359)--(-0.2121,1.233)--(-0.1364,1.111)--(-0.06061,0.9921)--(0.01515,0.8773)--(0.09091,0.7664)--(0.1667,0.6593)--(0.2424,0.5560)--(0.3182,0.4565)--(0.3939,0.3608)--(0.4697,0.2690)--(0.5455,0.1810)--(0.6212,0.09682)--(0.6970,0.01647)--(0.7727,-0.06006)--(0.8485,-0.1328)--(0.9242,-0.2016)--(1.000,-0.2667)--(1.076,-0.3279)--(1.152,-0.3853)--(1.227,-0.4388)--(1.303,-0.4886)--(1.379,-0.5345)--(1.455,-0.5766)--(1.530,-0.6148)--(1.606,-0.6493)--(1.682,-0.6799)--(1.758,-0.7067)--(1.833,-0.7296)--(1.909,-0.7488)--(1.985,-0.7641)--(2.061,-0.7755)--(2.136,-0.7832)--(2.212,-0.7870)--(2.288,-0.7870)--(2.364,-0.7832)--(2.439,-0.7755)--(2.515,-0.7641)--(2.591,-0.7488)--(2.667,-0.7296)--(2.742,-0.7067)--(2.818,-0.6799)--(2.894,-0.6493)--(2.970,-0.6148)--(3.045,-0.5766)--(3.121,-0.5345)--(3.197,-0.4886)--(3.273,-0.4388)--(3.348,-0.3853)--(3.424,-0.3279)--(3.500,-0.2667)--(3.576,-0.2016)--(3.652,-0.1328)--(3.727,-0.06006)--(3.803,0.01647)--(3.879,0.09682)--(3.955,0.1810)--(4.030,0.2690)--(4.106,0.3608)--(4.182,0.4565)--(4.258,0.5560)--(4.333,0.6593)--(4.409,0.7664)--(4.485,0.8773)--(4.561,0.9921)--(4.636,1.111)--(4.712,1.233)--(4.788,1.359)--(4.864,1.490)--(4.939,1.623)--(5.015,1.761)--(5.091,1.903)--(5.167,2.048)--(5.242,2.197)--(5.318,2.350)--(5.394,2.507)--(5.470,2.668)--(5.545,2.833)--(5.621,3.001)--(5.697,3.173)--(5.773,3.349)--(5.849,3.529)--(5.924,3.713)--(6.000,3.900);
-\draw [color=red,style=dotted] (-0.900,0) -- (-0.900,2.52);
-\draw [color=green,style=dashed] (-1.20,3.15) -- (0.600,-0.630);
-\draw []  (-0.90000,0) node [rotate=0] {$\bullet$};
-\draw (-0.90000,-0.40595) node {$x_n$};
-\draw []  (0.30000,0) node [rotate=0] {$\bullet$};
-\draw (0.30000,-0.41918) node {$x_{n+1}$};
-\draw []  (-0.90000,2.5200) node [rotate=0] {$\bullet$};
-\draw (-0.50410,2.8462) node {$y_n$};
-\draw []  (0.71296,0) node [rotate=0] {$\bullet$};
-\draw (0.71296,0.40595) node {$r_0$};
-\draw []  (3.7870,0) node [rotate=0] {$\bullet$};
-\draw (3.7870,0.40595) node {$r_1$};
-\draw []  (2.2500,-0.78750) node [rotate=0] {$\bullet$};
+\draw [color=blue] (-1.5000,3.9000)--(-1.4242,3.7125)--(-1.3484,3.5288)--(-1.2727,3.3490)--(-1.1969,3.1730)--(-1.1212,3.0008)--(-1.0454,2.8325)--(-0.9696,2.6679)--(-0.8939,2.5072)--(-0.8181,2.3504)--(-0.7424,2.1973)--(-0.6666,2.0481)--(-0.5909,1.9027)--(-0.5151,1.7611)--(-0.4393,1.6234)--(-0.3636,1.4895)--(-0.2878,1.3594)--(-0.2121,1.2331)--(-0.1363,1.1107)--(-0.0606,0.9921)--(0.0151,0.8773)--(0.0909,0.7663)--(0.1666,0.6592)--(0.2424,0.5559)--(0.3181,0.4564)--(0.3939,0.3608)--(0.4696,0.2689)--(0.5454,0.1809)--(0.6212,0.0968)--(0.6969,0.0164)--(0.7727,-0.0600)--(0.8484,-0.1327)--(0.9242,-0.2016)--(1.0000,-0.2666)--(1.0757,-0.3278)--(1.1515,-0.3852)--(1.2272,-0.4388)--(1.3030,-0.4885)--(1.3787,-0.5344)--(1.4545,-0.5765)--(1.5303,-0.6148)--(1.6060,-0.6492)--(1.6818,-0.6798)--(1.7575,-0.7066)--(1.8333,-0.7296)--(1.9090,-0.7487)--(1.9848,-0.7640)--(2.0606,-0.7755)--(2.1363,-0.7831)--(2.2121,-0.7870)--(2.2878,-0.7870)--(2.3636,-0.7831)--(2.4393,-0.7755)--(2.5151,-0.7640)--(2.5909,-0.7487)--(2.6666,-0.7296)--(2.7424,-0.7066)--(2.8181,-0.6798)--(2.8939,-0.6492)--(2.9696,-0.6148)--(3.0454,-0.5765)--(3.1212,-0.5344)--(3.1969,-0.4885)--(3.2727,-0.4388)--(3.3484,-0.3852)--(3.4242,-0.3278)--(3.5000,-0.2666)--(3.5757,-0.2016)--(3.6515,-0.1327)--(3.7272,-0.0600)--(3.8030,0.0164)--(3.8787,0.0968)--(3.9545,0.1809)--(4.0303,0.2689)--(4.1060,0.3608)--(4.1818,0.4564)--(4.2575,0.5559)--(4.3333,0.6592)--(4.4090,0.7663)--(4.4848,0.8773)--(4.5606,0.9921)--(4.6363,1.1107)--(4.7121,1.2331)--(4.7878,1.3594)--(4.8636,1.4895)--(4.9393,1.6234)--(5.0151,1.7611)--(5.0909,1.9027)--(5.1666,2.0481)--(5.2424,2.1973)--(5.3181,2.3504)--(5.3939,2.5072)--(5.4696,2.6679)--(5.5454,2.8325)--(5.6212,3.0008)--(5.6969,3.1730)--(5.7727,3.3490)--(5.8484,3.5288)--(5.9242,3.7125)--(6.0000,3.9000);
+\draw [color=red,style=dotted] (-0.9000,0.0000) -- (-0.9000,2.5200);
+\draw [color=green,style=dashed] (-1.2000,3.1500) -- (0.6000,-0.6300);
+\draw []  (-0.9000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9000,-0.4059) node {$x_n$};
+\draw []  (0.3000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.3000,-0.4191) node {$x_{n+1}$};
+\draw []  (-0.9000,2.5200) node [rotate=0] {$\bullet$};
+\draw (-0.5041,2.8461) node {$y_n$};
+\draw []  (0.7129,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.7129,0.4059) node {$r_0$};
+\draw []  (3.7870,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.7870,0.4059) node {$r_1$};
+\draw []  (2.2500,-0.7875) node [rotate=0] {$\bullet$};
 \draw (2.2500,-1.2122) node {$S$};
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.0000,-0.31492) node {$ 2 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 2 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_MomentForce.pstricks.recall b/src_phystricks/Fig_MomentForce.pstricks.recall
index 6e2f5e73e..3cf09ebb3 100644
--- a/src_phystricks/Fig_MomentForce.pstricks.recall
+++ b/src_phystricks/Fig_MomentForce.pstricks.recall
@@ -81,15 +81,15 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.42471) node {$O$};
-\draw [,->,>=latex] (0,0) -- (-1.0000,-1.0000);
-\draw (-1.0000,-0.44196) node {$\overline{ R }$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.4247) node {$O$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (-1.0000,-1.0000);
+\draw (-1.0000,-0.4419) node {$\overline{ R }$};
 \draw [,->,>=latex] (-1.0000,-1.0000) -- (-3.0000,-1.5000);
-\draw (-3.0000,-1.0420) node {$\overline{ F }$};
-\draw [color=blue,style=dotted] (0,0) -- (0.176,-0.706);
-\draw (0.47427,-0.15344) node {$d$};
-\draw [color=brown,style=dashed] (-1.00,-1.00) -- (0.467,-0.633);
+\draw (-3.0000,-1.0419) node {$\overline{ F }$};
+\draw [color=blue,style=dotted] (0.0000,0.0000) -- (0.1764,-0.7058);
+\draw (0.4742,-0.1534) node {$d$};
+\draw [color=brown,style=dashed] (-1.0000,-1.0000) -- (0.4675,-0.6331);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_NEtAchr.pstricks.recall b/src_phystricks/Fig_NEtAchr.pstricks.recall
index eb02886ee..b0d2b1846 100644
--- a/src_phystricks/Fig_NEtAchr.pstricks.recall
+++ b/src_phystricks/Fig_NEtAchr.pstricks.recall
@@ -69,11 +69,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=brown] plot [smooth,tension=1] coordinates {(-1.50,-0.500)(0.500,-0.300)(2.00,1.00)(3.50,1.50)(5.00,2.70)(5.80,2.70)};
-\draw []  (-1.5000,-0.50000) node [rotate=0] {$\bullet$};
-\draw (-1.5000,-0.17897) node {\( +\)};
-\draw []  (0.50000,-0.30000) node [rotate=0] {$\bullet$};
-\draw (0.50000,0.021030) node {\( +\)};
+\draw [color=brown] plot [smooth,tension=1] coordinates {(-1.5000,-0.5000)(0.5000,-0.3000)(2.0000,1.0000)(3.5000,1.5000)(5.0000,2.7000)(5.8000,2.7000)};
+\draw []  (-1.5000,-0.5000) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.1789) node {\( +\)};
+\draw []  (0.5000,-0.3000) node [rotate=0] {$\bullet$};
+\draw (0.5000,0.0210) node {\( +\)};
 \draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw (2.0000,1.3210) node {\( +\)};
 \draw []  (3.5000,1.5000) node [rotate=0] {$\bullet$};
diff --git a/src_phystricks/Fig_NiveauHyperbole.pstricks.recall b/src_phystricks/Fig_NiveauHyperbole.pstricks.recall
index 342639491..216897fa5 100644
--- a/src_phystricks/Fig_NiveauHyperbole.pstricks.recall
+++ b/src_phystricks/Fig_NiveauHyperbole.pstricks.recall
@@ -95,47 +95,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (1.000,0)--(1.020,0.2020)--(1.040,0.2871)--(1.061,0.3534)--(1.081,0.4101)--(1.101,0.4607)--(1.121,0.5071)--(1.141,0.5503)--(1.162,0.5911)--(1.182,0.6298)--(1.202,0.6670)--(1.222,0.7027)--(1.242,0.7373)--(1.263,0.7709)--(1.283,0.8035)--(1.303,0.8354)--(1.323,0.8666)--(1.343,0.8971)--(1.364,0.9271)--(1.384,0.9566)--(1.404,0.9856)--(1.424,1.014)--(1.444,1.042)--(1.465,1.070)--(1.485,1.098)--(1.505,1.125)--(1.525,1.152)--(1.545,1.178)--(1.566,1.205)--(1.586,1.231)--(1.606,1.257)--(1.626,1.282)--(1.646,1.308)--(1.667,1.333)--(1.687,1.359)--(1.707,1.383)--(1.727,1.408)--(1.747,1.433)--(1.768,1.458)--(1.788,1.482)--(1.808,1.506)--(1.828,1.531)--(1.848,1.555)--(1.869,1.579)--(1.889,1.602)--(1.909,1.626)--(1.929,1.650)--(1.949,1.673)--(1.970,1.697)--(1.990,1.720)--(2.010,1.744)--(2.030,1.767)--(2.051,1.790)--(2.071,1.813)--(2.091,1.836)--(2.111,1.859)--(2.131,1.882)--(2.152,1.905)--(2.172,1.928)--(2.192,1.951)--(2.212,1.973)--(2.232,1.996)--(2.253,2.018)--(2.273,2.041)--(2.293,2.063)--(2.313,2.086)--(2.333,2.108)--(2.354,2.131)--(2.374,2.153)--(2.394,2.175)--(2.414,2.197)--(2.434,2.219)--(2.455,2.242)--(2.475,2.264)--(2.495,2.286)--(2.515,2.308)--(2.535,2.330)--(2.556,2.352)--(2.576,2.374)--(2.596,2.396)--(2.616,2.418)--(2.636,2.439)--(2.657,2.461)--(2.677,2.483)--(2.697,2.505)--(2.717,2.526)--(2.737,2.548)--(2.758,2.570)--(2.778,2.592)--(2.798,2.613)--(2.818,2.635)--(2.838,2.656)--(2.859,2.678)--(2.879,2.700)--(2.899,2.721)--(2.919,2.743)--(2.939,2.764)--(2.960,2.786)--(2.980,2.807)--(3.000,2.828);
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+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall b/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall
index 9dc95b22d..24dbed392 100644
--- a/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall
+++ b/src_phystricks/Fig_NiveauHyperboleDeux.pstricks.recall
@@ -95,43 +95,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.6622) -- (0.0000,3.6622);
 %DEFAULT
 
-\draw [color=red] (-3.000,3.162)--(-2.939,3.105)--(-2.879,3.048)--(-2.818,2.990)--(-2.758,2.933)--(-2.697,2.876)--(-2.636,2.820)--(-2.576,2.763)--(-2.515,2.707)--(-2.455,2.650)--(-2.394,2.594)--(-2.333,2.539)--(-2.273,2.483)--(-2.212,2.428)--(-2.152,2.373)--(-2.091,2.318)--(-2.030,2.263)--(-1.970,2.209)--(-1.909,2.155)--(-1.848,2.102)--(-1.788,2.049)--(-1.727,1.996)--(-1.667,1.944)--(-1.606,1.892)--(-1.545,1.841)--(-1.485,1.790)--(-1.424,1.740)--(-1.364,1.691)--(-1.303,1.643)--(-1.242,1.595)--(-1.182,1.548)--(-1.121,1.502)--(-1.061,1.458)--(-1.000,1.414)--(-0.9394,1.372)--(-0.8788,1.331)--(-0.8182,1.292)--(-0.7576,1.255)--(-0.6970,1.219)--(-0.6364,1.185)--(-0.5758,1.154)--(-0.5152,1.125)--(-0.4545,1.098)--(-0.3939,1.075)--(-0.3333,1.054)--(-0.2727,1.037)--(-0.2121,1.022)--(-0.1515,1.011)--(-0.09091,1.004)--(-0.03030,1.000)--(0.03030,1.000)--(0.09091,1.004)--(0.1515,1.011)--(0.2121,1.022)--(0.2727,1.037)--(0.3333,1.054)--(0.3939,1.075)--(0.4545,1.098)--(0.5152,1.125)--(0.5758,1.154)--(0.6364,1.185)--(0.6970,1.219)--(0.7576,1.255)--(0.8182,1.292)--(0.8788,1.331)--(0.9394,1.372)--(1.000,1.414)--(1.061,1.458)--(1.121,1.502)--(1.182,1.548)--(1.242,1.595)--(1.303,1.643)--(1.364,1.691)--(1.424,1.740)--(1.485,1.790)--(1.545,1.841)--(1.606,1.892)--(1.667,1.944)--(1.727,1.996)--(1.788,2.049)--(1.848,2.102)--(1.909,2.155)--(1.970,2.209)--(2.030,2.263)--(2.091,2.318)--(2.152,2.373)--(2.212,2.428)--(2.273,2.483)--(2.333,2.539)--(2.394,2.594)--(2.455,2.650)--(2.515,2.707)--(2.576,2.763)--(2.636,2.820)--(2.697,2.876)--(2.758,2.933)--(2.818,2.990)--(2.879,3.048)--(2.939,3.105)--(3.000,3.162);
+\draw [color=red] (-3.0000,3.1622)--(-2.9393,3.1048)--(-2.8787,3.0475)--(-2.8181,2.9903)--(-2.7575,2.9332)--(-2.6969,2.8763)--(-2.6363,2.8196)--(-2.5757,2.7630)--(-2.5151,2.7066)--(-2.4545,2.6504)--(-2.3939,2.5944)--(-2.3333,2.5385)--(-2.2727,2.4830)--(-2.2121,2.4276)--(-2.1515,2.3725)--(-2.0909,2.3177)--(-2.0303,2.2632)--(-1.9696,2.2090)--(-1.9090,2.1551)--(-1.8484,2.1016)--(-1.7878,2.0485)--(-1.7272,1.9958)--(-1.6666,1.9436)--(-1.6060,1.8919)--(-1.5454,1.8407)--(-1.4848,1.7901)--(-1.4242,1.7402)--(-1.3636,1.6910)--(-1.3030,1.6425)--(-1.2424,1.5948)--(-1.1818,1.5481)--(-1.1212,1.5023)--(-1.0606,1.4576)--(-1.0000,1.4142)--(-0.9393,1.3720)--(-0.8787,1.3312)--(-0.8181,1.2920)--(-0.7575,1.2545)--(-0.6969,1.2189)--(-0.6363,1.1853)--(-0.5757,1.1539)--(-0.5151,1.1248)--(-0.4545,1.0984)--(-0.3939,1.0747)--(-0.3333,1.0540)--(-0.2727,1.0365)--(-0.2121,1.0222)--(-0.1515,1.0114)--(-0.0909,1.0041)--(-0.0303,1.0004)--(0.0303,1.0004)--(0.0909,1.0041)--(0.1515,1.0114)--(0.2121,1.0222)--(0.2727,1.0365)--(0.3333,1.0540)--(0.3939,1.0747)--(0.4545,1.0984)--(0.5151,1.1248)--(0.5757,1.1539)--(0.6363,1.1853)--(0.6969,1.2189)--(0.7575,1.2545)--(0.8181,1.2920)--(0.8787,1.3312)--(0.9393,1.3720)--(1.0000,1.4142)--(1.0606,1.4576)--(1.1212,1.5023)--(1.1818,1.5481)--(1.2424,1.5948)--(1.3030,1.6425)--(1.3636,1.6910)--(1.4242,1.7402)--(1.4848,1.7901)--(1.5454,1.8407)--(1.6060,1.8919)--(1.6666,1.9436)--(1.7272,1.9958)--(1.7878,2.0485)--(1.8484,2.1016)--(1.9090,2.1551)--(1.9696,2.2090)--(2.0303,2.2632)--(2.0909,2.3177)--(2.1515,2.3725)--(2.2121,2.4276)--(2.2727,2.4830)--(2.3333,2.5385)--(2.3939,2.5944)--(2.4545,2.6504)--(2.5151,2.7066)--(2.5757,2.7630)--(2.6363,2.8196)--(2.6969,2.8763)--(2.7575,2.9332)--(2.8181,2.9903)--(2.8787,3.0475)--(2.9393,3.1048)--(3.0000,3.1622);
 
-\draw [color=red] (-3.000,-3.162)--(-2.939,-3.105)--(-2.879,-3.048)--(-2.818,-2.990)--(-2.758,-2.933)--(-2.697,-2.876)--(-2.636,-2.820)--(-2.576,-2.763)--(-2.515,-2.707)--(-2.455,-2.650)--(-2.394,-2.594)--(-2.333,-2.539)--(-2.273,-2.483)--(-2.212,-2.428)--(-2.152,-2.373)--(-2.091,-2.318)--(-2.030,-2.263)--(-1.970,-2.209)--(-1.909,-2.155)--(-1.848,-2.102)--(-1.788,-2.049)--(-1.727,-1.996)--(-1.667,-1.944)--(-1.606,-1.892)--(-1.545,-1.841)--(-1.485,-1.790)--(-1.424,-1.740)--(-1.364,-1.691)--(-1.303,-1.643)--(-1.242,-1.595)--(-1.182,-1.548)--(-1.121,-1.502)--(-1.061,-1.458)--(-1.000,-1.414)--(-0.9394,-1.372)--(-0.8788,-1.331)--(-0.8182,-1.292)--(-0.7576,-1.255)--(-0.6970,-1.219)--(-0.6364,-1.185)--(-0.5758,-1.154)--(-0.5152,-1.125)--(-0.4545,-1.098)--(-0.3939,-1.075)--(-0.3333,-1.054)--(-0.2727,-1.037)--(-0.2121,-1.022)--(-0.1515,-1.011)--(-0.09091,-1.004)--(-0.03030,-1.000)--(0.03030,-1.000)--(0.09091,-1.004)--(0.1515,-1.011)--(0.2121,-1.022)--(0.2727,-1.037)--(0.3333,-1.054)--(0.3939,-1.075)--(0.4545,-1.098)--(0.5152,-1.125)--(0.5758,-1.154)--(0.6364,-1.185)--(0.6970,-1.219)--(0.7576,-1.255)--(0.8182,-1.292)--(0.8788,-1.331)--(0.9394,-1.372)--(1.000,-1.414)--(1.061,-1.458)--(1.121,-1.502)--(1.182,-1.548)--(1.242,-1.595)--(1.303,-1.643)--(1.364,-1.691)--(1.424,-1.740)--(1.485,-1.790)--(1.545,-1.841)--(1.606,-1.892)--(1.667,-1.944)--(1.727,-1.996)--(1.788,-2.049)--(1.848,-2.102)--(1.909,-2.155)--(1.970,-2.209)--(2.030,-2.263)--(2.091,-2.318)--(2.152,-2.373)--(2.212,-2.428)--(2.273,-2.483)--(2.333,-2.539)--(2.394,-2.594)--(2.455,-2.650)--(2.515,-2.707)--(2.576,-2.763)--(2.636,-2.820)--(2.697,-2.876)--(2.758,-2.933)--(2.818,-2.990)--(2.879,-3.048)--(2.939,-3.105)--(3.000,-3.162);
-\draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
-\draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw (0.21069,0.80458) node {$R$};
-\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
-\draw (0.19314,-0.80458) node {$S$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (-3.0000,-3.1622)--(-2.9393,-3.1048)--(-2.8787,-3.0475)--(-2.8181,-2.9903)--(-2.7575,-2.9332)--(-2.6969,-2.8763)--(-2.6363,-2.8196)--(-2.5757,-2.7630)--(-2.5151,-2.7066)--(-2.4545,-2.6504)--(-2.3939,-2.5944)--(-2.3333,-2.5385)--(-2.2727,-2.4830)--(-2.2121,-2.4276)--(-2.1515,-2.3725)--(-2.0909,-2.3177)--(-2.0303,-2.2632)--(-1.9696,-2.2090)--(-1.9090,-2.1551)--(-1.8484,-2.1016)--(-1.7878,-2.0485)--(-1.7272,-1.9958)--(-1.6666,-1.9436)--(-1.6060,-1.8919)--(-1.5454,-1.8407)--(-1.4848,-1.7901)--(-1.4242,-1.7402)--(-1.3636,-1.6910)--(-1.3030,-1.6425)--(-1.2424,-1.5948)--(-1.1818,-1.5481)--(-1.1212,-1.5023)--(-1.0606,-1.4576)--(-1.0000,-1.4142)--(-0.9393,-1.3720)--(-0.8787,-1.3312)--(-0.8181,-1.2920)--(-0.7575,-1.2545)--(-0.6969,-1.2189)--(-0.6363,-1.1853)--(-0.5757,-1.1539)--(-0.5151,-1.1248)--(-0.4545,-1.0984)--(-0.3939,-1.0747)--(-0.3333,-1.0540)--(-0.2727,-1.0365)--(-0.2121,-1.0222)--(-0.1515,-1.0114)--(-0.0909,-1.0041)--(-0.0303,-1.0004)--(0.0303,-1.0004)--(0.0909,-1.0041)--(0.1515,-1.0114)--(0.2121,-1.0222)--(0.2727,-1.0365)--(0.3333,-1.0540)--(0.3939,-1.0747)--(0.4545,-1.0984)--(0.5151,-1.1248)--(0.5757,-1.1539)--(0.6363,-1.1853)--(0.6969,-1.2189)--(0.7575,-1.2545)--(0.8181,-1.2920)--(0.8787,-1.3312)--(0.9393,-1.3720)--(1.0000,-1.4142)--(1.0606,-1.4576)--(1.1212,-1.5023)--(1.1818,-1.5481)--(1.2424,-1.5948)--(1.3030,-1.6425)--(1.3636,-1.6910)--(1.4242,-1.7402)--(1.4848,-1.7901)--(1.5454,-1.8407)--(1.6060,-1.8919)--(1.6666,-1.9436)--(1.7272,-1.9958)--(1.7878,-2.0485)--(1.8484,-2.1016)--(1.9090,-2.1551)--(1.9696,-2.2090)--(2.0303,-2.2632)--(2.0909,-2.3177)--(2.1515,-2.3725)--(2.2121,-2.4276)--(2.2727,-2.4830)--(2.3333,-2.5385)--(2.3939,-2.5944)--(2.4545,-2.6504)--(2.5151,-2.7066)--(2.5757,-2.7630)--(2.6363,-2.8196)--(2.6969,-2.8763)--(2.7575,-2.9332)--(2.8181,-2.9903)--(2.8787,-3.0475)--(2.9393,-3.1048)--(3.0000,-3.1622);
+\draw [color=brown] (-3.0000,3.0000) -- (3.0000,-3.0000);
+\draw [color=brown] (-3.0000,-3.0000) -- (3.0000,3.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.2106,0.8045) node {$R$};
+\draw []  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.1931,-0.8045) node {$S$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall b/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
index 292a4a5f5..c82750494 100644
--- a/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
+++ b/src_phystricks/Fig_OQTEoodIwAPfZE.pstricks.recall
@@ -116,39 +116,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.9000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-2.8131) -- (0,3.6993);
+\draw [,->,>=latex] (-4.9000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.8130) -- (0.0000,3.6993);
 %DEFAULT
 
-\draw [color=blue] (-4.4000,-2.3131)--(-4.3051,-1.7786)--(-4.2101,-1.3073)--(-4.1152,-0.89319)--(-4.0202,-0.53075)--(-3.9253,-0.21496)--(-3.8303,0.058716)--(-3.7354,0.29443)--(-3.6404,0.49595)--(-3.5455,0.66669)--(-3.4505,0.80977)--(-3.3556,0.92802)--(-3.2606,1.0240)--(-3.1657,1.1001)--(-3.0707,1.1584)--(-2.9758,1.2009)--(-2.8808,1.2293)--(-2.7859,1.2452)--(-2.6909,1.2501)--(-2.5960,1.2452)--(-2.5010,1.2319)--(-2.4061,1.2112)--(-2.3111,1.1841)--(-2.2162,1.1515)--(-2.1212,1.1143)--(-2.0263,1.0731)--(-1.9313,1.0287)--(-1.8364,0.98162)--(-1.7414,0.93253)--(-1.6465,0.88190)--(-1.5515,0.83018)--(-1.4566,0.77782)--(-1.3616,0.72520)--(-1.2667,0.67266)--(-1.1717,0.62052)--(-1.0768,0.56908)--(-0.98182,0.51859)--(-0.88687,0.46930)--(-0.79192,0.42143)--(-0.69697,0.37518)--(-0.60202,0.33071)--(-0.50707,0.28821)--(-0.41212,0.24782)--(-0.31717,0.20966)--(-0.22222,0.17388)--(-0.12727,0.14058)--(-0.032323,0.10985)--(0.062626,0.081805)--(0.15758,0.056515)--(0.25253,0.034060)--(0.34747,0.014511)--(0.44242,-0.0020696)--(0.53737,-0.015623)--(0.63232,-0.026098)--(0.72727,-0.033446)--(0.82222,-0.037624)--(0.91717,-0.038593)--(1.0121,-0.036315)--(1.1071,-0.030760)--(1.2020,-0.021896)--(1.2970,-0.0096977)--(1.3919,0.0058604)--(1.4869,0.024800)--(1.5818,0.047142)--(1.6768,0.072905)--(1.7717,0.10211)--(1.8667,0.13476)--(1.9616,0.17088)--(2.0566,0.21048)--(2.1515,0.25357)--(2.2465,0.30016)--(2.3414,0.35026)--(2.4364,0.40388)--(2.5313,0.46103)--(2.6263,0.52171)--(2.7212,0.58593)--(2.8162,0.65370)--(2.9111,0.72503)--(3.0061,0.79991)--(3.1010,0.87835)--(3.1960,0.96035)--(3.2909,1.0459)--(3.3859,1.1351)--(3.4808,1.2278)--(3.5758,1.3241)--(3.6707,1.4240)--(3.7657,1.5275)--(3.8606,1.6345)--(3.9556,1.7451)--(4.0505,1.8594)--(4.1455,1.9772)--(4.2404,2.0986)--(4.3354,2.2236)--(4.4303,2.3522)--(4.5253,2.4844)--(4.6202,2.6202)--(4.7151,2.7596)--(4.8101,2.9026)--(4.9051,3.0491)--(5.0000,3.1993);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.52441,-2.0000) node {$ -20 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.52441,-1.0000) node {$ -10 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.38250,1.0000) node {$ 10 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.38250,2.0000) node {$ 20 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.38250,3.0000) node {$ 30 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-4.4000,-2.3130)--(-4.3050,-1.7785)--(-4.2101,-1.3073)--(-4.1151,-0.8931)--(-4.0202,-0.5307)--(-3.9252,-0.2149)--(-3.8303,0.0587)--(-3.7353,0.2944)--(-3.6404,0.4959)--(-3.5454,0.6666)--(-3.4505,0.8097)--(-3.3555,0.9280)--(-3.2606,1.0240)--(-3.1656,1.1001)--(-3.0707,1.1584)--(-2.9757,1.2008)--(-2.8808,1.2292)--(-2.7858,1.2451)--(-2.6909,1.2500)--(-2.5959,1.2452)--(-2.5010,1.2319)--(-2.4060,1.2112)--(-2.3111,1.1841)--(-2.2161,1.1515)--(-2.1212,1.1142)--(-2.0262,1.0730)--(-1.9313,1.0286)--(-1.8363,0.9816)--(-1.7414,0.9325)--(-1.6464,0.8818)--(-1.5515,0.8301)--(-1.4565,0.7778)--(-1.3616,0.7251)--(-1.2666,0.6726)--(-1.1717,0.6205)--(-1.0767,0.5690)--(-0.9818,0.5185)--(-0.8868,0.4693)--(-0.7919,0.4214)--(-0.6969,0.3751)--(-0.6020,0.3307)--(-0.5070,0.2882)--(-0.4121,0.2478)--(-0.3171,0.2096)--(-0.2222,0.1738)--(-0.1272,0.1405)--(-0.0323,0.1098)--(0.0626,0.0818)--(0.1575,0.0565)--(0.2525,0.0340)--(0.3474,0.0145)--(0.4424,-0.0020)--(0.5373,-0.0156)--(0.6323,-0.0260)--(0.7272,-0.0334)--(0.8222,-0.0376)--(0.9171,-0.0385)--(1.0121,-0.0363)--(1.1070,-0.0307)--(1.2020,-0.0218)--(1.2969,-0.0096)--(1.3919,0.0058)--(1.4868,0.0248)--(1.5818,0.0471)--(1.6767,0.0729)--(1.7717,0.1021)--(1.8666,0.1347)--(1.9616,0.1708)--(2.0565,0.2104)--(2.1515,0.2535)--(2.2464,0.3001)--(2.3414,0.3502)--(2.4363,0.4038)--(2.5313,0.4610)--(2.6262,0.5217)--(2.7212,0.5859)--(2.8161,0.6537)--(2.9111,0.7250)--(3.0060,0.7999)--(3.1010,0.8783)--(3.1959,0.9603)--(3.2909,1.0459)--(3.3858,1.1350)--(3.4808,1.2278)--(3.5757,1.3241)--(3.6707,1.4239)--(3.7656,1.5274)--(3.8606,1.6345)--(3.9555,1.7451)--(4.0505,1.8593)--(4.1454,1.9771)--(4.2404,2.0986)--(4.3353,2.2236)--(4.4303,2.3522)--(4.5252,2.4843)--(4.6202,2.6201)--(4.7151,2.7595)--(4.8101,2.9025)--(4.9050,3.0491)--(5.0000,3.1993);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.5244,-2.0000) node {$ -20 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.5244,-1.0000) node {$ -10 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.3824,1.0000) node {$ 10 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.3824,2.0000) node {$ 20 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.3824,3.0000) node {$ 30 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall b/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall
index 2cc978bad..8db4a826a 100644
--- a/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall
+++ b/src_phystricks/Fig_PLTWoocPNeiZir.pstricks.recall
@@ -67,15 +67,15 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [color=red] (2.582,2.300)--(2.579,2.369)--(2.573,2.437)--(2.562,2.505)--(2.547,2.572)--(2.528,2.638)--(2.504,2.702)--(2.477,2.765)--(2.445,2.826)--(2.410,2.885)--(2.371,2.941)--(2.329,2.995)--(2.283,3.046)--(2.234,3.095)--(2.182,3.140)--(2.127,3.181)--(2.070,3.219)--(2.011,3.253)--(1.949,3.284)--(1.886,3.310)--(1.821,3.333)--(1.755,3.351)--(1.688,3.365)--(1.620,3.375)--(1.551,3.380)--(1.483,3.382)--(1.414,3.378)--(1.346,3.371)--(1.278,3.359)--(1.212,3.343)--(1.146,3.322)--(1.082,3.298)--(1.020,3.269)--(0.9592,3.237)--(0.9008,3.201)--(0.8449,3.161)--(0.7917,3.117)--(0.7412,3.071)--(0.6939,3.021)--(0.6498,2.969)--(0.6091,2.913)--(0.5720,2.856)--(0.5386,2.796)--(0.5091,2.734)--(0.4836,2.670)--(0.4622,2.605)--(0.4449,2.538)--(0.4319,2.471)--(0.4232,2.403)--(0.4189,2.334)--(0.4189,2.266)--(0.4232,2.197)--(0.4319,2.129)--(0.4449,2.062)--(0.4622,1.995)--(0.4836,1.930)--(0.5091,1.866)--(0.5386,1.804)--(0.5720,1.744)--(0.6091,1.687)--(0.6498,1.631)--(0.6939,1.579)--(0.7412,1.529)--(0.7917,1.483)--(0.8449,1.439)--(0.9008,1.399)--(0.9592,1.363)--(1.020,1.331)--(1.082,1.302)--(1.146,1.278)--(1.212,1.257)--(1.278,1.241)--(1.346,1.229)--(1.414,1.222)--(1.483,1.218)--(1.551,1.220)--(1.620,1.225)--(1.688,1.235)--(1.755,1.249)--(1.821,1.267)--(1.886,1.290)--(1.949,1.316)--(2.011,1.347)--(2.070,1.381)--(2.127,1.419)--(2.182,1.460)--(2.234,1.505)--(2.283,1.554)--(2.329,1.605)--(2.371,1.659)--(2.410,1.715)--(2.445,1.774)--(2.477,1.835)--(2.504,1.898)--(2.528,1.962)--(2.547,2.028)--(2.562,2.095)--(2.573,2.163)--(2.579,2.231)--(2.582,2.300);
+\draw [color=red] (2.5816,2.3000)--(2.5794,2.3686)--(2.5729,2.4369)--(2.5621,2.5047)--(2.5469,2.5716)--(2.5276,2.6375)--(2.5041,2.7020)--(2.4766,2.7648)--(2.4452,2.8259)--(2.4099,2.8847)--(2.3710,2.9413)--(2.3286,2.9952)--(2.2828,3.0464)--(2.2339,3.0945)--(2.1820,3.1395)--(2.1274,3.1810)--(2.0702,3.2191)--(2.0108,3.2534)--(1.9493,3.2839)--(1.8860,3.3104)--(1.8211,3.3328)--(1.7550,3.3511)--(1.6878,3.3652)--(1.6198,3.3750)--(1.5514,3.3804)--(1.4828,3.3815)--(1.4142,3.3782)--(1.3460,3.3706)--(1.2784,3.3587)--(1.2117,3.3425)--(1.1462,3.3221)--(1.0821,3.2976)--(1.0196,3.2691)--(0.9591,3.2367)--(0.9008,3.2005)--(0.8449,3.1607)--(0.7916,3.1174)--(0.7412,3.0708)--(0.6938,3.0212)--(0.6497,2.9686)--(0.6090,2.9133)--(0.5719,2.8556)--(0.5385,2.7956)--(0.5090,2.7336)--(0.4835,2.6699)--(0.4621,2.6047)--(0.4449,2.5383)--(0.4319,2.4709)--(0.4232,2.4028)--(0.4188,2.3343)--(0.4188,2.2656)--(0.4232,2.1971)--(0.4319,2.1290)--(0.4449,2.0616)--(0.4621,1.9952)--(0.4835,1.9300)--(0.5090,1.8663)--(0.5385,1.8043)--(0.5719,1.7443)--(0.6090,1.6866)--(0.6497,1.6313)--(0.6938,1.5787)--(0.7412,1.5291)--(0.7916,1.4825)--(0.8449,1.4392)--(0.9008,1.3994)--(0.9591,1.3632)--(1.0196,1.3308)--(1.0821,1.3023)--(1.1462,1.2778)--(1.2117,1.2574)--(1.2784,1.2412)--(1.3460,1.2293)--(1.4142,1.2217)--(1.4828,1.2184)--(1.5514,1.2195)--(1.6198,1.2249)--(1.6878,1.2347)--(1.7550,1.2488)--(1.8211,1.2671)--(1.8860,1.2895)--(1.9493,1.3160)--(2.0108,1.3465)--(2.0702,1.3808)--(2.1274,1.4189)--(2.1820,1.4604)--(2.2339,1.5054)--(2.2828,1.5535)--(2.3286,1.6047)--(2.3710,1.6586)--(2.4099,1.7152)--(2.4452,1.7740)--(2.4766,1.8351)--(2.5041,1.8979)--(2.5276,1.9624)--(2.5469,2.0283)--(2.5621,2.0952)--(2.5729,2.1630)--(2.5794,2.2313)--(2.5816,2.3000);
 
-\draw [color=red] (2.903,2.900)--(2.899,3.014)--(2.888,3.128)--(2.870,3.241)--(2.845,3.353)--(2.813,3.463)--(2.774,3.570)--(2.728,3.675)--(2.675,3.776)--(2.617,3.875)--(2.552,3.969)--(2.481,4.059)--(2.405,4.144)--(2.323,4.224)--(2.237,4.299)--(2.146,4.368)--(2.050,4.432)--(1.951,4.489)--(1.849,4.540)--(1.743,4.584)--(1.635,4.621)--(1.525,4.652)--(1.413,4.675)--(1.300,4.692)--(1.186,4.701)--(1.071,4.703)--(0.9571,4.697)--(0.8434,4.684)--(0.7308,4.665)--(0.6196,4.638)--(0.5104,4.604)--(0.4035,4.563)--(0.2994,4.515)--(0.1986,4.461)--(0.1014,4.401)--(0.008222,4.335)--(-0.08057,4.262)--(-0.1646,4.185)--(-0.2435,4.102)--(-0.3171,4.014)--(-0.3849,3.922)--(-0.4468,3.826)--(-0.5024,3.726)--(-0.5515,3.623)--(-0.5941,3.517)--(-0.6298,3.408)--(-0.6585,3.297)--(-0.6801,3.185)--(-0.6946,3.071)--(-0.7019,2.957)--(-0.7019,2.843)--(-0.6946,2.729)--(-0.6801,2.615)--(-0.6585,2.503)--(-0.6298,2.392)--(-0.5941,2.283)--(-0.5515,2.177)--(-0.5024,2.074)--(-0.4468,1.974)--(-0.3849,1.878)--(-0.3171,1.786)--(-0.2435,1.698)--(-0.1646,1.615)--(-0.08057,1.538)--(0.008222,1.465)--(0.1014,1.399)--(0.1986,1.339)--(0.2994,1.285)--(0.4035,1.237)--(0.5104,1.196)--(0.6196,1.162)--(0.7308,1.135)--(0.8434,1.116)--(0.9571,1.103)--(1.071,1.097)--(1.186,1.099)--(1.300,1.108)--(1.413,1.125)--(1.525,1.148)--(1.635,1.179)--(1.743,1.216)--(1.849,1.260)--(1.951,1.311)--(2.050,1.368)--(2.146,1.432)--(2.237,1.501)--(2.323,1.576)--(2.405,1.656)--(2.481,1.741)--(2.552,1.831)--(2.617,1.925)--(2.675,2.023)--(2.728,2.125)--(2.774,2.230)--(2.813,2.337)--(2.845,2.447)--(2.870,2.559)--(2.888,2.672)--(2.899,2.786)--(2.903,2.900);
+\draw [color=red] (2.9027,2.9000)--(2.8991,3.0143)--(2.8882,3.1282)--(2.8701,3.2411)--(2.8449,3.3527)--(2.8127,3.4625)--(2.7736,3.5700)--(2.7277,3.6748)--(2.6753,3.7765)--(2.6165,3.8746)--(2.5517,3.9688)--(2.4810,4.0588)--(2.4047,4.1440)--(2.3232,4.2243)--(2.2367,4.2992)--(2.1457,4.3684)--(2.0504,4.4318)--(1.9513,4.4890)--(1.8489,4.5398)--(1.7433,4.5840)--(1.6352,4.6214)--(1.5250,4.6519)--(1.4130,4.6753)--(1.2998,4.6916)--(1.1857,4.7007)--(1.0713,4.7025)--(0.9571,4.6971)--(0.8434,4.6844)--(0.7307,4.6645)--(0.6196,4.6375)--(0.5103,4.6036)--(0.4035,4.5627)--(0.2994,4.5152)--(0.1986,4.4612)--(0.1014,4.4009)--(0.0082,4.3345)--(-0.0805,4.2624)--(-0.1646,4.1848)--(-0.2435,4.1020)--(-0.3170,4.0144)--(-0.3849,3.9222)--(-0.4467,3.8260)--(-0.5023,3.7260)--(-0.5515,3.6227)--(-0.5940,3.5165)--(-0.6297,3.4079)--(-0.6584,3.2971)--(-0.6801,3.1848)--(-0.6946,3.0713)--(-0.7018,2.9571)--(-0.7018,2.8428)--(-0.6946,2.7286)--(-0.6801,2.6151)--(-0.6584,2.5028)--(-0.6297,2.3920)--(-0.5940,2.2834)--(-0.5515,2.1772)--(-0.5023,2.0739)--(-0.4467,1.9739)--(-0.3849,1.8777)--(-0.3170,1.7855)--(-0.2435,1.6979)--(-0.1646,1.6151)--(-0.0805,1.5375)--(0.0082,1.4654)--(0.1014,1.3990)--(0.1986,1.3387)--(0.2994,1.2847)--(0.4035,1.2372)--(0.5103,1.1963)--(0.6196,1.1624)--(0.7307,1.1354)--(0.8434,1.1155)--(0.9571,1.1028)--(1.0713,1.0974)--(1.1857,1.0992)--(1.2998,1.1083)--(1.4130,1.1246)--(1.5250,1.1480)--(1.6352,1.1785)--(1.7433,1.2159)--(1.8489,1.2601)--(1.9513,1.3109)--(2.0504,1.3681)--(2.1457,1.4315)--(2.2367,1.5007)--(2.3232,1.5756)--(2.4047,1.6559)--(2.4810,1.7411)--(2.5517,1.8311)--(2.6165,1.9253)--(2.6753,2.0234)--(2.7277,2.1251)--(2.7736,2.2299)--(2.8127,2.3374)--(2.8449,2.4472)--(2.8701,2.5588)--(2.8882,2.6717)--(2.8991,2.7856)--(2.9027,2.9000);
 
-\draw [color=red] (3.224,3.500)--(3.219,3.660)--(3.204,3.820)--(3.178,3.978)--(3.143,4.134)--(3.098,4.288)--(3.043,4.438)--(2.979,4.585)--(2.905,4.727)--(2.823,4.865)--(2.732,4.996)--(2.633,5.122)--(2.527,5.242)--(2.412,5.354)--(2.291,5.459)--(2.164,5.556)--(2.031,5.645)--(1.892,5.725)--(1.748,5.796)--(1.601,5.858)--(1.449,5.910)--(1.295,5.953)--(1.138,5.986)--(0.9797,6.008)--(0.8201,6.021)--(0.6600,6.024)--(0.5000,6.016)--(0.3408,5.998)--(0.1831,5.970)--(0.02745,5.933)--(-0.1255,5.885)--(-0.2751,5.828)--(-0.4208,5.761)--(-0.5619,5.686)--(-0.6980,5.601)--(-0.8285,5.508)--(-0.9528,5.407)--(-1.070,5.299)--(-1.181,5.183)--(-1.284,5.060)--(-1.379,4.931)--(-1.465,4.796)--(-1.543,4.656)--(-1.612,4.512)--(-1.672,4.363)--(-1.722,4.211)--(-1.762,4.056)--(-1.792,3.899)--(-1.812,3.740)--(-1.823,3.580)--(-1.823,3.420)--(-1.812,3.260)--(-1.792,3.101)--(-1.762,2.944)--(-1.722,2.789)--(-1.672,2.637)--(-1.612,2.488)--(-1.543,2.343)--(-1.465,2.204)--(-1.379,2.069)--(-1.284,1.940)--(-1.181,1.817)--(-1.070,1.701)--(-0.9528,1.593)--(-0.8285,1.492)--(-0.6980,1.399)--(-0.5619,1.314)--(-0.4208,1.239)--(-0.2751,1.172)--(-0.1255,1.115)--(0.02745,1.067)--(0.1831,1.030)--(0.3408,1.002)--(0.5000,0.9841)--(0.6600,0.9764)--(0.8201,0.9790)--(0.9797,0.9917)--(1.138,1.014)--(1.295,1.047)--(1.449,1.090)--(1.601,1.142)--(1.748,1.204)--(1.892,1.275)--(2.031,1.355)--(2.164,1.444)--(2.291,1.541)--(2.412,1.646)--(2.527,1.758)--(2.633,1.878)--(2.732,2.004)--(2.823,2.135)--(2.905,2.273)--(2.979,2.415)--(3.043,2.562)--(3.098,2.712)--(3.143,2.866)--(3.178,3.022)--(3.204,3.180)--(3.219,3.340)--(3.224,3.500);
+\draw [color=red] (3.2238,3.5000)--(3.2188,3.6600)--(3.2035,3.8195)--(3.1782,3.9776)--(3.1429,4.1338)--(3.0978,4.2875)--(3.0430,4.4380)--(2.9788,4.5847)--(2.9054,4.7271)--(2.8232,4.8645)--(2.7324,4.9964)--(2.6334,5.1223)--(2.5266,5.2416)--(2.4124,5.3540)--(2.2914,5.4589)--(2.1639,5.5558)--(2.0306,5.6446)--(1.8919,5.7246)--(1.7484,5.7958)--(1.6007,5.8576)--(1.4493,5.9100)--(1.2950,5.9527)--(1.1382,5.9855)--(0.9797,6.0083)--(0.8200,6.0210)--(0.6599,6.0235)--(0.4999,6.0159)--(0.3408,5.9981)--(0.1830,5.9703)--(0.0274,5.9326)--(-0.1254,5.8850)--(-0.2750,5.8279)--(-0.4207,5.7613)--(-0.5619,5.6857)--(-0.6980,5.6013)--(-0.8284,5.5084)--(-0.9527,5.4074)--(-1.0704,5.2987)--(-1.1809,5.1828)--(-1.2839,5.0601)--(-1.3788,4.9311)--(-1.4654,4.7964)--(-1.5433,4.6565)--(-1.6121,4.5119)--(-1.6716,4.3632)--(-1.7216,4.2110)--(-1.7618,4.0560)--(-1.7921,3.8987)--(-1.8124,3.7399)--(-1.8226,3.5800)--(-1.8226,3.4199)--(-1.8124,3.2600)--(-1.7921,3.1012)--(-1.7618,2.9439)--(-1.7216,2.7889)--(-1.6716,2.6367)--(-1.6121,2.4880)--(-1.5433,2.3434)--(-1.4654,2.2035)--(-1.3788,2.0688)--(-1.2839,1.9398)--(-1.1809,1.8171)--(-1.0704,1.7012)--(-0.9527,1.5925)--(-0.8284,1.4915)--(-0.6980,1.3986)--(-0.5619,1.3142)--(-0.4207,1.2386)--(-0.2750,1.1720)--(-0.1254,1.1149)--(0.0274,1.0673)--(0.1830,1.0296)--(0.3408,1.0018)--(0.4999,0.9840)--(0.6599,0.9764)--(0.8200,0.9789)--(0.9797,0.9916)--(1.1382,1.0144)--(1.2950,1.0472)--(1.4493,1.0899)--(1.6007,1.1423)--(1.7484,1.2041)--(1.8919,1.2753)--(2.0306,1.3553)--(2.1639,1.4441)--(2.2914,1.5410)--(2.4124,1.6459)--(2.5266,1.7583)--(2.6334,1.8776)--(2.7324,2.0035)--(2.8232,2.1354)--(2.9054,2.2728)--(2.9788,2.4152)--(3.0430,2.5619)--(3.0978,2.7124)--(3.1429,2.8661)--(3.1782,3.0223)--(3.2035,3.1804)--(3.2188,3.3399)--(3.2238,3.5000);
 \draw []  (2.1000,1.4000) node [rotate=0] {$\bullet$};
-\draw (2.3535,1.1089) node {\( P\)};
-\draw [] (0,0) -- (3.00,2.00);
-\draw [style=dashed] (2.10,1.40) -- (0.100,4.40);
+\draw (2.3534,1.1088) node {\( P\)};
+\draw [] (0.0000,0.0000) -- (3.0000,2.0000);
+\draw [style=dashed] (2.1000,1.4000) -- (0.0999,4.4000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_PONXooXYjEot.pstricks.recall b/src_phystricks/Fig_PONXooXYjEot.pstricks.recall
index 3adcf2189..20c383857 100644
--- a/src_phystricks/Fig_PONXooXYjEot.pstricks.recall
+++ b/src_phystricks/Fig_PONXooXYjEot.pstricks.recall
@@ -79,23 +79,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4998,0) -- (2.4998,0);
-\draw [,->,>=latex] (0,-1.2068) -- (0,1.2068);
+\draw [,->,>=latex] (-2.4998,0.0000) -- (2.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2068) -- (0.0000,1.2068);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.2559,-0.2479)--(0.3673,-0.3447)--(0.4563,-0.4148)--(0.5341,-0.4702)--(0.6049,-0.5158)--(0.6709,-0.5540)--(0.7333,-0.5861)--(0.7927,-0.6133)--(0.8498,-0.6362)--(0.9049,-0.6552)--(0.9581,-0.6709)--(1.010,-0.6834)--(1.060,-0.6931)--(1.109,-0.7001)--(1.156,-0.7046)--(1.202,-0.7068)--(1.247,-0.7068)--(1.291,-0.7048)--(1.333,-0.7007)--(1.374,-0.6948)--(1.414,-0.6871)--(1.453,-0.6777)--(1.491,-0.6666)--(1.527,-0.6540)--(1.562,-0.6399)--(1.596,-0.6244)--(1.629,-0.6076)--(1.660,-0.5895)--(1.691,-0.5701)--(1.719,-0.5496)--(1.747,-0.5280)--(1.773,-0.5054)--(1.798,-0.4817)--(1.821,-0.4571)--(1.843,-0.4317)--(1.864,-0.4055)--(1.883,-0.3785)--(1.901,-0.3508)--(1.917,-0.3224)--(1.932,-0.2935)--(1.946,-0.2640)--(1.958,-0.2341)--(1.968,-0.2037)--(1.977,-0.1730)--(1.985,-0.1420)--(1.991,-0.1107)--(1.995,-0.07919)--(1.998,-0.04757)--(2.000,-0.01587)--(2.000,0.01587)--(1.998,0.04757)--(1.995,0.07919)--(1.991,0.1107)--(1.985,0.1420)--(1.977,0.1730)--(1.968,0.2037)--(1.958,0.2341)--(1.946,0.2640)--(1.932,0.2935)--(1.917,0.3224)--(1.901,0.3508)--(1.883,0.3785)--(1.864,0.4055)--(1.843,0.4317)--(1.821,0.4571)--(1.798,0.4817)--(1.773,0.5054)--(1.747,0.5280)--(1.719,0.5496)--(1.691,0.5701)--(1.660,0.5895)--(1.629,0.6076)--(1.596,0.6244)--(1.562,0.6399)--(1.527,0.6540)--(1.491,0.6666)--(1.453,0.6777)--(1.414,0.6871)--(1.374,0.6948)--(1.333,0.7007)--(1.291,0.7048)--(1.247,0.7068)--(1.202,0.7068)--(1.156,0.7046)--(1.109,0.7001)--(1.060,0.6931)--(1.010,0.6834)--(0.9581,0.6709)--(0.9049,0.6552)--(0.8498,0.6362)--(0.7927,0.6133)--(0.7333,0.5861)--(0.6709,0.5540)--(0.6049,0.5158)--(0.5341,0.4702)--(0.4563,0.4148)--(0.3673,0.3447)--(0.2559,0.2479)--(0,0);
-\draw [color=blue] (0,0)--(-0.2559,0.2479)--(-0.3673,0.3447)--(-0.4563,0.4148)--(-0.5341,0.4702)--(-0.6049,0.5158)--(-0.6709,0.5540)--(-0.7333,0.5861)--(-0.7927,0.6133)--(-0.8498,0.6362)--(-0.9049,0.6552)--(-0.9581,0.6709)--(-1.010,0.6834)--(-1.060,0.6931)--(-1.109,0.7001)--(-1.156,0.7046)--(-1.202,0.7068)--(-1.247,0.7068)--(-1.291,0.7048)--(-1.333,0.7007)--(-1.374,0.6948)--(-1.414,0.6871)--(-1.453,0.6777)--(-1.491,0.6666)--(-1.527,0.6540)--(-1.562,0.6399)--(-1.596,0.6244)--(-1.629,0.6076)--(-1.660,0.5895)--(-1.691,0.5701)--(-1.719,0.5496)--(-1.747,0.5280)--(-1.773,0.5054)--(-1.798,0.4817)--(-1.821,0.4571)--(-1.843,0.4317)--(-1.864,0.4055)--(-1.883,0.3785)--(-1.901,0.3508)--(-1.917,0.3224)--(-1.932,0.2935)--(-1.946,0.2640)--(-1.958,0.2341)--(-1.968,0.2037)--(-1.977,0.1730)--(-1.985,0.1420)--(-1.991,0.1107)--(-1.995,0.07919)--(-1.998,0.04757)--(-2.000,0.01587)--(-2.000,-0.01587)--(-1.998,-0.04757)--(-1.995,-0.07919)--(-1.991,-0.1107)--(-1.985,-0.1420)--(-1.977,-0.1730)--(-1.968,-0.2037)--(-1.958,-0.2341)--(-1.946,-0.2640)--(-1.932,-0.2935)--(-1.917,-0.3224)--(-1.901,-0.3508)--(-1.883,-0.3785)--(-1.864,-0.4055)--(-1.843,-0.4317)--(-1.821,-0.4571)--(-1.798,-0.4817)--(-1.773,-0.5054)--(-1.747,-0.5280)--(-1.719,-0.5496)--(-1.691,-0.5701)--(-1.660,-0.5895)--(-1.629,-0.6076)--(-1.596,-0.6244)--(-1.562,-0.6399)--(-1.527,-0.6540)--(-1.491,-0.6666)--(-1.453,-0.6777)--(-1.414,-0.6871)--(-1.374,-0.6948)--(-1.333,-0.7007)--(-1.291,-0.7048)--(-1.247,-0.7068)--(-1.202,-0.7068)--(-1.156,-0.7046)--(-1.109,-0.7001)--(-1.060,-0.6931)--(-1.010,-0.6834)--(-0.9581,-0.6709)--(-0.9049,-0.6552)--(-0.8498,-0.6362)--(-0.7927,-0.6133)--(-0.7333,-0.5861)--(-0.6709,-0.5540)--(-0.6049,-0.5158)--(-0.5341,-0.4702)--(-0.4563,-0.4148)--(-0.3673,-0.3447)--(-0.2559,-0.2479)--(0,0);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.0000,0.0000)--(0.2558,-0.2478)--(0.3672,-0.3446)--(0.4562,-0.4147)--(0.5340,-0.4702)--(0.6049,-0.5158)--(0.6709,-0.5539)--(0.7332,-0.5861)--(0.7927,-0.6133)--(0.8498,-0.6361)--(0.9048,-0.6552)--(0.9581,-0.6708)--(1.0097,-0.6834)--(1.0598,-0.6930)--(1.1086,-0.7001)--(1.1560,-0.7046)--(1.2021,-0.7068)--(1.2470,-0.7068)--(1.2906,-0.7047)--(1.3331,-0.7007)--(1.3743,-0.6947)--(1.4144,-0.6870)--(1.4532,-0.6776)--(1.4908,-0.6666)--(1.5272,-0.6540)--(1.5624,-0.6399)--(1.5963,-0.6244)--(1.6290,-0.6076)--(1.6604,-0.5894)--(1.6905,-0.5701)--(1.7193,-0.5496)--(1.7468,-0.5280)--(1.7729,-0.5053)--(1.7977,-0.4817)--(1.8212,-0.4571)--(1.8432,-0.4317)--(1.8639,-0.4054)--(1.8831,-0.3784)--(1.9009,-0.3507)--(1.9173,-0.3224)--(1.9322,-0.2934)--(1.9456,-0.2640)--(1.9576,-0.2340)--(1.9681,-0.2037)--(1.9771,-0.1729)--(1.9847,-0.1419)--(1.9907,-0.1106)--(1.9952,-0.0791)--(1.9983,-0.0475)--(1.9998,-0.0158)--(1.9998,0.0158)--(1.9983,0.0475)--(1.9952,0.0791)--(1.9907,0.1106)--(1.9847,0.1419)--(1.9771,0.1729)--(1.9681,0.2037)--(1.9576,0.2340)--(1.9456,0.2640)--(1.9322,0.2934)--(1.9173,0.3224)--(1.9009,0.3507)--(1.8831,0.3784)--(1.8639,0.4054)--(1.8432,0.4317)--(1.8212,0.4571)--(1.7977,0.4817)--(1.7729,0.5053)--(1.7468,0.5280)--(1.7193,0.5496)--(1.6905,0.5701)--(1.6604,0.5894)--(1.6290,0.6076)--(1.5963,0.6244)--(1.5624,0.6399)--(1.5272,0.6540)--(1.4908,0.6666)--(1.4532,0.6776)--(1.4144,0.6870)--(1.3743,0.6947)--(1.3331,0.7007)--(1.2906,0.7047)--(1.2470,0.7068)--(1.2021,0.7068)--(1.1560,0.7046)--(1.1086,0.7001)--(1.0598,0.6930)--(1.0097,0.6834)--(0.9581,0.6708)--(0.9048,0.6552)--(0.8498,0.6361)--(0.7927,0.6133)--(0.7332,0.5861)--(0.6709,0.5539)--(0.6049,0.5158)--(0.5340,0.4702)--(0.4562,0.4147)--(0.3672,0.3446)--(0.2558,0.2478)--(0.0000,0.0000);
+\draw [color=blue] (0.0000,0.0000)--(-0.2558,0.2478)--(-0.3672,0.3446)--(-0.4562,0.4147)--(-0.5340,0.4702)--(-0.6049,0.5158)--(-0.6709,0.5539)--(-0.7332,0.5861)--(-0.7927,0.6133)--(-0.8498,0.6361)--(-0.9048,0.6552)--(-0.9581,0.6708)--(-1.0097,0.6834)--(-1.0598,0.6930)--(-1.1086,0.7001)--(-1.1560,0.7046)--(-1.2021,0.7068)--(-1.2470,0.7068)--(-1.2906,0.7047)--(-1.3331,0.7007)--(-1.3743,0.6947)--(-1.4144,0.6870)--(-1.4532,0.6776)--(-1.4908,0.6666)--(-1.5272,0.6540)--(-1.5624,0.6399)--(-1.5963,0.6244)--(-1.6290,0.6076)--(-1.6604,0.5894)--(-1.6905,0.5701)--(-1.7193,0.5496)--(-1.7468,0.5280)--(-1.7729,0.5053)--(-1.7977,0.4817)--(-1.8212,0.4571)--(-1.8432,0.4317)--(-1.8639,0.4054)--(-1.8831,0.3784)--(-1.9009,0.3507)--(-1.9173,0.3224)--(-1.9322,0.2934)--(-1.9456,0.2640)--(-1.9576,0.2340)--(-1.9681,0.2037)--(-1.9771,0.1729)--(-1.9847,0.1419)--(-1.9907,0.1106)--(-1.9952,0.0791)--(-1.9983,0.0475)--(-1.9998,0.0158)--(-1.9998,-0.0158)--(-1.9983,-0.0475)--(-1.9952,-0.0791)--(-1.9907,-0.1106)--(-1.9847,-0.1419)--(-1.9771,-0.1729)--(-1.9681,-0.2037)--(-1.9576,-0.2340)--(-1.9456,-0.2640)--(-1.9322,-0.2934)--(-1.9173,-0.3224)--(-1.9009,-0.3507)--(-1.8831,-0.3784)--(-1.8639,-0.4054)--(-1.8432,-0.4317)--(-1.8212,-0.4571)--(-1.7977,-0.4817)--(-1.7729,-0.5053)--(-1.7468,-0.5280)--(-1.7193,-0.5496)--(-1.6905,-0.5701)--(-1.6604,-0.5894)--(-1.6290,-0.6076)--(-1.5963,-0.6244)--(-1.5624,-0.6399)--(-1.5272,-0.6540)--(-1.4908,-0.6666)--(-1.4532,-0.6776)--(-1.4144,-0.6870)--(-1.3743,-0.6947)--(-1.3331,-0.7007)--(-1.2906,-0.7047)--(-1.2470,-0.7068)--(-1.2021,-0.7068)--(-1.1560,-0.7046)--(-1.1086,-0.7001)--(-1.0598,-0.6930)--(-1.0097,-0.6834)--(-0.9581,-0.6708)--(-0.9048,-0.6552)--(-0.8498,-0.6361)--(-0.7927,-0.6133)--(-0.7332,-0.5861)--(-0.6709,-0.5539)--(-0.6049,-0.5158)--(-0.5340,-0.4702)--(-0.4562,-0.4147)--(-0.3672,-0.3446)--(-0.2558,-0.2478)--(0.0000,0.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall b/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall
index e0da8fb27..212064ed3 100644
--- a/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall
+++ b/src_phystricks/Fig_ParallelogrammeOM.pstricks.recall
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.0000,0);
-\draw (3.3086,-0.29071) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw [,->,>=latex] (0.0000,0.0000) -- (3.0000,0.0000);
+\draw (3.3085,-0.2907) node {$a$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (2.0000,2.0000);
 \draw (2.0000,2.4267) node {$b$};
 \draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
-\draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
-\draw [style=dashed] (2.00,2.00) -- (2.00,0);
+\draw [color=blue,style=dotted] (2.0000,2.0000) -- (5.0000,2.0000);
+\draw [color=blue,style=dotted] (3.0000,0.0000) -- (5.0000,2.0000);
+\draw [style=dashed] (2.0000,2.0000) -- (2.0000,0.0000);
 \draw (2.3051,1.0000) node {$h$};
-\draw (0.80616,0.31808) node {$\theta$};
+\draw (0.8061,0.3180) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw [] (0.5000,0.0000)--(0.4999,0.0039)--(0.4999,0.0079)--(0.4998,0.0118)--(0.4997,0.0158)--(0.4996,0.0198)--(0.4994,0.0237)--(0.4992,0.0277)--(0.4989,0.0317)--(0.4987,0.0356)--(0.4984,0.0396)--(0.4980,0.0435)--(0.4977,0.0475)--(0.4973,0.0514)--(0.4969,0.0554)--(0.4964,0.0593)--(0.4959,0.0632)--(0.4954,0.0672)--(0.4949,0.0711)--(0.4943,0.0750)--(0.4937,0.0790)--(0.4930,0.0829)--(0.4924,0.0868)--(0.4916,0.0907)--(0.4909,0.0946)--(0.4901,0.0985)--(0.4894,0.1024)--(0.4885,0.1062)--(0.4877,0.1101)--(0.4868,0.1140)--(0.4859,0.1178)--(0.4849,0.1217)--(0.4839,0.1255)--(0.4829,0.1294)--(0.4819,0.1332)--(0.4808,0.1370)--(0.4797,0.1408)--(0.4786,0.1446)--(0.4774,0.1484)--(0.4762,0.1522)--(0.4750,0.1560)--(0.4737,0.1597)--(0.4725,0.1635)--(0.4711,0.1672)--(0.4698,0.1710)--(0.4684,0.1747)--(0.4670,0.1784)--(0.4656,0.1821)--(0.4641,0.1858)--(0.4626,0.1895)--(0.4611,0.1931)--(0.4596,0.1968)--(0.4580,0.2004)--(0.4564,0.2040)--(0.4548,0.2077)--(0.4531,0.2113)--(0.4514,0.2148)--(0.4497,0.2184)--(0.4479,0.2220)--(0.4462,0.2255)--(0.4444,0.2291)--(0.4425,0.2326)--(0.4407,0.2361)--(0.4388,0.2396)--(0.4369,0.2430)--(0.4349,0.2465)--(0.4330,0.2500)--(0.4310,0.2534)--(0.4289,0.2568)--(0.4269,0.2602)--(0.4248,0.2636)--(0.4227,0.2669)--(0.4206,0.2703)--(0.4184,0.2736)--(0.4162,0.2769)--(0.4140,0.2802)--(0.4118,0.2835)--(0.4095,0.2867)--(0.4072,0.2900)--(0.4049,0.2932)--(0.4026,0.2964)--(0.4002,0.2996)--(0.3978,0.3028)--(0.3954,0.3059)--(0.3930,0.3090)--(0.3905,0.3121)--(0.3880,0.3152)--(0.3855,0.3183)--(0.3830,0.3213)--(0.3804,0.3244)--(0.3778,0.3274)--(0.3752,0.3304)--(0.3726,0.3333)--(0.3699,0.3363)--(0.3672,0.3392)--(0.3645,0.3421)--(0.3618,0.3450)--(0.3591,0.3478)--(0.3563,0.3507)--(0.3535,0.3535);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ParamTangente.pstricks.recall b/src_phystricks/Fig_ParamTangente.pstricks.recall
index 7a28085b0..1c7965993 100644
--- a/src_phystricks/Fig_ParamTangente.pstricks.recall
+++ b/src_phystricks/Fig_ParamTangente.pstricks.recall
@@ -87,36 +87,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0424,0) -- (3.5561,0);
-\draw [,->,>=latex] (0,-3.8750) -- (0,3.9454);
+\draw [,->,>=latex] (-1.0423,0.0000) -- (3.5561,0.0000);
+\draw [,->,>=latex] (0.0000,-3.8750) -- (0.0000,3.9453);
 %DEFAULT
-\draw [color=blue] (3.000,-3.375)--(2.939,-3.175)--(2.879,-2.982)--(2.818,-2.798)--(2.758,-2.621)--(2.697,-2.452)--(2.636,-2.290)--(2.576,-2.136)--(2.515,-1.989)--(2.455,-1.849)--(2.394,-1.715)--(2.333,-1.588)--(2.273,-1.467)--(2.212,-1.353)--(2.152,-1.245)--(2.091,-1.143)--(2.030,-1.046)--(1.970,-0.9552)--(1.909,-0.8697)--(1.848,-0.7895)--(1.788,-0.7144)--(1.727,-0.6442)--(1.667,-0.5787)--(1.606,-0.5178)--(1.545,-0.4614)--(1.485,-0.4092)--(1.424,-0.3611)--(1.364,-0.3170)--(1.303,-0.2766)--(1.242,-0.2397)--(1.182,-0.2063)--(1.121,-0.1762)--(1.061,-0.1491)--(1.000,-0.1250)--(0.9394,-0.1036)--(0.8788,-0.08483)--(0.8182,-0.06846)--(0.7576,-0.05435)--(0.6970,-0.04232)--(0.6364,-0.03221)--(0.5758,-0.02386)--(0.5152,-0.01709)--(0.4545,-0.01174)--(0.3939,-0.007642)--(0.3333,-0.004630)--(0.2727,-0.002536)--(0.2121,-0.001193)--(0.1515,0)--(0.09091,0)--(0.03030,0)--(0.03030,0)--(0.09091,0)--(0.1515,0)--(0.2121,0.001193)--(0.2727,0.002536)--(0.3333,0.004630)--(0.3939,0.007642)--(0.4545,0.01174)--(0.5152,0.01709)--(0.5758,0.02386)--(0.6364,0.03221)--(0.6970,0.04232)--(0.7576,0.05435)--(0.8182,0.06846)--(0.8788,0.08483)--(0.9394,0.1036)--(1.000,0.1250)--(1.061,0.1491)--(1.121,0.1762)--(1.182,0.2063)--(1.242,0.2397)--(1.303,0.2766)--(1.364,0.3170)--(1.424,0.3611)--(1.485,0.4092)--(1.545,0.4614)--(1.606,0.5178)--(1.667,0.5787)--(1.727,0.6442)--(1.788,0.7144)--(1.848,0.7895)--(1.909,0.8697)--(1.970,0.9552)--(2.030,1.046)--(2.091,1.143)--(2.152,1.245)--(2.212,1.353)--(2.273,1.467)--(2.333,1.588)--(2.394,1.715)--(2.455,1.849)--(2.515,1.989)--(2.576,2.136)--(2.636,2.290)--(2.697,2.452)--(2.758,2.621)--(2.818,2.798)--(2.879,2.982)--(2.939,3.175)--(3.000,3.375);
-\draw [color=red,->,>=latex] (2.5078,-1.9715) -- (2.1174,-1.0508);
-\draw [color=red,->,>=latex] (1.7548,-0.67544) -- (1.1001,0.080499);
-\draw [color=red,->,>=latex] (0.45465,-0.011748) -- (-0.54236,0.065536);
-\draw [color=red,->,>=latex] (1.0298,0.13650) -- (1.9590,0.50601);
-\draw [color=red,->,>=latex] (2.0918,1.1442) -- (2.6122,1.9981);
-\draw [color=red,->,>=latex] (2.7162,2.5049) -- (3.0561,3.4454);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (3.0000,-3.3750)--(2.9393,-3.1745)--(2.8787,-2.9822)--(2.8181,-2.7978)--(2.7575,-2.6211)--(2.6969,-2.4521)--(2.6363,-2.2904)--(2.5757,-2.1361)--(2.5151,-1.9888)--(2.4545,-1.8485)--(2.3939,-1.7149)--(2.3333,-1.5879)--(2.2727,-1.4674)--(2.2121,-1.3531)--(2.1515,-1.2449)--(2.0909,-1.1426)--(2.0303,-1.0461)--(1.9696,-0.9552)--(1.9090,-0.8697)--(1.8484,-0.7895)--(1.7878,-0.7143)--(1.7272,-0.6441)--(1.6666,-0.5787)--(1.6060,-0.5178)--(1.5454,-0.4614)--(1.4848,-0.4092)--(1.4242,-0.3611)--(1.3636,-0.3169)--(1.3030,-0.2765)--(1.2424,-0.2397)--(1.1818,-0.2063)--(1.1212,-0.1761)--(1.0606,-0.1491)--(1.0000,-0.1250)--(0.9393,-0.1036)--(0.8787,-0.0848)--(0.8181,-0.0684)--(0.7575,-0.0543)--(0.6969,-0.0423)--(0.6363,-0.0322)--(0.5757,-0.0238)--(0.5151,-0.0170)--(0.4545,-0.0117)--(0.3939,-0.0076)--(0.3333,-0.0046)--(0.2727,-0.0025)--(0.2121,-0.0011)--(0.1515,0.0000)--(0.0909,0.0000)--(0.0303,0.0000)--(0.0303,0.0000)--(0.0909,0.0000)--(0.1515,0.0000)--(0.2121,0.0011)--(0.2727,0.0025)--(0.3333,0.0046)--(0.3939,0.0076)--(0.4545,0.0117)--(0.5151,0.0170)--(0.5757,0.0238)--(0.6363,0.0322)--(0.6969,0.0423)--(0.7575,0.0543)--(0.8181,0.0684)--(0.8787,0.0848)--(0.9393,0.1036)--(1.0000,0.1250)--(1.0606,0.1491)--(1.1212,0.1761)--(1.1818,0.2063)--(1.2424,0.2397)--(1.3030,0.2765)--(1.3636,0.3169)--(1.4242,0.3611)--(1.4848,0.4092)--(1.5454,0.4614)--(1.6060,0.5178)--(1.6666,0.5787)--(1.7272,0.6441)--(1.7878,0.7143)--(1.8484,0.7895)--(1.9090,0.8697)--(1.9696,0.9552)--(2.0303,1.0461)--(2.0909,1.1426)--(2.1515,1.2449)--(2.2121,1.3531)--(2.2727,1.4674)--(2.3333,1.5879)--(2.3939,1.7149)--(2.4545,1.8485)--(2.5151,1.9888)--(2.5757,2.1361)--(2.6363,2.2904)--(2.6969,2.4521)--(2.7575,2.6211)--(2.8181,2.7978)--(2.8787,2.9822)--(2.9393,3.1745)--(3.0000,3.3750);
+\draw [color=red,->,>=latex] (2.5078,-1.9714) -- (2.1174,-1.0508);
+\draw [color=red,->,>=latex] (1.7547,-0.6754) -- (1.1001,0.0804);
+\draw [color=red,->,>=latex] (0.4546,-0.0117) -- (-0.5423,0.0655);
+\draw [color=red,->,>=latex] (1.0297,0.1364) -- (1.9589,0.5060);
+\draw [color=red,->,>=latex] (2.0918,1.1441) -- (2.6122,1.9981);
+\draw [color=red,->,>=latex] (2.7162,2.5049) -- (3.0561,3.4453);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_PartieEntiere.pstricks.recall b/src_phystricks/Fig_PartieEntiere.pstricks.recall
index 5820d4ebf..3b5006c38 100644
--- a/src_phystricks/Fig_PartieEntiere.pstricks.recall
+++ b/src_phystricks/Fig_PartieEntiere.pstricks.recall
@@ -83,32 +83,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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.9739,1.0000)--(1.9789,1.0000)--(1.9839,1.0000)--(1.9889,1.0000)--(1.9939,1.0000)--(1.9989,1.0000)--(2.0040,2.0000)--(2.0090,2.0000)--(2.0140,2.0000)--(2.0190,2.0000)--(2.0240,2.0000)--(2.0290,2.0000)--(2.0340,2.0000)--(2.0390,2.0000)--(2.0440,2.0000)--(2.0490,2.0000)--(2.0540,2.0000)--(2.0590,2.0000)--(2.0640,2.0000)--(2.0690,2.0000)--(2.0740,2.0000)--(2.0790,2.0000)--(2.0840,2.0000)--(2.0890,2.0000)--(2.0940,2.0000)--(2.0990,2.0000)--(2.1041,2.0000)--(2.1091,2.0000)--(2.1141,2.0000)--(2.1191,2.0000)--(2.1241,2.0000)--(2.1291,2.0000)--(2.1341,2.0000)--(2.1391,2.0000)--(2.1441,2.0000)--(2.1491,2.0000)--(2.1541,2.0000)--(2.1591,2.0000)--(2.1641,2.0000)--(2.1691,2.0000)--(2.1741,2.0000)--(2.1791,2.0000)--(2.1841,2.0000)--(2.1891,2.0000)--(2.1941,2.0000)--(2.1991,2.0000)--(2.2042,2.0000)--(2.2092,2.0000)--(2.2142,2.0000)--(2.2192,2.0000)--(2.2242,2.0000)--(2.2292,2.0000)--(2.2342,2.0000)--(2.2392,2.0000)--(2.2442,2.0000)--(2.2492,2.0000)--(2.2542,2.0000)--(2.2592,2.0000)--(2.2642,2.0000)--(2.2692,2.0000)--(2.2742,2.0000)--(2.2792,2.0000)--(2.2842,2.0000)--(2.2892,2.0000)--(2.2942,2.0000)--(2.2992,2.0000)--(2.3043,2.0000)--(2.3093,2.0000)--(2.3143,2.0000)--(2.3193,2.0000)--(2.3243,2.0000)--(2.3293,2.0000)--(2.3343,2.0000)--(2.3393,2.0000)--(2.3443,2.0000)--(2.3493,2.0000)--(2.3543,2.0000)--(2.3593,2.0000)--(2.3643,2.0000)--(2.3693,2.0000)--(2.3743,2.0000)--(2.3793,2.0000)--(2.3843,2.0000)--(2.3893,2.0000)--(2.3943,2.0000)--(2.3993,2.0000)--(2.4044,2.0000)--(2.4094,2.0000)--(2.4144,2.0000)--(2.4194,2.0000)--(2.4244,2.0000)--(2.4294,2.0000)--(2.4344,2.0000)--(2.4394,2.0000)--(2.4444,2.0000)--(2.4494,2.0000)--(2.4544,2.0000)--(2.4594,2.0000)--(2.4644,2.0000)--(2.4694,2.0000)--(2.4744,2.0000)--(2.4794,2.0000)--(2.4844,2.0000)--(2.4894,2.0000)--(2.4944,2.0000)--(2.4994,2.0000)--(2.5045,2.0000)--(2.5095,2.0000)--(2.5145,2.0000)--(2.5195,2.0000)--(2.5245,2.0000)--(2.5295,2.0000)--(2.5345,2.0000)--(2.5395,2.0000)--(2.5445,2.0000)--(2.5495,2.0000)--(2.5545,2.0000)--(2.5595,2.0000)--(2.5645,2.0000)--(2.5695,2.0000)--(2.5745,2.0000)--(2.5795,2.0000)--(2.5845,2.0000)--(2.5895,2.0000)--(2.5945,2.0000)--(2.5995,2.0000)--(2.6046,2.0000)--(2.6096,2.0000)--(2.6146,2.0000)--(2.6196,2.0000)--(2.6246,2.0000)--(2.6296,2.0000)--(2.6346,2.0000)--(2.6396,2.0000)--(2.6446,2.0000)--(2.6496,2.0000)--(2.6546,2.0000)--(2.6596,2.0000)--(2.6646,2.0000)--(2.6696,2.0000)--(2.6746,2.0000)--(2.6796,2.0000)--(2.6846,2.0000)--(2.6896,2.0000)--(2.6946,2.0000)--(2.6996,2.0000)--(2.7047,2.0000)--(2.7097,2.0000)--(2.7147,2.0000)--(2.7197,2.0000)--(2.7247,2.0000)--(2.7297,2.0000)--(2.7347,2.0000)--(2.7397,2.0000)--(2.7447,2.0000)--(2.7497,2.0000)--(2.7547,2.0000)--(2.7597,2.0000)--(2.7647,2.0000)--(2.7697,2.0000)--(2.7747,2.0000)--(2.7797,2.0000)--(2.7847,2.0000)--(2.7897,2.0000)--(2.7947,2.0000)--(2.7997,2.0000)--(2.8048,2.0000)--(2.8098,2.0000)--(2.8148,2.0000)--(2.8198,2.0000)--(2.8248,2.0000)--(2.8298,2.0000)--(2.8348,2.0000)--(2.8398,2.0000)--(2.8448,2.0000)--(2.8498,2.0000)--(2.8548,2.0000)--(2.8598,2.0000)--(2.8648,2.0000)--(2.8698,2.0000)--(2.8748,2.0000)--(2.8798,2.0000)--(2.8848,2.0000)--(2.8898,2.0000)--(2.8948,2.0000)--(2.8998,2.0000)--(2.9049,2.0000)--(2.9099,2.0000)--(2.9149,2.0000)--(2.9199,2.0000)--(2.9249,2.0000)--(2.9299,2.0000)--(2.9349,2.0000)--(2.9399,2.0000)--(2.9449,2.0000)--(2.9499,2.0000)--(2.9549,2.0000)--(2.9599,2.0000)--(2.9649,2.0000)--(2.9699,2.0000)--(2.9749,2.0000)--(2.9799,2.0000)--(2.9849,2.0000)--(2.9899,2.0000)--(2.9949,2.0000)--(3.0000,3.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_Polirettangolo.pstricks.recall b/src_phystricks/Fig_Polirettangolo.pstricks.recall
index 8d5164762..0eb98b05f 100644
--- a/src_phystricks/Fig_Polirettangolo.pstricks.recall
+++ b/src_phystricks/Fig_Polirettangolo.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -89,11 +89,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,2.00) -- (1.50,2.00) -- (1.50,2.00) -- (1.50,1.00) -- (1.50,1.00) -- (0,1.00) -- (0,1.00) -- (0,2.00) --  cycle;
-\draw [style=dotted] (0,2.00) -- (1.50,2.00);
-\draw [style=dotted] (1.50,2.00) -- (1.50,1.00);
-\draw [style=dotted] (1.50,1.00) -- (0,1.00);
-\draw [style=dotted] (0,1.00) -- (0,2.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.0000,2.0000) -- (1.5000,2.0000) -- (1.5000,2.0000) -- (1.5000,1.0000) -- (1.5000,1.0000) -- (0.0000,1.0000) -- (0.0000,1.0000) -- (0.0000,2.0000) --  cycle;
+\draw [style=dotted] (0.0000,2.0000) -- (1.5000,2.0000);
+\draw [style=dotted] (1.5000,2.0000) -- (1.5000,1.0000);
+\draw [style=dotted] (1.5000,1.0000) -- (0.0000,1.0000);
+\draw [style=dotted] (0.0000,1.0000) -- (0.0000,2.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -101,11 +101,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.500,2.50) -- (2.00,2.50) -- (2.00,2.50) -- (2.00,2.00) -- (2.00,2.00) -- (0.500,2.00) -- (0.500,2.00) -- (0.500,2.50) --  cycle;
-\draw [style=dotted] (0.500,2.50) -- (2.00,2.50);
-\draw [style=dotted] (2.00,2.50) -- (2.00,2.00);
-\draw [style=dotted] (2.00,2.00) -- (0.500,2.00);
-\draw [style=dotted] (0.500,2.00) -- (0.500,2.50);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.5000,2.5000) -- (2.0000,2.5000) -- (2.0000,2.5000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (0.5000,2.0000) -- (0.5000,2.0000) -- (0.5000,2.5000) --  cycle;
+\draw [style=dotted] (0.5000,2.5000) -- (2.0000,2.5000);
+\draw [style=dotted] (2.0000,2.5000) -- (2.0000,2.0000);
+\draw [style=dotted] (2.0000,2.0000) -- (0.5000,2.0000);
+\draw [style=dotted] (0.5000,2.0000) -- (0.5000,2.5000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -113,11 +113,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,1.00) -- (3.00,1.00) -- (3.00,1.00) -- (3.00,0) -- (3.00,0) -- (2.00,0) -- (2.00,0) -- (2.00,1.00) --  cycle;
-\draw [style=dotted] (2.00,1.00) -- (3.00,1.00);
-\draw [style=dotted] (3.00,1.00) -- (3.00,0);
-\draw [style=dotted] (3.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (2.00,1.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.0000,1.0000) -- (3.0000,1.0000) -- (3.0000,1.0000) -- (3.0000,0.0000) -- (3.0000,0.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (2.0000,1.0000) --  cycle;
+\draw [style=dotted] (2.0000,1.0000) -- (3.0000,1.0000);
+\draw [style=dotted] (3.0000,1.0000) -- (3.0000,0.0000);
+\draw [style=dotted] (3.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (2.0000,1.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -125,27 +125,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,3.50) -- (3.50,3.50) -- (3.50,3.50) -- (3.50,1.50) -- (3.50,1.50) -- (2.00,1.50) -- (2.00,1.50) -- (2.00,3.50) --  cycle;
-\draw [style=dotted] (2.00,3.50) -- (3.50,3.50);
-\draw [style=dotted] (3.50,3.50) -- (3.50,1.50);
-\draw [style=dotted] (3.50,1.50) -- (2.00,1.50);
-\draw [style=dotted] (2.00,1.50) -- (2.00,3.50);
-\draw (1.0000,-0.31492) node {$ 2 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 4 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 6 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 8 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.0000,3.5000) -- (3.5000,3.5000) -- (3.5000,3.5000) -- (3.5000,1.5000) -- (3.5000,1.5000) -- (2.0000,1.5000) -- (2.0000,1.5000) -- (2.0000,3.5000) --  cycle;
+\draw [style=dotted] (2.0000,3.5000) -- (3.5000,3.5000);
+\draw [style=dotted] (3.5000,3.5000) -- (3.5000,1.5000);
+\draw [style=dotted] (3.5000,1.5000) -- (2.0000,1.5000);
+\draw [style=dotted] (2.0000,1.5000) -- (2.0000,3.5000);
+\draw (1.0000,-0.3149) node {$ 2 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 4 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 6 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 8 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ProjectionScalaire.pstricks.recall b/src_phystricks/Fig_ProjectionScalaire.pstricks.recall
index aa94342af..ce42fcd82 100644
--- a/src_phystricks/Fig_ProjectionScalaire.pstricks.recall
+++ b/src_phystricks/Fig_ProjectionScalaire.pstricks.recall
@@ -75,19 +75,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue,->,>=latex] (0,0) -- (1.5000,2.0000);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (1.5000,2.0000);
 \draw (1.8776,2.3368) node {$X$};
-\draw [color=blue,->,>=latex] (0,0) -- (2.5000,0);
-\draw (2.5000,0.32471) node {$Y$};
-\draw []  (1.5000,0) node [rotate=0] {$\bullet$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (0.75000,-0.30000) -- (0,-0.30000);
-\draw [,->,>=latex] (0.75000,-0.30000) -- (1.5000,-0.30000);
-\draw (0.75000,-0.67858) node {$x$};
-\draw [style=dotted] (1.50,2.00) -- (1.50,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (2.5000,0.0000);
+\draw (2.5000,0.3247) node {$Y$};
+\draw []  (1.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (0.7500,-0.3000) -- (0.0000,-0.3000);
+\draw [,->,>=latex] (0.7500,-0.3000) -- (1.5000,-0.3000);
+\draw (0.7500,-0.6785) node {$x$};
+\draw [style=dotted] (1.5000,2.0000) -- (1.5000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QCb.pstricks.recall b/src_phystricks/Fig_QCb.pstricks.recall
index 389c36fef..3a23fea8c 100644
--- a/src_phystricks/Fig_QCb.pstricks.recall
+++ b/src_phystricks/Fig_QCb.pstricks.recall
@@ -71,15 +71,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw (-0.79967,1.0000) node {\( xy\)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw (-0.7996,1.0000) node {\( xy\)};
 \draw (1.5676,1.0000) node {\( \sin(xy)\)};
 \draw (1.2003,-1.0000) node {\( xy\)};
-\draw (-0.79967,-1.0000) node {\( xy\)};
+\draw (-0.7996,-1.0000) node {\( xy\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall b/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall
index 25a73757d..f6e6c8bba 100644
--- a/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall
+++ b/src_phystricks/Fig_QMWKooRRulrgcH.pstricks.recall
@@ -80,23 +80,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
-\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-1.0000,-0.32471) node {\( A\)};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.32471) node {\( B\)};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0.28570,-0.26613) node {\( O\)};
-\draw []  (0,2.8284) node [rotate=0] {$\bullet$};
-\draw (0.23597,3.0946) node {\( I\)};
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.3247) node {\( A\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3247) node {\( B\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.2856,-0.2661) node {\( O\)};
+\draw []  (0.0000,2.8284) node [rotate=0] {$\bullet$};
+\draw (0.2359,3.0945) node {\( I\)};
 
-\draw [] (2.000,0)--(1.994,0.1903)--(1.976,0.3798)--(1.946,0.5678)--(1.904,0.7534)--(1.850,0.9361)--(1.785,1.115)--(1.709,1.289)--(1.622,1.459)--(1.524,1.622)--(1.416,1.779)--(1.298,1.928)--(1.171,2.070)--(1.036,2.204)--(0.8917,2.328)--(0.7402,2.444)--(0.5817,2.549)--(0.4168,2.644)--(0.2462,2.729)--(0.07066,2.802)--(-0.1092,2.865)--(-0.2927,2.915)--(-0.4791,2.954)--(-0.6675,2.982)--(-0.8573,2.997)--(-1.048,3.000)--(-1.238,2.991)--(-1.427,2.969)--(-1.614,2.936)--(-1.799,2.892)--(-1.981,2.835)--(-2.159,2.767)--(-2.332,2.688)--(-2.500,2.598)--(-2.662,2.498)--(-2.817,2.387)--(-2.965,2.267)--(-3.104,2.138)--(-3.236,2.000)--(-3.358,1.854)--(-3.471,1.701)--(-3.574,1.541)--(-3.667,1.375)--(-3.748,1.203)--(-3.819,1.026)--(-3.878,0.8452)--(-3.926,0.6609)--(-3.962,0.4740)--(-3.986,0.2852)--(-3.999,0.09518)--(-3.999,-0.09518)--(-3.986,-0.2852)--(-3.962,-0.4740)--(-3.926,-0.6609)--(-3.878,-0.8452)--(-3.819,-1.026)--(-3.748,-1.203)--(-3.667,-1.375)--(-3.574,-1.541)--(-3.471,-1.701)--(-3.358,-1.854)--(-3.236,-2.000)--(-3.104,-2.138)--(-2.965,-2.267)--(-2.817,-2.387)--(-2.662,-2.498)--(-2.500,-2.598)--(-2.332,-2.688)--(-2.159,-2.767)--(-1.981,-2.835)--(-1.799,-2.892)--(-1.614,-2.936)--(-1.427,-2.969)--(-1.238,-2.991)--(-1.048,-3.000)--(-0.8573,-2.997)--(-0.6675,-2.982)--(-0.4791,-2.954)--(-0.2927,-2.915)--(-0.1092,-2.865)--(0.07066,-2.802)--(0.2462,-2.729)--(0.4168,-2.644)--(0.5817,-2.549)--(0.7402,-2.444)--(0.8917,-2.328)--(1.036,-2.204)--(1.171,-2.070)--(1.298,-1.928)--(1.416,-1.779)--(1.524,-1.622)--(1.622,-1.459)--(1.709,-1.289)--(1.785,-1.115)--(1.850,-0.9361)--(1.904,-0.7534)--(1.946,-0.5678)--(1.976,-0.3798)--(1.994,-0.1903)--(2.000,0);
+\draw [] (2.0000,0.0000)--(1.9939,0.1902)--(1.9758,0.3797)--(1.9457,0.5677)--(1.9038,0.7534)--(1.8502,0.9361)--(1.7851,1.1149)--(1.7087,1.2893)--(1.6215,1.4585)--(1.5237,1.6219)--(1.4158,1.7787)--(1.2981,1.9283)--(1.1712,2.0702)--(1.0355,2.2037)--(0.8916,2.3284)--(0.7401,2.4437)--(0.5816,2.5491)--(0.4168,2.6443)--(0.2462,2.7288)--(0.0706,2.8024)--(-0.1092,2.8647)--(-0.2927,2.9154)--(-0.4790,2.9544)--(-0.6674,2.9815)--(-0.8572,2.9966)--(-1.0475,2.9996)--(-1.2377,2.9905)--(-1.4269,2.9694)--(-1.6144,2.9364)--(-1.7994,2.8915)--(-1.9812,2.8350)--(-2.1590,2.7670)--(-2.3321,2.6879)--(-2.5000,2.5980)--(-2.6617,2.4977)--(-2.8168,2.3872)--(-2.9645,2.2672)--(-3.1044,2.1380)--(-3.2357,2.0003)--(-3.3581,1.8544)--(-3.4710,1.7011)--(-3.5739,1.5410)--(-3.6665,1.3746)--(-3.7483,1.2027)--(-3.8190,1.0260)--(-3.8784,0.8451)--(-3.9262,0.6609)--(-3.9623,0.4740)--(-3.9864,0.2851)--(-3.9984,0.0951)--(-3.9984,-0.0951)--(-3.9864,-0.2851)--(-3.9623,-0.4740)--(-3.9262,-0.6609)--(-3.8784,-0.8451)--(-3.8190,-1.0260)--(-3.7483,-1.2027)--(-3.6665,-1.3746)--(-3.5739,-1.5410)--(-3.4710,-1.7011)--(-3.3581,-1.8544)--(-3.2357,-2.0003)--(-3.1044,-2.1380)--(-2.9645,-2.2672)--(-2.8168,-2.3872)--(-2.6617,-2.4977)--(-2.5000,-2.5980)--(-2.3321,-2.6879)--(-2.1590,-2.7670)--(-1.9812,-2.8350)--(-1.7994,-2.8915)--(-1.6144,-2.9364)--(-1.4269,-2.9694)--(-1.2377,-2.9905)--(-1.0475,-2.9996)--(-0.8572,-2.9966)--(-0.6674,-2.9815)--(-0.4790,-2.9544)--(-0.2927,-2.9154)--(-0.1092,-2.8647)--(0.0706,-2.8024)--(0.2462,-2.7288)--(0.4168,-2.6443)--(0.5816,-2.5491)--(0.7401,-2.4437)--(0.8916,-2.3284)--(1.0355,-2.2037)--(1.1712,-2.0702)--(1.2981,-1.9283)--(1.4158,-1.7787)--(1.5237,-1.6219)--(1.6215,-1.4585)--(1.7087,-1.2893)--(1.7851,-1.1149)--(1.8502,-0.9361)--(1.9038,-0.7534)--(1.9457,-0.5677)--(1.9758,-0.3797)--(1.9939,-0.1902)--(2.0000,0.0000);
 
-\draw [] (4.000,0)--(3.994,0.1903)--(3.976,0.3798)--(3.946,0.5678)--(3.904,0.7534)--(3.850,0.9361)--(3.785,1.115)--(3.709,1.289)--(3.622,1.459)--(3.524,1.622)--(3.416,1.779)--(3.298,1.928)--(3.171,2.070)--(3.036,2.204)--(2.892,2.328)--(2.740,2.444)--(2.582,2.549)--(2.417,2.644)--(2.246,2.729)--(2.071,2.802)--(1.891,2.865)--(1.707,2.915)--(1.521,2.954)--(1.333,2.982)--(1.143,2.997)--(0.9524,3.000)--(0.7623,2.991)--(0.5731,2.969)--(0.3856,2.936)--(0.2006,2.892)--(0.01880,2.835)--(-0.1590,2.767)--(-0.3322,2.688)--(-0.5000,2.598)--(-0.6618,2.498)--(-0.8168,2.387)--(-0.9646,2.267)--(-1.104,2.138)--(-1.236,2.000)--(-1.358,1.854)--(-1.471,1.701)--(-1.574,1.541)--(-1.667,1.375)--(-1.748,1.203)--(-1.819,1.026)--(-1.878,0.8452)--(-1.926,0.6609)--(-1.962,0.4740)--(-1.986,0.2852)--(-1.998,0.09518)--(-1.998,-0.09518)--(-1.986,-0.2852)--(-1.962,-0.4740)--(-1.926,-0.6609)--(-1.878,-0.8452)--(-1.819,-1.026)--(-1.748,-1.203)--(-1.667,-1.375)--(-1.574,-1.541)--(-1.471,-1.701)--(-1.358,-1.854)--(-1.236,-2.000)--(-1.104,-2.138)--(-0.9646,-2.267)--(-0.8168,-2.387)--(-0.6618,-2.498)--(-0.5000,-2.598)--(-0.3322,-2.688)--(-0.1590,-2.767)--(0.01880,-2.835)--(0.2006,-2.892)--(0.3856,-2.936)--(0.5731,-2.969)--(0.7623,-2.991)--(0.9524,-3.000)--(1.143,-2.997)--(1.333,-2.982)--(1.521,-2.954)--(1.707,-2.915)--(1.891,-2.865)--(2.071,-2.802)--(2.246,-2.729)--(2.417,-2.644)--(2.582,-2.549)--(2.740,-2.444)--(2.892,-2.328)--(3.036,-2.204)--(3.171,-2.070)--(3.298,-1.928)--(3.416,-1.779)--(3.524,-1.622)--(3.622,-1.459)--(3.709,-1.289)--(3.785,-1.115)--(3.850,-0.9361)--(3.904,-0.7534)--(3.946,-0.5678)--(3.976,-0.3798)--(3.994,-0.1903)--(4.000,0);
-\draw []  (-0.70000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-0.98570,1.3016) node {\( Q\)};
+\draw [] (4.0000,0.0000)--(3.9939,0.1902)--(3.9758,0.3797)--(3.9457,0.5677)--(3.9038,0.7534)--(3.8502,0.9361)--(3.7851,1.1149)--(3.7087,1.2893)--(3.6215,1.4585)--(3.5237,1.6219)--(3.4158,1.7787)--(3.2981,1.9283)--(3.1712,2.0702)--(3.0355,2.2037)--(2.8916,2.3284)--(2.7401,2.4437)--(2.5816,2.5491)--(2.4168,2.6443)--(2.2462,2.7288)--(2.0706,2.8024)--(1.8907,2.8647)--(1.7072,2.9154)--(1.5209,2.9544)--(1.3325,2.9815)--(1.1427,2.9966)--(0.9524,2.9996)--(0.7622,2.9905)--(0.5730,2.9694)--(0.3855,2.9364)--(0.2005,2.8915)--(0.0187,2.8350)--(-0.1590,2.7670)--(-0.3321,2.6879)--(-0.5000,2.5980)--(-0.6617,2.4977)--(-0.8168,2.3872)--(-0.9645,2.2672)--(-1.1044,2.1380)--(-1.2357,2.0003)--(-1.3581,1.8544)--(-1.4710,1.7011)--(-1.5739,1.5410)--(-1.6665,1.3746)--(-1.7483,1.2027)--(-1.8190,1.0260)--(-1.8784,0.8451)--(-1.9262,0.6609)--(-1.9623,0.4740)--(-1.9864,0.2851)--(-1.9984,0.0951)--(-1.9984,-0.0951)--(-1.9864,-0.2851)--(-1.9623,-0.4740)--(-1.9262,-0.6609)--(-1.8784,-0.8451)--(-1.8190,-1.0260)--(-1.7483,-1.2027)--(-1.6665,-1.3746)--(-1.5739,-1.5410)--(-1.4710,-1.7011)--(-1.3581,-1.8544)--(-1.2357,-2.0003)--(-1.1044,-2.1380)--(-0.9645,-2.2672)--(-0.8168,-2.3872)--(-0.6617,-2.4977)--(-0.4999,-2.5980)--(-0.3321,-2.6879)--(-0.1590,-2.7670)--(0.0187,-2.8350)--(0.2005,-2.8915)--(0.3855,-2.9364)--(0.5730,-2.9694)--(0.7622,-2.9905)--(0.9524,-2.9996)--(1.1427,-2.9966)--(1.3325,-2.9815)--(1.5209,-2.9544)--(1.7072,-2.9154)--(1.8907,-2.8647)--(2.0706,-2.8024)--(2.2462,-2.7288)--(2.4168,-2.6443)--(2.5816,-2.5491)--(2.7401,-2.4437)--(2.8916,-2.3284)--(3.0355,-2.2037)--(3.1712,-2.0702)--(3.2981,-1.9283)--(3.4158,-1.7787)--(3.5237,-1.6219)--(3.6215,-1.4585)--(3.7087,-1.2893)--(3.7851,-1.1149)--(3.8502,-0.9361)--(3.9038,-0.7534)--(3.9457,-0.5677)--(3.9758,-0.3797)--(3.9939,-0.1902)--(4.0000,0.0000);
+\draw []  (-0.7000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9856,1.3016) node {\( Q\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QPcdHwP.pstricks.recall b/src_phystricks/Fig_QPcdHwP.pstricks.recall
index bc69167a5..4fde4f320 100644
--- a/src_phystricks/Fig_QPcdHwP.pstricks.recall
+++ b/src_phystricks/Fig_QPcdHwP.pstricks.recall
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.30595) node {\( \alpha_2\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,-0.30595) node {\( \alpha_4\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3059) node {\( \alpha_2\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,-0.3059) node {\( \alpha_4\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall b/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall
index af9c3b2cf..f9bd83158 100644
--- a/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall
+++ b/src_phystricks/Fig_QSKDooujUbDCsu.pstricks.recall
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-2.10,0.700) -- (2.10,0.700);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.38245) node {\( \pi(b_1)\)};
-\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
-\draw (0.70000,1.0825) node {\( \pi(b_2)\)};
+\draw [] (-2.1000,0.7000) -- (2.1000,0.7000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3824) node {\( \pi(b_1)\)};
+\draw []  (0.7000,0.7000) node [rotate=0] {$\bullet$};
+\draw (0.7000,1.0824) node {\( \pi(b_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_RGjjpwF.pstricks.recall b/src_phystricks/Fig_RGjjpwF.pstricks.recall
index 055d4425c..2af1d6006 100644
--- a/src_phystricks/Fig_RGjjpwF.pstricks.recall
+++ b/src_phystricks/Fig_RGjjpwF.pstricks.recall
@@ -70,20 +70,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.31492) node {$1$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.20000) node {};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.20000) node {};
-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.20000) node {};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.20000) node {};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3149) node {$1$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2000) node {};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.2000) node {};
+\draw []  (3.0000,0.0000) node [rotate=0] {$o$};
+\draw (3.0000,0.2000) node {};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.2000) node {};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall b/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall
index 204852b2e..f8a93bbca 100644
--- a/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall
+++ b/src_phystricks/Fig_ROAOooPgUZIt.pstricks.recall
@@ -71,18 +71,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
 
-\draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [color=brown] (1.0000,3.0000)--(0.9979,3.0634)--(0.9919,3.1265)--(0.9819,3.1892)--(0.9679,3.2511)--(0.9500,3.3120)--(0.9283,3.3716)--(0.9029,3.4297)--(0.8738,3.4861)--(0.8412,3.5406)--(0.8052,3.5929)--(0.7660,3.6427)--(0.7237,3.6900)--(0.6785,3.7345)--(0.6305,3.7761)--(0.5800,3.8145)--(0.5272,3.8497)--(0.4722,3.8814)--(0.4154,3.9096)--(0.3568,3.9341)--(0.2969,3.9549)--(0.2357,3.9718)--(0.1736,3.9848)--(0.1108,3.9938)--(0.0475,3.9988)--(-0.0158,3.9998)--(-0.0792,3.9968)--(-0.1423,3.9898)--(-0.2048,3.9788)--(-0.2664,3.9638)--(-0.3270,3.9450)--(-0.3863,3.9223)--(-0.4440,3.8959)--(-0.5000,3.8660)--(-0.5539,3.8325)--(-0.6056,3.7957)--(-0.6548,3.7557)--(-0.7014,3.7126)--(-0.7452,3.6667)--(-0.7860,3.6181)--(-0.8236,3.5670)--(-0.8579,3.5136)--(-0.8888,3.4582)--(-0.9161,3.4009)--(-0.9396,3.3420)--(-0.9594,3.2817)--(-0.9754,3.2203)--(-0.9874,3.1580)--(-0.9954,3.0950)--(-0.9994,3.0317)--(-0.9994,2.9682)--(-0.9954,2.9049)--(-0.9874,2.8419)--(-0.9754,2.7796)--(-0.9594,2.7182)--(-0.9396,2.6579)--(-0.9161,2.5990)--(-0.8888,2.5417)--(-0.8579,2.4863)--(-0.8236,2.4329)--(-0.7860,2.3818)--(-0.7452,2.3332)--(-0.7014,2.2873)--(-0.6548,2.2442)--(-0.6056,2.2042)--(-0.5539,2.1674)--(-0.5000,2.1339)--(-0.4440,2.1040)--(-0.3863,2.0776)--(-0.3270,2.0549)--(-0.2664,2.0361)--(-0.2048,2.0211)--(-0.1423,2.0101)--(-0.0792,2.0031)--(-0.0158,2.0001)--(0.0475,2.0011)--(0.1108,2.0061)--(0.1736,2.0151)--(0.2357,2.0281)--(0.2969,2.0450)--(0.3568,2.0658)--(0.4154,2.0903)--(0.4722,2.1185)--(0.5272,2.1502)--(0.5800,2.1854)--(0.6305,2.2238)--(0.6785,2.2654)--(0.7237,2.3099)--(0.7660,2.3572)--(0.8052,2.4070)--(0.8412,2.4593)--(0.8738,2.5138)--(0.9029,2.5702)--(0.9283,2.6283)--(0.9500,2.6879)--(0.9679,2.7488)--(0.9819,2.8107)--(0.9919,2.8734)--(0.9979,2.9365)--(1.0000,3.0000);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0.0000);
 \draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
-\draw (1.8965,1.5000) node {$a$};
-\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
-\draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.14303,3.6345) node {$R$};
-\draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
+\draw (1.8964,1.5000) node {$a$};
+\draw []  (0.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw [color=red] (0.0000,3.0000) -- (0.8660,3.5000);
+\draw (0.1430,3.6345) node {$R$};
+\draw [color=blue,style=dotted] (0.0000,3.0000) -- (1.5000,3.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_RQsQKTl.pstricks.recall b/src_phystricks/Fig_RQsQKTl.pstricks.recall
index adc854ce0..35aef726a 100644
--- a/src_phystricks/Fig_RQsQKTl.pstricks.recall
+++ b/src_phystricks/Fig_RQsQKTl.pstricks.recall
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.31492) node {\( 1\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3149) node {\( 1\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_Refraction.pstricks.recall b/src_phystricks/Fig_Refraction.pstricks.recall
index 9e1e8d206..357938ef3 100644
--- a/src_phystricks/Fig_Refraction.pstricks.recall
+++ b/src_phystricks/Fig_Refraction.pstricks.recall
@@ -77,17 +77,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-3.00,0) -- (3.00,0);
-\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
-\draw (0.46653,2.0000) node {$\overline{ N }$};
-\draw (0.35618,0.94573) node {$\theta_1$};
+\draw [] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0000) -- (0.0000,2.0000);
+\draw (0.4665,2.0000) node {$\overline{ N }$};
+\draw (0.3561,0.9457) node {$\theta_1$};
 
-\draw [] (0.354,0.354)--(0.351,0.356)--(0.348,0.359)--(0.345,0.362)--(0.342,0.365)--(0.339,0.367)--(0.336,0.370)--(0.333,0.373)--(0.330,0.375)--(0.327,0.378)--(0.324,0.380)--(0.321,0.383)--(0.318,0.386)--(0.315,0.388)--(0.312,0.391)--(0.309,0.393)--(0.306,0.395)--(0.303,0.398)--(0.300,0.400)--(0.296,0.403)--(0.293,0.405)--(0.290,0.407)--(0.287,0.410)--(0.284,0.412)--(0.280,0.414)--(0.277,0.416)--(0.274,0.418)--(0.270,0.421)--(0.267,0.423)--(0.264,0.425)--(0.260,0.427)--(0.257,0.429)--(0.253,0.431)--(0.250,0.433)--(0.247,0.435)--(0.243,0.437)--(0.240,0.439)--(0.236,0.441)--(0.233,0.443)--(0.229,0.444)--(0.226,0.446)--(0.222,0.448)--(0.218,0.450)--(0.215,0.451)--(0.211,0.453)--(0.208,0.455)--(0.204,0.456)--(0.200,0.458)--(0.197,0.460)--(0.193,0.461)--(0.190,0.463)--(0.186,0.464)--(0.182,0.466)--(0.178,0.467)--(0.175,0.468)--(0.171,0.470)--(0.167,0.471)--(0.164,0.473)--(0.160,0.474)--(0.156,0.475)--(0.152,0.476)--(0.148,0.477)--(0.145,0.479)--(0.141,0.480)--(0.137,0.481)--(0.133,0.482)--(0.129,0.483)--(0.126,0.484)--(0.122,0.485)--(0.118,0.486)--(0.114,0.487)--(0.110,0.488)--(0.106,0.489)--(0.102,0.489)--(0.0985,0.490)--(0.0946,0.491)--(0.0907,0.492)--(0.0868,0.492)--(0.0829,0.493)--(0.0790,0.494)--(0.0751,0.494)--(0.0712,0.495)--(0.0672,0.495)--(0.0633,0.496)--(0.0594,0.496)--(0.0554,0.497)--(0.0515,0.497)--(0.0475,0.498)--(0.0436,0.498)--(0.0396,0.498)--(0.0357,0.499)--(0.0317,0.499)--(0.0278,0.499)--(0.0238,0.499)--(0.0198,0.500)--(0.0159,0.500)--(0.0119,0.500)--(0.00793,0.500)--(0.00397,0.500)--(0,0.500);
-\draw (-0.42770,-0.74428) node {$\theta_2$};
+\draw [] (0.3535,0.3535)--(0.3507,0.3563)--(0.3478,0.3591)--(0.3450,0.3618)--(0.3421,0.3645)--(0.3392,0.3672)--(0.3363,0.3699)--(0.3333,0.3726)--(0.3304,0.3752)--(0.3274,0.3778)--(0.3244,0.3804)--(0.3213,0.3830)--(0.3183,0.3855)--(0.3152,0.3880)--(0.3121,0.3905)--(0.3090,0.3930)--(0.3059,0.3954)--(0.3028,0.3978)--(0.2996,0.4002)--(0.2964,0.4026)--(0.2932,0.4049)--(0.2900,0.4072)--(0.2867,0.4095)--(0.2835,0.4118)--(0.2802,0.4140)--(0.2769,0.4162)--(0.2736,0.4184)--(0.2703,0.4206)--(0.2669,0.4227)--(0.2636,0.4248)--(0.2602,0.4269)--(0.2568,0.4289)--(0.2534,0.4310)--(0.2500,0.4330)--(0.2465,0.4349)--(0.2430,0.4369)--(0.2396,0.4388)--(0.2361,0.4407)--(0.2326,0.4425)--(0.2291,0.4444)--(0.2255,0.4462)--(0.2220,0.4479)--(0.2184,0.4497)--(0.2148,0.4514)--(0.2113,0.4531)--(0.2077,0.4548)--(0.2040,0.4564)--(0.2004,0.4580)--(0.1968,0.4596)--(0.1931,0.4611)--(0.1895,0.4626)--(0.1858,0.4641)--(0.1821,0.4656)--(0.1784,0.4670)--(0.1747,0.4684)--(0.1710,0.4698)--(0.1672,0.4711)--(0.1635,0.4725)--(0.1597,0.4737)--(0.1560,0.4750)--(0.1522,0.4762)--(0.1484,0.4774)--(0.1446,0.4786)--(0.1408,0.4797)--(0.1370,0.4808)--(0.1332,0.4819)--(0.1294,0.4829)--(0.1255,0.4839)--(0.1217,0.4849)--(0.1178,0.4859)--(0.1140,0.4868)--(0.1101,0.4877)--(0.1062,0.4885)--(0.1024,0.4894)--(0.0985,0.4901)--(0.0946,0.4909)--(0.0907,0.4916)--(0.0868,0.4924)--(0.0829,0.4930)--(0.0790,0.4937)--(0.0750,0.4943)--(0.0711,0.4949)--(0.0672,0.4954)--(0.0632,0.4959)--(0.0593,0.4964)--(0.0554,0.4969)--(0.0514,0.4973)--(0.0475,0.4977)--(0.0435,0.4980)--(0.0396,0.4984)--(0.0356,0.4987)--(0.0317,0.4989)--(0.0277,0.4992)--(0.0237,0.4994)--(0.0198,0.4996)--(0.0158,0.4997)--(0.0118,0.4998)--(0.0079,0.4999)--(0.0039,0.4999)--(0.0000,0.5000);
+\draw (-0.4277,-0.7442) node {$\theta_2$};
 
-\draw [] (-0.447,-0.224)--(-0.445,-0.229)--(-0.442,-0.234)--(-0.439,-0.238)--(-0.437,-0.243)--(-0.434,-0.248)--(-0.431,-0.253)--(-0.428,-0.258)--(-0.425,-0.263)--(-0.422,-0.267)--(-0.419,-0.272)--(-0.416,-0.277)--(-0.413,-0.281)--(-0.410,-0.286)--(-0.407,-0.291)--(-0.404,-0.295)--(-0.400,-0.300)--(-0.397,-0.304)--(-0.393,-0.309)--(-0.390,-0.313)--(-0.386,-0.317)--(-0.383,-0.322)--(-0.379,-0.326)--(-0.376,-0.330)--(-0.372,-0.334)--(-0.368,-0.338)--(-0.364,-0.342)--(-0.360,-0.346)--(-0.357,-0.350)--(-0.353,-0.354)--(-0.349,-0.358)--(-0.345,-0.362)--(-0.341,-0.366)--(-0.336,-0.370)--(-0.332,-0.374)--(-0.328,-0.377)--(-0.324,-0.381)--(-0.320,-0.385)--(-0.315,-0.388)--(-0.311,-0.392)--(-0.306,-0.395)--(-0.302,-0.398)--(-0.298,-0.402)--(-0.293,-0.405)--(-0.289,-0.408)--(-0.284,-0.412)--(-0.279,-0.415)--(-0.275,-0.418)--(-0.270,-0.421)--(-0.265,-0.424)--(-0.260,-0.427)--(-0.256,-0.430)--(-0.251,-0.432)--(-0.246,-0.435)--(-0.241,-0.438)--(-0.236,-0.441)--(-0.231,-0.443)--(-0.226,-0.446)--(-0.221,-0.448)--(-0.216,-0.451)--(-0.211,-0.453)--(-0.206,-0.456)--(-0.201,-0.458)--(-0.196,-0.460)--(-0.191,-0.462)--(-0.186,-0.464)--(-0.180,-0.466)--(-0.175,-0.468)--(-0.170,-0.470)--(-0.165,-0.472)--(-0.159,-0.474)--(-0.154,-0.476)--(-0.149,-0.477)--(-0.143,-0.479)--(-0.138,-0.481)--(-0.133,-0.482)--(-0.127,-0.484)--(-0.122,-0.485)--(-0.116,-0.486)--(-0.111,-0.488)--(-0.105,-0.489)--(-0.100,-0.490)--(-0.0945,-0.491)--(-0.0890,-0.492)--(-0.0835,-0.493)--(-0.0780,-0.494)--(-0.0724,-0.495)--(-0.0669,-0.496)--(-0.0614,-0.496)--(-0.0558,-0.497)--(-0.0502,-0.497)--(-0.0447,-0.498)--(-0.0391,-0.498)--(-0.0335,-0.499)--(-0.0279,-0.499)--(-0.0224,-0.500)--(-0.0168,-0.500)--(-0.0112,-0.500)--(-0.00559,-0.500)--(0,-0.500);
-\draw [color=red,->,>=latex] (1.0000,1.0000) -- (0,0);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.2649,-0.63246);
+\draw [] (-0.4472,-0.2236)--(-0.4446,-0.2285)--(-0.4421,-0.2335)--(-0.4394,-0.2384)--(-0.4367,-0.2433)--(-0.4340,-0.2482)--(-0.4312,-0.2530)--(-0.4283,-0.2578)--(-0.4254,-0.2626)--(-0.4224,-0.2674)--(-0.4194,-0.2721)--(-0.4163,-0.2767)--(-0.4132,-0.2814)--(-0.4101,-0.2860)--(-0.4068,-0.2906)--(-0.4036,-0.2951)--(-0.4002,-0.2996)--(-0.3969,-0.3040)--(-0.3934,-0.3085)--(-0.3899,-0.3128)--(-0.3864,-0.3172)--(-0.3829,-0.3215)--(-0.3792,-0.3257)--(-0.3756,-0.3300)--(-0.3719,-0.3341)--(-0.3681,-0.3383)--(-0.3643,-0.3424)--(-0.3604,-0.3464)--(-0.3565,-0.3504)--(-0.3526,-0.3544)--(-0.3486,-0.3583)--(-0.3446,-0.3622)--(-0.3405,-0.3660)--(-0.3364,-0.3698)--(-0.3322,-0.3736)--(-0.3280,-0.3773)--(-0.3238,-0.3809)--(-0.3195,-0.3845)--(-0.3152,-0.3880)--(-0.3108,-0.3916)--(-0.3064,-0.3950)--(-0.3020,-0.3984)--(-0.2975,-0.4018)--(-0.2930,-0.4051)--(-0.2885,-0.4083)--(-0.2839,-0.4115)--(-0.2793,-0.4147)--(-0.2746,-0.4178)--(-0.2699,-0.4208)--(-0.2652,-0.4238)--(-0.2604,-0.4267)--(-0.2556,-0.4296)--(-0.2508,-0.4325)--(-0.2460,-0.4352)--(-0.2411,-0.4380)--(-0.2362,-0.4406)--(-0.2312,-0.4432)--(-0.2263,-0.4458)--(-0.2213,-0.4483)--(-0.2162,-0.4508)--(-0.2112,-0.4531)--(-0.2061,-0.4555)--(-0.2010,-0.4578)--(-0.1959,-0.4600)--(-0.1907,-0.4621)--(-0.1855,-0.4642)--(-0.1803,-0.4663)--(-0.1751,-0.4683)--(-0.1698,-0.4702)--(-0.1646,-0.4721)--(-0.1593,-0.4739)--(-0.1540,-0.4756)--(-0.1486,-0.4773)--(-0.1433,-0.4790)--(-0.1379,-0.4805)--(-0.1325,-0.4820)--(-0.1271,-0.4835)--(-0.1217,-0.4849)--(-0.1163,-0.4862)--(-0.1109,-0.4875)--(-0.1054,-0.4887)--(-0.0999,-0.4899)--(-0.0944,-0.4909)--(-0.0889,-0.4920)--(-0.0834,-0.4929)--(-0.0779,-0.4938)--(-0.0724,-0.4947)--(-0.0668,-0.4955)--(-0.0613,-0.4962)--(-0.0558,-0.4968)--(-0.0502,-0.4974)--(-0.0446,-0.4980)--(-0.0391,-0.4984)--(-0.0335,-0.4988)--(-0.0279,-0.4992)--(-0.0223,-0.4994)--(-0.0167,-0.4997)--(-0.0111,-0.4998)--(-0.0055,-0.4999)--(0.0000,-0.5000);
+\draw [color=red,->,>=latex] (1.0000,1.0000) -- (0.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.2649,-0.6324);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_SBTooEasQsT.pstricks.recall b/src_phystricks/Fig_SBTooEasQsT.pstricks.recall
index a50ca2224..c29caad53 100644
--- a/src_phystricks/Fig_SBTooEasQsT.pstricks.recall
+++ b/src_phystricks/Fig_SBTooEasQsT.pstricks.recall
@@ -118,115 +118,115 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-4.00,-4.00) -- (-4.00,4.00);
-\draw [color=gray,style=solid] (-3.00,-4.00) -- (-3.00,4.00);
-\draw [color=gray,style=solid] (-2.00,-4.00) -- (-2.00,4.00);
-\draw [color=gray,style=solid] (-1.00,-4.00) -- (-1.00,4.00);
-\draw [color=gray,style=solid] (0,-4.00) -- (0,4.00);
-\draw [color=gray,style=solid] (1.00,-4.00) -- (1.00,4.00);
-\draw [color=gray,style=solid] (2.00,-4.00) -- (2.00,4.00);
-\draw [color=gray,style=dotted] (-3.50,-4.00) -- (-3.50,4.00);
-\draw [color=gray,style=dotted] (-2.50,-4.00) -- (-2.50,4.00);
-\draw [color=gray,style=dotted] (-1.50,-4.00) -- (-1.50,4.00);
-\draw [color=gray,style=dotted] (-0.500,-4.00) -- (-0.500,4.00);
-\draw [color=gray,style=dotted] (0.500,-4.00) -- (0.500,4.00);
-\draw [color=gray,style=dotted] (1.50,-4.00) -- (1.50,4.00);
-\draw [color=gray,style=dotted] (-4.00,-3.75) -- (2.00,-3.75);
-\draw [color=gray,style=dotted] (-4.00,-3.25) -- (2.00,-3.25);
-\draw [color=gray,style=dotted] (-4.00,-2.75) -- (2.00,-2.75);
-\draw [color=gray,style=dotted] (-4.00,-2.25) -- (2.00,-2.25);
-\draw [color=gray,style=dotted] (-4.00,-1.75) -- (2.00,-1.75);
-\draw [color=gray,style=dotted] (-4.00,-1.25) -- (2.00,-1.25);
-\draw [color=gray,style=dotted] (-4.00,-0.750) -- (2.00,-0.750);
-\draw [color=gray,style=dotted] (-4.00,-0.250) -- (2.00,-0.250);
-\draw [color=gray,style=dotted] (-4.00,0.250) -- (2.00,0.250);
-\draw [color=gray,style=dotted] (-4.00,0.750) -- (2.00,0.750);
-\draw [color=gray,style=dotted] (-4.00,1.25) -- (2.00,1.25);
-\draw [color=gray,style=dotted] (-4.00,1.75) -- (2.00,1.75);
-\draw [color=gray,style=dotted] (-4.00,2.25) -- (2.00,2.25);
-\draw [color=gray,style=dotted] (-4.00,2.75) -- (2.00,2.75);
-\draw [color=gray,style=dotted] (-4.00,3.25) -- (2.00,3.25);
-\draw [color=gray,style=dotted] (-4.00,3.75) -- (2.00,3.75);
-\draw [color=gray,style=solid] (-4.00,-4.00) -- (2.00,-4.00);
-\draw [color=gray,style=solid] (-4.00,-3.50) -- (2.00,-3.50);
-\draw [color=gray,style=solid] (-4.00,-3.00) -- (2.00,-3.00);
-\draw [color=gray,style=solid] (-4.00,-2.50) -- (2.00,-2.50);
-\draw [color=gray,style=solid] (-4.00,-2.00) -- (2.00,-2.00);
-\draw [color=gray,style=solid] (-4.00,-1.50) -- (2.00,-1.50);
-\draw [color=gray,style=solid] (-4.00,-1.00) -- (2.00,-1.00);
-\draw [color=gray,style=solid] (-4.00,-0.500) -- (2.00,-0.500);
-\draw [color=gray,style=solid] (-4.00,0) -- (2.00,0);
-\draw [color=gray,style=solid] (-4.00,0.500) -- (2.00,0.500);
-\draw [color=gray,style=solid] (-4.00,1.00) -- (2.00,1.00);
-\draw [color=gray,style=solid] (-4.00,1.50) -- (2.00,1.50);
-\draw [color=gray,style=solid] (-4.00,2.00) -- (2.00,2.00);
-\draw [color=gray,style=solid] (-4.00,2.50) -- (2.00,2.50);
-\draw [color=gray,style=solid] (-4.00,3.00) -- (2.00,3.00);
-\draw [color=gray,style=solid] (-4.00,3.50) -- (2.00,3.50);
-\draw [color=gray,style=solid] (-4.00,4.00) -- (2.00,4.00);
+\draw [color=gray,style=solid] (-4.0000,-4.0000) -- (-4.0000,4.0000);
+\draw [color=gray,style=solid] (-3.0000,-4.0000) -- (-3.0000,4.0000);
+\draw [color=gray,style=solid] (-2.0000,-4.0000) -- (-2.0000,4.0000);
+\draw [color=gray,style=solid] (-1.0000,-4.0000) -- (-1.0000,4.0000);
+\draw [color=gray,style=solid] (0.0000,-4.0000) -- (0.0000,4.0000);
+\draw [color=gray,style=solid] (1.0000,-4.0000) -- (1.0000,4.0000);
+\draw [color=gray,style=solid] (2.0000,-4.0000) -- (2.0000,4.0000);
+\draw [color=gray,style=dotted] (-3.5000,-4.0000) -- (-3.5000,4.0000);
+\draw [color=gray,style=dotted] (-2.5000,-4.0000) -- (-2.5000,4.0000);
+\draw [color=gray,style=dotted] (-1.5000,-4.0000) -- (-1.5000,4.0000);
+\draw [color=gray,style=dotted] (-0.5000,-4.0000) -- (-0.5000,4.0000);
+\draw [color=gray,style=dotted] (0.5000,-4.0000) -- (0.5000,4.0000);
+\draw [color=gray,style=dotted] (1.5000,-4.0000) -- (1.5000,4.0000);
+\draw [color=gray,style=dotted] (-4.0000,-3.7500) -- (2.0000,-3.7500);
+\draw [color=gray,style=dotted] (-4.0000,-3.2500) -- (2.0000,-3.2500);
+\draw [color=gray,style=dotted] (-4.0000,-2.7500) -- (2.0000,-2.7500);
+\draw [color=gray,style=dotted] (-4.0000,-2.2500) -- (2.0000,-2.2500);
+\draw [color=gray,style=dotted] (-4.0000,-1.7500) -- (2.0000,-1.7500);
+\draw [color=gray,style=dotted] (-4.0000,-1.2500) -- (2.0000,-1.2500);
+\draw [color=gray,style=dotted] (-4.0000,-0.7500) -- (2.0000,-0.7500);
+\draw [color=gray,style=dotted] (-4.0000,-0.2500) -- (2.0000,-0.2500);
+\draw [color=gray,style=dotted] (-4.0000,0.2500) -- (2.0000,0.2500);
+\draw [color=gray,style=dotted] (-4.0000,0.7500) -- (2.0000,0.7500);
+\draw [color=gray,style=dotted] (-4.0000,1.2500) -- (2.0000,1.2500);
+\draw [color=gray,style=dotted] (-4.0000,1.7500) -- (2.0000,1.7500);
+\draw [color=gray,style=dotted] (-4.0000,2.2500) -- (2.0000,2.2500);
+\draw [color=gray,style=dotted] (-4.0000,2.7500) -- (2.0000,2.7500);
+\draw [color=gray,style=dotted] (-4.0000,3.2500) -- (2.0000,3.2500);
+\draw [color=gray,style=dotted] (-4.0000,3.7500) -- (2.0000,3.7500);
+\draw [color=gray,style=solid] (-4.0000,-4.0000) -- (2.0000,-4.0000);
+\draw [color=gray,style=solid] (-4.0000,-3.5000) -- (2.0000,-3.5000);
+\draw [color=gray,style=solid] (-4.0000,-3.0000) -- (2.0000,-3.0000);
+\draw [color=gray,style=solid] (-4.0000,-2.5000) -- (2.0000,-2.5000);
+\draw [color=gray,style=solid] (-4.0000,-2.0000) -- (2.0000,-2.0000);
+\draw [color=gray,style=solid] (-4.0000,-1.5000) -- (2.0000,-1.5000);
+\draw [color=gray,style=solid] (-4.0000,-1.0000) -- (2.0000,-1.0000);
+\draw [color=gray,style=solid] (-4.0000,-0.5000) -- (2.0000,-0.5000);
+\draw [color=gray,style=solid] (-4.0000,0.0000) -- (2.0000,0.0000);
+\draw [color=gray,style=solid] (-4.0000,0.5000) -- (2.0000,0.5000);
+\draw [color=gray,style=solid] (-4.0000,1.0000) -- (2.0000,1.0000);
+\draw [color=gray,style=solid] (-4.0000,1.5000) -- (2.0000,1.5000);
+\draw [color=gray,style=solid] (-4.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=gray,style=solid] (-4.0000,2.5000) -- (2.0000,2.5000);
+\draw [color=gray,style=solid] (-4.0000,3.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (-4.0000,3.5000) -- (2.0000,3.5000);
+\draw [color=gray,style=solid] (-4.0000,4.0000) -- (2.0000,4.0000);
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,4.5000);
 %DEFAULT
 
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-\draw [color=blue] (-4.0000,0.0091578)--(-3.9394,0.0097300)--(-3.8788,0.010338)--(-3.8182,0.010984)--(-3.7576,0.011670)--(-3.6970,0.012399)--(-3.6364,0.013174)--(-3.5758,0.013997)--(-3.5152,0.014872)--(-3.4545,0.015801)--(-3.3939,0.016788)--(-3.3333,0.017837)--(-3.2727,0.018951)--(-3.2121,0.020136)--(-3.1515,0.021394)--(-3.0909,0.022730)--(-3.0303,0.024150)--(-2.9697,0.025659)--(-2.9091,0.027263)--(-2.8485,0.028966)--(-2.7879,0.030776)--(-2.7273,0.032699)--(-2.6667,0.034742)--(-2.6061,0.036912)--(-2.5455,0.039219)--(-2.4848,0.041669)--(-2.4242,0.044273)--(-2.3636,0.047039)--(-2.3030,0.049978)--(-2.2424,0.053100)--(-2.1818,0.056418)--(-2.1212,0.059943)--(-2.0606,0.063688)--(-2.0000,0.067668)--(-1.9394,0.071895)--(-1.8788,0.076388)--(-1.8182,0.081160)--(-1.7576,0.086231)--(-1.6970,0.091619)--(-1.6364,0.097343)--(-1.5758,0.10343)--(-1.5152,0.10989)--(-1.4545,0.11675)--(-1.3939,0.12405)--(-1.3333,0.13180)--(-1.2727,0.14003)--(-1.2121,0.14878)--(-1.1515,0.15808)--(-1.0909,0.16796)--(-1.0303,0.17845)--(-0.96970,0.18960)--(-0.90909,0.20145)--(-0.84848,0.21403)--(-0.78788,0.22740)--(-0.72727,0.24161)--(-0.66667,0.25671)--(-0.60606,0.27275)--(-0.54545,0.28979)--(-0.48485,0.30790)--(-0.42424,0.32713)--(-0.36364,0.34757)--(-0.30303,0.36929)--(-0.24242,0.39236)--(-0.18182,0.41688)--(-0.12121,0.44292)--(-0.060606,0.47060)--(0,0.50000)--(0.060606,0.53124)--(0.12121,0.56443)--(0.18182,0.59970)--(0.24242,0.63717)--(0.30303,0.67698)--(0.36364,0.71928)--(0.42424,0.76422)--(0.48485,0.81196)--(0.54545,0.86270)--(0.60606,0.91660)--(0.66667,0.97387)--(0.72727,1.0347)--(0.78788,1.0994)--(0.84848,1.1681)--(0.90909,1.2410)--(0.96970,1.3186)--(1.0303,1.4010)--(1.0909,1.4885)--(1.1515,1.5815)--(1.2121,1.6803)--(1.2727,1.7853)--(1.3333,1.8968)--(1.3939,2.0154)--(1.4545,2.1413)--(1.5152,2.2751)--(1.5758,2.4172)--(1.6364,2.5682)--(1.6970,2.7287)--(1.7576,2.8992)--(1.8182,3.0803)--(1.8788,3.2728)--(1.9394,3.4773)--(2.0000,3.6945);
+\draw [color=blue] (-4.0000,0.0091)--(-3.9393,0.0097)--(-3.8787,0.0103)--(-3.8181,0.0109)--(-3.7575,0.0116)--(-3.6969,0.0123)--(-3.6363,0.0131)--(-3.5757,0.0139)--(-3.5151,0.0148)--(-3.4545,0.0158)--(-3.3939,0.0167)--(-3.3333,0.0178)--(-3.2727,0.0189)--(-3.2121,0.0201)--(-3.1515,0.0213)--(-3.0909,0.0227)--(-3.0303,0.0241)--(-2.9696,0.0256)--(-2.9090,0.0272)--(-2.8484,0.0289)--(-2.7878,0.0307)--(-2.7272,0.0326)--(-2.6666,0.0347)--(-2.6060,0.0369)--(-2.5454,0.0392)--(-2.4848,0.0416)--(-2.4242,0.0442)--(-2.3636,0.0470)--(-2.3030,0.0499)--(-2.2424,0.0531)--(-2.1818,0.0564)--(-2.1212,0.0599)--(-2.0606,0.0636)--(-2.0000,0.0676)--(-1.9393,0.0718)--(-1.8787,0.0763)--(-1.8181,0.0811)--(-1.7575,0.0862)--(-1.6969,0.0916)--(-1.6363,0.0973)--(-1.5757,0.1034)--(-1.5151,0.1098)--(-1.4545,0.1167)--(-1.3939,0.1240)--(-1.3333,0.1317)--(-1.2727,0.1400)--(-1.2121,0.1487)--(-1.1515,0.1580)--(-1.0909,0.1679)--(-1.0303,0.1784)--(-0.9696,0.1895)--(-0.9090,0.2014)--(-0.8484,0.2140)--(-0.7878,0.2274)--(-0.7272,0.2416)--(-0.6666,0.2567)--(-0.6060,0.2727)--(-0.5454,0.2897)--(-0.4848,0.3078)--(-0.4242,0.3271)--(-0.3636,0.3475)--(-0.3030,0.3692)--(-0.2424,0.3923)--(-0.1818,0.4168)--(-0.1212,0.4429)--(-0.0606,0.4705)--(0.0000,0.5000)--(0.0606,0.5312)--(0.1212,0.5644)--(0.1818,0.5996)--(0.2424,0.6371)--(0.3030,0.6769)--(0.3636,0.7192)--(0.4242,0.7642)--(0.4848,0.8119)--(0.5454,0.8626)--(0.6060,0.9165)--(0.6666,0.9738)--(0.7272,1.0347)--(0.7878,1.0993)--(0.8484,1.1680)--(0.9090,1.2410)--(0.9696,1.3185)--(1.0303,1.4009)--(1.0909,1.4884)--(1.1515,1.5814)--(1.2121,1.6803)--(1.2727,1.7852)--(1.3333,1.8968)--(1.3939,2.0153)--(1.4545,2.1412)--(1.5151,2.2750)--(1.5757,2.4172)--(1.6363,2.5682)--(1.6969,2.7286)--(1.7575,2.8991)--(1.8181,3.0803)--(1.8787,3.2727)--(1.9393,3.4772)--(2.0000,3.6945);
 
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.52441,-4.0000) node {$ -40 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.52441,-3.0000) node {$ -30 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.52441,-2.0000) node {$ -20 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.52441,-1.0000) node {$ -10 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.38250,1.0000) node {$ 10 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.38250,2.0000) node {$ 20 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.38250,3.0000) node {$ 30 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.38250,4.0000) node {$ 40 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.5244,-4.0000) node {$ -40 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.5244,-3.0000) node {$ -30 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.5244,-2.0000) node {$ -20 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.5244,-1.0000) node {$ -10 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.3824,1.0000) node {$ 10 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.3824,2.0000) node {$ 20 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.3824,3.0000) node {$ 30 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.3824,4.0000) node {$ 40 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall b/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall
index 8f3b3168c..68f960673 100644
--- a/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall
+++ b/src_phystricks/Fig_SQNPooPTrLRQ.pstricks.recall
@@ -65,228 +65,228 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.0000,-4.0000) -- (-4.0442,-4.0442);
-\draw [,->,>=latex] (-4.0000,-3.4286) -- (-4.0547,-3.4755);
-\draw [,->,>=latex] (-4.0000,-2.8571) -- (-4.0674,-2.9053);
-\draw [,->,>=latex] (-4.0000,-2.2857) -- (-4.0818,-2.3325);
-\draw [,->,>=latex] (-4.0000,-1.7143) -- (-4.0971,-1.7559);
-\draw [,->,>=latex] (-4.0000,-1.1429) -- (-4.1111,-1.1746);
-\draw [,->,>=latex] (-4.0000,-0.57143) -- (-4.1213,-0.58875);
-\draw [,->,>=latex] (-4.0000,0) -- (-4.1250,0);
-\draw [,->,>=latex] (-4.0000,0.57143) -- (-4.1213,0.58875);
-\draw [,->,>=latex] (-4.0000,1.1429) -- (-4.1111,1.1746);
-\draw [,->,>=latex] (-4.0000,1.7143) -- (-4.0971,1.7559);
-\draw [,->,>=latex] (-4.0000,2.2857) -- (-4.0818,2.3325);
-\draw [,->,>=latex] (-4.0000,2.8571) -- (-4.0674,2.9053);
-\draw [,->,>=latex] (-4.0000,3.4286) -- (-4.0547,3.4755);
-\draw [,->,>=latex] (-4.0000,4.0000) -- (-4.0442,4.0442);
-\draw [,->,>=latex] (-3.4286,-4.0000) -- (-3.4755,-4.0547);
-\draw [,->,>=latex] (-3.4286,-3.4286) -- (-3.4887,-3.4887);
-\draw [,->,>=latex] (-3.4286,-2.8571) -- (-3.5057,-2.9214);
-\draw [,->,>=latex] (-3.4286,-2.2857) -- (-3.5266,-2.3511);
-\draw [,->,>=latex] (-3.4286,-1.7143) -- (-3.5503,-1.7752);
-\draw [,->,>=latex] (-3.4286,-1.1429) -- (-3.5738,-1.1913);
-\draw [,->,>=latex] (-3.4286,-0.57143) -- (-3.5919,-0.59864);
-\draw [,->,>=latex] (-3.4286,0) -- (-3.5987,0);
-\draw [,->,>=latex] (-3.4286,0.57143) -- (-3.5919,0.59864);
-\draw [,->,>=latex] (-3.4286,1.1429) -- (-3.5738,1.1913);
-\draw [,->,>=latex] (-3.4286,1.7143) -- (-3.5503,1.7752);
-\draw [,->,>=latex] (-3.4286,2.2857) -- (-3.5266,2.3511);
-\draw [,->,>=latex] (-3.4286,2.8571) -- (-3.5057,2.9214);
-\draw [,->,>=latex] (-3.4286,3.4286) -- (-3.4887,3.4887);
-\draw [,->,>=latex] (-3.4286,4.0000) -- (-3.4755,4.0547);
-\draw [,->,>=latex] (-2.8571,-4.0000) -- (-2.9053,-4.0674);
-\draw [,->,>=latex] (-2.8571,-3.4286) -- (-2.9214,-3.5057);
-\draw [,->,>=latex] (-2.8571,-2.8571) -- (-2.9438,-2.9438);
-\draw [,->,>=latex] (-2.8571,-2.2857) -- (-2.9738,-2.3790);
-\draw [,->,>=latex] (-2.8571,-1.7143) -- (-3.0116,-1.8070);
-\draw [,->,>=latex] (-2.8571,-1.1429) -- (-3.0532,-1.2213);
-\draw [,->,>=latex] (-2.8571,-0.57143) -- (-3.0881,-0.61763);
-\draw [,->,>=latex] (-2.8571,0) -- (-3.1021,0);
-\draw [,->,>=latex] (-2.8571,0.57143) -- (-3.0881,0.61763);
-\draw [,->,>=latex] (-2.8571,1.1429) -- (-3.0532,1.2213);
-\draw [,->,>=latex] (-2.8571,1.7143) -- (-3.0116,1.8070);
-\draw [,->,>=latex] (-2.8571,2.2857) -- (-2.9738,2.3790);
-\draw [,->,>=latex] (-2.8571,2.8571) -- (-2.9438,2.9438);
-\draw [,->,>=latex] (-2.8571,3.4286) -- (-2.9214,3.5057);
-\draw [,->,>=latex] (-2.8571,4.0000) -- (-2.9053,4.0674);
-\draw [,->,>=latex] (-2.2857,-4.0000) -- (-2.3325,-4.0818);
-\draw [,->,>=latex] (-2.2857,-3.4286) -- (-2.3511,-3.5266);
-\draw [,->,>=latex] (-2.2857,-2.8571) -- (-2.3790,-2.9738);
-\draw [,->,>=latex] (-2.2857,-2.2857) -- (-2.4211,-2.4211);
-\draw [,->,>=latex] (-2.2857,-1.7143) -- (-2.4817,-1.8613);
-\draw [,->,>=latex] (-2.2857,-1.1429) -- (-2.5596,-1.2798);
-\draw [,->,>=latex] (-2.2857,-0.57143) -- (-2.6353,-0.65881);
-\draw [,->,>=latex] (-2.2857,0) -- (-2.6685,0);
-\draw [,->,>=latex] (-2.2857,0.57143) -- (-2.6353,0.65881);
-\draw [,->,>=latex] (-2.2857,1.1429) -- (-2.5596,1.2798);
-\draw [,->,>=latex] (-2.2857,1.7143) -- (-2.4817,1.8613);
-\draw [,->,>=latex] (-2.2857,2.2857) -- (-2.4211,2.4211);
-\draw [,->,>=latex] (-2.2857,2.8571) -- (-2.3790,2.9738);
-\draw [,->,>=latex] (-2.2857,3.4286) -- (-2.3511,3.5266);
-\draw [,->,>=latex] (-2.2857,4.0000) -- (-2.3325,4.0818);
-\draw [,->,>=latex] (-1.7143,-4.0000) -- (-1.7559,-4.0971);
-\draw [,->,>=latex] (-1.7143,-3.4286) -- (-1.7752,-3.5503);
-\draw [,->,>=latex] (-1.7143,-2.8571) -- (-1.8070,-3.0116);
-\draw [,->,>=latex] (-1.7143,-2.2857) -- (-1.8613,-2.4817);
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_STdyNTH.pstricks.recall b/src_phystricks/Fig_STdyNTH.pstricks.recall
index b283a16d4..43e8bc3e2 100644
--- a/src_phystricks/Fig_STdyNTH.pstricks.recall
+++ b/src_phystricks/Fig_STdyNTH.pstricks.recall
@@ -70,20 +70,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.20000) node {};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.20000) node {};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.20000) node {};
-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.20000) node {};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.31492) node {$1$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2000) node {};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2000) node {};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.2000) node {};
+\draw []  (3.0000,0.0000) node [rotate=0] {$o$};
+\draw (3.0000,0.2000) node {};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.3149) node {$1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall b/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall
index bb5921328..5229e3cbd 100644
--- a/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall
+++ b/src_phystricks/Fig_SYNKooZBuEWsWw.pstricks.recall
@@ -119,49 +119,49 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-3.5251) -- (0,3.5661);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5251) -- (0.0000,3.5661);
 %DEFAULT
 
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-\draw [color=blue] (-5.000,3.066)--(-4.899,2.917)--(-4.798,2.776)--(-4.697,2.641)--(-4.596,2.514)--(-4.495,2.392)--(-4.394,2.277)--(-4.293,2.168)--(-4.192,2.064)--(-4.091,1.965)--(-3.990,1.872)--(-3.889,1.783)--(-3.788,1.699)--(-3.687,1.619)--(-3.586,1.543)--(-3.485,1.472)--(-3.384,1.404)--(-3.283,1.339)--(-3.182,1.278)--(-3.081,1.220)--(-2.980,1.166)--(-2.879,1.114)--(-2.778,1.065)--(-2.677,1.019)--(-2.576,0.9752)--(-2.475,0.9342)--(-2.374,0.8955)--(-2.273,0.8591)--(-2.172,0.8249)--(-2.071,0.7928)--(-1.970,0.7627)--(-1.869,0.7346)--(-1.768,0.7083)--(-1.667,0.6839)--(-1.566,0.6612)--(-1.465,0.6402)--(-1.364,0.6208)--(-1.263,0.6030)--(-1.162,0.5867)--(-1.061,0.5720)--(-0.9596,0.5587)--(-0.8586,0.5468)--(-0.7576,0.5363)--(-0.6566,0.5272)--(-0.5556,0.5194)--(-0.4545,0.5130)--(-0.3535,0.5078)--(-0.2525,0.5040)--(-0.1515,0.5014)--(-0.05051,0.5002)--(0.05051,0.5002)--(0.1515,0.5014)--(0.2525,0.5040)--(0.3535,0.5078)--(0.4545,0.5130)--(0.5556,0.5194)--(0.6566,0.5272)--(0.7576,0.5363)--(0.8586,0.5468)--(0.9596,0.5587)--(1.061,0.5720)--(1.162,0.5867)--(1.263,0.6030)--(1.364,0.6208)--(1.465,0.6402)--(1.566,0.6612)--(1.667,0.6839)--(1.768,0.7083)--(1.869,0.7346)--(1.970,0.7627)--(2.071,0.7928)--(2.172,0.8249)--(2.273,0.8591)--(2.374,0.8955)--(2.475,0.9342)--(2.576,0.9752)--(2.677,1.019)--(2.778,1.065)--(2.879,1.114)--(2.980,1.166)--(3.081,1.220)--(3.182,1.278)--(3.283,1.339)--(3.384,1.404)--(3.485,1.472)--(3.586,1.543)--(3.687,1.619)--(3.788,1.699)--(3.889,1.783)--(3.990,1.872)--(4.091,1.965)--(4.192,2.064)--(4.293,2.168)--(4.394,2.277)--(4.495,2.392)--(4.596,2.514)--(4.697,2.641)--(4.798,2.776)--(4.899,2.917)--(5.000,3.066);
-\draw (-4.0000,-0.32983) node {$ -2 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -1 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.0000,-0.31492) node {$ 2 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.5000) node {$ -7 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-3.0000) node {$ -6 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.5000) node {$ -5 $};
-\draw [] (-0.100,-2.50) -- (0.100,-2.50);
-\draw (-0.43316,-2.0000) node {$ -4 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.5000) node {$ -3 $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.43316,-1.0000) node {$ -2 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.43316,-0.50000) node {$ -1 $};
-\draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.29125,0.50000) node {$ 1 $};
-\draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,1.5000) node {$ 3 $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,2.5000) node {$ 5 $};
-\draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,3.5000) node {$ 7 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=blue] (-5.0000,3.0661)--(-4.8989,2.9172)--(-4.7979,2.7757)--(-4.6969,2.6413)--(-4.5959,2.5136)--(-4.4949,2.3923)--(-4.3939,2.2772)--(-4.2929,2.1678)--(-4.1919,2.0640)--(-4.0909,1.9654)--(-3.9898,1.8719)--(-3.8888,1.7832)--(-3.7878,1.6989)--(-3.6868,1.6191)--(-3.5858,1.5433)--(-3.4848,1.4715)--(-3.3838,1.4035)--(-3.2828,1.3390)--(-3.1818,1.2779)--(-3.0808,1.2201)--(-2.9797,1.1655)--(-2.8787,1.1138)--(-2.7777,1.0649)--(-2.6767,1.0187)--(-2.5757,0.9752)--(-2.4747,0.9341)--(-2.3737,0.8954)--(-2.2727,0.8591)--(-2.1717,0.8248)--(-2.0707,0.7928)--(-1.9696,0.7627)--(-1.8686,0.7345)--(-1.7676,0.7083)--(-1.6666,0.6838)--(-1.5656,0.6611)--(-1.4646,0.6401)--(-1.3636,0.6207)--(-1.2626,0.6029)--(-1.1616,0.5867)--(-1.0606,0.5719)--(-0.9595,0.5586)--(-0.8585,0.5467)--(-0.7575,0.5363)--(-0.6565,0.5271)--(-0.5555,0.5194)--(-0.4545,0.5129)--(-0.3535,0.5078)--(-0.2525,0.5039)--(-0.1515,0.5014)--(-0.0505,0.5001)--(0.0505,0.5001)--(0.1515,0.5014)--(0.2525,0.5039)--(0.3535,0.5078)--(0.4545,0.5129)--(0.5555,0.5194)--(0.6565,0.5271)--(0.7575,0.5363)--(0.8585,0.5467)--(0.9595,0.5586)--(1.0606,0.5719)--(1.1616,0.5867)--(1.2626,0.6029)--(1.3636,0.6207)--(1.4646,0.6401)--(1.5656,0.6611)--(1.6666,0.6838)--(1.7676,0.7083)--(1.8686,0.7345)--(1.9696,0.7627)--(2.0707,0.7928)--(2.1717,0.8248)--(2.2727,0.8591)--(2.3737,0.8954)--(2.4747,0.9341)--(2.5757,0.9752)--(2.6767,1.0187)--(2.7777,1.0649)--(2.8787,1.1138)--(2.9797,1.1655)--(3.0808,1.2201)--(3.1818,1.2779)--(3.2828,1.3390)--(3.3838,1.4035)--(3.4848,1.4715)--(3.5858,1.5433)--(3.6868,1.6191)--(3.7878,1.6989)--(3.8888,1.7832)--(3.9898,1.8719)--(4.0909,1.9654)--(4.1919,2.0640)--(4.2929,2.1678)--(4.3939,2.2772)--(4.4949,2.3923)--(4.5959,2.5136)--(4.6969,2.6413)--(4.7979,2.7757)--(4.8989,2.9172)--(5.0000,3.0661);
+\draw (-4.0000,-0.3298) node {$ -2 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -1 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 2 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -7 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-3.0000) node {$ -6 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.5000) node {$ -5 $};
+\draw [] (-0.1000,-2.5000) -- (0.1000,-2.5000);
+\draw (-0.4331,-2.0000) node {$ -4 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.5000) node {$ -3 $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.4331,-1.0000) node {$ -2 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.4331,-0.5000) node {$ -1 $};
+\draw [] (-0.1000,-0.5000) -- (0.1000,-0.5000);
+\draw (-0.2912,0.5000) node {$ 1 $};
+\draw [] (-0.1000,0.5000) -- (0.1000,0.5000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,1.5000) node {$ 3 $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,2.5000) node {$ 5 $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,3.5000) node {$ 7 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall b/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall
index cb4d2a63a..229eea3ad 100644
--- a/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall
+++ b/src_phystricks/Fig_SolsEqDiffSin.pstricks.recall
@@ -79,23 +79,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0808,0) -- (2.0808,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-2.0807,0.0000) -- (2.0807,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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+\draw [color=red] (-1.5807,-0.9999)--(-1.5488,-0.9997)--(-1.5169,-0.9985)--(-1.4849,-0.9963)--(-1.4530,-0.9930)--(-1.4211,-0.9888)--(-1.3891,-0.9835)--(-1.3572,-0.9772)--(-1.3253,-0.9700)--(-1.2933,-0.9617)--(-1.2614,-0.9525)--(-1.2295,-0.9423)--(-1.1975,-0.9311)--(-1.1656,-0.9190)--(-1.1337,-0.9059)--(-1.1017,-0.8920)--(-1.0698,-0.8771)--(-1.0378,-0.8613)--(-1.0059,-0.8446)--(-0.9740,-0.8271)--(-0.9420,-0.8087)--(-0.9101,-0.7895)--(-0.8782,-0.7696)--(-0.8462,-0.7488)--(-0.8143,-0.7272)--(-0.7824,-0.7049)--(-0.7504,-0.6819)--(-0.7185,-0.6582)--(-0.6866,-0.6339)--(-0.6546,-0.6089)--(-0.6227,-0.5832)--(-0.5908,-0.5570)--(-0.5588,-0.5302)--(-0.5269,-0.5028)--(-0.4949,-0.4750)--(-0.4630,-0.4466)--(-0.4311,-0.4178)--(-0.3991,-0.3886)--(-0.3672,-0.3590)--(-0.3353,-0.3290)--(-0.3033,-0.2987)--(-0.2714,-0.2681)--(-0.2395,-0.2372)--(-0.2075,-0.2060)--(-0.1756,-0.1747)--(-0.1437,-0.1432)--(-0.1117,-0.1115)--(-0.0798,-0.0797)--(-0.0479,-0.0478)--(-0.0159,-0.0159)--(0.0159,0.0159)--(0.0479,0.0478)--(0.0798,0.0797)--(0.1117,0.1115)--(0.1437,0.1432)--(0.1756,0.1747)--(0.2075,0.2060)--(0.2395,0.2372)--(0.2714,0.2681)--(0.3033,0.2987)--(0.3353,0.3290)--(0.3672,0.3590)--(0.3991,0.3886)--(0.4311,0.4178)--(0.4630,0.4466)--(0.4949,0.4750)--(0.5269,0.5028)--(0.5588,0.5302)--(0.5908,0.5570)--(0.6227,0.5832)--(0.6546,0.6089)--(0.6866,0.6339)--(0.7185,0.6582)--(0.7504,0.6819)--(0.7824,0.7049)--(0.8143,0.7272)--(0.8462,0.7488)--(0.8782,0.7696)--(0.9101,0.7895)--(0.9420,0.8087)--(0.9740,0.8271)--(1.0059,0.8446)--(1.0378,0.8613)--(1.0698,0.8771)--(1.1017,0.8920)--(1.1337,0.9059)--(1.1656,0.9190)--(1.1975,0.9311)--(1.2295,0.9423)--(1.2614,0.9525)--(1.2933,0.9617)--(1.3253,0.9700)--(1.3572,0.9772)--(1.3891,0.9835)--(1.4211,0.9888)--(1.4530,0.9930)--(1.4849,0.9963)--(1.5169,0.9985)--(1.5488,0.9997)--(1.5807,0.9999);
 
-\draw [color=blue] (-1.581,1.000)--(-1.549,1.000)--(-1.517,1.000)--(-1.485,1.000)--(-1.453,1.000)--(-1.421,1.000)--(-1.389,1.000)--(-1.357,1.000)--(-1.325,1.000)--(-1.293,1.000)--(-1.261,1.000)--(-1.230,1.000)--(-1.198,1.000)--(-1.166,1.000)--(-1.134,1.000)--(-1.102,1.000)--(-1.070,1.000)--(-1.038,1.000)--(-1.006,1.000)--(-0.9740,1.000)--(-0.9421,1.000)--(-0.9102,1.000)--(-0.8782,1.000)--(-0.8463,1.000)--(-0.8143,1.000)--(-0.7824,1.000)--(-0.7505,1.000)--(-0.7185,1.000)--(-0.6866,1.000)--(-0.6547,1.000)--(-0.6227,1.000)--(-0.5908,1.000)--(-0.5589,1.000)--(-0.5269,1.000)--(-0.4950,1.000)--(-0.4631,1.000)--(-0.4311,1.000)--(-0.3992,1.000)--(-0.3673,1.000)--(-0.3353,1.000)--(-0.3034,1.000)--(-0.2715,1.000)--(-0.2395,1.000)--(-0.2076,1.000)--(-0.1756,1.000)--(-0.1437,1.000)--(-0.1118,1.000)--(-0.07984,1.000)--(-0.04790,1.000)--(-0.01597,1.000)--(0.01597,1.000)--(0.04790,1.000)--(0.07984,1.000)--(0.1118,1.000)--(0.1437,1.000)--(0.1756,1.000)--(0.2076,1.000)--(0.2395,1.000)--(0.2715,1.000)--(0.3034,1.000)--(0.3353,1.000)--(0.3673,1.000)--(0.3992,1.000)--(0.4311,1.000)--(0.4631,1.000)--(0.4950,1.000)--(0.5269,1.000)--(0.5589,1.000)--(0.5908,1.000)--(0.6227,1.000)--(0.6547,1.000)--(0.6866,1.000)--(0.7185,1.000)--(0.7505,1.000)--(0.7824,1.000)--(0.8143,1.000)--(0.8463,1.000)--(0.8782,1.000)--(0.9102,1.000)--(0.9421,1.000)--(0.9740,1.000)--(1.006,1.000)--(1.038,1.000)--(1.070,1.000)--(1.102,1.000)--(1.134,1.000)--(1.166,1.000)--(1.198,1.000)--(1.230,1.000)--(1.261,1.000)--(1.293,1.000)--(1.325,1.000)--(1.357,1.000)--(1.389,1.000)--(1.421,1.000)--(1.453,1.000)--(1.485,1.000)--(1.517,1.000)--(1.549,1.000)--(1.581,1.000);
+\draw [color=blue] (-1.5807,1.0000)--(-1.5488,1.0000)--(-1.5169,1.0000)--(-1.4849,1.0000)--(-1.4530,1.0000)--(-1.4211,1.0000)--(-1.3891,1.0000)--(-1.3572,1.0000)--(-1.3253,1.0000)--(-1.2933,1.0000)--(-1.2614,1.0000)--(-1.2295,1.0000)--(-1.1975,1.0000)--(-1.1656,1.0000)--(-1.1337,1.0000)--(-1.1017,1.0000)--(-1.0698,1.0000)--(-1.0378,1.0000)--(-1.0059,1.0000)--(-0.9740,1.0000)--(-0.9420,1.0000)--(-0.9101,1.0000)--(-0.8782,1.0000)--(-0.8462,1.0000)--(-0.8143,1.0000)--(-0.7824,1.0000)--(-0.7504,1.0000)--(-0.7185,1.0000)--(-0.6866,1.0000)--(-0.6546,1.0000)--(-0.6227,1.0000)--(-0.5908,1.0000)--(-0.5588,1.0000)--(-0.5269,1.0000)--(-0.4949,1.0000)--(-0.4630,1.0000)--(-0.4311,1.0000)--(-0.3991,1.0000)--(-0.3672,1.0000)--(-0.3353,1.0000)--(-0.3033,1.0000)--(-0.2714,1.0000)--(-0.2395,1.0000)--(-0.2075,1.0000)--(-0.1756,1.0000)--(-0.1437,1.0000)--(-0.1117,1.0000)--(-0.0798,1.0000)--(-0.0479,1.0000)--(-0.0159,1.0000)--(0.0159,1.0000)--(0.0479,1.0000)--(0.0798,1.0000)--(0.1117,1.0000)--(0.1437,1.0000)--(0.1756,1.0000)--(0.2075,1.0000)--(0.2395,1.0000)--(0.2714,1.0000)--(0.3033,1.0000)--(0.3353,1.0000)--(0.3672,1.0000)--(0.3991,1.0000)--(0.4311,1.0000)--(0.4630,1.0000)--(0.4949,1.0000)--(0.5269,1.0000)--(0.5588,1.0000)--(0.5908,1.0000)--(0.6227,1.0000)--(0.6546,1.0000)--(0.6866,1.0000)--(0.7185,1.0000)--(0.7504,1.0000)--(0.7824,1.0000)--(0.8143,1.0000)--(0.8462,1.0000)--(0.8782,1.0000)--(0.9101,1.0000)--(0.9420,1.0000)--(0.9740,1.0000)--(1.0059,1.0000)--(1.0378,1.0000)--(1.0698,1.0000)--(1.1017,1.0000)--(1.1337,1.0000)--(1.1656,1.0000)--(1.1975,1.0000)--(1.2295,1.0000)--(1.2614,1.0000)--(1.2933,1.0000)--(1.3253,1.0000)--(1.3572,1.0000)--(1.3891,1.0000)--(1.4211,1.0000)--(1.4530,1.0000)--(1.4849,1.0000)--(1.5169,1.0000)--(1.5488,1.0000)--(1.5807,1.0000);
 
-\draw [color=green] (-1.581,-1.000)--(-1.549,-1.000)--(-1.517,-1.000)--(-1.485,-1.000)--(-1.453,-1.000)--(-1.421,-1.000)--(-1.389,-1.000)--(-1.357,-1.000)--(-1.325,-1.000)--(-1.293,-1.000)--(-1.261,-1.000)--(-1.230,-1.000)--(-1.198,-1.000)--(-1.166,-1.000)--(-1.134,-1.000)--(-1.102,-1.000)--(-1.070,-1.000)--(-1.038,-1.000)--(-1.006,-1.000)--(-0.9740,-1.000)--(-0.9421,-1.000)--(-0.9102,-1.000)--(-0.8782,-1.000)--(-0.8463,-1.000)--(-0.8143,-1.000)--(-0.7824,-1.000)--(-0.7505,-1.000)--(-0.7185,-1.000)--(-0.6866,-1.000)--(-0.6547,-1.000)--(-0.6227,-1.000)--(-0.5908,-1.000)--(-0.5589,-1.000)--(-0.5269,-1.000)--(-0.4950,-1.000)--(-0.4631,-1.000)--(-0.4311,-1.000)--(-0.3992,-1.000)--(-0.3673,-1.000)--(-0.3353,-1.000)--(-0.3034,-1.000)--(-0.2715,-1.000)--(-0.2395,-1.000)--(-0.2076,-1.000)--(-0.1756,-1.000)--(-0.1437,-1.000)--(-0.1118,-1.000)--(-0.07984,-1.000)--(-0.04790,-1.000)--(-0.01597,-1.000)--(0.01597,-1.000)--(0.04790,-1.000)--(0.07984,-1.000)--(0.1118,-1.000)--(0.1437,-1.000)--(0.1756,-1.000)--(0.2076,-1.000)--(0.2395,-1.000)--(0.2715,-1.000)--(0.3034,-1.000)--(0.3353,-1.000)--(0.3673,-1.000)--(0.3992,-1.000)--(0.4311,-1.000)--(0.4631,-1.000)--(0.4950,-1.000)--(0.5269,-1.000)--(0.5589,-1.000)--(0.5908,-1.000)--(0.6227,-1.000)--(0.6547,-1.000)--(0.6866,-1.000)--(0.7185,-1.000)--(0.7505,-1.000)--(0.7824,-1.000)--(0.8143,-1.000)--(0.8463,-1.000)--(0.8782,-1.000)--(0.9102,-1.000)--(0.9421,-1.000)--(0.9740,-1.000)--(1.006,-1.000)--(1.038,-1.000)--(1.070,-1.000)--(1.102,-1.000)--(1.134,-1.000)--(1.166,-1.000)--(1.198,-1.000)--(1.230,-1.000)--(1.261,-1.000)--(1.293,-1.000)--(1.325,-1.000)--(1.357,-1.000)--(1.389,-1.000)--(1.421,-1.000)--(1.453,-1.000)--(1.485,-1.000)--(1.517,-1.000)--(1.549,-1.000)--(1.581,-1.000);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=green] (-1.5807,-1.0000)--(-1.5488,-1.0000)--(-1.5169,-1.0000)--(-1.4849,-1.0000)--(-1.4530,-1.0000)--(-1.4211,-1.0000)--(-1.3891,-1.0000)--(-1.3572,-1.0000)--(-1.3253,-1.0000)--(-1.2933,-1.0000)--(-1.2614,-1.0000)--(-1.2295,-1.0000)--(-1.1975,-1.0000)--(-1.1656,-1.0000)--(-1.1337,-1.0000)--(-1.1017,-1.0000)--(-1.0698,-1.0000)--(-1.0378,-1.0000)--(-1.0059,-1.0000)--(-0.9740,-1.0000)--(-0.9420,-1.0000)--(-0.9101,-1.0000)--(-0.8782,-1.0000)--(-0.8462,-1.0000)--(-0.8143,-1.0000)--(-0.7824,-1.0000)--(-0.7504,-1.0000)--(-0.7185,-1.0000)--(-0.6866,-1.0000)--(-0.6546,-1.0000)--(-0.6227,-1.0000)--(-0.5908,-1.0000)--(-0.5588,-1.0000)--(-0.5269,-1.0000)--(-0.4949,-1.0000)--(-0.4630,-1.0000)--(-0.4311,-1.0000)--(-0.3991,-1.0000)--(-0.3672,-1.0000)--(-0.3353,-1.0000)--(-0.3033,-1.0000)--(-0.2714,-1.0000)--(-0.2395,-1.0000)--(-0.2075,-1.0000)--(-0.1756,-1.0000)--(-0.1437,-1.0000)--(-0.1117,-1.0000)--(-0.0798,-1.0000)--(-0.0479,-1.0000)--(-0.0159,-1.0000)--(0.0159,-1.0000)--(0.0479,-1.0000)--(0.0798,-1.0000)--(0.1117,-1.0000)--(0.1437,-1.0000)--(0.1756,-1.0000)--(0.2075,-1.0000)--(0.2395,-1.0000)--(0.2714,-1.0000)--(0.3033,-1.0000)--(0.3353,-1.0000)--(0.3672,-1.0000)--(0.3991,-1.0000)--(0.4311,-1.0000)--(0.4630,-1.0000)--(0.4949,-1.0000)--(0.5269,-1.0000)--(0.5588,-1.0000)--(0.5908,-1.0000)--(0.6227,-1.0000)--(0.6546,-1.0000)--(0.6866,-1.0000)--(0.7185,-1.0000)--(0.7504,-1.0000)--(0.7824,-1.0000)--(0.8143,-1.0000)--(0.8462,-1.0000)--(0.8782,-1.0000)--(0.9101,-1.0000)--(0.9420,-1.0000)--(0.9740,-1.0000)--(1.0059,-1.0000)--(1.0378,-1.0000)--(1.0698,-1.0000)--(1.1017,-1.0000)--(1.1337,-1.0000)--(1.1656,-1.0000)--(1.1975,-1.0000)--(1.2295,-1.0000)--(1.2614,-1.0000)--(1.2933,-1.0000)--(1.3253,-1.0000)--(1.3572,-1.0000)--(1.3891,-1.0000)--(1.4211,-1.0000)--(1.4530,-1.0000)--(1.4849,-1.0000)--(1.5169,-1.0000)--(1.5488,-1.0000)--(1.5807,-1.0000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall b/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall
index ad7c6baeb..d01245efa 100644
--- a/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall
+++ b/src_phystricks/Fig_SuiteInverseAlterne.pstricks.recall
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,2.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,2.0000);
 %DEFAULT
 \draw []  (1.0000,-3.0000) node [rotate=0] {$\bullet$};
 \draw (1.0000,-3.4298) node {$-1$};
 \draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw (2.0000,1.9825) node {$1/2$};
+\draw (2.0000,1.9824) node {$1/2$};
 \draw []  (3.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw (3.0000,-1.4825) node {$-1/3$};
-\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
-\draw (4.0000,1.2325) node {$1/4$};
-\draw []  (5.0000,-0.60000) node [rotate=0] {$\bullet$};
-\draw (5.0000,-1.0825) node {$-1/5$};
-\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
-\draw (6.0000,0.98246) node {$1/6$};
-\draw []  (7.0000,-0.42857) node [rotate=0] {$\bullet$};
-\draw (7.0000,-0.91103) node {$-1/7$};
-\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
-\draw (8.0000,0.85746) node {$1/8$};
-\draw []  (9.0000,-0.33333) node [rotate=0] {$\bullet$};
-\draw (9.0000,-0.81579) node {$-1/9$};
-\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
-\draw (10.000,0.78246) node {$1/10$};
+\draw (3.0000,-1.4824) node {$-1/3$};
+\draw []  (4.0000,0.7500) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2324) node {$1/4$};
+\draw []  (5.0000,-0.6000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-1.0824) node {$-1/5$};
+\draw []  (6.0000,0.5000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.9824) node {$1/6$};
+\draw []  (7.0000,-0.4285) node [rotate=0] {$\bullet$};
+\draw (7.0000,-0.9110) node {$-1/7$};
+\draw []  (8.0000,0.3750) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.8574) node {$1/8$};
+\draw []  (9.0000,-0.3333) node [rotate=0] {$\bullet$};
+\draw (9.0000,-0.8157) node {$-1/9$};
+\draw []  (10.000,0.3000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.7824) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SuiteUnSurn.pstricks.recall b/src_phystricks/Fig_SuiteUnSurn.pstricks.recall
index ad00f9bf4..7be6ac65d 100644
--- a/src_phystricks/Fig_SuiteUnSurn.pstricks.recall
+++ b/src_phystricks/Fig_SuiteUnSurn.pstricks.recall
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 \draw []  (1.0000,3.0000) node [rotate=0] {$\bullet$};
 \draw (1.0000,3.4149) node {$1$};
 \draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw (2.0000,1.9825) node {$1/2$};
+\draw (2.0000,1.9824) node {$1/2$};
 \draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.0000,1.4825) node {$1/3$};
-\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
-\draw (4.0000,1.2325) node {$1/4$};
-\draw []  (5.0000,0.60000) node [rotate=0] {$\bullet$};
-\draw (5.0000,1.0825) node {$1/5$};
-\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
-\draw (6.0000,0.98246) node {$1/6$};
-\draw []  (7.0000,0.42857) node [rotate=0] {$\bullet$};
-\draw (7.0000,0.91103) node {$1/7$};
-\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
-\draw (8.0000,0.85746) node {$1/8$};
-\draw []  (9.0000,0.33333) node [rotate=0] {$\bullet$};
-\draw (9.0000,0.81579) node {$1/9$};
-\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
-\draw (10.000,0.78246) node {$1/10$};
+\draw (3.0000,1.4824) node {$1/3$};
+\draw []  (4.0000,0.7500) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2324) node {$1/4$};
+\draw []  (5.0000,0.6000) node [rotate=0] {$\bullet$};
+\draw (5.0000,1.0824) node {$1/5$};
+\draw []  (6.0000,0.5000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.9824) node {$1/6$};
+\draw []  (7.0000,0.4285) node [rotate=0] {$\bullet$};
+\draw (7.0000,0.9110) node {$1/7$};
+\draw []  (8.0000,0.3750) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.8574) node {$1/8$};
+\draw []  (9.0000,0.3333) node [rotate=0] {$\bullet$};
+\draw (9.0000,0.8157) node {$1/9$};
+\draw []  (10.000,0.3000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.7824) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SurfaceCercle.pstricks.recall b/src_phystricks/Fig_SurfaceCercle.pstricks.recall
index 79caff591..63c498a41 100644
--- a/src_phystricks/Fig_SurfaceCercle.pstricks.recall
+++ b/src_phystricks/Fig_SurfaceCercle.pstricks.recall
@@ -63,13 +63,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4998) -- (0.0000,2.4998);
 %DEFAULT
 
-\draw [color=blue] (-2.000,0)--(-1.960,0.4000)--(-1.919,0.5628)--(-1.879,0.6857)--(-1.838,0.7876)--(-1.798,0.8759)--(-1.758,0.9544)--(-1.717,1.025)--(-1.677,1.090)--(-1.636,1.150)--(-1.596,1.205)--(-1.556,1.257)--(-1.515,1.305)--(-1.475,1.351)--(-1.434,1.394)--(-1.394,1.434)--(-1.354,1.472)--(-1.313,1.509)--(-1.273,1.543)--(-1.232,1.575)--(-1.192,1.606)--(-1.152,1.635)--(-1.111,1.663)--(-1.071,1.689)--(-1.030,1.714)--(-0.9899,1.738)--(-0.9495,1.760)--(-0.9091,1.781)--(-0.8687,1.801)--(-0.8283,1.820)--(-0.7879,1.838)--(-0.7475,1.855)--(-0.7071,1.871)--(-0.6667,1.886)--(-0.6263,1.899)--(-0.5859,1.912)--(-0.5455,1.924)--(-0.5051,1.935)--(-0.4646,1.945)--(-0.4242,1.954)--(-0.3838,1.963)--(-0.3434,1.970)--(-0.3030,1.977)--(-0.2626,1.983)--(-0.2222,1.988)--(-0.1818,1.992)--(-0.1414,1.995)--(-0.1010,1.997)--(-0.06061,1.999)--(-0.02020,2.000)--(0.02020,2.000)--(0.06061,1.999)--(0.1010,1.997)--(0.1414,1.995)--(0.1818,1.992)--(0.2222,1.988)--(0.2626,1.983)--(0.3030,1.977)--(0.3434,1.970)--(0.3838,1.963)--(0.4242,1.954)--(0.4646,1.945)--(0.5051,1.935)--(0.5455,1.924)--(0.5859,1.912)--(0.6263,1.899)--(0.6667,1.886)--(0.7071,1.871)--(0.7475,1.855)--(0.7879,1.838)--(0.8283,1.820)--(0.8687,1.801)--(0.9091,1.781)--(0.9495,1.760)--(0.9899,1.738)--(1.030,1.714)--(1.071,1.689)--(1.111,1.663)--(1.152,1.635)--(1.192,1.606)--(1.232,1.575)--(1.273,1.543)--(1.313,1.509)--(1.354,1.472)--(1.394,1.434)--(1.434,1.394)--(1.475,1.351)--(1.515,1.305)--(1.556,1.257)--(1.596,1.205)--(1.636,1.150)--(1.677,1.090)--(1.717,1.025)--(1.758,0.9544)--(1.798,0.8759)--(1.838,0.7876)--(1.879,0.6857)--(1.919,0.5628)--(1.960,0.4000)--(2.000,0);
+\draw [color=blue] (-2.0000,0.0000)--(-1.9595,0.3999)--(-1.9191,0.5627)--(-1.8787,0.6856)--(-1.8383,0.7876)--(-1.7979,0.8759)--(-1.7575,0.9544)--(-1.7171,1.0253)--(-1.6767,1.0901)--(-1.6363,1.1499)--(-1.5959,1.2053)--(-1.5555,1.2570)--(-1.5151,1.3054)--(-1.4747,1.3509)--(-1.4343,1.3937)--(-1.3939,1.4342)--(-1.3535,1.4723)--(-1.3131,1.5085)--(-1.2727,1.5427)--(-1.2323,1.5752)--(-1.1919,1.6060)--(-1.1515,1.6352)--(-1.1111,1.6629)--(-1.0707,1.6892)--(-1.0303,1.7141)--(-0.9898,1.7378)--(-0.9494,1.7602)--(-0.9090,1.7814)--(-0.8686,1.8014)--(-0.8282,1.8204)--(-0.7878,1.8382)--(-0.7474,1.8550)--(-0.7070,1.8708)--(-0.6666,1.8856)--(-0.6262,1.8994)--(-0.5858,1.9122)--(-0.5454,1.9241)--(-0.5050,1.9351)--(-0.4646,1.9452)--(-0.4242,1.9544)--(-0.3838,1.9628)--(-0.3434,1.9702)--(-0.3030,1.9769)--(-0.2626,1.9826)--(-0.2222,1.9876)--(-0.1818,1.9917)--(-0.1414,1.9949)--(-0.1010,1.9974)--(-0.0606,1.9990)--(-0.0202,1.9998)--(0.0202,1.9998)--(0.0606,1.9990)--(0.1010,1.9974)--(0.1414,1.9949)--(0.1818,1.9917)--(0.2222,1.9876)--(0.2626,1.9826)--(0.3030,1.9769)--(0.3434,1.9702)--(0.3838,1.9628)--(0.4242,1.9544)--(0.4646,1.9452)--(0.5050,1.9351)--(0.5454,1.9241)--(0.5858,1.9122)--(0.6262,1.8994)--(0.6666,1.8856)--(0.7070,1.8708)--(0.7474,1.8550)--(0.7878,1.8382)--(0.8282,1.8204)--(0.8686,1.8014)--(0.9090,1.7814)--(0.9494,1.7602)--(0.9898,1.7378)--(1.0303,1.7141)--(1.0707,1.6892)--(1.1111,1.6629)--(1.1515,1.6352)--(1.1919,1.6060)--(1.2323,1.5752)--(1.2727,1.5427)--(1.3131,1.5085)--(1.3535,1.4723)--(1.3939,1.4342)--(1.4343,1.3937)--(1.4747,1.3509)--(1.5151,1.3054)--(1.5555,1.2570)--(1.5959,1.2053)--(1.6363,1.1499)--(1.6767,1.0901)--(1.7171,1.0253)--(1.7575,0.9544)--(1.7979,0.8759)--(1.8383,0.7876)--(1.8787,0.6856)--(1.9191,0.5627)--(1.9595,0.3999)--(2.0000,0.0000);
 
-\draw [color=red] (-2.000,0)--(-1.960,-0.4000)--(-1.919,-0.5628)--(-1.879,-0.6857)--(-1.838,-0.7876)--(-1.798,-0.8759)--(-1.758,-0.9544)--(-1.717,-1.025)--(-1.677,-1.090)--(-1.636,-1.150)--(-1.596,-1.205)--(-1.556,-1.257)--(-1.515,-1.305)--(-1.475,-1.351)--(-1.434,-1.394)--(-1.394,-1.434)--(-1.354,-1.472)--(-1.313,-1.509)--(-1.273,-1.543)--(-1.232,-1.575)--(-1.192,-1.606)--(-1.152,-1.635)--(-1.111,-1.663)--(-1.071,-1.689)--(-1.030,-1.714)--(-0.9899,-1.738)--(-0.9495,-1.760)--(-0.9091,-1.781)--(-0.8687,-1.801)--(-0.8283,-1.820)--(-0.7879,-1.838)--(-0.7475,-1.855)--(-0.7071,-1.871)--(-0.6667,-1.886)--(-0.6263,-1.899)--(-0.5859,-1.912)--(-0.5455,-1.924)--(-0.5051,-1.935)--(-0.4646,-1.945)--(-0.4242,-1.954)--(-0.3838,-1.963)--(-0.3434,-1.970)--(-0.3030,-1.977)--(-0.2626,-1.983)--(-0.2222,-1.988)--(-0.1818,-1.992)--(-0.1414,-1.995)--(-0.1010,-1.997)--(-0.06061,-1.999)--(-0.02020,-2.000)--(0.02020,-2.000)--(0.06061,-1.999)--(0.1010,-1.997)--(0.1414,-1.995)--(0.1818,-1.992)--(0.2222,-1.988)--(0.2626,-1.983)--(0.3030,-1.977)--(0.3434,-1.970)--(0.3838,-1.963)--(0.4242,-1.954)--(0.4646,-1.945)--(0.5051,-1.935)--(0.5455,-1.924)--(0.5859,-1.912)--(0.6263,-1.899)--(0.6667,-1.886)--(0.7071,-1.871)--(0.7475,-1.855)--(0.7879,-1.838)--(0.8283,-1.820)--(0.8687,-1.801)--(0.9091,-1.781)--(0.9495,-1.760)--(0.9899,-1.738)--(1.030,-1.714)--(1.071,-1.689)--(1.111,-1.663)--(1.152,-1.635)--(1.192,-1.606)--(1.232,-1.575)--(1.273,-1.543)--(1.313,-1.509)--(1.354,-1.472)--(1.394,-1.434)--(1.434,-1.394)--(1.475,-1.351)--(1.515,-1.305)--(1.556,-1.257)--(1.596,-1.205)--(1.636,-1.150)--(1.677,-1.090)--(1.717,-1.025)--(1.758,-0.9544)--(1.798,-0.8759)--(1.838,-0.7876)--(1.879,-0.6857)--(1.919,-0.5628)--(1.960,-0.4000)--(2.000,0);
+\draw [color=red] (-2.0000,0.0000)--(-1.9595,-0.3999)--(-1.9191,-0.5627)--(-1.8787,-0.6856)--(-1.8383,-0.7876)--(-1.7979,-0.8759)--(-1.7575,-0.9544)--(-1.7171,-1.0253)--(-1.6767,-1.0901)--(-1.6363,-1.1499)--(-1.5959,-1.2053)--(-1.5555,-1.2570)--(-1.5151,-1.3054)--(-1.4747,-1.3509)--(-1.4343,-1.3937)--(-1.3939,-1.4342)--(-1.3535,-1.4723)--(-1.3131,-1.5085)--(-1.2727,-1.5427)--(-1.2323,-1.5752)--(-1.1919,-1.6060)--(-1.1515,-1.6352)--(-1.1111,-1.6629)--(-1.0707,-1.6892)--(-1.0303,-1.7141)--(-0.9898,-1.7378)--(-0.9494,-1.7602)--(-0.9090,-1.7814)--(-0.8686,-1.8014)--(-0.8282,-1.8204)--(-0.7878,-1.8382)--(-0.7474,-1.8550)--(-0.7070,-1.8708)--(-0.6666,-1.8856)--(-0.6262,-1.8994)--(-0.5858,-1.9122)--(-0.5454,-1.9241)--(-0.5050,-1.9351)--(-0.4646,-1.9452)--(-0.4242,-1.9544)--(-0.3838,-1.9628)--(-0.3434,-1.9702)--(-0.3030,-1.9769)--(-0.2626,-1.9826)--(-0.2222,-1.9876)--(-0.1818,-1.9917)--(-0.1414,-1.9949)--(-0.1010,-1.9974)--(-0.0606,-1.9990)--(-0.0202,-1.9998)--(0.0202,-1.9998)--(0.0606,-1.9990)--(0.1010,-1.9974)--(0.1414,-1.9949)--(0.1818,-1.9917)--(0.2222,-1.9876)--(0.2626,-1.9826)--(0.3030,-1.9769)--(0.3434,-1.9702)--(0.3838,-1.9628)--(0.4242,-1.9544)--(0.4646,-1.9452)--(0.5050,-1.9351)--(0.5454,-1.9241)--(0.5858,-1.9122)--(0.6262,-1.8994)--(0.6666,-1.8856)--(0.7070,-1.8708)--(0.7474,-1.8550)--(0.7878,-1.8382)--(0.8282,-1.8204)--(0.8686,-1.8014)--(0.9090,-1.7814)--(0.9494,-1.7602)--(0.9898,-1.7378)--(1.0303,-1.7141)--(1.0707,-1.6892)--(1.1111,-1.6629)--(1.1515,-1.6352)--(1.1919,-1.6060)--(1.2323,-1.5752)--(1.2727,-1.5427)--(1.3131,-1.5085)--(1.3535,-1.4723)--(1.3939,-1.4342)--(1.4343,-1.3937)--(1.4747,-1.3509)--(1.5151,-1.3054)--(1.5555,-1.2570)--(1.5959,-1.2053)--(1.6363,-1.1499)--(1.6767,-1.0901)--(1.7171,-1.0253)--(1.7575,-0.9544)--(1.7979,-0.8759)--(1.8383,-0.7876)--(1.8787,-0.6856)--(1.9191,-0.5627)--(1.9595,-0.3999)--(2.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall b/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall
index 067a5b0f8..304bc7c92 100644
--- a/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall
+++ b/src_phystricks/Fig_TIMYoochXZZNGP.pstricks.recall
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-2.10,0.700) -- (2.10,0.700);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.38245) node {\( \pi(e_1)\)};
-\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
-\draw (0.70000,1.0825) node {\( \pi(e_2)\)};
+\draw [] (-2.1000,0.7000) -- (2.1000,0.7000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3824) node {\( \pi(e_1)\)};
+\draw []  (0.7000,0.7000) node [rotate=0] {$\bullet$};
+\draw (0.7000,1.0824) node {\( \pi(e_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall b/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
index feed9e843..b2fe57901 100644
--- a/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
+++ b/src_phystricks/Fig_TVXooWoKkqV.pstricks.recall
@@ -88,29 +88,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.61131) -- (0,2.5278);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.6113) -- (0.0000,2.5277);
 %DEFAULT
 
-\draw [color=blue] (-2.000,1.127)--(-1.929,1.000)--(-1.859,0.8813)--(-1.788,0.7700)--(-1.717,0.6663)--(-1.646,0.5699)--(-1.576,0.4807)--(-1.505,0.3985)--(-1.434,0.3231)--(-1.364,0.2544)--(-1.293,0.1922)--(-1.222,0.1363)--(-1.152,0.08657)--(-1.081,0.04284)--(-1.010,0.004948)--(-0.9394,-0.02729)--(-0.8687,-0.05404)--(-0.7980,-0.07546)--(-0.7273,-0.09173)--(-0.6566,-0.1030)--(-0.5859,-0.1095)--(-0.5152,-0.1113)--(-0.4444,-0.1087)--(-0.3737,-0.1017)--(-0.3030,-0.09057)--(-0.2323,-0.07548)--(-0.1616,-0.05658)--(-0.09091,-0.03405)--(-0.02020,-0.008044)--(0.05051,0.02126)--(0.1212,0.05370)--(0.1919,0.08911)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5455,0.3047)--(0.6162,0.3544)--(0.6869,0.4058)--(0.7576,0.4589)--(0.8283,0.5134)--(0.8990,0.5692)--(0.9697,0.6261)--(1.040,0.6840)--(1.111,0.7426)--(1.182,0.8018)--(1.253,0.8615)--(1.323,0.9215)--(1.394,0.9815)--(1.465,1.042)--(1.535,1.101)--(1.606,1.161)--(1.677,1.219)--(1.747,1.277)--(1.818,1.335)--(1.889,1.391)--(1.960,1.445)--(2.030,1.499)--(2.101,1.551)--(2.172,1.601)--(2.242,1.649)--(2.313,1.695)--(2.384,1.739)--(2.455,1.780)--(2.525,1.819)--(2.596,1.855)--(2.667,1.888)--(2.737,1.918)--(2.808,1.945)--(2.879,1.968)--(2.949,1.988)--(3.020,2.004)--(3.091,2.016)--(3.162,2.024)--(3.232,2.028)--(3.303,2.027)--(3.374,2.022)--(3.444,2.011)--(3.515,1.996)--(3.586,1.976)--(3.657,1.951)--(3.727,1.920)--(3.798,1.883)--(3.869,1.841)--(3.939,1.792)--(4.010,1.738)--(4.081,1.677)--(4.151,1.610)--(4.222,1.536)--(4.293,1.455)--(4.364,1.368)--(4.434,1.273)--(4.505,1.171)--(4.576,1.062)--(4.646,0.9444)--(4.717,0.8194)--(4.788,0.6865)--(4.859,0.5454)--(4.929,0.3960)--(5.000,0.2381);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (-2.0000,1.1269)--(-1.9292,1.0002)--(-1.8585,0.8812)--(-1.7878,0.7700)--(-1.7171,0.6663)--(-1.6464,0.5699)--(-1.5757,0.4806)--(-1.5050,0.3984)--(-1.4343,0.3230)--(-1.3636,0.2543)--(-1.2929,0.1921)--(-1.2222,0.1362)--(-1.1515,0.0865)--(-1.0808,0.0428)--(-1.0101,0.0049)--(-0.9393,-0.0272)--(-0.8686,-0.0540)--(-0.7979,-0.0754)--(-0.7272,-0.0917)--(-0.6565,-0.1030)--(-0.5858,-0.1094)--(-0.5151,-0.1113)--(-0.4444,-0.1086)--(-0.3737,-0.1016)--(-0.3030,-0.0905)--(-0.2323,-0.0754)--(-0.1616,-0.0565)--(-0.0909,-0.0340)--(-0.0202,-0.0080)--(0.0505,0.0212)--(0.1212,0.0537)--(0.1919,0.0891)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5454,0.3047)--(0.6161,0.3543)--(0.6868,0.4058)--(0.7575,0.4588)--(0.8282,0.5133)--(0.8989,0.5691)--(0.9696,0.6261)--(1.0404,0.6839)--(1.1111,0.7425)--(1.1818,0.8018)--(1.2525,0.8615)--(1.3232,0.9214)--(1.3939,0.9815)--(1.4646,1.0415)--(1.5353,1.1012)--(1.6060,1.1606)--(1.6767,1.2194)--(1.7474,1.2774)--(1.8181,1.3345)--(1.8888,1.3906)--(1.9595,1.4454)--(2.0303,1.4988)--(2.1010,1.5507)--(2.1717,1.6008)--(2.2424,1.6489)--(2.3131,1.6950)--(2.3838,1.7388)--(2.4545,1.7802)--(2.5252,1.8191)--(2.5959,1.8551)--(2.6666,1.8883)--(2.7373,1.9183)--(2.8080,1.9451)--(2.8787,1.9684)--(2.9494,1.9881)--(3.0202,2.0041)--(3.0909,2.0162)--(3.1616,2.0241)--(3.2323,2.0277)--(3.3030,2.0270)--(3.3737,2.0216)--(3.4444,2.0114)--(3.5151,1.9963)--(3.5858,1.9761)--(3.6565,1.9505)--(3.7272,1.9196)--(3.7979,1.8830)--(3.8686,1.8406)--(3.9393,1.7923)--(4.0101,1.7378)--(4.0808,1.6771)--(4.1515,1.6099)--(4.2222,1.5360)--(4.2929,1.4554)--(4.3636,1.3678)--(4.4343,1.2730)--(4.5050,1.1710)--(4.5757,1.0615)--(4.6464,0.9443)--(4.7171,0.8194)--(4.7878,0.6864)--(4.8585,0.5453)--(4.9292,0.3959)--(5.0000,0.2380);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_TangenteQuestion.pstricks.recall b/src_phystricks/Fig_TangenteQuestion.pstricks.recall
index ad59a76cd..f84f922ac 100644
--- a/src_phystricks/Fig_TangenteQuestion.pstricks.recall
+++ b/src_phystricks/Fig_TangenteQuestion.pstricks.recall
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5800);
 %DEFAULT
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall b/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall
index 411ca3541..20797a306 100644
--- a/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall
+++ b/src_phystricks/Fig_TangenteQuestionOM.pstricks.recall
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5800);
 %DEFAULT
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ToreRevolution.pstricks.recall b/src_phystricks/Fig_ToreRevolution.pstricks.recall
index 044f09832..fbd66806d 100644
--- a/src_phystricks/Fig_ToreRevolution.pstricks.recall
+++ b/src_phystricks/Fig_ToreRevolution.pstricks.recall
@@ -71,18 +71,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
 
-\draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [color=brown] (1.0000,3.0000)--(0.9979,3.0634)--(0.9919,3.1265)--(0.9819,3.1892)--(0.9679,3.2511)--(0.9500,3.3120)--(0.9283,3.3716)--(0.9029,3.4297)--(0.8738,3.4861)--(0.8412,3.5406)--(0.8052,3.5929)--(0.7660,3.6427)--(0.7237,3.6900)--(0.6785,3.7345)--(0.6305,3.7761)--(0.5800,3.8145)--(0.5272,3.8497)--(0.4722,3.8814)--(0.4154,3.9096)--(0.3568,3.9341)--(0.2969,3.9549)--(0.2357,3.9718)--(0.1736,3.9848)--(0.1108,3.9938)--(0.0475,3.9988)--(-0.0158,3.9998)--(-0.0792,3.9968)--(-0.1423,3.9898)--(-0.2048,3.9788)--(-0.2664,3.9638)--(-0.3270,3.9450)--(-0.3863,3.9223)--(-0.4440,3.8959)--(-0.5000,3.8660)--(-0.5539,3.8325)--(-0.6056,3.7957)--(-0.6548,3.7557)--(-0.7014,3.7126)--(-0.7452,3.6667)--(-0.7860,3.6181)--(-0.8236,3.5670)--(-0.8579,3.5136)--(-0.8888,3.4582)--(-0.9161,3.4009)--(-0.9396,3.3420)--(-0.9594,3.2817)--(-0.9754,3.2203)--(-0.9874,3.1580)--(-0.9954,3.0950)--(-0.9994,3.0317)--(-0.9994,2.9682)--(-0.9954,2.9049)--(-0.9874,2.8419)--(-0.9754,2.7796)--(-0.9594,2.7182)--(-0.9396,2.6579)--(-0.9161,2.5990)--(-0.8888,2.5417)--(-0.8579,2.4863)--(-0.8236,2.4329)--(-0.7860,2.3818)--(-0.7452,2.3332)--(-0.7014,2.2873)--(-0.6548,2.2442)--(-0.6056,2.2042)--(-0.5539,2.1674)--(-0.5000,2.1339)--(-0.4440,2.1040)--(-0.3863,2.0776)--(-0.3270,2.0549)--(-0.2664,2.0361)--(-0.2048,2.0211)--(-0.1423,2.0101)--(-0.0792,2.0031)--(-0.0158,2.0001)--(0.0475,2.0011)--(0.1108,2.0061)--(0.1736,2.0151)--(0.2357,2.0281)--(0.2969,2.0450)--(0.3568,2.0658)--(0.4154,2.0903)--(0.4722,2.1185)--(0.5272,2.1502)--(0.5800,2.1854)--(0.6305,2.2238)--(0.6785,2.2654)--(0.7237,2.3099)--(0.7660,2.3572)--(0.8052,2.4070)--(0.8412,2.4593)--(0.8738,2.5138)--(0.9029,2.5702)--(0.9283,2.6283)--(0.9500,2.6879)--(0.9679,2.7488)--(0.9819,2.8107)--(0.9919,2.8734)--(0.9979,2.9365)--(1.0000,3.0000);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0.0000);
 \draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
-\draw (1.8965,1.5000) node {$a$};
-\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
-\draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.14303,3.6345) node {$R$};
-\draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
+\draw (1.8964,1.5000) node {$a$};
+\draw []  (0.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw [color=red] (0.0000,3.0000) -- (0.8660,3.5000);
+\draw (0.1430,3.6345) node {$R$};
+\draw [color=blue,style=dotted] (0.0000,3.0000) -- (1.5000,3.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TracerUn.pstricks.recall b/src_phystricks/Fig_TracerUn.pstricks.recall
index 5b92fa7bb..c8bfd6298 100644
--- a/src_phystricks/Fig_TracerUn.pstricks.recall
+++ b/src_phystricks/Fig_TracerUn.pstricks.recall
@@ -81,43 +81,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
-\draw [,->,>=latex] (0,-4.3056) -- (0,4.3056);
+\draw [,->,>=latex] (-1.9000,0.0000) -- (1.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.3055) -- (0.0000,4.3055);
 %DEFAULT
 
-\draw [color=blue] (-1.400,-3.806)--(-1.372,-3.729)--(-1.343,-3.652)--(-1.315,-3.575)--(-1.287,-3.498)--(-1.259,-3.421)--(-1.230,-3.344)--(-1.202,-3.267)--(-1.174,-3.191)--(-1.145,-3.114)--(-1.117,-3.037)--(-1.089,-2.960)--(-1.061,-2.883)--(-1.032,-2.806)--(-1.004,-2.729)--(-0.9758,-2.652)--(-0.9475,-2.576)--(-0.9192,-2.499)--(-0.8909,-2.422)--(-0.8626,-2.345)--(-0.8343,-2.268)--(-0.8061,-2.191)--(-0.7778,-2.114)--(-0.7495,-2.037)--(-0.7212,-1.960)--(-0.6929,-1.884)--(-0.6646,-1.807)--(-0.6364,-1.730)--(-0.6081,-1.653)--(-0.5798,-1.576)--(-0.5515,-1.499)--(-0.5232,-1.422)--(-0.4949,-1.345)--(-0.4667,-1.269)--(-0.4384,-1.192)--(-0.4101,-1.115)--(-0.3818,-1.038)--(-0.3535,-0.9610)--(-0.3253,-0.8841)--(-0.2970,-0.8072)--(-0.2687,-0.7304)--(-0.2404,-0.6535)--(-0.2121,-0.5766)--(-0.1838,-0.4997)--(-0.1556,-0.4228)--(-0.1273,-0.3460)--(-0.09899,-0.2691)--(-0.07071,-0.1922)--(-0.04242,-0.1153)--(-0.01414,-0.03844)--(0.01414,0.03844)--(0.04242,0.1153)--(0.07071,0.1922)--(0.09899,0.2691)--(0.1273,0.3460)--(0.1556,0.4228)--(0.1838,0.4997)--(0.2121,0.5766)--(0.2404,0.6535)--(0.2687,0.7304)--(0.2970,0.8072)--(0.3253,0.8841)--(0.3535,0.9610)--(0.3818,1.038)--(0.4101,1.115)--(0.4384,1.192)--(0.4667,1.269)--(0.4949,1.345)--(0.5232,1.422)--(0.5515,1.499)--(0.5798,1.576)--(0.6081,1.653)--(0.6364,1.730)--(0.6646,1.807)--(0.6929,1.884)--(0.7212,1.960)--(0.7495,2.037)--(0.7778,2.114)--(0.8061,2.191)--(0.8343,2.268)--(0.8626,2.345)--(0.8909,2.422)--(0.9192,2.499)--(0.9475,2.576)--(0.9758,2.652)--(1.004,2.729)--(1.032,2.806)--(1.061,2.883)--(1.089,2.960)--(1.117,3.037)--(1.145,3.114)--(1.174,3.191)--(1.202,3.267)--(1.230,3.344)--(1.259,3.421)--(1.287,3.498)--(1.315,3.575)--(1.343,3.652)--(1.372,3.729)--(1.400,3.806);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.43316,-4.2000) node {$ -6 $};
-\draw [] (-0.100,-4.20) -- (0.100,-4.20);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.29125,4.2000) node {$ 6 $};
-\draw [] (-0.100,4.20) -- (0.100,4.20);
+\draw [color=blue] (-1.4000,-3.8055)--(-1.3717,-3.7287)--(-1.3434,-3.6518)--(-1.3151,-3.5749)--(-1.2868,-3.4980)--(-1.2585,-3.4211)--(-1.2303,-3.3443)--(-1.2020,-3.2674)--(-1.1737,-3.1905)--(-1.1454,-3.1136)--(-1.1171,-3.0367)--(-1.0888,-2.9599)--(-1.0606,-2.8830)--(-1.0323,-2.8061)--(-1.0040,-2.7292)--(-0.9757,-2.6523)--(-0.9474,-2.5755)--(-0.9191,-2.4986)--(-0.8909,-2.4217)--(-0.8626,-2.3448)--(-0.8343,-2.2679)--(-0.8060,-2.1910)--(-0.7777,-2.1142)--(-0.7494,-2.0373)--(-0.7212,-1.9604)--(-0.6929,-1.8835)--(-0.6646,-1.8066)--(-0.6363,-1.7298)--(-0.6080,-1.6529)--(-0.5797,-1.5760)--(-0.5515,-1.4991)--(-0.5232,-1.4222)--(-0.4949,-1.3454)--(-0.4666,-1.2685)--(-0.4383,-1.1916)--(-0.4101,-1.1147)--(-0.3818,-1.0378)--(-0.3535,-0.9610)--(-0.3252,-0.8841)--(-0.2969,-0.8072)--(-0.2686,-0.7303)--(-0.2404,-0.6534)--(-0.2121,-0.5766)--(-0.1838,-0.4997)--(-0.1555,-0.4228)--(-0.1272,-0.3459)--(-0.0989,-0.2690)--(-0.0707,-0.1922)--(-0.0424,-0.1153)--(-0.0141,-0.0384)--(0.0141,0.0384)--(0.0424,0.1153)--(0.0707,0.1922)--(0.0989,0.2690)--(0.1272,0.3459)--(0.1555,0.4228)--(0.1838,0.4997)--(0.2121,0.5766)--(0.2404,0.6534)--(0.2686,0.7303)--(0.2969,0.8072)--(0.3252,0.8841)--(0.3535,0.9610)--(0.3818,1.0378)--(0.4101,1.1147)--(0.4383,1.1916)--(0.4666,1.2685)--(0.4949,1.3454)--(0.5232,1.4222)--(0.5515,1.4991)--(0.5797,1.5760)--(0.6080,1.6529)--(0.6363,1.7298)--(0.6646,1.8066)--(0.6929,1.8835)--(0.7212,1.9604)--(0.7494,2.0373)--(0.7777,2.1142)--(0.8060,2.1910)--(0.8343,2.2679)--(0.8626,2.3448)--(0.8909,2.4217)--(0.9191,2.4986)--(0.9474,2.5755)--(0.9757,2.6523)--(1.0040,2.7292)--(1.0323,2.8061)--(1.0606,2.8830)--(1.0888,2.9599)--(1.1171,3.0367)--(1.1454,3.1136)--(1.1737,3.1905)--(1.2020,3.2674)--(1.2303,3.3443)--(1.2585,3.4211)--(1.2868,3.4980)--(1.3151,3.5749)--(1.3434,3.6518)--(1.3717,3.7287)--(1.4000,3.8055);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.4331,-4.2000) node {$ -6 $};
+\draw [] (-0.1000,-4.2000) -- (0.1000,-4.2000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
+\draw (-0.2912,4.2000) node {$ 6 $};
+\draw [] (-0.1000,4.2000) -- (0.1000,4.2000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -163,23 +163,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (-2.000,2.000)--(-1.960,1.960)--(-1.919,1.919)--(-1.879,1.879)--(-1.838,1.838)--(-1.798,1.798)--(-1.758,1.758)--(-1.717,1.717)--(-1.677,1.677)--(-1.636,1.636)--(-1.596,1.596)--(-1.556,1.556)--(-1.515,1.515)--(-1.475,1.475)--(-1.434,1.434)--(-1.394,1.394)--(-1.354,1.354)--(-1.313,1.313)--(-1.273,1.273)--(-1.232,1.232)--(-1.192,1.192)--(-1.152,1.152)--(-1.111,1.111)--(-1.071,1.071)--(-1.030,1.030)--(-0.9899,0.9899)--(-0.9495,0.9495)--(-0.9091,0.9091)--(-0.8687,0.8687)--(-0.8283,0.8283)--(-0.7879,0.7879)--(-0.7475,0.7475)--(-0.7071,0.7071)--(-0.6667,0.6667)--(-0.6263,0.6263)--(-0.5859,0.5859)--(-0.5455,0.5455)--(-0.5051,0.5051)--(-0.4646,0.4646)--(-0.4242,0.4242)--(-0.3838,0.3838)--(-0.3434,0.3434)--(-0.3030,0.3030)--(-0.2626,0.2626)--(-0.2222,0.2222)--(-0.1818,0.1818)--(-0.1414,0.1414)--(-0.1010,0.1010)--(-0.06061,0.06061)--(-0.02020,0.02020)--(0.02020,0.02020)--(0.06061,0.06061)--(0.1010,0.1010)--(0.1414,0.1414)--(0.1818,0.1818)--(0.2222,0.2222)--(0.2626,0.2626)--(0.3030,0.3030)--(0.3434,0.3434)--(0.3838,0.3838)--(0.4242,0.4242)--(0.4646,0.4646)--(0.5051,0.5051)--(0.5455,0.5455)--(0.5859,0.5859)--(0.6263,0.6263)--(0.6667,0.6667)--(0.7071,0.7071)--(0.7475,0.7475)--(0.7879,0.7879)--(0.8283,0.8283)--(0.8687,0.8687)--(0.9091,0.9091)--(0.9495,0.9495)--(0.9899,0.9899)--(1.030,1.030)--(1.071,1.071)--(1.111,1.111)--(1.152,1.152)--(1.192,1.192)--(1.232,1.232)--(1.273,1.273)--(1.313,1.313)--(1.354,1.354)--(1.394,1.394)--(1.434,1.434)--(1.475,1.475)--(1.515,1.515)--(1.556,1.556)--(1.596,1.596)--(1.636,1.636)--(1.677,1.677)--(1.717,1.717)--(1.758,1.758)--(1.798,1.798)--(1.838,1.838)--(1.879,1.879)--(1.919,1.919)--(1.960,1.960)--(2.000,2.000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (-2.0000,2.0000)--(-1.9595,1.9595)--(-1.9191,1.9191)--(-1.8787,1.8787)--(-1.8383,1.8383)--(-1.7979,1.7979)--(-1.7575,1.7575)--(-1.7171,1.7171)--(-1.6767,1.6767)--(-1.6363,1.6363)--(-1.5959,1.5959)--(-1.5555,1.5555)--(-1.5151,1.5151)--(-1.4747,1.4747)--(-1.4343,1.4343)--(-1.3939,1.3939)--(-1.3535,1.3535)--(-1.3131,1.3131)--(-1.2727,1.2727)--(-1.2323,1.2323)--(-1.1919,1.1919)--(-1.1515,1.1515)--(-1.1111,1.1111)--(-1.0707,1.0707)--(-1.0303,1.0303)--(-0.9898,0.9898)--(-0.9494,0.9494)--(-0.9090,0.9090)--(-0.8686,0.8686)--(-0.8282,0.8282)--(-0.7878,0.7878)--(-0.7474,0.7474)--(-0.7070,0.7070)--(-0.6666,0.6666)--(-0.6262,0.6262)--(-0.5858,0.5858)--(-0.5454,0.5454)--(-0.5050,0.5050)--(-0.4646,0.4646)--(-0.4242,0.4242)--(-0.3838,0.3838)--(-0.3434,0.3434)--(-0.3030,0.3030)--(-0.2626,0.2626)--(-0.2222,0.2222)--(-0.1818,0.1818)--(-0.1414,0.1414)--(-0.1010,0.1010)--(-0.0606,0.0606)--(-0.0202,0.0202)--(0.0202,0.0202)--(0.0606,0.0606)--(0.1010,0.1010)--(0.1414,0.1414)--(0.1818,0.1818)--(0.2222,0.2222)--(0.2626,0.2626)--(0.3030,0.3030)--(0.3434,0.3434)--(0.3838,0.3838)--(0.4242,0.4242)--(0.4646,0.4646)--(0.5050,0.5050)--(0.5454,0.5454)--(0.5858,0.5858)--(0.6262,0.6262)--(0.6666,0.6666)--(0.7070,0.7070)--(0.7474,0.7474)--(0.7878,0.7878)--(0.8282,0.8282)--(0.8686,0.8686)--(0.9090,0.9090)--(0.9494,0.9494)--(0.9898,0.9898)--(1.0303,1.0303)--(1.0707,1.0707)--(1.1111,1.1111)--(1.1515,1.1515)--(1.1919,1.1919)--(1.2323,1.2323)--(1.2727,1.2727)--(1.3131,1.3131)--(1.3535,1.3535)--(1.3939,1.3939)--(1.4343,1.4343)--(1.4747,1.4747)--(1.5151,1.5151)--(1.5555,1.5555)--(1.5959,1.5959)--(1.6363,1.6363)--(1.6767,1.6767)--(1.7171,1.7171)--(1.7575,1.7575)--(1.7979,1.7979)--(1.8383,1.8383)--(1.8787,1.8787)--(1.9191,1.9191)--(1.9595,1.9595)--(2.0000,2.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -253,39 +253,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.8000);
+\draw [,->,>=latex] (-1.9000,0.0000) -- (1.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.8000);
 %DEFAULT
 
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+\draw [color=blue] (-1.4000,3.5000)--(-1.3858,3.4437)--(-1.3717,3.3880)--(-1.3575,3.3328)--(-1.3434,3.2783)--(-1.3292,3.2243)--(-1.3151,3.1708)--(-1.3010,3.1180)--(-1.2868,3.0657)--(-1.2727,3.0140)--(-1.2585,2.9629)--(-1.2444,2.9123)--(-1.2303,2.8623)--(-1.2161,2.8129)--(-1.2020,2.7640)--(-1.1878,2.7157)--(-1.1737,2.6680)--(-1.1595,2.6209)--(-1.1454,2.5743)--(-1.1313,2.5283)--(-1.1171,2.4829)--(-1.1030,2.4381)--(-1.0888,2.3938)--(-1.0747,2.3501)--(-1.0606,2.3069)--(-1.0464,2.2644)--(-1.0323,2.2224)--(-1.0181,2.1809)--(-1.0040,2.1401)--(-0.9898,2.0998)--(-0.9757,2.0601)--(-0.9616,2.0210)--(-0.9474,1.9824)--(-0.9333,1.9444)--(-0.9191,1.9070)--(-0.9050,1.8701)--(-0.8909,1.8338)--(-0.8767,1.7981)--(-0.8626,1.7630)--(-0.8484,1.7284)--(-0.8343,1.6944)--(-0.8202,1.6610)--(-0.8060,1.6281)--(-0.7919,1.5959)--(-0.7777,1.5641)--(-0.7636,1.5330)--(-0.7494,1.5024)--(-0.7353,1.4724)--(-0.7212,1.4430)--(-0.7070,1.4142)--(-0.6929,1.3859)--(-0.6787,1.3582)--(-0.6646,1.3310)--(-0.6505,1.3045)--(-0.6363,1.2785)--(-0.6222,1.2530)--(-0.6080,1.2282)--(-0.5939,1.2039)--(-0.5797,1.1802)--(-0.5656,1.1570)--(-0.5515,1.1345)--(-0.5373,1.1125)--(-0.5232,1.0911)--(-0.5090,1.0702)--(-0.4949,1.0499)--(-0.4808,1.0302)--(-0.4666,1.0111)--(-0.4525,0.9925)--(-0.4383,0.9745)--(-0.4242,0.9571)--(-0.4101,0.9402)--(-0.3959,0.9239)--(-0.3818,0.9082)--(-0.3676,0.8931)--(-0.3535,0.8785)--(-0.3393,0.8645)--(-0.3252,0.8511)--(-0.3111,0.8382)--(-0.2969,0.8259)--(-0.2828,0.8142)--(-0.2686,0.8031)--(-0.2545,0.7925)--(-0.2404,0.7825)--(-0.2262,0.7731)--(-0.2121,0.7642)--(-0.1979,0.7559)--(-0.1838,0.7482)--(-0.1696,0.7411)--(-0.1555,0.7345)--(-0.1414,0.7285)--(-0.1272,0.7231)--(-0.1131,0.7182)--(-0.0989,0.7139)--(-0.0848,0.7102)--(-0.0707,0.7071)--(-0.0565,0.7045)--(-0.0424,0.7025)--(-0.0282,0.7011)--(-0.0141,0.7002)--(0.0000,0.7000);
 
-\draw [color=blue] (0,0.7000)--(0.01414,0.7286)--(0.02828,0.7577)--(0.04242,0.7874)--(0.05657,0.8177)--(0.07071,0.8486)--(0.08485,0.8800)--(0.09899,0.9120)--(0.1131,0.9445)--(0.1273,0.9777)--(0.1414,1.011)--(0.1556,1.046)--(0.1697,1.081)--(0.1838,1.116)--(0.1980,1.152)--(0.2121,1.189)--(0.2263,1.226)--(0.2404,1.263)--(0.2545,1.302)--(0.2687,1.340)--(0.2828,1.380)--(0.2970,1.420)--(0.3111,1.460)--(0.3253,1.502)--(0.3394,1.543)--(0.3535,1.586)--(0.3677,1.628)--(0.3818,1.672)--(0.3960,1.716)--(0.4101,1.760)--(0.4242,1.806)--(0.4384,1.851)--(0.4525,1.898)--(0.4667,1.944)--(0.4808,1.992)--(0.4949,2.040)--(0.5091,2.088)--(0.5232,2.138)--(0.5374,2.187)--(0.5515,2.238)--(0.5657,2.288)--(0.5798,2.340)--(0.5939,2.392)--(0.6081,2.444)--(0.6222,2.498)--(0.6364,2.551)--(0.6505,2.606)--(0.6646,2.660)--(0.6788,2.716)--(0.6929,2.772)--(0.7071,2.828)--(0.7212,2.885)--(0.7354,2.943)--(0.7495,3.001)--(0.7636,3.060)--(0.7778,3.120)--(0.7919,3.180)--(0.8061,3.240)--(0.8202,3.301)--(0.8343,3.363)--(0.8485,3.425)--(0.8626,3.488)--(0.8768,3.552)--(0.8909,3.616)--(0.9051,3.680)--(0.9192,3.745)--(0.9333,3.811)--(0.9475,3.877)--(0.9616,3.944)--(0.9758,4.012)--(0.9899,4.080)--(1.004,4.148)--(1.018,4.217)--(1.032,4.287)--(1.046,4.357)--(1.061,4.428)--(1.075,4.500)--(1.089,4.572)--(1.103,4.644)--(1.117,4.717)--(1.131,4.791)--(1.145,4.865)--(1.160,4.940)--(1.174,5.016)--(1.188,5.092)--(1.202,5.168)--(1.216,5.245)--(1.230,5.323)--(1.244,5.401)--(1.259,5.480)--(1.273,5.560)--(1.287,5.640)--(1.301,5.720)--(1.315,5.801)--(1.329,5.883)--(1.343,5.965)--(1.358,6.048)--(1.372,6.131)--(1.386,6.215)--(1.400,6.300);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.29125,4.2000) node {$ 6 $};
-\draw [] (-0.100,4.20) -- (0.100,4.20);
-\draw (-0.29125,4.9000) node {$ 7 $};
-\draw [] (-0.100,4.90) -- (0.100,4.90);
-\draw (-0.29125,5.6000) node {$ 8 $};
-\draw [] (-0.100,5.60) -- (0.100,5.60);
-\draw (-0.29125,6.3000) node {$ 9 $};
-\draw [] (-0.100,6.30) -- (0.100,6.30);
+\draw [color=blue] (0.0000,0.7000)--(0.0141,0.7285)--(0.0282,0.7577)--(0.0424,0.7874)--(0.0565,0.8177)--(0.0707,0.8485)--(0.0848,0.8799)--(0.0989,0.9119)--(0.1131,0.9445)--(0.1272,0.9776)--(0.1414,1.0113)--(0.1555,1.0456)--(0.1696,1.0805)--(0.1838,1.1159)--(0.1979,1.1519)--(0.2121,1.1885)--(0.2262,1.2256)--(0.2404,1.2633)--(0.2545,1.3016)--(0.2686,1.3405)--(0.2828,1.3799)--(0.2969,1.4199)--(0.3111,1.4604)--(0.3252,1.5016)--(0.3393,1.5433)--(0.3535,1.5856)--(0.3676,1.6284)--(0.3818,1.6719)--(0.3959,1.7158)--(0.4101,1.7604)--(0.4242,1.8056)--(0.4383,1.8513)--(0.4525,1.8975)--(0.4666,1.9444)--(0.4808,1.9918)--(0.4949,2.0398)--(0.5090,2.0884)--(0.5232,2.1375)--(0.5373,2.1872)--(0.5515,2.2375)--(0.5656,2.2884)--(0.5797,2.3398)--(0.5939,2.3918)--(0.6080,2.4443)--(0.6222,2.4975)--(0.6363,2.5512)--(0.6505,2.6055)--(0.6646,2.6603)--(0.6787,2.7157)--(0.6929,2.7717)--(0.7070,2.8283)--(0.7212,2.8854)--(0.7353,2.9431)--(0.7494,3.0014)--(0.7636,3.0603)--(0.7777,3.1197)--(0.7919,3.1797)--(0.8060,3.2403)--(0.8202,3.3014)--(0.8343,3.3631)--(0.8484,3.4254)--(0.8626,3.4882)--(0.8767,3.5517)--(0.8909,3.6157)--(0.9050,3.6802)--(0.9191,3.7454)--(0.9333,3.8111)--(0.9474,3.8773)--(0.9616,3.9442)--(0.9757,4.0116)--(0.9898,4.0796)--(1.0040,4.1482)--(1.0181,4.2173)--(1.0323,4.2870)--(1.0464,4.3573)--(1.0606,4.4281)--(1.0747,4.4996)--(1.0888,4.5716)--(1.1030,4.6441)--(1.1171,4.7173)--(1.1313,4.7910)--(1.1454,4.8652)--(1.1595,4.9401)--(1.1737,5.0155)--(1.1878,5.0915)--(1.2020,5.1681)--(1.2161,5.2452)--(1.2303,5.3229)--(1.2444,5.4012)--(1.2585,5.4800)--(1.2727,5.5595)--(1.2868,5.6394)--(1.3010,5.7200)--(1.3151,5.8011)--(1.3292,5.8828)--(1.3434,5.9651)--(1.3575,6.0480)--(1.3717,6.1314)--(1.3858,6.2154)--(1.4000,6.3000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
+\draw (-0.2912,4.2000) node {$ 6 $};
+\draw [] (-0.1000,4.2000) -- (0.1000,4.2000);
+\draw (-0.2912,4.9000) node {$ 7 $};
+\draw [] (-0.1000,4.9000) -- (0.1000,4.9000);
+\draw (-0.2912,5.6000) node {$ 8 $};
+\draw [] (-0.1000,5.6000) -- (0.1000,5.6000);
+\draw (-0.2912,6.3000) node {$ 9 $};
+\draw [] (-0.1000,6.3000) -- (0.1000,6.3000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -335,27 +335,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
-\draw [,->,>=latex] (0,-2.6000) -- (0,1.2000);
+\draw [,->,>=latex] (-1.9000,0.0000) -- (1.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6000) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-1.400,-2.100)--(-1.372,-2.072)--(-1.343,-2.043)--(-1.315,-2.015)--(-1.287,-1.987)--(-1.259,-1.959)--(-1.230,-1.930)--(-1.202,-1.902)--(-1.174,-1.874)--(-1.145,-1.845)--(-1.117,-1.817)--(-1.089,-1.789)--(-1.061,-1.761)--(-1.032,-1.732)--(-1.004,-1.704)--(-0.9758,-1.676)--(-0.9475,-1.647)--(-0.9192,-1.619)--(-0.8909,-1.591)--(-0.8626,-1.563)--(-0.8343,-1.534)--(-0.8061,-1.506)--(-0.7778,-1.478)--(-0.7495,-1.449)--(-0.7212,-1.421)--(-0.6929,-1.393)--(-0.6646,-1.365)--(-0.6364,-1.336)--(-0.6081,-1.308)--(-0.5798,-1.280)--(-0.5515,-1.252)--(-0.5232,-1.223)--(-0.4949,-1.195)--(-0.4667,-1.167)--(-0.4384,-1.138)--(-0.4101,-1.110)--(-0.3818,-1.082)--(-0.3535,-1.054)--(-0.3253,-1.025)--(-0.2970,-0.9970)--(-0.2687,-0.9687)--(-0.2404,-0.9404)--(-0.2121,-0.9121)--(-0.1838,-0.8838)--(-0.1556,-0.8556)--(-0.1273,-0.8273)--(-0.09899,-0.7990)--(-0.07071,-0.7707)--(-0.04242,-0.7424)--(-0.01414,-0.7141)--(0.01414,-0.6859)--(0.04242,-0.6576)--(0.07071,-0.6293)--(0.09899,-0.6010)--(0.1273,-0.5727)--(0.1556,-0.5444)--(0.1838,-0.5162)--(0.2121,-0.4879)--(0.2404,-0.4596)--(0.2687,-0.4313)--(0.2970,-0.4030)--(0.3253,-0.3747)--(0.3535,-0.3465)--(0.3818,-0.3182)--(0.4101,-0.2899)--(0.4384,-0.2616)--(0.4667,-0.2333)--(0.4949,-0.2051)--(0.5232,-0.1768)--(0.5515,-0.1485)--(0.5798,-0.1202)--(0.6081,-0.09192)--(0.6364,-0.06364)--(0.6646,-0.03535)--(0.6929,-0.007071)--(0.7212,0.02121)--(0.7495,0.04949)--(0.7778,0.07778)--(0.8061,0.1061)--(0.8343,0.1343)--(0.8626,0.1626)--(0.8909,0.1909)--(0.9192,0.2192)--(0.9475,0.2475)--(0.9758,0.2758)--(1.004,0.3040)--(1.032,0.3323)--(1.061,0.3606)--(1.089,0.3889)--(1.117,0.4172)--(1.145,0.4455)--(1.174,0.4737)--(1.202,0.5020)--(1.230,0.5303)--(1.259,0.5586)--(1.287,0.5869)--(1.315,0.6152)--(1.343,0.6434)--(1.372,0.6717)--(1.400,0.7000);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-1.4000,-2.1000)--(-1.3717,-2.0717)--(-1.3434,-2.0434)--(-1.3151,-2.0151)--(-1.2868,-1.9868)--(-1.2585,-1.9585)--(-1.2303,-1.9303)--(-1.2020,-1.9020)--(-1.1737,-1.8737)--(-1.1454,-1.8454)--(-1.1171,-1.8171)--(-1.0888,-1.7888)--(-1.0606,-1.7606)--(-1.0323,-1.7323)--(-1.0040,-1.7040)--(-0.9757,-1.6757)--(-0.9474,-1.6474)--(-0.9191,-1.6191)--(-0.8909,-1.5909)--(-0.8626,-1.5626)--(-0.8343,-1.5343)--(-0.8060,-1.5060)--(-0.7777,-1.4777)--(-0.7494,-1.4494)--(-0.7212,-1.4212)--(-0.6929,-1.3929)--(-0.6646,-1.3646)--(-0.6363,-1.3363)--(-0.6080,-1.3080)--(-0.5797,-1.2797)--(-0.5515,-1.2515)--(-0.5232,-1.2232)--(-0.4949,-1.1949)--(-0.4666,-1.1666)--(-0.4383,-1.1383)--(-0.4101,-1.1101)--(-0.3818,-1.0818)--(-0.3535,-1.0535)--(-0.3252,-1.0252)--(-0.2969,-0.9969)--(-0.2686,-0.9686)--(-0.2404,-0.9404)--(-0.2121,-0.9121)--(-0.1838,-0.8838)--(-0.1555,-0.8555)--(-0.1272,-0.8272)--(-0.0989,-0.7989)--(-0.0707,-0.7707)--(-0.0424,-0.7424)--(-0.0141,-0.7141)--(0.0141,-0.6858)--(0.0424,-0.6575)--(0.0707,-0.6292)--(0.0989,-0.6010)--(0.1272,-0.5727)--(0.1555,-0.5444)--(0.1838,-0.5161)--(0.2121,-0.4878)--(0.2404,-0.4595)--(0.2686,-0.4313)--(0.2969,-0.4030)--(0.3252,-0.3747)--(0.3535,-0.3464)--(0.3818,-0.3181)--(0.4101,-0.2898)--(0.4383,-0.2616)--(0.4666,-0.2333)--(0.4949,-0.2050)--(0.5232,-0.1767)--(0.5515,-0.1484)--(0.5797,-0.1202)--(0.6080,-0.0919)--(0.6363,-0.0636)--(0.6646,-0.0353)--(0.6929,-0.0070)--(0.7212,0.0212)--(0.7494,0.0494)--(0.7777,0.0777)--(0.8060,0.1060)--(0.8343,0.1343)--(0.8626,0.1626)--(0.8909,0.1909)--(0.9191,0.2191)--(0.9474,0.2474)--(0.9757,0.2757)--(1.0040,0.3040)--(1.0323,0.3323)--(1.0606,0.3606)--(1.0888,0.3888)--(1.1171,0.4171)--(1.1454,0.4454)--(1.1737,0.4737)--(1.2020,0.5020)--(1.2303,0.5303)--(1.2585,0.5585)--(1.2868,0.5868)--(1.3151,0.6151)--(1.3434,0.6434)--(1.3717,0.6717)--(1.4000,0.7000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_Trajs.pstricks.recall b/src_phystricks/Fig_Trajs.pstricks.recall
index d6f17fd84..271b0587e 100644
--- a/src_phystricks/Fig_Trajs.pstricks.recall
+++ b/src_phystricks/Fig_Trajs.pstricks.recall
@@ -75,29 +75,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6426,0) -- (2.7232,0);
-\draw [,->,>=latex] (0,-1.9142) -- (0,2.7360);
+\draw [,->,>=latex] (-1.6426,0.0000) -- (2.7232,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9141) -- (0.0000,2.7359);
 %DEFAULT
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+\draw [color=red] (1.3817,0.3011)--(1.3875,0.2731)--(1.3928,0.2451)--(1.3974,0.2169)--(1.4015,0.1886)--(1.4050,0.1602)--(1.4080,0.1318)--(1.4104,0.1034)--(1.4122,0.0748)--(1.4134,0.0463)--(1.4141,0.0177)--(1.4141,-0.0107)--(1.4136,-0.0393)--(1.4125,-0.0678)--(1.4109,-0.0964)--(1.4086,-0.1248)--(1.4058,-0.1533)--(1.4024,-0.1816)--(1.3985,-0.2099)--(1.3940,-0.2381)--(1.3889,-0.2663)--(1.3832,-0.2943)--(1.3770,-0.3221)--(1.3702,-0.3499)--(1.3628,-0.3775)--(1.3549,-0.4050)--(1.3465,-0.4322)--(1.3375,-0.4594)--(1.3279,-0.4863)--(1.3178,-0.5130)--(1.3072,-0.5395)--(1.2960,-0.5658)--(1.2843,-0.5919)--(1.2721,-0.6177)--(1.2594,-0.6433)--(1.2461,-0.6686)--(1.2323,-0.6936)--(1.2181,-0.7184)--(1.2033,-0.7428)--(1.1881,-0.7670)--(1.1723,-0.7908)--(1.1561,-0.8144)--(1.1394,-0.8376)--(1.1223,-0.8604)--(1.1047,-0.8829)--(1.0866,-0.9050)--(1.0681,-0.9268)--(1.0492,-0.9482)--(1.0298,-0.9692)--(1.0100,-0.9898)--(0.9898,-1.0100)--(0.9692,-1.0298)--(0.9482,-1.0492)--(0.9268,-1.0681)--(0.9050,-1.0866)--(0.8829,-1.1047)--(0.8604,-1.1223)--(0.8376,-1.1394)--(0.8144,-1.1561)--(0.7908,-1.1723)--(0.7670,-1.1881)--(0.7428,-1.2033)--(0.7184,-1.2181)--(0.6936,-1.2323)--(0.6686,-1.2461)--(0.6433,-1.2594)--(0.6177,-1.2721)--(0.5919,-1.2843)--(0.5658,-1.2960)--(0.5395,-1.3072)--(0.5130,-1.3178)--(0.4863,-1.3279)--(0.4594,-1.3375)--(0.4322,-1.3465)--(0.4050,-1.3549)--(0.3775,-1.3628)--(0.3499,-1.3702)--(0.3221,-1.3770)--(0.2943,-1.3832)--(0.2663,-1.3889)--(0.2381,-1.3940)--(0.2099,-1.3985)--(0.1816,-1.4024)--(0.1533,-1.4058)--(0.1248,-1.4086)--(0.0964,-1.4109)--(0.0678,-1.4125)--(0.0393,-1.4136)--(0.0107,-1.4141)--(-0.0177,-1.4141)--(-0.0463,-1.4134)--(-0.0748,-1.4122)--(-0.1034,-1.4104)--(-0.1318,-1.4080)--(-0.1602,-1.4050)--(-0.1886,-1.4015)--(-0.2169,-1.3974)--(-0.2451,-1.3928)--(-0.2731,-1.3875)--(-0.3011,-1.3817);
 \draw [color=brown]  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw [color=green] (0.540,0.841)--(0.557,0.830)--(0.574,0.819)--(0.590,0.807)--(0.606,0.795)--(0.622,0.783)--(0.638,0.770)--(0.654,0.757)--(0.669,0.744)--(0.684,0.730)--(0.698,0.716)--(0.712,0.702)--(0.727,0.687)--(0.740,0.672)--(0.754,0.657)--(0.767,0.642)--(0.780,0.626)--(0.792,0.610)--(0.804,0.594)--(0.816,0.578)--(0.828,0.561)--(0.839,0.544)--(0.850,0.527)--(0.860,0.510)--(0.870,0.493)--(0.880,0.475)--(0.889,0.457)--(0.898,0.439)--(0.907,0.421)--(0.915,0.402)--(0.923,0.384)--(0.931,0.365)--(0.938,0.346)--(0.945,0.327)--(0.951,0.308)--(0.957,0.289)--(0.963,0.269)--(0.968,0.250)--(0.973,0.230)--(0.978,0.211)--(0.982,0.191)--(0.985,0.171)--(0.989,0.151)--(0.991,0.131)--(0.994,0.111)--(0.996,0.0908)--(0.997,0.0706)--(0.999,0.0505)--(1.00,0.0303)--(1.00,0.0101)--(1.00,-0.0101)--(1.00,-0.0303)--(0.999,-0.0505)--(0.997,-0.0706)--(0.996,-0.0908)--(0.994,-0.111)--(0.991,-0.131)--(0.989,-0.151)--(0.985,-0.171)--(0.982,-0.191)--(0.978,-0.211)--(0.973,-0.230)--(0.968,-0.250)--(0.963,-0.269)--(0.957,-0.289)--(0.951,-0.308)--(0.945,-0.327)--(0.938,-0.346)--(0.931,-0.365)--(0.923,-0.384)--(0.915,-0.402)--(0.907,-0.421)--(0.898,-0.439)--(0.889,-0.457)--(0.880,-0.475)--(0.870,-0.493)--(0.860,-0.510)--(0.850,-0.527)--(0.839,-0.544)--(0.828,-0.561)--(0.816,-0.578)--(0.804,-0.594)--(0.792,-0.610)--(0.780,-0.626)--(0.767,-0.642)--(0.754,-0.657)--(0.740,-0.672)--(0.727,-0.687)--(0.712,-0.702)--(0.698,-0.716)--(0.684,-0.730)--(0.669,-0.744)--(0.654,-0.757)--(0.638,-0.770)--(0.622,-0.783)--(0.606,-0.795)--(0.590,-0.807)--(0.574,-0.819)--(0.557,-0.830)--(0.540,-0.841);
-\draw [color=brown]  (1.0000,0) node [rotate=0] {$\bullet$};
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+\draw [color=green] (0.5403,0.8414)--(0.5571,0.8303)--(0.5738,0.8189)--(0.5902,0.8072)--(0.6064,0.7951)--(0.6224,0.7826)--(0.6380,0.7699)--(0.6535,0.7569)--(0.6686,0.7435)--(0.6835,0.7299)--(0.6981,0.7159)--(0.7124,0.7016)--(0.7265,0.6871)--(0.7402,0.6723)--(0.7536,0.6572)--(0.7667,0.6418)--(0.7796,0.6262)--(0.7920,0.6104)--(0.8042,0.5942)--(0.8161,0.5779)--(0.8276,0.5613)--(0.8387,0.5444)--(0.8496,0.5274)--(0.8600,0.5101)--(0.8702,0.4926)--(0.8799,0.4749)--(0.8894,0.4571)--(0.8984,0.4390)--(0.9071,0.4208)--(0.9154,0.4024)--(0.9234,0.3838)--(0.9309,0.3650)--(0.9381,0.3462)--(0.9449,0.3271)--(0.9513,0.3080)--(0.9574,0.2887)--(0.9630,0.2693)--(0.9682,0.2498)--(0.9731,0.2302)--(0.9775,0.2105)--(0.9816,0.1907)--(0.9852,0.1708)--(0.9885,0.1509)--(0.9913,0.1309)--(0.9938,0.1108)--(0.9958,0.0907)--(0.9975,0.0706)--(0.9987,0.0504)--(0.9995,0.0302)--(0.9999,0.0101)--(0.9999,-0.0101)--(0.9995,-0.0302)--(0.9987,-0.0504)--(0.9975,-0.0706)--(0.9958,-0.0907)--(0.9938,-0.1108)--(0.9913,-0.1309)--(0.9885,-0.1509)--(0.9852,-0.1708)--(0.9816,-0.1907)--(0.9775,-0.2105)--(0.9731,-0.2302)--(0.9682,-0.2498)--(0.9630,-0.2693)--(0.9574,-0.2887)--(0.9513,-0.3080)--(0.9449,-0.3271)--(0.9381,-0.3462)--(0.9309,-0.3650)--(0.9234,-0.3838)--(0.9154,-0.4024)--(0.9071,-0.4208)--(0.8984,-0.4390)--(0.8894,-0.4571)--(0.8799,-0.4749)--(0.8702,-0.4926)--(0.8600,-0.5101)--(0.8496,-0.5274)--(0.8387,-0.5444)--(0.8276,-0.5613)--(0.8161,-0.5779)--(0.8042,-0.5942)--(0.7920,-0.6104)--(0.7796,-0.6262)--(0.7667,-0.6418)--(0.7536,-0.6572)--(0.7402,-0.6723)--(0.7265,-0.6871)--(0.7124,-0.7016)--(0.6981,-0.7159)--(0.6835,-0.7299)--(0.6686,-0.7435)--(0.6535,-0.7569)--(0.6380,-0.7699)--(0.6224,-0.7826)--(0.6064,-0.7951)--(0.5902,-0.8072)--(0.5738,-0.8189)--(0.5571,-0.8303)--(0.5403,-0.8414);
+\draw [color=brown]  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=blue] (-0.3011,1.3817)--(-0.2731,1.3875)--(-0.2451,1.3928)--(-0.2169,1.3974)--(-0.1886,1.4015)--(-0.1602,1.4050)--(-0.1318,1.4080)--(-0.1034,1.4104)--(-0.0748,1.4122)--(-0.0463,1.4134)--(-0.0177,1.4141)--(0.0107,1.4141)--(0.0393,1.4136)--(0.0678,1.4125)--(0.0964,1.4109)--(0.1248,1.4086)--(0.1533,1.4058)--(0.1816,1.4024)--(0.2099,1.3985)--(0.2381,1.3940)--(0.2663,1.3889)--(0.2943,1.3832)--(0.3221,1.3770)--(0.3499,1.3702)--(0.3775,1.3628)--(0.4050,1.3549)--(0.4322,1.3465)--(0.4594,1.3375)--(0.4863,1.3279)--(0.5130,1.3178)--(0.5395,1.3072)--(0.5658,1.2960)--(0.5919,1.2843)--(0.6177,1.2721)--(0.6433,1.2594)--(0.6686,1.2461)--(0.6936,1.2323)--(0.7184,1.2181)--(0.7428,1.2033)--(0.7670,1.1881)--(0.7908,1.1723)--(0.8144,1.1561)--(0.8376,1.1394)--(0.8604,1.1223)--(0.8829,1.1047)--(0.9050,1.0866)--(0.9268,1.0681)--(0.9482,1.0492)--(0.9692,1.0298)--(0.9898,1.0100)--(1.0100,0.9898)--(1.0298,0.9692)--(1.0492,0.9482)--(1.0681,0.9268)--(1.0866,0.9050)--(1.1047,0.8829)--(1.1223,0.8604)--(1.1394,0.8376)--(1.1561,0.8144)--(1.1723,0.7908)--(1.1881,0.7670)--(1.2033,0.7428)--(1.2181,0.7184)--(1.2323,0.6936)--(1.2461,0.6686)--(1.2594,0.6433)--(1.2721,0.6177)--(1.2843,0.5919)--(1.2960,0.5658)--(1.3072,0.5395)--(1.3178,0.5130)--(1.3279,0.4863)--(1.3375,0.4594)--(1.3465,0.4322)--(1.3549,0.4050)--(1.3628,0.3775)--(1.3702,0.3499)--(1.3770,0.3221)--(1.3832,0.2943)--(1.3889,0.2663)--(1.3940,0.2381)--(1.3985,0.2099)--(1.4024,0.1816)--(1.4058,0.1533)--(1.4086,0.1248)--(1.4109,0.0964)--(1.4125,0.0678)--(1.4136,0.0393)--(1.4141,0.0107)--(1.4141,-0.0177)--(1.4134,-0.0463)--(1.4122,-0.0748)--(1.4104,-0.1034)--(1.4080,-0.1318)--(1.4050,-0.1602)--(1.4015,-0.1886)--(1.3974,-0.2169)--(1.3928,-0.2451)--(1.3875,-0.2731)--(1.3817,-0.3011);
 \draw [color=brown]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=cyan] (-1.143,1.922)--(-1.104,1.945)--(-1.064,1.967)--(-1.024,1.988)--(-0.9838,2.008)--(-0.9430,2.027)--(-0.9018,2.046)--(-0.8603,2.064)--(-0.8185,2.081)--(-0.7763,2.097)--(-0.7337,2.112)--(-0.6909,2.127)--(-0.6478,2.140)--(-0.6045,2.153)--(-0.5608,2.165)--(-0.5170,2.175)--(-0.4729,2.185)--(-0.4287,2.195)--(-0.3843,2.203)--(-0.3397,2.210)--(-0.2950,2.217)--(-0.2502,2.222)--(-0.2052,2.227)--(-0.1602,2.230)--(-0.1151,2.233)--(-0.06998,2.235)--(-0.02482,2.236)--(0.02035,2.236)--(0.06552,2.235)--(0.1107,2.233)--(0.1557,2.231)--(0.2008,2.227)--(0.2457,2.223)--(0.2906,2.217)--(0.3353,2.211)--(0.3799,2.204)--(0.4243,2.195)--(0.4686,2.186)--(0.5127,2.177)--(0.5565,2.166)--(0.6002,2.154)--(0.6435,2.141)--(0.6867,2.128)--(0.7295,2.114)--(0.7721,2.099)--(0.8143,2.083)--(0.8562,2.066)--(0.8978,2.048)--(0.9389,2.029)--(0.9797,2.010)--(1.020,1.990)--(1.060,1.969)--(1.100,1.947)--(1.139,1.924)--(1.177,1.901)--(1.216,1.877)--(1.253,1.852)--(1.290,1.826)--(1.327,1.800)--(1.363,1.773)--(1.399,1.745)--(1.434,1.716)--(1.468,1.687)--(1.502,1.657)--(1.535,1.626)--(1.567,1.595)--(1.599,1.563)--(1.631,1.530)--(1.661,1.497)--(1.691,1.463)--(1.720,1.429)--(1.749,1.393)--(1.777,1.358)--(1.804,1.322)--(1.830,1.285)--(1.856,1.248)--(1.880,1.210)--(1.904,1.172)--(1.928,1.133)--(1.950,1.094)--(1.972,1.054)--(1.993,1.014)--(2.013,0.9738)--(2.032,0.9329)--(2.051,0.8917)--(2.068,0.8501)--(2.085,0.8081)--(2.101,0.7658)--(2.116,0.7233)--(2.130,0.6804)--(2.143,0.6372)--(2.156,0.5938)--(2.167,0.5501)--(2.178,0.5062)--(2.188,0.4621)--(2.197,0.4178)--(2.205,0.3734)--(2.212,0.3287)--(2.218,0.2840)--(2.223,0.2391);
+\draw [color=cyan] (-1.1426,1.9220)--(-1.1035,1.9447)--(-1.0640,1.9666)--(-1.0241,1.9877)--(-0.9837,2.0080)--(-0.9429,2.0275)--(-0.9018,2.0461)--(-0.8603,2.0639)--(-0.8184,2.0808)--(-0.7762,2.0970)--(-0.7337,2.1122)--(-0.6909,2.1266)--(-0.6478,2.1401)--(-0.6044,2.1528)--(-0.5608,2.1645)--(-0.5170,2.1754)--(-0.4729,2.1854)--(-0.4287,2.1945)--(-0.3842,2.2027)--(-0.3397,2.2101)--(-0.2949,2.2165)--(-0.2501,2.2220)--(-0.2052,2.2266)--(-0.1602,2.2303)--(-0.1151,2.2331)--(-0.0699,2.2349)--(-0.0248,2.2359)--(0.0203,2.2359)--(0.0655,2.2351)--(0.1106,2.2333)--(0.1557,2.2306)--(0.2007,2.2270)--(0.2457,2.2225)--(0.2905,2.2171)--(0.3352,2.2107)--(0.3798,2.2035)--(0.4243,2.1954)--(0.4685,2.1864)--(0.5126,2.1765)--(0.5565,2.1657)--(0.6001,2.1540)--(0.6435,2.1414)--(0.6866,2.1280)--(0.7295,2.1137)--(0.7720,2.0985)--(0.8143,2.0825)--(0.8562,2.0656)--(0.8977,2.0479)--(0.9389,2.0293)--(0.9797,2.0099)--(1.0201,1.9897)--(1.0601,1.9687)--(1.0996,1.9469)--(1.1387,1.9243)--(1.1774,1.9009)--(1.2155,1.8767)--(1.2532,1.8518)--(1.2904,1.8261)--(1.3270,1.7997)--(1.3631,1.7725)--(1.3986,1.7446)--(1.4336,1.7160)--(1.4679,1.6867)--(1.5017,1.6567)--(1.5349,1.6260)--(1.5674,1.5947)--(1.5993,1.5627)--(1.6305,1.5300)--(1.6611,1.4968)--(1.6910,1.4629)--(1.7202,1.4285)--(1.7487,1.3934)--(1.7765,1.3578)--(1.8036,1.3217)--(1.8299,1.2849)--(1.8555,1.2477)--(1.8803,1.2100)--(1.9044,1.1717)--(1.9277,1.1330)--(1.9502,1.0939)--(1.9719,1.0542)--(1.9928,1.0142)--(2.0128,0.9737)--(2.0321,0.9329)--(2.0505,0.8916)--(2.0681,0.8500)--(2.0849,0.8081)--(2.1008,0.7658)--(2.1158,0.7232)--(2.1300,0.6803)--(2.1433,0.6371)--(2.1557,0.5937)--(2.1673,0.5500)--(2.1780,0.5062)--(2.1877,0.4621)--(2.1966,0.4178)--(2.2046,0.3733)--(2.2117,0.3287)--(2.2179,0.2839)--(2.2232,0.2391);
 \draw [color=brown]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_TriangleUV.pstricks.recall b/src_phystricks/Fig_TriangleUV.pstricks.recall
index 69dd3cc1b..d76fff3b4 100644
--- a/src_phystricks/Fig_TriangleUV.pstricks.recall
+++ b/src_phystricks/Fig_TriangleUV.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -89,16 +89,16 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,0) -- (0,3.00) -- (0,3.00) -- (3.00,0) -- (3.00,0) -- (0,0) --  cycle;
+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.0000,0.0000) -- (0.0000,3.0000) -- (0.0000,3.0000) -- (3.0000,0.0000) -- (3.0000,0.0000) -- (0.0000,0.0000) --  cycle;
 
-\draw [color=green,->,>=latex] (0,0) -- (1.0000,0);
-\draw (1.0000,-0.20595) node {\( e_u\)};
-\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
-\draw (-0.26708,1.0000) node {\( e_v\)};
+\draw [color=green,->,>=latex] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw (1.0000,-0.2059) node {\( e_u\)};
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (0.0000,1.0000);
+\draw (-0.2670,1.0000) node {\( e_v\)};
 \draw [color=green,->,>=latex] (1.7121,1.7121) -- (2.4192,2.4192);
-\draw (2.2468,2.5685) node {\( \nu\)};
+\draw (2.2467,2.5685) node {\( \nu\)};
 \draw [color=red,->,>=latex] (1.7121,1.7121) -- (1.0050,2.4192);
-\draw (0.80232,2.6147) node {\( T\)};
+\draw (0.8023,2.6146) node {\( T\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall b/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall
index a903ec27e..923951dc0 100644
--- a/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall
+++ b/src_phystricks/Fig_UGCFooQoCihh.pstricks.recall
@@ -96,37 +96,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.2067);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (9.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.2067);
 %DEFAULT
-\draw []  (0,1.3534) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,1.3533) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,2.7067) node [rotate=0] {$\bullet$};
 \draw []  (2.0000,2.7067) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,1.3534) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (7.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (8.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (9.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
+\draw []  (3.0000,1.3533) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall b/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall
index 55b304951..f5f510f2a 100644
--- a/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall
+++ b/src_phystricks/Fig_UIEHooSlbzIJ.pstricks.recall
@@ -116,52 +116,52 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (14.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (14.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
-\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,4.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,2.4000) node [rotate=0] {$\bullet$};
 \draw []  (2.0000,1.4400) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0.86400) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,0.51840) node [rotate=0] {$\bullet$};
-\draw []  (5.0000,0.31104) node [rotate=0] {$\bullet$};
-\draw []  (6.0000,0.18662) node [rotate=0] {$\bullet$};
-\draw []  (7.0000,0.11197) node [rotate=0] {$\bullet$};
-\draw []  (8.0000,0.067185) node [rotate=0] {$\bullet$};
-\draw []  (9.0000,0.040311) node [rotate=0] {$\bullet$};
-\draw []  (10.000,0.024186) node [rotate=0] {$\bullet$};
-\draw []  (11.000,0.014512) node [rotate=0] {$\bullet$};
-\draw []  (12.000,0.0087071) node [rotate=0] {$\bullet$};
-\draw []  (13.000,0.0052243) node [rotate=0] {$\bullet$};
-\draw []  (14.000,0.0031346) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.000,-0.31492) node {$ 11 $};
-\draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.000,-0.31492) node {$ 12 $};
-\draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.000,-0.31492) node {$ 13 $};
-\draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (14.000,-0.31492) node {$ 14 $};
-\draw [] (14.0,-0.100) -- (14.0,0.100);
+\draw []  (3.0000,0.8640) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.5184) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0.3110) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0.1866) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0.1119) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0.0671) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0.0403) node [rotate=0] {$\bullet$};
+\draw []  (10.000,0.0241) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0.0145) node [rotate=0] {$\bullet$};
+\draw []  (12.000,0.0087) node [rotate=0] {$\bullet$};
+\draw []  (13.000,0.0052) node [rotate=0] {$\bullet$};
+\draw []  (14.000,0.0031) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (11.000,-0.3149) node {$ 11 $};
+\draw [] (11.000,-0.1000) -- (11.000,0.1000);
+\draw (12.000,-0.3149) node {$ 12 $};
+\draw [] (12.000,-0.1000) -- (12.000,0.1000);
+\draw (13.000,-0.3149) node {$ 13 $};
+\draw [] (13.000,-0.1000) -- (13.000,0.1000);
+\draw (14.000,-0.3149) node {$ 14 $};
+\draw [] (14.000,-0.1000) -- (14.000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall b/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall
index 003cc3578..0d2922eb9 100644
--- a/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall
+++ b/src_phystricks/Fig_UMEBooVTMyfD.pstricks.recall
@@ -100,35 +100,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
-\draw [color=blue] (0,6.0000)--(0.10101,4.9025)--(0.20202,4.0057)--(0.30303,3.2730)--(0.40404,2.6743)--(0.50505,2.1851)--(0.60606,1.7854)--(0.70707,1.4588)--(0.80808,1.1920)--(0.90909,0.97392)--(1.0101,0.79577)--(1.1111,0.65021)--(1.2121,0.53127)--(1.3131,0.43409)--(1.4141,0.35469)--(1.5152,0.28981)--(1.6162,0.23679)--(1.7172,0.19348)--(1.8182,0.15809)--(1.9192,0.12917)--(2.0202,0.10554)--(2.1212,0.086236)--(2.2222,0.070462)--(2.3232,0.057573)--(2.4242,0.047041)--(2.5253,0.038437)--(2.6263,0.031406)--(2.7273,0.025661)--(2.8283,0.020967)--(2.9293,0.017132)--(3.0303,0.013998)--(3.1313,0.011437)--(3.2323,0.0093452)--(3.3333,0.0076358)--(3.4343,0.0062390)--(3.5354,0.0050978)--(3.6364,0.0041653)--(3.7374,0.0034034)--(3.8384,0.0027808)--(3.9394,0.0022722)--(4.0404,0.0018565)--(4.1414,0.0015169)--(4.2424,0.0012394)--(4.3434,0.0010127)--(4.4444,0)--(4.5455,0)--(4.6465,0)--(4.7475,0)--(4.8485,0)--(4.9495,0)--(5.0505,0)--(5.1515,0)--(5.2525,0)--(5.3535,0)--(5.4545,0)--(5.5556,0)--(5.6566,0)--(5.7576,0)--(5.8586,0)--(5.9596,0)--(6.0606,0)--(6.1616,0)--(6.2626,0)--(6.3636,0)--(6.4646,0)--(6.5657,0)--(6.6667,0)--(6.7677,0)--(6.8687,0)--(6.9697,0)--(7.0707,0)--(7.1717,0)--(7.2727,0)--(7.3737,0)--(7.4747,0)--(7.5758,0)--(7.6768,0)--(7.7778,0)--(7.8788,0)--(7.9798,0)--(8.0808,0)--(8.1818,0)--(8.2828,0)--(8.3838,0)--(8.4848,0)--(8.5859,0)--(8.6869,0)--(8.7879,0)--(8.8889,0)--(8.9899,0)--(9.0909,0)--(9.1919,0)--(9.2929,0)--(9.3939,0)--(9.4949,0)--(9.5960,0)--(9.6970,0)--(9.7980,0)--(9.8990,0)--(10.000,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,6.0000) node {$ 2 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw [color=blue] (0.0000,6.0000)--(0.1010,4.9024)--(0.2020,4.0057)--(0.3030,3.2729)--(0.4040,2.6742)--(0.5050,2.1850)--(0.6060,1.7853)--(0.7070,1.4588)--(0.8080,1.1919)--(0.9090,0.9739)--(1.0101,0.7957)--(1.1111,0.6502)--(1.2121,0.5312)--(1.3131,0.4340)--(1.4141,0.3546)--(1.5151,0.2898)--(1.6161,0.2367)--(1.7171,0.1934)--(1.8181,0.1580)--(1.9191,0.1291)--(2.0202,0.1055)--(2.1212,0.0862)--(2.2222,0.0704)--(2.3232,0.0575)--(2.4242,0.0470)--(2.5252,0.0384)--(2.6262,0.0314)--(2.7272,0.0256)--(2.8282,0.0209)--(2.9292,0.0171)--(3.0303,0.0139)--(3.1313,0.0114)--(3.2323,0.0093)--(3.3333,0.0076)--(3.4343,0.0062)--(3.5353,0.0050)--(3.6363,0.0041)--(3.7373,0.0034)--(3.8383,0.0027)--(3.9393,0.0022)--(4.0404,0.0018)--(4.1414,0.0015)--(4.2424,0.0012)--(4.3434,0.0010)--(4.4444,0.0000)--(4.5454,0.0000)--(4.6464,0.0000)--(4.7474,0.0000)--(4.8484,0.0000)--(4.9494,0.0000)--(5.0505,0.0000)--(5.1515,0.0000)--(5.2525,0.0000)--(5.3535,0.0000)--(5.4545,0.0000)--(5.5555,0.0000)--(5.6565,0.0000)--(5.7575,0.0000)--(5.8585,0.0000)--(5.9595,0.0000)--(6.0606,0.0000)--(6.1616,0.0000)--(6.2626,0.0000)--(6.3636,0.0000)--(6.4646,0.0000)--(6.5656,0.0000)--(6.6666,0.0000)--(6.7676,0.0000)--(6.8686,0.0000)--(6.9696,0.0000)--(7.0707,0.0000)--(7.1717,0.0000)--(7.2727,0.0000)--(7.3737,0.0000)--(7.4747,0.0000)--(7.5757,0.0000)--(7.6767,0.0000)--(7.7777,0.0000)--(7.8787,0.0000)--(7.9797,0.0000)--(8.0808,0.0000)--(8.1818,0.0000)--(8.2828,0.0000)--(8.3838,0.0000)--(8.4848,0.0000)--(8.5858,0.0000)--(8.6868,0.0000)--(8.7878,0.0000)--(8.8888,0.0000)--(8.9898,0.0000)--(9.0909,0.0000)--(9.1919,0.0000)--(9.2929,0.0000)--(9.3939,0.0000)--(9.4949,0.0000)--(9.5959,0.0000)--(9.6969,0.0000)--(9.7979,0.0000)--(9.8989,0.0000)--(10.000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,6.0000) node {$ 2 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall b/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall
index 8310c82c8..46b815d49 100644
--- a/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall
+++ b/src_phystricks/Fig_UNVooMsXxHa.pstricks.recall
@@ -107,83 +107,83 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-3.00,-5.00) -- (-3.00,5.00);
-\draw [color=gray,style=solid] (-2.00,-5.00) -- (-2.00,5.00);
-\draw [color=gray,style=solid] (-1.00,-5.00) -- (-1.00,5.00);
-\draw [color=gray,style=solid] (0,-5.00) -- (0,5.00);
-\draw [color=gray,style=solid] (1.00,-5.00) -- (1.00,5.00);
-\draw [color=gray,style=solid] (2.00,-5.00) -- (2.00,5.00);
-\draw [color=gray,style=solid] (3.00,-5.00) -- (3.00,5.00);
-\draw [color=gray,style=dotted] (-2.50,-5.00) -- (-2.50,5.00);
-\draw [color=gray,style=dotted] (-1.50,-5.00) -- (-1.50,5.00);
-\draw [color=gray,style=dotted] (-0.500,-5.00) -- (-0.500,5.00);
-\draw [color=gray,style=dotted] (0.500,-5.00) -- (0.500,5.00);
-\draw [color=gray,style=dotted] (1.50,-5.00) -- (1.50,5.00);
-\draw [color=gray,style=dotted] (2.50,-5.00) -- (2.50,5.00);
-\draw [color=gray,style=dotted] (-3.00,-4.50) -- (3.00,-4.50);
-\draw [color=gray,style=dotted] (-3.00,-3.50) -- (3.00,-3.50);
-\draw [color=gray,style=dotted] (-3.00,-2.50) -- (3.00,-2.50);
-\draw [color=gray,style=dotted] (-3.00,-1.50) -- (3.00,-1.50);
-\draw [color=gray,style=dotted] (-3.00,-0.500) -- (3.00,-0.500);
-\draw [color=gray,style=dotted] (-3.00,0.500) -- (3.00,0.500);
-\draw [color=gray,style=dotted] (-3.00,1.50) -- (3.00,1.50);
-\draw [color=gray,style=dotted] (-3.00,2.50) -- (3.00,2.50);
-\draw [color=gray,style=dotted] (-3.00,3.50) -- (3.00,3.50);
-\draw [color=gray,style=dotted] (-3.00,4.50) -- (3.00,4.50);
-\draw [color=gray,style=solid] (-3.00,-5.00) -- (3.00,-5.00);
-\draw [color=gray,style=solid] (-3.00,-4.00) -- (3.00,-4.00);
-\draw [color=gray,style=solid] (-3.00,-3.00) -- (3.00,-3.00);
-\draw [color=gray,style=solid] (-3.00,-2.00) -- (3.00,-2.00);
-\draw [color=gray,style=solid] (-3.00,-1.00) -- (3.00,-1.00);
-\draw [color=gray,style=solid] (-3.00,0) -- (3.00,0);
-\draw [color=gray,style=solid] (-3.00,1.00) -- (3.00,1.00);
-\draw [color=gray,style=solid] (-3.00,2.00) -- (3.00,2.00);
-\draw [color=gray,style=solid] (-3.00,3.00) -- (3.00,3.00);
-\draw [color=gray,style=solid] (-3.00,4.00) -- (3.00,4.00);
-\draw [color=gray,style=solid] (-3.00,5.00) -- (3.00,5.00);
+\draw [color=gray,style=solid] (-3.0000,-5.0000) -- (-3.0000,5.0000);
+\draw [color=gray,style=solid] (-2.0000,-5.0000) -- (-2.0000,5.0000);
+\draw [color=gray,style=solid] (-1.0000,-5.0000) -- (-1.0000,5.0000);
+\draw [color=gray,style=solid] (0.0000,-5.0000) -- (0.0000,5.0000);
+\draw [color=gray,style=solid] (1.0000,-5.0000) -- (1.0000,5.0000);
+\draw [color=gray,style=solid] (2.0000,-5.0000) -- (2.0000,5.0000);
+\draw [color=gray,style=solid] (3.0000,-5.0000) -- (3.0000,5.0000);
+\draw [color=gray,style=dotted] (-2.5000,-5.0000) -- (-2.5000,5.0000);
+\draw [color=gray,style=dotted] (-1.5000,-5.0000) -- (-1.5000,5.0000);
+\draw [color=gray,style=dotted] (-0.5000,-5.0000) -- (-0.5000,5.0000);
+\draw [color=gray,style=dotted] (0.5000,-5.0000) -- (0.5000,5.0000);
+\draw [color=gray,style=dotted] (1.5000,-5.0000) -- (1.5000,5.0000);
+\draw [color=gray,style=dotted] (2.5000,-5.0000) -- (2.5000,5.0000);
+\draw [color=gray,style=dotted] (-3.0000,-4.5000) -- (3.0000,-4.5000);
+\draw [color=gray,style=dotted] (-3.0000,-3.5000) -- (3.0000,-3.5000);
+\draw [color=gray,style=dotted] (-3.0000,-2.5000) -- (3.0000,-2.5000);
+\draw [color=gray,style=dotted] (-3.0000,-1.5000) -- (3.0000,-1.5000);
+\draw [color=gray,style=dotted] (-3.0000,-0.5000) -- (3.0000,-0.5000);
+\draw [color=gray,style=dotted] (-3.0000,0.5000) -- (3.0000,0.5000);
+\draw [color=gray,style=dotted] (-3.0000,1.5000) -- (3.0000,1.5000);
+\draw [color=gray,style=dotted] (-3.0000,2.5000) -- (3.0000,2.5000);
+\draw [color=gray,style=dotted] (-3.0000,3.5000) -- (3.0000,3.5000);
+\draw [color=gray,style=dotted] (-3.0000,4.5000) -- (3.0000,4.5000);
+\draw [color=gray,style=solid] (-3.0000,-5.0000) -- (3.0000,-5.0000);
+\draw [color=gray,style=solid] (-3.0000,-4.0000) -- (3.0000,-4.0000);
+\draw [color=gray,style=solid] (-3.0000,-3.0000) -- (3.0000,-3.0000);
+\draw [color=gray,style=solid] (-3.0000,-2.0000) -- (3.0000,-2.0000);
+\draw [color=gray,style=solid] (-3.0000,-1.0000) -- (3.0000,-1.0000);
+\draw [color=gray,style=solid] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [color=gray,style=solid] (-3.0000,1.0000) -- (3.0000,1.0000);
+\draw [color=gray,style=solid] (-3.0000,2.0000) -- (3.0000,2.0000);
+\draw [color=gray,style=solid] (-3.0000,3.0000) -- (3.0000,3.0000);
+\draw [color=gray,style=solid] (-3.0000,4.0000) -- (3.0000,4.0000);
+\draw [color=gray,style=solid] (-3.0000,5.0000) -- (3.0000,5.0000);
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-5.5000) -- (0,5.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-5.5000) -- (0.0000,5.5000);
 %DEFAULT
 \draw (-3.4183,4.9138) node {\( y=cosh(x)\)};
 
-\draw [color=blue] (-2.275,4.914)--(-2.229,4.698)--(-2.183,4.492)--(-2.137,4.295)--(-2.091,4.108)--(-2.045,3.929)--(-1.999,3.758)--(-1.953,3.596)--(-1.907,3.441)--(-1.861,3.293)--(-1.815,3.152)--(-1.769,3.018)--(-1.723,2.891)--(-1.677,2.769)--(-1.631,2.653)--(-1.585,2.543)--(-1.539,2.438)--(-1.493,2.339)--(-1.448,2.244)--(-1.402,2.154)--(-1.356,2.068)--(-1.310,1.987)--(-1.264,1.911)--(-1.218,1.838)--(-1.172,1.769)--(-1.126,1.704)--(-1.080,1.642)--(-1.034,1.584)--(-0.9880,1.529)--(-0.9420,1.478)--(-0.8961,1.429)--(-0.8501,1.384)--(-0.8042,1.341)--(-0.7582,1.301)--(-0.7123,1.265)--(-0.6663,1.230)--(-0.6204,1.199)--(-0.5744,1.170)--(-0.5285,1.143)--(-0.4825,1.119)--(-0.4366,1.097)--(-0.3906,1.077)--(-0.3446,1.060)--(-0.2987,1.045)--(-0.2527,1.032)--(-0.2068,1.021)--(-0.1608,1.013)--(-0.1149,1.007)--(-0.06893,1.002)--(-0.02298,1.000)--(0.02298,1.000)--(0.06893,1.002)--(0.1149,1.007)--(0.1608,1.013)--(0.2068,1.021)--(0.2527,1.032)--(0.2987,1.045)--(0.3446,1.060)--(0.3906,1.077)--(0.4366,1.097)--(0.4825,1.119)--(0.5285,1.143)--(0.5744,1.170)--(0.6204,1.199)--(0.6663,1.230)--(0.7123,1.265)--(0.7582,1.301)--(0.8042,1.341)--(0.8501,1.384)--(0.8961,1.429)--(0.9420,1.478)--(0.9880,1.529)--(1.034,1.584)--(1.080,1.642)--(1.126,1.704)--(1.172,1.769)--(1.218,1.838)--(1.264,1.911)--(1.310,1.987)--(1.356,2.068)--(1.402,2.154)--(1.448,2.244)--(1.493,2.339)--(1.539,2.438)--(1.585,2.543)--(1.631,2.653)--(1.677,2.769)--(1.723,2.891)--(1.769,3.018)--(1.815,3.152)--(1.861,3.293)--(1.907,3.441)--(1.953,3.596)--(1.999,3.758)--(2.045,3.929)--(2.091,4.108)--(2.137,4.295)--(2.183,4.492)--(2.229,4.698)--(2.275,4.914);
-\draw (-3.4233,-4.8110) node {\( y=sinh(x)\)};
+\draw [color=blue] (-2.2746,4.9138)--(-2.2287,4.6978)--(-2.1827,4.4917)--(-2.1368,4.2952)--(-2.0908,4.1077)--(-2.0449,3.9289)--(-1.9989,3.7584)--(-1.9530,3.5958)--(-1.9070,3.4408)--(-1.8611,3.2931)--(-1.8151,3.1523)--(-1.7691,3.0182)--(-1.7232,2.8905)--(-1.6772,2.7689)--(-1.6313,2.6531)--(-1.5853,2.5430)--(-1.5394,2.4382)--(-1.4934,2.3385)--(-1.4475,2.2438)--(-1.4015,2.1538)--(-1.3556,2.0684)--(-1.3096,1.9874)--(-1.2637,1.9105)--(-1.2177,1.8377)--(-1.1718,1.7688)--(-1.1258,1.7036)--(-1.0798,1.6420)--(-1.0339,1.5838)--(-0.9879,1.5290)--(-0.9420,1.4775)--(-0.8960,1.4290)--(-0.8501,1.3836)--(-0.8041,1.3411)--(-0.7582,1.3014)--(-0.7122,1.2645)--(-0.6663,1.2303)--(-0.6203,1.1986)--(-0.5744,1.1695)--(-0.5284,1.1429)--(-0.4825,1.1186)--(-0.4365,1.0968)--(-0.3906,1.0772)--(-0.3446,1.0599)--(-0.2986,1.0449)--(-0.2527,1.0321)--(-0.2067,1.0214)--(-0.1608,1.0129)--(-0.1148,1.0066)--(-0.0689,1.0023)--(-0.0229,1.0002)--(0.0229,1.0002)--(0.0689,1.0023)--(0.1148,1.0066)--(0.1608,1.0129)--(0.2067,1.0214)--(0.2527,1.0321)--(0.2986,1.0449)--(0.3446,1.0599)--(0.3906,1.0772)--(0.4365,1.0968)--(0.4825,1.1186)--(0.5284,1.1429)--(0.5744,1.1695)--(0.6203,1.1986)--(0.6663,1.2303)--(0.7122,1.2645)--(0.7582,1.3014)--(0.8041,1.3411)--(0.8501,1.3836)--(0.8960,1.4290)--(0.9420,1.4775)--(0.9879,1.5290)--(1.0339,1.5838)--(1.0798,1.6420)--(1.1258,1.7036)--(1.1718,1.7688)--(1.2177,1.8377)--(1.2637,1.9105)--(1.3096,1.9874)--(1.3556,2.0684)--(1.4015,2.1538)--(1.4475,2.2438)--(1.4934,2.3385)--(1.5394,2.4382)--(1.5853,2.5430)--(1.6313,2.6531)--(1.6772,2.7689)--(1.7232,2.8905)--(1.7691,3.0182)--(1.8151,3.1523)--(1.8611,3.2931)--(1.9070,3.4408)--(1.9530,3.5958)--(1.9989,3.7584)--(2.0449,3.9289)--(2.0908,4.1077)--(2.1368,4.2952)--(2.1827,4.4917)--(2.2287,4.6978)--(2.2746,4.9138);
+\draw (-3.4233,-4.8109) node {\( y=sinh(x)\)};
 
-\draw [color=blue] (-2.275,-4.811)--(-2.229,-4.590)--(-2.183,-4.379)--(-2.137,-4.177)--(-2.091,-3.984)--(-2.045,-3.800)--(-1.999,-3.623)--(-1.953,-3.454)--(-1.907,-3.292)--(-1.861,-3.138)--(-1.815,-2.990)--(-1.769,-2.848)--(-1.723,-2.712)--(-1.677,-2.582)--(-1.631,-2.458)--(-1.585,-2.338)--(-1.539,-2.224)--(-1.493,-2.114)--(-1.448,-2.009)--(-1.402,-1.908)--(-1.356,-1.811)--(-1.310,-1.718)--(-1.264,-1.628)--(-1.218,-1.542)--(-1.172,-1.459)--(-1.126,-1.379)--(-1.080,-1.302)--(-1.034,-1.228)--(-0.9880,-1.157)--(-0.9420,-1.088)--(-0.8961,-1.021)--(-0.8501,-0.9563)--(-0.8042,-0.8937)--(-0.7582,-0.8330)--(-0.7123,-0.7740)--(-0.6663,-0.7167)--(-0.6204,-0.6609)--(-0.5744,-0.6065)--(-0.5285,-0.5534)--(-0.4825,-0.5014)--(-0.4366,-0.4506)--(-0.3906,-0.4006)--(-0.3446,-0.3515)--(-0.2987,-0.3032)--(-0.2527,-0.2554)--(-0.2068,-0.2083)--(-0.1608,-0.1615)--(-0.1149,-0.1151)--(-0.06893,-0.06898)--(-0.02298,-0.02298)--(0.02298,0.02298)--(0.06893,0.06898)--(0.1149,0.1151)--(0.1608,0.1615)--(0.2068,0.2083)--(0.2527,0.2554)--(0.2987,0.3032)--(0.3446,0.3515)--(0.3906,0.4006)--(0.4366,0.4506)--(0.4825,0.5014)--(0.5285,0.5534)--(0.5744,0.6065)--(0.6204,0.6609)--(0.6663,0.7167)--(0.7123,0.7740)--(0.7582,0.8330)--(0.8042,0.8937)--(0.8501,0.9563)--(0.8961,1.021)--(0.9420,1.088)--(0.9880,1.157)--(1.034,1.228)--(1.080,1.302)--(1.126,1.379)--(1.172,1.459)--(1.218,1.542)--(1.264,1.628)--(1.310,1.718)--(1.356,1.811)--(1.402,1.908)--(1.448,2.009)--(1.493,2.114)--(1.539,2.224)--(1.585,2.338)--(1.631,2.458)--(1.677,2.582)--(1.723,2.712)--(1.769,2.848)--(1.815,2.990)--(1.861,3.138)--(1.907,3.292)--(1.953,3.454)--(1.999,3.623)--(2.045,3.800)--(2.091,3.984)--(2.137,4.177)--(2.183,4.379)--(2.229,4.590)--(2.275,4.811);
+\draw [color=blue] (-2.2746,-4.8109)--(-2.2287,-4.5901)--(-2.1827,-4.3790)--(-2.1368,-4.1772)--(-2.0908,-3.9841)--(-2.0449,-3.7995)--(-1.9989,-3.6229)--(-1.9530,-3.4540)--(-1.9070,-3.2923)--(-1.8611,-3.1376)--(-1.8151,-2.9895)--(-1.7691,-2.8478)--(-1.7232,-2.7120)--(-1.6772,-2.5820)--(-1.6313,-2.4575)--(-1.5853,-2.3381)--(-1.5394,-2.2237)--(-1.4934,-2.1139)--(-1.4475,-2.0087)--(-1.4015,-1.9076)--(-1.3556,-1.8106)--(-1.3096,-1.7175)--(-1.2637,-1.6279)--(-1.2177,-1.5418)--(-1.1718,-1.4590)--(-1.1258,-1.3792)--(-1.0798,-1.3023)--(-1.0339,-1.2282)--(-0.9879,-1.1567)--(-0.9420,-1.0876)--(-0.8960,-1.0209)--(-0.8501,-0.9562)--(-0.8041,-0.8937)--(-0.7582,-0.8329)--(-0.7122,-0.7740)--(-0.6663,-0.7167)--(-0.6203,-0.6609)--(-0.5744,-0.6065)--(-0.5284,-0.5534)--(-0.4825,-0.5014)--(-0.4365,-0.4505)--(-0.3906,-0.4006)--(-0.3446,-0.3515)--(-0.2986,-0.3031)--(-0.2527,-0.2554)--(-0.2067,-0.2082)--(-0.1608,-0.1615)--(-0.1148,-0.1151)--(-0.0689,-0.0689)--(-0.0229,-0.0229)--(0.0229,0.0229)--(0.0689,0.0689)--(0.1148,0.1151)--(0.1608,0.1615)--(0.2067,0.2082)--(0.2527,0.2554)--(0.2986,0.3031)--(0.3446,0.3515)--(0.3906,0.4006)--(0.4365,0.4505)--(0.4825,0.5014)--(0.5284,0.5534)--(0.5744,0.6065)--(0.6203,0.6609)--(0.6663,0.7167)--(0.7122,0.7740)--(0.7582,0.8329)--(0.8041,0.8937)--(0.8501,0.9562)--(0.8960,1.0209)--(0.9420,1.0876)--(0.9879,1.1567)--(1.0339,1.2282)--(1.0798,1.3023)--(1.1258,1.3792)--(1.1718,1.4590)--(1.2177,1.5418)--(1.2637,1.6279)--(1.3096,1.7175)--(1.3556,1.8106)--(1.4015,1.9076)--(1.4475,2.0087)--(1.4934,2.1139)--(1.5394,2.2237)--(1.5853,2.3381)--(1.6313,2.4575)--(1.6772,2.5820)--(1.7232,2.7120)--(1.7691,2.8478)--(1.8151,2.9895)--(1.8611,3.1376)--(1.9070,3.2923)--(1.9530,3.4540)--(1.9989,3.6229)--(2.0449,3.7995)--(2.0908,3.9841)--(2.1368,4.1772)--(2.1827,4.3790)--(2.2287,4.5901)--(2.2746,4.8109);
 
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-5.0000) node {$ -5 $};
-\draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-5.0000) node {$ -5 $};
+\draw [] (-0.1000,-5.0000) -- (0.1000,-5.0000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall b/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall
index a266f3275..0a5257a4f 100644
--- a/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall
+++ b/src_phystricks/Fig_UQZooGFLNEq.pstricks.recall
@@ -108,37 +108,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-2.0708) -- (0,2.0708);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0707) -- (0.0000,2.0707);
 %DEFAULT
 
-\draw [color=blue] (-5.000,-1.373)--(-4.899,-1.369)--(-4.798,-1.365)--(-4.697,-1.361)--(-4.596,-1.357)--(-4.495,-1.352)--(-4.394,-1.347)--(-4.293,-1.342)--(-4.192,-1.337)--(-4.091,-1.331)--(-3.990,-1.325)--(-3.889,-1.319)--(-3.788,-1.313)--(-3.687,-1.306)--(-3.586,-1.299)--(-3.485,-1.291)--(-3.384,-1.283)--(-3.283,-1.275)--(-3.182,-1.266)--(-3.081,-1.257)--(-2.980,-1.247)--(-2.879,-1.236)--(-2.778,-1.225)--(-2.677,-1.213)--(-2.576,-1.200)--(-2.475,-1.187)--(-2.374,-1.172)--(-2.273,-1.156)--(-2.172,-1.139)--(-2.071,-1.121)--(-1.970,-1.101)--(-1.869,-1.079)--(-1.768,-1.056)--(-1.667,-1.030)--(-1.566,-1.002)--(-1.465,-0.9717)--(-1.364,-0.9380)--(-1.263,-0.9010)--(-1.162,-0.8600)--(-1.061,-0.8148)--(-0.9596,-0.7648)--(-0.8586,-0.7095)--(-0.7576,-0.6483)--(-0.6566,-0.5810)--(-0.5556,-0.5071)--(-0.4545,-0.4266)--(-0.3535,-0.3398)--(-0.2525,-0.2474)--(-0.1515,-0.1504)--(-0.05051,-0.05046)--(0.05051,0.05046)--(0.1515,0.1504)--(0.2525,0.2474)--(0.3535,0.3398)--(0.4545,0.4266)--(0.5556,0.5071)--(0.6566,0.5810)--(0.7576,0.6483)--(0.8586,0.7095)--(0.9596,0.7648)--(1.061,0.8148)--(1.162,0.8600)--(1.263,0.9010)--(1.364,0.9380)--(1.465,0.9717)--(1.566,1.002)--(1.667,1.030)--(1.768,1.056)--(1.869,1.079)--(1.970,1.101)--(2.071,1.121)--(2.172,1.139)--(2.273,1.156)--(2.374,1.172)--(2.475,1.187)--(2.576,1.200)--(2.677,1.213)--(2.778,1.225)--(2.879,1.236)--(2.980,1.247)--(3.081,1.257)--(3.182,1.266)--(3.283,1.275)--(3.384,1.283)--(3.485,1.291)--(3.586,1.299)--(3.687,1.306)--(3.788,1.313)--(3.889,1.319)--(3.990,1.325)--(4.091,1.331)--(4.192,1.337)--(4.293,1.342)--(4.394,1.347)--(4.495,1.352)--(4.596,1.357)--(4.697,1.361)--(4.798,1.365)--(4.899,1.369)--(5.000,1.373);
-\draw [color=red,style=dashed] (-5.00,1.57) -- (5.00,1.57);
-\draw [color=red,style=dashed] (-5.00,-1.57) -- (5.00,-1.57);
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
-\draw [] (-0.100,1.57) -- (0.100,1.57);
+\draw [color=blue] (-5.0000,-1.3734)--(-4.8989,-1.3694)--(-4.7979,-1.3653)--(-4.6969,-1.3610)--(-4.5959,-1.3565)--(-4.4949,-1.3518)--(-4.3939,-1.3470)--(-4.2929,-1.3419)--(-4.1919,-1.3366)--(-4.0909,-1.3310)--(-3.9898,-1.3252)--(-3.8888,-1.3191)--(-3.7878,-1.3126)--(-3.6868,-1.3059)--(-3.5858,-1.2988)--(-3.4848,-1.2913)--(-3.3838,-1.2834)--(-3.2828,-1.2751)--(-3.1818,-1.2662)--(-3.0808,-1.2569)--(-2.9797,-1.2470)--(-2.8787,-1.2364)--(-2.7777,-1.2252)--(-2.6767,-1.2132)--(-2.5757,-1.2004)--(-2.4747,-1.1867)--(-2.3737,-1.1720)--(-2.2727,-1.1562)--(-2.1717,-1.1392)--(-2.0707,-1.1209)--(-1.9696,-1.1010)--(-1.8686,-1.0794)--(-1.7676,-1.0559)--(-1.6666,-1.0303)--(-1.5656,-1.0023)--(-1.4646,-0.9717)--(-1.3636,-0.9380)--(-1.2626,-0.9009)--(-1.1616,-0.8600)--(-1.0606,-0.8148)--(-0.9595,-0.7647)--(-0.8585,-0.7094)--(-0.7575,-0.6483)--(-0.6565,-0.5809)--(-0.5555,-0.5070)--(-0.4545,-0.4266)--(-0.3535,-0.3398)--(-0.2525,-0.2473)--(-0.1515,-0.1503)--(-0.0505,-0.0504)--(0.0505,0.0504)--(0.1515,0.1503)--(0.2525,0.2473)--(0.3535,0.3398)--(0.4545,0.4266)--(0.5555,0.5070)--(0.6565,0.5809)--(0.7575,0.6483)--(0.8585,0.7094)--(0.9595,0.7647)--(1.0606,0.8148)--(1.1616,0.8600)--(1.2626,0.9009)--(1.3636,0.9380)--(1.4646,0.9717)--(1.5656,1.0023)--(1.6666,1.0303)--(1.7676,1.0559)--(1.8686,1.0794)--(1.9696,1.1010)--(2.0707,1.1209)--(2.1717,1.1392)--(2.2727,1.1562)--(2.3737,1.1720)--(2.4747,1.1867)--(2.5757,1.2004)--(2.6767,1.2132)--(2.7777,1.2252)--(2.8787,1.2364)--(2.9797,1.2470)--(3.0808,1.2569)--(3.1818,1.2662)--(3.2828,1.2751)--(3.3838,1.2834)--(3.4848,1.2913)--(3.5858,1.2988)--(3.6868,1.3059)--(3.7878,1.3126)--(3.8888,1.3191)--(3.9898,1.3252)--(4.0909,1.3310)--(4.1919,1.3366)--(4.2929,1.3419)--(4.3939,1.3470)--(4.4949,1.3518)--(4.5959,1.3565)--(4.6969,1.3610)--(4.7979,1.3653)--(4.8989,1.3694)--(5.0000,1.3734);
+\draw [color=red,style=dashed] (-5.0000,1.5707) -- (5.0000,1.5707);
+\draw [color=red,style=dashed] (-5.0000,-1.5707) -- (5.0000,-1.5707);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.5937,-1.5707) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-0.1000,-1.5707) -- (0.1000,-1.5707);
+\draw (-0.4518,1.5707) node {$ \frac{1}{2} \, \pi $};
+\draw [] (-0.1000,1.5707) -- (0.1000,1.5707);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UneCellule.pstricks.recall b/src_phystricks/Fig_UneCellule.pstricks.recall
index 5b9b823d9..9890cc16b 100644
--- a/src_phystricks/Fig_UneCellule.pstricks.recall
+++ b/src_phystricks/Fig_UneCellule.pstricks.recall
@@ -103,54 +103,54 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.41406) node {$a_1=y_{10}$};
-\draw [style=dotted] (1.00,0) -- (1.00,2.00);
-\draw [] (1.00,2.00) -- (1.00,5.00);
-\draw []  (2.2000,0) node [rotate=0] {$\bullet$};
-\draw (2.2000,-0.71406) node {$y_{11}$};
-\draw [style=dotted] (2.20,0) -- (2.20,2.00);
-\draw [] (2.20,2.00) -- (2.20,5.00);
-\draw []  (3.7000,0) node [rotate=0] {$\bullet$};
-\draw (3.7000,-0.41406) node {$y_{12}$};
-\draw [style=dotted] (3.70,0) -- (3.70,2.00);
-\draw [] (3.70,2.00) -- (3.70,5.00);
-\draw []  (4.2000,0) node [rotate=0] {$\bullet$};
-\draw (4.2000,-0.71406) node {$y_{13}$};
-\draw [style=dotted] (4.20,0) -- (4.20,2.00);
-\draw [] (4.20,2.00) -- (4.20,5.00);
-\draw []  (5.2000,0) node [rotate=0] {$\bullet$};
-\draw (5.2000,-0.41406) node {$y_{14}$};
-\draw [style=dotted] (5.20,0) -- (5.20,2.00);
-\draw [] (5.20,2.00) -- (5.20,5.00);
-\draw []  (6.2000,0) node [rotate=0] {$\bullet$};
-\draw (6.2000,-0.76222) node {$b_1=y_{15}$};
-\draw [style=dotted] (6.20,0) -- (6.20,2.00);
-\draw [] (6.20,2.00) -- (6.20,5.00);
-\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
-\draw (-0.95841,2.0000) node {$a_2=y_{20}$};
-\draw [style=dotted] (0,2.00) -- (1.00,2.00);
-\draw [] (1.00,2.00) -- (6.20,2.00);
-\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
-\draw (-0.53948,2.5000) node {$y_{21}$};
-\draw [style=dotted] (0,2.50) -- (1.00,2.50);
-\draw [] (1.00,2.50) -- (6.20,2.50);
-\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
-\draw (-0.53948,4.0000) node {$y_{22}$};
-\draw [style=dotted] (0,4.00) -- (1.00,4.00);
-\draw [] (1.00,4.00) -- (6.20,4.00);
-\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
-\draw (-0.94026,5.0000) node {$b_2=y_{23}$};
-\draw [style=dotted] (0,5.00) -- (1.00,5.00);
-\draw [] (1.00,5.00) -- (6.20,5.00);
-\fill [color=lightgray] (4.20,4.00) -- (5.20,4.00) -- (5.20,4.00) -- (5.20,2.50) -- (5.20,2.50) -- (4.20,2.50) -- (4.20,2.50) -- (4.20,4.00) --  cycle;
-\draw [] (4.20,4.00) -- (5.20,4.00);
-\draw [] (5.20,4.00) -- (5.20,2.50);
-\draw [] (5.20,2.50) -- (4.20,2.50);
-\draw [] (4.20,2.50) -- (4.20,4.00);
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.4140) node {$a_1=y_{10}$};
+\draw [style=dotted] (1.0000,0.0000) -- (1.0000,2.0000);
+\draw [] (1.0000,2.0000) -- (1.0000,5.0000);
+\draw []  (2.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.2000,-0.7140) node {$y_{11}$};
+\draw [style=dotted] (2.2000,0.0000) -- (2.2000,2.0000);
+\draw [] (2.2000,2.0000) -- (2.2000,5.0000);
+\draw []  (3.7000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.7000,-0.4140) node {$y_{12}$};
+\draw [style=dotted] (3.7000,0.0000) -- (3.7000,2.0000);
+\draw [] (3.7000,2.0000) -- (3.7000,5.0000);
+\draw []  (4.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.2000,-0.7140) node {$y_{13}$};
+\draw [style=dotted] (4.2000,0.0000) -- (4.2000,2.0000);
+\draw [] (4.2000,2.0000) -- (4.2000,5.0000);
+\draw []  (5.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.2000,-0.4140) node {$y_{14}$};
+\draw [style=dotted] (5.2000,0.0000) -- (5.2000,2.0000);
+\draw [] (5.2000,2.0000) -- (5.2000,5.0000);
+\draw []  (6.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.2000,-0.7622) node {$b_1=y_{15}$};
+\draw [style=dotted] (6.2000,0.0000) -- (6.2000,2.0000);
+\draw [] (6.2000,2.0000) -- (6.2000,5.0000);
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9584,2.0000) node {$a_2=y_{20}$};
+\draw [style=dotted] (0.0000,2.0000) -- (1.0000,2.0000);
+\draw [] (1.0000,2.0000) -- (6.2000,2.0000);
+\draw []  (0.0000,2.5000) node [rotate=0] {$\bullet$};
+\draw (-0.5394,2.5000) node {$y_{21}$};
+\draw [style=dotted] (0.0000,2.5000) -- (1.0000,2.5000);
+\draw [] (1.0000,2.5000) -- (6.2000,2.5000);
+\draw []  (0.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (-0.5394,4.0000) node {$y_{22}$};
+\draw [style=dotted] (0.0000,4.0000) -- (1.0000,4.0000);
+\draw [] (1.0000,4.0000) -- (6.2000,4.0000);
+\draw []  (0.0000,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9402,5.0000) node {$b_2=y_{23}$};
+\draw [style=dotted] (0.0000,5.0000) -- (1.0000,5.0000);
+\draw [] (1.0000,5.0000) -- (6.2000,5.0000);
+\fill [color=lightgray] (4.2000,4.0000) -- (5.2000,4.0000) -- (5.2000,4.0000) -- (5.2000,2.5000) -- (5.2000,2.5000) -- (4.2000,2.5000) -- (4.2000,2.5000) -- (4.2000,4.0000) --  cycle;
+\draw [] (4.2000,4.0000) -- (5.2000,4.0000);
+\draw [] (5.2000,4.0000) -- (5.2000,2.5000);
+\draw [] (5.2000,2.5000) -- (4.2000,2.5000);
+\draw [] (4.2000,2.5000) -- (4.2000,4.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall b/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall
index 1b0b6770c..699fe6f83 100644
--- a/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall
+++ b/src_phystricks/Fig_VBOIooRHhKOH.pstricks.recall
@@ -91,31 +91,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
-\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
+\draw [,->,>=latex] (-1.4000,0.0000) -- (1.4000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5250) -- (0.0000,4.5500);
 %DEFAULT
 
-\draw [color=blue] (-0.9000,-2.025)--(-0.8818,-1.905)--(-0.8636,-1.789)--(-0.8455,-1.679)--(-0.8273,-1.573)--(-0.8091,-1.471)--(-0.7909,-1.374)--(-0.7727,-1.282)--(-0.7545,-1.193)--(-0.7364,-1.109)--(-0.7182,-1.029)--(-0.7000,-0.9528)--(-0.6818,-0.8804)--(-0.6636,-0.8119)--(-0.6455,-0.7470)--(-0.6273,-0.6856)--(-0.6091,-0.6277)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5364,-0.4286)--(-0.5182,-0.3865)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4455,-0.2455)--(-0.4273,-0.2167)--(-0.4091,-0.1902)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1238)--(-0.3364,-0.1057)--(-0.3182,-0.08948)--(-0.3000,-0.07500)--(-0.2818,-0.06217)--(-0.2636,-0.05090)--(-0.2455,-0.04108)--(-0.2273,-0.03261)--(-0.2091,-0.02539)--(-0.1909,-0.01933)--(-0.1727,-0.01431)--(-0.1545,-0.01025)--(-0.1364,-0.007044)--(-0.1182,-0.004585)--(-0.1000,-0.002778)--(-0.08182,-0.001521)--(-0.06364,0)--(-0.04545,0)--(-0.02727,0)--(-0.009091,0)--(0.009091,0)--(0.02727,0)--(0.04545,0)--(0.06364,0)--(0.08182,0.001521)--(0.1000,0.002778)--(0.1182,0.004585)--(0.1364,0.007044)--(0.1545,0.01025)--(0.1727,0.01431)--(0.1909,0.01933)--(0.2091,0.02539)--(0.2273,0.03261)--(0.2455,0.04108)--(0.2636,0.05090)--(0.2818,0.06217)--(0.3000,0.07500)--(0.3182,0.08948)--(0.3364,0.1057)--(0.3545,0.1238)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4091,0.1902)--(0.4273,0.2167)--(0.4455,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5182,0.3865)--(0.5364,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6091,0.6277)--(0.6273,0.6856)--(0.6455,0.7470)--(0.6636,0.8119)--(0.6818,0.8804)--(0.7000,0.9528)--(0.7182,1.029)--(0.7364,1.109)--(0.7545,1.193)--(0.7727,1.282)--(0.7909,1.374)--(0.8091,1.471)--(0.8273,1.573)--(0.8455,1.679)--(0.8636,1.789)--(0.8818,1.905)--(0.9000,2.025);
+\draw [color=blue] (-0.9000,-2.0250)--(-0.8818,-1.9047)--(-0.8636,-1.7893)--(-0.8454,-1.6786)--(-0.8272,-1.5726)--(-0.8090,-1.4712)--(-0.7909,-1.3742)--(-0.7727,-1.2816)--(-0.7545,-1.1933)--(-0.7363,-1.1091)--(-0.7181,-1.0289)--(-0.7000,-0.9527)--(-0.6818,-0.8804)--(-0.6636,-0.8118)--(-0.6454,-0.7469)--(-0.6272,-0.6855)--(-0.6090,-0.6276)--(-0.5909,-0.5731)--(-0.5727,-0.5218)--(-0.5545,-0.4737)--(-0.5363,-0.4286)--(-0.5181,-0.3864)--(-0.5000,-0.3472)--(-0.4818,-0.3107)--(-0.4636,-0.2768)--(-0.4454,-0.2455)--(-0.4272,-0.2166)--(-0.4090,-0.1901)--(-0.3909,-0.1659)--(-0.3727,-0.1438)--(-0.3545,-0.1237)--(-0.3363,-0.1057)--(-0.3181,-0.0894)--(-0.3000,-0.0750)--(-0.2818,-0.0621)--(-0.2636,-0.0508)--(-0.2454,-0.0410)--(-0.2272,-0.0326)--(-0.2090,-0.0253)--(-0.1909,-0.0193)--(-0.1727,-0.0143)--(-0.1545,-0.0102)--(-0.1363,-0.0070)--(-0.1181,-0.0045)--(-0.0999,-0.0027)--(-0.0818,-0.0015)--(-0.0636,0.0000)--(-0.0454,0.0000)--(-0.0272,0.0000)--(-0.0090,0.0000)--(0.0090,0.0000)--(0.0272,0.0000)--(0.0454,0.0000)--(0.0636,0.0000)--(0.0818,0.0015)--(0.1000,0.0027)--(0.1181,0.0045)--(0.1363,0.0070)--(0.1545,0.0102)--(0.1727,0.0143)--(0.1909,0.0193)--(0.2090,0.0253)--(0.2272,0.0326)--(0.2454,0.0410)--(0.2636,0.0508)--(0.2818,0.0621)--(0.3000,0.0750)--(0.3181,0.0894)--(0.3363,0.1057)--(0.3545,0.1237)--(0.3727,0.1438)--(0.3909,0.1659)--(0.4090,0.1901)--(0.4272,0.2166)--(0.4454,0.2455)--(0.4636,0.2768)--(0.4818,0.3107)--(0.5000,0.3472)--(0.5181,0.3864)--(0.5363,0.4286)--(0.5545,0.4737)--(0.5727,0.5218)--(0.5909,0.5731)--(0.6090,0.6276)--(0.6272,0.6855)--(0.6454,0.7469)--(0.6636,0.8118)--(0.6818,0.8804)--(0.7000,0.9527)--(0.7181,1.0289)--(0.7363,1.1091)--(0.7545,1.1933)--(0.7727,1.2816)--(0.7909,1.3742)--(0.8090,1.4712)--(0.8272,1.5726)--(0.8454,1.6786)--(0.8636,1.7893)--(0.8818,1.9047)--(0.9000,2.0250);
 
-\draw [color=red] (-0.9000,4.050)--(-0.8818,3.888)--(-0.8636,3.729)--(-0.8455,3.574)--(-0.8273,3.422)--(-0.8091,3.273)--(-0.7909,3.128)--(-0.7727,2.986)--(-0.7545,2.847)--(-0.7364,2.711)--(-0.7182,2.579)--(-0.7000,2.450)--(-0.6818,2.324)--(-0.6636,2.202)--(-0.6455,2.083)--(-0.6273,1.967)--(-0.6091,1.855)--(-0.5909,1.746)--(-0.5727,1.640)--(-0.5545,1.538)--(-0.5364,1.438)--(-0.5182,1.343)--(-0.5000,1.250)--(-0.4818,1.161)--(-0.4636,1.075)--(-0.4455,0.9921)--(-0.4273,0.9128)--(-0.4091,0.8368)--(-0.3909,0.7641)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3364,0.5657)--(-0.3182,0.5062)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2455,0.3012)--(-0.2273,0.2583)--(-0.2091,0.2186)--(-0.1909,0.1822)--(-0.1727,0.1492)--(-0.1545,0.1194)--(-0.1364,0.09298)--(-0.1182,0.06983)--(-0.1000,0.05000)--(-0.08182,0.03347)--(-0.06364,0.02025)--(-0.04545,0.01033)--(-0.02727,0.003719)--(-0.009091,0)--(0.009091,0)--(0.02727,0.003719)--(0.04545,0.01033)--(0.06364,0.02025)--(0.08182,0.03347)--(0.1000,0.05000)--(0.1182,0.06983)--(0.1364,0.09298)--(0.1545,0.1194)--(0.1727,0.1492)--(0.1909,0.1822)--(0.2091,0.2186)--(0.2273,0.2583)--(0.2455,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3182,0.5062)--(0.3364,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7641)--(0.4091,0.8368)--(0.4273,0.9128)--(0.4455,0.9921)--(0.4636,1.075)--(0.4818,1.161)--(0.5000,1.250)--(0.5182,1.343)--(0.5364,1.438)--(0.5545,1.538)--(0.5727,1.640)--(0.5909,1.746)--(0.6091,1.855)--(0.6273,1.967)--(0.6455,2.083)--(0.6636,2.202)--(0.6818,2.324)--(0.7000,2.450)--(0.7182,2.579)--(0.7364,2.711)--(0.7545,2.847)--(0.7727,2.986)--(0.7909,3.128)--(0.8091,3.273)--(0.8273,3.422)--(0.8455,3.574)--(0.8636,3.729)--(0.8818,3.888)--(0.9000,4.050);
-\draw (-1.2000,-0.32983) node {$ -2 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (-0.60000,-0.32983) node {$ -1 $};
-\draw [] (-0.600,-0.100) -- (-0.600,0.100);
-\draw (0.60000,-0.31492) node {$ 1 $};
-\draw [] (0.600,-0.100) -- (0.600,0.100);
-\draw (1.2000,-0.31492) node {$ 2 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (-0.43316,-2.4000) node {$ -4 $};
-\draw [] (-0.100,-2.40) -- (0.100,-2.40);
-\draw (-0.43316,-1.2000) node {$ -2 $};
-\draw [] (-0.100,-1.20) -- (0.100,-1.20);
-\draw (-0.29125,1.2000) node {$ 2 $};
-\draw [] (-0.100,1.20) -- (0.100,1.20);
-\draw (-0.29125,2.4000) node {$ 4 $};
-\draw [] (-0.100,2.40) -- (0.100,2.40);
-\draw (-0.29125,3.6000) node {$ 6 $};
-\draw [] (-0.100,3.60) -- (0.100,3.60);
+\draw [color=red] (-0.9000,4.0500)--(-0.8818,3.8880)--(-0.8636,3.7293)--(-0.8454,3.5739)--(-0.8272,3.4219)--(-0.8090,3.2731)--(-0.7909,3.1276)--(-0.7727,2.9855)--(-0.7545,2.8466)--(-0.7363,2.7111)--(-0.7181,2.5789)--(-0.7000,2.4500)--(-0.6818,2.3243)--(-0.6636,2.2020)--(-0.6454,2.0830)--(-0.6272,1.9673)--(-0.6090,1.8549)--(-0.5909,1.7458)--(-0.5727,1.6400)--(-0.5545,1.5376)--(-0.5363,1.4384)--(-0.5181,1.3425)--(-0.5000,1.2500)--(-0.4818,1.1607)--(-0.4636,1.0747)--(-0.4454,0.9921)--(-0.4272,0.9128)--(-0.4090,0.8367)--(-0.3909,0.7640)--(-0.3727,0.6946)--(-0.3545,0.6285)--(-0.3363,0.5657)--(-0.3181,0.5061)--(-0.3000,0.4500)--(-0.2818,0.3971)--(-0.2636,0.3475)--(-0.2454,0.3012)--(-0.2272,0.2582)--(-0.2090,0.2185)--(-0.1909,0.1822)--(-0.1727,0.1491)--(-0.1545,0.1194)--(-0.1363,0.0929)--(-0.1181,0.0698)--(-0.0999,0.0499)--(-0.0818,0.0334)--(-0.0636,0.0202)--(-0.0454,0.0103)--(-0.0272,0.0037)--(-0.0090,0.0000)--(0.0090,0.0000)--(0.0272,0.0037)--(0.0454,0.0103)--(0.0636,0.0202)--(0.0818,0.0334)--(0.1000,0.0500)--(0.1181,0.0698)--(0.1363,0.0929)--(0.1545,0.1194)--(0.1727,0.1491)--(0.1909,0.1822)--(0.2090,0.2185)--(0.2272,0.2582)--(0.2454,0.3012)--(0.2636,0.3475)--(0.2818,0.3971)--(0.3000,0.4500)--(0.3181,0.5061)--(0.3363,0.5657)--(0.3545,0.6285)--(0.3727,0.6946)--(0.3909,0.7640)--(0.4090,0.8367)--(0.4272,0.9128)--(0.4454,0.9921)--(0.4636,1.0747)--(0.4818,1.1607)--(0.5000,1.2500)--(0.5181,1.3425)--(0.5363,1.4384)--(0.5545,1.5376)--(0.5727,1.6400)--(0.5909,1.7458)--(0.6090,1.8549)--(0.6272,1.9673)--(0.6454,2.0830)--(0.6636,2.2020)--(0.6818,2.3243)--(0.7000,2.4500)--(0.7181,2.5789)--(0.7363,2.7111)--(0.7545,2.8466)--(0.7727,2.9855)--(0.7909,3.1276)--(0.8090,3.2731)--(0.8272,3.4219)--(0.8454,3.5739)--(0.8636,3.7293)--(0.8818,3.8880)--(0.9000,4.0500);
+\draw (-1.2000,-0.3298) node {$ -2 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (-0.6000,-0.3298) node {$ -1 $};
+\draw [] (-0.6000,-0.1000) -- (-0.6000,0.1000);
+\draw (0.6000,-0.3149) node {$ 1 $};
+\draw [] (0.6000,-0.1000) -- (0.6000,0.1000);
+\draw (1.2000,-0.3149) node {$ 2 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (-0.4331,-2.4000) node {$ -4 $};
+\draw [] (-0.1000,-2.4000) -- (0.1000,-2.4000);
+\draw (-0.4331,-1.2000) node {$ -2 $};
+\draw [] (-0.1000,-1.2000) -- (0.1000,-1.2000);
+\draw (-0.2912,1.2000) node {$ 2 $};
+\draw [] (-0.1000,1.2000) -- (0.1000,1.2000);
+\draw (-0.2912,2.4000) node {$ 4 $};
+\draw [] (-0.1000,2.4000) -- (0.1000,2.4000);
+\draw (-0.2912,3.6000) node {$ 6 $};
+\draw [] (-0.1000,3.6000) -- (0.1000,3.6000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall b/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall
index 34b8ef2f9..7c55e9baa 100644
--- a/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall
+++ b/src_phystricks/Fig_VDFMooHMmFZr.pstricks.recall
@@ -65,15 +65,15 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
-\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
-\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
-\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
-\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
-\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
-\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
-\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
+\draw [color=blue] (1.0000,0.0000)--(1.0611,0.0674)--(1.1165,0.1424)--(1.1644,0.2244)--(1.2032,0.3122)--(1.2317,0.4045)--(1.2486,0.4999)--(1.2533,0.5965)--(1.2451,0.6927)--(1.2238,0.7865)--(1.1897,0.8759)--(1.1432,0.9592)--(1.0851,1.0347)--(1.0166,1.1007)--(0.9391,1.1559)--(0.8541,1.1994)--(0.7634,1.2303)--(0.6688,1.2483)--(0.5723,1.2533)--(0.4758,1.2455)--(0.3811,1.2256)--(0.2897,1.1944)--(0.2033,1.1532)--(0.1230,1.1033)--(0.0498,1.0463)--(-0.0156,0.9840)--(-0.0729,0.9181)--(-0.1222,0.8503)--(-0.1637,0.7825)--(-0.1980,0.7162)--(-0.2259,0.6529)--(-0.2486,0.5936)--(-0.2673,0.5394)--(-0.2834,0.4910)--(-0.2984,0.4486)--(-0.3137,0.4122)--(-0.3307,0.3817)--(-0.3507,0.3563)--(-0.3749,0.3354)--(-0.4041,0.3177)--(-0.4389,0.3022)--(-0.4798,0.2872)--(-0.5268,0.2715)--(-0.5796,0.2536)--(-0.6376,0.2320)--(-0.7001,0.2055)--(-0.7658,0.1729)--(-0.8333,0.1333)--(-0.9012,0.0860)--(-0.9678,0.0307)--(-1.0311,-0.0327)--(-1.0896,-0.1040)--(-1.1414,-0.1826)--(-1.1850,-0.2676)--(-1.2188,-0.3578)--(-1.2417,-0.4519)--(-1.2525,-0.5481)--(-1.2508,-0.6448)--(-1.2361,-0.7400)--(-1.2083,-0.8319)--(-1.1680,-0.9185)--(-1.1156,-0.9980)--(-1.0521,-1.0689)--(-0.9789,-1.1297)--(-0.8974,-1.1792)--(-0.8093,-1.2165)--(-0.7165,-1.2410)--(-0.6207,-1.2524)--(-0.5240,-1.2510)--(-0.4281,-1.2370)--(-0.3349,-1.2113)--(-0.2458,-1.1750)--(-0.1623,-1.1292)--(-0.0855,-1.0756)--(-0.0161,-1.0157)--(0.0453,-0.9513)--(0.0986,-0.8843)--(0.1439,-0.8163)--(0.1817,-0.7491)--(0.2127,-0.6841)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2756,-0.5145)--(0.2910,-0.4690)--(0.3059,-0.4296)--(0.3219,-0.3962)--(0.3403,-0.3684)--(0.3622,-0.3454)--(0.3888,-0.3262)--(0.4207,-0.3098)--(0.4586,-0.2947)--(0.5025,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1900)--(0.7994,-0.1540)--(0.8673,-0.1106)--(0.9348,-0.0594)--(1.0000,0.0000);
+\draw [,->,>=latex] (1.2468,0.6811) -- (1.2454,0.6910);
+\draw [,->,>=latex] (0.3575,1.2187) -- (0.3480,1.2158);
+\draw [,->,>=latex] (-0.2777,0.5083) -- (-0.2808,0.4988);
+\draw [,->,>=latex] (-0.7003,0.2054) -- (-0.7094,0.2013);
+\draw [,->,>=latex] (-1.2468,-0.6811) -- (-1.2454,-0.6910);
+\draw [,->,>=latex] (-0.3575,-1.2187) -- (-0.3480,-1.2158);
+\draw [,->,>=latex] (0.2777,-0.5083) -- (0.2808,-0.4988);
+\draw [,->,>=latex] (0.7003,-0.2054) -- (0.7094,-0.2013);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall b/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall
index d8e38e460..abfb5217b 100644
--- a/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall
+++ b/src_phystricks/Fig_VNBGooSqMsGU.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,19 +81,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (0.0000,0.0000) -- (0.0000,2.0000);
+\draw [color=red,style=solid] (1.0000,2.0000) -- (1.0000,0.0000);
 
-\draw [color=red] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=red] (0.0000,2.0000)--(0.0101,2.0000)--(0.0202,2.0000)--(0.0303,2.0000)--(0.0404,2.0000)--(0.0505,2.0000)--(0.0606,2.0000)--(0.0707,2.0000)--(0.0808,2.0000)--(0.0909,2.0000)--(0.1010,2.0000)--(0.1111,2.0000)--(0.1212,2.0000)--(0.1313,2.0000)--(0.1414,2.0000)--(0.1515,2.0000)--(0.1616,2.0000)--(0.1717,2.0000)--(0.1818,2.0000)--(0.1919,2.0000)--(0.2020,2.0000)--(0.2121,2.0000)--(0.2222,2.0000)--(0.2323,2.0000)--(0.2424,2.0000)--(0.2525,2.0000)--(0.2626,2.0000)--(0.2727,2.0000)--(0.2828,2.0000)--(0.2929,2.0000)--(0.3030,2.0000)--(0.3131,2.0000)--(0.3232,2.0000)--(0.3333,2.0000)--(0.3434,2.0000)--(0.3535,2.0000)--(0.3636,2.0000)--(0.3737,2.0000)--(0.3838,2.0000)--(0.3939,2.0000)--(0.4040,2.0000)--(0.4141,2.0000)--(0.4242,2.0000)--(0.4343,2.0000)--(0.4444,2.0000)--(0.4545,2.0000)--(0.4646,2.0000)--(0.4747,2.0000)--(0.4848,2.0000)--(0.4949,2.0000)--(0.5050,2.0000)--(0.5151,2.0000)--(0.5252,2.0000)--(0.5353,2.0000)--(0.5454,2.0000)--(0.5555,2.0000)--(0.5656,2.0000)--(0.5757,2.0000)--(0.5858,2.0000)--(0.5959,2.0000)--(0.6060,2.0000)--(0.6161,2.0000)--(0.6262,2.0000)--(0.6363,2.0000)--(0.6464,2.0000)--(0.6565,2.0000)--(0.6666,2.0000)--(0.6767,2.0000)--(0.6868,2.0000)--(0.6969,2.0000)--(0.7070,2.0000)--(0.7171,2.0000)--(0.7272,2.0000)--(0.7373,2.0000)--(0.7474,2.0000)--(0.7575,2.0000)--(0.7676,2.0000)--(0.7777,2.0000)--(0.7878,2.0000)--(0.7979,2.0000)--(0.8080,2.0000)--(0.8181,2.0000)--(0.8282,2.0000)--(0.8383,2.0000)--(0.8484,2.0000)--(0.8585,2.0000)--(0.8686,2.0000)--(0.8787,2.0000)--(0.8888,2.0000)--(0.8989,2.0000)--(0.9090,2.0000)--(0.9191,2.0000)--(0.9292,2.0000)--(0.9393,2.0000)--(0.9494,2.0000)--(0.9595,2.0000)--(0.9696,2.0000)--(0.9797,2.0000)--(0.9898,2.0000)--(1.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall b/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall
index ba9700c1b..fdb12865a 100644
--- a/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall
+++ b/src_phystricks/Fig_VWFLooPSrOqz.pstricks.recall
@@ -83,13 +83,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
-\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
+\draw [,->,>=latex] (-1.6500,0.0000) -- (3.6500,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6500) -- (0.0000,2.6500);
 %DEFAULT
-\draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
-\draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
-\draw [color=red] (2.15,-2.15) -- (-1.15,1.15);
-\draw [color=red] (-1.15,-1.15) -- (2.15,2.15);
+\draw [color=red] (-0.1500,2.1500) -- (3.1500,-1.1500);
+\draw [color=red] (3.1500,1.1500) -- (-0.1500,-2.1500);
+\draw [color=red] (2.1500,-2.1500) -- (-1.1500,1.1500);
+\draw [color=red] (-1.1500,-1.1500) -- (2.1500,2.1500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,27 +97,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,1.00) -- (2.00,0) -- (2.00,0) -- (1.00,-1.00) -- (1.00,-1.00) -- (0,0) -- (0,0) -- (1.00,1.00) --  cycle;
-\draw [color=blue] (1.00,1.00) -- (2.00,0);
-\draw [color=blue] (2.00,0) -- (1.00,-1.00);
-\draw [color=blue] (1.00,-1.00) -- (0,0);
-\draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,1.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (1.0000,-1.0000) -- (1.0000,-1.0000) -- (0.0000,0.0000) -- (0.0000,0.0000) -- (1.0000,1.0000) --  cycle;
+\draw [color=blue] (1.0000,1.0000) -- (2.0000,0.0000);
+\draw [color=blue] (2.0000,0.0000) -- (1.0000,-1.0000);
+\draw [color=blue] (1.0000,-1.0000) -- (0.0000,0.0000);
+\draw [color=blue] (0.0000,0.0000) -- (1.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_WUYooCISzeB.pstricks.recall b/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
index b138a1fb5..279ac1275 100644
--- a/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
+++ b/src_phystricks/Fig_WUYooCISzeB.pstricks.recall
@@ -61,29 +61,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.99185) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.9918) -- (0.0000,4.0275);
 %DEFAULT
 
-\draw [color=red] (-0.8400,2.700)--(-0.8013,2.668)--(-0.7626,2.636)--(-0.7239,2.603)--(-0.6852,2.571)--(-0.6466,2.539)--(-0.6079,2.507)--(-0.5692,2.474)--(-0.5305,2.442)--(-0.4918,2.410)--(-0.4531,2.378)--(-0.4144,2.345)--(-0.3757,2.313)--(-0.3370,2.281)--(-0.2984,2.249)--(-0.2597,2.216)--(-0.2210,2.184)--(-0.1823,2.152)--(-0.1436,2.120)--(-0.1049,2.087)--(-0.06622,2.055)--(-0.02753,2.023)--(0.01116,1.991)--(0.04985,1.958)--(0.08854,1.926)--(0.1272,1.894)--(0.1659,1.862)--(0.2046,1.829)--(0.2433,1.797)--(0.2820,1.765)--(0.3207,1.733)--(0.3594,1.701)--(0.3980,1.668)--(0.4367,1.636)--(0.4754,1.604)--(0.5141,1.572)--(0.5528,1.539)--(0.5915,1.507)--(0.6302,1.475)--(0.6689,1.443)--(0.7076,1.410)--(0.7463,1.378)--(0.7849,1.346)--(0.8236,1.314)--(0.8623,1.281)--(0.9010,1.249)--(0.9397,1.217)--(0.9784,1.185)--(1.017,1.152)--(1.056,1.120)--(1.094,1.088)--(1.133,1.056)--(1.172,1.023)--(1.211,0.9912)--(1.249,0.9590)--(1.288,0.9268)--(1.327,0.8945)--(1.365,0.8623)--(1.404,0.8300)--(1.443,0.7978)--(1.481,0.7655)--(1.520,0.7333)--(1.559,0.7011)--(1.597,0.6688)--(1.636,0.6366)--(1.675,0.6043)--(1.713,0.5721)--(1.752,0.5399)--(1.791,0.5076)--(1.830,0.4754)--(1.868,0.4431)--(1.907,0.4109)--(1.946,0.3787)--(1.984,0.3464)--(2.023,0.3142)--(2.062,0.2819)--(2.100,0.2497)--(2.139,0.2175)--(2.178,0.1852)--(2.216,0.1530)--(2.255,0.1207)--(2.294,0.08849)--(2.332,0.05625)--(2.371,0.02401)--(2.410,-0.008233)--(2.449,-0.04047)--(2.487,-0.07271)--(2.526,-0.1050)--(2.565,-0.1372)--(2.603,-0.1694)--(2.642,-0.2017)--(2.681,-0.2339)--(2.719,-0.2662)--(2.758,-0.2984)--(2.797,-0.3306)--(2.835,-0.3629)--(2.874,-0.3951)--(2.913,-0.4274)--(2.952,-0.4596)--(2.990,-0.4918);
+\draw [color=red] (-0.8400,2.7000)--(-0.8013,2.6677)--(-0.7626,2.6355)--(-0.7239,2.6032)--(-0.6852,2.5710)--(-0.6465,2.5387)--(-0.6078,2.5065)--(-0.5691,2.4743)--(-0.5304,2.4420)--(-0.4917,2.4098)--(-0.4531,2.3775)--(-0.4144,2.3453)--(-0.3757,2.3131)--(-0.3370,2.2808)--(-0.2983,2.2486)--(-0.2596,2.2163)--(-0.2209,2.1841)--(-0.1822,2.1519)--(-0.1435,2.1196)--(-0.1049,2.0874)--(-0.0662,2.0551)--(-0.0275,2.0229)--(0.0111,1.9907)--(0.0498,1.9584)--(0.0885,1.9262)--(0.1272,1.8939)--(0.1659,1.8617)--(0.2046,1.8294)--(0.2432,1.7972)--(0.2819,1.7650)--(0.3206,1.7327)--(0.3593,1.7005)--(0.3980,1.6682)--(0.4367,1.6360)--(0.4754,1.6038)--(0.5141,1.5715)--(0.5528,1.5393)--(0.5914,1.5070)--(0.6301,1.4748)--(0.6688,1.4426)--(0.7075,1.4103)--(0.7462,1.3781)--(0.7849,1.3458)--(0.8236,1.3136)--(0.8623,1.2814)--(0.9010,1.2491)--(0.9396,1.2169)--(0.9783,1.1846)--(1.0170,1.1524)--(1.0557,1.1201)--(1.0944,1.0879)--(1.1331,1.0557)--(1.1718,1.0234)--(1.2105,0.9912)--(1.2492,0.9589)--(1.2878,0.9267)--(1.3265,0.8945)--(1.3652,0.8622)--(1.4039,0.8300)--(1.4426,0.7977)--(1.4813,0.7655)--(1.5200,0.7333)--(1.5587,0.7010)--(1.5974,0.6688)--(1.6360,0.6365)--(1.6747,0.6043)--(1.7134,0.5721)--(1.7521,0.5398)--(1.7908,0.5076)--(1.8295,0.4753)--(1.8682,0.4431)--(1.9069,0.4108)--(1.9456,0.3786)--(1.9842,0.3464)--(2.0229,0.3141)--(2.0616,0.2819)--(2.1003,0.2496)--(2.1390,0.2174)--(2.1777,0.1852)--(2.2164,0.1529)--(2.2551,0.1207)--(2.2938,0.0884)--(2.3325,0.0562)--(2.3711,0.0240)--(2.4098,-0.0082)--(2.4485,-0.0404)--(2.4872,-0.0727)--(2.5259,-0.1049)--(2.5646,-0.1371)--(2.6033,-0.1694)--(2.6420,-0.2016)--(2.6807,-0.2339)--(2.7193,-0.2661)--(2.7580,-0.2984)--(2.7967,-0.3306)--(2.8354,-0.3628)--(2.8741,-0.3951)--(2.9128,-0.4273)--(2.9515,-0.4596)--(2.9902,-0.4918);
 
-\draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -141,29 +141,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0275);
 %DEFAULT
 
-\draw [color=red] (-0.8400,3.190)--(-0.8096,3.130)--(-0.7792,3.071)--(-0.7488,3.013)--(-0.7184,2.957)--(-0.6879,2.902)--(-0.6575,2.848)--(-0.6271,2.796)--(-0.5967,2.745)--(-0.5663,2.695)--(-0.5359,2.646)--(-0.5055,2.599)--(-0.4751,2.553)--(-0.4446,2.508)--(-0.4142,2.464)--(-0.3838,2.422)--(-0.3534,2.381)--(-0.3230,2.342)--(-0.2926,2.303)--(-0.2622,2.266)--(-0.2318,2.230)--(-0.2013,2.196)--(-0.1709,2.163)--(-0.1405,2.131)--(-0.1101,2.100)--(-0.07969,2.071)--(-0.04928,2.043)--(-0.01887,2.016)--(0.01154,1.990)--(0.04196,1.966)--(0.07237,1.943)--(0.1028,1.922)--(0.1332,1.901)--(0.1636,1.882)--(0.1940,1.864)--(0.2244,1.848)--(0.2548,1.833)--(0.2853,1.819)--(0.3157,1.806)--(0.3461,1.795)--(0.3765,1.785)--(0.4069,1.776)--(0.4373,1.768)--(0.4677,1.762)--(0.4981,1.757)--(0.5285,1.754)--(0.5590,1.751)--(0.5894,1.750)--(0.6198,1.750)--(0.6502,1.752)--(0.6806,1.755)--(0.7110,1.759)--(0.7414,1.764)--(0.7719,1.771)--(0.8023,1.778)--(0.8327,1.788)--(0.8631,1.798)--(0.8935,1.810)--(0.9239,1.823)--(0.9543,1.837)--(0.9847,1.853)--(1.015,1.870)--(1.046,1.888)--(1.076,1.907)--(1.106,1.928)--(1.137,1.950)--(1.167,1.973)--(1.198,1.998)--(1.228,2.024)--(1.258,2.051)--(1.289,2.080)--(1.319,2.109)--(1.350,2.140)--(1.380,2.173)--(1.411,2.206)--(1.441,2.241)--(1.471,2.277)--(1.502,2.315)--(1.532,2.353)--(1.563,2.393)--(1.593,2.435)--(1.623,2.477)--(1.654,2.521)--(1.684,2.566)--(1.715,2.613)--(1.745,2.660)--(1.775,2.710)--(1.806,2.760)--(1.836,2.811)--(1.867,2.864)--(1.897,2.918)--(1.928,2.974)--(1.958,3.031)--(1.988,3.089)--(2.019,3.148)--(2.049,3.208)--(2.080,3.270)--(2.110,3.333)--(2.140,3.398)--(2.171,3.464);
+\draw [color=red] (-0.8400,3.1900)--(-0.8095,3.1298)--(-0.7791,3.0709)--(-0.7487,3.0133)--(-0.7183,2.9569)--(-0.6879,2.9019)--(-0.6575,2.8481)--(-0.6271,2.7957)--(-0.5967,2.7445)--(-0.5662,2.6946)--(-0.5358,2.6459)--(-0.5054,2.5986)--(-0.4750,2.5525)--(-0.4446,2.5078)--(-0.4142,2.4643)--(-0.3838,2.4221)--(-0.3534,2.3812)--(-0.3229,2.3416)--(-0.2925,2.3032)--(-0.2621,2.2662)--(-0.2317,2.2304)--(-0.2013,2.1959)--(-0.1709,2.1627)--(-0.1405,2.1308)--(-0.1101,2.1001)--(-0.0796,2.0708)--(-0.0492,2.0427)--(-0.0188,2.0159)--(0.0115,1.9904)--(0.0419,1.9662)--(0.0723,1.9433)--(0.1027,1.9216)--(0.1331,1.9013)--(0.1636,1.8822)--(0.1940,1.8644)--(0.2244,1.8479)--(0.2548,1.8327)--(0.2852,1.8187)--(0.3156,1.8061)--(0.3460,1.7947)--(0.3764,1.7846)--(0.4069,1.7758)--(0.4373,1.7683)--(0.4677,1.7621)--(0.4981,1.7572)--(0.5285,1.7535)--(0.5589,1.7511)--(0.5893,1.7500)--(0.6197,1.7502)--(0.6502,1.7517)--(0.6806,1.7545)--(0.7110,1.7585)--(0.7414,1.7638)--(0.7718,1.7705)--(0.8022,1.7784)--(0.8326,1.7875)--(0.8630,1.7980)--(0.8934,1.8098)--(0.9239,1.8228)--(0.9543,1.8371)--(0.9847,1.8527)--(1.0151,1.8696)--(1.0455,1.8878)--(1.0759,1.9073)--(1.1063,1.9280)--(1.1367,1.9501)--(1.1672,1.9734)--(1.1976,1.9980)--(1.2280,2.0239)--(1.2584,2.0510)--(1.2888,2.0795)--(1.3192,2.1092)--(1.3496,2.1402)--(1.3800,2.1726)--(1.4105,2.2061)--(1.4409,2.2410)--(1.4713,2.2772)--(1.5017,2.3146)--(1.5321,2.3534)--(1.5625,2.3934)--(1.5929,2.4347)--(1.6233,2.4773)--(1.6538,2.5211)--(1.6842,2.5663)--(1.7146,2.6127)--(1.7450,2.6605)--(1.7754,2.7095)--(1.8058,2.7598)--(1.8362,2.8113)--(1.8666,2.8642)--(1.8971,2.9183)--(1.9275,2.9738)--(1.9579,3.0305)--(1.9883,3.0885)--(2.0187,3.1478)--(2.0491,3.2083)--(2.0795,3.2702)--(2.1099,3.3333)--(2.1404,3.3978)--(2.1708,3.4635);
 
-\draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -221,29 +221,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.98318) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.9831) -- (0.0000,4.0275);
 %DEFAULT
 
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-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -301,29 +301,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0275);
 %DEFAULT
 
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-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall b/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall
index a1dcf7284..f90b8f02b 100644
--- a/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall
+++ b/src_phystricks/Fig_XJMooCQTlNL.pstricks.recall
@@ -84,35 +84,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-3.7958) -- (0,3.7958);
+\draw [,->,>=latex] (-6.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7958) -- (0.0000,3.7958);
 %DEFAULT
 
-\draw [color=blue] (-6.000,-3.296)--(-5.879,-3.169)--(-5.758,-3.044)--(-5.636,-2.920)--(-5.515,-2.797)--(-5.394,-2.676)--(-5.273,-2.556)--(-5.151,-2.437)--(-5.030,-2.320)--(-4.909,-2.204)--(-4.788,-2.090)--(-4.667,-1.977)--(-4.545,-1.866)--(-4.424,-1.756)--(-4.303,-1.648)--(-4.182,-1.542)--(-4.061,-1.438)--(-3.939,-1.335)--(-3.818,-1.234)--(-3.697,-1.136)--(-3.576,-1.039)--(-3.455,-0.9440)--(-3.333,-0.8514)--(-3.212,-0.7609)--(-3.091,-0.6728)--(-2.970,-0.5870)--(-2.848,-0.5037)--(-2.727,-0.4229)--(-2.606,-0.3449)--(-2.485,-0.2697)--(-2.364,-0.1974)--(-2.242,-0.1283)--(-2.121,-0.06241)--(-2.000,0)--(-1.879,0.05873)--(-1.758,0.1135)--(-1.636,0.1642)--(-1.515,0.2103)--(-1.394,0.2516)--(-1.273,0.2876)--(-1.152,0.3179)--(-1.030,0.3417)--(-0.9091,0.3584)--(-0.7879,0.3670)--(-0.6667,0.3662)--(-0.5455,0.3544)--(-0.4242,0.3289)--(-0.3030,0.2859)--(-0.1818,0.2180)--(-0.06061,0.1060)--(0.06061,-0.1060)--(0.1818,-0.2180)--(0.3030,-0.2859)--(0.4242,-0.3289)--(0.5455,-0.3544)--(0.6667,-0.3662)--(0.7879,-0.3670)--(0.9091,-0.3584)--(1.030,-0.3417)--(1.152,-0.3179)--(1.273,-0.2876)--(1.394,-0.2516)--(1.515,-0.2103)--(1.636,-0.1642)--(1.758,-0.1135)--(1.879,-0.05873)--(2.000,0)--(2.121,0.06241)--(2.242,0.1283)--(2.364,0.1974)--(2.485,0.2697)--(2.606,0.3449)--(2.727,0.4229)--(2.848,0.5037)--(2.970,0.5870)--(3.091,0.6728)--(3.212,0.7609)--(3.333,0.8514)--(3.455,0.9440)--(3.576,1.039)--(3.697,1.136)--(3.818,1.234)--(3.939,1.335)--(4.061,1.438)--(4.182,1.542)--(4.303,1.648)--(4.424,1.756)--(4.545,1.866)--(4.667,1.977)--(4.788,2.090)--(4.909,2.204)--(5.030,2.320)--(5.151,2.437)--(5.273,2.556)--(5.394,2.676)--(5.515,2.797)--(5.636,2.920)--(5.758,3.044)--(5.879,3.169)--(6.000,3.296);
-\draw (-6.0000,-0.32983) node {$ -3 $};
-\draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -2 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -1 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.0000,-0.31492) node {$ 2 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.0000,-0.31492) node {$ 3 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-6.0000,-3.2958)--(-5.8787,-3.1692)--(-5.7575,-3.0439)--(-5.6363,-2.9198)--(-5.5151,-2.7971)--(-5.3939,-2.6757)--(-5.2727,-2.5556)--(-5.1515,-2.4370)--(-5.0303,-2.3198)--(-4.9090,-2.2040)--(-4.7878,-2.0897)--(-4.6666,-1.9770)--(-4.5454,-1.8658)--(-4.4242,-1.7563)--(-4.3030,-1.6484)--(-4.1818,-1.5422)--(-4.0606,-1.4378)--(-3.9393,-1.3352)--(-3.8181,-1.2344)--(-3.6969,-1.1356)--(-3.5757,-1.0388)--(-3.4545,-0.9440)--(-3.3333,-0.8513)--(-3.2121,-0.7609)--(-3.0909,-0.6727)--(-2.9696,-0.5869)--(-2.8484,-0.5036)--(-2.7272,-0.4229)--(-2.6060,-0.3449)--(-2.4848,-0.2696)--(-2.3636,-0.1974)--(-2.2424,-0.1282)--(-2.1212,-0.0624)--(-2.0000,0.0000)--(-1.8787,0.0587)--(-1.7575,0.1135)--(-1.6363,0.1641)--(-1.5151,0.2103)--(-1.3939,0.2516)--(-1.2727,0.2876)--(-1.1515,0.3178)--(-1.0303,0.3416)--(-0.9090,0.3583)--(-0.7878,0.3669)--(-0.6666,0.3662)--(-0.5454,0.3543)--(-0.4242,0.3289)--(-0.3030,0.2859)--(-0.1818,0.2179)--(-0.0606,0.1059)--(0.0606,-0.1059)--(0.1818,-0.2179)--(0.3030,-0.2859)--(0.4242,-0.3289)--(0.5454,-0.3543)--(0.6666,-0.3662)--(0.7878,-0.3669)--(0.9090,-0.3583)--(1.0303,-0.3416)--(1.1515,-0.3178)--(1.2727,-0.2876)--(1.3939,-0.2516)--(1.5151,-0.2103)--(1.6363,-0.1641)--(1.7575,-0.1135)--(1.8787,-0.0587)--(2.0000,0.0000)--(2.1212,0.0624)--(2.2424,0.1282)--(2.3636,0.1974)--(2.4848,0.2696)--(2.6060,0.3449)--(2.7272,0.4229)--(2.8484,0.5036)--(2.9696,0.5869)--(3.0909,0.6727)--(3.2121,0.7609)--(3.3333,0.8513)--(3.4545,0.9440)--(3.5757,1.0388)--(3.6969,1.1356)--(3.8181,1.2344)--(3.9393,1.3352)--(4.0606,1.4378)--(4.1818,1.5422)--(4.3030,1.6484)--(4.4242,1.7563)--(4.5454,1.8658)--(4.6666,1.9770)--(4.7878,2.0897)--(4.9090,2.2040)--(5.0303,2.3198)--(5.1515,2.4370)--(5.2727,2.5556)--(5.3939,2.6757)--(5.5151,2.7971)--(5.6363,2.9198)--(5.7575,3.0439)--(5.8787,3.1692)--(6.0000,3.2958);
+\draw (-6.0000,-0.3298) node {$ -3 $};
+\draw [] (-6.0000,-0.1000) -- (-6.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -2 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -1 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 2 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 3 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
index 75b352840..9cd54e3f6 100644
--- a/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
+++ b/src_phystricks/Fig_XOLBooGcrjiwoU.pstricks.recall
@@ -120,41 +120,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-4.5252) -- (0,4.0659);
+\draw [,->,>=latex] (-6.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5251) -- (0.0000,4.0658);
 %DEFAULT
 
-\draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
-\draw (-6.0000,-0.32983) node {$ -10 $};
-\draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.8000,-0.32983) node {$ -8 $};
-\draw [] (-4.80,-0.100) -- (-4.80,0.100);
-\draw (-3.6000,-0.32983) node {$ -6 $};
-\draw [] (-3.60,-0.100) -- (-3.60,0.100);
-\draw (-2.4000,-0.32983) node {$ -4 $};
-\draw [] (-2.40,-0.100) -- (-2.40,0.100);
-\draw (-1.2000,-0.32983) node {$ -2 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 2 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 4 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 6 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (4.8000,-0.31492) node {$ 8 $};
-\draw [] (4.80,-0.100) -- (4.80,0.100);
-\draw (6.0000,-0.31492) node {$ 10 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.59441,-4.5000) node {$ -\frac{3}{200} $};
-\draw [] (-0.100,-4.50) -- (0.100,-4.50);
-\draw (-0.59441,-3.0000) node {$ -\frac{1}{100} $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.59441,-1.5000) node {$ -\frac{1}{200} $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.45250,1.5000) node {$ \frac{1}{200} $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.45250,3.0000) node {$ \frac{1}{100} $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-6.0000,3.5658)--(-5.8787,1.9708)--(-5.7575,1.0758)--(-5.6363,0.5798)--(-5.5151,0.3083)--(-5.3939,0.1616)--(-5.2727,0.0835)--(-5.1515,0.0424)--(-5.0303,0.0212)--(-4.9090,0.0104)--(-4.7878,0.0050)--(-4.6666,0.0000)--(-4.5454,0.0000)--(-4.4242,0.0000)--(-4.3030,0.0000)--(-4.1818,0.0000)--(-4.0606,0.0000)--(-3.9393,0.0000)--(-3.8181,0.0000)--(-3.6969,0.0000)--(-3.5757,0.0000)--(-3.4545,0.0000)--(-3.3333,0.0000)--(-3.2121,0.0000)--(-3.0909,0.0000)--(-2.9696,0.0000)--(-2.8484,0.0000)--(-2.7272,0.0000)--(-2.6060,0.0000)--(-2.4848,0.0000)--(-2.3636,0.0000)--(-2.2424,0.0000)--(-2.1212,0.0000)--(-2.0000,0.0000)--(-1.8787,0.0000)--(-1.7575,0.0000)--(-1.6363,0.0000)--(-1.5151,0.0000)--(-1.3939,0.0000)--(-1.2727,0.0000)--(-1.1515,0.0000)--(-1.0303,0.0000)--(-0.9090,0.0000)--(-0.7878,0.0000)--(-0.6666,0.0000)--(-0.5454,0.0000)--(-0.4242,0.0000)--(-0.3030,0.0000)--(-0.1818,0.0000)--(-0.0606,0.0000)--(0.0606,0.0000)--(0.1818,0.0000)--(0.3030,0.0000)--(0.4242,0.0000)--(0.5454,0.0000)--(0.6666,0.0000)--(0.7878,0.0000)--(0.9090,0.0000)--(1.0303,0.0000)--(1.1515,0.0000)--(1.2727,0.0000)--(1.3939,0.0000)--(1.5151,0.0000)--(1.6363,0.0000)--(1.7575,0.0000)--(1.8787,0.0000)--(2.0000,0.0000)--(2.1212,0.0000)--(2.2424,0.0000)--(2.3636,0.0000)--(2.4848,0.0000)--(2.6060,0.0000)--(2.7272,0.0000)--(2.8484,0.0000)--(2.9696,0.0000)--(3.0909,0.0000)--(3.2121,0.0000)--(3.3333,0.0000)--(3.4545,0.0000)--(3.5757,0.0000)--(3.6969,0.0000)--(3.8181,0.0000)--(3.9393,0.0000)--(4.0606,0.0000)--(4.1818,0.0000)--(4.3030,0.0000)--(4.4242,0.0000)--(4.5454,0.0000)--(4.6666,0.0000)--(4.7878,-0.0054)--(4.9090,-0.0112)--(5.0303,-0.0229)--(5.1515,-0.0459)--(5.2727,-0.0908)--(5.3939,-0.1768)--(5.5151,-0.3392)--(5.6363,-0.6418)--(5.7575,-1.1984)--(5.8787,-2.2097)--(6.0000,-4.0251);
+\draw (-6.0000,-0.3298) node {$ -10 $};
+\draw [] (-6.0000,-0.1000) -- (-6.0000,0.1000);
+\draw (-4.8000,-0.3298) node {$ -8 $};
+\draw [] (-4.8000,-0.1000) -- (-4.8000,0.1000);
+\draw (-3.6000,-0.3298) node {$ -6 $};
+\draw [] (-3.6000,-0.1000) -- (-3.6000,0.1000);
+\draw (-2.4000,-0.3298) node {$ -4 $};
+\draw [] (-2.4000,-0.1000) -- (-2.4000,0.1000);
+\draw (-1.2000,-0.3298) node {$ -2 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 2 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 4 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 6 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (4.8000,-0.3149) node {$ 8 $};
+\draw [] (4.8000,-0.1000) -- (4.8000,0.1000);
+\draw (6.0000,-0.3149) node {$ 10 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.5944,-4.5000) node {$ -\frac{3}{200} $};
+\draw [] (-0.1000,-4.5000) -- (0.1000,-4.5000);
+\draw (-0.5944,-3.0000) node {$ -\frac{1}{100} $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.5944,-1.5000) node {$ -\frac{1}{200} $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.4525,1.5000) node {$ \frac{1}{200} $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
+\draw (-0.4525,3.0000) node {$ \frac{1}{100} $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall b/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall
index 766d698d6..96456fb8a 100644
--- a/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall
+++ b/src_phystricks/Fig_XTGooSFFtPu.pstricks.recall
@@ -107,93 +107,93 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-5.00,-4.00) -- (-5.00,3.00);
-\draw [color=gray,style=solid] (-4.00,-4.00) -- (-4.00,3.00);
-\draw [color=gray,style=solid] (-3.00,-4.00) -- (-3.00,3.00);
-\draw [color=gray,style=solid] (-2.00,-4.00) -- (-2.00,3.00);
-\draw [color=gray,style=solid] (-1.00,-4.00) -- (-1.00,3.00);
-\draw [color=gray,style=solid] (0,-4.00) -- (0,3.00);
-\draw [color=gray,style=solid] (1.00,-4.00) -- (1.00,3.00);
-\draw [color=gray,style=solid] (2.00,-4.00) -- (2.00,3.00);
-\draw [color=gray,style=solid] (3.00,-4.00) -- (3.00,3.00);
-\draw [color=gray,style=solid] (4.00,-4.00) -- (4.00,3.00);
-\draw [color=gray,style=solid] (5.00,-4.00) -- (5.00,3.00);
-\draw [color=gray,style=dotted] (-4.50,-4.00) -- (-4.50,3.00);
-\draw [color=gray,style=dotted] (-3.50,-4.00) -- (-3.50,3.00);
-\draw [color=gray,style=dotted] (-2.50,-4.00) -- (-2.50,3.00);
-\draw [color=gray,style=dotted] (-1.50,-4.00) -- (-1.50,3.00);
-\draw [color=gray,style=dotted] (-0.500,-4.00) -- (-0.500,3.00);
-\draw [color=gray,style=dotted] (0.500,-4.00) -- (0.500,3.00);
-\draw [color=gray,style=dotted] (1.50,-4.00) -- (1.50,3.00);
-\draw [color=gray,style=dotted] (2.50,-4.00) -- (2.50,3.00);
-\draw [color=gray,style=dotted] (3.50,-4.00) -- (3.50,3.00);
-\draw [color=gray,style=dotted] (4.50,-4.00) -- (4.50,3.00);
-\draw [color=gray,style=dotted] (-5.00,-3.50) -- (5.00,-3.50);
-\draw [color=gray,style=dotted] (-5.00,-2.50) -- (5.00,-2.50);
-\draw [color=gray,style=dotted] (-5.00,-1.50) -- (5.00,-1.50);
-\draw [color=gray,style=dotted] (-5.00,-0.500) -- (5.00,-0.500);
-\draw [color=gray,style=dotted] (-5.00,0.500) -- (5.00,0.500);
-\draw [color=gray,style=dotted] (-5.00,1.50) -- (5.00,1.50);
-\draw [color=gray,style=dotted] (-5.00,2.50) -- (5.00,2.50);
-\draw [color=gray,style=solid] (-5.00,-4.00) -- (5.00,-4.00);
-\draw [color=gray,style=solid] (-5.00,-3.00) -- (5.00,-3.00);
-\draw [color=gray,style=solid] (-5.00,-2.00) -- (5.00,-2.00);
-\draw [color=gray,style=solid] (-5.00,-1.00) -- (5.00,-1.00);
-\draw [color=gray,style=solid] (-5.00,0) -- (5.00,0);
-\draw [color=gray,style=solid] (-5.00,1.00) -- (5.00,1.00);
-\draw [color=gray,style=solid] (-5.00,2.00) -- (5.00,2.00);
-\draw [color=gray,style=solid] (-5.00,3.00) -- (5.00,3.00);
+\draw [color=gray,style=solid] (-5.0000,-4.0000) -- (-5.0000,3.0000);
+\draw [color=gray,style=solid] (-4.0000,-4.0000) -- (-4.0000,3.0000);
+\draw [color=gray,style=solid] (-3.0000,-4.0000) -- (-3.0000,3.0000);
+\draw [color=gray,style=solid] (-2.0000,-4.0000) -- (-2.0000,3.0000);
+\draw [color=gray,style=solid] (-1.0000,-4.0000) -- (-1.0000,3.0000);
+\draw [color=gray,style=solid] (0.0000,-4.0000) -- (0.0000,3.0000);
+\draw [color=gray,style=solid] (1.0000,-4.0000) -- (1.0000,3.0000);
+\draw [color=gray,style=solid] (2.0000,-4.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (3.0000,-4.0000) -- (3.0000,3.0000);
+\draw [color=gray,style=solid] (4.0000,-4.0000) -- (4.0000,3.0000);
+\draw [color=gray,style=solid] (5.0000,-4.0000) -- (5.0000,3.0000);
+\draw [color=gray,style=dotted] (-4.5000,-4.0000) -- (-4.5000,3.0000);
+\draw [color=gray,style=dotted] (-3.5000,-4.0000) -- (-3.5000,3.0000);
+\draw [color=gray,style=dotted] (-2.5000,-4.0000) -- (-2.5000,3.0000);
+\draw [color=gray,style=dotted] (-1.5000,-4.0000) -- (-1.5000,3.0000);
+\draw [color=gray,style=dotted] (-0.5000,-4.0000) -- (-0.5000,3.0000);
+\draw [color=gray,style=dotted] (0.5000,-4.0000) -- (0.5000,3.0000);
+\draw [color=gray,style=dotted] (1.5000,-4.0000) -- (1.5000,3.0000);
+\draw [color=gray,style=dotted] (2.5000,-4.0000) -- (2.5000,3.0000);
+\draw [color=gray,style=dotted] (3.5000,-4.0000) -- (3.5000,3.0000);
+\draw [color=gray,style=dotted] (4.5000,-4.0000) -- (4.5000,3.0000);
+\draw [color=gray,style=dotted] (-5.0000,-3.5000) -- (5.0000,-3.5000);
+\draw [color=gray,style=dotted] (-5.0000,-2.5000) -- (5.0000,-2.5000);
+\draw [color=gray,style=dotted] (-5.0000,-1.5000) -- (5.0000,-1.5000);
+\draw [color=gray,style=dotted] (-5.0000,-0.5000) -- (5.0000,-0.5000);
+\draw [color=gray,style=dotted] (-5.0000,0.5000) -- (5.0000,0.5000);
+\draw [color=gray,style=dotted] (-5.0000,1.5000) -- (5.0000,1.5000);
+\draw [color=gray,style=dotted] (-5.0000,2.5000) -- (5.0000,2.5000);
+\draw [color=gray,style=solid] (-5.0000,-4.0000) -- (5.0000,-4.0000);
+\draw [color=gray,style=solid] (-5.0000,-3.0000) -- (5.0000,-3.0000);
+\draw [color=gray,style=solid] (-5.0000,-2.0000) -- (5.0000,-2.0000);
+\draw [color=gray,style=solid] (-5.0000,-1.0000) -- (5.0000,-1.0000);
+\draw [color=gray,style=solid] (-5.0000,0.0000) -- (5.0000,0.0000);
+\draw [color=gray,style=solid] (-5.0000,1.0000) -- (5.0000,1.0000);
+\draw [color=gray,style=solid] (-5.0000,2.0000) -- (5.0000,2.0000);
+\draw [color=gray,style=solid] (-5.0000,3.0000) -- (5.0000,3.0000);
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (-5.000,-4.000)--(-4.960,-3.960)--(-4.919,-3.919)--(-4.879,-3.879)--(-4.838,-3.838)--(-4.798,-3.798)--(-4.758,-3.758)--(-4.717,-3.717)--(-4.677,-3.677)--(-4.636,-3.636)--(-4.596,-3.596)--(-4.556,-3.556)--(-4.515,-3.515)--(-4.475,-3.475)--(-4.434,-3.434)--(-4.394,-3.394)--(-4.354,-3.354)--(-4.313,-3.313)--(-4.273,-3.273)--(-4.232,-3.232)--(-4.192,-3.192)--(-4.151,-3.152)--(-4.111,-3.111)--(-4.071,-3.071)--(-4.030,-3.030)--(-3.990,-2.990)--(-3.949,-2.949)--(-3.909,-2.909)--(-3.869,-2.869)--(-3.828,-2.828)--(-3.788,-2.788)--(-3.747,-2.747)--(-3.707,-2.707)--(-3.667,-2.667)--(-3.626,-2.626)--(-3.586,-2.586)--(-3.545,-2.545)--(-3.505,-2.505)--(-3.465,-2.465)--(-3.424,-2.424)--(-3.384,-2.384)--(-3.343,-2.343)--(-3.303,-2.303)--(-3.263,-2.263)--(-3.222,-2.222)--(-3.182,-2.182)--(-3.141,-2.141)--(-3.101,-2.101)--(-3.061,-2.061)--(-3.020,-2.020)--(-2.980,-1.980)--(-2.939,-1.939)--(-2.899,-1.899)--(-2.859,-1.859)--(-2.818,-1.818)--(-2.778,-1.778)--(-2.737,-1.737)--(-2.697,-1.697)--(-2.657,-1.657)--(-2.616,-1.616)--(-2.576,-1.576)--(-2.535,-1.535)--(-2.495,-1.495)--(-2.455,-1.455)--(-2.414,-1.414)--(-2.374,-1.374)--(-2.333,-1.333)--(-2.293,-1.293)--(-2.253,-1.253)--(-2.212,-1.212)--(-2.172,-1.172)--(-2.131,-1.131)--(-2.091,-1.091)--(-2.051,-1.051)--(-2.010,-1.010)--(-1.970,-0.9697)--(-1.929,-0.9293)--(-1.889,-0.8889)--(-1.848,-0.8485)--(-1.808,-0.8081)--(-1.768,-0.7677)--(-1.727,-0.7273)--(-1.687,-0.6869)--(-1.646,-0.6465)--(-1.606,-0.6061)--(-1.566,-0.5657)--(-1.525,-0.5253)--(-1.485,-0.4848)--(-1.444,-0.4444)--(-1.404,-0.4040)--(-1.364,-0.3636)--(-1.323,-0.3232)--(-1.283,-0.2828)--(-1.242,-0.2424)--(-1.202,-0.2020)--(-1.162,-0.1616)--(-1.121,-0.1212)--(-1.081,-0.08081)--(-1.040,-0.04040)--(-1.000,0);
+\draw [color=blue] (-5.0000,-4.0000)--(-4.9595,-3.9595)--(-4.9191,-3.9191)--(-4.8787,-3.8787)--(-4.8383,-3.8383)--(-4.7979,-3.7979)--(-4.7575,-3.7575)--(-4.7171,-3.7171)--(-4.6767,-3.6767)--(-4.6363,-3.6363)--(-4.5959,-3.5959)--(-4.5555,-3.5555)--(-4.5151,-3.5151)--(-4.4747,-3.4747)--(-4.4343,-3.4343)--(-4.3939,-3.3939)--(-4.3535,-3.3535)--(-4.3131,-3.3131)--(-4.2727,-3.2727)--(-4.2323,-3.2323)--(-4.1919,-3.1919)--(-4.1515,-3.1515)--(-4.1111,-3.1111)--(-4.0707,-3.0707)--(-4.0303,-3.0303)--(-3.9898,-2.9898)--(-3.9494,-2.9494)--(-3.9090,-2.9090)--(-3.8686,-2.8686)--(-3.8282,-2.8282)--(-3.7878,-2.7878)--(-3.7474,-2.7474)--(-3.7070,-2.7070)--(-3.6666,-2.6666)--(-3.6262,-2.6262)--(-3.5858,-2.5858)--(-3.5454,-2.5454)--(-3.5050,-2.5050)--(-3.4646,-2.4646)--(-3.4242,-2.4242)--(-3.3838,-2.3838)--(-3.3434,-2.3434)--(-3.3030,-2.3030)--(-3.2626,-2.2626)--(-3.2222,-2.2222)--(-3.1818,-2.1818)--(-3.1414,-2.1414)--(-3.1010,-2.1010)--(-3.0606,-2.0606)--(-3.0202,-2.0202)--(-2.9797,-1.9797)--(-2.9393,-1.9393)--(-2.8989,-1.8989)--(-2.8585,-1.8585)--(-2.8181,-1.8181)--(-2.7777,-1.7777)--(-2.7373,-1.7373)--(-2.6969,-1.6969)--(-2.6565,-1.6565)--(-2.6161,-1.6161)--(-2.5757,-1.5757)--(-2.5353,-1.5353)--(-2.4949,-1.4949)--(-2.4545,-1.4545)--(-2.4141,-1.4141)--(-2.3737,-1.3737)--(-2.3333,-1.3333)--(-2.2929,-1.2929)--(-2.2525,-1.2525)--(-2.2121,-1.2121)--(-2.1717,-1.1717)--(-2.1313,-1.1313)--(-2.0909,-1.0909)--(-2.0505,-1.0505)--(-2.0101,-1.0101)--(-1.9696,-0.9696)--(-1.9292,-0.9292)--(-1.8888,-0.8888)--(-1.8484,-0.8484)--(-1.8080,-0.8080)--(-1.7676,-0.7676)--(-1.7272,-0.7272)--(-1.6868,-0.6868)--(-1.6464,-0.6464)--(-1.6060,-0.6060)--(-1.5656,-0.5656)--(-1.5252,-0.5252)--(-1.4848,-0.4848)--(-1.4444,-0.4444)--(-1.4040,-0.4040)--(-1.3636,-0.3636)--(-1.3232,-0.3232)--(-1.2828,-0.2828)--(-1.2424,-0.2424)--(-1.2020,-0.2020)--(-1.1616,-0.1616)--(-1.1212,-0.1212)--(-1.0808,-0.0808)--(-1.0404,-0.0404)--(-1.0000,0.0000);
 
-\draw [color=blue] (-1.000,-2.000)--(-0.9798,-1.960)--(-0.9596,-1.919)--(-0.9394,-1.879)--(-0.9192,-1.838)--(-0.8990,-1.798)--(-0.8788,-1.758)--(-0.8586,-1.717)--(-0.8384,-1.677)--(-0.8182,-1.636)--(-0.7980,-1.596)--(-0.7778,-1.556)--(-0.7576,-1.515)--(-0.7374,-1.475)--(-0.7172,-1.434)--(-0.6970,-1.394)--(-0.6768,-1.354)--(-0.6566,-1.313)--(-0.6364,-1.273)--(-0.6162,-1.232)--(-0.5960,-1.192)--(-0.5758,-1.152)--(-0.5556,-1.111)--(-0.5354,-1.071)--(-0.5152,-1.030)--(-0.4949,-0.9899)--(-0.4747,-0.9495)--(-0.4545,-0.9091)--(-0.4343,-0.8687)--(-0.4141,-0.8283)--(-0.3939,-0.7879)--(-0.3737,-0.7475)--(-0.3535,-0.7071)--(-0.3333,-0.6667)--(-0.3131,-0.6263)--(-0.2929,-0.5859)--(-0.2727,-0.5455)--(-0.2525,-0.5051)--(-0.2323,-0.4646)--(-0.2121,-0.4242)--(-0.1919,-0.3838)--(-0.1717,-0.3434)--(-0.1515,-0.3030)--(-0.1313,-0.2626)--(-0.1111,-0.2222)--(-0.09091,-0.1818)--(-0.07071,-0.1414)--(-0.05051,-0.1010)--(-0.03030,-0.06061)--(-0.01010,-0.02020)--(0.01010,0.02020)--(0.03030,0.06061)--(0.05051,0.1010)--(0.07071,0.1414)--(0.09091,0.1818)--(0.1111,0.2222)--(0.1313,0.2626)--(0.1515,0.3030)--(0.1717,0.3434)--(0.1919,0.3838)--(0.2121,0.4242)--(0.2323,0.4646)--(0.2525,0.5051)--(0.2727,0.5455)--(0.2929,0.5859)--(0.3131,0.6263)--(0.3333,0.6667)--(0.3535,0.7071)--(0.3737,0.7475)--(0.3939,0.7879)--(0.4141,0.8283)--(0.4343,0.8687)--(0.4545,0.9091)--(0.4747,0.9495)--(0.4949,0.9899)--(0.5152,1.030)--(0.5354,1.071)--(0.5556,1.111)--(0.5758,1.152)--(0.5960,1.192)--(0.6162,1.232)--(0.6364,1.273)--(0.6566,1.313)--(0.6768,1.354)--(0.6970,1.394)--(0.7172,1.434)--(0.7374,1.475)--(0.7576,1.515)--(0.7778,1.556)--(0.7980,1.596)--(0.8182,1.636)--(0.8384,1.677)--(0.8586,1.717)--(0.8788,1.758)--(0.8990,1.798)--(0.9192,1.838)--(0.9394,1.879)--(0.9596,1.919)--(0.9798,1.960)--(1.000,2.000);
+\draw [color=blue] (-1.0000,-2.0000)--(-0.9797,-1.9595)--(-0.9595,-1.9191)--(-0.9393,-1.8787)--(-0.9191,-1.8383)--(-0.8989,-1.7979)--(-0.8787,-1.7575)--(-0.8585,-1.7171)--(-0.8383,-1.6767)--(-0.8181,-1.6363)--(-0.7979,-1.5959)--(-0.7777,-1.5555)--(-0.7575,-1.5151)--(-0.7373,-1.4747)--(-0.7171,-1.4343)--(-0.6969,-1.3939)--(-0.6767,-1.3535)--(-0.6565,-1.3131)--(-0.6363,-1.2727)--(-0.6161,-1.2323)--(-0.5959,-1.1919)--(-0.5757,-1.1515)--(-0.5555,-1.1111)--(-0.5353,-1.0707)--(-0.5151,-1.0303)--(-0.4949,-0.9898)--(-0.4747,-0.9494)--(-0.4545,-0.9090)--(-0.4343,-0.8686)--(-0.4141,-0.8282)--(-0.3939,-0.7878)--(-0.3737,-0.7474)--(-0.3535,-0.7070)--(-0.3333,-0.6666)--(-0.3131,-0.6262)--(-0.2929,-0.5858)--(-0.2727,-0.5454)--(-0.2525,-0.5050)--(-0.2323,-0.4646)--(-0.2121,-0.4242)--(-0.1919,-0.3838)--(-0.1717,-0.3434)--(-0.1515,-0.3030)--(-0.1313,-0.2626)--(-0.1111,-0.2222)--(-0.0909,-0.1818)--(-0.0707,-0.1414)--(-0.0505,-0.1010)--(-0.0303,-0.0606)--(-0.0101,-0.0202)--(0.0101,0.0202)--(0.0303,0.0606)--(0.0505,0.1010)--(0.0707,0.1414)--(0.0909,0.1818)--(0.1111,0.2222)--(0.1313,0.2626)--(0.1515,0.3030)--(0.1717,0.3434)--(0.1919,0.3838)--(0.2121,0.4242)--(0.2323,0.4646)--(0.2525,0.5050)--(0.2727,0.5454)--(0.2929,0.5858)--(0.3131,0.6262)--(0.3333,0.6666)--(0.3535,0.7070)--(0.3737,0.7474)--(0.3939,0.7878)--(0.4141,0.8282)--(0.4343,0.8686)--(0.4545,0.9090)--(0.4747,0.9494)--(0.4949,0.9898)--(0.5151,1.0303)--(0.5353,1.0707)--(0.5555,1.1111)--(0.5757,1.1515)--(0.5959,1.1919)--(0.6161,1.2323)--(0.6363,1.2727)--(0.6565,1.3131)--(0.6767,1.3535)--(0.6969,1.3939)--(0.7171,1.4343)--(0.7373,1.4747)--(0.7575,1.5151)--(0.7777,1.5555)--(0.7979,1.5959)--(0.8181,1.6363)--(0.8383,1.6767)--(0.8585,1.7171)--(0.8787,1.7575)--(0.8989,1.7979)--(0.9191,1.8383)--(0.9393,1.8787)--(0.9595,1.9191)--(0.9797,1.9595)--(1.0000,2.0000);
 
-\draw [color=blue] (1.000,3.000)--(1.040,2.960)--(1.081,2.919)--(1.121,2.879)--(1.162,2.838)--(1.202,2.798)--(1.242,2.758)--(1.283,2.717)--(1.323,2.677)--(1.364,2.636)--(1.404,2.596)--(1.444,2.556)--(1.485,2.515)--(1.525,2.475)--(1.566,2.434)--(1.606,2.394)--(1.646,2.354)--(1.687,2.313)--(1.727,2.273)--(1.768,2.232)--(1.808,2.192)--(1.848,2.152)--(1.889,2.111)--(1.929,2.071)--(1.970,2.030)--(2.010,1.990)--(2.051,1.949)--(2.091,1.909)--(2.131,1.869)--(2.172,1.828)--(2.212,1.788)--(2.253,1.747)--(2.293,1.707)--(2.333,1.667)--(2.374,1.626)--(2.414,1.586)--(2.455,1.545)--(2.495,1.505)--(2.535,1.465)--(2.576,1.424)--(2.616,1.384)--(2.657,1.343)--(2.697,1.303)--(2.737,1.263)--(2.778,1.222)--(2.818,1.182)--(2.859,1.141)--(2.899,1.101)--(2.939,1.061)--(2.980,1.020)--(3.020,0.9798)--(3.061,0.9394)--(3.101,0.8990)--(3.141,0.8586)--(3.182,0.8182)--(3.222,0.7778)--(3.263,0.7374)--(3.303,0.6970)--(3.343,0.6566)--(3.384,0.6162)--(3.424,0.5758)--(3.465,0.5354)--(3.505,0.4949)--(3.545,0.4545)--(3.586,0.4141)--(3.626,0.3737)--(3.667,0.3333)--(3.707,0.2929)--(3.747,0.2525)--(3.788,0.2121)--(3.828,0.1717)--(3.869,0.1313)--(3.909,0.09091)--(3.949,0.05051)--(3.990,0.01010)--(4.030,-0.03030)--(4.071,-0.07071)--(4.111,-0.1111)--(4.151,-0.1515)--(4.192,-0.1919)--(4.232,-0.2323)--(4.273,-0.2727)--(4.313,-0.3131)--(4.354,-0.3535)--(4.394,-0.3939)--(4.434,-0.4343)--(4.475,-0.4747)--(4.515,-0.5152)--(4.556,-0.5556)--(4.596,-0.5960)--(4.636,-0.6364)--(4.677,-0.6768)--(4.717,-0.7172)--(4.758,-0.7576)--(4.798,-0.7980)--(4.838,-0.8384)--(4.879,-0.8788)--(4.919,-0.9192)--(4.960,-0.9596)--(5.000,-1.000);
+\draw [color=blue] (1.0000,3.0000)--(1.0404,2.9595)--(1.0808,2.9191)--(1.1212,2.8787)--(1.1616,2.8383)--(1.2020,2.7979)--(1.2424,2.7575)--(1.2828,2.7171)--(1.3232,2.6767)--(1.3636,2.6363)--(1.4040,2.5959)--(1.4444,2.5555)--(1.4848,2.5151)--(1.5252,2.4747)--(1.5656,2.4343)--(1.6060,2.3939)--(1.6464,2.3535)--(1.6868,2.3131)--(1.7272,2.2727)--(1.7676,2.2323)--(1.8080,2.1919)--(1.8484,2.1515)--(1.8888,2.1111)--(1.9292,2.0707)--(1.9696,2.0303)--(2.0101,1.9898)--(2.0505,1.9494)--(2.0909,1.9090)--(2.1313,1.8686)--(2.1717,1.8282)--(2.2121,1.7878)--(2.2525,1.7474)--(2.2929,1.7070)--(2.3333,1.6666)--(2.3737,1.6262)--(2.4141,1.5858)--(2.4545,1.5454)--(2.4949,1.5050)--(2.5353,1.4646)--(2.5757,1.4242)--(2.6161,1.3838)--(2.6565,1.3434)--(2.6969,1.3030)--(2.7373,1.2626)--(2.7777,1.2222)--(2.8181,1.1818)--(2.8585,1.1414)--(2.8989,1.1010)--(2.9393,1.0606)--(2.9797,1.0202)--(3.0202,0.9797)--(3.0606,0.9393)--(3.1010,0.8989)--(3.1414,0.8585)--(3.1818,0.8181)--(3.2222,0.7777)--(3.2626,0.7373)--(3.3030,0.6969)--(3.3434,0.6565)--(3.3838,0.6161)--(3.4242,0.5757)--(3.4646,0.5353)--(3.5050,0.4949)--(3.5454,0.4545)--(3.5858,0.4141)--(3.6262,0.3737)--(3.6666,0.3333)--(3.7070,0.2929)--(3.7474,0.2525)--(3.7878,0.2121)--(3.8282,0.1717)--(3.8686,0.1313)--(3.9090,0.0909)--(3.9494,0.0505)--(3.9898,0.0101)--(4.0303,-0.0303)--(4.0707,-0.0707)--(4.1111,-0.1111)--(4.1515,-0.1515)--(4.1919,-0.1919)--(4.2323,-0.2323)--(4.2727,-0.2727)--(4.3131,-0.3131)--(4.3535,-0.3535)--(4.3939,-0.3939)--(4.4343,-0.4343)--(4.4747,-0.4747)--(4.5151,-0.5151)--(4.5555,-0.5555)--(4.5959,-0.5959)--(4.6363,-0.6363)--(4.6767,-0.6767)--(4.7171,-0.7171)--(4.7575,-0.7575)--(4.7979,-0.7979)--(4.8383,-0.8383)--(4.8787,-0.8787)--(4.9191,-0.9191)--(4.9595,-0.9595)--(5.0000,-1.0000);
 \draw [color=blue]  (-1.0000,-2.0000) node [rotate=0] {$\bullet$};
 \draw [color=blue]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw []  (0.50000,1.0000) node [rotate=0] {$\bullet$};
-\draw (0.84517,0.70650) node {\( Z_1\)};
+\draw []  (0.5000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.8451,0.7064) node {\( Z_1\)};
 \draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.3452,1.2935) node {\( Z_2\)};
+\draw (3.3451,1.2935) node {\( Z_2\)};
 
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall b/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall
index 824be3771..97af61aa9 100644
--- a/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall
+++ b/src_phystricks/Fig_YHJYooTEXLLn.pstricks.recall
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.02020,0.03030)--(0.04040,0.06061)--(0.06061,0.09091)--(0.08081,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5152)--(0.3636,0.5455)--(0.3838,0.5758)--(0.4040,0.6061)--(0.4242,0.6364)--(0.4444,0.6667)--(0.4646,0.6970)--(0.4848,0.7273)--(0.5051,0.7576)--(0.5253,0.7879)--(0.5455,0.8182)--(0.5657,0.8485)--(0.5859,0.8788)--(0.6061,0.9091)--(0.6263,0.9394)--(0.6465,0.9697)--(0.6667,1.000)--(0.6869,1.030)--(0.7071,1.061)--(0.7273,1.091)--(0.7475,1.121)--(0.7677,1.152)--(0.7879,1.182)--(0.8081,1.212)--(0.8283,1.242)--(0.8485,1.273)--(0.8687,1.303)--(0.8889,1.333)--(0.9091,1.364)--(0.9293,1.394)--(0.9495,1.424)--(0.9697,1.455)--(0.9899,1.485)--(1.010,1.515)--(1.030,1.545)--(1.051,1.576)--(1.071,1.606)--(1.091,1.636)--(1.111,1.667)--(1.131,1.697)--(1.152,1.727)--(1.172,1.758)--(1.192,1.788)--(1.212,1.818)--(1.232,1.848)--(1.253,1.879)--(1.273,1.909)--(1.293,1.939)--(1.313,1.970)--(1.333,2.000)--(1.354,2.030)--(1.374,2.061)--(1.394,2.091)--(1.414,2.121)--(1.434,2.152)--(1.455,2.182)--(1.475,2.212)--(1.495,2.242)--(1.515,2.273)--(1.535,2.303)--(1.556,2.333)--(1.576,2.364)--(1.596,2.394)--(1.616,2.424)--(1.636,2.455)--(1.657,2.485)--(1.677,2.515)--(1.697,2.545)--(1.717,2.576)--(1.737,2.606)--(1.758,2.636)--(1.778,2.667)--(1.798,2.697)--(1.818,2.727)--(1.838,2.758)--(1.859,2.788)--(1.879,2.818)--(1.899,2.848)--(1.919,2.879)--(1.939,2.909)--(1.960,2.939)--(1.980,2.970)--(2.000,3.000);
+\draw [color=blue] (0.0000,0.0000)--(0.0202,0.0303)--(0.0404,0.0606)--(0.0606,0.0909)--(0.0808,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5151)--(0.3636,0.5454)--(0.3838,0.5757)--(0.4040,0.6060)--(0.4242,0.6363)--(0.4444,0.6666)--(0.4646,0.6969)--(0.4848,0.7272)--(0.5050,0.7575)--(0.5252,0.7878)--(0.5454,0.8181)--(0.5656,0.8484)--(0.5858,0.8787)--(0.6060,0.9090)--(0.6262,0.9393)--(0.6464,0.9696)--(0.6666,1.0000)--(0.6868,1.0303)--(0.7070,1.0606)--(0.7272,1.0909)--(0.7474,1.1212)--(0.7676,1.1515)--(0.7878,1.1818)--(0.8080,1.2121)--(0.8282,1.2424)--(0.8484,1.2727)--(0.8686,1.3030)--(0.8888,1.3333)--(0.9090,1.3636)--(0.9292,1.3939)--(0.9494,1.4242)--(0.9696,1.4545)--(0.9898,1.4848)--(1.0101,1.5151)--(1.0303,1.5454)--(1.0505,1.5757)--(1.0707,1.6060)--(1.0909,1.6363)--(1.1111,1.6666)--(1.1313,1.6969)--(1.1515,1.7272)--(1.1717,1.7575)--(1.1919,1.7878)--(1.2121,1.8181)--(1.2323,1.8484)--(1.2525,1.8787)--(1.2727,1.9090)--(1.2929,1.9393)--(1.3131,1.9696)--(1.3333,2.0000)--(1.3535,2.0303)--(1.3737,2.0606)--(1.3939,2.0909)--(1.4141,2.1212)--(1.4343,2.1515)--(1.4545,2.1818)--(1.4747,2.2121)--(1.4949,2.2424)--(1.5151,2.2727)--(1.5353,2.3030)--(1.5555,2.3333)--(1.5757,2.3636)--(1.5959,2.3939)--(1.6161,2.4242)--(1.6363,2.4545)--(1.6565,2.4848)--(1.6767,2.5151)--(1.6969,2.5454)--(1.7171,2.5757)--(1.7373,2.6060)--(1.7575,2.6363)--(1.7777,2.6666)--(1.7979,2.6969)--(1.8181,2.7272)--(1.8383,2.7575)--(1.8585,2.7878)--(1.8787,2.8181)--(1.8989,2.8484)--(1.9191,2.8787)--(1.9393,2.9090)--(1.9595,2.9393)--(1.9797,2.9696)--(2.0000,3.0000);
 \draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
-\draw (2.5389,3.2532) node {$(R,h)$};
-\draw (0.72367,0.33595) node {$\alpha$};
+\draw (2.5388,3.2531) node {$(R,h)$};
+\draw (0.7236,0.3359) node {$\alpha$};
 
-\draw [color=red] (0.500,0)--(0.500,0.00496)--(0.500,0.00993)--(0.500,0.0149)--(0.500,0.0198)--(0.499,0.0248)--(0.499,0.0298)--(0.499,0.0347)--(0.498,0.0397)--(0.498,0.0446)--(0.498,0.0496)--(0.497,0.0545)--(0.496,0.0594)--(0.496,0.0643)--(0.495,0.0693)--(0.494,0.0742)--(0.494,0.0791)--(0.493,0.0840)--(0.492,0.0889)--(0.491,0.0938)--(0.490,0.0986)--(0.489,0.103)--(0.488,0.108)--(0.487,0.113)--(0.486,0.118)--(0.485,0.123)--(0.483,0.128)--(0.482,0.132)--(0.481,0.137)--(0.479,0.142)--(0.478,0.147)--(0.477,0.151)--(0.475,0.156)--(0.473,0.161)--(0.472,0.166)--(0.470,0.170)--(0.468,0.175)--(0.467,0.180)--(0.465,0.184)--(0.463,0.189)--(0.461,0.193)--(0.459,0.198)--(0.457,0.202)--(0.455,0.207)--(0.453,0.212)--(0.451,0.216)--(0.449,0.220)--(0.447,0.225)--(0.444,0.229)--(0.442,0.234)--(0.440,0.238)--(0.437,0.242)--(0.435,0.247)--(0.432,0.251)--(0.430,0.255)--(0.427,0.260)--(0.425,0.264)--(0.422,0.268)--(0.419,0.272)--(0.417,0.276)--(0.414,0.281)--(0.411,0.285)--(0.408,0.289)--(0.405,0.293)--(0.402,0.297)--(0.399,0.301)--(0.396,0.305)--(0.393,0.309)--(0.390,0.312)--(0.387,0.316)--(0.384,0.320)--(0.381,0.324)--(0.378,0.328)--(0.374,0.331)--(0.371,0.335)--(0.368,0.339)--(0.364,0.342)--(0.361,0.346)--(0.357,0.350)--(0.354,0.353)--(0.350,0.357)--(0.347,0.360)--(0.343,0.364)--(0.340,0.367)--(0.336,0.370)--(0.332,0.374)--(0.329,0.377)--(0.325,0.380)--(0.321,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.306,0.396)--(0.302,0.399)--(0.298,0.402)--(0.294,0.405)--(0.290,0.408)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416);
-\draw (2.0000,-0.32572) node {$\mathit{R}$};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.30273,3.0000) node {$\mathit{h}$};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (0.5000,0.0000)--(0.4999,0.0049)--(0.4999,0.0099)--(0.4997,0.0148)--(0.4996,0.0198)--(0.4993,0.0248)--(0.4991,0.0297)--(0.4987,0.0347)--(0.4984,0.0396)--(0.4980,0.0446)--(0.4975,0.0495)--(0.4970,0.0544)--(0.4964,0.0594)--(0.4958,0.0643)--(0.4951,0.0692)--(0.4944,0.0741)--(0.4937,0.0790)--(0.4928,0.0839)--(0.4920,0.0888)--(0.4911,0.0937)--(0.4901,0.0986)--(0.4891,0.1034)--(0.4881,0.1083)--(0.4870,0.1131)--(0.4858,0.1180)--(0.4846,0.1228)--(0.4834,0.1276)--(0.4821,0.1324)--(0.4808,0.1371)--(0.4794,0.1419)--(0.4779,0.1467)--(0.4765,0.1514)--(0.4749,0.1561)--(0.4734,0.1608)--(0.4717,0.1655)--(0.4701,0.1702)--(0.4684,0.1749)--(0.4666,0.1795)--(0.4648,0.1841)--(0.4629,0.1887)--(0.4610,0.1933)--(0.4591,0.1979)--(0.4571,0.2024)--(0.4551,0.2070)--(0.4530,0.2115)--(0.4509,0.2160)--(0.4487,0.2204)--(0.4465,0.2249)--(0.4443,0.2293)--(0.4420,0.2337)--(0.4396,0.2381)--(0.4372,0.2424)--(0.4348,0.2467)--(0.4323,0.2511)--(0.4298,0.2553)--(0.4273,0.2596)--(0.4247,0.2638)--(0.4220,0.2680)--(0.4193,0.2722)--(0.4166,0.2763)--(0.4138,0.2805)--(0.4110,0.2846)--(0.4082,0.2886)--(0.4053,0.2927)--(0.4024,0.2967)--(0.3994,0.3007)--(0.3964,0.3046)--(0.3934,0.3085)--(0.3903,0.3124)--(0.3872,0.3163)--(0.3840,0.3201)--(0.3808,0.3239)--(0.3776,0.3277)--(0.3743,0.3314)--(0.3710,0.3351)--(0.3676,0.3388)--(0.3643,0.3424)--(0.3609,0.3460)--(0.3574,0.3496)--(0.3539,0.3531)--(0.3504,0.3566)--(0.3468,0.3601)--(0.3432,0.3635)--(0.3396,0.3669)--(0.3360,0.3702)--(0.3323,0.3735)--(0.3285,0.3768)--(0.3248,0.3801)--(0.3210,0.3833)--(0.3172,0.3864)--(0.3133,0.3896)--(0.3094,0.3927)--(0.3055,0.3957)--(0.3016,0.3987)--(0.2976,0.4017)--(0.2936,0.4046)--(0.2896,0.4075)--(0.2855,0.4104)--(0.2814,0.4132)--(0.2773,0.4160);
+\draw (2.0000,-0.3257) node {$\mathit{R}$};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.3027,3.0000) node {$\mathit{h}$};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall b/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall
index 9465f44db..b3a9e0cc9 100644
--- a/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall
+++ b/src_phystricks/Fig_YQIDooBqpAdbIM.pstricks.recall
@@ -74,24 +74,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.31799,-0.14894) node {\( A \)};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.3287,-0.15396) node {\( B \)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3179,-0.1489) node {\( A \)};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.3287,-0.1539) node {\( B \)};
 \draw []  (2.5000,3.0000) node [rotate=0] {$\bullet$};
 \draw (2.5000,3.3247) node {\( O \)};
-\draw [] (0,0) -- (5.00,0);
-\draw [] (5.00,0) -- (2.50,3.00);
-\draw [] (2.50,3.00) -- (0,0);
-\draw [color=red,->,>=latex] (0.32009,0.38411) -- (0.31241,0.39051);
+\draw [] (0.0000,0.0000) -- (5.0000,0.0000);
+\draw [] (5.0000,0.0000) -- (2.5000,3.0000);
+\draw [] (2.5000,3.0000) -- (0.0000,0.0000);
+\draw [color=red,->,>=latex] (0.3200,0.3841) -- (0.3124,0.3905);
 
-\draw [color=red] (0.500,0)--(0.500,0.00442)--(0.500,0.00885)--(0.500,0.0133)--(0.500,0.0177)--(0.500,0.0221)--(0.499,0.0265)--(0.499,0.0310)--(0.499,0.0354)--(0.498,0.0398)--(0.498,0.0442)--(0.498,0.0486)--(0.497,0.0530)--(0.497,0.0574)--(0.496,0.0618)--(0.496,0.0662)--(0.495,0.0706)--(0.494,0.0749)--(0.494,0.0793)--(0.493,0.0837)--(0.492,0.0880)--(0.491,0.0924)--(0.491,0.0967)--(0.490,0.101)--(0.489,0.105)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.123)--(0.484,0.127)--(0.482,0.131)--(0.481,0.135)--(0.480,0.140)--(0.479,0.144)--(0.478,0.148)--(0.476,0.152)--(0.475,0.157)--(0.473,0.161)--(0.472,0.165)--(0.471,0.169)--(0.469,0.173)--(0.467,0.177)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.194)--(0.459,0.198)--(0.457,0.202)--(0.456,0.206)--(0.454,0.210)--(0.452,0.214)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.230)--(0.442,0.234)--(0.440,0.238)--(0.438,0.242)--(0.436,0.245)--(0.433,0.249)--(0.431,0.253)--(0.429,0.257)--(0.427,0.261)--(0.424,0.265)--(0.422,0.268)--(0.420,0.272)--(0.417,0.276)--(0.415,0.279)--(0.412,0.283)--(0.410,0.287)--(0.407,0.290)--(0.405,0.294)--(0.402,0.297)--(0.399,0.301)--(0.397,0.305)--(0.394,0.308)--(0.391,0.311)--(0.388,0.315)--(0.386,0.318)--(0.383,0.322)--(0.380,0.325)--(0.377,0.328)--(0.374,0.332)--(0.371,0.335)--(0.368,0.338)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.353,0.354)--(0.350,0.357)--(0.346,0.361)--(0.343,0.364)--(0.340,0.367)--(0.337,0.370)--(0.333,0.373)--(0.330,0.375)--(0.327,0.378)--(0.323,0.381)--(0.320,0.384);
-\draw [color=red,->,>=latex] (2.8201,2.6159) -- (2.8278,2.6223);
+\draw [color=red] (0.5000,0.0000)--(0.4999,0.0044)--(0.4999,0.0088)--(0.4998,0.0132)--(0.4996,0.0176)--(0.4995,0.0221)--(0.4992,0.0265)--(0.4990,0.0309)--(0.4987,0.0353)--(0.4984,0.0397)--(0.4980,0.0441)--(0.4976,0.0485)--(0.4971,0.0529)--(0.4966,0.0573)--(0.4961,0.0617)--(0.4956,0.0661)--(0.4949,0.0705)--(0.4943,0.0749)--(0.4936,0.0793)--(0.4929,0.0836)--(0.4921,0.0880)--(0.4913,0.0923)--(0.4905,0.0967)--(0.4896,0.1010)--(0.4887,0.1053)--(0.4878,0.1097)--(0.4868,0.1140)--(0.4857,0.1183)--(0.4847,0.1226)--(0.4836,0.1269)--(0.4824,0.1311)--(0.4813,0.1354)--(0.4800,0.1397)--(0.4788,0.1439)--(0.4775,0.1481)--(0.4762,0.1523)--(0.4748,0.1566)--(0.4734,0.1607)--(0.4719,0.1649)--(0.4705,0.1691)--(0.4690,0.1733)--(0.4674,0.1774)--(0.4658,0.1815)--(0.4642,0.1856)--(0.4625,0.1897)--(0.4608,0.1938)--(0.4591,0.1979)--(0.4573,0.2020)--(0.4555,0.2060)--(0.4537,0.2100)--(0.4518,0.2140)--(0.4499,0.2180)--(0.4479,0.2220)--(0.4460,0.2259)--(0.4439,0.2299)--(0.4419,0.2338)--(0.4398,0.2377)--(0.4377,0.2416)--(0.4355,0.2455)--(0.4333,0.2493)--(0.4311,0.2531)--(0.4289,0.2569)--(0.4266,0.2607)--(0.4242,0.2645)--(0.4219,0.2682)--(0.4195,0.2719)--(0.4171,0.2756)--(0.4146,0.2793)--(0.4121,0.2830)--(0.4096,0.2866)--(0.4071,0.2902)--(0.4045,0.2938)--(0.4019,0.2974)--(0.3992,0.3009)--(0.3965,0.3045)--(0.3938,0.3080)--(0.3911,0.3114)--(0.3883,0.3149)--(0.3855,0.3183)--(0.3827,0.3217)--(0.3798,0.3251)--(0.3769,0.3284)--(0.3740,0.3318)--(0.3710,0.3350)--(0.3681,0.3383)--(0.3651,0.3416)--(0.3620,0.3448)--(0.3590,0.3480)--(0.3559,0.3511)--(0.3527,0.3543)--(0.3496,0.3574)--(0.3464,0.3605)--(0.3432,0.3635)--(0.3400,0.3665)--(0.3367,0.3695)--(0.3334,0.3725)--(0.3301,0.3754)--(0.3268,0.3783)--(0.3234,0.3812)--(0.3200,0.3841);
+\draw [color=red,->,>=latex] (2.8200,2.6158) -- (2.8277,2.6222);
 
-\draw [color=red] (2.180,2.616)--(2.185,2.611)--(2.191,2.607)--(2.196,2.603)--(2.202,2.599)--(2.208,2.594)--(2.213,2.590)--(2.219,2.586)--(2.225,2.582)--(2.231,2.579)--(2.237,2.575)--(2.243,2.571)--(2.249,2.568)--(2.255,2.564)--(2.261,2.561)--(2.267,2.557)--(2.273,2.554)--(2.280,2.551)--(2.286,2.548)--(2.292,2.545)--(2.299,2.542)--(2.305,2.539)--(2.312,2.537)--(2.318,2.534)--(2.325,2.532)--(2.331,2.529)--(2.338,2.527)--(2.345,2.525)--(2.351,2.523)--(2.358,2.521)--(2.365,2.519)--(2.372,2.517)--(2.378,2.515)--(2.385,2.513)--(2.392,2.512)--(2.399,2.510)--(2.406,2.509)--(2.413,2.508)--(2.420,2.507)--(2.427,2.505)--(2.434,2.504)--(2.440,2.504)--(2.447,2.503)--(2.454,2.502)--(2.461,2.501)--(2.468,2.501)--(2.475,2.501)--(2.482,2.500)--(2.489,2.500)--(2.496,2.500)--(2.504,2.500)--(2.511,2.500)--(2.518,2.500)--(2.525,2.501)--(2.532,2.501)--(2.539,2.501)--(2.546,2.502)--(2.553,2.503)--(2.560,2.504)--(2.566,2.504)--(2.573,2.505)--(2.580,2.507)--(2.587,2.508)--(2.594,2.509)--(2.601,2.510)--(2.608,2.512)--(2.615,2.513)--(2.622,2.515)--(2.628,2.517)--(2.635,2.519)--(2.642,2.521)--(2.649,2.523)--(2.655,2.525)--(2.662,2.527)--(2.669,2.529)--(2.675,2.532)--(2.682,2.534)--(2.688,2.537)--(2.695,2.539)--(2.701,2.542)--(2.708,2.545)--(2.714,2.548)--(2.720,2.551)--(2.727,2.554)--(2.733,2.557)--(2.739,2.561)--(2.745,2.564)--(2.751,2.568)--(2.757,2.571)--(2.763,2.575)--(2.769,2.579)--(2.775,2.582)--(2.781,2.586)--(2.787,2.590)--(2.792,2.594)--(2.798,2.599)--(2.804,2.603)--(2.809,2.607)--(2.815,2.611)--(2.820,2.616);
-\draw [color=red,->,>=latex] (4.5000,0) -- (4.5000,-0.010000);
+\draw [color=red] (2.1799,2.6158)--(2.1853,2.6114)--(2.1908,2.6070)--(2.1963,2.6027)--(2.2019,2.5985)--(2.2076,2.5943)--(2.2133,2.5903)--(2.2191,2.5863)--(2.2249,2.5824)--(2.2308,2.5786)--(2.2367,2.5748)--(2.2427,2.5712)--(2.2488,2.5676)--(2.2549,2.5641)--(2.2610,2.5607)--(2.2672,2.5574)--(2.2734,2.5542)--(2.2797,2.5511)--(2.2860,2.5480)--(2.2924,2.5451)--(2.2988,2.5422)--(2.3052,2.5394)--(2.3117,2.5367)--(2.3182,2.5341)--(2.3248,2.5316)--(2.3314,2.5292)--(2.3380,2.5269)--(2.3447,2.5247)--(2.3514,2.5225)--(2.3581,2.5205)--(2.3648,2.5186)--(2.3716,2.5167)--(2.3784,2.5150)--(2.3852,2.5133)--(2.3920,2.5117)--(2.3989,2.5103)--(2.4058,2.5089)--(2.4127,2.5076)--(2.4196,2.5064)--(2.4265,2.5054)--(2.4335,2.5044)--(2.4404,2.5035)--(2.4474,2.5027)--(2.4544,2.5020)--(2.4614,2.5014)--(2.4684,2.5009)--(2.4754,2.5006)--(2.4824,2.5003)--(2.4894,2.5001)--(2.4964,2.5000)--(2.5035,2.5000)--(2.5105,2.5001)--(2.5175,2.5003)--(2.5245,2.5006)--(2.5315,2.5009)--(2.5385,2.5014)--(2.5455,2.5020)--(2.5525,2.5027)--(2.5595,2.5035)--(2.5664,2.5044)--(2.5734,2.5054)--(2.5803,2.5064)--(2.5872,2.5076)--(2.5941,2.5089)--(2.6010,2.5103)--(2.6079,2.5117)--(2.6147,2.5133)--(2.6215,2.5150)--(2.6283,2.5167)--(2.6351,2.5186)--(2.6418,2.5205)--(2.6485,2.5225)--(2.6552,2.5247)--(2.6619,2.5269)--(2.6685,2.5292)--(2.6751,2.5316)--(2.6817,2.5341)--(2.6882,2.5367)--(2.6947,2.5394)--(2.7011,2.5422)--(2.7075,2.5451)--(2.7139,2.5480)--(2.7202,2.5511)--(2.7265,2.5542)--(2.7327,2.5574)--(2.7389,2.5607)--(2.7450,2.5641)--(2.7511,2.5676)--(2.7572,2.5712)--(2.7632,2.5748)--(2.7691,2.5786)--(2.7750,2.5824)--(2.7808,2.5863)--(2.7866,2.5903)--(2.7923,2.5943)--(2.7980,2.5985)--(2.8036,2.6027)--(2.8091,2.6070)--(2.8146,2.6114)--(2.8200,2.6158);
+\draw [color=red,->,>=latex] (4.5000,0.0000) -- (4.5000,-0.0100);
 
-\draw [color=red] (4.680,0.3841)--(4.677,0.3813)--(4.673,0.3784)--(4.670,0.3755)--(4.667,0.3725)--(4.663,0.3696)--(4.660,0.3666)--(4.657,0.3636)--(4.654,0.3605)--(4.650,0.3574)--(4.647,0.3543)--(4.644,0.3512)--(4.641,0.3480)--(4.638,0.3448)--(4.635,0.3416)--(4.632,0.3384)--(4.629,0.3351)--(4.626,0.3318)--(4.623,0.3285)--(4.620,0.3251)--(4.617,0.3218)--(4.614,0.3184)--(4.612,0.3149)--(4.609,0.3115)--(4.606,0.3080)--(4.603,0.3045)--(4.601,0.3010)--(4.598,0.2974)--(4.595,0.2939)--(4.593,0.2903)--(4.590,0.2867)--(4.588,0.2830)--(4.585,0.2794)--(4.583,0.2757)--(4.580,0.2720)--(4.578,0.2683)--(4.576,0.2645)--(4.573,0.2608)--(4.571,0.2570)--(4.569,0.2532)--(4.567,0.2493)--(4.564,0.2455)--(4.562,0.2416)--(4.560,0.2378)--(4.558,0.2339)--(4.556,0.2299)--(4.554,0.2260)--(4.552,0.2220)--(4.550,0.2181)--(4.548,0.2141)--(4.546,0.2101)--(4.544,0.2060)--(4.543,0.2020)--(4.541,0.1980)--(4.539,0.1939)--(4.537,0.1898)--(4.536,0.1857)--(4.534,0.1816)--(4.533,0.1775)--(4.531,0.1733)--(4.529,0.1692)--(4.528,0.1650)--(4.527,0.1608)--(4.525,0.1566)--(4.524,0.1524)--(4.522,0.1482)--(4.521,0.1439)--(4.520,0.1397)--(4.519,0.1354)--(4.518,0.1312)--(4.516,0.1269)--(4.515,0.1226)--(4.514,0.1183)--(4.513,0.1140)--(4.512,0.1097)--(4.511,0.1054)--(4.510,0.1011)--(4.509,0.09673)--(4.509,0.09238)--(4.508,0.08803)--(4.507,0.08367)--(4.506,0.07931)--(4.506,0.07493)--(4.505,0.07056)--(4.504,0.06617)--(4.504,0.06179)--(4.503,0.05739)--(4.503,0.05299)--(4.502,0.04859)--(4.502,0.04419)--(4.502,0.03978)--(4.501,0.03537)--(4.501,0.03095)--(4.501,0.02653)--(4.500,0.02212)--(4.500,0.01769)--(4.500,0.01327)--(4.500,0.008849)--(4.500,0.004424)--(4.500,0);
+\draw [color=red] (4.6799,0.3841)--(4.6765,0.3812)--(4.6731,0.3783)--(4.6698,0.3754)--(4.6665,0.3725)--(4.6632,0.3695)--(4.6599,0.3665)--(4.6567,0.3635)--(4.6535,0.3605)--(4.6503,0.3574)--(4.6472,0.3543)--(4.6440,0.3511)--(4.6409,0.3480)--(4.6379,0.3448)--(4.6348,0.3416)--(4.6318,0.3383)--(4.6289,0.3350)--(4.6259,0.3318)--(4.6230,0.3284)--(4.6201,0.3251)--(4.6172,0.3217)--(4.6144,0.3183)--(4.6116,0.3149)--(4.6088,0.3114)--(4.6061,0.3080)--(4.6034,0.3045)--(4.6007,0.3009)--(4.5980,0.2974)--(4.5954,0.2938)--(4.5928,0.2902)--(4.5903,0.2866)--(4.5878,0.2830)--(4.5853,0.2793)--(4.5828,0.2756)--(4.5804,0.2719)--(4.5780,0.2682)--(4.5757,0.2645)--(4.5733,0.2607)--(4.5710,0.2569)--(4.5688,0.2531)--(4.5666,0.2493)--(4.5644,0.2455)--(4.5622,0.2416)--(4.5601,0.2377)--(4.5580,0.2338)--(4.5560,0.2299)--(4.5539,0.2259)--(4.5520,0.2220)--(4.5500,0.2180)--(4.5481,0.2140)--(4.5462,0.2100)--(4.5444,0.2060)--(4.5426,0.2020)--(4.5408,0.1979)--(4.5391,0.1938)--(4.5374,0.1897)--(4.5357,0.1856)--(4.5341,0.1815)--(4.5325,0.1774)--(4.5309,0.1733)--(4.5294,0.1691)--(4.5280,0.1649)--(4.5265,0.1607)--(4.5251,0.1566)--(4.5237,0.1523)--(4.5224,0.1481)--(4.5211,0.1439)--(4.5199,0.1397)--(4.5186,0.1354)--(4.5175,0.1311)--(4.5163,0.1269)--(4.5152,0.1226)--(4.5142,0.1183)--(4.5131,0.1140)--(4.5121,0.1097)--(4.5112,0.1053)--(4.5103,0.1010)--(4.5094,0.0967)--(4.5086,0.0923)--(4.5078,0.0880)--(4.5070,0.0836)--(4.5063,0.0793)--(4.5056,0.0749)--(4.5050,0.0705)--(4.5043,0.0661)--(4.5038,0.0617)--(4.5033,0.0573)--(4.5028,0.0529)--(4.5023,0.0485)--(4.5019,0.0441)--(4.5015,0.0397)--(4.5012,0.0353)--(4.5009,0.0309)--(4.5007,0.0265)--(4.5004,0.0221)--(4.5003,0.0176)--(4.5001,0.0132)--(4.5000,0.0088)--(4.5000,0.0044)--(4.5000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall b/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall
index 0ddf6c696..1fdc9e711 100644
--- a/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall
+++ b/src_phystricks/Fig_YQVHooYsGLHQ.pstricks.recall
@@ -65,230 +65,230 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (-4.0000,-4.0000) -- (-4.2357,-3.7643);
-\draw [,->,>=latex] (-4.0000,-3.4286) -- (-4.2169,-3.1755);
-\draw [,->,>=latex] (-4.0000,-2.8571) -- (-4.1937,-2.5859);
-\draw [,->,>=latex] (-4.0000,-2.2857) -- (-4.1654,-1.9963);
-\draw [,->,>=latex] (-4.0000,-1.7143) -- (-4.1313,-1.4079);
-\draw [,->,>=latex] (-4.0000,-1.1429) -- (-4.0916,-0.82235);
-\draw [,->,>=latex] (-4.0000,-0.57143) -- (-4.0471,-0.24145);
-\draw [,->,>=latex] (-4.0000,0) -- (-4.0000,0.33333);
-\draw [,->,>=latex] (-4.0000,0.57143) -- (-3.9529,0.90141);
-\draw [,->,>=latex] (-4.0000,1.1429) -- (-3.9084,1.4634);
-\draw [,->,>=latex] (-4.0000,1.7143) -- (-3.8687,2.0207);
+\draw [,->,>=latex] (-4.0000,-4.0000) -- (-4.2357,-3.7642);
+\draw [,->,>=latex] (-4.0000,-3.4285) -- (-4.2169,-3.1754);
+\draw [,->,>=latex] (-4.0000,-2.8571) -- (-4.1937,-2.5858);
+\draw [,->,>=latex] (-4.0000,-2.2857) -- (-4.1653,-1.9962);
+\draw [,->,>=latex] (-4.0000,-1.7142) -- (-4.1313,-1.4079);
+\draw [,->,>=latex] (-4.0000,-1.1428) -- (-4.0915,-0.8223);
+\draw [,->,>=latex] (-4.0000,-0.5714) -- (-4.0471,-0.2414);
+\draw [,->,>=latex] (-4.0000,0.0000) -- (-4.0000,0.3333);
+\draw [,->,>=latex] (-4.0000,0.5714) -- (-3.9528,0.9014);
+\draw [,->,>=latex] (-4.0000,1.1428) -- (-3.9084,1.4633);
+\draw [,->,>=latex] (-4.0000,1.7142) -- (-3.8686,2.0206);
 \draw [,->,>=latex] (-4.0000,2.2857) -- (-3.8346,2.5751);
-\draw [,->,>=latex] (-4.0000,2.8571) -- (-3.8063,3.1284);
-\draw [,->,>=latex] (-4.0000,3.4286) -- (-3.7831,3.6817);
-\draw [,->,>=latex] (-4.0000,4.0000) -- (-3.7643,4.2357);
-\draw [,->,>=latex] (-3.4286,-4.0000) -- (-3.6817,-3.7831);
-\draw [,->,>=latex] (-3.4286,-3.4286) -- (-3.6643,-3.1929);
-\draw [,->,>=latex] (-3.4286,-2.8571) -- (-3.6420,-2.6011);
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-\draw [,->,>=latex] (-3.4286,-1.7143) -- (-3.5776,-1.4161);
-\draw [,->,>=latex] (-3.4286,-1.1429) -- (-3.5340,-0.82663);
-\draw [,->,>=latex] (-3.4286,-0.57143) -- (-3.4834,-0.24263);
-\draw [,->,>=latex] (-3.4286,0) -- (-3.4286,0.33333);
-\draw [,->,>=latex] (-3.4286,0.57143) -- (-3.3738,0.90023);
-\draw [,->,>=latex] (-3.4286,1.1429) -- (-3.3232,1.4591);
-\draw [,->,>=latex] (-3.4286,1.7143) -- (-3.2795,2.0124);
-\draw [,->,>=latex] (-3.4286,2.2857) -- (-3.2437,2.5631);
-\draw [,->,>=latex] (-3.4286,2.8571) -- (-3.2152,3.1132);
-\draw [,->,>=latex] (-3.4286,3.4286) -- (-3.1929,3.6643);
-\draw [,->,>=latex] (-3.4286,4.0000) -- (-3.1755,4.2169);
-\draw [,->,>=latex] (-2.8571,-4.0000) -- (-3.1284,-3.8063);
-\draw [,->,>=latex] (-2.8571,-3.4286) -- (-3.1132,-3.2152);
+\draw [,->,>=latex] (-4.0000,2.8571) -- (-3.8062,3.1283);
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+\draw [,->,>=latex] (-4.0000,4.0000) -- (-3.7642,4.2357);
+\draw [,->,>=latex] (-3.4285,-4.0000) -- (-3.6816,-3.7830);
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+\draw [,->,>=latex] (-3.4285,-2.8571) -- (-3.6419,-2.6010);
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+\draw [,->,>=latex] (-3.4285,0.0000) -- (-3.4285,0.3333);
+\draw [,->,>=latex] (-3.4285,0.5714) -- (-3.3737,0.9002);
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+\draw [,->,>=latex] (-3.4285,2.2857) -- (-3.2436,2.5630);
+\draw [,->,>=latex] (-3.4285,2.8571) -- (-3.2151,3.1132);
+\draw [,->,>=latex] (-3.4285,3.4285) -- (-3.1928,3.6642);
+\draw [,->,>=latex] (-3.4285,4.0000) -- (-3.1754,4.2169);
+\draw [,->,>=latex] (-2.8571,-4.0000) -- (-3.1283,-3.8062);
+\draw [,->,>=latex] (-2.8571,-3.4285) -- (-3.1132,-3.2151);
 \draw [,->,>=latex] (-2.8571,-2.8571) -- (-3.0928,-2.6214);
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+\draw [,->,>=latex] (-2.8571,1.7142) -- (-2.6856,2.0001);
 \draw [,->,>=latex] (-2.8571,2.2857) -- (-2.6489,2.5460);
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-\draw [,->,>=latex] (-2.8571,4.0000) -- (-2.5859,4.1937);
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+\draw [,->,>=latex] (-2.8571,4.0000) -- (-2.5858,4.1937);
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-\draw [,->,>=latex] (-2.2857,-3.4286) -- (-2.5631,-3.2437);
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 \draw [,->,>=latex] (-2.2857,-2.8571) -- (-2.5460,-2.6489);
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 \draw [,->,>=latex] (2.2857,4.0000) -- (2.5751,3.8346);
-\draw [,->,>=latex] (2.8571,-4.0000) -- (2.5859,-4.1937);
-\draw [,->,>=latex] (2.8571,-3.4286) -- (2.6011,-3.6420);
+\draw [,->,>=latex] (2.8571,-4.0000) -- (2.5858,-4.1937);
+\draw [,->,>=latex] (2.8571,-3.4285) -- (2.6010,-3.6419);
 \draw [,->,>=latex] (2.8571,-2.8571) -- (2.6214,-3.0928);
 \draw [,->,>=latex] (2.8571,-2.2857) -- (2.6489,-2.5460);
-\draw [,->,>=latex] (2.8571,-1.7143) -- (2.6856,-2.0001);
-\draw [,->,>=latex] (2.8571,-1.1429) -- (2.7333,-1.4523);
-\draw [,->,>=latex] (2.8571,-0.57143) -- (2.7918,-0.89829);
-\draw [,->,>=latex] (2.8571,0) -- (2.8571,-0.33333);
-\draw [,->,>=latex] (2.8571,0.57143) -- (2.9225,0.24457);
-\draw [,->,>=latex] (2.8571,1.1429) -- (2.9809,0.83336);
-\draw [,->,>=latex] (2.8571,1.7143) -- (3.0286,1.4285);
-\draw [,->,>=latex] (2.8571,2.2857) -- (3.0654,2.0254);
+\draw [,->,>=latex] (2.8571,-1.7142) -- (2.6856,-2.0001);
+\draw [,->,>=latex] (2.8571,-1.1428) -- (2.7333,-1.4523);
+\draw [,->,>=latex] (2.8571,-0.5714) -- (2.7917,-0.8982);
+\draw [,->,>=latex] (2.8571,0.0000) -- (2.8571,-0.3333);
+\draw [,->,>=latex] (2.8571,0.5714) -- (2.9225,0.2445);
+\draw [,->,>=latex] (2.8571,1.1428) -- (2.9809,0.8333);
+\draw [,->,>=latex] (2.8571,1.7142) -- (3.0286,1.4284);
+\draw [,->,>=latex] (2.8571,2.2857) -- (3.0653,2.0254);
 \draw [,->,>=latex] (2.8571,2.8571) -- (3.0928,2.6214);
-\draw [,->,>=latex] (2.8571,3.4286) -- (3.1132,3.2152);
-\draw [,->,>=latex] (2.8571,4.0000) -- (3.1284,3.8063);
-\draw [,->,>=latex] (3.4286,-4.0000) -- (3.1755,-4.2169);
-\draw [,->,>=latex] (3.4286,-3.4286) -- (3.1929,-3.6643);
-\draw [,->,>=latex] (3.4286,-2.8571) -- (3.2152,-3.1132);
-\draw [,->,>=latex] (3.4286,-2.2857) -- (3.2437,-2.5631);
-\draw [,->,>=latex] (3.4286,-1.7143) -- (3.2795,-2.0124);
-\draw [,->,>=latex] (3.4286,-1.1429) -- (3.3232,-1.4591);
-\draw [,->,>=latex] (3.4286,-0.57143) -- (3.3738,-0.90023);
-\draw [,->,>=latex] (3.4286,0) -- (3.4286,-0.33333);
-\draw [,->,>=latex] (3.4286,0.57143) -- (3.4834,0.24263);
-\draw [,->,>=latex] (3.4286,1.1429) -- (3.5340,0.82663);
-\draw [,->,>=latex] (3.4286,1.7143) -- (3.5776,1.4161);
-\draw [,->,>=latex] (3.4286,2.2857) -- (3.6135,2.0084);
-\draw [,->,>=latex] (3.4286,2.8571) -- (3.6420,2.6011);
-\draw [,->,>=latex] (3.4286,3.4286) -- (3.6643,3.1929);
-\draw [,->,>=latex] (3.4286,4.0000) -- (3.6817,3.7831);
-\draw [,->,>=latex] (4.0000,-4.0000) -- (3.7643,-4.2357);
-\draw [,->,>=latex] (4.0000,-3.4286) -- (3.7831,-3.6817);
-\draw [,->,>=latex] (4.0000,-2.8571) -- (3.8063,-3.1284);
+\draw [,->,>=latex] (2.8571,3.4285) -- (3.1132,3.2151);
+\draw [,->,>=latex] (2.8571,4.0000) -- (3.1283,3.8062);
+\draw [,->,>=latex] (3.4285,-4.0000) -- (3.1754,-4.2169);
+\draw [,->,>=latex] (3.4285,-3.4285) -- (3.1928,-3.6642);
+\draw [,->,>=latex] (3.4285,-2.8571) -- (3.2151,-3.1132);
+\draw [,->,>=latex] (3.4285,-2.2857) -- (3.2436,-2.5630);
+\draw [,->,>=latex] (3.4285,-1.7142) -- (3.2795,-2.0124);
+\draw [,->,>=latex] (3.4285,-1.1428) -- (3.3231,-1.4590);
+\draw [,->,>=latex] (3.4285,-0.5714) -- (3.3737,-0.9002);
+\draw [,->,>=latex] (3.4285,0.0000) -- (3.4285,-0.3333);
+\draw [,->,>=latex] (3.4285,0.5714) -- (3.4833,0.2426);
+\draw [,->,>=latex] (3.4285,1.1428) -- (3.5339,0.8266);
+\draw [,->,>=latex] (3.4285,1.7142) -- (3.5776,1.4161);
+\draw [,->,>=latex] (3.4285,2.2857) -- (3.6134,2.0083);
+\draw [,->,>=latex] (3.4285,2.8571) -- (3.6419,2.6010);
+\draw [,->,>=latex] (3.4285,3.4285) -- (3.6642,3.1928);
+\draw [,->,>=latex] (3.4285,4.0000) -- (3.6816,3.7830);
+\draw [,->,>=latex] (4.0000,-4.0000) -- (3.7642,-4.2357);
+\draw [,->,>=latex] (4.0000,-3.4285) -- (3.7830,-3.6816);
+\draw [,->,>=latex] (4.0000,-2.8571) -- (3.8062,-3.1283);
 \draw [,->,>=latex] (4.0000,-2.2857) -- (3.8346,-2.5751);
-\draw [,->,>=latex] (4.0000,-1.7143) -- (3.8687,-2.0207);
-\draw [,->,>=latex] (4.0000,-1.1429) -- (3.9084,-1.4634);
-\draw [,->,>=latex] (4.0000,-0.57143) -- (3.9529,-0.90141);
-\draw [,->,>=latex] (4.0000,0) -- (4.0000,-0.33333);
-\draw [,->,>=latex] (4.0000,0.57143) -- (4.0471,0.24145);
-\draw [,->,>=latex] (4.0000,1.1429) -- (4.0916,0.82235);
-\draw [,->,>=latex] (4.0000,1.7143) -- (4.1313,1.4079);
-\draw [,->,>=latex] (4.0000,2.2857) -- (4.1654,1.9963);
-\draw [,->,>=latex] (4.0000,2.8571) -- (4.1937,2.5859);
-\draw [,->,>=latex] (4.0000,3.4286) -- (4.2169,3.1755);
-\draw [,->,>=latex] (4.0000,4.0000) -- (4.2357,3.7643);
+\draw [,->,>=latex] (4.0000,-1.7142) -- (3.8686,-2.0206);
+\draw [,->,>=latex] (4.0000,-1.1428) -- (3.9084,-1.4633);
+\draw [,->,>=latex] (4.0000,-0.5714) -- (3.9528,-0.9014);
+\draw [,->,>=latex] (4.0000,0.0000) -- (4.0000,-0.3333);
+\draw [,->,>=latex] (4.0000,0.5714) -- (4.0471,0.2414);
+\draw [,->,>=latex] (4.0000,1.1428) -- (4.0915,0.8223);
+\draw [,->,>=latex] (4.0000,1.7142) -- (4.1313,1.4079);
+\draw [,->,>=latex] (4.0000,2.2857) -- (4.1653,1.9962);
+\draw [,->,>=latex] (4.0000,2.8571) -- (4.1937,2.5858);
+\draw [,->,>=latex] (4.0000,3.4285) -- (4.2169,3.1754);
+\draw [,->,>=latex] (4.0000,4.0000) -- (4.2357,3.7642);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall b/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall
index 6b17f4219..48f128d39 100644
--- a/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall
+++ b/src_phystricks/Fig_YYECooQlnKtD.pstricks.recall
@@ -104,43 +104,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (11.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (11.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.1682);
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw []  (1.0000,0.0013778) node [rotate=0] {$\bullet$};
-\draw []  (2.0000,0.014467) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0.090017) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,0.36757) node [rotate=0] {$\bullet$};
-\draw []  (5.0000,1.0292) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0.0013) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0.0144) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.0900) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.3675) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,1.0291) node [rotate=0] {$\bullet$};
 \draw []  (6.0000,2.0012) node [rotate=0] {$\bullet$};
-\draw []  (7.0000,2.6683) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,2.6682) node [rotate=0] {$\bullet$};
 \draw []  (8.0000,2.3347) node [rotate=0] {$\bullet$};
 \draw []  (9.0000,1.2106) node [rotate=0] {$\bullet$};
-\draw []  (10.000,0.28248) node [rotate=0] {$\bullet$};
-\draw []  (11.000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.000,-0.31492) node {$ 11 $};
-\draw [] (11.0,-0.100) -- (11.0,0.100);
+\draw []  (10.000,0.2824) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (11.000,-0.3149) node {$ 11 $};
+\draw [] (11.000,-0.1000) -- (11.000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall b/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall
index 9a04c9614..575874fa4 100644
--- a/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall
+++ b/src_phystricks/Fig_ZTTooXtHkci.pstricks.recall
@@ -49,8 +49,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,0.50000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,0.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -59,19 +59,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,0) -- (2.00,0) -- (2.00,0) -- (0,-2.00) -- (0,-2.00) -- (0,-1.00) -- (0,-1.00) -- (1.00,0) --  cycle;
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (0,-2.00);
-\draw [] (0,-2.00) -- (0,-1.00);
-\draw [] (0,-1.00) -- (1.00,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,0.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (0.0000,-2.0000) -- (0.0000,-2.0000) -- (0.0000,-1.0000) -- (0.0000,-1.0000) -- (1.0000,0.0000) --  cycle;
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (0.0000,-2.0000);
+\draw [] (0.0000,-2.0000) -- (0.0000,-1.0000);
+\draw [] (0.0000,-1.0000) -- (1.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -117,8 +117,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -127,23 +127,23 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-2.00,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (1.00,1.00) -- (1.00,1.00) -- (-1.00,1.00) -- (-1.00,1.00) -- (-2.00,2.00) --  cycle;
-\draw [] (-2.00,2.00) -- (2.00,2.00);
-\draw [] (2.00,2.00) -- (1.00,1.00);
-\draw [] (1.00,1.00) -- (-1.00,1.00);
-\draw [] (-1.00,1.00) -- (-2.00,2.00);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (1.0000,1.0000) -- (1.0000,1.0000) -- (-1.0000,1.0000) -- (-1.0000,1.0000) -- (-2.0000,2.0000) --  cycle;
+\draw [] (-2.0000,2.0000) -- (2.0000,2.0000);
+\draw [] (2.0000,2.0000) -- (1.0000,1.0000);
+\draw [] (1.0000,1.0000) -- (-1.0000,1.0000);
+\draw [] (-1.0000,1.0000) -- (-2.0000,2.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_examsseptii.pstricks.recall b/src_phystricks/Fig_examsseptii.pstricks.recall
index 32328027d..df5b5e8b6 100644
--- a/src_phystricks/Fig_examsseptii.pstricks.recall
+++ b/src_phystricks/Fig_examsseptii.pstricks.recall
@@ -107,39 +107,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (9.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue] (0,-2.00) -- (0,2.00);
-\draw [color=blue] (1.00,-2.00) -- (1.00,2.00);
-\draw [color=blue] (4.00,-2.00) -- (4.00,2.00);
-\draw [color=blue] (9.00,-2.00) -- (9.00,2.00);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,-2.0000) -- (0.0000,2.0000);
+\draw [color=blue] (1.0000,-2.0000) -- (1.0000,2.0000);
+\draw [color=blue] (4.0000,-2.0000) -- (4.0000,2.0000);
+\draw [color=blue] (9.0000,-2.0000) -- (9.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_senotopologo.pstricks.recall b/src_phystricks/Fig_senotopologo.pstricks.recall
index 180a3c929..c21f663cb 100644
--- a/src_phystricks/Fig_senotopologo.pstricks.recall
+++ b/src_phystricks/Fig_senotopologo.pstricks.recall
@@ -95,34 +95,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.5000,0) -- (8.5000,0);
-\draw [,->,>=latex] (0,-2.2025) -- (0,2.2025);
+\draw [,->,>=latex] (-8.5000,0.0000) -- (8.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.2024) -- (0.0000,2.2024);
 %DEFAULT
 
-\draw [color=blue] 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+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_trigoWedd.pstricks.recall b/src_phystricks/Fig_trigoWedd.pstricks.recall
index 899d83cd8..c96b2e88e 100644
--- a/src_phystricks/Fig_trigoWedd.pstricks.recall
+++ b/src_phystricks/Fig_trigoWedd.pstricks.recall
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
-\draw (1.2899,0.75595) node {$z_0$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.21406) node {$q$};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$1$};
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw []  (0.8660,0.5000) node [rotate=0] {$\bullet$};
+\draw (1.2898,0.7559) node {$z_0$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.2140) node {$q$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$1$};
 
 %OTHER STUFF
 %END PSPICTURE

From a63b8f6c8c41892b9d22fa87e52eb839e4c0c461 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 03:13:46 +0200
Subject: [PATCH 30/64] (pictures) Remove the graduation on the axes of the
 pictures 'UEGE'

---
 src_phystricks/phystricksUEGEooHEDIJVPn.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src_phystricks/phystricksUEGEooHEDIJVPn.py b/src_phystricks/phystricksUEGEooHEDIJVPn.py
index 2e11155db..16877b8a5 100644
--- a/src_phystricks/phystricksUEGEooHEDIJVPn.py
+++ b/src_phystricks/phystricksUEGEooHEDIJVPn.py
@@ -29,7 +29,7 @@ def UEGEooHEDIJVPn():
         segV.parameters.style="dashed"
         pspict.DrawGraphs(P,Q,S,segH,segV)
 
-    pspict.axes.no_numbering()
+    pspict.axes.no_graduation()
     pspict.DrawDefaultAxes()
     fig.no_figure()
     fig.conclude()

From 15eb2b72bc25886111b97fa30c2f0fdbf1ef765f Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 03:22:53 +0200
Subject: [PATCH 31/64] (pictures) Let 'phystricks' auto determine the distance
 for the angle mark 'theta' in the picture 'AMDU'

---
 src_phystricks/phystricksAMDUooZZUOqa.py | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src_phystricks/phystricksAMDUooZZUOqa.py b/src_phystricks/phystricksAMDUooZZUOqa.py
index f8fba0b8b..826352bf5 100644
--- a/src_phystricks/phystricksAMDUooZZUOqa.py
+++ b/src_phystricks/phystricksAMDUooZZUOqa.py
@@ -24,7 +24,8 @@ def AMDUooZZUOqa():
     M.put_mark(0.3,M.advised_mark_angle(pspict),"$\sigma(t)$",pspict=pspict)
 
     angle=AngleAOB(P,O,Q,r=0.5)
-    angle.put_mark(0.2,None,r"$\theta$",pspict=pspict)
+    #angle.put_mark(0.2,None,r"$\theta$",pspict=pspict)
+    angle.put_mark(text=r"$\theta$",pspict=pspict)
     seg_theta=Segment(O,P)
     seg_sigma=Segment(O,Q)
     

From da512b173e60e25cac21311e911da2d5d8d53b8f Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 03:39:01 +0200
Subject: [PATCH 32/64] (picture) Make the angle and the dilatation larger in
 picture 'BIF'.

---
 src_phystricks/phystricksBIFooDsvVHb.py | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/src_phystricks/phystricksBIFooDsvVHb.py b/src_phystricks/phystricksBIFooDsvVHb.py
index 1057cf0dd..e92a1e71e 100644
--- a/src_phystricks/phystricksBIFooDsvVHb.py
+++ b/src_phystricks/phystricksBIFooDsvVHb.py
@@ -2,12 +2,11 @@
 from phystricks import *
 def BIFooDsvVHb():
     pspict,fig = SinglePicture("BIFooDsvVHb")
-    pspict.dilatation_X(1.5)
-    pspict.dilatation_Y(1.5)
+    pspict.dilatation(2)
 
     O=Point(0,0)
     cercle=Circle(  O,1 )
-    P=cercle.get_point(30)
+    P=cercle.get_point(55)
     Y=Point(0,P.y)
     X=Point(P.x,0)
 

From bb8f79d20d5fef693980ec747946195cf0d865e4 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 03:47:26 +0200
Subject: [PATCH 33/64] (picture) Make some pictures slightly better looking.

---
 src_phystricks/phystricksSurfaceEntreCourbes.py | 17 ++++++-----------
 src_phystricks/phystricksratrap.py              |  8 ++++----
 2 files changed, 10 insertions(+), 15 deletions(-)

diff --git a/src_phystricks/phystricksSurfaceEntreCourbes.py b/src_phystricks/phystricksSurfaceEntreCourbes.py
index 5ea8b17bb..4894ce04a 100644
--- a/src_phystricks/phystricksSurfaceEntreCourbes.py
+++ b/src_phystricks/phystricksSurfaceEntreCourbes.py
@@ -13,10 +13,10 @@ def SurfaceEntreCourbes():
     pspicts[2].mother.caption=u"La surface entre les deux fonctions."
 
     x=var('x')
-    mx=-0.5
-    Mx=2.5
-    f1 = phyFunction(3*(x-1)**2+3).graph(mx,Mx)
-    f2 = phyFunction(-3*(x-1)**2+8).graph(mx,Mx)
+    mx=0.1
+    Mx=2.3
+    f1 = phyFunction(3*(x-1.25)**2+3).graph(mx,Mx)
+    f2 = phyFunction(-3*(x-1.25)**2+8).graph(mx,Mx)
 
     intersection=Intersection(f1,f2)
     i1=intersection[0].x
@@ -40,14 +40,9 @@ def SurfaceEntreCourbes():
     moyenne_surface.parameters.fill.color="red"
 
 
-    #grande_surface.parameters.color="brown"
-    #petite_surface.parameters.color="cyan"
-    #moyenne_surface.parameters.color="red"
-
-
-    pspicts[0].DrawGraphs(f1,f2,grande_surface)
+    pspicts[0].DrawGraphs(grande_surface,f1,f2)
     pspicts[1].DrawGraphs(f1,f2,petite_surface)
-    pspicts[2].DrawGraphs(f1,f2,moyenne_surface)
+    pspicts[2].DrawGraphs(moyenne_surface,f1,f2)
 
     for psp in pspicts :
         psp.DrawGraphs(A,B)
diff --git a/src_phystricks/phystricksratrap.py b/src_phystricks/phystricksratrap.py
index b609aafc9..bed941413 100644
--- a/src_phystricks/phystricksratrap.py
+++ b/src_phystricks/phystricksratrap.py
@@ -22,10 +22,10 @@ def ratrap():
     Ay = Point(0,a)
     By = Point(0,b)
 
-    Ax.put_mark(0.1,text="$a$",pspict=pspict,position="N")
-    Bx.put_mark(0.1,text="$b$",pspict=pspict,position="N")
-    Ay.put_mark(0.1,text="$a$",pspict=pspict,position="E")
-    By.put_mark(0.1,text="$b$",pspict=pspict,position="E")
+    Ax.put_mark(0.2,text="$a$",pspict=pspict,position="N")
+    Bx.put_mark(0.2,text="$b$",pspict=pspict,position="N")
+    Ay.put_mark(0.2,text="$a$",pspict=pspict,position="E")
+    By.put_mark(0.2,text="$b$",pspict=pspict,position="E")
 
     pspict.DrawGraphs(surface,Ax,Ay,Bx,By)
     pspict.axes.no_graduation()

From 0cb52c67191c28f5ce8c6be9667d5e9dbd0410d7 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 03:57:32 +0200
Subject: [PATCH 34/64] (pictures) Update many pictures and their '.recall'
 files.

---
 auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks    | 210 ++++-----
 auto/pictures_tex/Fig_ADUGmRRA.pstricks       |  10 +-
 auto/pictures_tex/Fig_ADUGmRRB.pstricks       |  10 +-
 auto/pictures_tex/Fig_ADUGmRRC.pstricks       |  10 +-
 auto/pictures_tex/Fig_AIFsOQO.pstricks        |  90 ++--
 auto/pictures_tex/Fig_ALIzHFm.pstricks        |  16 +-
 auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks   |   2 +-
 auto/pictures_tex/Fig_ASHYooUVHkak.pstricks   |  28 +-
 .../Fig_AccumulationIsole.pstricks            |  18 +-
 auto/pictures_tex/Fig_AdhIntFr.pstricks       |  58 +--
 auto/pictures_tex/Fig_AdhIntFrDeux.pstricks   |  64 +--
 auto/pictures_tex/Fig_AdhIntFrSix.pstricks    | 216 ++++-----
 auto/pictures_tex/Fig_AdhIntFrTrois.pstricks  | 108 ++---
 auto/pictures_tex/Fig_AireParabole.pstricks   |  52 +--
 .../Fig_ArcLongueurFinesse.pstricks           |  34 +-
 auto/pictures_tex/Fig_BEHTooWsdrys.pstricks   |  88 ++--
 auto/pictures_tex/Fig_BIFooDsvVHb.pstricks    |  24 +-
 auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks   |  18 +-
 auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks   |  38 +-
 auto/pictures_tex/Fig_Bateau.pstricks         |  32 +-
 auto/pictures_tex/Fig_BiaisOuPas.pstricks     |  46 +-
 auto/pictures_tex/Fig_BoulePtLoin.pstricks    |  12 +-
 auto/pictures_tex/Fig_CELooGVvzMc.pstricks    |  50 +-
 auto/pictures_tex/Fig_CFMooGzvfRP.pstricks    |  18 +-
 auto/pictures_tex/Fig_CMMAooQegASg.pstricks   |   8 +-
 auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks   |  72 +--
 auto/pictures_tex/Fig_CSCii.pstricks          |  28 +-
 auto/pictures_tex/Fig_CSCiii.pstricks         |  82 ++--
 auto/pictures_tex/Fig_CSCiv.pstricks          |  52 +--
 auto/pictures_tex/Fig_CSCv.pstricks           |  62 +--
 auto/pictures_tex/Fig_CSCvi.pstricks          |  78 ++--
 auto/pictures_tex/Fig_CURGooXvruWV.pstricks   |  34 +-
 auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks |  14 +-
 auto/pictures_tex/Fig_Cardioid.pstricks       |   2 +-
 auto/pictures_tex/Fig_Cardioideexo.pstricks   |  52 +--
 auto/pictures_tex/Fig_CbCartTui.pstricks      | 106 ++---
 auto/pictures_tex/Fig_CbCartTuii.pstricks     |  10 +-
 auto/pictures_tex/Fig_CbCartTuiii.pstricks    |  22 +-
 .../pictures_tex/Fig_CercleImplicite.pstricks |  18 +-
 auto/pictures_tex/Fig_CercleTnu.pstricks      |  18 +-
 auto/pictures_tex/Fig_CercleTrigono.pstricks  |  34 +-
 .../pictures_tex/Fig_ChampGraviation.pstricks | 442 +++++++++---------
 auto/pictures_tex/Fig_CheminFresnel.pstricks  |  22 +-
 auto/pictures_tex/Fig_ChiSquared.pstricks     |  38 +-
 .../Fig_ChiSquaresQuantile.pstricks           |  58 +--
 auto/pictures_tex/Fig_ChoixInfini.pstricks    |  66 +--
 auto/pictures_tex/Fig_CoinPasVar.pstricks     |  16 +-
 auto/pictures_tex/Fig_ConeRevolution.pstricks |  20 +-
 auto/pictures_tex/Fig_ContourGreen.pstricks   |  18 +-
 auto/pictures_tex/Fig_ContourSqL.pstricks     |  26 +-
 .../Fig_ContourTgNDivergence.pstricks         |  50 +-
 auto/pictures_tex/Fig_CoordPolaires.pstricks  |  24 +-
 auto/pictures_tex/Fig_CornetGlace.pstricks    |  24 +-
 .../Fig_CourbeRectifiable.pstricks            |  30 +-
 auto/pictures_tex/Fig_CouroneExam.pstricks    |  30 +-
 .../Fig_CurvilignesPolaires.pstricks          |  22 +-
 auto/pictures_tex/Fig_CycloideA.pstricks      |  66 +--
 auto/pictures_tex/Fig_DDCTooYscVzA.pstricks   |  50 +-
 auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks |  20 +-
 auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks |  66 +--
 auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks |  42 +-
 auto/pictures_tex/Fig_DTIYKkP.pstricks        |   6 +-
 auto/pictures_tex/Fig_DZVooQZLUtf.pstricks    | 338 +++++++-------
 .../Fig_DefinitionCartesiennes.pstricks       |  78 ++--
 .../pictures_tex/Fig_DerivTangenteOM.pstricks |  38 +-
 auto/pictures_tex/Fig_DessinLim.pstricks      |  30 +-
 auto/pictures_tex/Fig_DeuxCercles.pstricks    |  22 +-
 auto/pictures_tex/Fig_Differentielle.pstricks |  36 +-
 auto/pictures_tex/Fig_DisqueConv.pstricks     |  10 +-
 .../Fig_DistanceEnsemble.pstricks             |  16 +-
 .../pictures_tex/Fig_DistanceEuclide.pstricks |  40 +-
 auto/pictures_tex/Fig_DivergenceDeux.pstricks | 416 ++++++++---------
 .../pictures_tex/Fig_DivergenceTrois.pstricks |  90 ++--
 auto/pictures_tex/Fig_DivergenceUn.pstricks   |  88 ++--
 auto/pictures_tex/Fig_DynkinNUtPJx.pstricks   | 110 ++---
 auto/pictures_tex/Fig_DynkinpWjUbE.pstricks   |   8 +-
 auto/pictures_tex/Fig_DynkinqlgIQl.pstricks   |   8 +-
 auto/pictures_tex/Fig_DynkinrjbHIu.pstricks   |   8 +-
 auto/pictures_tex/Fig_EELKooMwkockxB.pstricks |  32 +-
 auto/pictures_tex/Fig_EHDooGDwfjC.pstricks    |  22 +-
 auto/pictures_tex/Fig_EJRsWXw.pstricks        |  26 +-
 auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks |  50 +-
 auto/pictures_tex/Fig_ExPolygone.pstricks     |  54 +--
 auto/pictures_tex/Fig_ExSinLarge.pstricks     |  38 +-
 .../Fig_ExampleIntegration.pstricks           |  42 +-
 .../Fig_ExampleIntegrationdeux.pstricks       | 102 ++--
 .../pictures_tex/Fig_ExempleArcParam.pstricks |  68 +--
 auto/pictures_tex/Fig_ExempleNonRang.pstricks |   8 +-
 .../Fig_ExerciceGraphesbis.pstricks           | 326 ++++++-------
 auto/pictures_tex/Fig_ExoCUd.pstricks         |  48 +-
 auto/pictures_tex/Fig_ExoMagnetique.pstricks  |  12 +-
 auto/pictures_tex/Fig_ExoParamCD.pstricks     |  32 +-
 auto/pictures_tex/Fig_ExoPolaire.pstricks     |  18 +-
 auto/pictures_tex/Fig_ExoProjection.pstricks  |  18 +-
 .../Fig_ExoUnSurxPolaire.pstricks             |  80 ++--
 auto/pictures_tex/Fig_ExoVarj.pstricks        |  16 +-
 auto/pictures_tex/Fig_ExoXLVL.pstricks        |  38 +-
 auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks |  66 +--
 auto/pictures_tex/Fig_FGRooDhFkch.pstricks    |  58 +--
 auto/pictures_tex/Fig_FGWjJBX.pstricks        |  32 +-
 auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks   |  14 +-
 auto/pictures_tex/Fig_FWJuNhU.pstricks        |  40 +-
 auto/pictures_tex/Fig_FXVooJYAfif.pstricks    |  10 +-
 auto/pictures_tex/Fig_FnCosApprox.pstricks    |  34 +-
 .../Fig_FonctionEtDeriveOM.pstricks           |  64 +--
 .../Fig_FonctionXtroisOM.pstricks             |  44 +-
 auto/pictures_tex/Fig_GBnUivi.pstricks        | 106 ++---
 auto/pictures_tex/Fig_GCNooKEbjWB.pstricks    |  32 +-
 auto/pictures_tex/Fig_GMIooJvcCXg.pstricks    |  84 ++--
 auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks   |  26 +-
 auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks   |  38 +-
 auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks |  22 +-
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 lst_actu.py                                   |   2 -
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 388 files changed, 9994 insertions(+), 10014 deletions(-)

diff --git a/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks b/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
index 974934d32..ce0f78fb5 100644
--- a/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
+++ b/auto/pictures_tex/Fig_ACUooQwcDMZ.pstricks
@@ -49,21 +49,21 @@
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-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.1931);
 %DEFAULT
 
-\draw [color=blue] (0.1353,-2.000)--(0.1542,-1.870)--(0.1730,-1.754)--(0.1918,-1.651)--(0.2107,-1.557)--(0.2295,-1.472)--(0.2483,-1.393)--(0.2672,-1.320)--(0.2860,-1.252)--(0.3049,-1.188)--(0.3237,-1.128)--(0.3425,-1.071)--(0.3614,-1.018)--(0.3802,-0.9671)--(0.3990,-0.9187)--(0.4179,-0.8726)--(0.4367,-0.8285)--(0.4555,-0.7863)--(0.4744,-0.7458)--(0.4932,-0.7068)--(0.5120,-0.6694)--(0.5309,-0.6332)--(0.5497,-0.5984)--(0.5685,-0.5647)--(0.5874,-0.5321)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6439,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3834)--(0.7004,-0.3561)--(0.7192,-0.3296)--(0.7381,-0.3037)--(0.7569,-0.2785)--(0.7757,-0.2540)--(0.7946,-0.2300)--(0.8134,-0.2065)--(0.8322,-0.1836)--(0.8511,-0.1613)--(0.8699,-0.1394)--(0.8887,-0.1180)--(0.9076,-0.09698)--(0.9264,-0.07644)--(0.9452,-0.05632)--(0.9641,-0.03659)--(0.9829,-0.01724)--(1.002,0.001744)--(1.021,0.02037)--(1.039,0.03866)--(1.058,0.05662)--(1.077,0.07426)--(1.096,0.09159)--(1.115,0.1086)--(1.134,0.1254)--(1.152,0.1419)--(1.171,0.1581)--(1.190,0.1740)--(1.209,0.1897)--(1.228,0.2052)--(1.247,0.2204)--(1.265,0.2354)--(1.284,0.2502)--(1.303,0.2648)--(1.322,0.2791)--(1.341,0.2932)--(1.360,0.3072)--(1.378,0.3210)--(1.397,0.3345)--(1.416,0.3479)--(1.435,0.3611)--(1.454,0.3742)--(1.473,0.3870)--(1.491,0.3998)--(1.510,0.4123)--(1.529,0.4247)--(1.548,0.4369)--(1.567,0.4490)--(1.586,0.4610)--(1.604,0.4728)--(1.623,0.4845)--(1.642,0.4960)--(1.661,0.5074)--(1.680,0.5187)--(1.699,0.5298)--(1.717,0.5409)--(1.736,0.5518)--(1.755,0.5626)--(1.774,0.5732)--(1.793,0.5838)--(1.812,0.5942)--(1.830,0.6046)--(1.849,0.6148)--(1.868,0.6250)--(1.887,0.6350)--(1.906,0.6449)--(1.925,0.6547)--(1.943,0.6645)--(1.962,0.6741)--(1.981,0.6837)--(2.000,0.6931);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.1353,-2.0000)--(0.1541,-1.8696)--(0.1730,-1.7544)--(0.1918,-1.6510)--(0.2106,-1.5574)--(0.2295,-1.4718)--(0.2483,-1.3929)--(0.2671,-1.3198)--(0.2860,-1.2517)--(0.3048,-1.1879)--(0.3236,-1.1279)--(0.3425,-1.0714)--(0.3613,-1.0178)--(0.3801,-0.9670)--(0.3990,-0.9187)--(0.4178,-0.8726)--(0.4366,-0.8285)--(0.4555,-0.7862)--(0.4743,-0.7457)--(0.4932,-0.7068)--(0.5120,-0.6693)--(0.5308,-0.6332)--(0.5497,-0.5983)--(0.5685,-0.5646)--(0.5873,-0.5320)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6438,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3833)--(0.7003,-0.3561)--(0.7192,-0.3295)--(0.7380,-0.3037)--(0.7568,-0.2785)--(0.7757,-0.2539)--(0.7945,-0.2299)--(0.8133,-0.2065)--(0.8322,-0.1836)--(0.8510,-0.1612)--(0.8699,-0.1393)--(0.8887,-0.1179)--(0.9075,-0.0969)--(0.9264,-0.0764)--(0.9452,-0.0563)--(0.9640,-0.0365)--(0.9829,-0.0172)--(1.0017,0.0017)--(1.0205,0.0203)--(1.0394,0.0386)--(1.0582,0.0566)--(1.0770,0.0742)--(1.0959,0.0915)--(1.1147,0.1086)--(1.1335,0.1253)--(1.1524,0.1418)--(1.1712,0.1580)--(1.1900,0.1740)--(1.2089,0.1897)--(1.2277,0.2051)--(1.2466,0.2204)--(1.2654,0.2354)--(1.2842,0.2501)--(1.3031,0.2647)--(1.3219,0.2791)--(1.3407,0.2932)--(1.3596,0.3071)--(1.3784,0.3209)--(1.3972,0.3345)--(1.4161,0.3479)--(1.4349,0.3611)--(1.4537,0.3741)--(1.4726,0.3870)--(1.4914,0.3997)--(1.5102,0.4123)--(1.5291,0.4246)--(1.5479,0.4369)--(1.5667,0.4490)--(1.5856,0.4609)--(1.6044,0.4727)--(1.6233,0.4844)--(1.6421,0.4959)--(1.6609,0.5074)--(1.6798,0.5186)--(1.6986,0.5298)--(1.7174,0.5408)--(1.7363,0.5517)--(1.7551,0.5625)--(1.7739,0.5732)--(1.7928,0.5837)--(1.8116,0.5942)--(1.8304,0.6045)--(1.8493,0.6148)--(1.8681,0.6249)--(1.8869,0.6349)--(1.9058,0.6449)--(1.9246,0.6547)--(1.9434,0.6644)--(1.9623,0.6741)--(1.9811,0.6836)--(2.0000,0.6931);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -109,27 +109,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.1931);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.1931);
 %DEFAULT
 
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+\draw [color=blue] (0.1353,-2.0000)--(0.1541,-1.8696)--(0.1730,-1.7544)--(0.1918,-1.6510)--(0.2106,-1.5574)--(0.2295,-1.4718)--(0.2483,-1.3929)--(0.2671,-1.3198)--(0.2860,-1.2517)--(0.3048,-1.1879)--(0.3236,-1.1279)--(0.3425,-1.0714)--(0.3613,-1.0178)--(0.3801,-0.9670)--(0.3990,-0.9187)--(0.4178,-0.8726)--(0.4366,-0.8285)--(0.4555,-0.7862)--(0.4743,-0.7457)--(0.4932,-0.7068)--(0.5120,-0.6693)--(0.5308,-0.6332)--(0.5497,-0.5983)--(0.5685,-0.5646)--(0.5873,-0.5320)--(0.6062,-0.5005)--(0.6250,-0.4699)--(0.6438,-0.4402)--(0.6627,-0.4114)--(0.6815,-0.3833)--(0.7003,-0.3561)--(0.7192,-0.3295)--(0.7380,-0.3037)--(0.7568,-0.2785)--(0.7757,-0.2539)--(0.7945,-0.2299)--(0.8133,-0.2065)--(0.8322,-0.1836)--(0.8510,-0.1612)--(0.8699,-0.1393)--(0.8887,-0.1179)--(0.9075,-0.0969)--(0.9264,-0.0764)--(0.9452,-0.0563)--(0.9640,-0.0365)--(0.9829,-0.0172)--(1.0017,0.0017)--(1.0205,0.0203)--(1.0394,0.0386)--(1.0582,0.0566)--(1.0770,0.0742)--(1.0959,0.0915)--(1.1147,0.1086)--(1.1335,0.1253)--(1.1524,0.1418)--(1.1712,0.1580)--(1.1900,0.1740)--(1.2089,0.1897)--(1.2277,0.2051)--(1.2466,0.2204)--(1.2654,0.2354)--(1.2842,0.2501)--(1.3031,0.2647)--(1.3219,0.2791)--(1.3407,0.2932)--(1.3596,0.3071)--(1.3784,0.3209)--(1.3972,0.3345)--(1.4161,0.3479)--(1.4349,0.3611)--(1.4537,0.3741)--(1.4726,0.3870)--(1.4914,0.3997)--(1.5102,0.4123)--(1.5291,0.4246)--(1.5479,0.4369)--(1.5667,0.4490)--(1.5856,0.4609)--(1.6044,0.4727)--(1.6233,0.4844)--(1.6421,0.4959)--(1.6609,0.5074)--(1.6798,0.5186)--(1.6986,0.5298)--(1.7174,0.5408)--(1.7363,0.5517)--(1.7551,0.5625)--(1.7739,0.5732)--(1.7928,0.5837)--(1.8116,0.5942)--(1.8304,0.6045)--(1.8493,0.6148)--(1.8681,0.6249)--(1.8869,0.6349)--(1.9058,0.6449)--(1.9246,0.6547)--(1.9434,0.6644)--(1.9623,0.6741)--(1.9811,0.6836)--(2.0000,0.6931);
 
-\draw [color=blue] (-2.000,0.6931)--(-1.981,0.6837)--(-1.962,0.6741)--(-1.943,0.6645)--(-1.925,0.6547)--(-1.906,0.6449)--(-1.887,0.6350)--(-1.868,0.6250)--(-1.849,0.6148)--(-1.830,0.6046)--(-1.812,0.5942)--(-1.793,0.5838)--(-1.774,0.5732)--(-1.755,0.5626)--(-1.736,0.5518)--(-1.717,0.5409)--(-1.699,0.5298)--(-1.680,0.5187)--(-1.661,0.5074)--(-1.642,0.4960)--(-1.623,0.4845)--(-1.604,0.4728)--(-1.586,0.4610)--(-1.567,0.4490)--(-1.548,0.4369)--(-1.529,0.4247)--(-1.510,0.4123)--(-1.491,0.3998)--(-1.473,0.3870)--(-1.454,0.3742)--(-1.435,0.3611)--(-1.416,0.3479)--(-1.397,0.3345)--(-1.378,0.3210)--(-1.360,0.3072)--(-1.341,0.2932)--(-1.322,0.2791)--(-1.303,0.2648)--(-1.284,0.2502)--(-1.265,0.2354)--(-1.247,0.2204)--(-1.228,0.2052)--(-1.209,0.1897)--(-1.190,0.1740)--(-1.171,0.1581)--(-1.152,0.1419)--(-1.134,0.1254)--(-1.115,0.1086)--(-1.096,0.09159)--(-1.077,0.07426)--(-1.058,0.05662)--(-1.039,0.03866)--(-1.021,0.02037)--(-1.002,0.001744)--(-0.9829,-0.01724)--(-0.9641,-0.03659)--(-0.9452,-0.05632)--(-0.9264,-0.07644)--(-0.9076,-0.09698)--(-0.8887,-0.1180)--(-0.8699,-0.1394)--(-0.8511,-0.1613)--(-0.8322,-0.1836)--(-0.8134,-0.2065)--(-0.7946,-0.2300)--(-0.7757,-0.2540)--(-0.7569,-0.2785)--(-0.7381,-0.3037)--(-0.7192,-0.3296)--(-0.7004,-0.3561)--(-0.6815,-0.3834)--(-0.6627,-0.4114)--(-0.6439,-0.4402)--(-0.6250,-0.4699)--(-0.6062,-0.5005)--(-0.5874,-0.5321)--(-0.5685,-0.5647)--(-0.5497,-0.5984)--(-0.5309,-0.6332)--(-0.5120,-0.6694)--(-0.4932,-0.7068)--(-0.4744,-0.7458)--(-0.4555,-0.7863)--(-0.4367,-0.8285)--(-0.4179,-0.8726)--(-0.3990,-0.9187)--(-0.3802,-0.9671)--(-0.3614,-1.018)--(-0.3425,-1.071)--(-0.3237,-1.128)--(-0.3049,-1.188)--(-0.2860,-1.252)--(-0.2672,-1.320)--(-0.2483,-1.393)--(-0.2295,-1.472)--(-0.2107,-1.557)--(-0.1918,-1.651)--(-0.1730,-1.754)--(-0.1542,-1.870)--(-0.1353,-2.000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-2.0000,0.6931)--(-1.9811,0.6836)--(-1.9623,0.6741)--(-1.9434,0.6644)--(-1.9246,0.6547)--(-1.9058,0.6449)--(-1.8869,0.6349)--(-1.8681,0.6249)--(-1.8493,0.6148)--(-1.8304,0.6045)--(-1.8116,0.5942)--(-1.7928,0.5837)--(-1.7739,0.5732)--(-1.7551,0.5625)--(-1.7363,0.5517)--(-1.7174,0.5408)--(-1.6986,0.5298)--(-1.6798,0.5186)--(-1.6609,0.5074)--(-1.6421,0.4959)--(-1.6233,0.4844)--(-1.6044,0.4727)--(-1.5856,0.4609)--(-1.5667,0.4490)--(-1.5479,0.4369)--(-1.5291,0.4246)--(-1.5102,0.4123)--(-1.4914,0.3997)--(-1.4726,0.3870)--(-1.4537,0.3741)--(-1.4349,0.3611)--(-1.4161,0.3479)--(-1.3972,0.3345)--(-1.3784,0.3209)--(-1.3596,0.3071)--(-1.3407,0.2932)--(-1.3219,0.2791)--(-1.3031,0.2647)--(-1.2842,0.2501)--(-1.2654,0.2354)--(-1.2466,0.2204)--(-1.2277,0.2051)--(-1.2089,0.1897)--(-1.1900,0.1740)--(-1.1712,0.1580)--(-1.1524,0.1418)--(-1.1335,0.1253)--(-1.1147,0.1086)--(-1.0959,0.0915)--(-1.0770,0.0742)--(-1.0582,0.0566)--(-1.0394,0.0386)--(-1.0205,0.0203)--(-1.0017,0.0017)--(-0.9829,-0.0172)--(-0.9640,-0.0365)--(-0.9452,-0.0563)--(-0.9264,-0.0764)--(-0.9075,-0.0969)--(-0.8887,-0.1179)--(-0.8699,-0.1393)--(-0.8510,-0.1612)--(-0.8322,-0.1836)--(-0.8133,-0.2065)--(-0.7945,-0.2299)--(-0.7757,-0.2539)--(-0.7568,-0.2785)--(-0.7380,-0.3037)--(-0.7192,-0.3295)--(-0.7003,-0.3561)--(-0.6815,-0.3833)--(-0.6627,-0.4114)--(-0.6438,-0.4402)--(-0.6250,-0.4699)--(-0.6062,-0.5005)--(-0.5873,-0.5320)--(-0.5685,-0.5646)--(-0.5497,-0.5983)--(-0.5308,-0.6332)--(-0.5120,-0.6693)--(-0.4932,-0.7068)--(-0.4743,-0.7457)--(-0.4555,-0.7862)--(-0.4366,-0.8285)--(-0.4178,-0.8726)--(-0.3990,-0.9187)--(-0.3801,-0.9670)--(-0.3613,-1.0178)--(-0.3425,-1.0714)--(-0.3236,-1.1279)--(-0.3048,-1.1879)--(-0.2860,-1.2517)--(-0.2671,-1.3198)--(-0.2483,-1.3929)--(-0.2295,-1.4718)--(-0.2106,-1.5574)--(-0.1918,-1.6510)--(-0.1730,-1.7544)--(-0.1541,-1.8696)--(-0.1353,-2.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -175,23 +175,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3647,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.5986);
+\draw [,->,>=latex] (-1.3646,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.5986);
 %DEFAULT
 
-\draw [color=blue] (-0.8647,-2.000)--(-0.8357,-1.806)--(-0.8068,-1.644)--(-0.7779,-1.504)--(-0.7489,-1.382)--(-0.7200,-1.273)--(-0.6910,-1.175)--(-0.6621,-1.085)--(-0.6332,-1.003)--(-0.6042,-0.9269)--(-0.5753,-0.8564)--(-0.5464,-0.7905)--(-0.5174,-0.7286)--(-0.4885,-0.6704)--(-0.4596,-0.6154)--(-0.4306,-0.5632)--(-0.4017,-0.5136)--(-0.3728,-0.4664)--(-0.3438,-0.4213)--(-0.3149,-0.3782)--(-0.2859,-0.3368)--(-0.2570,-0.2971)--(-0.2281,-0.2589)--(-0.1991,-0.2221)--(-0.1702,-0.1866)--(-0.1413,-0.1523)--(-0.1123,-0.1192)--(-0.08339,-0.08708)--(-0.05446,-0.05600)--(-0.02552,-0.02585)--(0.003415,0.003410)--(0.03235,0.03184)--(0.06129,0.05948)--(0.09022,0.08638)--(0.1192,0.1126)--(0.1481,0.1381)--(0.1770,0.1630)--(0.2060,0.1873)--(0.2349,0.2110)--(0.2638,0.2342)--(0.2928,0.2568)--(0.3217,0.2789)--(0.3506,0.3006)--(0.3796,0.3218)--(0.4085,0.3425)--(0.4375,0.3629)--(0.4664,0.3828)--(0.4953,0.4023)--(0.5243,0.4215)--(0.5532,0.4403)--(0.5821,0.4588)--(0.6111,0.4769)--(0.6400,0.4947)--(0.6689,0.5122)--(0.6979,0.5294)--(0.7268,0.5463)--(0.7558,0.5629)--(0.7847,0.5792)--(0.8136,0.5953)--(0.8426,0.6112)--(0.8715,0.6267)--(0.9004,0.6421)--(0.9294,0.6572)--(0.9583,0.6721)--(0.9872,0.6867)--(1.016,0.7012)--(1.045,0.7155)--(1.074,0.7295)--(1.103,0.7434)--(1.132,0.7570)--(1.161,0.7705)--(1.190,0.7838)--(1.219,0.7969)--(1.248,0.8099)--(1.277,0.8227)--(1.306,0.8353)--(1.334,0.8478)--(1.363,0.8601)--(1.392,0.8723)--(1.421,0.8843)--(1.450,0.8962)--(1.479,0.9079)--(1.508,0.9195)--(1.537,0.9310)--(1.566,0.9423)--(1.595,0.9535)--(1.624,0.9646)--(1.653,0.9756)--(1.682,0.9865)--(1.711,0.9972)--(1.740,1.008)--(1.769,1.018)--(1.797,1.029)--(1.826,1.039)--(1.855,1.049)--(1.884,1.059)--(1.913,1.069)--(1.942,1.079)--(1.971,1.089)--(2.000,1.099);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-0.8646,-2.0000)--(-0.8357,-1.8062)--(-0.8067,-1.6439)--(-0.7778,-1.5044)--(-0.7489,-1.3819)--(-0.7199,-1.2729)--(-0.6910,-1.1745)--(-0.6621,-1.0850)--(-0.6331,-1.0028)--(-0.6042,-0.9269)--(-0.5753,-0.8563)--(-0.5463,-0.7904)--(-0.5174,-0.7286)--(-0.4884,-0.6704)--(-0.4595,-0.6153)--(-0.4306,-0.5632)--(-0.4016,-0.5136)--(-0.3727,-0.4664)--(-0.3438,-0.4213)--(-0.3148,-0.3781)--(-0.2859,-0.3367)--(-0.2570,-0.2970)--(-0.2280,-0.2588)--(-0.1991,-0.2220)--(-0.1702,-0.1865)--(-0.1412,-0.1522)--(-0.1123,-0.1191)--(-0.0833,-0.0870)--(-0.0544,-0.0559)--(-0.0255,-0.0258)--(0.0034,0.0034)--(0.0323,0.0318)--(0.0612,0.0594)--(0.0902,0.0863)--(0.1191,0.1125)--(0.1480,0.1381)--(0.1770,0.1629)--(0.2059,0.1872)--(0.2349,0.2109)--(0.2638,0.2341)--(0.2927,0.2567)--(0.3217,0.2789)--(0.3506,0.3005)--(0.3795,0.3217)--(0.4085,0.3425)--(0.4374,0.3628)--(0.4663,0.3828)--(0.4953,0.4023)--(0.5242,0.4215)--(0.5531,0.4403)--(0.5821,0.4587)--(0.6110,0.4768)--(0.6400,0.4947)--(0.6689,0.5121)--(0.6978,0.5293)--(0.7268,0.5462)--(0.7557,0.5628)--(0.7846,0.5792)--(0.8136,0.5953)--(0.8425,0.6111)--(0.8714,0.6267)--(0.9004,0.6420)--(0.9293,0.6571)--(0.9583,0.6720)--(0.9872,0.6867)--(1.0161,0.7012)--(1.0451,0.7154)--(1.0740,0.7295)--(1.1029,0.7433)--(1.1319,0.7570)--(1.1608,0.7705)--(1.1897,0.7838)--(1.2187,0.7969)--(1.2476,0.8098)--(1.2765,0.8226)--(1.3055,0.8353)--(1.3344,0.8477)--(1.3634,0.8601)--(1.3923,0.8722)--(1.4212,0.8842)--(1.4502,0.8961)--(1.4791,0.9079)--(1.5080,0.9195)--(1.5370,0.9309)--(1.5659,0.9423)--(1.5948,0.9535)--(1.6238,0.9646)--(1.6527,0.9756)--(1.6817,0.9864)--(1.7106,0.9971)--(1.7395,1.0078)--(1.7685,1.0183)--(1.7974,1.0287)--(1.8263,1.0389)--(1.8553,1.0491)--(1.8842,1.0592)--(1.9131,1.0692)--(1.9421,1.0791)--(1.9710,1.0889)--(2.0000,1.0986);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -233,21 +233,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,2.1931);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.1931);
 %DEFAULT
 
-\draw [color=blue] (0.1353,-1.000)--(0.1542,-0.8697)--(0.1730,-0.7544)--(0.1918,-0.6511)--(0.2107,-0.5574)--(0.2295,-0.4718)--(0.2483,-0.3929)--(0.2672,-0.3198)--(0.2860,-0.2517)--(0.3049,-0.1879)--(0.3237,-0.1280)--(0.3425,-0.07142)--(0.3614,-0.01789)--(0.3802,0.03292)--(0.3990,0.08127)--(0.4179,0.1274)--(0.4367,0.1715)--(0.4555,0.2137)--(0.4744,0.2542)--(0.4932,0.2932)--(0.5120,0.3306)--(0.5309,0.3668)--(0.5497,0.4016)--(0.5685,0.4353)--(0.5874,0.4679)--(0.6062,0.4995)--(0.6250,0.5301)--(0.6439,0.5598)--(0.6627,0.5886)--(0.6815,0.6166)--(0.7004,0.6439)--(0.7192,0.6704)--(0.7381,0.6963)--(0.7569,0.7215)--(0.7757,0.7460)--(0.7946,0.7700)--(0.8134,0.7935)--(0.8322,0.8164)--(0.8511,0.8387)--(0.8699,0.8606)--(0.8887,0.8820)--(0.9076,0.9030)--(0.9264,0.9236)--(0.9452,0.9437)--(0.9641,0.9634)--(0.9829,0.9828)--(1.002,1.002)--(1.021,1.020)--(1.039,1.039)--(1.058,1.057)--(1.077,1.074)--(1.096,1.092)--(1.115,1.109)--(1.134,1.125)--(1.152,1.142)--(1.171,1.158)--(1.190,1.174)--(1.209,1.190)--(1.228,1.205)--(1.247,1.220)--(1.265,1.235)--(1.284,1.250)--(1.303,1.265)--(1.322,1.279)--(1.341,1.293)--(1.360,1.307)--(1.378,1.321)--(1.397,1.335)--(1.416,1.348)--(1.435,1.361)--(1.454,1.374)--(1.473,1.387)--(1.491,1.400)--(1.510,1.412)--(1.529,1.425)--(1.548,1.437)--(1.567,1.449)--(1.586,1.461)--(1.604,1.473)--(1.623,1.484)--(1.642,1.496)--(1.661,1.507)--(1.680,1.519)--(1.699,1.530)--(1.717,1.541)--(1.736,1.552)--(1.755,1.563)--(1.774,1.573)--(1.793,1.584)--(1.812,1.594)--(1.830,1.605)--(1.849,1.615)--(1.868,1.625)--(1.887,1.635)--(1.906,1.645)--(1.925,1.655)--(1.943,1.664)--(1.962,1.674)--(1.981,1.684)--(2.000,1.693);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.1353,-1.0000)--(0.1541,-0.8696)--(0.1730,-0.7544)--(0.1918,-0.6510)--(0.2106,-0.5574)--(0.2295,-0.4718)--(0.2483,-0.3929)--(0.2671,-0.3198)--(0.2860,-0.2517)--(0.3048,-0.1879)--(0.3236,-0.1279)--(0.3425,-0.0714)--(0.3613,-0.0178)--(0.3801,0.0329)--(0.3990,0.0812)--(0.4178,0.1273)--(0.4366,0.1714)--(0.4555,0.2137)--(0.4743,0.2542)--(0.4932,0.2931)--(0.5120,0.3306)--(0.5308,0.3667)--(0.5497,0.4016)--(0.5685,0.4353)--(0.5873,0.4679)--(0.6062,0.4994)--(0.6250,0.5300)--(0.6438,0.5597)--(0.6627,0.5885)--(0.6815,0.6166)--(0.7003,0.6438)--(0.7192,0.6704)--(0.7380,0.6962)--(0.7568,0.7214)--(0.7757,0.7460)--(0.7945,0.7700)--(0.8133,0.7934)--(0.8322,0.8163)--(0.8510,0.8387)--(0.8699,0.8606)--(0.8887,0.8820)--(0.9075,0.9030)--(0.9264,0.9235)--(0.9452,0.9436)--(0.9640,0.9634)--(0.9829,0.9827)--(1.0017,1.0017)--(1.0205,1.0203)--(1.0394,1.0386)--(1.0582,1.0566)--(1.0770,1.0742)--(1.0959,1.0915)--(1.1147,1.1086)--(1.1335,1.1253)--(1.1524,1.1418)--(1.1712,1.1580)--(1.1900,1.1740)--(1.2089,1.1897)--(1.2277,1.2051)--(1.2466,1.2204)--(1.2654,1.2354)--(1.2842,1.2501)--(1.3031,1.2647)--(1.3219,1.2791)--(1.3407,1.2932)--(1.3596,1.3071)--(1.3784,1.3209)--(1.3972,1.3345)--(1.4161,1.3479)--(1.4349,1.3611)--(1.4537,1.3741)--(1.4726,1.3870)--(1.4914,1.3997)--(1.5102,1.4123)--(1.5291,1.4246)--(1.5479,1.4369)--(1.5667,1.4490)--(1.5856,1.4609)--(1.6044,1.4727)--(1.6233,1.4844)--(1.6421,1.4959)--(1.6609,1.5074)--(1.6798,1.5186)--(1.6986,1.5298)--(1.7174,1.5408)--(1.7363,1.5517)--(1.7551,1.5625)--(1.7739,1.5732)--(1.7928,1.5837)--(1.8116,1.5942)--(1.8304,1.6045)--(1.8493,1.6148)--(1.8681,1.6249)--(1.8869,1.6349)--(1.9058,1.6449)--(1.9246,1.6547)--(1.9434,1.6644)--(1.9623,1.6741)--(1.9811,1.6836)--(2.0000,1.6931);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -289,21 +289,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (0.1353,2.000)--(0.1643,1.806)--(0.1932,1.644)--(0.2221,1.504)--(0.2511,1.382)--(0.2800,1.273)--(0.3090,1.175)--(0.3379,1.085)--(0.3668,1.003)--(0.3958,0.9269)--(0.4247,0.8564)--(0.4536,0.7905)--(0.4826,0.7286)--(0.5115,0.6704)--(0.5404,0.6154)--(0.5694,0.5632)--(0.5983,0.5136)--(0.6273,0.4664)--(0.6562,0.4213)--(0.6851,0.3782)--(0.7141,0.3368)--(0.7430,0.2971)--(0.7719,0.2589)--(0.8009,0.2221)--(0.8298,0.1866)--(0.8587,0.1523)--(0.8877,0.1192)--(0.9166,0.08708)--(0.9455,0.05600)--(0.9745,0.02585)--(1.003,0.003410)--(1.032,0.03184)--(1.061,0.05948)--(1.090,0.08638)--(1.119,0.1126)--(1.148,0.1381)--(1.177,0.1630)--(1.206,0.1873)--(1.235,0.2110)--(1.264,0.2342)--(1.293,0.2568)--(1.322,0.2789)--(1.351,0.3006)--(1.380,0.3218)--(1.409,0.3425)--(1.437,0.3629)--(1.466,0.3828)--(1.495,0.4023)--(1.524,0.4215)--(1.553,0.4403)--(1.582,0.4588)--(1.611,0.4769)--(1.640,0.4947)--(1.669,0.5122)--(1.698,0.5294)--(1.727,0.5463)--(1.756,0.5629)--(1.785,0.5792)--(1.814,0.5953)--(1.843,0.6112)--(1.871,0.6267)--(1.900,0.6421)--(1.929,0.6572)--(1.958,0.6721)--(1.987,0.6867)--(2.016,0.7012)--(2.045,0.7155)--(2.074,0.7295)--(2.103,0.7434)--(2.132,0.7570)--(2.161,0.7705)--(2.190,0.7838)--(2.219,0.7969)--(2.248,0.8099)--(2.277,0.8227)--(2.306,0.8353)--(2.334,0.8478)--(2.363,0.8601)--(2.392,0.8723)--(2.421,0.8843)--(2.450,0.8962)--(2.479,0.9079)--(2.508,0.9195)--(2.537,0.9310)--(2.566,0.9423)--(2.595,0.9535)--(2.624,0.9646)--(2.653,0.9756)--(2.682,0.9865)--(2.711,0.9972)--(2.740,1.008)--(2.769,1.018)--(2.797,1.029)--(2.826,1.039)--(2.855,1.049)--(2.884,1.059)--(2.913,1.069)--(2.942,1.079)--(2.971,1.089)--(3.000,1.099);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.1353,2.0000)--(0.1642,1.8062)--(0.1932,1.6439)--(0.2221,1.5044)--(0.2510,1.3819)--(0.2800,1.2729)--(0.3089,1.1745)--(0.3378,1.0850)--(0.3668,1.0028)--(0.3957,0.9269)--(0.4246,0.8563)--(0.4536,0.7904)--(0.4825,0.7286)--(0.5115,0.6704)--(0.5404,0.6153)--(0.5693,0.5632)--(0.5983,0.5136)--(0.6272,0.4664)--(0.6561,0.4213)--(0.6851,0.3781)--(0.7140,0.3367)--(0.7429,0.2970)--(0.7719,0.2588)--(0.8008,0.2220)--(0.8297,0.1865)--(0.8587,0.1522)--(0.8876,0.1191)--(0.9166,0.0870)--(0.9455,0.0559)--(0.9744,0.0258)--(1.0034,0.0034)--(1.0323,0.0318)--(1.0612,0.0594)--(1.0902,0.0863)--(1.1191,0.1125)--(1.1480,0.1381)--(1.1770,0.1629)--(1.2059,0.1872)--(1.2349,0.2109)--(1.2638,0.2341)--(1.2927,0.2567)--(1.3217,0.2789)--(1.3506,0.3005)--(1.3795,0.3217)--(1.4085,0.3425)--(1.4374,0.3628)--(1.4663,0.3828)--(1.4953,0.4023)--(1.5242,0.4215)--(1.5531,0.4403)--(1.5821,0.4587)--(1.6110,0.4768)--(1.6400,0.4947)--(1.6689,0.5121)--(1.6978,0.5293)--(1.7268,0.5462)--(1.7557,0.5628)--(1.7846,0.5792)--(1.8136,0.5953)--(1.8425,0.6111)--(1.8714,0.6267)--(1.9004,0.6420)--(1.9293,0.6571)--(1.9583,0.6720)--(1.9872,0.6867)--(2.0161,0.7012)--(2.0451,0.7154)--(2.0740,0.7295)--(2.1029,0.7433)--(2.1319,0.7570)--(2.1608,0.7705)--(2.1897,0.7838)--(2.2187,0.7969)--(2.2476,0.8098)--(2.2765,0.8226)--(2.3055,0.8353)--(2.3344,0.8477)--(2.3634,0.8601)--(2.3923,0.8722)--(2.4212,0.8842)--(2.4502,0.8961)--(2.4791,0.9079)--(2.5080,0.9195)--(2.5370,0.9309)--(2.5659,0.9423)--(2.5948,0.9535)--(2.6238,0.9646)--(2.6527,0.9756)--(2.6817,0.9864)--(2.7106,0.9971)--(2.7395,1.0078)--(2.7685,1.0183)--(2.7974,1.0287)--(2.8263,1.0389)--(2.8553,1.0491)--(2.8842,1.0592)--(2.9131,1.0692)--(2.9421,1.0791)--(2.9710,1.0889)--(3.0000,1.0986);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -345,19 +345,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5481);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5481);
 %DEFAULT
 
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-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
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+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -411,31 +411,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,1.8863);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,1.8862);
 %DEFAULT
 
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+\draw [color=blue] (0.1353,-4.0000)--(0.1541,-3.7393)--(0.1730,-3.5088)--(0.1918,-3.3021)--(0.2106,-3.1148)--(0.2295,-2.9436)--(0.2483,-2.7858)--(0.2671,-2.6396)--(0.2860,-2.5034)--(0.3048,-2.3758)--(0.3236,-2.2559)--(0.3425,-2.1428)--(0.3613,-2.0357)--(0.3801,-1.9341)--(0.3990,-1.8374)--(0.4178,-1.7452)--(0.4366,-1.6570)--(0.4555,-1.5725)--(0.4743,-1.4915)--(0.4932,-1.4136)--(0.5120,-1.3387)--(0.5308,-1.2664)--(0.5497,-1.1967)--(0.5685,-1.1293)--(0.5873,-1.0641)--(0.6062,-1.0010)--(0.6250,-0.9398)--(0.6438,-0.8804)--(0.6627,-0.8228)--(0.6815,-0.7667)--(0.7003,-0.7122)--(0.7192,-0.6591)--(0.7380,-0.6074)--(0.7568,-0.5570)--(0.7757,-0.5079)--(0.7945,-0.4599)--(0.8133,-0.4130)--(0.8322,-0.3672)--(0.8510,-0.3225)--(0.8699,-0.2787)--(0.8887,-0.2359)--(0.9075,-0.1939)--(0.9264,-0.1528)--(0.9452,-0.1126)--(0.9640,-0.0731)--(0.9829,-0.0344)--(1.0017,0.0034)--(1.0205,0.0407)--(1.0394,0.0773)--(1.0582,0.1132)--(1.0770,0.1485)--(1.0959,0.1831)--(1.1147,0.2172)--(1.1335,0.2507)--(1.1524,0.2837)--(1.1712,0.3161)--(1.1900,0.3480)--(1.2089,0.3794)--(1.2277,0.4103)--(1.2466,0.4408)--(1.2654,0.4708)--(1.2842,0.5003)--(1.3031,0.5294)--(1.3219,0.5582)--(1.3407,0.5864)--(1.3596,0.6143)--(1.3784,0.6419)--(1.3972,0.6690)--(1.4161,0.6958)--(1.4349,0.7222)--(1.4537,0.7483)--(1.4726,0.7740)--(1.4914,0.7995)--(1.5102,0.8246)--(1.5291,0.8493)--(1.5479,0.8738)--(1.5667,0.8980)--(1.5856,0.9219)--(1.6044,0.9455)--(1.6233,0.9689)--(1.6421,0.9919)--(1.6609,1.0148)--(1.6798,1.0373)--(1.6986,1.0596)--(1.7174,1.0817)--(1.7363,1.1035)--(1.7551,1.1251)--(1.7739,1.1464)--(1.7928,1.1675)--(1.8116,1.1884)--(1.8304,1.2091)--(1.8493,1.2296)--(1.8681,1.2499)--(1.8869,1.2699)--(1.9058,1.2898)--(1.9246,1.3094)--(1.9434,1.3289)--(1.9623,1.3482)--(1.9811,1.3673)--(2.0000,1.3862);
 
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-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-2.0000,1.3862)--(-1.9811,1.3673)--(-1.9623,1.3482)--(-1.9434,1.3289)--(-1.9246,1.3094)--(-1.9058,1.2898)--(-1.8869,1.2699)--(-1.8681,1.2499)--(-1.8493,1.2296)--(-1.8304,1.2091)--(-1.8116,1.1884)--(-1.7928,1.1675)--(-1.7739,1.1464)--(-1.7551,1.1251)--(-1.7363,1.1035)--(-1.7174,1.0817)--(-1.6986,1.0596)--(-1.6798,1.0373)--(-1.6609,1.0148)--(-1.6421,0.9919)--(-1.6233,0.9689)--(-1.6044,0.9455)--(-1.5856,0.9219)--(-1.5667,0.8980)--(-1.5479,0.8738)--(-1.5291,0.8493)--(-1.5102,0.8246)--(-1.4914,0.7995)--(-1.4726,0.7740)--(-1.4537,0.7483)--(-1.4349,0.7222)--(-1.4161,0.6958)--(-1.3972,0.6690)--(-1.3784,0.6419)--(-1.3596,0.6143)--(-1.3407,0.5864)--(-1.3219,0.5582)--(-1.3031,0.5294)--(-1.2842,0.5003)--(-1.2654,0.4708)--(-1.2466,0.4408)--(-1.2277,0.4103)--(-1.2089,0.3794)--(-1.1900,0.3480)--(-1.1712,0.3161)--(-1.1524,0.2837)--(-1.1335,0.2507)--(-1.1147,0.2172)--(-1.0959,0.1831)--(-1.0770,0.1485)--(-1.0582,0.1132)--(-1.0394,0.0773)--(-1.0205,0.0407)--(-1.0017,0.0034)--(-0.9829,-0.0344)--(-0.9640,-0.0731)--(-0.9452,-0.1126)--(-0.9264,-0.1528)--(-0.9075,-0.1939)--(-0.8887,-0.2359)--(-0.8699,-0.2787)--(-0.8510,-0.3225)--(-0.8322,-0.3672)--(-0.8133,-0.4130)--(-0.7945,-0.4599)--(-0.7757,-0.5079)--(-0.7568,-0.5570)--(-0.7380,-0.6074)--(-0.7192,-0.6591)--(-0.7003,-0.7122)--(-0.6815,-0.7667)--(-0.6627,-0.8228)--(-0.6438,-0.8804)--(-0.6250,-0.9398)--(-0.6062,-1.0010)--(-0.5873,-1.0641)--(-0.5685,-1.1293)--(-0.5497,-1.1967)--(-0.5308,-1.2664)--(-0.5120,-1.3387)--(-0.4932,-1.4136)--(-0.4743,-1.4915)--(-0.4555,-1.5725)--(-0.4366,-1.6570)--(-0.4178,-1.7452)--(-0.3990,-1.8374)--(-0.3801,-1.9341)--(-0.3613,-2.0357)--(-0.3425,-2.1428)--(-0.3236,-2.2559)--(-0.3048,-2.3758)--(-0.2860,-2.5034)--(-0.2671,-2.6396)--(-0.2483,-2.7858)--(-0.2295,-2.9436)--(-0.2106,-3.1148)--(-0.1918,-3.3021)--(-0.1730,-3.5088)--(-0.1541,-3.7393)--(-0.1353,-4.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ADUGmRRA.pstricks b/auto/pictures_tex/Fig_ADUGmRRA.pstricks
index 3eafef08f..16ac4b910 100644
--- a/auto/pictures_tex/Fig_ADUGmRRA.pstricks
+++ b/auto/pictures_tex/Fig_ADUGmRRA.pstricks
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.20595) node {\( \alpha_i\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2059) node {\( \alpha_i\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2191) node {\( \alpha_{i+1}\)};
 %END PSPICTURE
 \end{tikzpicture}
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diff --git a/auto/pictures_tex/Fig_ADUGmRRB.pstricks b/auto/pictures_tex/Fig_ADUGmRRB.pstricks
index 70dd143f3..4932d0624 100644
--- a/auto/pictures_tex/Fig_ADUGmRRB.pstricks
+++ b/auto/pictures_tex/Fig_ADUGmRRB.pstricks
@@ -70,11 +70,11 @@
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-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
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 %END PSPICTURE
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diff --git a/auto/pictures_tex/Fig_ADUGmRRC.pstricks b/auto/pictures_tex/Fig_ADUGmRRC.pstricks
index 8ffd11439..e384890b1 100644
--- a/auto/pictures_tex/Fig_ADUGmRRC.pstricks
+++ b/auto/pictures_tex/Fig_ADUGmRRC.pstricks
@@ -70,11 +70,11 @@
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-\draw []  (0,0) node [rotate=0] {$\bullet$};
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-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.21918) node {\( \alpha_{i+1}\)};
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+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
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+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
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 %END PSPICTURE
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diff --git a/auto/pictures_tex/Fig_AIFsOQO.pstricks b/auto/pictures_tex/Fig_AIFsOQO.pstricks
index 598522bfa..1c0661d43 100644
--- a/auto/pictures_tex/Fig_AIFsOQO.pstricks
+++ b/auto/pictures_tex/Fig_AIFsOQO.pstricks
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diff --git a/auto/pictures_tex/Fig_ALIzHFm.pstricks b/auto/pictures_tex/Fig_ALIzHFm.pstricks
index e5623dafb..1e5c03fa2 100644
--- a/auto/pictures_tex/Fig_ALIzHFm.pstricks
+++ b/auto/pictures_tex/Fig_ALIzHFm.pstricks
@@ -74,14 +74,14 @@
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diff --git a/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks b/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
index 37bbbdd11..4c1689b9c 100644
--- a/auto/pictures_tex/Fig_AMDUooZZUOqa.pstricks
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-\draw (0.6519,0.1894) node {$\theta$};
+\draw (0.7513,0.1784) node {$\theta$};
 
 \draw [] (0.4924,-0.0868)--(0.4934,-0.0807)--(0.4943,-0.0746)--(0.4952,-0.0685)--(0.4960,-0.0624)--(0.4968,-0.0562)--(0.4974,-0.0501)--(0.4980,-0.0440)--(0.4985,-0.0378)--(0.4989,-0.0317)--(0.4993,-0.0255)--(0.4996,-0.0193)--(0.4998,-0.0132)--(0.4999,-0.0070)--(0.4999,0.0000)--(0.4999,0.0052)--(0.4998,0.0114)--(0.4996,0.0176)--(0.4994,0.0237)--(0.4991,0.0299)--(0.4986,0.0361)--(0.4982,0.0422)--(0.4976,0.0484)--(0.4970,0.0545)--(0.4963,0.0606)--(0.4955,0.0667)--(0.4946,0.0729)--(0.4937,0.0790)--(0.4927,0.0850)--(0.4916,0.0911)--(0.4904,0.0972)--(0.4892,0.1032)--(0.4879,0.1092)--(0.4865,0.1153)--(0.4850,0.1213)--(0.4835,0.1272)--(0.4819,0.1332)--(0.4802,0.1391)--(0.4784,0.1450)--(0.4766,0.1509)--(0.4747,0.1568)--(0.4727,0.1627)--(0.4707,0.1685)--(0.4686,0.1743)--(0.4664,0.1800)--(0.4641,0.1858)--(0.4618,0.1915)--(0.4594,0.1972)--(0.4569,0.2028)--(0.4544,0.2085)--(0.4518,0.2141)--(0.4491,0.2196)--(0.4464,0.2251)--(0.4436,0.2306)--(0.4407,0.2361)--(0.4377,0.2415)--(0.4347,0.2469)--(0.4316,0.2522)--(0.4285,0.2575)--(0.4253,0.2628)--(0.4220,0.2680)--(0.4187,0.2732)--(0.4153,0.2784)--(0.4118,0.2835)--(0.4083,0.2885)--(0.4047,0.2936)--(0.4010,0.2985)--(0.3973,0.3035)--(0.3935,0.3083)--(0.3897,0.3132)--(0.3858,0.3180)--(0.3818,0.3227)--(0.3778,0.3274)--(0.3738,0.3320)--(0.3696,0.3366)--(0.3654,0.3411)--(0.3612,0.3456)--(0.3569,0.3501)--(0.3526,0.3544)--(0.3482,0.3588)--(0.3437,0.3630)--(0.3392,0.3672)--(0.3346,0.3714)--(0.3300,0.3755)--(0.3254,0.3796)--(0.3207,0.3835)--(0.3159,0.3875)--(0.3111,0.3913)--(0.3063,0.3951)--(0.3014,0.3989)--(0.2964,0.4026)--(0.2914,0.4062)--(0.2864,0.4098)--(0.2813,0.4133)--(0.2762,0.4167)--(0.2710,0.4201)--(0.2658,0.4234)--(0.2606,0.4267)--(0.2553,0.4298)--(0.2500,0.4330);
 \draw [] (0.0000,0.0000) -- (1.9696,-0.3472);
diff --git a/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks b/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks
index 4e8788b1b..10f8b5696 100644
--- a/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks
+++ b/auto/pictures_tex/Fig_ASHYooUVHkak.pstricks
@@ -72,21 +72,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.9000);
+\draw [,->,>=latex] (-2.4000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.9000);
 %DEFAULT
-\draw [style=dashed] (-1.90,1.20) -- (3.00,1.20);
-\draw []  (0,1.2000) node [rotate=0] {$\bullet$};
-\draw (0.22944,1.4682) node {\( \delta\)};
-\draw [] (-0.300,1.40) -- (0.300,1.00);
-\draw [] (1.70,1.40) -- (2.30,1.00);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.33963) node {\( t_1\)};
-\draw []  (-1.5000,0) node [rotate=0] {$\bullet$};
-\draw (-1.5000,-0.33963) node {\( t_2\)};
-\draw [style=dotted] (2.00,0) -- (2.00,1.20);
-\draw [style=dotted] (-1.50,0) -- (-1.50,1.20);
-\draw [] (-1.80,1.40) -- (-1.20,1.00);
+\draw [style=dashed] (-1.9000,1.2000) -- (3.0000,1.2000);
+\draw []  (0.0000,1.2000) node [rotate=0] {$\bullet$};
+\draw (0.2294,1.4681) node {\( \delta\)};
+\draw [] (-0.3000,1.4000) -- (0.3000,1.0000);
+\draw [] (1.7000,1.4000) -- (2.3000,1.0000);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3396) node {\( t_1\)};
+\draw []  (-1.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.3396) node {\( t_2\)};
+\draw [style=dotted] (2.0000,0.0000) -- (2.0000,1.2000);
+\draw [style=dotted] (-1.5000,0.0000) -- (-1.5000,1.2000);
+\draw [] (-1.8000,1.4000) -- (-1.2000,1.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_AccumulationIsole.pstricks b/auto/pictures_tex/Fig_AccumulationIsole.pstricks
index 3b674d22b..9277d4a0b 100644
--- a/auto/pictures_tex/Fig_AccumulationIsole.pstricks
+++ b/auto/pictures_tex/Fig_AccumulationIsole.pstricks
@@ -77,20 +77,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw (1.5425,1.0000) node {$P$};
-\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-1.5443,0) node {$Q$};
-\draw []  (0.50000,0.50000) node [rotate=0] {$\bullet$};
-\draw (0.16544,0.16316) node {$S$};
+\draw [color=red]  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5442,0.0000) node {$Q$};
+\draw []  (0.5000,0.5000) node [rotate=0] {$\bullet$};
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_AdhIntFr.pstricks b/auto/pictures_tex/Fig_AdhIntFr.pstricks
index a1650bae7..857863ea3 100644
--- a/auto/pictures_tex/Fig_AdhIntFr.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFr.pstricks
@@ -83,13 +83,13 @@
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 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,32 +97,32 @@
 % setting the default values
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          hatchthickness=0.4pt}
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+\draw [] (1.0000,2.0000) -- (1.0000,2.0000);
+\draw [] (4.0000,2.0000) -- (4.0000,2.0000);
+\draw []  (3.5000,0.7500) node [rotate=0] {$\bullet$};
 
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-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
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+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks b/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks
index ff57c58d7..272312662 100644
--- a/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFrDeux.pstricks
@@ -91,13 +91,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,6.5000);
 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -105,36 +105,36 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (1.0000,2.0000) -- (1.0000,2.0000);
+\draw [] (3.0000,4.0000) -- (3.0000,6.0000);
 \draw []  (1.0000,2.0000) node [rotate=0] {$\bullet$};
 
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-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.29125,6.0000) node {$ 6 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw [] (1.7000,2.0000)--(1.6985,2.0443)--(1.6943,2.0886)--(1.6873,2.1324)--(1.6775,2.1758)--(1.6650,2.2184)--(1.6498,2.2601)--(1.6320,2.3008)--(1.6116,2.3403)--(1.5888,2.3784)--(1.5636,2.4150)--(1.5362,2.4499)--(1.5066,2.4830)--(1.4749,2.5142)--(1.4413,2.5433)--(1.4060,2.5702)--(1.3690,2.5948)--(1.3305,2.6170)--(1.2907,2.6367)--(1.2498,2.6539)--(1.2078,2.6684)--(1.1650,2.6802)--(1.1215,2.6893)--(1.0775,2.6956)--(1.0333,2.6992)--(0.9888,2.6999)--(0.9445,2.6977)--(0.9003,2.6928)--(0.8566,2.6851)--(0.8134,2.6746)--(0.7710,2.6615)--(0.7295,2.6456)--(0.6891,2.6271)--(0.6500,2.6062)--(0.6122,2.5827)--(0.5760,2.5570)--(0.5415,2.5290)--(0.5089,2.4988)--(0.4783,2.4667)--(0.4497,2.4327)--(0.4234,2.3969)--(0.3994,2.3595)--(0.3778,2.3207)--(0.3587,2.2806)--(0.3422,2.2394)--(0.3283,2.1972)--(0.3171,2.1542)--(0.3087,2.1106)--(0.3031,2.0665)--(0.3003,2.0222)--(0.3003,1.9777)--(0.3031,1.9334)--(0.3087,1.8893)--(0.3171,1.8457)--(0.3283,1.8027)--(0.3422,1.7605)--(0.3587,1.7193)--(0.3778,1.6792)--(0.3994,1.6404)--(0.4234,1.6030)--(0.4497,1.5672)--(0.4783,1.5332)--(0.5089,1.5011)--(0.5415,1.4709)--(0.5760,1.4429)--(0.6122,1.4172)--(0.6500,1.3937)--(0.6891,1.3728)--(0.7295,1.3543)--(0.7710,1.3384)--(0.8134,1.3253)--(0.8566,1.3148)--(0.9003,1.3071)--(0.9445,1.3022)--(0.9888,1.3000)--(1.0333,1.3007)--(1.0775,1.3043)--(1.1215,1.3106)--(1.1650,1.3197)--(1.2078,1.3315)--(1.2498,1.3460)--(1.2907,1.3632)--(1.3305,1.3829)--(1.3690,1.4051)--(1.4060,1.4297)--(1.4413,1.4566)--(1.4749,1.4857)--(1.5066,1.5169)--(1.5362,1.5500)--(1.5636,1.5849)--(1.5888,1.6215)--(1.6116,1.6596)--(1.6320,1.6991)--(1.6498,1.7398)--(1.6650,1.7815)--(1.6775,1.8241)--(1.6873,1.8675)--(1.6943,1.9113)--(1.6985,1.9556)--(1.7000,2.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_AdhIntFrSix.pstricks b/auto/pictures_tex/Fig_AdhIntFrSix.pstricks
index 855693c3c..c04f51d28 100644
--- a/auto/pictures_tex/Fig_AdhIntFrSix.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFrSix.pstricks
@@ -67,116 +67,116 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.1000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.1000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
-\draw [color=blue] (4.00,0) -- (4.00,4.00);
-\draw [color=blue] (2.00,0) -- (2.00,4.00);
-\draw [color=blue] (1.33,0) -- (1.33,4.00);
-\draw [color=blue] (1.00,0) -- (1.00,4.00);
-\draw [color=blue] (0.800,0) -- (0.800,4.00);
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+\draw (4.0000,-0.3149) node {$ 1 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,4.0000) node {$ 1 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks b/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks
index 0ccc5fe2d..cc82200c7 100644
--- a/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks
+++ b/auto/pictures_tex/Fig_AdhIntFrTrois.pstricks
@@ -79,35 +79,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (8.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (8.5000,0.0000);
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+\draw [color=gray,style=dashed] 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+\draw [color=gray,style=dashed] 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+\draw [color=gray,style=dashed] 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+\draw [color=gray,style=dashed] 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
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -115,37 +115,37 @@
 % setting the default values
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-\draw [] (-0.100,3.00) -- (0.100,3.00);
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+\draw (-4.0000,-0.3298) node {$ -1 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 1 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 2 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_AireParabole.pstricks b/auto/pictures_tex/Fig_AireParabole.pstricks
index 69294f452..0d48af449 100644
--- a/auto/pictures_tex/Fig_AireParabole.pstricks
+++ b/auto/pictures_tex/Fig_AireParabole.pstricks
@@ -91,34 +91,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,6.5000);
 %DEFAULT
 
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-\draw [] (2.00,-1.00) -- (2.00,3.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.29125,6.0000) node {$ 6 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw [color=blue] (-1.0000,2.0000)--(-0.9595,1.8804)--(-0.9191,1.7641)--(-0.8787,1.6510)--(-0.8383,1.5412)--(-0.7979,1.4347)--(-0.7575,1.3314)--(-0.7171,1.2315)--(-0.6767,1.1347)--(-0.6363,1.0413)--(-0.5959,0.9511)--(-0.5555,0.8641)--(-0.5151,0.7805)--(-0.4747,0.7001)--(-0.4343,0.6229)--(-0.3939,0.5491)--(-0.3535,0.4785)--(-0.3131,0.4111)--(-0.2727,0.3471)--(-0.2323,0.2862)--(-0.1919,0.2287)--(-0.1515,0.1744)--(-0.1111,0.1234)--(-0.0707,0.0757)--(-0.0303,0.0312)--(0.0101,-0.0099)--(0.0505,-0.0479)--(0.0909,-0.0826)--(0.1313,-0.1140)--(0.1717,-0.1422)--(0.2121,-0.1671)--(0.2525,-0.1887)--(0.2929,-0.2071)--(0.3333,-0.2222)--(0.3737,-0.2340)--(0.4141,-0.2426)--(0.4545,-0.2479)--(0.4949,-0.2499)--(0.5353,-0.2487)--(0.5757,-0.2442)--(0.6161,-0.2365)--(0.6565,-0.2254)--(0.6969,-0.2112)--(0.7373,-0.1936)--(0.7777,-0.1728)--(0.8181,-0.1487)--(0.8585,-0.1214)--(0.8989,-0.0908)--(0.9393,-0.0569)--(0.9797,-0.0197)--(1.0202,0.0206)--(1.0606,0.0642)--(1.1010,0.1112)--(1.1414,0.1614)--(1.1818,0.2148)--(1.2222,0.2716)--(1.2626,0.3315)--(1.3030,0.3948)--(1.3434,0.4613)--(1.3838,0.5311)--(1.4242,0.6042)--(1.4646,0.6805)--(1.5050,0.7601)--(1.5454,0.8429)--(1.5858,0.9290)--(1.6262,1.0184)--(1.6666,1.1111)--(1.7070,1.2070)--(1.7474,1.3061)--(1.7878,1.4086)--(1.8282,1.5143)--(1.8686,1.6233)--(1.9090,1.7355)--(1.9494,1.8510)--(1.9898,1.9697)--(2.0303,2.0918)--(2.0707,2.2171)--(2.1111,2.3456)--(2.1515,2.4775)--(2.1919,2.6125)--(2.2323,2.7509)--(2.2727,2.8925)--(2.3131,3.0374)--(2.3535,3.1855)--(2.3939,3.3370)--(2.4343,3.4916)--(2.4747,3.6496)--(2.5151,3.8108)--(2.5555,3.9753)--(2.5959,4.1430)--(2.6363,4.3140)--(2.6767,4.4883)--(2.7171,4.6658)--(2.7575,4.8466)--(2.7979,5.0307)--(2.8383,5.2180)--(2.8787,5.4086)--(2.9191,5.6024)--(2.9595,5.7996)--(3.0000,6.0000);
+\draw [] (2.0000,-1.0000) -- (2.0000,3.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ArcLongueurFinesse.pstricks b/auto/pictures_tex/Fig_ArcLongueurFinesse.pstricks
index 3ce96b5bd..8ba02ddc0 100644
--- a/auto/pictures_tex/Fig_ArcLongueurFinesse.pstricks
+++ b/auto/pictures_tex/Fig_ArcLongueurFinesse.pstricks
@@ -65,23 +65,23 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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-\draw [color=red] (2.64,-0.566) -- (5.28,0.216);
-\draw [color=red] (5.28,0.216) -- (7.92,0.217);
-\draw [color=red] (7.92,0.217) -- (10.6,-0.566);
-\draw [color=red] (10.6,-0.566) -- (13.2,0.700);
-\draw [color=green] (0,0.700) -- (1.20,-0.0996);
-\draw [color=green] (1.20,-0.0996) -- (2.40,-0.672);
-\draw [color=green] (2.40,-0.672) -- (3.60,0.291);
-\draw [color=green] (3.60,0.291) -- (4.80,0.589);
-\draw [color=green] (4.80,0.589) -- (6.00,-0.459);
-\draw [color=green] (6.00,-0.459) -- (7.20,-0.459);
-\draw [color=green] (7.20,-0.459) -- (8.40,0.589);
-\draw [color=green] (8.40,0.589) -- (9.60,0.291);
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+\draw [color=blue,style=dashed] (0.0000,0.7000)--(0.1332,0.6873)--(0.2665,0.6498)--(0.3998,0.5888)--(0.5331,0.5066)--(0.6663,0.4060)--(0.7996,0.2907)--(0.9329,0.1650)--(1.0662,0.0333)--(1.1995,-0.0996)--(1.3327,-0.2289)--(1.4660,-0.3500)--(1.5993,-0.4584)--(1.7326,-0.5502)--(1.8659,-0.6221)--(1.9991,-0.6716)--(2.1324,-0.6968)--(2.2657,-0.6968)--(2.3990,-0.6716)--(2.5323,-0.6221)--(2.6655,-0.5502)--(2.7988,-0.4584)--(2.9321,-0.3500)--(3.0654,-0.2289)--(3.1987,-0.0996)--(3.3319,0.0333)--(3.4652,0.1650)--(3.5985,0.2907)--(3.7318,0.4060)--(3.8651,0.5066)--(3.9983,0.5888)--(4.1316,0.6498)--(4.2649,0.6873)--(4.3982,0.7000)--(4.5315,0.6873)--(4.6647,0.6498)--(4.7980,0.5888)--(4.9313,0.5066)--(5.0646,0.4060)--(5.1979,0.2907)--(5.3311,0.1650)--(5.4644,0.0333)--(5.5977,-0.0996)--(5.7310,-0.2289)--(5.8643,-0.3499)--(5.9975,-0.4584)--(6.1308,-0.5502)--(6.2641,-0.6221)--(6.3974,-0.6716)--(6.5307,-0.6968)--(6.6639,-0.6968)--(6.7972,-0.6716)--(6.9305,-0.6221)--(7.0638,-0.5502)--(7.1971,-0.4584)--(7.3303,-0.3500)--(7.4636,-0.2289)--(7.5969,-0.0996)--(7.7302,0.0333)--(7.8635,0.1650)--(7.9967,0.2907)--(8.1300,0.4060)--(8.2633,0.5066)--(8.3966,0.5888)--(8.5299,0.6498)--(8.6631,0.6873)--(8.7964,0.7000)--(8.9297,0.6873)--(9.0630,0.6498)--(9.1962,0.5888)--(9.3295,0.5066)--(9.4628,0.4060)--(9.5961,0.2907)--(9.7294,0.1650)--(9.8626,0.0333)--(9.9959,-0.0996)--(10.129,-0.2289)--(10.262,-0.3499)--(10.395,-0.4584)--(10.529,-0.5502)--(10.662,-0.6221)--(10.795,-0.6716)--(10.928,-0.6968)--(11.062,-0.6968)--(11.195,-0.6716)--(11.328,-0.6221)--(11.462,-0.5502)--(11.595,-0.4584)--(11.728,-0.3500)--(11.861,-0.2289)--(11.995,-0.0996)--(12.128,0.0333)--(12.261,0.1650)--(12.395,0.2907)--(12.528,0.4060)--(12.661,0.5066)--(12.794,0.5888)--(12.928,0.6498)--(13.061,0.6873)--(13.194,0.7000);
+\draw [color=red] (0.0000,0.7000) -- (2.6389,-0.5663);
+\draw [color=red] (2.6389,-0.5663) -- (5.2778,0.2163);
+\draw [color=red] (5.2778,0.2163) -- (7.9168,0.2163);
+\draw [color=red] (7.9168,0.2163) -- (10.555,-0.5663);
+\draw [color=red] (10.555,-0.5663) -- (13.194,0.7000);
+\draw [color=green] (0.0000,0.7000) -- (1.1995,-0.0996);
+\draw [color=green] (1.1995,-0.0996) -- (2.3990,-0.6716);
+\draw [color=green] (2.3990,-0.6716) -- (3.5985,0.2907);
+\draw [color=green] (3.5985,0.2907) -- (4.7980,0.5888);
+\draw [color=green] (4.7980,0.5888) -- (5.9975,-0.4584);
+\draw [color=green] (5.9975,-0.4584) -- (7.1971,-0.4584);
+\draw [color=green] (7.1971,-0.4584) -- (8.3966,0.5888);
+\draw [color=green] (8.3966,0.5888) -- (9.5961,0.2907);
+\draw [color=green] (9.5961,0.2907) -- (10.795,-0.6716);
+\draw [color=green] (10.795,-0.6716) -- (11.995,-0.0996);
+\draw [color=green] (11.995,-0.0996) -- (13.194,0.7000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks b/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
index a64d6dad8..5fddf7ead 100644
--- a/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
+++ b/auto/pictures_tex/Fig_BEHTooWsdrys.pstricks
@@ -65,58 +65,58 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
+\fill [color=cyan] (1.6000,-1.2000) -- (3.0666,-1.2000) -- (3.0666,-1.2000) -- (3.0666,1.2000) -- (3.0666,1.2000) -- (1.6000,1.2000) -- (1.6000,1.2000) -- (1.6000,-1.2000) --  cycle;
 
 
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-\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
-\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
-\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
-\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,-0.6666) -- (2.2250,-0.6666);
+\draw [,->,>=latex] (1.6000,-0.3333) -- (2.2250,-0.3333);
+\draw [,->,>=latex] (1.6000,0.0000) -- (2.2250,0.0000);
+\draw [,->,>=latex] (1.6000,0.3333) -- (2.2250,0.3333);
+\draw [,->,>=latex] (1.6000,0.6666) -- (2.2250,0.6666);
 \draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
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-\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
-\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
-\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
-\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
-\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,-0.6666) -- (2.7619,-0.6666);
+\draw [,->,>=latex] (2.3333,-0.3333) -- (2.7619,-0.3333);
+\draw [,->,>=latex] (2.3333,0.0000) -- (2.7619,0.0000);
+\draw [,->,>=latex] (2.3333,0.3333) -- (2.7619,0.3333);
+\draw [,->,>=latex] (2.3333,0.6666) -- (2.7619,0.6666);
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-\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
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-\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
-\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
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-\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
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-\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
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+\draw [,->,>=latex] (3.0666,-0.6666) -- (3.3927,-0.6666);
+\draw [,->,>=latex] (3.0666,-0.3333) -- (3.3927,-0.3333);
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+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0631,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.6666) -- (4.0631,-0.6666);
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+\draw [,->,>=latex] (3.8000,0.0000) -- (4.0631,0.0000);
+\draw [,->,>=latex] (3.8000,0.3333) -- (4.0631,0.3333);
+\draw [,->,>=latex] (3.8000,0.6666) -- (4.0631,0.6666);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0631,1.0000);
 \draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
-\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
-\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
-\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
-\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
-\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,-0.6666) -- (4.7539,-0.6666);
+\draw [,->,>=latex] (4.5333,-0.3333) -- (4.7539,-0.3333);
+\draw [,->,>=latex] (4.5333,0.0000) -- (4.7539,0.0000);
+\draw [,->,>=latex] (4.5333,0.3333) -- (4.7539,0.3333);
+\draw [,->,>=latex] (4.5333,0.6666) -- (4.7539,0.6666);
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-\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
-\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
-\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
-\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
-\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
-\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
-\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
-\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
-\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
-\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
-\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
-\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
-\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
-\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
+\draw [,->,>=latex] (5.2666,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2666,-0.6666) -- (5.4565,-0.6666);
+\draw [,->,>=latex] (5.2666,-0.3333) -- (5.4565,-0.3333);
+\draw [,->,>=latex] (5.2666,0.0000) -- (5.4565,0.0000);
+\draw [,->,>=latex] (5.2666,0.3333) -- (5.4565,0.3333);
+\draw [,->,>=latex] (5.2666,0.6666) -- (5.4565,0.6666);
+\draw [,->,>=latex] (5.2666,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1666,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.6666) -- (6.1666,-0.6666);
+\draw [,->,>=latex] (6.0000,-0.3333) -- (6.1666,-0.3333);
+\draw [,->,>=latex] (6.0000,0.0000) -- (6.1666,0.0000);
+\draw [,->,>=latex] (6.0000,0.3333) -- (6.1666,0.3333);
+\draw [,->,>=latex] (6.0000,0.6666) -- (6.1666,0.6666);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1666,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks b/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
index 7ac338be2..5d8af556c 100644
--- a/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
+++ b/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
@@ -72,21 +72,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [] (1.50,0)--(1.50,0.0951)--(1.49,0.190)--(1.47,0.284)--(1.45,0.377)--(1.43,0.468)--(1.39,0.557)--(1.35,0.645)--(1.31,0.729)--(1.26,0.811)--(1.21,0.889)--(1.15,0.964)--(1.09,1.04)--(1.02,1.10)--(0.946,1.16)--(0.870,1.22)--(0.791,1.27)--(0.708,1.32)--(0.623,1.36)--(0.535,1.40)--(0.445,1.43)--(0.354,1.46)--(0.260,1.48)--(0.166,1.49)--(0.0714,1.50)--(-0.0238,1.50)--(-0.119,1.50)--(-0.213,1.48)--(-0.307,1.47)--(-0.400,1.45)--(-0.491,1.42)--(-0.580,1.38)--(-0.666,1.34)--(-0.750,1.30)--(-0.831,1.25)--(-0.908,1.19)--(-0.982,1.13)--(-1.05,1.07)--(-1.12,1.00)--(-1.18,0.927)--(-1.24,0.851)--(-1.29,0.771)--(-1.33,0.687)--(-1.37,0.601)--(-1.41,0.513)--(-1.44,0.423)--(-1.46,0.330)--(-1.48,0.237)--(-1.49,0.143)--(-1.50,0.0476)--(-1.50,-0.0476)--(-1.49,-0.143)--(-1.48,-0.237)--(-1.46,-0.330)--(-1.44,-0.423)--(-1.41,-0.513)--(-1.37,-0.601)--(-1.33,-0.687)--(-1.29,-0.771)--(-1.24,-0.851)--(-1.18,-0.927)--(-1.12,-1.00)--(-1.05,-1.07)--(-0.982,-1.13)--(-0.908,-1.19)--(-0.831,-1.25)--(-0.750,-1.30)--(-0.666,-1.34)--(-0.580,-1.38)--(-0.491,-1.42)--(-0.400,-1.45)--(-0.307,-1.47)--(-0.213,-1.48)--(-0.119,-1.50)--(-0.0238,-1.50)--(0.0714,-1.50)--(0.166,-1.49)--(0.260,-1.48)--(0.354,-1.46)--(0.445,-1.43)--(0.535,-1.40)--(0.623,-1.36)--(0.708,-1.32)--(0.791,-1.27)--(0.870,-1.22)--(0.946,-1.16)--(1.02,-1.10)--(1.09,-1.04)--(1.15,-0.964)--(1.21,-0.889)--(1.26,-0.811)--(1.31,-0.729)--(1.35,-0.645)--(1.39,-0.557)--(1.43,-0.468)--(1.45,-0.377)--(1.47,-0.284)--(1.49,-0.190)--(1.50,-0.0951)--(1.50,0);
-\draw (1.0127,0.25615) node {\( \theta\)};
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw (0.7147,0.3590) node {\( \theta\)};
 
-\draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw []  (1.2990,0) node [rotate=0] {$\bullet$};
-\draw (1.2990,-0.27858) node {\( x\)};
-\draw []  (0,0.75000) node [rotate=0] {$\bullet$};
-\draw (-0.29602,0.75000) node {\( y\)};
-\draw [] (0,0) -- (1.30,0.750);
-\draw [style=dashed] (1.30,0.750) -- (1.30,0);
-\draw [style=dashed] (1.30,0.750) -- (0,0.750);
+\draw [] (0.5000,0.0000)--(0.4999,0.0048)--(0.4999,0.0096)--(0.4997,0.0145)--(0.4996,0.0193)--(0.4994,0.0242)--(0.4991,0.0290)--(0.4988,0.0339)--(0.4984,0.0387)--(0.4980,0.0435)--(0.4976,0.0484)--(0.4971,0.0532)--(0.4966,0.0580)--(0.4960,0.0628)--(0.4954,0.0676)--(0.4947,0.0724)--(0.4939,0.0772)--(0.4932,0.0820)--(0.4924,0.0868)--(0.4915,0.0915)--(0.4906,0.0963)--(0.4896,0.1011)--(0.4886,0.1058)--(0.4876,0.1105)--(0.4865,0.1153)--(0.4853,0.1200)--(0.4841,0.1247)--(0.4829,0.1294)--(0.4816,0.1340)--(0.4803,0.1387)--(0.4789,0.1434)--(0.4775,0.1480)--(0.4761,0.1526)--(0.4746,0.1572)--(0.4730,0.1618)--(0.4714,0.1664)--(0.4698,0.1710)--(0.4681,0.1755)--(0.4664,0.1800)--(0.4646,0.1846)--(0.4628,0.1890)--(0.4610,0.1935)--(0.4591,0.1980)--(0.4571,0.2024)--(0.4551,0.2069)--(0.4531,0.2113)--(0.4510,0.2156)--(0.4489,0.2200)--(0.4468,0.2243)--(0.4446,0.2287)--(0.4423,0.2330)--(0.4401,0.2373)--(0.4377,0.2415)--(0.4354,0.2457)--(0.4330,0.2500)--(0.4305,0.2541)--(0.4280,0.2583)--(0.4255,0.2624)--(0.4229,0.2666)--(0.4203,0.2706)--(0.4177,0.2747)--(0.4150,0.2787)--(0.4123,0.2828)--(0.4095,0.2867)--(0.4067,0.2907)--(0.4039,0.2946)--(0.4010,0.2985)--(0.3981,0.3024)--(0.3951,0.3063)--(0.3922,0.3101)--(0.3891,0.3139)--(0.3861,0.3176)--(0.3830,0.3213)--(0.3798,0.3250)--(0.3767,0.3287)--(0.3735,0.3323)--(0.3702,0.3360)--(0.3669,0.3395)--(0.3636,0.3431)--(0.3603,0.3466)--(0.3569,0.3501)--(0.3535,0.3535)--(0.3501,0.3569)--(0.3466,0.3603)--(0.3431,0.3636)--(0.3395,0.3669)--(0.3360,0.3702)--(0.3323,0.3735)--(0.3287,0.3767)--(0.3250,0.3798)--(0.3213,0.3830)--(0.3176,0.3861)--(0.3139,0.3891)--(0.3101,0.3922)--(0.3063,0.3951)--(0.3024,0.3981)--(0.2985,0.4010)--(0.2946,0.4039)--(0.2907,0.4067)--(0.2867,0.4095);
+\draw []  (1.1471,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.1471,-0.2785) node {\( x\)};
+\draw []  (0.0000,1.6383) node [rotate=0] {$\bullet$};
+\draw (-0.2960,1.6383) node {\( y\)};
+\draw [] (0.0000,0.0000) -- (1.1471,1.6383);
+\draw [style=dashed] (1.1471,1.6383) -- (1.1471,0.0000);
+\draw [style=dashed] (1.1471,1.6383) -- (0.0000,1.6383);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks b/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks
index a72e93213..6c66b5674 100644
--- a/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks
+++ b/auto/pictures_tex/Fig_BNHLooLDxdPA.pstricks
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.0000,0);
-\draw (3.3086,-0.29071) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw [,->,>=latex] (0.0000,0.0000) -- (3.0000,0.0000);
+\draw (3.3085,-0.2907) node {$a$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (2.0000,2.0000);
 \draw (2.0000,2.4267) node {$b$};
 \draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
-\draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
-\draw [style=dashed] (2.00,2.00) -- (2.00,0);
+\draw [color=blue,style=dotted] (2.0000,2.0000) -- (5.0000,2.0000);
+\draw [color=blue,style=dotted] (3.0000,0.0000) -- (5.0000,2.0000);
+\draw [style=dashed] (2.0000,2.0000) -- (2.0000,0.0000);
 \draw (2.3051,1.0000) node {$h$};
-\draw (0.80915,0.31918) node {$\theta$};
+\draw (0.8091,0.3191) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw [] (0.5000,0.0000)--(0.4999,0.0039)--(0.4999,0.0079)--(0.4998,0.0118)--(0.4997,0.0158)--(0.4996,0.0198)--(0.4994,0.0237)--(0.4992,0.0277)--(0.4989,0.0317)--(0.4987,0.0356)--(0.4984,0.0396)--(0.4980,0.0435)--(0.4977,0.0475)--(0.4973,0.0514)--(0.4969,0.0554)--(0.4964,0.0593)--(0.4959,0.0632)--(0.4954,0.0672)--(0.4949,0.0711)--(0.4943,0.0750)--(0.4937,0.0790)--(0.4930,0.0829)--(0.4924,0.0868)--(0.4916,0.0907)--(0.4909,0.0946)--(0.4901,0.0985)--(0.4894,0.1024)--(0.4885,0.1062)--(0.4877,0.1101)--(0.4868,0.1140)--(0.4859,0.1178)--(0.4849,0.1217)--(0.4839,0.1255)--(0.4829,0.1294)--(0.4819,0.1332)--(0.4808,0.1370)--(0.4797,0.1408)--(0.4786,0.1446)--(0.4774,0.1484)--(0.4762,0.1522)--(0.4750,0.1560)--(0.4737,0.1597)--(0.4725,0.1635)--(0.4711,0.1672)--(0.4698,0.1710)--(0.4684,0.1747)--(0.4670,0.1784)--(0.4656,0.1821)--(0.4641,0.1858)--(0.4626,0.1895)--(0.4611,0.1931)--(0.4596,0.1968)--(0.4580,0.2004)--(0.4564,0.2040)--(0.4548,0.2077)--(0.4531,0.2113)--(0.4514,0.2148)--(0.4497,0.2184)--(0.4479,0.2220)--(0.4462,0.2255)--(0.4444,0.2291)--(0.4425,0.2326)--(0.4407,0.2361)--(0.4388,0.2396)--(0.4369,0.2430)--(0.4349,0.2465)--(0.4330,0.2500)--(0.4310,0.2534)--(0.4289,0.2568)--(0.4269,0.2602)--(0.4248,0.2636)--(0.4227,0.2669)--(0.4206,0.2703)--(0.4184,0.2736)--(0.4162,0.2769)--(0.4140,0.2802)--(0.4118,0.2835)--(0.4095,0.2867)--(0.4072,0.2900)--(0.4049,0.2932)--(0.4026,0.2964)--(0.4002,0.2996)--(0.3978,0.3028)--(0.3954,0.3059)--(0.3930,0.3090)--(0.3905,0.3121)--(0.3880,0.3152)--(0.3855,0.3183)--(0.3830,0.3213)--(0.3804,0.3244)--(0.3778,0.3274)--(0.3752,0.3304)--(0.3726,0.3333)--(0.3699,0.3363)--(0.3672,0.3392)--(0.3645,0.3421)--(0.3618,0.3450)--(0.3591,0.3478)--(0.3563,0.3507)--(0.3535,0.3535);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks b/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
index 3f5418461..39102d176 100644
--- a/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
+++ b/auto/pictures_tex/Fig_BQXKooPqSEMN.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.2499);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (0.0000,0.0000) -- (0.0000,0.9722);
+\draw [] (5.0000,2.6388) -- (5.0000,0.0000);
 
-\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3485,0.6490)--(-0.2727,0.7215)--(-0.1970,0.7928)--(-0.1212,0.8628)--(-0.04545,0.9316)--(0.03030,0.9991)--(0.1061,1.065)--(0.1818,1.130)--(0.2576,1.194)--(0.3333,1.256)--(0.4091,1.317)--(0.4848,1.377)--(0.5606,1.436)--(0.6364,1.493)--(0.7121,1.549)--(0.7879,1.604)--(0.8636,1.657)--(0.9394,1.709)--(1.015,1.760)--(1.091,1.810)--(1.167,1.858)--(1.242,1.905)--(1.318,1.951)--(1.394,1.995)--(1.470,2.039)--(1.545,2.081)--(1.621,2.121)--(1.697,2.161)--(1.773,2.199)--(1.848,2.236)--(1.924,2.271)--(2.000,2.306)--(2.076,2.339)--(2.152,2.370)--(2.227,2.401)--(2.303,2.430)--(2.379,2.458)--(2.455,2.485)--(2.530,2.510)--(2.606,2.534)--(2.682,2.557)--(2.758,2.578)--(2.833,2.599)--(2.909,2.618)--(2.985,2.635)--(3.061,2.652)--(3.136,2.667)--(3.212,2.681)--(3.288,2.694)--(3.364,2.705)--(3.439,2.715)--(3.515,2.724)--(3.591,2.731)--(3.667,2.738)--(3.742,2.743)--(3.818,2.746)--(3.894,2.749)--(3.970,2.750)--(4.045,2.750)--(4.121,2.748)--(4.197,2.746)--(4.273,2.742)--(4.349,2.737)--(4.424,2.730)--(4.500,2.722)--(4.576,2.713)--(4.651,2.703)--(4.727,2.691)--(4.803,2.678)--(4.879,2.664)--(4.955,2.649)--(5.030,2.632)--(5.106,2.614)--(5.182,2.595)--(5.258,2.574)--(5.333,2.552)--(5.409,2.529)--(5.485,2.505)--(5.561,2.479)--(5.636,2.452)--(5.712,2.424)--(5.788,2.395)--(5.864,2.364)--(5.939,2.332)--(6.015,2.299)--(6.091,2.264)--(6.167,2.228)--(6.242,2.191)--(6.318,2.153)--(6.394,2.113)--(6.470,2.072)--(6.545,2.030)--(6.621,1.987)--(6.697,1.942)--(6.773,1.896)--(6.849,1.848)--(6.924,1.800)--(7.000,1.750);
+\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3484,0.6489)--(-0.2727,0.7215)--(-0.1969,0.7928)--(-0.1212,0.8628)--(-0.0454,0.9315)--(0.0303,0.9990)--(0.1060,1.0652)--(0.1818,1.1301)--(0.2575,1.1938)--(0.3333,1.2561)--(0.4090,1.3172)--(0.4848,1.3770)--(0.5606,1.4356)--(0.6363,1.4928)--(0.7121,1.5488)--(0.7878,1.6035)--(0.8636,1.6570)--(0.9393,1.7091)--(1.0151,1.7600)--(1.0909,1.8096)--(1.1666,1.8580)--(1.2424,1.9050)--(1.3181,1.9508)--(1.3939,1.9953)--(1.4696,2.0386)--(1.5454,2.0805)--(1.6212,2.1212)--(1.6969,2.1606)--(1.7727,2.1988)--(1.8484,2.2356)--(1.9242,2.2712)--(2.0000,2.3055)--(2.0757,2.3385)--(2.1515,2.3703)--(2.2272,2.4008)--(2.3030,2.4300)--(2.3787,2.4579)--(2.4545,2.4846)--(2.5303,2.5099)--(2.6060,2.5341)--(2.6818,2.5569)--(2.7575,2.5784)--(2.8333,2.5987)--(2.9090,2.6177)--(2.9848,2.6354)--(3.0606,2.6519)--(3.1363,2.6671)--(3.2121,2.6810)--(3.2878,2.6936)--(3.3636,2.7050)--(3.4393,2.7150)--(3.5151,2.7238)--(3.5909,2.7314)--(3.6666,2.7376)--(3.7424,2.7426)--(3.8181,2.7463)--(3.8939,2.7487)--(3.9696,2.7498)--(4.0454,2.7497)--(4.1212,2.7483)--(4.1969,2.7456)--(4.2727,2.7417)--(4.3484,2.7365)--(4.4242,2.7300)--(4.5000,2.7222)--(4.5757,2.7131)--(4.6515,2.7028)--(4.7272,2.6912)--(4.8030,2.6783)--(4.8787,2.6641)--(4.9545,2.6487)--(5.0303,2.6320)--(5.1060,2.6140)--(5.1818,2.5948)--(5.2575,2.5742)--(5.3333,2.5524)--(5.4090,2.5293)--(5.4848,2.5050)--(5.5606,2.4793)--(5.6363,2.4524)--(5.7121,2.4242)--(5.7878,2.3948)--(5.8636,2.3640)--(5.9393,2.3320)--(6.0151,2.2987)--(6.0909,2.2642)--(6.1666,2.2283)--(6.2424,2.1912)--(6.3181,2.1528)--(6.3939,2.1132)--(6.4696,2.0722)--(6.5454,2.0300)--(6.6212,1.9865)--(6.6969,1.9418)--(6.7727,1.8957)--(6.8484,1.8484)--(6.9242,1.7998)--(7.0000,1.7500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
-\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
-\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
-\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
-\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
-\draw (5.4420,3.2118) node {$f(x)$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.27858) node {$x$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.34557) node {$x+\Delta x$};
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,2.6388) -- (6.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,0.0000) --  cycle;
+\draw [color=red,style=dashed] (5.0000,0.0000) -- (6.0000,0.0000);
+\draw [color=red,style=dashed] (6.0000,0.0000) -- (6.0000,2.6388);
+\draw [color=red,style=dashed] (6.0000,2.6388) -- (5.0000,2.6388);
+\draw [color=red,style=dashed] (5.0000,2.6388) -- (5.0000,0.0000);
+\draw []  (5.0000,2.6388) node [rotate=0] {$\bullet$};
+\draw (5.4419,3.2118) node {$f(x)$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.2785) node {$x$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.3455) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Bateau.pstricks b/auto/pictures_tex/Fig_Bateau.pstricks
index 117a93c45..0ad5f1e6d 100644
--- a/auto/pictures_tex/Fig_Bateau.pstricks
+++ b/auto/pictures_tex/Fig_Bateau.pstricks
@@ -97,29 +97,29 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-1.00,0) -- (5.00,0);
-\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
-\draw (0,2.9247) node {$A$};
-\draw [,->,>=latex] (2.0000,2.5000) -- (0,2.5000);
+\draw [] (-1.0000,0.0000) -- (5.0000,0.0000);
+\draw []  (0.0000,2.5000) node [rotate=0] {$\bullet$};
+\draw (0.0000,2.9247) node {$A$};
+\draw [,->,>=latex] (2.0000,2.5000) -- (0.0000,2.5000);
 \draw [,->,>=latex] (2.0000,2.5000) -- (4.0000,2.5000);
 \draw (2.0000,2.7257) node {$\unit{4}{\kilo\meter}$};
 \draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,4.5000);
-\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,0);
+\draw [,->,>=latex] (4.2000,2.2500) -- (4.2000,0.0000);
 \draw (4.6714,2.2500) node {$\unit{9}{\kilo\meter}$};
-\draw [,->,>=latex] (0,1.2500) -- (0,2.5000);
-\draw [,->,>=latex] (0,1.2500) -- (0,0);
-\draw (-0.47143,1.2500) node {$\unit{3}{\kilo\meter}$};
-\draw []  (1.4286,0) node [rotate=0] {$\bullet$};
-\draw (1.7352,0.33684) node {$I$};
-\draw [color=brown,style=dashed] (4.00,4.50) -- (4.00,-4.50);
-\draw [color=blue,style=dashed] (0,2.50) -- (4.00,-4.50);
+\draw [,->,>=latex] (0.0000,1.2500) -- (0.0000,2.5000);
+\draw [,->,>=latex] (0.0000,1.2500) -- (0.0000,0.0000);
+\draw (-0.4714,1.2500) node {$\unit{3}{\kilo\meter}$};
+\draw []  (1.4285,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.7352,0.3368) node {$I$};
+\draw [color=brown,style=dashed] (4.0000,4.5000) -- (4.0000,-4.5000);
+\draw [color=blue,style=dashed] (0.0000,2.5000) -- (4.0000,-4.5000);
 \draw []  (4.0000,4.5000) node [rotate=0] {$\bullet$};
 \draw (4.0000,4.9247) node {$B$};
 \draw []  (4.0000,-4.5000) node [rotate=0] {$\bullet$};
-\draw (4.0000,-4.9408) node {$B'$};
-\draw [,->,>=latex] (0.71429,-0.20000) -- (0,-0.20000);
-\draw [,->,>=latex] (0.71429,-0.20000) -- (1.4286,-0.20000);
-\draw (0.71429,-0.42572) node {$x\kilo\meter$};
+\draw (4.0000,-4.9407) node {$B'$};
+\draw [,->,>=latex] (0.7142,-0.2000) -- (0.0000,-0.2000);
+\draw [,->,>=latex] (0.7142,-0.2000) -- (1.4285,-0.2000);
+\draw (0.7142,-0.4257) node {$x\kilo\meter$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_BiaisOuPas.pstricks b/auto/pictures_tex/Fig_BiaisOuPas.pstricks
index 5a720b6dc..b2ab27c07 100644
--- a/auto/pictures_tex/Fig_BiaisOuPas.pstricks
+++ b/auto/pictures_tex/Fig_BiaisOuPas.pstricks
@@ -91,32 +91,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.0000,0) -- (8.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4685);
+\draw [,->,>=latex] (-8.0000,0.0000) -- (8.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4684);
 %DEFAULT
 
-\draw [color=blue] (-7.500,0)--(-7.349,0)--(-7.197,0)--(-7.045,0)--(-6.894,0)--(-6.742,0)--(-6.591,0)--(-6.439,0.001311)--(-6.288,0.001784)--(-6.136,0.002412)--(-5.985,0.003235)--(-5.833,0.004309)--(-5.682,0.005696)--(-5.530,0.007475)--(-5.379,0.009739)--(-5.227,0.01259)--(-5.076,0.01617)--(-4.924,0.02060)--(-4.773,0.02606)--(-4.621,0.03273)--(-4.470,0.04080)--(-4.318,0.05048)--(-4.167,0.06201)--(-4.015,0.07562)--(-3.864,0.09153)--(-3.712,0.1100)--(-3.561,0.1312)--(-3.409,0.1553)--(-3.258,0.1826)--(-3.106,0.2130)--(-2.955,0.2468)--(-2.803,0.2837)--(-2.652,0.3238)--(-2.500,0.3669)--(-2.348,0.4127)--(-2.197,0.4607)--(-2.045,0.5107)--(-1.894,0.5618)--(-1.742,0.6136)--(-1.591,0.6652)--(-1.439,0.7160)--(-1.288,0.7649)--(-1.136,0.8112)--(-0.9848,0.8540)--(-0.8333,0.8925)--(-0.6818,0.9259)--(-0.5303,0.9535)--(-0.3788,0.9747)--(-0.2273,0.9891)--(-0.07576,0.9964)--(0.07576,0.9964)--(0.2273,0.9891)--(0.3788,0.9747)--(0.5303,0.9535)--(0.6818,0.9259)--(0.8333,0.8925)--(0.9848,0.8540)--(1.136,0.8112)--(1.288,0.7649)--(1.439,0.7160)--(1.591,0.6652)--(1.742,0.6136)--(1.894,0.5618)--(2.045,0.5107)--(2.197,0.4607)--(2.348,0.4127)--(2.500,0.3669)--(2.652,0.3238)--(2.803,0.2837)--(2.955,0.2468)--(3.106,0.2130)--(3.258,0.1826)--(3.409,0.1553)--(3.561,0.1312)--(3.712,0.1100)--(3.864,0.09153)--(4.015,0.07562)--(4.167,0.06201)--(4.318,0.05048)--(4.470,0.04080)--(4.621,0.03273)--(4.773,0.02606)--(4.924,0.02060)--(5.076,0.01617)--(5.227,0.01259)--(5.379,0.009739)--(5.530,0.007475)--(5.682,0.005696)--(5.833,0.004309)--(5.985,0.003235)--(6.136,0.002412)--(6.288,0.001784)--(6.439,0.001311)--(6.591,0)--(6.742,0)--(6.894,0)--(7.045,0)--(7.197,0)--(7.349,0)--(7.500,0);
+\draw [color=blue] (-7.5000,0.0000)--(-7.3484,0.0000)--(-7.1969,0.0000)--(-7.0454,0.0000)--(-6.8939,0.0000)--(-6.7424,0.0000)--(-6.5909,0.0000)--(-6.4393,0.0013)--(-6.2878,0.0017)--(-6.1363,0.0024)--(-5.9848,0.0032)--(-5.8333,0.0043)--(-5.6818,0.0056)--(-5.5303,0.0074)--(-5.3787,0.0097)--(-5.2272,0.0125)--(-5.0757,0.0161)--(-4.9242,0.0206)--(-4.7727,0.0260)--(-4.6212,0.0327)--(-4.4696,0.0407)--(-4.3181,0.0504)--(-4.1666,0.0620)--(-4.0151,0.0756)--(-3.8636,0.0915)--(-3.7121,0.1099)--(-3.5606,0.1311)--(-3.4090,0.1553)--(-3.2575,0.1825)--(-3.1060,0.2130)--(-2.9545,0.2467)--(-2.8030,0.2837)--(-2.6515,0.3238)--(-2.5000,0.3669)--(-2.3484,0.4126)--(-2.1969,0.4607)--(-2.0454,0.5106)--(-1.8939,0.5618)--(-1.7424,0.6135)--(-1.5909,0.6652)--(-1.4393,0.7159)--(-1.2878,0.7648)--(-1.1363,0.8111)--(-0.9848,0.8539)--(-0.8333,0.8924)--(-0.6818,0.9258)--(-0.5303,0.9534)--(-0.3787,0.9747)--(-0.2272,0.9891)--(-0.0757,0.9964)--(0.0757,0.9964)--(0.2272,0.9891)--(0.3787,0.9747)--(0.5303,0.9534)--(0.6818,0.9258)--(0.8333,0.8924)--(0.9848,0.8539)--(1.1363,0.8111)--(1.2878,0.7648)--(1.4393,0.7159)--(1.5909,0.6652)--(1.7424,0.6135)--(1.8939,0.5618)--(2.0454,0.5106)--(2.1969,0.4607)--(2.3484,0.4126)--(2.5000,0.3669)--(2.6515,0.3238)--(2.8030,0.2837)--(2.9545,0.2467)--(3.1060,0.2130)--(3.2575,0.1825)--(3.4090,0.1553)--(3.5606,0.1311)--(3.7121,0.1099)--(3.8636,0.0915)--(4.0151,0.0756)--(4.1666,0.0620)--(4.3181,0.0504)--(4.4696,0.0407)--(4.6212,0.0327)--(4.7727,0.0260)--(4.9242,0.0206)--(5.0757,0.0161)--(5.2272,0.0125)--(5.3787,0.0097)--(5.5303,0.0074)--(5.6818,0.0056)--(5.8333,0.0043)--(5.9848,0.0032)--(6.1363,0.0024)--(6.2878,0.0017)--(6.4393,0.0013)--(6.5909,0.0000)--(6.7424,0.0000)--(6.8939,0.0000)--(7.0454,0.0000)--(7.1969,0.0000)--(7.3484,0.0000)--(7.5000,0.0000);
 
-\draw [color=red] (-7.500,0)--(-7.349,0)--(-7.197,0)--(-7.045,0)--(-6.894,0)--(-6.742,0)--(-6.591,0)--(-6.439,0)--(-6.288,0)--(-6.136,0)--(-5.985,0)--(-5.833,0)--(-5.682,0)--(-5.530,0)--(-5.379,0)--(-5.227,0)--(-5.076,0)--(-4.924,0)--(-4.773,0)--(-4.621,0)--(-4.470,0)--(-4.318,0)--(-4.167,0)--(-4.015,0)--(-3.864,0)--(-3.712,0)--(-3.561,0)--(-3.409,0)--(-3.258,0)--(-3.106,0)--(-2.955,0)--(-2.803,0)--(-2.652,0)--(-2.500,0)--(-2.348,0)--(-2.197,0)--(-2.045,0)--(-1.894,0.002101)--(-1.742,0.01038)--(-1.591,0.04270)--(-1.439,0.1462)--(-1.288,0.4164)--(-1.136,0.9870)--(-0.9848,1.947)--(-0.8333,3.197)--(-0.6818,4.369)--(-0.5303,4.969)--(-0.3788,4.702)--(-0.2273,3.703)--(-0.07576,2.427)--(0.07576,1.324)--(0.2273,0.6012)--(0.3788,0.2271)--(0.5303,0.07141)--(0.6818,0.01869)--(0.8333,0.004069)--(0.9848,0)--(1.136,0)--(1.288,0)--(1.439,0)--(1.591,0)--(1.742,0)--(1.894,0)--(2.045,0)--(2.197,0)--(2.348,0)--(2.500,0)--(2.652,0)--(2.803,0)--(2.955,0)--(3.106,0)--(3.258,0)--(3.409,0)--(3.561,0)--(3.712,0)--(3.864,0)--(4.015,0)--(4.167,0)--(4.318,0)--(4.470,0)--(4.621,0)--(4.773,0)--(4.924,0)--(5.076,0)--(5.227,0)--(5.379,0)--(5.530,0)--(5.682,0)--(5.833,0)--(5.985,0)--(6.136,0)--(6.288,0)--(6.439,0)--(6.591,0)--(6.742,0)--(6.894,0)--(7.045,0)--(7.197,0)--(7.349,0)--(7.500,0);
-\draw [color=cyan,->,>=latex] (0,-0.50000) -- (-1.2500,-0.50000);
-\draw [color=cyan,->,>=latex] (0,-0.50000) -- (1.2500,-0.50000);
-\draw (0,-0.92471) node {\( I\)};
-\draw (-7.5000,-0.32983) node {$ -3 $};
-\draw [] (-7.50,-0.100) -- (-7.50,0.100);
-\draw (-5.0000,-0.32983) node {$ -2 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-2.5000,-0.32983) node {$ -1 $};
-\draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (2.5000,-0.31492) node {$ 1 $};
-\draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (5.0000,-0.31492) node {$ 2 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (7.5000,-0.31492) node {$ 3 $};
-\draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (-0.29125,2.5000) node {$ 1 $};
-\draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.29125,5.0000) node {$ 2 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=red] (-7.5000,0.0000)--(-7.3484,0.0000)--(-7.1969,0.0000)--(-7.0454,0.0000)--(-6.8939,0.0000)--(-6.7424,0.0000)--(-6.5909,0.0000)--(-6.4393,0.0000)--(-6.2878,0.0000)--(-6.1363,0.0000)--(-5.9848,0.0000)--(-5.8333,0.0000)--(-5.6818,0.0000)--(-5.5303,0.0000)--(-5.3787,0.0000)--(-5.2272,0.0000)--(-5.0757,0.0000)--(-4.9242,0.0000)--(-4.7727,0.0000)--(-4.6212,0.0000)--(-4.4696,0.0000)--(-4.3181,0.0000)--(-4.1666,0.0000)--(-4.0151,0.0000)--(-3.8636,0.0000)--(-3.7121,0.0000)--(-3.5606,0.0000)--(-3.4090,0.0000)--(-3.2575,0.0000)--(-3.1060,0.0000)--(-2.9545,0.0000)--(-2.8030,0.0000)--(-2.6515,0.0000)--(-2.5000,0.0000)--(-2.3484,0.0000)--(-2.1969,0.0000)--(-2.0454,0.0000)--(-1.8939,0.0021)--(-1.7424,0.0103)--(-1.5909,0.0427)--(-1.4393,0.1461)--(-1.2878,0.4163)--(-1.1363,0.9870)--(-0.9848,1.9473)--(-0.8333,3.1974)--(-0.6818,4.3691)--(-0.5303,4.9684)--(-0.3787,4.7021)--(-0.2272,3.7034)--(-0.0757,2.4275)--(0.0757,1.3241)--(0.2272,0.6011)--(0.3787,0.2271)--(0.5303,0.0714)--(0.6818,0.0186)--(0.8333,0.0040)--(0.9848,0.0000)--(1.1363,0.0000)--(1.2878,0.0000)--(1.4393,0.0000)--(1.5909,0.0000)--(1.7424,0.0000)--(1.8939,0.0000)--(2.0454,0.0000)--(2.1969,0.0000)--(2.3484,0.0000)--(2.5000,0.0000)--(2.6515,0.0000)--(2.8030,0.0000)--(2.9545,0.0000)--(3.1060,0.0000)--(3.2575,0.0000)--(3.4090,0.0000)--(3.5606,0.0000)--(3.7121,0.0000)--(3.8636,0.0000)--(4.0151,0.0000)--(4.1666,0.0000)--(4.3181,0.0000)--(4.4696,0.0000)--(4.6212,0.0000)--(4.7727,0.0000)--(4.9242,0.0000)--(5.0757,0.0000)--(5.2272,0.0000)--(5.3787,0.0000)--(5.5303,0.0000)--(5.6818,0.0000)--(5.8333,0.0000)--(5.9848,0.0000)--(6.1363,0.0000)--(6.2878,0.0000)--(6.4393,0.0000)--(6.5909,0.0000)--(6.7424,0.0000)--(6.8939,0.0000)--(7.0454,0.0000)--(7.1969,0.0000)--(7.3484,0.0000)--(7.5000,0.0000);
+\draw [color=cyan,->,>=latex] (0.0000,-0.5000) -- (-1.2500,-0.5000);
+\draw [color=cyan,->,>=latex] (0.0000,-0.5000) -- (1.2500,-0.5000);
+\draw (0.0000,-0.9247) node {\( I\)};
+\draw (-7.5000,-0.3298) node {$ -3 $};
+\draw [] (-7.5000,-0.1000) -- (-7.5000,0.1000);
+\draw (-5.0000,-0.3298) node {$ -2 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-2.5000,-0.3298) node {$ -1 $};
+\draw [] (-2.5000,-0.1000) -- (-2.5000,0.1000);
+\draw (2.5000,-0.3149) node {$ 1 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (5.0000,-0.3149) node {$ 2 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (7.5000,-0.3149) node {$ 3 $};
+\draw [] (7.5000,-0.1000) -- (7.5000,0.1000);
+\draw (-0.2912,2.5000) node {$ 1 $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.2912,5.0000) node {$ 2 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_BoulePtLoin.pstricks b/auto/pictures_tex/Fig_BoulePtLoin.pstricks
index 7e8a51e8e..c0058deeb 100644
--- a/auto/pictures_tex/Fig_BoulePtLoin.pstricks
+++ b/auto/pictures_tex/Fig_BoulePtLoin.pstricks
@@ -77,14 +77,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [style=dashed] (1.41,1.41) -- (1.94,1.94);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.30860,0.29071) node {$a$};
+\draw [style=dashed] (1.4142,1.4142) -- (1.9445,1.9445);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3085,0.2907) node {$a$};
 \draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.0501,1.6428) node {$x$};
-\draw [,->,>=latex] (0,0) -- (1.4142,1.4142);
+\draw (1.0501,1.6427) node {$x$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (1.4142,1.4142);
 
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 \draw []  (1.9445,1.9445) node [rotate=0] {$\bullet$};
 \draw (1.9445,1.5198) node {$P$};
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CELooGVvzMc.pstricks b/auto/pictures_tex/Fig_CELooGVvzMc.pstricks
index 3ceb434c7..1f9995268 100644
--- a/auto/pictures_tex/Fig_CELooGVvzMc.pstricks
+++ b/auto/pictures_tex/Fig_CELooGVvzMc.pstricks
@@ -80,34 +80,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.7900,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.7900);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.7900,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.7900);
 %DEFAULT
 
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+\draw [color=blue] (0.0000,0.0000)--(0.0232,0.0000)--(0.0464,0.0021)--(0.0696,0.0048)--(0.0929,0.0086)--(0.1161,0.0134)--(0.1393,0.0194)--(0.1626,0.0264)--(0.1858,0.0345)--(0.2090,0.0437)--(0.2323,0.0539)--(0.2555,0.0653)--(0.2787,0.0777)--(0.3020,0.0912)--(0.3252,0.1057)--(0.3484,0.1214)--(0.3717,0.1381)--(0.3949,0.1559)--(0.4181,0.1748)--(0.4414,0.1948)--(0.4646,0.2158)--(0.4878,0.2380)--(0.5111,0.2612)--(0.5343,0.2855)--(0.5575,0.3108)--(0.5808,0.3373)--(0.6040,0.3648)--(0.6272,0.3934)--(0.6505,0.4231)--(0.6737,0.4539)--(0.6969,0.4857)--(0.7202,0.5186)--(0.7434,0.5526)--(0.7666,0.5877)--(0.7898,0.6239)--(0.8131,0.6611)--(0.8363,0.6995)--(0.8595,0.7389)--(0.8828,0.7793)--(0.9060,0.8209)--(0.9292,0.8635)--(0.9525,0.9073)--(0.9757,0.9521)--(0.9989,0.9979)--(1.0222,1.0449)--(1.0454,1.0929)--(1.0686,1.1420)--(1.0919,1.1922)--(1.1151,1.2435)--(1.1383,1.2959)--(1.1616,1.3493)--(1.1848,1.4038)--(1.2080,1.4594)--(1.2313,1.5161)--(1.2545,1.5738)--(1.2777,1.6327)--(1.3010,1.6926)--(1.3242,1.7536)--(1.3474,1.8156)--(1.3707,1.8788)--(1.3939,1.9430)--(1.4171,2.0083)--(1.4404,2.0747)--(1.4636,2.1422)--(1.4868,2.2107)--(1.5101,2.2804)--(1.5333,2.3511)--(1.5565,2.4228)--(1.5797,2.4957)--(1.6030,2.5697)--(1.6262,2.6447)--(1.6494,2.7208)--(1.6727,2.7980)--(1.6959,2.8762)--(1.7191,2.9556)--(1.7424,3.0360)--(1.7656,3.1175)--(1.7888,3.2001)--(1.8121,3.2837)--(1.8353,3.3685)--(1.8585,3.4543)--(1.8818,3.5412)--(1.9050,3.6292)--(1.9282,3.7182)--(1.9515,3.8084)--(1.9747,3.8996)--(1.9979,3.9919)--(2.0212,4.0852)--(2.0444,4.1797)--(2.0676,4.2752)--(2.0909,4.3719)--(2.1141,4.4695)--(2.1373,4.5683)--(2.1606,4.6682)--(2.1838,4.7691)--(2.2070,4.8711)--(2.2303,4.9742)--(2.2535,5.0784)--(2.2767,5.1836)--(2.3000,5.2900);
 
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-\draw [style=dashed] (0,0) -- (3.79,3.79);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0534,0.2311)--(0.1068,0.3269)--(0.1603,0.4003)--(0.2137,0.4623)--(0.2671,0.5168)--(0.3206,0.5662)--(0.3740,0.6115)--(0.4274,0.6538)--(0.4809,0.6934)--(0.5343,0.7309)--(0.5877,0.7666)--(0.6412,0.8007)--(0.6946,0.8334)--(0.7480,0.8649)--(0.8015,0.8952)--(0.8549,0.9246)--(0.9083,0.9530)--(0.9618,0.9807)--(1.0152,1.0075)--(1.0686,1.0337)--(1.1221,1.0593)--(1.1755,1.0842)--(1.2289,1.1085)--(1.2824,1.1324)--(1.3358,1.1557)--(1.3892,1.1786)--(1.4427,1.2011)--(1.4961,1.2231)--(1.5495,1.2448)--(1.6030,1.2661)--(1.6564,1.2870)--(1.7098,1.3076)--(1.7633,1.3279)--(1.8167,1.3478)--(1.8702,1.3675)--(1.9236,1.3869)--(1.9770,1.4060)--(2.0305,1.4249)--(2.0839,1.4435)--(2.1373,1.4619)--(2.1908,1.4801)--(2.2442,1.4980)--(2.2976,1.5158)--(2.3511,1.5333)--(2.4045,1.5506)--(2.4579,1.5677)--(2.5114,1.5847)--(2.5648,1.6015)--(2.6182,1.6181)--(2.6717,1.6345)--(2.7251,1.6508)--(2.7785,1.6669)--(2.8320,1.6828)--(2.8854,1.6986)--(2.9388,1.7143)--(2.9923,1.7298)--(3.0457,1.7452)--(3.0991,1.7604)--(3.1526,1.7755)--(3.2060,1.7905)--(3.2594,1.8054)--(3.3129,1.8201)--(3.3663,1.8347)--(3.4197,1.8492)--(3.4732,1.8636)--(3.5266,1.8779)--(3.5801,1.8921)--(3.6335,1.9061)--(3.6869,1.9201)--(3.7404,1.9340)--(3.7938,1.9477)--(3.8472,1.9614)--(3.9007,1.9750)--(3.9541,1.9885)--(4.0075,2.0018)--(4.0610,2.0151)--(4.1144,2.0284)--(4.1678,2.0415)--(4.2213,2.0545)--(4.2747,2.0675)--(4.3281,2.0804)--(4.3816,2.0932)--(4.4350,2.1059)--(4.4884,2.1186)--(4.5419,2.1311)--(4.5953,2.1436)--(4.6487,2.1561)--(4.7022,2.1684)--(4.7556,2.1807)--(4.8090,2.1929)--(4.8625,2.2051)--(4.9159,2.2171)--(4.9693,2.2292)--(5.0228,2.2411)--(5.0762,2.2530)--(5.1296,2.2648)--(5.1831,2.2766)--(5.2365,2.2883)--(5.2900,2.3000);
+\draw [style=dashed] (0.0000,0.0000) -- (3.7950,3.7950);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks b/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
index 0eb3ad650..31f37cf34 100644
--- a/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
+++ b/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
@@ -64,21 +64,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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-\draw [] (0,0) -- (1.73,1.00);
-\draw [] (0,0) -- (-1.73,1.00);
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+\draw [] (0.0000,0.0000) -- (1.7320,1.0000);
+\draw [] (0.0000,0.0000) -- (-1.7320,1.0000);
 \draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (-1.7320,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3949,0.31186) node {\( \pi/6\)};
+\draw (1.3948,0.3118) node {\( \pi/6\)};
 
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-\draw (-1.3949,0.31186) node {\( \pi/6\)};
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CMMAooQegASg.pstricks b/auto/pictures_tex/Fig_CMMAooQegASg.pstricks
index 6b8a9b57d..ebf8af8aa 100644
--- a/auto/pictures_tex/Fig_CMMAooQegASg.pstricks
+++ b/auto/pictures_tex/Fig_CMMAooQegASg.pstricks
@@ -63,13 +63,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4998) -- (0.0000,2.4998);
 %DEFAULT
 
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks b/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks
index 90633a3ea..36f0bf98c 100644
--- a/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks
+++ b/auto/pictures_tex/Fig_CQIXooBEDnfK.pstricks
@@ -95,47 +95,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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+\draw [color=brown] (-3.0000,3.0000) -- (3.0000,-3.0000);
+\draw [color=brown] (-3.0000,-3.0000) -- (3.0000,3.0000);
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.7867,0.1954) node {$P$};
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.7850,0.2309) node {$Q$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CSCii.pstricks b/auto/pictures_tex/Fig_CSCii.pstricks
index fe4d62089..2997d784d 100644
--- a/auto/pictures_tex/Fig_CSCii.pstricks
+++ b/auto/pictures_tex/Fig_CSCii.pstricks
@@ -71,22 +71,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-1.2698) -- (0,1.5000);
+\draw [,->,>=latex] (-1.2697,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2697) -- (0.0000,1.5000);
 %DEFAULT
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+\draw [color=lightgray] (1.0000,1.0000) -- (1.0000,0.0000);
+\draw [color=lightgray] (1.0000,1.0000) -- (0.0000,1.0000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CSCiii.pstricks b/auto/pictures_tex/Fig_CSCiii.pstricks
index ebdb1078f..a027cc720 100644
--- a/auto/pictures_tex/Fig_CSCiii.pstricks
+++ b/auto/pictures_tex/Fig_CSCiii.pstricks
@@ -41,16 +41,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.6734);
+\draw [,->,>=latex] (-1.1515,0.0000) -- (3.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.6734);
 %DEFAULT
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+\draw (1.5000,-0.4207) node {$ \frac{1}{2} $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.3108,1.5000) node {$ \frac{1}{2} $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -92,19 +92,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-0.50000) -- (0,4.3345);
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+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.3344);
 %DEFAULT
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+\draw (-1.5000,-0.4207) node {$ -\frac{1}{20} $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (1.5000,-0.4207) node {$ \frac{1}{20} $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (-0.3816,3.0000) node {$ \frac{1}{10} $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -146,27 +146,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
-\draw [,->,>=latex] (0,-2.6734) -- (0,2.6734);
+\draw [,->,>=latex] (-1.1515,0.0000) -- (3.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6734) -- (0.0000,2.6734);
 %DEFAULT
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+\draw (1.5000,-0.4207) node {$ \frac{1}{2} $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4527,-1.5000) node {$ -\frac{1}{2} $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.3108,1.5000) node {$ \frac{1}{2} $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CSCiv.pstricks b/auto/pictures_tex/Fig_CSCiv.pstricks
index 42c742b85..14477e968 100644
--- a/auto/pictures_tex/Fig_CSCiv.pstricks
+++ b/auto/pictures_tex/Fig_CSCiv.pstricks
@@ -91,33 +91,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4935,0) -- (1.8801,0);
-\draw [,->,>=latex] (0,-5.4875) -- (0,2.0459);
+\draw [,->,>=latex] (-2.4935,0.0000) -- (1.8801,0.0000);
+\draw [,->,>=latex] (0.0000,-5.4874) -- (0.0000,2.0458);
 %DEFAULT
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+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CSCv.pstricks b/auto/pictures_tex/Fig_CSCv.pstricks
index eafafce59..1065eab20 100644
--- a/auto/pictures_tex/Fig_CSCv.pstricks
+++ b/auto/pictures_tex/Fig_CSCv.pstricks
@@ -65,29 +65,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
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+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -125,17 +125,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CSCvi.pstricks b/auto/pictures_tex/Fig_CSCvi.pstricks
index 8a0a3f7b3..8dc9d59f3 100644
--- a/auto/pictures_tex/Fig_CSCvi.pstricks
+++ b/auto/pictures_tex/Fig_CSCvi.pstricks
@@ -61,30 +61,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (2.0708,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4332);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (2.0707,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4331);
 %DEFAULT
 
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-\draw [color=lightgray,style=dashed] (-1.57,0) -- (-1.57,4.93);
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-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
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-\draw (-0.29125,2.0000) node {$ 2 $};
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-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
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+\draw [color=lightgray,style=dashed] (-1.5707,0.0000) -- (-1.5707,4.9331);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -142,24 +142,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.4211,0);
-\draw [,->,>=latex] (0,-5.0437) -- (0,0.80024);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.4210,0.0000);
+\draw [,->,>=latex] (0.0000,-5.0437) -- (0.0000,0.8002);
 %DEFAULT
-\draw [color=blue] (1.921,-4.544)--(1.910,-4.190)--(1.898,-3.876)--(1.886,-3.595)--(1.872,-3.341)--(1.858,-3.111)--(1.844,-2.901)--(1.829,-2.709)--(1.813,-2.531)--(1.797,-2.367)--(1.779,-2.214)--(1.762,-2.072)--(1.744,-1.939)--(1.725,-1.815)--(1.705,-1.698)--(1.686,-1.587)--(1.665,-1.483)--(1.644,-1.385)--(1.623,-1.292)--(1.601,-1.204)--(1.579,-1.120)--(1.556,-1.040)--(1.533,-0.9641)--(1.509,-0.8919)--(1.485,-0.8231)--(1.460,-0.7575)--(1.436,-0.6949)--(1.411,-0.6352)--(1.385,-0.5781)--(1.359,-0.5237)--(1.333,-0.4717)--(1.307,-0.4221)--(1.281,-0.3747)--(1.254,-0.3295)--(1.227,-0.2864)--(1.200,-0.2452)--(1.173,-0.2060)--(1.146,-0.1686)--(1.118,-0.1331)--(1.091,-0.09928)--(1.063,-0.06715)--(1.035,-0.03666)--(1.008,-0.007756)--(0.9800,0.01960)--(0.9523,0.04545)--(0.9247,0.06984)--(0.8971,0.09280)--(0.8696,0.1144)--(0.8422,0.1346)--(0.8149,0.1535)--(0.7878,0.1711)--(0.7608,0.1874)--(0.7340,0.2025)--(0.7074,0.2164)--(0.6811,0.2292)--(0.6549,0.2408)--(0.6291,0.2513)--(0.6035,0.2607)--(0.5782,0.2690)--(0.5533,0.2763)--(0.5287,0.2825)--(0.5044,0.2878)--(0.4806,0.2921)--(0.4571,0.2955)--(0.4341,0.2980)--(0.4115,0.2995)--(0.3893,0.3002)--(0.3676,0.3001)--(0.3464,0.2991)--(0.3257,0.2974)--(0.3055,0.2949)--(0.2859,0.2916)--(0.2668,0.2877)--(0.2482,0.2830)--(0.2302,0.2776)--(0.2129,0.2717)--(0.1961,0.2650)--(0.1799,0.2578)--(0.1644,0.2501)--(0.1495,0.2417)--(0.1353,0.2329)--(0.1217,0.2236)--(0.1088,0.2137)--(0.09657,0.2035)--(0.08504,0.1928)--(0.07422,0.1817)--(0.06411,0.1703)--(0.05471,0.1585)--(0.04604,0.1464)--(0.03810,0.1340)--(0.03090,0.1214)--(0.02444,0.1085)--(0.01873,0.09541)--(0.01377,0.08212)--(0.009571,0.06868)--(0.006129,0.05510)--(0.003449,0.04142)--(0.001533,0.02766)--(0,0.01384)--(0,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-5.0000) node {$ -5 $};
-\draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
+\draw [color=blue] (1.9210,-4.5437)--(1.9099,-4.1899)--(1.8980,-3.8759)--(1.8855,-3.5947)--(1.8723,-3.3412)--(1.8584,-3.1112)--(1.8439,-2.9013)--(1.8287,-2.7087)--(1.8129,-2.5312)--(1.7965,-2.3669)--(1.7795,-2.2144)--(1.7618,-2.0722)--(1.7436,-1.9393)--(1.7248,-1.8147)--(1.7054,-1.6976)--(1.6855,-1.5873)--(1.6651,-1.4833)--(1.6442,-1.3849)--(1.6228,-1.2918)--(1.6009,-1.2035)--(1.5785,-1.1196)--(1.5557,-1.0399)--(1.5325,-0.9641)--(1.5088,-0.8919)--(1.4848,-0.8230)--(1.4604,-0.7574)--(1.4356,-0.6948)--(1.4105,-0.6351)--(1.3851,-0.5781)--(1.3594,-0.5237)--(1.3334,-0.4717)--(1.3072,-0.4220)--(1.2807,-0.3747)--(1.2541,-0.3295)--(1.2272,-0.2863)--(1.2001,-0.2452)--(1.1729,-0.2060)--(1.1456,-0.1686)--(1.1181,-0.1330)--(1.0906,-0.0992)--(1.0630,-0.0671)--(1.0353,-0.0366)--(1.0076,-0.0077)--(0.9800,0.0195)--(0.9523,0.0454)--(0.9246,0.0698)--(0.8971,0.0927)--(0.8696,0.1143)--(0.8422,0.1345)--(0.8149,0.1534)--(0.7877,0.1710)--(0.7608,0.1874)--(0.7340,0.2025)--(0.7074,0.2164)--(0.6810,0.2291)--(0.6549,0.2407)--(0.6290,0.2512)--(0.6035,0.2606)--(0.5782,0.2689)--(0.5532,0.2762)--(0.5286,0.2825)--(0.5044,0.2878)--(0.4805,0.2921)--(0.4571,0.2954)--(0.4340,0.2979)--(0.4114,0.2995)--(0.3893,0.3002)--(0.3676,0.3001)--(0.3464,0.2991)--(0.3257,0.2974)--(0.3055,0.2948)--(0.2858,0.2916)--(0.2667,0.2876)--(0.2482,0.2829)--(0.2302,0.2776)--(0.2128,0.2716)--(0.1960,0.2650)--(0.1799,0.2578)--(0.1643,0.2500)--(0.1494,0.2417)--(0.1352,0.2328)--(0.1216,0.2235)--(0.1087,0.2137)--(0.0965,0.2034)--(0.0850,0.1928)--(0.0742,0.1817)--(0.0641,0.1703)--(0.0547,0.1585)--(0.0460,0.1464)--(0.0381,0.1340)--(0.0308,0.1213)--(0.0244,0.1085)--(0.0187,0.0954)--(0.0137,0.0821)--(0.0095,0.0686)--(0.0061,0.0551)--(0.0034,0.0414)--(0.0015,0.0276)--(0.0000,0.0138)--(0.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-5.0000) node {$ -5 $};
+\draw [] (-0.1000,-5.0000) -- (0.1000,-5.0000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CURGooXvruWV.pstricks b/auto/pictures_tex/Fig_CURGooXvruWV.pstricks
index 8d44d1bc7..4389d953c 100644
--- a/auto/pictures_tex/Fig_CURGooXvruWV.pstricks
+++ b/auto/pictures_tex/Fig_CURGooXvruWV.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,23 +81,23 @@
 % setting the default values
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+\draw [color=green] (0.0000,0.0000)--(0.0202,0.0202)--(0.0404,0.0404)--(0.0606,0.0606)--(0.0808,0.0808)--(0.1010,0.1010)--(0.1212,0.1212)--(0.1414,0.1414)--(0.1616,0.1616)--(0.1818,0.1818)--(0.2020,0.2020)--(0.2222,0.2222)--(0.2424,0.2424)--(0.2626,0.2626)--(0.2828,0.2828)--(0.3030,0.3030)--(0.3232,0.3232)--(0.3434,0.3434)--(0.3636,0.3636)--(0.3838,0.3838)--(0.4040,0.4040)--(0.4242,0.4242)--(0.4444,0.4444)--(0.4646,0.4646)--(0.4848,0.4848)--(0.5050,0.5050)--(0.5252,0.5252)--(0.5454,0.5454)--(0.5656,0.5656)--(0.5858,0.5858)--(0.6060,0.6060)--(0.6262,0.6262)--(0.6464,0.6464)--(0.6666,0.6666)--(0.6868,0.6868)--(0.7070,0.7070)--(0.7272,0.7272)--(0.7474,0.7474)--(0.7676,0.7676)--(0.7878,0.7878)--(0.8080,0.8080)--(0.8282,0.8282)--(0.8484,0.8484)--(0.8686,0.8686)--(0.8888,0.8888)--(0.9090,0.9090)--(0.9292,0.9292)--(0.9494,0.9494)--(0.9696,0.9696)--(0.9898,0.9898)--(1.0101,1.0101)--(1.0303,1.0303)--(1.0505,1.0505)--(1.0707,1.0707)--(1.0909,1.0909)--(1.1111,1.1111)--(1.1313,1.1313)--(1.1515,1.1515)--(1.1717,1.1717)--(1.1919,1.1919)--(1.2121,1.2121)--(1.2323,1.2323)--(1.2525,1.2525)--(1.2727,1.2727)--(1.2929,1.2929)--(1.3131,1.3131)--(1.3333,1.3333)--(1.3535,1.3535)--(1.3737,1.3737)--(1.3939,1.3939)--(1.4141,1.4141)--(1.4343,1.4343)--(1.4545,1.4545)--(1.4747,1.4747)--(1.4949,1.4949)--(1.5151,1.5151)--(1.5353,1.5353)--(1.5555,1.5555)--(1.5757,1.5757)--(1.5959,1.5959)--(1.6161,1.6161)--(1.6363,1.6363)--(1.6565,1.6565)--(1.6767,1.6767)--(1.6969,1.6969)--(1.7171,1.7171)--(1.7373,1.7373)--(1.7575,1.7575)--(1.7777,1.7777)--(1.7979,1.7979)--(1.8181,1.8181)--(1.8383,1.8383)--(1.8585,1.8585)--(1.8787,1.8787)--(1.8989,1.8989)--(1.9191,1.9191)--(1.9393,1.9393)--(1.9595,1.9595)--(1.9797,1.9797)--(2.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
index b289ededc..53a4f6010 100644
--- a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
+++ b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
@@ -68,16 +68,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (1.0000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,0.50000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (1.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,0.5000);
 %DEFAULT
-\draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0);
-\draw [] (0,0) -- (-3.00,-1.00);
+\draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (-3.0000,-1.0000);
 \draw []  (-3.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw (-3.8283,-1.2049) node {\( -x+\lambda i\)};
-\draw (0.53123,-0.70911) node {\( \arg(z)\)};
+\draw (-3.8283,-1.2048) node {\( -x+\lambda i\)};
+\draw (0.5312,-0.7091) node {\( \arg(z)\)};
 
-\draw [] (-0.474,-0.158)--(-0.470,-0.172)--(-0.465,-0.185)--(-0.459,-0.198)--(-0.453,-0.211)--(-0.447,-0.224)--(-0.441,-0.236)--(-0.434,-0.249)--(-0.426,-0.261)--(-0.419,-0.273)--(-0.411,-0.285)--(-0.403,-0.297)--(-0.394,-0.308)--(-0.385,-0.319)--(-0.376,-0.330)--(-0.366,-0.340)--(-0.356,-0.351)--(-0.346,-0.361)--(-0.336,-0.370)--(-0.325,-0.380)--(-0.314,-0.389)--(-0.303,-0.398)--(-0.292,-0.406)--(-0.280,-0.414)--(-0.268,-0.422)--(-0.256,-0.430)--(-0.243,-0.437)--(-0.231,-0.444)--(-0.218,-0.450)--(-0.205,-0.456)--(-0.192,-0.462)--(-0.179,-0.467)--(-0.166,-0.472)--(-0.152,-0.476)--(-0.138,-0.480)--(-0.125,-0.484)--(-0.111,-0.488)--(-0.0970,-0.491)--(-0.0830,-0.493)--(-0.0689,-0.495)--(-0.0547,-0.497)--(-0.0406,-0.498)--(-0.0264,-0.499)--(-0.0121,-0.500)--(0.00211,-0.500)--(0.0163,-0.500)--(0.0306,-0.499)--(0.0448,-0.498)--(0.0589,-0.497)--(0.0731,-0.495)--(0.0871,-0.492)--(0.101,-0.490)--(0.115,-0.487)--(0.129,-0.483)--(0.143,-0.479)--(0.156,-0.475)--(0.170,-0.470)--(0.183,-0.465)--(0.196,-0.460)--(0.209,-0.454)--(0.222,-0.448)--(0.235,-0.442)--(0.247,-0.435)--(0.259,-0.427)--(0.271,-0.420)--(0.283,-0.412)--(0.295,-0.404)--(0.306,-0.395)--(0.317,-0.386)--(0.328,-0.377)--(0.339,-0.368)--(0.349,-0.358)--(0.359,-0.348)--(0.369,-0.337)--(0.379,-0.327)--(0.388,-0.316)--(0.396,-0.305)--(0.405,-0.293)--(0.413,-0.282)--(0.421,-0.270)--(0.429,-0.258)--(0.436,-0.245)--(0.443,-0.233)--(0.449,-0.220)--(0.455,-0.207)--(0.461,-0.194)--(0.466,-0.181)--(0.471,-0.168)--(0.476,-0.154)--(0.480,-0.141)--(0.484,-0.127)--(0.487,-0.113)--(0.490,-0.0990)--(0.493,-0.0850)--(0.495,-0.0710)--(0.497,-0.0568)--(0.498,-0.0427)--(0.499,-0.0285)--(0.500,-0.0142)--(0.500,0);
+\draw [] (-0.4743,-0.1581)--(-0.4696,-0.1715)--(-0.4645,-0.1848)--(-0.4591,-0.1980)--(-0.4532,-0.2110)--(-0.4470,-0.2238)--(-0.4405,-0.2364)--(-0.4336,-0.2489)--(-0.4263,-0.2611)--(-0.4187,-0.2732)--(-0.4108,-0.2850)--(-0.4025,-0.2966)--(-0.3939,-0.3079)--(-0.3849,-0.3190)--(-0.3757,-0.3298)--(-0.3661,-0.3404)--(-0.3563,-0.3507)--(-0.3462,-0.3607)--(-0.3357,-0.3704)--(-0.3251,-0.3798)--(-0.3141,-0.3889)--(-0.3029,-0.3977)--(-0.2914,-0.4062)--(-0.2798,-0.4143)--(-0.2678,-0.4221)--(-0.2557,-0.4296)--(-0.2434,-0.4367)--(-0.2308,-0.4434)--(-0.2181,-0.4498)--(-0.2052,-0.4559)--(-0.1921,-0.4615)--(-0.1789,-0.4668)--(-0.1656,-0.4717)--(-0.1520,-0.4763)--(-0.1384,-0.4804)--(-0.1247,-0.4841)--(-0.1108,-0.4875)--(-0.0969,-0.4905)--(-0.0829,-0.4930)--(-0.0688,-0.4952)--(-0.0547,-0.4969)--(-0.0405,-0.4983)--(-0.0263,-0.4993)--(-0.0121,-0.4998)--(0.0021,-0.4999)--(0.0163,-0.4997)--(0.0305,-0.4990)--(0.0447,-0.4979)--(0.0589,-0.4965)--(0.0730,-0.4946)--(0.0871,-0.4923)--(0.1010,-0.4896)--(0.1150,-0.4865)--(0.1288,-0.4831)--(0.1425,-0.4792)--(0.1561,-0.4750)--(0.1695,-0.4703)--(0.1829,-0.4653)--(0.1960,-0.4599)--(0.2091,-0.4541)--(0.2219,-0.4480)--(0.2346,-0.4415)--(0.2471,-0.4346)--(0.2593,-0.4274)--(0.2714,-0.4198)--(0.2832,-0.4119)--(0.2949,-0.4037)--(0.3062,-0.3952)--(0.3174,-0.3863)--(0.3282,-0.3771)--(0.3389,-0.3676)--(0.3492,-0.3578)--(0.3592,-0.3477)--(0.3690,-0.3373)--(0.3785,-0.3267)--(0.3876,-0.3157)--(0.3964,-0.3046)--(0.4050,-0.2932)--(0.4131,-0.2815)--(0.4210,-0.2696)--(0.4285,-0.2575)--(0.4357,-0.2452)--(0.4425,-0.2327)--(0.4489,-0.2200)--(0.4550,-0.2071)--(0.4607,-0.1941)--(0.4661,-0.1809)--(0.4710,-0.1675)--(0.4756,-0.1541)--(0.4798,-0.1404)--(0.4836,-0.1267)--(0.4870,-0.1129)--(0.4900,-0.0990)--(0.4927,-0.0850)--(0.4949,-0.0709)--(0.4967,-0.0568)--(0.4981,-0.0426)--(0.4991,-0.0284)--(0.4997,-0.0142)--(0.5000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_Cardioid.pstricks b/auto/pictures_tex/Fig_Cardioid.pstricks
index aa20450b4..dc102ab60 100644
--- a/auto/pictures_tex/Fig_Cardioid.pstricks
+++ b/auto/pictures_tex/Fig_Cardioid.pstricks
@@ -65,7 +65,7 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red] (3.000,0)--(2.991,0.1901)--(2.964,0.3783)--(2.919,0.5626)--(2.857,0.7414)--(2.779,0.9127)--(2.685,1.075)--(2.577,1.227)--(2.456,1.367)--(2.323,1.493)--(2.181,1.606)--(2.029,1.703)--(1.871,1.784)--(1.708,1.850)--(1.542,1.898)--(1.375,1.931)--(1.208,1.947)--(1.043,1.947)--(0.8820,1.931)--(0.7264,1.901)--(0.5776,1.858)--(0.4370,1.801)--(0.3057,1.734)--(0.1847,1.656)--(0.07477,1.570)--(-0.02342,1.476)--(-0.1095,1.377)--(-0.1831,1.273)--(-0.2443,1.168)--(-0.2932,1.061)--(-0.3301,0.9539)--(-0.3556,0.8490)--(-0.3703,0.7472)--(-0.3750,0.6495)--(-0.3706,0.5571)--(-0.3583,0.4708)--(-0.3390,0.3913)--(-0.3141,0.3191)--(-0.2848,0.2548)--(-0.2523,0.1984)--(-0.2178,0.1500)--(-0.1828,0.1094)--(-0.1482,0.07641)--(-0.1153,0.05045)--(-0.08501,0.03094)--(-0.05830,0.01712)--(-0.03595,0.008120)--(-0.01860,0.002977)--(-0.006761,0)--(0,0)--(0,0)--(-0.006761,0)--(-0.01860,-0.002977)--(-0.03595,-0.008120)--(-0.05830,-0.01712)--(-0.08501,-0.03094)--(-0.1153,-0.05045)--(-0.1482,-0.07641)--(-0.1828,-0.1094)--(-0.2178,-0.1500)--(-0.2523,-0.1984)--(-0.2848,-0.2548)--(-0.3141,-0.3191)--(-0.3390,-0.3913)--(-0.3583,-0.4708)--(-0.3706,-0.5571)--(-0.3750,-0.6495)--(-0.3703,-0.7472)--(-0.3556,-0.8490)--(-0.3301,-0.9539)--(-0.2932,-1.061)--(-0.2443,-1.168)--(-0.1831,-1.273)--(-0.1095,-1.377)--(-0.02342,-1.476)--(0.07477,-1.570)--(0.1847,-1.656)--(0.3057,-1.734)--(0.4370,-1.801)--(0.5776,-1.858)--(0.7264,-1.901)--(0.8820,-1.931)--(1.043,-1.947)--(1.208,-1.947)--(1.375,-1.931)--(1.542,-1.898)--(1.708,-1.850)--(1.871,-1.784)--(2.029,-1.703)--(2.181,-1.606)--(2.323,-1.493)--(2.456,-1.367)--(2.577,-1.227)--(2.685,-1.075)--(2.779,-0.9127)--(2.857,-0.7414)--(2.919,-0.5626)--(2.964,-0.3783)--(2.991,-0.1901)--(3.000,0);
+\draw [color=red] (3.0000,0.0000)--(2.9909,0.1900)--(2.9638,0.3782)--(2.9191,0.5626)--(2.8573,0.7413)--(2.7790,0.9127)--(2.6853,1.0750)--(2.5773,1.2268)--(2.4561,1.3665)--(2.3234,1.4931)--(2.1805,1.6055)--(2.0293,1.7027)--(1.8712,1.7842)--(1.7083,1.8495)--(1.5422,1.8983)--(1.3747,1.9306)--(1.2077,1.9465)--(1.0429,1.9466)--(0.8819,1.9312)--(0.7263,1.9012)--(0.5776,1.8576)--(0.4370,1.8013)--(0.3057,1.7337)--(0.1846,1.6559)--(0.0747,1.5695)--(-0.0234,1.4760)--(-0.1094,1.3767)--(-0.1830,1.2734)--(-0.2442,1.1675)--(-0.2931,1.0605)--(-0.3301,0.9538)--(-0.3556,0.8490)--(-0.3703,0.7471)--(-0.3750,0.6495)--(-0.3706,0.5570)--(-0.3582,0.4707)--(-0.3390,0.3912)--(-0.3141,0.3191)--(-0.2847,0.2547)--(-0.2522,0.1983)--(-0.2178,0.1499)--(-0.1827,0.1094)--(-0.1482,0.0764)--(-0.1152,0.0504)--(-0.0850,0.0309)--(-0.0582,0.0171)--(-0.0359,0.0081)--(-0.0186,0.0029)--(-0.0067,0.0000)--(0.0000,0.0000)--(0.0000,0.0000)--(-0.0067,0.0000)--(-0.0186,-0.0029)--(-0.0359,-0.0081)--(-0.0582,-0.0171)--(-0.0850,-0.0309)--(-0.1152,-0.0504)--(-0.1482,-0.0764)--(-0.1827,-0.1094)--(-0.2178,-0.1499)--(-0.2522,-0.1983)--(-0.2847,-0.2547)--(-0.3141,-0.3191)--(-0.3390,-0.3912)--(-0.3582,-0.4707)--(-0.3706,-0.5570)--(-0.3750,-0.6495)--(-0.3703,-0.7471)--(-0.3556,-0.8490)--(-0.3301,-0.9538)--(-0.2931,-1.0605)--(-0.2442,-1.1675)--(-0.1830,-1.2734)--(-0.1094,-1.3767)--(-0.0234,-1.4760)--(0.0747,-1.5695)--(0.1846,-1.6559)--(0.3057,-1.7337)--(0.4370,-1.8013)--(0.5776,-1.8576)--(0.7263,-1.9012)--(0.8819,-1.9312)--(1.0429,-1.9466)--(1.2077,-1.9465)--(1.3747,-1.9306)--(1.5422,-1.8983)--(1.7083,-1.8495)--(1.8712,-1.7842)--(2.0293,-1.7027)--(2.1805,-1.6055)--(2.3234,-1.4931)--(2.4561,-1.3665)--(2.5773,-1.2268)--(2.6853,-1.0750)--(2.7790,-0.9127)--(2.8573,-0.7413)--(2.9191,-0.5626)--(2.9638,-0.3782)--(2.9909,-0.1900)--(3.0000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_Cardioideexo.pstricks b/auto/pictures_tex/Fig_Cardioideexo.pstricks
index 1b2d525b7..87bc7a7ef 100644
--- a/auto/pictures_tex/Fig_Cardioideexo.pstricks
+++ b/auto/pictures_tex/Fig_Cardioideexo.pstricks
@@ -79,37 +79,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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-\draw [color=lightgray] (1.00,1.00) -- (1.00,0);
-\draw [color=lightgray] (1.00,0) -- (0,0);
-\draw [color=lightgray] (0,0) -- (0,1.00);
-\draw [color=lightgray,style=dashed] (0,0) -- (1.21,1.21);
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+\draw [color=lightgray] (0.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=lightgray] (1.0000,1.0000) -- (1.0000,0.0000);
+\draw [color=lightgray] (1.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=lightgray] (0.0000,0.0000) -- (0.0000,1.0000);
+\draw [color=lightgray,style=dashed] (0.0000,0.0000) -- (1.2071,1.2071);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.2071,1.2071) node [rotate=0] {$\bullet$};
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CbCartTui.pstricks b/auto/pictures_tex/Fig_CbCartTui.pstricks
index ad4c00516..6771f4c1f 100644
--- a/auto/pictures_tex/Fig_CbCartTui.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTui.pstricks
@@ -103,61 +103,61 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.1400,0) -- (4.1400,0);
-\draw [,->,>=latex] (0,-3.9720) -- (0,4.0280);
+\draw [,->,>=latex] (-4.1400,0.0000) -- (4.1400,0.0000);
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+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CbCartTuii.pstricks b/auto/pictures_tex/Fig_CbCartTuii.pstricks
index f3a5e95ac..912d80e0f 100644
--- a/auto/pictures_tex/Fig_CbCartTuii.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTuii.pstricks
@@ -67,12 +67,12 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.1489) -- (0,1.1489);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1488) -- (0.0000,1.1488);
 %DEFAULT
-\draw [color=blue] (2.00,0)--(1.99,0.126)--(1.97,0.247)--(1.93,0.358)--(1.87,0.456)--(1.81,0.535)--(1.72,0.595)--(1.63,0.633)--(1.53,0.649)--(1.42,0.644)--(1.30,0.619)--(1.17,0.578)--(1.05,0.523)--(0.921,0.459)--(0.795,0.389)--(0.673,0.318)--(0.556,0.249)--(0.446,0.186)--(0.345,0.130)--(0.255,0.0849)--(0.176,0.0500)--(0.111,0.0255)--(0.0603,0.0103)--(0.0246,0.00271)--(0.00453,0)--(0,0)--(0.0126,0)--(0.0405,-0.00571)--(0.0839,-0.0168)--(0.142,-0.0365)--(0.214,-0.0661)--(0.299,-0.106)--(0.394,-0.157)--(0.500,-0.217)--(0.614,-0.283)--(0.734,-0.354)--(0.858,-0.424)--(0.984,-0.492)--(1.11,-0.552)--(1.24,-0.600)--(1.36,-0.634)--(1.47,-0.649)--(1.58,-0.644)--(1.68,-0.617)--(1.77,-0.568)--(1.84,-0.498)--(1.90,-0.409)--(1.95,-0.304)--(1.98,-0.188)--(2.00,-0.0634)--(2.00,0.0634)--(1.98,0.188)--(1.95,0.304)--(1.90,0.409)--(1.84,0.498)--(1.77,0.568)--(1.68,0.617)--(1.58,0.644)--(1.47,0.649)--(1.36,0.634)--(1.24,0.600)--(1.11,0.552)--(0.984,0.492)--(0.858,0.424)--(0.734,0.354)--(0.614,0.283)--(0.500,0.217)--(0.394,0.157)--(0.299,0.106)--(0.214,0.0661)--(0.142,0.0365)--(0.0839,0.0168)--(0.0405,0.00571)--(0.0126,0)--(0,0)--(0.00453,0)--(0.0246,-0.00271)--(0.0603,-0.0103)--(0.111,-0.0255)--(0.176,-0.0500)--(0.255,-0.0849)--(0.345,-0.130)--(0.446,-0.186)--(0.556,-0.249)--(0.673,-0.318)--(0.795,-0.389)--(0.921,-0.459)--(1.05,-0.523)--(1.17,-0.578)--(1.30,-0.619)--(1.42,-0.644)--(1.53,-0.649)--(1.63,-0.633)--(1.72,-0.595)--(1.81,-0.535)--(1.87,-0.456)--(1.93,-0.358)--(1.97,-0.247)--(1.99,-0.126)--(2.00,0);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
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+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CbCartTuiii.pstricks b/auto/pictures_tex/Fig_CbCartTuiii.pstricks
index 8b557fe13..c83988743 100644
--- a/auto/pictures_tex/Fig_CbCartTuiii.pstricks
+++ b/auto/pictures_tex/Fig_CbCartTuiii.pstricks
@@ -71,18 +71,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4997,0) -- (2.4997,0);
-\draw [,->,>=latex] (0,-2.4977) -- (0,2.4977);
+\draw [,->,>=latex] (-2.4997,0.0000) -- (2.4997,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4977) -- (0.0000,2.4977);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.253,0.379)--(0.502,0.743)--(0.743,1.08)--(0.972,1.38)--(1.19,1.63)--(1.38,1.82)--(1.55,1.94)--(1.70,2.00)--(1.82,1.98)--(1.91,1.89)--(1.97,1.73)--(2.00,1.51)--(1.99,1.24)--(1.96,0.916)--(1.89,0.563)--(1.79,0.190)--(1.67,-0.190)--(1.51,-0.563)--(1.33,-0.916)--(1.13,-1.24)--(0.916,-1.51)--(0.684,-1.73)--(0.441,-1.89)--(0.190,-1.98)--(-0.0635,-2.00)--(-0.316,-1.94)--(-0.563,-1.82)--(-0.802,-1.63)--(-1.03,-1.38)--(-1.24,-1.08)--(-1.43,-0.743)--(-1.59,-0.379)--(-1.73,0)--(-1.84,0.379)--(-1.93,0.743)--(-1.98,1.08)--(-2.00,1.38)--(-1.99,1.63)--(-1.94,1.82)--(-1.87,1.94)--(-1.76,2.00)--(-1.63,1.98)--(-1.47,1.89)--(-1.29,1.73)--(-1.08,1.51)--(-0.860,1.24)--(-0.624,0.916)--(-0.379,0.563)--(-0.127,0.190)--(0.127,-0.190)--(0.379,-0.563)--(0.624,-0.916)--(0.860,-1.24)--(1.08,-1.51)--(1.29,-1.73)--(1.47,-1.89)--(1.63,-1.98)--(1.76,-2.00)--(1.87,-1.94)--(1.94,-1.82)--(1.99,-1.63)--(2.00,-1.38)--(1.98,-1.08)--(1.93,-0.743)--(1.84,-0.379)--(1.73,0)--(1.59,0.379)--(1.43,0.743)--(1.24,1.08)--(1.03,1.38)--(0.802,1.63)--(0.563,1.82)--(0.316,1.94)--(0.0635,2.00)--(-0.190,1.98)--(-0.441,1.89)--(-0.684,1.73)--(-0.916,1.51)--(-1.13,1.24)--(-1.33,0.916)--(-1.51,0.563)--(-1.67,0.190)--(-1.79,-0.190)--(-1.89,-0.563)--(-1.96,-0.916)--(-1.99,-1.24)--(-2.00,-1.51)--(-1.97,-1.73)--(-1.91,-1.89)--(-1.82,-1.98)--(-1.70,-2.00)--(-1.55,-1.94)--(-1.38,-1.82)--(-1.19,-1.63)--(-0.972,-1.38)--(-0.743,-1.08)--(-0.502,-0.743)--(-0.253,-0.379)--(0,0);
-\draw (-2.0000,-0.32983) node {$ -1 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -1 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.2531,0.3785)--(0.5022,0.7433)--(0.7433,1.0812)--(0.9723,1.3801)--(1.1858,1.6291)--(1.3801,1.8192)--(1.5522,1.9436)--(1.6994,1.9977)--(1.8192,1.9796)--(1.9098,1.8900)--(1.9696,1.7320)--(1.9977,1.5114)--(1.9937,1.2363)--(1.9576,0.9164)--(1.8900,0.5634)--(1.7919,0.1901)--(1.6651,-0.1901)--(1.5114,-0.5634)--(1.3335,-0.9164)--(1.1341,-1.2363)--(0.9164,-1.5114)--(0.6840,-1.7320)--(0.4406,-1.8900)--(0.1901,-1.9796)--(-0.0634,-1.9977)--(-0.3160,-1.9436)--(-0.5634,-1.8192)--(-0.8018,-1.6291)--(-1.0273,-1.3801)--(-1.2363,-1.0812)--(-1.4253,-0.7433)--(-1.5915,-0.3785)--(-1.7320,0.0000)--(-1.8447,0.3785)--(-1.9276,0.7433)--(-1.9796,1.0812)--(-1.9997,1.3801)--(-1.9876,1.6291)--(-1.9436,1.8192)--(-1.8682,1.9436)--(-1.7629,1.9977)--(-1.6291,1.9796)--(-1.4691,1.8900)--(-1.2855,1.7320)--(-1.0812,1.5114)--(-0.8595,1.2363)--(-0.6240,0.9164)--(-0.3785,0.5634)--(-0.1268,0.1901)--(0.1268,-0.1901)--(0.3785,-0.5634)--(0.6240,-0.9164)--(0.8595,-1.2363)--(1.0812,-1.5114)--(1.2855,-1.7320)--(1.4691,-1.8900)--(1.6291,-1.9796)--(1.7629,-1.9977)--(1.8682,-1.9436)--(1.9436,-1.8192)--(1.9876,-1.6291)--(1.9997,-1.3801)--(1.9796,-1.0812)--(1.9276,-0.7433)--(1.8447,-0.3785)--(1.7320,0.0000)--(1.5915,0.3785)--(1.4253,0.7433)--(1.2363,1.0812)--(1.0273,1.3801)--(0.8018,1.6291)--(0.5634,1.8192)--(0.3160,1.9436)--(0.0634,1.9977)--(-0.1901,1.9796)--(-0.4406,1.8900)--(-0.6840,1.7320)--(-0.9164,1.5114)--(-1.1341,1.2363)--(-1.3335,0.9164)--(-1.5114,0.5634)--(-1.6651,0.1901)--(-1.7919,-0.1901)--(-1.8900,-0.5634)--(-1.9576,-0.9164)--(-1.9937,-1.2363)--(-1.9977,-1.5114)--(-1.9696,-1.7320)--(-1.9098,-1.8900)--(-1.8192,-1.9796)--(-1.6994,-1.9977)--(-1.5522,-1.9436)--(-1.3801,-1.8192)--(-1.1858,-1.6291)--(-0.9723,-1.3801)--(-0.7433,-1.0812)--(-0.5022,-0.7433)--(-0.2531,-0.3785)--(0.0000,0.0000);
+\draw (-2.0000,-0.3298) node {$ -1 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CercleImplicite.pstricks b/auto/pictures_tex/Fig_CercleImplicite.pstricks
index 65090a7cf..570ddd92b 100644
--- a/auto/pictures_tex/Fig_CercleImplicite.pstricks
+++ b/auto/pictures_tex/Fig_CercleImplicite.pstricks
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 \draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.7689,1.7511) node {$P$};
+\draw (1.7688,1.7510) node {$P$};
 \draw []  (1.4142,-1.4142) node [rotate=0] {$\bullet$};
 \draw (1.8158,-1.7671) node {\( P'\)};
-\draw []  (-2.0000,0) node [rotate=0] {$\bullet$};
-\draw (-2.3564,0.37233) node {\( Q\)};
-\draw []  (1.4142,0) node [rotate=0] {$\bullet$};
-\draw (1.0978,-0.29071) node {\( x\)};
-\draw [color=red,style=dotted] (1.41,1.41) -- (1.41,-1.41);
+\draw []  (-2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-2.3564,0.3723) node {\( Q\)};
+\draw []  (1.4142,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0977,-0.2907) node {\( x\)};
+\draw [color=red,style=dotted] (1.4142,1.4142) -- (1.4142,-1.4142);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_CercleTnu.pstricks b/auto/pictures_tex/Fig_CercleTnu.pstricks
index 3feef34ae..4b6a8499f 100644
--- a/auto/pictures_tex/Fig_CercleTnu.pstricks
+++ b/auto/pictures_tex/Fig_CercleTnu.pstricks
@@ -73,14 +73,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=green,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
-\draw (2.3385,1.8384) node {\( n\)};
+\draw [color=green,->,>=latex] (1.7320,1.0000) -- (2.5980,1.5000);
+\draw (2.3385,1.8383) node {\( n\)};
 \draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
-\draw (1.6552,2.1243) node {\( e_{\theta}\)};
-\draw [color=green,->,>=latex] (-0.68404,1.8794) -- (-1.0261,2.8191);
-\draw (-1.4175,2.6379) node {\( n\)};
-\draw [color=red,->,>=latex] (-0.68404,1.8794) -- (-1.6237,1.5374);
-\draw (-1.8897,1.9276) node {\( e_{\theta}\)};
+\draw (1.6551,2.1243) node {\( e_{\theta}\)};
+\draw [color=green,->,>=latex] (-0.6840,1.8793) -- (-1.0260,2.8190);
+\draw (-1.4175,2.6378) node {\( n\)};
+\draw [color=red,->,>=latex] (-0.6840,1.8793) -- (-1.6237,1.5373);
+\draw (-1.8896,1.9275) node {\( e_{\theta}\)};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -88,8 +88,8 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=brown] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CercleTrigono.pstricks b/auto/pictures_tex/Fig_CercleTrigono.pstricks
index 391fc1759..bb88ff553 100644
--- a/auto/pictures_tex/Fig_CercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_CercleTrigono.pstricks
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
-\draw [color=brown,style=dotted] (1.73,1.00) -- (1.73,0);
-\draw [color=brown,style=dotted] (1.73,1.00) -- (0,1.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
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-\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,0);
-\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,1.0000);
-\draw (-0.75804,0.50000) node {$\sin(\theta)$};
-\draw [,->,>=latex] (0.86602,-0.20000) -- (0,-0.20000);
-\draw [,->,>=latex] (0.86602,-0.20000) -- (1.7320,-0.20000);
-\draw (0.86602,-0.48246) node {$\cos(\theta)$};
-\draw [color=brown,->,>=latex] (0,0) -- (1.7320,1.0000);
-\draw (0.91614,0.23026) node {$\theta$};
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw [color=brown,style=dotted] (1.7320,1.0000) -- (1.7320,0.0000);
+\draw [color=brown,style=dotted] (1.7320,1.0000) -- (0.0000,1.0000);
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+\draw [,->,>=latex] (-0.2000,0.5000) -- (-0.2000,0.0000);
+\draw [,->,>=latex] (-0.2000,0.5000) -- (-0.2000,1.0000);
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+\draw [,->,>=latex] (0.8660,-0.2000) -- (1.7320,-0.2000);
+\draw (0.8660,-0.4824) node {$\cos(\theta)$};
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diff --git a/auto/pictures_tex/Fig_ChampGraviation.pstricks b/auto/pictures_tex/Fig_ChampGraviation.pstricks
index 518a4dcc8..6adcbc888 100644
--- a/auto/pictures_tex/Fig_ChampGraviation.pstricks
+++ b/auto/pictures_tex/Fig_ChampGraviation.pstricks
@@ -65,228 +65,228 @@
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CheminFresnel.pstricks b/auto/pictures_tex/Fig_CheminFresnel.pstricks
index 5589fe1bc..67f8f4c4b 100644
--- a/auto/pictures_tex/Fig_CheminFresnel.pstricks
+++ b/auto/pictures_tex/Fig_CheminFresnel.pstricks
@@ -75,18 +75,18 @@
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+\draw [,->,>=latex] (0.7071,0.7071) -- (0.7141,0.7141);
+\draw (1.0000,-0.2140) node {\( \gamma_1\)};
+\draw (2.1137,0.9176) node {\( \gamma_2\)};
+\draw (0.4627,0.8918) node {\( \gamma_3\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ChiSquared.pstricks b/auto/pictures_tex/Fig_ChiSquared.pstricks
index 77a4f4ecd..50b7f9efb 100644
--- a/auto/pictures_tex/Fig_ChiSquared.pstricks
+++ b/auto/pictures_tex/Fig_ChiSquared.pstricks
@@ -95,27 +95,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (15.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.3819);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (15.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.3819);
 %DEFAULT
 
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-\draw (2.5000,-0.31492) node {$ 5 $};
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-\draw (5.0000,-0.31492) node {$ 10 $};
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-\draw (7.5000,-0.31492) node {$ 15 $};
-\draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (10.000,-0.31492) node {$ 20 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (12.500,-0.31492) node {$ 25 $};
-\draw [] (12.5,-0.100) -- (12.5,0.100);
-\draw (15.000,-0.31492) node {$ 30 $};
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-\draw (-0.38167,5.0000) node {$ \frac{1}{10} $};
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+\draw (2.5000,-0.3149) node {$ 5 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (5.0000,-0.3149) node {$ 10 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (7.5000,-0.3149) node {$ 15 $};
+\draw [] (7.5000,-0.1000) -- (7.5000,0.1000);
+\draw (10.000,-0.3149) node {$ 20 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (12.500,-0.3149) node {$ 25 $};
+\draw [] (12.500,-0.1000) -- (12.500,0.1000);
+\draw (15.000,-0.3149) node {$ 30 $};
+\draw [] (15.000,-0.1000) -- (15.000,0.1000);
+\draw (-0.3816,2.5000) node {$ \frac{1}{20} $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.3816,5.0000) node {$ \frac{1}{10} $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks b/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
index 8e28a13e5..aac167aaf 100644
--- a/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
+++ b/auto/pictures_tex/Fig_ChiSquaresQuantile.pstricks
@@ -95,11 +95,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (0,0) -- (9.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.3842);
+\draw [,->,>=latex] (0.0000,0.0000) -- (9.5000,0.0000);
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 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -107,11 +107,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -119,27 +119,27 @@
 % setting the default values
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+\draw [color=blue] (4.8000,0.0000)--(4.8424,0.0000)--(4.8848,0.0000)--(4.9272,0.0000)--(4.9696,0.0000)--(5.0121,0.0000)--(5.0545,0.0000)--(5.0969,0.0000)--(5.1393,0.0000)--(5.1818,0.0000)--(5.2242,0.0000)--(5.2666,0.0000)--(5.3090,0.0000)--(5.3515,0.0000)--(5.3939,0.0000)--(5.4363,0.0000)--(5.4787,0.0000)--(5.5212,0.0000)--(5.5636,0.0000)--(5.6060,0.0000)--(5.6484,0.0000)--(5.6909,0.0000)--(5.7333,0.0000)--(5.7757,0.0000)--(5.8181,0.0000)--(5.8606,0.0000)--(5.9030,0.0000)--(5.9454,0.0000)--(5.9878,0.0000)--(6.0303,0.0000)--(6.0727,0.0000)--(6.1151,0.0000)--(6.1575,0.0000)--(6.2000,0.0000)--(6.2424,0.0000)--(6.2848,0.0000)--(6.3272,0.0000)--(6.3696,0.0000)--(6.4121,0.0000)--(6.4545,0.0000)--(6.4969,0.0000)--(6.5393,0.0000)--(6.5818,0.0000)--(6.6242,0.0000)--(6.6666,0.0000)--(6.7090,0.0000)--(6.7515,0.0000)--(6.7939,0.0000)--(6.8363,0.0000)--(6.8787,0.0000)--(6.9212,0.0000)--(6.9636,0.0000)--(7.0060,0.0000)--(7.0484,0.0000)--(7.0909,0.0000)--(7.1333,0.0000)--(7.1757,0.0000)--(7.2181,0.0000)--(7.2606,0.0000)--(7.3030,0.0000)--(7.3454,0.0000)--(7.3878,0.0000)--(7.4303,0.0000)--(7.4727,0.0000)--(7.5151,0.0000)--(7.5575,0.0000)--(7.6000,0.0000)--(7.6424,0.0000)--(7.6848,0.0000)--(7.7272,0.0000)--(7.7696,0.0000)--(7.8121,0.0000)--(7.8545,0.0000)--(7.8969,0.0000)--(7.9393,0.0000)--(7.9818,0.0000)--(8.0242,0.0000)--(8.0666,0.0000)--(8.1090,0.0000)--(8.1515,0.0000)--(8.1939,0.0000)--(8.2363,0.0000)--(8.2787,0.0000)--(8.3212,0.0000)--(8.3636,0.0000)--(8.4060,0.0000)--(8.4484,0.0000)--(8.4909,0.0000)--(8.5333,0.0000)--(8.5757,0.0000)--(8.6181,0.0000)--(8.6606,0.0000)--(8.7030,0.0000)--(8.7454,0.0000)--(8.7878,0.0000)--(8.8303,0.0000)--(8.8727,0.0000)--(8.9151,0.0000)--(8.9575,0.0000)--(9.0000,0.0000);
+\draw [] (4.8000,0.0000) -- (4.8000,1.4313);
+\draw [] (9.0000,0.0161) -- (9.0000,0.0000);
+\draw (1.5000,-0.3149) node {$ 5 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 10 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.5000,-0.3149) node {$ 15 $};
+\draw [] (4.5000,-0.1000) -- (4.5000,0.1000);
+\draw (6.0000,-0.3149) node {$ 20 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.5000,-0.3149) node {$ 25 $};
+\draw [] (7.5000,-0.1000) -- (7.5000,0.1000);
+\draw (9.0000,-0.3149) node {$ 30 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (-0.3816,2.5000) node {$ \frac{1}{20} $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.3816,5.0000) node {$ \frac{1}{10} $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ChoixInfini.pstricks b/auto/pictures_tex/Fig_ChoixInfini.pstricks
index 40c07f5d0..c75de4bed 100644
--- a/auto/pictures_tex/Fig_ChoixInfini.pstricks
+++ b/auto/pictures_tex/Fig_ChoixInfini.pstricks
@@ -57,25 +57,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw [color=blue] (-3.00,1.00) -- (3.00,1.00);
-\draw [color=blue]  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-3.0000,1.0000) -- (3.0000,1.0000);
+\draw [color=blue]  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -121,23 +121,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue] (-2.00,0) -- (2.00,2.00);
+\draw [color=blue] (-2.0000,0.0000) -- (2.0000,2.0000);
 \draw [color=blue]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CoinPasVar.pstricks b/auto/pictures_tex/Fig_CoinPasVar.pstricks
index 471cee114..83860cb3d 100644
--- a/auto/pictures_tex/Fig_CoinPasVar.pstricks
+++ b/auto/pictures_tex/Fig_CoinPasVar.pstricks
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue] (-2.00,0) -- (0,2.00);
-\draw [color=blue] (2.00,0) -- (0,2.00);
-\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
-\draw (0.23724,2.1954) node {\( N\)};
+\draw [color=blue] (-2.0000,0.0000) -- (0.0000,2.0000);
+\draw [color=blue] (2.0000,0.0000) -- (0.0000,2.0000);
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (0.2372,2.1954) node {\( N\)};
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.2158,1.2103) node {\( t_1\)};
+\draw (-1.2157,1.2103) node {\( t_1\)};
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.2158,1.2103) node {\( t_2\)};
+\draw (1.2157,1.2103) node {\( t_2\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ConeRevolution.pstricks b/auto/pictures_tex/Fig_ConeRevolution.pstricks
index 1c9a11ba0..8f2baeb65 100644
--- a/auto/pictures_tex/Fig_ConeRevolution.pstricks
+++ b/auto/pictures_tex/Fig_ConeRevolution.pstricks
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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 \draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
-\draw (2.5389,3.2532) node {$(R,h)$};
-\draw (0.72367,0.33595) node {$\alpha$};
+\draw (2.5388,3.2531) node {$(R,h)$};
+\draw (0.7236,0.3359) node {$\alpha$};
 
-\draw [color=red] (0.500,0)--(0.500,0.00496)--(0.500,0.00993)--(0.500,0.0149)--(0.500,0.0198)--(0.499,0.0248)--(0.499,0.0298)--(0.499,0.0347)--(0.498,0.0397)--(0.498,0.0446)--(0.498,0.0496)--(0.497,0.0545)--(0.496,0.0594)--(0.496,0.0643)--(0.495,0.0693)--(0.494,0.0742)--(0.494,0.0791)--(0.493,0.0840)--(0.492,0.0889)--(0.491,0.0938)--(0.490,0.0986)--(0.489,0.103)--(0.488,0.108)--(0.487,0.113)--(0.486,0.118)--(0.485,0.123)--(0.483,0.128)--(0.482,0.132)--(0.481,0.137)--(0.479,0.142)--(0.478,0.147)--(0.477,0.151)--(0.475,0.156)--(0.473,0.161)--(0.472,0.166)--(0.470,0.170)--(0.468,0.175)--(0.467,0.180)--(0.465,0.184)--(0.463,0.189)--(0.461,0.193)--(0.459,0.198)--(0.457,0.202)--(0.455,0.207)--(0.453,0.212)--(0.451,0.216)--(0.449,0.220)--(0.447,0.225)--(0.444,0.229)--(0.442,0.234)--(0.440,0.238)--(0.437,0.242)--(0.435,0.247)--(0.432,0.251)--(0.430,0.255)--(0.427,0.260)--(0.425,0.264)--(0.422,0.268)--(0.419,0.272)--(0.417,0.276)--(0.414,0.281)--(0.411,0.285)--(0.408,0.289)--(0.405,0.293)--(0.402,0.297)--(0.399,0.301)--(0.396,0.305)--(0.393,0.309)--(0.390,0.312)--(0.387,0.316)--(0.384,0.320)--(0.381,0.324)--(0.378,0.328)--(0.374,0.331)--(0.371,0.335)--(0.368,0.339)--(0.364,0.342)--(0.361,0.346)--(0.357,0.350)--(0.354,0.353)--(0.350,0.357)--(0.347,0.360)--(0.343,0.364)--(0.340,0.367)--(0.336,0.370)--(0.332,0.374)--(0.329,0.377)--(0.325,0.380)--(0.321,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.306,0.396)--(0.302,0.399)--(0.298,0.402)--(0.294,0.405)--(0.290,0.408)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416);
-\draw (2.0000,-0.32572) node {$\mathit{R}$};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.30273,3.0000) node {$\mathit{h}$};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (0.5000,0.0000)--(0.4999,0.0049)--(0.4999,0.0099)--(0.4997,0.0148)--(0.4996,0.0198)--(0.4993,0.0248)--(0.4991,0.0297)--(0.4987,0.0347)--(0.4984,0.0396)--(0.4980,0.0446)--(0.4975,0.0495)--(0.4970,0.0544)--(0.4964,0.0594)--(0.4958,0.0643)--(0.4951,0.0692)--(0.4944,0.0741)--(0.4937,0.0790)--(0.4928,0.0839)--(0.4920,0.0888)--(0.4911,0.0937)--(0.4901,0.0986)--(0.4891,0.1034)--(0.4881,0.1083)--(0.4870,0.1131)--(0.4858,0.1180)--(0.4846,0.1228)--(0.4834,0.1276)--(0.4821,0.1324)--(0.4808,0.1371)--(0.4794,0.1419)--(0.4779,0.1467)--(0.4765,0.1514)--(0.4749,0.1561)--(0.4734,0.1608)--(0.4717,0.1655)--(0.4701,0.1702)--(0.4684,0.1749)--(0.4666,0.1795)--(0.4648,0.1841)--(0.4629,0.1887)--(0.4610,0.1933)--(0.4591,0.1979)--(0.4571,0.2024)--(0.4551,0.2070)--(0.4530,0.2115)--(0.4509,0.2160)--(0.4487,0.2204)--(0.4465,0.2249)--(0.4443,0.2293)--(0.4420,0.2337)--(0.4396,0.2381)--(0.4372,0.2424)--(0.4348,0.2467)--(0.4323,0.2511)--(0.4298,0.2553)--(0.4273,0.2596)--(0.4247,0.2638)--(0.4220,0.2680)--(0.4193,0.2722)--(0.4166,0.2763)--(0.4138,0.2805)--(0.4110,0.2846)--(0.4082,0.2886)--(0.4053,0.2927)--(0.4024,0.2967)--(0.3994,0.3007)--(0.3964,0.3046)--(0.3934,0.3085)--(0.3903,0.3124)--(0.3872,0.3163)--(0.3840,0.3201)--(0.3808,0.3239)--(0.3776,0.3277)--(0.3743,0.3314)--(0.3710,0.3351)--(0.3676,0.3388)--(0.3643,0.3424)--(0.3609,0.3460)--(0.3574,0.3496)--(0.3539,0.3531)--(0.3504,0.3566)--(0.3468,0.3601)--(0.3432,0.3635)--(0.3396,0.3669)--(0.3360,0.3702)--(0.3323,0.3735)--(0.3285,0.3768)--(0.3248,0.3801)--(0.3210,0.3833)--(0.3172,0.3864)--(0.3133,0.3896)--(0.3094,0.3927)--(0.3055,0.3957)--(0.3016,0.3987)--(0.2976,0.4017)--(0.2936,0.4046)--(0.2896,0.4075)--(0.2855,0.4104)--(0.2814,0.4132)--(0.2773,0.4160);
+\draw (2.0000,-0.3257) node {$\mathit{R}$};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.3027,3.0000) node {$\mathit{h}$};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ContourGreen.pstricks b/auto/pictures_tex/Fig_ContourGreen.pstricks
index 6990892da..355cff092 100644
--- a/auto/pictures_tex/Fig_ContourGreen.pstricks
+++ b/auto/pictures_tex/Fig_ContourGreen.pstricks
@@ -65,15 +65,15 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (1.000,0)--(1.061,0.06744)--(1.117,0.1425)--(1.164,0.2244)--(1.203,0.3122)--(1.232,0.4045)--(1.249,0.4999)--(1.253,0.5966)--(1.245,0.6928)--(1.224,0.7865)--(1.190,0.8760)--(1.143,0.9593)--(1.085,1.035)--(1.017,1.101)--(0.9391,1.156)--(0.8541,1.199)--(0.7634,1.230)--(0.6689,1.248)--(0.5724,1.253)--(0.4759,1.246)--(0.3811,1.226)--(0.2898,1.194)--(0.2033,1.153)--(0.1230,1.103)--(0.04984,1.046)--(-0.01561,0.9840)--(-0.07299,0.9181)--(-0.1223,0.8504)--(-0.1637,0.7826)--(-0.1980,0.7163)--(-0.2260,0.6529)--(-0.2487,0.5937)--(-0.2674,0.5395)--(-0.2835,0.4910)--(-0.2985,0.4486)--(-0.3138,0.4123)--(-0.3308,0.3817)--(-0.3508,0.3564)--(-0.3749,0.3354)--(-0.4041,0.3178)--(-0.4390,0.3022)--(-0.4798,0.2873)--(-0.5268,0.2716)--(-0.5796,0.2537)--(-0.6377,0.2321)--(-0.7001,0.2056)--(-0.7658,0.1730)--(-0.8334,0.1334)--(-0.9013,0.08606)--(-0.9678,0.03072)--(-1.031,-0.03273)--(-1.090,-0.1041)--(-1.141,-0.1827)--(-1.185,-0.2677)--(-1.219,-0.3579)--(-1.242,-0.4519)--(-1.253,-0.5482)--(-1.251,-0.6449)--(-1.236,-0.7401)--(-1.208,-0.8319)--(-1.168,-0.9185)--(-1.116,-0.9981)--(-1.052,-1.069)--(-0.9790,-1.130)--(-0.8975,-1.179)--(-0.8094,-1.217)--(-0.7165,-1.241)--(-0.6208,-1.252)--(-0.5240,-1.251)--(-0.4282,-1.237)--(-0.3349,-1.211)--(-0.2459,-1.175)--(-0.1624,-1.129)--(-0.08551,-1.076)--(-0.01612,-1.016)--(0.04532,-0.9514)--(0.09863,-0.8844)--(0.1440,-0.8164)--(0.1817,-0.7492)--(0.2127,-0.6842)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2757,-0.5145)--(0.2910,-0.4691)--(0.3060,-0.4297)--(0.3220,-0.3963)--(0.3403,-0.3685)--(0.3623,-0.3454)--(0.3888,-0.3263)--(0.4208,-0.3098)--(0.4586,-0.2948)--(0.5026,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1901)--(0.7995,-0.1541)--(0.8674,-0.1107)--(0.9348,-0.05941)--(1.000,0);
-\draw [,->,>=latex] (1.2468,0.68114) -- (1.2455,0.69104);
-\draw [,->,>=latex] (0.35760,1.2188) -- (0.34804,1.2158);
-\draw [,->,>=latex] (-0.27771,0.50835) -- (-0.28086,0.49886);
-\draw [,->,>=latex] (-0.70033,0.20548) -- (-0.70942,0.20132);
-\draw [,->,>=latex] (-1.2468,-0.68114) -- (-1.2455,-0.69105);
-\draw [,->,>=latex] (-0.35759,-1.2188) -- (-0.34804,-1.2158);
-\draw [,->,>=latex] (0.27771,-0.50835) -- (0.28086,-0.49886);
-\draw [,->,>=latex] (0.70033,-0.20548) -- (0.70942,-0.20132);
+\draw [color=blue] (1.0000,0.0000)--(1.0611,0.0674)--(1.1165,0.1424)--(1.1644,0.2244)--(1.2032,0.3122)--(1.2317,0.4045)--(1.2486,0.4999)--(1.2533,0.5965)--(1.2451,0.6927)--(1.2238,0.7865)--(1.1897,0.8759)--(1.1432,0.9592)--(1.0851,1.0347)--(1.0166,1.1007)--(0.9391,1.1559)--(0.8541,1.1994)--(0.7634,1.2303)--(0.6688,1.2483)--(0.5723,1.2533)--(0.4758,1.2455)--(0.3811,1.2256)--(0.2897,1.1944)--(0.2033,1.1532)--(0.1230,1.1033)--(0.0498,1.0463)--(-0.0156,0.9840)--(-0.0729,0.9181)--(-0.1222,0.8503)--(-0.1637,0.7825)--(-0.1980,0.7162)--(-0.2259,0.6529)--(-0.2486,0.5936)--(-0.2673,0.5394)--(-0.2834,0.4910)--(-0.2984,0.4486)--(-0.3137,0.4122)--(-0.3307,0.3817)--(-0.3507,0.3563)--(-0.3749,0.3354)--(-0.4041,0.3177)--(-0.4389,0.3022)--(-0.4798,0.2872)--(-0.5268,0.2715)--(-0.5796,0.2536)--(-0.6376,0.2320)--(-0.7001,0.2055)--(-0.7658,0.1729)--(-0.8333,0.1333)--(-0.9012,0.0860)--(-0.9678,0.0307)--(-1.0311,-0.0327)--(-1.0896,-0.1040)--(-1.1414,-0.1826)--(-1.1850,-0.2676)--(-1.2188,-0.3578)--(-1.2417,-0.4519)--(-1.2525,-0.5481)--(-1.2508,-0.6448)--(-1.2361,-0.7400)--(-1.2083,-0.8319)--(-1.1680,-0.9185)--(-1.1156,-0.9980)--(-1.0521,-1.0689)--(-0.9789,-1.1297)--(-0.8974,-1.1792)--(-0.8093,-1.2165)--(-0.7165,-1.2410)--(-0.6207,-1.2524)--(-0.5240,-1.2510)--(-0.4281,-1.2370)--(-0.3349,-1.2113)--(-0.2458,-1.1750)--(-0.1623,-1.1292)--(-0.0855,-1.0756)--(-0.0161,-1.0157)--(0.0453,-0.9513)--(0.0986,-0.8843)--(0.1439,-0.8163)--(0.1817,-0.7491)--(0.2127,-0.6841)--(0.2379,-0.6227)--(0.2584,-0.5659)--(0.2756,-0.5145)--(0.2910,-0.4690)--(0.3059,-0.4296)--(0.3219,-0.3962)--(0.3403,-0.3684)--(0.3622,-0.3454)--(0.3888,-0.3262)--(0.4207,-0.3098)--(0.4586,-0.2947)--(0.5025,-0.2796)--(0.5525,-0.2630)--(0.6080,-0.2434)--(0.6684,-0.2195)--(0.7326,-0.1900)--(0.7994,-0.1540)--(0.8673,-0.1106)--(0.9348,-0.0594)--(1.0000,0.0000);
+\draw [,->,>=latex] (1.2468,0.6811) -- (1.2454,0.6910);
+\draw [,->,>=latex] (0.3575,1.2187) -- (0.3480,1.2158);
+\draw [,->,>=latex] (-0.2777,0.5083) -- (-0.2808,0.4988);
+\draw [,->,>=latex] (-0.7003,0.2054) -- (-0.7094,0.2013);
+\draw [,->,>=latex] (-1.2468,-0.6811) -- (-1.2454,-0.6910);
+\draw [,->,>=latex] (-0.3575,-1.2187) -- (-0.3480,-1.2158);
+\draw [,->,>=latex] (0.2777,-0.5083) -- (0.2808,-0.4988);
+\draw [,->,>=latex] (0.7003,-0.2054) -- (0.7094,-0.2013);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ContourSqL.pstricks b/auto/pictures_tex/Fig_ContourSqL.pstricks
index 766a75511..4aaf950f6 100644
--- a/auto/pictures_tex/Fig_ContourSqL.pstricks
+++ b/auto/pictures_tex/Fig_ContourSqL.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -77,17 +77,17 @@
 % setting the default values
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diff --git a/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks b/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks
index e2b22fef2..7c6762cc9 100644
--- a/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks
+++ b/auto/pictures_tex/Fig_ContourTgNDivergence.pstricks
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diff --git a/auto/pictures_tex/Fig_CoordPolaires.pstricks b/auto/pictures_tex/Fig_CoordPolaires.pstricks
index 827399ee2..e35320430 100644
--- a/auto/pictures_tex/Fig_CoordPolaires.pstricks
+++ b/auto/pictures_tex/Fig_CoordPolaires.pstricks
@@ -83,21 +83,21 @@
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-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [] (0.5000,0.0000)--(0.4999,0.0055)--(0.4998,0.0111)--(0.4997,0.0167)--(0.4994,0.0223)--(0.4992,0.0279)--(0.4988,0.0335)--(0.4984,0.0391)--(0.4980,0.0446)--(0.4974,0.0502)--(0.4968,0.0558)--(0.4962,0.0613)--(0.4955,0.0668)--(0.4947,0.0724)--(0.4938,0.0779)--(0.4929,0.0834)--(0.4920,0.0889)--(0.4909,0.0944)--(0.4899,0.0999)--(0.4887,0.1054)--(0.4875,0.1109)--(0.4862,0.1163)--(0.4849,0.1217)--(0.4835,0.1271)--(0.4820,0.1325)--(0.4805,0.1379)--(0.4790,0.1433)--(0.4773,0.1486)--(0.4756,0.1540)--(0.4739,0.1593)--(0.4721,0.1646)--(0.4702,0.1698)--(0.4683,0.1751)--(0.4663,0.1803)--(0.4642,0.1855)--(0.4621,0.1907)--(0.4600,0.1959)--(0.4578,0.2010)--(0.4555,0.2061)--(0.4531,0.2112)--(0.4508,0.2162)--(0.4483,0.2213)--(0.4458,0.2263)--(0.4432,0.2312)--(0.4406,0.2362)--(0.4380,0.2411)--(0.4352,0.2460)--(0.4325,0.2508)--(0.4296,0.2556)--(0.4267,0.2604)--(0.4238,0.2652)--(0.4208,0.2699)--(0.4178,0.2746)--(0.4147,0.2793)--(0.4115,0.2839)--(0.4083,0.2885)--(0.4051,0.2930)--(0.4018,0.2975)--(0.3984,0.3020)--(0.3950,0.3064)--(0.3916,0.3108)--(0.3880,0.3152)--(0.3845,0.3195)--(0.3809,0.3238)--(0.3773,0.3280)--(0.3736,0.3322)--(0.3698,0.3364)--(0.3660,0.3405)--(0.3622,0.3446)--(0.3583,0.3486)--(0.3544,0.3526)--(0.3504,0.3565)--(0.3464,0.3604)--(0.3424,0.3643)--(0.3383,0.3681)--(0.3341,0.3719)--(0.3300,0.3756)--(0.3257,0.3792)--(0.3215,0.3829)--(0.3172,0.3864)--(0.3128,0.3899)--(0.3085,0.3934)--(0.3040,0.3969)--(0.2996,0.4002)--(0.2951,0.4036)--(0.2906,0.4068)--(0.2860,0.4101)--(0.2814,0.4132)--(0.2767,0.4163)--(0.2721,0.4194)--(0.2674,0.4224)--(0.2626,0.4254)--(0.2578,0.4283)--(0.2530,0.4312)--(0.2482,0.4340)--(0.2433,0.4367)--(0.2384,0.4394)--(0.2335,0.4421)--(0.2285,0.4446)--(0.2236,0.4472);
+\draw (0.2337,1.1680) node {$r$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CornetGlace.pstricks b/auto/pictures_tex/Fig_CornetGlace.pstricks
index a8a07a6d4..532fd1e8c 100644
--- a/auto/pictures_tex/Fig_CornetGlace.pstricks
+++ b/auto/pictures_tex/Fig_CornetGlace.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.7000,0) -- (1.7000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.7000);
+\draw [,->,>=latex] (-1.7000,0.0000) -- (1.7000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.7000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -80,19 +80,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=blue] (0.0000,0.0000)--(0.0121,0.0121)--(0.0242,0.0242)--(0.0363,0.0363)--(0.0484,0.0484)--(0.0606,0.0606)--(0.0727,0.0727)--(0.0848,0.0848)--(0.0969,0.0969)--(0.1090,0.1090)--(0.1212,0.1212)--(0.1333,0.1333)--(0.1454,0.1454)--(0.1575,0.1575)--(0.1696,0.1696)--(0.1818,0.1818)--(0.1939,0.1939)--(0.2060,0.2060)--(0.2181,0.2181)--(0.2303,0.2303)--(0.2424,0.2424)--(0.2545,0.2545)--(0.2666,0.2666)--(0.2787,0.2787)--(0.2909,0.2909)--(0.3030,0.3030)--(0.3151,0.3151)--(0.3272,0.3272)--(0.3393,0.3393)--(0.3515,0.3515)--(0.3636,0.3636)--(0.3757,0.3757)--(0.3878,0.3878)--(0.4000,0.4000)--(0.4121,0.4121)--(0.4242,0.4242)--(0.4363,0.4363)--(0.4484,0.4484)--(0.4606,0.4606)--(0.4727,0.4727)--(0.4848,0.4848)--(0.4969,0.4969)--(0.5090,0.5090)--(0.5212,0.5212)--(0.5333,0.5333)--(0.5454,0.5454)--(0.5575,0.5575)--(0.5696,0.5696)--(0.5818,0.5818)--(0.5939,0.5939)--(0.6060,0.6060)--(0.6181,0.6181)--(0.6303,0.6303)--(0.6424,0.6424)--(0.6545,0.6545)--(0.6666,0.6666)--(0.6787,0.6787)--(0.6909,0.6909)--(0.7030,0.7030)--(0.7151,0.7151)--(0.7272,0.7272)--(0.7393,0.7393)--(0.7515,0.7515)--(0.7636,0.7636)--(0.7757,0.7757)--(0.7878,0.7878)--(0.8000,0.8000)--(0.8121,0.8121)--(0.8242,0.8242)--(0.8363,0.8363)--(0.8484,0.8484)--(0.8606,0.8606)--(0.8727,0.8727)--(0.8848,0.8848)--(0.8969,0.8969)--(0.9090,0.9090)--(0.9212,0.9212)--(0.9333,0.9333)--(0.9454,0.9454)--(0.9575,0.9575)--(0.9696,0.9696)--(0.9818,0.9818)--(0.9939,0.9939)--(1.0060,1.0060)--(1.0181,1.0181)--(1.0303,1.0303)--(1.0424,1.0424)--(1.0545,1.0545)--(1.0666,1.0666)--(1.0787,1.0787)--(1.0909,1.0909)--(1.1030,1.1030)--(1.1151,1.1151)--(1.1272,1.1272)--(1.1393,1.1393)--(1.1515,1.1515)--(1.1636,1.1636)--(1.1757,1.1757)--(1.1878,1.1878)--(1.2000,1.2000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CourbeRectifiable.pstricks b/auto/pictures_tex/Fig_CourbeRectifiable.pstricks
index c87a9cf28..06e3bd8f9 100644
--- a/auto/pictures_tex/Fig_CourbeRectifiable.pstricks
+++ b/auto/pictures_tex/Fig_CourbeRectifiable.pstricks
@@ -85,21 +85,21 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red] (-14.0,0) -- (-13.6,1.40);
-\draw [color=red] (-13.6,1.40) -- (-12.3,0);
-\draw [color=red] (-12.3,0) -- (-10.2,-1.40);
-\draw [color=red] (-10.2,-1.40) -- (-7.56,0);
-\draw []  (-14.000,0) node [rotate=0] {$\bullet$};
-\draw (-14.592,0) node {$\gamma(t_{0})$};
-\draw []  (-13.565,1.4000) node [rotate=0] {$\bullet$};
-\draw (-13.565,1.7825) node {$\gamma(t_{1})$};
-\draw []  (-12.286,0) node [rotate=0] {$\bullet$};
-\draw (-12.837,-0.30378) node {$\gamma(t_{2})$};
-\draw []  (-10.244,-1.4000) node [rotate=0] {$\bullet$};
-\draw (-10.244,-1.7825) node {$\gamma(t_{3})$};
-\draw []  (-7.5642,0) node [rotate=0] {$\bullet$};
-\draw (-8.0755,0.34271) node {$\gamma(t_{4})$};
-\draw [color=blue] (-14.000,0)--(-13.999,0.088794)--(-13.997,0.17723)--(-13.994,0.26495)--(-13.989,0.35161)--(-13.982,0.43685)--(-13.974,0.52033)--(-13.965,0.60171)--(-13.954,0.68068)--(-13.942,0.75690)--(-13.929,0.83007)--(-13.914,0.89990)--(-13.897,0.96611)--(-13.879,1.0284)--(-13.860,1.0866)--(-13.840,1.1404)--(-13.818,1.1896)--(-13.794,1.2340)--(-13.769,1.2735)--(-13.743,1.3078)--(-13.715,1.3369)--(-13.686,1.3605)--(-13.656,1.3787)--(-13.624,1.3914)--(-13.591,1.3984)--(-13.556,1.3998)--(-13.520,1.3956)--(-13.483,1.3857)--(-13.444,1.3703)--(-13.404,1.3494)--(-13.362,1.3230)--(-13.319,1.2913)--(-13.275,1.2544)--(-13.229,1.2124)--(-13.182,1.1656)--(-13.134,1.1141)--(-13.085,1.0581)--(-13.034,0.99777)--(-12.981,0.93348)--(-12.928,0.86542)--(-12.873,0.79388)--(-12.816,0.71915)--(-12.759,0.64152)--(-12.700,0.56130)--(-12.640,0.47883)--(-12.578,0.39443)--(-12.516,0.30843)--(-12.452,0.22120)--(-12.386,0.13308)--(-12.320,0.044419)--(-12.252,-0.044419)--(-12.183,-0.13308)--(-12.113,-0.22120)--(-12.041,-0.30843)--(-11.968,-0.39443)--(-11.895,-0.47883)--(-11.819,-0.56130)--(-11.743,-0.64152)--(-11.665,-0.71915)--(-11.587,-0.79388)--(-11.507,-0.86542)--(-11.425,-0.93348)--(-11.343,-0.99777)--(-11.260,-1.0581)--(-11.175,-1.1141)--(-11.089,-1.1656)--(-11.002,-1.2124)--(-10.914,-1.2544)--(-10.825,-1.2913)--(-10.735,-1.3230)--(-10.644,-1.3494)--(-10.551,-1.3703)--(-10.458,-1.3857)--(-10.363,-1.3956)--(-10.268,-1.3998)--(-10.171,-1.3984)--(-10.073,-1.3914)--(-9.9746,-1.3787)--(-9.8749,-1.3605)--(-9.7742,-1.3369)--(-9.6724,-1.3078)--(-9.5697,-1.2735)--(-9.4660,-1.2340)--(-9.3613,-1.1896)--(-9.2557,-1.1404)--(-9.1491,-1.0866)--(-9.0416,-1.0284)--(-8.9332,-0.96611)--(-8.8239,-0.89990)--(-8.7136,-0.83007)--(-8.6025,-0.75690)--(-8.4905,-0.68068)--(-8.3776,-0.60171)--(-8.2639,-0.52033)--(-8.1493,-0.43685)--(-8.0339,-0.35161)--(-7.9177,-0.26495)--(-7.8007,-0.17723)--(-7.6828,-0.088794)--(-7.5642,0);
+\draw [color=red] (-14.000,0.0000) -- (-13.564,1.4000);
+\draw [color=red] (-13.564,1.4000) -- (-12.286,0.0000);
+\draw [color=red] (-12.286,0.0000) -- (-10.243,-1.4000);
+\draw [color=red] (-10.243,-1.4000) -- (-7.5642,0.0000);
+\draw []  (-14.000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-14.591,0.0000) node {$\gamma(t_{0})$};
+\draw []  (-13.564,1.4000) node [rotate=0] {$\bullet$};
+\draw (-13.564,1.7824) node {$\gamma(t_{1})$};
+\draw []  (-12.286,0.0000) node [rotate=0] {$\bullet$};
+\draw (-12.836,-0.3037) node {$\gamma(t_{2})$};
+\draw []  (-10.243,-1.4000) node [rotate=0] {$\bullet$};
+\draw (-10.243,-1.7824) node {$\gamma(t_{3})$};
+\draw []  (-7.5642,0.0000) node [rotate=0] {$\bullet$};
+\draw (-8.0754,0.3427) node {$\gamma(t_{4})$};
+\draw [color=blue] (-14.000,0.0000)--(-13.999,0.0887)--(-13.997,0.1772)--(-13.993,0.2649)--(-13.988,0.3516)--(-13.982,0.4368)--(-13.974,0.5203)--(-13.965,0.6017)--(-13.954,0.6806)--(-13.942,0.7568)--(-13.928,0.8300)--(-13.913,0.8999)--(-13.897,0.9661)--(-13.879,1.0284)--(-13.860,1.0866)--(-13.839,1.1404)--(-13.817,1.1896)--(-13.794,1.2340)--(-13.769,1.2734)--(-13.742,1.3078)--(-13.715,1.3368)--(-13.686,1.3605)--(-13.655,1.3787)--(-13.623,1.3913)--(-13.590,1.3984)--(-13.555,1.3998)--(-13.519,1.3955)--(-13.482,1.3857)--(-13.443,1.3703)--(-13.403,1.3493)--(-13.362,1.3230)--(-13.319,1.2912)--(-13.274,1.2543)--(-13.229,1.2124)--(-13.182,1.1655)--(-13.134,1.1140)--(-13.084,1.0580)--(-13.033,0.9977)--(-12.981,0.9334)--(-12.927,0.8654)--(-12.872,0.7938)--(-12.816,0.7191)--(-12.758,0.6415)--(-12.700,0.5613)--(-12.639,0.4788)--(-12.578,0.3944)--(-12.515,0.3084)--(-12.451,0.2212)--(-12.386,0.1330)--(-12.319,0.0444)--(-12.252,-0.0444)--(-12.183,-0.1330)--(-12.112,-0.2212)--(-12.041,-0.3084)--(-11.968,-0.3944)--(-11.894,-0.4788)--(-11.819,-0.5613)--(-11.742,-0.6415)--(-11.665,-0.7191)--(-11.586,-0.7938)--(-11.506,-0.8654)--(-11.425,-0.9334)--(-11.343,-0.9977)--(-11.259,-1.0580)--(-11.175,-1.1140)--(-11.089,-1.1655)--(-11.002,-1.2124)--(-10.914,-1.2543)--(-10.825,-1.2912)--(-10.735,-1.3230)--(-10.643,-1.3493)--(-10.551,-1.3703)--(-10.457,-1.3857)--(-10.363,-1.3955)--(-10.267,-1.3998)--(-10.171,-1.3984)--(-10.073,-1.3913)--(-9.9746,-1.3787)--(-9.8749,-1.3605)--(-9.7741,-1.3368)--(-9.6724,-1.3078)--(-9.5696,-1.2734)--(-9.4659,-1.2340)--(-9.3613,-1.1896)--(-9.2556,-1.1404)--(-9.1491,-1.0866)--(-9.0416,-1.0284)--(-8.9331,-0.9661)--(-8.8238,-0.8999)--(-8.7136,-0.8300)--(-8.6024,-0.7568)--(-8.4904,-0.6806)--(-8.3776,-0.6017)--(-8.2638,-0.5203)--(-8.1493,-0.4368)--(-8.0339,-0.3516)--(-7.9176,-0.2649)--(-7.8006,-0.1772)--(-7.6828,-0.0887)--(-7.5642,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CouroneExam.pstricks b/auto/pictures_tex/Fig_CouroneExam.pstricks
index 33009d868..6c2bbbb83 100644
--- a/auto/pictures_tex/Fig_CouroneExam.pstricks
+++ b/auto/pictures_tex/Fig_CouroneExam.pstricks
@@ -71,22 +71,22 @@
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 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
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+\draw [color=blue] (0.0000,2.0000) -- (0.0000,1.0000);
+\draw [color=blue] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks b/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks
index 401de7a4e..2c06f37c0 100644
--- a/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks
+++ b/auto/pictures_tex/Fig_CurvilignesPolaires.pstricks
@@ -44,12 +44,12 @@
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+\draw [color=brown,style=dashed] (1.2990,0.7500) -- (3.0310,1.7500);
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5980,1.5000);
+\draw (2.2869,1.8657) node {$e_{r}$};
 \draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
-\draw (1.6075,2.1865) node {$e_{\theta}$};
+\draw (1.6075,2.1864) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -89,12 +89,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
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-\draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
-\draw (-2.0855,-1.2429) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
-\draw (-0.69205,-1.1323) node {$e_{\theta}$};
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+\draw [color=brown,style=dashed] (-0.9396,-0.3420) -- (-2.8190,-1.0260);
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-2.3492,-0.8550);
+\draw (-2.0854,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-1.0675,-1.4527);
+\draw (-0.6920,-1.1322) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_CycloideA.pstricks b/auto/pictures_tex/Fig_CycloideA.pstricks
index 22616691c..9fe6d18e5 100644
--- a/auto/pictures_tex/Fig_CycloideA.pstricks
+++ b/auto/pictures_tex/Fig_CycloideA.pstricks
@@ -115,40 +115,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (13.066,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.4995);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (13.066,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.4994);
 %DEFAULT
-\draw [color=blue] (0,0)--(0,0.0080452)--(0.0027181,0.032051)--(0.0091366,0.071632)--(0.021535,0.12615)--(0.041757,0.19473)--(0.071519,0.27627)--(0.11238,0.36945)--(0.16574,0.47277)--(0.23277,0.58459)--(0.31443,0.70308)--(0.41146,0.82635)--(0.52433,0.95242)--(0.65327,1.0793)--(0.79826,1.2048)--(0.95899,1.3271)--(1.1349,1.4441)--(1.3253,1.5539)--(1.5290,1.6549)--(1.7450,1.7453)--(1.9716,1.8237)--(2.2074,1.8888)--(2.4505,1.9397)--(2.6992,1.9754)--(2.9513,1.9955)--(3.2051,1.9995)--(3.4583,1.9874)--(3.7089,1.9595)--(3.9551,1.9161)--(4.1947,1.8580)--(4.4261,1.7861)--(4.6476,1.7015)--(4.8576,1.6056)--(5.0548,1.5000)--(5.2381,1.3863)--(5.4065,1.2665)--(5.5594,1.1423)--(5.6964,1.0159)--(5.8173,0.88916)--(5.9222,0.76424)--(6.0115,0.64311)--(6.0857,0.52773)--(6.1458,0.41994)--(6.1927,0.32149)--(6.2278,0.23396)--(6.2526,0.15875)--(6.2687,0.097073)--(6.2779,0.049929)--(6.2820,0.018071)--(6.2831,0.0020133)--(6.2832,0.0020133)--(6.2843,0.018071)--(6.2885,0.049929)--(6.2977,0.097073)--(6.3137,0.15875)--(6.3385,0.23396)--(6.3737,0.32149)--(6.4206,0.41994)--(6.4807,0.52773)--(6.5549,0.64311)--(6.6442,0.76424)--(6.7491,0.88916)--(6.8700,1.0159)--(7.0070,1.1423)--(7.1599,1.2665)--(7.3283,1.3863)--(7.5116,1.5000)--(7.7087,1.6056)--(7.9188,1.7015)--(8.1402,1.7861)--(8.3716,1.8580)--(8.6113,1.9161)--(8.8575,1.9595)--(9.1081,1.9874)--(9.3613,1.9995)--(9.6150,1.9955)--(9.8672,1.9754)--(10.116,1.9397)--(10.359,1.8888)--(10.595,1.8237)--(10.821,1.7453)--(11.037,1.6549)--(11.241,1.5539)--(11.431,1.4441)--(11.607,1.3271)--(11.768,1.2048)--(11.913,1.0793)--(12.042,0.95242)--(12.155,0.82635)--(12.252,0.70308)--(12.334,0.58459)--(12.401,0.47277)--(12.454,0.36945)--(12.495,0.27627)--(12.525,0.19473)--(12.545,0.12615)--(12.557,0.071632)--(12.564,0.032051)--(12.566,0.0080452)--(12.566,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.000,-0.31492) node {$ 11 $};
-\draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.000,-0.31492) node {$ 12 $};
-\draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.000,-0.31492) node {$ 13 $};
-\draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0000,0.0080)--(0.0027,0.0320)--(0.0091,0.0716)--(0.0215,0.1261)--(0.0417,0.1947)--(0.0715,0.2762)--(0.1123,0.3694)--(0.1657,0.4727)--(0.2327,0.5845)--(0.3144,0.7030)--(0.4114,0.8263)--(0.5243,0.9524)--(0.6532,1.0792)--(0.7982,1.2048)--(0.9589,1.3270)--(1.1349,1.4440)--(1.3252,1.5539)--(1.5290,1.6548)--(1.7449,1.7452)--(1.9716,1.8236)--(2.2073,1.8888)--(2.4505,1.9396)--(2.6991,1.9754)--(2.9513,1.9954)--(3.2050,1.9994)--(3.4582,1.9874)--(3.7089,1.9594)--(3.9550,1.9161)--(4.1947,1.8579)--(4.4261,1.7860)--(4.6476,1.7014)--(4.8576,1.6056)--(5.0548,1.5000)--(5.2380,1.3863)--(5.4064,1.2664)--(5.5594,1.1423)--(5.6963,1.0158)--(5.8172,0.8891)--(5.9221,0.7642)--(6.0114,0.6431)--(6.0857,0.5277)--(6.1457,0.4199)--(6.1927,0.3214)--(6.2278,0.2339)--(6.2526,0.1587)--(6.2687,0.0970)--(6.2778,0.0499)--(6.2820,0.0180)--(6.2831,0.0020)--(6.2832,0.0020)--(6.2843,0.0180)--(6.2884,0.0499)--(6.2976,0.0970)--(6.3137,0.1587)--(6.3385,0.2339)--(6.3736,0.3214)--(6.4206,0.4199)--(6.4806,0.5277)--(6.5549,0.6431)--(6.6441,0.7642)--(6.7490,0.8891)--(6.8699,1.0158)--(7.0069,1.1423)--(7.1598,1.2664)--(7.3282,1.3863)--(7.5115,1.5000)--(7.7087,1.6056)--(7.9187,1.7014)--(8.1402,1.7860)--(8.3716,1.8579)--(8.6113,1.9161)--(8.8574,1.9594)--(9.1081,1.9874)--(9.3613,1.9994)--(9.6150,1.9954)--(9.8672,1.9754)--(10.115,1.9396)--(10.359,1.8888)--(10.594,1.8236)--(10.821,1.7452)--(11.037,1.6548)--(11.241,1.5539)--(11.431,1.4440)--(11.607,1.3270)--(11.768,1.2048)--(11.913,1.0792)--(12.042,0.9524)--(12.154,0.8263)--(12.251,0.7030)--(12.333,0.5845)--(12.400,0.4727)--(12.453,0.3694)--(12.494,0.2762)--(12.524,0.1947)--(12.544,0.1261)--(12.557,0.0716)--(12.563,0.0320)--(12.566,0.0080)--(12.566,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (11.000,-0.3149) node {$ 11 $};
+\draw [] (11.000,-0.1000) -- (11.000,0.1000);
+\draw (12.000,-0.3149) node {$ 12 $};
+\draw [] (12.000,-0.1000) -- (12.000,0.1000);
+\draw (13.000,-0.3149) node {$ 13 $};
+\draw [] (13.000,-0.1000) -- (13.000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks b/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks
index bb8a4f9fd..190c40404 100644
--- a/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks
+++ b/auto/pictures_tex/Fig_DDCTooYscVzA.pstricks
@@ -65,31 +65,31 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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-\draw [color=green,->,>=latex] (2.0000,0) -- (2.7071,-0.70711);
-\draw [color=red,->,>=latex] (2.3414,1.8273) -- (1.8616,2.7046);
-\draw [color=green,->,>=latex] (2.3414,1.8273) -- (3.2188,2.3071);
-\draw [color=red,->,>=latex] (0.61190,2.4018) -- (-0.32151,2.0429);
-\draw [color=green,->,>=latex] (0.61190,2.4018) -- (0.25309,3.3352);
-\draw [color=red,->,>=latex] (-0.59548,0.90107) -- (-0.94670,-0.035225);
-\draw [color=green,->,>=latex] (-0.59548,0.90107) -- (-1.5318,1.2523);
-\draw [color=red,->,>=latex] (-2.1766,-0.20446) -- (-2.7726,-1.0075);
-\draw [color=green,->,>=latex] (-2.1766,-0.20446) -- (-2.9796,0.39153);
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-\draw [color=green,->,>=latex] (-2.1728,-2.0668) -- (-2.9199,-2.7314);
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-\draw [color=green,->,>=latex] (0.77682,-0.65307) -- (1.2955,-1.5081);
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+\draw [,->,>=latex] (2.0000,0.0000) -- (2.0141,0.0141);
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+\draw [,->,>=latex] (0.7768,-0.6530) -- (0.7939,-0.6427);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks b/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks
index 28b9aef27..dc2f78bc4 100644
--- a/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks
+++ b/auto/pictures_tex/Fig_DGFSooWgbuuMoB.pstricks
@@ -60,21 +60,21 @@
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 %PSTRICKS CODE
 %AXES
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-\draw [] (1.25,-0.100) -- (1.25,0.100);
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-\draw [] (3.75,-0.100) -- (3.75,0.100);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
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+\draw [] (1.2500,-0.1000) -- (1.2500,0.1000);
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw [] (3.7500,-0.1000) -- (3.7500,0.1000);
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks b/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
index 7432ae77f..2e7cbdaeb 100644
--- a/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
+++ b/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
@@ -107,45 +107,45 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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-\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-1.3270,0.29553) node {\( \lambda_1\)};
+\draw [] (5.0000,0.0000)--(4.9939,0.1902)--(4.9758,0.3797)--(4.9457,0.5677)--(4.9038,0.7534)--(4.8502,0.9361)--(4.7851,1.1149)--(4.7087,1.2893)--(4.6215,1.4585)--(4.5237,1.6219)--(4.4158,1.7787)--(4.2981,1.9283)--(4.1712,2.0702)--(4.0355,2.2037)--(3.8916,2.3284)--(3.7401,2.4437)--(3.5816,2.5491)--(3.4168,2.6443)--(3.2462,2.7288)--(3.0706,2.8024)--(2.8907,2.8647)--(2.7072,2.9154)--(2.5209,2.9544)--(2.3325,2.9815)--(2.1427,2.9966)--(1.9524,2.9996)--(1.7622,2.9905)--(1.5730,2.9694)--(1.3855,2.9364)--(1.2005,2.8915)--(1.0187,2.8350)--(0.8409,2.7670)--(0.6678,2.6879)--(0.4999,2.5980)--(0.3382,2.4977)--(0.1831,2.3872)--(0.0354,2.2672)--(-0.1044,2.1380)--(-0.2357,2.0003)--(-0.3581,1.8544)--(-0.4710,1.7011)--(-0.5739,1.5410)--(-0.6665,1.3746)--(-0.7483,1.2027)--(-0.8190,1.0260)--(-0.8784,0.8451)--(-0.9262,0.6609)--(-0.9623,0.4740)--(-0.9864,0.2851)--(-0.9984,0.0951)--(-0.9984,-0.0951)--(-0.9864,-0.2851)--(-0.9623,-0.4740)--(-0.9262,-0.6609)--(-0.8784,-0.8451)--(-0.8190,-1.0260)--(-0.7483,-1.2027)--(-0.6665,-1.3746)--(-0.5739,-1.5410)--(-0.4710,-1.7011)--(-0.3581,-1.8544)--(-0.2357,-2.0003)--(-0.1044,-2.1380)--(0.0354,-2.2672)--(0.1831,-2.3872)--(0.3382,-2.4977)--(0.5000,-2.5980)--(0.6678,-2.6879)--(0.8409,-2.7670)--(1.0187,-2.8350)--(1.2005,-2.8915)--(1.3855,-2.9364)--(1.5730,-2.9694)--(1.7622,-2.9905)--(1.9524,-2.9996)--(2.1427,-2.9966)--(2.3325,-2.9815)--(2.5209,-2.9544)--(2.7072,-2.9154)--(2.8907,-2.8647)--(3.0706,-2.8024)--(3.2462,-2.7288)--(3.4168,-2.6443)--(3.5816,-2.5491)--(3.7401,-2.4437)--(3.8916,-2.3284)--(4.0355,-2.2037)--(4.1712,-2.0702)--(4.2981,-1.9283)--(4.4158,-1.7787)--(4.5237,-1.6219)--(4.6215,-1.4585)--(4.7087,-1.2893)--(4.7851,-1.1149)--(4.8502,-0.9361)--(4.9038,-0.7534)--(4.9457,-0.5677)--(4.9758,-0.3797)--(4.9939,-0.1902)--(5.0000,0.0000);
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.3270,0.2955) node {\( \lambda_1\)};
 \draw []  (2.0000,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.6730,1.1187) node {\( \lambda_2\)};
+\draw (1.6729,1.1186) node {\( \lambda_2\)};
 \draw []  (2.0000,-1.4142) node [rotate=0] {$\bullet$};
-\draw (1.6730,-1.1187) node {\( \lambda_3\)};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (1.6729,-1.1186) node {\( \lambda_3\)};
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks b/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
index 86b9be42e..83dad0aaf 100644
--- a/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
+++ b/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
@@ -95,33 +95,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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+\draw [] (3.0000,0.0000)--(2.9979,0.0634)--(2.9919,0.1265)--(2.9819,0.1892)--(2.9679,0.2511)--(2.9500,0.3120)--(2.9283,0.3716)--(2.9029,0.4297)--(2.8738,0.4861)--(2.8412,0.5406)--(2.8052,0.5929)--(2.7660,0.6427)--(2.7237,0.6900)--(2.6785,0.7345)--(2.6305,0.7761)--(2.5800,0.8145)--(2.5272,0.8497)--(2.4722,0.8814)--(2.4154,0.9096)--(2.3568,0.9341)--(2.2969,0.9549)--(2.2357,0.9718)--(2.1736,0.9848)--(2.1108,0.9938)--(2.0475,0.9988)--(1.9841,0.9998)--(1.9207,0.9968)--(1.8576,0.9898)--(1.7951,0.9788)--(1.7335,0.9638)--(1.6729,0.9450)--(1.6136,0.9223)--(1.5559,0.8959)--(1.5000,0.8660)--(1.4460,0.8325)--(1.3943,0.7957)--(1.3451,0.7557)--(1.2985,0.7126)--(1.2547,0.6667)--(1.2139,0.6181)--(1.1763,0.5670)--(1.1420,0.5136)--(1.1111,0.4582)--(1.0838,0.4009)--(1.0603,0.3420)--(1.0405,0.2817)--(1.0245,0.2203)--(1.0125,0.1580)--(1.0045,0.0950)--(1.0005,0.0317)--(1.0005,-0.0317)--(1.0045,-0.0950)--(1.0125,-0.1580)--(1.0245,-0.2203)--(1.0405,-0.2817)--(1.0603,-0.3420)--(1.0838,-0.4009)--(1.1111,-0.4582)--(1.1420,-0.5136)--(1.1763,-0.5670)--(1.2139,-0.6181)--(1.2547,-0.6667)--(1.2985,-0.7126)--(1.3451,-0.7557)--(1.3943,-0.7957)--(1.4460,-0.8325)--(1.5000,-0.8660)--(1.5559,-0.8959)--(1.6136,-0.9223)--(1.6729,-0.9450)--(1.7335,-0.9638)--(1.7951,-0.9788)--(1.8576,-0.9898)--(1.9207,-0.9968)--(1.9841,-0.9998)--(2.0475,-0.9988)--(2.1108,-0.9938)--(2.1736,-0.9848)--(2.2357,-0.9718)--(2.2969,-0.9549)--(2.3568,-0.9341)--(2.4154,-0.9096)--(2.4722,-0.8814)--(2.5272,-0.8497)--(2.5800,-0.8145)--(2.6305,-0.7761)--(2.6785,-0.7345)--(2.7237,-0.6900)--(2.7660,-0.6427)--(2.8052,-0.5929)--(2.8412,-0.5406)--(2.8738,-0.4861)--(2.9029,-0.4297)--(2.9283,-0.3716)--(2.9500,-0.3120)--(2.9679,-0.2511)--(2.9819,-0.1892)--(2.9919,-0.1265)--(2.9979,-0.0634)--(3.0000,0.0000);
 
-\draw [] (8.000,0)--(7.996,0.1268)--(7.984,0.2532)--(7.964,0.3785)--(7.936,0.5023)--(7.900,0.6241)--(7.857,0.7433)--(7.806,0.8596)--(7.748,0.9724)--(7.682,1.081)--(7.611,1.186)--(7.532,1.286)--(7.447,1.380)--(7.357,1.469)--(7.261,1.552)--(7.160,1.629)--(7.054,1.699)--(6.945,1.763)--(6.831,1.819)--(6.714,1.868)--(6.594,1.910)--(6.471,1.944)--(6.347,1.970)--(6.222,1.988)--(6.095,1.998)--(5.968,2.000)--(5.841,1.994)--(5.715,1.980)--(5.590,1.958)--(5.467,1.928)--(5.346,1.890)--(5.227,1.845)--(5.112,1.792)--(5.000,1.732)--(4.892,1.665)--(4.789,1.592)--(4.690,1.512)--(4.597,1.425)--(4.509,1.334)--(4.428,1.236)--(4.353,1.134)--(4.284,1.027)--(4.222,0.9165)--(4.168,0.8019)--(4.121,0.6840)--(4.081,0.5635)--(4.049,0.4406)--(4.025,0.3160)--(4.009,0.1901)--(4.001,0.06346)--(4.001,-0.06346)--(4.009,-0.1901)--(4.025,-0.3160)--(4.049,-0.4406)--(4.081,-0.5635)--(4.121,-0.6840)--(4.168,-0.8019)--(4.222,-0.9165)--(4.284,-1.027)--(4.353,-1.134)--(4.428,-1.236)--(4.509,-1.334)--(4.597,-1.425)--(4.690,-1.512)--(4.789,-1.592)--(4.892,-1.665)--(5.000,-1.732)--(5.112,-1.792)--(5.227,-1.845)--(5.346,-1.890)--(5.467,-1.928)--(5.590,-1.958)--(5.715,-1.980)--(5.841,-1.994)--(5.968,-2.000)--(6.095,-1.998)--(6.222,-1.988)--(6.347,-1.970)--(6.471,-1.944)--(6.594,-1.910)--(6.714,-1.868)--(6.831,-1.819)--(6.945,-1.763)--(7.054,-1.699)--(7.160,-1.629)--(7.261,-1.552)--(7.357,-1.469)--(7.447,-1.380)--(7.532,-1.286)--(7.611,-1.186)--(7.682,-1.081)--(7.748,-0.9724)--(7.806,-0.8596)--(7.857,-0.7433)--(7.900,-0.6241)--(7.936,-0.5023)--(7.964,-0.3785)--(7.984,-0.2532)--(7.996,-0.1268)--(8.000,0);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.6730,0.29553) node {\( \lambda_1\)};
+\draw [] (8.0000,0.0000)--(7.9959,0.1268)--(7.9839,0.2531)--(7.9638,0.3785)--(7.9358,0.5022)--(7.9001,0.6240)--(7.8567,0.7433)--(7.8058,0.8595)--(7.7476,0.9723)--(7.6825,1.0812)--(7.6105,1.1858)--(7.5320,1.2855)--(7.4474,1.3801)--(7.3570,1.4691)--(7.2611,1.5522)--(7.1601,1.6291)--(7.0544,1.6994)--(6.9445,1.7629)--(6.8308,1.8192)--(6.7137,1.8682)--(6.5938,1.9098)--(6.4715,1.9436)--(6.3472,1.9696)--(6.2216,1.9876)--(6.0951,1.9977)--(5.9682,1.9997)--(5.8415,1.9937)--(5.7153,1.9796)--(5.5903,1.9576)--(5.4670,1.9276)--(5.3458,1.8900)--(5.2273,1.8447)--(5.1118,1.7919)--(5.0000,1.7320)--(4.8921,1.6651)--(4.7887,1.5915)--(4.6902,1.5114)--(4.5970,1.4253)--(4.5094,1.3335)--(4.4278,1.2363)--(4.3526,1.1341)--(4.2840,1.0273)--(4.2223,0.9164)--(4.1677,0.8018)--(4.1206,0.6840)--(4.0810,0.5634)--(4.0491,0.4406)--(4.0251,0.3160)--(4.0090,0.1901)--(4.0010,0.0634)--(4.0010,-0.0634)--(4.0090,-0.1901)--(4.0251,-0.3160)--(4.0491,-0.4406)--(4.0810,-0.5634)--(4.1206,-0.6840)--(4.1677,-0.8018)--(4.2223,-0.9164)--(4.2840,-1.0273)--(4.3526,-1.1341)--(4.4278,-1.2363)--(4.5094,-1.3335)--(4.5970,-1.4253)--(4.6902,-1.5114)--(4.7887,-1.5915)--(4.8921,-1.6651)--(5.0000,-1.7320)--(5.1118,-1.7919)--(5.2273,-1.8447)--(5.3458,-1.8900)--(5.4670,-1.9276)--(5.5903,-1.9576)--(5.7153,-1.9796)--(5.8415,-1.9937)--(5.9682,-1.9997)--(6.0951,-1.9977)--(6.2216,-1.9876)--(6.3472,-1.9696)--(6.4715,-1.9436)--(6.5938,-1.9098)--(6.7137,-1.8682)--(6.8308,-1.8192)--(6.9445,-1.7629)--(7.0544,-1.6994)--(7.1601,-1.6291)--(7.2611,-1.5522)--(7.3570,-1.4691)--(7.4474,-1.3801)--(7.5320,-1.2855)--(7.6105,-1.1858)--(7.6825,-1.0812)--(7.7476,-0.9723)--(7.8058,-0.8595)--(7.8567,-0.7433)--(7.9001,-0.6240)--(7.9358,-0.5022)--(7.9638,-0.3785)--(7.9839,-0.2531)--(7.9959,-0.1268)--(8.0000,0.0000);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.6729,0.2955) node {\( \lambda_1\)};
 \draw []  (5.0000,1.7320) node [rotate=0] {$\bullet$};
-\draw (5.0000,2.0862) node {\( \lambda_2\)};
+\draw (5.0000,2.0861) node {\( \lambda_2\)};
 \draw []  (5.0000,-1.7320) node [rotate=0] {$\bullet$};
-\draw (5.0000,-2.0862) node {\( \lambda_3\)};
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.0000,-0.31492) node {$ 2 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.0000,-0.31492) node {$ 3 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (8.0000,-0.31492) node {$ 4 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -1 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (5.0000,-2.0861) node {\( \lambda_3\)};
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 2 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 3 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 4 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DTIYKkP.pstricks b/auto/pictures_tex/Fig_DTIYKkP.pstricks
index 2166b0fcf..78fbacbb2 100644
--- a/auto/pictures_tex/Fig_DTIYKkP.pstricks
+++ b/auto/pictures_tex/Fig_DTIYKkP.pstricks
@@ -82,13 +82,13 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [color=blue] (-5.000,0)--(-4.970,0.1206)--(-4.939,0.2400)--(-4.909,0.3581)--(-4.879,0.4751)--(-4.849,0.5908)--(-4.818,0.7052)--(-4.788,0.8185)--(-4.758,0.9305)--(-4.727,1.041)--(-4.697,1.151)--(-4.667,1.259)--(-4.636,1.366)--(-4.606,1.472)--(-4.576,1.577)--(-4.545,1.680)--(-4.515,1.783)--(-4.485,1.884)--(-4.455,1.983)--(-4.424,2.082)--(-4.394,2.179)--(-4.364,2.275)--(-4.333,2.370)--(-4.303,2.464)--(-4.273,2.556)--(-4.242,2.648)--(-4.212,2.738)--(-4.182,2.826)--(-4.151,2.914)--(-4.121,3.000)--(-4.091,3.085)--(-4.061,3.169)--(-4.030,3.252)--(-4.000,3.333)--(-3.970,3.414)--(-3.939,3.492)--(-3.909,3.570)--(-3.879,3.647)--(-3.848,3.722)--(-3.818,3.796)--(-3.788,3.869)--(-3.758,3.941)--(-3.727,4.011)--(-3.697,4.080)--(-3.667,4.148)--(-3.636,4.215)--(-3.606,4.280)--(-3.576,4.345)--(-3.545,4.408)--(-3.515,4.470)--(-3.485,4.530)--(-3.455,4.590)--(-3.424,4.648)--(-3.394,4.705)--(-3.364,4.760)--(-3.333,4.815)--(-3.303,4.868)--(-3.273,4.920)--(-3.242,4.971)--(-3.212,5.021)--(-3.182,5.069)--(-3.152,5.116)--(-3.121,5.162)--(-3.091,5.207)--(-3.061,5.250)--(-3.030,5.292)--(-3.000,5.333)--(-2.970,5.373)--(-2.939,5.412)--(-2.909,5.449)--(-2.879,5.485)--(-2.848,5.520)--(-2.818,5.554)--(-2.788,5.586)--(-2.758,5.617)--(-2.727,5.647)--(-2.697,5.676)--(-2.667,5.704)--(-2.636,5.730)--(-2.606,5.755)--(-2.576,5.779)--(-2.545,5.802)--(-2.515,5.823)--(-2.485,5.843)--(-2.455,5.862)--(-2.424,5.880)--(-2.394,5.897)--(-2.364,5.912)--(-2.333,5.926)--(-2.303,5.939)--(-2.273,5.950)--(-2.242,5.961)--(-2.212,5.970)--(-2.182,5.978)--(-2.152,5.985)--(-2.121,5.990)--(-2.091,5.995)--(-2.061,5.998)--(-2.030,5.999)--(-2.000,6.000);
+\draw [color=blue] (-5.0000,0.0000)--(-4.9696,0.1205)--(-4.9393,0.2399)--(-4.9090,0.3581)--(-4.8787,0.4750)--(-4.8484,0.5907)--(-4.8181,0.7052)--(-4.7878,0.8184)--(-4.7575,0.9305)--(-4.7272,1.0413)--(-4.6969,1.1509)--(-4.6666,1.2592)--(-4.6363,1.3663)--(-4.6060,1.4722)--(-4.5757,1.5769)--(-4.5454,1.6804)--(-4.5151,1.7826)--(-4.4848,1.8836)--(-4.4545,1.9834)--(-4.4242,2.0820)--(-4.3939,2.1793)--(-4.3636,2.2754)--(-4.3333,2.3703)--(-4.3030,2.4640)--(-4.2727,2.5564)--(-4.2424,2.6476)--(-4.2121,2.7376)--(-4.1818,2.8264)--(-4.1515,2.9139)--(-4.1212,3.0003)--(-4.0909,3.0853)--(-4.0606,3.1692)--(-4.0303,3.2519)--(-4.0000,3.3333)--(-3.9696,3.4135)--(-3.9393,3.4925)--(-3.9090,3.5702)--(-3.8787,3.6467)--(-3.8484,3.7220)--(-3.8181,3.7961)--(-3.7878,3.8689)--(-3.7575,3.9406)--(-3.7272,4.0110)--(-3.6969,4.0801)--(-3.6666,4.1481)--(-3.6363,4.2148)--(-3.6060,4.2803)--(-3.5757,4.3446)--(-3.5454,4.4077)--(-3.5151,4.4695)--(-3.4848,4.5301)--(-3.4545,4.5895)--(-3.4242,4.6476)--(-3.3939,4.7046)--(-3.3636,4.7603)--(-3.3333,4.8148)--(-3.3030,4.8680)--(-3.2727,4.9201)--(-3.2424,4.9709)--(-3.2121,5.0205)--(-3.1818,5.0688)--(-3.1515,5.1160)--(-3.1212,5.1619)--(-3.0909,5.2066)--(-3.0606,5.2500)--(-3.0303,5.2923)--(-3.0000,5.3333)--(-2.9696,5.3731)--(-2.9393,5.4116)--(-2.9090,5.4490)--(-2.8787,5.4851)--(-2.8484,5.5200)--(-2.8181,5.5537)--(-2.7878,5.5861)--(-2.7575,5.6173)--(-2.7272,5.6473)--(-2.6969,5.6761)--(-2.6666,5.7037)--(-2.6363,5.7300)--(-2.6060,5.7551)--(-2.5757,5.7790)--(-2.5454,5.8016)--(-2.5151,5.8230)--(-2.4848,5.8432)--(-2.4545,5.8622)--(-2.4242,5.8800)--(-2.3939,5.8965)--(-2.3636,5.9118)--(-2.3333,5.9259)--(-2.3030,5.9387)--(-2.2727,5.9504)--(-2.2424,5.9608)--(-2.2121,5.9700)--(-2.1818,5.9779)--(-2.1515,5.9846)--(-2.1212,5.9902)--(-2.0909,5.9944)--(-2.0606,5.9975)--(-2.0303,5.9993)--(-2.0000,6.0000);
 \draw [color=brown]  (-4.0000,3.3333) node [rotate=0] {$\bullet$};
-\draw (-4.7275,3.6358) node {$o=[\mtu]$};
+\draw (-4.7274,3.6357) node {$o=[\mtu]$};
 \draw (-1.1978,6.0000) node {$[\SO(2)]$};
 \draw [color=cyan,->,>=latex] (-4.5000,1.8333) -- (-2.5000,2.3333);
 \draw (-1.4416,1.9799) node {$[ e^{sE(w)} e^{xq_0}]$};
-\draw (-5.1029,2.1187) node {$[ e^{xq_0}]$};
+\draw (-5.1029,2.1186) node {$[ e^{xq_0}]$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks b/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
index aca707323..d2188f5a2 100644
--- a/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
+++ b/auto/pictures_tex/Fig_DZVooQZLUtf.pstricks
@@ -53,29 +53,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7236);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7236);
 %DEFAULT
 
-\draw [color=blue] (0.01000,3.224)--(0.05030,2.093)--(0.09061,1.681)--(0.1309,1.423)--(0.1712,1.235)--(0.2115,1.087)--(0.2518,0.9653)--(0.2921,0.8614)--(0.3324,0.7709)--(0.3727,0.6908)--(0.4130,0.6190)--(0.4533,0.5538)--(0.4936,0.4942)--(0.5339,0.4392)--(0.5742,0.3883)--(0.6145,0.3408)--(0.6548,0.2963)--(0.6952,0.2545)--(0.7355,0.2151)--(0.7758,0.1777)--(0.8161,0.1423)--(0.8564,0.1085)--(0.8967,0.07635)--(0.9370,0.04557)--(0.9773,0.01609)--(1.018,0.01220)--(1.058,0.03939)--(1.098,0.06556)--(1.138,0.09079)--(1.179,0.1151)--(1.219,0.1387)--(1.259,0.1614)--(1.300,0.1835)--(1.340,0.2049)--(1.380,0.2256)--(1.421,0.2458)--(1.461,0.2653)--(1.501,0.2844)--(1.542,0.3029)--(1.582,0.3210)--(1.622,0.3386)--(1.662,0.3558)--(1.703,0.3726)--(1.743,0.3889)--(1.783,0.4049)--(1.824,0.4206)--(1.864,0.4359)--(1.904,0.4509)--(1.945,0.4655)--(1.985,0.4799)--(2.025,0.4939)--(2.065,0.5077)--(2.106,0.5213)--(2.146,0.5345)--(2.186,0.5476)--(2.227,0.5604)--(2.267,0.5729)--(2.307,0.5852)--(2.348,0.5974)--(2.388,0.6093)--(2.428,0.6210)--(2.468,0.6325)--(2.509,0.6439)--(2.549,0.6550)--(2.589,0.6660)--(2.630,0.6768)--(2.670,0.6875)--(2.710,0.6979)--(2.751,0.7083)--(2.791,0.7185)--(2.831,0.7285)--(2.872,0.7384)--(2.912,0.7481)--(2.952,0.7578)--(2.992,0.7673)--(3.033,0.7766)--(3.073,0.7859)--(3.113,0.7950)--(3.154,0.8040)--(3.194,0.8129)--(3.234,0.8217)--(3.275,0.8303)--(3.315,0.8389)--(3.355,0.8474)--(3.395,0.8557)--(3.436,0.8640)--(3.476,0.8721)--(3.516,0.8802)--(3.557,0.8882)--(3.597,0.8961)--(3.637,0.9039)--(3.678,0.9116)--(3.718,0.9192)--(3.758,0.9268)--(3.798,0.9342)--(3.839,0.9416)--(3.879,0.9489)--(3.919,0.9562)--(3.960,0.9633)--(4.000,0.9704);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=blue] (0.0100,3.2236)--(0.0503,2.0927)--(0.0906,1.6808)--(0.1309,1.4232)--(0.1712,1.2353)--(0.2115,1.0874)--(0.2518,0.9653)--(0.2921,0.8614)--(0.3324,0.7709)--(0.3727,0.6908)--(0.4130,0.6189)--(0.4533,0.5537)--(0.4936,0.4941)--(0.5339,0.4392)--(0.5742,0.3882)--(0.6145,0.3408)--(0.6548,0.2963)--(0.6951,0.2545)--(0.7354,0.2150)--(0.7757,0.1777)--(0.8160,0.1422)--(0.8563,0.1085)--(0.8966,0.0763)--(0.9369,0.0455)--(0.9772,0.0160)--(1.0175,0.0121)--(1.0578,0.0393)--(1.0981,0.0655)--(1.1384,0.0907)--(1.1787,0.1151)--(1.2190,0.1386)--(1.2593,0.1614)--(1.2996,0.1834)--(1.3400,0.2048)--(1.3803,0.2256)--(1.4206,0.2457)--(1.4609,0.2653)--(1.5012,0.2843)--(1.5415,0.3029)--(1.5818,0.3210)--(1.6221,0.3386)--(1.6624,0.3557)--(1.7027,0.3725)--(1.7430,0.3889)--(1.7833,0.4049)--(1.8236,0.4205)--(1.8639,0.4358)--(1.9042,0.4508)--(1.9445,0.4655)--(1.9848,0.4798)--(2.0251,0.4939)--(2.0654,0.5077)--(2.1057,0.5212)--(2.1460,0.5345)--(2.1863,0.5475)--(2.2266,0.5603)--(2.2669,0.5729)--(2.3072,0.5852)--(2.3475,0.5973)--(2.3878,0.6092)--(2.4281,0.6209)--(2.4684,0.6325)--(2.5087,0.6438)--(2.5490,0.6550)--(2.5893,0.6659)--(2.6296,0.6768)--(2.6700,0.6874)--(2.7103,0.6979)--(2.7506,0.7082)--(2.7909,0.7184)--(2.8312,0.7284)--(2.8715,0.7383)--(2.9118,0.7481)--(2.9521,0.7577)--(2.9924,0.7672)--(3.0327,0.7766)--(3.0730,0.7858)--(3.1133,0.7949)--(3.1536,0.8039)--(3.1939,0.8128)--(3.2342,0.8216)--(3.2745,0.8303)--(3.3148,0.8388)--(3.3551,0.8473)--(3.3954,0.8557)--(3.4357,0.8639)--(3.4760,0.8721)--(3.5163,0.8801)--(3.5566,0.8881)--(3.5969,0.8960)--(3.6372,0.9038)--(3.6775,0.9115)--(3.7178,0.9192)--(3.7581,0.9267)--(3.7984,0.9342)--(3.8387,0.9416)--(3.8790,0.9489)--(3.9193,0.9561)--(3.9596,0.9633)--(4.0000,0.9704);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -141,33 +141,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.7236) -- (0,1.4704);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7236) -- (0.0000,1.4704);
 %DEFAULT
 
-\draw [color=blue] (0.01000,-3.224)--(0.05030,-2.093)--(0.09061,-1.681)--(0.1309,-1.423)--(0.1712,-1.235)--(0.2115,-1.087)--(0.2518,-0.9653)--(0.2921,-0.8614)--(0.3324,-0.7709)--(0.3727,-0.6908)--(0.4130,-0.6190)--(0.4533,-0.5538)--(0.4936,-0.4942)--(0.5339,-0.4392)--(0.5742,-0.3883)--(0.6145,-0.3408)--(0.6548,-0.2963)--(0.6952,-0.2545)--(0.7355,-0.2151)--(0.7758,-0.1777)--(0.8161,-0.1423)--(0.8564,-0.1085)--(0.8967,-0.07635)--(0.9370,-0.04557)--(0.9773,-0.01609)--(1.018,0.01220)--(1.058,0.03939)--(1.098,0.06556)--(1.138,0.09079)--(1.179,0.1151)--(1.219,0.1387)--(1.259,0.1614)--(1.300,0.1835)--(1.340,0.2049)--(1.380,0.2256)--(1.421,0.2458)--(1.461,0.2653)--(1.501,0.2844)--(1.542,0.3029)--(1.582,0.3210)--(1.622,0.3386)--(1.662,0.3558)--(1.703,0.3726)--(1.743,0.3889)--(1.783,0.4049)--(1.824,0.4206)--(1.864,0.4359)--(1.904,0.4509)--(1.945,0.4655)--(1.985,0.4799)--(2.025,0.4939)--(2.065,0.5077)--(2.106,0.5213)--(2.146,0.5345)--(2.186,0.5476)--(2.227,0.5604)--(2.267,0.5729)--(2.307,0.5852)--(2.348,0.5974)--(2.388,0.6093)--(2.428,0.6210)--(2.468,0.6325)--(2.509,0.6439)--(2.549,0.6550)--(2.589,0.6660)--(2.630,0.6768)--(2.670,0.6875)--(2.710,0.6979)--(2.751,0.7083)--(2.791,0.7185)--(2.831,0.7285)--(2.872,0.7384)--(2.912,0.7481)--(2.952,0.7578)--(2.992,0.7673)--(3.033,0.7766)--(3.073,0.7859)--(3.113,0.7950)--(3.154,0.8040)--(3.194,0.8129)--(3.234,0.8217)--(3.275,0.8303)--(3.315,0.8389)--(3.355,0.8474)--(3.395,0.8557)--(3.436,0.8640)--(3.476,0.8721)--(3.516,0.8802)--(3.557,0.8882)--(3.597,0.8961)--(3.637,0.9039)--(3.678,0.9116)--(3.718,0.9192)--(3.758,0.9268)--(3.798,0.9342)--(3.839,0.9416)--(3.879,0.9489)--(3.919,0.9562)--(3.960,0.9633)--(4.000,0.9704);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
+\draw [color=blue] (0.0100,-3.2236)--(0.0503,-2.0927)--(0.0906,-1.6808)--(0.1309,-1.4232)--(0.1712,-1.2353)--(0.2115,-1.0874)--(0.2518,-0.9653)--(0.2921,-0.8614)--(0.3324,-0.7709)--(0.3727,-0.6908)--(0.4130,-0.6189)--(0.4533,-0.5537)--(0.4936,-0.4941)--(0.5339,-0.4392)--(0.5742,-0.3882)--(0.6145,-0.3408)--(0.6548,-0.2963)--(0.6951,-0.2545)--(0.7354,-0.2150)--(0.7757,-0.1777)--(0.8160,-0.1422)--(0.8563,-0.1085)--(0.8966,-0.0763)--(0.9369,-0.0455)--(0.9772,-0.0160)--(1.0175,0.0121)--(1.0578,0.0393)--(1.0981,0.0655)--(1.1384,0.0907)--(1.1787,0.1151)--(1.2190,0.1386)--(1.2593,0.1614)--(1.2996,0.1834)--(1.3400,0.2048)--(1.3803,0.2256)--(1.4206,0.2457)--(1.4609,0.2653)--(1.5012,0.2843)--(1.5415,0.3029)--(1.5818,0.3210)--(1.6221,0.3386)--(1.6624,0.3557)--(1.7027,0.3725)--(1.7430,0.3889)--(1.7833,0.4049)--(1.8236,0.4205)--(1.8639,0.4358)--(1.9042,0.4508)--(1.9445,0.4655)--(1.9848,0.4798)--(2.0251,0.4939)--(2.0654,0.5077)--(2.1057,0.5212)--(2.1460,0.5345)--(2.1863,0.5475)--(2.2266,0.5603)--(2.2669,0.5729)--(2.3072,0.5852)--(2.3475,0.5973)--(2.3878,0.6092)--(2.4281,0.6209)--(2.4684,0.6325)--(2.5087,0.6438)--(2.5490,0.6550)--(2.5893,0.6659)--(2.6296,0.6768)--(2.6700,0.6874)--(2.7103,0.6979)--(2.7506,0.7082)--(2.7909,0.7184)--(2.8312,0.7284)--(2.8715,0.7383)--(2.9118,0.7481)--(2.9521,0.7577)--(2.9924,0.7672)--(3.0327,0.7766)--(3.0730,0.7858)--(3.1133,0.7949)--(3.1536,0.8039)--(3.1939,0.8128)--(3.2342,0.8216)--(3.2745,0.8303)--(3.3148,0.8388)--(3.3551,0.8473)--(3.3954,0.8557)--(3.4357,0.8639)--(3.4760,0.8721)--(3.5163,0.8801)--(3.5566,0.8881)--(3.5969,0.8960)--(3.6372,0.9038)--(3.6775,0.9115)--(3.7178,0.9192)--(3.7581,0.9267)--(3.7984,0.9342)--(3.8387,0.9416)--(3.8790,0.9489)--(3.9193,0.9561)--(3.9596,0.9633)--(4.0000,0.9704);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -229,33 +229,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.0236) -- (0,2.1704);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0236) -- (0.0000,2.1704);
 %DEFAULT
 
-\draw [color=blue] (0.01000,-2.524)--(0.05030,-1.393)--(0.09061,-0.9809)--(0.1309,-0.7233)--(0.1712,-0.5354)--(0.2115,-0.3874)--(0.2518,-0.2653)--(0.2921,-0.1614)--(0.3324,-0.07094)--(0.3727,0.009164)--(0.4130,0.08104)--(0.4533,0.1462)--(0.4936,0.2058)--(0.5339,0.2608)--(0.5742,0.3117)--(0.6145,0.3592)--(0.6548,0.4037)--(0.6952,0.4455)--(0.7355,0.4849)--(0.7758,0.5223)--(0.8161,0.5577)--(0.8564,0.5915)--(0.8967,0.6236)--(0.9370,0.6544)--(0.9773,0.6839)--(1.018,0.7122)--(1.058,0.7394)--(1.098,0.7656)--(1.138,0.7908)--(1.179,0.8151)--(1.219,0.8387)--(1.259,0.8614)--(1.300,0.8835)--(1.340,0.9049)--(1.380,0.9256)--(1.421,0.9458)--(1.461,0.9653)--(1.501,0.9844)--(1.542,1.003)--(1.582,1.021)--(1.622,1.039)--(1.662,1.056)--(1.703,1.073)--(1.743,1.089)--(1.783,1.105)--(1.824,1.121)--(1.864,1.136)--(1.904,1.151)--(1.945,1.166)--(1.985,1.180)--(2.025,1.194)--(2.065,1.208)--(2.106,1.221)--(2.146,1.235)--(2.186,1.248)--(2.227,1.260)--(2.267,1.273)--(2.307,1.285)--(2.348,1.297)--(2.388,1.309)--(2.428,1.321)--(2.468,1.333)--(2.509,1.344)--(2.549,1.355)--(2.589,1.366)--(2.630,1.377)--(2.670,1.387)--(2.710,1.398)--(2.751,1.408)--(2.791,1.418)--(2.831,1.428)--(2.872,1.438)--(2.912,1.448)--(2.952,1.458)--(2.992,1.467)--(3.033,1.477)--(3.073,1.486)--(3.113,1.495)--(3.154,1.504)--(3.194,1.513)--(3.234,1.522)--(3.275,1.530)--(3.315,1.539)--(3.355,1.547)--(3.395,1.556)--(3.436,1.564)--(3.476,1.572)--(3.516,1.580)--(3.557,1.588)--(3.597,1.596)--(3.637,1.604)--(3.678,1.612)--(3.718,1.619)--(3.758,1.627)--(3.798,1.634)--(3.839,1.642)--(3.879,1.649)--(3.919,1.656)--(3.960,1.663)--(4.000,1.670);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
+\draw [color=blue] (0.0100,-2.5236)--(0.0503,-1.3927)--(0.0906,-0.9808)--(0.1309,-0.7232)--(0.1712,-0.5353)--(0.2115,-0.3874)--(0.2518,-0.2653)--(0.2921,-0.1614)--(0.3324,-0.0709)--(0.3727,0.0091)--(0.4130,0.0810)--(0.4533,0.1462)--(0.4936,0.2058)--(0.5339,0.2607)--(0.5742,0.3117)--(0.6145,0.3591)--(0.6548,0.4036)--(0.6951,0.4454)--(0.7354,0.4849)--(0.7757,0.5222)--(0.8160,0.5577)--(0.8563,0.5914)--(0.8966,0.6236)--(0.9369,0.6544)--(0.9772,0.6839)--(1.0175,0.7121)--(1.0578,0.7393)--(1.0981,0.7655)--(1.1384,0.7907)--(1.1787,0.8151)--(1.2190,0.8386)--(1.2593,0.8614)--(1.2996,0.8834)--(1.3400,0.9048)--(1.3803,0.9256)--(1.4206,0.9457)--(1.4609,0.9653)--(1.5012,0.9843)--(1.5415,1.0029)--(1.5818,1.0210)--(1.6221,1.0386)--(1.6624,1.0557)--(1.7027,1.0725)--(1.7430,1.0889)--(1.7833,1.1049)--(1.8236,1.1205)--(1.8639,1.1358)--(1.9042,1.1508)--(1.9445,1.1655)--(1.9848,1.1798)--(2.0251,1.1939)--(2.0654,1.2077)--(2.1057,1.2212)--(2.1460,1.2345)--(2.1863,1.2475)--(2.2266,1.2603)--(2.2669,1.2729)--(2.3072,1.2852)--(2.3475,1.2973)--(2.3878,1.3092)--(2.4281,1.3209)--(2.4684,1.3325)--(2.5087,1.3438)--(2.5490,1.3550)--(2.5893,1.3659)--(2.6296,1.3768)--(2.6700,1.3874)--(2.7103,1.3979)--(2.7506,1.4082)--(2.7909,1.4184)--(2.8312,1.4284)--(2.8715,1.4383)--(2.9118,1.4481)--(2.9521,1.4577)--(2.9924,1.4672)--(3.0327,1.4766)--(3.0730,1.4858)--(3.1133,1.4949)--(3.1536,1.5039)--(3.1939,1.5128)--(3.2342,1.5216)--(3.2745,1.5303)--(3.3148,1.5388)--(3.3551,1.5473)--(3.3954,1.5557)--(3.4357,1.5639)--(3.4760,1.5721)--(3.5163,1.5801)--(3.5566,1.5881)--(3.5969,1.5960)--(3.6372,1.6038)--(3.6775,1.6115)--(3.7178,1.6192)--(3.7581,1.6267)--(3.7984,1.6342)--(3.8387,1.6416)--(3.8790,1.6489)--(3.9193,1.6561)--(3.9596,1.6633)--(4.0000,1.6704);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -301,21 +301,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.3242);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.3241);
 %DEFAULT
 
-\draw [color=blue] (1.000,0)--(1.030,0.1209)--(1.061,0.1698)--(1.091,0.2065)--(1.121,0.2368)--(1.152,0.2629)--(1.182,0.2861)--(1.212,0.3070)--(1.242,0.3261)--(1.273,0.3438)--(1.303,0.3601)--(1.333,0.3755)--(1.364,0.3898)--(1.394,0.4034)--(1.424,0.4163)--(1.455,0.4285)--(1.485,0.4401)--(1.515,0.4512)--(1.545,0.4618)--(1.576,0.4720)--(1.606,0.4818)--(1.636,0.4912)--(1.667,0.5003)--(1.697,0.5090)--(1.727,0.5175)--(1.758,0.5257)--(1.788,0.5336)--(1.818,0.5412)--(1.848,0.5487)--(1.879,0.5559)--(1.909,0.5629)--(1.939,0.5697)--(1.970,0.5763)--(2.000,0.5828)--(2.030,0.5891)--(2.061,0.5952)--(2.091,0.6012)--(2.121,0.6070)--(2.152,0.6127)--(2.182,0.6183)--(2.212,0.6237)--(2.242,0.6291)--(2.273,0.6343)--(2.303,0.6394)--(2.333,0.6443)--(2.364,0.6492)--(2.394,0.6540)--(2.424,0.6587)--(2.455,0.6633)--(2.485,0.6678)--(2.515,0.6723)--(2.545,0.6766)--(2.576,0.6809)--(2.606,0.6851)--(2.636,0.6892)--(2.667,0.6933)--(2.697,0.6972)--(2.727,0.7012)--(2.758,0.7050)--(2.788,0.7088)--(2.818,0.7125)--(2.848,0.7162)--(2.879,0.7198)--(2.909,0.7234)--(2.939,0.7269)--(2.970,0.7303)--(3.000,0.7337)--(3.030,0.7371)--(3.061,0.7403)--(3.091,0.7436)--(3.121,0.7468)--(3.152,0.7500)--(3.182,0.7531)--(3.212,0.7562)--(3.242,0.7592)--(3.273,0.7622)--(3.303,0.7652)--(3.333,0.7681)--(3.364,0.7710)--(3.394,0.7738)--(3.424,0.7766)--(3.455,0.7794)--(3.485,0.7821)--(3.515,0.7848)--(3.545,0.7875)--(3.576,0.7902)--(3.606,0.7928)--(3.636,0.7953)--(3.667,0.7979)--(3.697,0.8004)--(3.727,0.8029)--(3.758,0.8054)--(3.788,0.8078)--(3.818,0.8102)--(3.848,0.8126)--(3.879,0.8150)--(3.909,0.8173)--(3.939,0.8196)--(3.970,0.8219)--(4.000,0.8242);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (1.0000,0.0000)--(1.0303,0.1209)--(1.0606,0.1697)--(1.0909,0.2064)--(1.1212,0.2367)--(1.1515,0.2629)--(1.1818,0.2861)--(1.2121,0.3070)--(1.2424,0.3261)--(1.2727,0.3437)--(1.3030,0.3601)--(1.3333,0.3754)--(1.3636,0.3898)--(1.3939,0.4034)--(1.4242,0.4162)--(1.4545,0.4284)--(1.4848,0.4401)--(1.5151,0.4512)--(1.5454,0.4618)--(1.5757,0.4720)--(1.6060,0.4818)--(1.6363,0.4912)--(1.6666,0.5003)--(1.6969,0.5090)--(1.7272,0.5175)--(1.7575,0.5256)--(1.7878,0.5335)--(1.8181,0.5412)--(1.8484,0.5486)--(1.8787,0.5558)--(1.9090,0.5628)--(1.9393,0.5697)--(1.9696,0.5763)--(2.0000,0.5827)--(2.0303,0.5890)--(2.0606,0.5952)--(2.0909,0.6011)--(2.1212,0.6070)--(2.1515,0.6127)--(2.1818,0.6182)--(2.2121,0.6237)--(2.2424,0.6290)--(2.2727,0.6342)--(2.3030,0.6393)--(2.3333,0.6443)--(2.3636,0.6492)--(2.3939,0.6540)--(2.4242,0.6587)--(2.4545,0.6633)--(2.4848,0.6678)--(2.5151,0.6722)--(2.5454,0.6766)--(2.5757,0.6808)--(2.6060,0.6850)--(2.6363,0.6892)--(2.6666,0.6932)--(2.6969,0.6972)--(2.7272,0.7011)--(2.7575,0.7050)--(2.7878,0.7087)--(2.8181,0.7125)--(2.8484,0.7161)--(2.8787,0.7197)--(2.9090,0.7233)--(2.9393,0.7268)--(2.9696,0.7303)--(3.0000,0.7337)--(3.0303,0.7370)--(3.0606,0.7403)--(3.0909,0.7436)--(3.1212,0.7468)--(3.1515,0.7499)--(3.1818,0.7530)--(3.2121,0.7561)--(3.2424,0.7592)--(3.2727,0.7622)--(3.3030,0.7651)--(3.3333,0.7680)--(3.3636,0.7709)--(3.3939,0.7738)--(3.4242,0.7766)--(3.4545,0.7793)--(3.4848,0.7821)--(3.5151,0.7848)--(3.5454,0.7875)--(3.5757,0.7901)--(3.6060,0.7927)--(3.6363,0.7953)--(3.6666,0.7979)--(3.6969,0.8004)--(3.7272,0.8029)--(3.7575,0.8053)--(3.7878,0.8078)--(3.8181,0.8102)--(3.8484,0.8126)--(3.8787,0.8149)--(3.9090,0.8173)--(3.9393,0.8196)--(3.9696,0.8219)--(4.0000,0.8241);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -381,35 +381,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.4900,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-3.7236) -- (0,1.5529);
+\draw [,->,>=latex] (-1.4900,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7236) -- (0.0000,1.5528);
 %DEFAULT
 
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-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
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-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
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+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -471,39 +471,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-3.7725) -- (0,1.7828);
+\draw [,->,>=latex] (-2.2500,0.0000) -- (2.2500,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7725) -- (0.0000,1.7828);
 %DEFAULT
 
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-\draw (0.70000,-0.31492) node {$ 1 $};
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-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
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-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
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-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
+\draw [color=blue] (0.0675,-3.2725)--(0.0845,-2.9585)--(0.1015,-2.7022)--(0.1185,-2.4857)--(0.1355,-2.2982)--(0.1525,-2.1328)--(0.1695,-1.9850)--(0.1865,-1.8513)--(0.2035,-1.7292)--(0.2205,-1.6169)--(0.2375,-1.5130)--(0.2545,-1.4163)--(0.2715,-1.3258)--(0.2885,-1.2408)--(0.3055,-1.1607)--(0.3225,-1.0849)--(0.3395,-1.0130)--(0.3564,-0.9446)--(0.3734,-0.8794)--(0.3904,-0.8171)--(0.4074,-0.7575)--(0.4244,-0.7003)--(0.4414,-0.6453)--(0.4584,-0.5924)--(0.4754,-0.5415)--(0.4924,-0.4923)--(0.5094,-0.4448)--(0.5264,-0.3989)--(0.5434,-0.3544)--(0.5604,-0.3113)--(0.5774,-0.2695)--(0.5944,-0.2289)--(0.6114,-0.1894)--(0.6283,-0.1510)--(0.6453,-0.1137)--(0.6623,-0.0773)--(0.6793,-0.0418)--(0.6963,-0.0072)--(0.7133,0.0264)--(0.7303,0.0594)--(0.7473,0.0916)--(0.7643,0.1231)--(0.7813,0.1539)--(0.7983,0.1840)--(0.8153,0.2135)--(0.8323,0.2424)--(0.8493,0.2706)--(0.8663,0.2984)--(0.8833,0.3256)--(0.9003,0.3523)--(0.9172,0.3784)--(0.9342,0.4041)--(0.9512,0.4294)--(0.9682,0.4542)--(0.9852,0.4785)--(1.0022,0.5025)--(1.0192,0.5260)--(1.0362,0.5492)--(1.0532,0.5719)--(1.0702,0.5943)--(1.0872,0.6164)--(1.1042,0.6381)--(1.1212,0.6595)--(1.1382,0.6805)--(1.1552,0.7013)--(1.1722,0.7217)--(1.1891,0.7419)--(1.2061,0.7618)--(1.2231,0.7813)--(1.2401,0.8007)--(1.2571,0.8197)--(1.2741,0.8385)--(1.2911,0.8571)--(1.3081,0.8754)--(1.3251,0.8934)--(1.3421,0.9113)--(1.3591,0.9289)--(1.3761,0.9463)--(1.3931,0.9635)--(1.4101,0.9804)--(1.4271,0.9972)--(1.4441,1.0138)--(1.4611,1.0302)--(1.4780,1.0464)--(1.4950,1.0624)--(1.5120,1.0782)--(1.5290,1.0938)--(1.5460,1.1093)--(1.5630,1.1246)--(1.5800,1.1397)--(1.5970,1.1547)--(1.6140,1.1695)--(1.6310,1.1842)--(1.6480,1.1987)--(1.6650,1.2131)--(1.6820,1.2273)--(1.6990,1.2414)--(1.7160,1.2553)--(1.7330,1.2691)--(1.7500,1.2828);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -565,37 +565,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6000,0) -- (2.6000,0);
-\draw [,->,>=latex] (0,-3.5467) -- (0,1.2690);
+\draw [,->,>=latex] (-2.6000,0.0000) -- (2.6000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5466) -- (0.0000,1.2690);
 %DEFAULT
 
-\draw [color=blue] (-2.100,0.7690)--(-2.079,0.7620)--(-2.058,0.7548)--(-2.037,0.7476)--(-2.016,0.7403)--(-1.994,0.7329)--(-1.973,0.7255)--(-1.952,0.7179)--(-1.931,0.7103)--(-1.910,0.7026)--(-1.889,0.6948)--(-1.868,0.6870)--(-1.847,0.6790)--(-1.825,0.6709)--(-1.804,0.6628)--(-1.783,0.6546)--(-1.762,0.6462)--(-1.741,0.6378)--(-1.720,0.6292)--(-1.699,0.6206)--(-1.678,0.6118)--(-1.656,0.6030)--(-1.635,0.5940)--(-1.614,0.5849)--(-1.593,0.5756)--(-1.572,0.5663)--(-1.551,0.5568)--(-1.530,0.5472)--(-1.509,0.5375)--(-1.487,0.5276)--(-1.466,0.5176)--(-1.445,0.5075)--(-1.424,0.4972)--(-1.403,0.4867)--(-1.382,0.4761)--(-1.361,0.4653)--(-1.340,0.4544)--(-1.319,0.4432)--(-1.297,0.4319)--(-1.276,0.4204)--(-1.255,0.4088)--(-1.234,0.3969)--(-1.213,0.3848)--(-1.192,0.3725)--(-1.171,0.3600)--(-1.150,0.3472)--(-1.128,0.3343)--(-1.107,0.3210)--(-1.086,0.3075)--(-1.065,0.2938)--(-1.044,0.2798)--(-1.023,0.2655)--(-1.002,0.2509)--(-0.9806,0.2359)--(-0.9595,0.2207)--(-0.9383,0.2051)--(-0.9172,0.1892)--(-0.8961,0.1729)--(-0.8750,0.1562)--(-0.8539,0.1391)--(-0.8327,0.1215)--(-0.8116,0.1036)--(-0.7905,0.08510)--(-0.7694,0.06615)--(-0.7483,0.04666)--(-0.7271,0.02662)--(-0.7060,0.005983)--(-0.6849,-0.01528)--(-0.6638,-0.03721)--(-0.6426,-0.05984)--(-0.6215,-0.08323)--(-0.6004,-0.1074)--(-0.5793,-0.1325)--(-0.5582,-0.1585)--(-0.5370,-0.1855)--(-0.5159,-0.2136)--(-0.4948,-0.2429)--(-0.4737,-0.2734)--(-0.4526,-0.3053)--(-0.4314,-0.3388)--(-0.4103,-0.3739)--(-0.3892,-0.4109)--(-0.3681,-0.4500)--(-0.3470,-0.4913)--(-0.3258,-0.5353)--(-0.3047,-0.5822)--(-0.2836,-0.6325)--(-0.2625,-0.6867)--(-0.2413,-0.7454)--(-0.2202,-0.8095)--(-0.1991,-0.8801)--(-0.1780,-0.9586)--(-0.1569,-1.047)--(-0.1357,-1.148)--(-0.1146,-1.267)--(-0.09350,-1.409)--(-0.07238,-1.588)--(-0.05126,-1.830)--(-0.03013,-2.202)--(-0.009013,-3.047);
+\draw [color=blue] (-2.1000,0.7690)--(-2.0788,0.7619)--(-2.0577,0.7548)--(-2.0366,0.7475)--(-2.0155,0.7402)--(-1.9943,0.7329)--(-1.9732,0.7254)--(-1.9521,0.7179)--(-1.9310,0.7103)--(-1.9099,0.7026)--(-1.8887,0.6948)--(-1.8676,0.6869)--(-1.8465,0.6789)--(-1.8254,0.6709)--(-1.8043,0.6627)--(-1.7831,0.6545)--(-1.7620,0.6462)--(-1.7409,0.6377)--(-1.7198,0.6292)--(-1.6986,0.6205)--(-1.6775,0.6118)--(-1.6564,0.6029)--(-1.6353,0.5939)--(-1.6142,0.5848)--(-1.5930,0.5756)--(-1.5719,0.5663)--(-1.5508,0.5568)--(-1.5297,0.5472)--(-1.5086,0.5375)--(-1.4874,0.5276)--(-1.4663,0.5176)--(-1.4452,0.5074)--(-1.4241,0.4971)--(-1.4030,0.4867)--(-1.3818,0.4760)--(-1.3607,0.4653)--(-1.3396,0.4543)--(-1.3185,0.4432)--(-1.2973,0.4319)--(-1.2762,0.4204)--(-1.2551,0.4087)--(-1.2340,0.3968)--(-1.2129,0.3847)--(-1.1917,0.3724)--(-1.1706,0.3599)--(-1.1495,0.3472)--(-1.1284,0.3342)--(-1.1073,0.3210)--(-1.0861,0.3075)--(-1.0650,0.2937)--(-1.0439,0.2797)--(-1.0228,0.2654)--(-1.0017,0.2508)--(-0.9805,0.2359)--(-0.9594,0.2207)--(-0.9383,0.2051)--(-0.9172,0.1891)--(-0.8960,0.1728)--(-0.8749,0.1561)--(-0.8538,0.1390)--(-0.8327,0.1215)--(-0.8116,0.1035)--(-0.7904,0.0851)--(-0.7693,0.0661)--(-0.7482,0.0466)--(-0.7271,0.0266)--(-0.7060,0.0059)--(-0.6848,-0.0152)--(-0.6637,-0.0372)--(-0.6426,-0.0598)--(-0.6215,-0.0832)--(-0.6004,-0.1074)--(-0.5792,-0.1325)--(-0.5581,-0.1585)--(-0.5370,-0.1855)--(-0.5159,-0.2135)--(-0.4947,-0.2428)--(-0.4736,-0.2733)--(-0.4525,-0.3053)--(-0.4314,-0.3387)--(-0.4103,-0.3739)--(-0.3891,-0.4109)--(-0.3680,-0.4499)--(-0.3469,-0.4913)--(-0.3258,-0.5352)--(-0.3047,-0.5822)--(-0.2835,-0.6324)--(-0.2624,-0.6866)--(-0.2413,-0.7453)--(-0.2202,-0.8095)--(-0.1991,-0.8800)--(-0.1779,-0.9585)--(-0.1568,-1.0470)--(-0.1357,-1.1482)--(-0.1146,-1.2666)--(-0.0934,-1.4092)--(-0.0723,-1.5884)--(-0.0512,-1.8299)--(-0.0301,-2.2017)--(-0.0090,-3.0466);
 
-\draw [color=blue] (0.009013,-3.047)--(0.03013,-2.202)--(0.05126,-1.830)--(0.07238,-1.588)--(0.09350,-1.409)--(0.1146,-1.267)--(0.1357,-1.148)--(0.1569,-1.047)--(0.1780,-0.9586)--(0.1991,-0.8801)--(0.2202,-0.8095)--(0.2413,-0.7454)--(0.2625,-0.6867)--(0.2836,-0.6325)--(0.3047,-0.5822)--(0.3258,-0.5353)--(0.3470,-0.4913)--(0.3681,-0.4500)--(0.3892,-0.4109)--(0.4103,-0.3739)--(0.4314,-0.3388)--(0.4526,-0.3053)--(0.4737,-0.2734)--(0.4948,-0.2429)--(0.5159,-0.2136)--(0.5370,-0.1855)--(0.5582,-0.1585)--(0.5793,-0.1325)--(0.6004,-0.1074)--(0.6215,-0.08323)--(0.6426,-0.05984)--(0.6638,-0.03721)--(0.6849,-0.01528)--(0.7060,0.005983)--(0.7271,0.02662)--(0.7483,0.04666)--(0.7694,0.06615)--(0.7905,0.08510)--(0.8116,0.1036)--(0.8327,0.1215)--(0.8539,0.1391)--(0.8750,0.1562)--(0.8961,0.1729)--(0.9172,0.1892)--(0.9383,0.2051)--(0.9595,0.2207)--(0.9806,0.2359)--(1.002,0.2509)--(1.023,0.2655)--(1.044,0.2798)--(1.065,0.2938)--(1.086,0.3075)--(1.107,0.3210)--(1.128,0.3343)--(1.150,0.3472)--(1.171,0.3600)--(1.192,0.3725)--(1.213,0.3848)--(1.234,0.3969)--(1.255,0.4088)--(1.276,0.4204)--(1.297,0.4319)--(1.319,0.4432)--(1.340,0.4544)--(1.361,0.4653)--(1.382,0.4761)--(1.403,0.4867)--(1.424,0.4972)--(1.445,0.5075)--(1.466,0.5176)--(1.487,0.5276)--(1.509,0.5375)--(1.530,0.5472)--(1.551,0.5568)--(1.572,0.5663)--(1.593,0.5756)--(1.614,0.5849)--(1.635,0.5940)--(1.656,0.6030)--(1.678,0.6118)--(1.699,0.6206)--(1.720,0.6292)--(1.741,0.6378)--(1.762,0.6462)--(1.783,0.6546)--(1.804,0.6628)--(1.825,0.6709)--(1.847,0.6790)--(1.868,0.6870)--(1.889,0.6948)--(1.910,0.7026)--(1.931,0.7103)--(1.952,0.7179)--(1.973,0.7255)--(1.994,0.7329)--(2.016,0.7403)--(2.037,0.7476)--(2.058,0.7548)--(2.079,0.7620)--(2.100,0.7690);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (0.0090,-3.0466)--(0.0301,-2.2017)--(0.0512,-1.8299)--(0.0723,-1.5884)--(0.0934,-1.4092)--(0.1146,-1.2666)--(0.1357,-1.1482)--(0.1568,-1.0470)--(0.1779,-0.9585)--(0.1991,-0.8800)--(0.2202,-0.8095)--(0.2413,-0.7453)--(0.2624,-0.6866)--(0.2835,-0.6324)--(0.3047,-0.5822)--(0.3258,-0.5352)--(0.3469,-0.4913)--(0.3680,-0.4499)--(0.3891,-0.4109)--(0.4103,-0.3739)--(0.4314,-0.3387)--(0.4525,-0.3053)--(0.4736,-0.2733)--(0.4947,-0.2428)--(0.5159,-0.2135)--(0.5370,-0.1855)--(0.5581,-0.1585)--(0.5792,-0.1325)--(0.6004,-0.1074)--(0.6215,-0.0832)--(0.6426,-0.0598)--(0.6637,-0.0372)--(0.6848,-0.0152)--(0.7060,0.0059)--(0.7271,0.0266)--(0.7482,0.0466)--(0.7693,0.0661)--(0.7904,0.0851)--(0.8116,0.1035)--(0.8327,0.1215)--(0.8538,0.1390)--(0.8749,0.1561)--(0.8960,0.1728)--(0.9172,0.1891)--(0.9383,0.2051)--(0.9594,0.2207)--(0.9805,0.2359)--(1.0017,0.2508)--(1.0228,0.2654)--(1.0439,0.2797)--(1.0650,0.2937)--(1.0861,0.3075)--(1.1073,0.3210)--(1.1284,0.3342)--(1.1495,0.3472)--(1.1706,0.3599)--(1.1917,0.3724)--(1.2129,0.3847)--(1.2340,0.3968)--(1.2551,0.4087)--(1.2762,0.4204)--(1.2973,0.4319)--(1.3185,0.4432)--(1.3396,0.4543)--(1.3607,0.4653)--(1.3818,0.4760)--(1.4030,0.4867)--(1.4241,0.4971)--(1.4452,0.5074)--(1.4663,0.5176)--(1.4874,0.5276)--(1.5086,0.5375)--(1.5297,0.5472)--(1.5508,0.5568)--(1.5719,0.5663)--(1.5930,0.5756)--(1.6142,0.5848)--(1.6353,0.5939)--(1.6564,0.6029)--(1.6775,0.6118)--(1.6986,0.6205)--(1.7198,0.6292)--(1.7409,0.6377)--(1.7620,0.6462)--(1.7831,0.6545)--(1.8043,0.6627)--(1.8254,0.6709)--(1.8465,0.6789)--(1.8676,0.6869)--(1.8887,0.6948)--(1.9099,0.7026)--(1.9310,0.7103)--(1.9521,0.7179)--(1.9732,0.7254)--(1.9943,0.7329)--(2.0155,0.7402)--(2.0366,0.7475)--(2.0577,0.7548)--(2.0788,0.7619)--(2.1000,0.7690);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks b/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks
index 0ed1188bd..75c750781 100644
--- a/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks
+++ b/auto/pictures_tex/Fig_DefinitionCartesiennes.pstricks
@@ -103,47 +103,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-2.0000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
-\draw [color=blue,style=dashed] (3.00,1.00) -- (3.00,0);
-\draw [color=blue,style=dashed] (3.00,1.00) -- (0,1.00);
-\draw (3.5004,1.2141) node {$(3,1)$};
-\draw [color=blue,->,>=latex] (0,0) -- (3.0000,1.0000);
-\draw [color=green,style=dashed] (-1.50,-2.50) -- (-1.50,0);
-\draw [color=green,style=dashed] (-1.50,-2.50) -- (0,-2.50);
-\draw (-2.5247,-2.7682) node {$(-1.5,-2.5)$};
-\draw [color=green,->,>=latex] (0,0) -- (-1.5000,-2.5000);
-\draw [color=brown,style=dashed] (-1.00,2.50) -- (-1.00,0);
-\draw [color=brown,style=dashed] (-1.00,2.50) -- (0,2.50);
+\draw [color=blue,style=dashed] (3.0000,1.0000) -- (3.0000,0.0000);
+\draw [color=blue,style=dashed] (3.0000,1.0000) -- (0.0000,1.0000);
+\draw (3.5003,1.2140) node {$(3,1)$};
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (3.0000,1.0000);
+\draw [color=green,style=dashed] (-1.5000,-2.5000) -- (-1.5000,0.0000);
+\draw [color=green,style=dashed] (-1.5000,-2.5000) -- (0.0000,-2.5000);
+\draw (-2.5246,-2.7682) node {$(-1.5,-2.5)$};
+\draw [color=green,->,>=latex] (0.0000,0.0000) -- (-1.5000,-2.5000);
+\draw [color=brown,style=dashed] (-1.0000,2.5000) -- (-1.0000,0.0000);
+\draw [color=brown,style=dashed] (-1.0000,2.5000) -- (0.0000,2.5000);
 \draw (-1.7265,2.7753) node {$(-1,2.5)$};
-\draw [color=brown,->,>=latex] (0,0) -- (-1.0000,2.5000);
-\draw [color=cyan,style=dashed] (1.50,-1.00) -- (1.50,0);
-\draw [color=cyan,style=dashed] (1.50,-1.00) -- (0,-1.00);
-\draw (2.2726,-1.2379) node {$(1.5,-1)$};
-\draw [color=cyan,->,>=latex] (0,0) -- (1.5000,-1.0000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=brown,->,>=latex] (0.0000,0.0000) -- (-1.0000,2.5000);
+\draw [color=cyan,style=dashed] (1.5000,-1.0000) -- (1.5000,0.0000);
+\draw [color=cyan,style=dashed] (1.5000,-1.0000) -- (0.0000,-1.0000);
+\draw (2.2725,-1.2379) node {$(1.5,-1)$};
+\draw [color=cyan,->,>=latex] (0.0000,0.0000) -- (1.5000,-1.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DerivTangenteOM.pstricks b/auto/pictures_tex/Fig_DerivTangenteOM.pstricks
index a0c5b4ece..606108c8a 100644
--- a/auto/pictures_tex/Fig_DerivTangenteOM.pstricks
+++ b/auto/pictures_tex/Fig_DerivTangenteOM.pstricks
@@ -87,33 +87,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.8750,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,7.8518);
 %DEFAULT
-\draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
-\draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
-\draw [color=green,style=dashed] (3.00,1.50) -- (3.00,0);
-\draw [color=green,style=dashed] (6.50,6.09) -- (0,6.09);
-\draw [color=green,style=dashed] (3.00,1.50) -- (0,1.50);
-\draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
+\draw [color=cyan] (2.1250,0.3535) -- (7.3750,7.2320);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (6.5000,0.0000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (3.0000,0.0000);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (0.0000,6.0856);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (0.0000,1.5000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (6.5000,1.5000);
 
-\draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
+\draw [color=blue] (1.0000,1.0185)--(1.0606,1.0220)--(1.1212,1.0261)--(1.1818,1.0305)--(1.2424,1.0355)--(1.3030,1.0409)--(1.3636,1.0469)--(1.4242,1.0535)--(1.4848,1.0606)--(1.5454,1.0683)--(1.6060,1.0767)--(1.6666,1.0857)--(1.7272,1.0954)--(1.7878,1.1058)--(1.8484,1.1169)--(1.9090,1.1288)--(1.9696,1.1415)--(2.0303,1.1549)--(2.0909,1.1692)--(2.1515,1.1844)--(2.2121,1.2004)--(2.2727,1.2173)--(2.3333,1.2352)--(2.3939,1.2540)--(2.4545,1.2738)--(2.5151,1.2946)--(2.5757,1.3164)--(2.6363,1.3393)--(2.6969,1.3632)--(2.7575,1.3883)--(2.8181,1.4144)--(2.8787,1.4418)--(2.9393,1.4703)--(3.0000,1.5000)--(3.0606,1.5309)--(3.1212,1.5630)--(3.1818,1.5965)--(3.2424,1.6312)--(3.3030,1.6673)--(3.3636,1.7047)--(3.4242,1.7435)--(3.4848,1.7837)--(3.5454,1.8253)--(3.6060,1.8683)--(3.6666,1.9128)--(3.7272,1.9589)--(3.7878,2.0064)--(3.8484,2.0555)--(3.9090,2.1061)--(3.9696,2.1584)--(4.0303,2.2123)--(4.0909,2.2678)--(4.1515,2.3250)--(4.2121,2.3839)--(4.2727,2.4445)--(4.3333,2.5068)--(4.3939,2.5709)--(4.4545,2.6368)--(4.5151,2.7046)--(4.5757,2.7741)--(4.6363,2.8456)--(4.6969,2.9189)--(4.7575,2.9941)--(4.8181,3.0713)--(4.8787,3.1505)--(4.9393,3.2316)--(5.0000,3.3148)--(5.0606,3.4000)--(5.1212,3.4872)--(5.1818,3.5766)--(5.2424,3.6681)--(5.3030,3.7617)--(5.3636,3.8574)--(5.4242,3.9554)--(5.4848,4.0556)--(5.5454,4.1580)--(5.6060,4.2627)--(5.6666,4.3696)--(5.7272,4.4789)--(5.7878,4.5905)--(5.8484,4.7045)--(5.9090,4.8209)--(5.9696,4.9396)--(6.0303,5.0609)--(6.0909,5.1845)--(6.1515,5.3107)--(6.2121,5.4394)--(6.2727,5.5706)--(6.3333,5.7043)--(6.3939,5.8407)--(6.4545,5.9797)--(6.5151,6.1212)--(6.5757,6.2655)--(6.6363,6.4124)--(6.6969,6.5621)--(6.7575,6.7145)--(6.8181,6.8696)--(6.8787,7.0275)--(6.9393,7.1882)--(7.0000,7.3518);
 \draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,-0.27858) node {$a$};
-\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
-\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.2785) node {$a$};
+\draw []  (0.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.4473,1.5000) node {$f(a)$};
 \draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
-\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
-\draw (6.5000,-0.27858) node {$x$};
-\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
-\draw (-0.45521,6.0856) node {$f(x)$};
+\draw []  (6.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.2785) node {$x$};
+\draw []  (0.0000,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.4552,6.0856) node {$f(x)$};
 \draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
 \draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
-\draw (4.7500,0.97897) node {$x-a$};
+\draw (4.7500,0.9789) node {$x-a$};
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
-\draw (7.8256,3.7928) node {$f(x)-f(a)$};
+\draw (7.8255,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DessinLim.pstricks b/auto/pictures_tex/Fig_DessinLim.pstricks
index 59d0386bb..509b71d62 100644
--- a/auto/pictures_tex/Fig_DessinLim.pstricks
+++ b/auto/pictures_tex/Fig_DessinLim.pstricks
@@ -87,25 +87,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.8000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.8000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.8000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8000);
 %DEFAULT
 
-\draw [color=blue] (2.300,0)--(2.300,0.03649)--(2.299,0.07297)--(2.297,0.1094)--(2.295,0.1459)--(2.293,0.1823)--(2.290,0.2186)--(2.286,0.2549)--(2.281,0.2912)--(2.277,0.3273)--(2.271,0.3634)--(2.265,0.3994)--(2.258,0.4353)--(2.251,0.4711)--(2.243,0.5067)--(2.235,0.5422)--(2.226,0.5776)--(2.217,0.6129)--(2.207,0.6480)--(2.196,0.6829)--(2.185,0.7177)--(2.173,0.7523)--(2.161,0.7866)--(2.149,0.8208)--(2.135,0.8548)--(2.121,0.8886)--(2.107,0.9221)--(2.092,0.9555)--(2.077,0.9885)--(2.061,1.021)--(2.044,1.054)--(2.027,1.086)--(2.010,1.118)--(1.992,1.150)--(1.973,1.181)--(1.954,1.213)--(1.935,1.243)--(1.915,1.274)--(1.894,1.304)--(1.874,1.334)--(1.852,1.364)--(1.830,1.393)--(1.808,1.422)--(1.785,1.450)--(1.762,1.478)--(1.738,1.506)--(1.714,1.534)--(1.690,1.561)--(1.665,1.587)--(1.639,1.613)--(1.613,1.639)--(1.587,1.665)--(1.561,1.690)--(1.534,1.714)--(1.506,1.738)--(1.478,1.762)--(1.450,1.785)--(1.422,1.808)--(1.393,1.830)--(1.364,1.852)--(1.334,1.874)--(1.304,1.894)--(1.274,1.915)--(1.243,1.935)--(1.213,1.954)--(1.181,1.973)--(1.150,1.992)--(1.118,2.010)--(1.086,2.027)--(1.054,2.044)--(1.021,2.061)--(0.9885,2.077)--(0.9555,2.092)--(0.9221,2.107)--(0.8886,2.121)--(0.8548,2.135)--(0.8208,2.149)--(0.7866,2.161)--(0.7523,2.173)--(0.7177,2.185)--(0.6829,2.196)--(0.6480,2.207)--(0.6129,2.217)--(0.5776,2.226)--(0.5422,2.235)--(0.5067,2.243)--(0.4711,2.251)--(0.4353,2.258)--(0.3994,2.265)--(0.3634,2.271)--(0.3273,2.277)--(0.2912,2.281)--(0.2549,2.286)--(0.2186,2.290)--(0.1823,2.293)--(0.1459,2.295)--(0.1094,2.297)--(0.07297,2.299)--(0.03649,2.300)--(0,2.300);
-\draw [] (0,0) -- (2.30,2.30);
-\draw [style=dashed] (0,1.63) -- (1.63,1.63);
-\draw [style=dashed] (1.63,0) -- (1.63,1.63);
-\draw [] (2.30,2.30) -- (2.30,0);
-\draw []  (0,1.6263) node [rotate=0] {$\bullet$};
-\draw (-0.57160,1.6263) node {\( \sin(x)\)};
-\draw []  (1.6263,0) node [rotate=0] {$\bullet$};
-\draw (1.6263,-0.28245) node {\( \cos(x)\)};
-\draw []  (2.3000,0) node [rotate=0] {$\bullet$};
-\draw (2.4869,-0.21131) node {\( A\)};
+\draw [color=blue] (2.3000,0.0000)--(2.2997,0.0364)--(2.2988,0.0729)--(2.2973,0.1094)--(2.2953,0.1458)--(2.2927,0.1822)--(2.2895,0.2186)--(2.2858,0.2549)--(2.2814,0.2911)--(2.2765,0.3273)--(2.2711,0.3634)--(2.2650,0.3993)--(2.2584,0.4352)--(2.2512,0.4710)--(2.2434,0.5067)--(2.2351,0.5422)--(2.2262,0.5776)--(2.2168,0.6128)--(2.2068,0.6479)--(2.1962,0.6829)--(2.1851,0.7176)--(2.1735,0.7522)--(2.1612,0.7866)--(2.1485,0.8208)--(2.1352,0.8548)--(2.1214,0.8885)--(2.1070,0.9221)--(2.0921,0.9554)--(2.0767,0.9885)--(2.0607,1.0213)--(2.0443,1.0539)--(2.0273,1.0862)--(2.0098,1.1182)--(1.9918,1.1500)--(1.9733,1.1814)--(1.9543,1.2126)--(1.9348,1.2434)--(1.9149,1.2740)--(1.8944,1.3042)--(1.8735,1.3341)--(1.8521,1.3636)--(1.8302,1.3929)--(1.8079,1.4217)--(1.7851,1.4502)--(1.7619,1.4784)--(1.7382,1.5061)--(1.7141,1.5335)--(1.6895,1.5605)--(1.6645,1.5871)--(1.6391,1.6133)--(1.6133,1.6391)--(1.5871,1.6645)--(1.5605,1.6895)--(1.5335,1.7141)--(1.5061,1.7382)--(1.4784,1.7619)--(1.4502,1.7851)--(1.4217,1.8079)--(1.3929,1.8302)--(1.3636,1.8521)--(1.3341,1.8735)--(1.3042,1.8944)--(1.2740,1.9149)--(1.2434,1.9348)--(1.2126,1.9543)--(1.1814,1.9733)--(1.1500,1.9918)--(1.1182,2.0098)--(1.0862,2.0273)--(1.0539,2.0443)--(1.0213,2.0607)--(0.9885,2.0767)--(0.9554,2.0921)--(0.9221,2.1070)--(0.8885,2.1214)--(0.8548,2.1352)--(0.8208,2.1485)--(0.7866,2.1612)--(0.7522,2.1735)--(0.7176,2.1851)--(0.6829,2.1962)--(0.6479,2.2068)--(0.6128,2.2168)--(0.5776,2.2262)--(0.5422,2.2351)--(0.5067,2.2434)--(0.4710,2.2512)--(0.4352,2.2584)--(0.3993,2.2650)--(0.3634,2.2711)--(0.3273,2.2765)--(0.2911,2.2814)--(0.2549,2.2858)--(0.2186,2.2895)--(0.1822,2.2927)--(0.1458,2.2953)--(0.1094,2.2973)--(0.0729,2.2988)--(0.0364,2.2997)--(0.0000,2.3000);
+\draw [] (0.0000,0.0000) -- (2.3000,2.3000);
+\draw [style=dashed] (0.0000,1.6263) -- (1.6263,1.6263);
+\draw [style=dashed] (1.6263,0.0000) -- (1.6263,1.6263);
+\draw [] (2.3000,2.3000) -- (2.3000,0.0000);
+\draw []  (0.0000,1.6263) node [rotate=0] {$\bullet$};
+\draw (-0.5715,1.6263) node {\( \sin(x)\)};
+\draw []  (1.6263,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.6263,-0.2824) node {\( \cos(x)\)};
+\draw []  (2.3000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.4868,-0.2113) node {\( A\)};
 \draw []  (2.3000,2.3000) node [rotate=0] {$\bullet$};
-\draw (2.5320,2.3000) node {\( T\)};
+\draw (2.5319,2.3000) node {\( T\)};
 \draw []  (1.6263,1.6263) node [rotate=0] {$\bullet$};
-\draw (2.0689,1.6263) node {\( P\)};
+\draw (2.0688,1.6263) node {\( P\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DeuxCercles.pstricks b/auto/pictures_tex/Fig_DeuxCercles.pstricks
index b1136c0f4..c6ffd32e5 100644
--- a/auto/pictures_tex/Fig_DeuxCercles.pstricks
+++ b/auto/pictures_tex/Fig_DeuxCercles.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -76,17 +76,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_Differentielle.pstricks b/auto/pictures_tex/Fig_Differentielle.pstricks
index 00d72fbeb..311716e53 100644
--- a/auto/pictures_tex/Fig_Differentielle.pstricks
+++ b/auto/pictures_tex/Fig_Differentielle.pstricks
@@ -91,36 +91,36 @@
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-\draw [,->,>=latex] (0,-0.50000) -- (0,4.7000);
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+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.7000);
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+\draw [style=dotted] (2.0000,2.0000) -- (4.0000,2.0000);
+\draw [style=dotted] (4.0000,2.0000) -- (4.0000,4.0000);
 
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+\draw [color=red] (1.0000,1.0000)--(1.0323,1.0323)--(1.0646,1.0646)--(1.0969,1.0969)--(1.1292,1.1292)--(1.1616,1.1616)--(1.1939,1.1939)--(1.2262,1.2262)--(1.2585,1.2585)--(1.2909,1.2909)--(1.3232,1.3232)--(1.3555,1.3555)--(1.3878,1.3878)--(1.4202,1.4202)--(1.4525,1.4525)--(1.4848,1.4848)--(1.5171,1.5171)--(1.5494,1.5494)--(1.5818,1.5818)--(1.6141,1.6141)--(1.6464,1.6464)--(1.6787,1.6787)--(1.7111,1.7111)--(1.7434,1.7434)--(1.7757,1.7757)--(1.8080,1.8080)--(1.8404,1.8404)--(1.8727,1.8727)--(1.9050,1.9050)--(1.9373,1.9373)--(1.9696,1.9696)--(2.0020,2.0020)--(2.0343,2.0343)--(2.0666,2.0666)--(2.0989,2.0989)--(2.1313,2.1313)--(2.1636,2.1636)--(2.1959,2.1959)--(2.2282,2.2282)--(2.2606,2.2606)--(2.2929,2.2929)--(2.3252,2.3252)--(2.3575,2.3575)--(2.3898,2.3898)--(2.4222,2.4222)--(2.4545,2.4545)--(2.4868,2.4868)--(2.5191,2.5191)--(2.5515,2.5515)--(2.5838,2.5838)--(2.6161,2.6161)--(2.6484,2.6484)--(2.6808,2.6808)--(2.7131,2.7131)--(2.7454,2.7454)--(2.7777,2.7777)--(2.8101,2.8101)--(2.8424,2.8424)--(2.8747,2.8747)--(2.9070,2.9070)--(2.9393,2.9393)--(2.9717,2.9717)--(3.0040,3.0040)--(3.0363,3.0363)--(3.0686,3.0686)--(3.1010,3.1010)--(3.1333,3.1333)--(3.1656,3.1656)--(3.1979,3.1979)--(3.2303,3.2303)--(3.2626,3.2626)--(3.2949,3.2949)--(3.3272,3.3272)--(3.3595,3.3595)--(3.3919,3.3919)--(3.4242,3.4242)--(3.4565,3.4565)--(3.4888,3.4888)--(3.5212,3.5212)--(3.5535,3.5535)--(3.5858,3.5858)--(3.6181,3.6181)--(3.6505,3.6505)--(3.6828,3.6828)--(3.7151,3.7151)--(3.7474,3.7474)--(3.7797,3.7797)--(3.8121,3.8121)--(3.8444,3.8444)--(3.8767,3.8767)--(3.9090,3.9090)--(3.9414,3.9414)--(3.9737,3.9737)--(4.0060,4.0060)--(4.0383,4.0383)--(4.0707,4.0707)--(4.1030,4.1030)--(4.1353,4.1353)--(4.1676,4.1676)--(4.2000,4.2000);
 
-\draw [color=blue] (1.000,0.6137)--(1.032,0.6773)--(1.065,0.7390)--(1.097,0.7988)--(1.129,0.8569)--(1.162,0.9133)--(1.194,0.9682)--(1.226,1.022)--(1.259,1.074)--(1.291,1.124)--(1.323,1.174)--(1.356,1.222)--(1.388,1.269)--(1.420,1.315)--(1.453,1.360)--(1.485,1.404)--(1.517,1.447)--(1.549,1.490)--(1.582,1.531)--(1.614,1.571)--(1.646,1.611)--(1.679,1.650)--(1.711,1.688)--(1.743,1.725)--(1.776,1.762)--(1.808,1.798)--(1.840,1.834)--(1.873,1.868)--(1.905,1.903)--(1.937,1.936)--(1.970,1.969)--(2.002,2.002)--(2.034,2.034)--(2.067,2.066)--(2.099,2.097)--(2.131,2.127)--(2.164,2.157)--(2.196,2.187)--(2.228,2.216)--(2.261,2.245)--(2.293,2.273)--(2.325,2.301)--(2.358,2.329)--(2.390,2.356)--(2.422,2.383)--(2.455,2.410)--(2.487,2.436)--(2.519,2.462)--(2.552,2.487)--(2.584,2.512)--(2.616,2.537)--(2.648,2.562)--(2.681,2.586)--(2.713,2.610)--(2.745,2.634)--(2.778,2.657)--(2.810,2.680)--(2.842,2.703)--(2.875,2.726)--(2.907,2.748)--(2.939,2.770)--(2.972,2.792)--(3.004,2.814)--(3.036,2.835)--(3.069,2.856)--(3.101,2.877)--(3.133,2.898)--(3.166,2.918)--(3.198,2.939)--(3.230,2.959)--(3.263,2.979)--(3.295,2.998)--(3.327,3.018)--(3.360,3.037)--(3.392,3.056)--(3.424,3.075)--(3.457,3.094)--(3.489,3.113)--(3.521,3.131)--(3.554,3.150)--(3.586,3.168)--(3.618,3.186)--(3.651,3.203)--(3.683,3.221)--(3.715,3.239)--(3.747,3.256)--(3.780,3.273)--(3.812,3.290)--(3.844,3.307)--(3.877,3.324)--(3.909,3.340)--(3.941,3.357)--(3.974,3.373)--(4.006,3.389)--(4.038,3.405)--(4.071,3.421)--(4.103,3.437)--(4.135,3.453)--(4.168,3.468)--(4.200,3.484);
-\draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,3.3863);
+\draw [color=blue] (1.0000,0.6137)--(1.0323,0.6773)--(1.0646,0.7389)--(1.0969,0.7988)--(1.1292,0.8568)--(1.1616,0.9133)--(1.1939,0.9682)--(1.2262,1.0216)--(1.2585,1.0736)--(1.2909,1.1243)--(1.3232,1.1738)--(1.3555,1.2221)--(1.3878,1.2692)--(1.4202,1.3153)--(1.4525,1.3603)--(1.4848,1.4043)--(1.5171,1.4474)--(1.5494,1.4895)--(1.5818,1.5308)--(1.6141,1.5713)--(1.6464,1.6109)--(1.6787,1.6498)--(1.7111,1.6879)--(1.7434,1.7254)--(1.7757,1.7621)--(1.8080,1.7982)--(1.8404,1.8336)--(1.8727,1.8684)--(1.9050,1.9027)--(1.9373,1.9363)--(1.9696,1.9694)--(2.0020,2.0020)--(2.0343,2.0340)--(2.0666,2.0655)--(2.0989,2.0966)--(2.1313,2.1271)--(2.1636,2.1572)--(2.1959,2.1869)--(2.2282,2.2161)--(2.2606,2.2449)--(2.2929,2.2733)--(2.3252,2.3013)--(2.3575,2.3289)--(2.3898,2.3562)--(2.4222,2.3830)--(2.4545,2.4095)--(2.4868,2.4357)--(2.5191,2.4615)--(2.5515,2.4870)--(2.5838,2.5122)--(2.6161,2.5371)--(2.6484,2.5616)--(2.6808,2.5859)--(2.7131,2.6099)--(2.7454,2.6335)--(2.7777,2.6570)--(2.8101,2.6801)--(2.8424,2.7030)--(2.8747,2.7256)--(2.9070,2.7479)--(2.9393,2.7701)--(2.9717,2.7919)--(3.0040,2.8136)--(3.0363,2.8350)--(3.0686,2.8562)--(3.1010,2.8771)--(3.1333,2.8979)--(3.1656,2.9184)--(3.1979,2.9387)--(3.2303,2.9588)--(3.2626,2.9787)--(3.2949,2.9984)--(3.3272,3.0180)--(3.3595,3.0373)--(3.3919,3.0564)--(3.4242,3.0754)--(3.4565,3.0942)--(3.4888,3.1128)--(3.5212,3.1313)--(3.5535,3.1495)--(3.5858,3.1677)--(3.6181,3.1856)--(3.6505,3.2034)--(3.6828,3.2210)--(3.7151,3.2385)--(3.7474,3.2558)--(3.7797,3.2730)--(3.8121,3.2900)--(3.8444,3.3069)--(3.8767,3.3237)--(3.9090,3.3403)--(3.9414,3.3567)--(3.9737,3.3731)--(4.0060,3.3893)--(4.0383,3.4053)--(4.0707,3.4213)--(4.1030,3.4371)--(4.1353,3.4528)--(4.1676,3.4684)--(4.2000,3.4838);
+\draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,3.3862);
 \draw [,->,>=latex] (4.3000,3.6931) -- (4.3000,4.0000);
 \draw (5.1211,3.6931) node {$\epsilon(h)$};
 \draw [,->,>=latex] (5.5000,3.0000) -- (5.5000,2.0000);
 \draw [,->,>=latex] (5.5000,3.0000) -- (5.5000,4.0000);
-\draw (6.3791,3.0000) node {$T(h)$};
+\draw (6.3790,3.0000) node {$T(h)$};
 \draw [,->,>=latex] (3.0000,1.5000) -- (2.0000,1.5000);
 \draw [,->,>=latex] (3.0000,1.5000) -- (4.0000,1.5000);
-\draw (3.0000,1.0733) node {$h$};
+\draw (3.0000,1.0732) node {$h$};
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (1.2991,2.5360) node {$f(a)$};
+\draw (1.2990,2.5360) node {$f(a)$};
 \draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,3.3863) node [rotate=0] {$\bullet$};
-\draw (4.8502,2.7089) node {$f(x)$};
+\draw []  (4.0000,3.3862) node [rotate=0] {$\bullet$};
+\draw (4.8501,2.7088) node {$f(x)$};
 \draw []  (4.0000,4.0000) node [rotate=0] {$\bullet$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.37858) node {$a$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.37858) node {$x$};
-\draw [style=dotted] (2.00,2.00) -- (2.00,0);
-\draw [style=dotted] (4.00,2.00) -- (4.00,0);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3785) node {$a$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.3785) node {$x$};
+\draw [style=dotted] (2.0000,2.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (4.0000,2.0000) -- (4.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DisqueConv.pstricks b/auto/pictures_tex/Fig_DisqueConv.pstricks
index f268bbcad..3daf3e818 100644
--- a/auto/pictures_tex/Fig_DisqueConv.pstricks
+++ b/auto/pictures_tex/Fig_DisqueConv.pstricks
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (1.7653,1.8233) node {$z_0$};
+\draw (1.7652,1.8233) node {$z_0$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -79,8 +79,8 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DistanceEnsemble.pstricks b/auto/pictures_tex/Fig_DistanceEnsemble.pstricks
index fb287b1fd..9b810198e 100644
--- a/auto/pictures_tex/Fig_DistanceEnsemble.pstricks
+++ b/auto/pictures_tex/Fig_DistanceEnsemble.pstricks
@@ -77,16 +77,16 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (1.7632,1.7511) node {$A$};
-\draw [] (-3.98,-2.30) -- (-1.73,-1.00);
-\draw [style=dotted] (-3.98,-2.30) -- (-0.347,1.97);
-\draw [style=dotted] (-3.98,-2.30) -- (0,-1.40);
+\draw (1.7632,1.7510) node {$A$};
+\draw [] (-3.9837,-2.3000) -- (-1.7320,-1.0000);
+\draw [style=dotted] (-3.9837,-2.3000) -- (-0.3472,1.9696);
+\draw [style=dotted] (-3.9837,-2.3000) -- (0.0000,-1.4000);
 
-\draw [] (2.00,0)--(2.00,0.127)--(1.98,0.253)--(1.96,0.379)--(1.94,0.502)--(1.90,0.624)--(1.86,0.743)--(1.81,0.860)--(1.75,0.972)--(1.68,1.08)--(1.61,1.19)--(1.53,1.29)--(1.45,1.38)--(1.36,1.47)--(1.26,1.55)--(1.16,1.63)--(1.05,1.70)--(0.945,1.76)--(0.831,1.82)--(0.714,1.87)--(0.594,1.91)--(0.472,1.94)--(0.347,1.97)--(0.222,1.99)--(0.0952,2.00)--(-0.0317,2.00)--(-0.158,1.99)--(-0.285,1.98)--(-0.410,1.96)--(-0.533,1.93)--(-0.654,1.89)--(-0.773,1.84)--(-0.888,1.79)--(-1.00,1.73)--(-1.11,1.67)--(-1.21,1.59)--(-1.31,1.51)--(-1.40,1.43)--(-1.49,1.33)--(-1.57,1.24)--(-1.65,1.13)--(-1.72,1.03)--(-1.78,0.916)--(-1.83,0.802)--(-1.88,0.684)--(-1.92,0.563)--(-1.95,0.441)--(-1.97,0.316)--(-1.99,0.190)--(-2.00,0.0635)--(-2.00,-0.0635)--(-1.99,-0.190)--(-1.97,-0.316)--(-1.95,-0.441)--(-1.92,-0.563)--(-1.88,-0.684)--(-1.83,-0.802)--(-1.78,-0.916)--(-1.72,-1.03)--(-1.65,-1.13)--(-1.57,-1.24)--(-1.49,-1.33)--(-1.40,-1.43)--(-1.31,-1.51)--(-1.21,-1.59)--(-1.11,-1.67)--(-1.00,-1.73)--(-0.888,-1.79)--(-0.773,-1.84)--(-0.654,-1.89)--(-0.533,-1.93)--(-0.410,-1.96)--(-0.285,-1.98)--(-0.158,-1.99)--(-0.0317,-2.00)--(0.0952,-2.00)--(0.222,-1.99)--(0.347,-1.97)--(0.472,-1.94)--(0.594,-1.91)--(0.714,-1.87)--(0.831,-1.82)--(0.945,-1.76)--(1.05,-1.70)--(1.16,-1.63)--(1.26,-1.55)--(1.36,-1.47)--(1.45,-1.38)--(1.53,-1.29)--(1.61,-1.19)--(1.68,-1.08)--(1.75,-0.972)--(1.81,-0.860)--(1.86,-0.743)--(1.90,-0.624)--(1.94,-0.502)--(1.96,-0.379)--(1.98,-0.253)--(2.00,-0.127)--(2.00,0);
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 \draw []  (-1.7320,-1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.7320,-1.6141) node {$p$};
-\draw []  (0,-1.4000) node [rotate=0] {$\bullet$};
-\draw []  (-0.34729,1.9696) node [rotate=0] {$\bullet$};
+\draw (-1.7320,-1.6140) node {$p$};
+\draw []  (0.0000,-1.4000) node [rotate=0] {$\bullet$};
+\draw []  (-0.3472,1.9696) node [rotate=0] {$\bullet$};
 \draw []  (-3.9837,-2.3000) node [rotate=0] {$\bullet$};
 \draw (-4.3880,-2.3000) node {$x$};
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_DistanceEuclide.pstricks b/auto/pictures_tex/Fig_DistanceEuclide.pstricks
index de1bd21a0..c4015af51 100644
--- a/auto/pictures_tex/Fig_DistanceEuclide.pstricks
+++ b/auto/pictures_tex/Fig_DistanceEuclide.pstricks
@@ -91,32 +91,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
 \draw []  (1.0000,4.0000) node [rotate=0] {$\bullet$};
 \draw (1.0000,4.4901) node {$(A_x,A_y)$};
 \draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.9711,1.0000) node {$(B_x,B_y)$};
+\draw (3.9710,1.0000) node {$(B_x,B_y)$};
 \draw []  (3.0000,4.0000) node [rotate=0] {$\bullet$};
 \draw (3.3556,4.3368) node {$C$};
-\draw [] (1.00,4.00) -- (3.00,1.00);
-\draw [] (1.00,4.00) -- (3.00,4.00);
-\draw [] (3.00,1.00) -- (3.00,4.00);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [] (1.0000,4.0000) -- (3.0000,1.0000);
+\draw [] (1.0000,4.0000) -- (3.0000,4.0000);
+\draw [] (3.0000,1.0000) -- (3.0000,4.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_DivergenceDeux.pstricks b/auto/pictures_tex/Fig_DivergenceDeux.pstricks
index 1b692ca44..86dad7615 100644
--- a/auto/pictures_tex/Fig_DivergenceDeux.pstricks
+++ b/auto/pictures_tex/Fig_DivergenceDeux.pstricks
@@ -65,230 +65,230 @@
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diff --git a/auto/pictures_tex/Fig_DivergenceTrois.pstricks b/auto/pictures_tex/Fig_DivergenceTrois.pstricks
index 8a25fb77c..1c983862e 100644
--- a/auto/pictures_tex/Fig_DivergenceTrois.pstricks
+++ b/auto/pictures_tex/Fig_DivergenceTrois.pstricks
@@ -65,51 +65,51 @@
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+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2337,2.3504);
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+\draw [,->,>=latex] (-2.1953,0.1422) -- (-2.6489,0.1715);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DivergenceUn.pstricks b/auto/pictures_tex/Fig_DivergenceUn.pstricks
index 8f1c5afca..eb4722e04 100644
--- a/auto/pictures_tex/Fig_DivergenceUn.pstricks
+++ b/auto/pictures_tex/Fig_DivergenceUn.pstricks
@@ -65,58 +65,58 @@
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 %PSTRICKS CODE
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-\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
-\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
-\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,-0.6666) -- (4.7539,-0.6666);
+\draw [,->,>=latex] (4.5333,-0.3333) -- (4.7539,-0.3333);
+\draw [,->,>=latex] (4.5333,0.0000) -- (4.7539,0.0000);
+\draw [,->,>=latex] (4.5333,0.3333) -- (4.7539,0.3333);
+\draw [,->,>=latex] (4.5333,0.6666) -- (4.7539,0.6666);
 \draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
-\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
-\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
-\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
-\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
-\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
-\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
-\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
-\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
-\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
-\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
-\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
-\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
-\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
-\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
+\draw [,->,>=latex] (5.2666,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2666,-0.6666) -- (5.4565,-0.6666);
+\draw [,->,>=latex] (5.2666,-0.3333) -- (5.4565,-0.3333);
+\draw [,->,>=latex] (5.2666,0.0000) -- (5.4565,0.0000);
+\draw [,->,>=latex] (5.2666,0.3333) -- (5.4565,0.3333);
+\draw [,->,>=latex] (5.2666,0.6666) -- (5.4565,0.6666);
+\draw [,->,>=latex] (5.2666,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1666,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.6666) -- (6.1666,-0.6666);
+\draw [,->,>=latex] (6.0000,-0.3333) -- (6.1666,-0.3333);
+\draw [,->,>=latex] (6.0000,0.0000) -- (6.1666,0.0000);
+\draw [,->,>=latex] (6.0000,0.3333) -- (6.1666,0.3333);
+\draw [,->,>=latex] (6.0000,0.6666) -- (6.1666,0.6666);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1666,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks b/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks
index 40c268f53..d26af81de 100644
--- a/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks
+++ b/auto/pictures_tex/Fig_DynkinNUtPJx.pstricks
@@ -39,17 +39,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.31492) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3149) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -84,17 +84,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.31492) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.3149) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -129,17 +129,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.31492) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.3149) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -174,17 +174,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw (3.2327,-0.24365) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw (3.2326,-0.2436) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -219,17 +219,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,-0.500);
-\draw [] (2.00,0) -- (3.00,0.500);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw []  (3.0000,-0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw []  (3.0000,0.50000) node [rotate=0] {$o$};
-\draw (3.2327,0.75635) node {\(1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,-0.5000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (3.0000,-0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw []  (3.0000,0.5000) node [rotate=0] {$o$};
+\draw (3.2326,0.7563) node {\(1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks b/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks
index aad544245..10a89f5ff 100644
--- a/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks
+++ b/auto/pictures_tex/Fig_DynkinpWjUbE.pstricks
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.26444) node {\( 1\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2644) node {\( 1\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks b/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks
index 569d4db5b..9dd3d2fc7 100644
--- a/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks
+++ b/auto/pictures_tex/Fig_DynkinqlgIQl.pstricks
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.26444) node {\( 2\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2644) node {\( 2\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks b/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks
index de90bcf5a..61a64d0bf 100644
--- a/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks
+++ b/auto/pictures_tex/Fig_DynkinrjbHIu.pstricks
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.26444) node {\( 1\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2644) node {\( 1\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks b/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks
index ad2371f2a..129d6ad96 100644
--- a/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks
+++ b/auto/pictures_tex/Fig_EELKooMwkockxB.pstricks
@@ -94,29 +94,29 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (8.00,2.00);
-\draw []  (1.6000,0.40000) node [rotate=0] {$\bullet$};
-\draw (1.7450,0.12740) node {\( a\)};
+\draw [] (0.0000,0.0000) -- (8.0000,2.0000);
+\draw []  (1.6000,0.4000) node [rotate=0] {$\bullet$};
+\draw (1.7449,0.1273) node {\( a\)};
 \draw []  (4.8000,1.2000) node [rotate=0] {$\bullet$};
-\draw (4.9268,0.87924) node {\( b\)};
+\draw (4.9268,0.8792) node {\( b\)};
 \draw []  (6.4000,1.6000) node [rotate=0] {$\bullet$};
-\draw (6.5275,1.3274) node {\( c\)};
+\draw (6.5274,1.3273) node {\( c\)};
 \draw []  (4.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (4.1528,0.72740) node {\( x\)};
-\draw [] (1.60,0.400) -- (2.60,3.40);
-\draw [] (4.80,1.20) -- (2.60,3.40);
+\draw (4.1528,0.7273) node {\( x\)};
+\draw [] (1.6000,0.4000) -- (2.6000,3.4000);
+\draw [] (4.8000,1.2000) -- (2.6000,3.4000);
 \draw []  (2.2000,2.2000) node [rotate=0] {$\bullet$};
 \draw (1.9184,2.3773) node {\( p\)};
-\draw []  (4.0667,1.9333) node [rotate=0] {$\bullet$};
-\draw (4.3353,2.1333) node {\( q\)};
-\draw [] (6.40,1.60) -- (0.940,2.38);
+\draw []  (4.0666,1.9333) node [rotate=0] {$\bullet$};
+\draw (4.3353,2.1332) node {\( q\)};
+\draw [] (6.4000,1.6000) -- (0.9400,2.3800);
 \draw []  (2.6000,3.4000) node [rotate=0] {$\bullet$};
 \draw (2.8817,3.6374) node {\( m\)};
-\draw [] (4.07,1.93) -- (1.60,0.400);
-\draw [] (2.20,2.20) -- (4.80,1.20);
-\draw []  (3.6182,1.6545) node [rotate=0] {$\bullet$};
-\draw (3.3268,1.4928) node {\( n\)};
-\draw [] (2.60,3.40) -- (4.00,1.00);
+\draw [] (4.0666,1.9333) -- (1.6000,0.4000);
+\draw [] (2.2000,2.2000) -- (4.8000,1.2000);
+\draw []  (3.6181,1.6545) node [rotate=0] {$\bullet$};
+\draw (3.3267,1.4927) node {\( n\)};
+\draw [] (2.6000,3.4000) -- (4.0000,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks b/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks
index 61e7d4c9d..9925ad92e 100644
--- a/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks
+++ b/auto/pictures_tex/Fig_EHDooGDwfjC.pstricks
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.32471) node {\( A\)};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.2890,0.26613) node {\( B\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.3247) node {\( A\)};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.2890,0.2661) node {\( B\)};
 \draw []  (12.000,-3.0000) node [rotate=0] {$\bullet$};
-\draw (12.285,-3.2661) node {\( C\)};
+\draw (12.284,-3.2661) node {\( C\)};
 \draw []  (4.0000,-1.0000) node [rotate=0] {$\bullet$};
 \draw (3.6905,-1.2661) node {\( K\)};
-\draw []  (1.3333,0) node [rotate=0] {$\bullet$};
-\draw (1.3333,0.32471) node {\( L\)};
-\draw [] (0,0) -- (4.00,0);
-\draw [] (0,0) -- (12.0,-3.00);
-\draw [style=dashed] (12.0,-3.00) -- (4.00,0);
-\draw [style=dashed] (4.00,-1.00) -- (1.33,0);
+\draw []  (1.3333,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.3333,0.3247) node {\( L\)};
+\draw [] (0.0000,0.0000) -- (4.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (12.000,-3.0000);
+\draw [style=dashed] (12.000,-3.0000) -- (4.0000,0.0000);
+\draw [style=dashed] (4.0000,-1.0000) -- (1.3333,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_EJRsWXw.pstricks b/auto/pictures_tex/Fig_EJRsWXw.pstricks
index c26000343..552d43246 100644
--- a/auto/pictures_tex/Fig_EJRsWXw.pstricks
+++ b/auto/pictures_tex/Fig_EJRsWXw.pstricks
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,17 +81,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-1.00,1.00) -- (-2.00,2.00) -- (-2.00,2.00) -- (1.00,3.00) -- (1.00,3.00) -- (-1.00,1.00) --  cycle;
-\draw [] (-1.00,1.00) -- (-2.00,2.00);
-\draw [] (-2.00,2.00) -- (1.00,3.00);
-\draw [] (1.00,3.00) -- (-1.00,1.00);
-\draw [color=green] (-1.00,1.00) -- (-2.00,2.00);
-\draw [color=green] (-1.00,1.00) -- (1.00,3.00);
-\draw [color=red] (-2.00,2.00) -- (1.00,3.00);
-\draw (1.5000,-0.27858) node {\( x\)};
-\draw (1.5000,-0.27858) node {\( x\)};
-\draw (0.26590,3.5000) node {\( t\)};
-\draw (0.26590,3.5000) node {\( t\)};
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-1.0000,1.0000) -- (-2.0000,2.0000) -- (-2.0000,2.0000) -- (1.0000,3.0000) -- (1.0000,3.0000) -- (-1.0000,1.0000) --  cycle;
+\draw [] (-1.0000,1.0000) -- (-2.0000,2.0000);
+\draw [] (-2.0000,2.0000) -- (1.0000,3.0000);
+\draw [] (1.0000,3.0000) -- (-1.0000,1.0000);
+\draw [color=green] (-1.0000,1.0000) -- (-2.0000,2.0000);
+\draw [color=green] (-1.0000,1.0000) -- (1.0000,3.0000);
+\draw [color=red] (-2.0000,2.0000) -- (1.0000,3.0000);
+\draw (1.5000,-0.2785) node {\( x\)};
+\draw (1.5000,-0.2785) node {\( x\)};
+\draw (0.2659,3.5000) node {\( t\)};
+\draw (0.2659,3.5000) node {\( t\)};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
index 9a27504e5..662c87d49 100644
--- a/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
+++ b/auto/pictures_tex/Fig_ERPMooZibfNOiU.pstricks
@@ -78,33 +78,33 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (2.0868,1.7595) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.4102,2.9353);
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (3.4102,2.9353);
 \draw []  (1.7350,1.4643) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.34725,0.36520);
-\draw []  (2.6575,2.4377) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.2478,3.0982);
-\draw []  (0.96811,1.0201) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.21509,0.55343);
-\draw []  (2.8537,2.7936) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.1558,3.1756);
-\draw []  (0.58356,0.88873) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.15485,0.65749);
-\draw []  (3.0136,3.3088) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.9997,3.2873);
-\draw []  (0.048375,0.82073) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.071918,0.83064);
-\draw []  (2.8961,4.0071) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.6718,3.4583);
-\draw []  (-0.61889,1.0577) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (-0.039497,1.1833);
-\draw [] (-3.00,0) -- (3.00,0);
-\draw [] (0,0) -- (3.83,3.21);
-\draw (0.77455,0.24959) node {\( \alpha\)};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.3472,0.3652);
+\draw []  (2.6575,2.4376) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (3.2478,3.0982);
+\draw []  (0.9681,1.0200) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.2150,0.5534);
+\draw []  (2.8536,2.7935) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (3.1558,3.1756);
+\draw []  (0.5835,0.8887) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.1548,0.6574);
+\draw []  (3.0135,3.3088) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (2.9997,3.2873);
+\draw []  (0.0483,0.8207) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.0719,0.8306);
+\draw []  (2.8960,4.0070) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (2.6717,3.4583);
+\draw []  (-0.6188,1.0576) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (-0.0394,1.1832);
+\draw [] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (3.8302,3.2139);
+\draw (0.7745,0.2495) node {\( \alpha\)};
 
-\draw [] (0.500,0)--(0.500,0.00353)--(0.500,0.00705)--(0.500,0.0106)--(0.500,0.0141)--(0.500,0.0176)--(0.500,0.0211)--(0.499,0.0247)--(0.499,0.0282)--(0.499,0.0317)--(0.499,0.0352)--(0.499,0.0387)--(0.498,0.0423)--(0.498,0.0458)--(0.498,0.0493)--(0.497,0.0528)--(0.497,0.0563)--(0.496,0.0598)--(0.496,0.0633)--(0.496,0.0668)--(0.495,0.0703)--(0.495,0.0738)--(0.494,0.0773)--(0.493,0.0807)--(0.493,0.0842)--(0.492,0.0877)--(0.492,0.0912)--(0.491,0.0946)--(0.490,0.0981)--(0.490,0.102)--(0.489,0.105)--(0.488,0.108)--(0.487,0.112)--(0.487,0.115)--(0.486,0.119)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.132)--(0.481,0.136)--(0.480,0.139)--(0.479,0.143)--(0.478,0.146)--(0.477,0.149)--(0.476,0.153)--(0.475,0.156)--(0.474,0.159)--(0.473,0.163)--(0.472,0.166)--(0.470,0.169)--(0.469,0.173)--(0.468,0.176)--(0.467,0.179)--(0.465,0.183)--(0.464,0.186)--(0.463,0.189)--(0.462,0.192)--(0.460,0.196)--(0.459,0.199)--(0.457,0.202)--(0.456,0.205)--(0.454,0.209)--(0.453,0.212)--(0.451,0.215)--(0.450,0.218)--(0.448,0.221)--(0.447,0.224)--(0.445,0.228)--(0.444,0.231)--(0.442,0.234)--(0.440,0.237)--(0.439,0.240)--(0.437,0.243)--(0.435,0.246)--(0.433,0.249)--(0.432,0.252)--(0.430,0.255)--(0.428,0.258)--(0.426,0.261)--(0.424,0.264)--(0.423,0.267)--(0.421,0.270)--(0.419,0.273)--(0.417,0.276)--(0.415,0.279)--(0.413,0.282)--(0.411,0.285)--(0.409,0.288)--(0.407,0.291)--(0.405,0.294)--(0.403,0.296)--(0.401,0.299)--(0.398,0.302)--(0.396,0.305)--(0.394,0.308)--(0.392,0.310)--(0.390,0.313)--(0.388,0.316)--(0.385,0.319)--(0.383,0.321);
-\draw []  (1.9151,1.6070) node [rotate=0] {$\bullet$};
-\draw (2.1862,1.3291) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.38)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw [] (0.5000,0.0000)--(0.4999,0.0035)--(0.4999,0.0070)--(0.4998,0.0105)--(0.4998,0.0141)--(0.4996,0.0176)--(0.4995,0.0211)--(0.4993,0.0246)--(0.4992,0.0281)--(0.4989,0.0317)--(0.4987,0.0352)--(0.4984,0.0387)--(0.4982,0.0422)--(0.4979,0.0457)--(0.4975,0.0492)--(0.4972,0.0527)--(0.4968,0.0562)--(0.4964,0.0597)--(0.4959,0.0632)--(0.4955,0.0667)--(0.4950,0.0702)--(0.4945,0.0737)--(0.4939,0.0772)--(0.4934,0.0807)--(0.4928,0.0842)--(0.4922,0.0876)--(0.4916,0.0911)--(0.4909,0.0946)--(0.4902,0.0980)--(0.4895,0.1015)--(0.4888,0.1049)--(0.4881,0.1084)--(0.4873,0.1118)--(0.4865,0.1153)--(0.4856,0.1187)--(0.4848,0.1221)--(0.4839,0.1255)--(0.4830,0.1289)--(0.4821,0.1323)--(0.4812,0.1357)--(0.4802,0.1391)--(0.4792,0.1425)--(0.4782,0.1459)--(0.4771,0.1493)--(0.4761,0.1526)--(0.4750,0.1560)--(0.4739,0.1593)--(0.4727,0.1627)--(0.4716,0.1660)--(0.4704,0.1693)--(0.4692,0.1726)--(0.4680,0.1759)--(0.4667,0.1792)--(0.4654,0.1825)--(0.4641,0.1858)--(0.4628,0.1890)--(0.4615,0.1923)--(0.4601,0.1956)--(0.4587,0.1988)--(0.4573,0.2020)--(0.4559,0.2052)--(0.4544,0.2085)--(0.4529,0.2117)--(0.4514,0.2148)--(0.4499,0.2180)--(0.4483,0.2212)--(0.4468,0.2243)--(0.4452,0.2275)--(0.4436,0.2306)--(0.4419,0.2338)--(0.4403,0.2369)--(0.4386,0.2400)--(0.4369,0.2430)--(0.4351,0.2461)--(0.4334,0.2492)--(0.4316,0.2522)--(0.4298,0.2553)--(0.4280,0.2583)--(0.4262,0.2613)--(0.4243,0.2643)--(0.4225,0.2673)--(0.4206,0.2703)--(0.4187,0.2732)--(0.4167,0.2762)--(0.4148,0.2791)--(0.4128,0.2820)--(0.4108,0.2849)--(0.4088,0.2878)--(0.4067,0.2907)--(0.4047,0.2936)--(0.4026,0.2964)--(0.4005,0.2992)--(0.3984,0.3021)--(0.3962,0.3049)--(0.3941,0.3076)--(0.3919,0.3104)--(0.3897,0.3132)--(0.3875,0.3159)--(0.3852,0.3186)--(0.3830,0.3213);
+\draw []  (1.9151,1.6069) node [rotate=0] {$\bullet$};
+\draw (2.1861,1.3290) node {\( P\)};
+\draw [] plot [smooth,tension=1] coordinates {(2.8413,4.0812)(2.7848,4.1414)(2.7214,4.1954)(2.6514,4.2429)(2.5750,4.2839)(2.4926,4.3181)(2.4046,4.3454)(2.3112,4.3657)(2.2128,4.3789)(2.1099,4.3849)(2.0028,4.3837)(1.8920,4.3754)(1.7779,4.3600)(1.6611,4.3374)(1.5418,4.3079)(1.4208,4.2715)(1.2983,4.2284)(1.1750,4.1788)(1.0512,4.1228)(0.9276,4.0607)(0.8046,3.9927)(0.6828,3.9191)(0.5625,3.8402)(0.4442,3.7564)(0.3286,3.6679)(0.2160,3.5751)(0.1068,3.4783)(0.0016,3.3781)(-0.0992,3.2747)(-0.1954,3.1687)(-0.2864,3.0603)(-0.3719,2.9501)(-0.4516,2.8385)(-0.5251,2.7260)(-0.5921,2.6130)(-0.6524,2.4999)(-0.7058,2.3872)(-0.7519,2.2755)(-0.7907,2.1650)(-0.8220,2.0563)(-0.8455,1.9498)(-0.8614,1.8459)(-0.8694,1.7450)(-0.8696,1.6477)(-0.8619,1.5541)(-0.8464,1.4648)(-0.8232,1.3801)(-0.7923,1.3003)(-0.7538,1.2257)(-0.7080,1.1567)(-0.6550,1.0935)(-0.5950,1.0364)(-0.5282,0.9856)(-0.4550,0.9413)(-0.3756,0.9037)(-0.2903,0.8729)(-0.1995,0.8491)(-0.1036,0.8324)(-0.0029,0.8228)(0.1020,0.8203)(0.2110,0.8251)(0.3235,0.8370)(0.4391,0.8560)(0.5572,0.8820)(0.6774,0.9150)(0.7992,0.9547)(0.9222,1.0011)(1.0457,1.0540)(1.1695,1.1131)(1.2928,1.1781)(1.4153,1.2490)(1.5365,1.3252)(1.6558,1.4067)(1.7728,1.4929)(1.8870,1.5836)(1.9979,1.6784)(2.1052,1.7769)(2.2083,1.8788)(2.3069,1.9835)(2.4005,2.0908)(2.4888,2.2001)(2.5715,2.3111)(2.6481,2.4232)(2.7184,2.5360)(2.7821,2.6491)(2.8390,2.7621)(2.8887,2.8743)(2.9312,2.9855)(2.9663,3.0952)(2.9937,3.2028)(3.0134,3.3081)(3.0254,3.4105)(3.0295,3.5097)(3.0257,3.6052)(3.0141,3.6966)(2.9947,3.7837)(2.9676,3.8660)(2.9329,3.9432)(2.8908,4.0151)(2.8413,4.0812)};
 \draw (4.1391,3.5225) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExPolygone.pstricks b/auto/pictures_tex/Fig_ExPolygone.pstricks
index c729554c1..fec2fe5fb 100644
--- a/auto/pictures_tex/Fig_ExPolygone.pstricks
+++ b/auto/pictures_tex/Fig_ExPolygone.pstricks
@@ -83,13 +83,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
-\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
+\draw [,->,>=latex] (-1.6500,0.0000) -- (3.6500,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6500) -- (0.0000,2.6500);
 %DEFAULT
-\draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
-\draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
-\draw [color=red] (2.15,-2.15) -- (-1.15,1.15);
-\draw [color=red] (-1.15,-1.15) -- (2.15,2.15);
+\draw [color=red] (-0.1500,2.1500) -- (3.1500,-1.1500);
+\draw [color=red] (3.1500,1.1500) -- (-0.1500,-2.1500);
+\draw [color=red] (2.1500,-2.1500) -- (-1.1500,1.1500);
+\draw [color=red] (-1.1500,-1.1500) -- (2.1500,2.1500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,27 +97,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,1.00) -- (2.00,0) -- (2.00,0) -- (1.00,-1.00) -- (1.00,-1.00) -- (0,0) -- (0,0) -- (1.00,1.00) --  cycle;
-\draw [color=blue] (1.00,1.00) -- (2.00,0);
-\draw [color=blue] (2.00,0) -- (1.00,-1.00);
-\draw [color=blue] (1.00,-1.00) -- (0,0);
-\draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,1.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (1.0000,-1.0000) -- (1.0000,-1.0000) -- (0.0000,0.0000) -- (0.0000,0.0000) -- (1.0000,1.0000) --  cycle;
+\draw [color=blue] (1.0000,1.0000) -- (2.0000,0.0000);
+\draw [color=blue] (2.0000,0.0000) -- (1.0000,-1.0000);
+\draw [color=blue] (1.0000,-1.0000) -- (0.0000,0.0000);
+\draw [color=blue] (0.0000,0.0000) -- (1.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExSinLarge.pstricks b/auto/pictures_tex/Fig_ExSinLarge.pstricks
index 306d962f8..c2542d003 100644
--- a/auto/pictures_tex/Fig_ExSinLarge.pstricks
+++ b/auto/pictures_tex/Fig_ExSinLarge.pstricks
@@ -75,26 +75,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.6416,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4999);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.6415,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4998);
 %DEFAULT
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+\draw [color=blue,style=solid] (0.0000,2.0000)--(0.0317,2.0317)--(0.0634,2.0634)--(0.0951,2.0950)--(0.1269,2.1265)--(0.1586,2.1580)--(0.1903,2.1892)--(0.2221,2.2203)--(0.2538,2.2511)--(0.2855,2.2817)--(0.3173,2.3120)--(0.3490,2.3420)--(0.3807,2.3716)--(0.4125,2.4009)--(0.4442,2.4297)--(0.4759,2.4582)--(0.5077,2.4861)--(0.5394,2.5136)--(0.5711,2.5406)--(0.6029,2.5670)--(0.6346,2.5929)--(0.6663,2.6181)--(0.6981,2.6427)--(0.7298,2.6667)--(0.7615,2.6900)--(0.7933,2.7126)--(0.8250,2.7345)--(0.8567,2.7557)--(0.8885,2.7761)--(0.9202,2.7957)--(0.9519,2.8145)--(0.9837,2.8325)--(1.0154,2.8497)--(1.0471,2.8660)--(1.0789,2.8814)--(1.1106,2.8959)--(1.1423,2.9096)--(1.1741,2.9223)--(1.2058,2.9341)--(1.2375,2.9450)--(1.2693,2.9549)--(1.3010,2.9638)--(1.3327,2.9718)--(1.3645,2.9788)--(1.3962,2.9848)--(1.4279,2.9898)--(1.4597,2.9938)--(1.4914,2.9968)--(1.5231,2.9988)--(1.5549,2.9998)--(1.5866,2.9998)--(1.6183,2.9988)--(1.6501,2.9968)--(1.6818,2.9938)--(1.7135,2.9898)--(1.7453,2.9848)--(1.7770,2.9788)--(1.8087,2.9718)--(1.8405,2.9638)--(1.8722,2.9549)--(1.9039,2.9450)--(1.9357,2.9341)--(1.9674,2.9223)--(1.9991,2.9096)--(2.0309,2.8959)--(2.0626,2.8814)--(2.0943,2.8660)--(2.1261,2.8497)--(2.1578,2.8325)--(2.1895,2.8145)--(2.2213,2.7957)--(2.2530,2.7761)--(2.2847,2.7557)--(2.3165,2.7345)--(2.3482,2.7126)--(2.3799,2.6900)--(2.4117,2.6667)--(2.4434,2.6427)--(2.4751,2.6181)--(2.5069,2.5929)--(2.5386,2.5670)--(2.5703,2.5406)--(2.6021,2.5136)--(2.6338,2.4861)--(2.6655,2.4582)--(2.6973,2.4297)--(2.7290,2.4009)--(2.7607,2.3716)--(2.7925,2.3420)--(2.8242,2.3120)--(2.8559,2.2817)--(2.8877,2.2511)--(2.9194,2.2203)--(2.9511,2.1892)--(2.9829,2.1580)--(3.0146,2.1265)--(3.0463,2.0950)--(3.0781,2.0634)--(3.1098,2.0317)--(3.1415,2.0000);
+\draw [] (0.0000,2.0000) -- (0.0000,1.0000);
+\draw [] (3.1415,1.0000) -- (3.1415,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExampleIntegration.pstricks b/auto/pictures_tex/Fig_ExampleIntegration.pstricks
index 63012a8e2..82c7f4343 100644
--- a/auto/pictures_tex/Fig_ExampleIntegration.pstricks
+++ b/auto/pictures_tex/Fig_ExampleIntegration.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -89,27 +89,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
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+\draw [color=blue,style=solid] (0.0000,-1.0000)--(0.0202,-0.9995)--(0.0404,-0.9983)--(0.0606,-0.9963)--(0.0808,-0.9934)--(0.1010,-0.9897)--(0.1212,-0.9853)--(0.1414,-0.9800)--(0.1616,-0.9738)--(0.1818,-0.9669)--(0.2020,-0.9591)--(0.2222,-0.9506)--(0.2424,-0.9412)--(0.2626,-0.9310)--(0.2828,-0.9200)--(0.3030,-0.9081)--(0.3232,-0.8955)--(0.3434,-0.8820)--(0.3636,-0.8677)--(0.3838,-0.8526)--(0.4040,-0.8367)--(0.4242,-0.8200)--(0.4444,-0.8024)--(0.4646,-0.7841)--(0.4848,-0.7649)--(0.5050,-0.7449)--(0.5252,-0.7241)--(0.5454,-0.7024)--(0.5656,-0.6800)--(0.5858,-0.6567)--(0.6060,-0.6326)--(0.6262,-0.6077)--(0.6464,-0.5820)--(0.6666,-0.5555)--(0.6868,-0.5282)--(0.7070,-0.5000)--(0.7272,-0.4710)--(0.7474,-0.4412)--(0.7676,-0.4106)--(0.7878,-0.3792)--(0.8080,-0.3470)--(0.8282,-0.3139)--(0.8484,-0.2800)--(0.8686,-0.2453)--(0.8888,-0.2098)--(0.9090,-0.1735)--(0.9292,-0.1364)--(0.9494,-0.0984)--(0.9696,-0.0596)--(0.9898,-0.0200)--(1.0101,0.0203)--(1.0303,0.0615)--(1.0505,0.1035)--(1.0707,0.1464)--(1.0909,0.1900)--(1.1111,0.2345)--(1.1313,0.2798)--(1.1515,0.3259)--(1.1717,0.3729)--(1.1919,0.4206)--(1.2121,0.4692)--(1.2323,0.5186)--(1.2525,0.5688)--(1.2727,0.6198)--(1.2929,0.6716)--(1.3131,0.7243)--(1.3333,0.7777)--(1.3535,0.8320)--(1.3737,0.8871)--(1.3939,0.9430)--(1.4141,0.9997)--(1.4343,1.0573)--(1.4545,1.1157)--(1.4747,1.1748)--(1.4949,1.2348)--(1.5151,1.2956)--(1.5353,1.3573)--(1.5555,1.4197)--(1.5757,1.4830)--(1.5959,1.5470)--(1.6161,1.6119)--(1.6363,1.6776)--(1.6565,1.7442)--(1.6767,1.8115)--(1.6969,1.8797)--(1.7171,1.9486)--(1.7373,2.0184)--(1.7575,2.0890)--(1.7777,2.1604)--(1.7979,2.2327)--(1.8181,2.3057)--(1.8383,2.3796)--(1.8585,2.4543)--(1.8787,2.5298)--(1.8989,2.6061)--(1.9191,2.6832)--(1.9393,2.7612)--(1.9595,2.8400)--(1.9797,2.9196)--(2.0000,3.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks b/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks
index 8e4adfdc0..cec7cc8f0 100644
--- a/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks
+++ b/auto/pictures_tex/Fig_ExampleIntegrationdeux.pstricks
@@ -103,8 +103,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,5.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,5.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -113,11 +113,11 @@
 % setting the default values
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          hatchthickness=0.4pt}
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+\draw [] (-1.0000,2.0000) -- (-1.0000,-2.0000);
+\draw [] (5.0000,4.0000) -- (5.0000,4.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -125,53 +125,53 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-3.0000,-0.32983) node {$ -3 $};
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-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
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-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
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-\draw (-0.29125,2.0000) node {$ 2 $};
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-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue,style=solid] (-3.0000,0.0000)--(-2.9696,-0.2461)--(-2.9393,-0.3481)--(-2.9090,-0.4264)--(-2.8787,-0.4923)--(-2.8484,-0.5504)--(-2.8181,-0.6030)--(-2.7878,-0.6513)--(-2.7575,-0.6963)--(-2.7272,-0.7385)--(-2.6969,-0.7784)--(-2.6666,-0.8164)--(-2.6363,-0.8528)--(-2.6060,-0.8876)--(-2.5757,-0.9211)--(-2.5454,-0.9534)--(-2.5151,-0.9847)--(-2.4848,-1.0150)--(-2.4545,-1.0444)--(-2.4242,-1.0730)--(-2.3939,-1.1009)--(-2.3636,-1.1281)--(-2.3333,-1.1547)--(-2.3030,-1.1806)--(-2.2727,-1.2060)--(-2.2424,-1.2309)--(-2.2121,-1.2552)--(-2.1818,-1.2792)--(-2.1515,-1.3026)--(-2.1212,-1.3257)--(-2.0909,-1.3483)--(-2.0606,-1.3706)--(-2.0303,-1.3926)--(-2.0000,-1.4142)--(-1.9696,-1.4354)--(-1.9393,-1.4564)--(-1.9090,-1.4770)--(-1.8787,-1.4974)--(-1.8484,-1.5175)--(-1.8181,-1.5374)--(-1.7878,-1.5569)--(-1.7575,-1.5763)--(-1.7272,-1.5954)--(-1.6969,-1.6143)--(-1.6666,-1.6329)--(-1.6363,-1.6514)--(-1.6060,-1.6696)--(-1.5757,-1.6877)--(-1.5454,-1.7056)--(-1.5151,-1.7232)--(-1.4848,-1.7407)--(-1.4545,-1.7580)--(-1.4242,-1.7752)--(-1.3939,-1.7922)--(-1.3636,-1.8090)--(-1.3333,-1.8257)--(-1.3030,-1.8422)--(-1.2727,-1.8586)--(-1.2424,-1.8748)--(-1.2121,-1.8909)--(-1.1818,-1.9069)--(-1.1515,-1.9227)--(-1.1212,-1.9384)--(-1.0909,-1.9540)--(-1.0606,-1.9694)--(-1.0303,-1.9847)--(-1.0000,-2.0000)--(-0.9696,-2.0150)--(-0.9393,-2.0300)--(-0.9090,-2.0449)--(-0.8787,-2.0597)--(-0.8484,-2.0743)--(-0.8181,-2.0889)--(-0.7878,-2.1033)--(-0.7575,-2.1177)--(-0.7272,-2.1320)--(-0.6969,-2.1461)--(-0.6666,-2.1602)--(-0.6363,-2.1742)--(-0.6060,-2.1881)--(-0.5757,-2.2019)--(-0.5454,-2.2156)--(-0.5151,-2.2292)--(-0.4848,-2.2428)--(-0.4545,-2.2563)--(-0.4242,-2.2696)--(-0.3939,-2.2830)--(-0.3636,-2.2962)--(-0.3333,-2.3094)--(-0.3030,-2.3224)--(-0.2727,-2.3354)--(-0.2424,-2.3484)--(-0.2121,-2.3613)--(-0.1818,-2.3741)--(-0.1515,-2.3868)--(-0.1212,-2.3994)--(-0.0909,-2.4120)--(-0.0606,-2.4246)--(-0.0303,-2.4370)--(0.0000,-2.4494);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExempleArcParam.pstricks b/auto/pictures_tex/Fig_ExempleArcParam.pstricks
index 3f874e1c9..9be72d754 100644
--- a/auto/pictures_tex/Fig_ExempleArcParam.pstricks
+++ b/auto/pictures_tex/Fig_ExempleArcParam.pstricks
@@ -65,30 +65,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.8998,0) -- (1.8998,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.8982);
+\draw [,->,>=latex] (-1.8998,0.0000) -- (1.8998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.8982);
 %DEFAULT
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-\draw (-1.4000,-0.32983) node {$ -2 $};
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-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.29125,4.2000) node {$ 6 $};
-\draw [] (-0.100,4.20) -- (0.100,4.20);
+\draw [color=blue] (0.0000,0.0000)--(0.0887,0.0444)--(0.1772,0.0888)--(0.2649,0.1332)--(0.3516,0.1777)--(0.4368,0.2221)--(0.5203,0.2665)--(0.6017,0.3109)--(0.6806,0.3554)--(0.7568,0.3998)--(0.8300,0.4442)--(0.8999,0.4886)--(0.9661,0.5331)--(1.0284,0.5775)--(1.0866,0.6219)--(1.1404,0.6663)--(1.1896,0.7108)--(1.2340,0.7552)--(1.2734,0.7996)--(1.3078,0.8441)--(1.3368,0.8885)--(1.3605,0.9329)--(1.3787,0.9773)--(1.3913,1.0218)--(1.3984,1.0662)--(1.3998,1.1106)--(1.3955,1.1550)--(1.3857,1.1995)--(1.3703,1.2439)--(1.3493,1.2883)--(1.3230,1.3327)--(1.2912,1.3772)--(1.2543,1.4216)--(1.2124,1.4660)--(1.1655,1.5105)--(1.1140,1.5549)--(1.0580,1.5993)--(0.9977,1.6437)--(0.9334,1.6882)--(0.8654,1.7326)--(0.7938,1.7770)--(0.7191,1.8214)--(0.6415,1.8659)--(0.5613,1.9103)--(0.4788,1.9547)--(0.3944,1.9991)--(0.3084,2.0436)--(0.2212,2.0880)--(0.1330,2.1324)--(0.0444,2.1769)--(-0.0444,2.2213)--(-0.1330,2.2657)--(-0.2212,2.3101)--(-0.3084,2.3546)--(-0.3944,2.3990)--(-0.4788,2.4434)--(-0.5613,2.4878)--(-0.6415,2.5323)--(-0.7191,2.5767)--(-0.7938,2.6211)--(-0.8654,2.6655)--(-0.9334,2.7100)--(-0.9977,2.7544)--(-1.0580,2.7988)--(-1.1140,2.8433)--(-1.1655,2.8877)--(-1.2124,2.9321)--(-1.2543,2.9765)--(-1.2912,3.0210)--(-1.3230,3.0654)--(-1.3493,3.1098)--(-1.3703,3.1542)--(-1.3857,3.1987)--(-1.3955,3.2431)--(-1.3998,3.2875)--(-1.3984,3.3319)--(-1.3913,3.3764)--(-1.3787,3.4208)--(-1.3605,3.4652)--(-1.3368,3.5096)--(-1.3078,3.5541)--(-1.2734,3.5985)--(-1.2340,3.6429)--(-1.1896,3.6874)--(-1.1404,3.7318)--(-1.0866,3.7762)--(-1.0284,3.8206)--(-0.9661,3.8651)--(-0.8999,3.9095)--(-0.8300,3.9539)--(-0.7568,3.9983)--(-0.6806,4.0428)--(-0.6017,4.0872)--(-0.5203,4.1316)--(-0.4368,4.1760)--(-0.3516,4.2205)--(-0.2649,4.2649)--(-0.1772,4.3093)--(-0.0887,4.3538)--(0.0000,4.3982);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
+\draw (-0.2912,4.2000) node {$ 6 $};
+\draw [] (-0.1000,4.2000) -- (0.1000,4.2000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -126,18 +126,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9998,0) -- (1.9998,0);
-\draw [,->,>=latex] (0,-1.9998) -- (0,1.9998);
+\draw [,->,>=latex] (-1.9998,0.0000) -- (1.9998,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9998) -- (0.0000,1.9998);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.190,-0.0951)--(0.377,-0.190)--(0.557,-0.284)--(0.729,-0.377)--(0.889,-0.468)--(1.04,-0.557)--(1.16,-0.645)--(1.27,-0.729)--(1.36,-0.811)--(1.43,-0.889)--(1.48,-0.964)--(1.50,-1.04)--(1.50,-1.10)--(1.47,-1.16)--(1.42,-1.22)--(1.34,-1.27)--(1.25,-1.32)--(1.13,-1.36)--(1.00,-1.40)--(0.851,-1.43)--(0.687,-1.46)--(0.513,-1.48)--(0.330,-1.49)--(0.143,-1.50)--(-0.0476,-1.50)--(-0.237,-1.50)--(-0.423,-1.48)--(-0.601,-1.47)--(-0.771,-1.45)--(-0.927,-1.42)--(-1.07,-1.38)--(-1.19,-1.34)--(-1.30,-1.30)--(-1.38,-1.25)--(-1.45,-1.19)--(-1.48,-1.13)--(-1.50,-1.07)--(-1.49,-1.00)--(-1.46,-0.927)--(-1.40,-0.851)--(-1.32,-0.771)--(-1.22,-0.687)--(-1.10,-0.601)--(-0.964,-0.513)--(-0.811,-0.423)--(-0.645,-0.330)--(-0.468,-0.237)--(-0.284,-0.143)--(-0.0951,-0.0476)--(0.0951,0.0476)--(0.284,0.143)--(0.468,0.237)--(0.645,0.330)--(0.811,0.423)--(0.964,0.513)--(1.10,0.601)--(1.22,0.687)--(1.32,0.771)--(1.40,0.851)--(1.46,0.927)--(1.49,1.00)--(1.50,1.07)--(1.48,1.13)--(1.45,1.19)--(1.38,1.25)--(1.30,1.30)--(1.19,1.34)--(1.07,1.38)--(0.927,1.42)--(0.771,1.45)--(0.601,1.47)--(0.423,1.48)--(0.237,1.50)--(0.0476,1.50)--(-0.143,1.50)--(-0.330,1.49)--(-0.513,1.48)--(-0.687,1.46)--(-0.851,1.43)--(-1.00,1.40)--(-1.13,1.36)--(-1.25,1.32)--(-1.34,1.27)--(-1.42,1.22)--(-1.47,1.16)--(-1.50,1.10)--(-1.50,1.04)--(-1.48,0.964)--(-1.43,0.889)--(-1.36,0.811)--(-1.27,0.729)--(-1.16,0.645)--(-1.04,0.557)--(-0.889,0.468)--(-0.729,0.377)--(-0.557,0.284)--(-0.377,0.190)--(-0.190,0.0951)--(0,0);
-\draw (-1.5000,-0.32983) node {$ -1 $};
-\draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (1.5000,-0.31492) node {$ 1 $};
-\draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (-0.43316,-1.5000) node {$ -1 $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.29125,1.5000) node {$ 1 $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
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+\draw (-1.5000,-0.3298) node {$ -1 $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (1.5000,-0.3149) node {$ 1 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (-0.4331,-1.5000) node {$ -1 $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.2912,1.5000) node {$ 1 $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExempleNonRang.pstricks b/auto/pictures_tex/Fig_ExempleNonRang.pstricks
index 89185f4ea..b0f5de877 100644
--- a/auto/pictures_tex/Fig_ExempleNonRang.pstricks
+++ b/auto/pictures_tex/Fig_ExempleNonRang.pstricks
@@ -63,13 +63,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.3284) -- (0,3.3284);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.3284) -- (0.0000,3.3284);
 %DEFAULT
 
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
index 637a3a657..8571a0820 100644
--- a/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
+++ b/auto/pictures_tex/Fig_ExerciceGraphesbis.pstricks
@@ -65,31 +65,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.2000) -- (0,1.1996);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2000) -- (0.0000,1.1996);
 %DEFAULT
 
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-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
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-\draw [] (-0.100,0.700) -- (0.100,0.700);
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+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -151,29 +151,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.2000);
 %DEFAULT
 
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-\draw (-2.1000,-0.32983) node {$ -3 $};
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-\draw (-1.4000,-0.32983) node {$ -2 $};
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-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-2.1991,0.7000)--(-2.1435,0.6977)--(-2.0880,0.6912)--(-2.0325,0.6802)--(-1.9769,0.6650)--(-1.9214,0.6456)--(-1.8659,0.6221)--(-1.8103,0.5948)--(-1.7548,0.5636)--(-1.6993,0.5290)--(-1.6437,0.4910)--(-1.5882,0.4499)--(-1.5327,0.4060)--(-1.4771,0.3595)--(-1.4216,0.3108)--(-1.3661,0.2601)--(-1.3105,0.2078)--(-1.2550,0.1542)--(-1.1995,0.0996)--(-1.1439,0.0443)--(-1.0884,0.0111)--(-1.0329,0.0665)--(-0.9773,0.1215)--(-0.9218,0.1758)--(-0.8663,0.2289)--(-0.8107,0.2806)--(-0.7552,0.3305)--(-0.6997,0.3784)--(-0.6441,0.4239)--(-0.5886,0.4667)--(-0.5331,0.5066)--(-0.4775,0.5433)--(-0.4220,0.5765)--(-0.3665,0.6062)--(-0.3109,0.6320)--(-0.2554,0.6539)--(-0.1999,0.6716)--(-0.1443,0.6851)--(-0.0888,0.6943)--(-0.0333,0.6992)--(0.0222,0.6996)--(0.0777,0.6956)--(0.1332,0.6873)--(0.1888,0.6746)--(0.2443,0.6577)--(0.2998,0.6367)--(0.3554,0.6116)--(0.4109,0.5827)--(0.4664,0.5502)--(0.5220,0.5142)--(0.5775,0.4749)--(0.6330,0.4327)--(0.6886,0.3877)--(0.7441,0.3403)--(0.7996,0.2907)--(0.8552,0.2394)--(0.9107,0.1865)--(0.9662,0.1324)--(1.0218,0.0775)--(1.0773,0.0222)--(1.1328,0.0333)--(1.1884,0.0886)--(1.2439,0.1433)--(1.2994,0.1972)--(1.3550,0.2498)--(1.4105,0.3008)--(1.4660,0.3500)--(1.5216,0.3969)--(1.5771,0.4413)--(1.6326,0.4830)--(1.6882,0.5216)--(1.7437,0.5570)--(1.7992,0.5888)--(1.8548,0.6170)--(1.9103,0.6412)--(1.9658,0.6615)--(2.0214,0.6775)--(2.0769,0.6893)--(2.1324,0.6968)--(2.1880,0.6999)--(2.2435,0.6985)--(2.2990,0.6928)--(2.3546,0.6828)--(2.4101,0.6684)--(2.4656,0.6498)--(2.5212,0.6271)--(2.5767,0.6005)--(2.6322,0.5702)--(2.6878,0.5362)--(2.7433,0.4988)--(2.7988,0.4584)--(2.8544,0.4150)--(2.9099,0.3690)--(2.9654,0.3207)--(3.0210,0.2704)--(3.0765,0.2184)--(3.1320,0.1650)--(3.1876,0.1106)--(3.2431,0.0554)--(3.2986,0.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -239,35 +239,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (2.8562,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4997);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (2.8561,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4997);
 %DEFAULT
 
-\draw [color=blue] (-1.571,0)--(-1.531,0.001573)--(-1.491,0.006281)--(-1.452,0.01409)--(-1.412,0.02496)--(-1.372,0.03882)--(-1.333,0.05558)--(-1.293,0.07514)--(-1.253,0.09736)--(-1.214,0.1221)--(-1.174,0.1493)--(-1.134,0.1786)--(-1.095,0.2100)--(-1.055,0.2432)--(-1.015,0.2780)--(-0.9758,0.3142)--(-0.9361,0.3515)--(-0.8965,0.3898)--(-0.8568,0.4288)--(-0.8171,0.4683)--(-0.7775,0.5079)--(-0.7378,0.5475)--(-0.6981,0.5868)--(-0.6585,0.6256)--(-0.6188,0.6635)--(-0.5791,0.7005)--(-0.5395,0.7361)--(-0.4998,0.7703)--(-0.4601,0.8028)--(-0.4205,0.8334)--(-0.3808,0.8619)--(-0.3411,0.8881)--(-0.3015,0.9118)--(-0.2618,0.9330)--(-0.2221,0.9515)--(-0.1825,0.9671)--(-0.1428,0.9797)--(-0.1031,0.9894)--(-0.06347,0.9960)--(-0.02380,0.9994)--(0.01587,0.9997)--(0.05553,0.9969)--(0.09520,0.9910)--(0.1349,0.9819)--(0.1745,0.9698)--(0.2142,0.9548)--(0.2539,0.9369)--(0.2935,0.9163)--(0.3332,0.8930)--(0.3729,0.8673)--(0.4125,0.8393)--(0.4522,0.8091)--(0.4919,0.7770)--(0.5315,0.7431)--(0.5712,0.7077)--(0.6109,0.6710)--(0.6505,0.6332)--(0.6902,0.5946)--(0.7299,0.5554)--(0.7695,0.5159)--(0.8092,0.4762)--(0.8489,0.4367)--(0.8885,0.3976)--(0.9282,0.3591)--(0.9679,0.3216)--(1.008,0.2851)--(1.047,0.2500)--(1.087,0.2165)--(1.127,0.1847)--(1.166,0.1550)--(1.206,0.1274)--(1.246,0.1021)--(1.285,0.07937)--(1.325,0.05927)--(1.365,0.04195)--(1.404,0.02750)--(1.444,0.01603)--(1.484,0.007596)--(1.523,0.002264)--(1.563,0)--(1.603,0.001007)--(1.642,0.005089)--(1.682,0.01229)--(1.722,0.02255)--(1.761,0.03582)--(1.801,0.05200)--(1.841,0.07101)--(1.880,0.09271)--(1.920,0.1170)--(1.960,0.1437)--(1.999,0.1726)--(2.039,0.2035)--(2.079,0.2364)--(2.118,0.2709)--(2.158,0.3068)--(2.198,0.3440)--(2.237,0.3821)--(2.277,0.4210)--(2.317,0.4604)--(2.356,0.5000);
-\draw (-2.0000,-0.32983) node {$ -4 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.5000,-0.32983) node {$ -3 $};
-\draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.0000,-0.32983) node {$ -2 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.50000,-0.32983) node {$ -1 $};
-\draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.50000,-0.31492) node {$ 1 $};
-\draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.0000,-0.31492) node {$ 2 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.5000,-0.31492) node {$ 3 $};
-\draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.0000,-0.31492) node {$ 4 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.5000,-0.31492) node {$ 5 $};
-\draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (-0.43316,-0.50000) node {$ -1 $};
-\draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.29125,0.50000) node {$ 1 $};
-\draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-1.5707,0.0000)--(-1.5311,0.0015)--(-1.4914,0.0062)--(-1.4517,0.0140)--(-1.4121,0.0249)--(-1.3724,0.0388)--(-1.3327,0.0555)--(-1.2931,0.0751)--(-1.2534,0.0973)--(-1.2137,0.1221)--(-1.1741,0.1492)--(-1.1344,0.1786)--(-1.0947,0.2099)--(-1.0551,0.2431)--(-1.0154,0.2779)--(-0.9757,0.3141)--(-0.9361,0.3515)--(-0.8964,0.3898)--(-0.8567,0.4288)--(-0.8171,0.4682)--(-0.7774,0.5079)--(-0.7377,0.5475)--(-0.6981,0.5868)--(-0.6584,0.6255)--(-0.6187,0.6635)--(-0.5791,0.7004)--(-0.5394,0.7361)--(-0.4997,0.7703)--(-0.4601,0.8028)--(-0.4204,0.8333)--(-0.3807,0.8618)--(-0.3411,0.8880)--(-0.3014,0.9118)--(-0.2617,0.9330)--(-0.2221,0.9514)--(-0.1824,0.9670)--(-0.1427,0.9797)--(-0.1031,0.9894)--(-0.0634,0.9959)--(-0.0237,0.9994)--(0.0158,0.9997)--(0.0555,0.9969)--(0.0951,0.9909)--(0.1348,0.9819)--(0.1745,0.9698)--(0.2141,0.9548)--(0.2538,0.9369)--(0.2935,0.9162)--(0.3331,0.8930)--(0.3728,0.8672)--(0.4125,0.8392)--(0.4521,0.8090)--(0.4918,0.7769)--(0.5315,0.7430)--(0.5711,0.7077)--(0.6108,0.6710)--(0.6505,0.6332)--(0.6901,0.5946)--(0.7298,0.5554)--(0.7695,0.5158)--(0.8091,0.4762)--(0.8488,0.4367)--(0.8885,0.3975)--(0.9281,0.3591)--(0.9678,0.3215)--(1.0075,0.2851)--(1.0471,0.2500)--(1.0868,0.2164)--(1.1265,0.1847)--(1.1661,0.1549)--(1.2058,0.1273)--(1.2455,0.1021)--(1.2851,0.0793)--(1.3248,0.0592)--(1.3645,0.0419)--(1.4041,0.0274)--(1.4438,0.0160)--(1.4835,0.0075)--(1.5231,0.0022)--(1.5628,0.0000)--(1.6025,0.0010)--(1.6421,0.0050)--(1.6818,0.0122)--(1.7215,0.0225)--(1.7611,0.0358)--(1.8008,0.0520)--(1.8405,0.0710)--(1.8801,0.0927)--(1.9198,0.1169)--(1.9595,0.1436)--(1.9991,0.1725)--(2.0388,0.2035)--(2.0785,0.2363)--(2.1181,0.2708)--(2.1578,0.3068)--(2.1975,0.3439)--(2.2371,0.3821)--(2.2768,0.4209)--(2.3165,0.4603)--(2.3561,0.5000);
+\draw (-2.0000,-0.3298) node {$ -4 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.5000,-0.3298) node {$ -3 $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -2 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (-0.5000,-0.3298) node {$ -1 $};
+\draw [] (-0.5000,-0.1000) -- (-0.5000,0.1000);
+\draw (0.5000,-0.3149) node {$ 1 $};
+\draw [] (0.5000,-0.1000) -- (0.5000,0.1000);
+\draw (1.0000,-0.3149) node {$ 2 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (1.5000,-0.3149) node {$ 3 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (2.0000,-0.3149) node {$ 4 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (2.5000,-0.3149) node {$ 5 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (-0.4331,-0.5000) node {$ -1 $};
+\draw [] (-0.1000,-0.5000) -- (0.1000,-0.5000);
+\draw (-0.2912,0.5000) node {$ 1 $};
+\draw [] (-0.1000,0.5000) -- (0.1000,0.5000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -329,31 +329,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1996) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0)--(-2.144,0.05547)--(-2.088,0.1106)--(-2.033,0.1650)--(-1.977,0.2184)--(-1.921,0.2704)--(-1.866,0.3208)--(-1.810,0.3691)--(-1.755,0.4150)--(-1.699,0.4584)--(-1.644,0.4989)--(-1.588,0.5362)--(-1.533,0.5702)--(-1.477,0.6006)--(-1.422,0.6272)--(-1.366,0.6499)--(-1.311,0.6684)--(-1.255,0.6828)--(-1.200,0.6929)--(-1.144,0.6986)--(-1.088,0.6999)--(-1.033,0.6968)--(-0.9774,0.6894)--(-0.9219,0.6776)--(-0.8663,0.6615)--(-0.8108,0.6413)--(-0.7552,0.6170)--(-0.6997,0.5889)--(-0.6442,0.5570)--(-0.5887,0.5217)--(-0.5331,0.4831)--(-0.4776,0.4414)--(-0.4221,0.3969)--(-0.3665,0.3500)--(-0.3110,0.3009)--(-0.2555,0.2498)--(-0.1999,0.1972)--(-0.1444,0.1434)--(-0.08885,0.08861)--(-0.03332,0.03331)--(0.02221,-0.02221)--(0.07775,-0.07759)--(0.1333,-0.1325)--(0.1888,-0.1865)--(0.2443,-0.2394)--(0.2999,-0.2908)--(0.3554,-0.3403)--(0.4109,-0.3877)--(0.4665,-0.4327)--(0.5220,-0.4750)--(0.5775,-0.5142)--(0.6331,-0.5502)--(0.6886,-0.5828)--(0.7441,-0.6117)--(0.7997,-0.6367)--(0.8552,-0.6578)--(0.9107,-0.6747)--(0.9663,-0.6873)--(1.022,-0.6957)--(1.077,-0.6996)--(1.133,-0.6992)--(1.188,-0.6944)--(1.244,-0.6852)--(1.299,-0.6716)--(1.355,-0.6539)--(1.411,-0.6320)--(1.466,-0.6062)--(1.522,-0.5766)--(1.577,-0.5433)--(1.633,-0.5066)--(1.688,-0.4667)--(1.744,-0.4239)--(1.799,-0.3784)--(1.855,-0.3306)--(1.910,-0.2807)--(1.966,-0.2289)--(2.021,-0.1758)--(2.077,-0.1216)--(2.132,-0.06654)--(2.188,-0.01111)--(2.244,0.04440)--(2.299,0.09962)--(2.355,0.1542)--(2.410,0.2078)--(2.466,0.2602)--(2.521,0.3108)--(2.577,0.3596)--(2.632,0.4060)--(2.688,0.4500)--(2.743,0.4910)--(2.799,0.5290)--(2.854,0.5637)--(2.910,0.5948)--(2.965,0.6222)--(3.021,0.6456)--(3.077,0.6650)--(3.132,0.6803)--(3.188,0.6912)--(3.243,0.6978)--(3.299,0.7000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-2.1991,0.0000)--(-2.1435,0.0554)--(-2.0880,0.1106)--(-2.0325,0.1650)--(-1.9769,0.2184)--(-1.9214,0.2704)--(-1.8659,0.3207)--(-1.8103,0.3690)--(-1.7548,0.4150)--(-1.6993,0.4584)--(-1.6437,0.4988)--(-1.5882,0.5362)--(-1.5327,0.5702)--(-1.4771,0.6005)--(-1.4216,0.6271)--(-1.3661,0.6498)--(-1.3105,0.6684)--(-1.2550,0.6828)--(-1.1995,0.6928)--(-1.1439,0.6985)--(-1.0884,0.6999)--(-1.0329,0.6968)--(-0.9773,0.6893)--(-0.9218,0.6775)--(-0.8663,0.6615)--(-0.8107,0.6412)--(-0.7552,0.6170)--(-0.6997,0.5888)--(-0.6441,0.5570)--(-0.5886,0.5216)--(-0.5331,0.4830)--(-0.4775,0.4413)--(-0.4220,0.3969)--(-0.3665,0.3500)--(-0.3109,0.3008)--(-0.2554,0.2498)--(-0.1999,0.1972)--(-0.1443,0.1433)--(-0.0888,0.0886)--(-0.0333,0.0333)--(0.0222,-0.0222)--(0.0777,-0.0775)--(0.1332,-0.1324)--(0.1888,-0.1865)--(0.2443,-0.2394)--(0.2998,-0.2907)--(0.3554,-0.3403)--(0.4109,-0.3877)--(0.4664,-0.4327)--(0.5220,-0.4749)--(0.5775,-0.5142)--(0.6330,-0.5502)--(0.6886,-0.5827)--(0.7441,-0.6116)--(0.7996,-0.6367)--(0.8552,-0.6577)--(0.9107,-0.6746)--(0.9662,-0.6873)--(1.0218,-0.6956)--(1.0773,-0.6996)--(1.1328,-0.6992)--(1.1884,-0.6943)--(1.2439,-0.6851)--(1.2994,-0.6716)--(1.3550,-0.6539)--(1.4105,-0.6320)--(1.4660,-0.6062)--(1.5216,-0.5765)--(1.5771,-0.5433)--(1.6326,-0.5066)--(1.6882,-0.4667)--(1.7437,-0.4239)--(1.7992,-0.3784)--(1.8548,-0.3305)--(1.9103,-0.2806)--(1.9658,-0.2289)--(2.0214,-0.1758)--(2.0769,-0.1215)--(2.1324,-0.0665)--(2.1880,-0.0111)--(2.2435,0.0443)--(2.2990,0.0996)--(2.3546,0.1542)--(2.4101,0.2078)--(2.4656,0.2601)--(2.5212,0.3108)--(2.5767,0.3595)--(2.6322,0.4060)--(2.6878,0.4499)--(2.7433,0.4910)--(2.7988,0.5290)--(2.8544,0.5636)--(2.9099,0.5948)--(2.9654,0.6221)--(3.0210,0.6456)--(3.0765,0.6650)--(3.1320,0.6802)--(3.1876,0.6912)--(3.2431,0.6977)--(3.2986,0.7000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -415,32 +415,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.2000) -- (0,1.1999);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1999) -- (0.0000,1.1999);
 %DEFAULT
 
-\draw [color=blue] 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+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -502,31 +502,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5996,0) -- (4.8982,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.2000);
+\draw [,->,>=latex] (-1.5995,0.0000) -- (4.8982,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-1.100,0)--(-1.077,0.1247)--(-1.055,0.1763)--(-1.033,0.2158)--(-1.011,0.2491)--(-0.9885,0.2782)--(-0.9663,0.3045)--(-0.9441,0.3286)--(-0.9219,0.3508)--(-0.8996,0.3715)--(-0.8774,0.3910)--(-0.8552,0.4094)--(-0.8330,0.4268)--(-0.8108,0.4432)--(-0.7886,0.4589)--(-0.7664,0.4738)--(-0.7441,0.4881)--(-0.7219,0.5017)--(-0.6997,0.5147)--(-0.6775,0.5271)--(-0.6553,0.5390)--(-0.6331,0.5504)--(-0.6109,0.5612)--(-0.5887,0.5716)--(-0.5664,0.5815)--(-0.5442,0.5910)--(-0.5220,0.6000)--(-0.4998,0.6085)--(-0.4776,0.6167)--(-0.4554,0.6244)--(-0.4332,0.6318)--(-0.4109,0.6387)--(-0.3887,0.6453)--(-0.3665,0.6514)--(-0.3443,0.6572)--(-0.3221,0.6626)--(-0.2999,0.6676)--(-0.2777,0.6723)--(-0.2555,0.6766)--(-0.2332,0.6805)--(-0.2110,0.6840)--(-0.1888,0.6872)--(-0.1666,0.6901)--(-0.1444,0.6925)--(-0.1222,0.6947)--(-0.09996,0.6964)--(-0.07775,0.6978)--(-0.05553,0.6989)--(-0.03332,0.6996)--(-0.01111,0.7000)--(0.01111,0.7000)--(0.03332,0.6996)--(0.05553,0.6989)--(0.07775,0.6978)--(0.09996,0.6964)--(0.1222,0.6947)--(0.1444,0.6925)--(0.1666,0.6901)--(0.1888,0.6872)--(0.2110,0.6840)--(0.2332,0.6805)--(0.2555,0.6766)--(0.2777,0.6723)--(0.2999,0.6676)--(0.3221,0.6626)--(0.3443,0.6572)--(0.3665,0.6514)--(0.3887,0.6453)--(0.4109,0.6387)--(0.4332,0.6318)--(0.4554,0.6244)--(0.4776,0.6167)--(0.4998,0.6085)--(0.5220,0.6000)--(0.5442,0.5910)--(0.5664,0.5815)--(0.5887,0.5716)--(0.6109,0.5612)--(0.6331,0.5504)--(0.6553,0.5390)--(0.6775,0.5271)--(0.6997,0.5147)--(0.7219,0.5017)--(0.7441,0.4881)--(0.7664,0.4738)--(0.7886,0.4589)--(0.8108,0.4432)--(0.8330,0.4268)--(0.8552,0.4094)--(0.8774,0.3910)--(0.8996,0.3715)--(0.9219,0.3508)--(0.9441,0.3286)--(0.9663,0.3045)--(0.9885,0.2782)--(1.011,0.2491)--(1.033,0.2158)--(1.055,0.1763)--(1.077,0.1247)--(1.100,0);
+\draw [color=blue] (-1.0995,0.0000)--(-1.0773,0.1246)--(-1.0551,0.1762)--(-1.0329,0.2158)--(-1.0107,0.2490)--(-0.9884,0.2782)--(-0.9662,0.3045)--(-0.9440,0.3285)--(-0.9218,0.3508)--(-0.8996,0.3715)--(-0.8774,0.3910)--(-0.8552,0.4093)--(-0.8329,0.4267)--(-0.8107,0.4432)--(-0.7885,0.4589)--(-0.7663,0.4738)--(-0.7441,0.4880)--(-0.7219,0.5016)--(-0.6997,0.5146)--(-0.6775,0.5271)--(-0.6552,0.5390)--(-0.6330,0.5503)--(-0.6108,0.5612)--(-0.5886,0.5715)--(-0.5664,0.5814)--(-0.5442,0.5909)--(-0.5220,0.5999)--(-0.4997,0.6085)--(-0.4775,0.6166)--(-0.4553,0.6244)--(-0.4331,0.6317)--(-0.4109,0.6387)--(-0.3887,0.6452)--(-0.3665,0.6514)--(-0.3443,0.6572)--(-0.3220,0.6625)--(-0.2998,0.6676)--(-0.2776,0.6722)--(-0.2554,0.6765)--(-0.2332,0.6804)--(-0.2110,0.6840)--(-0.1888,0.6872)--(-0.1665,0.6900)--(-0.1443,0.6925)--(-0.1221,0.6946)--(-0.0999,0.6964)--(-0.0777,0.6978)--(-0.0555,0.6988)--(-0.0333,0.6996)--(-0.0111,0.6999)--(0.0111,0.6999)--(0.0333,0.6996)--(0.0555,0.6988)--(0.0777,0.6978)--(0.0999,0.6964)--(0.1221,0.6946)--(0.1443,0.6925)--(0.1665,0.6900)--(0.1888,0.6872)--(0.2110,0.6840)--(0.2332,0.6804)--(0.2554,0.6765)--(0.2776,0.6722)--(0.2998,0.6676)--(0.3220,0.6625)--(0.3443,0.6572)--(0.3665,0.6514)--(0.3887,0.6452)--(0.4109,0.6387)--(0.4331,0.6317)--(0.4553,0.6244)--(0.4775,0.6166)--(0.4997,0.6085)--(0.5220,0.5999)--(0.5442,0.5909)--(0.5664,0.5814)--(0.5886,0.5715)--(0.6108,0.5612)--(0.6330,0.5503)--(0.6552,0.5390)--(0.6775,0.5271)--(0.6997,0.5146)--(0.7219,0.5016)--(0.7441,0.4880)--(0.7663,0.4738)--(0.7885,0.4589)--(0.8107,0.4432)--(0.8329,0.4267)--(0.8552,0.4093)--(0.8774,0.3910)--(0.8996,0.3715)--(0.9218,0.3508)--(0.9440,0.3285)--(0.9662,0.3045)--(0.9884,0.2782)--(1.0107,0.2490)--(1.0329,0.2158)--(1.0551,0.1762)--(1.0773,0.1246)--(1.0995,0.0000);
 
-\draw [color=blue] (3.299,0)--(3.310,0.08817)--(3.321,0.1247)--(3.332,0.1527)--(3.343,0.1763)--(3.354,0.1971)--(3.365,0.2158)--(3.376,0.2330)--(3.388,0.2491)--(3.399,0.2641)--(3.410,0.2782)--(3.421,0.2917)--(3.432,0.3045)--(3.443,0.3168)--(3.454,0.3286)--(3.465,0.3399)--(3.476,0.3508)--(3.487,0.3613)--(3.499,0.3715)--(3.510,0.3814)--(3.521,0.3910)--(3.532,0.4003)--(3.543,0.4094)--(3.554,0.4182)--(3.565,0.4268)--(3.576,0.4351)--(3.587,0.4432)--(3.599,0.4512)--(3.610,0.4589)--(3.621,0.4665)--(3.632,0.4738)--(3.643,0.4811)--(3.654,0.4881)--(3.665,0.4950)--(3.676,0.5017)--(3.687,0.5083)--(3.699,0.5147)--(3.710,0.5210)--(3.721,0.5271)--(3.732,0.5331)--(3.743,0.5390)--(3.754,0.5447)--(3.765,0.5504)--(3.776,0.5559)--(3.787,0.5612)--(3.798,0.5665)--(3.810,0.5716)--(3.821,0.5766)--(3.832,0.5815)--(3.843,0.5863)--(3.854,0.5910)--(3.865,0.5955)--(3.876,0.6000)--(3.887,0.6043)--(3.898,0.6085)--(3.910,0.6127)--(3.921,0.6167)--(3.932,0.6206)--(3.943,0.6244)--(3.954,0.6282)--(3.965,0.6318)--(3.976,0.6353)--(3.987,0.6387)--(3.998,0.6420)--(4.010,0.6453)--(4.021,0.6484)--(4.032,0.6514)--(4.043,0.6544)--(4.054,0.6572)--(4.065,0.6600)--(4.076,0.6626)--(4.087,0.6652)--(4.098,0.6676)--(4.109,0.6700)--(4.121,0.6723)--(4.132,0.6745)--(4.143,0.6766)--(4.154,0.6786)--(4.165,0.6805)--(4.176,0.6823)--(4.187,0.6840)--(4.198,0.6857)--(4.209,0.6872)--(4.221,0.6887)--(4.232,0.6901)--(4.243,0.6913)--(4.254,0.6925)--(4.265,0.6936)--(4.276,0.6947)--(4.287,0.6956)--(4.298,0.6964)--(4.309,0.6972)--(4.320,0.6978)--(4.332,0.6984)--(4.343,0.6989)--(4.354,0.6993)--(4.365,0.6996)--(4.376,0.6998)--(4.387,0.7000)--(4.398,0.7000);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (4.2000,-0.31492) node {$ 6 $};
-\draw [] (4.20,-0.100) -- (4.20,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (3.2986,0.0000)--(3.3097,0.0881)--(3.3208,0.1246)--(3.3319,0.1526)--(3.3430,0.1762)--(3.3542,0.1970)--(3.3653,0.2158)--(3.3764,0.2330)--(3.3875,0.2490)--(3.3986,0.2640)--(3.4097,0.2782)--(3.4208,0.2916)--(3.4319,0.3045)--(3.4430,0.3167)--(3.4541,0.3285)--(3.4652,0.3398)--(3.4763,0.3508)--(3.4874,0.3613)--(3.4985,0.3715)--(3.5096,0.3814)--(3.5208,0.3910)--(3.5319,0.4003)--(3.5430,0.4093)--(3.5541,0.4181)--(3.5652,0.4267)--(3.5763,0.4350)--(3.5874,0.4432)--(3.5985,0.4511)--(3.6096,0.4589)--(3.6207,0.4664)--(3.6318,0.4738)--(3.6429,0.4810)--(3.6540,0.4880)--(3.6651,0.4949)--(3.6762,0.5016)--(3.6874,0.5082)--(3.6985,0.5146)--(3.7096,0.5209)--(3.7207,0.5271)--(3.7318,0.5331)--(3.7429,0.5390)--(3.7540,0.5447)--(3.7651,0.5503)--(3.7762,0.5558)--(3.7873,0.5612)--(3.7984,0.5664)--(3.8095,0.5715)--(3.8206,0.5766)--(3.8317,0.5814)--(3.8428,0.5862)--(3.8540,0.5909)--(3.8651,0.5955)--(3.8762,0.5999)--(3.8873,0.6043)--(3.8984,0.6085)--(3.9095,0.6126)--(3.9206,0.6166)--(3.9317,0.6206)--(3.9428,0.6244)--(3.9539,0.6281)--(3.9650,0.6317)--(3.9761,0.6352)--(3.9872,0.6387)--(3.9983,0.6420)--(4.0094,0.6452)--(4.0206,0.6483)--(4.0317,0.6514)--(4.0428,0.6543)--(4.0539,0.6572)--(4.0650,0.6599)--(4.0761,0.6625)--(4.0872,0.6651)--(4.0983,0.6676)--(4.1094,0.6699)--(4.1205,0.6722)--(4.1316,0.6744)--(4.1427,0.6765)--(4.1538,0.6785)--(4.1649,0.6804)--(4.1760,0.6823)--(4.1872,0.6840)--(4.1983,0.6856)--(4.2094,0.6872)--(4.2205,0.6886)--(4.2316,0.6900)--(4.2427,0.6913)--(4.2538,0.6925)--(4.2649,0.6936)--(4.2760,0.6946)--(4.2871,0.6955)--(4.2982,0.6964)--(4.3093,0.6971)--(4.3204,0.6978)--(4.3315,0.6984)--(4.3426,0.6988)--(4.3538,0.6992)--(4.3649,0.6996)--(4.3760,0.6998)--(4.3871,0.6999)--(4.3982,0.7000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (4.2000,-0.3149) node {$ 6 $};
+\draw [] (4.2000,-0.1000) -- (4.2000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -588,31 +588,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (3.7987,0);
-\draw [,->,>=latex] (0,-1.1996) -- (0,1.2000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (3.7986,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1996) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0.7000)--(-2.144,0.6650)--(-2.088,0.5637)--(-2.033,0.4060)--(-1.977,0.2078)--(-1.921,-0.01111)--(-1.866,-0.2289)--(-1.810,-0.4239)--(-1.755,-0.5766)--(-1.699,-0.6716)--(-1.644,-0.6996)--(-1.588,-0.6578)--(-1.533,-0.5502)--(-1.477,-0.3877)--(-1.422,-0.1865)--(-1.366,0.03331)--(-1.311,0.2498)--(-1.255,0.4414)--(-1.200,0.5889)--(-1.144,0.6776)--(-1.088,0.6986)--(-1.033,0.6499)--(-0.9774,0.5362)--(-0.9219,0.3691)--(-0.8663,0.1650)--(-0.8108,-0.05547)--(-0.7552,-0.2704)--(-0.6997,-0.4584)--(-0.6442,-0.6006)--(-0.5887,-0.6828)--(-0.5331,-0.6968)--(-0.4776,-0.6413)--(-0.4221,-0.5217)--(-0.3665,-0.3500)--(-0.3110,-0.1434)--(-0.2555,0.07759)--(-0.1999,0.2908)--(-0.1444,0.4750)--(-0.08885,0.6117)--(-0.03332,0.6873)--(0.02221,0.6944)--(0.07775,0.6320)--(0.1333,0.5066)--(0.1888,0.3306)--(0.2443,0.1216)--(0.2999,-0.09962)--(0.3554,-0.3108)--(0.4109,-0.4910)--(0.4665,-0.6222)--(0.5220,-0.6912)--(0.5775,-0.6912)--(0.6331,-0.6222)--(0.6886,-0.4910)--(0.7441,-0.3108)--(0.7997,-0.09962)--(0.8552,0.1216)--(0.9107,0.3306)--(0.9663,0.5066)--(1.022,0.6320)--(1.077,0.6944)--(1.133,0.6873)--(1.188,0.6117)--(1.244,0.4750)--(1.299,0.2908)--(1.355,0.07759)--(1.411,-0.1434)--(1.466,-0.3500)--(1.522,-0.5217)--(1.577,-0.6413)--(1.633,-0.6968)--(1.688,-0.6828)--(1.744,-0.6006)--(1.799,-0.4584)--(1.855,-0.2704)--(1.910,-0.05547)--(1.966,0.1650)--(2.021,0.3691)--(2.077,0.5362)--(2.132,0.6499)--(2.188,0.6986)--(2.244,0.6776)--(2.299,0.5889)--(2.355,0.4414)--(2.410,0.2498)--(2.466,0.03331)--(2.521,-0.1865)--(2.577,-0.3877)--(2.632,-0.5502)--(2.688,-0.6578)--(2.743,-0.6996)--(2.799,-0.6716)--(2.854,-0.5766)--(2.910,-0.4239)--(2.965,-0.2289)--(3.021,-0.01111)--(3.077,0.2078)--(3.132,0.4060)--(3.188,0.5637)--(3.243,0.6650)--(3.299,0.7000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (2.8000,-0.31492) node {$ 4 $};
-\draw [] (2.80,-0.100) -- (2.80,0.100);
-\draw (3.5000,-0.31492) node {$ 5 $};
-\draw [] (3.50,-0.100) -- (3.50,0.100);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
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+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (2.8000,-0.3149) node {$ 4 $};
+\draw [] (2.8000,-0.1000) -- (2.8000,0.1000);
+\draw (3.5000,-0.3149) node {$ 5 $};
+\draw [] (3.5000,-0.1000) -- (3.5000,0.1000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExoCUd.pstricks b/auto/pictures_tex/Fig_ExoCUd.pstricks
index 271934b9a..3dac3bca8 100644
--- a/auto/pictures_tex/Fig_ExoCUd.pstricks
+++ b/auto/pictures_tex/Fig_ExoCUd.pstricks
@@ -75,34 +75,34 @@
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 %AXES
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-\draw [,->,>=latex] (0,-1.0000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.0000) -- (0.0000,4.5000);
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-\draw []  (0,2.5600) node [rotate=0] {$\bullet$};
-\draw (0.30816,2.8862) node {$y$};
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+\draw [color=gray,style=dashed] (1.0000,-0.5000) -- (1.0000,4.0000);
+\draw []  (0.0000,2.5600) node [rotate=0] {$\bullet$};
+\draw (0.3081,2.8861) node {$y$};
 \draw []  (2.6000,2.5600) node [rotate=0] {$\bullet$};
-\draw []  (-0.60000,2.5600) node [rotate=0] {$\bullet$};
-\draw []  (2.6000,0) node [rotate=0] {$\bullet$};
-\draw (2.6000,-0.41918) node {$x_+$};
-\draw []  (-0.60000,0) node [rotate=0] {$\bullet$};
-\draw (-0.60000,-0.41918) node {$x_-$};
-\draw [style=dashed] (2.60,2.56) -- (2.60,0);
-\draw [style=dashed] (-0.600,2.56) -- (-0.600,0);
-\draw [style=dotted] (2.60,2.56) -- (-0.600,2.56);
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw []  (-0.6000,2.5600) node [rotate=0] {$\bullet$};
+\draw []  (2.6000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.6000,-0.4191) node {$x_+$};
+\draw []  (-0.6000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.6000,-0.4191) node {$x_-$};
+\draw [style=dashed] (2.6000,2.5600) -- (2.6000,0.0000);
+\draw [style=dashed] (-0.6000,2.5600) -- (-0.6000,0.0000);
+\draw [style=dotted] (2.6000,2.5600) -- (-0.6000,2.5600);
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExoMagnetique.pstricks b/auto/pictures_tex/Fig_ExoMagnetique.pstricks
index 6466d25a8..ad8e03dc3 100644
--- a/auto/pictures_tex/Fig_ExoMagnetique.pstricks
+++ b/auto/pictures_tex/Fig_ExoMagnetique.pstricks
@@ -77,13 +77,13 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [style=dashed] (0,-2.00) -- (0,2.00);
-\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
-\draw (0.39455,1.0000) node {$I$};
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-2.0000,1.0000);
-\draw (-1.7636,0.73184) node {$d$};
+\draw [style=dashed] (0.0000,-2.0000) -- (0.0000,2.0000);
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (0.0000,1.0000);
+\draw (0.3945,1.0000) node {$I$};
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-2.0000,1.0000);
+\draw (-1.7635,0.7318) node {$d$};
 \draw []  (-2.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-2.7065,1.3239) node {$(r,\theta,z)$};
+\draw (-2.7064,1.3238) node {$(r,\theta,z)$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ExoParamCD.pstricks b/auto/pictures_tex/Fig_ExoParamCD.pstricks
index 126669a8f..9e60d2dd7 100644
--- a/auto/pictures_tex/Fig_ExoParamCD.pstricks
+++ b/auto/pictures_tex/Fig_ExoParamCD.pstricks
@@ -71,26 +71,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
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-\draw [,->,>=latex] (0,-3.0000) -- (-0.030000,-3.0000);
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-\draw [color=red] (0,0)--(0.190,-0.285)--(0.380,-0.568)--(0.568,-0.845)--(0.753,-1.11)--(0.936,-1.37)--(1.11,-1.62)--(1.29,-1.85)--(1.46,-2.07)--(1.62,-2.27)--(1.78,-2.44)--(1.93,-2.60)--(2.07,-2.73)--(2.20,-2.83)--(2.33,-2.92)--(2.44,-2.97)--(2.55,-3.00)--(2.64,-3.00)--(2.73,-2.97)--(2.80,-2.92)--(2.86,-2.83)--(2.92,-2.73)--(2.95,-2.60)--(2.98,-2.44)--(3.00,-2.27)--(3.00,-2.07)--(2.99,-1.85)--(2.97,-1.62)--(2.94,-1.37)--(2.89,-1.11)--(2.83,-0.845)--(2.77,-0.568)--(2.69,-0.285)--(2.60,0)--(2.50,0.285)--(2.39,0.568)--(2.27,0.845)--(2.14,1.11)--(2.00,1.37)--(1.85,1.62)--(1.70,1.85)--(1.54,2.07)--(1.37,2.27)--(1.20,2.44)--(1.03,2.60)--(0.845,2.73)--(0.661,2.83)--(0.474,2.92)--(0.285,2.97)--(0.0952,3.00)--(-0.0952,3.00)--(-0.285,2.97)--(-0.474,2.92)--(-0.661,2.83)--(-0.845,2.73)--(-1.03,2.60)--(-1.20,2.44)--(-1.37,2.27)--(-1.54,2.07)--(-1.70,1.85)--(-1.85,1.62)--(-2.00,1.37)--(-2.14,1.11)--(-2.27,0.845)--(-2.39,0.568)--(-2.50,0.285)--(-2.60,0)--(-2.69,-0.285)--(-2.77,-0.568)--(-2.83,-0.845)--(-2.89,-1.11)--(-2.94,-1.37)--(-2.97,-1.62)--(-2.99,-1.85)--(-3.00,-2.07)--(-3.00,-2.27)--(-2.98,-2.44)--(-2.95,-2.60)--(-2.92,-2.73)--(-2.86,-2.83)--(-2.80,-2.92)--(-2.73,-2.97)--(-2.64,-3.00)--(-2.55,-3.00)--(-2.44,-2.97)--(-2.33,-2.92)--(-2.20,-2.83)--(-2.07,-2.73)--(-1.93,-2.60)--(-1.78,-2.44)--(-1.62,-2.27)--(-1.46,-2.07)--(-1.29,-1.85)--(-1.11,-1.62)--(-0.936,-1.37)--(-0.753,-1.11)--(-0.568,-0.845)--(-0.380,-0.568)--(-0.190,-0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.016641,-0.024962);
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+\draw [,->,>=latex] (0.0000,0.0000) -- (0.0166,-0.0249);
 \draw [,->,>=latex] (3.0000,-2.1213) -- (3.0000,-2.0913);
-\draw [,->,>=latex] (0,3.0000) -- (-0.030000,3.0000);
-\draw (-3.0000,-0.32983) node {$ -1 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -1 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [,->,>=latex] (0.0000,3.0000) -- (-0.0300,3.0000);
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -1 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExoPolaire.pstricks b/auto/pictures_tex/Fig_ExoPolaire.pstricks
index 6980c9bb4..f17182b99 100644
--- a/auto/pictures_tex/Fig_ExoPolaire.pstricks
+++ b/auto/pictures_tex/Fig_ExoPolaire.pstricks
@@ -75,18 +75,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.2321,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.2320,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
 \draw (2.3896,1.0000) node {$(\sqrt{3},1)$};
-\draw (0.65798,0.88654) node {$l$};
-\draw (1.0127,0.25615) node {$\theta$};
+\draw (0.6579,0.8865) node {$l$};
+\draw (1.0127,0.2561) node {$\theta$};
 
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-\draw [,->,>=latex] (0,0) -- (1.7320,1.0000);
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
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+\draw [,->,>=latex] (0.0000,0.0000) -- (1.7320,1.0000);
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExoProjection.pstricks b/auto/pictures_tex/Fig_ExoProjection.pstricks
index 7d7fed300..fb7d5e3ae 100644
--- a/auto/pictures_tex/Fig_ExoProjection.pstricks
+++ b/auto/pictures_tex/Fig_ExoProjection.pstricks
@@ -79,18 +79,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6000,0) -- (3.3000,0);
-\draw [,->,>=latex] (0,-1.5500) -- (0,3.5000);
+\draw [,->,>=latex] (-2.6000,0.0000) -- (3.3000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5500) -- (0.0000,3.5000);
 %DEFAULT
-\draw [style=dashed] (2.80,1.40) -- (-2.10,-1.05);
+\draw [style=dashed] (2.8000,1.4000) -- (-2.1000,-1.0500);
 \draw []  (2.4000,1.2000) node [rotate=0] {$\bullet$};
-\draw (3.2463,0.94356) node {$\pr_w(A)$};
+\draw (3.2463,0.9435) node {$\pr_w(A)$};
 \draw [color=red,->,>=latex] (1.5000,3.0000) -- (2.4000,1.2000);
-\draw [color=blue,->,>=latex] (0,0) -- (1.4000,0.70000);
-\draw (1.0716,1.0084) node {$w$};
-\draw []  (-1.0000,-0.50000) node [rotate=0] {$\bullet$};
-\draw (-1.3909,-0.78246) node {$P(\lambda)$};
-\draw [color=cyan,->,>=latex] (1.5000,3.0000) -- (-1.0000,-0.50000);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (1.4000,0.7000);
+\draw (1.0715,1.0083) node {$w$};
+\draw []  (-1.0000,-0.5000) node [rotate=0] {$\bullet$};
+\draw (-1.3908,-0.7824) node {$P(\lambda)$};
+\draw [color=cyan,->,>=latex] (1.5000,3.0000) -- (-1.0000,-0.5000);
 \draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
 \draw (1.5000,3.4247) node {$A$};
 
diff --git a/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks b/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks
index c9969ffe2..917499bce 100644
--- a/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks
+++ b/auto/pictures_tex/Fig_ExoUnSurxPolaire.pstricks
@@ -103,49 +103,49 @@
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+\draw (-0.4331,-3.0000) node {$ -3 $};
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+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExoVarj.pstricks b/auto/pictures_tex/Fig_ExoVarj.pstricks
index 9f183c7db..0bde7f009 100644
--- a/auto/pictures_tex/Fig_ExoVarj.pstricks
+++ b/auto/pictures_tex/Fig_ExoVarj.pstricks
@@ -71,17 +71,17 @@
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 %DEFAULT
 
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+\draw [color=blue] (-3.0000,1.4576)--(-2.9393,1.4696)--(-2.8787,1.4796)--(-2.8181,1.4875)--(-2.7575,1.4935)--(-2.6969,1.4975)--(-2.6363,1.4996)--(-2.5757,1.4998)--(-2.5151,1.4982)--(-2.4545,1.4949)--(-2.3939,1.4897)--(-2.3333,1.4828)--(-2.2727,1.4742)--(-2.2121,1.4638)--(-2.1515,1.4517)--(-2.0909,1.4380)--(-2.0303,1.4225)--(-1.9696,1.4054)--(-1.9090,1.3866)--(-1.8484,1.3661)--(-1.7878,1.3439)--(-1.7272,1.3200)--(-1.6666,1.2945)--(-1.6060,1.2673)--(-1.5454,1.2384)--(-1.4848,1.2079)--(-1.4242,1.1756)--(-1.3636,1.1417)--(-1.3030,1.1062)--(-1.2424,1.0689)--(-1.1818,1.0300)--(-1.1212,0.9895)--(-1.0606,0.9474)--(-1.0000,0.9036)--(-0.9393,0.8583)--(-0.8787,0.8115)--(-0.8181,0.7631)--(-0.7575,0.7132)--(-0.6969,0.6620)--(-0.6363,0.6094)--(-0.5757,0.5556)--(-0.5151,0.5005)--(-0.4545,0.4444)--(-0.3939,0.3873)--(-0.3333,0.3292)--(-0.2727,0.2705)--(-0.2121,0.2110)--(-0.1515,0.1511)--(-0.0909,0.0908)--(-0.0303,0.0302)--(0.0303,0.0302)--(0.0909,0.0908)--(0.1515,0.1511)--(0.2121,0.2110)--(0.2727,0.2705)--(0.3333,0.3292)--(0.3939,0.3873)--(0.4545,0.4444)--(0.5151,0.5005)--(0.5757,0.5556)--(0.6363,0.6094)--(0.6969,0.6620)--(0.7575,0.7132)--(0.8181,0.7631)--(0.8787,0.8115)--(0.9393,0.8583)--(1.0000,0.9036)--(1.0606,0.9474)--(1.1212,0.9895)--(1.1818,1.0300)--(1.2424,1.0689)--(1.3030,1.1062)--(1.3636,1.1417)--(1.4242,1.1756)--(1.4848,1.2079)--(1.5454,1.2384)--(1.6060,1.2673)--(1.6666,1.2945)--(1.7272,1.3200)--(1.7878,1.3439)--(1.8484,1.3661)--(1.9090,1.3866)--(1.9696,1.4054)--(2.0303,1.4225)--(2.0909,1.4380)--(2.1515,1.4517)--(2.2121,1.4638)--(2.2727,1.4742)--(2.3333,1.4828)--(2.3939,1.4897)--(2.4545,1.4949)--(2.5151,1.4982)--(2.5757,1.4998)--(2.6363,1.4996)--(2.6969,1.4975)--(2.7575,1.4935)--(2.8181,1.4875)--(2.8787,1.4796)--(2.9393,1.4696)--(3.0000,1.4576);
 
-\draw [color=red] (-3.00,-1.46)--(-2.94,-1.47)--(-2.88,-1.48)--(-2.82,-1.49)--(-2.76,-1.49)--(-2.70,-1.50)--(-2.64,-1.50)--(-2.58,-1.50)--(-2.52,-1.50)--(-2.45,-1.49)--(-2.39,-1.49)--(-2.33,-1.48)--(-2.27,-1.47)--(-2.21,-1.46)--(-2.15,-1.45)--(-2.09,-1.44)--(-2.03,-1.42)--(-1.97,-1.41)--(-1.91,-1.39)--(-1.85,-1.37)--(-1.79,-1.34)--(-1.73,-1.32)--(-1.67,-1.29)--(-1.61,-1.27)--(-1.55,-1.24)--(-1.48,-1.21)--(-1.42,-1.18)--(-1.36,-1.14)--(-1.30,-1.11)--(-1.24,-1.07)--(-1.18,-1.03)--(-1.12,-0.990)--(-1.06,-0.947)--(-1.00,-0.904)--(-0.939,-0.858)--(-0.879,-0.812)--(-0.818,-0.763)--(-0.758,-0.713)--(-0.697,-0.662)--(-0.636,-0.609)--(-0.576,-0.556)--(-0.515,-0.501)--(-0.455,-0.444)--(-0.394,-0.387)--(-0.333,-0.329)--(-0.273,-0.271)--(-0.212,-0.211)--(-0.152,-0.151)--(-0.0909,-0.0908)--(-0.0303,-0.0303)--(0.0303,-0.0303)--(0.0909,-0.0908)--(0.152,-0.151)--(0.212,-0.211)--(0.273,-0.271)--(0.333,-0.329)--(0.394,-0.387)--(0.455,-0.444)--(0.515,-0.501)--(0.576,-0.556)--(0.636,-0.609)--(0.697,-0.662)--(0.758,-0.713)--(0.818,-0.763)--(0.879,-0.812)--(0.939,-0.858)--(1.00,-0.904)--(1.06,-0.947)--(1.12,-0.990)--(1.18,-1.03)--(1.24,-1.07)--(1.30,-1.11)--(1.36,-1.14)--(1.42,-1.18)--(1.48,-1.21)--(1.55,-1.24)--(1.61,-1.27)--(1.67,-1.29)--(1.73,-1.32)--(1.79,-1.34)--(1.85,-1.37)--(1.91,-1.39)--(1.97,-1.41)--(2.03,-1.42)--(2.09,-1.44)--(2.15,-1.45)--(2.21,-1.46)--(2.27,-1.47)--(2.33,-1.48)--(2.39,-1.49)--(2.45,-1.49)--(2.52,-1.50)--(2.58,-1.50)--(2.64,-1.50)--(2.70,-1.50)--(2.76,-1.49)--(2.82,-1.49)--(2.88,-1.48)--(2.94,-1.47)--(3.00,-1.46);
-\draw (-3.0000,-0.32983) node {$ -1 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
+\draw [color=red] (-3.0000,-1.4576)--(-2.9393,-1.4696)--(-2.8787,-1.4796)--(-2.8181,-1.4875)--(-2.7575,-1.4935)--(-2.6969,-1.4975)--(-2.6363,-1.4996)--(-2.5757,-1.4998)--(-2.5151,-1.4982)--(-2.4545,-1.4949)--(-2.3939,-1.4897)--(-2.3333,-1.4828)--(-2.2727,-1.4742)--(-2.2121,-1.4638)--(-2.1515,-1.4517)--(-2.0909,-1.4380)--(-2.0303,-1.4225)--(-1.9696,-1.4054)--(-1.9090,-1.3866)--(-1.8484,-1.3661)--(-1.7878,-1.3439)--(-1.7272,-1.3200)--(-1.6666,-1.2945)--(-1.6060,-1.2673)--(-1.5454,-1.2384)--(-1.4848,-1.2079)--(-1.4242,-1.1756)--(-1.3636,-1.1417)--(-1.3030,-1.1062)--(-1.2424,-1.0689)--(-1.1818,-1.0300)--(-1.1212,-0.9895)--(-1.0606,-0.9474)--(-1.0000,-0.9036)--(-0.9393,-0.8583)--(-0.8787,-0.8115)--(-0.8181,-0.7631)--(-0.7575,-0.7132)--(-0.6969,-0.6620)--(-0.6363,-0.6094)--(-0.5757,-0.5556)--(-0.5151,-0.5005)--(-0.4545,-0.4444)--(-0.3939,-0.3873)--(-0.3333,-0.3292)--(-0.2727,-0.2705)--(-0.2121,-0.2110)--(-0.1515,-0.1511)--(-0.0909,-0.0908)--(-0.0303,-0.0302)--(0.0303,-0.0302)--(0.0909,-0.0908)--(0.1515,-0.1511)--(0.2121,-0.2110)--(0.2727,-0.2705)--(0.3333,-0.3292)--(0.3939,-0.3873)--(0.4545,-0.4444)--(0.5151,-0.5005)--(0.5757,-0.5556)--(0.6363,-0.6094)--(0.6969,-0.6620)--(0.7575,-0.7132)--(0.8181,-0.7631)--(0.8787,-0.8115)--(0.9393,-0.8583)--(1.0000,-0.9036)--(1.0606,-0.9474)--(1.1212,-0.9895)--(1.1818,-1.0300)--(1.2424,-1.0689)--(1.3030,-1.1062)--(1.3636,-1.1417)--(1.4242,-1.1756)--(1.4848,-1.2079)--(1.5454,-1.2384)--(1.6060,-1.2673)--(1.6666,-1.2945)--(1.7272,-1.3200)--(1.7878,-1.3439)--(1.8484,-1.3661)--(1.9090,-1.3866)--(1.9696,-1.4054)--(2.0303,-1.4225)--(2.0909,-1.4380)--(2.1515,-1.4517)--(2.2121,-1.4638)--(2.2727,-1.4742)--(2.3333,-1.4828)--(2.3939,-1.4897)--(2.4545,-1.4949)--(2.5151,-1.4982)--(2.5757,-1.4998)--(2.6363,-1.4996)--(2.6969,-1.4975)--(2.7575,-1.4935)--(2.8181,-1.4875)--(2.8787,-1.4796)--(2.9393,-1.4696)--(3.0000,-1.4576);
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ExoXLVL.pstricks b/auto/pictures_tex/Fig_ExoXLVL.pstricks
index 9dd1f83e3..0ac411b31 100644
--- a/auto/pictures_tex/Fig_ExoXLVL.pstricks
+++ b/auto/pictures_tex/Fig_ExoXLVL.pstricks
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
-\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red] (0,2.50) -- (2.50,2.50);
-\draw [color=red] (2.50,2.50) -- (2.50,0);
-\draw [color=red] (2.50,0) -- (0,0);
-\draw [color=red] (0,0) -- (0,2.50);
-\draw [color=cyan,style=dashed] (-0.100,-2.50) -- (-2.50,-2.50);
-\draw [color=cyan,style=dashed] (-2.50,-2.50) -- (-2.50,-0.100);
-\draw [color=cyan,style=dashed] (-2.50,-0.100) -- (-0.100,-0.100);
-\draw [color=cyan,style=dashed] (-0.100,-0.100) -- (-0.100,-2.50);
-\draw [color=green] (0,-2.50) -- (2.50,-2.50);
-\draw [color=green] (2.50,-2.50) -- (2.50,0);
-\draw [color=green] (2.50,0) -- (0,0);
-\draw [color=green] (0,0) -- (0,-2.50);
-\draw (-1.0997,1.3000) node {\( xy\)};
+\draw [color=blue,style=dashed] (-2.5000,0.1000) -- (-0.1000,0.1000);
+\draw [color=blue,style=dashed] (-0.1000,0.1000) -- (-0.1000,2.5000);
+\draw [color=blue,style=dashed] (-0.1000,2.5000) -- (-2.5000,2.5000);
+\draw [color=blue,style=dashed] (-2.5000,2.5000) -- (-2.5000,0.1000);
+\draw [color=red] (0.0000,2.5000) -- (2.5000,2.5000);
+\draw [color=red] (2.5000,2.5000) -- (2.5000,0.0000);
+\draw [color=red] (2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=red] (0.0000,0.0000) -- (0.0000,2.5000);
+\draw [color=cyan,style=dashed] (-0.1000,-2.5000) -- (-2.5000,-2.5000);
+\draw [color=cyan,style=dashed] (-2.5000,-2.5000) -- (-2.5000,-0.1000);
+\draw [color=cyan,style=dashed] (-2.5000,-0.1000) -- (-0.1000,-0.1000);
+\draw [color=cyan,style=dashed] (-0.1000,-0.1000) -- (-0.1000,-2.5000);
+\draw [color=green] (0.0000,-2.5000) -- (2.5000,-2.5000);
+\draw [color=green] (2.5000,-2.5000) -- (2.5000,0.0000);
+\draw [color=green] (2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=green] (0.0000,0.0000) -- (0.0000,-2.5000);
+\draw (-1.0996,1.3000) node {\( xy\)};
 \draw (1.6733,1.2500) node {\( x-y\)};
 \draw (-1.0205,-1.3000) node {\( x^2y\)};
 \draw (1.6733,-1.2500) node {\( x+y\)};
diff --git a/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks b/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks
index aacb17a58..7d8c5c8c6 100644
--- a/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks
+++ b/auto/pictures_tex/Fig_FCUEooTpEPFoeQ.pstricks
@@ -99,7 +99,7 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (1.4562,-1.1844) node {$
+\draw (1.4562,-1.1843) node {$
         \begin{pmatrix}
         \phantom{
         \begin{matrix}
@@ -108,40 +108,40 @@
         }
         \end{pmatrix}$
         };
-\draw (0.29125,-0.23688) node {*};
-\draw (0.87375,-0.23688) node {*};
-\draw (1.4562,-0.23688) node {*};
-\draw (2.0387,-0.23688) node {*};
-\draw (2.6213,-0.23688) node {*};
-\draw (0.29125,-0.71062) node {0};
-\draw (0.87375,-0.71062) node {*};
-\draw (1.4562,-0.71062) node {*};
-\draw (2.0387,-0.71062) node {*};
-\draw (2.6213,-0.71062) node {*};
-\draw (0.29125,-1.1844) node {0};
-\draw (0.87375,-1.1844) node {0};
-\draw (1.4562,-1.1844) node {*};
-\draw (2.0387,-1.1844) node {*};
-\draw (2.6213,-1.1844) node {*};
-\draw (0.29125,-1.6581) node {0};
-\draw (0.87375,-1.6581) node {0};
+\draw (0.2912,-0.2368) node {*};
+\draw (0.8737,-0.2368) node {*};
+\draw (1.4562,-0.2368) node {*};
+\draw (2.0387,-0.2368) node {*};
+\draw (2.6212,-0.2368) node {*};
+\draw (0.2912,-0.7106) node {0};
+\draw (0.8737,-0.7106) node {*};
+\draw (1.4562,-0.7106) node {*};
+\draw (2.0387,-0.7106) node {*};
+\draw (2.6212,-0.7106) node {*};
+\draw (0.2912,-1.1843) node {0};
+\draw (0.8737,-1.1843) node {0};
+\draw (1.4562,-1.1843) node {*};
+\draw (2.0387,-1.1843) node {*};
+\draw (2.6212,-1.1843) node {*};
+\draw (0.2912,-1.6581) node {0};
+\draw (0.8737,-1.6581) node {0};
 \draw (1.4562,-1.6581) node {*};
 \draw (2.0387,-1.6581) node {*};
-\draw (2.6213,-1.6581) node {*};
-\draw (0.29125,-2.1319) node {0};
-\draw (0.87375,-2.1319) node {0};
-\draw (1.4562,-2.1319) node {*};
-\draw (2.0387,-2.1319) node {*};
-\draw (2.6213,-2.1319) node {*};
-\draw [color=red] (0.100,-0.0500) -- (1.06,-0.0500);
-\draw [color=red] (1.06,-0.0500) -- (1.06,-0.898);
-\draw [color=red] (1.06,-0.898) -- (0.100,-0.898);
-\draw [color=red] (0.100,-0.898) -- (0.100,-0.0500);
-\draw [color=blue] (1.27,-0.997) -- (2.81,-0.997);
-\draw [color=blue] (2.81,-0.997) -- (2.81,-2.32);
-\draw [color=blue] (2.81,-2.32) -- (1.27,-2.32);
-\draw [color=blue] (1.27,-2.32) -- (1.27,-0.997);
-\draw (1.2124,0.22642) node {\( \Delta_k(A_2)\)};
+\draw (2.6212,-1.6581) node {*};
+\draw (0.2912,-2.1318) node {0};
+\draw (0.8737,-2.1318) node {0};
+\draw (1.4562,-2.1318) node {*};
+\draw (2.0387,-2.1318) node {*};
+\draw (2.6212,-2.1318) node {*};
+\draw [color=red] (0.1000,-0.0500) -- (1.0649,-0.0500);
+\draw [color=red] (1.0649,-0.0500) -- (1.0649,-0.8975);
+\draw [color=red] (1.0649,-0.8975) -- (0.1000,-0.8975);
+\draw [color=red] (0.1000,-0.8975) -- (0.1000,-0.0500);
+\draw [color=blue] (1.2649,-0.9975) -- (2.8124,-0.9975);
+\draw [color=blue] (2.8124,-0.9975) -- (2.8124,-2.3187);
+\draw [color=blue] (2.8124,-2.3187) -- (1.2649,-2.3187);
+\draw [color=blue] (1.2649,-2.3187) -- (1.2649,-0.9975);
+\draw (1.2123,0.2264) node {\( \Delta_k(A_2)\)};
 \draw (2.0387,-2.6012) node {\( \Omega_{k+1}(A_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_FGRooDhFkch.pstricks b/auto/pictures_tex/Fig_FGRooDhFkch.pstricks
index f4993c771..935fc87f8 100644
--- a/auto/pictures_tex/Fig_FGRooDhFkch.pstricks
+++ b/auto/pictures_tex/Fig_FGRooDhFkch.pstricks
@@ -86,39 +86,39 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-3.00,-3.14) -- (-3.00,3.14);
-\draw [color=gray,style=solid] (0,-3.14) -- (0,3.14);
-\draw [color=gray,style=solid] (3.00,-3.14) -- (3.00,3.14);
-\draw [color=gray,style=dotted] (-1.50,-3.14) -- (-1.50,3.14);
-\draw [color=gray,style=dotted] (1.50,-3.14) -- (1.50,3.14);
-\draw [color=gray,style=dotted] (-3.00,-2.36) -- (3.00,-2.36);
-\draw [color=gray,style=dotted] (-3.00,-0.785) -- (3.00,-0.785);
-\draw [color=gray,style=dotted] (-3.00,0.785) -- (3.00,0.785);
-\draw [color=gray,style=dotted] (-3.00,2.36) -- (3.00,2.36);
-\draw [color=gray,style=solid] (-3.00,-3.14) -- (3.00,-3.14);
-\draw [color=gray,style=solid] (-3.00,-1.57) -- (3.00,-1.57);
-\draw [color=gray,style=solid] (-3.00,0) -- (3.00,0);
-\draw [color=gray,style=solid] (-3.00,1.57) -- (3.00,1.57);
-\draw [color=gray,style=solid] (-3.00,3.14) -- (3.00,3.14);
+\draw [color=gray,style=solid] (-3.0000,-3.1415) -- (-3.0000,3.1415);
+\draw [color=gray,style=solid] (0.0000,-3.1415) -- (0.0000,3.1415);
+\draw [color=gray,style=solid] (3.0000,-3.1415) -- (3.0000,3.1415);
+\draw [color=gray,style=dotted] (-1.5000,-3.1415) -- (-1.5000,3.1415);
+\draw [color=gray,style=dotted] (1.5000,-3.1415) -- (1.5000,3.1415);
+\draw [color=gray,style=dotted] (-3.0000,-2.3561) -- (3.0000,-2.3561);
+\draw [color=gray,style=dotted] (-3.0000,-0.7853) -- (3.0000,-0.7853);
+\draw [color=gray,style=dotted] (-3.0000,0.7853) -- (3.0000,0.7853);
+\draw [color=gray,style=dotted] (-3.0000,2.3561) -- (3.0000,2.3561);
+\draw [color=gray,style=solid] (-3.0000,-3.1415) -- (3.0000,-3.1415);
+\draw [color=gray,style=solid] (-3.0000,-1.5707) -- (3.0000,-1.5707);
+\draw [color=gray,style=solid] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [color=gray,style=solid] (-3.0000,1.5707) -- (3.0000,1.5707);
+\draw [color=gray,style=solid] (-3.0000,3.1415) -- (3.0000,3.1415);
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.6416) -- (0,3.6416);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.6415) -- (0.0000,3.6415);
 %DEFAULT
 
-\draw [color=blue] (-3.000,-1.571)--(-2.939,-1.369)--(-2.879,-1.286)--(-2.818,-1.221)--(-2.758,-1.166)--(-2.697,-1.117)--(-2.636,-1.073)--(-2.576,-1.033)--(-2.515,-0.9943)--(-2.455,-0.9582)--(-2.394,-0.9239)--(-2.333,-0.8911)--(-2.273,-0.8596)--(-2.212,-0.8292)--(-2.152,-0.7997)--(-2.091,-0.7712)--(-2.030,-0.7434)--(-1.970,-0.7163)--(-1.909,-0.6898)--(-1.848,-0.6639)--(-1.788,-0.6385)--(-1.727,-0.6135)--(-1.667,-0.5890)--(-1.606,-0.5649)--(-1.545,-0.5412)--(-1.485,-0.5178)--(-1.424,-0.4947)--(-1.364,-0.4719)--(-1.303,-0.4493)--(-1.242,-0.4270)--(-1.182,-0.4049)--(-1.121,-0.3830)--(-1.061,-0.3613)--(-1.000,-0.3398)--(-0.9394,-0.3185)--(-0.8788,-0.2973)--(-0.8182,-0.2762)--(-0.7576,-0.2553)--(-0.6970,-0.2345)--(-0.6364,-0.2137)--(-0.5758,-0.1931)--(-0.5152,-0.1726)--(-0.4545,-0.1521)--(-0.3939,-0.1317)--(-0.3333,-0.1113)--(-0.2727,-0.09103)--(-0.2121,-0.07077)--(-0.1515,-0.05053)--(-0.09091,-0.03031)--(-0.03030,-0.01010)--(0.03030,0.01010)--(0.09091,0.03031)--(0.1515,0.05053)--(0.2121,0.07077)--(0.2727,0.09103)--(0.3333,0.1113)--(0.3939,0.1317)--(0.4545,0.1521)--(0.5152,0.1726)--(0.5758,0.1931)--(0.6364,0.2137)--(0.6970,0.2345)--(0.7576,0.2553)--(0.8182,0.2762)--(0.8788,0.2973)--(0.9394,0.3185)--(1.000,0.3398)--(1.061,0.3613)--(1.121,0.3830)--(1.182,0.4049)--(1.242,0.4270)--(1.303,0.4493)--(1.364,0.4719)--(1.424,0.4947)--(1.485,0.5178)--(1.545,0.5412)--(1.606,0.5649)--(1.667,0.5890)--(1.727,0.6135)--(1.788,0.6385)--(1.848,0.6639)--(1.909,0.6898)--(1.970,0.7163)--(2.030,0.7434)--(2.091,0.7712)--(2.152,0.7997)--(2.212,0.8292)--(2.273,0.8596)--(2.333,0.8911)--(2.394,0.9239)--(2.455,0.9582)--(2.515,0.9943)--(2.576,1.033)--(2.636,1.073)--(2.697,1.117)--(2.758,1.166)--(2.818,1.221)--(2.879,1.286)--(2.939,1.369)--(3.000,1.571);
+\draw [color=blue] (-3.0000,-1.5707)--(-2.9393,-1.3694)--(-2.8787,-1.2855)--(-2.8181,-1.2208)--(-2.7575,-1.1660)--(-2.6969,-1.1174)--(-2.6363,-1.0733)--(-2.5757,-1.0325)--(-2.5151,-0.9943)--(-2.4545,-0.9582)--(-2.3939,-0.9239)--(-2.3333,-0.8911)--(-2.2727,-0.8595)--(-2.2121,-0.8291)--(-2.1515,-0.7997)--(-2.0909,-0.7711)--(-2.0303,-0.7433)--(-1.9696,-0.7162)--(-1.9090,-0.6897)--(-1.8484,-0.6638)--(-1.7878,-0.6384)--(-1.7272,-0.6135)--(-1.6666,-0.5890)--(-1.6060,-0.5649)--(-1.5454,-0.5411)--(-1.4848,-0.5177)--(-1.4242,-0.4946)--(-1.3636,-0.4718)--(-1.3030,-0.4493)--(-1.2424,-0.4269)--(-1.1818,-0.4049)--(-1.1212,-0.3830)--(-1.0606,-0.3613)--(-1.0000,-0.3398)--(-0.9393,-0.3184)--(-0.8787,-0.2972)--(-0.8181,-0.2762)--(-0.7575,-0.2552)--(-0.6969,-0.2344)--(-0.6363,-0.2137)--(-0.5757,-0.1931)--(-0.5151,-0.1725)--(-0.4545,-0.1521)--(-0.3939,-0.1316)--(-0.3333,-0.1113)--(-0.2727,-0.0910)--(-0.2121,-0.0707)--(-0.1515,-0.0505)--(-0.0909,-0.0303)--(-0.0303,-0.0101)--(0.0303,0.0101)--(0.0909,0.0303)--(0.1515,0.0505)--(0.2121,0.0707)--(0.2727,0.0910)--(0.3333,0.1113)--(0.3939,0.1316)--(0.4545,0.1521)--(0.5151,0.1725)--(0.5757,0.1931)--(0.6363,0.2137)--(0.6969,0.2344)--(0.7575,0.2552)--(0.8181,0.2762)--(0.8787,0.2972)--(0.9393,0.3184)--(1.0000,0.3398)--(1.0606,0.3613)--(1.1212,0.3830)--(1.1818,0.4049)--(1.2424,0.4269)--(1.3030,0.4493)--(1.3636,0.4718)--(1.4242,0.4946)--(1.4848,0.5177)--(1.5454,0.5411)--(1.6060,0.5649)--(1.6666,0.5890)--(1.7272,0.6135)--(1.7878,0.6384)--(1.8484,0.6638)--(1.9090,0.6897)--(1.9696,0.7162)--(2.0303,0.7433)--(2.0909,0.7711)--(2.1515,0.7997)--(2.2121,0.8291)--(2.2727,0.8595)--(2.3333,0.8911)--(2.3939,0.9239)--(2.4545,0.9582)--(2.5151,0.9943)--(2.5757,1.0325)--(2.6363,1.0733)--(2.6969,1.1174)--(2.7575,1.1660)--(2.8181,1.2208)--(2.8787,1.2855)--(2.9393,1.3694)--(3.0000,1.5707);
 
-\draw (-3.0000,-0.32983) node {$ -1 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.45249,-3.1416) node {$ -\pi $};
-\draw [] (-0.100,-3.14) -- (0.100,-3.14);
-\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
-\draw [] (-0.100,1.57) -- (0.100,1.57);
-\draw (-0.31058,3.1416) node {$ \pi $};
-\draw [] (-0.100,3.14) -- (0.100,3.14);
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4524,-3.1415) node {$ -\pi $};
+\draw [] (-0.1000,-3.1415) -- (0.1000,-3.1415);
+\draw (-0.5937,-1.5707) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-0.1000,-1.5707) -- (0.1000,-1.5707);
+\draw (-0.4518,1.5707) node {$ \frac{1}{2} \, \pi $};
+\draw [] (-0.1000,1.5707) -- (0.1000,1.5707);
+\draw (-0.3105,3.1415) node {$ \pi $};
+\draw [] (-0.1000,3.1415) -- (0.1000,3.1415);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_FGWjJBX.pstricks b/auto/pictures_tex/Fig_FGWjJBX.pstricks
index 78a04f112..a3aed9a03 100644
--- a/auto/pictures_tex/Fig_FGWjJBX.pstricks
+++ b/auto/pictures_tex/Fig_FGWjJBX.pstricks
@@ -82,22 +82,22 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [style=dotted] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0.500);
-\draw [] (3.00,0) -- (4.00,-0.500);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.30595) node {\( \alpha_1\)};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,0.30595) node {\( \alpha_2\)};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,0.32154) node {\( \alpha_{l-2}\)};
-\draw []  (4.0000,0.50000) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.82154) node {\( \alpha_{l-1}\)};
-\draw []  (4.0000,-0.50000) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.19169) node {\( \alpha_l\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [style=dotted] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.5000);
+\draw [] (3.0000,0.0000) -- (4.0000,-0.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.3059) node {\( \alpha_1\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,0.3059) node {\( \alpha_2\)};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,0.3215) node {\( \alpha_{l-2}\)};
+\draw []  (4.0000,0.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.8215) node {\( \alpha_{l-1}\)};
+\draw []  (4.0000,-0.5000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.1916) node {\( \alpha_l\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks b/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks
index 1957bbd5f..bab9a7af9 100644
--- a/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks
+++ b/auto/pictures_tex/Fig_FNBQooYgkAmS.pstricks
@@ -74,13 +74,13 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (3.000,0)--(2.994,0.06342)--(2.976,0.1266)--(2.946,0.1893)--(2.904,0.2511)--(2.850,0.3120)--(2.785,0.3717)--(2.709,0.4298)--(2.622,0.4862)--(2.524,0.5406)--(2.416,0.5929)--(2.298,0.6428)--(2.171,0.6901)--(2.036,0.7346)--(1.892,0.7761)--(1.740,0.8146)--(1.582,0.8497)--(1.417,0.8815)--(1.246,0.9096)--(1.071,0.9342)--(0.8908,0.9549)--(0.7073,0.9718)--(0.5209,0.9848)--(0.3325,0.9938)--(0.1427,0.9989)--(-0.04760,0.9999)--(-0.2377,0.9969)--(-0.4269,0.9898)--(-0.6144,0.9788)--(-0.7994,0.9638)--(-0.9812,0.9450)--(-1.159,0.9224)--(-1.332,0.8960)--(-1.500,0.8660)--(-1.662,0.8326)--(-1.817,0.7958)--(-1.965,0.7558)--(-2.104,0.7127)--(-2.236,0.6668)--(-2.358,0.6182)--(-2.471,0.5671)--(-2.574,0.5137)--(-2.667,0.4582)--(-2.748,0.4009)--(-2.819,0.3420)--(-2.878,0.2817)--(-2.926,0.2203)--(-2.962,0.1580)--(-2.986,0.09506)--(-2.999,0.03173)--(-2.999,-0.03173)--(-2.986,-0.09506)--(-2.962,-0.1580)--(-2.926,-0.2203)--(-2.878,-0.2817)--(-2.819,-0.3420)--(-2.748,-0.4009)--(-2.667,-0.4582)--(-2.574,-0.5137)--(-2.471,-0.5671)--(-2.358,-0.6182)--(-2.236,-0.6668)--(-2.104,-0.7127)--(-1.965,-0.7558)--(-1.817,-0.7958)--(-1.662,-0.8326)--(-1.500,-0.8660)--(-1.332,-0.8960)--(-1.159,-0.9224)--(-0.9812,-0.9450)--(-0.7994,-0.9638)--(-0.6144,-0.9788)--(-0.4269,-0.9898)--(-0.2377,-0.9969)--(-0.04760,-0.9999)--(0.1427,-0.9989)--(0.3325,-0.9938)--(0.5209,-0.9848)--(0.7073,-0.9718)--(0.8908,-0.9549)--(1.071,-0.9342)--(1.246,-0.9096)--(1.417,-0.8815)--(1.582,-0.8497)--(1.740,-0.8146)--(1.892,-0.7761)--(2.036,-0.7346)--(2.171,-0.6901)--(2.298,-0.6428)--(2.416,-0.5929)--(2.524,-0.5406)--(2.622,-0.4862)--(2.709,-0.4298)--(2.785,-0.3717)--(2.850,-0.3120)--(2.904,-0.2511)--(2.946,-0.1893)--(2.976,-0.1266)--(2.994,-0.06342)--(3.000,0);
-\draw []  (2.1213,0.70711) node [rotate=0] {$\bullet$};
-\draw (2.0116,0.98561) node {\( x\)};
-\draw [,->,>=latex] (2.1213,0.70711) -- (2.4375,1.6558);
-\draw (1.7615,1.9015) node {\( \nabla q(x)\)};
-\draw [,->,>=latex] (2.1213,0.70711) -- (2.1486,-0.29252);
-\draw (2.5897,-0.16235) node {\( Ax\)};
+\draw [color=blue] (3.0000,0.0000)--(2.9939,0.0634)--(2.9758,0.1265)--(2.9457,0.1892)--(2.9038,0.2511)--(2.8502,0.3120)--(2.7851,0.3716)--(2.7087,0.4297)--(2.6215,0.4861)--(2.5237,0.5406)--(2.4158,0.5929)--(2.2981,0.6427)--(2.1712,0.6900)--(2.0355,0.7345)--(1.8916,0.7761)--(1.7401,0.8145)--(1.5816,0.8497)--(1.4168,0.8814)--(1.2462,0.9096)--(1.0706,0.9341)--(0.8907,0.9549)--(0.7072,0.9718)--(0.5209,0.9848)--(0.3325,0.9938)--(0.1427,0.9988)--(-0.0475,0.9998)--(-0.2377,0.9968)--(-0.4269,0.9898)--(-0.6144,0.9788)--(-0.7994,0.9638)--(-0.9812,0.9450)--(-1.1590,0.9223)--(-1.3321,0.8959)--(-1.5000,0.8660)--(-1.6617,0.8325)--(-1.8168,0.7957)--(-1.9645,0.7557)--(-2.1044,0.7126)--(-2.2357,0.6667)--(-2.3581,0.6181)--(-2.4710,0.5670)--(-2.5739,0.5136)--(-2.6665,0.4582)--(-2.7483,0.4009)--(-2.8190,0.3420)--(-2.8784,0.2817)--(-2.9262,0.2203)--(-2.9623,0.1580)--(-2.9864,0.0950)--(-2.9984,0.0317)--(-2.9984,-0.0317)--(-2.9864,-0.0950)--(-2.9623,-0.1580)--(-2.9262,-0.2203)--(-2.8784,-0.2817)--(-2.8190,-0.3420)--(-2.7483,-0.4009)--(-2.6665,-0.4582)--(-2.5739,-0.5136)--(-2.4710,-0.5670)--(-2.3581,-0.6181)--(-2.2357,-0.6667)--(-2.1044,-0.7126)--(-1.9645,-0.7557)--(-1.8168,-0.7957)--(-1.6617,-0.8325)--(-1.5000,-0.8660)--(-1.3321,-0.8959)--(-1.1590,-0.9223)--(-0.9812,-0.9450)--(-0.7994,-0.9638)--(-0.6144,-0.9788)--(-0.4269,-0.9898)--(-0.2377,-0.9968)--(-0.0475,-0.9998)--(0.1427,-0.9988)--(0.3325,-0.9938)--(0.5209,-0.9848)--(0.7072,-0.9718)--(0.8907,-0.9549)--(1.0706,-0.9341)--(1.2462,-0.9096)--(1.4168,-0.8814)--(1.5816,-0.8497)--(1.7401,-0.8145)--(1.8916,-0.7761)--(2.0355,-0.7345)--(2.1712,-0.6900)--(2.2981,-0.6427)--(2.4158,-0.5929)--(2.5237,-0.5406)--(2.6215,-0.4861)--(2.7087,-0.4297)--(2.7851,-0.3716)--(2.8502,-0.3120)--(2.9038,-0.2511)--(2.9457,-0.1892)--(2.9758,-0.1265)--(2.9939,-0.0634)--(3.0000,0.0000);
+\draw []  (2.1213,0.7071) node [rotate=0] {$\bullet$};
+\draw (2.0115,0.9856) node {\( x\)};
+\draw [,->,>=latex] (2.1213,0.7071) -- (2.4375,1.6557);
+\draw (1.7614,1.9014) node {\( \nabla q(x)\)};
+\draw [,->,>=latex] (2.1213,0.7071) -- (2.1486,-0.2925);
+\draw (2.5897,-0.1623) node {\( Ax\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_FWJuNhU.pstricks b/auto/pictures_tex/Fig_FWJuNhU.pstricks
index c5cad1966..2dae2d4c7 100644
--- a/auto/pictures_tex/Fig_FWJuNhU.pstricks
+++ b/auto/pictures_tex/Fig_FWJuNhU.pstricks
@@ -79,28 +79,28 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
-\draw [color=blue,style=dashed] (-2.50,0.100) -- (-0.100,0.100);
-\draw [color=blue,style=dashed] (-0.100,0.100) -- (-0.100,2.50);
-\draw [color=blue,style=dashed] (-0.100,2.50) -- (-2.50,2.50);
-\draw [color=blue,style=dashed] (-2.50,2.50) -- (-2.50,0.100);
-\draw [color=red,style=dashed] (0,2.50) -- (2.50,2.50);
-\draw [color=red,style=dashed] (2.50,2.50) -- (2.50,0);
-\draw [color=red,style=dashed] (2.50,0) -- (0,0);
-\draw [color=red,style=dashed] (0,0) -- (0,2.50);
-\draw [color=cyan] (0,-2.50) -- (-2.50,-2.50);
-\draw [color=cyan] (-2.50,-2.50) -- (-2.50,0);
-\draw [color=cyan] (-2.50,0) -- (0,0);
-\draw [color=cyan] (0,0) -- (0,-2.50);
-\draw [color=green,style=dashed] (0.100,-2.50) -- (2.50,-2.50);
-\draw [color=green,style=dashed] (2.50,-2.50) -- (2.50,-0.100);
-\draw [color=green,style=dashed] (2.50,-0.100) -- (0.100,-0.100);
-\draw [color=green,style=dashed] (0.100,-0.100) -- (0.100,-2.50);
-\draw (-1.0997,1.3000) node {\( xy\)};
+\draw [color=blue,style=dashed] (-2.5000,0.1000) -- (-0.1000,0.1000);
+\draw [color=blue,style=dashed] (-0.1000,0.1000) -- (-0.1000,2.5000);
+\draw [color=blue,style=dashed] (-0.1000,2.5000) -- (-2.5000,2.5000);
+\draw [color=blue,style=dashed] (-2.5000,2.5000) -- (-2.5000,0.1000);
+\draw [color=red,style=dashed] (0.0000,2.5000) -- (2.5000,2.5000);
+\draw [color=red,style=dashed] (2.5000,2.5000) -- (2.5000,0.0000);
+\draw [color=red,style=dashed] (2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=red,style=dashed] (0.0000,0.0000) -- (0.0000,2.5000);
+\draw [color=cyan] (0.0000,-2.5000) -- (-2.5000,-2.5000);
+\draw [color=cyan] (-2.5000,-2.5000) -- (-2.5000,0.0000);
+\draw [color=cyan] (-2.5000,0.0000) -- (0.0000,0.0000);
+\draw [color=cyan] (0.0000,0.0000) -- (0.0000,-2.5000);
+\draw [color=green,style=dashed] (0.1000,-2.5000) -- (2.5000,-2.5000);
+\draw [color=green,style=dashed] (2.5000,-2.5000) -- (2.5000,-0.1000);
+\draw [color=green,style=dashed] (2.5000,-0.1000) -- (0.1000,-0.1000);
+\draw [color=green,style=dashed] (0.1000,-0.1000) -- (0.1000,-2.5000);
+\draw (-1.0996,1.3000) node {\( xy\)};
 \draw (1.6733,1.2500) node {\( x-y\)};
-\draw (-0.97051,-1.2500) node {\( x^2y\)};
+\draw (-0.9705,-1.2500) node {\( x^2y\)};
 \draw (1.7233,-1.3000) node {\( x+y\)};
 
 %OTHER STUFF
diff --git a/auto/pictures_tex/Fig_FXVooJYAfif.pstricks b/auto/pictures_tex/Fig_FXVooJYAfif.pstricks
index d5c572a19..b4d41699b 100644
--- a/auto/pictures_tex/Fig_FXVooJYAfif.pstricks
+++ b/auto/pictures_tex/Fig_FXVooJYAfif.pstricks
@@ -60,13 +60,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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-\draw [color=red] (-4.00,1.00) -- (4.00,1.00);
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+\draw [color=red] (-4.0000,1.0000) -- (4.0000,1.0000);
+\draw [color=red] (-4.0000,-1.0000) -- (4.0000,-1.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_FnCosApprox.pstricks b/auto/pictures_tex/Fig_FnCosApprox.pstricks
index 890159f3e..6f99255f5 100644
--- a/auto/pictures_tex/Fig_FnCosApprox.pstricks
+++ b/auto/pictures_tex/Fig_FnCosApprox.pstricks
@@ -91,25 +91,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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-\draw []  (1.5708,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.7999,1.6614) node {$P$};
-\draw (1.5708,-0.42071) node {$ \frac{1}{4} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{4} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.27858) node {$ \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.43316,-2.0000) node {$ -1 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,2.0000)--(0.0634,1.9989)--(0.1269,1.9959)--(0.1903,1.9909)--(0.2538,1.9839)--(0.3173,1.9748)--(0.3807,1.9638)--(0.4442,1.9508)--(0.5077,1.9358)--(0.5711,1.9189)--(0.6346,1.9001)--(0.6981,1.8793)--(0.7615,1.8567)--(0.8250,1.8322)--(0.8885,1.8058)--(0.9519,1.7776)--(1.0154,1.7476)--(1.0789,1.7159)--(1.1423,1.6825)--(1.2058,1.6473)--(1.2693,1.6105)--(1.3327,1.5721)--(1.3962,1.5320)--(1.4597,1.4905)--(1.5231,1.4474)--(1.5866,1.4029)--(1.6501,1.3570)--(1.7135,1.3097)--(1.7770,1.2611)--(1.8405,1.2112)--(1.9039,1.1601)--(1.9674,1.1078)--(2.0309,1.0544)--(2.0943,1.0000)--(2.1578,0.9445)--(2.2213,0.8881)--(2.2847,0.8308)--(2.3482,0.7726)--(2.4117,0.7137)--(2.4751,0.6541)--(2.5386,0.5938)--(2.6021,0.5329)--(2.6655,0.4715)--(2.7290,0.4096)--(2.7925,0.3472)--(2.8559,0.2846)--(2.9194,0.2216)--(2.9829,0.1584)--(3.0463,0.0951)--(3.1098,0.0317)--(3.1733,-0.0317)--(3.2367,-0.0951)--(3.3002,-0.1584)--(3.3637,-0.2216)--(3.4271,-0.2846)--(3.4906,-0.3472)--(3.5541,-0.4096)--(3.6175,-0.4715)--(3.6810,-0.5329)--(3.7445,-0.5938)--(3.8079,-0.6541)--(3.8714,-0.7137)--(3.9349,-0.7726)--(3.9983,-0.8308)--(4.0618,-0.8881)--(4.1253,-0.9445)--(4.1887,-1.0000)--(4.2522,-1.0544)--(4.3157,-1.1078)--(4.3791,-1.1601)--(4.4426,-1.2112)--(4.5061,-1.2611)--(4.5695,-1.3097)--(4.6330,-1.3570)--(4.6965,-1.4029)--(4.7599,-1.4474)--(4.8234,-1.4905)--(4.8869,-1.5320)--(4.9503,-1.5721)--(5.0138,-1.6105)--(5.0773,-1.6473)--(5.1407,-1.6825)--(5.2042,-1.7159)--(5.2677,-1.7476)--(5.3311,-1.7776)--(5.3946,-1.8058)--(5.4581,-1.8322)--(5.5215,-1.8567)--(5.5850,-1.8793)--(5.6485,-1.9001)--(5.7119,-1.9189)--(5.7754,-1.9358)--(5.8389,-1.9508)--(5.9023,-1.9638)--(5.9658,-1.9748)--(6.0293,-1.9839)--(6.0927,-1.9909)--(6.1562,-1.9959)--(6.2197,-1.9989)--(6.2831,-2.0000);
+\draw []  (1.5707,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.7999,1.6613) node {$P$};
+\draw (1.5707,-0.4207) node {$ \frac{1}{4} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{4} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.2785) node {$ \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks b/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
index 5fcac2505..3571a0d1c 100644
--- a/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
+++ b/auto/pictures_tex/Fig_FonctionEtDeriveOM.pstricks
@@ -119,41 +119,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
+\draw [,->,>=latex] (-4.0000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.7995) -- (0.0000,4.0547);
 %DEFAULT
 
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+\draw [color=blue] (-3.5000,-0.9928)--(-3.4292,-0.6362)--(-3.3585,-0.2871)--(-3.2878,0.0506)--(-3.2171,0.3737)--(-3.1464,0.6787)--(-3.0757,0.9630)--(-3.0050,1.2238)--(-2.9343,1.4592)--(-2.8636,1.6673)--(-2.7929,1.8468)--(-2.7222,1.9968)--(-2.6515,2.1167)--(-2.5808,2.2065)--(-2.5101,2.2664)--(-2.4393,2.2970)--(-2.3686,2.2995)--(-2.2979,2.2750)--(-2.2272,2.2254)--(-2.1565,2.1525)--(-2.0858,2.0586)--(-2.0151,1.9459)--(-1.9444,1.8171)--(-1.8737,1.6749)--(-1.8030,1.5220)--(-1.7323,1.3612)--(-1.6616,1.1953)--(-1.5909,1.0272)--(-1.5202,0.8595)--(-1.4494,0.6948)--(-1.3787,0.5355)--(-1.3080,0.3839)--(-1.2373,0.2420)--(-1.1666,0.1116)--(-1.0959,-0.0056)--(-1.0252,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2680)--(-0.8131,-0.3235)--(-0.7424,-0.3625)--(-0.6717,-0.3854)--(-0.6010,-0.3927)--(-0.5303,-0.3852)--(-0.4595,-0.3640)--(-0.3888,-0.3304)--(-0.3181,-0.2858)--(-0.2474,-0.2321)--(-0.1767,-0.1711)--(-0.1060,-0.1048)--(-0.0353,-0.0353)--(0.0353,0.0353)--(0.1060,0.1048)--(0.1767,0.1711)--(0.2474,0.2321)--(0.3181,0.2858)--(0.3888,0.3304)--(0.4595,0.3640)--(0.5303,0.3852)--(0.6010,0.3927)--(0.6717,0.3854)--(0.7424,0.3625)--(0.8131,0.3235)--(0.8838,0.2680)--(0.9545,0.1963)--(1.0252,0.1086)--(1.0959,0.0056)--(1.1666,-0.1116)--(1.2373,-0.2420)--(1.3080,-0.3839)--(1.3787,-0.5355)--(1.4494,-0.6948)--(1.5202,-0.8595)--(1.5909,-1.0272)--(1.6616,-1.1953)--(1.7323,-1.3612)--(1.8030,-1.5220)--(1.8737,-1.6749)--(1.9444,-1.8171)--(2.0151,-1.9459)--(2.0858,-2.0586)--(2.1565,-2.1525)--(2.2272,-2.2254)--(2.2979,-2.2750)--(2.3686,-2.2995)--(2.4393,-2.2970)--(2.5101,-2.2664)--(2.5808,-2.2065)--(2.6515,-2.1167)--(2.7222,-1.9968)--(2.7929,-1.8468)--(2.8636,-1.6673)--(2.9343,-1.4592)--(3.0050,-1.2238)--(3.0757,-0.9630)--(3.1464,-0.6787)--(3.2171,-0.3737)--(3.2878,-0.0506)--(3.3585,0.2871)--(3.4292,0.6362)--(3.5000,0.9928);
 
-\draw [color=red] (-3.500,3.555)--(-3.429,3.500)--(-3.359,3.406)--(-3.288,3.277)--(-3.217,3.114)--(-3.146,2.921)--(-3.076,2.702)--(-3.005,2.459)--(-2.934,2.198)--(-2.864,1.921)--(-2.793,1.632)--(-2.722,1.337)--(-2.652,1.038)--(-2.581,0.7401)--(-2.510,0.4468)--(-2.439,0.1618)--(-2.369,-0.1114)--(-2.298,-0.3696)--(-2.227,-0.6099)--(-2.157,-0.8297)--(-2.086,-1.027)--(-2.015,-1.199)--(-1.944,-1.346)--(-1.874,-1.466)--(-1.803,-1.558)--(-1.732,-1.622)--(-1.662,-1.658)--(-1.591,-1.667)--(-1.520,-1.650)--(-1.449,-1.608)--(-1.379,-1.542)--(-1.308,-1.456)--(-1.237,-1.350)--(-1.167,-1.228)--(-1.096,-1.092)--(-1.025,-0.9453)--(-0.9545,-0.7902)--(-0.8838,-0.6299)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.002540)--(-0.5303,0.1441)--(-0.4596,0.2739)--(-0.3889,0.3896)--(-0.3182,0.4892)--(-0.2475,0.5710)--(-0.1768,0.6336)--(-0.1061,0.6760)--(-0.03535,0.6973)--(0.03535,0.6973)--(0.1061,0.6760)--(0.1768,0.6336)--(0.2475,0.5710)--(0.3182,0.4892)--(0.3889,0.3896)--(0.4596,0.2739)--(0.5303,0.1441)--(0.6010,0.002540)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6299)--(0.9545,-0.7902)--(1.025,-0.9453)--(1.096,-1.092)--(1.167,-1.228)--(1.237,-1.350)--(1.308,-1.456)--(1.379,-1.542)--(1.449,-1.608)--(1.520,-1.650)--(1.591,-1.667)--(1.662,-1.658)--(1.732,-1.622)--(1.803,-1.558)--(1.874,-1.466)--(1.944,-1.346)--(2.015,-1.199)--(2.086,-1.027)--(2.157,-0.8297)--(2.227,-0.6099)--(2.298,-0.3696)--(2.369,-0.1114)--(2.439,0.1618)--(2.510,0.4468)--(2.581,0.7401)--(2.652,1.038)--(2.722,1.337)--(2.793,1.632)--(2.864,1.921)--(2.934,2.198)--(3.005,2.459)--(3.076,2.702)--(3.146,2.921)--(3.217,3.114)--(3.288,3.277)--(3.359,3.406)--(3.429,3.500)--(3.500,3.555);
-\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
-\draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
-\draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=red] (-3.5000,3.5547)--(-3.4292,3.4996)--(-3.3585,3.4061)--(-3.2878,3.2766)--(-3.2171,3.1140)--(-3.1464,2.9213)--(-3.0757,2.7019)--(-3.0050,2.4594)--(-2.9343,2.1976)--(-2.8636,1.9206)--(-2.7929,1.6322)--(-2.7222,1.3367)--(-2.6515,1.0379)--(-2.5808,0.7400)--(-2.5101,0.4467)--(-2.4393,0.1618)--(-2.3686,-0.1113)--(-2.2979,-0.3695)--(-2.2272,-0.6098)--(-2.1565,-0.8297)--(-2.0858,-1.0268)--(-2.0151,-1.1994)--(-1.9444,-1.3460)--(-1.8737,-1.4656)--(-1.8030,-1.5575)--(-1.7323,-1.6215)--(-1.6616,-1.6577)--(-1.5909,-1.6667)--(-1.5202,-1.6496)--(-1.4494,-1.6076)--(-1.3787,-1.5424)--(-1.3080,-1.4559)--(-1.2373,-1.3503)--(-1.1666,-1.2283)--(-1.0959,-1.0923)--(-1.0252,-0.9453)--(-0.9545,-0.7901)--(-0.8838,-0.6298)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.0025)--(-0.5303,0.1441)--(-0.4595,0.2739)--(-0.3888,0.3896)--(-0.3181,0.4892)--(-0.2474,0.5710)--(-0.1767,0.6336)--(-0.1060,0.6759)--(-0.0353,0.6973)--(0.0353,0.6973)--(0.1060,0.6759)--(0.1767,0.6336)--(0.2474,0.5710)--(0.3181,0.4892)--(0.3888,0.3896)--(0.4595,0.2739)--(0.5303,0.1441)--(0.6010,0.0025)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6298)--(0.9545,-0.7901)--(1.0252,-0.9453)--(1.0959,-1.0923)--(1.1666,-1.2283)--(1.2373,-1.3503)--(1.3080,-1.4559)--(1.3787,-1.5424)--(1.4494,-1.6076)--(1.5202,-1.6496)--(1.5909,-1.6667)--(1.6616,-1.6577)--(1.7323,-1.6215)--(1.8030,-1.5575)--(1.8737,-1.4656)--(1.9444,-1.3460)--(2.0151,-1.1994)--(2.0858,-1.0268)--(2.1565,-0.8297)--(2.2272,-0.6098)--(2.2979,-0.3695)--(2.3686,-0.1113)--(2.4393,0.1618)--(2.5101,0.4467)--(2.5808,0.7400)--(2.6515,1.0379)--(2.7222,1.3367)--(2.7929,1.6322)--(2.8636,1.9206)--(2.9343,2.1976)--(3.0050,2.4594)--(3.0757,2.7019)--(3.1464,2.9213)--(3.2171,3.1140)--(3.2878,3.2766)--(3.3585,3.4061)--(3.4292,3.4996)--(3.5000,3.5547);
+\draw (-3.2986,-0.4207) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (-3.2986,-0.1000) -- (-3.2986,0.1000);
+\draw (-2.1991,-0.3210) node {$-\mathit{\pi}$};
+\draw [] (-2.1991,-0.1000) -- (-2.1991,0.1000);
+\draw (-1.0995,-0.4207) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (-1.0995,-0.1000) -- (-1.0995,0.1000);
+\draw (1.0995,-0.4207) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (1.0995,-0.1000) -- (1.0995,0.1000);
+\draw (2.1991,-0.2785) node {$\mathit{\pi}$};
+\draw [] (2.1991,-0.1000) -- (2.1991,0.1000);
+\draw (3.2986,-0.4207) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (3.2986,-0.1000) -- (3.2986,0.1000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks b/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks
index 8c1bda278..0211c51c8 100644
--- a/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks
+++ b/auto/pictures_tex/Fig_FonctionXtroisOM.pstricks
@@ -91,31 +91,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.4000,0) -- (1.4000,0);
-\draw [,->,>=latex] (0,-2.5250) -- (0,4.5500);
+\draw [,->,>=latex] (-1.4000,0.0000) -- (1.4000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5250) -- (0.0000,4.5500);
 %DEFAULT
 
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diff --git a/auto/pictures_tex/Fig_GBnUivi.pstricks b/auto/pictures_tex/Fig_GBnUivi.pstricks
index 447475bac..eb4ed3200 100644
--- a/auto/pictures_tex/Fig_GBnUivi.pstricks
+++ b/auto/pictures_tex/Fig_GBnUivi.pstricks
@@ -102,61 +102,61 @@
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diff --git a/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks b/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks
index 115bccffb..bec0dac16 100644
--- a/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks
+++ b/auto/pictures_tex/Fig_GCNooKEbjWB.pstricks
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-\draw [color=gray,style=solid] (0,3.00) -- (3.00,3.00);
+\draw [color=gray,style=solid] (0.0000,0.0000) -- (0.0000,3.0000);
+\draw [color=gray,style=solid] (3.0000,0.0000) -- (3.0000,3.0000);
+\draw [color=gray,style=dotted] (1.5000,0.0000) -- (1.5000,3.0000);
+\draw [color=gray,style=dotted] (0.0000,1.5000) -- (3.0000,1.5000);
+\draw [color=gray,style=solid] (0.0000,0.0000) -- (3.0000,0.0000);
+\draw [color=gray,style=solid] (0.0000,3.0000) -- (3.0000,3.0000);
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.0152,0.0152)--(0.0303,0.0303)--(0.0455,0.0455)--(0.0606,0.0606)--(0.0758,0.0758)--(0.0909,0.0909)--(0.106,0.106)--(0.121,0.121)--(0.136,0.136)--(0.152,0.152)--(0.167,0.167)--(0.182,0.182)--(0.197,0.197)--(0.212,0.212)--(0.227,0.227)--(0.242,0.242)--(0.258,0.258)--(0.273,0.273)--(0.288,0.288)--(0.303,0.303)--(0.318,0.318)--(0.333,0.333)--(0.348,0.348)--(0.364,0.364)--(0.379,0.379)--(0.394,0.394)--(0.409,0.409)--(0.424,0.424)--(0.439,0.439)--(0.455,0.455)--(0.470,0.470)--(0.485,0.485)--(0.500,0.500)--(0.515,0.515)--(0.530,0.530)--(0.545,0.545)--(0.561,0.561)--(0.576,0.576)--(0.591,0.591)--(0.606,0.606)--(0.621,0.621)--(0.636,0.636)--(0.651,0.651)--(0.667,0.667)--(0.682,0.682)--(0.697,0.697)--(0.712,0.712)--(0.727,0.727)--(0.742,0.742)--(0.758,0.758)--(0.773,0.773)--(0.788,0.788)--(0.803,0.803)--(0.818,0.818)--(0.833,0.833)--(0.849,0.849)--(0.864,0.864)--(0.879,0.879)--(0.894,0.894)--(0.909,0.909)--(0.924,0.924)--(0.939,0.939)--(0.955,0.955)--(0.970,0.970)--(0.985,0.985)--(1.00,1.00)--(1.02,1.02)--(1.03,1.03)--(1.05,1.05)--(1.06,1.06)--(1.08,1.08)--(1.09,1.09)--(1.11,1.11)--(1.12,1.12)--(1.14,1.14)--(1.15,1.15)--(1.17,1.17)--(1.18,1.18)--(1.20,1.20)--(1.21,1.21)--(1.23,1.23)--(1.24,1.24)--(1.26,1.26)--(1.27,1.27)--(1.29,1.29)--(1.30,1.30)--(1.32,1.32)--(1.33,1.33)--(1.35,1.35)--(1.36,1.36)--(1.38,1.38)--(1.39,1.39)--(1.41,1.41)--(1.42,1.42)--(1.44,1.44)--(1.45,1.45)--(1.47,1.47)--(1.48,1.48)--(1.50,1.50);
+\draw [color=blue] (0.0000,0.0000)--(0.0151,0.0151)--(0.0303,0.0303)--(0.0454,0.0454)--(0.0606,0.0606)--(0.0757,0.0757)--(0.0909,0.0909)--(0.1060,0.1060)--(0.1212,0.1212)--(0.1363,0.1363)--(0.1515,0.1515)--(0.1666,0.1666)--(0.1818,0.1818)--(0.1969,0.1969)--(0.2121,0.2121)--(0.2272,0.2272)--(0.2424,0.2424)--(0.2575,0.2575)--(0.2727,0.2727)--(0.2878,0.2878)--(0.3030,0.3030)--(0.3181,0.3181)--(0.3333,0.3333)--(0.3484,0.3484)--(0.3636,0.3636)--(0.3787,0.3787)--(0.3939,0.3939)--(0.4090,0.4090)--(0.4242,0.4242)--(0.4393,0.4393)--(0.4545,0.4545)--(0.4696,0.4696)--(0.4848,0.4848)--(0.5000,0.5000)--(0.5151,0.5151)--(0.5303,0.5303)--(0.5454,0.5454)--(0.5606,0.5606)--(0.5757,0.5757)--(0.5909,0.5909)--(0.6060,0.6060)--(0.6212,0.6212)--(0.6363,0.6363)--(0.6515,0.6515)--(0.6666,0.6666)--(0.6818,0.6818)--(0.6969,0.6969)--(0.7121,0.7121)--(0.7272,0.7272)--(0.7424,0.7424)--(0.7575,0.7575)--(0.7727,0.7727)--(0.7878,0.7878)--(0.8030,0.8030)--(0.8181,0.8181)--(0.8333,0.8333)--(0.8484,0.8484)--(0.8636,0.8636)--(0.8787,0.8787)--(0.8939,0.8939)--(0.9090,0.9090)--(0.9242,0.9242)--(0.9393,0.9393)--(0.9545,0.9545)--(0.9696,0.9696)--(0.9848,0.9848)--(1.0000,1.0000)--(1.0151,1.0151)--(1.0303,1.0303)--(1.0454,1.0454)--(1.0606,1.0606)--(1.0757,1.0757)--(1.0909,1.0909)--(1.1060,1.1060)--(1.1212,1.1212)--(1.1363,1.1363)--(1.1515,1.1515)--(1.1666,1.1666)--(1.1818,1.1818)--(1.1969,1.1969)--(1.2121,1.2121)--(1.2272,1.2272)--(1.2424,1.2424)--(1.2575,1.2575)--(1.2727,1.2727)--(1.2878,1.2878)--(1.3030,1.3030)--(1.3181,1.3181)--(1.3333,1.3333)--(1.3484,1.3484)--(1.3636,1.3636)--(1.3787,1.3787)--(1.3939,1.3939)--(1.4090,1.4090)--(1.4242,1.4242)--(1.4393,1.4393)--(1.4545,1.4545)--(1.4696,1.4696)--(1.4848,1.4848)--(1.5000,1.5000);
 
-\draw [color=blue] (1.50,3.00)--(1.52,2.98)--(1.53,2.97)--(1.55,2.95)--(1.56,2.94)--(1.58,2.92)--(1.59,2.91)--(1.61,2.89)--(1.62,2.88)--(1.64,2.86)--(1.65,2.85)--(1.67,2.83)--(1.68,2.82)--(1.70,2.80)--(1.71,2.79)--(1.73,2.77)--(1.74,2.76)--(1.76,2.74)--(1.77,2.73)--(1.79,2.71)--(1.80,2.70)--(1.82,2.68)--(1.83,2.67)--(1.85,2.65)--(1.86,2.64)--(1.88,2.62)--(1.89,2.61)--(1.91,2.59)--(1.92,2.58)--(1.94,2.56)--(1.95,2.55)--(1.97,2.53)--(1.98,2.52)--(2.00,2.50)--(2.02,2.48)--(2.03,2.47)--(2.05,2.45)--(2.06,2.44)--(2.08,2.42)--(2.09,2.41)--(2.11,2.39)--(2.12,2.38)--(2.14,2.36)--(2.15,2.35)--(2.17,2.33)--(2.18,2.32)--(2.20,2.30)--(2.21,2.29)--(2.23,2.27)--(2.24,2.26)--(2.26,2.24)--(2.27,2.23)--(2.29,2.21)--(2.30,2.20)--(2.32,2.18)--(2.33,2.17)--(2.35,2.15)--(2.36,2.14)--(2.38,2.12)--(2.39,2.11)--(2.41,2.09)--(2.42,2.08)--(2.44,2.06)--(2.45,2.05)--(2.47,2.03)--(2.48,2.02)--(2.50,2.00)--(2.52,1.98)--(2.53,1.97)--(2.55,1.95)--(2.56,1.94)--(2.58,1.92)--(2.59,1.91)--(2.61,1.89)--(2.62,1.88)--(2.64,1.86)--(2.65,1.85)--(2.67,1.83)--(2.68,1.82)--(2.70,1.80)--(2.71,1.79)--(2.73,1.77)--(2.74,1.76)--(2.76,1.74)--(2.77,1.73)--(2.79,1.71)--(2.80,1.70)--(2.82,1.68)--(2.83,1.67)--(2.85,1.65)--(2.86,1.64)--(2.88,1.62)--(2.89,1.61)--(2.91,1.59)--(2.92,1.58)--(2.94,1.56)--(2.95,1.55)--(2.97,1.53)--(2.98,1.52)--(3.00,1.50);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
+\draw [color=blue] (1.5000,3.0000)--(1.5151,2.9848)--(1.5303,2.9696)--(1.5454,2.9545)--(1.5606,2.9393)--(1.5757,2.9242)--(1.5909,2.9090)--(1.6060,2.8939)--(1.6212,2.8787)--(1.6363,2.8636)--(1.6515,2.8484)--(1.6666,2.8333)--(1.6818,2.8181)--(1.6969,2.8030)--(1.7121,2.7878)--(1.7272,2.7727)--(1.7424,2.7575)--(1.7575,2.7424)--(1.7727,2.7272)--(1.7878,2.7121)--(1.8030,2.6969)--(1.8181,2.6818)--(1.8333,2.6666)--(1.8484,2.6515)--(1.8636,2.6363)--(1.8787,2.6212)--(1.8939,2.6060)--(1.9090,2.5909)--(1.9242,2.5757)--(1.9393,2.5606)--(1.9545,2.5454)--(1.9696,2.5303)--(1.9848,2.5151)--(2.0000,2.5000)--(2.0151,2.4848)--(2.0303,2.4696)--(2.0454,2.4545)--(2.0606,2.4393)--(2.0757,2.4242)--(2.0909,2.4090)--(2.1060,2.3939)--(2.1212,2.3787)--(2.1363,2.3636)--(2.1515,2.3484)--(2.1666,2.3333)--(2.1818,2.3181)--(2.1969,2.3030)--(2.2121,2.2878)--(2.2272,2.2727)--(2.2424,2.2575)--(2.2575,2.2424)--(2.2727,2.2272)--(2.2878,2.2121)--(2.3030,2.1969)--(2.3181,2.1818)--(2.3333,2.1666)--(2.3484,2.1515)--(2.3636,2.1363)--(2.3787,2.1212)--(2.3939,2.1060)--(2.4090,2.0909)--(2.4242,2.0757)--(2.4393,2.0606)--(2.4545,2.0454)--(2.4696,2.0303)--(2.4848,2.0151)--(2.5000,2.0000)--(2.5151,1.9848)--(2.5303,1.9696)--(2.5454,1.9545)--(2.5606,1.9393)--(2.5757,1.9242)--(2.5909,1.9090)--(2.6060,1.8939)--(2.6212,1.8787)--(2.6363,1.8636)--(2.6515,1.8484)--(2.6666,1.8333)--(2.6818,1.8181)--(2.6969,1.8030)--(2.7121,1.7878)--(2.7272,1.7727)--(2.7424,1.7575)--(2.7575,1.7424)--(2.7727,1.7272)--(2.7878,1.7121)--(2.8030,1.6969)--(2.8181,1.6818)--(2.8333,1.6666)--(2.8484,1.6515)--(2.8636,1.6363)--(2.8787,1.6212)--(2.8939,1.6060)--(2.9090,1.5909)--(2.9242,1.5757)--(2.9393,1.5606)--(2.9545,1.5454)--(2.9696,1.5303)--(2.9848,1.5151)--(3.0000,1.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.5000,1.5000) node [rotate=0] {$o$};
 \draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
 \draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw [style=dashed] (1.50,1.50) -- (1.50,3.00);
+\draw [style=dashed] (1.5000,1.5000) -- (1.5000,3.0000);
 
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks b/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks
index 83e1fe1b2..379aaa659 100644
--- a/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks
+++ b/auto/pictures_tex/Fig_GMIooJvcCXg.pstricks
@@ -95,53 +95,53 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-1.00,-1.57) -- (-1.00,4.71);
-\draw [color=gray,style=solid] (0,-1.57) -- (0,4.71);
-\draw [color=gray,style=solid] (1.00,-1.57) -- (1.00,4.71);
-\draw [color=gray,style=solid] (2.00,-1.57) -- (2.00,4.71);
-\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,4.71);
-\draw [color=gray,style=solid] (4.00,-1.57) -- (4.00,4.71);
-\draw [color=gray,style=dotted] (-0.500,-1.57) -- (-0.500,4.71);
-\draw [color=gray,style=dotted] (0.500,-1.57) -- (0.500,4.71);
-\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,4.71);
-\draw [color=gray,style=dotted] (2.50,-1.57) -- (2.50,4.71);
-\draw [color=gray,style=dotted] (3.50,-1.57) -- (3.50,4.71);
-\draw [color=gray,style=dotted] (-1.00,-0.785) -- (4.00,-0.785);
-\draw [color=gray,style=dotted] (-1.00,0.785) -- (4.00,0.785);
-\draw [color=gray,style=dotted] (-1.00,2.36) -- (4.00,2.36);
-\draw [color=gray,style=dotted] (-1.00,3.93) -- (4.00,3.93);
-\draw [color=gray,style=solid] (-1.00,-1.57) -- (4.00,-1.57);
-\draw [color=gray,style=solid] (-1.00,0) -- (4.00,0);
-\draw [color=gray,style=solid] (-1.00,1.57) -- (4.00,1.57);
-\draw [color=gray,style=solid] (-1.00,3.14) -- (4.00,3.14);
-\draw [color=gray,style=solid] (-1.00,4.71) -- (4.00,4.71);
+\draw [color=gray,style=solid] (-1.0000,-1.5707) -- (-1.0000,4.7123);
+\draw [color=gray,style=solid] (0.0000,-1.5707) -- (0.0000,4.7123);
+\draw [color=gray,style=solid] (1.0000,-1.5707) -- (1.0000,4.7123);
+\draw [color=gray,style=solid] (2.0000,-1.5707) -- (2.0000,4.7123);
+\draw [color=gray,style=solid] (3.0000,-1.5707) -- (3.0000,4.7123);
+\draw [color=gray,style=solid] (4.0000,-1.5707) -- (4.0000,4.7123);
+\draw [color=gray,style=dotted] (-0.5000,-1.5707) -- (-0.5000,4.7123);
+\draw [color=gray,style=dotted] (0.5000,-1.5707) -- (0.5000,4.7123);
+\draw [color=gray,style=dotted] (1.5000,-1.5707) -- (1.5000,4.7123);
+\draw [color=gray,style=dotted] (2.5000,-1.5707) -- (2.5000,4.7123);
+\draw [color=gray,style=dotted] (3.5000,-1.5707) -- (3.5000,4.7123);
+\draw [color=gray,style=dotted] (-1.0000,-0.7853) -- (4.0000,-0.7853);
+\draw [color=gray,style=dotted] (-1.0000,0.7853) -- (4.0000,0.7853);
+\draw [color=gray,style=dotted] (-1.0000,2.3561) -- (4.0000,2.3561);
+\draw [color=gray,style=dotted] (-1.0000,3.9269) -- (4.0000,3.9269);
+\draw [color=gray,style=solid] (-1.0000,-1.5707) -- (4.0000,-1.5707);
+\draw [color=gray,style=solid] (-1.0000,0.0000) -- (4.0000,0.0000);
+\draw [color=gray,style=solid] (-1.0000,1.5707) -- (4.0000,1.5707);
+\draw [color=gray,style=solid] (-1.0000,3.1415) -- (4.0000,3.1415);
+\draw [color=gray,style=solid] (-1.0000,4.7123) -- (4.0000,4.7123);
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-2.0708) -- (0,5.2124);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0707) -- (0.0000,5.2123);
 %DEFAULT
 
-\draw [color=blue] (0,1.000)--(0.03173,0.9995)--(0.06347,0.9980)--(0.09520,0.9955)--(0.1269,0.9920)--(0.1587,0.9874)--(0.1904,0.9819)--(0.2221,0.9754)--(0.2539,0.9679)--(0.2856,0.9595)--(0.3173,0.9501)--(0.3491,0.9397)--(0.3808,0.9284)--(0.4125,0.9161)--(0.4443,0.9029)--(0.4760,0.8888)--(0.5077,0.8738)--(0.5395,0.8580)--(0.5712,0.8413)--(0.6029,0.8237)--(0.6347,0.8053)--(0.6664,0.7861)--(0.6981,0.7660)--(0.7299,0.7453)--(0.7616,0.7237)--(0.7933,0.7015)--(0.8251,0.6785)--(0.8568,0.6549)--(0.8885,0.6306)--(0.9203,0.6056)--(0.9520,0.5801)--(0.9837,0.5539)--(1.015,0.5272)--(1.047,0.5000)--(1.079,0.4723)--(1.111,0.4441)--(1.142,0.4154)--(1.174,0.3863)--(1.206,0.3569)--(1.238,0.3271)--(1.269,0.2969)--(1.301,0.2665)--(1.333,0.2358)--(1.365,0.2048)--(1.396,0.1736)--(1.428,0.1423)--(1.460,0.1108)--(1.491,0.07925)--(1.523,0.04758)--(1.555,0.01587)--(1.587,-0.01587)--(1.618,-0.04758)--(1.650,-0.07925)--(1.682,-0.1108)--(1.714,-0.1423)--(1.745,-0.1736)--(1.777,-0.2048)--(1.809,-0.2358)--(1.841,-0.2665)--(1.872,-0.2969)--(1.904,-0.3271)--(1.936,-0.3569)--(1.967,-0.3863)--(1.999,-0.4154)--(2.031,-0.4441)--(2.063,-0.4723)--(2.094,-0.5000)--(2.126,-0.5272)--(2.158,-0.5539)--(2.190,-0.5801)--(2.221,-0.6056)--(2.253,-0.6306)--(2.285,-0.6549)--(2.317,-0.6785)--(2.348,-0.7015)--(2.380,-0.7237)--(2.412,-0.7453)--(2.443,-0.7660)--(2.475,-0.7861)--(2.507,-0.8053)--(2.539,-0.8237)--(2.570,-0.8413)--(2.602,-0.8580)--(2.634,-0.8738)--(2.666,-0.8888)--(2.697,-0.9029)--(2.729,-0.9161)--(2.761,-0.9284)--(2.793,-0.9397)--(2.824,-0.9501)--(2.856,-0.9595)--(2.888,-0.9679)--(2.919,-0.9754)--(2.951,-0.9819)--(2.983,-0.9874)--(3.015,-0.9920)--(3.046,-0.9955)--(3.078,-0.9980)--(3.110,-0.9995)--(3.142,-1.000);
+\draw [color=blue] (0.0000,1.0000)--(0.0317,0.9994)--(0.0634,0.9979)--(0.0951,0.9954)--(0.1269,0.9919)--(0.1586,0.9874)--(0.1903,0.9819)--(0.2221,0.9754)--(0.2538,0.9679)--(0.2855,0.9594)--(0.3173,0.9500)--(0.3490,0.9396)--(0.3807,0.9283)--(0.4125,0.9161)--(0.4442,0.9029)--(0.4759,0.8888)--(0.5077,0.8738)--(0.5394,0.8579)--(0.5711,0.8412)--(0.6029,0.8236)--(0.6346,0.8052)--(0.6663,0.7860)--(0.6981,0.7660)--(0.7298,0.7452)--(0.7615,0.7237)--(0.7933,0.7014)--(0.8250,0.6785)--(0.8567,0.6548)--(0.8885,0.6305)--(0.9202,0.6056)--(0.9519,0.5800)--(0.9837,0.5539)--(1.0154,0.5272)--(1.0471,0.5000)--(1.0789,0.4722)--(1.1106,0.4440)--(1.1423,0.4154)--(1.1741,0.3863)--(1.2058,0.3568)--(1.2375,0.3270)--(1.2693,0.2969)--(1.3010,0.2664)--(1.3327,0.2357)--(1.3645,0.2048)--(1.3962,0.1736)--(1.4279,0.1423)--(1.4597,0.1108)--(1.4914,0.0792)--(1.5231,0.0475)--(1.5549,0.0158)--(1.5866,-0.0158)--(1.6183,-0.0475)--(1.6501,-0.0792)--(1.6818,-0.1108)--(1.7135,-0.1423)--(1.7453,-0.1736)--(1.7770,-0.2048)--(1.8087,-0.2357)--(1.8405,-0.2664)--(1.8722,-0.2969)--(1.9039,-0.3270)--(1.9357,-0.3568)--(1.9674,-0.3863)--(1.9991,-0.4154)--(2.0309,-0.4440)--(2.0626,-0.4722)--(2.0943,-0.5000)--(2.1261,-0.5272)--(2.1578,-0.5539)--(2.1895,-0.5800)--(2.2213,-0.6056)--(2.2530,-0.6305)--(2.2847,-0.6548)--(2.3165,-0.6785)--(2.3482,-0.7014)--(2.3799,-0.7237)--(2.4117,-0.7452)--(2.4434,-0.7660)--(2.4751,-0.7860)--(2.5069,-0.8052)--(2.5386,-0.8236)--(2.5703,-0.8412)--(2.6021,-0.8579)--(2.6338,-0.8738)--(2.6655,-0.8888)--(2.6973,-0.9029)--(2.7290,-0.9161)--(2.7607,-0.9283)--(2.7925,-0.9396)--(2.8242,-0.9500)--(2.8559,-0.9594)--(2.8877,-0.9679)--(2.9194,-0.9754)--(2.9511,-0.9819)--(2.9829,-0.9874)--(3.0146,-0.9919)--(3.0463,-0.9954)--(3.0781,-0.9979)--(3.1098,-0.9994)--(3.1415,-1.0000);
 
-\draw [color=blue] (-1.000,3.142)--(-0.9798,2.940)--(-0.9596,2.856)--(-0.9394,2.792)--(-0.9192,2.737)--(-0.8990,2.688)--(-0.8788,2.644)--(-0.8586,2.603)--(-0.8384,2.565)--(-0.8182,2.529)--(-0.7980,2.495)--(-0.7778,2.462)--(-0.7576,2.430)--(-0.7374,2.400)--(-0.7172,2.371)--(-0.6970,2.342)--(-0.6768,2.314)--(-0.6566,2.287)--(-0.6364,2.261)--(-0.6162,2.235)--(-0.5960,2.209)--(-0.5758,2.184)--(-0.5556,2.160)--(-0.5354,2.136)--(-0.5152,2.112)--(-0.4949,2.089)--(-0.4747,2.065)--(-0.4545,2.043)--(-0.4343,2.020)--(-0.4141,1.998)--(-0.3939,1.976)--(-0.3737,1.954)--(-0.3535,1.932)--(-0.3333,1.911)--(-0.3131,1.889)--(-0.2929,1.868)--(-0.2727,1.847)--(-0.2525,1.826)--(-0.2323,1.805)--(-0.2121,1.785)--(-0.1919,1.764)--(-0.1717,1.743)--(-0.1515,1.723)--(-0.1313,1.702)--(-0.1111,1.682)--(-0.09091,1.662)--(-0.07071,1.642)--(-0.05051,1.621)--(-0.03030,1.601)--(-0.01010,1.581)--(0.01010,1.561)--(0.03030,1.540)--(0.05051,1.520)--(0.07071,1.500)--(0.09091,1.480)--(0.1111,1.459)--(0.1313,1.439)--(0.1515,1.419)--(0.1717,1.398)--(0.1919,1.378)--(0.2121,1.357)--(0.2323,1.336)--(0.2525,1.316)--(0.2727,1.295)--(0.2929,1.274)--(0.3131,1.252)--(0.3333,1.231)--(0.3535,1.209)--(0.3737,1.188)--(0.3939,1.166)--(0.4141,1.144)--(0.4343,1.121)--(0.4545,1.099)--(0.4747,1.076)--(0.4949,1.053)--(0.5152,1.030)--(0.5354,1.006)--(0.5556,0.9818)--(0.5758,0.9573)--(0.5960,0.9323)--(0.6162,0.9069)--(0.6364,0.8810)--(0.6566,0.8545)--(0.6768,0.8274)--(0.6970,0.7996)--(0.7172,0.7711)--(0.7374,0.7416)--(0.7576,0.7112)--(0.7778,0.6797)--(0.7980,0.6469)--(0.8182,0.6126)--(0.8384,0.5765)--(0.8586,0.5383)--(0.8788,0.4975)--(0.8990,0.4533)--(0.9192,0.4048)--(0.9394,0.3499)--(0.9596,0.2852)--(0.9798,0.2013)--(1.000,0);
+\draw [color=blue] (-1.0000,3.1415)--(-0.9797,2.9402)--(-0.9595,2.8563)--(-0.9393,2.7916)--(-0.9191,2.7368)--(-0.8989,2.6882)--(-0.8787,2.6441)--(-0.8585,2.6033)--(-0.8383,2.5651)--(-0.8181,2.5290)--(-0.7979,2.4947)--(-0.7777,2.4619)--(-0.7575,2.4303)--(-0.7373,2.3999)--(-0.7171,2.3705)--(-0.6969,2.3419)--(-0.6767,2.3141)--(-0.6565,2.2870)--(-0.6363,2.2605)--(-0.6161,2.2346)--(-0.5959,2.2092)--(-0.5757,2.1843)--(-0.5555,2.1598)--(-0.5353,2.1357)--(-0.5151,2.1119)--(-0.4949,2.0885)--(-0.4747,2.0654)--(-0.4545,2.0426)--(-0.4343,2.0201)--(-0.4141,1.9977)--(-0.3939,1.9757)--(-0.3737,1.9538)--(-0.3535,1.9321)--(-0.3333,1.9106)--(-0.3131,1.8892)--(-0.2929,1.8680)--(-0.2727,1.8470)--(-0.2525,1.8260)--(-0.2323,1.8052)--(-0.2121,1.7845)--(-0.1919,1.7639)--(-0.1717,1.7433)--(-0.1515,1.7228)--(-0.1313,1.7024)--(-0.1111,1.6821)--(-0.0909,1.6618)--(-0.0707,1.6415)--(-0.0505,1.6213)--(-0.0303,1.6011)--(-0.0101,1.5808)--(0.0101,1.5606)--(0.0303,1.5404)--(0.0505,1.5202)--(0.0707,1.5000)--(0.0909,1.4797)--(0.1111,1.4594)--(0.1313,1.4391)--(0.1515,1.4186)--(0.1717,1.3982)--(0.1919,1.3776)--(0.2121,1.3570)--(0.2323,1.3363)--(0.2525,1.3155)--(0.2727,1.2945)--(0.2929,1.2735)--(0.3131,1.2523)--(0.3333,1.2309)--(0.3535,1.2094)--(0.3737,1.1877)--(0.3939,1.1658)--(0.4141,1.1437)--(0.4343,1.1214)--(0.4545,1.0989)--(0.4747,1.0761)--(0.4949,1.0530)--(0.5151,1.0296)--(0.5353,1.0058)--(0.5555,0.9817)--(0.5757,0.9572)--(0.5959,0.9323)--(0.6161,0.9069)--(0.6363,0.8810)--(0.6565,0.8545)--(0.6767,0.8274)--(0.6969,0.7996)--(0.7171,0.7710)--(0.7373,0.7416)--(0.7575,0.7112)--(0.7777,0.6796)--(0.7979,0.6468)--(0.8181,0.6125)--(0.8383,0.5764)--(0.8585,0.5382)--(0.8787,0.4974)--(0.8989,0.4533)--(0.9191,0.4047)--(0.9393,0.3499)--(0.9595,0.2852)--(0.9797,0.2013)--(1.0000,0.0000);
 
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
-\draw [] (-0.100,1.57) -- (0.100,1.57);
-\draw (-0.31058,3.1416) node {$ \pi $};
-\draw [] (-0.100,3.14) -- (0.100,3.14);
-\draw (-0.45183,4.7124) node {$ \frac{3}{2} \, \pi $};
-\draw [] (-0.100,4.71) -- (0.100,4.71);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.5937,-1.5707) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-0.1000,-1.5707) -- (0.1000,-1.5707);
+\draw (-0.4518,1.5707) node {$ \frac{1}{2} \, \pi $};
+\draw [] (-0.1000,1.5707) -- (0.1000,1.5707);
+\draw (-0.3105,3.1415) node {$ \pi $};
+\draw [] (-0.1000,3.1415) -- (0.1000,3.1415);
+\draw (-0.4518,4.7123) node {$ \frac{3}{2} \, \pi $};
+\draw [] (-0.1000,4.7123) -- (0.1000,4.7123);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks b/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks
index 02a0a4b23..63d8b55e5 100644
--- a/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks
+++ b/auto/pictures_tex/Fig_GVDJooYzMxLW.pstricks
@@ -89,24 +89,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (2.00,3.46) -- (0,0);
-\draw [] (2.00,3.46) -- (4.00,0);
-\draw [] (4.00,0) -- (0,0);
-\draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.2518,0.36492) node {$60$};
+\draw [] (2.0000,3.4641) -- (0.0000,0.0000);
+\draw [] (2.0000,3.4641) -- (4.0000,0.0000);
+\draw [] (4.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=blue,style=dotted] (2.0000,3.4641) -- (2.0000,0.0000);
+\draw (3.2517,0.3649) node {$60$};
 
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+\draw [color=red] (3.7500,0.4330)--(3.7454,0.4303)--(3.7408,0.4276)--(3.7363,0.4248)--(3.7319,0.4220)--(3.7274,0.4191)--(3.7230,0.4162)--(3.7186,0.4133)--(3.7142,0.4103)--(3.7099,0.4072)--(3.7056,0.4041)--(3.7014,0.4010)--(3.6971,0.3978)--(3.6930,0.3946)--(3.6888,0.3913)--(3.6847,0.3880)--(3.6806,0.3847)--(3.6765,0.3813)--(3.6725,0.3778)--(3.6685,0.3743)--(3.6646,0.3708)--(3.6607,0.3672)--(3.6568,0.3636)--(3.6530,0.3600)--(3.6492,0.3563)--(3.6455,0.3526)--(3.6418,0.3488)--(3.6381,0.3450)--(3.6345,0.3411)--(3.6309,0.3373)--(3.6273,0.3333)--(3.6238,0.3294)--(3.6203,0.3254)--(3.6169,0.3213)--(3.6135,0.3173)--(3.6102,0.3132)--(3.6069,0.3090)--(3.6037,0.3049)--(3.6005,0.3006)--(3.5973,0.2964)--(3.5942,0.2921)--(3.5911,0.2878)--(3.5881,0.2835)--(3.5851,0.2791)--(3.5822,0.2747)--(3.5793,0.2703)--(3.5765,0.2658)--(3.5737,0.2613)--(3.5710,0.2568)--(3.5683,0.2522)--(3.5656,0.2477)--(3.5630,0.2430)--(3.5605,0.2384)--(3.5580,0.2338)--(3.5555,0.2291)--(3.5531,0.2243)--(3.5508,0.2196)--(3.5485,0.2148)--(3.5462,0.2101)--(3.5440,0.2052)--(3.5419,0.2004)--(3.5398,0.1956)--(3.5378,0.1907)--(3.5358,0.1858)--(3.5338,0.1809)--(3.5319,0.1759)--(3.5301,0.1710)--(3.5283,0.1660)--(3.5266,0.1610)--(3.5249,0.1560)--(3.5233,0.1509)--(3.5217,0.1459)--(3.5202,0.1408)--(3.5187,0.1357)--(3.5173,0.1306)--(3.5160,0.1255)--(3.5147,0.1204)--(3.5134,0.1153)--(3.5122,0.1101)--(3.5111,0.1049)--(3.5100,0.0998)--(3.5090,0.0946)--(3.5080,0.0894)--(3.5071,0.0842)--(3.5062,0.0790)--(3.5054,0.0737)--(3.5047,0.0685)--(3.5040,0.0632)--(3.5033,0.0580)--(3.5027,0.0527)--(3.5022,0.0475)--(3.5017,0.0422)--(3.5013,0.0369)--(3.5010,0.0317)--(3.5006,0.0264)--(3.5004,0.0211)--(3.5002,0.0158)--(3.5001,0.0105)--(3.5000,0.0052)--(3.5000,0.0000);
 \draw (2.3119,2.2340) node {$30$};
 
-\draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
+\draw [color=cyan] (2.0000,2.9641)--(2.0026,2.9641)--(2.0052,2.9641)--(2.0079,2.9641)--(2.0105,2.9642)--(2.0132,2.9642)--(2.0158,2.9643)--(2.0185,2.9644)--(2.0211,2.9645)--(2.0237,2.9646)--(2.0264,2.9648)--(2.0290,2.9649)--(2.0317,2.9651)--(2.0343,2.9652)--(2.0369,2.9654)--(2.0396,2.9656)--(2.0422,2.9658)--(2.0448,2.9661)--(2.0475,2.9663)--(2.0501,2.9666)--(2.0527,2.9668)--(2.0554,2.9671)--(2.0580,2.9674)--(2.0606,2.9677)--(2.0632,2.9681)--(2.0659,2.9684)--(2.0685,2.9688)--(2.0711,2.9691)--(2.0737,2.9695)--(2.0763,2.9699)--(2.0790,2.9703)--(2.0816,2.9708)--(2.0842,2.9712)--(2.0868,2.9716)--(2.0894,2.9721)--(2.0920,2.9726)--(2.0946,2.9731)--(2.0972,2.9736)--(2.0998,2.9741)--(2.1024,2.9747)--(2.1049,2.9752)--(2.1075,2.9758)--(2.1101,2.9763)--(2.1127,2.9769)--(2.1153,2.9775)--(2.1178,2.9781)--(2.1204,2.9788)--(2.1230,2.9794)--(2.1255,2.9801)--(2.1281,2.9807)--(2.1306,2.9814)--(2.1332,2.9821)--(2.1357,2.9828)--(2.1383,2.9836)--(2.1408,2.9843)--(2.1434,2.9851)--(2.1459,2.9858)--(2.1484,2.9866)--(2.1509,2.9874)--(2.1535,2.9882)--(2.1560,2.9890)--(2.1585,2.9898)--(2.1610,2.9907)--(2.1635,2.9916)--(2.1660,2.9924)--(2.1685,2.9933)--(2.1710,2.9942)--(2.1734,2.9951)--(2.1759,2.9960)--(2.1784,2.9970)--(2.1809,2.9979)--(2.1833,2.9989)--(2.1858,2.9999)--(2.1882,3.0009)--(2.1907,3.0019)--(2.1931,3.0029)--(2.1956,3.0039)--(2.1980,3.0049)--(2.2004,3.0060)--(2.2028,3.0071)--(2.2052,3.0081)--(2.2077,3.0092)--(2.2101,3.0103)--(2.2125,3.0115)--(2.2148,3.0126)--(2.2172,3.0137)--(2.2196,3.0149)--(2.2220,3.0161)--(2.2243,3.0172)--(2.2267,3.0184)--(2.2291,3.0196)--(2.2314,3.0209)--(2.2338,3.0221)--(2.2361,3.0233)--(2.2384,3.0246)--(2.2407,3.0258)--(2.2430,3.0271)--(2.2454,3.0284)--(2.2477,3.0297)--(2.2500,3.0310);
 \draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
 \draw (2.0000,3.8888) node {$A$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.44758,0) node {$B$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.4435,0) node {$C$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.42471) node {$H$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.4475,0.0000) node {$B$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.4434,0.0000) node {$C$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.4247) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks b/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks
index bbb7404a6..1fcfa15f5 100644
--- a/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks
+++ b/auto/pictures_tex/Fig_GWOYooRxHKSm.pstricks
@@ -87,33 +87,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.8750,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,7.8518);
 %DEFAULT
-\draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
-\draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
-\draw [color=green,style=dashed] (3.00,1.50) -- (3.00,0);
-\draw [color=green,style=dashed] (6.50,6.09) -- (0,6.09);
-\draw [color=green,style=dashed] (3.00,1.50) -- (0,1.50);
-\draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
+\draw [color=cyan] (2.1250,0.3535) -- (7.3750,7.2320);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (6.5000,0.0000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (3.0000,0.0000);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (0.0000,6.0856);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (0.0000,1.5000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (6.5000,1.5000);
 
-\draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
+\draw [color=blue] (1.0000,1.0185)--(1.0606,1.0220)--(1.1212,1.0261)--(1.1818,1.0305)--(1.2424,1.0355)--(1.3030,1.0409)--(1.3636,1.0469)--(1.4242,1.0535)--(1.4848,1.0606)--(1.5454,1.0683)--(1.6060,1.0767)--(1.6666,1.0857)--(1.7272,1.0954)--(1.7878,1.1058)--(1.8484,1.1169)--(1.9090,1.1288)--(1.9696,1.1415)--(2.0303,1.1549)--(2.0909,1.1692)--(2.1515,1.1844)--(2.2121,1.2004)--(2.2727,1.2173)--(2.3333,1.2352)--(2.3939,1.2540)--(2.4545,1.2738)--(2.5151,1.2946)--(2.5757,1.3164)--(2.6363,1.3393)--(2.6969,1.3632)--(2.7575,1.3883)--(2.8181,1.4144)--(2.8787,1.4418)--(2.9393,1.4703)--(3.0000,1.5000)--(3.0606,1.5309)--(3.1212,1.5630)--(3.1818,1.5965)--(3.2424,1.6312)--(3.3030,1.6673)--(3.3636,1.7047)--(3.4242,1.7435)--(3.4848,1.7837)--(3.5454,1.8253)--(3.6060,1.8683)--(3.6666,1.9128)--(3.7272,1.9589)--(3.7878,2.0064)--(3.8484,2.0555)--(3.9090,2.1061)--(3.9696,2.1584)--(4.0303,2.2123)--(4.0909,2.2678)--(4.1515,2.3250)--(4.2121,2.3839)--(4.2727,2.4445)--(4.3333,2.5068)--(4.3939,2.5709)--(4.4545,2.6368)--(4.5151,2.7046)--(4.5757,2.7741)--(4.6363,2.8456)--(4.6969,2.9189)--(4.7575,2.9941)--(4.8181,3.0713)--(4.8787,3.1505)--(4.9393,3.2316)--(5.0000,3.3148)--(5.0606,3.4000)--(5.1212,3.4872)--(5.1818,3.5766)--(5.2424,3.6681)--(5.3030,3.7617)--(5.3636,3.8574)--(5.4242,3.9554)--(5.4848,4.0556)--(5.5454,4.1580)--(5.6060,4.2627)--(5.6666,4.3696)--(5.7272,4.4789)--(5.7878,4.5905)--(5.8484,4.7045)--(5.9090,4.8209)--(5.9696,4.9396)--(6.0303,5.0609)--(6.0909,5.1845)--(6.1515,5.3107)--(6.2121,5.4394)--(6.2727,5.5706)--(6.3333,5.7043)--(6.3939,5.8407)--(6.4545,5.9797)--(6.5151,6.1212)--(6.5757,6.2655)--(6.6363,6.4124)--(6.6969,6.5621)--(6.7575,6.7145)--(6.8181,6.8696)--(6.8787,7.0275)--(6.9393,7.1882)--(7.0000,7.3518);
 \draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,-0.27858) node {$a$};
-\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
-\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.2785) node {$a$};
+\draw []  (0.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.4473,1.5000) node {$f(a)$};
 \draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
-\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
-\draw (6.5000,-0.27858) node {$x$};
-\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
-\draw (-0.45521,6.0856) node {$f(x)$};
+\draw []  (6.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.2785) node {$x$};
+\draw []  (0.0000,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.4552,6.0856) node {$f(x)$};
 \draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
 \draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
-\draw (4.7500,0.97897) node {$x-a$};
+\draw (4.7500,0.9789) node {$x-a$};
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
-\draw (7.8256,3.7928) node {$f(x)-f(a)$};
+\draw (7.8255,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks b/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks
index b6547c9ce..44e397bc3 100644
--- a/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks
+++ b/auto/pictures_tex/Fig_GYODoojTiGZSkJ.pstricks
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.42400,-0.12838) node {\(A\)};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.2820,-0.36117) node {\(B\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.4240,-0.1283) node {\(A\)};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.2819,-0.3611) node {\(B\)};
 \draw []  (6.0000,4.0000) node [rotate=0] {$\bullet$};
-\draw (6.2607,4.3699) node {\(C\)};
-\draw [] (0,0) -- (5.00,0);
-\draw [] (5.00,0) -- (6.00,4.00);
-\draw [] (6.00,4.00) -- (0,0);
-\draw [] (0,0) -- (5.60,2.40);
+\draw (6.2606,4.3699) node {\(C\)};
+\draw [] (0.0000,0.0000) -- (5.0000,0.0000);
+\draw [] (5.0000,0.0000) -- (6.0000,4.0000);
+\draw [] (6.0000,4.0000) -- (0.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (5.6000,2.4000);
 \draw []  (4.4800,1.9200) node [rotate=0] {$\bullet$};
-\draw (4.7253,1.6115) node {\( N\)};
+\draw (4.7253,1.6114) node {\( N\)};
 \draw []  (5.6000,2.4000) node [rotate=0] {$\bullet$};
-\draw (5.9365,2.2268) node {\( P\)};
+\draw (5.9365,2.2267) node {\( P\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks b/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks
index 61cae4865..51d91f6b5 100644
--- a/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks
+++ b/auto/pictures_tex/Fig_Grapheunsurunmoinsx.pstricks
@@ -107,54 +107,54 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-5.2619) -- (0,5.2619);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-5.2619) -- (0.0000,5.2619);
 %DEFAULT
 
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-\draw [style=dotted] (1.00,-4.76) -- (1.00,4.76);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.43316,-5.0000) node {$ -5 $};
-\draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue] (1.2100,-4.7619)--(1.2583,-3.8702)--(1.3067,-3.2597)--(1.3551,-2.8156)--(1.4035,-2.4780)--(1.4519,-2.2127)--(1.5003,-1.9987)--(1.5486,-1.8225)--(1.5970,-1.6748)--(1.6454,-1.5492)--(1.6938,-1.4412)--(1.7422,-1.3473)--(1.7906,-1.2648)--(1.8389,-1.1919)--(1.8873,-1.1269)--(1.9357,-1.0686)--(1.9841,-1.0161)--(2.0325,-0.9684)--(2.0809,-0.9251)--(2.1292,-0.8855)--(2.1776,-0.8491)--(2.2260,-0.8156)--(2.2744,-0.7846)--(2.3228,-0.7559)--(2.3712,-0.7292)--(2.4195,-0.7044)--(2.4679,-0.6812)--(2.5163,-0.6594)--(2.5647,-0.6390)--(2.6131,-0.6199)--(2.6615,-0.6018)--(2.7098,-0.5848)--(2.7582,-0.5687)--(2.8066,-0.5535)--(2.8550,-0.5390)--(2.9034,-0.5253)--(2.9518,-0.5123)--(3.0002,-0.4999)--(3.0485,-0.4881)--(3.0969,-0.4768)--(3.1453,-0.4661)--(3.1937,-0.4558)--(3.2421,-0.4460)--(3.2905,-0.4365)--(3.3388,-0.4275)--(3.3872,-0.4188)--(3.4356,-0.4105)--(3.4840,-0.4025)--(3.5324,-0.3948)--(3.5808,-0.3874)--(3.6291,-0.3803)--(3.6775,-0.3734)--(3.7259,-0.3668)--(3.7743,-0.3604)--(3.8227,-0.3542)--(3.8711,-0.3482)--(3.9194,-0.3425)--(3.9678,-0.3369)--(4.0162,-0.3315)--(4.0646,-0.3263)--(4.1130,-0.3212)--(4.1614,-0.3163)--(4.2097,-0.3115)--(4.2581,-0.3069)--(4.3065,-0.3024)--(4.3549,-0.2980)--(4.4033,-0.2938)--(4.4517,-0.2897)--(4.5001,-0.2857)--(4.5484,-0.2818)--(4.5968,-0.2780)--(4.6452,-0.2743)--(4.6936,-0.2707)--(4.7420,-0.2672)--(4.7904,-0.2638)--(4.8387,-0.2604)--(4.8871,-0.2572)--(4.9355,-0.2540)--(4.9839,-0.2510)--(5.0323,-0.2479)--(5.0807,-0.2450)--(5.1290,-0.2421)--(5.1774,-0.2393)--(5.2258,-0.2366)--(5.2742,-0.2339)--(5.3226,-0.2313)--(5.3710,-0.2287)--(5.4193,-0.2262)--(5.4677,-0.2238)--(5.5161,-0.2214)--(5.5645,-0.2190)--(5.6129,-0.2167)--(5.6613,-0.2145)--(5.7096,-0.2123)--(5.7580,-0.2101)--(5.8064,-0.2080)--(5.8548,-0.2059)--(5.9032,-0.2039)--(5.9516,-0.2019)--(6.0000,-0.2000);
+\draw [style=dotted] (1.0000,-4.7619) -- (1.0000,4.7619);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-5.0000) node {$ -5 $};
+\draw [] (-0.1000,-5.0000) -- (0.1000,-5.0000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks b/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks
index 1d7103b76..24c9873b6 100644
--- a/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks
+++ b/auto/pictures_tex/Fig_HCJPooHsaTgI.pstricks
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4998);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -51,25 +51,25 @@
 % setting the default values
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+\draw [color=blue] (1.0000,1.6347)--(1.0505,1.5624)--(1.1010,1.4970)--(1.1515,1.4392)--(1.2020,1.3895)--(1.2525,1.3484)--(1.3030,1.3161)--(1.3535,1.2928)--(1.4040,1.2785)--(1.4545,1.2731)--(1.5050,1.2763)--(1.5555,1.2878)--(1.6060,1.3071)--(1.6565,1.3337)--(1.7070,1.3669)--(1.7575,1.4059)--(1.8080,1.4499)--(1.8585,1.4980)--(1.9090,1.5493)--(1.9595,1.6029)--(2.0101,1.6578)--(2.0606,1.7132)--(2.1111,1.7680)--(2.1616,1.8214)--(2.2121,1.8726)--(2.2626,1.9208)--(2.3131,1.9653)--(2.3636,2.0056)--(2.4141,2.0411)--(2.4646,2.0714)--(2.5151,2.0963)--(2.5656,2.1155)--(2.6161,2.1290)--(2.6666,2.1368)--(2.7171,2.1391)--(2.7676,2.1362)--(2.8181,2.1284)--(2.8686,2.1162)--(2.9191,2.1001)--(2.9696,2.0808)--(3.0202,2.0588)--(3.0707,2.0350)--(3.1212,2.0101)--(3.1717,1.9849)--(3.2222,1.9602)--(3.2727,1.9367)--(3.3232,1.9152)--(3.3737,1.8964)--(3.4242,1.8810)--(3.4747,1.8696)--(3.5252,1.8627)--(3.5757,1.8608)--(3.6262,1.8642)--(3.6767,1.8731)--(3.7272,1.8877)--(3.7777,1.9080)--(3.8282,1.9339)--(3.8787,1.9653)--(3.9292,2.0017)--(3.9797,2.0428)--(4.0303,2.0881)--(4.0808,2.1370)--(4.1313,2.1887)--(4.1818,2.2424)--(4.2323,2.2974)--(4.2828,2.3527)--(4.3333,2.4075)--(4.3838,2.4607)--(4.4343,2.5115)--(4.4848,2.5589)--(4.5353,2.6020)--(4.5858,2.6399)--(4.6363,2.6718)--(4.6868,2.6971)--(4.7373,2.7149)--(4.7878,2.7249)--(4.8383,2.7265)--(4.8888,2.7194)--(4.9393,2.7033)--(4.9898,2.6783)--(5.0404,2.6442)--(5.0909,2.6014)--(5.1414,2.5502)--(5.1919,2.4909)--(5.2424,2.4241)--(5.2929,2.3505)--(5.3434,2.2708)--(5.3939,2.1861)--(5.4444,2.0972)--(5.4949,2.0052)--(5.5454,1.9112)--(5.5959,1.8163)--(5.6464,1.7217)--(5.6969,1.6286)--(5.7474,1.5381)--(5.7979,1.4515)--(5.8484,1.3698)--(5.8989,1.2941)--(5.9494,1.2254)--(6.0000,1.1645);
 
-\draw [color=cyan] (1.00,5.00) -- (6.00,5.00);
-\draw [color=cyan] (6.00,5.00) -- (6.00,1.16);
-\draw [color=cyan] (6.00,1.16) -- (1.00,1.16);
-\draw [color=cyan] (1.00,1.16) -- (1.00,5.00);
+\draw [color=cyan] (1.0000,4.9998) -- (6.0000,4.9998);
+\draw [color=cyan] (6.0000,4.9998) -- (6.0000,1.1645);
+\draw [color=cyan] (6.0000,1.1645) -- (1.0000,1.1645);
+\draw [color=cyan] (1.0000,1.1645) -- (1.0000,4.9998);
 
 %OTHER STUFF
 %END PSPICTURE
@@ -108,8 +108,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -118,24 +118,24 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-0.27898,1.0000) node {$c$};
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-\draw [color=cyan] (1.16,6.00) -- (5.00,6.00);
-\draw [color=cyan] (5.00,6.00) -- (5.00,1.00);
-\draw [color=cyan] (5.00,1.00) -- (1.16,1.00);
-\draw [color=cyan] (1.16,1.00) -- (1.16,6.00);
+\draw [color=cyan] (1.1645,6.0000) -- (4.9998,6.0000);
+\draw [color=cyan] (4.9998,6.0000) -- (4.9998,1.0000);
+\draw [color=cyan] (4.9998,1.0000) -- (1.1645,1.0000);
+\draw [color=cyan] (1.1645,1.0000) -- (1.1645,6.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks b/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks
index bf237b1bd..f19c097bb 100644
--- a/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks
+++ b/auto/pictures_tex/Fig_HFAYooOrfMAA.pstricks
@@ -67,21 +67,21 @@
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 %PSTRICKS CODE
 %AXES
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+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.0000) -- (0.0000,2.5000);
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+\draw [] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (1.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=red] (-0.5000,-0.5000) -- (1.5000,1.5000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3546,0.66316) node {$P$};
+\draw (1.3546,0.6631) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks b/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks
index feb15aa45..07f9cdb05 100644
--- a/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks
+++ b/auto/pictures_tex/Fig_HGQPooKrRtAN.pstricks
@@ -44,12 +44,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
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-\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
-\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=brown,style=dashed] (2.0000,0.0000)--(1.9998,0.0211)--(1.9995,0.0423)--(1.9989,0.0634)--(1.9982,0.0845)--(1.9972,0.1057)--(1.9959,0.1268)--(1.9945,0.1479)--(1.9928,0.1690)--(1.9909,0.1901)--(1.9888,0.2111)--(1.9864,0.2321)--(1.9839,0.2531)--(1.9811,0.2741)--(1.9781,0.2950)--(1.9748,0.3160)--(1.9714,0.3368)--(1.9677,0.3577)--(1.9638,0.3785)--(1.9597,0.3992)--(1.9554,0.4199)--(1.9508,0.4406)--(1.9460,0.4612)--(1.9411,0.4817)--(1.9358,0.5022)--(1.9304,0.5227)--(1.9248,0.5431)--(1.9189,0.5634)--(1.9129,0.5837)--(1.9066,0.6039)--(1.9001,0.6240)--(1.8934,0.6441)--(1.8865,0.6641)--(1.8793,0.6840)--(1.8720,0.7038)--(1.8644,0.7236)--(1.8567,0.7433)--(1.8487,0.7629)--(1.8405,0.7824)--(1.8322,0.8018)--(1.8236,0.8211)--(1.8148,0.8404)--(1.8058,0.8595)--(1.7966,0.8786)--(1.7872,0.8975)--(1.7776,0.9164)--(1.7678,0.9352)--(1.7578,0.9538)--(1.7476,0.9723)--(1.7373,0.9908)--(1.7267,1.0091)--(1.7159,1.0273)--(1.7050,1.0454)--(1.6938,1.0634)--(1.6825,1.0812)--(1.6709,1.0990)--(1.6592,1.1166)--(1.6473,1.1341)--(1.6352,1.1514)--(1.6229,1.1687)--(1.6105,1.1858)--(1.5979,1.2027)--(1.5850,1.2196)--(1.5721,1.2363)--(1.5589,1.2528)--(1.5456,1.2692)--(1.5320,1.2855)--(1.5184,1.3017)--(1.5045,1.3176)--(1.4905,1.3335)--(1.4763,1.3492)--(1.4619,1.3647)--(1.4474,1.3801)--(1.4327,1.3953)--(1.4179,1.4104)--(1.4029,1.4253)--(1.3877,1.4401)--(1.3724,1.4547)--(1.3570,1.4691)--(1.3414,1.4834)--(1.3256,1.4975)--(1.3097,1.5114)--(1.2936,1.5252)--(1.2774,1.5388)--(1.2611,1.5522)--(1.2446,1.5655)--(1.2279,1.5786)--(1.2112,1.5915)--(1.1943,1.6042)--(1.1772,1.6167)--(1.1601,1.6291)--(1.1428,1.6413)--(1.1253,1.6533)--(1.1078,1.6651)--(1.0901,1.6767)--(1.0723,1.6882)--(1.0544,1.6994)--(1.0364,1.7105)--(1.0182,1.7213)--(1.0000,1.7320);
+\draw [color=brown,style=dashed] (1.2990,0.7500) -- (3.0310,1.7500);
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5980,1.5000);
+\draw (2.2869,1.8657) node {$e_{r}$};
 \draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
-\draw (1.6075,2.1865) node {$e_{\theta}$};
+\draw (1.6075,2.1864) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -89,12 +89,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
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-\draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
-\draw (-2.0855,-1.2429) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
-\draw (-0.69205,-1.1323) node {$e_{\theta}$};
+\draw [color=brown,style=dashed] (-1.4772,0.2604)--(-1.4798,0.2448)--(-1.4823,0.2291)--(-1.4847,0.2134)--(-1.4869,0.1977)--(-1.4889,0.1820)--(-1.4907,0.1662)--(-1.4924,0.1504)--(-1.4939,0.1346)--(-1.4952,0.1188)--(-1.4964,0.1030)--(-1.4974,0.0872)--(-1.4983,0.0713)--(-1.4989,0.0555)--(-1.4994,0.0396)--(-1.4998,0.0237)--(-1.4999,0.0079)--(-1.4999,-0.0079)--(-1.4998,-0.0237)--(-1.4994,-0.0396)--(-1.4989,-0.0555)--(-1.4983,-0.0713)--(-1.4974,-0.0872)--(-1.4964,-0.1030)--(-1.4952,-0.1188)--(-1.4939,-0.1346)--(-1.4924,-0.1504)--(-1.4907,-0.1662)--(-1.4889,-0.1820)--(-1.4869,-0.1977)--(-1.4847,-0.2134)--(-1.4823,-0.2291)--(-1.4798,-0.2448)--(-1.4772,-0.2604)--(-1.4743,-0.2760)--(-1.4713,-0.2916)--(-1.4682,-0.3072)--(-1.4648,-0.3227)--(-1.4613,-0.3381)--(-1.4577,-0.3536)--(-1.4538,-0.3690)--(-1.4499,-0.3843)--(-1.4457,-0.3997)--(-1.4414,-0.4149)--(-1.4369,-0.4302)--(-1.4323,-0.4453)--(-1.4275,-0.4605)--(-1.4226,-0.4755)--(-1.4175,-0.4906)--(-1.4122,-0.5055)--(-1.4068,-0.5204)--(-1.4012,-0.5353)--(-1.3954,-0.5501)--(-1.3895,-0.5648)--(-1.3835,-0.5795)--(-1.3773,-0.5941)--(-1.3709,-0.6086)--(-1.3644,-0.6231)--(-1.3577,-0.6375)--(-1.3509,-0.6518)--(-1.3439,-0.6660)--(-1.3368,-0.6802)--(-1.3295,-0.6943)--(-1.3221,-0.7084)--(-1.3146,-0.7223)--(-1.3068,-0.7362)--(-1.2990,-0.7500)--(-1.2910,-0.7636)--(-1.2828,-0.7773)--(-1.2745,-0.7908)--(-1.2661,-0.8042)--(-1.2575,-0.8176)--(-1.2488,-0.8308)--(-1.2399,-0.8440)--(-1.2309,-0.8571)--(-1.2218,-0.8700)--(-1.2125,-0.8829)--(-1.2031,-0.8957)--(-1.1936,-0.9084)--(-1.1839,-0.9209)--(-1.1741,-0.9334)--(-1.1642,-0.9458)--(-1.1541,-0.9580)--(-1.1439,-0.9702)--(-1.1336,-0.9822)--(-1.1231,-0.9942)--(-1.1125,-1.0060)--(-1.1018,-1.0177)--(-1.0910,-1.0293)--(-1.0801,-1.0408)--(-1.0690,-1.0522)--(-1.0578,-1.0634)--(-1.0465,-1.0745)--(-1.0351,-1.0856)--(-1.0235,-1.0964)--(-1.0119,-1.1072)--(-1.0001,-1.1178)--(-0.9882,-1.1284)--(-0.9762,-1.1388)--(-0.9641,-1.1490);
+\draw [color=brown,style=dashed] (-0.9396,-0.3420) -- (-2.8190,-1.0260);
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-2.3492,-0.8550);
+\draw (-2.0854,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-1.0675,-1.4527);
+\draw (-0.6920,-1.1322) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks b/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks
index d8605c418..5a89561a2 100644
--- a/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks
+++ b/auto/pictures_tex/Fig_HLJooGDZnqF.pstricks
@@ -103,87 +103,87 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
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-\draw [color=gray,style=solid] (-1.00,-3.00) -- (-1.00,6.00);
-\draw [color=gray,style=solid] (0,-3.00) -- (0,6.00);
-\draw [color=gray,style=solid] (1.00,-3.00) -- (1.00,6.00);
-\draw [color=gray,style=solid] (2.00,-3.00) -- (2.00,6.00);
-\draw [color=gray,style=dotted] (-4.50,-3.00) -- (-4.50,6.00);
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+\draw [color=gray,style=dotted] (-3.5000,-3.0000) -- (-3.5000,6.0000);
+\draw [color=gray,style=dotted] (-2.5000,-3.0000) -- (-2.5000,6.0000);
+\draw [color=gray,style=dotted] (-1.5000,-3.0000) -- (-1.5000,6.0000);
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+\draw [color=gray,style=dotted] (0.5000,-3.0000) -- (0.5000,6.0000);
+\draw [color=gray,style=dotted] (1.5000,-3.0000) -- (1.5000,6.0000);
+\draw [color=gray,style=dotted] (-5.0000,-2.5000) -- (2.0000,-2.5000);
+\draw [color=gray,style=dotted] (-5.0000,-1.5000) -- (2.0000,-1.5000);
+\draw [color=gray,style=dotted] (-5.0000,-0.5000) -- (2.0000,-0.5000);
+\draw [color=gray,style=dotted] (-5.0000,0.5000) -- (2.0000,0.5000);
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+\draw [color=gray,style=dotted] (-5.0000,2.5000) -- (2.0000,2.5000);
+\draw [color=gray,style=dotted] (-5.0000,3.5000) -- (2.0000,3.5000);
+\draw [color=gray,style=dotted] (-5.0000,4.5000) -- (2.0000,4.5000);
+\draw [color=gray,style=dotted] (-5.0000,5.5000) -- (2.0000,5.5000);
+\draw [color=gray,style=solid] (-5.0000,-3.0000) -- (2.0000,-3.0000);
+\draw [color=gray,style=solid] (-5.0000,-2.0000) -- (2.0000,-2.0000);
+\draw [color=gray,style=solid] (-5.0000,-1.0000) -- (2.0000,-1.0000);
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+\draw [color=gray,style=solid] (-5.0000,4.0000) -- (2.0000,4.0000);
+\draw [color=gray,style=solid] (-5.0000,5.0000) -- (2.0000,5.0000);
+\draw [color=gray,style=solid] (-5.0000,6.0000) -- (2.0000,6.0000);
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-\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,6.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,6.5000);
 %DEFAULT
 
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+\draw [color=blue] (0.0000,3.0000)--(0.0101,2.9696)--(0.0202,2.9393)--(0.0303,2.9090)--(0.0404,2.8787)--(0.0505,2.8484)--(0.0606,2.8181)--(0.0707,2.7878)--(0.0808,2.7575)--(0.0909,2.7272)--(0.1010,2.6969)--(0.1111,2.6666)--(0.1212,2.6363)--(0.1313,2.6060)--(0.1414,2.5757)--(0.1515,2.5454)--(0.1616,2.5151)--(0.1717,2.4848)--(0.1818,2.4545)--(0.1919,2.4242)--(0.2020,2.3939)--(0.2121,2.3636)--(0.2222,2.3333)--(0.2323,2.3030)--(0.2424,2.2727)--(0.2525,2.2424)--(0.2626,2.2121)--(0.2727,2.1818)--(0.2828,2.1515)--(0.2929,2.1212)--(0.3030,2.0909)--(0.3131,2.0606)--(0.3232,2.0303)--(0.3333,2.0000)--(0.3434,1.9696)--(0.3535,1.9393)--(0.3636,1.9090)--(0.3737,1.8787)--(0.3838,1.8484)--(0.3939,1.8181)--(0.4040,1.7878)--(0.4141,1.7575)--(0.4242,1.7272)--(0.4343,1.6969)--(0.4444,1.6666)--(0.4545,1.6363)--(0.4646,1.6060)--(0.4747,1.5757)--(0.4848,1.5454)--(0.4949,1.5151)--(0.5050,1.4848)--(0.5151,1.4545)--(0.5252,1.4242)--(0.5353,1.3939)--(0.5454,1.3636)--(0.5555,1.3333)--(0.5656,1.3030)--(0.5757,1.2727)--(0.5858,1.2424)--(0.5959,1.2121)--(0.6060,1.1818)--(0.6161,1.1515)--(0.6262,1.1212)--(0.6363,1.0909)--(0.6464,1.0606)--(0.6565,1.0303)--(0.6666,1.0000)--(0.6767,0.9696)--(0.6868,0.9393)--(0.6969,0.9090)--(0.7070,0.8787)--(0.7171,0.8484)--(0.7272,0.8181)--(0.7373,0.7878)--(0.7474,0.7575)--(0.7575,0.7272)--(0.7676,0.6969)--(0.7777,0.6666)--(0.7878,0.6363)--(0.7979,0.6060)--(0.8080,0.5757)--(0.8181,0.5454)--(0.8282,0.5151)--(0.8383,0.4848)--(0.8484,0.4545)--(0.8585,0.4242)--(0.8686,0.3939)--(0.8787,0.3636)--(0.8888,0.3333)--(0.8989,0.3030)--(0.9090,0.2727)--(0.9191,0.2424)--(0.9292,0.2121)--(0.9393,0.1818)--(0.9494,0.1515)--(0.9595,0.1212)--(0.9696,0.0909)--(0.9797,0.0606)--(0.9898,0.0303)--(1.0000,0.0000);
 
-\draw [color=blue] (1.000,3.000)--(1.010,3.030)--(1.020,3.061)--(1.030,3.091)--(1.040,3.121)--(1.051,3.152)--(1.061,3.182)--(1.071,3.212)--(1.081,3.242)--(1.091,3.273)--(1.101,3.303)--(1.111,3.333)--(1.121,3.364)--(1.131,3.394)--(1.141,3.424)--(1.152,3.455)--(1.162,3.485)--(1.172,3.515)--(1.182,3.545)--(1.192,3.576)--(1.202,3.606)--(1.212,3.636)--(1.222,3.667)--(1.232,3.697)--(1.242,3.727)--(1.253,3.758)--(1.263,3.788)--(1.273,3.818)--(1.283,3.848)--(1.293,3.879)--(1.303,3.909)--(1.313,3.939)--(1.323,3.970)--(1.333,4.000)--(1.343,4.030)--(1.354,4.061)--(1.364,4.091)--(1.374,4.121)--(1.384,4.151)--(1.394,4.182)--(1.404,4.212)--(1.414,4.242)--(1.424,4.273)--(1.434,4.303)--(1.444,4.333)--(1.455,4.364)--(1.465,4.394)--(1.475,4.424)--(1.485,4.455)--(1.495,4.485)--(1.505,4.515)--(1.515,4.545)--(1.525,4.576)--(1.535,4.606)--(1.545,4.636)--(1.556,4.667)--(1.566,4.697)--(1.576,4.727)--(1.586,4.758)--(1.596,4.788)--(1.606,4.818)--(1.616,4.849)--(1.626,4.879)--(1.636,4.909)--(1.646,4.939)--(1.657,4.970)--(1.667,5.000)--(1.677,5.030)--(1.687,5.061)--(1.697,5.091)--(1.707,5.121)--(1.717,5.151)--(1.727,5.182)--(1.737,5.212)--(1.747,5.242)--(1.758,5.273)--(1.768,5.303)--(1.778,5.333)--(1.788,5.364)--(1.798,5.394)--(1.808,5.424)--(1.818,5.455)--(1.828,5.485)--(1.838,5.515)--(1.848,5.545)--(1.859,5.576)--(1.869,5.606)--(1.879,5.636)--(1.889,5.667)--(1.899,5.697)--(1.909,5.727)--(1.919,5.758)--(1.929,5.788)--(1.939,5.818)--(1.949,5.849)--(1.960,5.879)--(1.970,5.909)--(1.980,5.939)--(1.990,5.970)--(2.000,6.000);
-\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (0,0) node [rotate=0] {$o$};
+\draw [color=blue] (1.0000,3.0000)--(1.0101,3.0303)--(1.0202,3.0606)--(1.0303,3.0909)--(1.0404,3.1212)--(1.0505,3.1515)--(1.0606,3.1818)--(1.0707,3.2121)--(1.0808,3.2424)--(1.0909,3.2727)--(1.1010,3.3030)--(1.1111,3.3333)--(1.1212,3.3636)--(1.1313,3.3939)--(1.1414,3.4242)--(1.1515,3.4545)--(1.1616,3.4848)--(1.1717,3.5151)--(1.1818,3.5454)--(1.1919,3.5757)--(1.2020,3.6060)--(1.2121,3.6363)--(1.2222,3.6666)--(1.2323,3.6969)--(1.2424,3.7272)--(1.2525,3.7575)--(1.2626,3.7878)--(1.2727,3.8181)--(1.2828,3.8484)--(1.2929,3.8787)--(1.3030,3.9090)--(1.3131,3.9393)--(1.3232,3.9696)--(1.3333,4.0000)--(1.3434,4.0303)--(1.3535,4.0606)--(1.3636,4.0909)--(1.3737,4.1212)--(1.3838,4.1515)--(1.3939,4.1818)--(1.4040,4.2121)--(1.4141,4.2424)--(1.4242,4.2727)--(1.4343,4.3030)--(1.4444,4.3333)--(1.4545,4.3636)--(1.4646,4.3939)--(1.4747,4.4242)--(1.4848,4.4545)--(1.4949,4.4848)--(1.5050,4.5151)--(1.5151,4.5454)--(1.5252,4.5757)--(1.5353,4.6060)--(1.5454,4.6363)--(1.5555,4.6666)--(1.5656,4.6969)--(1.5757,4.7272)--(1.5858,4.7575)--(1.5959,4.7878)--(1.6060,4.8181)--(1.6161,4.8484)--(1.6262,4.8787)--(1.6363,4.9090)--(1.6464,4.9393)--(1.6565,4.9696)--(1.6666,5.0000)--(1.6767,5.0303)--(1.6868,5.0606)--(1.6969,5.0909)--(1.7070,5.1212)--(1.7171,5.1515)--(1.7272,5.1818)--(1.7373,5.2121)--(1.7474,5.2424)--(1.7575,5.2727)--(1.7676,5.3030)--(1.7777,5.3333)--(1.7878,5.3636)--(1.7979,5.3939)--(1.8080,5.4242)--(1.8181,5.4545)--(1.8282,5.4848)--(1.8383,5.5151)--(1.8484,5.5454)--(1.8585,5.5757)--(1.8686,5.6060)--(1.8787,5.6363)--(1.8888,5.6666)--(1.8989,5.6969)--(1.9090,5.7272)--(1.9191,5.7575)--(1.9292,5.7878)--(1.9393,5.8181)--(1.9494,5.8484)--(1.9595,5.8787)--(1.9696,5.9090)--(1.9797,5.9393)--(1.9898,5.9696)--(2.0000,6.0000);
+\draw []  (0.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
 \draw []  (1.0000,3.0000) node [rotate=0] {$o$};
 
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.29125,6.0000) node {$ 6 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_HNxitLj.pstricks b/auto/pictures_tex/Fig_HNxitLj.pstricks
index ea0ee8ecf..9c98b55bc 100644
--- a/auto/pictures_tex/Fig_HNxitLj.pstricks
+++ b/auto/pictures_tex/Fig_HNxitLj.pstricks
@@ -75,27 +75,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw []  (-1.0000,0) node [rotate=0] {$\diamondsuit$};
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\diamondsuit$};
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\diamondsuit$};
 \draw []  (-1.0000,-1.0000) node [rotate=0] {$\diamondsuit$};
-\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
-\draw (1.5000,-0.38249) node {\( \sA^*_{\sH}\)};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw (1.5000,-0.3824) node {\( \sA^*_{\sH}\)};
+\draw (1.5000,-0.3824) node {\( \sA^*_{\sH}\)};
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_HasseAGdfdy.pstricks b/auto/pictures_tex/Fig_HasseAGdfdy.pstricks
index 95b7c5107..2da7f01c3 100644
--- a/auto/pictures_tex/Fig_HasseAGdfdy.pstricks
+++ b/auto/pictures_tex/Fig_HasseAGdfdy.pstricks
@@ -86,24 +86,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.27858) node {\( \alpha\)};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.36222) node {\( \beta\)};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.31406) node {\( \gamma\)};
-\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
-\draw (0,2.2786) node {\( a\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.2785) node {\( \alpha\)};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3622) node {\( \beta\)};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.3140) node {\( \gamma\)};
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,2.2785) node {\( a\)};
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
 \draw (2.0000,2.3267) node {\( b\)};
 \draw []  (4.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (4.0000,2.2786) node {\( c\)};
-\draw [] (0,0) -- (0,2.00);
-\draw [] (2.00,0) -- (2.00,2.00);
-\draw [] (4.00,0) -- (4.00,2.00);
-\draw [] (0,2.00) -- (2.00,0);
-\draw [] (2.00,2.00) -- (4.00,0);
-\draw [] (0,0) -- (4.00,2.00);
+\draw (4.0000,2.2785) node {\( c\)};
+\draw [] (0.0000,0.0000) -- (0.0000,2.0000);
+\draw [] (2.0000,0.0000) -- (2.0000,2.0000);
+\draw [] (4.0000,0.0000) -- (4.0000,2.0000);
+\draw [] (0.0000,2.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,2.0000) -- (4.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (4.0000,2.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks b/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks
index b7cae7f35..b7851a4a4 100644
--- a/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks
+++ b/auto/pictures_tex/Fig_IOCTooePeHGCXH.pstricks
@@ -95,24 +95,24 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [] (2.000,2.000)--(1.996,2.127)--(1.984,2.253)--(1.964,2.379)--(1.936,2.502)--(1.900,2.624)--(1.857,2.743)--(1.806,2.860)--(1.748,2.972)--(1.683,3.081)--(1.611,3.186)--(1.532,3.286)--(1.447,3.380)--(1.357,3.469)--(1.261,3.552)--(1.160,3.629)--(1.054,3.699)--(0.9445,3.763)--(0.8308,3.819)--(0.7138,3.868)--(0.5938,3.910)--(0.4715,3.944)--(0.3473,3.970)--(0.2217,3.988)--(0.09516,3.998)--(-0.03173,4.000)--(-0.1585,3.994)--(-0.2846,3.980)--(-0.4096,3.958)--(-0.5330,3.928)--(-0.6541,3.890)--(-0.7727,3.845)--(-0.8881,3.792)--(-1.000,3.732)--(-1.108,3.665)--(-1.211,3.592)--(-1.310,3.512)--(-1.403,3.425)--(-1.491,3.334)--(-1.572,3.236)--(-1.647,3.134)--(-1.716,3.027)--(-1.778,2.916)--(-1.832,2.802)--(-1.879,2.684)--(-1.919,2.563)--(-1.951,2.441)--(-1.975,2.316)--(-1.991,2.190)--(-1.999,2.063)--(-1.999,1.937)--(-1.991,1.810)--(-1.975,1.684)--(-1.951,1.559)--(-1.919,1.437)--(-1.879,1.316)--(-1.832,1.198)--(-1.778,1.084)--(-1.716,0.9726)--(-1.647,0.8659)--(-1.572,0.7637)--(-1.491,0.6665)--(-1.403,0.5746)--(-1.310,0.4885)--(-1.211,0.4085)--(-1.108,0.3349)--(-1.000,0.2679)--(-0.8881,0.2080)--(-0.7727,0.1553)--(-0.6541,0.1100)--(-0.5330,0.07232)--(-0.4096,0.04240)--(-0.2846,0.02036)--(-0.1585,0.006290)--(-0.03173,0)--(0.09516,0.002265)--(0.2217,0.01232)--(0.3473,0.03038)--(0.4715,0.05638)--(0.5938,0.09020)--(0.7138,0.1317)--(0.8308,0.1807)--(0.9445,0.2371)--(1.054,0.3005)--(1.160,0.3708)--(1.261,0.4477)--(1.357,0.5308)--(1.447,0.6198)--(1.532,0.7144)--(1.611,0.8142)--(1.683,0.9187)--(1.748,1.028)--(1.806,1.140)--(1.857,1.257)--(1.900,1.376)--(1.936,1.498)--(1.964,1.621)--(1.984,1.747)--(1.996,1.873)--(2.000,2.000);
+\draw [] (2.0000,2.0000)--(1.9959,2.1268)--(1.9839,2.2531)--(1.9638,2.3785)--(1.9358,2.5022)--(1.9001,2.6240)--(1.8567,2.7433)--(1.8058,2.8595)--(1.7476,2.9723)--(1.6825,3.0812)--(1.6105,3.1858)--(1.5320,3.2855)--(1.4474,3.3801)--(1.3570,3.4691)--(1.2611,3.5522)--(1.1601,3.6291)--(1.0544,3.6994)--(0.9445,3.7629)--(0.8308,3.8192)--(0.7137,3.8682)--(0.5938,3.9098)--(0.4715,3.9436)--(0.3472,3.9696)--(0.2216,3.9876)--(0.0951,3.9977)--(-0.0317,3.9997)--(-0.1584,3.9937)--(-0.2846,3.9796)--(-0.4096,3.9576)--(-0.5329,3.9276)--(-0.6541,3.8900)--(-0.7726,3.8447)--(-0.8881,3.7919)--(-1.0000,3.7320)--(-1.1078,3.6651)--(-1.2112,3.5915)--(-1.3097,3.5114)--(-1.4029,3.4253)--(-1.4905,3.3335)--(-1.5721,3.2363)--(-1.6473,3.1341)--(-1.7159,3.0273)--(-1.7776,2.9164)--(-1.8322,2.8018)--(-1.8793,2.6840)--(-1.9189,2.5634)--(-1.9508,2.4406)--(-1.9748,2.3160)--(-1.9909,2.1901)--(-1.9989,2.0634)--(-1.9989,1.9365)--(-1.9909,1.8098)--(-1.9748,1.6839)--(-1.9508,1.5593)--(-1.9189,1.4365)--(-1.8793,1.3159)--(-1.8322,1.1981)--(-1.7776,1.0835)--(-1.7159,0.9726)--(-1.6473,0.8658)--(-1.5721,0.7636)--(-1.4905,0.6664)--(-1.4029,0.5746)--(-1.3097,0.4885)--(-1.2112,0.4084)--(-1.1078,0.3348)--(-0.9999,0.2679)--(-0.8881,0.2080)--(-0.7726,0.1552)--(-0.6541,0.1099)--(-0.5329,0.0723)--(-0.4096,0.0423)--(-0.2846,0.0203)--(-0.1584,0.0062)--(-0.0317,0.0000)--(0.0951,0.0022)--(0.2216,0.0123)--(0.3472,0.0303)--(0.4715,0.0563)--(0.5938,0.0901)--(0.7137,0.1317)--(0.8308,0.1807)--(0.9445,0.2370)--(1.0544,0.3005)--(1.1601,0.3708)--(1.2611,0.4477)--(1.3570,0.5308)--(1.4474,0.6198)--(1.5320,0.7144)--(1.6105,0.8141)--(1.6825,0.9187)--(1.7476,1.0276)--(1.8058,1.1404)--(1.8567,1.2566)--(1.9001,1.3759)--(1.9358,1.4977)--(1.9638,1.6214)--(1.9839,1.7468)--(1.9959,1.8731)--(2.0000,2.0000);
 
-\draw [] (7.000,3.000)--(6.994,3.190)--(6.976,3.380)--(6.946,3.568)--(6.904,3.753)--(6.850,3.936)--(6.785,4.115)--(6.709,4.289)--(6.622,4.459)--(6.524,4.622)--(6.416,4.779)--(6.298,4.928)--(6.171,5.070)--(6.036,5.204)--(5.892,5.328)--(5.740,5.444)--(5.582,5.549)--(5.417,5.644)--(5.246,5.729)--(5.071,5.802)--(4.891,5.865)--(4.707,5.915)--(4.521,5.954)--(4.333,5.982)--(4.143,5.997)--(3.952,6.000)--(3.762,5.991)--(3.573,5.969)--(3.386,5.936)--(3.201,5.892)--(3.019,5.835)--(2.841,5.767)--(2.668,5.688)--(2.500,5.598)--(2.338,5.498)--(2.183,5.387)--(2.035,5.267)--(1.896,5.138)--(1.764,5.000)--(1.642,4.854)--(1.529,4.701)--(1.426,4.541)--(1.333,4.375)--(1.252,4.203)--(1.181,4.026)--(1.122,3.845)--(1.074,3.661)--(1.038,3.474)--(1.014,3.285)--(1.002,3.095)--(1.002,2.905)--(1.014,2.715)--(1.038,2.526)--(1.074,2.339)--(1.122,2.155)--(1.181,1.974)--(1.252,1.797)--(1.333,1.625)--(1.426,1.459)--(1.529,1.299)--(1.642,1.146)--(1.764,0.9997)--(1.896,0.8619)--(2.035,0.7327)--(2.183,0.6127)--(2.338,0.5023)--(2.500,0.4019)--(2.668,0.3120)--(2.841,0.2329)--(3.019,0.1650)--(3.201,0.1085)--(3.386,0.06359)--(3.573,0.03054)--(3.762,0.009436)--(3.952,0)--(4.143,0.003398)--(4.333,0.01848)--(4.521,0.04558)--(4.707,0.08457)--(4.891,0.1353)--(5.071,0.1976)--(5.246,0.2711)--(5.417,0.3556)--(5.582,0.4508)--(5.740,0.5563)--(5.892,0.6716)--(6.036,0.7962)--(6.171,0.9298)--(6.298,1.072)--(6.416,1.221)--(6.524,1.378)--(6.622,1.541)--(6.709,1.711)--(6.785,1.885)--(6.850,2.064)--(6.904,2.247)--(6.946,2.432)--(6.976,2.620)--(6.994,2.810)--(7.000,3.000);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.50870) node {\( \tilde \phi(a)\)};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.5172,-0.33726) node {\( \tilde\phi(c)\)};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.50870) node {\( \tilde \phi(b)\)};
-\draw [] (-4.00,0) -- (8.00,0);
-\draw []  (1.0790,3.6840) node [rotate=0] {$\bullet$};
-\draw (1.6899,3.6840) node {\( \tilde\phi(m)\)};
-\draw (-2.2928,3.3087) node {\( \tilde \phi(A)\)};
-\draw (6.6610,5.4714) node {\( \tilde \phi(B)\)};
+\draw [] (7.0000,3.0000)--(6.9939,3.1902)--(6.9758,3.3797)--(6.9457,3.5677)--(6.9038,3.7534)--(6.8502,3.9361)--(6.7851,4.1149)--(6.7087,4.2893)--(6.6215,4.4585)--(6.5237,4.6219)--(6.4158,4.7787)--(6.2981,4.9283)--(6.1712,5.0702)--(6.0355,5.2037)--(5.8916,5.3284)--(5.7401,5.4437)--(5.5816,5.5491)--(5.4168,5.6443)--(5.2462,5.7288)--(5.0706,5.8024)--(4.8907,5.8647)--(4.7072,5.9154)--(4.5209,5.9544)--(4.3325,5.9815)--(4.1427,5.9966)--(3.9524,5.9996)--(3.7622,5.9905)--(3.5730,5.9694)--(3.3855,5.9364)--(3.2005,5.8915)--(3.0187,5.8350)--(2.8409,5.7670)--(2.6678,5.6879)--(2.5000,5.5980)--(2.3382,5.4977)--(2.1831,5.3872)--(2.0354,5.2672)--(1.8955,5.1380)--(1.7642,5.0003)--(1.6418,4.8544)--(1.5289,4.7011)--(1.4260,4.5410)--(1.3334,4.3746)--(1.2516,4.2027)--(1.1809,4.0260)--(1.1215,3.8451)--(1.0737,3.6609)--(1.0376,3.4740)--(1.0135,3.2851)--(1.0015,3.0951)--(1.0015,2.9048)--(1.0135,2.7148)--(1.0376,2.5259)--(1.0737,2.3390)--(1.1215,2.1548)--(1.1809,1.9739)--(1.2516,1.7972)--(1.3334,1.6253)--(1.4260,1.4589)--(1.5289,1.2988)--(1.6418,1.1455)--(1.7642,0.9996)--(1.8955,0.8619)--(2.0354,0.7327)--(2.1831,0.6127)--(2.3382,0.5022)--(2.5000,0.4019)--(2.6678,0.3120)--(2.8409,0.2329)--(3.0187,0.1649)--(3.2005,0.1084)--(3.3855,0.0635)--(3.5730,0.0305)--(3.7622,0.0094)--(3.9524,0.0000)--(4.1427,0.0033)--(4.3325,0.0184)--(4.5209,0.0455)--(4.7072,0.0845)--(4.8907,0.1352)--(5.0706,0.1975)--(5.2462,0.2711)--(5.4168,0.3556)--(5.5816,0.4508)--(5.7401,0.5562)--(5.8916,0.6715)--(6.0355,0.7962)--(6.1712,0.9297)--(6.2981,1.0716)--(6.4158,1.2212)--(6.5237,1.3780)--(6.6215,1.5414)--(6.7087,1.7106)--(6.7851,1.8850)--(6.8502,2.0638)--(6.9038,2.2465)--(6.9457,2.4322)--(6.9758,2.6202)--(6.9939,2.8097)--(7.0000,3.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.5086) node {\( \tilde \phi(a)\)};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.5171,-0.3372) node {\( \tilde\phi(c)\)};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.5086) node {\( \tilde \phi(b)\)};
+\draw [] (-4.0000,0.0000) -- (8.0000,0.0000);
+\draw []  (1.0790,3.6839) node [rotate=0] {$\bullet$};
+\draw (1.6898,3.6839) node {\( \tilde\phi(m)\)};
+\draw (-2.2927,3.3086) node {\( \tilde \phi(A)\)};
+\draw (6.6609,5.4714) node {\( \tilde \phi(B)\)};
 \draw []  (1.7445,1.0219) node [rotate=0] {$\bullet$};
-\draw (2.0358,1.0219) node {\( 0\)};
-\draw [] (0.746,5.02) -- (2.08,-0.309);
-\draw (8.5574,0) node {\( \tilde\phi(\mC)\)};
+\draw (2.0357,1.0219) node {\( 0\)};
+\draw [] (0.7462,5.0149) -- (2.0772,-0.3091);
+\draw (8.5573,0.0000) node {\( \tilde\phi(\mC)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_IWuPxFc.pstricks b/auto/pictures_tex/Fig_IWuPxFc.pstricks
index a40e3d90c..ed922425f 100644
--- a/auto/pictures_tex/Fig_IWuPxFc.pstricks
+++ b/auto/pictures_tex/Fig_IWuPxFc.pstricks
@@ -95,40 +95,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4989,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-3.1457) -- (0,3.1457);
+\draw [,->,>=latex] (-2.4988,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.1456) -- (0.0000,3.1456);
 %DEFAULT
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diff --git a/auto/pictures_tex/Fig_IYAvSvI.pstricks b/auto/pictures_tex/Fig_IYAvSvI.pstricks
index 9bf4d113f..23d56cc1b 100644
--- a/auto/pictures_tex/Fig_IYAvSvI.pstricks
+++ b/auto/pictures_tex/Fig_IYAvSvI.pstricks
@@ -78,26 +78,26 @@
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diff --git a/auto/pictures_tex/Fig_IntBoutCercle.pstricks b/auto/pictures_tex/Fig_IntBoutCercle.pstricks
index 01492c84e..b55e125c4 100644
--- a/auto/pictures_tex/Fig_IntBoutCercle.pstricks
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+\draw [] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (1.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=red] (-0.5000,-0.5000) -- (1.5000,1.5000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3546,0.66316) node {$P$};
+\draw (1.3546,0.6631) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntDeuxCarres.pstricks b/auto/pictures_tex/Fig_IntDeuxCarres.pstricks
index d22b20139..b1fbacb81 100644
--- a/auto/pictures_tex/Fig_IntDeuxCarres.pstricks
+++ b/auto/pictures_tex/Fig_IntDeuxCarres.pstricks
@@ -72,16 +72,16 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-2.00,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,-2.00) -- (2.00,-2.00) -- (-2.00,-2.00) -- (-2.00,-2.00) -- (-2.00,2.00) --  cycle;
-\draw [color=red] (-2.00,2.00) -- (2.00,2.00);
-\draw [color=red] (2.00,2.00) -- (2.00,-2.00);
-\draw [color=red] (2.00,-2.00) -- (-2.00,-2.00);
-\draw [color=red] (-2.00,-2.00) -- (-2.00,2.00);
-\fill [color=white] (-1.00,1.00) -- (1.00,1.00) -- (1.00,1.00) -- (1.00,-1.00) -- (1.00,-1.00) -- (-1.00,-1.00) -- (-1.00,-1.00) -- (-1.00,1.00) --  cycle;
-\draw [color=red] (-1.00,1.00) -- (1.00,1.00);
-\draw [color=red] (1.00,1.00) -- (1.00,-1.00);
-\draw [color=red] (1.00,-1.00) -- (-1.00,-1.00);
-\draw [color=red] (-1.00,-1.00) -- (-1.00,1.00);
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,-2.0000) -- (2.0000,-2.0000) -- (-2.0000,-2.0000) -- (-2.0000,-2.0000) -- (-2.0000,2.0000) --  cycle;
+\draw [color=red] (-2.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=red] (2.0000,2.0000) -- (2.0000,-2.0000);
+\draw [color=red] (2.0000,-2.0000) -- (-2.0000,-2.0000);
+\draw [color=red] (-2.0000,-2.0000) -- (-2.0000,2.0000);
+\fill [color=white] (-1.0000,1.0000) -- (1.0000,1.0000) -- (1.0000,1.0000) -- (1.0000,-1.0000) -- (1.0000,-1.0000) -- (-1.0000,-1.0000) -- (-1.0000,-1.0000) -- (-1.0000,1.0000) --  cycle;
+\draw [color=red] (-1.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=red] (1.0000,1.0000) -- (1.0000,-1.0000);
+\draw [color=red] (1.0000,-1.0000) -- (-1.0000,-1.0000);
+\draw [color=red] (-1.0000,-1.0000) -- (-1.0000,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntEcourbe.pstricks b/auto/pictures_tex/Fig_IntEcourbe.pstricks
index f9fb48682..ee8fb43b6 100644
--- a/auto/pictures_tex/Fig_IntEcourbe.pstricks
+++ b/auto/pictures_tex/Fig_IntEcourbe.pstricks
@@ -79,13 +79,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0000);
 %DEFAULT
 
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-\draw [color=blue] (1.000,1.341)--(1.020,1.352)--(1.040,1.363)--(1.061,1.373)--(1.081,1.382)--(1.101,1.392)--(1.121,1.401)--(1.141,1.409)--(1.162,1.417)--(1.182,1.425)--(1.202,1.433)--(1.222,1.440)--(1.242,1.447)--(1.263,1.453)--(1.283,1.459)--(1.303,1.464)--(1.323,1.470)--(1.343,1.474)--(1.364,1.479)--(1.384,1.483)--(1.404,1.486)--(1.424,1.489)--(1.444,1.492)--(1.465,1.494)--(1.485,1.496)--(1.505,1.498)--(1.525,1.499)--(1.545,1.500)--(1.566,1.500)--(1.586,1.500)--(1.606,1.499)--(1.626,1.498)--(1.646,1.497)--(1.667,1.495)--(1.687,1.493)--(1.707,1.491)--(1.727,1.488)--(1.747,1.484)--(1.768,1.481)--(1.788,1.477)--(1.808,1.472)--(1.828,1.467)--(1.848,1.462)--(1.869,1.456)--(1.889,1.450)--(1.909,1.443)--(1.929,1.436)--(1.949,1.429)--(1.970,1.421)--(1.990,1.413)--(2.010,1.405)--(2.030,1.396)--(2.051,1.387)--(2.071,1.378)--(2.091,1.368)--(2.111,1.358)--(2.131,1.347)--(2.152,1.336)--(2.172,1.325)--(2.192,1.313)--(2.212,1.301)--(2.232,1.289)--(2.253,1.276)--(2.273,1.264)--(2.293,1.250)--(2.313,1.237)--(2.333,1.223)--(2.354,1.209)--(2.374,1.195)--(2.394,1.180)--(2.414,1.165)--(2.434,1.150)--(2.455,1.134)--(2.475,1.119)--(2.495,1.103)--(2.515,1.086)--(2.535,1.070)--(2.556,1.053)--(2.576,1.036)--(2.596,1.019)--(2.616,1.002)--(2.636,0.9840)--(2.657,0.9662)--(2.677,0.9483)--(2.697,0.9301)--(2.717,0.9118)--(2.737,0.8933)--(2.758,0.8746)--(2.778,0.8558)--(2.798,0.8369)--(2.818,0.8178)--(2.838,0.7986)--(2.859,0.7792)--(2.879,0.7598)--(2.899,0.7402)--(2.919,0.7206)--(2.939,0.7008)--(2.960,0.6810)--(2.980,0.6611)--(3.000,0.6411);
+\draw [color=blue] (1.0000,1.3414)--(1.0202,1.3522)--(1.0404,1.3626)--(1.0606,1.3726)--(1.0808,1.3823)--(1.1010,1.3916)--(1.1212,1.4006)--(1.1414,1.4092)--(1.1616,1.4174)--(1.1818,1.4252)--(1.2020,1.4327)--(1.2222,1.4398)--(1.2424,1.4465)--(1.2626,1.4528)--(1.2828,1.4588)--(1.3030,1.4643)--(1.3232,1.4695)--(1.3434,1.4742)--(1.3636,1.4786)--(1.3838,1.4825)--(1.4040,1.4861)--(1.4242,1.4892)--(1.4444,1.4920)--(1.4646,1.4943)--(1.4848,1.4963)--(1.5050,1.4978)--(1.5252,1.4989)--(1.5454,1.4996)--(1.5656,1.4999)--(1.5858,1.4998)--(1.6060,1.4993)--(1.6262,1.4984)--(1.6464,1.4971)--(1.6666,1.4954)--(1.6868,1.4932)--(1.7070,1.4907)--(1.7272,1.4877)--(1.7474,1.4844)--(1.7676,1.4806)--(1.7878,1.4765)--(1.8080,1.4719)--(1.8282,1.4670)--(1.8484,1.4616)--(1.8686,1.4559)--(1.8888,1.4498)--(1.9090,1.4433)--(1.9292,1.4364)--(1.9494,1.4291)--(1.9696,1.4214)--(1.9898,1.4134)--(2.0101,1.4050)--(2.0303,1.3962)--(2.0505,1.3871)--(2.0707,1.3776)--(2.0909,1.3677)--(2.1111,1.3575)--(2.1313,1.3469)--(2.1515,1.3360)--(2.1717,1.3248)--(2.1919,1.3132)--(2.2121,1.3013)--(2.2323,1.2890)--(2.2525,1.2764)--(2.2727,1.2635)--(2.2929,1.2503)--(2.3131,1.2368)--(2.3333,1.2230)--(2.3535,1.2089)--(2.3737,1.1945)--(2.3939,1.1799)--(2.4141,1.1649)--(2.4343,1.1497)--(2.4545,1.1342)--(2.4747,1.1185)--(2.4949,1.1025)--(2.5151,1.0862)--(2.5353,1.0697)--(2.5555,1.0530)--(2.5757,1.0361)--(2.5959,1.0189)--(2.6161,1.0015)--(2.6363,0.9840)--(2.6565,0.9662)--(2.6767,0.9482)--(2.6969,0.9301)--(2.7171,0.9117)--(2.7373,0.8933)--(2.7575,0.8746)--(2.7777,0.8558)--(2.7979,0.8368)--(2.8181,0.8178)--(2.8383,0.7985)--(2.8585,0.7792)--(2.8787,0.7597)--(2.8989,0.7402)--(2.9191,0.7205)--(2.9393,0.7008)--(2.9595,0.6809)--(2.9797,0.6610)--(3.0000,0.6411);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -93,30 +93,30 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [] (1.00,1.34) -- (1.00,3.50);
-\draw [] (3.00,3.50) -- (3.00,0.641);
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+\draw [color=red,style=solid] (1.0000,3.5000)--(1.0202,3.4600)--(1.0404,3.4208)--(1.0606,3.3824)--(1.0808,3.3449)--(1.1010,3.3081)--(1.1212,3.2722)--(1.1414,3.2371)--(1.1616,3.2028)--(1.1818,3.1694)--(1.2020,3.1367)--(1.2222,3.1049)--(1.2424,3.0739)--(1.2626,3.0437)--(1.2828,3.0143)--(1.3030,2.9857)--(1.3232,2.9580)--(1.3434,2.9310)--(1.3636,2.9049)--(1.3838,2.8796)--(1.4040,2.8551)--(1.4242,2.8314)--(1.4444,2.8086)--(1.4646,2.7866)--(1.4848,2.7653)--(1.5050,2.7449)--(1.5252,2.7253)--(1.5454,2.7066)--(1.5656,2.6886)--(1.5858,2.6715)--(1.6060,2.6551)--(1.6262,2.6396)--(1.6464,2.6249)--(1.6666,2.6111)--(1.6868,2.5980)--(1.7070,2.5858)--(1.7272,2.5743)--(1.7474,2.5637)--(1.7676,2.5539)--(1.7878,2.5449)--(1.8080,2.5368)--(1.8282,2.5294)--(1.8484,2.5229)--(1.8686,2.5172)--(1.8888,2.5123)--(1.9090,2.5082)--(1.9292,2.5049)--(1.9494,2.5025)--(1.9696,2.5009)--(1.9898,2.5001)--(2.0101,2.5001)--(2.0303,2.5009)--(2.0505,2.5025)--(2.0707,2.5049)--(2.0909,2.5082)--(2.1111,2.5123)--(2.1313,2.5172)--(2.1515,2.5229)--(2.1717,2.5294)--(2.1919,2.5368)--(2.2121,2.5449)--(2.2323,2.5539)--(2.2525,2.5637)--(2.2727,2.5743)--(2.2929,2.5858)--(2.3131,2.5980)--(2.3333,2.6111)--(2.3535,2.6249)--(2.3737,2.6396)--(2.3939,2.6551)--(2.4141,2.6715)--(2.4343,2.6886)--(2.4545,2.7066)--(2.4747,2.7253)--(2.4949,2.7449)--(2.5151,2.7653)--(2.5353,2.7866)--(2.5555,2.8086)--(2.5757,2.8314)--(2.5959,2.8551)--(2.6161,2.8796)--(2.6363,2.9049)--(2.6565,2.9310)--(2.6767,2.9580)--(2.6969,2.9857)--(2.7171,3.0143)--(2.7373,3.0437)--(2.7575,3.0739)--(2.7777,3.1049)--(2.7979,3.1367)--(2.8181,3.1694)--(2.8383,3.2028)--(2.8585,3.2371)--(2.8787,3.2722)--(2.8989,3.3081)--(2.9191,3.3449)--(2.9393,3.3824)--(2.9595,3.4208)--(2.9797,3.4600)--(3.0000,3.5000);
+\draw [color=red,style=solid] (1.0000,1.3414)--(1.0202,1.3522)--(1.0404,1.3626)--(1.0606,1.3726)--(1.0808,1.3823)--(1.1010,1.3916)--(1.1212,1.4006)--(1.1414,1.4092)--(1.1616,1.4174)--(1.1818,1.4252)--(1.2020,1.4327)--(1.2222,1.4398)--(1.2424,1.4465)--(1.2626,1.4528)--(1.2828,1.4588)--(1.3030,1.4643)--(1.3232,1.4695)--(1.3434,1.4742)--(1.3636,1.4786)--(1.3838,1.4825)--(1.4040,1.4861)--(1.4242,1.4892)--(1.4444,1.4920)--(1.4646,1.4943)--(1.4848,1.4963)--(1.5050,1.4978)--(1.5252,1.4989)--(1.5454,1.4996)--(1.5656,1.4999)--(1.5858,1.4998)--(1.6060,1.4993)--(1.6262,1.4984)--(1.6464,1.4971)--(1.6666,1.4954)--(1.6868,1.4932)--(1.7070,1.4907)--(1.7272,1.4877)--(1.7474,1.4844)--(1.7676,1.4806)--(1.7878,1.4765)--(1.8080,1.4719)--(1.8282,1.4670)--(1.8484,1.4616)--(1.8686,1.4559)--(1.8888,1.4498)--(1.9090,1.4433)--(1.9292,1.4364)--(1.9494,1.4291)--(1.9696,1.4214)--(1.9898,1.4134)--(2.0101,1.4050)--(2.0303,1.3962)--(2.0505,1.3871)--(2.0707,1.3776)--(2.0909,1.3677)--(2.1111,1.3575)--(2.1313,1.3469)--(2.1515,1.3360)--(2.1717,1.3248)--(2.1919,1.3132)--(2.2121,1.3013)--(2.2323,1.2890)--(2.2525,1.2764)--(2.2727,1.2635)--(2.2929,1.2503)--(2.3131,1.2368)--(2.3333,1.2230)--(2.3535,1.2089)--(2.3737,1.1945)--(2.3939,1.1799)--(2.4141,1.1649)--(2.4343,1.1497)--(2.4545,1.1342)--(2.4747,1.1185)--(2.4949,1.1025)--(2.5151,1.0862)--(2.5353,1.0697)--(2.5555,1.0530)--(2.5757,1.0361)--(2.5959,1.0189)--(2.6161,1.0015)--(2.6363,0.9840)--(2.6565,0.9662)--(2.6767,0.9482)--(2.6969,0.9301)--(2.7171,0.9117)--(2.7373,0.8933)--(2.7575,0.8746)--(2.7777,0.8558)--(2.7979,0.8368)--(2.8181,0.8178)--(2.8383,0.7985)--(2.8585,0.7792)--(2.8787,0.7597)--(2.8989,0.7402)--(2.9191,0.7205)--(2.9393,0.7008)--(2.9595,0.6809)--(2.9797,0.6610)--(3.0000,0.6411);
+\draw [] (1.0000,1.3414) -- (1.0000,3.5000);
+\draw [] (3.0000,3.5000) -- (3.0000,0.6411);
 
-\draw [color=cyan] (1.00,3.50) -- (3.00,3.50);
-\draw [color=cyan] (3.00,3.50) -- (3.00,0.641);
-\draw [color=cyan] (3.00,0.641) -- (1.00,0.641);
-\draw [color=cyan] (1.00,0.641) -- (1.00,3.50);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=cyan] (1.0000,3.5000) -- (3.0000,3.5000);
+\draw [color=cyan] (3.0000,3.5000) -- (3.0000,0.6411);
+\draw [color=cyan] (3.0000,0.6411) -- (1.0000,0.6411);
+\draw [color=cyan] (1.0000,0.6411) -- (1.0000,3.5000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_IntRectangle.pstricks b/auto/pictures_tex/Fig_IntRectangle.pstricks
index 1ac58ccc5..21ba95b9f 100644
--- a/auto/pictures_tex/Fig_IntRectangle.pstricks
+++ b/auto/pictures_tex/Fig_IntRectangle.pstricks
@@ -71,22 +71,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
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-\draw (-0.29125,1.0000) node {$ 1 $};
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-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
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+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_IntTriangle.pstricks b/auto/pictures_tex/Fig_IntTriangle.pstricks
index 5831092a4..7c812e549 100644
--- a/auto/pictures_tex/Fig_IntTriangle.pstricks
+++ b/auto/pictures_tex/Fig_IntTriangle.pstricks
@@ -71,8 +71,8 @@
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 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
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 % declaring the keys in tikz
@@ -81,23 +81,23 @@
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+\draw [color=green] (0.0000,0.0000)--(0.0202,0.0202)--(0.0404,0.0404)--(0.0606,0.0606)--(0.0808,0.0808)--(0.1010,0.1010)--(0.1212,0.1212)--(0.1414,0.1414)--(0.1616,0.1616)--(0.1818,0.1818)--(0.2020,0.2020)--(0.2222,0.2222)--(0.2424,0.2424)--(0.2626,0.2626)--(0.2828,0.2828)--(0.3030,0.3030)--(0.3232,0.3232)--(0.3434,0.3434)--(0.3636,0.3636)--(0.3838,0.3838)--(0.4040,0.4040)--(0.4242,0.4242)--(0.4444,0.4444)--(0.4646,0.4646)--(0.4848,0.4848)--(0.5050,0.5050)--(0.5252,0.5252)--(0.5454,0.5454)--(0.5656,0.5656)--(0.5858,0.5858)--(0.6060,0.6060)--(0.6262,0.6262)--(0.6464,0.6464)--(0.6666,0.6666)--(0.6868,0.6868)--(0.7070,0.7070)--(0.7272,0.7272)--(0.7474,0.7474)--(0.7676,0.7676)--(0.7878,0.7878)--(0.8080,0.8080)--(0.8282,0.8282)--(0.8484,0.8484)--(0.8686,0.8686)--(0.8888,0.8888)--(0.9090,0.9090)--(0.9292,0.9292)--(0.9494,0.9494)--(0.9696,0.9696)--(0.9898,0.9898)--(1.0101,1.0101)--(1.0303,1.0303)--(1.0505,1.0505)--(1.0707,1.0707)--(1.0909,1.0909)--(1.1111,1.1111)--(1.1313,1.1313)--(1.1515,1.1515)--(1.1717,1.1717)--(1.1919,1.1919)--(1.2121,1.2121)--(1.2323,1.2323)--(1.2525,1.2525)--(1.2727,1.2727)--(1.2929,1.2929)--(1.3131,1.3131)--(1.3333,1.3333)--(1.3535,1.3535)--(1.3737,1.3737)--(1.3939,1.3939)--(1.4141,1.4141)--(1.4343,1.4343)--(1.4545,1.4545)--(1.4747,1.4747)--(1.4949,1.4949)--(1.5151,1.5151)--(1.5353,1.5353)--(1.5555,1.5555)--(1.5757,1.5757)--(1.5959,1.5959)--(1.6161,1.6161)--(1.6363,1.6363)--(1.6565,1.6565)--(1.6767,1.6767)--(1.6969,1.6969)--(1.7171,1.7171)--(1.7373,1.7373)--(1.7575,1.7575)--(1.7777,1.7777)--(1.7979,1.7979)--(1.8181,1.8181)--(1.8383,1.8383)--(1.8585,1.8585)--(1.8787,1.8787)--(1.8989,1.8989)--(1.9191,1.9191)--(1.9393,1.9393)--(1.9595,1.9595)--(1.9797,1.9797)--(2.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_IntTrois.pstricks b/auto/pictures_tex/Fig_IntTrois.pstricks
index 654f2a3b0..5afdcb38a 100644
--- a/auto/pictures_tex/Fig_IntTrois.pstricks
+++ b/auto/pictures_tex/Fig_IntTrois.pstricks
@@ -37,20 +37,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\fill [color=lightgray] (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,0) --  cycle;
-\draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
-\draw [] (1.73,1.00) -- (1.73,2.00);
-\draw [color=red] (0,2.00) -- (2.00,2.00);
-\draw [color=red] (2.00,2.00) -- (2.00,0);
+\fill [color=lightgray] (2.0000,0.0000) -- (1.9997,0.0317) -- (1.9989,0.0634) -- (1.9977,0.0951) -- (1.9959,0.1268) -- (1.9937,0.1584) -- (1.9909,0.1901) -- (1.9876,0.2216) -- (1.9839,0.2531) -- (1.9796,0.2846) -- (1.9748,0.3160) -- (1.9696,0.3472) -- (1.9638,0.3785) -- (1.9576,0.4096) -- (1.9508,0.4406) -- (1.9436,0.4715) -- (1.9358,0.5022) -- (1.9276,0.5329) -- (1.9189,0.5634) -- (1.9098,0.5938) -- (1.9001,0.6240) -- (1.8900,0.6541) -- (1.8793,0.6840) -- (1.8682,0.7137) -- (1.8567,0.7433) -- (1.8447,0.7726) -- (1.8322,0.8018) -- (1.8192,0.8308) -- (1.8058,0.8595) -- (1.7919,0.8881) -- (1.7776,0.9164) -- (1.7629,0.9445) -- (1.7476,0.9723) -- (1.7320,1.0000) -- (1.7159,1.0273) -- (1.6994,1.0544) -- (1.6825,1.0812) -- (1.6651,1.1078) -- (1.6473,1.1341) -- (1.6291,1.1601) -- (1.6105,1.1858) -- (1.5915,1.2112) -- (1.5721,1.2363) -- (1.5522,1.2611) -- (1.5320,1.2855) -- (1.5114,1.3097) -- (1.4905,1.3335) -- (1.4691,1.3570) -- (1.4474,1.3801) -- (1.4253,1.4029) -- (1.4029,1.4253) -- (1.3801,1.4474) -- (1.3570,1.4691) -- (1.3335,1.4905) -- (1.3097,1.5114) -- (1.2855,1.5320) -- (1.2611,1.5522) -- (1.2363,1.5721) -- (1.2112,1.5915) -- (1.1858,1.6105) -- (1.1601,1.6291) -- (1.1341,1.6473) -- (1.1078,1.6651) -- (1.0812,1.6825) -- (1.0544,1.6994) -- (1.0273,1.7159) -- (1.0000,1.7320) -- (0.9723,1.7476) -- (0.9445,1.7629) -- (0.9164,1.7776) -- (0.8881,1.7919) -- (0.8595,1.8058) -- (0.8308,1.8192) -- (0.8018,1.8322) -- (0.7726,1.8447) -- (0.7433,1.8567) -- (0.7137,1.8682) -- (0.6840,1.8793) -- (0.6541,1.8900) -- (0.6240,1.9001) -- (0.5938,1.9098) -- (0.5634,1.9189) -- (0.5329,1.9276) -- (0.5022,1.9358) -- (0.4715,1.9436) -- (0.4406,1.9508) -- (0.4096,1.9576) -- (0.3785,1.9638) -- (0.3472,1.9696) -- (0.3160,1.9748) -- (0.2846,1.9796) -- (0.2531,1.9839) -- (0.2216,1.9876) -- (0.1901,1.9909) -- (0.1584,1.9937) -- (0.1268,1.9959) -- (0.0951,1.9977) -- (0.0634,1.9989) -- (0.0317,1.9997) -- (0.0000,2.0000) -- (0.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,0.0000) --  cycle;
+\draw [color=red] (2.0000,0.0000)--(1.9997,0.0317)--(1.9989,0.0634)--(1.9977,0.0951)--(1.9959,0.1268)--(1.9937,0.1584)--(1.9909,0.1901)--(1.9876,0.2216)--(1.9839,0.2531)--(1.9796,0.2846)--(1.9748,0.3160)--(1.9696,0.3472)--(1.9638,0.3785)--(1.9576,0.4096)--(1.9508,0.4406)--(1.9436,0.4715)--(1.9358,0.5022)--(1.9276,0.5329)--(1.9189,0.5634)--(1.9098,0.5938)--(1.9001,0.6240)--(1.8900,0.6541)--(1.8793,0.6840)--(1.8682,0.7137)--(1.8567,0.7433)--(1.8447,0.7726)--(1.8322,0.8018)--(1.8192,0.8308)--(1.8058,0.8595)--(1.7919,0.8881)--(1.7776,0.9164)--(1.7629,0.9445)--(1.7476,0.9723)--(1.7320,1.0000)--(1.7159,1.0273)--(1.6994,1.0544)--(1.6825,1.0812)--(1.6651,1.1078)--(1.6473,1.1341)--(1.6291,1.1601)--(1.6105,1.1858)--(1.5915,1.2112)--(1.5721,1.2363)--(1.5522,1.2611)--(1.5320,1.2855)--(1.5114,1.3097)--(1.4905,1.3335)--(1.4691,1.3570)--(1.4474,1.3801)--(1.4253,1.4029)--(1.4029,1.4253)--(1.3801,1.4474)--(1.3570,1.4691)--(1.3335,1.4905)--(1.3097,1.5114)--(1.2855,1.5320)--(1.2611,1.5522)--(1.2363,1.5721)--(1.2112,1.5915)--(1.1858,1.6105)--(1.1601,1.6291)--(1.1341,1.6473)--(1.1078,1.6651)--(1.0812,1.6825)--(1.0544,1.6994)--(1.0273,1.7159)--(1.0000,1.7320)--(0.9723,1.7476)--(0.9445,1.7629)--(0.9164,1.7776)--(0.8881,1.7919)--(0.8595,1.8058)--(0.8308,1.8192)--(0.8018,1.8322)--(0.7726,1.8447)--(0.7433,1.8567)--(0.7137,1.8682)--(0.6840,1.8793)--(0.6541,1.8900)--(0.6240,1.9001)--(0.5938,1.9098)--(0.5634,1.9189)--(0.5329,1.9276)--(0.5022,1.9358)--(0.4715,1.9436)--(0.4406,1.9508)--(0.4096,1.9576)--(0.3785,1.9638)--(0.3472,1.9696)--(0.3160,1.9748)--(0.2846,1.9796)--(0.2531,1.9839)--(0.2216,1.9876)--(0.1901,1.9909)--(0.1584,1.9937)--(0.1268,1.9959)--(0.0951,1.9977)--(0.0634,1.9989)--(0.0317,1.9997)--(0.0000,2.0000);
+\draw [] (1.7320,1.0000) -- (1.7320,2.0000);
+\draw [color=red] (0.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=red] (2.0000,2.0000) -- (2.0000,0.0000);
 \draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.7320,2.0000) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -84,25 +84,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [style=dotted] (0,0) -- (1.64,1.15);
-\draw [style=dotted] (0,0) -- (1.15,1.64);
-\fill [color=lightgray] (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,0) --  cycle;
-\draw [color=red] (0,2.00) -- (2.00,2.00);
-\draw [color=red] (2.00,2.00) -- (2.00,0);
-\draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
-\draw [] (1.64,1.15) -- (2.00,1.40);
-\draw []  (1.6383,1.1472) node [rotate=0] {$\bullet$};
+\draw [style=dotted] (0.0000,0.0000) -- (1.6383,1.1471);
+\draw [style=dotted] (0.0000,0.0000) -- (1.1471,1.6383);
+\fill [color=lightgray] (2.0000,0.0000) -- (1.9997,0.0317) -- (1.9989,0.0634) -- (1.9977,0.0951) -- (1.9959,0.1268) -- (1.9937,0.1584) -- (1.9909,0.1901) -- (1.9876,0.2216) -- (1.9839,0.2531) -- (1.9796,0.2846) -- (1.9748,0.3160) -- (1.9696,0.3472) -- (1.9638,0.3785) -- (1.9576,0.4096) -- (1.9508,0.4406) -- (1.9436,0.4715) -- (1.9358,0.5022) -- (1.9276,0.5329) -- (1.9189,0.5634) -- (1.9098,0.5938) -- (1.9001,0.6240) -- (1.8900,0.6541) -- (1.8793,0.6840) -- (1.8682,0.7137) -- (1.8567,0.7433) -- (1.8447,0.7726) -- (1.8322,0.8018) -- (1.8192,0.8308) -- (1.8058,0.8595) -- (1.7919,0.8881) -- (1.7776,0.9164) -- (1.7629,0.9445) -- (1.7476,0.9723) -- (1.7320,1.0000) -- (1.7159,1.0273) -- (1.6994,1.0544) -- (1.6825,1.0812) -- (1.6651,1.1078) -- (1.6473,1.1341) -- (1.6291,1.1601) -- (1.6105,1.1858) -- (1.5915,1.2112) -- (1.5721,1.2363) -- (1.5522,1.2611) -- (1.5320,1.2855) -- (1.5114,1.3097) -- (1.4905,1.3335) -- (1.4691,1.3570) -- (1.4474,1.3801) -- (1.4253,1.4029) -- (1.4029,1.4253) -- (1.3801,1.4474) -- (1.3570,1.4691) -- (1.3335,1.4905) -- (1.3097,1.5114) -- (1.2855,1.5320) -- (1.2611,1.5522) -- (1.2363,1.5721) -- (1.2112,1.5915) -- (1.1858,1.6105) -- (1.1601,1.6291) -- (1.1341,1.6473) -- (1.1078,1.6651) -- (1.0812,1.6825) -- (1.0544,1.6994) -- (1.0273,1.7159) -- (1.0000,1.7320) -- (0.9723,1.7476) -- (0.9445,1.7629) -- (0.9164,1.7776) -- (0.8881,1.7919) -- (0.8595,1.8058) -- (0.8308,1.8192) -- (0.8018,1.8322) -- (0.7726,1.8447) -- (0.7433,1.8567) -- (0.7137,1.8682) -- (0.6840,1.8793) -- (0.6541,1.8900) -- (0.6240,1.9001) -- (0.5938,1.9098) -- (0.5634,1.9189) -- (0.5329,1.9276) -- (0.5022,1.9358) -- (0.4715,1.9436) -- (0.4406,1.9508) -- (0.4096,1.9576) -- (0.3785,1.9638) -- (0.3472,1.9696) -- (0.3160,1.9748) -- (0.2846,1.9796) -- (0.2531,1.9839) -- (0.2216,1.9876) -- (0.1901,1.9909) -- (0.1584,1.9937) -- (0.1268,1.9959) -- (0.0951,1.9977) -- (0.0634,1.9989) -- (0.0317,1.9997) -- (0.0000,2.0000) -- (0.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,0.0000) --  cycle;
+\draw [color=red] (0.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=red] (2.0000,2.0000) -- (2.0000,0.0000);
+\draw [color=red] (2.0000,0.0000)--(1.9997,0.0317)--(1.9989,0.0634)--(1.9977,0.0951)--(1.9959,0.1268)--(1.9937,0.1584)--(1.9909,0.1901)--(1.9876,0.2216)--(1.9839,0.2531)--(1.9796,0.2846)--(1.9748,0.3160)--(1.9696,0.3472)--(1.9638,0.3785)--(1.9576,0.4096)--(1.9508,0.4406)--(1.9436,0.4715)--(1.9358,0.5022)--(1.9276,0.5329)--(1.9189,0.5634)--(1.9098,0.5938)--(1.9001,0.6240)--(1.8900,0.6541)--(1.8793,0.6840)--(1.8682,0.7137)--(1.8567,0.7433)--(1.8447,0.7726)--(1.8322,0.8018)--(1.8192,0.8308)--(1.8058,0.8595)--(1.7919,0.8881)--(1.7776,0.9164)--(1.7629,0.9445)--(1.7476,0.9723)--(1.7320,1.0000)--(1.7159,1.0273)--(1.6994,1.0544)--(1.6825,1.0812)--(1.6651,1.1078)--(1.6473,1.1341)--(1.6291,1.1601)--(1.6105,1.1858)--(1.5915,1.2112)--(1.5721,1.2363)--(1.5522,1.2611)--(1.5320,1.2855)--(1.5114,1.3097)--(1.4905,1.3335)--(1.4691,1.3570)--(1.4474,1.3801)--(1.4253,1.4029)--(1.4029,1.4253)--(1.3801,1.4474)--(1.3570,1.4691)--(1.3335,1.4905)--(1.3097,1.5114)--(1.2855,1.5320)--(1.2611,1.5522)--(1.2363,1.5721)--(1.2112,1.5915)--(1.1858,1.6105)--(1.1601,1.6291)--(1.1341,1.6473)--(1.1078,1.6651)--(1.0812,1.6825)--(1.0544,1.6994)--(1.0273,1.7159)--(1.0000,1.7320)--(0.9723,1.7476)--(0.9445,1.7629)--(0.9164,1.7776)--(0.8881,1.7919)--(0.8595,1.8058)--(0.8308,1.8192)--(0.8018,1.8322)--(0.7726,1.8447)--(0.7433,1.8567)--(0.7137,1.8682)--(0.6840,1.8793)--(0.6541,1.8900)--(0.6240,1.9001)--(0.5938,1.9098)--(0.5634,1.9189)--(0.5329,1.9276)--(0.5022,1.9358)--(0.4715,1.9436)--(0.4406,1.9508)--(0.4096,1.9576)--(0.3785,1.9638)--(0.3472,1.9696)--(0.3160,1.9748)--(0.2846,1.9796)--(0.2531,1.9839)--(0.2216,1.9876)--(0.1901,1.9909)--(0.1584,1.9937)--(0.1268,1.9959)--(0.0951,1.9977)--(0.0634,1.9989)--(0.0317,1.9997)--(0.0000,2.0000);
+\draw [] (1.6383,1.1471) -- (2.0000,1.4004);
+\draw []  (1.6383,1.1471) node [rotate=0] {$\bullet$};
 \draw []  (2.0000,1.4004) node [rotate=0] {$\bullet$};
-\draw []  (1.1472,1.6383) node [rotate=0] {$\bullet$};
+\draw []  (1.1471,1.6383) node [rotate=0] {$\bullet$};
 \draw []  (1.4004,2.0000) node [rotate=0] {$\bullet$};
-\draw [] (1.15,1.64) -- (1.40,2.00);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [] (1.1471,1.6383) -- (1.4004,2.0000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_IntegraleSimple.pstricks b/auto/pictures_tex/Fig_IntegraleSimple.pstricks
index 50af4c229..0d91ab0cb 100644
--- a/auto/pictures_tex/Fig_IntegraleSimple.pstricks
+++ b/auto/pictures_tex/Fig_IntegraleSimple.pstricks
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4994);
 %DEFAULT
-\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
-\draw (-1.5708,-0.27858) node {$a$};
-\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
-\draw (6.2832,-0.32674) node {$b$};
+\draw []  (-1.5707,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5707,-0.2785) node {$a$};
+\draw []  (6.2831,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.2831,-0.3267) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [style=dashed] (-1.5707,0.0000) -- (-1.5707,1.0000);
+\draw [style=dashed] (6.2831,2.0000) -- (6.2831,0.0000);
 
-\draw [color=blue] (-1.571,1.000)--(-1.491,1.003)--(-1.412,1.013)--(-1.333,1.028)--(-1.253,1.050)--(-1.174,1.078)--(-1.095,1.111)--(-1.015,1.150)--(-0.9361,1.195)--(-0.8568,1.244)--(-0.7775,1.299)--(-0.6981,1.357)--(-0.6188,1.420)--(-0.5395,1.486)--(-0.4601,1.556)--(-0.3808,1.628)--(-0.3015,1.703)--(-0.2221,1.780)--(-0.1428,1.858)--(-0.06347,1.937)--(0.01587,2.016)--(0.09520,2.095)--(0.1745,2.174)--(0.2539,2.251)--(0.3332,2.327)--(0.4125,2.401)--(0.4919,2.472)--(0.5712,2.541)--(0.6505,2.606)--(0.7299,2.667)--(0.8092,2.724)--(0.8885,2.776)--(0.9679,2.824)--(1.047,2.866)--(1.127,2.903)--(1.206,2.934)--(1.285,2.960)--(1.365,2.979)--(1.444,2.992)--(1.523,2.999)--(1.603,3.000)--(1.682,2.994)--(1.761,2.982)--(1.841,2.964)--(1.920,2.940)--(1.999,2.910)--(2.079,2.874)--(2.158,2.833)--(2.237,2.786)--(2.317,2.735)--(2.396,2.678)--(2.475,2.618)--(2.555,2.554)--(2.634,2.486)--(2.713,2.415)--(2.793,2.342)--(2.872,2.266)--(2.951,2.189)--(3.031,2.111)--(3.110,2.032)--(3.189,1.952)--(3.269,1.873)--(3.348,1.795)--(3.427,1.718)--(3.507,1.643)--(3.586,1.570)--(3.665,1.500)--(3.745,1.433)--(3.824,1.369)--(3.903,1.310)--(3.983,1.255)--(4.062,1.204)--(4.141,1.159)--(4.221,1.119)--(4.300,1.084)--(4.379,1.055)--(4.458,1.032)--(4.538,1.015)--(4.617,1.005)--(4.697,1.000)--(4.776,1.002)--(4.855,1.010)--(4.935,1.025)--(5.014,1.045)--(5.093,1.072)--(5.173,1.104)--(5.252,1.142)--(5.331,1.185)--(5.411,1.234)--(5.490,1.287)--(5.569,1.345)--(5.648,1.407)--(5.728,1.473)--(5.807,1.542)--(5.887,1.614)--(5.966,1.688)--(6.045,1.764)--(6.125,1.842)--(6.204,1.921)--(6.283,2.000);
+\draw [color=blue] (-1.5707,1.0000)--(-1.4914,1.0031)--(-1.4121,1.0125)--(-1.3327,1.0281)--(-1.2534,1.0499)--(-1.1741,1.0776)--(-1.0947,1.1111)--(-1.0154,1.1502)--(-0.9361,1.1947)--(-0.8567,1.2442)--(-0.7774,1.2985)--(-0.6981,1.3572)--(-0.6187,1.4199)--(-0.5394,1.4863)--(-0.4601,1.5559)--(-0.3807,1.6283)--(-0.3014,1.7030)--(-0.2221,1.7796)--(-0.1427,1.8576)--(-0.0634,1.9365)--(0.0158,2.0158)--(0.0951,2.0950)--(0.1745,2.1736)--(0.2538,2.2511)--(0.3331,2.3270)--(0.4125,2.4009)--(0.4918,2.4722)--(0.5711,2.5406)--(0.6505,2.6056)--(0.7298,2.6667)--(0.8091,2.7237)--(0.8885,2.7761)--(0.9678,2.8236)--(1.0471,2.8660)--(1.1265,2.9029)--(1.2058,2.9341)--(1.2851,2.9594)--(1.3645,2.9788)--(1.4438,2.9919)--(1.5231,2.9988)--(1.6025,2.9994)--(1.6818,2.9938)--(1.7611,2.9819)--(1.8405,2.9638)--(1.9198,2.9396)--(1.9991,2.9096)--(2.0785,2.8738)--(2.1578,2.8325)--(2.2371,2.7860)--(2.3165,2.7345)--(2.3958,2.6785)--(2.4751,2.6181)--(2.5545,2.5539)--(2.6338,2.4861)--(2.7131,2.4154)--(2.7925,2.3420)--(2.8718,2.2664)--(2.9511,2.1892)--(3.0305,2.1108)--(3.1098,2.0317)--(3.1891,1.9524)--(3.2685,1.8734)--(3.3478,1.7951)--(3.4271,1.7182)--(3.5065,1.6431)--(3.5858,1.5702)--(3.6651,1.5000)--(3.7445,1.4329)--(3.8238,1.3694)--(3.9031,1.3099)--(3.9825,1.2547)--(4.0618,1.2042)--(4.1411,1.1587)--(4.2205,1.1185)--(4.2998,1.0838)--(4.3791,1.0549)--(4.4585,1.0320)--(4.5378,1.0151)--(4.6171,1.0045)--(4.6965,1.0001)--(4.7758,1.0020)--(4.8551,1.0101)--(4.9345,1.0245)--(5.0138,1.0450)--(5.0931,1.0716)--(5.1725,1.1040)--(5.2518,1.1420)--(5.3311,1.1854)--(5.4105,1.2339)--(5.4898,1.2873)--(5.5691,1.3451)--(5.6485,1.4070)--(5.7278,1.4727)--(5.8071,1.5417)--(5.8865,1.6136)--(5.9658,1.6879)--(6.0451,1.7642)--(6.1245,1.8419)--(6.2038,1.9207)--(6.2831,2.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_IntervalleUn.pstricks b/auto/pictures_tex/Fig_IntervalleUn.pstricks
index 27457d70b..e9032c9e1 100644
--- a/auto/pictures_tex/Fig_IntervalleUn.pstricks
+++ b/auto/pictures_tex/Fig_IntervalleUn.pstricks
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-1.50,0) -- (4.50,0);
-\draw []  (0.90000,0) node [rotate=0] {$\bullet$};
-\draw (0.90000,-0.37858) node {$a$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.41492) node {$0$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,-0.41492) node {$1$};
-\draw [,->,>=latex] (0.45000,0.30000) -- (0,0.30000);
-\draw [,->,>=latex] (0.45000,0.30000) -- (0.90000,0.30000);
-\draw (0.45000,0.67858) node {$a$};
-\draw [,->,>=latex] (1.9500,0.30000) -- (0.90000,0.30000);
-\draw [,->,>=latex] (1.9500,0.30000) -- (3.0000,0.30000);
-\draw (1.9500,0.72983) node {$1-a$};
+\draw [] (-1.5000,0.0000) -- (4.5000,0.0000);
+\draw []  (0.9000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.9000,-0.3785) node {$a$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.4149) node {$0$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.4149) node {$1$};
+\draw [,->,>=latex] (0.4500,0.3000) -- (0.0000,0.3000);
+\draw [,->,>=latex] (0.4500,0.3000) -- (0.9000,0.3000);
+\draw (0.4500,0.6785) node {$a$};
+\draw [,->,>=latex] (1.9500,0.3000) -- (0.9000,0.3000);
+\draw [,->,>=latex] (1.9500,0.3000) -- (3.0000,0.3000);
+\draw (1.9500,0.7298) node {$1-a$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_IsomCarre.pstricks b/auto/pictures_tex/Fig_IsomCarre.pstricks
index 52c75fa0f..6d9487b98 100644
--- a/auto/pictures_tex/Fig_IsomCarre.pstricks
+++ b/auto/pictures_tex/Fig_IsomCarre.pstricks
@@ -85,20 +85,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue] (-1.00,-1.00) -- (1.00,-1.00);
-\draw [color=blue] (1.00,-1.00) -- (1.00,1.00);
-\draw [color=blue] (1.00,1.00) -- (-1.00,1.00);
-\draw [color=blue] (-1.00,1.00) -- (-1.00,-1.00);
-\draw [] (0,-1.50) -- (0,1.50);
+\draw [color=blue] (-1.0000,-1.0000) -- (1.0000,-1.0000);
+\draw [color=blue] (1.0000,-1.0000) -- (1.0000,1.0000);
+\draw [color=blue] (1.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue] (-1.0000,1.0000) -- (-1.0000,-1.0000);
+\draw [] (0.0000,-1.5000) -- (0.0000,1.5000);
 \draw []  (-1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.2076,1.1954) node {\( A\)};
+\draw (-1.2075,1.1954) node {\( A\)};
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.2183,1.1954) node {\( B\)};
+\draw (1.2182,1.1954) node {\( B\)};
 \draw []  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
 \draw (1.2142,-1.1954) node {\( C\)};
 \draw []  (-1.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.2269,-1.1954) node {\( D\)};
-\draw (0.15626,1.6493) node {\( s\)};
+\draw (-1.2268,-1.1954) node {\( D\)};
+\draw (0.1562,1.6492) node {\( s\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_JGuKEjH.pstricks b/auto/pictures_tex/Fig_JGuKEjH.pstricks
index a14c0796d..3f97fb0e7 100644
--- a/auto/pictures_tex/Fig_JGuKEjH.pstricks
+++ b/auto/pictures_tex/Fig_JGuKEjH.pstricks
@@ -63,21 +63,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
          hatchthickness/.code={\setlength{\hatchthickness}{#1}}}
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,0) -- (-0.643,0.766) -- (-0.643,0.766) -- (-0.669,0.743) -- (-0.695,0.719) -- (-0.719,0.695) -- (-0.743,0.669) -- (-0.766,0.643) -- (-0.788,0.616) -- (-0.809,0.588) -- (-0.829,0.559) -- (-0.848,0.530) -- (-0.866,0.500) -- (-0.883,0.470) -- (-0.899,0.438) -- (-0.914,0.407) -- (-0.927,0.374) -- (-0.940,0.342) -- (-0.951,0.309) -- (-0.961,0.276) -- (-0.970,0.242) -- (-0.978,0.208) -- (-0.985,0.174) -- (-0.990,0.139) -- (-0.995,0.105) -- (-0.998,0.0698) -- (-0.999,0.0349) -- (-1.00,0) -- (-0.999,-0.0349) -- (-0.998,-0.0698) -- (-0.995,-0.105) -- (-0.990,-0.139) -- (-0.985,-0.174) -- (-0.978,-0.208) -- (-0.970,-0.242) -- (-0.961,-0.276) -- (-0.951,-0.309) -- (-0.940,-0.342) -- (-0.927,-0.374) -- (-0.914,-0.407) -- (-0.899,-0.438) -- (-0.883,-0.469) -- (-0.866,-0.500) -- (-0.848,-0.530) -- (-0.829,-0.559) -- (-0.809,-0.588) -- (-0.788,-0.616) -- (-0.766,-0.643) -- (-0.743,-0.669) -- (-0.719,-0.695) -- (-0.695,-0.719) -- (-0.669,-0.743) -- (1.00,0) -- (-0.643,-0.766) --  cycle;
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw [color=lightgray] (1.00,0) -- (-0.643,0.766);
-\draw [color=lightgray] (1.00,0) -- (-0.643,-0.766);
+\fill [color=lightgray,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,0.0000) -- (-0.6427,0.7660) -- (-0.6427,0.7660) -- (-0.6691,0.7431) -- (-0.6946,0.7193) -- (-0.7193,0.6946) -- (-0.7431,0.6691) -- (-0.7660,0.6427) -- (-0.7880,0.6156) -- (-0.8090,0.5877) -- (-0.8290,0.5591) -- (-0.8480,0.5299) -- (-0.8660,0.5000) -- (-0.8829,0.4694) -- (-0.8987,0.4383) -- (-0.9135,0.4067) -- (-0.9271,0.3746) -- (-0.9396,0.3420) -- (-0.9510,0.3090) -- (-0.9612,0.2756) -- (-0.9702,0.2419) -- (-0.9781,0.2079) -- (-0.9848,0.1736) -- (-0.9902,0.1391) -- (-0.9945,0.1045) -- (-0.9975,0.0697) -- (-0.9993,0.0348) -- (-1.0000,0.0000) -- (-0.9993,-0.0348) -- (-0.9975,-0.0697) -- (-0.9945,-0.1045) -- (-0.9902,-0.1391) -- (-0.9848,-0.1736) -- (-0.9781,-0.2079) -- (-0.9702,-0.2419) -- (-0.9612,-0.2756) -- (-0.9510,-0.3090) -- (-0.9396,-0.3420) -- (-0.9271,-0.3746) -- (-0.9135,-0.4067) -- (-0.8987,-0.4383) -- (-0.8829,-0.4694) -- (-0.8660,-0.5000) -- (-0.8480,-0.5299) -- (-0.8290,-0.5591) -- (-0.8090,-0.5877) -- (-0.7880,-0.6156) -- (-0.7660,-0.6427) -- (-0.7431,-0.6691) -- (-0.7193,-0.6946) -- (-0.6946,-0.7193) -- (-0.6691,-0.7431) -- (1.0000,0.0000) -- (-0.6427,-0.7660) --  cycle;
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=lightgray] (1.0000,0.0000) -- (-0.6427,0.7660);
+\draw [color=lightgray] (1.0000,0.0000) -- (-0.6427,-0.7660);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_JJAooWpimYW.pstricks b/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
index 9f0dd8da9..79ade3f77 100644
--- a/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
+++ b/auto/pictures_tex/Fig_JJAooWpimYW.pstricks
@@ -108,35 +108,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
-\draw [,->,>=latex] (0,-1.4989) -- (0,1.4999);
+\draw [,->,>=latex] (-8.3539,0.0000) -- (8.3539,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4988) -- (0.0000,1.4998);
 %DEFAULT
 
-\draw [color=blue] (-7.854,0)--(-7.695,0.1580)--(-7.537,0.3120)--(-7.378,0.4582)--(-7.219,0.5929)--(-7.061,0.7127)--(-6.902,0.8146)--(-6.743,0.8960)--(-6.585,0.9549)--(-6.426,0.9898)--(-6.267,0.9999)--(-6.109,0.9848)--(-5.950,0.9450)--(-5.791,0.8815)--(-5.633,0.7958)--(-5.474,0.6901)--(-5.315,0.5671)--(-5.157,0.4298)--(-4.998,0.2817)--(-4.839,0.1266)--(-4.681,-0.03173)--(-4.522,-0.1893)--(-4.363,-0.3420)--(-4.205,-0.4862)--(-4.046,-0.6182)--(-3.887,-0.7346)--(-3.729,-0.8326)--(-3.570,-0.9096)--(-3.411,-0.9638)--(-3.253,-0.9938)--(-3.094,-0.9989)--(-2.935,-0.9788)--(-2.777,-0.9342)--(-2.618,-0.8660)--(-2.459,-0.7761)--(-2.301,-0.6668)--(-2.142,-0.5406)--(-1.983,-0.4009)--(-1.825,-0.2511)--(-1.666,-0.09506)--(-1.507,0.06342)--(-1.349,0.2203)--(-1.190,0.3717)--(-1.031,0.5137)--(-0.8727,0.6428)--(-0.7140,0.7558)--(-0.5553,0.8497)--(-0.3967,0.9224)--(-0.2380,0.9718)--(-0.07933,0.9969)--(0.07933,0.9969)--(0.2380,0.9718)--(0.3967,0.9224)--(0.5553,0.8497)--(0.7140,0.7558)--(0.8727,0.6428)--(1.031,0.5137)--(1.190,0.3717)--(1.349,0.2203)--(1.507,0.06342)--(1.666,-0.09506)--(1.825,-0.2511)--(1.983,-0.4009)--(2.142,-0.5406)--(2.301,-0.6668)--(2.459,-0.7761)--(2.618,-0.8660)--(2.777,-0.9342)--(2.935,-0.9788)--(3.094,-0.9989)--(3.253,-0.9938)--(3.411,-0.9638)--(3.570,-0.9096)--(3.729,-0.8326)--(3.887,-0.7346)--(4.046,-0.6182)--(4.205,-0.4862)--(4.363,-0.3420)--(4.522,-0.1893)--(4.681,-0.03173)--(4.839,0.1266)--(4.998,0.2817)--(5.157,0.4298)--(5.315,0.5671)--(5.474,0.6901)--(5.633,0.7958)--(5.791,0.8815)--(5.950,0.9450)--(6.109,0.9848)--(6.267,0.9999)--(6.426,0.9898)--(6.585,0.9549)--(6.743,0.8960)--(6.902,0.8146)--(7.061,0.7127)--(7.219,0.5929)--(7.378,0.4582)--(7.537,0.3120)--(7.695,0.1580)--(7.854,0);
-\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
-\draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
-\draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-7.8539,0.0000)--(-7.6953,0.1580)--(-7.5366,0.3120)--(-7.3779,0.4582)--(-7.2193,0.5929)--(-7.0606,0.7126)--(-6.9019,0.8145)--(-6.7433,0.8959)--(-6.5846,0.9549)--(-6.4259,0.9898)--(-6.2673,0.9998)--(-6.1086,0.9848)--(-5.9499,0.9450)--(-5.7913,0.8814)--(-5.6326,0.7957)--(-5.4739,0.6900)--(-5.3153,0.5670)--(-5.1566,0.4297)--(-4.9979,0.2817)--(-4.8393,0.1265)--(-4.6806,-0.0317)--(-4.5219,-0.1892)--(-4.3633,-0.3420)--(-4.2046,-0.4861)--(-4.0459,-0.6181)--(-3.8873,-0.7345)--(-3.7286,-0.8325)--(-3.5699,-0.9096)--(-3.4113,-0.9638)--(-3.2526,-0.9938)--(-3.0939,-0.9988)--(-2.9353,-0.9788)--(-2.7766,-0.9341)--(-2.6179,-0.8660)--(-2.4593,-0.7761)--(-2.3006,-0.6667)--(-2.1419,-0.5406)--(-1.9833,-0.4009)--(-1.8246,-0.2511)--(-1.6659,-0.0950)--(-1.5073,0.0634)--(-1.3486,0.2203)--(-1.1899,0.3716)--(-1.0313,0.5136)--(-0.8726,0.6427)--(-0.7139,0.7557)--(-0.5553,0.8497)--(-0.3966,0.9223)--(-0.2379,0.9718)--(-0.0793,0.9968)--(0.0793,0.9968)--(0.2379,0.9718)--(0.3966,0.9223)--(0.5553,0.8497)--(0.7139,0.7557)--(0.8726,0.6427)--(1.0313,0.5136)--(1.1899,0.3716)--(1.3486,0.2203)--(1.5073,0.0634)--(1.6659,-0.0950)--(1.8246,-0.2511)--(1.9833,-0.4009)--(2.1419,-0.5406)--(2.3006,-0.6667)--(2.4593,-0.7761)--(2.6179,-0.8660)--(2.7766,-0.9341)--(2.9353,-0.9788)--(3.0939,-0.9988)--(3.2526,-0.9938)--(3.4113,-0.9638)--(3.5699,-0.9096)--(3.7286,-0.8325)--(3.8873,-0.7345)--(4.0459,-0.6181)--(4.2046,-0.4861)--(4.3633,-0.3420)--(4.5219,-0.1892)--(4.6806,-0.0317)--(4.8393,0.1265)--(4.9979,0.2817)--(5.1566,0.4297)--(5.3153,0.5670)--(5.4739,0.6900)--(5.6326,0.7957)--(5.7913,0.8814)--(5.9499,0.9450)--(6.1086,0.9848)--(6.2673,0.9998)--(6.4259,0.9898)--(6.5846,0.9549)--(6.7433,0.8959)--(6.9019,0.8145)--(7.0606,0.7126)--(7.2193,0.5929)--(7.3779,0.4582)--(7.5366,0.3120)--(7.6953,0.1580)--(7.8539,0.0000);
+\draw (-7.8539,-0.4207) node {$ -\frac{5}{2} \, \pi $};
+\draw [] (-7.8539,-0.1000) -- (-7.8539,0.1000);
+\draw (-6.2831,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-6.2831,-0.1000) -- (-6.2831,0.1000);
+\draw (-4.7123,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-4.7123,-0.1000) -- (-4.7123,0.1000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (7.8539,-0.4207) node {$ \frac{5}{2} \, \pi $};
+\draw [] (7.8539,-0.1000) -- (7.8539,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks b/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
index ab72b83d5..e2b94607d 100644
--- a/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
+++ b/auto/pictures_tex/Fig_JSLooFJWXtB.pstricks
@@ -92,8 +92,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.2124,0) -- (5.2124,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-5.2123,0.0000) -- (5.2123,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -102,11 +102,11 @@
 % setting the default values
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+\draw [] (-3.1415,0.0000) -- (-3.1415,0.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -114,11 +114,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -126,11 +126,11 @@
 % setting the default values
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          hatchthickness=0.4pt}
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -138,29 +138,29 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (3.1415,0.0000) -- (3.1415,0.0000);
+\draw [] (4.7123,-1.0000) -- (4.7123,0.0000);
 
-\draw [color=blue] (-4.712,1.000)--(-4.617,0.9955)--(-4.522,0.9819)--(-4.427,0.9595)--(-4.332,0.9284)--(-4.236,0.8888)--(-4.141,0.8413)--(-4.046,0.7861)--(-3.951,0.7237)--(-3.856,0.6549)--(-3.760,0.5801)--(-3.665,0.5000)--(-3.570,0.4154)--(-3.475,0.3271)--(-3.380,0.2358)--(-3.284,0.1423)--(-3.189,0.04758)--(-3.094,-0.04758)--(-2.999,-0.1423)--(-2.904,-0.2358)--(-2.808,-0.3271)--(-2.713,-0.4154)--(-2.618,-0.5000)--(-2.523,-0.5801)--(-2.428,-0.6549)--(-2.332,-0.7237)--(-2.237,-0.7861)--(-2.142,-0.8413)--(-2.047,-0.8888)--(-1.952,-0.9284)--(-1.856,-0.9595)--(-1.761,-0.9819)--(-1.666,-0.9955)--(-1.571,-1.000)--(-1.476,-0.9955)--(-1.380,-0.9819)--(-1.285,-0.9595)--(-1.190,-0.9284)--(-1.095,-0.8888)--(-0.9996,-0.8413)--(-0.9044,-0.7861)--(-0.8092,-0.7237)--(-0.7140,-0.6549)--(-0.6188,-0.5801)--(-0.5236,-0.5000)--(-0.4284,-0.4154)--(-0.3332,-0.3271)--(-0.2380,-0.2358)--(-0.1428,-0.1423)--(-0.04760,-0.04758)--(0.04760,0.04758)--(0.1428,0.1423)--(0.2380,0.2358)--(0.3332,0.3271)--(0.4284,0.4154)--(0.5236,0.5000)--(0.6188,0.5801)--(0.7140,0.6549)--(0.8092,0.7237)--(0.9044,0.7861)--(0.9996,0.8413)--(1.095,0.8888)--(1.190,0.9284)--(1.285,0.9595)--(1.380,0.9819)--(1.476,0.9955)--(1.571,1.000)--(1.666,0.9955)--(1.761,0.9819)--(1.856,0.9595)--(1.952,0.9284)--(2.047,0.8888)--(2.142,0.8413)--(2.237,0.7861)--(2.332,0.7237)--(2.428,0.6549)--(2.523,0.5801)--(2.618,0.5000)--(2.713,0.4154)--(2.808,0.3271)--(2.904,0.2358)--(2.999,0.1423)--(3.094,0.04758)--(3.189,-0.04758)--(3.284,-0.1423)--(3.380,-0.2358)--(3.475,-0.3271)--(3.570,-0.4154)--(3.665,-0.5000)--(3.760,-0.5801)--(3.856,-0.6549)--(3.951,-0.7237)--(4.046,-0.7861)--(4.141,-0.8413)--(4.236,-0.8888)--(4.332,-0.9284)--(4.427,-0.9595)--(4.522,-0.9819)--(4.617,-0.9955)--(4.712,-1.000);
-\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
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+\draw (-4.7123,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-4.7123,-0.1000) -- (-4.7123,0.1000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks b/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
index 649f91548..66eb5ab83 100644
--- a/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
+++ b/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
@@ -83,21 +83,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 \draw (1.5233,2.0000) node {$(x,y)$};
-\draw [,->,>=latex] (0,0) -- (1.0000,2.0000);
-\draw (0.68452,0.41391) node {$\theta$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (1.0000,2.0000);
+\draw (0.6845,0.4139) node {$\theta$};
 
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-\draw (0.23371,1.1680) node {$r$};
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-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
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+\draw (0.2337,1.1680) node {$r$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks b/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks
index 18f70b22a..abc3a7474 100644
--- a/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks
+++ b/auto/pictures_tex/Fig_KGQXooZFNVnW.pstricks
@@ -95,43 +95,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.6622) -- (0.0000,3.6622);
 %DEFAULT
 
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+\draw [color=red] (-3.0000,3.1622)--(-2.9393,3.1048)--(-2.8787,3.0475)--(-2.8181,2.9903)--(-2.7575,2.9332)--(-2.6969,2.8763)--(-2.6363,2.8196)--(-2.5757,2.7630)--(-2.5151,2.7066)--(-2.4545,2.6504)--(-2.3939,2.5944)--(-2.3333,2.5385)--(-2.2727,2.4830)--(-2.2121,2.4276)--(-2.1515,2.3725)--(-2.0909,2.3177)--(-2.0303,2.2632)--(-1.9696,2.2090)--(-1.9090,2.1551)--(-1.8484,2.1016)--(-1.7878,2.0485)--(-1.7272,1.9958)--(-1.6666,1.9436)--(-1.6060,1.8919)--(-1.5454,1.8407)--(-1.4848,1.7901)--(-1.4242,1.7402)--(-1.3636,1.6910)--(-1.3030,1.6425)--(-1.2424,1.5948)--(-1.1818,1.5481)--(-1.1212,1.5023)--(-1.0606,1.4576)--(-1.0000,1.4142)--(-0.9393,1.3720)--(-0.8787,1.3312)--(-0.8181,1.2920)--(-0.7575,1.2545)--(-0.6969,1.2189)--(-0.6363,1.1853)--(-0.5757,1.1539)--(-0.5151,1.1248)--(-0.4545,1.0984)--(-0.3939,1.0747)--(-0.3333,1.0540)--(-0.2727,1.0365)--(-0.2121,1.0222)--(-0.1515,1.0114)--(-0.0909,1.0041)--(-0.0303,1.0004)--(0.0303,1.0004)--(0.0909,1.0041)--(0.1515,1.0114)--(0.2121,1.0222)--(0.2727,1.0365)--(0.3333,1.0540)--(0.3939,1.0747)--(0.4545,1.0984)--(0.5151,1.1248)--(0.5757,1.1539)--(0.6363,1.1853)--(0.6969,1.2189)--(0.7575,1.2545)--(0.8181,1.2920)--(0.8787,1.3312)--(0.9393,1.3720)--(1.0000,1.4142)--(1.0606,1.4576)--(1.1212,1.5023)--(1.1818,1.5481)--(1.2424,1.5948)--(1.3030,1.6425)--(1.3636,1.6910)--(1.4242,1.7402)--(1.4848,1.7901)--(1.5454,1.8407)--(1.6060,1.8919)--(1.6666,1.9436)--(1.7272,1.9958)--(1.7878,2.0485)--(1.8484,2.1016)--(1.9090,2.1551)--(1.9696,2.2090)--(2.0303,2.2632)--(2.0909,2.3177)--(2.1515,2.3725)--(2.2121,2.4276)--(2.2727,2.4830)--(2.3333,2.5385)--(2.3939,2.5944)--(2.4545,2.6504)--(2.5151,2.7066)--(2.5757,2.7630)--(2.6363,2.8196)--(2.6969,2.8763)--(2.7575,2.9332)--(2.8181,2.9903)--(2.8787,3.0475)--(2.9393,3.1048)--(3.0000,3.1622);
 
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-\draw [color=brown] (-3.00,3.00) -- (3.00,-3.00);
-\draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw (0.21069,0.80458) node {$R$};
-\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
-\draw (0.19314,-0.80458) node {$S$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (-3.0000,-3.1622)--(-2.9393,-3.1048)--(-2.8787,-3.0475)--(-2.8181,-2.9903)--(-2.7575,-2.9332)--(-2.6969,-2.8763)--(-2.6363,-2.8196)--(-2.5757,-2.7630)--(-2.5151,-2.7066)--(-2.4545,-2.6504)--(-2.3939,-2.5944)--(-2.3333,-2.5385)--(-2.2727,-2.4830)--(-2.2121,-2.4276)--(-2.1515,-2.3725)--(-2.0909,-2.3177)--(-2.0303,-2.2632)--(-1.9696,-2.2090)--(-1.9090,-2.1551)--(-1.8484,-2.1016)--(-1.7878,-2.0485)--(-1.7272,-1.9958)--(-1.6666,-1.9436)--(-1.6060,-1.8919)--(-1.5454,-1.8407)--(-1.4848,-1.7901)--(-1.4242,-1.7402)--(-1.3636,-1.6910)--(-1.3030,-1.6425)--(-1.2424,-1.5948)--(-1.1818,-1.5481)--(-1.1212,-1.5023)--(-1.0606,-1.4576)--(-1.0000,-1.4142)--(-0.9393,-1.3720)--(-0.8787,-1.3312)--(-0.8181,-1.2920)--(-0.7575,-1.2545)--(-0.6969,-1.2189)--(-0.6363,-1.1853)--(-0.5757,-1.1539)--(-0.5151,-1.1248)--(-0.4545,-1.0984)--(-0.3939,-1.0747)--(-0.3333,-1.0540)--(-0.2727,-1.0365)--(-0.2121,-1.0222)--(-0.1515,-1.0114)--(-0.0909,-1.0041)--(-0.0303,-1.0004)--(0.0303,-1.0004)--(0.0909,-1.0041)--(0.1515,-1.0114)--(0.2121,-1.0222)--(0.2727,-1.0365)--(0.3333,-1.0540)--(0.3939,-1.0747)--(0.4545,-1.0984)--(0.5151,-1.1248)--(0.5757,-1.1539)--(0.6363,-1.1853)--(0.6969,-1.2189)--(0.7575,-1.2545)--(0.8181,-1.2920)--(0.8787,-1.3312)--(0.9393,-1.3720)--(1.0000,-1.4142)--(1.0606,-1.4576)--(1.1212,-1.5023)--(1.1818,-1.5481)--(1.2424,-1.5948)--(1.3030,-1.6425)--(1.3636,-1.6910)--(1.4242,-1.7402)--(1.4848,-1.7901)--(1.5454,-1.8407)--(1.6060,-1.8919)--(1.6666,-1.9436)--(1.7272,-1.9958)--(1.7878,-2.0485)--(1.8484,-2.1016)--(1.9090,-2.1551)--(1.9696,-2.2090)--(2.0303,-2.2632)--(2.0909,-2.3177)--(2.1515,-2.3725)--(2.2121,-2.4276)--(2.2727,-2.4830)--(2.3333,-2.5385)--(2.3939,-2.5944)--(2.4545,-2.6504)--(2.5151,-2.7066)--(2.5757,-2.7630)--(2.6363,-2.8196)--(2.6969,-2.8763)--(2.7575,-2.9332)--(2.8181,-2.9903)--(2.8787,-3.0475)--(2.9393,-3.1048)--(3.0000,-3.1622);
+\draw [color=brown] (-3.0000,3.0000) -- (3.0000,-3.0000);
+\draw [color=brown] (-3.0000,-3.0000) -- (3.0000,3.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.2106,0.8045) node {$R$};
+\draw []  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.1931,-0.8045) node {$S$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks b/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks
index e54d5c080..9723d869d 100644
--- a/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks
+++ b/auto/pictures_tex/Fig_KKJAooubQzgBgP.pstricks
@@ -72,19 +72,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4487);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4486);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.03030,0.03029)--(0.06061,0.06049)--(0.09091,0.09054)--(0.1212,0.1203)--(0.1515,0.1498)--(0.1818,0.1789)--(0.2121,0.2075)--(0.2424,0.2356)--(0.2727,0.2631)--(0.3030,0.2900)--(0.3333,0.3162)--(0.3636,0.3417)--(0.3939,0.3665)--(0.4242,0.3905)--(0.4545,0.4138)--(0.4848,0.4363)--(0.5152,0.4580)--(0.5455,0.4789)--(0.5758,0.4990)--(0.6061,0.5183)--(0.6364,0.5369)--(0.6667,0.5547)--(0.6970,0.5718)--(0.7273,0.5882)--(0.7576,0.6039)--(0.7879,0.6189)--(0.8182,0.6332)--(0.8485,0.6470)--(0.8788,0.6601)--(0.9091,0.6727)--(0.9394,0.6847)--(0.9697,0.6961)--(1.000,0.7071)--(1.030,0.7176)--(1.061,0.7276)--(1.091,0.7372)--(1.121,0.7463)--(1.152,0.7550)--(1.182,0.7634)--(1.212,0.7714)--(1.242,0.7790)--(1.273,0.7863)--(1.303,0.7933)--(1.333,0.8000)--(1.364,0.8064)--(1.394,0.8125)--(1.424,0.8184)--(1.455,0.8240)--(1.485,0.8294)--(1.515,0.8346)--(1.545,0.8396)--(1.576,0.8443)--(1.606,0.8489)--(1.636,0.8533)--(1.667,0.8575)--(1.697,0.8615)--(1.727,0.8654)--(1.758,0.8692)--(1.788,0.8728)--(1.818,0.8762)--(1.848,0.8795)--(1.879,0.8827)--(1.909,0.8858)--(1.939,0.8888)--(1.970,0.8917)--(2.000,0.8944)--(2.030,0.8971)--(2.061,0.8997)--(2.091,0.9021)--(2.121,0.9045)--(2.152,0.9068)--(2.182,0.9091)--(2.212,0.9112)--(2.242,0.9133)--(2.273,0.9153)--(2.303,0.9173)--(2.333,0.9191)--(2.364,0.9210)--(2.394,0.9227)--(2.424,0.9244)--(2.455,0.9261)--(2.485,0.9277)--(2.515,0.9292)--(2.545,0.9307)--(2.576,0.9322)--(2.606,0.9336)--(2.636,0.9350)--(2.667,0.9363)--(2.697,0.9376)--(2.727,0.9389)--(2.758,0.9401)--(2.788,0.9413)--(2.818,0.9424)--(2.848,0.9435)--(2.879,0.9446)--(2.909,0.9457)--(2.939,0.9467)--(2.970,0.9477)--(3.000,0.9487);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0303,0.0302)--(0.0606,0.0604)--(0.0909,0.0905)--(0.1212,0.1203)--(0.1515,0.1498)--(0.1818,0.1788)--(0.2121,0.2075)--(0.2424,0.2356)--(0.2727,0.2631)--(0.3030,0.2900)--(0.3333,0.3162)--(0.3636,0.3417)--(0.3939,0.3665)--(0.4242,0.3905)--(0.4545,0.4138)--(0.4848,0.4362)--(0.5151,0.4579)--(0.5454,0.4788)--(0.5757,0.4989)--(0.6060,0.5183)--(0.6363,0.5368)--(0.6666,0.5547)--(0.6969,0.5717)--(0.7272,0.5881)--(0.7575,0.6038)--(0.7878,0.6188)--(0.8181,0.6332)--(0.8484,0.6469)--(0.8787,0.6601)--(0.9090,0.6726)--(0.9393,0.6846)--(0.9696,0.6961)--(1.0000,0.7071)--(1.0303,0.7175)--(1.0606,0.7275)--(1.0909,0.7371)--(1.1212,0.7462)--(1.1515,0.7550)--(1.1818,0.7633)--(1.2121,0.7713)--(1.2424,0.7790)--(1.2727,0.7863)--(1.3030,0.7933)--(1.3333,0.8000)--(1.3636,0.8064)--(1.3939,0.8125)--(1.4242,0.8184)--(1.4545,0.8240)--(1.4848,0.8294)--(1.5151,0.8346)--(1.5454,0.8395)--(1.5757,0.8443)--(1.6060,0.8488)--(1.6363,0.8532)--(1.6666,0.8574)--(1.6969,0.8615)--(1.7272,0.8654)--(1.7575,0.8691)--(1.7878,0.8727)--(1.8181,0.8762)--(1.8484,0.8795)--(1.8787,0.8827)--(1.9090,0.8858)--(1.9393,0.8888)--(1.9696,0.8916)--(2.0000,0.8944)--(2.0303,0.8970)--(2.0606,0.8996)--(2.0909,0.9021)--(2.1212,0.9045)--(2.1515,0.9068)--(2.1818,0.9090)--(2.2121,0.9112)--(2.2424,0.9133)--(2.2727,0.9153)--(2.3030,0.9172)--(2.3333,0.9191)--(2.3636,0.9209)--(2.3939,0.9227)--(2.4242,0.9244)--(2.4545,0.9260)--(2.4848,0.9276)--(2.5151,0.9292)--(2.5454,0.9307)--(2.5757,0.9322)--(2.6060,0.9336)--(2.6363,0.9349)--(2.6666,0.9363)--(2.6969,0.9376)--(2.7272,0.9388)--(2.7575,0.9400)--(2.7878,0.9412)--(2.8181,0.9424)--(2.8484,0.9435)--(2.8787,0.9446)--(2.9090,0.9456)--(2.9393,0.9467)--(2.9696,0.9477)--(3.0000,0.9486);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks b/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks
index 92685399c..f9cdf1b0f 100644
--- a/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks
+++ b/auto/pictures_tex/Fig_KKLooMbjxdI.pstricks
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4994);
 %DEFAULT
-\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
-\draw (-1.5708,-0.27858) node {$a$};
-\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
-\draw (6.2832,-0.32674) node {$b$};
+\draw []  (-1.5707,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5707,-0.2785) node {$a$};
+\draw []  (6.2831,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.2831,-0.3267) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [style=dashed] (6.2831,2.0000) -- (6.2831,0.0000);
 
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 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_KKRooHseDzC.pstricks b/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
index 62e8d054f..0537c7bdf 100644
--- a/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
+++ b/auto/pictures_tex/Fig_KKRooHseDzC.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.3750);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.3749);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
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+\draw [] (0.0000,0.0000) -- (0.0000,1.9861);
+\draw [] (5.0000,2.8194) -- (5.0000,0.0000);
 
-\draw [color=black] (-0.5000,1.750)--(-0.4242,1.788)--(-0.3485,1.824)--(-0.2727,1.861)--(-0.1970,1.896)--(-0.1212,1.931)--(-0.04545,1.966)--(0.03030,2.000)--(0.1061,2.033)--(0.1818,2.065)--(0.2576,2.097)--(0.3333,2.128)--(0.4091,2.159)--(0.4848,2.189)--(0.5606,2.218)--(0.6364,2.246)--(0.7121,2.274)--(0.7879,2.302)--(0.8636,2.329)--(0.9394,2.355)--(1.015,2.380)--(1.091,2.405)--(1.167,2.429)--(1.242,2.453)--(1.318,2.475)--(1.394,2.498)--(1.470,2.519)--(1.545,2.540)--(1.621,2.561)--(1.697,2.580)--(1.773,2.599)--(1.848,2.618)--(1.924,2.636)--(2.000,2.653)--(2.076,2.669)--(2.152,2.685)--(2.227,2.700)--(2.303,2.715)--(2.379,2.729)--(2.455,2.742)--(2.530,2.755)--(2.606,2.767)--(2.682,2.778)--(2.758,2.789)--(2.833,2.799)--(2.909,2.809)--(2.985,2.818)--(3.061,2.826)--(3.136,2.834)--(3.212,2.841)--(3.288,2.847)--(3.364,2.853)--(3.439,2.858)--(3.515,2.862)--(3.591,2.866)--(3.667,2.869)--(3.742,2.871)--(3.818,2.873)--(3.894,2.874)--(3.970,2.875)--(4.045,2.875)--(4.121,2.874)--(4.197,2.873)--(4.273,2.871)--(4.349,2.868)--(4.424,2.865)--(4.500,2.861)--(4.576,2.857)--(4.651,2.851)--(4.727,2.846)--(4.803,2.839)--(4.879,2.832)--(4.955,2.824)--(5.030,2.816)--(5.106,2.807)--(5.182,2.797)--(5.258,2.787)--(5.333,2.776)--(5.409,2.765)--(5.485,2.753)--(5.561,2.740)--(5.636,2.726)--(5.712,2.712)--(5.788,2.697)--(5.864,2.682)--(5.939,2.666)--(6.015,2.649)--(6.091,2.632)--(6.167,2.614)--(6.242,2.596)--(6.318,2.576)--(6.394,2.557)--(6.470,2.536)--(6.545,2.515)--(6.621,2.493)--(6.697,2.471)--(6.773,2.448)--(6.849,2.424)--(6.924,2.400)--(7.000,2.375);
+\draw [color=black] (-0.5000,1.7500)--(-0.4242,1.7875)--(-0.3484,1.8244)--(-0.2727,1.8607)--(-0.1969,1.8964)--(-0.1212,1.9314)--(-0.0454,1.9657)--(0.0303,1.9995)--(0.1060,2.0326)--(0.1818,2.0650)--(0.2575,2.0969)--(0.3333,2.1280)--(0.4090,2.1586)--(0.4848,2.1885)--(0.5606,2.2178)--(0.6363,2.2464)--(0.7121,2.2744)--(0.7878,2.3017)--(0.8636,2.3285)--(0.9393,2.3545)--(1.0151,2.3800)--(1.0909,2.4048)--(1.1666,2.4290)--(1.2424,2.4525)--(1.3181,2.4754)--(1.3939,2.4976)--(1.4696,2.5193)--(1.5454,2.5402)--(1.6212,2.5606)--(1.6969,2.5803)--(1.7727,2.5994)--(1.8484,2.6178)--(1.9242,2.6356)--(2.0000,2.6527)--(2.0757,2.6692)--(2.1515,2.6851)--(2.2272,2.7004)--(2.3030,2.7150)--(2.3787,2.7289)--(2.4545,2.7423)--(2.5303,2.7549)--(2.6060,2.7670)--(2.6818,2.7784)--(2.7575,2.7892)--(2.8333,2.7993)--(2.9090,2.8088)--(2.9848,2.8177)--(3.0606,2.8259)--(3.1363,2.8335)--(3.2121,2.8405)--(3.2878,2.8468)--(3.3636,2.8525)--(3.4393,2.8575)--(3.5151,2.8619)--(3.5909,2.8657)--(3.6666,2.8688)--(3.7424,2.8713)--(3.8181,2.8731)--(3.8939,2.8743)--(3.9696,2.8749)--(4.0454,2.8748)--(4.1212,2.8741)--(4.1969,2.8728)--(4.2727,2.8708)--(4.3484,2.8682)--(4.4242,2.8650)--(4.5000,2.8611)--(4.5757,2.8565)--(4.6515,2.8514)--(4.7272,2.8456)--(4.8030,2.8391)--(4.8787,2.8320)--(4.9545,2.8243)--(5.0303,2.8160)--(5.1060,2.8070)--(5.1818,2.7974)--(5.2575,2.7871)--(5.3333,2.7762)--(5.4090,2.7646)--(5.4848,2.7525)--(5.5606,2.7396)--(5.6363,2.7262)--(5.7121,2.7121)--(5.7878,2.6974)--(5.8636,2.6820)--(5.9393,2.6660)--(6.0151,2.6493)--(6.0909,2.6321)--(6.1666,2.6141)--(6.2424,2.5956)--(6.3181,2.5764)--(6.3939,2.5566)--(6.4696,2.5361)--(6.5454,2.5150)--(6.6212,2.4932)--(6.6969,2.4709)--(6.7727,2.4478)--(6.8484,2.4242)--(6.9242,2.3999)--(7.0000,2.3750);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.82) -- (6.00,2.82) -- (5.00,2.82) -- (5.00,2.82) -- (5.00,0) --  cycle;
-\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.82);
-\draw [color=red,style=dashed] (6.00,2.82) -- (5.00,2.82);
-\draw [color=red,style=dashed] (5.00,2.82) -- (5.00,0);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,2.8194) -- (6.0000,2.8194) -- (5.0000,2.8194) -- (5.0000,2.8194) -- (5.0000,0.0000) --  cycle;
+\draw [color=red,style=dashed] (5.0000,0.0000) -- (6.0000,0.0000);
+\draw [color=red,style=dashed] (6.0000,0.0000) -- (6.0000,2.8194);
+\draw [color=red,style=dashed] (6.0000,2.8194) -- (5.0000,2.8194);
+\draw [color=red,style=dashed] (5.0000,2.8194) -- (5.0000,0.0000);
 \draw []  (5.0000,2.8194) node [rotate=0] {$\bullet$};
-\draw (5.4420,3.3924) node {$f(x)$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.27858) node {$x$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.34557) node {$x+\Delta x$};
+\draw (5.4419,3.3923) node {$f(x)$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.2785) node {$x$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.3455) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_KScolorD.pstricks b/auto/pictures_tex/Fig_KScolorD.pstricks
index 091738bd0..92eee38ae 100644
--- a/auto/pictures_tex/Fig_KScolorD.pstricks
+++ b/auto/pictures_tex/Fig_KScolorD.pstricks
@@ -63,17 +63,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6969,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.6940);
+\draw [,->,>=latex] (-1.6968,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.6940);
 %DEFAULT
 
-\draw [color=blue] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
+\draw [color=blue] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
 
-\draw [color=black] plot [smooth,tension=1] coordinates {(0,1.00)(-0.119,1.19)(-0.159,0.784)(-0.354,1.15)(-0.311,0.737)(-0.575,1.05)(-0.452,0.660)(-0.773,0.918)(-0.574,0.557)(-0.940,0.746)(-0.673,0.433)(-1.07,0.544)(-0.745,0.290)(-1.16,0.322)(-0.788,0.137)(-1.20,0.0869)(-1.00,0)};
+\draw [color=black] plot [smooth,tension=1] coordinates {(0.0000,1.0000)(-0.1194,1.1940)(-0.1591,0.7840)(-0.3538,1.1466)(-0.3105,0.7372)(-0.5746,1.0534)(-0.4515,0.6603)(-0.7733,0.9175)(-0.5742,0.5569)(-0.9399,0.7460)(-0.6729,0.4325)(-1.0694,0.5443)(-0.7454,0.2904)(-1.1560,0.3216)(-0.7882,0.1367)(-1.1968,0.0868)(-1.0000,0.0000)};
 
-\draw [color=black] (0,-1.00)--(0.0159,-1.00)--(0.0317,-1.00)--(0.0476,-0.999)--(0.0634,-0.998)--(0.0792,-0.997)--(0.0951,-0.995)--(0.111,-0.994)--(0.127,-0.992)--(0.142,-0.990)--(0.158,-0.987)--(0.174,-0.985)--(0.189,-0.982)--(0.205,-0.979)--(0.220,-0.975)--(0.236,-0.972)--(0.251,-0.968)--(0.266,-0.964)--(0.282,-0.959)--(0.297,-0.955)--(0.312,-0.950)--(0.327,-0.945)--(0.342,-0.940)--(0.357,-0.934)--(0.372,-0.928)--(0.386,-0.922)--(0.401,-0.916)--(0.415,-0.910)--(0.430,-0.903)--(0.444,-0.896)--(0.458,-0.889)--(0.472,-0.881)--(0.486,-0.874)--(0.500,-0.866)--(0.514,-0.858)--(0.527,-0.850)--(0.541,-0.841)--(0.554,-0.833)--(0.567,-0.824)--(0.580,-0.815)--(0.593,-0.805)--(0.606,-0.796)--(0.618,-0.786)--(0.631,-0.776)--(0.643,-0.766)--(0.655,-0.756)--(0.667,-0.745)--(0.679,-0.735)--(0.690,-0.724)--(0.701,-0.713)--(0.713,-0.701)--(0.724,-0.690)--(0.735,-0.679)--(0.745,-0.667)--(0.756,-0.655)--(0.766,-0.643)--(0.776,-0.631)--(0.786,-0.618)--(0.796,-0.606)--(0.805,-0.593)--(0.815,-0.580)--(0.824,-0.567)--(0.833,-0.554)--(0.841,-0.541)--(0.850,-0.527)--(0.858,-0.514)--(0.866,-0.500)--(0.874,-0.486)--(0.881,-0.472)--(0.889,-0.458)--(0.896,-0.444)--(0.903,-0.430)--(0.910,-0.415)--(0.916,-0.401)--(0.922,-0.386)--(0.928,-0.372)--(0.934,-0.357)--(0.940,-0.342)--(0.945,-0.327)--(0.950,-0.312)--(0.955,-0.297)--(0.959,-0.282)--(0.964,-0.266)--(0.968,-0.251)--(0.972,-0.236)--(0.975,-0.220)--(0.979,-0.205)--(0.982,-0.189)--(0.985,-0.174)--(0.987,-0.158)--(0.990,-0.142)--(0.992,-0.127)--(0.994,-0.111)--(0.995,-0.0951)--(0.997,-0.0792)--(0.998,-0.0634)--(0.999,-0.0476)--(1.00,-0.0317)--(1.00,-0.0159)--(1.00,0);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw [color=black] (0.0000,-1.0000)--(0.0158,-0.9998)--(0.0317,-0.9994)--(0.0475,-0.9988)--(0.0634,-0.9979)--(0.0792,-0.9968)--(0.0950,-0.9954)--(0.1108,-0.9938)--(0.1265,-0.9919)--(0.1423,-0.9898)--(0.1580,-0.9874)--(0.1736,-0.9848)--(0.1892,-0.9819)--(0.2048,-0.9788)--(0.2203,-0.9754)--(0.2357,-0.9718)--(0.2511,-0.9679)--(0.2664,-0.9638)--(0.2817,-0.9594)--(0.2969,-0.9549)--(0.3120,-0.9500)--(0.3270,-0.9450)--(0.3420,-0.9396)--(0.3568,-0.9341)--(0.3716,-0.9283)--(0.3863,-0.9223)--(0.4009,-0.9161)--(0.4154,-0.9096)--(0.4297,-0.9029)--(0.4440,-0.8959)--(0.4582,-0.8888)--(0.4722,-0.8814)--(0.4861,-0.8738)--(0.5000,-0.8660)--(0.5136,-0.8579)--(0.5272,-0.8497)--(0.5406,-0.8412)--(0.5539,-0.8325)--(0.5670,-0.8236)--(0.5800,-0.8145)--(0.5929,-0.8052)--(0.6056,-0.7957)--(0.6181,-0.7860)--(0.6305,-0.7761)--(0.6427,-0.7660)--(0.6548,-0.7557)--(0.6667,-0.7452)--(0.6785,-0.7345)--(0.6900,-0.7237)--(0.7014,-0.7126)--(0.7126,-0.7014)--(0.7237,-0.6900)--(0.7345,-0.6785)--(0.7452,-0.6667)--(0.7557,-0.6548)--(0.7660,-0.6427)--(0.7761,-0.6305)--(0.7860,-0.6181)--(0.7957,-0.6056)--(0.8052,-0.5929)--(0.8145,-0.5800)--(0.8236,-0.5670)--(0.8325,-0.5539)--(0.8412,-0.5406)--(0.8497,-0.5272)--(0.8579,-0.5136)--(0.8660,-0.5000)--(0.8738,-0.4861)--(0.8814,-0.4722)--(0.8888,-0.4582)--(0.8959,-0.4440)--(0.9029,-0.4297)--(0.9096,-0.4154)--(0.9161,-0.4009)--(0.9223,-0.3863)--(0.9283,-0.3716)--(0.9341,-0.3568)--(0.9396,-0.3420)--(0.9450,-0.3270)--(0.9500,-0.3120)--(0.9549,-0.2969)--(0.9594,-0.2817)--(0.9638,-0.2664)--(0.9679,-0.2511)--(0.9718,-0.2357)--(0.9754,-0.2203)--(0.9788,-0.2048)--(0.9819,-0.1892)--(0.9848,-0.1736)--(0.9874,-0.1580)--(0.9898,-0.1423)--(0.9919,-0.1265)--(0.9938,-0.1108)--(0.9954,-0.0950)--(0.9968,-0.0792)--(0.9979,-0.0634)--(0.9988,-0.0475)--(0.9994,-0.0317)--(0.9998,-0.0158)--(1.0000,0.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LAfWmaN.pstricks b/auto/pictures_tex/Fig_LAfWmaN.pstricks
index 45c4fc2fa..10fefba50 100644
--- a/auto/pictures_tex/Fig_LAfWmaN.pstricks
+++ b/auto/pictures_tex/Fig_LAfWmaN.pstricks
@@ -91,33 +91,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.6200,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,1.1000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (5.6200,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,1.1000);
 %DEFAULT
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+\draw [color=blue] (-1.0000,-1.5000)--(-0.9393,-1.5000)--(-0.8787,-1.5000)--(-0.8181,-1.5000)--(-0.7575,-1.5000)--(-0.6969,-1.5000)--(-0.6363,-1.5000)--(-0.5757,-1.5000)--(-0.5151,-1.5000)--(-0.4545,-1.5000)--(-0.3939,-1.5000)--(-0.3333,-1.5000)--(-0.2727,-1.5000)--(-0.2121,-1.5000)--(-0.1515,-1.5000)--(-0.0909,-1.5000)--(-0.0303,-1.5000)--(0.0303,-1.5000)--(0.0909,-1.5000)--(0.1515,-1.5000)--(0.2121,-1.5000)--(0.2727,-1.5000)--(0.3333,-1.5000)--(0.3939,-1.5000)--(0.4545,-1.5000)--(0.5151,-1.5000)--(0.5757,-1.5000)--(0.6363,-1.5000)--(0.6969,-1.5000)--(0.7575,-1.5000)--(0.8181,-1.5000)--(0.8787,-1.5000)--(0.9393,-1.5000)--(1.0000,-1.5000)--(1.0606,-1.5000)--(1.1212,-1.5000)--(1.1818,-1.5000)--(1.2424,-1.5000)--(1.3030,-1.5000)--(1.3636,-1.5000)--(1.4242,-1.5000)--(1.4848,-1.5000)--(1.5454,-1.5000)--(1.6060,-1.5000)--(1.6666,-1.5000)--(1.7272,-1.5000)--(1.7878,-1.5000)--(1.8484,-1.5000)--(1.9090,-1.5000)--(1.9696,-1.5000)--(2.0303,-1.5000)--(2.0909,-1.5000)--(2.1515,-1.5000)--(2.2121,-1.5000)--(2.2727,-1.5000)--(2.3333,-1.5000)--(2.3939,-1.5000)--(2.4545,-1.5000)--(2.5151,-1.5000)--(2.5757,-1.5000)--(2.6363,-1.5000)--(2.6969,-1.5000)--(2.7575,-1.5000)--(2.8181,-1.5000)--(2.8787,-1.5000)--(2.9393,-1.5000)--(3.0000,-1.5000)--(3.0606,-1.5000)--(3.1212,-1.5000)--(3.1818,-1.5000)--(3.2424,-1.5000)--(3.3030,-1.5000)--(3.3636,-1.5000)--(3.4242,-1.5000)--(3.4848,-1.5000)--(3.5454,-1.5000)--(3.6060,-1.5000)--(3.6666,-1.5000)--(3.7272,-1.5000)--(3.7878,-1.5000)--(3.8484,-1.5000)--(3.9090,-1.5000)--(3.9696,-1.5000)--(4.0303,-1.5000)--(4.0909,-1.5000)--(4.1515,-1.5000)--(4.2121,-1.5000)--(4.2727,-1.5000)--(4.3333,-1.5000)--(4.3939,-1.5000)--(4.4545,-1.5000)--(4.5151,-1.5000)--(4.5757,-1.5000)--(4.6363,-1.5000)--(4.6969,-1.5000)--(4.7575,-1.5000)--(4.8181,-1.5000)--(4.8787,-1.5000)--(4.9393,-1.5000)--(5.0000,-1.5000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_LBGooAdteCt.pstricks b/auto/pictures_tex/Fig_LBGooAdteCt.pstricks
index 5bc36bb3b..1b9248e31 100644
--- a/auto/pictures_tex/Fig_LBGooAdteCt.pstricks
+++ b/auto/pictures_tex/Fig_LBGooAdteCt.pstricks
@@ -99,63 +99,63 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-5.00,0) -- (-5.00,5.00);
-\draw [color=gray,style=solid] (-4.00,0) -- (-4.00,5.00);
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-\draw [color=gray,style=solid] (-2.00,0) -- (-2.00,5.00);
-\draw [color=gray,style=solid] (-1.00,0) -- (-1.00,5.00);
-\draw [color=gray,style=solid] (0,0) -- (0,5.00);
-\draw [color=gray,style=solid] (1.00,0) -- (1.00,5.00);
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-\draw [color=gray,style=dotted] (-4.50,0) -- (-4.50,5.00);
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-\draw [color=gray,style=solid] (-5.00,5.00) -- (2.00,5.00);
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+\draw [color=gray,style=dotted] (-0.5000,0.0000) -- (-0.5000,5.0000);
+\draw [color=gray,style=dotted] (0.5000,0.0000) -- (0.5000,5.0000);
+\draw [color=gray,style=dotted] (1.5000,0.0000) -- (1.5000,5.0000);
+\draw [color=gray,style=dotted] (-5.0000,0.5000) -- (2.0000,0.5000);
+\draw [color=gray,style=dotted] (-5.0000,1.5000) -- (2.0000,1.5000);
+\draw [color=gray,style=dotted] (-5.0000,2.5000) -- (2.0000,2.5000);
+\draw [color=gray,style=dotted] (-5.0000,3.5000) -- (2.0000,3.5000);
+\draw [color=gray,style=dotted] (-5.0000,4.5000) -- (2.0000,4.5000);
+\draw [color=gray,style=solid] (-5.0000,0.0000) -- (2.0000,0.0000);
+\draw [color=gray,style=solid] (-5.0000,1.0000) -- (2.0000,1.0000);
+\draw [color=gray,style=solid] (-5.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=gray,style=solid] (-5.0000,3.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (-5.0000,4.0000) -- (2.0000,4.0000);
+\draw [color=gray,style=solid] (-5.0000,5.0000) -- (2.0000,5.0000);
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
 
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+\draw [color=blue] (-5.0000,0.0067)--(-4.9336,0.0072)--(-4.8673,0.0076)--(-4.8010,0.0082)--(-4.7346,0.0087)--(-4.6683,0.0093)--(-4.6020,0.0100)--(-4.5357,0.0107)--(-4.4693,0.0114)--(-4.4030,0.0122)--(-4.3367,0.0130)--(-4.2703,0.0139)--(-4.2040,0.0149)--(-4.1377,0.0159)--(-4.0714,0.0170)--(-4.0050,0.0182)--(-3.9387,0.0194)--(-3.8724,0.0208)--(-3.8060,0.0222)--(-3.7397,0.0237)--(-3.6734,0.0253)--(-3.6071,0.0271)--(-3.5407,0.0289)--(-3.4744,0.0309)--(-3.4081,0.0331)--(-3.3417,0.0353)--(-3.2754,0.0377)--(-3.2091,0.0403)--(-3.1428,0.0431)--(-3.0764,0.0461)--(-3.0101,0.0492)--(-2.9438,0.0526)--(-2.8774,0.0562)--(-2.8111,0.0601)--(-2.7448,0.0642)--(-2.6785,0.0686)--(-2.6121,0.0733)--(-2.5458,0.0784)--(-2.4795,0.0837)--(-2.4131,0.0895)--(-2.3468,0.0956)--(-2.2805,0.1022)--(-2.2142,0.1092)--(-2.1478,0.1167)--(-2.0815,0.1247)--(-2.0152,0.1332)--(-1.9488,0.1424)--(-1.8825,0.1522)--(-1.8162,0.1626)--(-1.7499,0.1737)--(-1.6835,0.1857)--(-1.6172,0.1984)--(-1.5509,0.2120)--(-1.4845,0.2265)--(-1.4182,0.2421)--(-1.3519,0.2587)--(-1.2856,0.2764)--(-1.2192,0.2954)--(-1.1529,0.3157)--(-1.0866,0.3373)--(-1.0202,0.3604)--(-0.9539,0.3852)--(-0.8876,0.4116)--(-0.8213,0.4398)--(-0.7549,0.4700)--(-0.6886,0.5022)--(-0.6223,0.5366)--(-0.5559,0.5735)--(-0.4896,0.6128)--(-0.4233,0.6548)--(-0.3570,0.6997)--(-0.2906,0.7477)--(-0.2243,0.7990)--(-0.1580,0.8538)--(-0.0916,0.9123)--(-0.0253,0.9749)--(0.0409,1.0418)--(0.1072,1.1132)--(0.1736,1.1896)--(0.2399,1.2711)--(0.3062,1.3583)--(0.3726,1.4515)--(0.4389,1.5510)--(0.5052,1.6574)--(0.5715,1.7710)--(0.6379,1.8925)--(0.7042,2.0223)--(0.7705,2.1610)--(0.8369,2.3092)--(0.9032,2.4675)--(0.9695,2.6368)--(1.0358,2.8176)--(1.1022,3.0108)--(1.1685,3.2173)--(1.2348,3.4379)--(1.3012,3.6737)--(1.3675,3.9256)--(1.4338,4.1948)--(1.5001,4.4825)--(1.5665,4.7899);
 
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_LEJNDxI.pstricks b/auto/pictures_tex/Fig_LEJNDxI.pstricks
index 57274a81d..598ad3f88 100644
--- a/auto/pictures_tex/Fig_LEJNDxI.pstricks
+++ b/auto/pictures_tex/Fig_LEJNDxI.pstricks
@@ -66,20 +66,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.20000) node {};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.20000) node {};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.20000) node {};
-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.20000) node {};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.20000) node {};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2000) node {};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2000) node {};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.2000) node {};
+\draw []  (3.0000,0.0000) node [rotate=0] {$o$};
+\draw (3.0000,0.2000) node {};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.2000) node {};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks b/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks
index 03400f4b5..2a7bc9ce4 100644
--- a/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks
+++ b/auto/pictures_tex/Fig_LLVMooWOkvAB.pstricks
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -77,17 +77,17 @@
 % setting the default values
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+\draw [,->,>=latex] (1.5000,0.7500) -- (1.5212,0.7712);
+\draw [color=green] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=green] (3.0000,3.0000) -- (3.0000,3.0000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks b/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks
index d1387e358..a2739b882 100644
--- a/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks
+++ b/auto/pictures_tex/Fig_LMHMooCscXNNdU.pstricks
@@ -100,38 +100,38 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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-\draw (-4.0000,-0.32983) node {$ -4 $};
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-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
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-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
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-\draw [] (-0.100,2.00) -- (0.100,2.00);
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+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_Laurin.pstricks b/auto/pictures_tex/Fig_Laurin.pstricks
index bd70fac13..5e2d5a7bc 100644
--- a/auto/pictures_tex/Fig_Laurin.pstricks
+++ b/auto/pictures_tex/Fig_Laurin.pstricks
@@ -99,41 +99,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,7.8891);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,7.8890);
 %DEFAULT
 
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-\draw (-0.29125,7.0000) node {$ 7 $};
-\draw [] (-0.100,7.00) -- (0.100,7.00);
+\draw [color=red] (-2.0000,1.0000)--(-1.9595,0.9604)--(-1.9191,0.9224)--(-1.8787,0.8861)--(-1.8383,0.8514)--(-1.7979,0.8183)--(-1.7575,0.7869)--(-1.7171,0.7571)--(-1.6767,0.7290)--(-1.6363,0.7024)--(-1.5959,0.6775)--(-1.5555,0.6543)--(-1.5151,0.6326)--(-1.4747,0.6126)--(-1.4343,0.5943)--(-1.3939,0.5775)--(-1.3535,0.5624)--(-1.3131,0.5490)--(-1.2727,0.5371)--(-1.2323,0.5269)--(-1.1919,0.5184)--(-1.1515,0.5114)--(-1.1111,0.5061)--(-1.0707,0.5024)--(-1.0303,0.5004)--(-0.9898,0.5000)--(-0.9494,0.5012)--(-0.9090,0.5041)--(-0.8686,0.5086)--(-0.8282,0.5147)--(-0.7878,0.5224)--(-0.7474,0.5318)--(-0.7070,0.5429)--(-0.6666,0.5555)--(-0.6262,0.5698)--(-0.5858,0.5857)--(-0.5454,0.6033)--(-0.5050,0.6224)--(-0.4646,0.6433)--(-0.4242,0.6657)--(-0.3838,0.6898)--(-0.3434,0.7155)--(-0.3030,0.7428)--(-0.2626,0.7718)--(-0.2222,0.8024)--(-0.1818,0.8347)--(-0.1414,0.8685)--(-0.1010,0.9040)--(-0.0606,0.9412)--(-0.0202,0.9800)--(0.0202,1.0204)--(0.0606,1.0624)--(0.1010,1.1061)--(0.1414,1.1514)--(0.1818,1.1983)--(0.2222,1.2469)--(0.2626,1.2971)--(0.3030,1.3489)--(0.3434,1.4024)--(0.3838,1.4575)--(0.4242,1.5142)--(0.4646,1.5725)--(0.5050,1.6325)--(0.5454,1.6942)--(0.5858,1.7574)--(0.6262,1.8223)--(0.6666,1.8888)--(0.7070,1.9570)--(0.7474,2.0268)--(0.7878,2.0982)--(0.8282,2.1713)--(0.8686,2.2459)--(0.9090,2.3223)--(0.9494,2.4002)--(0.9898,2.4798)--(1.0303,2.5610)--(1.0707,2.6439)--(1.1111,2.7283)--(1.1515,2.8145)--(1.1919,2.9022)--(1.2323,2.9916)--(1.2727,3.0826)--(1.3131,3.1752)--(1.3535,3.2695)--(1.3939,3.3654)--(1.4343,3.4630)--(1.4747,3.5621)--(1.5151,3.6629)--(1.5555,3.7654)--(1.5959,3.8695)--(1.6363,3.9752)--(1.6767,4.0825)--(1.7171,4.1915)--(1.7575,4.3021)--(1.7979,4.4143)--(1.8383,4.5282)--(1.8787,4.6437)--(1.9191,4.7608)--(1.9595,4.8796)--(2.0000,5.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
+\draw (-0.2912,7.0000) node {$ 7 $};
+\draw [] (-0.1000,7.0000) -- (0.1000,7.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_LesSpheres.pstricks b/auto/pictures_tex/Fig_LesSpheres.pstricks
index 3858e3937..cd673c46c 100644
--- a/auto/pictures_tex/Fig_LesSpheres.pstricks
+++ b/auto/pictures_tex/Fig_LesSpheres.pstricks
@@ -41,21 +41,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw [] (0,1.00) -- (-1.00,0);
-\draw [] (-1.00,0) -- (0,-1.00);
-\draw [] (0,-1.00) -- (1.00,0);
-\draw [] (1.00,0) -- (0,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (0.0000,1.0000) -- (-1.0000,0.0000);
+\draw [] (-1.0000,0.0000) -- (0.0000,-1.0000);
+\draw [] (0.0000,-1.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (0.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -93,19 +93,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -143,21 +143,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
-\draw [] (1.00,1.00) -- (-1.00,1.00);
-\draw [] (-1.00,1.00) -- (-1.00,-1.00);
-\draw [] (-1.00,-1.00) -- (1.00,-1.00);
-\draw [] (1.00,-1.00) -- (1.00,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [] (1.0000,1.0000) -- (-1.0000,1.0000);
+\draw [] (-1.0000,1.0000) -- (-1.0000,-1.0000);
+\draw [] (-1.0000,-1.0000) -- (1.0000,-1.0000);
+\draw [] (1.0000,-1.0000) -- (1.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_LesSubFigures.pstricks b/auto/pictures_tex/Fig_LesSubFigures.pstricks
index 29f58eaae..6757fb5ec 100644
--- a/auto/pictures_tex/Fig_LesSubFigures.pstricks
+++ b/auto/pictures_tex/Fig_LesSubFigures.pstricks
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.7141,2.0547) -- (3.2758,3.1849);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
 \draw (3.0787,2.5199) node {$Q_{0}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.5492,1.9345) -- (3.0382,3.2170);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
-\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (2.3975,2.8868) node [rotate=0] {$\bullet$};
+\draw (2.6952,2.4360) node {$Q_{1}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.4022,1.7769) -- (2.7827,3.2509);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
-\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (1.9950,2.7631) node [rotate=0] {$\bullet$};
+\draw (2.3223,2.3215) node {$Q_{2}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7788);
 %DEFAULT
-\draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.2850,1.5627) -- (2.4974,3.2788);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
-\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw (1.9663,2.1571) node {$Q_{3}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks b/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks
index 27a5ac04b..3b29a8a32 100644
--- a/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks
+++ b/auto/pictures_tex/Fig_LesSubFiguresOM.pstricks
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.7141,2.0547) -- (3.2758,3.1849);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
 \draw (3.0787,2.5199) node {$Q_{0}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.5492,1.9345) -- (3.0382,3.2170);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
-\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (2.3975,2.8868) node [rotate=0] {$\bullet$};
+\draw (2.6952,2.4360) node {$Q_{1}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.4022,1.7769) -- (2.7827,3.2509);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
-\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (1.9950,2.7631) node [rotate=0] {$\bullet$};
+\draw (2.3223,2.3215) node {$Q_{2}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7788);
 %DEFAULT
-\draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.2850,1.5627) -- (2.4974,3.2788);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
-\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw (1.9663,2.1571) node {$Q_{3}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MCKyvdk.pstricks b/auto/pictures_tex/Fig_MCKyvdk.pstricks
index 8e7561c6c..0cb581bbd 100644
--- a/auto/pictures_tex/Fig_MCKyvdk.pstricks
+++ b/auto/pictures_tex/Fig_MCKyvdk.pstricks
@@ -95,26 +95,26 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (-0.27830,3.2661) node {\( A\)};
+\draw (-0.2782,3.2661) node {\( A\)};
 \draw (3.0000,3.3247) node {\( B\)};
-\draw (3.2849,-0.26613) node {\( C\)};
-\draw (-0.35616,0) node {\( D\)};
-\draw (1.0124,4.0161) node {\( E\)};
+\draw (3.2849,-0.2661) node {\( C\)};
+\draw (-0.3561,0.0000) node {\( D\)};
+\draw (1.0123,4.0161) node {\( E\)};
 \draw (4.6417,3.7500) node {\( F\)};
-\draw (4.6425,0.75000) node {\( G\)};
-\draw (0.99109,1.0161) node {\( H\)};
-\draw [] (0,3.00) -- (1.30,3.75);
-\draw [] (3.00,3.00) -- (4.30,3.75);
-\draw [] (3.00,0) -- (4.30,0.750);
-\draw [style=dashed] (0,0) -- (1.30,0.750);
-\draw [] (1.30,3.75) -- (4.30,3.75);
-\draw [] (4.30,3.75) -- (4.30,0.750);
-\draw [style=dashed] (4.30,0.750) -- (1.30,0.750);
-\draw [style=dashed] (1.30,0.750) -- (1.30,3.75);
-\draw [] (0,3.00) -- (3.00,3.00);
-\draw [] (3.00,3.00) -- (3.00,0);
-\draw [] (3.00,0) -- (0,0);
-\draw [] (0,0) -- (0,3.00);
+\draw (4.6425,0.7500) node {\( G\)};
+\draw (0.9910,1.0161) node {\( H\)};
+\draw [] (0.0000,3.0000) -- (1.2990,3.7500);
+\draw [] (3.0000,3.0000) -- (4.2990,3.7500);
+\draw [] (3.0000,0.0000) -- (4.2990,0.7500);
+\draw [style=dashed] (0.0000,0.0000) -- (1.2990,0.7500);
+\draw [] (1.2990,3.7500) -- (4.2990,3.7500);
+\draw [] (4.2990,3.7500) -- (4.2990,0.7500);
+\draw [style=dashed] (4.2990,0.7500) -- (1.2990,0.7500);
+\draw [style=dashed] (1.2990,0.7500) -- (1.2990,3.7500);
+\draw [] (0.0000,3.0000) -- (3.0000,3.0000);
+\draw [] (3.0000,3.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (0.0000,3.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_MCQueGF.pstricks b/auto/pictures_tex/Fig_MCQueGF.pstricks
index 8594e6364..89057aa14 100644
--- a/auto/pictures_tex/Fig_MCQueGF.pstricks
+++ b/auto/pictures_tex/Fig_MCQueGF.pstricks
@@ -87,37 +87,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
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+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_MNICGhR.pstricks b/auto/pictures_tex/Fig_MNICGhR.pstricks
index ee87719a4..902511998 100644
--- a/auto/pictures_tex/Fig_MNICGhR.pstricks
+++ b/auto/pictures_tex/Fig_MNICGhR.pstricks
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.30595) node {$\alpha_1$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.30595) node {$\alpha_2$};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.30595) node {$\alpha_3$};
-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.32154) node {$\alpha_{l-1}$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.30831) node {$\alpha_l$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3059) node {$\alpha_1$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.3059) node {$\alpha_2$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.3059) node {$\alpha_3$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$o$};
+\draw (3.0000,0.3215) node {$\alpha_{l-1}$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.3083) node {$\alpha_l$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_Mantisse.pstricks b/auto/pictures_tex/Fig_Mantisse.pstricks
index 1c275f207..c33f42fe7 100644
--- a/auto/pictures_tex/Fig_Mantisse.pstricks
+++ b/auto/pictures_tex/Fig_Mantisse.pstricks
@@ -83,29 +83,29 @@
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 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4990);
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+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4989);
 %DEFAULT
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+\draw [color=blue] 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+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_MaxVraissLp.pstricks b/auto/pictures_tex/Fig_MaxVraissLp.pstricks
index 4d9809e18..2acb70e7e 100644
--- a/auto/pictures_tex/Fig_MaxVraissLp.pstricks
+++ b/auto/pictures_tex/Fig_MaxVraissLp.pstricks
@@ -87,25 +87,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.1682);
 %DEFAULT
-\draw []  (3.0000,2.6683) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,2.6682) node [rotate=0] {$\bullet$};
 
-\draw [color=blue] (0,0)--(0.101,0.00115)--(0.202,0.00858)--(0.303,0.0269)--(0.404,0.0593)--(0.505,0.108)--(0.606,0.172)--(0.707,0.254)--(0.808,0.351)--(0.909,0.463)--(1.01,0.587)--(1.11,0.722)--(1.21,0.865)--(1.31,1.01)--(1.41,1.17)--(1.52,1.32)--(1.62,1.47)--(1.72,1.63)--(1.82,1.77)--(1.92,1.91)--(2.02,2.04)--(2.12,2.16)--(2.22,2.27)--(2.32,2.36)--(2.42,2.45)--(2.53,2.52)--(2.63,2.58)--(2.73,2.62)--(2.83,2.65)--(2.93,2.67)--(3.03,2.67)--(3.13,2.66)--(3.23,2.64)--(3.33,2.60)--(3.43,2.56)--(3.54,2.50)--(3.64,2.44)--(3.74,2.37)--(3.84,2.29)--(3.94,2.20)--(4.04,2.11)--(4.14,2.02)--(4.24,1.92)--(4.34,1.82)--(4.44,1.72)--(4.55,1.62)--(4.65,1.52)--(4.75,1.42)--(4.85,1.32)--(4.95,1.22)--(5.05,1.12)--(5.15,1.03)--(5.25,0.945)--(5.35,0.861)--(5.45,0.781)--(5.56,0.705)--(5.66,0.633)--(5.76,0.567)--(5.86,0.504)--(5.96,0.446)--(6.06,0.393)--(6.16,0.345)--(6.26,0.300)--(6.36,0.260)--(6.46,0.224)--(6.57,0.191)--(6.67,0.163)--(6.77,0.137)--(6.87,0.115)--(6.97,0.0953)--(7.07,0.0785)--(7.17,0.0641)--(7.27,0.0518)--(7.37,0.0415)--(7.47,0.0328)--(7.58,0.0257)--(7.68,0.0198)--(7.78,0.0151)--(7.88,0.0113)--(7.98,0.00837)--(8.08,0.00607)--(8.18,0.00432)--(8.28,0.00300)--(8.38,0.00204)--(8.48,0.00134)--(8.59,0)--(8.69,0)--(8.79,0)--(8.89,0)--(8.99,0)--(9.09,0)--(9.19,0)--(9.29,0)--(9.39,0)--(9.50,0)--(9.60,0)--(9.70,0)--(9.80,0)--(9.90,0)--(10.0,0);
-\draw [style=dotted] (3.00,2.67) -- (3.00,0);
-\draw (10.500,-0.41406) node {$p$};
-\draw (10.500,-0.41406) node {$p$};
-\draw (3.0000,-0.42071) node {$ \frac{3}{10} $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.0000,-0.42071) node {$ \frac{3}{5} $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (9.0000,-0.42071) node {$ \frac{9}{10} $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.65793,3.1683) node {$L(p)$};
-\draw (-0.65793,3.1683) node {$L(p)$};
-\draw (-0.31083,2.0000) node {$ \frac{1}{5} $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.1010,0.0011)--(0.2020,0.0085)--(0.3030,0.0269)--(0.4040,0.0593)--(0.5050,0.1075)--(0.6060,0.1724)--(0.7070,0.2538)--(0.8080,0.3510)--(0.9090,0.4626)--(1.0101,0.5868)--(1.1111,0.7217)--(1.2121,0.8649)--(1.3131,1.0142)--(1.4141,1.1672)--(1.5151,1.3214)--(1.6161,1.4748)--(1.7171,1.6250)--(1.8181,1.7702)--(1.9191,1.9086)--(2.0202,2.0384)--(2.1212,2.1584)--(2.2222,2.2674)--(2.3232,2.3643)--(2.4242,2.4484)--(2.5252,2.5192)--(2.6262,2.5763)--(2.7272,2.6196)--(2.8282,2.6491)--(2.9292,2.6650)--(3.0303,2.6676)--(3.1313,2.6575)--(3.2323,2.6351)--(3.3333,2.6012)--(3.4343,2.5565)--(3.5353,2.5020)--(3.6363,2.4384)--(3.7373,2.3668)--(3.8383,2.2881)--(3.9393,2.2033)--(4.0404,2.1133)--(4.1414,2.0191)--(4.2424,1.9217)--(4.3434,1.8220)--(4.4444,1.7207)--(4.5454,1.6189)--(4.6464,1.5171)--(4.7474,1.4162)--(4.8484,1.3168)--(4.9494,1.2195)--(5.0505,1.1248)--(5.1515,1.0333)--(5.2525,0.9452)--(5.3535,0.8609)--(5.4545,0.7807)--(5.5555,0.7048)--(5.6565,0.6333)--(5.7575,0.5664)--(5.8585,0.5041)--(5.9595,0.4464)--(6.0606,0.3933)--(6.1616,0.3445)--(6.2626,0.3002)--(6.3636,0.2599)--(6.4646,0.2237)--(6.5656,0.1913)--(6.6666,0.1625)--(6.7676,0.1371)--(6.8686,0.1147)--(6.9696,0.0953)--(7.0707,0.0785)--(7.1717,0.0640)--(7.2727,0.0518)--(7.3737,0.0414)--(7.4747,0.0328)--(7.5757,0.0256)--(7.6767,0.0198)--(7.7777,0.0151)--(7.8787,0.0113)--(7.9797,0.0083)--(8.0808,0.0060)--(8.1818,0.0043)--(8.2828,0.0030)--(8.3838,0.0020)--(8.4848,0.0013)--(8.5858,0.0000)--(8.6868,0.0000)--(8.7878,0.0000)--(8.8888,0.0000)--(8.9898,0.0000)--(9.0909,0.0000)--(9.1919,0.0000)--(9.2929,0.0000)--(9.3939,0.0000)--(9.4949,0.0000)--(9.5959,0.0000)--(9.6969,0.0000)--(9.7979,0.0000)--(9.8989,0.0000)--(10.000,0.0000);
+\draw [style=dotted] (3.0000,2.6682) -- (3.0000,0.0000);
+\draw (10.500,-0.4140) node {$p$};
+\draw (10.500,-0.4140) node {$p$};
+\draw (3.0000,-0.4207) node {$ \frac{3}{10} $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (6.0000,-0.4207) node {$ \frac{3}{5} $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (9.0000,-0.4207) node {$ \frac{9}{10} $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (-0.6579,3.1682) node {$L(p)$};
+\draw (-0.6579,3.1682) node {$L(p)$};
+\draw (-0.3108,2.0000) node {$ \frac{1}{5} $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_MethodeChemin.pstricks b/auto/pictures_tex/Fig_MethodeChemin.pstricks
index 8934d3069..51e3cf49d 100644
--- a/auto/pictures_tex/Fig_MethodeChemin.pstricks
+++ b/auto/pictures_tex/Fig_MethodeChemin.pstricks
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
+\draw [,->,>=latex] (-2.0000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0000) -- (0.0000,2.0000);
 %DEFAULT
-\draw [color=red,style=dashed] (-1.50,1.50) -- (1.50,-1.50);
-\draw [color=blue,style=dashed] (-1.50,-0.750) -- (1.50,0.750);
+\draw [color=red,style=dashed] (-1.5000,1.5000) -- (1.5000,-1.5000);
+\draw [color=blue,style=dashed] (-1.5000,-0.7500) -- (1.5000,0.7500);
 \draw (-1.5000,1.9416) node {$y=-x$};
-\draw (2.3251,1.1446) node {$y=x/2$};
+\draw (2.3250,1.1445) node {$y=x/2$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_MethodeNewton.pstricks b/auto/pictures_tex/Fig_MethodeNewton.pstricks
index c62cf7e8b..b9a349019 100644
--- a/auto/pictures_tex/Fig_MethodeNewton.pstricks
+++ b/auto/pictures_tex/Fig_MethodeNewton.pstricks
@@ -95,31 +95,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-1.2875) -- (0,4.4000);
+\draw [,->,>=latex] (-2.0000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2875) -- (0.0000,4.4000);
 %DEFAULT
 
-\draw [color=blue] (-1.500,3.900)--(-1.424,3.713)--(-1.348,3.529)--(-1.273,3.349)--(-1.197,3.173)--(-1.121,3.001)--(-1.045,2.833)--(-0.9697,2.668)--(-0.8939,2.507)--(-0.8182,2.350)--(-0.7424,2.197)--(-0.6667,2.048)--(-0.5909,1.903)--(-0.5152,1.761)--(-0.4394,1.623)--(-0.3636,1.490)--(-0.2879,1.359)--(-0.2121,1.233)--(-0.1364,1.111)--(-0.06061,0.9921)--(0.01515,0.8773)--(0.09091,0.7664)--(0.1667,0.6593)--(0.2424,0.5560)--(0.3182,0.4565)--(0.3939,0.3608)--(0.4697,0.2690)--(0.5455,0.1810)--(0.6212,0.09682)--(0.6970,0.01647)--(0.7727,-0.06006)--(0.8485,-0.1328)--(0.9242,-0.2016)--(1.000,-0.2667)--(1.076,-0.3279)--(1.152,-0.3853)--(1.227,-0.4388)--(1.303,-0.4886)--(1.379,-0.5345)--(1.455,-0.5766)--(1.530,-0.6148)--(1.606,-0.6493)--(1.682,-0.6799)--(1.758,-0.7067)--(1.833,-0.7296)--(1.909,-0.7488)--(1.985,-0.7641)--(2.061,-0.7755)--(2.136,-0.7832)--(2.212,-0.7870)--(2.288,-0.7870)--(2.364,-0.7832)--(2.439,-0.7755)--(2.515,-0.7641)--(2.591,-0.7488)--(2.667,-0.7296)--(2.742,-0.7067)--(2.818,-0.6799)--(2.894,-0.6493)--(2.970,-0.6148)--(3.045,-0.5766)--(3.121,-0.5345)--(3.197,-0.4886)--(3.273,-0.4388)--(3.348,-0.3853)--(3.424,-0.3279)--(3.500,-0.2667)--(3.576,-0.2016)--(3.652,-0.1328)--(3.727,-0.06006)--(3.803,0.01647)--(3.879,0.09682)--(3.955,0.1810)--(4.030,0.2690)--(4.106,0.3608)--(4.182,0.4565)--(4.258,0.5560)--(4.333,0.6593)--(4.409,0.7664)--(4.485,0.8773)--(4.561,0.9921)--(4.636,1.111)--(4.712,1.233)--(4.788,1.359)--(4.864,1.490)--(4.939,1.623)--(5.015,1.761)--(5.091,1.903)--(5.167,2.048)--(5.242,2.197)--(5.318,2.350)--(5.394,2.507)--(5.470,2.668)--(5.545,2.833)--(5.621,3.001)--(5.697,3.173)--(5.773,3.349)--(5.849,3.529)--(5.924,3.713)--(6.000,3.900);
-\draw [color=red,style=dotted] (-0.900,0) -- (-0.900,2.52);
-\draw [color=green,style=dashed] (-1.20,3.15) -- (0.600,-0.630);
-\draw []  (-0.90000,0) node [rotate=0] {$\bullet$};
-\draw (-0.90000,-0.40595) node {$x_n$};
-\draw []  (0.30000,0) node [rotate=0] {$\bullet$};
-\draw (0.30000,-0.41918) node {$x_{n+1}$};
-\draw []  (-0.90000,2.5200) node [rotate=0] {$\bullet$};
-\draw (-0.50410,2.8462) node {$y_n$};
-\draw []  (0.71296,0) node [rotate=0] {$\bullet$};
-\draw (0.71296,0.40595) node {$r_0$};
-\draw []  (3.7870,0) node [rotate=0] {$\bullet$};
-\draw (3.7870,0.40595) node {$r_1$};
-\draw []  (2.2500,-0.78750) node [rotate=0] {$\bullet$};
+\draw [color=blue] (-1.5000,3.9000)--(-1.4242,3.7125)--(-1.3484,3.5288)--(-1.2727,3.3490)--(-1.1969,3.1730)--(-1.1212,3.0008)--(-1.0454,2.8325)--(-0.9696,2.6679)--(-0.8939,2.5072)--(-0.8181,2.3504)--(-0.7424,2.1973)--(-0.6666,2.0481)--(-0.5909,1.9027)--(-0.5151,1.7611)--(-0.4393,1.6234)--(-0.3636,1.4895)--(-0.2878,1.3594)--(-0.2121,1.2331)--(-0.1363,1.1107)--(-0.0606,0.9921)--(0.0151,0.8773)--(0.0909,0.7663)--(0.1666,0.6592)--(0.2424,0.5559)--(0.3181,0.4564)--(0.3939,0.3608)--(0.4696,0.2689)--(0.5454,0.1809)--(0.6212,0.0968)--(0.6969,0.0164)--(0.7727,-0.0600)--(0.8484,-0.1327)--(0.9242,-0.2016)--(1.0000,-0.2666)--(1.0757,-0.3278)--(1.1515,-0.3852)--(1.2272,-0.4388)--(1.3030,-0.4885)--(1.3787,-0.5344)--(1.4545,-0.5765)--(1.5303,-0.6148)--(1.6060,-0.6492)--(1.6818,-0.6798)--(1.7575,-0.7066)--(1.8333,-0.7296)--(1.9090,-0.7487)--(1.9848,-0.7640)--(2.0606,-0.7755)--(2.1363,-0.7831)--(2.2121,-0.7870)--(2.2878,-0.7870)--(2.3636,-0.7831)--(2.4393,-0.7755)--(2.5151,-0.7640)--(2.5909,-0.7487)--(2.6666,-0.7296)--(2.7424,-0.7066)--(2.8181,-0.6798)--(2.8939,-0.6492)--(2.9696,-0.6148)--(3.0454,-0.5765)--(3.1212,-0.5344)--(3.1969,-0.4885)--(3.2727,-0.4388)--(3.3484,-0.3852)--(3.4242,-0.3278)--(3.5000,-0.2666)--(3.5757,-0.2016)--(3.6515,-0.1327)--(3.7272,-0.0600)--(3.8030,0.0164)--(3.8787,0.0968)--(3.9545,0.1809)--(4.0303,0.2689)--(4.1060,0.3608)--(4.1818,0.4564)--(4.2575,0.5559)--(4.3333,0.6592)--(4.4090,0.7663)--(4.4848,0.8773)--(4.5606,0.9921)--(4.6363,1.1107)--(4.7121,1.2331)--(4.7878,1.3594)--(4.8636,1.4895)--(4.9393,1.6234)--(5.0151,1.7611)--(5.0909,1.9027)--(5.1666,2.0481)--(5.2424,2.1973)--(5.3181,2.3504)--(5.3939,2.5072)--(5.4696,2.6679)--(5.5454,2.8325)--(5.6212,3.0008)--(5.6969,3.1730)--(5.7727,3.3490)--(5.8484,3.5288)--(5.9242,3.7125)--(6.0000,3.9000);
+\draw [color=red,style=dotted] (-0.9000,0.0000) -- (-0.9000,2.5200);
+\draw [color=green,style=dashed] (-1.2000,3.1500) -- (0.6000,-0.6300);
+\draw []  (-0.9000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9000,-0.4059) node {$x_n$};
+\draw []  (0.3000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.3000,-0.4191) node {$x_{n+1}$};
+\draw []  (-0.9000,2.5200) node [rotate=0] {$\bullet$};
+\draw (-0.5041,2.8461) node {$y_n$};
+\draw []  (0.7129,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.7129,0.4059) node {$r_0$};
+\draw []  (3.7870,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.7870,0.4059) node {$r_1$};
+\draw []  (2.2500,-0.7875) node [rotate=0] {$\bullet$};
 \draw (2.2500,-1.2122) node {$S$};
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.0000,-0.31492) node {$ 2 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 2 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_MomentForce.pstricks b/auto/pictures_tex/Fig_MomentForce.pstricks
index 6e2f5e73e..3cf09ebb3 100644
--- a/auto/pictures_tex/Fig_MomentForce.pstricks
+++ b/auto/pictures_tex/Fig_MomentForce.pstricks
@@ -81,15 +81,15 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.42471) node {$O$};
-\draw [,->,>=latex] (0,0) -- (-1.0000,-1.0000);
-\draw (-1.0000,-0.44196) node {$\overline{ R }$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.4247) node {$O$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (-1.0000,-1.0000);
+\draw (-1.0000,-0.4419) node {$\overline{ R }$};
 \draw [,->,>=latex] (-1.0000,-1.0000) -- (-3.0000,-1.5000);
-\draw (-3.0000,-1.0420) node {$\overline{ F }$};
-\draw [color=blue,style=dotted] (0,0) -- (0.176,-0.706);
-\draw (0.47427,-0.15344) node {$d$};
-\draw [color=brown,style=dashed] (-1.00,-1.00) -- (0.467,-0.633);
+\draw (-3.0000,-1.0419) node {$\overline{ F }$};
+\draw [color=blue,style=dotted] (0.0000,0.0000) -- (0.1764,-0.7058);
+\draw (0.4742,-0.1534) node {$d$};
+\draw [color=brown,style=dashed] (-1.0000,-1.0000) -- (0.4675,-0.6331);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_MoulinEau.pstricks b/auto/pictures_tex/Fig_MoulinEau.pstricks
index a61cb53e6..cc264ec62 100644
--- a/auto/pictures_tex/Fig_MoulinEau.pstricks
+++ b/auto/pictures_tex/Fig_MoulinEau.pstricks
@@ -35,14 +35,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (0.0000,0.0000);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
@@ -75,16 +75,16 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (-1.29,0.468);
-\draw [color=red,->,>=latex] (-0.64279,1.2340) -- (-1.2296,1.7264);
-\draw [color=green,->,>=latex] (-0.64279,1.2340) -- (-1.0560,0.74155);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (-1.2855,0.4679);
+\draw [color=red,->,>=latex] (-0.6427,1.2339) -- (-1.2296,1.7263);
+\draw [color=green,->,>=latex] (-0.6427,1.2339) -- (-1.0559,0.7415);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_NEtAchr.pstricks b/auto/pictures_tex/Fig_NEtAchr.pstricks
index eb02886ee..b0d2b1846 100644
--- a/auto/pictures_tex/Fig_NEtAchr.pstricks
+++ b/auto/pictures_tex/Fig_NEtAchr.pstricks
@@ -69,11 +69,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=brown] plot [smooth,tension=1] coordinates {(-1.50,-0.500)(0.500,-0.300)(2.00,1.00)(3.50,1.50)(5.00,2.70)(5.80,2.70)};
-\draw []  (-1.5000,-0.50000) node [rotate=0] {$\bullet$};
-\draw (-1.5000,-0.17897) node {\( +\)};
-\draw []  (0.50000,-0.30000) node [rotate=0] {$\bullet$};
-\draw (0.50000,0.021030) node {\( +\)};
+\draw [color=brown] plot [smooth,tension=1] coordinates {(-1.5000,-0.5000)(0.5000,-0.3000)(2.0000,1.0000)(3.5000,1.5000)(5.0000,2.7000)(5.8000,2.7000)};
+\draw []  (-1.5000,-0.5000) node [rotate=0] {$\bullet$};
+\draw (-1.5000,-0.1789) node {\( +\)};
+\draw []  (0.5000,-0.3000) node [rotate=0] {$\bullet$};
+\draw (0.5000,0.0210) node {\( +\)};
 \draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw (2.0000,1.3210) node {\( +\)};
 \draw []  (3.5000,1.5000) node [rotate=0] {$\bullet$};
diff --git a/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks b/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks
index fc28bdea2..307497c31 100644
--- a/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks
+++ b/auto/pictures_tex/Fig_NOCGooYRHLCn.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.2499);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
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+\draw [color=blue] (0.0000,0.9722)--(0.0505,1.0168)--(0.1010,1.0608)--(0.1515,1.1043)--(0.2020,1.1472)--(0.2525,1.1896)--(0.3030,1.2313)--(0.3535,1.2725)--(0.4040,1.3132)--(0.4545,1.3533)--(0.5050,1.3928)--(0.5555,1.4317)--(0.6060,1.4701)--(0.6565,1.5079)--(0.7070,1.5451)--(0.7575,1.5818)--(0.8080,1.6179)--(0.8585,1.6535)--(0.9090,1.6884)--(0.9595,1.7228)--(1.0101,1.7567)--(1.0606,1.7899)--(1.1111,1.8227)--(1.1616,1.8548)--(1.2121,1.8864)--(1.2626,1.9174)--(1.3131,1.9478)--(1.3636,1.9777)--(1.4141,2.0070)--(1.4646,2.0357)--(1.5151,2.0639)--(1.5656,2.0915)--(1.6161,2.1185)--(1.6666,2.1450)--(1.7171,2.1709)--(1.7676,2.1963)--(1.8181,2.2210)--(1.8686,2.2452)--(1.9191,2.2689)--(1.9696,2.2919)--(2.0202,2.3144)--(2.0707,2.3364)--(2.1212,2.3577)--(2.1717,2.3785)--(2.2222,2.3988)--(2.2727,2.4185)--(2.3232,2.4376)--(2.3737,2.4561)--(2.4242,2.4741)--(2.4747,2.4915)--(2.5252,2.5083)--(2.5757,2.5246)--(2.6262,2.5403)--(2.6767,2.5554)--(2.7272,2.5700)--(2.7777,2.5840)--(2.8282,2.5974)--(2.8787,2.6103)--(2.9292,2.6226)--(2.9797,2.6343)--(3.0303,2.6455)--(3.0808,2.6561)--(3.1313,2.6661)--(3.1818,2.6756)--(3.2323,2.6845)--(3.2828,2.6928)--(3.3333,2.7006)--(3.3838,2.7078)--(3.4343,2.7144)--(3.4848,2.7205)--(3.5353,2.7260)--(3.5858,2.7309)--(3.6363,2.7353)--(3.6868,2.7391)--(3.7373,2.7423)--(3.7878,2.7450)--(3.8383,2.7470)--(3.8888,2.7486)--(3.9393,2.7495)--(3.9898,2.7499)--(4.0404,2.7498)--(4.0909,2.7490)--(4.1414,2.7477)--(4.1919,2.7459)--(4.2424,2.7434)--(4.2929,2.7404)--(4.3434,2.7368)--(4.3939,2.7327)--(4.4444,2.7280)--(4.4949,2.7227)--(4.5454,2.7169)--(4.5959,2.7105)--(4.6464,2.7035)--(4.6969,2.6960)--(4.7474,2.6879)--(4.7979,2.6792)--(4.8484,2.6700)--(4.8989,2.6602)--(4.9494,2.6498)--(5.0000,2.6388);
+\draw [color=blue] (0.0000,0.0000)--(0.0505,0.0000)--(0.1010,0.0000)--(0.1515,0.0000)--(0.2020,0.0000)--(0.2525,0.0000)--(0.3030,0.0000)--(0.3535,0.0000)--(0.4040,0.0000)--(0.4545,0.0000)--(0.5050,0.0000)--(0.5555,0.0000)--(0.6060,0.0000)--(0.6565,0.0000)--(0.7070,0.0000)--(0.7575,0.0000)--(0.8080,0.0000)--(0.8585,0.0000)--(0.9090,0.0000)--(0.9595,0.0000)--(1.0101,0.0000)--(1.0606,0.0000)--(1.1111,0.0000)--(1.1616,0.0000)--(1.2121,0.0000)--(1.2626,0.0000)--(1.3131,0.0000)--(1.3636,0.0000)--(1.4141,0.0000)--(1.4646,0.0000)--(1.5151,0.0000)--(1.5656,0.0000)--(1.6161,0.0000)--(1.6666,0.0000)--(1.7171,0.0000)--(1.7676,0.0000)--(1.8181,0.0000)--(1.8686,0.0000)--(1.9191,0.0000)--(1.9696,0.0000)--(2.0202,0.0000)--(2.0707,0.0000)--(2.1212,0.0000)--(2.1717,0.0000)--(2.2222,0.0000)--(2.2727,0.0000)--(2.3232,0.0000)--(2.3737,0.0000)--(2.4242,0.0000)--(2.4747,0.0000)--(2.5252,0.0000)--(2.5757,0.0000)--(2.6262,0.0000)--(2.6767,0.0000)--(2.7272,0.0000)--(2.7777,0.0000)--(2.8282,0.0000)--(2.8787,0.0000)--(2.9292,0.0000)--(2.9797,0.0000)--(3.0303,0.0000)--(3.0808,0.0000)--(3.1313,0.0000)--(3.1818,0.0000)--(3.2323,0.0000)--(3.2828,0.0000)--(3.3333,0.0000)--(3.3838,0.0000)--(3.4343,0.0000)--(3.4848,0.0000)--(3.5353,0.0000)--(3.5858,0.0000)--(3.6363,0.0000)--(3.6868,0.0000)--(3.7373,0.0000)--(3.7878,0.0000)--(3.8383,0.0000)--(3.8888,0.0000)--(3.9393,0.0000)--(3.9898,0.0000)--(4.0404,0.0000)--(4.0909,0.0000)--(4.1414,0.0000)--(4.1919,0.0000)--(4.2424,0.0000)--(4.2929,0.0000)--(4.3434,0.0000)--(4.3939,0.0000)--(4.4444,0.0000)--(4.4949,0.0000)--(4.5454,0.0000)--(4.5959,0.0000)--(4.6464,0.0000)--(4.6969,0.0000)--(4.7474,0.0000)--(4.7979,0.0000)--(4.8484,0.0000)--(4.8989,0.0000)--(4.9494,0.0000)--(5.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (0.0000,0.9722);
+\draw [] (5.0000,2.6388) -- (5.0000,0.0000);
 
-\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3485,0.6490)--(-0.2727,0.7215)--(-0.1970,0.7928)--(-0.1212,0.8628)--(-0.04545,0.9316)--(0.03030,0.9991)--(0.1061,1.065)--(0.1818,1.130)--(0.2576,1.194)--(0.3333,1.256)--(0.4091,1.317)--(0.4848,1.377)--(0.5606,1.436)--(0.6364,1.493)--(0.7121,1.549)--(0.7879,1.604)--(0.8636,1.657)--(0.9394,1.709)--(1.015,1.760)--(1.091,1.810)--(1.167,1.858)--(1.242,1.905)--(1.318,1.951)--(1.394,1.995)--(1.470,2.039)--(1.545,2.081)--(1.621,2.121)--(1.697,2.161)--(1.773,2.199)--(1.848,2.236)--(1.924,2.271)--(2.000,2.306)--(2.076,2.339)--(2.152,2.370)--(2.227,2.401)--(2.303,2.430)--(2.379,2.458)--(2.455,2.485)--(2.530,2.510)--(2.606,2.534)--(2.682,2.557)--(2.758,2.578)--(2.833,2.599)--(2.909,2.618)--(2.985,2.635)--(3.061,2.652)--(3.136,2.667)--(3.212,2.681)--(3.288,2.694)--(3.364,2.705)--(3.439,2.715)--(3.515,2.724)--(3.591,2.731)--(3.667,2.738)--(3.742,2.743)--(3.818,2.746)--(3.894,2.749)--(3.970,2.750)--(4.045,2.750)--(4.121,2.748)--(4.197,2.746)--(4.273,2.742)--(4.349,2.737)--(4.424,2.730)--(4.500,2.722)--(4.576,2.713)--(4.651,2.703)--(4.727,2.691)--(4.803,2.678)--(4.879,2.664)--(4.955,2.649)--(5.030,2.632)--(5.106,2.614)--(5.182,2.595)--(5.258,2.574)--(5.333,2.552)--(5.409,2.529)--(5.485,2.505)--(5.561,2.479)--(5.636,2.452)--(5.712,2.424)--(5.788,2.395)--(5.864,2.364)--(5.939,2.332)--(6.015,2.299)--(6.091,2.264)--(6.167,2.228)--(6.242,2.191)--(6.318,2.153)--(6.394,2.113)--(6.470,2.072)--(6.545,2.030)--(6.621,1.987)--(6.697,1.942)--(6.773,1.896)--(6.849,1.848)--(6.924,1.800)--(7.000,1.750);
+\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3484,0.6489)--(-0.2727,0.7215)--(-0.1969,0.7928)--(-0.1212,0.8628)--(-0.0454,0.9315)--(0.0303,0.9990)--(0.1060,1.0652)--(0.1818,1.1301)--(0.2575,1.1938)--(0.3333,1.2561)--(0.4090,1.3172)--(0.4848,1.3770)--(0.5606,1.4356)--(0.6363,1.4928)--(0.7121,1.5488)--(0.7878,1.6035)--(0.8636,1.6570)--(0.9393,1.7091)--(1.0151,1.7600)--(1.0909,1.8096)--(1.1666,1.8580)--(1.2424,1.9050)--(1.3181,1.9508)--(1.3939,1.9953)--(1.4696,2.0386)--(1.5454,2.0805)--(1.6212,2.1212)--(1.6969,2.1606)--(1.7727,2.1988)--(1.8484,2.2356)--(1.9242,2.2712)--(2.0000,2.3055)--(2.0757,2.3385)--(2.1515,2.3703)--(2.2272,2.4008)--(2.3030,2.4300)--(2.3787,2.4579)--(2.4545,2.4846)--(2.5303,2.5099)--(2.6060,2.5341)--(2.6818,2.5569)--(2.7575,2.5784)--(2.8333,2.5987)--(2.9090,2.6177)--(2.9848,2.6354)--(3.0606,2.6519)--(3.1363,2.6671)--(3.2121,2.6810)--(3.2878,2.6936)--(3.3636,2.7050)--(3.4393,2.7150)--(3.5151,2.7238)--(3.5909,2.7314)--(3.6666,2.7376)--(3.7424,2.7426)--(3.8181,2.7463)--(3.8939,2.7487)--(3.9696,2.7498)--(4.0454,2.7497)--(4.1212,2.7483)--(4.1969,2.7456)--(4.2727,2.7417)--(4.3484,2.7365)--(4.4242,2.7300)--(4.5000,2.7222)--(4.5757,2.7131)--(4.6515,2.7028)--(4.7272,2.6912)--(4.8030,2.6783)--(4.8787,2.6641)--(4.9545,2.6487)--(5.0303,2.6320)--(5.1060,2.6140)--(5.1818,2.5948)--(5.2575,2.5742)--(5.3333,2.5524)--(5.4090,2.5293)--(5.4848,2.5050)--(5.5606,2.4793)--(5.6363,2.4524)--(5.7121,2.4242)--(5.7878,2.3948)--(5.8636,2.3640)--(5.9393,2.3320)--(6.0151,2.2987)--(6.0909,2.2642)--(6.1666,2.2283)--(6.2424,2.1912)--(6.3181,2.1528)--(6.3939,2.1132)--(6.4696,2.0722)--(6.5454,2.0300)--(6.6212,1.9865)--(6.6969,1.9418)--(6.7727,1.8957)--(6.8484,1.8484)--(6.9242,1.7998)--(7.0000,1.7500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
-\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
-\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
-\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
-\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
-\draw (5.4420,3.2118) node {$f(x)$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.27858) node {$x$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.34557) node {$x+\Delta x$};
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,2.6388) -- (6.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,0.0000) --  cycle;
+\draw [color=red,style=dashed] (5.0000,0.0000) -- (6.0000,0.0000);
+\draw [color=red,style=dashed] (6.0000,0.0000) -- (6.0000,2.6388);
+\draw [color=red,style=dashed] (6.0000,2.6388) -- (5.0000,2.6388);
+\draw [color=red,style=dashed] (5.0000,2.6388) -- (5.0000,0.0000);
+\draw []  (5.0000,2.6388) node [rotate=0] {$\bullet$};
+\draw (5.4419,3.2118) node {$f(x)$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.2785) node {$x$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.3455) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_NWDooOObSHB.pstricks b/auto/pictures_tex/Fig_NWDooOObSHB.pstricks
index 0769eba3a..ddce63f29 100644
--- a/auto/pictures_tex/Fig_NWDooOObSHB.pstricks
+++ b/auto/pictures_tex/Fig_NWDooOObSHB.pstricks
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.3000,0) -- (7.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.7812);
+\draw [,->,>=latex] (-8.3000,0.0000) -- (7.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.7812);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,11 +85,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (-7.8000,0.0000) -- (-7.8000,1.1250);
+\draw [] (-2.6000,2.1250) -- (-2.6000,0.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,17 +97,17 @@
 % setting the default values
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+\draw [] (-2.6000,0.0000) -- (-2.6000,2.1250);
+\draw [] (6.5000,6.2812) -- (6.5000,0.0000);
+\draw []  (-7.8000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-7.8000,-0.2785) node {\( a\)};
+\draw []  (-2.6000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-2.6000,-0.3267) node {\( b\)};
+\draw []  (6.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.2785) node {\( c\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_NiveauHyperbole.pstricks b/auto/pictures_tex/Fig_NiveauHyperbole.pstricks
index 342639491..216897fa5 100644
--- a/auto/pictures_tex/Fig_NiveauHyperbole.pstricks
+++ b/auto/pictures_tex/Fig_NiveauHyperbole.pstricks
@@ -95,47 +95,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks b/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks
index 9dc95b22d..24dbed392 100644
--- a/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks
+++ b/auto/pictures_tex/Fig_NiveauHyperboleDeux.pstricks
@@ -95,43 +95,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-3.6623) -- (0,3.6623);
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+\draw [,->,>=latex] (0.0000,-3.6622) -- (0.0000,3.6622);
 %DEFAULT
 
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-\draw [color=brown] (-3.00,-3.00) -- (3.00,3.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw (0.21069,0.80458) node {$R$};
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-\draw (-1.0000,-0.32983) node {$ -1 $};
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-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
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-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
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-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
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+\draw [color=brown] (-3.0000,3.0000) -- (3.0000,-3.0000);
+\draw [color=brown] (-3.0000,-3.0000) -- (3.0000,3.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.2106,0.8045) node {$R$};
+\draw []  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (0.1931,-0.8045) node {$S$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks b/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
index 292a4a5f5..c82750494 100644
--- a/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
+++ b/auto/pictures_tex/Fig_OQTEoodIwAPfZE.pstricks
@@ -116,39 +116,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.9000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-2.8131) -- (0,3.6993);
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+\draw [,->,>=latex] (0.0000,-2.8130) -- (0.0000,3.6993);
 %DEFAULT
 
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-\draw (-0.38250,3.0000) node {$ 30 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-4.4000,-2.3130)--(-4.3050,-1.7785)--(-4.2101,-1.3073)--(-4.1151,-0.8931)--(-4.0202,-0.5307)--(-3.9252,-0.2149)--(-3.8303,0.0587)--(-3.7353,0.2944)--(-3.6404,0.4959)--(-3.5454,0.6666)--(-3.4505,0.8097)--(-3.3555,0.9280)--(-3.2606,1.0240)--(-3.1656,1.1001)--(-3.0707,1.1584)--(-2.9757,1.2008)--(-2.8808,1.2292)--(-2.7858,1.2451)--(-2.6909,1.2500)--(-2.5959,1.2452)--(-2.5010,1.2319)--(-2.4060,1.2112)--(-2.3111,1.1841)--(-2.2161,1.1515)--(-2.1212,1.1142)--(-2.0262,1.0730)--(-1.9313,1.0286)--(-1.8363,0.9816)--(-1.7414,0.9325)--(-1.6464,0.8818)--(-1.5515,0.8301)--(-1.4565,0.7778)--(-1.3616,0.7251)--(-1.2666,0.6726)--(-1.1717,0.6205)--(-1.0767,0.5690)--(-0.9818,0.5185)--(-0.8868,0.4693)--(-0.7919,0.4214)--(-0.6969,0.3751)--(-0.6020,0.3307)--(-0.5070,0.2882)--(-0.4121,0.2478)--(-0.3171,0.2096)--(-0.2222,0.1738)--(-0.1272,0.1405)--(-0.0323,0.1098)--(0.0626,0.0818)--(0.1575,0.0565)--(0.2525,0.0340)--(0.3474,0.0145)--(0.4424,-0.0020)--(0.5373,-0.0156)--(0.6323,-0.0260)--(0.7272,-0.0334)--(0.8222,-0.0376)--(0.9171,-0.0385)--(1.0121,-0.0363)--(1.1070,-0.0307)--(1.2020,-0.0218)--(1.2969,-0.0096)--(1.3919,0.0058)--(1.4868,0.0248)--(1.5818,0.0471)--(1.6767,0.0729)--(1.7717,0.1021)--(1.8666,0.1347)--(1.9616,0.1708)--(2.0565,0.2104)--(2.1515,0.2535)--(2.2464,0.3001)--(2.3414,0.3502)--(2.4363,0.4038)--(2.5313,0.4610)--(2.6262,0.5217)--(2.7212,0.5859)--(2.8161,0.6537)--(2.9111,0.7250)--(3.0060,0.7999)--(3.1010,0.8783)--(3.1959,0.9603)--(3.2909,1.0459)--(3.3858,1.1350)--(3.4808,1.2278)--(3.5757,1.3241)--(3.6707,1.4239)--(3.7656,1.5274)--(3.8606,1.6345)--(3.9555,1.7451)--(4.0505,1.8593)--(4.1454,1.9771)--(4.2404,2.0986)--(4.3353,2.2236)--(4.4303,2.3522)--(4.5252,2.4843)--(4.6202,2.6201)--(4.7151,2.7595)--(4.8101,2.9025)--(4.9050,3.0491)--(5.0000,3.1993);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.5244,-2.0000) node {$ -20 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.5244,-1.0000) node {$ -10 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.3824,1.0000) node {$ 10 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.3824,2.0000) node {$ 20 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.3824,3.0000) node {$ 30 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_Osculateur.pstricks b/auto/pictures_tex/Fig_Osculateur.pstricks
index 3136c04e9..c76dc0bf1 100644
--- a/auto/pictures_tex/Fig_Osculateur.pstricks
+++ b/auto/pictures_tex/Fig_Osculateur.pstricks
@@ -67,17 +67,17 @@
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+\draw []  (-15.296,-1.9021) node [rotate=0] {$\bullet$};
 
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 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_PHTVjfk.pstricks b/auto/pictures_tex/Fig_PHTVjfk.pstricks
index 833669663..e9326c40f 100644
--- a/auto/pictures_tex/Fig_PHTVjfk.pstricks
+++ b/auto/pictures_tex/Fig_PHTVjfk.pstricks
@@ -79,29 +79,29 @@
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-\draw [,->,>=latex] (0,-2.2315) -- (0,2.2315);
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+\draw [,->,>=latex] (0.0000,-2.2314) -- (0.0000,2.2314);
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+\draw [] plot [smooth,tension=1] coordinates {(-0.0335,-0.7062)(0.0335,-0.7062)(0.0505,-0.7046)(0.1006,-0.6996)(0.1677,-0.6862)(0.2348,-0.6661)(0.3020,-0.6392)(0.3054,-0.6375)(0.3691,-0.6021)(0.4166,-0.5704)(0.4362,-0.5558)(0.4961,-0.5033)(0.5033,-0.4961)(0.5558,-0.4362)(0.5704,-0.4166)(0.6021,-0.3691)(0.6375,-0.3054)(0.6392,-0.3020)(0.6661,-0.2348)(0.6862,-0.1677)(0.6996,-0.1006)(0.7046,-0.0505)(0.7062,-0.0335)(0.7062,0.0335)(0.7046,0.0505)(0.6996,0.1006)(0.6862,0.1677)(0.6661,0.2348)(0.6392,0.3020)(0.6375,0.3054)(0.6021,0.3691)(0.5704,0.4166)(0.5558,0.4362)(0.5033,0.4961)(0.4961,0.5033)(0.4362,0.5558)(0.4166,0.5704)(0.3691,0.6021)(0.3054,0.6375)(0.3020,0.6392)(0.2348,0.6661)(0.1677,0.6862)(0.1006,0.6996)(0.0505,0.7046)(0.0335,0.7062)(-0.0335,0.7062)(-0.0505,0.7046)(-0.1006,0.6996)(-0.1677,0.6862)(-0.2348,0.6661)(-0.3020,0.6392)(-0.3054,0.6375)(-0.3691,0.6021)(-0.4166,0.5704)(-0.4362,0.5558)(-0.4961,0.5033)(-0.5033,0.4961)(-0.5558,0.4362)(-0.5704,0.4166)(-0.6021,0.3691)(-0.6375,0.3054)(-0.6392,0.3020)(-0.6661,0.2348)(-0.6862,0.1677)(-0.6996,0.1006)(-0.7046,0.0505)(-0.7062,0.0335)(-0.7062,-0.0335)(-0.7046,-0.0505)(-0.6996,-0.1006)(-0.6862,-0.1677)(-0.6661,-0.2348)(-0.6392,-0.3020)(-0.6375,-0.3054)(-0.6021,-0.3691)(-0.5704,-0.4166)(-0.5558,-0.4362)(-0.5033,-0.4961)(-0.4961,-0.5033)(-0.4362,-0.5558)(-0.4166,-0.5704)(-0.3691,-0.6021)(-0.3054,-0.6375)(-0.3020,-0.6392)(-0.2348,-0.6661)(-0.1677,-0.6862)(-0.1006,-0.6996)(-0.0505,-0.7046)(-0.0335,-0.7062)};
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks b/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks
index 2cc978bad..8db4a826a 100644
--- a/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks
+++ b/auto/pictures_tex/Fig_PLTWoocPNeiZir.pstricks
@@ -67,15 +67,15 @@
 %PSTRICKS CODE
 %DEFAULT
 
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+\draw [color=red] (2.9027,2.9000)--(2.8991,3.0143)--(2.8882,3.1282)--(2.8701,3.2411)--(2.8449,3.3527)--(2.8127,3.4625)--(2.7736,3.5700)--(2.7277,3.6748)--(2.6753,3.7765)--(2.6165,3.8746)--(2.5517,3.9688)--(2.4810,4.0588)--(2.4047,4.1440)--(2.3232,4.2243)--(2.2367,4.2992)--(2.1457,4.3684)--(2.0504,4.4318)--(1.9513,4.4890)--(1.8489,4.5398)--(1.7433,4.5840)--(1.6352,4.6214)--(1.5250,4.6519)--(1.4130,4.6753)--(1.2998,4.6916)--(1.1857,4.7007)--(1.0713,4.7025)--(0.9571,4.6971)--(0.8434,4.6844)--(0.7307,4.6645)--(0.6196,4.6375)--(0.5103,4.6036)--(0.4035,4.5627)--(0.2994,4.5152)--(0.1986,4.4612)--(0.1014,4.4009)--(0.0082,4.3345)--(-0.0805,4.2624)--(-0.1646,4.1848)--(-0.2435,4.1020)--(-0.3170,4.0144)--(-0.3849,3.9222)--(-0.4467,3.8260)--(-0.5023,3.7260)--(-0.5515,3.6227)--(-0.5940,3.5165)--(-0.6297,3.4079)--(-0.6584,3.2971)--(-0.6801,3.1848)--(-0.6946,3.0713)--(-0.7018,2.9571)--(-0.7018,2.8428)--(-0.6946,2.7286)--(-0.6801,2.6151)--(-0.6584,2.5028)--(-0.6297,2.3920)--(-0.5940,2.2834)--(-0.5515,2.1772)--(-0.5023,2.0739)--(-0.4467,1.9739)--(-0.3849,1.8777)--(-0.3170,1.7855)--(-0.2435,1.6979)--(-0.1646,1.6151)--(-0.0805,1.5375)--(0.0082,1.4654)--(0.1014,1.3990)--(0.1986,1.3387)--(0.2994,1.2847)--(0.4035,1.2372)--(0.5103,1.1963)--(0.6196,1.1624)--(0.7307,1.1354)--(0.8434,1.1155)--(0.9571,1.1028)--(1.0713,1.0974)--(1.1857,1.0992)--(1.2998,1.1083)--(1.4130,1.1246)--(1.5250,1.1480)--(1.6352,1.1785)--(1.7433,1.2159)--(1.8489,1.2601)--(1.9513,1.3109)--(2.0504,1.3681)--(2.1457,1.4315)--(2.2367,1.5007)--(2.3232,1.5756)--(2.4047,1.6559)--(2.4810,1.7411)--(2.5517,1.8311)--(2.6165,1.9253)--(2.6753,2.0234)--(2.7277,2.1251)--(2.7736,2.2299)--(2.8127,2.3374)--(2.8449,2.4472)--(2.8701,2.5588)--(2.8882,2.6717)--(2.8991,2.7856)--(2.9027,2.9000);
 
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+\draw [color=red] (3.2238,3.5000)--(3.2188,3.6600)--(3.2035,3.8195)--(3.1782,3.9776)--(3.1429,4.1338)--(3.0978,4.2875)--(3.0430,4.4380)--(2.9788,4.5847)--(2.9054,4.7271)--(2.8232,4.8645)--(2.7324,4.9964)--(2.6334,5.1223)--(2.5266,5.2416)--(2.4124,5.3540)--(2.2914,5.4589)--(2.1639,5.5558)--(2.0306,5.6446)--(1.8919,5.7246)--(1.7484,5.7958)--(1.6007,5.8576)--(1.4493,5.9100)--(1.2950,5.9527)--(1.1382,5.9855)--(0.9797,6.0083)--(0.8200,6.0210)--(0.6599,6.0235)--(0.4999,6.0159)--(0.3408,5.9981)--(0.1830,5.9703)--(0.0274,5.9326)--(-0.1254,5.8850)--(-0.2750,5.8279)--(-0.4207,5.7613)--(-0.5619,5.6857)--(-0.6980,5.6013)--(-0.8284,5.5084)--(-0.9527,5.4074)--(-1.0704,5.2987)--(-1.1809,5.1828)--(-1.2839,5.0601)--(-1.3788,4.9311)--(-1.4654,4.7964)--(-1.5433,4.6565)--(-1.6121,4.5119)--(-1.6716,4.3632)--(-1.7216,4.2110)--(-1.7618,4.0560)--(-1.7921,3.8987)--(-1.8124,3.7399)--(-1.8226,3.5800)--(-1.8226,3.4199)--(-1.8124,3.2600)--(-1.7921,3.1012)--(-1.7618,2.9439)--(-1.7216,2.7889)--(-1.6716,2.6367)--(-1.6121,2.4880)--(-1.5433,2.3434)--(-1.4654,2.2035)--(-1.3788,2.0688)--(-1.2839,1.9398)--(-1.1809,1.8171)--(-1.0704,1.7012)--(-0.9527,1.5925)--(-0.8284,1.4915)--(-0.6980,1.3986)--(-0.5619,1.3142)--(-0.4207,1.2386)--(-0.2750,1.1720)--(-0.1254,1.1149)--(0.0274,1.0673)--(0.1830,1.0296)--(0.3408,1.0018)--(0.4999,0.9840)--(0.6599,0.9764)--(0.8200,0.9789)--(0.9797,0.9916)--(1.1382,1.0144)--(1.2950,1.0472)--(1.4493,1.0899)--(1.6007,1.1423)--(1.7484,1.2041)--(1.8919,1.2753)--(2.0306,1.3553)--(2.1639,1.4441)--(2.2914,1.5410)--(2.4124,1.6459)--(2.5266,1.7583)--(2.6334,1.8776)--(2.7324,2.0035)--(2.8232,2.1354)--(2.9054,2.2728)--(2.9788,2.4152)--(3.0430,2.5619)--(3.0978,2.7124)--(3.1429,2.8661)--(3.1782,3.0223)--(3.2035,3.1804)--(3.2188,3.3399)--(3.2238,3.5000);
 \draw []  (2.1000,1.4000) node [rotate=0] {$\bullet$};
-\draw (2.3535,1.1089) node {\( P\)};
-\draw [] (0,0) -- (3.00,2.00);
-\draw [style=dashed] (2.10,1.40) -- (0.100,4.40);
+\draw (2.3534,1.1088) node {\( P\)};
+\draw [] (0.0000,0.0000) -- (3.0000,2.0000);
+\draw [style=dashed] (2.1000,1.4000) -- (0.0999,4.4000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_PONXooXYjEot.pstricks b/auto/pictures_tex/Fig_PONXooXYjEot.pstricks
index 3adcf2189..20c383857 100644
--- a/auto/pictures_tex/Fig_PONXooXYjEot.pstricks
+++ b/auto/pictures_tex/Fig_PONXooXYjEot.pstricks
@@ -79,23 +79,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4998,0) -- (2.4998,0);
-\draw [,->,>=latex] (0,-1.2068) -- (0,1.2068);
+\draw [,->,>=latex] (-2.4998,0.0000) -- (2.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-1.2068) -- (0.0000,1.2068);
 %DEFAULT
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-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
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-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
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+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks b/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks
index ab49695de..9cb321f63 100644
--- a/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks
+++ b/auto/pictures_tex/Fig_PVJooJDyNAg.pstricks
@@ -143,65 +143,65 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.9978,0) -- (5.9978,0);
-\draw [,->,>=latex] (0,-4.0000) -- (0,4.0000);
+\draw [,->,>=latex] (-5.9977,0.0000) -- (5.9977,0.0000);
+\draw [,->,>=latex] (0.0000,-4.0000) -- (0.0000,4.0000);
 %DEFAULT
 
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-\draw [style=dashed] (-5.50,-3.50) -- (-5.50,3.50);
-\draw [style=dashed] (-3.30,-3.50) -- (-3.30,3.50);
-\draw [style=dashed] (-1.10,-3.50) -- (-1.10,3.50);
-\draw [style=dashed] (1.10,-3.50) -- (1.10,3.50);
-\draw [style=dashed] (3.30,-3.50) -- (3.30,3.50);
-\draw [style=dashed] (5.50,-3.50) -- (5.50,3.50);
-\draw (-5.4978,-0.42071) node {$ -\frac{5}{2} \, \pi $};
-\draw [] (-5.50,-0.100) -- (-5.50,0.100);
-\draw (-4.3982,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-4.40,-0.100) -- (-4.40,0.100);
-\draw (-3.2987,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.1991,-0.32103) node {$ -\pi $};
-\draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.0996,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.0996,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.1991,-0.27858) node {$ \pi $};
-\draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.2987,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (4.3982,-0.31492) node {$ 2 \, \pi $};
-\draw [] (4.40,-0.100) -- (4.40,0.100);
-\draw (5.4978,-0.42071) node {$ \frac{5}{2} \, \pi $};
-\draw [] (5.50,-0.100) -- (5.50,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=blue] (3.4685,-2.8273)--(3.4876,-2.5299)--(3.5066,-2.2857)--(3.5257,-2.0814)--(3.5448,-1.9078)--(3.5638,-1.7582)--(3.5829,-1.6277)--(3.6020,-1.5128)--(3.6210,-1.4107)--(3.6401,-1.3191)--(3.6592,-1.2366)--(3.6783,-1.1616)--(3.6973,-1.0931)--(3.7164,-1.0302)--(3.7355,-0.9721)--(3.7545,-0.9183)--(3.7736,-0.8682)--(3.7927,-0.8214)--(3.8117,-0.7774)--(3.8308,-0.7361)--(3.8499,-0.6970)--(3.8689,-0.6601)--(3.8880,-0.6249)--(3.9071,-0.5915)--(3.9261,-0.5595)--(3.9452,-0.5289)--(3.9643,-0.4996)--(3.9833,-0.4713)--(4.0024,-0.4441)--(4.0215,-0.4178)--(4.0405,-0.3924)--(4.0596,-0.3677)--(4.0787,-0.3437)--(4.0977,-0.3203)--(4.1168,-0.2975)--(4.1359,-0.2753)--(4.1549,-0.2535)--(4.1740,-0.2321)--(4.1931,-0.2111)--(4.2121,-0.1905)--(4.2312,-0.1702)--(4.2503,-0.1501)--(4.2693,-0.1303)--(4.2884,-0.1106)--(4.3075,-0.0912)--(4.3265,-0.0718)--(4.3456,-0.0526)--(4.3647,-0.0335)--(4.3837,-0.0144)--(4.4028,0.0046)--(4.4219,0.0237)--(4.4409,0.0428)--(4.4600,0.0619)--(4.4791,0.0812)--(4.4981,0.1006)--(4.5172,0.1201)--(4.5363,0.1399)--(4.5553,0.1598)--(4.5744,0.1800)--(4.5935,0.2005)--(4.6125,0.2213)--(4.6316,0.2424)--(4.6507,0.2640)--(4.6697,0.2860)--(4.6888,0.3085)--(4.7079,0.3316)--(4.7269,0.3552)--(4.7460,0.3796)--(4.7651,0.4046)--(4.7841,0.4305)--(4.8032,0.4572)--(4.8223,0.4849)--(4.8413,0.5137)--(4.8604,0.5436)--(4.8795,0.5748)--(4.8986,0.6075)--(4.9176,0.6418)--(4.9367,0.6778)--(4.9558,0.7157)--(4.9748,0.7558)--(4.9939,0.7984)--(5.0130,0.8437)--(5.0320,0.8921)--(5.0511,0.9439)--(5.0702,0.9997)--(5.0892,1.0601)--(5.1083,1.1256)--(5.1274,1.1971)--(5.1464,1.2756)--(5.1655,1.3623)--(5.1846,1.4587)--(5.2036,1.5668)--(5.2227,1.6888)--(5.2418,1.8281)--(5.2608,1.9887)--(5.2799,2.1762)--(5.2990,2.3984)--(5.3180,2.6664)--(5.3371,2.9965)--(5.3562,3.4137);
+\draw [style=dashed] (-5.4977,-3.5000) -- (-5.4977,3.5000);
+\draw [style=dashed] (-3.2986,-3.5000) -- (-3.2986,3.5000);
+\draw [style=dashed] (-1.0995,-3.5000) -- (-1.0995,3.5000);
+\draw [style=dashed] (1.0995,-3.5000) -- (1.0995,3.5000);
+\draw [style=dashed] (3.2986,-3.5000) -- (3.2986,3.5000);
+\draw [style=dashed] (5.4977,-3.5000) -- (5.4977,3.5000);
+\draw (-5.4977,-0.4207) node {$ -\frac{5}{2} \, \pi $};
+\draw [] (-5.4977,-0.1000) -- (-5.4977,0.1000);
+\draw (-4.3982,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-4.3982,-0.1000) -- (-4.3982,0.1000);
+\draw (-3.2986,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-3.2986,-0.1000) -- (-3.2986,0.1000);
+\draw (-2.1991,-0.3210) node {$ -\pi $};
+\draw [] (-2.1991,-0.1000) -- (-2.1991,0.1000);
+\draw (-1.0995,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.0995,-0.1000) -- (-1.0995,0.1000);
+\draw (1.0995,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.0995,-0.1000) -- (1.0995,0.1000);
+\draw (2.1991,-0.2785) node {$ \pi $};
+\draw [] (2.1991,-0.1000) -- (2.1991,0.1000);
+\draw (3.2986,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (3.2986,-0.1000) -- (3.2986,0.1000);
+\draw (4.3982,-0.3149) node {$ 2 \, \pi $};
+\draw [] (4.3982,-0.1000) -- (4.3982,0.1000);
+\draw (5.4977,-0.4207) node {$ \frac{5}{2} \, \pi $};
+\draw [] (5.4977,-0.1000) -- (5.4977,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks b/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks
index e0da8fb27..212064ed3 100644
--- a/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks
+++ b/auto/pictures_tex/Fig_ParallelogrammeOM.pstricks
@@ -81,19 +81,19 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [,->,>=latex] (0,0) -- (3.0000,0);
-\draw (3.3086,-0.29071) node {$a$};
-\draw [,->,>=latex] (0,0) -- (2.0000,2.0000);
+\draw [,->,>=latex] (0.0000,0.0000) -- (3.0000,0.0000);
+\draw (3.3085,-0.2907) node {$a$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (2.0000,2.0000);
 \draw (2.0000,2.4267) node {$b$};
 \draw []  (5.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw [color=blue,style=dotted] (2.00,2.00) -- (5.00,2.00);
-\draw [color=blue,style=dotted] (3.00,0) -- (5.00,2.00);
-\draw [style=dashed] (2.00,2.00) -- (2.00,0);
+\draw [color=blue,style=dotted] (2.0000,2.0000) -- (5.0000,2.0000);
+\draw [color=blue,style=dotted] (3.0000,0.0000) -- (5.0000,2.0000);
+\draw [style=dashed] (2.0000,2.0000) -- (2.0000,0.0000);
 \draw (2.3051,1.0000) node {$h$};
-\draw (0.80616,0.31808) node {$\theta$};
+\draw (0.8061,0.3180) node {$\theta$};
 
-\draw [] (0.500,0)--(0.500,0.00397)--(0.500,0.00793)--(0.500,0.0119)--(0.500,0.0159)--(0.500,0.0198)--(0.499,0.0238)--(0.499,0.0278)--(0.499,0.0317)--(0.499,0.0357)--(0.498,0.0396)--(0.498,0.0436)--(0.498,0.0475)--(0.497,0.0515)--(0.497,0.0554)--(0.496,0.0594)--(0.496,0.0633)--(0.495,0.0672)--(0.495,0.0712)--(0.494,0.0751)--(0.494,0.0790)--(0.493,0.0829)--(0.492,0.0868)--(0.492,0.0907)--(0.491,0.0946)--(0.490,0.0985)--(0.489,0.102)--(0.489,0.106)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.133)--(0.481,0.137)--(0.480,0.141)--(0.479,0.145)--(0.477,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.473,0.164)--(0.471,0.167)--(0.470,0.171)--(0.468,0.175)--(0.467,0.178)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.193)--(0.460,0.197)--(0.458,0.200)--(0.456,0.204)--(0.455,0.208)--(0.453,0.211)--(0.451,0.215)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.229)--(0.443,0.233)--(0.441,0.236)--(0.439,0.240)--(0.437,0.243)--(0.435,0.247)--(0.433,0.250)--(0.431,0.253)--(0.429,0.257)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.421,0.270)--(0.418,0.274)--(0.416,0.277)--(0.414,0.280)--(0.412,0.284)--(0.410,0.287)--(0.407,0.290)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.398,0.303)--(0.395,0.306)--(0.393,0.309)--(0.391,0.312)--(0.388,0.315)--(0.386,0.318)--(0.383,0.321)--(0.380,0.324)--(0.378,0.327)--(0.375,0.330)--(0.373,0.333)--(0.370,0.336)--(0.367,0.339)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.354,0.354);
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
+\draw [] (0.5000,0.0000)--(0.4999,0.0039)--(0.4999,0.0079)--(0.4998,0.0118)--(0.4997,0.0158)--(0.4996,0.0198)--(0.4994,0.0237)--(0.4992,0.0277)--(0.4989,0.0317)--(0.4987,0.0356)--(0.4984,0.0396)--(0.4980,0.0435)--(0.4977,0.0475)--(0.4973,0.0514)--(0.4969,0.0554)--(0.4964,0.0593)--(0.4959,0.0632)--(0.4954,0.0672)--(0.4949,0.0711)--(0.4943,0.0750)--(0.4937,0.0790)--(0.4930,0.0829)--(0.4924,0.0868)--(0.4916,0.0907)--(0.4909,0.0946)--(0.4901,0.0985)--(0.4894,0.1024)--(0.4885,0.1062)--(0.4877,0.1101)--(0.4868,0.1140)--(0.4859,0.1178)--(0.4849,0.1217)--(0.4839,0.1255)--(0.4829,0.1294)--(0.4819,0.1332)--(0.4808,0.1370)--(0.4797,0.1408)--(0.4786,0.1446)--(0.4774,0.1484)--(0.4762,0.1522)--(0.4750,0.1560)--(0.4737,0.1597)--(0.4725,0.1635)--(0.4711,0.1672)--(0.4698,0.1710)--(0.4684,0.1747)--(0.4670,0.1784)--(0.4656,0.1821)--(0.4641,0.1858)--(0.4626,0.1895)--(0.4611,0.1931)--(0.4596,0.1968)--(0.4580,0.2004)--(0.4564,0.2040)--(0.4548,0.2077)--(0.4531,0.2113)--(0.4514,0.2148)--(0.4497,0.2184)--(0.4479,0.2220)--(0.4462,0.2255)--(0.4444,0.2291)--(0.4425,0.2326)--(0.4407,0.2361)--(0.4388,0.2396)--(0.4369,0.2430)--(0.4349,0.2465)--(0.4330,0.2500)--(0.4310,0.2534)--(0.4289,0.2568)--(0.4269,0.2602)--(0.4248,0.2636)--(0.4227,0.2669)--(0.4206,0.2703)--(0.4184,0.2736)--(0.4162,0.2769)--(0.4140,0.2802)--(0.4118,0.2835)--(0.4095,0.2867)--(0.4072,0.2900)--(0.4049,0.2932)--(0.4026,0.2964)--(0.4002,0.2996)--(0.3978,0.3028)--(0.3954,0.3059)--(0.3930,0.3090)--(0.3905,0.3121)--(0.3880,0.3152)--(0.3855,0.3183)--(0.3830,0.3213)--(0.3804,0.3244)--(0.3778,0.3274)--(0.3752,0.3304)--(0.3726,0.3333)--(0.3699,0.3363)--(0.3672,0.3392)--(0.3645,0.3421)--(0.3618,0.3450)--(0.3591,0.3478)--(0.3563,0.3507)--(0.3535,0.3535);
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ParamTangente.pstricks b/auto/pictures_tex/Fig_ParamTangente.pstricks
index 7a28085b0..1c7965993 100644
--- a/auto/pictures_tex/Fig_ParamTangente.pstricks
+++ b/auto/pictures_tex/Fig_ParamTangente.pstricks
@@ -87,36 +87,36 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0424,0) -- (3.5561,0);
-\draw [,->,>=latex] (0,-3.8750) -- (0,3.9454);
+\draw [,->,>=latex] (-1.0423,0.0000) -- (3.5561,0.0000);
+\draw [,->,>=latex] (0.0000,-3.8750) -- (0.0000,3.9453);
 %DEFAULT
-\draw [color=blue] (3.000,-3.375)--(2.939,-3.175)--(2.879,-2.982)--(2.818,-2.798)--(2.758,-2.621)--(2.697,-2.452)--(2.636,-2.290)--(2.576,-2.136)--(2.515,-1.989)--(2.455,-1.849)--(2.394,-1.715)--(2.333,-1.588)--(2.273,-1.467)--(2.212,-1.353)--(2.152,-1.245)--(2.091,-1.143)--(2.030,-1.046)--(1.970,-0.9552)--(1.909,-0.8697)--(1.848,-0.7895)--(1.788,-0.7144)--(1.727,-0.6442)--(1.667,-0.5787)--(1.606,-0.5178)--(1.545,-0.4614)--(1.485,-0.4092)--(1.424,-0.3611)--(1.364,-0.3170)--(1.303,-0.2766)--(1.242,-0.2397)--(1.182,-0.2063)--(1.121,-0.1762)--(1.061,-0.1491)--(1.000,-0.1250)--(0.9394,-0.1036)--(0.8788,-0.08483)--(0.8182,-0.06846)--(0.7576,-0.05435)--(0.6970,-0.04232)--(0.6364,-0.03221)--(0.5758,-0.02386)--(0.5152,-0.01709)--(0.4545,-0.01174)--(0.3939,-0.007642)--(0.3333,-0.004630)--(0.2727,-0.002536)--(0.2121,-0.001193)--(0.1515,0)--(0.09091,0)--(0.03030,0)--(0.03030,0)--(0.09091,0)--(0.1515,0)--(0.2121,0.001193)--(0.2727,0.002536)--(0.3333,0.004630)--(0.3939,0.007642)--(0.4545,0.01174)--(0.5152,0.01709)--(0.5758,0.02386)--(0.6364,0.03221)--(0.6970,0.04232)--(0.7576,0.05435)--(0.8182,0.06846)--(0.8788,0.08483)--(0.9394,0.1036)--(1.000,0.1250)--(1.061,0.1491)--(1.121,0.1762)--(1.182,0.2063)--(1.242,0.2397)--(1.303,0.2766)--(1.364,0.3170)--(1.424,0.3611)--(1.485,0.4092)--(1.545,0.4614)--(1.606,0.5178)--(1.667,0.5787)--(1.727,0.6442)--(1.788,0.7144)--(1.848,0.7895)--(1.909,0.8697)--(1.970,0.9552)--(2.030,1.046)--(2.091,1.143)--(2.152,1.245)--(2.212,1.353)--(2.273,1.467)--(2.333,1.588)--(2.394,1.715)--(2.455,1.849)--(2.515,1.989)--(2.576,2.136)--(2.636,2.290)--(2.697,2.452)--(2.758,2.621)--(2.818,2.798)--(2.879,2.982)--(2.939,3.175)--(3.000,3.375);
-\draw [color=red,->,>=latex] (2.5078,-1.9715) -- (2.1174,-1.0508);
-\draw [color=red,->,>=latex] (1.7548,-0.67544) -- (1.1001,0.080499);
-\draw [color=red,->,>=latex] (0.45465,-0.011748) -- (-0.54236,0.065536);
-\draw [color=red,->,>=latex] (1.0298,0.13650) -- (1.9590,0.50601);
-\draw [color=red,->,>=latex] (2.0918,1.1442) -- (2.6122,1.9981);
-\draw [color=red,->,>=latex] (2.7162,2.5049) -- (3.0561,3.4454);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (3.0000,-3.3750)--(2.9393,-3.1745)--(2.8787,-2.9822)--(2.8181,-2.7978)--(2.7575,-2.6211)--(2.6969,-2.4521)--(2.6363,-2.2904)--(2.5757,-2.1361)--(2.5151,-1.9888)--(2.4545,-1.8485)--(2.3939,-1.7149)--(2.3333,-1.5879)--(2.2727,-1.4674)--(2.2121,-1.3531)--(2.1515,-1.2449)--(2.0909,-1.1426)--(2.0303,-1.0461)--(1.9696,-0.9552)--(1.9090,-0.8697)--(1.8484,-0.7895)--(1.7878,-0.7143)--(1.7272,-0.6441)--(1.6666,-0.5787)--(1.6060,-0.5178)--(1.5454,-0.4614)--(1.4848,-0.4092)--(1.4242,-0.3611)--(1.3636,-0.3169)--(1.3030,-0.2765)--(1.2424,-0.2397)--(1.1818,-0.2063)--(1.1212,-0.1761)--(1.0606,-0.1491)--(1.0000,-0.1250)--(0.9393,-0.1036)--(0.8787,-0.0848)--(0.8181,-0.0684)--(0.7575,-0.0543)--(0.6969,-0.0423)--(0.6363,-0.0322)--(0.5757,-0.0238)--(0.5151,-0.0170)--(0.4545,-0.0117)--(0.3939,-0.0076)--(0.3333,-0.0046)--(0.2727,-0.0025)--(0.2121,-0.0011)--(0.1515,0.0000)--(0.0909,0.0000)--(0.0303,0.0000)--(0.0303,0.0000)--(0.0909,0.0000)--(0.1515,0.0000)--(0.2121,0.0011)--(0.2727,0.0025)--(0.3333,0.0046)--(0.3939,0.0076)--(0.4545,0.0117)--(0.5151,0.0170)--(0.5757,0.0238)--(0.6363,0.0322)--(0.6969,0.0423)--(0.7575,0.0543)--(0.8181,0.0684)--(0.8787,0.0848)--(0.9393,0.1036)--(1.0000,0.1250)--(1.0606,0.1491)--(1.1212,0.1761)--(1.1818,0.2063)--(1.2424,0.2397)--(1.3030,0.2765)--(1.3636,0.3169)--(1.4242,0.3611)--(1.4848,0.4092)--(1.5454,0.4614)--(1.6060,0.5178)--(1.6666,0.5787)--(1.7272,0.6441)--(1.7878,0.7143)--(1.8484,0.7895)--(1.9090,0.8697)--(1.9696,0.9552)--(2.0303,1.0461)--(2.0909,1.1426)--(2.1515,1.2449)--(2.2121,1.3531)--(2.2727,1.4674)--(2.3333,1.5879)--(2.3939,1.7149)--(2.4545,1.8485)--(2.5151,1.9888)--(2.5757,2.1361)--(2.6363,2.2904)--(2.6969,2.4521)--(2.7575,2.6211)--(2.8181,2.7978)--(2.8787,2.9822)--(2.9393,3.1745)--(3.0000,3.3750);
+\draw [color=red,->,>=latex] (2.5078,-1.9714) -- (2.1174,-1.0508);
+\draw [color=red,->,>=latex] (1.7547,-0.6754) -- (1.1001,0.0804);
+\draw [color=red,->,>=latex] (0.4546,-0.0117) -- (-0.5423,0.0655);
+\draw [color=red,->,>=latex] (1.0297,0.1364) -- (1.9589,0.5060);
+\draw [color=red,->,>=latex] (2.0918,1.1441) -- (2.6122,1.9981);
+\draw [color=red,->,>=latex] (2.7162,2.5049) -- (3.0561,3.4453);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_PartieEntiere.pstricks b/auto/pictures_tex/Fig_PartieEntiere.pstricks
index 5820d4ebf..3b5006c38 100644
--- a/auto/pictures_tex/Fig_PartieEntiere.pstricks
+++ b/auto/pictures_tex/Fig_PartieEntiere.pstricks
@@ -83,32 +83,32 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,3.5000);
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.9739,1.0000)--(1.9789,1.0000)--(1.9839,1.0000)--(1.9889,1.0000)--(1.9939,1.0000)--(1.9989,1.0000)--(2.0040,2.0000)--(2.0090,2.0000)--(2.0140,2.0000)--(2.0190,2.0000)--(2.0240,2.0000)--(2.0290,2.0000)--(2.0340,2.0000)--(2.0390,2.0000)--(2.0440,2.0000)--(2.0490,2.0000)--(2.0540,2.0000)--(2.0590,2.0000)--(2.0640,2.0000)--(2.0690,2.0000)--(2.0740,2.0000)--(2.0790,2.0000)--(2.0840,2.0000)--(2.0890,2.0000)--(2.0940,2.0000)--(2.0990,2.0000)--(2.1041,2.0000)--(2.1091,2.0000)--(2.1141,2.0000)--(2.1191,2.0000)--(2.1241,2.0000)--(2.1291,2.0000)--(2.1341,2.0000)--(2.1391,2.0000)--(2.1441,2.0000)--(2.1491,2.0000)--(2.1541,2.0000)--(2.1591,2.0000)--(2.1641,2.0000)--(2.1691,2.0000)--(2.1741,2.0000)--(2.1791,2.0000)--(2.1841,2.0000)--(2.1891,2.0000)--(2.1941,2.0000)--(2.1991,2.0000)--(2.2042,2.0000)--(2.2092,2.0000)--(2.2142,2.0000)--(2.2192,2.0000)--(2.2242,2.0000)--(2.2292,2.0000)--(2.2342,2.0000)--(2.2392,2.0000)--(2.2442,2.0000)--(2.2492,2.0000)--(2.2542,2.0000)--(2.2592,2.0000)--(2.2642,2.0000)--(2.2692,2.0000)--(2.2742,2.0000)--(2.2792,2.0000)--(2.2842,2.0000)--(2.2892,2.0000)--(2.2942,2.0000)--(2.2992,2.0000)--(2.3043,2.0000)--(2.3093,2.0000)--(2.3143,2.0000)--(2.3193,2.0000)--(2.3243,2.0000)--(2.3293,2.0000)--(2.3343,2.0000)--(2.3393,2.0000)--(2.3443,2.0000)--(2.3493,2.0000)--(2.3543,2.0000)--(2.3593,2.0000)--(2.3643,2.0000)--(2.3693,2.0000)--(2.3743,2.0000)--(2.3793,2.0000)--(2.3843,2.0000)--(2.3893,2.0000)--(2.3943,2.0000)--(2.3993,2.0000)--(2.4044,2.0000)--(2.4094,2.0000)--(2.4144,2.0000)--(2.4194,2.0000)--(2.4244,2.0000)--(2.4294,2.0000)--(2.4344,2.0000)--(2.4394,2.0000)--(2.4444,2.0000)--(2.4494,2.0000)--(2.4544,2.0000)--(2.4594,2.0000)--(2.4644,2.0000)--(2.4694,2.0000)--(2.4744,2.0000)--(2.4794,2.0000)--(2.4844,2.0000)--(2.4894,2.0000)--(2.4944,2.0000)--(2.4994,2.0000)--(2.5045,2.0000)--(2.5095,2.0000)--(2.5145,2.0000)--(2.5195,2.0000)--(2.5245,2.0000)--(2.5295,2.0000)--(2.5345,2.0000)--(2.5395,2.0000)--(2.5445,2.0000)--(2.5495,2.0000)--(2.5545,2.0000)--(2.5595,2.0000)--(2.5645,2.0000)--(2.5695,2.0000)--(2.5745,2.0000)--(2.5795,2.0000)--(2.5845,2.0000)--(2.5895,2.0000)--(2.5945,2.0000)--(2.5995,2.0000)--(2.6046,2.0000)--(2.6096,2.0000)--(2.6146,2.0000)--(2.6196,2.0000)--(2.6246,2.0000)--(2.6296,2.0000)--(2.6346,2.0000)--(2.6396,2.0000)--(2.6446,2.0000)--(2.6496,2.0000)--(2.6546,2.0000)--(2.6596,2.0000)--(2.6646,2.0000)--(2.6696,2.0000)--(2.6746,2.0000)--(2.6796,2.0000)--(2.6846,2.0000)--(2.6896,2.0000)--(2.6946,2.0000)--(2.6996,2.0000)--(2.7047,2.0000)--(2.7097,2.0000)--(2.7147,2.0000)--(2.7197,2.0000)--(2.7247,2.0000)--(2.7297,2.0000)--(2.7347,2.0000)--(2.7397,2.0000)--(2.7447,2.0000)--(2.7497,2.0000)--(2.7547,2.0000)--(2.7597,2.0000)--(2.7647,2.0000)--(2.7697,2.0000)--(2.7747,2.0000)--(2.7797,2.0000)--(2.7847,2.0000)--(2.7897,2.0000)--(2.7947,2.0000)--(2.7997,2.0000)--(2.8048,2.0000)--(2.8098,2.0000)--(2.8148,2.0000)--(2.8198,2.0000)--(2.8248,2.0000)--(2.8298,2.0000)--(2.8348,2.0000)--(2.8398,2.0000)--(2.8448,2.0000)--(2.8498,2.0000)--(2.8548,2.0000)--(2.8598,2.0000)--(2.8648,2.0000)--(2.8698,2.0000)--(2.8748,2.0000)--(2.8798,2.0000)--(2.8848,2.0000)--(2.8898,2.0000)--(2.8948,2.0000)--(2.8998,2.0000)--(2.9049,2.0000)--(2.9099,2.0000)--(2.9149,2.0000)--(2.9199,2.0000)--(2.9249,2.0000)--(2.9299,2.0000)--(2.9349,2.0000)--(2.9399,2.0000)--(2.9449,2.0000)--(2.9499,2.0000)--(2.9549,2.0000)--(2.9599,2.0000)--(2.9649,2.0000)--(2.9699,2.0000)--(2.9749,2.0000)--(2.9799,2.0000)--(2.9849,2.0000)--(2.9899,2.0000)--(2.9949,2.0000)--(3.0000,3.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_Polirettangolo.pstricks b/auto/pictures_tex/Fig_Polirettangolo.pstricks
index 8d5164762..0eb98b05f 100644
--- a/auto/pictures_tex/Fig_Polirettangolo.pstricks
+++ b/auto/pictures_tex/Fig_Polirettangolo.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -89,11 +89,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,2.00) -- (1.50,2.00) -- (1.50,2.00) -- (1.50,1.00) -- (1.50,1.00) -- (0,1.00) -- (0,1.00) -- (0,2.00) --  cycle;
-\draw [style=dotted] (0,2.00) -- (1.50,2.00);
-\draw [style=dotted] (1.50,2.00) -- (1.50,1.00);
-\draw [style=dotted] (1.50,1.00) -- (0,1.00);
-\draw [style=dotted] (0,1.00) -- (0,2.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.0000,2.0000) -- (1.5000,2.0000) -- (1.5000,2.0000) -- (1.5000,1.0000) -- (1.5000,1.0000) -- (0.0000,1.0000) -- (0.0000,1.0000) -- (0.0000,2.0000) --  cycle;
+\draw [style=dotted] (0.0000,2.0000) -- (1.5000,2.0000);
+\draw [style=dotted] (1.5000,2.0000) -- (1.5000,1.0000);
+\draw [style=dotted] (1.5000,1.0000) -- (0.0000,1.0000);
+\draw [style=dotted] (0.0000,1.0000) -- (0.0000,2.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -101,11 +101,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.500,2.50) -- (2.00,2.50) -- (2.00,2.50) -- (2.00,2.00) -- (2.00,2.00) -- (0.500,2.00) -- (0.500,2.00) -- (0.500,2.50) --  cycle;
-\draw [style=dotted] (0.500,2.50) -- (2.00,2.50);
-\draw [style=dotted] (2.00,2.50) -- (2.00,2.00);
-\draw [style=dotted] (2.00,2.00) -- (0.500,2.00);
-\draw [style=dotted] (0.500,2.00) -- (0.500,2.50);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.5000,2.5000) -- (2.0000,2.5000) -- (2.0000,2.5000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (0.5000,2.0000) -- (0.5000,2.0000) -- (0.5000,2.5000) --  cycle;
+\draw [style=dotted] (0.5000,2.5000) -- (2.0000,2.5000);
+\draw [style=dotted] (2.0000,2.5000) -- (2.0000,2.0000);
+\draw [style=dotted] (2.0000,2.0000) -- (0.5000,2.0000);
+\draw [style=dotted] (0.5000,2.0000) -- (0.5000,2.5000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -113,11 +113,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,1.00) -- (3.00,1.00) -- (3.00,1.00) -- (3.00,0) -- (3.00,0) -- (2.00,0) -- (2.00,0) -- (2.00,1.00) --  cycle;
-\draw [style=dotted] (2.00,1.00) -- (3.00,1.00);
-\draw [style=dotted] (3.00,1.00) -- (3.00,0);
-\draw [style=dotted] (3.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (2.00,1.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.0000,1.0000) -- (3.0000,1.0000) -- (3.0000,1.0000) -- (3.0000,0.0000) -- (3.0000,0.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (2.0000,1.0000) --  cycle;
+\draw [style=dotted] (2.0000,1.0000) -- (3.0000,1.0000);
+\draw [style=dotted] (3.0000,1.0000) -- (3.0000,0.0000);
+\draw [style=dotted] (3.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (2.0000,1.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -125,27 +125,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,3.50) -- (3.50,3.50) -- (3.50,3.50) -- (3.50,1.50) -- (3.50,1.50) -- (2.00,1.50) -- (2.00,1.50) -- (2.00,3.50) --  cycle;
-\draw [style=dotted] (2.00,3.50) -- (3.50,3.50);
-\draw [style=dotted] (3.50,3.50) -- (3.50,1.50);
-\draw [style=dotted] (3.50,1.50) -- (2.00,1.50);
-\draw [style=dotted] (2.00,1.50) -- (2.00,3.50);
-\draw (1.0000,-0.31492) node {$ 2 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 4 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 6 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 8 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.0000,3.5000) -- (3.5000,3.5000) -- (3.5000,3.5000) -- (3.5000,1.5000) -- (3.5000,1.5000) -- (2.0000,1.5000) -- (2.0000,1.5000) -- (2.0000,3.5000) --  cycle;
+\draw [style=dotted] (2.0000,3.5000) -- (3.5000,3.5000);
+\draw [style=dotted] (3.5000,3.5000) -- (3.5000,1.5000);
+\draw [style=dotted] (3.5000,1.5000) -- (2.0000,1.5000);
+\draw [style=dotted] (2.0000,1.5000) -- (2.0000,3.5000);
+\draw (1.0000,-0.3149) node {$ 2 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 4 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 6 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 8 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ProjPoly.pstricks b/auto/pictures_tex/Fig_ProjPoly.pstricks
index 2b8360aeb..4abb9a4fc 100644
--- a/auto/pictures_tex/Fig_ProjPoly.pstricks
+++ b/auto/pictures_tex/Fig_ProjPoly.pstricks
@@ -111,57 +111,57 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.8117,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-2.8116,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
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-\draw (-2.5000,-0.32983) node {$ -5 $};
-\draw [] (-2.50,-0.100) -- (-2.50,0.100);
-\draw (-2.0000,-0.32983) node {$ -4 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.5000,-0.32983) node {$ -3 $};
-\draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (-1.0000,-0.32983) node {$ -2 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (-0.50000,-0.32983) node {$ -1 $};
-\draw [] (-0.500,-0.100) -- (-0.500,0.100);
-\draw (0.50000,-0.31492) node {$ 1 $};
-\draw [] (0.500,-0.100) -- (0.500,0.100);
-\draw (1.0000,-0.31492) node {$ 2 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (1.5000,-0.31492) node {$ 3 $};
-\draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (2.0000,-0.31492) node {$ 4 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (2.5000,-0.31492) node {$ 5 $};
-\draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (3.0000,-0.31492) node {$ 6 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -6 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.5000) node {$ -5 $};
-\draw [] (-0.100,-2.50) -- (0.100,-2.50);
-\draw (-0.43316,-2.0000) node {$ -4 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.5000) node {$ -3 $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.43316,-1.0000) node {$ -2 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.43316,-0.50000) node {$ -1 $};
-\draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.29125,0.50000) node {$ 1 $};
-\draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,1.5000) node {$ 3 $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,2.5000) node {$ 5 $};
-\draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
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+\draw [] plot [smooth,tension=1] coordinates {(2.5000,2.1793)(2.4853,2.1644)(2.4664,2.1447)(2.4529,2.1308)(2.4328,2.1100)(2.4204,2.0973)(2.3993,2.0752)(2.3880,2.0637)(2.3657,2.0405)(2.3557,2.0302)(2.3322,2.0057)(2.3233,1.9966)(2.2986,1.9708)(2.2910,1.9630)(2.2651,1.9359)(2.2588,1.9295)(2.2315,1.9010)(2.2266,1.8959)(2.1979,1.8660)(2.1944,1.8624)(2.1644,1.8309)(2.1623,1.8288)(2.1308,1.7958)(2.1303,1.7953)(2.0983,1.7617)(2.0973,1.7606)(2.0663,1.7281)(2.0637,1.7254)(2.0344,1.6946)(2.0302,1.6901)(2.0025,1.6610)(1.9966,1.6547)(1.9707,1.6275)(1.9630,1.6192)(1.9390,1.5939)(1.9295,1.5836)(1.9074,1.5604)(1.8959,1.5480)(1.8758,1.5268)(1.8624,1.5123)(1.8444,1.4932)(1.8288,1.4764)(1.8130,1.4597)(1.7953,1.4405)(1.7817,1.4261)(1.7617,1.4044)(1.7505,1.3926)(1.7281,1.3683)(1.7194,1.3590)(1.6946,1.3320)(1.6885,1.3255)(1.6610,1.2955)(1.6576,1.2919)(1.6275,1.2589)(1.6269,1.2583)(1.5963,1.2248)(1.5939,1.2221)(1.5658,1.1912)(1.5604,1.1851)(1.5355,1.1577)(1.5268,1.1478)(1.5053,1.1241)(1.4932,1.1104)(1.4754,1.0906)(1.4597,1.0726)(1.4456,1.0570)(1.4261,1.0347)(1.4161,1.0234)(1.3926,0.9964)(1.3867,0.9899)(1.3590,0.9578)(1.3577,0.9563)(1.3288,0.9228)(1.3255,0.9187)(1.3002,0.8892)(1.2919,0.8791)(1.2719,0.8557)(1.2583,0.8390)(1.2440,0.8221)(1.2248,0.7984)(1.2165,0.7885)(1.1912,0.7572)(1.1894,0.7550)(1.1626,0.7214)(1.1577,0.7150)(1.1362,0.6879)(1.1241,0.6718)(1.1104,0.6543)(1.0906,0.6276)(1.0853,0.6208)(1.0607,0.5872)(1.0570,0.5819)(1.0366,0.5536)(1.0234,0.5343)(1.0134,0.5201)(0.9910,0.4865)(0.9899,0.4847)(0.9693,0.4530)(0.9563,0.4315)(0.9487,0.4194)(0.9291,0.3859)(0.9228,0.3740)(0.9105,0.3523)(0.8934,0.3187)(0.8892,0.3097)(0.8773,0.2852)(0.8629,0.2516)(0.8557,0.2325)(0.8499,0.2181)(0.8384,0.1845)(0.8288,0.1510)(0.8221,0.1213)(0.8211,0.1174)(0.8151,0.0838)(0.8110,0.0503)(0.8090,0.0167)(0.8090,-0.0167)(0.8110,-0.0503)(0.8151,-0.0838)(0.8211,-0.1174)(0.8221,-0.1213)(0.8288,-0.1510)(0.8384,-0.1845)(0.8499,-0.2181)(0.8557,-0.2325)(0.8629,-0.2516)(0.8773,-0.2852)(0.8892,-0.3097)(0.8934,-0.3187)(0.9105,-0.3523)(0.9228,-0.3740)(0.9291,-0.3859)(0.9487,-0.4194)(0.9563,-0.4315)(0.9693,-0.4530)(0.9899,-0.4847)(0.9910,-0.4865)(1.0134,-0.5201)(1.0234,-0.5343)(1.0366,-0.5536)(1.0570,-0.5819)(1.0607,-0.5872)(1.0853,-0.6208)(1.0906,-0.6276)(1.1104,-0.6543)(1.1241,-0.6718)(1.1362,-0.6879)(1.1577,-0.7150)(1.1626,-0.7214)(1.1894,-0.7550)(1.1912,-0.7572)(1.2165,-0.7885)(1.2248,-0.7984)(1.2440,-0.8221)(1.2583,-0.8390)(1.2719,-0.8557)(1.2919,-0.8791)(1.3002,-0.8892)(1.3255,-0.9187)(1.3288,-0.9228)(1.3577,-0.9563)(1.3590,-0.9578)(1.3867,-0.9899)(1.3926,-0.9964)(1.4161,-1.0234)(1.4261,-1.0347)(1.4456,-1.0570)(1.4597,-1.0726)(1.4754,-1.0906)(1.4932,-1.1104)(1.5053,-1.1241)(1.5268,-1.1478)(1.5355,-1.1577)(1.5604,-1.1851)(1.5658,-1.1912)(1.5939,-1.2221)(1.5963,-1.2248)(1.6269,-1.2583)(1.6275,-1.2589)(1.6576,-1.2919)(1.6610,-1.2955)(1.6885,-1.3255)(1.6946,-1.3320)(1.7194,-1.3590)(1.7281,-1.3683)(1.7505,-1.3926)(1.7617,-1.4044)(1.7817,-1.4261)(1.7953,-1.4405)(1.8130,-1.4597)(1.8288,-1.4764)(1.8444,-1.4932)(1.8624,-1.5123)(1.8758,-1.5268)(1.8959,-1.5480)(1.9074,-1.5604)(1.9295,-1.5836)(1.9390,-1.5939)(1.9630,-1.6192)(1.9707,-1.6275)(1.9966,-1.6547)(2.0025,-1.6610)(2.0302,-1.6901)(2.0344,-1.6946)(2.0637,-1.7254)(2.0663,-1.7281)(2.0973,-1.7606)(2.0983,-1.7617)(2.1303,-1.7953)(2.1308,-1.7958)(2.1623,-1.8288)(2.1644,-1.8309)(2.1944,-1.8624)(2.1979,-1.8660)(2.2266,-1.8959)(2.2315,-1.9010)(2.2588,-1.9295)(2.2651,-1.9359)(2.2910,-1.9630)(2.2986,-1.9708)(2.3233,-1.9966)(2.3322,-2.0057)(2.3557,-2.0302)(2.3657,-2.0405)(2.3880,-2.0637)(2.3993,-2.0752)(2.4204,-2.0973)(2.4328,-2.1100)(2.4529,-2.1308)(2.4664,-2.1447)(2.4853,-2.1644)(2.5000,-2.1793)};
+\draw (-2.5000,-0.3298) node {$ -5 $};
+\draw [] (-2.5000,-0.1000) -- (-2.5000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -4 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.5000,-0.3298) node {$ -3 $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -2 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (-0.5000,-0.3298) node {$ -1 $};
+\draw [] (-0.5000,-0.1000) -- (-0.5000,0.1000);
+\draw (0.5000,-0.3149) node {$ 1 $};
+\draw [] (0.5000,-0.1000) -- (0.5000,0.1000);
+\draw (1.0000,-0.3149) node {$ 2 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (1.5000,-0.3149) node {$ 3 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (2.0000,-0.3149) node {$ 4 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (2.5000,-0.3149) node {$ 5 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 6 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -6 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.5000) node {$ -5 $};
+\draw [] (-0.1000,-2.5000) -- (0.1000,-2.5000);
+\draw (-0.4331,-2.0000) node {$ -4 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.5000) node {$ -3 $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.4331,-1.0000) node {$ -2 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.4331,-0.5000) node {$ -1 $};
+\draw [] (-0.1000,-0.5000) -- (0.1000,-0.5000);
+\draw (-0.2912,0.5000) node {$ 1 $};
+\draw [] (-0.1000,0.5000) -- (0.1000,0.5000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,1.5000) node {$ 3 $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,2.5000) node {$ 5 $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ProjectionScalaire.pstricks b/auto/pictures_tex/Fig_ProjectionScalaire.pstricks
index aa94342af..ce42fcd82 100644
--- a/auto/pictures_tex/Fig_ProjectionScalaire.pstricks
+++ b/auto/pictures_tex/Fig_ProjectionScalaire.pstricks
@@ -75,19 +75,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue,->,>=latex] (0,0) -- (1.5000,2.0000);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (1.5000,2.0000);
 \draw (1.8776,2.3368) node {$X$};
-\draw [color=blue,->,>=latex] (0,0) -- (2.5000,0);
-\draw (2.5000,0.32471) node {$Y$};
-\draw []  (1.5000,0) node [rotate=0] {$\bullet$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (0.75000,-0.30000) -- (0,-0.30000);
-\draw [,->,>=latex] (0.75000,-0.30000) -- (1.5000,-0.30000);
-\draw (0.75000,-0.67858) node {$x$};
-\draw [style=dotted] (1.50,2.00) -- (1.50,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (2.5000,0.0000);
+\draw (2.5000,0.3247) node {$Y$};
+\draw []  (1.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (0.7500,-0.3000) -- (0.0000,-0.3000);
+\draw [,->,>=latex] (0.7500,-0.3000) -- (1.5000,-0.3000);
+\draw (0.7500,-0.6785) node {$x$};
+\draw [style=dotted] (1.5000,2.0000) -- (1.5000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QCb.pstricks b/auto/pictures_tex/Fig_QCb.pstricks
index 389c36fef..3a23fea8c 100644
--- a/auto/pictures_tex/Fig_QCb.pstricks
+++ b/auto/pictures_tex/Fig_QCb.pstricks
@@ -71,15 +71,15 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw (-0.79967,1.0000) node {\( xy\)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw (-0.7996,1.0000) node {\( xy\)};
 \draw (1.5676,1.0000) node {\( \sin(xy)\)};
 \draw (1.2003,-1.0000) node {\( xy\)};
-\draw (-0.79967,-1.0000) node {\( xy\)};
+\draw (-0.7996,-1.0000) node {\( xy\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks b/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks
index 3b06e3eda..7a7dd07e9 100644
--- a/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks
+++ b/auto/pictures_tex/Fig_QIZooQNQSJj.pstricks
@@ -79,17 +79,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw (-1.4116,1.5804) node {\( A\)};
-\draw (-1.9800,-0.79165) node {\( B\)};
-\draw (1.9759,0.79165) node {\( C\)};
+\draw (-1.9799,-0.7916) node {\( B\)};
+\draw (1.9758,0.7916) node {\( C\)};
 
-\draw [] (1.750,0)--(1.746,0.1110)--(1.736,0.2215)--(1.718,0.3312)--(1.694,0.4395)--(1.663,0.5461)--(1.625,0.6504)--(1.580,0.7521)--(1.529,0.8508)--(1.472,0.9461)--(1.409,1.038)--(1.341,1.125)--(1.267,1.208)--(1.187,1.286)--(1.103,1.358)--(1.015,1.426)--(0.9226,1.487)--(0.8265,1.543)--(0.7270,1.592)--(0.6245,1.635)--(0.5196,1.671)--(0.4126,1.701)--(0.3039,1.723)--(0.1940,1.739)--(0.08327,1.748)--(-0.02777,1.750)--(-0.1387,1.744)--(-0.2491,1.732)--(-0.3584,1.713)--(-0.4663,1.687)--(-0.5724,1.654)--(-0.6761,1.614)--(-0.7771,1.568)--(-0.8750,1.516)--(-0.9694,1.457)--(-1.060,1.393)--(-1.146,1.323)--(-1.228,1.247)--(-1.304,1.167)--(-1.376,1.082)--(-1.441,0.9924)--(-1.501,0.8989)--(-1.555,0.8019)--(-1.603,0.7016)--(-1.644,0.5985)--(-1.679,0.4930)--(-1.707,0.3855)--(-1.728,0.2765)--(-1.742,0.1663)--(-1.749,0.05552)--(-1.749,-0.05552)--(-1.742,-0.1663)--(-1.728,-0.2765)--(-1.707,-0.3855)--(-1.679,-0.4930)--(-1.644,-0.5985)--(-1.603,-0.7016)--(-1.555,-0.8019)--(-1.501,-0.8989)--(-1.441,-0.9924)--(-1.376,-1.082)--(-1.304,-1.167)--(-1.228,-1.247)--(-1.146,-1.323)--(-1.060,-1.393)--(-0.9694,-1.457)--(-0.8750,-1.516)--(-0.7771,-1.568)--(-0.6761,-1.614)--(-0.5724,-1.654)--(-0.4663,-1.687)--(-0.3584,-1.713)--(-0.2491,-1.732)--(-0.1387,-1.744)--(-0.02777,-1.750)--(0.08327,-1.748)--(0.1940,-1.739)--(0.3039,-1.723)--(0.4126,-1.701)--(0.5196,-1.671)--(0.6245,-1.635)--(0.7270,-1.592)--(0.8265,-1.543)--(0.9226,-1.487)--(1.015,-1.426)--(1.103,-1.358)--(1.187,-1.286)--(1.267,-1.208)--(1.341,-1.125)--(1.409,-1.038)--(1.472,-0.9461)--(1.529,-0.8508)--(1.580,-0.7521)--(1.625,-0.6504)--(1.663,-0.5461)--(1.694,-0.4395)--(1.718,-0.3312)--(1.736,-0.2215)--(1.746,-0.1110)--(1.750,0);
-\draw (-0.42285,-0.55206) node {\( H\)};
-\draw [] (-1.21,1.27) -- (-0.658,-0.239);
-\draw [] (-1.21,1.27) -- (-1.64,-0.599);
-\draw [] (-1.64,-0.599) -- (1.64,0.599);
-\draw [] (1.64,0.599) -- (-1.21,1.27);
-\draw [] (-0.478,0.145) -- (-0.760,0.0425);
-\draw [] (-0.478,0.145) -- (-0.376,-0.137);
+\draw [] (1.7500,0.0000)--(1.7464,0.1109)--(1.7359,0.2215)--(1.7183,0.3311)--(1.6939,0.4395)--(1.6626,0.5460)--(1.6246,0.6504)--(1.5801,0.7521)--(1.5292,0.8508)--(1.4721,0.9461)--(1.4092,1.0375)--(1.3405,1.1248)--(1.2665,1.2076)--(1.1873,1.2855)--(1.1034,1.3582)--(1.0150,1.4255)--(0.9226,1.4870)--(0.8264,1.5425)--(0.7269,1.5918)--(0.6245,1.6347)--(0.5196,1.6710)--(0.4125,1.7006)--(0.3038,1.7234)--(0.1939,1.7392)--(0.0832,1.7480)--(-0.0277,1.7497)--(-0.1386,1.7444)--(-0.2490,1.7321)--(-0.3584,1.7129)--(-0.4663,1.6867)--(-0.5723,1.6537)--(-0.6761,1.6141)--(-0.7771,1.5679)--(-0.8750,1.5155)--(-0.9693,1.4569)--(-1.0598,1.3925)--(-1.1460,1.3225)--(-1.2275,1.2472)--(-1.3042,1.1668)--(-1.3755,1.0817)--(-1.4414,0.9923)--(-1.5014,0.8989)--(-1.5554,0.8018)--(-1.6031,0.7016)--(-1.6444,0.5985)--(-1.6791,0.4930)--(-1.7070,0.3855)--(-1.7280,0.2765)--(-1.7420,0.1663)--(-1.7491,0.0555)--(-1.7491,-0.0555)--(-1.7420,-0.1663)--(-1.7280,-0.2765)--(-1.7070,-0.3855)--(-1.6791,-0.4930)--(-1.6444,-0.5985)--(-1.6031,-0.7016)--(-1.5554,-0.8018)--(-1.5014,-0.8989)--(-1.4414,-0.9923)--(-1.3755,-1.0817)--(-1.3042,-1.1668)--(-1.2275,-1.2472)--(-1.1460,-1.3225)--(-1.0598,-1.3925)--(-0.9693,-1.4569)--(-0.8749,-1.5155)--(-0.7771,-1.5679)--(-0.6761,-1.6141)--(-0.5723,-1.6537)--(-0.4663,-1.6867)--(-0.3584,-1.7129)--(-0.2490,-1.7321)--(-0.1386,-1.7444)--(-0.0277,-1.7497)--(0.0832,-1.7480)--(0.1939,-1.7392)--(0.3038,-1.7234)--(0.4125,-1.7006)--(0.5196,-1.6710)--(0.6245,-1.6347)--(0.7269,-1.5918)--(0.8264,-1.5425)--(0.9226,-1.4870)--(1.0150,-1.4255)--(1.1034,-1.3582)--(1.1873,-1.2855)--(1.2665,-1.2076)--(1.3405,-1.1248)--(1.4092,-1.0375)--(1.4721,-0.9461)--(1.5292,-0.8508)--(1.5801,-0.7521)--(1.6246,-0.6504)--(1.6626,-0.5460)--(1.6939,-0.4395)--(1.7183,-0.3311)--(1.7359,-0.2215)--(1.7464,-0.1109)--(1.7500,0.0000);
+\draw (-0.4228,-0.5520) node {\( H\)};
+\draw [] (-1.2063,1.2677) -- (-0.6577,-0.2394);
+\draw [] (-1.2063,1.2677) -- (-1.6444,-0.5985);
+\draw [] (-1.6444,-0.5985) -- (1.6444,0.5985);
+\draw [] (1.6444,0.5985) -- (-1.2063,1.2677);
+\draw [] (-0.4784,0.1450) -- (-0.7603,0.0424);
+\draw [] (-0.4784,0.1450) -- (-0.3758,-0.1368);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks b/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
index 25a73757d..f6e6c8bba 100644
--- a/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
+++ b/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
@@ -80,23 +80,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
-\draw []  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw (-1.0000,-0.32471) node {\( A\)};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.32471) node {\( B\)};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0.28570,-0.26613) node {\( O\)};
-\draw []  (0,2.8284) node [rotate=0] {$\bullet$};
-\draw (0.23597,3.0946) node {\( I\)};
+\draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.0000,-0.3247) node {\( A\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3247) node {\( B\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.2856,-0.2661) node {\( O\)};
+\draw []  (0.0000,2.8284) node [rotate=0] {$\bullet$};
+\draw (0.2359,3.0945) node {\( I\)};
 
-\draw [] (2.000,0)--(1.994,0.1903)--(1.976,0.3798)--(1.946,0.5678)--(1.904,0.7534)--(1.850,0.9361)--(1.785,1.115)--(1.709,1.289)--(1.622,1.459)--(1.524,1.622)--(1.416,1.779)--(1.298,1.928)--(1.171,2.070)--(1.036,2.204)--(0.8917,2.328)--(0.7402,2.444)--(0.5817,2.549)--(0.4168,2.644)--(0.2462,2.729)--(0.07066,2.802)--(-0.1092,2.865)--(-0.2927,2.915)--(-0.4791,2.954)--(-0.6675,2.982)--(-0.8573,2.997)--(-1.048,3.000)--(-1.238,2.991)--(-1.427,2.969)--(-1.614,2.936)--(-1.799,2.892)--(-1.981,2.835)--(-2.159,2.767)--(-2.332,2.688)--(-2.500,2.598)--(-2.662,2.498)--(-2.817,2.387)--(-2.965,2.267)--(-3.104,2.138)--(-3.236,2.000)--(-3.358,1.854)--(-3.471,1.701)--(-3.574,1.541)--(-3.667,1.375)--(-3.748,1.203)--(-3.819,1.026)--(-3.878,0.8452)--(-3.926,0.6609)--(-3.962,0.4740)--(-3.986,0.2852)--(-3.999,0.09518)--(-3.999,-0.09518)--(-3.986,-0.2852)--(-3.962,-0.4740)--(-3.926,-0.6609)--(-3.878,-0.8452)--(-3.819,-1.026)--(-3.748,-1.203)--(-3.667,-1.375)--(-3.574,-1.541)--(-3.471,-1.701)--(-3.358,-1.854)--(-3.236,-2.000)--(-3.104,-2.138)--(-2.965,-2.267)--(-2.817,-2.387)--(-2.662,-2.498)--(-2.500,-2.598)--(-2.332,-2.688)--(-2.159,-2.767)--(-1.981,-2.835)--(-1.799,-2.892)--(-1.614,-2.936)--(-1.427,-2.969)--(-1.238,-2.991)--(-1.048,-3.000)--(-0.8573,-2.997)--(-0.6675,-2.982)--(-0.4791,-2.954)--(-0.2927,-2.915)--(-0.1092,-2.865)--(0.07066,-2.802)--(0.2462,-2.729)--(0.4168,-2.644)--(0.5817,-2.549)--(0.7402,-2.444)--(0.8917,-2.328)--(1.036,-2.204)--(1.171,-2.070)--(1.298,-1.928)--(1.416,-1.779)--(1.524,-1.622)--(1.622,-1.459)--(1.709,-1.289)--(1.785,-1.115)--(1.850,-0.9361)--(1.904,-0.7534)--(1.946,-0.5678)--(1.976,-0.3798)--(1.994,-0.1903)--(2.000,0);
+\draw [] (2.0000,0.0000)--(1.9939,0.1902)--(1.9758,0.3797)--(1.9457,0.5677)--(1.9038,0.7534)--(1.8502,0.9361)--(1.7851,1.1149)--(1.7087,1.2893)--(1.6215,1.4585)--(1.5237,1.6219)--(1.4158,1.7787)--(1.2981,1.9283)--(1.1712,2.0702)--(1.0355,2.2037)--(0.8916,2.3284)--(0.7401,2.4437)--(0.5816,2.5491)--(0.4168,2.6443)--(0.2462,2.7288)--(0.0706,2.8024)--(-0.1092,2.8647)--(-0.2927,2.9154)--(-0.4790,2.9544)--(-0.6674,2.9815)--(-0.8572,2.9966)--(-1.0475,2.9996)--(-1.2377,2.9905)--(-1.4269,2.9694)--(-1.6144,2.9364)--(-1.7994,2.8915)--(-1.9812,2.8350)--(-2.1590,2.7670)--(-2.3321,2.6879)--(-2.5000,2.5980)--(-2.6617,2.4977)--(-2.8168,2.3872)--(-2.9645,2.2672)--(-3.1044,2.1380)--(-3.2357,2.0003)--(-3.3581,1.8544)--(-3.4710,1.7011)--(-3.5739,1.5410)--(-3.6665,1.3746)--(-3.7483,1.2027)--(-3.8190,1.0260)--(-3.8784,0.8451)--(-3.9262,0.6609)--(-3.9623,0.4740)--(-3.9864,0.2851)--(-3.9984,0.0951)--(-3.9984,-0.0951)--(-3.9864,-0.2851)--(-3.9623,-0.4740)--(-3.9262,-0.6609)--(-3.8784,-0.8451)--(-3.8190,-1.0260)--(-3.7483,-1.2027)--(-3.6665,-1.3746)--(-3.5739,-1.5410)--(-3.4710,-1.7011)--(-3.3581,-1.8544)--(-3.2357,-2.0003)--(-3.1044,-2.1380)--(-2.9645,-2.2672)--(-2.8168,-2.3872)--(-2.6617,-2.4977)--(-2.5000,-2.5980)--(-2.3321,-2.6879)--(-2.1590,-2.7670)--(-1.9812,-2.8350)--(-1.7994,-2.8915)--(-1.6144,-2.9364)--(-1.4269,-2.9694)--(-1.2377,-2.9905)--(-1.0475,-2.9996)--(-0.8572,-2.9966)--(-0.6674,-2.9815)--(-0.4790,-2.9544)--(-0.2927,-2.9154)--(-0.1092,-2.8647)--(0.0706,-2.8024)--(0.2462,-2.7288)--(0.4168,-2.6443)--(0.5816,-2.5491)--(0.7401,-2.4437)--(0.8916,-2.3284)--(1.0355,-2.2037)--(1.1712,-2.0702)--(1.2981,-1.9283)--(1.4158,-1.7787)--(1.5237,-1.6219)--(1.6215,-1.4585)--(1.7087,-1.2893)--(1.7851,-1.1149)--(1.8502,-0.9361)--(1.9038,-0.7534)--(1.9457,-0.5677)--(1.9758,-0.3797)--(1.9939,-0.1902)--(2.0000,0.0000);
 
-\draw [] (4.000,0)--(3.994,0.1903)--(3.976,0.3798)--(3.946,0.5678)--(3.904,0.7534)--(3.850,0.9361)--(3.785,1.115)--(3.709,1.289)--(3.622,1.459)--(3.524,1.622)--(3.416,1.779)--(3.298,1.928)--(3.171,2.070)--(3.036,2.204)--(2.892,2.328)--(2.740,2.444)--(2.582,2.549)--(2.417,2.644)--(2.246,2.729)--(2.071,2.802)--(1.891,2.865)--(1.707,2.915)--(1.521,2.954)--(1.333,2.982)--(1.143,2.997)--(0.9524,3.000)--(0.7623,2.991)--(0.5731,2.969)--(0.3856,2.936)--(0.2006,2.892)--(0.01880,2.835)--(-0.1590,2.767)--(-0.3322,2.688)--(-0.5000,2.598)--(-0.6618,2.498)--(-0.8168,2.387)--(-0.9646,2.267)--(-1.104,2.138)--(-1.236,2.000)--(-1.358,1.854)--(-1.471,1.701)--(-1.574,1.541)--(-1.667,1.375)--(-1.748,1.203)--(-1.819,1.026)--(-1.878,0.8452)--(-1.926,0.6609)--(-1.962,0.4740)--(-1.986,0.2852)--(-1.998,0.09518)--(-1.998,-0.09518)--(-1.986,-0.2852)--(-1.962,-0.4740)--(-1.926,-0.6609)--(-1.878,-0.8452)--(-1.819,-1.026)--(-1.748,-1.203)--(-1.667,-1.375)--(-1.574,-1.541)--(-1.471,-1.701)--(-1.358,-1.854)--(-1.236,-2.000)--(-1.104,-2.138)--(-0.9646,-2.267)--(-0.8168,-2.387)--(-0.6618,-2.498)--(-0.5000,-2.598)--(-0.3322,-2.688)--(-0.1590,-2.767)--(0.01880,-2.835)--(0.2006,-2.892)--(0.3856,-2.936)--(0.5731,-2.969)--(0.7623,-2.991)--(0.9524,-3.000)--(1.143,-2.997)--(1.333,-2.982)--(1.521,-2.954)--(1.707,-2.915)--(1.891,-2.865)--(2.071,-2.802)--(2.246,-2.729)--(2.417,-2.644)--(2.582,-2.549)--(2.740,-2.444)--(2.892,-2.328)--(3.036,-2.204)--(3.171,-2.070)--(3.298,-1.928)--(3.416,-1.779)--(3.524,-1.622)--(3.622,-1.459)--(3.709,-1.289)--(3.785,-1.115)--(3.850,-0.9361)--(3.904,-0.7534)--(3.946,-0.5678)--(3.976,-0.3798)--(3.994,-0.1903)--(4.000,0);
-\draw []  (-0.70000,1.0000) node [rotate=0] {$\bullet$};
-\draw (-0.98570,1.3016) node {\( Q\)};
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+\draw []  (-0.7000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9856,1.3016) node {\( Q\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks b/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks
index 21bc4eded..bfe913e1b 100644
--- a/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks
+++ b/auto/pictures_tex/Fig_QOBAooZZZOrl.pstricks
@@ -41,22 +41,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4250);
 %DEFAULT
 
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+\draw [] (2.1742,0.0000) -- (2.1742,1.6500);
+\draw [] (5.8257,1.6500) -- (5.8257,0.0000);
+\draw []  (2.1742,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.1742,-0.3785) node {$a$};
+\draw []  (5.8257,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,22 +95,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4250);
 %DEFAULT
 
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+\draw [] (2.1742,0.0000) -- (2.1742,1.6500);
+\draw [] (5.8257,1.6500) -- (5.8257,0.0000);
+\draw []  (2.1742,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.1742,-0.3785) node {$a$};
+\draw []  (5.8257,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,22 +149,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4250);
 %DEFAULT
 
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+\draw [] (2.1742,1.6500) -- (2.1742,1.6500);
+\draw [] (5.8257,1.6500) -- (5.8257,1.6500);
+\draw []  (2.1742,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.1742,-0.3785) node {$a$};
+\draw []  (5.8257,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QPcdHwP.pstricks b/auto/pictures_tex/Fig_QPcdHwP.pstricks
index bc69167a5..4fde4f320 100644
--- a/auto/pictures_tex/Fig_QPcdHwP.pstricks
+++ b/auto/pictures_tex/Fig_QPcdHwP.pstricks
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.30595) node {\( \alpha_2\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,-0.30595) node {\( \alpha_4\)};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3059) node {\( \alpha_2\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,-0.3059) node {\( \alpha_4\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_QQa.pstricks b/auto/pictures_tex/Fig_QQa.pstricks
index b9cb733d7..d4ad1e802 100644
--- a/auto/pictures_tex/Fig_QQa.pstricks
+++ b/auto/pictures_tex/Fig_QQa.pstricks
@@ -33,10 +33,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (0.50000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (0.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,-2.0000)(0.0700,-1.9000)(-0.0700,-1.8000)(0.0700,-1.7000)(-0.0700,-1.6000)(0.0700,-1.5000)(-0.0700,-1.4000)(0.0700,-1.3000)(-0.0700,-1.2000)(0.0700,-1.1000)(-0.0700,-1.0000)(0.0700,-0.9000)(-0.0700,-0.8000)(0.0700,-0.7000)(-0.0700,-0.6000)(0.0700,-0.5000)(-0.0700,-0.4000)(0.0700,-0.3000)(-0.0700,-0.2000)(0.0700,-0.1000)(-0.0700,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -102,10 +102,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,-2.00)(-1.98,-1.88)(-1.81,-1.91)(-1.84,-1.74)(-1.66,-1.76)(-1.69,-1.59)(-1.52,-1.62)(-1.55,-1.45)(-1.38,-1.48)(-1.41,-1.31)(-1.24,-1.34)(-1.26,-1.16)(-1.09,-1.19)(-1.12,-1.02)(-0.951,-1.05)(-0.978,-0.879)(-0.808,-0.907)(-0.835,-0.736)(-0.665,-0.764)(-0.692,-0.593)(-0.522,-0.621)(-0.549,-0.451)(-0.379,-0.478)(-0.407,-0.308)(-0.236,-0.335)(-0.264,-0.165)(-0.0934,-0.192)(-0.121,-0.0219)(0.0495,-0.0495)(0.0219,0.121)(0.192,0.0934)(0.165,0.264)(0.335,0.236)(0.308,0.407)(0.478,0.379)(0.451,0.549)(0.621,0.522)(0.593,0.692)(0.764,0.665)(0.736,0.835)(0.907,0.808)(0.879,0.978)(1.05,0.951)(1.02,1.12)(1.19,1.09)(1.16,1.26)(1.34,1.24)(1.31,1.41)(1.48,1.38)(1.45,1.55)(1.62,1.52)(1.59,1.69)(1.76,1.66)(1.74,1.84)(1.91,1.81)(1.88,1.98)(2.00,2.00)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(-2.0000,-2.0000)(-1.9780,-1.8790)(-1.8076,-1.9066)(-1.8352,-1.7362)(-1.6647,-1.7637)(-1.6923,-1.5933)(-1.5219,-1.6209)(-1.5494,-1.4505)(-1.3790,-1.4780)(-1.4066,-1.3076)(-1.2362,-1.3352)(-1.2637,-1.1647)(-1.0933,-1.1923)(-1.1209,-1.0219)(-0.9505,-1.0494)(-0.9780,-0.8790)(-0.8076,-0.9066)(-0.8352,-0.7362)(-0.6647,-0.7637)(-0.6923,-0.5933)(-0.5219,-0.6209)(-0.5494,-0.4505)(-0.3790,-0.4780)(-0.4066,-0.3076)(-0.2362,-0.3352)(-0.2637,-0.1647)(-0.0933,-0.1923)(-0.1209,-0.0219)(0.0494,-0.0494)(0.0219,0.1209)(0.1923,0.0933)(0.1647,0.2637)(0.3352,0.2362)(0.3076,0.4066)(0.4780,0.3790)(0.4505,0.5494)(0.6209,0.5219)(0.5933,0.6923)(0.7637,0.6647)(0.7362,0.8352)(0.9066,0.8076)(0.8790,0.9780)(1.0494,0.9505)(1.0219,1.1209)(1.1923,1.0933)(1.1647,1.2637)(1.3352,1.2362)(1.3076,1.4066)(1.4780,1.3790)(1.4505,1.5494)(1.6209,1.5219)(1.5933,1.6923)(1.7637,1.6647)(1.7362,1.8352)(1.9066,1.8076)(1.8790,1.9780)(2.0000,2.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -136,11 +136,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.8026) -- (0,1.2352);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.8025) -- (0.0000,1.2351);
 %DEFAULT
 
-\draw [color=red] plot [smooth,tension=1] coordinates {(0.100,-2.30)(0.0409,-2.20)(0.191,-2.11)(0.0654,-1.99)(0.218,-1.91)(0.0952,-1.79)(0.250,-1.72)(0.131,-1.60)(0.289,-1.53)(0.175,-1.40)(0.336,-1.33)(0.228,-1.20)(0.391,-1.15)(0.291,-1.01)(0.457,-0.962)(0.365,-0.817)(0.534,-0.783)(0.453,-0.631)(0.624,-0.609)(0.554,-0.452)(0.726,-0.443)(0.668,-0.281)(0.840,-0.286)(0.794,-0.120)(0.966,-0.137)(0.933,0.0318)(1.10,0.00217)(1.08,0.173)(1.25,0.133)(1.24,0.304)(1.40,0.253)(1.40,0.425)(1.57,0.365)(1.58,0.537)(1.73,0.469)(1.75,0.640)(1.90,0.565)(1.93,0.735)(2.00,0.693)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.1000,-2.3025)(0.0408,-2.1956)(0.1914,-2.1124)(0.0654,-1.9945)(0.2181,-1.9146)(0.0952,-1.7947)(0.2501,-1.7199)(0.1313,-1.5958)(0.2889,-1.5263)(0.1751,-1.3972)(0.3357,-1.3345)(0.2280,-1.2004)(0.3912,-1.1464)(0.2906,-1.0076)(0.4570,-0.9622)(0.3652,-0.8172)(0.5343,-0.7827)(0.4528,-0.6314)(0.6239,-0.6089)(0.5537,-0.4520)(0.7258,-0.4429)(0.6681,-0.2807)(0.8396,-0.2857)(0.7941,-0.1201)(0.9656,-0.1372)(0.9329,0.0317)(1.1025,0.0021)(1.0818,0.1730)(1.2495,0.1325)(1.2394,0.3040)(1.4038,0.2528)(1.4047,0.4253)(1.5653,0.3647)(1.5751,0.5369)(1.7324,0.4684)(1.7490,0.6395)(1.9043,0.5649)(1.9279,0.7351)(2.0000,0.6931)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -171,11 +171,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,-2.0000)(0.0700,-1.9000)(-0.0700,-1.8000)(0.0700,-1.7000)(-0.0700,-1.6000)(0.0700,-1.5000)(-0.0700,-1.4000)(0.0700,-1.3000)(-0.0700,-1.2000)(0.0700,-1.1000)(-0.0700,-1.0000)(0.0700,-0.9000)(-0.0700,-0.8000)(0.0700,-0.7000)(-0.0700,-0.6000)(0.0700,-0.5000)(-0.0700,-0.4000)(0.0700,-0.3000)(-0.0700,-0.2000)(0.0700,-0.1000)(-0.0700,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(-2.0000,0.0000)(-1.9000,0.0700)(-1.8000,-0.0700)(-1.7000,0.0700)(-1.6000,-0.0700)(-1.5000,0.0700)(-1.4000,-0.0700)(-1.3000,0.0700)(-1.2000,-0.0700)(-1.1000,0.0700)(-1.0000,-0.0700)(-0.9000,0.0700)(-0.8000,-0.0700)(-0.7000,0.0700)(-0.6000,-0.0700)(-0.5000,0.0700)(-0.4000,-0.0700)(-0.3000,0.0700)(-0.2000,-0.0700)(-0.1000,0.0700)(0.0000,-0.0700)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -206,11 +206,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,-2.0000)(0.0700,-1.9000)(-0.0700,-1.8000)(0.0700,-1.7000)(-0.0700,-1.6000)(0.0700,-1.5000)(-0.0700,-1.4000)(0.0700,-1.3000)(-0.0700,-1.2000)(0.0700,-1.1000)(-0.0700,-1.0000)(0.0700,-0.9000)(-0.0700,-0.8000)(0.0700,-0.7000)(-0.0700,-0.6000)(0.0700,-0.5000)(-0.0700,-0.4000)(0.0700,-0.3000)(-0.0700,-0.2000)(0.0700,-0.1000)(-0.0700,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(-2.0000,0.0000)(-1.9000,0.0700)(-1.8000,-0.0700)(-1.7000,0.0700)(-1.6000,-0.0700)(-1.5000,0.0700)(-1.4000,-0.0700)(-1.3000,0.0700)(-1.2000,-0.0700)(-1.1000,0.0700)(-1.0000,-0.0700)(-0.9000,0.0700)(-0.8000,-0.0700)(-0.7000,0.0700)(-0.6000,-0.0700)(-0.5000,0.0700)(-0.4000,-0.0700)(-0.3000,0.0700)(-0.2000,-0.0700)(-0.1000,0.0700)(0.0000,-0.0700)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks b/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks
index af9c3b2cf..f9bd83158 100644
--- a/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks
+++ b/auto/pictures_tex/Fig_QSKDooujUbDCsu.pstricks
@@ -70,11 +70,11 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-2.10,0.700) -- (2.10,0.700);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.38245) node {\( \pi(b_1)\)};
-\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
-\draw (0.70000,1.0825) node {\( \pi(b_2)\)};
+\draw [] (-2.1000,0.7000) -- (2.1000,0.7000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3824) node {\( \pi(b_1)\)};
+\draw []  (0.7000,0.7000) node [rotate=0] {$\bullet$};
+\draw (0.7000,1.0824) node {\( \pi(b_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_QXyVaKD.pstricks b/auto/pictures_tex/Fig_QXyVaKD.pstricks
index c8f85cd29..5d96b472f 100644
--- a/auto/pictures_tex/Fig_QXyVaKD.pstricks
+++ b/auto/pictures_tex/Fig_QXyVaKD.pstricks
@@ -67,21 +67,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
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 %DEFAULT
 
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+\draw [color=red] (-0.5000,-0.5000) -- (1.5000,1.5000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3546,0.66316) node {$P$};
+\draw (1.3546,0.6631) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_QuelCote.pstricks b/auto/pictures_tex/Fig_QuelCote.pstricks
index 314ffc1de..62f4d980b 100644
--- a/auto/pictures_tex/Fig_QuelCote.pstricks
+++ b/auto/pictures_tex/Fig_QuelCote.pstricks
@@ -85,18 +85,18 @@
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-\draw (-0.10311,0.39686) node {$n(t)$};
-\draw [color=green,style=dashed,->,>=latex] (-2.0667,0.29444) -- (-4.0302,0.67449);
+\draw [color=blue] (-3.0000,-1.7500)--(-2.9803,-1.7204)--(-2.9607,-1.6907)--(-2.9413,-1.6610)--(-2.9219,-1.6312)--(-2.9026,-1.6014)--(-2.8834,-1.5714)--(-2.8643,-1.5414)--(-2.8453,-1.5114)--(-2.8264,-1.4812)--(-2.8076,-1.4510)--(-2.7889,-1.4206)--(-2.7703,-1.3902)--(-2.7519,-1.3597)--(-2.7335,-1.3291)--(-2.7153,-1.2984)--(-2.6971,-1.2676)--(-2.6791,-1.2367)--(-2.6612,-1.2057)--(-2.6435,-1.1746)--(-2.6259,-1.1434)--(-2.6084,-1.1121)--(-2.5910,-1.0806)--(-2.5738,-1.0491)--(-2.5567,-1.0174)--(-2.5398,-0.9856)--(-2.5230,-0.9536)--(-2.5063,-0.9215)--(-2.4898,-0.8893)--(-2.4735,-0.8569)--(-2.4574,-0.8244)--(-2.4414,-0.7917)--(-2.4255,-0.7589)--(-2.4099,-0.7259)--(-2.3944,-0.6927)--(-2.3791,-0.6593)--(-2.3640,-0.6258)--(-2.3491,-0.5921)--(-2.3344,-0.5581)--(-2.3199,-0.5240)--(-2.3056,-0.4896)--(-2.2915,-0.4551)--(-2.2777,-0.4203)--(-2.2640,-0.3853)--(-2.2506,-0.3500)--(-2.2375,-0.3145)--(-2.2246,-0.2787)--(-2.2119,-0.2427)--(-2.1995,-0.2063)--(-2.1874,-0.1697)--(-2.1755,-0.1327)--(-2.1640,-0.0955)--(-2.1527,-0.0579)--(-2.1417,-0.0200)--(-2.1311,0.0182)--(-2.1207,0.0569)--(-2.1107,0.0959)--(-2.1010,0.1354)--(-2.0917,0.1752)--(-2.0828,0.2156)--(-2.0742,0.2563)--(-2.0660,0.2976)--(-2.0582,0.3393)--(-2.0508,0.3816)--(-2.0439,0.4244)--(-2.0374,0.4678)--(-2.0313,0.5118)--(-2.0257,0.5565)--(-2.0207,0.6017)--(-2.0161,0.6477)--(-2.0120,0.6944)--(-2.0085,0.7418)--(-2.0056,0.7901)--(-2.0032,0.8392)--(-2.0015,0.8891)--(-2.0004,0.9400)--(-2.0000,0.9919)--(-2.0002,1.0448)--(-2.0012,1.0987)--(-2.0029,1.1539)--(-2.0053,1.2102)--(-2.0086,1.2678)--(-2.0127,1.3267)--(-2.0177,1.3871)--(-2.0236,1.4489)--(-2.0305,1.5124)--(-2.0383,1.5776)--(-2.0473,1.6445)--(-2.0573,1.7134)--(-2.0684,1.7843)--(-2.0808,1.8574)--(-2.0944,1.9328)--(-2.1093,2.0107)--(-2.1256,2.0912)--(-2.1434,2.1745)--(-2.1627,2.2608)--(-2.1837,2.3503)--(-2.2063,2.4433)--(-2.2307,2.5400)--(-2.2571,2.6408);
+\draw [color=brown,style=dashed] (-2.4467,-1.6691) -- (-1.6866,2.2580);
+\draw [color=brown,->,>=latex] (-2.0666,0.2944) -- (-1.6866,2.2580);
+\draw (-0.8272,2.2580) node {$\gamma'(t)$};
+\draw [,->,>=latex] (-2.0666,0.2944) -- (-3.3160,1.8561);
+\draw (-2.7117,2.2547) node {$\gamma''(t)$};
+\draw [color=green,->,>=latex] (-2.0666,0.2944) -- (-0.1031,-0.0855);
+\draw (-0.1031,0.3968) node {$n(t)$};
+\draw [color=green,style=dashed,->,>=latex] (-2.0666,0.2944) -- (-4.0302,0.6744);
 \draw (-4.0302,1.1569) node {$-n(t)$};
-\draw []  (-2.0667,0.29444) node [rotate=0] {$\bullet$};
-\draw (-1.7120,-0.042396) node {$P$};
+\draw []  (-2.0666,0.2944) node [rotate=0] {$\bullet$};
+\draw (-1.7120,-0.0423) node {$P$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RGjjpwF.pstricks b/auto/pictures_tex/Fig_RGjjpwF.pstricks
index 055d4425c..2af1d6006 100644
--- a/auto/pictures_tex/Fig_RGjjpwF.pstricks
+++ b/auto/pictures_tex/Fig_RGjjpwF.pstricks
@@ -70,20 +70,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
-\draw [] (3.00,0) -- (4.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.31492) node {$1$};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
-\draw (1.0000,0.20000) node {};
-\draw []  (2.0000,0) node [rotate=0] {$o$};
-\draw (2.0000,0.20000) node {};
-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.20000) node {};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.20000) node {};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3149) node {$1$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
+\draw (1.0000,0.2000) node {};
+\draw []  (2.0000,0.0000) node [rotate=0] {$o$};
+\draw (2.0000,0.2000) node {};
+\draw []  (3.0000,0.0000) node [rotate=0] {$o$};
+\draw (3.0000,0.2000) node {};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,0.2000) node {};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RLuqsrr.pstricks b/auto/pictures_tex/Fig_RLuqsrr.pstricks
index f50e8f6d6..4afde3c45 100644
--- a/auto/pictures_tex/Fig_RLuqsrr.pstricks
+++ b/auto/pictures_tex/Fig_RLuqsrr.pstricks
@@ -87,25 +87,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-0.91381) -- (0,2.9142);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-0.9138) -- (0.0000,2.9141);
 %DEFAULT
 
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+\draw [color=red] (0.0000,0.0000)--(0.0634,0.1346)--(0.1269,0.2831)--(0.1903,0.4432)--(0.2538,0.6123)--(0.3173,0.7876)--(0.3807,0.9663)--(0.4442,1.1455)--(0.5077,1.3224)--(0.5711,1.4942)--(0.6346,1.6579)--(0.6981,1.8111)--(0.7615,1.9512)--(0.8250,2.0761)--(0.8885,2.1836)--(0.9519,2.2720)--(1.0154,2.3400)--(1.0789,2.3864)--(1.1423,2.4106)--(1.2058,2.4120)--(1.2693,2.3907)--(1.3327,2.3470)--(1.3962,2.2817)--(1.4597,2.1957)--(1.5231,2.0905)--(1.5866,1.9677)--(1.6501,1.8294)--(1.7135,1.6777)--(1.7770,1.5151)--(1.8405,1.3443)--(1.9039,1.1678)--(1.9674,0.9887)--(2.0309,0.8098)--(2.0943,0.6339)--(2.1578,0.4639)--(2.2213,0.3026)--(2.2847,0.1524)--(2.3482,0.0159)--(2.4117,-0.1046)--(2.4751,-0.2075)--(2.5386,-0.2910)--(2.6021,-0.3537)--(2.6655,-0.3946)--(2.7290,-0.4131)--(2.7925,-0.4088)--(2.8559,-0.3818)--(2.9194,-0.3327)--(2.9829,-0.2621)--(3.0463,-0.1711)--(3.1098,-0.0614)--(3.1733,0.0654)--(3.2367,0.2073)--(3.3002,0.3619)--(3.3637,0.5268)--(3.4271,0.6993)--(3.4906,0.8767)--(3.5541,1.0560)--(3.6175,1.2345)--(3.6810,1.4091)--(3.7445,1.5772)--(3.8079,1.7360)--(3.8714,1.8830)--(3.9349,2.0157)--(3.9983,2.1321)--(4.0618,2.2303)--(4.1253,2.3086)--(4.1887,2.3660)--(4.2522,2.4013)--(4.3157,2.4141)--(4.3791,2.4042)--(4.4426,2.3716)--(4.5061,2.3170)--(4.5695,2.2412)--(4.6330,2.1454)--(4.6965,2.0312)--(4.7599,1.9004)--(4.8234,1.7551)--(4.8869,1.5976)--(4.9503,1.4306)--(5.0138,1.2566)--(5.0773,1.0784)--(5.1407,0.8991)--(5.2042,0.7213)--(5.2677,0.5480)--(5.3311,0.3820)--(5.3946,0.2260)--(5.4581,0.0823)--(5.5215,-0.0464)--(5.5850,-0.1584)--(5.6485,-0.2518)--(5.7119,-0.3250)--(5.7754,-0.3769)--(5.8389,-0.4066)--(5.9023,-0.4138)--(5.9658,-0.3981)--(6.0293,-0.3600)--(6.0927,-0.3000)--(6.1562,-0.2190)--(6.2197,-0.1185)--(6.2831,0.0000);
 
-\draw [color=blue] (0,0)--(0.06347,0.008045)--(0.1269,0.03205)--(0.1904,0.07163)--(0.2539,0.1262)--(0.3173,0.1947)--(0.3808,0.2763)--(0.4443,0.3694)--(0.5077,0.4728)--(0.5712,0.5846)--(0.6347,0.7031)--(0.6981,0.8264)--(0.7616,0.9524)--(0.8251,1.079)--(0.8885,1.205)--(0.9520,1.327)--(1.015,1.444)--(1.079,1.554)--(1.142,1.655)--(1.206,1.745)--(1.269,1.824)--(1.333,1.889)--(1.396,1.940)--(1.460,1.975)--(1.523,1.995)--(1.587,1.999)--(1.650,1.987)--(1.714,1.959)--(1.777,1.916)--(1.841,1.858)--(1.904,1.786)--(1.967,1.701)--(2.031,1.606)--(2.094,1.500)--(2.158,1.386)--(2.221,1.266)--(2.285,1.142)--(2.348,1.016)--(2.412,0.8892)--(2.475,0.7642)--(2.539,0.6431)--(2.602,0.5277)--(2.666,0.4199)--(2.729,0.3215)--(2.793,0.2340)--(2.856,0.1587)--(2.919,0.09707)--(2.983,0.04993)--(3.046,0.01807)--(3.110,0.002013)--(3.173,0.002013)--(3.237,0.01807)--(3.300,0.04993)--(3.364,0.09707)--(3.427,0.1587)--(3.491,0.2340)--(3.554,0.3215)--(3.618,0.4199)--(3.681,0.5277)--(3.745,0.6431)--(3.808,0.7642)--(3.871,0.8892)--(3.935,1.016)--(3.998,1.142)--(4.062,1.266)--(4.125,1.386)--(4.189,1.500)--(4.252,1.606)--(4.316,1.701)--(4.379,1.786)--(4.443,1.858)--(4.506,1.916)--(4.570,1.959)--(4.633,1.987)--(4.697,1.999)--(4.760,1.995)--(4.823,1.975)--(4.887,1.940)--(4.950,1.889)--(5.014,1.824)--(5.077,1.745)--(5.141,1.655)--(5.204,1.554)--(5.268,1.444)--(5.331,1.327)--(5.395,1.205)--(5.458,1.079)--(5.522,0.9524)--(5.585,0.8264)--(5.648,0.7031)--(5.712,0.5846)--(5.775,0.4728)--(5.839,0.3694)--(5.902,0.2763)--(5.966,0.1947)--(6.029,0.1262)--(6.093,0.07163)--(6.156,0.03205)--(6.220,0.008045)--(6.283,0);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0634,0.0080)--(0.1269,0.0320)--(0.1903,0.0716)--(0.2538,0.1261)--(0.3173,0.1947)--(0.3807,0.2762)--(0.4442,0.3694)--(0.5077,0.4727)--(0.5711,0.5845)--(0.6346,0.7030)--(0.6981,0.8263)--(0.7615,0.9524)--(0.8250,1.0792)--(0.8885,1.2048)--(0.9519,1.3270)--(1.0154,1.4440)--(1.0789,1.5539)--(1.1423,1.6548)--(1.2058,1.7452)--(1.2693,1.8236)--(1.3327,1.8888)--(1.3962,1.9396)--(1.4597,1.9754)--(1.5231,1.9954)--(1.5866,1.9994)--(1.6501,1.9874)--(1.7135,1.9594)--(1.7770,1.9161)--(1.8405,1.8579)--(1.9039,1.7860)--(1.9674,1.7014)--(2.0309,1.6056)--(2.0943,1.5000)--(2.1578,1.3863)--(2.2213,1.2664)--(2.2847,1.1423)--(2.3482,1.0158)--(2.4117,0.8891)--(2.4751,0.7642)--(2.5386,0.6431)--(2.6021,0.5277)--(2.6655,0.4199)--(2.7290,0.3214)--(2.7925,0.2339)--(2.8559,0.1587)--(2.9194,0.0970)--(2.9829,0.0499)--(3.0463,0.0180)--(3.1098,0.0020)--(3.1733,0.0020)--(3.2367,0.0180)--(3.3002,0.0499)--(3.3637,0.0970)--(3.4271,0.1587)--(3.4906,0.2339)--(3.5541,0.3214)--(3.6175,0.4199)--(3.6810,0.5277)--(3.7445,0.6431)--(3.8079,0.7642)--(3.8714,0.8891)--(3.9349,1.0158)--(3.9983,1.1423)--(4.0618,1.2664)--(4.1253,1.3863)--(4.1887,1.5000)--(4.2522,1.6056)--(4.3157,1.7014)--(4.3791,1.7860)--(4.4426,1.8579)--(4.5061,1.9161)--(4.5695,1.9594)--(4.6330,1.9874)--(4.6965,1.9994)--(4.7599,1.9954)--(4.8234,1.9754)--(4.8869,1.9396)--(4.9503,1.8888)--(5.0138,1.8236)--(5.0773,1.7452)--(5.1407,1.6548)--(5.2042,1.5539)--(5.2677,1.4440)--(5.3311,1.3270)--(5.3946,1.2048)--(5.4581,1.0792)--(5.5215,0.9524)--(5.5850,0.8263)--(5.6485,0.7030)--(5.7119,0.5845)--(5.7754,0.4727)--(5.8389,0.3694)--(5.9023,0.2762)--(5.9658,0.1947)--(6.0293,0.1261)--(6.0927,0.0716)--(6.1562,0.0320)--(6.2197,0.0080)--(6.2831,0.0000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks b/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks
index 204852b2e..f8a93bbca 100644
--- a/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks
+++ b/auto/pictures_tex/Fig_ROAOooPgUZIt.pstricks
@@ -71,18 +71,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
 
-\draw [color=brown] (1.000,3.000)--(0.9980,3.063)--(0.9920,3.127)--(0.9819,3.189)--(0.9679,3.251)--(0.9501,3.312)--(0.9284,3.372)--(0.9029,3.430)--(0.8738,3.486)--(0.8413,3.541)--(0.8053,3.593)--(0.7660,3.643)--(0.7237,3.690)--(0.6785,3.735)--(0.6306,3.776)--(0.5801,3.815)--(0.5272,3.850)--(0.4723,3.881)--(0.4154,3.910)--(0.3569,3.934)--(0.2969,3.955)--(0.2358,3.972)--(0.1736,3.985)--(0.1108,3.994)--(0.04758,3.999)--(-0.01587,4.000)--(-0.07925,3.997)--(-0.1423,3.990)--(-0.2048,3.979)--(-0.2665,3.964)--(-0.3271,3.945)--(-0.3863,3.922)--(-0.4441,3.896)--(-0.5000,3.866)--(-0.5539,3.833)--(-0.6056,3.796)--(-0.6549,3.756)--(-0.7015,3.713)--(-0.7453,3.667)--(-0.7861,3.618)--(-0.8237,3.567)--(-0.8580,3.514)--(-0.8888,3.458)--(-0.9161,3.401)--(-0.9397,3.342)--(-0.9595,3.282)--(-0.9754,3.220)--(-0.9874,3.158)--(-0.9955,3.095)--(-0.9995,3.032)--(-0.9995,2.968)--(-0.9955,2.905)--(-0.9874,2.842)--(-0.9754,2.780)--(-0.9595,2.718)--(-0.9397,2.658)--(-0.9161,2.599)--(-0.8888,2.542)--(-0.8580,2.486)--(-0.8237,2.433)--(-0.7861,2.382)--(-0.7453,2.333)--(-0.7015,2.287)--(-0.6549,2.244)--(-0.6056,2.204)--(-0.5539,2.167)--(-0.5000,2.134)--(-0.4441,2.104)--(-0.3863,2.078)--(-0.3271,2.055)--(-0.2665,2.036)--(-0.2048,2.021)--(-0.1423,2.010)--(-0.07925,2.003)--(-0.01587,2.000)--(0.04758,2.001)--(0.1108,2.006)--(0.1736,2.015)--(0.2358,2.028)--(0.2969,2.045)--(0.3569,2.066)--(0.4154,2.090)--(0.4723,2.119)--(0.5272,2.150)--(0.5801,2.185)--(0.6306,2.224)--(0.6785,2.265)--(0.7237,2.310)--(0.7660,2.357)--(0.8053,2.407)--(0.8413,2.459)--(0.8738,2.514)--(0.9029,2.570)--(0.9284,2.628)--(0.9501,2.688)--(0.9679,2.749)--(0.9819,2.811)--(0.9920,2.873)--(0.9980,2.937)--(1.000,3.000);
-\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [color=brown] (1.0000,3.0000)--(0.9979,3.0634)--(0.9919,3.1265)--(0.9819,3.1892)--(0.9679,3.2511)--(0.9500,3.3120)--(0.9283,3.3716)--(0.9029,3.4297)--(0.8738,3.4861)--(0.8412,3.5406)--(0.8052,3.5929)--(0.7660,3.6427)--(0.7237,3.6900)--(0.6785,3.7345)--(0.6305,3.7761)--(0.5800,3.8145)--(0.5272,3.8497)--(0.4722,3.8814)--(0.4154,3.9096)--(0.3568,3.9341)--(0.2969,3.9549)--(0.2357,3.9718)--(0.1736,3.9848)--(0.1108,3.9938)--(0.0475,3.9988)--(-0.0158,3.9998)--(-0.0792,3.9968)--(-0.1423,3.9898)--(-0.2048,3.9788)--(-0.2664,3.9638)--(-0.3270,3.9450)--(-0.3863,3.9223)--(-0.4440,3.8959)--(-0.5000,3.8660)--(-0.5539,3.8325)--(-0.6056,3.7957)--(-0.6548,3.7557)--(-0.7014,3.7126)--(-0.7452,3.6667)--(-0.7860,3.6181)--(-0.8236,3.5670)--(-0.8579,3.5136)--(-0.8888,3.4582)--(-0.9161,3.4009)--(-0.9396,3.3420)--(-0.9594,3.2817)--(-0.9754,3.2203)--(-0.9874,3.1580)--(-0.9954,3.0950)--(-0.9994,3.0317)--(-0.9994,2.9682)--(-0.9954,2.9049)--(-0.9874,2.8419)--(-0.9754,2.7796)--(-0.9594,2.7182)--(-0.9396,2.6579)--(-0.9161,2.5990)--(-0.8888,2.5417)--(-0.8579,2.4863)--(-0.8236,2.4329)--(-0.7860,2.3818)--(-0.7452,2.3332)--(-0.7014,2.2873)--(-0.6548,2.2442)--(-0.6056,2.2042)--(-0.5539,2.1674)--(-0.5000,2.1339)--(-0.4440,2.1040)--(-0.3863,2.0776)--(-0.3270,2.0549)--(-0.2664,2.0361)--(-0.2048,2.0211)--(-0.1423,2.0101)--(-0.0792,2.0031)--(-0.0158,2.0001)--(0.0475,2.0011)--(0.1108,2.0061)--(0.1736,2.0151)--(0.2357,2.0281)--(0.2969,2.0450)--(0.3568,2.0658)--(0.4154,2.0903)--(0.4722,2.1185)--(0.5272,2.1502)--(0.5800,2.1854)--(0.6305,2.2238)--(0.6785,2.2654)--(0.7237,2.3099)--(0.7660,2.3572)--(0.8052,2.4070)--(0.8412,2.4593)--(0.8738,2.5138)--(0.9029,2.5702)--(0.9283,2.6283)--(0.9500,2.6879)--(0.9679,2.7488)--(0.9819,2.8107)--(0.9919,2.8734)--(0.9979,2.9365)--(1.0000,3.0000);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0.0000);
 \draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
-\draw (1.8965,1.5000) node {$a$};
-\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
-\draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.14303,3.6345) node {$R$};
-\draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
+\draw (1.8964,1.5000) node {$a$};
+\draw []  (0.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw [color=red] (0.0000,3.0000) -- (0.8660,3.5000);
+\draw (0.1430,3.6345) node {$R$};
+\draw [color=blue,style=dotted] (0.0000,3.0000) -- (1.5000,3.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks b/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks
index 9b902f047..733a2337b 100644
--- a/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks
+++ b/auto/pictures_tex/Fig_RPNooQXxpZZ.pstricks
@@ -92,27 +92,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-1.9457) -- (0,1.9457);
+\draw [,->,>=latex] (-6.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9456) -- (0.0000,1.9456);
 %DEFAULT
 
-\draw [color=blue] (-6.0000,0.059192)--(-5.8788,0.027135)--(-5.7576,0.0060480)--(-5.6364,0)--(-5.5152,0.011586)--(-5.3939,0.039487)--(-5.2727,0.080491)--(-5.1515,0.12891)--(-5.0303,0.17762)--(-4.9091,0.21910)--(-4.7879,0.24669)--(-4.6667,0.25565)--(-4.5455,0.24409)--(-4.4242,0.21334)--(-4.3030,0.16799)--(-4.1818,0.11529)--(-4.0606,0.064108)--(-3.9394,0.023666)--(-3.8182,0.0020298)--(-3.6970,0.0047735)--(-3.5758,0.033930)--(-3.4545,0.087448)--(-3.3333,0.15925)--(-3.2121,0.23994)--(-3.0909,0.31806)--(-2.9697,0.38179)--(-2.8485,0.42076)--(-2.7273,0.42785)--(-2.6061,0.40058)--(-2.4848,0.34187)--(-2.3636,0.26014)--(-2.2424,0.16851)--(-2.1212,0.083272)--(-2.0000,0.021790)--(-1.8788,0)--(-1.7576,0.030313)--(-1.6364,0.11884)--(-1.5152,0.26464)--(-1.3939,0.45879)--(-1.2727,0.68492)--(-1.1515,0.92065)--(-1.0303,1.1399)--(-0.90909,1.3159)--(-0.78788,1.4242)--(-0.66667,1.4457)--(-0.54545,1.3694)--(-0.42424,1.1936)--(-0.30303,0.92708)--(-0.18182,0.58774)--(-0.060606,0.20133)--(0.060606,-0.20133)--(0.18182,-0.58774)--(0.30303,-0.92708)--(0.42424,-1.1936)--(0.54545,-1.3694)--(0.66667,-1.4457)--(0.78788,-1.4242)--(0.90909,-1.3159)--(1.0303,-1.1399)--(1.1515,-0.92065)--(1.2727,-0.68492)--(1.3939,-0.45879)--(1.5152,-0.26464)--(1.6364,-0.11884)--(1.7576,-0.030313)--(1.8788,0)--(2.0000,-0.021790)--(2.1212,-0.083272)--(2.2424,-0.16851)--(2.3636,-0.26014)--(2.4848,-0.34187)--(2.6061,-0.40058)--(2.7273,-0.42785)--(2.8485,-0.42076)--(2.9697,-0.38179)--(3.0909,-0.31806)--(3.2121,-0.23994)--(3.3333,-0.15925)--(3.4545,-0.087448)--(3.5758,-0.033930)--(3.6970,-0.0047735)--(3.8182,-0.0020298)--(3.9394,-0.023666)--(4.0606,-0.064108)--(4.1818,-0.11529)--(4.3030,-0.16799)--(4.4242,-0.21334)--(4.5455,-0.24409)--(4.6667,-0.25565)--(4.7879,-0.24669)--(4.9091,-0.21910)--(5.0303,-0.17762)--(5.1515,-0.12891)--(5.2727,-0.080491)--(5.3939,-0.039487)--(5.5152,-0.011586)--(5.6364,0)--(5.7576,-0.0060480)--(5.8788,-0.027135)--(6.0000,-0.059192);
-\draw (-5.6549,-0.32983) node {$ -6 \, \pi $};
-\draw [] (-5.66,-0.100) -- (-5.66,0.100);
-\draw (-3.7699,-0.32983) node {$ -4 \, \pi $};
-\draw [] (-3.77,-0.100) -- (-3.77,0.100);
-\draw (-1.8850,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-1.88,-0.100) -- (-1.88,0.100);
-\draw (1.8850,-0.31492) node {$ 2 \, \pi $};
-\draw [] (1.88,-0.100) -- (1.88,0.100);
-\draw (3.7699,-0.31492) node {$ 4 \, \pi $};
-\draw [] (3.77,-0.100) -- (3.77,0.100);
-\draw (5.6549,-0.31492) node {$ 6 \, \pi $};
-\draw [] (5.66,-0.100) -- (5.66,0.100);
-\draw (-0.45274,-1.0000) node {$ -\frac{1}{2} $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.31083,1.0000) node {$ \frac{1}{2} $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-6.0000,0.0591)--(-5.8787,0.0271)--(-5.7575,0.0060)--(-5.6363,0.0000)--(-5.5151,0.0115)--(-5.3939,0.0394)--(-5.2727,0.0804)--(-5.1515,0.1289)--(-5.0303,0.1776)--(-4.9090,0.2191)--(-4.7878,0.2466)--(-4.6666,0.2556)--(-4.5454,0.2440)--(-4.4242,0.2133)--(-4.3030,0.1679)--(-4.1818,0.1152)--(-4.0606,0.0641)--(-3.9393,0.0236)--(-3.8181,0.0020)--(-3.6969,0.0047)--(-3.5757,0.0339)--(-3.4545,0.0874)--(-3.3333,0.1592)--(-3.2121,0.2399)--(-3.0909,0.3180)--(-2.9696,0.3817)--(-2.8484,0.4207)--(-2.7272,0.4278)--(-2.6060,0.4005)--(-2.4848,0.3418)--(-2.3636,0.2601)--(-2.2424,0.1685)--(-2.1212,0.0832)--(-2.0000,0.0217)--(-1.8787,0.0000)--(-1.7575,0.0303)--(-1.6363,0.1188)--(-1.5151,0.2646)--(-1.3939,0.4587)--(-1.2727,0.6849)--(-1.1515,0.9206)--(-1.0303,1.1399)--(-0.9090,1.3159)--(-0.7878,1.4241)--(-0.6666,1.4456)--(-0.5454,1.3693)--(-0.4242,1.1936)--(-0.3030,0.9270)--(-0.1818,0.5877)--(-0.0606,0.2013)--(0.0606,-0.2013)--(0.1818,-0.5877)--(0.3030,-0.9270)--(0.4242,-1.1936)--(0.5454,-1.3693)--(0.6666,-1.4456)--(0.7878,-1.4241)--(0.9090,-1.3159)--(1.0303,-1.1399)--(1.1515,-0.9206)--(1.2727,-0.6849)--(1.3939,-0.4587)--(1.5151,-0.2646)--(1.6363,-0.1188)--(1.7575,-0.0303)--(1.8787,0.0000)--(2.0000,-0.0217)--(2.1212,-0.0832)--(2.2424,-0.1685)--(2.3636,-0.2601)--(2.4848,-0.3418)--(2.6060,-0.4005)--(2.7272,-0.4278)--(2.8484,-0.4207)--(2.9696,-0.3817)--(3.0909,-0.3180)--(3.2121,-0.2399)--(3.3333,-0.1592)--(3.4545,-0.0874)--(3.5757,-0.0339)--(3.6969,-0.0047)--(3.8181,-0.0020)--(3.9393,-0.0236)--(4.0606,-0.0641)--(4.1818,-0.1152)--(4.3030,-0.1679)--(4.4242,-0.2133)--(4.5454,-0.2440)--(4.6666,-0.2556)--(4.7878,-0.2466)--(4.9090,-0.2191)--(5.0303,-0.1776)--(5.1515,-0.1289)--(5.2727,-0.0804)--(5.3939,-0.0394)--(5.5151,-0.0115)--(5.6363,0.0000)--(5.7575,-0.0060)--(5.8787,-0.0271)--(6.0000,-0.0591);
+\draw (-5.6548,-0.3298) node {$ -6 \, \pi $};
+\draw [] (-5.6548,-0.1000) -- (-5.6548,0.1000);
+\draw (-3.7699,-0.3298) node {$ -4 \, \pi $};
+\draw [] (-3.7699,-0.1000) -- (-3.7699,0.1000);
+\draw (-1.8849,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-1.8849,-0.1000) -- (-1.8849,0.1000);
+\draw (1.8849,-0.3149) node {$ 2 \, \pi $};
+\draw [] (1.8849,-0.1000) -- (1.8849,0.1000);
+\draw (3.7699,-0.3149) node {$ 4 \, \pi $};
+\draw [] (3.7699,-0.1000) -- (3.7699,0.1000);
+\draw (5.6548,-0.3149) node {$ 6 \, \pi $};
+\draw [] (5.6548,-0.1000) -- (5.6548,0.1000);
+\draw (-0.4527,-1.0000) node {$ -\frac{1}{2} $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.3108,1.0000) node {$ \frac{1}{2} $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_RQsQKTl.pstricks b/auto/pictures_tex/Fig_RQsQKTl.pstricks
index adc854ce0..35aef726a 100644
--- a/auto/pictures_tex/Fig_RQsQKTl.pstricks
+++ b/auto/pictures_tex/Fig_RQsQKTl.pstricks
@@ -66,10 +66,10 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.31492) node {\( 1\)};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.3149) node {\( 1\)};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_Refraction.pstricks b/auto/pictures_tex/Fig_Refraction.pstricks
index 9e1e8d206..357938ef3 100644
--- a/auto/pictures_tex/Fig_Refraction.pstricks
+++ b/auto/pictures_tex/Fig_Refraction.pstricks
@@ -77,17 +77,17 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (-3.00,0) -- (3.00,0);
-\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
-\draw (0.46653,2.0000) node {$\overline{ N }$};
-\draw (0.35618,0.94573) node {$\theta_1$};
+\draw [] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0000) -- (0.0000,2.0000);
+\draw (0.4665,2.0000) node {$\overline{ N }$};
+\draw (0.3561,0.9457) node {$\theta_1$};
 
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-\draw (-0.42770,-0.74428) node {$\theta_2$};
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+\draw [color=red,->,>=latex] (1.0000,1.0000) -- (0.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.2649,-0.6324);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks b/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks
index 27241d0e2..5c4449b10 100644
--- a/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks
+++ b/auto/pictures_tex/Fig_RegioniPrimoeSecondoTipo.pstricks
@@ -65,8 +65,8 @@
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 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
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@@ -75,23 +75,23 @@
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+\draw [color=red] (1.0000,5.0000)--(1.0303,5.0596)--(1.0606,5.1175)--(1.0909,5.1735)--(1.1212,5.2277)--(1.1515,5.2800)--(1.1818,5.3305)--(1.2121,5.3792)--(1.2424,5.4260)--(1.2727,5.4710)--(1.3030,5.5142)--(1.3333,5.5555)--(1.3636,5.5950)--(1.3939,5.6326)--(1.4242,5.6685)--(1.4545,5.7024)--(1.4848,5.7346)--(1.5151,5.7649)--(1.5454,5.7933)--(1.5757,5.8200)--(1.6060,5.8448)--(1.6363,5.8677)--(1.6666,5.8888)--(1.6969,5.9081)--(1.7272,5.9256)--(1.7575,5.9412)--(1.7878,5.9550)--(1.8181,5.9669)--(1.8484,5.9770)--(1.8787,5.9853)--(1.9090,5.9917)--(1.9393,5.9963)--(1.9696,5.9990)--(2.0000,6.0000)--(2.0303,5.9990)--(2.0606,5.9963)--(2.0909,5.9917)--(2.1212,5.9853)--(2.1515,5.9770)--(2.1818,5.9669)--(2.2121,5.9550)--(2.2424,5.9412)--(2.2727,5.9256)--(2.3030,5.9081)--(2.3333,5.8888)--(2.3636,5.8677)--(2.3939,5.8448)--(2.4242,5.8200)--(2.4545,5.7933)--(2.4848,5.7649)--(2.5151,5.7346)--(2.5454,5.7024)--(2.5757,5.6685)--(2.6060,5.6326)--(2.6363,5.5950)--(2.6666,5.5555)--(2.6969,5.5142)--(2.7272,5.4710)--(2.7575,5.4260)--(2.7878,5.3792)--(2.8181,5.3305)--(2.8484,5.2800)--(2.8787,5.2277)--(2.9090,5.1735)--(2.9393,5.1175)--(2.9696,5.0596)--(3.0000,5.0000)--(3.0303,4.9384)--(3.0606,4.8751)--(3.0909,4.8099)--(3.1212,4.7428)--(3.1515,4.6740)--(3.1818,4.6033)--(3.2121,4.5307)--(3.2424,4.4563)--(3.2727,4.3801)--(3.3030,4.3021)--(3.3333,4.2222)--(3.3636,4.1404)--(3.3939,4.0569)--(3.4242,3.9715)--(3.4545,3.8842)--(3.4848,3.7952)--(3.5151,3.7043)--(3.5454,3.6115)--(3.5757,3.5169)--(3.6060,3.4205)--(3.6363,3.3223)--(3.6666,3.2222)--(3.6969,3.1202)--(3.7272,3.0165)--(3.7575,2.9109)--(3.7878,2.8034)--(3.8181,2.6942)--(3.8484,2.5831)--(3.8787,2.4701)--(3.9090,2.3553)--(3.9393,2.2387)--(3.9696,2.1202)--(4.0000,2.0000);
 
 %OTHER STUFF
 %END PSPICTURE
@@ -154,8 +154,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -163,19 +163,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw (-0.37898,1.0000) node {$c$};
-\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
-\draw (-0.39499,5.0000) node {$d$};
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+\draw [color=blue] (2.0000,1.0000)--(2.0403,1.0404)--(2.0807,1.0808)--(2.1209,1.1212)--(2.1609,1.1616)--(2.2006,1.2020)--(2.2400,1.2424)--(2.2790,1.2828)--(2.3176,1.3232)--(2.3556,1.3636)--(2.3931,1.4040)--(2.4299,1.4444)--(2.4660,1.4848)--(2.5014,1.5252)--(2.5359,1.5656)--(2.5696,1.6060)--(2.6023,1.6464)--(2.6341,1.6868)--(2.6648,1.7272)--(2.6944,1.7676)--(2.7229,1.8080)--(2.7502,1.8484)--(2.7763,1.8888)--(2.8011,1.9292)--(2.8247,1.9696)--(2.8468,2.0101)--(2.8676,2.0505)--(2.8870,2.0909)--(2.9049,2.1313)--(2.9214,2.1717)--(2.9363,2.2121)--(2.9497,2.2525)--(2.9616,2.2929)--(2.9719,2.3333)--(2.9806,2.3737)--(2.9877,2.4141)--(2.9932,2.4545)--(2.9971,2.4949)--(2.9993,2.5353)--(2.9999,2.5757)--(2.9989,2.6161)--(2.9963,2.6565)--(2.9920,2.6969)--(2.9861,2.7373)--(2.9786,2.7777)--(2.9695,2.8181)--(2.9588,2.8585)--(2.9466,2.8989)--(2.9328,2.9393)--(2.9175,2.9797)--(2.9007,3.0202)--(2.8824,3.0606)--(2.8626,3.1010)--(2.8415,3.1414)--(2.8190,3.1818)--(2.7952,3.2222)--(2.7700,3.2626)--(2.7436,3.3030)--(2.7160,3.3434)--(2.6872,3.3838)--(2.6573,3.4242)--(2.6264,3.4646)--(2.5944,3.5050)--(2.5614,3.5454)--(2.5275,3.5858)--(2.4928,3.6262)--(2.4572,3.6666)--(2.4209,3.7070)--(2.3839,3.7474)--(2.3463,3.7878)--(2.3082,3.8282)--(2.2695,3.8686)--(2.2304,3.9090)--(2.1909,3.9494)--(2.1511,3.9898)--(2.1110,4.0303)--(2.0708,4.0707)--(2.0304,4.1111)--(1.9900,4.1515)--(1.9496,4.1919)--(1.9093,4.2323)--(1.8692,4.2727)--(1.8293,4.3131)--(1.7896,4.3535)--(1.7503,4.3939)--(1.7114,4.4343)--(1.6729,4.4747)--(1.6350,4.5151)--(1.5977,4.5555)--(1.5611,4.5959)--(1.5251,4.6363)--(1.4900,4.6767)--(1.4556,4.7171)--(1.4222,4.7575)--(1.3897,4.7979)--(1.3582,4.8383)--(1.3277,4.8787)--(1.2984,4.9191)--(1.2702,4.9595)--(1.2431,5.0000);
+\draw [style=dashed] (1.2431,5.0000) -- (6.0000,5.0000);
+\draw [style=dashed] (2.0000,1.0000) -- (5.1111,1.0000);
+\draw [color=blue] (5.1111,1.0000)--(5.1023,1.0404)--(5.0938,1.0808)--(5.0858,1.1212)--(5.0780,1.1616)--(5.0707,1.2020)--(5.0637,1.2424)--(5.0571,1.2828)--(5.0508,1.3232)--(5.0449,1.3636)--(5.0394,1.4040)--(5.0342,1.4444)--(5.0294,1.4848)--(5.0250,1.5252)--(5.0209,1.5656)--(5.0172,1.6060)--(5.0138,1.6464)--(5.0108,1.6868)--(5.0082,1.7272)--(5.0059,1.7676)--(5.0040,1.8080)--(5.0025,1.8484)--(5.0013,1.8888)--(5.0005,1.9292)--(5.0001,1.9696)--(5.0000,2.0101)--(5.0002,2.0505)--(5.0009,2.0909)--(5.0019,2.1313)--(5.0032,2.1717)--(5.0049,2.2121)--(5.0070,2.2525)--(5.0095,2.2929)--(5.0123,2.3333)--(5.0155,2.3737)--(5.0190,2.4141)--(5.0229,2.4545)--(5.0272,2.4949)--(5.0318,2.5353)--(5.0368,2.5757)--(5.0421,2.6161)--(5.0478,2.6565)--(5.0539,2.6969)--(5.0604,2.7373)--(5.0672,2.7777)--(5.0743,2.8181)--(5.0819,2.8585)--(5.0897,2.8989)--(5.0980,2.9393)--(5.1066,2.9797)--(5.1156,3.0202)--(5.1249,3.0606)--(5.1346,3.1010)--(5.1447,3.1414)--(5.1551,3.1818)--(5.1659,3.2222)--(5.1771,3.2626)--(5.1886,3.3030)--(5.2005,3.3434)--(5.2127,3.3838)--(5.2253,3.4242)--(5.2383,3.4646)--(5.2516,3.5050)--(5.2653,3.5454)--(5.2794,3.5858)--(5.2938,3.6262)--(5.3086,3.6666)--(5.3237,3.7070)--(5.3392,3.7474)--(5.3551,3.7878)--(5.3714,3.8282)--(5.3879,3.8686)--(5.4049,3.9090)--(5.4222,3.9494)--(5.4399,3.9898)--(5.4580,4.0303)--(5.4764,4.0707)--(5.4951,4.1111)--(5.5143,4.1515)--(5.5338,4.1919)--(5.5536,4.2323)--(5.5739,4.2727)--(5.5945,4.3131)--(5.6154,4.3535)--(5.6367,4.3939)--(5.6584,4.4343)--(5.6804,4.4747)--(5.7028,4.5151)--(5.7256,4.5555)--(5.7487,4.5959)--(5.7722,4.6363)--(5.7961,4.6767)--(5.8203,4.7171)--(5.8449,4.7575)--(5.8698,4.7979)--(5.8951,4.8383)--(5.9208,4.8787)--(5.9468,4.9191)--(5.9732,4.9595)--(6.0000,5.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3789,1.0000) node {$c$};
+\draw []  (0.0000,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3949,5.0000) node {$d$};
 \draw (2.4480,2.7306) node {$h_1$};
 \draw (5.5954,3.0000) node {$h_2$};
-\draw [style=dotted] (2.00,1.00) -- (0,1.00);
-\draw [style=dotted] (1.24,5.00) -- (0,5.00);
+\draw [style=dotted] (2.0000,1.0000) -- (0.0000,1.0000);
+\draw [style=dotted] (1.2431,5.0000) -- (0.0000,5.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SBTooEasQsT.pstricks b/auto/pictures_tex/Fig_SBTooEasQsT.pstricks
index a50ca2224..c29caad53 100644
--- a/auto/pictures_tex/Fig_SBTooEasQsT.pstricks
+++ b/auto/pictures_tex/Fig_SBTooEasQsT.pstricks
@@ -118,115 +118,115 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-4.00,-4.00) -- (-4.00,4.00);
-\draw [color=gray,style=solid] (-3.00,-4.00) -- (-3.00,4.00);
-\draw [color=gray,style=solid] (-2.00,-4.00) -- (-2.00,4.00);
-\draw [color=gray,style=solid] (-1.00,-4.00) -- (-1.00,4.00);
-\draw [color=gray,style=solid] (0,-4.00) -- (0,4.00);
-\draw [color=gray,style=solid] (1.00,-4.00) -- (1.00,4.00);
-\draw [color=gray,style=solid] (2.00,-4.00) -- (2.00,4.00);
-\draw [color=gray,style=dotted] (-3.50,-4.00) -- (-3.50,4.00);
-\draw [color=gray,style=dotted] (-2.50,-4.00) -- (-2.50,4.00);
-\draw [color=gray,style=dotted] (-1.50,-4.00) -- (-1.50,4.00);
-\draw [color=gray,style=dotted] (-0.500,-4.00) -- (-0.500,4.00);
-\draw [color=gray,style=dotted] (0.500,-4.00) -- (0.500,4.00);
-\draw [color=gray,style=dotted] (1.50,-4.00) -- (1.50,4.00);
-\draw [color=gray,style=dotted] (-4.00,-3.75) -- (2.00,-3.75);
-\draw [color=gray,style=dotted] (-4.00,-3.25) -- (2.00,-3.25);
-\draw [color=gray,style=dotted] (-4.00,-2.75) -- (2.00,-2.75);
-\draw [color=gray,style=dotted] (-4.00,-2.25) -- (2.00,-2.25);
-\draw [color=gray,style=dotted] (-4.00,-1.75) -- (2.00,-1.75);
-\draw [color=gray,style=dotted] (-4.00,-1.25) -- (2.00,-1.25);
-\draw [color=gray,style=dotted] (-4.00,-0.750) -- (2.00,-0.750);
-\draw [color=gray,style=dotted] (-4.00,-0.250) -- (2.00,-0.250);
-\draw [color=gray,style=dotted] (-4.00,0.250) -- (2.00,0.250);
-\draw [color=gray,style=dotted] (-4.00,0.750) -- (2.00,0.750);
-\draw [color=gray,style=dotted] (-4.00,1.25) -- (2.00,1.25);
-\draw [color=gray,style=dotted] (-4.00,1.75) -- (2.00,1.75);
-\draw [color=gray,style=dotted] (-4.00,2.25) -- (2.00,2.25);
-\draw [color=gray,style=dotted] (-4.00,2.75) -- (2.00,2.75);
-\draw [color=gray,style=dotted] (-4.00,3.25) -- (2.00,3.25);
-\draw [color=gray,style=dotted] (-4.00,3.75) -- (2.00,3.75);
-\draw [color=gray,style=solid] (-4.00,-4.00) -- (2.00,-4.00);
-\draw [color=gray,style=solid] (-4.00,-3.50) -- (2.00,-3.50);
-\draw [color=gray,style=solid] (-4.00,-3.00) -- (2.00,-3.00);
-\draw [color=gray,style=solid] (-4.00,-2.50) -- (2.00,-2.50);
-\draw [color=gray,style=solid] (-4.00,-2.00) -- (2.00,-2.00);
-\draw [color=gray,style=solid] (-4.00,-1.50) -- (2.00,-1.50);
-\draw [color=gray,style=solid] (-4.00,-1.00) -- (2.00,-1.00);
-\draw [color=gray,style=solid] (-4.00,-0.500) -- (2.00,-0.500);
-\draw [color=gray,style=solid] (-4.00,0) -- (2.00,0);
-\draw [color=gray,style=solid] (-4.00,0.500) -- (2.00,0.500);
-\draw [color=gray,style=solid] (-4.00,1.00) -- (2.00,1.00);
-\draw [color=gray,style=solid] (-4.00,1.50) -- (2.00,1.50);
-\draw [color=gray,style=solid] (-4.00,2.00) -- (2.00,2.00);
-\draw [color=gray,style=solid] (-4.00,2.50) -- (2.00,2.50);
-\draw [color=gray,style=solid] (-4.00,3.00) -- (2.00,3.00);
-\draw [color=gray,style=solid] (-4.00,3.50) -- (2.00,3.50);
-\draw [color=gray,style=solid] (-4.00,4.00) -- (2.00,4.00);
+\draw [color=gray,style=solid] (-4.0000,-4.0000) -- (-4.0000,4.0000);
+\draw [color=gray,style=solid] (-3.0000,-4.0000) -- (-3.0000,4.0000);
+\draw [color=gray,style=solid] (-2.0000,-4.0000) -- (-2.0000,4.0000);
+\draw [color=gray,style=solid] (-1.0000,-4.0000) -- (-1.0000,4.0000);
+\draw [color=gray,style=solid] (0.0000,-4.0000) -- (0.0000,4.0000);
+\draw [color=gray,style=solid] (1.0000,-4.0000) -- (1.0000,4.0000);
+\draw [color=gray,style=solid] (2.0000,-4.0000) -- (2.0000,4.0000);
+\draw [color=gray,style=dotted] (-3.5000,-4.0000) -- (-3.5000,4.0000);
+\draw [color=gray,style=dotted] (-2.5000,-4.0000) -- (-2.5000,4.0000);
+\draw [color=gray,style=dotted] (-1.5000,-4.0000) -- (-1.5000,4.0000);
+\draw [color=gray,style=dotted] (-0.5000,-4.0000) -- (-0.5000,4.0000);
+\draw [color=gray,style=dotted] (0.5000,-4.0000) -- (0.5000,4.0000);
+\draw [color=gray,style=dotted] (1.5000,-4.0000) -- (1.5000,4.0000);
+\draw [color=gray,style=dotted] (-4.0000,-3.7500) -- (2.0000,-3.7500);
+\draw [color=gray,style=dotted] (-4.0000,-3.2500) -- (2.0000,-3.2500);
+\draw [color=gray,style=dotted] (-4.0000,-2.7500) -- (2.0000,-2.7500);
+\draw [color=gray,style=dotted] (-4.0000,-2.2500) -- (2.0000,-2.2500);
+\draw [color=gray,style=dotted] (-4.0000,-1.7500) -- (2.0000,-1.7500);
+\draw [color=gray,style=dotted] (-4.0000,-1.2500) -- (2.0000,-1.2500);
+\draw [color=gray,style=dotted] (-4.0000,-0.7500) -- (2.0000,-0.7500);
+\draw [color=gray,style=dotted] (-4.0000,-0.2500) -- (2.0000,-0.2500);
+\draw [color=gray,style=dotted] (-4.0000,0.2500) -- (2.0000,0.2500);
+\draw [color=gray,style=dotted] (-4.0000,0.7500) -- (2.0000,0.7500);
+\draw [color=gray,style=dotted] (-4.0000,1.2500) -- (2.0000,1.2500);
+\draw [color=gray,style=dotted] (-4.0000,1.7500) -- (2.0000,1.7500);
+\draw [color=gray,style=dotted] (-4.0000,2.2500) -- (2.0000,2.2500);
+\draw [color=gray,style=dotted] (-4.0000,2.7500) -- (2.0000,2.7500);
+\draw [color=gray,style=dotted] (-4.0000,3.2500) -- (2.0000,3.2500);
+\draw [color=gray,style=dotted] (-4.0000,3.7500) -- (2.0000,3.7500);
+\draw [color=gray,style=solid] (-4.0000,-4.0000) -- (2.0000,-4.0000);
+\draw [color=gray,style=solid] (-4.0000,-3.5000) -- (2.0000,-3.5000);
+\draw [color=gray,style=solid] (-4.0000,-3.0000) -- (2.0000,-3.0000);
+\draw [color=gray,style=solid] (-4.0000,-2.5000) -- (2.0000,-2.5000);
+\draw [color=gray,style=solid] (-4.0000,-2.0000) -- (2.0000,-2.0000);
+\draw [color=gray,style=solid] (-4.0000,-1.5000) -- (2.0000,-1.5000);
+\draw [color=gray,style=solid] (-4.0000,-1.0000) -- (2.0000,-1.0000);
+\draw [color=gray,style=solid] (-4.0000,-0.5000) -- (2.0000,-0.5000);
+\draw [color=gray,style=solid] (-4.0000,0.0000) -- (2.0000,0.0000);
+\draw [color=gray,style=solid] (-4.0000,0.5000) -- (2.0000,0.5000);
+\draw [color=gray,style=solid] (-4.0000,1.0000) -- (2.0000,1.0000);
+\draw [color=gray,style=solid] (-4.0000,1.5000) -- (2.0000,1.5000);
+\draw [color=gray,style=solid] (-4.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=gray,style=solid] (-4.0000,2.5000) -- (2.0000,2.5000);
+\draw [color=gray,style=solid] (-4.0000,3.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (-4.0000,3.5000) -- (2.0000,3.5000);
+\draw [color=gray,style=solid] (-4.0000,4.0000) -- (2.0000,4.0000);
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,4.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (2.5000,0.0000);
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-\draw [color=blue] (-4.0000,0.0091578)--(-3.9394,0.0097300)--(-3.8788,0.010338)--(-3.8182,0.010984)--(-3.7576,0.011670)--(-3.6970,0.012399)--(-3.6364,0.013174)--(-3.5758,0.013997)--(-3.5152,0.014872)--(-3.4545,0.015801)--(-3.3939,0.016788)--(-3.3333,0.017837)--(-3.2727,0.018951)--(-3.2121,0.020136)--(-3.1515,0.021394)--(-3.0909,0.022730)--(-3.0303,0.024150)--(-2.9697,0.025659)--(-2.9091,0.027263)--(-2.8485,0.028966)--(-2.7879,0.030776)--(-2.7273,0.032699)--(-2.6667,0.034742)--(-2.6061,0.036912)--(-2.5455,0.039219)--(-2.4848,0.041669)--(-2.4242,0.044273)--(-2.3636,0.047039)--(-2.3030,0.049978)--(-2.2424,0.053100)--(-2.1818,0.056418)--(-2.1212,0.059943)--(-2.0606,0.063688)--(-2.0000,0.067668)--(-1.9394,0.071895)--(-1.8788,0.076388)--(-1.8182,0.081160)--(-1.7576,0.086231)--(-1.6970,0.091619)--(-1.6364,0.097343)--(-1.5758,0.10343)--(-1.5152,0.10989)--(-1.4545,0.11675)--(-1.3939,0.12405)--(-1.3333,0.13180)--(-1.2727,0.14003)--(-1.2121,0.14878)--(-1.1515,0.15808)--(-1.0909,0.16796)--(-1.0303,0.17845)--(-0.96970,0.18960)--(-0.90909,0.20145)--(-0.84848,0.21403)--(-0.78788,0.22740)--(-0.72727,0.24161)--(-0.66667,0.25671)--(-0.60606,0.27275)--(-0.54545,0.28979)--(-0.48485,0.30790)--(-0.42424,0.32713)--(-0.36364,0.34757)--(-0.30303,0.36929)--(-0.24242,0.39236)--(-0.18182,0.41688)--(-0.12121,0.44292)--(-0.060606,0.47060)--(0,0.50000)--(0.060606,0.53124)--(0.12121,0.56443)--(0.18182,0.59970)--(0.24242,0.63717)--(0.30303,0.67698)--(0.36364,0.71928)--(0.42424,0.76422)--(0.48485,0.81196)--(0.54545,0.86270)--(0.60606,0.91660)--(0.66667,0.97387)--(0.72727,1.0347)--(0.78788,1.0994)--(0.84848,1.1681)--(0.90909,1.2410)--(0.96970,1.3186)--(1.0303,1.4010)--(1.0909,1.4885)--(1.1515,1.5815)--(1.2121,1.6803)--(1.2727,1.7853)--(1.3333,1.8968)--(1.3939,2.0154)--(1.4545,2.1413)--(1.5152,2.2751)--(1.5758,2.4172)--(1.6364,2.5682)--(1.6970,2.7287)--(1.7576,2.8992)--(1.8182,3.0803)--(1.8788,3.2728)--(1.9394,3.4773)--(2.0000,3.6945);
+\draw [color=blue] (-4.0000,0.0091)--(-3.9393,0.0097)--(-3.8787,0.0103)--(-3.8181,0.0109)--(-3.7575,0.0116)--(-3.6969,0.0123)--(-3.6363,0.0131)--(-3.5757,0.0139)--(-3.5151,0.0148)--(-3.4545,0.0158)--(-3.3939,0.0167)--(-3.3333,0.0178)--(-3.2727,0.0189)--(-3.2121,0.0201)--(-3.1515,0.0213)--(-3.0909,0.0227)--(-3.0303,0.0241)--(-2.9696,0.0256)--(-2.9090,0.0272)--(-2.8484,0.0289)--(-2.7878,0.0307)--(-2.7272,0.0326)--(-2.6666,0.0347)--(-2.6060,0.0369)--(-2.5454,0.0392)--(-2.4848,0.0416)--(-2.4242,0.0442)--(-2.3636,0.0470)--(-2.3030,0.0499)--(-2.2424,0.0531)--(-2.1818,0.0564)--(-2.1212,0.0599)--(-2.0606,0.0636)--(-2.0000,0.0676)--(-1.9393,0.0718)--(-1.8787,0.0763)--(-1.8181,0.0811)--(-1.7575,0.0862)--(-1.6969,0.0916)--(-1.6363,0.0973)--(-1.5757,0.1034)--(-1.5151,0.1098)--(-1.4545,0.1167)--(-1.3939,0.1240)--(-1.3333,0.1317)--(-1.2727,0.1400)--(-1.2121,0.1487)--(-1.1515,0.1580)--(-1.0909,0.1679)--(-1.0303,0.1784)--(-0.9696,0.1895)--(-0.9090,0.2014)--(-0.8484,0.2140)--(-0.7878,0.2274)--(-0.7272,0.2416)--(-0.6666,0.2567)--(-0.6060,0.2727)--(-0.5454,0.2897)--(-0.4848,0.3078)--(-0.4242,0.3271)--(-0.3636,0.3475)--(-0.3030,0.3692)--(-0.2424,0.3923)--(-0.1818,0.4168)--(-0.1212,0.4429)--(-0.0606,0.4705)--(0.0000,0.5000)--(0.0606,0.5312)--(0.1212,0.5644)--(0.1818,0.5996)--(0.2424,0.6371)--(0.3030,0.6769)--(0.3636,0.7192)--(0.4242,0.7642)--(0.4848,0.8119)--(0.5454,0.8626)--(0.6060,0.9165)--(0.6666,0.9738)--(0.7272,1.0347)--(0.7878,1.0993)--(0.8484,1.1680)--(0.9090,1.2410)--(0.9696,1.3185)--(1.0303,1.4009)--(1.0909,1.4884)--(1.1515,1.5814)--(1.2121,1.6803)--(1.2727,1.7852)--(1.3333,1.8968)--(1.3939,2.0153)--(1.4545,2.1412)--(1.5151,2.2750)--(1.5757,2.4172)--(1.6363,2.5682)--(1.6969,2.7286)--(1.7575,2.8991)--(1.8181,3.0803)--(1.8787,3.2727)--(1.9393,3.4772)--(2.0000,3.6945);
 
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.52441,-4.0000) node {$ -40 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.52441,-3.0000) node {$ -30 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.52441,-2.0000) node {$ -20 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.52441,-1.0000) node {$ -10 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.38250,1.0000) node {$ 10 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.38250,2.0000) node {$ 20 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.38250,3.0000) node {$ 30 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.38250,4.0000) node {$ 40 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.5244,-4.0000) node {$ -40 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.5244,-3.0000) node {$ -30 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.5244,-2.0000) node {$ -20 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.5244,-1.0000) node {$ -10 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.3824,1.0000) node {$ 10 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.3824,2.0000) node {$ 20 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.3824,3.0000) node {$ 30 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.3824,4.0000) node {$ 40 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SFdgHdO.pstricks b/auto/pictures_tex/Fig_SFdgHdO.pstricks
index aa92dc4e9..3afa9a990 100644
--- a/auto/pictures_tex/Fig_SFdgHdO.pstricks
+++ b/auto/pictures_tex/Fig_SFdgHdO.pstricks
@@ -67,23 +67,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5390,0) -- (1.5474,0);
-\draw [,->,>=latex] (0,-1.5497) -- (0,1.5497);
+\draw [,->,>=latex] (-1.5389,0.0000) -- (1.5473,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5496) -- (0.0000,1.5496);
 %DEFAULT
 
-\draw [color=black] plot [smooth,tension=1] coordinates {(0.00100,1.00)(0.105,1.04)(0.189,0.931)(0.310,1.00)(0.370,0.875)(0.504,0.921)(0.537,0.784)(0.677,0.803)(0.682,0.661)(0.823,0.652)(0.800,0.513)(0.936,0.476)(0.886,0.344)(1.01,0.281)(0.936,0.162)(1.05,0.0740)(0.999,0.0447)};
+\draw [color=black] plot [smooth,tension=1] coordinates {(0.0010,0.9999)(0.1054,1.0446)(0.1887,0.9310)(0.3103,1.0030)(0.3695,0.8751)(0.5035,0.9213)(0.5365,0.7839)(0.6767,0.8028)(0.6820,0.6613)(0.8229,0.6520)(0.7996,0.5128)(0.9361,0.4756)(0.8856,0.3437)(1.0117,0.2809)(0.9361,0.1616)(1.0473,0.0739)(0.9990,0.0447)};
 
-\draw [color=green] plot [smooth,tension=1] coordinates {(-0.999,0.0447)(-1.04,0.152)(-0.922,0.230)(-0.988,0.356)(-0.857,0.410)(-0.898,0.545)(-0.759,0.571)(-0.772,0.712)(-0.630,0.711)(-0.615,0.851)(-0.476,0.822)(-0.433,0.957)(-0.303,0.900)(-0.233,1.02)(-0.118,0.943)(-0.0252,1.05)(-0.00100,1.00)};
+\draw [color=green] plot [smooth,tension=1] coordinates {(-0.9990,0.0447)(-1.0389,0.1517)(-0.9216,0.2302)(-0.9878,0.3557)(-0.8571,0.4095)(-0.8975,0.5449)(-0.7589,0.5713)(-0.7717,0.7119)(-0.6301,0.7109)(-0.6150,0.8510)(-0.4761,0.8220)(-0.4329,0.9565)(-0.3031,0.9003)(-0.2333,1.0237)(-0.1175,0.9427)(-0.0252,1.0496)(-0.0010,0.9999)};
 
-\draw [color=green] plot [smooth,tension=1] coordinates {(0.00100,-1.00)(0.105,-1.04)(0.189,-0.931)(0.310,-1.00)(0.370,-0.875)(0.504,-0.921)(0.537,-0.784)(0.677,-0.803)(0.682,-0.661)(0.823,-0.652)(0.800,-0.513)(0.936,-0.476)(0.886,-0.344)(1.01,-0.281)(0.936,-0.162)(1.05,-0.0740)(0.999,-0.0447)};
+\draw [color=green] plot [smooth,tension=1] coordinates {(0.0010,-0.9999)(0.1054,-1.0446)(0.1887,-0.9310)(0.3103,-1.0030)(0.3695,-0.8751)(0.5035,-0.9213)(0.5365,-0.7839)(0.6767,-0.8028)(0.6820,-0.6613)(0.8229,-0.6520)(0.7996,-0.5128)(0.9361,-0.4756)(0.8856,-0.3437)(1.0117,-0.2809)(0.9361,-0.1616)(1.0473,-0.0739)(0.9990,-0.0447)};
 
-\draw [color=black] plot [smooth,tension=1] coordinates {(-0.999,-0.0447)(-1.04,-0.152)(-0.922,-0.230)(-0.988,-0.356)(-0.857,-0.410)(-0.898,-0.545)(-0.759,-0.571)(-0.772,-0.712)(-0.630,-0.711)(-0.615,-0.851)(-0.476,-0.822)(-0.433,-0.957)(-0.303,-0.900)(-0.233,-1.02)(-0.118,-0.943)(-0.0252,-1.05)(-0.00100,-1.00)};
-\draw [color=red]  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,-1.0000) node [rotate=0] {$\bullet$};
-\draw (2.0495,-0.36509) node {$K_H$};
-\draw (2.0495,-0.36509) node {$K_H$};
+\draw [color=black] plot [smooth,tension=1] coordinates {(-0.9990,-0.0447)(-1.0389,-0.1517)(-0.9216,-0.2302)(-0.9878,-0.3557)(-0.8571,-0.4095)(-0.8975,-0.5449)(-0.7589,-0.5713)(-0.7717,-0.7119)(-0.6301,-0.7109)(-0.6150,-0.8510)(-0.4761,-0.8220)(-0.4329,-0.9565)(-0.3031,-0.9003)(-0.2333,-1.0237)(-0.1175,-0.9427)(-0.0252,-1.0496)(-0.0010,-0.9999)};
+\draw [color=red]  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=red]  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (2.0494,-0.3650) node {$K_H$};
+\draw (2.0494,-0.3650) node {$K_H$};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks b/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks
index d7811ff6e..657a0615e 100644
--- a/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks
+++ b/auto/pictures_tex/Fig_SJAWooRDGzIkrj.pstricks
@@ -87,21 +87,21 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.34202,0.93969) node [rotate=0] {$\bullet$};
-\draw (0.50689,1.2062) node {\( a\)};
-\draw []  (0.34202,-0.93969) node [rotate=0] {$\bullet$};
-\draw (0.48875,-1.2544) node {\( b\)};
-\draw []  (2.9238,0) node [rotate=0] {$\bullet$};
-\draw (3.4840,0) node {\( m\)};
-\draw [] (-0.303,1.17) -- (3.57,-0.235);
-\draw [] (-0.303,-1.17) -- (3.57,0.235);
-\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
-\draw (0.58852,0.32142) node {\( x\)};
-\draw []  (-0.54003,0.84164) node [rotate=0] {$\bullet$};
-\draw (-0.72702,1.0885) node {\( c\)};
-\draw [] (3.44,-0.126) -- (-1.06,0.968);
-\draw (-1.2344,-0.53513) node {\( \mC\)};
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw []  (0.3420,0.9396) node [rotate=0] {$\bullet$};
+\draw (0.5068,1.2062) node {\( a\)};
+\draw []  (0.3420,-0.9396) node [rotate=0] {$\bullet$};
+\draw (0.4887,-1.2543) node {\( b\)};
+\draw []  (2.9238,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.4840,0.0000) node {\( m\)};
+\draw [] (-0.3034,1.1746) -- (3.5692,-0.2349);
+\draw [] (-0.3034,-1.1746) -- (3.5692,0.2349);
+\draw []  (0.8660,0.5000) node [rotate=0] {$\bullet$};
+\draw (0.5885,0.3214) node {\( x\)};
+\draw []  (-0.5400,0.8416) node [rotate=0] {$\bullet$};
+\draw (-0.7270,1.0885) node {\( c\)};
+\draw [] (3.4433,-0.1262) -- (-1.0596,0.9678);
+\draw (-1.2343,-0.5351) node {\( \mC\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks b/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks
index 8f3b3168c..68f960673 100644
--- a/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks
+++ b/auto/pictures_tex/Fig_SQNPooPTrLRQ.pstricks
@@ -65,228 +65,228 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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-\draw [,->,>=latex] (-4.0000,-3.4286) -- (-4.0547,-3.4755);
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-\draw [,->,>=latex] (-4.0000,-1.7143) -- (-4.0971,-1.7559);
-\draw [,->,>=latex] (-4.0000,-1.1429) -- (-4.1111,-1.1746);
-\draw [,->,>=latex] (-4.0000,-0.57143) -- (-4.1213,-0.58875);
-\draw [,->,>=latex] (-4.0000,0) -- (-4.1250,0);
-\draw [,->,>=latex] (-4.0000,0.57143) -- (-4.1213,0.58875);
-\draw [,->,>=latex] (-4.0000,1.1429) -- (-4.1111,1.1746);
-\draw [,->,>=latex] (-4.0000,1.7143) -- (-4.0971,1.7559);
-\draw [,->,>=latex] (-4.0000,2.2857) -- (-4.0818,2.3325);
-\draw [,->,>=latex] (-4.0000,2.8571) -- (-4.0674,2.9053);
-\draw [,->,>=latex] (-4.0000,3.4286) -- (-4.0547,3.4755);
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-\draw [,->,>=latex] (-3.4286,-3.4286) -- (-3.4887,-3.4887);
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-\draw [,->,>=latex] (-2.8571,0.57143) -- (-3.0881,0.61763);
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-\draw [,->,>=latex] (-2.8571,2.2857) -- (-2.9738,2.3790);
-\draw [,->,>=latex] (-2.8571,2.8571) -- (-2.9438,2.9438);
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diff --git a/auto/pictures_tex/Fig_STdyNTH.pstricks b/auto/pictures_tex/Fig_STdyNTH.pstricks
index b283a16d4..43e8bc3e2 100644
--- a/auto/pictures_tex/Fig_STdyNTH.pstricks
+++ b/auto/pictures_tex/Fig_STdyNTH.pstricks
@@ -70,20 +70,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (1.00,0);
-\draw [] (1.00,0) -- (2.00,0);
-\draw [style=dotted] (2.00,0) -- (3.00,0);
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-\draw []  (0,0) node [rotate=0] {$o$};
-\draw (0,0.20000) node {};
-\draw []  (1.0000,0) node [rotate=0] {$o$};
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-\draw []  (2.0000,0) node [rotate=0] {$o$};
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-\draw []  (3.0000,0) node [rotate=0] {$o$};
-\draw (3.0000,0.20000) node {};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,0.31492) node {$1$};
+\draw [] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [style=dotted] (2.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (4.0000,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$o$};
+\draw (0.0000,0.2000) node {};
+\draw []  (1.0000,0.0000) node [rotate=0] {$o$};
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+\draw (3.0000,0.2000) node {};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
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diff --git a/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks b/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks
index bb5921328..5229e3cbd 100644
--- a/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks
+++ b/auto/pictures_tex/Fig_SYNKooZBuEWsWw.pstricks
@@ -119,49 +119,49 @@
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-\draw (-4.0000,-0.32983) node {$ -2 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -1 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.0000,-0.31492) node {$ 2 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.5000) node {$ -7 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-3.0000) node {$ -6 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.5000) node {$ -5 $};
-\draw [] (-0.100,-2.50) -- (0.100,-2.50);
-\draw (-0.43316,-2.0000) node {$ -4 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.5000) node {$ -3 $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.43316,-1.0000) node {$ -2 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.43316,-0.50000) node {$ -1 $};
-\draw [] (-0.100,-0.500) -- (0.100,-0.500);
-\draw (-0.29125,0.50000) node {$ 1 $};
-\draw [] (-0.100,0.500) -- (0.100,0.500);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,1.5000) node {$ 3 $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,2.5000) node {$ 5 $};
-\draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,3.5000) node {$ 7 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=blue] (-5.0000,3.0661)--(-4.8989,2.9172)--(-4.7979,2.7757)--(-4.6969,2.6413)--(-4.5959,2.5136)--(-4.4949,2.3923)--(-4.3939,2.2772)--(-4.2929,2.1678)--(-4.1919,2.0640)--(-4.0909,1.9654)--(-3.9898,1.8719)--(-3.8888,1.7832)--(-3.7878,1.6989)--(-3.6868,1.6191)--(-3.5858,1.5433)--(-3.4848,1.4715)--(-3.3838,1.4035)--(-3.2828,1.3390)--(-3.1818,1.2779)--(-3.0808,1.2201)--(-2.9797,1.1655)--(-2.8787,1.1138)--(-2.7777,1.0649)--(-2.6767,1.0187)--(-2.5757,0.9752)--(-2.4747,0.9341)--(-2.3737,0.8954)--(-2.2727,0.8591)--(-2.1717,0.8248)--(-2.0707,0.7928)--(-1.9696,0.7627)--(-1.8686,0.7345)--(-1.7676,0.7083)--(-1.6666,0.6838)--(-1.5656,0.6611)--(-1.4646,0.6401)--(-1.3636,0.6207)--(-1.2626,0.6029)--(-1.1616,0.5867)--(-1.0606,0.5719)--(-0.9595,0.5586)--(-0.8585,0.5467)--(-0.7575,0.5363)--(-0.6565,0.5271)--(-0.5555,0.5194)--(-0.4545,0.5129)--(-0.3535,0.5078)--(-0.2525,0.5039)--(-0.1515,0.5014)--(-0.0505,0.5001)--(0.0505,0.5001)--(0.1515,0.5014)--(0.2525,0.5039)--(0.3535,0.5078)--(0.4545,0.5129)--(0.5555,0.5194)--(0.6565,0.5271)--(0.7575,0.5363)--(0.8585,0.5467)--(0.9595,0.5586)--(1.0606,0.5719)--(1.1616,0.5867)--(1.2626,0.6029)--(1.3636,0.6207)--(1.4646,0.6401)--(1.5656,0.6611)--(1.6666,0.6838)--(1.7676,0.7083)--(1.8686,0.7345)--(1.9696,0.7627)--(2.0707,0.7928)--(2.1717,0.8248)--(2.2727,0.8591)--(2.3737,0.8954)--(2.4747,0.9341)--(2.5757,0.9752)--(2.6767,1.0187)--(2.7777,1.0649)--(2.8787,1.1138)--(2.9797,1.1655)--(3.0808,1.2201)--(3.1818,1.2779)--(3.2828,1.3390)--(3.3838,1.4035)--(3.4848,1.4715)--(3.5858,1.5433)--(3.6868,1.6191)--(3.7878,1.6989)--(3.8888,1.7832)--(3.9898,1.8719)--(4.0909,1.9654)--(4.1919,2.0640)--(4.2929,2.1678)--(4.3939,2.2772)--(4.4949,2.3923)--(4.5959,2.5136)--(4.6969,2.6413)--(4.7979,2.7757)--(4.8989,2.9172)--(5.0000,3.0661);
+\draw (-4.0000,-0.3298) node {$ -2 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -1 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 2 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.5000) node {$ -7 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-3.0000) node {$ -6 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.5000) node {$ -5 $};
+\draw [] (-0.1000,-2.5000) -- (0.1000,-2.5000);
+\draw (-0.4331,-2.0000) node {$ -4 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.5000) node {$ -3 $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.4331,-1.0000) node {$ -2 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.4331,-0.5000) node {$ -1 $};
+\draw [] (-0.1000,-0.5000) -- (0.1000,-0.5000);
+\draw (-0.2912,0.5000) node {$ 1 $};
+\draw [] (-0.1000,0.5000) -- (0.1000,0.5000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,1.5000) node {$ 3 $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,2.5000) node {$ 5 $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,3.5000) node {$ 7 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SenoTopologo.pstricks b/auto/pictures_tex/Fig_SenoTopologo.pstricks
index 18e3d2a7e..a1ade4ed6 100644
--- a/auto/pictures_tex/Fig_SenoTopologo.pstricks
+++ b/auto/pictures_tex/Fig_SenoTopologo.pstricks
@@ -63,14 +63,14 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-1.5862) -- (0,2.7732);
+\draw [,->,>=latex] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5861) -- (0.0000,2.7732);
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
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks b/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks
index cb4d2a63a..229eea3ad 100644
--- a/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks
+++ b/auto/pictures_tex/Fig_SolsEqDiffSin.pstricks
@@ -79,23 +79,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0808,0) -- (2.0808,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-2.0807,0.0000) -- (2.0807,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=green] (-1.5807,-1.0000)--(-1.5488,-1.0000)--(-1.5169,-1.0000)--(-1.4849,-1.0000)--(-1.4530,-1.0000)--(-1.4211,-1.0000)--(-1.3891,-1.0000)--(-1.3572,-1.0000)--(-1.3253,-1.0000)--(-1.2933,-1.0000)--(-1.2614,-1.0000)--(-1.2295,-1.0000)--(-1.1975,-1.0000)--(-1.1656,-1.0000)--(-1.1337,-1.0000)--(-1.1017,-1.0000)--(-1.0698,-1.0000)--(-1.0378,-1.0000)--(-1.0059,-1.0000)--(-0.9740,-1.0000)--(-0.9420,-1.0000)--(-0.9101,-1.0000)--(-0.8782,-1.0000)--(-0.8462,-1.0000)--(-0.8143,-1.0000)--(-0.7824,-1.0000)--(-0.7504,-1.0000)--(-0.7185,-1.0000)--(-0.6866,-1.0000)--(-0.6546,-1.0000)--(-0.6227,-1.0000)--(-0.5908,-1.0000)--(-0.5588,-1.0000)--(-0.5269,-1.0000)--(-0.4949,-1.0000)--(-0.4630,-1.0000)--(-0.4311,-1.0000)--(-0.3991,-1.0000)--(-0.3672,-1.0000)--(-0.3353,-1.0000)--(-0.3033,-1.0000)--(-0.2714,-1.0000)--(-0.2395,-1.0000)--(-0.2075,-1.0000)--(-0.1756,-1.0000)--(-0.1437,-1.0000)--(-0.1117,-1.0000)--(-0.0798,-1.0000)--(-0.0479,-1.0000)--(-0.0159,-1.0000)--(0.0159,-1.0000)--(0.0479,-1.0000)--(0.0798,-1.0000)--(0.1117,-1.0000)--(0.1437,-1.0000)--(0.1756,-1.0000)--(0.2075,-1.0000)--(0.2395,-1.0000)--(0.2714,-1.0000)--(0.3033,-1.0000)--(0.3353,-1.0000)--(0.3672,-1.0000)--(0.3991,-1.0000)--(0.4311,-1.0000)--(0.4630,-1.0000)--(0.4949,-1.0000)--(0.5269,-1.0000)--(0.5588,-1.0000)--(0.5908,-1.0000)--(0.6227,-1.0000)--(0.6546,-1.0000)--(0.6866,-1.0000)--(0.7185,-1.0000)--(0.7504,-1.0000)--(0.7824,-1.0000)--(0.8143,-1.0000)--(0.8462,-1.0000)--(0.8782,-1.0000)--(0.9101,-1.0000)--(0.9420,-1.0000)--(0.9740,-1.0000)--(1.0059,-1.0000)--(1.0378,-1.0000)--(1.0698,-1.0000)--(1.1017,-1.0000)--(1.1337,-1.0000)--(1.1656,-1.0000)--(1.1975,-1.0000)--(1.2295,-1.0000)--(1.2614,-1.0000)--(1.2933,-1.0000)--(1.3253,-1.0000)--(1.3572,-1.0000)--(1.3891,-1.0000)--(1.4211,-1.0000)--(1.4530,-1.0000)--(1.4849,-1.0000)--(1.5169,-1.0000)--(1.5488,-1.0000)--(1.5807,-1.0000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SolsSinpA.pstricks b/auto/pictures_tex/Fig_SolsSinpA.pstricks
index b23b7aff3..50f508c52 100644
--- a/auto/pictures_tex/Fig_SolsSinpA.pstricks
+++ b/auto/pictures_tex/Fig_SolsSinpA.pstricks
@@ -87,31 +87,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.6416,0) -- (3.6416,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-3.6415,0.0000) -- (3.6415,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=brown] (-3.1415,-1.0000)--(-3.1098,-0.9994)--(-3.0781,-0.9979)--(-3.0463,-0.9954)--(-3.0146,-0.9919)--(-2.9829,-0.9874)--(-2.9511,-0.9819)--(-2.9194,-0.9754)--(-2.8877,-0.9679)--(-2.8559,-0.9594)--(-2.8242,-0.9500)--(-2.7925,-0.9396)--(-2.7607,-0.9283)--(-2.7290,-0.9161)--(-2.6973,-0.9029)--(-2.6655,-0.8888)--(-2.6338,-0.8738)--(-2.6021,-0.8579)--(-2.5703,-0.8412)--(-2.5386,-0.8236)--(-2.5069,-0.8052)--(-2.4751,-0.7860)--(-2.4434,-0.7660)--(-2.4117,-0.7452)--(-2.3799,-0.7237)--(-2.3482,-0.7014)--(-2.3165,-0.6785)--(-2.2847,-0.6548)--(-2.2530,-0.6305)--(-2.2213,-0.6056)--(-2.1895,-0.5800)--(-2.1578,-0.5539)--(-2.1261,-0.5272)--(-2.0943,-0.5000)--(-2.0626,-0.4722)--(-2.0309,-0.4440)--(-1.9991,-0.4154)--(-1.9674,-0.3863)--(-1.9357,-0.3568)--(-1.9039,-0.3270)--(-1.8722,-0.2969)--(-1.8405,-0.2664)--(-1.8087,-0.2357)--(-1.7770,-0.2048)--(-1.7453,-0.1736)--(-1.7135,-0.1423)--(-1.6818,-0.1108)--(-1.6501,-0.0792)--(-1.6183,-0.0475)--(-1.5866,-0.0158)--(-1.5549,0.0158)--(-1.5231,0.0475)--(-1.4914,0.0792)--(-1.4597,0.1108)--(-1.4279,0.1423)--(-1.3962,0.1736)--(-1.3645,0.2048)--(-1.3327,0.2357)--(-1.3010,0.2664)--(-1.2693,0.2969)--(-1.2375,0.3270)--(-1.2058,0.3568)--(-1.1741,0.3863)--(-1.1423,0.4154)--(-1.1106,0.4440)--(-1.0789,0.4722)--(-1.0471,0.5000)--(-1.0154,0.5272)--(-0.9837,0.5539)--(-0.9519,0.5800)--(-0.9202,0.6056)--(-0.8885,0.6305)--(-0.8567,0.6548)--(-0.8250,0.6785)--(-0.7933,0.7014)--(-0.7615,0.7237)--(-0.7298,0.7452)--(-0.6981,0.7660)--(-0.6663,0.7860)--(-0.6346,0.8052)--(-0.6029,0.8236)--(-0.5711,0.8412)--(-0.5394,0.8579)--(-0.5077,0.8738)--(-0.4759,0.8888)--(-0.4442,0.9029)--(-0.4125,0.9161)--(-0.3807,0.9283)--(-0.3490,0.9396)--(-0.3173,0.9500)--(-0.2855,0.9594)--(-0.2538,0.9679)--(-0.2221,0.9754)--(-0.1903,0.9819)--(-0.1586,0.9874)--(-0.1269,0.9919)--(-0.0951,0.9954)--(-0.0634,0.9979)--(-0.0317,0.9994)--(0.0000,1.0000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SpiraleLimite.pstricks b/auto/pictures_tex/Fig_SpiraleLimite.pstricks
index 3908c2a7d..89c0ff412 100644
--- a/auto/pictures_tex/Fig_SpiraleLimite.pstricks
+++ b/auto/pictures_tex/Fig_SpiraleLimite.pstricks
@@ -63,11 +63,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.7107);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.7106);
 %DEFAULT
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-\draw [] (3.00,-0.100) -- (3.00,0.100);
+\draw [color=blue] (0.0000,0.0000)--(0.0214,0.0214)--(0.0428,0.0428)--(0.0642,0.0642)--(0.0857,0.0857)--(0.1071,0.1071)--(0.1285,0.1285)--(0.1500,0.1499)--(0.1714,0.1713)--(0.1929,0.1927)--(0.2143,0.2141)--(0.2358,0.2355)--(0.2573,0.2569)--(0.2788,0.2782)--(0.3004,0.2995)--(0.3219,0.3208)--(0.3435,0.3421)--(0.3651,0.3633)--(0.3868,0.3845)--(0.4085,0.4056)--(0.4303,0.4267)--(0.4521,0.4478)--(0.4739,0.4688)--(0.4959,0.4897)--(0.5179,0.5105)--(0.5399,0.5313)--(0.5621,0.5520)--(0.5843,0.5726)--(0.6067,0.5931)--(0.6291,0.6135)--(0.6517,0.6338)--(0.6743,0.6539)--(0.6971,0.6740)--(0.7200,0.6938)--(0.7431,0.7136)--(0.7663,0.7332)--(0.7897,0.7526)--(0.8132,0.7718)--(0.8369,0.7908)--(0.8608,0.8097)--(0.8849,0.8283)--(0.9091,0.8467)--(0.9336,0.8649)--(0.9583,0.8828)--(0.9833,0.9004)--(1.0084,0.9178)--(1.0339,0.9349)--(1.0596,0.9516)--(1.0855,0.9681)--(1.1117,0.9842)--(1.1382,0.9999)--(1.1651,1.0153)--(1.1922,1.0303)--(1.2196,1.0449)--(1.2474,1.0590)--(1.2755,1.0727)--(1.3040,1.0859)--(1.3328,1.0986)--(1.3620,1.1108)--(1.3916,1.1224)--(1.4215,1.1335)--(1.4519,1.1440)--(1.4827,1.1538)--(1.5139,1.1630)--(1.5455,1.1715)--(1.5776,1.1793)--(1.6101,1.1863)--(1.6431,1.1925)--(1.6766,1.1979)--(1.7105,1.2024)--(1.7450,1.2060)--(1.7799,1.2086)--(1.8154,1.2101)--(1.8514,1.2106)--(1.8879,1.2100)--(1.9249,1.2081)--(1.9625,1.2050)--(2.0007,1.2005)--(2.0395,1.1946)--(2.0788,1.1871)--(2.1187,1.1781)--(2.1592,1.1672)--(2.2003,1.1545)--(2.2420,1.1397)--(2.2844,1.1228)--(2.3274,1.1034)--(2.3710,1.0815)--(2.4153,1.0567)--(2.4602,1.0287)--(2.5058,0.9972)--(2.5520,0.9616)--(2.5990,0.9215)--(2.6466,0.8760)--(2.6949,0.8240)--(2.7440,0.7643)--(2.7937,0.6944)--(2.8442,0.6108)--(2.8954,0.5065)--(2.9473,0.3636)--(3.0000,0.0000);
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks b/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks
index 19a5c0484..c5e7152c3 100644
--- a/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks
+++ b/auto/pictures_tex/Fig_SubfiguresCDUTraceCycloide.pstricks
@@ -69,65 +69,65 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,3.1310);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.4902,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,3.1309);
 %DEFAULT
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.0000,1.0000) -- (0,1.0000);
-\draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw [color=green,->,>=latex] (1.0000,1.0000) -- (1.7071,0.29289);
-\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (2.1416,1.0000) -- (3.1416,1.0000);
-\draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw [color=green,->,>=latex] (2.1416,1.0000) -- (1.4345,0.29289);
-\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (4.7124,0) -- (4.7124,1.0000);
-\draw [color=brown] (3.71,0) -- (5.71,0);
-\draw [color=green,->,>=latex] (4.7124,0) -- (4.7124,-1.0000);
-\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (7.2832,1.0000) -- (6.2832,1.0000);
-\draw [color=brown] (6.58,0.293) -- (7.99,1.71);
-\draw [color=green,->,>=latex] (7.2832,1.0000) -- (7.9903,0.29289);
+\draw [,->,>=latex] (1.0000,1.0000) -- (0.0000,1.0000);
+\draw [color=brown] (0.2928,0.2928) -- (1.7071,1.7071);
+\draw [color=green,->,>=latex] (1.0000,1.0000) -- (1.7071,0.2928);
+\draw []  (2.1415,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (2.1415,1.0000) -- (3.1415,1.0000);
+\draw [color=brown] (1.4344,1.7071) -- (2.8486,0.2928);
+\draw [color=green,->,>=latex] (2.1415,1.0000) -- (1.4344,0.2928);
+\draw []  (4.7123,0.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (4.7123,0.0000) -- (4.7123,1.0000);
+\draw [color=brown] (3.7123,0.0000) -- (5.7123,0.0000);
+\draw [color=green,->,>=latex] (4.7123,0.0000) -- (4.7123,-1.0000);
+\draw []  (7.2831,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (7.2831,1.0000) -- (6.2831,1.0000);
+\draw [color=brown] (6.5760,0.2928) -- (7.9902,1.7071);
+\draw [color=green,->,>=latex] (7.2831,1.0000) -- (7.9902,0.2928);
 \draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.4925,1.7071) -- (0.78540,1.0000);
-\draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw [color=green,->,>=latex] (1.4925,1.7071) -- (2.4164,1.3244);
-\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.6491,1.7071) -- (2.3562,1.0000);
-\draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw [color=green,->,>=latex] (1.6491,1.7071) -- (0.72521,1.3244);
-\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (3.2199,0.29289) -- (3.9270,1.0000);
-\draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw [color=green,->,>=latex] (3.2199,0.29289) -- (2.8372,-0.63099);
-\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (6.2049,0.29289) -- (5.4978,1.0000);
-\draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
-\draw [color=green,->,>=latex] (6.2049,0.29289) -- (6.5876,-0.63099);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [,->,>=latex] (1.4925,1.7071) -- (0.7853,1.0000);
+\draw [color=brown] (1.1098,0.7832) -- (1.8751,2.6309);
+\draw [color=green,->,>=latex] (1.4925,1.7071) -- (2.4163,1.3244);
+\draw []  (1.6490,1.7071) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.6490,1.7071) -- (2.3561,1.0000);
+\draw [color=brown] (1.2664,2.6309) -- (2.0317,0.7832);
+\draw [color=green,->,>=latex] (1.6490,1.7071) -- (0.7252,1.3244);
+\draw []  (3.2198,0.2928) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (3.2198,0.2928) -- (3.9269,1.0000);
+\draw [color=brown] (2.2960,0.6755) -- (4.1437,-0.0897);
+\draw [color=green,->,>=latex] (3.2198,0.2928) -- (2.8372,-0.6309);
+\draw []  (6.2048,0.2928) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (6.2048,0.2928) -- (5.4977,1.0000);
+\draw [color=brown] (5.2810,-0.0897) -- (7.1287,0.6755);
+\draw [color=green,->,>=latex] (6.2048,0.2928) -- (6.5875,-0.6309);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -189,48 +189,48 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
-\draw [,->,>=latex] (0,-0.58979) -- (0,3.1310);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.4902,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5897) -- (0.0000,3.1309);
 %DEFAULT
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
-\draw [color=brown] (3.71,0) -- (5.71,0);
-\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown] (6.58,0.293) -- (7.99,1.71);
+\draw [color=brown] (0.2928,0.2928) -- (1.7071,1.7071);
+\draw []  (2.1415,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown] (1.4344,1.7071) -- (2.8486,0.2928);
+\draw []  (4.7123,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown] (3.7123,0.0000) -- (5.7123,0.0000);
+\draw []  (7.2831,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown] (6.5760,0.2928) -- (7.9902,1.7071);
 \draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
-\draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
-\draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
-\draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
-\draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
-\draw [color=blue,style=dashed] (1.000,1.000)--(1.061,1.063)--(1.119,1.127)--(1.172,1.189)--(1.222,1.251)--(1.267,1.312)--(1.309,1.372)--(1.347,1.430)--(1.382,1.486)--(1.412,1.541)--(1.440,1.593)--(1.464,1.643)--(1.485,1.690)--(1.504,1.735)--(1.519,1.776)--(1.532,1.815)--(1.543,1.850)--(1.551,1.881)--(1.558,1.910)--(1.563,1.934)--(1.566,1.955)--(1.569,1.972)--(1.570,1.985)--(1.571,1.994)--(1.571,1.999)--(1.571,2.000)--(1.571,1.997)--(1.571,1.990)--(1.572,1.979)--(1.574,1.964)--(1.577,1.945)--(1.581,1.922)--(1.587,1.896)--(1.594,1.866)--(1.604,1.833)--(1.616,1.796)--(1.630,1.756)--(1.647,1.713)--(1.666,1.667)--(1.689,1.618)--(1.715,1.567)--(1.744,1.514)--(1.777,1.458)--(1.813,1.401)--(1.853,1.342)--(1.896,1.282)--(1.944,1.220)--(1.995,1.158)--(2.051,1.095)--(2.110,1.032)--(2.174,0.9683)--(2.241,0.9049)--(2.313,0.8420)--(2.388,0.7797)--(2.468,0.7183)--(2.551,0.6580)--(2.638,0.5991)--(2.729,0.5418)--(2.823,0.4863)--(2.921,0.4329)--(3.022,0.3818)--(3.126,0.3332)--(3.233,0.2873)--(3.344,0.2443)--(3.456,0.2042)--(3.571,0.1674)--(3.689,0.1340)--(3.808,0.1040)--(3.929,0.07765)--(4.052,0.05500)--(4.176,0.03616)--(4.301,0.02120)--(4.427,0.01018)--(4.554,0.003145)--(4.681,0)--(4.808,0.001133)--(4.934,0.006162)--(5.061,0.01519)--(5.186,0.02819)--(5.311,0.04510)--(5.434,0.06585)--(5.556,0.09037)--(5.677,0.1185)--(5.795,0.1503)--(5.911,0.1854)--(6.025,0.2239)--(6.137,0.2654)--(6.245,0.3099)--(6.351,0.3572)--(6.454,0.4071)--(6.553,0.4594)--(6.649,0.5138)--(6.742,0.5702)--(6.831,0.6283)--(6.916,0.6880)--(6.997,0.7489)--(7.075,0.8107)--(7.148,0.8734)--(7.218,0.9366)--(7.283,1.000);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=brown] (1.1098,0.7832) -- (1.8751,2.6309);
+\draw []  (1.6490,1.7071) node [rotate=0] {$\bullet$};
+\draw [color=brown] (1.2664,2.6309) -- (2.0317,0.7832);
+\draw []  (3.2198,0.2928) node [rotate=0] {$\bullet$};
+\draw [color=brown] (2.2960,0.6755) -- (4.1437,-0.0897);
+\draw []  (6.2048,0.2928) node [rotate=0] {$\bullet$};
+\draw [color=brown] (5.2810,-0.0897) -- (7.1287,0.6755);
+\draw [color=blue,style=dashed] (1.0000,1.0000)--(1.0614,1.0634)--(1.1188,1.1265)--(1.1723,1.1892)--(1.2218,1.2511)--(1.2674,1.3120)--(1.3091,1.3716)--(1.3471,1.4297)--(1.3815,1.4861)--(1.4124,1.5406)--(1.4399,1.5929)--(1.4641,1.6427)--(1.4853,1.6900)--(1.5035,1.7345)--(1.5190,1.7761)--(1.5320,1.8145)--(1.5426,1.8497)--(1.5512,1.8814)--(1.5578,1.9096)--(1.5627,1.9341)--(1.5662,1.9549)--(1.5685,1.9718)--(1.5699,1.9848)--(1.5705,1.9938)--(1.5707,1.9988)--(1.5707,1.9998)--(1.5708,1.9968)--(1.5712,1.9898)--(1.5722,1.9788)--(1.5740,1.9638)--(1.5769,1.9450)--(1.5811,1.9223)--(1.5868,1.8959)--(1.5943,1.8660)--(1.6039,1.8325)--(1.6157,1.7957)--(1.6299,1.7557)--(1.6467,1.7126)--(1.6664,1.6667)--(1.6891,1.6181)--(1.7149,1.5670)--(1.7441,1.5136)--(1.7767,1.4582)--(1.8129,1.4009)--(1.8528,1.3420)--(1.8965,1.2817)--(1.9440,1.2203)--(1.9954,1.1580)--(2.0509,1.0950)--(2.1103,1.0317)--(2.1738,0.9682)--(2.2413,0.9049)--(2.3128,0.8419)--(2.3882,0.7796)--(2.4676,0.7182)--(2.5509,0.6579)--(2.6380,0.5990)--(2.7287,0.5417)--(2.8230,0.4863)--(2.9208,0.4329)--(3.0219,0.3818)--(3.1261,0.3332)--(3.2334,0.2873)--(3.3435,0.2442)--(3.4562,0.2042)--(3.5714,0.1674)--(3.6887,0.1339)--(3.8081,0.1040)--(3.9293,0.0776)--(4.0521,0.0549)--(4.1761,0.0361)--(4.3013,0.0211)--(4.4272,0.0101)--(4.5538,0.0031)--(4.6806,0.0000)--(4.8075,0.0011)--(4.9342,0.0061)--(5.0605,0.0151)--(5.1861,0.0281)--(5.3107,0.0450)--(5.4342,0.0658)--(5.5562,0.0903)--(5.6765,0.1185)--(5.7949,0.1502)--(5.9112,0.1854)--(6.0252,0.2238)--(6.1366,0.2654)--(6.2453,0.3099)--(6.3510,0.3572)--(6.4537,0.4070)--(6.5532,0.4593)--(6.6493,0.5138)--(6.7418,0.5702)--(6.8307,0.6283)--(6.9159,0.6879)--(6.9972,0.7488)--(7.0747,0.8107)--(7.1482,0.8734)--(7.2177,0.9365)--(7.2831,1.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks b/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks
index ad7c6baeb..d01245efa 100644
--- a/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks
+++ b/auto/pictures_tex/Fig_SuiteInverseAlterne.pstricks
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,2.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,2.0000);
 %DEFAULT
 \draw []  (1.0000,-3.0000) node [rotate=0] {$\bullet$};
 \draw (1.0000,-3.4298) node {$-1$};
 \draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw (2.0000,1.9825) node {$1/2$};
+\draw (2.0000,1.9824) node {$1/2$};
 \draw []  (3.0000,-1.0000) node [rotate=0] {$\bullet$};
-\draw (3.0000,-1.4825) node {$-1/3$};
-\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
-\draw (4.0000,1.2325) node {$1/4$};
-\draw []  (5.0000,-0.60000) node [rotate=0] {$\bullet$};
-\draw (5.0000,-1.0825) node {$-1/5$};
-\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
-\draw (6.0000,0.98246) node {$1/6$};
-\draw []  (7.0000,-0.42857) node [rotate=0] {$\bullet$};
-\draw (7.0000,-0.91103) node {$-1/7$};
-\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
-\draw (8.0000,0.85746) node {$1/8$};
-\draw []  (9.0000,-0.33333) node [rotate=0] {$\bullet$};
-\draw (9.0000,-0.81579) node {$-1/9$};
-\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
-\draw (10.000,0.78246) node {$1/10$};
+\draw (3.0000,-1.4824) node {$-1/3$};
+\draw []  (4.0000,0.7500) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2324) node {$1/4$};
+\draw []  (5.0000,-0.6000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-1.0824) node {$-1/5$};
+\draw []  (6.0000,0.5000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.9824) node {$1/6$};
+\draw []  (7.0000,-0.4285) node [rotate=0] {$\bullet$};
+\draw (7.0000,-0.9110) node {$-1/7$};
+\draw []  (8.0000,0.3750) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.8574) node {$1/8$};
+\draw []  (9.0000,-0.3333) node [rotate=0] {$\bullet$};
+\draw (9.0000,-0.8157) node {$-1/9$};
+\draw []  (10.000,0.3000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.7824) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SuiteUnSurn.pstricks b/auto/pictures_tex/Fig_SuiteUnSurn.pstricks
index ad00f9bf4..7be6ac65d 100644
--- a/auto/pictures_tex/Fig_SuiteUnSurn.pstricks
+++ b/auto/pictures_tex/Fig_SuiteUnSurn.pstricks
@@ -103,29 +103,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 \draw []  (1.0000,3.0000) node [rotate=0] {$\bullet$};
 \draw (1.0000,3.4149) node {$1$};
 \draw []  (2.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw (2.0000,1.9825) node {$1/2$};
+\draw (2.0000,1.9824) node {$1/2$};
 \draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.0000,1.4825) node {$1/3$};
-\draw []  (4.0000,0.75000) node [rotate=0] {$\bullet$};
-\draw (4.0000,1.2325) node {$1/4$};
-\draw []  (5.0000,0.60000) node [rotate=0] {$\bullet$};
-\draw (5.0000,1.0825) node {$1/5$};
-\draw []  (6.0000,0.50000) node [rotate=0] {$\bullet$};
-\draw (6.0000,0.98246) node {$1/6$};
-\draw []  (7.0000,0.42857) node [rotate=0] {$\bullet$};
-\draw (7.0000,0.91103) node {$1/7$};
-\draw []  (8.0000,0.37500) node [rotate=0] {$\bullet$};
-\draw (8.0000,0.85746) node {$1/8$};
-\draw []  (9.0000,0.33333) node [rotate=0] {$\bullet$};
-\draw (9.0000,0.81579) node {$1/9$};
-\draw []  (10.000,0.30000) node [rotate=0] {$\bullet$};
-\draw (10.000,0.78246) node {$1/10$};
+\draw (3.0000,1.4824) node {$1/3$};
+\draw []  (4.0000,0.7500) node [rotate=0] {$\bullet$};
+\draw (4.0000,1.2324) node {$1/4$};
+\draw []  (5.0000,0.6000) node [rotate=0] {$\bullet$};
+\draw (5.0000,1.0824) node {$1/5$};
+\draw []  (6.0000,0.5000) node [rotate=0] {$\bullet$};
+\draw (6.0000,0.9824) node {$1/6$};
+\draw []  (7.0000,0.4285) node [rotate=0] {$\bullet$};
+\draw (7.0000,0.9110) node {$1/7$};
+\draw []  (8.0000,0.3750) node [rotate=0] {$\bullet$};
+\draw (8.0000,0.8574) node {$1/8$};
+\draw []  (9.0000,0.3333) node [rotate=0] {$\bullet$};
+\draw (9.0000,0.8157) node {$1/9$};
+\draw []  (10.000,0.3000) node [rotate=0] {$\bullet$};
+\draw (10.000,0.7824) node {$1/10$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfaceCercle.pstricks b/auto/pictures_tex/Fig_SurfaceCercle.pstricks
index 79caff591..63c498a41 100644
--- a/auto/pictures_tex/Fig_SurfaceCercle.pstricks
+++ b/auto/pictures_tex/Fig_SurfaceCercle.pstricks
@@ -63,13 +63,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.4999) -- (0,2.4999);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4998) -- (0.0000,2.4998);
 %DEFAULT
 
-\draw [color=blue] (-2.000,0)--(-1.960,0.4000)--(-1.919,0.5628)--(-1.879,0.6857)--(-1.838,0.7876)--(-1.798,0.8759)--(-1.758,0.9544)--(-1.717,1.025)--(-1.677,1.090)--(-1.636,1.150)--(-1.596,1.205)--(-1.556,1.257)--(-1.515,1.305)--(-1.475,1.351)--(-1.434,1.394)--(-1.394,1.434)--(-1.354,1.472)--(-1.313,1.509)--(-1.273,1.543)--(-1.232,1.575)--(-1.192,1.606)--(-1.152,1.635)--(-1.111,1.663)--(-1.071,1.689)--(-1.030,1.714)--(-0.9899,1.738)--(-0.9495,1.760)--(-0.9091,1.781)--(-0.8687,1.801)--(-0.8283,1.820)--(-0.7879,1.838)--(-0.7475,1.855)--(-0.7071,1.871)--(-0.6667,1.886)--(-0.6263,1.899)--(-0.5859,1.912)--(-0.5455,1.924)--(-0.5051,1.935)--(-0.4646,1.945)--(-0.4242,1.954)--(-0.3838,1.963)--(-0.3434,1.970)--(-0.3030,1.977)--(-0.2626,1.983)--(-0.2222,1.988)--(-0.1818,1.992)--(-0.1414,1.995)--(-0.1010,1.997)--(-0.06061,1.999)--(-0.02020,2.000)--(0.02020,2.000)--(0.06061,1.999)--(0.1010,1.997)--(0.1414,1.995)--(0.1818,1.992)--(0.2222,1.988)--(0.2626,1.983)--(0.3030,1.977)--(0.3434,1.970)--(0.3838,1.963)--(0.4242,1.954)--(0.4646,1.945)--(0.5051,1.935)--(0.5455,1.924)--(0.5859,1.912)--(0.6263,1.899)--(0.6667,1.886)--(0.7071,1.871)--(0.7475,1.855)--(0.7879,1.838)--(0.8283,1.820)--(0.8687,1.801)--(0.9091,1.781)--(0.9495,1.760)--(0.9899,1.738)--(1.030,1.714)--(1.071,1.689)--(1.111,1.663)--(1.152,1.635)--(1.192,1.606)--(1.232,1.575)--(1.273,1.543)--(1.313,1.509)--(1.354,1.472)--(1.394,1.434)--(1.434,1.394)--(1.475,1.351)--(1.515,1.305)--(1.556,1.257)--(1.596,1.205)--(1.636,1.150)--(1.677,1.090)--(1.717,1.025)--(1.758,0.9544)--(1.798,0.8759)--(1.838,0.7876)--(1.879,0.6857)--(1.919,0.5628)--(1.960,0.4000)--(2.000,0);
+\draw [color=blue] (-2.0000,0.0000)--(-1.9595,0.3999)--(-1.9191,0.5627)--(-1.8787,0.6856)--(-1.8383,0.7876)--(-1.7979,0.8759)--(-1.7575,0.9544)--(-1.7171,1.0253)--(-1.6767,1.0901)--(-1.6363,1.1499)--(-1.5959,1.2053)--(-1.5555,1.2570)--(-1.5151,1.3054)--(-1.4747,1.3509)--(-1.4343,1.3937)--(-1.3939,1.4342)--(-1.3535,1.4723)--(-1.3131,1.5085)--(-1.2727,1.5427)--(-1.2323,1.5752)--(-1.1919,1.6060)--(-1.1515,1.6352)--(-1.1111,1.6629)--(-1.0707,1.6892)--(-1.0303,1.7141)--(-0.9898,1.7378)--(-0.9494,1.7602)--(-0.9090,1.7814)--(-0.8686,1.8014)--(-0.8282,1.8204)--(-0.7878,1.8382)--(-0.7474,1.8550)--(-0.7070,1.8708)--(-0.6666,1.8856)--(-0.6262,1.8994)--(-0.5858,1.9122)--(-0.5454,1.9241)--(-0.5050,1.9351)--(-0.4646,1.9452)--(-0.4242,1.9544)--(-0.3838,1.9628)--(-0.3434,1.9702)--(-0.3030,1.9769)--(-0.2626,1.9826)--(-0.2222,1.9876)--(-0.1818,1.9917)--(-0.1414,1.9949)--(-0.1010,1.9974)--(-0.0606,1.9990)--(-0.0202,1.9998)--(0.0202,1.9998)--(0.0606,1.9990)--(0.1010,1.9974)--(0.1414,1.9949)--(0.1818,1.9917)--(0.2222,1.9876)--(0.2626,1.9826)--(0.3030,1.9769)--(0.3434,1.9702)--(0.3838,1.9628)--(0.4242,1.9544)--(0.4646,1.9452)--(0.5050,1.9351)--(0.5454,1.9241)--(0.5858,1.9122)--(0.6262,1.8994)--(0.6666,1.8856)--(0.7070,1.8708)--(0.7474,1.8550)--(0.7878,1.8382)--(0.8282,1.8204)--(0.8686,1.8014)--(0.9090,1.7814)--(0.9494,1.7602)--(0.9898,1.7378)--(1.0303,1.7141)--(1.0707,1.6892)--(1.1111,1.6629)--(1.1515,1.6352)--(1.1919,1.6060)--(1.2323,1.5752)--(1.2727,1.5427)--(1.3131,1.5085)--(1.3535,1.4723)--(1.3939,1.4342)--(1.4343,1.3937)--(1.4747,1.3509)--(1.5151,1.3054)--(1.5555,1.2570)--(1.5959,1.2053)--(1.6363,1.1499)--(1.6767,1.0901)--(1.7171,1.0253)--(1.7575,0.9544)--(1.7979,0.8759)--(1.8383,0.7876)--(1.8787,0.6856)--(1.9191,0.5627)--(1.9595,0.3999)--(2.0000,0.0000);
 
-\draw [color=red] (-2.000,0)--(-1.960,-0.4000)--(-1.919,-0.5628)--(-1.879,-0.6857)--(-1.838,-0.7876)--(-1.798,-0.8759)--(-1.758,-0.9544)--(-1.717,-1.025)--(-1.677,-1.090)--(-1.636,-1.150)--(-1.596,-1.205)--(-1.556,-1.257)--(-1.515,-1.305)--(-1.475,-1.351)--(-1.434,-1.394)--(-1.394,-1.434)--(-1.354,-1.472)--(-1.313,-1.509)--(-1.273,-1.543)--(-1.232,-1.575)--(-1.192,-1.606)--(-1.152,-1.635)--(-1.111,-1.663)--(-1.071,-1.689)--(-1.030,-1.714)--(-0.9899,-1.738)--(-0.9495,-1.760)--(-0.9091,-1.781)--(-0.8687,-1.801)--(-0.8283,-1.820)--(-0.7879,-1.838)--(-0.7475,-1.855)--(-0.7071,-1.871)--(-0.6667,-1.886)--(-0.6263,-1.899)--(-0.5859,-1.912)--(-0.5455,-1.924)--(-0.5051,-1.935)--(-0.4646,-1.945)--(-0.4242,-1.954)--(-0.3838,-1.963)--(-0.3434,-1.970)--(-0.3030,-1.977)--(-0.2626,-1.983)--(-0.2222,-1.988)--(-0.1818,-1.992)--(-0.1414,-1.995)--(-0.1010,-1.997)--(-0.06061,-1.999)--(-0.02020,-2.000)--(0.02020,-2.000)--(0.06061,-1.999)--(0.1010,-1.997)--(0.1414,-1.995)--(0.1818,-1.992)--(0.2222,-1.988)--(0.2626,-1.983)--(0.3030,-1.977)--(0.3434,-1.970)--(0.3838,-1.963)--(0.4242,-1.954)--(0.4646,-1.945)--(0.5051,-1.935)--(0.5455,-1.924)--(0.5859,-1.912)--(0.6263,-1.899)--(0.6667,-1.886)--(0.7071,-1.871)--(0.7475,-1.855)--(0.7879,-1.838)--(0.8283,-1.820)--(0.8687,-1.801)--(0.9091,-1.781)--(0.9495,-1.760)--(0.9899,-1.738)--(1.030,-1.714)--(1.071,-1.689)--(1.111,-1.663)--(1.152,-1.635)--(1.192,-1.606)--(1.232,-1.575)--(1.273,-1.543)--(1.313,-1.509)--(1.354,-1.472)--(1.394,-1.434)--(1.434,-1.394)--(1.475,-1.351)--(1.515,-1.305)--(1.556,-1.257)--(1.596,-1.205)--(1.636,-1.150)--(1.677,-1.090)--(1.717,-1.025)--(1.758,-0.9544)--(1.798,-0.8759)--(1.838,-0.7876)--(1.879,-0.6857)--(1.919,-0.5628)--(1.960,-0.4000)--(2.000,0);
+\draw [color=red] (-2.0000,0.0000)--(-1.9595,-0.3999)--(-1.9191,-0.5627)--(-1.8787,-0.6856)--(-1.8383,-0.7876)--(-1.7979,-0.8759)--(-1.7575,-0.9544)--(-1.7171,-1.0253)--(-1.6767,-1.0901)--(-1.6363,-1.1499)--(-1.5959,-1.2053)--(-1.5555,-1.2570)--(-1.5151,-1.3054)--(-1.4747,-1.3509)--(-1.4343,-1.3937)--(-1.3939,-1.4342)--(-1.3535,-1.4723)--(-1.3131,-1.5085)--(-1.2727,-1.5427)--(-1.2323,-1.5752)--(-1.1919,-1.6060)--(-1.1515,-1.6352)--(-1.1111,-1.6629)--(-1.0707,-1.6892)--(-1.0303,-1.7141)--(-0.9898,-1.7378)--(-0.9494,-1.7602)--(-0.9090,-1.7814)--(-0.8686,-1.8014)--(-0.8282,-1.8204)--(-0.7878,-1.8382)--(-0.7474,-1.8550)--(-0.7070,-1.8708)--(-0.6666,-1.8856)--(-0.6262,-1.8994)--(-0.5858,-1.9122)--(-0.5454,-1.9241)--(-0.5050,-1.9351)--(-0.4646,-1.9452)--(-0.4242,-1.9544)--(-0.3838,-1.9628)--(-0.3434,-1.9702)--(-0.3030,-1.9769)--(-0.2626,-1.9826)--(-0.2222,-1.9876)--(-0.1818,-1.9917)--(-0.1414,-1.9949)--(-0.1010,-1.9974)--(-0.0606,-1.9990)--(-0.0202,-1.9998)--(0.0202,-1.9998)--(0.0606,-1.9990)--(0.1010,-1.9974)--(0.1414,-1.9949)--(0.1818,-1.9917)--(0.2222,-1.9876)--(0.2626,-1.9826)--(0.3030,-1.9769)--(0.3434,-1.9702)--(0.3838,-1.9628)--(0.4242,-1.9544)--(0.4646,-1.9452)--(0.5050,-1.9351)--(0.5454,-1.9241)--(0.5858,-1.9122)--(0.6262,-1.8994)--(0.6666,-1.8856)--(0.7070,-1.8708)--(0.7474,-1.8550)--(0.7878,-1.8382)--(0.8282,-1.8204)--(0.8686,-1.8014)--(0.9090,-1.7814)--(0.9494,-1.7602)--(0.9898,-1.7378)--(1.0303,-1.7141)--(1.0707,-1.6892)--(1.1111,-1.6629)--(1.1515,-1.6352)--(1.1919,-1.6060)--(1.2323,-1.5752)--(1.2727,-1.5427)--(1.3131,-1.5085)--(1.3535,-1.4723)--(1.3939,-1.4342)--(1.4343,-1.3937)--(1.4747,-1.3509)--(1.5151,-1.3054)--(1.5555,-1.2570)--(1.5959,-1.2053)--(1.6363,-1.1499)--(1.6767,-1.0901)--(1.7171,-1.0253)--(1.7575,-0.9544)--(1.7979,-0.8759)--(1.8383,-0.7876)--(1.8787,-0.6856)--(1.9191,-0.5627)--(1.9595,-0.3999)--(2.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks b/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks
index 3a69fd3a0..f9d0646be 100644
--- a/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks
+++ b/auto/pictures_tex/Fig_SurfaceEntreCourbes.pstricks
@@ -41,22 +41,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8999);
 %DEFAULT
+\fill [color=brown] (0.6742,1.6500) -- (0.7111,1.6799) -- (0.7480,1.7093) -- (0.7849,1.7381) -- (0.8217,1.7663) -- (0.8586,1.7938) -- (0.8955,1.8207) -- (0.9324,1.8471) -- (0.9693,1.8728) -- (1.0062,1.8979) -- (1.0430,1.9224) -- (1.0799,1.9462) -- (1.1168,1.9695) -- (1.1537,1.9922) -- (1.1906,2.0142) -- (1.2275,2.0356) -- (1.2643,2.0564) -- (1.3012,2.0766) -- (1.3381,2.0962) -- (1.3750,2.1152) -- (1.4119,2.1336) -- (1.4488,2.1513) -- (1.4856,2.1685) -- (1.5225,2.1850) -- (1.5594,2.2009) -- (1.5963,2.2162) -- (1.6332,2.2309) -- (1.6701,2.2450) -- (1.7070,2.2585) -- (1.7438,2.2713) -- (1.7807,2.2836) -- (1.8176,2.2952) -- (1.8545,2.3062) -- (1.8914,2.3166) -- (1.9283,2.3264) -- (1.9651,2.3356) -- (2.0020,2.3442) -- (2.0389,2.3521) -- (2.0758,2.3595) -- (2.1127,2.3662) -- (2.1496,2.3723) -- (2.1864,2.3778) -- (2.2233,2.3827) -- (2.2602,2.3870) -- (2.2971,2.3907) -- (2.3340,2.3938) -- (2.3709,2.3962) -- (2.4077,2.3980) -- (2.4446,2.3993) -- (2.4815,2.3999) -- (2.5184,2.3999) -- (2.5553,2.3993) -- (2.5922,2.3980) -- (2.6290,2.3962) -- (2.6659,2.3938) -- (2.7028,2.3907) -- (2.7397,2.3870) -- (2.7766,2.3827) -- (2.8135,2.3778) -- (2.8503,2.3723) -- (2.8872,2.3662) -- (2.9241,2.3595) -- (2.9610,2.3521) -- (2.9979,2.3442) -- (3.0348,2.3356) -- (3.0716,2.3264) -- (3.1085,2.3166) -- (3.1454,2.3062) -- (3.1823,2.2952) -- (3.2192,2.2836) -- (3.2561,2.2713) -- (3.2929,2.2585) -- (3.3298,2.2450) -- (3.3667,2.2309) -- (3.4036,2.2162) -- (3.4405,2.2009) -- (3.4774,2.1850) -- (3.5143,2.1685) -- (3.5511,2.1513) -- (3.5880,2.1336) -- (3.6249,2.1152) -- (3.6618,2.0962) -- (3.6987,2.0766) -- (3.7356,2.0564) -- (3.7724,2.0356) -- (3.8093,2.0142) -- (3.8462,1.9922) -- (3.8831,1.9695) -- (3.9200,1.9462) -- (3.9569,1.9224) -- (3.9937,1.8979) -- (4.0306,1.8728) -- (4.0675,1.8471) -- (4.1044,1.8207) -- (4.1413,1.7938) -- (4.1782,1.7663) -- (4.2150,1.7381) -- (4.2519,1.7093) -- (4.2888,1.6799) -- (4.3257,1.6500) -- (4.3257,1.6500) -- (4.3257,0.0000) -- (4.3257,0.0000) -- (4.2888,0.0000) -- (4.2519,0.0000) -- (4.2150,0.0000) -- (4.1782,0.0000) -- (4.1413,0.0000) -- (4.1044,0.0000) -- (4.0675,0.0000) -- (4.0306,0.0000) -- (3.9937,0.0000) -- (3.9569,0.0000) -- (3.9200,0.0000) -- (3.8831,0.0000) -- (3.8462,0.0000) -- (3.8093,0.0000) -- (3.7724,0.0000) -- (3.7356,0.0000) -- (3.6987,0.0000) -- (3.6618,0.0000) -- (3.6249,0.0000) -- (3.5880,0.0000) -- (3.5511,0.0000) -- (3.5143,0.0000) -- (3.4774,0.0000) -- (3.4405,0.0000) -- (3.4036,0.0000) -- (3.3667,0.0000) -- (3.3298,0.0000) -- (3.2929,0.0000) -- (3.2561,0.0000) -- (3.2192,0.0000) -- (3.1823,0.0000) -- (3.1454,0.0000) -- (3.1085,0.0000) -- (3.0716,0.0000) -- (3.0348,0.0000) -- (2.9979,0.0000) -- (2.9610,0.0000) -- (2.9241,0.0000) -- (2.8872,0.0000) -- (2.8503,0.0000) -- (2.8135,0.0000) -- (2.7766,0.0000) -- (2.7397,0.0000) -- (2.7028,0.0000) -- (2.6659,0.0000) -- (2.6290,0.0000) -- (2.5922,0.0000) -- (2.5553,0.0000) -- (2.5184,0.0000) -- (2.4815,0.0000) -- (2.4446,0.0000) -- (2.4077,0.0000) -- (2.3709,0.0000) -- (2.3340,0.0000) -- (2.2971,0.0000) -- (2.2602,0.0000) -- (2.2233,0.0000) -- (2.1864,0.0000) -- (2.1496,0.0000) -- (2.1127,0.0000) -- (2.0758,0.0000) -- (2.0389,0.0000) -- (2.0020,0.0000) -- (1.9651,0.0000) -- (1.9283,0.0000) -- (1.8914,0.0000) -- (1.8545,0.0000) -- (1.8176,0.0000) -- (1.7807,0.0000) -- (1.7438,0.0000) -- (1.7070,0.0000) -- (1.6701,0.0000) -- (1.6332,0.0000) -- (1.5963,0.0000) -- (1.5594,0.0000) -- (1.5225,0.0000) -- (1.4856,0.0000) -- (1.4488,0.0000) -- (1.4119,0.0000) -- (1.3750,0.0000) -- (1.3381,0.0000) -- (1.3012,0.0000) -- (1.2643,0.0000) -- (1.2275,0.0000) -- (1.1906,0.0000) -- (1.1537,0.0000) -- (1.1168,0.0000) -- (1.0799,0.0000) -- (1.0430,0.0000) -- (1.0062,0.0000) -- (0.9693,0.0000) -- (0.9324,0.0000) -- (0.8955,0.0000) -- (0.8586,0.0000) -- (0.8217,0.0000) -- (0.7849,0.0000) -- (0.7480,0.0000) -- (0.7111,0.0000) -- (0.6742,0.0000) -- (0.6742,0.0000) -- (0.6742,1.6500) --  cycle;
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+\draw [] (4.3257,1.6500) -- (4.3257,0.0000);
 
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+\draw []  (0.6742,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.6742,-0.3785) node {$a$};
+\draw []  (4.3257,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.3257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,22 +95,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.1000,0.0000);
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+\draw [] (0.6742,0.0000) -- (0.6742,1.6500);
+\draw [] (4.3257,1.6500) -- (4.3257,0.0000);
+\draw []  (0.6742,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.6742,-0.3785) node {$a$};
+\draw []  (4.3257,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.3257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,22 +149,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8999);
 %DEFAULT
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-\draw []  (3.8257,0) node [rotate=0] {$\bullet$};
-\draw (3.8257,-0.42674) node {$b$};
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+\draw []  (0.6742,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.6742,-0.3785) node {$a$};
+\draw []  (4.3257,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.3257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks b/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks
index a3b7cc8a7..e40581b9a 100644
--- a/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks
+++ b/auto/pictures_tex/Fig_SurfaceHorizVerti.pstricks
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4998);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -51,26 +51,26 @@
 % setting the default values
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+\draw [color=blue] (1.0000,1.6347)--(1.0505,1.5624)--(1.1010,1.4970)--(1.1515,1.4392)--(1.2020,1.3895)--(1.2525,1.3484)--(1.3030,1.3161)--(1.3535,1.2928)--(1.4040,1.2785)--(1.4545,1.2731)--(1.5050,1.2763)--(1.5555,1.2878)--(1.6060,1.3071)--(1.6565,1.3337)--(1.7070,1.3669)--(1.7575,1.4059)--(1.8080,1.4499)--(1.8585,1.4980)--(1.9090,1.5493)--(1.9595,1.6029)--(2.0101,1.6578)--(2.0606,1.7132)--(2.1111,1.7680)--(2.1616,1.8214)--(2.2121,1.8726)--(2.2626,1.9208)--(2.3131,1.9653)--(2.3636,2.0056)--(2.4141,2.0411)--(2.4646,2.0714)--(2.5151,2.0963)--(2.5656,2.1155)--(2.6161,2.1290)--(2.6666,2.1368)--(2.7171,2.1391)--(2.7676,2.1362)--(2.8181,2.1284)--(2.8686,2.1162)--(2.9191,2.1001)--(2.9696,2.0808)--(3.0202,2.0588)--(3.0707,2.0350)--(3.1212,2.0101)--(3.1717,1.9849)--(3.2222,1.9602)--(3.2727,1.9367)--(3.3232,1.9152)--(3.3737,1.8964)--(3.4242,1.8810)--(3.4747,1.8696)--(3.5252,1.8627)--(3.5757,1.8608)--(3.6262,1.8642)--(3.6767,1.8731)--(3.7272,1.8877)--(3.7777,1.9080)--(3.8282,1.9339)--(3.8787,1.9653)--(3.9292,2.0017)--(3.9797,2.0428)--(4.0303,2.0881)--(4.0808,2.1370)--(4.1313,2.1887)--(4.1818,2.2424)--(4.2323,2.2974)--(4.2828,2.3527)--(4.3333,2.4075)--(4.3838,2.4607)--(4.4343,2.5115)--(4.4848,2.5589)--(4.5353,2.6020)--(4.5858,2.6399)--(4.6363,2.6718)--(4.6868,2.6971)--(4.7373,2.7149)--(4.7878,2.7249)--(4.8383,2.7265)--(4.8888,2.7194)--(4.9393,2.7033)--(4.9898,2.6783)--(5.0404,2.6442)--(5.0909,2.6014)--(5.1414,2.5502)--(5.1919,2.4909)--(5.2424,2.4241)--(5.2929,2.3505)--(5.3434,2.2708)--(5.3939,2.1861)--(5.4444,2.0972)--(5.4949,2.0052)--(5.5454,1.9112)--(5.5959,1.8163)--(5.6464,1.7217)--(5.6969,1.6286)--(5.7474,1.5381)--(5.7979,1.4515)--(5.8484,1.3698)--(5.8989,1.2941)--(5.9494,1.2254)--(6.0000,1.1645);
 
-\draw [color=cyan] (1.00,5.00) -- (6.00,5.00);
-\draw [color=cyan] (6.00,5.00) -- (6.00,1.16);
-\draw [color=cyan] (6.00,1.16) -- (1.00,1.16);
-\draw [color=cyan] (1.00,1.16) -- (1.00,5.00);
+\draw [color=cyan] (1.0000,4.9998) -- (6.0000,4.9998);
+\draw [color=cyan] (6.0000,4.9998) -- (6.0000,1.1645);
+\draw [color=cyan] (6.0000,1.1645) -- (1.0000,1.1645);
+\draw [color=cyan] (1.0000,1.1645) -- (1.0000,4.9998);
 
 %OTHER STUFF
 %END PSPICTURE
@@ -109,8 +109,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -119,24 +119,24 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-0.27898,1.0000) node {$c$};
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+\draw [color=blue,style=solid] (1.6347,1.0000)--(1.5624,1.0505)--(1.4970,1.1010)--(1.4392,1.1515)--(1.3895,1.2020)--(1.3484,1.2525)--(1.3161,1.3030)--(1.2928,1.3535)--(1.2785,1.4040)--(1.2731,1.4545)--(1.2763,1.5050)--(1.2878,1.5555)--(1.3071,1.6060)--(1.3337,1.6565)--(1.3669,1.7070)--(1.4059,1.7575)--(1.4499,1.8080)--(1.4980,1.8585)--(1.5493,1.9090)--(1.6029,1.9595)--(1.6578,2.0101)--(1.7132,2.0606)--(1.7680,2.1111)--(1.8214,2.1616)--(1.8726,2.2121)--(1.9208,2.2626)--(1.9653,2.3131)--(2.0056,2.3636)--(2.0411,2.4141)--(2.0714,2.4646)--(2.0963,2.5151)--(2.1155,2.5656)--(2.1290,2.6161)--(2.1368,2.6666)--(2.1391,2.7171)--(2.1362,2.7676)--(2.1284,2.8181)--(2.1162,2.8686)--(2.1001,2.9191)--(2.0808,2.9696)--(2.0588,3.0202)--(2.0350,3.0707)--(2.0101,3.1212)--(1.9849,3.1717)--(1.9602,3.2222)--(1.9367,3.2727)--(1.9152,3.3232)--(1.8964,3.3737)--(1.8810,3.4242)--(1.8696,3.4747)--(1.8627,3.5252)--(1.8608,3.5757)--(1.8642,3.6262)--(1.8731,3.6767)--(1.8877,3.7272)--(1.9080,3.7777)--(1.9339,3.8282)--(1.9653,3.8787)--(2.0017,3.9292)--(2.0428,3.9797)--(2.0881,4.0303)--(2.1370,4.0808)--(2.1887,4.1313)--(2.2424,4.1818)--(2.2974,4.2323)--(2.3527,4.2828)--(2.4075,4.3333)--(2.4607,4.3838)--(2.5115,4.4343)--(2.5589,4.4848)--(2.6020,4.5353)--(2.6399,4.5858)--(2.6718,4.6363)--(2.6971,4.6868)--(2.7149,4.7373)--(2.7249,4.7878)--(2.7265,4.8383)--(2.7194,4.8888)--(2.7033,4.9393)--(2.6783,4.9898)--(2.6442,5.0404)--(2.6014,5.0909)--(2.5502,5.1414)--(2.4909,5.1919)--(2.4241,5.2424)--(2.3505,5.2929)--(2.2708,5.3434)--(2.1861,5.3939)--(2.0972,5.4444)--(2.0052,5.4949)--(1.9112,5.5454)--(1.8163,5.5959)--(1.7217,5.6464)--(1.6286,5.6969)--(1.5381,5.7474)--(1.4515,5.7979)--(1.3698,5.8484)--(1.2941,5.8989)--(1.2254,5.9494)--(1.1645,6.0000);
 
-\draw [color=cyan] (1.16,6.00) -- (5.00,6.00);
-\draw [color=cyan] (5.00,6.00) -- (5.00,1.00);
-\draw [color=cyan] (5.00,1.00) -- (1.16,1.00);
-\draw [color=cyan] (1.16,1.00) -- (1.16,6.00);
+\draw [color=cyan] (1.1645,6.0000) -- (4.9998,6.0000);
+\draw [color=cyan] (4.9998,6.0000) -- (4.9998,1.0000);
+\draw [color=cyan] (4.9998,1.0000) -- (1.1645,1.0000);
+\draw [color=cyan] (1.1645,1.0000) -- (1.1645,6.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks b/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks
index fd9c3b21d..0b50e3485 100644
--- a/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks
+++ b/auto/pictures_tex/Fig_SurfacePrimiteGeog.pstricks
@@ -79,11 +79,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.4000);
 %DEFAULT
 
-\draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
+\draw [color=red] (1.5000,1.5000)--(1.5606,1.6165)--(1.6212,1.7242)--(1.6818,1.8243)--(1.7424,1.9173)--(1.8030,2.0042)--(1.8636,2.0853)--(1.9242,2.1614)--(1.9848,2.2328)--(2.0454,2.3000)--(2.1060,2.3633)--(2.1666,2.4230)--(2.2272,2.4795)--(2.2878,2.5331)--(2.3484,2.5838)--(2.4090,2.6320)--(2.4696,2.6779)--(2.5303,2.7215)--(2.5909,2.7631)--(2.6515,2.8028)--(2.7121,2.8407)--(2.7727,2.8770)--(2.8333,2.9117)--(2.8939,2.9450)--(2.9545,2.9769)--(3.0151,3.0075)--(3.0757,3.0369)--(3.1363,3.0652)--(3.1969,3.0924)--(3.2575,3.1186)--(3.3181,3.1438)--(3.3787,3.1681)--(3.4393,3.1916)--(3.5000,3.2142)--(3.5606,3.2361)--(3.6212,3.2573)--(3.6818,3.2777)--(3.7424,3.2975)--(3.8030,3.3167)--(3.8636,3.3352)--(3.9242,3.3532)--(3.9848,3.3707)--(4.0454,3.3876)--(4.1060,3.4040)--(4.1666,3.4200)--(4.2272,3.4354)--(4.2878,3.4505)--(4.3484,3.4651)--(4.4090,3.4793)--(4.4696,3.4932)--(4.5303,3.5066)--(4.5909,3.5198)--(4.6515,3.5325)--(4.7121,3.5450)--(4.7727,3.5571)--(4.8333,3.5689)--(4.8939,3.5804)--(4.9545,3.5917)--(5.0151,3.6027)--(5.0757,3.6134)--(5.1363,3.6238)--(5.1969,3.6341)--(5.2575,3.6440)--(5.3181,3.6538)--(5.3787,3.6633)--(5.4393,3.6727)--(5.5000,3.6818)--(5.5606,3.6907)--(5.6212,3.6994)--(5.6818,3.7080)--(5.7424,3.7163)--(5.8030,3.7245)--(5.8636,3.7325)--(5.9242,3.7404)--(5.9848,3.7481)--(6.0454,3.7556)--(6.1060,3.7630)--(6.1666,3.7702)--(6.2272,3.7773)--(6.2878,3.7843)--(6.3484,3.7911)--(6.4090,3.7978)--(6.4696,3.8044)--(6.5303,3.8109)--(6.5909,3.8172)--(6.6515,3.8234)--(6.7121,3.8295)--(6.7727,3.8355)--(6.8333,3.8414)--(6.8939,3.8472)--(6.9545,3.8529)--(7.0151,3.8585)--(7.0757,3.8640)--(7.1363,3.8694)--(7.1969,3.8747)--(7.2575,3.8799)--(7.3181,3.8850)--(7.3787,3.8901)--(7.4393,3.8951)--(7.5000,3.9000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -91,15 +91,15 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw (6.0000,-0.3785) node {$x$};
 \draw (8.3552,3.9000) node {$f(x)$};
-\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (9.6701,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -107,11 +107,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [style=dashed] (3.0000,0.0000) -- (3.0000,3.0000);
+\draw [style=dashed] (6.0000,3.7500) -- (6.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TGdUoZR.pstricks b/auto/pictures_tex/Fig_TGdUoZR.pstricks
index feb371f98..5a30b9a8f 100644
--- a/auto/pictures_tex/Fig_TGdUoZR.pstricks
+++ b/auto/pictures_tex/Fig_TGdUoZR.pstricks
@@ -66,46 +66,46 @@
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+\draw [] (-0.2500,0.2500) -- (-0.2500,-0.2500);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks b/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks
index 067a5b0f8..304bc7c92 100644
--- a/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks
+++ b/auto/pictures_tex/Fig_TIMYoochXZZNGP.pstricks
@@ -70,11 +70,11 @@
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-\draw [] (-2.10,0.700) -- (2.10,0.700);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.38245) node {\( \pi(e_1)\)};
-\draw []  (0.70000,0.70000) node [rotate=0] {$\bullet$};
-\draw (0.70000,1.0825) node {\( \pi(e_2)\)};
+\draw [] (-2.1000,0.7000) -- (2.1000,0.7000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3824) node {\( \pi(e_1)\)};
+\draw []  (0.7000,0.7000) node [rotate=0] {$\bullet$};
+\draw (0.7000,1.0824) node {\( \pi(e_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
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diff --git a/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks b/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
index 4c1dc4011..22ee5927c 100644
--- a/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
+++ b/auto/pictures_tex/Fig_TKXZooLwXzjS.pstricks
@@ -65,58 +65,58 @@
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-\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
-\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
-\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
-\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
-\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
-\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
-\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
-\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
-\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
-\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (3.0666,-1.0000) -- (3.3927,-1.0000);
+\draw [,->,>=latex] (3.0666,-0.6666) -- (3.3927,-0.6666);
+\draw [,->,>=latex] (3.0666,-0.3333) -- (3.3927,-0.3333);
+\draw [,->,>=latex] (3.0666,0.0000) -- (3.3927,0.0000);
+\draw [,->,>=latex] (3.0666,0.3333) -- (3.3927,0.3333);
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+\draw [,->,>=latex] (3.8000,0.3333) -- (4.0631,0.3333);
+\draw [,->,>=latex] (3.8000,0.6666) -- (4.0631,0.6666);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0631,1.0000);
 \draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
-\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
-\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
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-\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,-0.6666) -- (4.7539,-0.6666);
+\draw [,->,>=latex] (4.5333,-0.3333) -- (4.7539,-0.3333);
+\draw [,->,>=latex] (4.5333,0.0000) -- (4.7539,0.0000);
+\draw [,->,>=latex] (4.5333,0.3333) -- (4.7539,0.3333);
+\draw [,->,>=latex] (4.5333,0.6666) -- (4.7539,0.6666);
 \draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
-\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
-\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
-\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
-\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
-\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
-\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
-\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
-\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
-\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
-\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
-\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
-\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
-\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
-\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
+\draw [,->,>=latex] (5.2666,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2666,-0.6666) -- (5.4565,-0.6666);
+\draw [,->,>=latex] (5.2666,-0.3333) -- (5.4565,-0.3333);
+\draw [,->,>=latex] (5.2666,0.0000) -- (5.4565,0.0000);
+\draw [,->,>=latex] (5.2666,0.3333) -- (5.4565,0.3333);
+\draw [,->,>=latex] (5.2666,0.6666) -- (5.4565,0.6666);
+\draw [,->,>=latex] (5.2666,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1666,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.6666) -- (6.1666,-0.6666);
+\draw [,->,>=latex] (6.0000,-0.3333) -- (6.1666,-0.3333);
+\draw [,->,>=latex] (6.0000,0.0000) -- (6.1666,0.0000);
+\draw [,->,>=latex] (6.0000,0.3333) -- (6.1666,0.3333);
+\draw [,->,>=latex] (6.0000,0.6666) -- (6.1666,0.6666);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1666,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks b/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
index feed9e843..b2fe57901 100644
--- a/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
+++ b/auto/pictures_tex/Fig_TVXooWoKkqV.pstricks
@@ -88,29 +88,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.61131) -- (0,2.5278);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.6113) -- (0.0000,2.5277);
 %DEFAULT
 
-\draw [color=blue] (-2.000,1.127)--(-1.929,1.000)--(-1.859,0.8813)--(-1.788,0.7700)--(-1.717,0.6663)--(-1.646,0.5699)--(-1.576,0.4807)--(-1.505,0.3985)--(-1.434,0.3231)--(-1.364,0.2544)--(-1.293,0.1922)--(-1.222,0.1363)--(-1.152,0.08657)--(-1.081,0.04284)--(-1.010,0.004948)--(-0.9394,-0.02729)--(-0.8687,-0.05404)--(-0.7980,-0.07546)--(-0.7273,-0.09173)--(-0.6566,-0.1030)--(-0.5859,-0.1095)--(-0.5152,-0.1113)--(-0.4444,-0.1087)--(-0.3737,-0.1017)--(-0.3030,-0.09057)--(-0.2323,-0.07548)--(-0.1616,-0.05658)--(-0.09091,-0.03405)--(-0.02020,-0.008044)--(0.05051,0.02126)--(0.1212,0.05370)--(0.1919,0.08911)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5455,0.3047)--(0.6162,0.3544)--(0.6869,0.4058)--(0.7576,0.4589)--(0.8283,0.5134)--(0.8990,0.5692)--(0.9697,0.6261)--(1.040,0.6840)--(1.111,0.7426)--(1.182,0.8018)--(1.253,0.8615)--(1.323,0.9215)--(1.394,0.9815)--(1.465,1.042)--(1.535,1.101)--(1.606,1.161)--(1.677,1.219)--(1.747,1.277)--(1.818,1.335)--(1.889,1.391)--(1.960,1.445)--(2.030,1.499)--(2.101,1.551)--(2.172,1.601)--(2.242,1.649)--(2.313,1.695)--(2.384,1.739)--(2.455,1.780)--(2.525,1.819)--(2.596,1.855)--(2.667,1.888)--(2.737,1.918)--(2.808,1.945)--(2.879,1.968)--(2.949,1.988)--(3.020,2.004)--(3.091,2.016)--(3.162,2.024)--(3.232,2.028)--(3.303,2.027)--(3.374,2.022)--(3.444,2.011)--(3.515,1.996)--(3.586,1.976)--(3.657,1.951)--(3.727,1.920)--(3.798,1.883)--(3.869,1.841)--(3.939,1.792)--(4.010,1.738)--(4.081,1.677)--(4.151,1.610)--(4.222,1.536)--(4.293,1.455)--(4.364,1.368)--(4.434,1.273)--(4.505,1.171)--(4.576,1.062)--(4.646,0.9444)--(4.717,0.8194)--(4.788,0.6865)--(4.859,0.5454)--(4.929,0.3960)--(5.000,0.2381);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (-2.0000,1.1269)--(-1.9292,1.0002)--(-1.8585,0.8812)--(-1.7878,0.7700)--(-1.7171,0.6663)--(-1.6464,0.5699)--(-1.5757,0.4806)--(-1.5050,0.3984)--(-1.4343,0.3230)--(-1.3636,0.2543)--(-1.2929,0.1921)--(-1.2222,0.1362)--(-1.1515,0.0865)--(-1.0808,0.0428)--(-1.0101,0.0049)--(-0.9393,-0.0272)--(-0.8686,-0.0540)--(-0.7979,-0.0754)--(-0.7272,-0.0917)--(-0.6565,-0.1030)--(-0.5858,-0.1094)--(-0.5151,-0.1113)--(-0.4444,-0.1086)--(-0.3737,-0.1016)--(-0.3030,-0.0905)--(-0.2323,-0.0754)--(-0.1616,-0.0565)--(-0.0909,-0.0340)--(-0.0202,-0.0080)--(0.0505,0.0212)--(0.1212,0.0537)--(0.1919,0.0891)--(0.2626,0.1273)--(0.3333,0.1681)--(0.4040,0.2114)--(0.4747,0.2570)--(0.5454,0.3047)--(0.6161,0.3543)--(0.6868,0.4058)--(0.7575,0.4588)--(0.8282,0.5133)--(0.8989,0.5691)--(0.9696,0.6261)--(1.0404,0.6839)--(1.1111,0.7425)--(1.1818,0.8018)--(1.2525,0.8615)--(1.3232,0.9214)--(1.3939,0.9815)--(1.4646,1.0415)--(1.5353,1.1012)--(1.6060,1.1606)--(1.6767,1.2194)--(1.7474,1.2774)--(1.8181,1.3345)--(1.8888,1.3906)--(1.9595,1.4454)--(2.0303,1.4988)--(2.1010,1.5507)--(2.1717,1.6008)--(2.2424,1.6489)--(2.3131,1.6950)--(2.3838,1.7388)--(2.4545,1.7802)--(2.5252,1.8191)--(2.5959,1.8551)--(2.6666,1.8883)--(2.7373,1.9183)--(2.8080,1.9451)--(2.8787,1.9684)--(2.9494,1.9881)--(3.0202,2.0041)--(3.0909,2.0162)--(3.1616,2.0241)--(3.2323,2.0277)--(3.3030,2.0270)--(3.3737,2.0216)--(3.4444,2.0114)--(3.5151,1.9963)--(3.5858,1.9761)--(3.6565,1.9505)--(3.7272,1.9196)--(3.7979,1.8830)--(3.8686,1.8406)--(3.9393,1.7923)--(4.0101,1.7378)--(4.0808,1.6771)--(4.1515,1.6099)--(4.2222,1.5360)--(4.2929,1.4554)--(4.3636,1.3678)--(4.4343,1.2730)--(4.5050,1.1710)--(4.5757,1.0615)--(4.6464,0.9443)--(4.7171,0.8194)--(4.7878,0.6864)--(4.8585,0.5453)--(4.9292,0.3959)--(5.0000,0.2380);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks b/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
index 66e9b1259..f15627b9e 100644
--- a/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
+++ b/auto/pictures_tex/Fig_TWHooJjXEtS.pstricks
@@ -108,35 +108,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-8.3539,0.0000) -- (8.3539,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [color=blue] (-7.854,-1.000)--(-7.695,-0.9874)--(-7.537,-0.9501)--(-7.378,-0.8888)--(-7.219,-0.8053)--(-7.061,-0.7015)--(-6.902,-0.5801)--(-6.743,-0.4441)--(-6.585,-0.2969)--(-6.426,-0.1423)--(-6.267,0.01587)--(-6.109,0.1736)--(-5.950,0.3271)--(-5.791,0.4723)--(-5.633,0.6056)--(-5.474,0.7237)--(-5.315,0.8237)--(-5.157,0.9029)--(-4.998,0.9595)--(-4.839,0.9920)--(-4.681,0.9995)--(-4.522,0.9819)--(-4.363,0.9397)--(-4.205,0.8738)--(-4.046,0.7861)--(-3.887,0.6785)--(-3.729,0.5539)--(-3.570,0.4154)--(-3.411,0.2665)--(-3.253,0.1108)--(-3.094,-0.04758)--(-2.935,-0.2048)--(-2.777,-0.3569)--(-2.618,-0.5000)--(-2.459,-0.6306)--(-2.301,-0.7453)--(-2.142,-0.8413)--(-1.983,-0.9161)--(-1.825,-0.9679)--(-1.666,-0.9955)--(-1.507,-0.9980)--(-1.349,-0.9754)--(-1.190,-0.9284)--(-1.031,-0.8580)--(-0.8727,-0.7660)--(-0.7140,-0.6549)--(-0.5553,-0.5272)--(-0.3967,-0.3863)--(-0.2380,-0.2358)--(-0.07933,-0.07925)--(0.07933,0.07925)--(0.2380,0.2358)--(0.3967,0.3863)--(0.5553,0.5272)--(0.7140,0.6549)--(0.8727,0.7660)--(1.031,0.8580)--(1.190,0.9284)--(1.349,0.9754)--(1.507,0.9980)--(1.666,0.9955)--(1.825,0.9679)--(1.983,0.9161)--(2.142,0.8413)--(2.301,0.7453)--(2.459,0.6306)--(2.618,0.5000)--(2.777,0.3569)--(2.935,0.2048)--(3.094,0.04758)--(3.253,-0.1108)--(3.411,-0.2665)--(3.570,-0.4154)--(3.729,-0.5539)--(3.887,-0.6785)--(4.046,-0.7861)--(4.205,-0.8738)--(4.363,-0.9397)--(4.522,-0.9819)--(4.681,-0.9995)--(4.839,-0.9920)--(4.998,-0.9595)--(5.157,-0.9029)--(5.315,-0.8237)--(5.474,-0.7237)--(5.633,-0.6056)--(5.791,-0.4723)--(5.950,-0.3271)--(6.109,-0.1736)--(6.267,-0.01587)--(6.426,0.1423)--(6.585,0.2969)--(6.743,0.4441)--(6.902,0.5801)--(7.061,0.7015)--(7.219,0.8053)--(7.378,0.8888)--(7.537,0.9501)--(7.695,0.9874)--(7.854,1.000);
-\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
-\draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
-\draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-7.8539,-1.0000)--(-7.6953,-0.9874)--(-7.5366,-0.9500)--(-7.3779,-0.8888)--(-7.2193,-0.8052)--(-7.0606,-0.7014)--(-6.9019,-0.5800)--(-6.7433,-0.4440)--(-6.5846,-0.2969)--(-6.4259,-0.1423)--(-6.2673,0.0158)--(-6.1086,0.1736)--(-5.9499,0.3270)--(-5.7913,0.4722)--(-5.6326,0.6056)--(-5.4739,0.7237)--(-5.3153,0.8236)--(-5.1566,0.9029)--(-4.9979,0.9594)--(-4.8393,0.9919)--(-4.6806,0.9994)--(-4.5219,0.9819)--(-4.3633,0.9396)--(-4.2046,0.8738)--(-4.0459,0.7860)--(-3.8873,0.6785)--(-3.7286,0.5539)--(-3.5699,0.4154)--(-3.4113,0.2664)--(-3.2526,0.1108)--(-3.0939,-0.0475)--(-2.9353,-0.2048)--(-2.7766,-0.3568)--(-2.6179,-0.5000)--(-2.4593,-0.6305)--(-2.3006,-0.7452)--(-2.1419,-0.8412)--(-1.9833,-0.9161)--(-1.8246,-0.9679)--(-1.6659,-0.9954)--(-1.5073,-0.9979)--(-1.3486,-0.9754)--(-1.1899,-0.9283)--(-1.0313,-0.8579)--(-0.8726,-0.7660)--(-0.7139,-0.6548)--(-0.5553,-0.5272)--(-0.3966,-0.3863)--(-0.2379,-0.2357)--(-0.0793,-0.0792)--(0.0793,0.0792)--(0.2379,0.2357)--(0.3966,0.3863)--(0.5553,0.5272)--(0.7139,0.6548)--(0.8726,0.7660)--(1.0313,0.8579)--(1.1899,0.9283)--(1.3486,0.9754)--(1.5073,0.9979)--(1.6659,0.9954)--(1.8246,0.9679)--(1.9833,0.9161)--(2.1419,0.8412)--(2.3006,0.7452)--(2.4593,0.6305)--(2.6179,0.5000)--(2.7766,0.3568)--(2.9353,0.2048)--(3.0939,0.0475)--(3.2526,-0.1108)--(3.4113,-0.2664)--(3.5699,-0.4154)--(3.7286,-0.5539)--(3.8873,-0.6785)--(4.0459,-0.7860)--(4.2046,-0.8738)--(4.3633,-0.9396)--(4.5219,-0.9819)--(4.6806,-0.9994)--(4.8393,-0.9919)--(4.9979,-0.9594)--(5.1566,-0.9029)--(5.3153,-0.8236)--(5.4739,-0.7237)--(5.6326,-0.6056)--(5.7913,-0.4722)--(5.9499,-0.3270)--(6.1086,-0.1736)--(6.2673,-0.0158)--(6.4259,0.1423)--(6.5846,0.2969)--(6.7433,0.4440)--(6.9019,0.5800)--(7.0606,0.7014)--(7.2193,0.8052)--(7.3779,0.8888)--(7.5366,0.9500)--(7.6953,0.9874)--(7.8539,1.0000);
+\draw (-7.8539,-0.4207) node {$ -\frac{5}{2} \, \pi $};
+\draw [] (-7.8539,-0.1000) -- (-7.8539,0.1000);
+\draw (-6.2831,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-6.2831,-0.1000) -- (-6.2831,0.1000);
+\draw (-4.7123,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-4.7123,-0.1000) -- (-4.7123,0.1000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (7.8539,-0.4207) node {$ \frac{5}{2} \, \pi $};
+\draw [] (7.8539,-0.1000) -- (7.8539,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_TZCISko.pstricks b/auto/pictures_tex/Fig_TZCISko.pstricks
index 068402609..fcf282bc4 100644
--- a/auto/pictures_tex/Fig_TZCISko.pstricks
+++ b/auto/pictures_tex/Fig_TZCISko.pstricks
@@ -71,20 +71,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4999) -- (0.0000,2.4999);
 %DEFAULT
 
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+\draw [color=blue] 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+\draw [color=blue] 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+\draw [color=blue] 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+\draw (10.000,-0.3149) node {$ 1 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_TangentSegment.pstricks b/auto/pictures_tex/Fig_TangentSegment.pstricks
index 7ff8d9539..d4da27183 100644
--- a/auto/pictures_tex/Fig_TangentSegment.pstricks
+++ b/auto/pictures_tex/Fig_TangentSegment.pstricks
@@ -103,45 +103,45 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.7038,0) -- (7.7832,0);
-\draw [,->,>=latex] (0,-3.0689) -- (0,2.5000);
+\draw [,->,>=latex] (-3.7037,0.0000) -- (7.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0689) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (-3.142,-2.000)--(-3.036,-1.997)--(-2.931,-1.989)--(-2.826,-1.975)--(-2.720,-1.956)--(-2.615,-1.931)--(-2.510,-1.901)--(-2.404,-1.866)--(-2.299,-1.825)--(-2.194,-1.780)--(-2.089,-1.729)--(-1.983,-1.674)--(-1.878,-1.614)--(-1.773,-1.550)--(-1.667,-1.481)--(-1.562,-1.408)--(-1.457,-1.331)--(-1.351,-1.251)--(-1.246,-1.167)--(-1.141,-1.080)--(-1.036,-0.9899)--(-0.9303,-0.8971)--(-0.8250,-0.8018)--(-0.7197,-0.7042)--(-0.6144,-0.6048)--(-0.5091,-0.5036)--(-0.4038,-0.4010)--(-0.2985,-0.2974)--(-0.1932,-0.1929)--(-0.08787,-0.08784)--(0.01743,0.01743)--(0.1227,0.1227)--(0.2280,0.2275)--(0.3333,0.3318)--(0.4386,0.4351)--(0.5439,0.5373)--(0.6492,0.6379)--(0.7545,0.7368)--(0.8598,0.8336)--(0.9651,0.9281)--(1.070,1.020)--(1.176,1.109)--(1.281,1.195)--(1.386,1.278)--(1.492,1.357)--(1.597,1.433)--(1.702,1.504)--(1.808,1.571)--(1.913,1.634)--(2.018,1.693)--(2.123,1.746)--(2.229,1.795)--(2.334,1.839)--(2.439,1.878)--(2.545,1.912)--(2.650,1.940)--(2.755,1.963)--(2.861,1.980)--(2.966,1.992)--(3.071,1.999)--(3.176,2.000)--(3.282,1.995)--(3.387,1.985)--(3.492,1.969)--(3.598,1.948)--(3.703,1.922)--(3.808,1.890)--(3.914,1.853)--(4.019,1.811)--(4.124,1.763)--(4.229,1.711)--(4.335,1.655)--(4.440,1.593)--(4.545,1.527)--(4.651,1.457)--(4.756,1.383)--(4.861,1.305)--(4.967,1.224)--(5.072,1.139)--(5.177,1.050)--(5.282,0.9595)--(5.388,0.8658)--(5.493,0.7697)--(5.598,0.6715)--(5.704,0.5714)--(5.809,0.4698)--(5.914,0.3668)--(6.020,0.2628)--(6.125,0.1581)--(6.230,0.05300)--(6.335,-0.05229)--(6.441,-0.1574)--(6.546,-0.2621)--(6.651,-0.3661)--(6.757,-0.4691)--(6.862,-0.5708)--(6.967,-0.6708)--(7.073,-0.7691)--(7.178,-0.8652)--(7.283,-0.9589);
-\draw [] (-3.20,-2.57) -- (0.0622,-0.259);
-\draw []  (-1.5708,-1.4142) node [rotate=0] {$\bullet$};
-\draw [] (1.14,2.00) -- (5.14,2.00);
-\draw []  (3.1416,2.0000) node [rotate=0] {$\bullet$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (-3.1415,-2.0000)--(-3.0362,-1.9972)--(-2.9309,-1.9889)--(-2.8256,-1.9751)--(-2.7203,-1.9558)--(-2.6150,-1.9310)--(-2.5097,-1.9010)--(-2.4044,-1.8656)--(-2.2991,-1.8251)--(-2.1938,-1.7796)--(-2.0885,-1.7291)--(-1.9832,-1.6738)--(-1.8779,-1.6139)--(-1.7726,-1.5495)--(-1.6673,-1.4808)--(-1.5620,-1.4080)--(-1.4567,-1.3313)--(-1.3514,-1.2509)--(-1.2461,-1.1670)--(-1.1408,-1.0800)--(-1.0355,-0.9899)--(-0.9302,-0.8970)--(-0.8249,-0.8017)--(-0.7196,-0.7042)--(-0.6143,-0.6047)--(-0.5090,-0.5035)--(-0.4037,-0.4010)--(-0.2984,-0.2973)--(-0.1931,-0.1928)--(-0.0878,-0.0878)--(0.0174,0.0174)--(0.1227,0.1226)--(0.2280,0.2275)--(0.3333,0.3317)--(0.4386,0.4351)--(0.5439,0.5372)--(0.6492,0.6378)--(0.7545,0.7367)--(0.8598,0.8335)--(0.9651,0.9281)--(1.0704,1.0200)--(1.1757,1.1091)--(1.2810,1.1952)--(1.3863,1.2779)--(1.4916,1.3571)--(1.5969,1.4325)--(1.7022,1.5040)--(1.8075,1.5713)--(1.9128,1.6342)--(2.0181,1.6926)--(2.1234,1.7463)--(2.2287,1.7952)--(2.3340,1.8391)--(2.4393,1.8779)--(2.5446,1.9115)--(2.6499,1.9398)--(2.7552,1.9628)--(2.8605,1.9802)--(2.9658,1.9922)--(3.0711,1.9987)--(3.1764,1.9996)--(3.2817,1.9950)--(3.3870,1.9849)--(3.4923,1.9693)--(3.5976,1.9482)--(3.7029,1.9217)--(3.8082,1.8899)--(3.9135,1.8528)--(4.0188,1.8106)--(4.1241,1.7634)--(4.2294,1.7113)--(4.3347,1.6545)--(4.4400,1.5930)--(4.5453,1.5272)--(4.6506,1.4571)--(4.7559,1.3830)--(4.8612,1.3051)--(4.9665,1.2235)--(5.0718,1.1386)--(5.1771,1.0504)--(5.2824,0.9594)--(5.3877,0.8657)--(5.4930,0.7697)--(5.5983,0.6715)--(5.7036,0.5714)--(5.8089,0.4697)--(5.9142,0.3668)--(6.0195,0.2628)--(6.1248,0.1581)--(6.2301,0.0530)--(6.3354,-0.0522)--(6.4407,-0.1574)--(6.5460,-0.2621)--(6.6513,-0.3661)--(6.7566,-0.4690)--(6.8619,-0.5707)--(6.9672,-0.6708)--(7.0725,-0.7690)--(7.1778,-0.8651)--(7.2831,-0.9588);
+\draw [] (-3.2037,-2.5689) -- (0.0621,-0.2595);
+\draw []  (-1.5707,-1.4142) node [rotate=0] {$\bullet$};
+\draw [] (1.1415,2.0000) -- (5.1415,2.0000);
+\draw []  (3.1415,2.0000) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_TangenteDetail.pstricks b/auto/pictures_tex/Fig_TangenteDetail.pstricks
index bfa327892..cbca04407 100644
--- a/auto/pictures_tex/Fig_TangenteDetail.pstricks
+++ b/auto/pictures_tex/Fig_TangenteDetail.pstricks
@@ -111,35 +111,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.1051);
 %DEFAULT
-\draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
-\draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
-\draw [color=green,style=dashed] (1.70,3.24) -- (1.70,0);
-\draw [color=green,style=dashed] (4.00,4.25) -- (0,4.25);
-\draw [color=green,style=dashed] (1.70,3.24) -- (0,3.24);
-\draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
+\draw [color=cyan] (0.8950,2.8801) -- (4.8050,4.6051);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (4.0000,0.0000);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (1.7000,0.0000);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (0.0000,4.2500);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (0.0000,3.2352);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (4.0000,3.2352);
 
-\draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw [color=blue] (0.7000,0.7142)--(0.7434,0.9646)--(0.7868,1.1874)--(0.8303,1.3868)--(0.8737,1.5664)--(0.9171,1.7290)--(0.9606,1.8769)--(1.0040,2.0120)--(1.0474,2.1359)--(1.0909,2.2500)--(1.1343,2.3552)--(1.1777,2.4528)--(1.2212,2.5434)--(1.2646,2.6277)--(1.3080,2.7065)--(1.3515,2.7802)--(1.3949,2.8493)--(1.4383,2.9143)--(1.4818,2.9754)--(1.5252,3.0331)--(1.5686,3.0875)--(1.6121,3.1390)--(1.6555,3.1879)--(1.6989,3.2342)--(1.7424,3.2782)--(1.7858,3.3201)--(1.8292,3.3600)--(1.8727,3.3980)--(1.9161,3.4343)--(1.9595,3.4690)--(2.0030,3.5022)--(2.0464,3.5340)--(2.0898,3.5645)--(2.1333,3.5937)--(2.1767,3.6218)--(2.2202,3.6487)--(2.2636,3.6746)--(2.3070,3.6996)--(2.3505,3.7236)--(2.3939,3.7468)--(2.4373,3.7691)--(2.4808,3.7907)--(2.5242,3.8115)--(2.5676,3.8316)--(2.6111,3.8510)--(2.6545,3.8698)--(2.6979,3.8880)--(2.7414,3.9056)--(2.7848,3.9227)--(2.8282,3.9392)--(2.8717,3.9553)--(2.9151,3.9708)--(2.9585,3.9860)--(3.0020,4.0006)--(3.0454,4.0149)--(3.0888,4.0287)--(3.1323,4.0422)--(3.1757,4.0553)--(3.2191,4.0680)--(3.2626,4.0804)--(3.3060,4.0925)--(3.3494,4.1043)--(3.3929,4.1158)--(3.4363,4.1269)--(3.4797,4.1378)--(3.5232,4.1485)--(3.5666,4.1588)--(3.6101,4.1689)--(3.6535,4.1788)--(3.6969,4.1885)--(3.7404,4.1979)--(3.7838,4.2071)--(3.8272,4.2161)--(3.8707,4.2249)--(3.9141,4.2335)--(3.9575,4.2419)--(4.0010,4.2501)--(4.0444,4.2582)--(4.0878,4.2661)--(4.1313,4.2738)--(4.1747,4.2813)--(4.2181,4.2887)--(4.2616,4.2960)--(4.3050,4.3031)--(4.3484,4.3101)--(4.3919,4.3169)--(4.4353,4.3236)--(4.4787,4.3301)--(4.5222,4.3366)--(4.5656,4.3429)--(4.6090,4.3491)--(4.6525,4.3551)--(4.6959,4.3611)--(4.7393,4.3670)--(4.7828,4.3727)--(4.8262,4.3784)--(4.8696,4.3839)--(4.9131,4.3893)--(4.9565,4.3947)--(5.0000,4.4000);
+\draw []  (1.7000,3.2352) node [rotate=0] {$\bullet$};
 \draw (1.3414,3.5681) node {$P$};
-\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
-\draw (1.7000,-0.27858) node {$a$};
-\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
-\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (1.7000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.2785) node {$a$};
+\draw []  (0.0000,3.2352) node [rotate=0] {$\bullet$};
+\draw (-0.4473,3.2352) node {$f(a)$};
 \draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
-\draw (3.8004,4.7051) node {$Q$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.27858) node {$x$};
-\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
-\draw (-0.45521,4.2500) node {$f(x)$};
-\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
-\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
-\draw (2.8500,2.7143) node {$x-a$};
+\draw (3.8004,4.7050) node {$Q$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.2785) node {$x$};
+\draw []  (0.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.4552,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0352) -- (1.7000,3.0352);
+\draw [,->,>=latex] (2.8500,3.0352) -- (4.0000,3.0352);
+\draw (2.8500,2.7142) node {$x-a$};
 \draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
-\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
-\draw (5.3256,3.7426) node {$f(x)-f(a)$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2352);
+\draw (5.3255,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteDetailOM.pstricks b/auto/pictures_tex/Fig_TangenteDetailOM.pstricks
index 4771ca36f..53f08374e 100644
--- a/auto/pictures_tex/Fig_TangenteDetailOM.pstricks
+++ b/auto/pictures_tex/Fig_TangenteDetailOM.pstricks
@@ -111,35 +111,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.1051);
 %DEFAULT
-\draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
-\draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
-\draw [color=green,style=dashed] (1.70,3.24) -- (1.70,0);
-\draw [color=green,style=dashed] (4.00,4.25) -- (0,4.25);
-\draw [color=green,style=dashed] (1.70,3.24) -- (0,3.24);
-\draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
+\draw [color=cyan] (0.8950,2.8801) -- (4.8050,4.6051);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (4.0000,0.0000);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (1.7000,0.0000);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (0.0000,4.2500);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (0.0000,3.2352);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (4.0000,3.2352);
 
-\draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
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+\draw []  (1.7000,3.2352) node [rotate=0] {$\bullet$};
 \draw (1.3414,3.5681) node {$P$};
-\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
-\draw (1.7000,-0.27858) node {$a$};
-\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
-\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (1.7000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.2785) node {$a$};
+\draw []  (0.0000,3.2352) node [rotate=0] {$\bullet$};
+\draw (-0.4473,3.2352) node {$f(a)$};
 \draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
-\draw (3.8004,4.7051) node {$Q$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.27858) node {$x$};
-\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
-\draw (-0.45521,4.2500) node {$f(x)$};
-\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
-\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
-\draw (2.8500,2.7143) node {$x-a$};
+\draw (3.8004,4.7050) node {$Q$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.2785) node {$x$};
+\draw []  (0.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.4552,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0352) -- (1.7000,3.0352);
+\draw [,->,>=latex] (2.8500,3.0352) -- (4.0000,3.0352);
+\draw (2.8500,2.7142) node {$x-a$};
 \draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
-\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
-\draw (5.3256,3.7426) node {$f(x)-f(a)$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2352);
+\draw (5.3255,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteQuestion.pstricks b/auto/pictures_tex/Fig_TangenteQuestion.pstricks
index ad59a76cd..f84f922ac 100644
--- a/auto/pictures_tex/Fig_TangenteQuestion.pstricks
+++ b/auto/pictures_tex/Fig_TangenteQuestion.pstricks
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5800);
 %DEFAULT
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks b/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks
index 411ca3541..20797a306 100644
--- a/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks
+++ b/auto/pictures_tex/Fig_TangenteQuestionOM.pstricks
@@ -111,13 +111,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5800);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5800);
 %DEFAULT
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
index 4b392a788..312cf3840 100644
--- a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
@@ -79,24 +79,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5393) -- (0,3.7681);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5392) -- (0.0000,3.7680);
 %DEFAULT
-\draw [color=red,style=dashed] (0,0) -- (2.00,2.38);
-\draw [color=cyan,style=dashed] (0,0) -- (2.00,-1.15);
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+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (1.2855,1.5320);
+\draw [color=cyan,->,>=latex] (0.0000,0.0000) -- (-1.7320,1.0000);
+\draw [] (2.0000,3.2680) -- (2.0000,-2.0392);
 \draw [color=red]  (2.0000,2.3835) node [rotate=0] {$\bullet$};
-\draw (2.6976,2.3835) node {$\tan(\theta)$};
+\draw (2.6975,2.3835) node {$\tan(\theta)$};
 \draw [color=cyan]  (2.0000,-1.1547) node [rotate=0] {$\bullet$};
 \draw (2.7262,-1.1547) node {$\tan(\varphi)$};
 
diff --git a/auto/pictures_tex/Fig_ToreRevolution.pstricks b/auto/pictures_tex/Fig_ToreRevolution.pstricks
index 044f09832..fbd66806d 100644
--- a/auto/pictures_tex/Fig_ToreRevolution.pstricks
+++ b/auto/pictures_tex/Fig_ToreRevolution.pstricks
@@ -71,18 +71,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
 
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-\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0);
+\draw [color=brown] (1.0000,3.0000)--(0.9979,3.0634)--(0.9919,3.1265)--(0.9819,3.1892)--(0.9679,3.2511)--(0.9500,3.3120)--(0.9283,3.3716)--(0.9029,3.4297)--(0.8738,3.4861)--(0.8412,3.5406)--(0.8052,3.5929)--(0.7660,3.6427)--(0.7237,3.6900)--(0.6785,3.7345)--(0.6305,3.7761)--(0.5800,3.8145)--(0.5272,3.8497)--(0.4722,3.8814)--(0.4154,3.9096)--(0.3568,3.9341)--(0.2969,3.9549)--(0.2357,3.9718)--(0.1736,3.9848)--(0.1108,3.9938)--(0.0475,3.9988)--(-0.0158,3.9998)--(-0.0792,3.9968)--(-0.1423,3.9898)--(-0.2048,3.9788)--(-0.2664,3.9638)--(-0.3270,3.9450)--(-0.3863,3.9223)--(-0.4440,3.8959)--(-0.5000,3.8660)--(-0.5539,3.8325)--(-0.6056,3.7957)--(-0.6548,3.7557)--(-0.7014,3.7126)--(-0.7452,3.6667)--(-0.7860,3.6181)--(-0.8236,3.5670)--(-0.8579,3.5136)--(-0.8888,3.4582)--(-0.9161,3.4009)--(-0.9396,3.3420)--(-0.9594,3.2817)--(-0.9754,3.2203)--(-0.9874,3.1580)--(-0.9954,3.0950)--(-0.9994,3.0317)--(-0.9994,2.9682)--(-0.9954,2.9049)--(-0.9874,2.8419)--(-0.9754,2.7796)--(-0.9594,2.7182)--(-0.9396,2.6579)--(-0.9161,2.5990)--(-0.8888,2.5417)--(-0.8579,2.4863)--(-0.8236,2.4329)--(-0.7860,2.3818)--(-0.7452,2.3332)--(-0.7014,2.2873)--(-0.6548,2.2442)--(-0.6056,2.2042)--(-0.5539,2.1674)--(-0.5000,2.1339)--(-0.4440,2.1040)--(-0.3863,2.0776)--(-0.3270,2.0549)--(-0.2664,2.0361)--(-0.2048,2.0211)--(-0.1423,2.0101)--(-0.0792,2.0031)--(-0.0158,2.0001)--(0.0475,2.0011)--(0.1108,2.0061)--(0.1736,2.0151)--(0.2357,2.0281)--(0.2969,2.0450)--(0.3568,2.0658)--(0.4154,2.0903)--(0.4722,2.1185)--(0.5272,2.1502)--(0.5800,2.1854)--(0.6305,2.2238)--(0.6785,2.2654)--(0.7237,2.3099)--(0.7660,2.3572)--(0.8052,2.4070)--(0.8412,2.4593)--(0.8738,2.5138)--(0.9029,2.5702)--(0.9283,2.6283)--(0.9500,2.6879)--(0.9679,2.7488)--(0.9819,2.8107)--(0.9919,2.8734)--(0.9979,2.9365)--(1.0000,3.0000);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,0.0000);
 \draw [,->,>=latex] (1.5000,1.5000) -- (1.5000,3.0000);
-\draw (1.8965,1.5000) node {$a$};
-\draw []  (0,3.0000) node [rotate=0] {$\bullet$};
-\draw [color=red] (0,3.00) -- (0.866,3.50);
-\draw (0.14303,3.6345) node {$R$};
-\draw [color=blue,style=dotted] (0,3.00) -- (1.50,3.00);
+\draw (1.8964,1.5000) node {$a$};
+\draw []  (0.0000,3.0000) node [rotate=0] {$\bullet$};
+\draw [color=red] (0.0000,3.0000) -- (0.8660,3.5000);
+\draw (0.1430,3.6345) node {$R$};
+\draw [color=blue,style=dotted] (0.0000,3.0000) -- (1.5000,3.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_TracerUn.pstricks b/auto/pictures_tex/Fig_TracerUn.pstricks
index 5b92fa7bb..c8bfd6298 100644
--- a/auto/pictures_tex/Fig_TracerUn.pstricks
+++ b/auto/pictures_tex/Fig_TracerUn.pstricks
@@ -81,43 +81,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
-\draw [,->,>=latex] (0,-4.3056) -- (0,4.3056);
+\draw [,->,>=latex] (-1.9000,0.0000) -- (1.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.3055) -- (0.0000,4.3055);
 %DEFAULT
 
-\draw [color=blue] (-1.400,-3.806)--(-1.372,-3.729)--(-1.343,-3.652)--(-1.315,-3.575)--(-1.287,-3.498)--(-1.259,-3.421)--(-1.230,-3.344)--(-1.202,-3.267)--(-1.174,-3.191)--(-1.145,-3.114)--(-1.117,-3.037)--(-1.089,-2.960)--(-1.061,-2.883)--(-1.032,-2.806)--(-1.004,-2.729)--(-0.9758,-2.652)--(-0.9475,-2.576)--(-0.9192,-2.499)--(-0.8909,-2.422)--(-0.8626,-2.345)--(-0.8343,-2.268)--(-0.8061,-2.191)--(-0.7778,-2.114)--(-0.7495,-2.037)--(-0.7212,-1.960)--(-0.6929,-1.884)--(-0.6646,-1.807)--(-0.6364,-1.730)--(-0.6081,-1.653)--(-0.5798,-1.576)--(-0.5515,-1.499)--(-0.5232,-1.422)--(-0.4949,-1.345)--(-0.4667,-1.269)--(-0.4384,-1.192)--(-0.4101,-1.115)--(-0.3818,-1.038)--(-0.3535,-0.9610)--(-0.3253,-0.8841)--(-0.2970,-0.8072)--(-0.2687,-0.7304)--(-0.2404,-0.6535)--(-0.2121,-0.5766)--(-0.1838,-0.4997)--(-0.1556,-0.4228)--(-0.1273,-0.3460)--(-0.09899,-0.2691)--(-0.07071,-0.1922)--(-0.04242,-0.1153)--(-0.01414,-0.03844)--(0.01414,0.03844)--(0.04242,0.1153)--(0.07071,0.1922)--(0.09899,0.2691)--(0.1273,0.3460)--(0.1556,0.4228)--(0.1838,0.4997)--(0.2121,0.5766)--(0.2404,0.6535)--(0.2687,0.7304)--(0.2970,0.8072)--(0.3253,0.8841)--(0.3535,0.9610)--(0.3818,1.038)--(0.4101,1.115)--(0.4384,1.192)--(0.4667,1.269)--(0.4949,1.345)--(0.5232,1.422)--(0.5515,1.499)--(0.5798,1.576)--(0.6081,1.653)--(0.6364,1.730)--(0.6646,1.807)--(0.6929,1.884)--(0.7212,1.960)--(0.7495,2.037)--(0.7778,2.114)--(0.8061,2.191)--(0.8343,2.268)--(0.8626,2.345)--(0.8909,2.422)--(0.9192,2.499)--(0.9475,2.576)--(0.9758,2.652)--(1.004,2.729)--(1.032,2.806)--(1.061,2.883)--(1.089,2.960)--(1.117,3.037)--(1.145,3.114)--(1.174,3.191)--(1.202,3.267)--(1.230,3.344)--(1.259,3.421)--(1.287,3.498)--(1.315,3.575)--(1.343,3.652)--(1.372,3.729)--(1.400,3.806);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.43316,-4.2000) node {$ -6 $};
-\draw [] (-0.100,-4.20) -- (0.100,-4.20);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.29125,4.2000) node {$ 6 $};
-\draw [] (-0.100,4.20) -- (0.100,4.20);
+\draw [color=blue] (-1.4000,-3.8055)--(-1.3717,-3.7287)--(-1.3434,-3.6518)--(-1.3151,-3.5749)--(-1.2868,-3.4980)--(-1.2585,-3.4211)--(-1.2303,-3.3443)--(-1.2020,-3.2674)--(-1.1737,-3.1905)--(-1.1454,-3.1136)--(-1.1171,-3.0367)--(-1.0888,-2.9599)--(-1.0606,-2.8830)--(-1.0323,-2.8061)--(-1.0040,-2.7292)--(-0.9757,-2.6523)--(-0.9474,-2.5755)--(-0.9191,-2.4986)--(-0.8909,-2.4217)--(-0.8626,-2.3448)--(-0.8343,-2.2679)--(-0.8060,-2.1910)--(-0.7777,-2.1142)--(-0.7494,-2.0373)--(-0.7212,-1.9604)--(-0.6929,-1.8835)--(-0.6646,-1.8066)--(-0.6363,-1.7298)--(-0.6080,-1.6529)--(-0.5797,-1.5760)--(-0.5515,-1.4991)--(-0.5232,-1.4222)--(-0.4949,-1.3454)--(-0.4666,-1.2685)--(-0.4383,-1.1916)--(-0.4101,-1.1147)--(-0.3818,-1.0378)--(-0.3535,-0.9610)--(-0.3252,-0.8841)--(-0.2969,-0.8072)--(-0.2686,-0.7303)--(-0.2404,-0.6534)--(-0.2121,-0.5766)--(-0.1838,-0.4997)--(-0.1555,-0.4228)--(-0.1272,-0.3459)--(-0.0989,-0.2690)--(-0.0707,-0.1922)--(-0.0424,-0.1153)--(-0.0141,-0.0384)--(0.0141,0.0384)--(0.0424,0.1153)--(0.0707,0.1922)--(0.0989,0.2690)--(0.1272,0.3459)--(0.1555,0.4228)--(0.1838,0.4997)--(0.2121,0.5766)--(0.2404,0.6534)--(0.2686,0.7303)--(0.2969,0.8072)--(0.3252,0.8841)--(0.3535,0.9610)--(0.3818,1.0378)--(0.4101,1.1147)--(0.4383,1.1916)--(0.4666,1.2685)--(0.4949,1.3454)--(0.5232,1.4222)--(0.5515,1.4991)--(0.5797,1.5760)--(0.6080,1.6529)--(0.6363,1.7298)--(0.6646,1.8066)--(0.6929,1.8835)--(0.7212,1.9604)--(0.7494,2.0373)--(0.7777,2.1142)--(0.8060,2.1910)--(0.8343,2.2679)--(0.8626,2.3448)--(0.8909,2.4217)--(0.9191,2.4986)--(0.9474,2.5755)--(0.9757,2.6523)--(1.0040,2.7292)--(1.0323,2.8061)--(1.0606,2.8830)--(1.0888,2.9599)--(1.1171,3.0367)--(1.1454,3.1136)--(1.1737,3.1905)--(1.2020,3.2674)--(1.2303,3.3443)--(1.2585,3.4211)--(1.2868,3.4980)--(1.3151,3.5749)--(1.3434,3.6518)--(1.3717,3.7287)--(1.4000,3.8055);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.4331,-4.2000) node {$ -6 $};
+\draw [] (-0.1000,-4.2000) -- (0.1000,-4.2000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
+\draw (-0.2912,4.2000) node {$ 6 $};
+\draw [] (-0.1000,4.2000) -- (0.1000,4.2000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -163,23 +163,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (-2.000,2.000)--(-1.960,1.960)--(-1.919,1.919)--(-1.879,1.879)--(-1.838,1.838)--(-1.798,1.798)--(-1.758,1.758)--(-1.717,1.717)--(-1.677,1.677)--(-1.636,1.636)--(-1.596,1.596)--(-1.556,1.556)--(-1.515,1.515)--(-1.475,1.475)--(-1.434,1.434)--(-1.394,1.394)--(-1.354,1.354)--(-1.313,1.313)--(-1.273,1.273)--(-1.232,1.232)--(-1.192,1.192)--(-1.152,1.152)--(-1.111,1.111)--(-1.071,1.071)--(-1.030,1.030)--(-0.9899,0.9899)--(-0.9495,0.9495)--(-0.9091,0.9091)--(-0.8687,0.8687)--(-0.8283,0.8283)--(-0.7879,0.7879)--(-0.7475,0.7475)--(-0.7071,0.7071)--(-0.6667,0.6667)--(-0.6263,0.6263)--(-0.5859,0.5859)--(-0.5455,0.5455)--(-0.5051,0.5051)--(-0.4646,0.4646)--(-0.4242,0.4242)--(-0.3838,0.3838)--(-0.3434,0.3434)--(-0.3030,0.3030)--(-0.2626,0.2626)--(-0.2222,0.2222)--(-0.1818,0.1818)--(-0.1414,0.1414)--(-0.1010,0.1010)--(-0.06061,0.06061)--(-0.02020,0.02020)--(0.02020,0.02020)--(0.06061,0.06061)--(0.1010,0.1010)--(0.1414,0.1414)--(0.1818,0.1818)--(0.2222,0.2222)--(0.2626,0.2626)--(0.3030,0.3030)--(0.3434,0.3434)--(0.3838,0.3838)--(0.4242,0.4242)--(0.4646,0.4646)--(0.5051,0.5051)--(0.5455,0.5455)--(0.5859,0.5859)--(0.6263,0.6263)--(0.6667,0.6667)--(0.7071,0.7071)--(0.7475,0.7475)--(0.7879,0.7879)--(0.8283,0.8283)--(0.8687,0.8687)--(0.9091,0.9091)--(0.9495,0.9495)--(0.9899,0.9899)--(1.030,1.030)--(1.071,1.071)--(1.111,1.111)--(1.152,1.152)--(1.192,1.192)--(1.232,1.232)--(1.273,1.273)--(1.313,1.313)--(1.354,1.354)--(1.394,1.394)--(1.434,1.434)--(1.475,1.475)--(1.515,1.515)--(1.556,1.556)--(1.596,1.596)--(1.636,1.636)--(1.677,1.677)--(1.717,1.717)--(1.758,1.758)--(1.798,1.798)--(1.838,1.838)--(1.879,1.879)--(1.919,1.919)--(1.960,1.960)--(2.000,2.000);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (-2.0000,2.0000)--(-1.9595,1.9595)--(-1.9191,1.9191)--(-1.8787,1.8787)--(-1.8383,1.8383)--(-1.7979,1.7979)--(-1.7575,1.7575)--(-1.7171,1.7171)--(-1.6767,1.6767)--(-1.6363,1.6363)--(-1.5959,1.5959)--(-1.5555,1.5555)--(-1.5151,1.5151)--(-1.4747,1.4747)--(-1.4343,1.4343)--(-1.3939,1.3939)--(-1.3535,1.3535)--(-1.3131,1.3131)--(-1.2727,1.2727)--(-1.2323,1.2323)--(-1.1919,1.1919)--(-1.1515,1.1515)--(-1.1111,1.1111)--(-1.0707,1.0707)--(-1.0303,1.0303)--(-0.9898,0.9898)--(-0.9494,0.9494)--(-0.9090,0.9090)--(-0.8686,0.8686)--(-0.8282,0.8282)--(-0.7878,0.7878)--(-0.7474,0.7474)--(-0.7070,0.7070)--(-0.6666,0.6666)--(-0.6262,0.6262)--(-0.5858,0.5858)--(-0.5454,0.5454)--(-0.5050,0.5050)--(-0.4646,0.4646)--(-0.4242,0.4242)--(-0.3838,0.3838)--(-0.3434,0.3434)--(-0.3030,0.3030)--(-0.2626,0.2626)--(-0.2222,0.2222)--(-0.1818,0.1818)--(-0.1414,0.1414)--(-0.1010,0.1010)--(-0.0606,0.0606)--(-0.0202,0.0202)--(0.0202,0.0202)--(0.0606,0.0606)--(0.1010,0.1010)--(0.1414,0.1414)--(0.1818,0.1818)--(0.2222,0.2222)--(0.2626,0.2626)--(0.3030,0.3030)--(0.3434,0.3434)--(0.3838,0.3838)--(0.4242,0.4242)--(0.4646,0.4646)--(0.5050,0.5050)--(0.5454,0.5454)--(0.5858,0.5858)--(0.6262,0.6262)--(0.6666,0.6666)--(0.7070,0.7070)--(0.7474,0.7474)--(0.7878,0.7878)--(0.8282,0.8282)--(0.8686,0.8686)--(0.9090,0.9090)--(0.9494,0.9494)--(0.9898,0.9898)--(1.0303,1.0303)--(1.0707,1.0707)--(1.1111,1.1111)--(1.1515,1.1515)--(1.1919,1.1919)--(1.2323,1.2323)--(1.2727,1.2727)--(1.3131,1.3131)--(1.3535,1.3535)--(1.3939,1.3939)--(1.4343,1.4343)--(1.4747,1.4747)--(1.5151,1.5151)--(1.5555,1.5555)--(1.5959,1.5959)--(1.6363,1.6363)--(1.6767,1.6767)--(1.7171,1.7171)--(1.7575,1.7575)--(1.7979,1.7979)--(1.8383,1.8383)--(1.8787,1.8787)--(1.9191,1.9191)--(1.9595,1.9595)--(2.0000,2.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -253,39 +253,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.8000);
+\draw [,->,>=latex] (-1.9000,0.0000) -- (1.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.8000);
 %DEFAULT
 
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+\draw [color=blue] (-1.4000,3.5000)--(-1.3858,3.4437)--(-1.3717,3.3880)--(-1.3575,3.3328)--(-1.3434,3.2783)--(-1.3292,3.2243)--(-1.3151,3.1708)--(-1.3010,3.1180)--(-1.2868,3.0657)--(-1.2727,3.0140)--(-1.2585,2.9629)--(-1.2444,2.9123)--(-1.2303,2.8623)--(-1.2161,2.8129)--(-1.2020,2.7640)--(-1.1878,2.7157)--(-1.1737,2.6680)--(-1.1595,2.6209)--(-1.1454,2.5743)--(-1.1313,2.5283)--(-1.1171,2.4829)--(-1.1030,2.4381)--(-1.0888,2.3938)--(-1.0747,2.3501)--(-1.0606,2.3069)--(-1.0464,2.2644)--(-1.0323,2.2224)--(-1.0181,2.1809)--(-1.0040,2.1401)--(-0.9898,2.0998)--(-0.9757,2.0601)--(-0.9616,2.0210)--(-0.9474,1.9824)--(-0.9333,1.9444)--(-0.9191,1.9070)--(-0.9050,1.8701)--(-0.8909,1.8338)--(-0.8767,1.7981)--(-0.8626,1.7630)--(-0.8484,1.7284)--(-0.8343,1.6944)--(-0.8202,1.6610)--(-0.8060,1.6281)--(-0.7919,1.5959)--(-0.7777,1.5641)--(-0.7636,1.5330)--(-0.7494,1.5024)--(-0.7353,1.4724)--(-0.7212,1.4430)--(-0.7070,1.4142)--(-0.6929,1.3859)--(-0.6787,1.3582)--(-0.6646,1.3310)--(-0.6505,1.3045)--(-0.6363,1.2785)--(-0.6222,1.2530)--(-0.6080,1.2282)--(-0.5939,1.2039)--(-0.5797,1.1802)--(-0.5656,1.1570)--(-0.5515,1.1345)--(-0.5373,1.1125)--(-0.5232,1.0911)--(-0.5090,1.0702)--(-0.4949,1.0499)--(-0.4808,1.0302)--(-0.4666,1.0111)--(-0.4525,0.9925)--(-0.4383,0.9745)--(-0.4242,0.9571)--(-0.4101,0.9402)--(-0.3959,0.9239)--(-0.3818,0.9082)--(-0.3676,0.8931)--(-0.3535,0.8785)--(-0.3393,0.8645)--(-0.3252,0.8511)--(-0.3111,0.8382)--(-0.2969,0.8259)--(-0.2828,0.8142)--(-0.2686,0.8031)--(-0.2545,0.7925)--(-0.2404,0.7825)--(-0.2262,0.7731)--(-0.2121,0.7642)--(-0.1979,0.7559)--(-0.1838,0.7482)--(-0.1696,0.7411)--(-0.1555,0.7345)--(-0.1414,0.7285)--(-0.1272,0.7231)--(-0.1131,0.7182)--(-0.0989,0.7139)--(-0.0848,0.7102)--(-0.0707,0.7071)--(-0.0565,0.7045)--(-0.0424,0.7025)--(-0.0282,0.7011)--(-0.0141,0.7002)--(0.0000,0.7000);
 
-\draw [color=blue] (0,0.7000)--(0.01414,0.7286)--(0.02828,0.7577)--(0.04242,0.7874)--(0.05657,0.8177)--(0.07071,0.8486)--(0.08485,0.8800)--(0.09899,0.9120)--(0.1131,0.9445)--(0.1273,0.9777)--(0.1414,1.011)--(0.1556,1.046)--(0.1697,1.081)--(0.1838,1.116)--(0.1980,1.152)--(0.2121,1.189)--(0.2263,1.226)--(0.2404,1.263)--(0.2545,1.302)--(0.2687,1.340)--(0.2828,1.380)--(0.2970,1.420)--(0.3111,1.460)--(0.3253,1.502)--(0.3394,1.543)--(0.3535,1.586)--(0.3677,1.628)--(0.3818,1.672)--(0.3960,1.716)--(0.4101,1.760)--(0.4242,1.806)--(0.4384,1.851)--(0.4525,1.898)--(0.4667,1.944)--(0.4808,1.992)--(0.4949,2.040)--(0.5091,2.088)--(0.5232,2.138)--(0.5374,2.187)--(0.5515,2.238)--(0.5657,2.288)--(0.5798,2.340)--(0.5939,2.392)--(0.6081,2.444)--(0.6222,2.498)--(0.6364,2.551)--(0.6505,2.606)--(0.6646,2.660)--(0.6788,2.716)--(0.6929,2.772)--(0.7071,2.828)--(0.7212,2.885)--(0.7354,2.943)--(0.7495,3.001)--(0.7636,3.060)--(0.7778,3.120)--(0.7919,3.180)--(0.8061,3.240)--(0.8202,3.301)--(0.8343,3.363)--(0.8485,3.425)--(0.8626,3.488)--(0.8768,3.552)--(0.8909,3.616)--(0.9051,3.680)--(0.9192,3.745)--(0.9333,3.811)--(0.9475,3.877)--(0.9616,3.944)--(0.9758,4.012)--(0.9899,4.080)--(1.004,4.148)--(1.018,4.217)--(1.032,4.287)--(1.046,4.357)--(1.061,4.428)--(1.075,4.500)--(1.089,4.572)--(1.103,4.644)--(1.117,4.717)--(1.131,4.791)--(1.145,4.865)--(1.160,4.940)--(1.174,5.016)--(1.188,5.092)--(1.202,5.168)--(1.216,5.245)--(1.230,5.323)--(1.244,5.401)--(1.259,5.480)--(1.273,5.560)--(1.287,5.640)--(1.301,5.720)--(1.315,5.801)--(1.329,5.883)--(1.343,5.965)--(1.358,6.048)--(1.372,6.131)--(1.386,6.215)--(1.400,6.300);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.29125,4.2000) node {$ 6 $};
-\draw [] (-0.100,4.20) -- (0.100,4.20);
-\draw (-0.29125,4.9000) node {$ 7 $};
-\draw [] (-0.100,4.90) -- (0.100,4.90);
-\draw (-0.29125,5.6000) node {$ 8 $};
-\draw [] (-0.100,5.60) -- (0.100,5.60);
-\draw (-0.29125,6.3000) node {$ 9 $};
-\draw [] (-0.100,6.30) -- (0.100,6.30);
+\draw [color=blue] (0.0000,0.7000)--(0.0141,0.7285)--(0.0282,0.7577)--(0.0424,0.7874)--(0.0565,0.8177)--(0.0707,0.8485)--(0.0848,0.8799)--(0.0989,0.9119)--(0.1131,0.9445)--(0.1272,0.9776)--(0.1414,1.0113)--(0.1555,1.0456)--(0.1696,1.0805)--(0.1838,1.1159)--(0.1979,1.1519)--(0.2121,1.1885)--(0.2262,1.2256)--(0.2404,1.2633)--(0.2545,1.3016)--(0.2686,1.3405)--(0.2828,1.3799)--(0.2969,1.4199)--(0.3111,1.4604)--(0.3252,1.5016)--(0.3393,1.5433)--(0.3535,1.5856)--(0.3676,1.6284)--(0.3818,1.6719)--(0.3959,1.7158)--(0.4101,1.7604)--(0.4242,1.8056)--(0.4383,1.8513)--(0.4525,1.8975)--(0.4666,1.9444)--(0.4808,1.9918)--(0.4949,2.0398)--(0.5090,2.0884)--(0.5232,2.1375)--(0.5373,2.1872)--(0.5515,2.2375)--(0.5656,2.2884)--(0.5797,2.3398)--(0.5939,2.3918)--(0.6080,2.4443)--(0.6222,2.4975)--(0.6363,2.5512)--(0.6505,2.6055)--(0.6646,2.6603)--(0.6787,2.7157)--(0.6929,2.7717)--(0.7070,2.8283)--(0.7212,2.8854)--(0.7353,2.9431)--(0.7494,3.0014)--(0.7636,3.0603)--(0.7777,3.1197)--(0.7919,3.1797)--(0.8060,3.2403)--(0.8202,3.3014)--(0.8343,3.3631)--(0.8484,3.4254)--(0.8626,3.4882)--(0.8767,3.5517)--(0.8909,3.6157)--(0.9050,3.6802)--(0.9191,3.7454)--(0.9333,3.8111)--(0.9474,3.8773)--(0.9616,3.9442)--(0.9757,4.0116)--(0.9898,4.0796)--(1.0040,4.1482)--(1.0181,4.2173)--(1.0323,4.2870)--(1.0464,4.3573)--(1.0606,4.4281)--(1.0747,4.4996)--(1.0888,4.5716)--(1.1030,4.6441)--(1.1171,4.7173)--(1.1313,4.7910)--(1.1454,4.8652)--(1.1595,4.9401)--(1.1737,5.0155)--(1.1878,5.0915)--(1.2020,5.1681)--(1.2161,5.2452)--(1.2303,5.3229)--(1.2444,5.4012)--(1.2585,5.4800)--(1.2727,5.5595)--(1.2868,5.6394)--(1.3010,5.7200)--(1.3151,5.8011)--(1.3292,5.8828)--(1.3434,5.9651)--(1.3575,6.0480)--(1.3717,6.1314)--(1.3858,6.2154)--(1.4000,6.3000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
+\draw (-0.2912,4.2000) node {$ 6 $};
+\draw [] (-0.1000,4.2000) -- (0.1000,4.2000);
+\draw (-0.2912,4.9000) node {$ 7 $};
+\draw [] (-0.1000,4.9000) -- (0.1000,4.9000);
+\draw (-0.2912,5.6000) node {$ 8 $};
+\draw [] (-0.1000,5.6000) -- (0.1000,5.6000);
+\draw (-0.2912,6.3000) node {$ 9 $};
+\draw [] (-0.1000,6.3000) -- (0.1000,6.3000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -335,27 +335,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9000,0) -- (1.9000,0);
-\draw [,->,>=latex] (0,-2.6000) -- (0,1.2000);
+\draw [,->,>=latex] (-1.9000,0.0000) -- (1.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6000) -- (0.0000,1.2000);
 %DEFAULT
 
-\draw [color=blue] (-1.400,-2.100)--(-1.372,-2.072)--(-1.343,-2.043)--(-1.315,-2.015)--(-1.287,-1.987)--(-1.259,-1.959)--(-1.230,-1.930)--(-1.202,-1.902)--(-1.174,-1.874)--(-1.145,-1.845)--(-1.117,-1.817)--(-1.089,-1.789)--(-1.061,-1.761)--(-1.032,-1.732)--(-1.004,-1.704)--(-0.9758,-1.676)--(-0.9475,-1.647)--(-0.9192,-1.619)--(-0.8909,-1.591)--(-0.8626,-1.563)--(-0.8343,-1.534)--(-0.8061,-1.506)--(-0.7778,-1.478)--(-0.7495,-1.449)--(-0.7212,-1.421)--(-0.6929,-1.393)--(-0.6646,-1.365)--(-0.6364,-1.336)--(-0.6081,-1.308)--(-0.5798,-1.280)--(-0.5515,-1.252)--(-0.5232,-1.223)--(-0.4949,-1.195)--(-0.4667,-1.167)--(-0.4384,-1.138)--(-0.4101,-1.110)--(-0.3818,-1.082)--(-0.3535,-1.054)--(-0.3253,-1.025)--(-0.2970,-0.9970)--(-0.2687,-0.9687)--(-0.2404,-0.9404)--(-0.2121,-0.9121)--(-0.1838,-0.8838)--(-0.1556,-0.8556)--(-0.1273,-0.8273)--(-0.09899,-0.7990)--(-0.07071,-0.7707)--(-0.04242,-0.7424)--(-0.01414,-0.7141)--(0.01414,-0.6859)--(0.04242,-0.6576)--(0.07071,-0.6293)--(0.09899,-0.6010)--(0.1273,-0.5727)--(0.1556,-0.5444)--(0.1838,-0.5162)--(0.2121,-0.4879)--(0.2404,-0.4596)--(0.2687,-0.4313)--(0.2970,-0.4030)--(0.3253,-0.3747)--(0.3535,-0.3465)--(0.3818,-0.3182)--(0.4101,-0.2899)--(0.4384,-0.2616)--(0.4667,-0.2333)--(0.4949,-0.2051)--(0.5232,-0.1768)--(0.5515,-0.1485)--(0.5798,-0.1202)--(0.6081,-0.09192)--(0.6364,-0.06364)--(0.6646,-0.03535)--(0.6929,-0.007071)--(0.7212,0.02121)--(0.7495,0.04949)--(0.7778,0.07778)--(0.8061,0.1061)--(0.8343,0.1343)--(0.8626,0.1626)--(0.8909,0.1909)--(0.9192,0.2192)--(0.9475,0.2475)--(0.9758,0.2758)--(1.004,0.3040)--(1.032,0.3323)--(1.061,0.3606)--(1.089,0.3889)--(1.117,0.4172)--(1.145,0.4455)--(1.174,0.4737)--(1.202,0.5020)--(1.230,0.5303)--(1.259,0.5586)--(1.287,0.5869)--(1.315,0.6152)--(1.343,0.6434)--(1.372,0.6717)--(1.400,0.7000);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
+\draw [color=blue] (-1.4000,-2.1000)--(-1.3717,-2.0717)--(-1.3434,-2.0434)--(-1.3151,-2.0151)--(-1.2868,-1.9868)--(-1.2585,-1.9585)--(-1.2303,-1.9303)--(-1.2020,-1.9020)--(-1.1737,-1.8737)--(-1.1454,-1.8454)--(-1.1171,-1.8171)--(-1.0888,-1.7888)--(-1.0606,-1.7606)--(-1.0323,-1.7323)--(-1.0040,-1.7040)--(-0.9757,-1.6757)--(-0.9474,-1.6474)--(-0.9191,-1.6191)--(-0.8909,-1.5909)--(-0.8626,-1.5626)--(-0.8343,-1.5343)--(-0.8060,-1.5060)--(-0.7777,-1.4777)--(-0.7494,-1.4494)--(-0.7212,-1.4212)--(-0.6929,-1.3929)--(-0.6646,-1.3646)--(-0.6363,-1.3363)--(-0.6080,-1.3080)--(-0.5797,-1.2797)--(-0.5515,-1.2515)--(-0.5232,-1.2232)--(-0.4949,-1.1949)--(-0.4666,-1.1666)--(-0.4383,-1.1383)--(-0.4101,-1.1101)--(-0.3818,-1.0818)--(-0.3535,-1.0535)--(-0.3252,-1.0252)--(-0.2969,-0.9969)--(-0.2686,-0.9686)--(-0.2404,-0.9404)--(-0.2121,-0.9121)--(-0.1838,-0.8838)--(-0.1555,-0.8555)--(-0.1272,-0.8272)--(-0.0989,-0.7989)--(-0.0707,-0.7707)--(-0.0424,-0.7424)--(-0.0141,-0.7141)--(0.0141,-0.6858)--(0.0424,-0.6575)--(0.0707,-0.6292)--(0.0989,-0.6010)--(0.1272,-0.5727)--(0.1555,-0.5444)--(0.1838,-0.5161)--(0.2121,-0.4878)--(0.2404,-0.4595)--(0.2686,-0.4313)--(0.2969,-0.4030)--(0.3252,-0.3747)--(0.3535,-0.3464)--(0.3818,-0.3181)--(0.4101,-0.2898)--(0.4383,-0.2616)--(0.4666,-0.2333)--(0.4949,-0.2050)--(0.5232,-0.1767)--(0.5515,-0.1484)--(0.5797,-0.1202)--(0.6080,-0.0919)--(0.6363,-0.0636)--(0.6646,-0.0353)--(0.6929,-0.0070)--(0.7212,0.0212)--(0.7494,0.0494)--(0.7777,0.0777)--(0.8060,0.1060)--(0.8343,0.1343)--(0.8626,0.1626)--(0.8909,0.1909)--(0.9191,0.2191)--(0.9474,0.2474)--(0.9757,0.2757)--(1.0040,0.3040)--(1.0323,0.3323)--(1.0606,0.3606)--(1.0888,0.3888)--(1.1171,0.4171)--(1.1454,0.4454)--(1.1737,0.4737)--(1.2020,0.5020)--(1.2303,0.5303)--(1.2585,0.5585)--(1.2868,0.5868)--(1.3151,0.6151)--(1.3434,0.6434)--(1.3717,0.6717)--(1.4000,0.7000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_Trajs.pstricks b/auto/pictures_tex/Fig_Trajs.pstricks
index d6f17fd84..271b0587e 100644
--- a/auto/pictures_tex/Fig_Trajs.pstricks
+++ b/auto/pictures_tex/Fig_Trajs.pstricks
@@ -75,29 +75,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6426,0) -- (2.7232,0);
-\draw [,->,>=latex] (0,-1.9142) -- (0,2.7360);
+\draw [,->,>=latex] (-1.6426,0.0000) -- (2.7232,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9141) -- (0.0000,2.7359);
 %DEFAULT
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 \draw [color=brown]  (1.0000,-1.0000) node [rotate=0] {$\bullet$};
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-\draw [color=brown]  (1.0000,0) node [rotate=0] {$\bullet$};
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+\draw [color=brown]  (1.0000,0.0000) node [rotate=0] {$\bullet$};
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 \draw [color=brown]  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=cyan] (-1.143,1.922)--(-1.104,1.945)--(-1.064,1.967)--(-1.024,1.988)--(-0.9838,2.008)--(-0.9430,2.027)--(-0.9018,2.046)--(-0.8603,2.064)--(-0.8185,2.081)--(-0.7763,2.097)--(-0.7337,2.112)--(-0.6909,2.127)--(-0.6478,2.140)--(-0.6045,2.153)--(-0.5608,2.165)--(-0.5170,2.175)--(-0.4729,2.185)--(-0.4287,2.195)--(-0.3843,2.203)--(-0.3397,2.210)--(-0.2950,2.217)--(-0.2502,2.222)--(-0.2052,2.227)--(-0.1602,2.230)--(-0.1151,2.233)--(-0.06998,2.235)--(-0.02482,2.236)--(0.02035,2.236)--(0.06552,2.235)--(0.1107,2.233)--(0.1557,2.231)--(0.2008,2.227)--(0.2457,2.223)--(0.2906,2.217)--(0.3353,2.211)--(0.3799,2.204)--(0.4243,2.195)--(0.4686,2.186)--(0.5127,2.177)--(0.5565,2.166)--(0.6002,2.154)--(0.6435,2.141)--(0.6867,2.128)--(0.7295,2.114)--(0.7721,2.099)--(0.8143,2.083)--(0.8562,2.066)--(0.8978,2.048)--(0.9389,2.029)--(0.9797,2.010)--(1.020,1.990)--(1.060,1.969)--(1.100,1.947)--(1.139,1.924)--(1.177,1.901)--(1.216,1.877)--(1.253,1.852)--(1.290,1.826)--(1.327,1.800)--(1.363,1.773)--(1.399,1.745)--(1.434,1.716)--(1.468,1.687)--(1.502,1.657)--(1.535,1.626)--(1.567,1.595)--(1.599,1.563)--(1.631,1.530)--(1.661,1.497)--(1.691,1.463)--(1.720,1.429)--(1.749,1.393)--(1.777,1.358)--(1.804,1.322)--(1.830,1.285)--(1.856,1.248)--(1.880,1.210)--(1.904,1.172)--(1.928,1.133)--(1.950,1.094)--(1.972,1.054)--(1.993,1.014)--(2.013,0.9738)--(2.032,0.9329)--(2.051,0.8917)--(2.068,0.8501)--(2.085,0.8081)--(2.101,0.7658)--(2.116,0.7233)--(2.130,0.6804)--(2.143,0.6372)--(2.156,0.5938)--(2.167,0.5501)--(2.178,0.5062)--(2.188,0.4621)--(2.197,0.4178)--(2.205,0.3734)--(2.212,0.3287)--(2.218,0.2840)--(2.223,0.2391);
+\draw [color=cyan] (-1.1426,1.9220)--(-1.1035,1.9447)--(-1.0640,1.9666)--(-1.0241,1.9877)--(-0.9837,2.0080)--(-0.9429,2.0275)--(-0.9018,2.0461)--(-0.8603,2.0639)--(-0.8184,2.0808)--(-0.7762,2.0970)--(-0.7337,2.1122)--(-0.6909,2.1266)--(-0.6478,2.1401)--(-0.6044,2.1528)--(-0.5608,2.1645)--(-0.5170,2.1754)--(-0.4729,2.1854)--(-0.4287,2.1945)--(-0.3842,2.2027)--(-0.3397,2.2101)--(-0.2949,2.2165)--(-0.2501,2.2220)--(-0.2052,2.2266)--(-0.1602,2.2303)--(-0.1151,2.2331)--(-0.0699,2.2349)--(-0.0248,2.2359)--(0.0203,2.2359)--(0.0655,2.2351)--(0.1106,2.2333)--(0.1557,2.2306)--(0.2007,2.2270)--(0.2457,2.2225)--(0.2905,2.2171)--(0.3352,2.2107)--(0.3798,2.2035)--(0.4243,2.1954)--(0.4685,2.1864)--(0.5126,2.1765)--(0.5565,2.1657)--(0.6001,2.1540)--(0.6435,2.1414)--(0.6866,2.1280)--(0.7295,2.1137)--(0.7720,2.0985)--(0.8143,2.0825)--(0.8562,2.0656)--(0.8977,2.0479)--(0.9389,2.0293)--(0.9797,2.0099)--(1.0201,1.9897)--(1.0601,1.9687)--(1.0996,1.9469)--(1.1387,1.9243)--(1.1774,1.9009)--(1.2155,1.8767)--(1.2532,1.8518)--(1.2904,1.8261)--(1.3270,1.7997)--(1.3631,1.7725)--(1.3986,1.7446)--(1.4336,1.7160)--(1.4679,1.6867)--(1.5017,1.6567)--(1.5349,1.6260)--(1.5674,1.5947)--(1.5993,1.5627)--(1.6305,1.5300)--(1.6611,1.4968)--(1.6910,1.4629)--(1.7202,1.4285)--(1.7487,1.3934)--(1.7765,1.3578)--(1.8036,1.3217)--(1.8299,1.2849)--(1.8555,1.2477)--(1.8803,1.2100)--(1.9044,1.1717)--(1.9277,1.1330)--(1.9502,1.0939)--(1.9719,1.0542)--(1.9928,1.0142)--(2.0128,0.9737)--(2.0321,0.9329)--(2.0505,0.8916)--(2.0681,0.8500)--(2.0849,0.8081)--(2.1008,0.7658)--(2.1158,0.7232)--(2.1300,0.6803)--(2.1433,0.6371)--(2.1557,0.5937)--(2.1673,0.5500)--(2.1780,0.5062)--(2.1877,0.4621)--(2.1966,0.4178)--(2.2046,0.3733)--(2.2117,0.3287)--(2.2179,0.2839)--(2.2232,0.2391);
 \draw [color=brown]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_TriangleRectangle.pstricks b/auto/pictures_tex/Fig_TriangleRectangle.pstricks
index 1807898e1..fb1989307 100644
--- a/auto/pictures_tex/Fig_TriangleRectangle.pstricks
+++ b/auto/pictures_tex/Fig_TriangleRectangle.pstricks
@@ -89,24 +89,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (2.00,3.46) -- (0,0);
-\draw [] (2.00,3.46) -- (4.00,0);
-\draw [] (4.00,0) -- (0,0);
-\draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.2518,0.36492) node {$60$};
+\draw [] (2.0000,3.4641) -- (0.0000,0.0000);
+\draw [] (2.0000,3.4641) -- (4.0000,0.0000);
+\draw [] (4.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=blue,style=dotted] (2.0000,3.4641) -- (2.0000,0.0000);
+\draw (3.2517,0.3649) node {$60$};
 
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+\draw [color=red] (3.7500,0.4330)--(3.7454,0.4303)--(3.7408,0.4276)--(3.7363,0.4248)--(3.7319,0.4220)--(3.7274,0.4191)--(3.7230,0.4162)--(3.7186,0.4133)--(3.7142,0.4103)--(3.7099,0.4072)--(3.7056,0.4041)--(3.7014,0.4010)--(3.6971,0.3978)--(3.6930,0.3946)--(3.6888,0.3913)--(3.6847,0.3880)--(3.6806,0.3847)--(3.6765,0.3813)--(3.6725,0.3778)--(3.6685,0.3743)--(3.6646,0.3708)--(3.6607,0.3672)--(3.6568,0.3636)--(3.6530,0.3600)--(3.6492,0.3563)--(3.6455,0.3526)--(3.6418,0.3488)--(3.6381,0.3450)--(3.6345,0.3411)--(3.6309,0.3373)--(3.6273,0.3333)--(3.6238,0.3294)--(3.6203,0.3254)--(3.6169,0.3213)--(3.6135,0.3173)--(3.6102,0.3132)--(3.6069,0.3090)--(3.6037,0.3049)--(3.6005,0.3006)--(3.5973,0.2964)--(3.5942,0.2921)--(3.5911,0.2878)--(3.5881,0.2835)--(3.5851,0.2791)--(3.5822,0.2747)--(3.5793,0.2703)--(3.5765,0.2658)--(3.5737,0.2613)--(3.5710,0.2568)--(3.5683,0.2522)--(3.5656,0.2477)--(3.5630,0.2430)--(3.5605,0.2384)--(3.5580,0.2338)--(3.5555,0.2291)--(3.5531,0.2243)--(3.5508,0.2196)--(3.5485,0.2148)--(3.5462,0.2101)--(3.5440,0.2052)--(3.5419,0.2004)--(3.5398,0.1956)--(3.5378,0.1907)--(3.5358,0.1858)--(3.5338,0.1809)--(3.5319,0.1759)--(3.5301,0.1710)--(3.5283,0.1660)--(3.5266,0.1610)--(3.5249,0.1560)--(3.5233,0.1509)--(3.5217,0.1459)--(3.5202,0.1408)--(3.5187,0.1357)--(3.5173,0.1306)--(3.5160,0.1255)--(3.5147,0.1204)--(3.5134,0.1153)--(3.5122,0.1101)--(3.5111,0.1049)--(3.5100,0.0998)--(3.5090,0.0946)--(3.5080,0.0894)--(3.5071,0.0842)--(3.5062,0.0790)--(3.5054,0.0737)--(3.5047,0.0685)--(3.5040,0.0632)--(3.5033,0.0580)--(3.5027,0.0527)--(3.5022,0.0475)--(3.5017,0.0422)--(3.5013,0.0369)--(3.5010,0.0317)--(3.5006,0.0264)--(3.5004,0.0211)--(3.5002,0.0158)--(3.5001,0.0105)--(3.5000,0.0052)--(3.5000,0.0000);
 \draw (2.3119,2.2340) node {$30$};
 
-\draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
+\draw [color=cyan] (2.0000,2.9641)--(2.0026,2.9641)--(2.0052,2.9641)--(2.0079,2.9641)--(2.0105,2.9642)--(2.0132,2.9642)--(2.0158,2.9643)--(2.0185,2.9644)--(2.0211,2.9645)--(2.0237,2.9646)--(2.0264,2.9648)--(2.0290,2.9649)--(2.0317,2.9651)--(2.0343,2.9652)--(2.0369,2.9654)--(2.0396,2.9656)--(2.0422,2.9658)--(2.0448,2.9661)--(2.0475,2.9663)--(2.0501,2.9666)--(2.0527,2.9668)--(2.0554,2.9671)--(2.0580,2.9674)--(2.0606,2.9677)--(2.0632,2.9681)--(2.0659,2.9684)--(2.0685,2.9688)--(2.0711,2.9691)--(2.0737,2.9695)--(2.0763,2.9699)--(2.0790,2.9703)--(2.0816,2.9708)--(2.0842,2.9712)--(2.0868,2.9716)--(2.0894,2.9721)--(2.0920,2.9726)--(2.0946,2.9731)--(2.0972,2.9736)--(2.0998,2.9741)--(2.1024,2.9747)--(2.1049,2.9752)--(2.1075,2.9758)--(2.1101,2.9763)--(2.1127,2.9769)--(2.1153,2.9775)--(2.1178,2.9781)--(2.1204,2.9788)--(2.1230,2.9794)--(2.1255,2.9801)--(2.1281,2.9807)--(2.1306,2.9814)--(2.1332,2.9821)--(2.1357,2.9828)--(2.1383,2.9836)--(2.1408,2.9843)--(2.1434,2.9851)--(2.1459,2.9858)--(2.1484,2.9866)--(2.1509,2.9874)--(2.1535,2.9882)--(2.1560,2.9890)--(2.1585,2.9898)--(2.1610,2.9907)--(2.1635,2.9916)--(2.1660,2.9924)--(2.1685,2.9933)--(2.1710,2.9942)--(2.1734,2.9951)--(2.1759,2.9960)--(2.1784,2.9970)--(2.1809,2.9979)--(2.1833,2.9989)--(2.1858,2.9999)--(2.1882,3.0009)--(2.1907,3.0019)--(2.1931,3.0029)--(2.1956,3.0039)--(2.1980,3.0049)--(2.2004,3.0060)--(2.2028,3.0071)--(2.2052,3.0081)--(2.2077,3.0092)--(2.2101,3.0103)--(2.2125,3.0115)--(2.2148,3.0126)--(2.2172,3.0137)--(2.2196,3.0149)--(2.2220,3.0161)--(2.2243,3.0172)--(2.2267,3.0184)--(2.2291,3.0196)--(2.2314,3.0209)--(2.2338,3.0221)--(2.2361,3.0233)--(2.2384,3.0246)--(2.2407,3.0258)--(2.2430,3.0271)--(2.2454,3.0284)--(2.2477,3.0297)--(2.2500,3.0310);
 \draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
 \draw (2.0000,3.8888) node {$A$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.44758,0) node {$B$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.4435,0) node {$C$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.42471) node {$H$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.4475,0.0000) node {$B$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.4434,0.0000) node {$C$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.4247) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_TriangleUV.pstricks b/auto/pictures_tex/Fig_TriangleUV.pstricks
index 69dd3cc1b..d76fff3b4 100644
--- a/auto/pictures_tex/Fig_TriangleUV.pstricks
+++ b/auto/pictures_tex/Fig_TriangleUV.pstricks
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -89,16 +89,16 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0,0) -- (0,3.00) -- (0,3.00) -- (3.00,0) -- (3.00,0) -- (0,0) --  cycle;
+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (0.0000,0.0000) -- (0.0000,3.0000) -- (0.0000,3.0000) -- (3.0000,0.0000) -- (3.0000,0.0000) -- (0.0000,0.0000) --  cycle;
 
-\draw [color=green,->,>=latex] (0,0) -- (1.0000,0);
-\draw (1.0000,-0.20595) node {\( e_u\)};
-\draw [color=red,->,>=latex] (0,0) -- (0,1.0000);
-\draw (-0.26708,1.0000) node {\( e_v\)};
+\draw [color=green,->,>=latex] (0.0000,0.0000) -- (1.0000,0.0000);
+\draw (1.0000,-0.2059) node {\( e_u\)};
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (0.0000,1.0000);
+\draw (-0.2670,1.0000) node {\( e_v\)};
 \draw [color=green,->,>=latex] (1.7121,1.7121) -- (2.4192,2.4192);
-\draw (2.2468,2.5685) node {\( \nu\)};
+\draw (2.2467,2.5685) node {\( \nu\)};
 \draw [color=red,->,>=latex] (1.7121,1.7121) -- (1.0050,2.4192);
-\draw (0.80232,2.6147) node {\( T\)};
+\draw (0.8023,2.6146) node {\( T\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks b/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks
index 039dac099..fb19a61e0 100644
--- a/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks
+++ b/auto/pictures_tex/Fig_UCDQooMCxpDszQ.pstricks
@@ -60,10 +60,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (0.50000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,0.50000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (0.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,0.5000);
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks b/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks
index 23fd0b1bb..b1653b7ec 100644
--- a/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks
+++ b/auto/pictures_tex/Fig_UEGEooHEDIJVPn.pstricks
@@ -96,46 +96,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
-\draw [] (0,0) -- (5.00,5.00);
+\draw [] (0.0000,0.0000) -- (5.0000,5.0000);
 
-\draw [color=blue] (0.1000,4.996)--(0.1495,4.594)--(0.1990,4.308)--(0.2485,4.086)--(0.2980,3.904)--(0.3475,3.750)--(0.3970,3.617)--(0.4465,3.500)--(0.4960,3.394)--(0.5455,3.299)--(0.5949,3.212)--(0.6444,3.133)--(0.6939,3.059)--(0.7434,2.990)--(0.7929,2.925)--(0.8424,2.865)--(0.8919,2.808)--(0.9414,2.754)--(0.9909,2.702)--(1.040,2.654)--(1.090,2.607)--(1.139,2.563)--(1.189,2.520)--(1.238,2.479)--(1.288,2.440)--(1.337,2.402)--(1.387,2.366)--(1.436,2.331)--(1.486,2.297)--(1.535,2.264)--(1.585,2.233)--(1.634,2.202)--(1.684,2.172)--(1.733,2.143)--(1.783,2.115)--(1.832,2.088)--(1.882,2.061)--(1.931,2.035)--(1.981,2.010)--(2.030,1.985)--(2.080,1.961)--(2.129,1.937)--(2.179,1.914)--(2.228,1.892)--(2.278,1.870)--(2.327,1.848)--(2.377,1.827)--(2.426,1.807)--(2.476,1.787)--(2.525,1.767)--(2.575,1.747)--(2.624,1.728)--(2.674,1.710)--(2.723,1.691)--(2.773,1.673)--(2.822,1.656)--(2.872,1.638)--(2.921,1.621)--(2.971,1.604)--(3.020,1.588)--(3.070,1.572)--(3.119,1.556)--(3.169,1.540)--(3.218,1.524)--(3.268,1.509)--(3.317,1.494)--(3.367,1.479)--(3.416,1.465)--(3.466,1.450)--(3.515,1.436)--(3.565,1.422)--(3.614,1.408)--(3.664,1.395)--(3.713,1.381)--(3.763,1.368)--(3.812,1.355)--(3.862,1.342)--(3.911,1.329)--(3.961,1.317)--(4.010,1.304)--(4.060,1.292)--(4.109,1.280)--(4.159,1.268)--(4.208,1.256)--(4.258,1.244)--(4.307,1.233)--(4.357,1.221)--(4.406,1.210)--(4.456,1.199)--(4.505,1.188)--(4.555,1.177)--(4.604,1.166)--(4.654,1.156)--(4.703,1.145)--(4.753,1.134)--(4.802,1.124)--(4.852,1.114)--(4.901,1.104)--(4.951,1.094)--(5.000,1.084);
+\draw [color=blue] (0.1000,4.9957)--(0.1494,4.5936)--(0.1989,4.3076)--(0.2484,4.0855)--(0.2979,3.9038)--(0.3474,3.7502)--(0.3969,3.6170)--(0.4464,3.4995)--(0.4959,3.3944)--(0.5454,3.2992)--(0.5949,3.2124)--(0.6444,3.1325)--(0.6939,3.0585)--(0.7434,2.9896)--(0.7929,2.9251)--(0.8424,2.8646)--(0.8919,2.8075)--(0.9414,2.7535)--(0.9909,2.7022)--(1.0404,2.6535)--(1.0898,2.6070)--(1.1393,2.5626)--(1.1888,2.5201)--(1.2383,2.4793)--(1.2878,2.4401)--(1.3373,2.4024)--(1.3868,2.3660)--(1.4363,2.3310)--(1.4858,2.2971)--(1.5353,2.2643)--(1.5848,2.2326)--(1.6343,2.2019)--(1.6838,2.1720)--(1.7333,2.1431)--(1.7828,2.1149)--(1.8323,2.0875)--(1.8818,2.0609)--(1.9313,2.0349)--(1.9808,2.0096)--(2.0303,1.9849)--(2.0797,1.9608)--(2.1292,1.9373)--(2.1787,1.9143)--(2.2282,1.8919)--(2.2777,1.8699)--(2.3272,1.8484)--(2.3767,1.8274)--(2.4262,1.8067)--(2.4757,1.7866)--(2.5252,1.7668)--(2.5747,1.7473)--(2.6242,1.7283)--(2.6737,1.7096)--(2.7232,1.6913)--(2.7727,1.6733)--(2.8222,1.6556)--(2.8717,1.6382)--(2.9212,1.6211)--(2.9707,1.6043)--(3.0202,1.5878)--(3.0696,1.5715)--(3.1191,1.5555)--(3.1686,1.5398)--(3.2181,1.5243)--(3.2676,1.5090)--(3.3171,1.4940)--(3.3666,1.4792)--(3.4161,1.4646)--(3.4656,1.4502)--(3.5151,1.4360)--(3.5646,1.4220)--(3.6141,1.4082)--(3.6636,1.3946)--(3.7131,1.3812)--(3.7626,1.3680)--(3.8121,1.3549)--(3.8616,1.3420)--(3.9111,1.3293)--(3.9606,1.3167)--(4.0101,1.3043)--(4.0595,1.2920)--(4.1090,1.2799)--(4.1585,1.2679)--(4.2080,1.2561)--(4.2575,1.2444)--(4.3070,1.2328)--(4.3565,1.2214)--(4.4060,1.2101)--(4.4555,1.1989)--(4.5050,1.1879)--(4.5545,1.1770)--(4.6040,1.1662)--(4.6535,1.1555)--(4.7030,1.1449)--(4.7525,1.1344)--(4.8020,1.1241)--(4.8515,1.1138)--(4.9010,1.1037)--(4.9505,1.0936)--(5.0000,1.0837);
 \draw []  (5.0000,1.0837) node [rotate=0] {$\bullet$};
-\draw (5.3918,1.2782) node {\( P_{ 0  }\)};
+\draw (5.3917,1.2781) node {\( P_{ 0  }\)};
 \draw []  (1.0837,1.0837) node [rotate=0] {$\bullet$};
-\draw (0.71885,1.3853) node {\( Q_{0}\)};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.30595) node {\( x_{ 0  }\)};
-\draw [color=cyan,style=dashed] (5.00,1.08) -- (1.08,1.08);
-\draw [color=red,style=dashed] (5.00,1.08) -- (5.00,0);
-\draw []  (1.0837,2.6128) node [rotate=0] {$\bullet$};
-\draw (1.3567,2.9496) node {\( P_{ 1  }\)};
-\draw []  (2.6128,2.6128) node [rotate=0] {$\bullet$};
-\draw (2.2479,2.9144) node {\( Q_{1}\)};
-\draw []  (1.0837,0) node [rotate=0] {$\bullet$};
-\draw (1.0837,-0.30595) node {\( x_{ 1  }\)};
-\draw [color=cyan,style=dashed] (1.08,2.61) -- (2.61,2.61);
-\draw [color=red,style=dashed] (1.08,2.61) -- (1.08,0);
-\draw []  (2.6128,1.7327) node [rotate=0] {$\bullet$};
-\draw (2.9758,1.9954) node {\( P_{ 2  }\)};
+\draw (0.7188,1.3853) node {\( Q_{0}\)};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.3059) node {\( x_{ 0  }\)};
+\draw [color=cyan,style=dashed] (5.0000,1.0837) -- (1.0837,1.0837);
+\draw [color=red,style=dashed] (5.0000,1.0837) -- (5.0000,0.0000);
+\draw []  (1.0837,2.6127) node [rotate=0] {$\bullet$};
+\draw (1.3566,2.9495) node {\( P_{ 1  }\)};
+\draw []  (2.6127,2.6127) node [rotate=0] {$\bullet$};
+\draw (2.2478,2.9143) node {\( Q_{1}\)};
+\draw []  (1.0837,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0837,-0.3059) node {\( x_{ 1  }\)};
+\draw [color=cyan,style=dashed] (1.0837,2.6127) -- (2.6127,2.6127);
+\draw [color=red,style=dashed] (1.0837,2.6127) -- (1.0837,0.0000);
+\draw []  (2.6127,1.7327) node [rotate=0] {$\bullet$};
+\draw (2.9757,1.9953) node {\( P_{ 2  }\)};
 \draw []  (1.7327,1.7327) node [rotate=0] {$\bullet$};
-\draw (1.3679,2.0344) node {\( Q_{2}\)};
-\draw []  (2.6128,0) node [rotate=0] {$\bullet$};
-\draw (2.6128,-0.30595) node {\( x_{ 2  }\)};
-\draw [color=cyan,style=dashed] (2.61,1.73) -- (1.73,1.73);
-\draw [color=red,style=dashed] (2.61,1.73) -- (2.61,0);
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw (1.3678,2.0343) node {\( Q_{2}\)};
+\draw []  (2.6127,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.6127,-0.3059) node {\( x_{ 2  }\)};
+\draw [color=cyan,style=dashed] (2.6127,1.7327) -- (1.7327,1.7327);
+\draw [color=red,style=dashed] (2.6127,1.7327) -- (2.6127,0.0000);
+
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks b/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks
index a903ec27e..923951dc0 100644
--- a/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks
+++ b/auto/pictures_tex/Fig_UGCFooQoCihh.pstricks
@@ -96,37 +96,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.2067);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (9.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.2067);
 %DEFAULT
-\draw []  (0,1.3534) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,1.3533) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,2.7067) node [rotate=0] {$\bullet$};
 \draw []  (2.0000,2.7067) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,1.3534) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (7.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (8.0000,0) node [rotate=0] {$\bullet$};
-\draw []  (9.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
+\draw []  (3.0000,1.3533) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks b/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks
index 55b304951..f5f510f2a 100644
--- a/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks
+++ b/auto/pictures_tex/Fig_UIEHooSlbzIJ.pstricks
@@ -116,52 +116,52 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (14.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (14.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
-\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,4.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.0000,2.4000) node [rotate=0] {$\bullet$};
 \draw []  (2.0000,1.4400) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0.86400) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,0.51840) node [rotate=0] {$\bullet$};
-\draw []  (5.0000,0.31104) node [rotate=0] {$\bullet$};
-\draw []  (6.0000,0.18662) node [rotate=0] {$\bullet$};
-\draw []  (7.0000,0.11197) node [rotate=0] {$\bullet$};
-\draw []  (8.0000,0.067185) node [rotate=0] {$\bullet$};
-\draw []  (9.0000,0.040311) node [rotate=0] {$\bullet$};
-\draw []  (10.000,0.024186) node [rotate=0] {$\bullet$};
-\draw []  (11.000,0.014512) node [rotate=0] {$\bullet$};
-\draw []  (12.000,0.0087071) node [rotate=0] {$\bullet$};
-\draw []  (13.000,0.0052243) node [rotate=0] {$\bullet$};
-\draw []  (14.000,0.0031346) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.000,-0.31492) node {$ 11 $};
-\draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.000,-0.31492) node {$ 12 $};
-\draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.000,-0.31492) node {$ 13 $};
-\draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (14.000,-0.31492) node {$ 14 $};
-\draw [] (14.0,-0.100) -- (14.0,0.100);
+\draw []  (3.0000,0.8640) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.5184) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,0.3110) node [rotate=0] {$\bullet$};
+\draw []  (6.0000,0.1866) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,0.1119) node [rotate=0] {$\bullet$};
+\draw []  (8.0000,0.0671) node [rotate=0] {$\bullet$};
+\draw []  (9.0000,0.0403) node [rotate=0] {$\bullet$};
+\draw []  (10.000,0.0241) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0.0145) node [rotate=0] {$\bullet$};
+\draw []  (12.000,0.0087) node [rotate=0] {$\bullet$};
+\draw []  (13.000,0.0052) node [rotate=0] {$\bullet$};
+\draw []  (14.000,0.0031) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (11.000,-0.3149) node {$ 11 $};
+\draw [] (11.000,-0.1000) -- (11.000,0.1000);
+\draw (12.000,-0.3149) node {$ 12 $};
+\draw [] (12.000,-0.1000) -- (12.000,0.1000);
+\draw (13.000,-0.3149) node {$ 13 $};
+\draw [] (13.000,-0.1000) -- (13.000,0.1000);
+\draw (14.000,-0.3149) node {$ 14 $};
+\draw [] (14.000,-0.1000) -- (14.000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks b/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks
index 003cc3578..0d2922eb9 100644
--- a/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks
+++ b/auto/pictures_tex/Fig_UMEBooVTMyfD.pstricks
@@ -100,35 +100,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
-\draw [color=blue] (0,6.0000)--(0.10101,4.9025)--(0.20202,4.0057)--(0.30303,3.2730)--(0.40404,2.6743)--(0.50505,2.1851)--(0.60606,1.7854)--(0.70707,1.4588)--(0.80808,1.1920)--(0.90909,0.97392)--(1.0101,0.79577)--(1.1111,0.65021)--(1.2121,0.53127)--(1.3131,0.43409)--(1.4141,0.35469)--(1.5152,0.28981)--(1.6162,0.23679)--(1.7172,0.19348)--(1.8182,0.15809)--(1.9192,0.12917)--(2.0202,0.10554)--(2.1212,0.086236)--(2.2222,0.070462)--(2.3232,0.057573)--(2.4242,0.047041)--(2.5253,0.038437)--(2.6263,0.031406)--(2.7273,0.025661)--(2.8283,0.020967)--(2.9293,0.017132)--(3.0303,0.013998)--(3.1313,0.011437)--(3.2323,0.0093452)--(3.3333,0.0076358)--(3.4343,0.0062390)--(3.5354,0.0050978)--(3.6364,0.0041653)--(3.7374,0.0034034)--(3.8384,0.0027808)--(3.9394,0.0022722)--(4.0404,0.0018565)--(4.1414,0.0015169)--(4.2424,0.0012394)--(4.3434,0.0010127)--(4.4444,0)--(4.5455,0)--(4.6465,0)--(4.7475,0)--(4.8485,0)--(4.9495,0)--(5.0505,0)--(5.1515,0)--(5.2525,0)--(5.3535,0)--(5.4545,0)--(5.5556,0)--(5.6566,0)--(5.7576,0)--(5.8586,0)--(5.9596,0)--(6.0606,0)--(6.1616,0)--(6.2626,0)--(6.3636,0)--(6.4646,0)--(6.5657,0)--(6.6667,0)--(6.7677,0)--(6.8687,0)--(6.9697,0)--(7.0707,0)--(7.1717,0)--(7.2727,0)--(7.3737,0)--(7.4747,0)--(7.5758,0)--(7.6768,0)--(7.7778,0)--(7.8788,0)--(7.9798,0)--(8.0808,0)--(8.1818,0)--(8.2828,0)--(8.3838,0)--(8.4848,0)--(8.5859,0)--(8.6869,0)--(8.7879,0)--(8.8889,0)--(8.9899,0)--(9.0909,0)--(9.1919,0)--(9.2929,0)--(9.3939,0)--(9.4949,0)--(9.5960,0)--(9.6970,0)--(9.7980,0)--(9.8990,0)--(10.000,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,6.0000) node {$ 2 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw [color=blue] (0.0000,6.0000)--(0.1010,4.9024)--(0.2020,4.0057)--(0.3030,3.2729)--(0.4040,2.6742)--(0.5050,2.1850)--(0.6060,1.7853)--(0.7070,1.4588)--(0.8080,1.1919)--(0.9090,0.9739)--(1.0101,0.7957)--(1.1111,0.6502)--(1.2121,0.5312)--(1.3131,0.4340)--(1.4141,0.3546)--(1.5151,0.2898)--(1.6161,0.2367)--(1.7171,0.1934)--(1.8181,0.1580)--(1.9191,0.1291)--(2.0202,0.1055)--(2.1212,0.0862)--(2.2222,0.0704)--(2.3232,0.0575)--(2.4242,0.0470)--(2.5252,0.0384)--(2.6262,0.0314)--(2.7272,0.0256)--(2.8282,0.0209)--(2.9292,0.0171)--(3.0303,0.0139)--(3.1313,0.0114)--(3.2323,0.0093)--(3.3333,0.0076)--(3.4343,0.0062)--(3.5353,0.0050)--(3.6363,0.0041)--(3.7373,0.0034)--(3.8383,0.0027)--(3.9393,0.0022)--(4.0404,0.0018)--(4.1414,0.0015)--(4.2424,0.0012)--(4.3434,0.0010)--(4.4444,0.0000)--(4.5454,0.0000)--(4.6464,0.0000)--(4.7474,0.0000)--(4.8484,0.0000)--(4.9494,0.0000)--(5.0505,0.0000)--(5.1515,0.0000)--(5.2525,0.0000)--(5.3535,0.0000)--(5.4545,0.0000)--(5.5555,0.0000)--(5.6565,0.0000)--(5.7575,0.0000)--(5.8585,0.0000)--(5.9595,0.0000)--(6.0606,0.0000)--(6.1616,0.0000)--(6.2626,0.0000)--(6.3636,0.0000)--(6.4646,0.0000)--(6.5656,0.0000)--(6.6666,0.0000)--(6.7676,0.0000)--(6.8686,0.0000)--(6.9696,0.0000)--(7.0707,0.0000)--(7.1717,0.0000)--(7.2727,0.0000)--(7.3737,0.0000)--(7.4747,0.0000)--(7.5757,0.0000)--(7.6767,0.0000)--(7.7777,0.0000)--(7.8787,0.0000)--(7.9797,0.0000)--(8.0808,0.0000)--(8.1818,0.0000)--(8.2828,0.0000)--(8.3838,0.0000)--(8.4848,0.0000)--(8.5858,0.0000)--(8.6868,0.0000)--(8.7878,0.0000)--(8.8888,0.0000)--(8.9898,0.0000)--(9.0909,0.0000)--(9.1919,0.0000)--(9.2929,0.0000)--(9.3939,0.0000)--(9.4949,0.0000)--(9.5959,0.0000)--(9.6969,0.0000)--(9.7979,0.0000)--(9.8989,0.0000)--(10.000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,6.0000) node {$ 2 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks b/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks
index 8310c82c8..46b815d49 100644
--- a/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks
+++ b/auto/pictures_tex/Fig_UNVooMsXxHa.pstricks
@@ -107,83 +107,83 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-3.00,-5.00) -- (-3.00,5.00);
-\draw [color=gray,style=solid] (-2.00,-5.00) -- (-2.00,5.00);
-\draw [color=gray,style=solid] (-1.00,-5.00) -- (-1.00,5.00);
-\draw [color=gray,style=solid] (0,-5.00) -- (0,5.00);
-\draw [color=gray,style=solid] (1.00,-5.00) -- (1.00,5.00);
-\draw [color=gray,style=solid] (2.00,-5.00) -- (2.00,5.00);
-\draw [color=gray,style=solid] (3.00,-5.00) -- (3.00,5.00);
-\draw [color=gray,style=dotted] (-2.50,-5.00) -- (-2.50,5.00);
-\draw [color=gray,style=dotted] (-1.50,-5.00) -- (-1.50,5.00);
-\draw [color=gray,style=dotted] (-0.500,-5.00) -- (-0.500,5.00);
-\draw [color=gray,style=dotted] (0.500,-5.00) -- (0.500,5.00);
-\draw [color=gray,style=dotted] (1.50,-5.00) -- (1.50,5.00);
-\draw [color=gray,style=dotted] (2.50,-5.00) -- (2.50,5.00);
-\draw [color=gray,style=dotted] (-3.00,-4.50) -- (3.00,-4.50);
-\draw [color=gray,style=dotted] (-3.00,-3.50) -- (3.00,-3.50);
-\draw [color=gray,style=dotted] (-3.00,-2.50) -- (3.00,-2.50);
-\draw [color=gray,style=dotted] (-3.00,-1.50) -- (3.00,-1.50);
-\draw [color=gray,style=dotted] (-3.00,-0.500) -- (3.00,-0.500);
-\draw [color=gray,style=dotted] (-3.00,0.500) -- (3.00,0.500);
-\draw [color=gray,style=dotted] (-3.00,1.50) -- (3.00,1.50);
-\draw [color=gray,style=dotted] (-3.00,2.50) -- (3.00,2.50);
-\draw [color=gray,style=dotted] (-3.00,3.50) -- (3.00,3.50);
-\draw [color=gray,style=dotted] (-3.00,4.50) -- (3.00,4.50);
-\draw [color=gray,style=solid] (-3.00,-5.00) -- (3.00,-5.00);
-\draw [color=gray,style=solid] (-3.00,-4.00) -- (3.00,-4.00);
-\draw [color=gray,style=solid] (-3.00,-3.00) -- (3.00,-3.00);
-\draw [color=gray,style=solid] (-3.00,-2.00) -- (3.00,-2.00);
-\draw [color=gray,style=solid] (-3.00,-1.00) -- (3.00,-1.00);
-\draw [color=gray,style=solid] (-3.00,0) -- (3.00,0);
-\draw [color=gray,style=solid] (-3.00,1.00) -- (3.00,1.00);
-\draw [color=gray,style=solid] (-3.00,2.00) -- (3.00,2.00);
-\draw [color=gray,style=solid] (-3.00,3.00) -- (3.00,3.00);
-\draw [color=gray,style=solid] (-3.00,4.00) -- (3.00,4.00);
-\draw [color=gray,style=solid] (-3.00,5.00) -- (3.00,5.00);
+\draw [color=gray,style=solid] (-3.0000,-5.0000) -- (-3.0000,5.0000);
+\draw [color=gray,style=solid] (-2.0000,-5.0000) -- (-2.0000,5.0000);
+\draw [color=gray,style=solid] (-1.0000,-5.0000) -- (-1.0000,5.0000);
+\draw [color=gray,style=solid] (0.0000,-5.0000) -- (0.0000,5.0000);
+\draw [color=gray,style=solid] (1.0000,-5.0000) -- (1.0000,5.0000);
+\draw [color=gray,style=solid] (2.0000,-5.0000) -- (2.0000,5.0000);
+\draw [color=gray,style=solid] (3.0000,-5.0000) -- (3.0000,5.0000);
+\draw [color=gray,style=dotted] (-2.5000,-5.0000) -- (-2.5000,5.0000);
+\draw [color=gray,style=dotted] (-1.5000,-5.0000) -- (-1.5000,5.0000);
+\draw [color=gray,style=dotted] (-0.5000,-5.0000) -- (-0.5000,5.0000);
+\draw [color=gray,style=dotted] (0.5000,-5.0000) -- (0.5000,5.0000);
+\draw [color=gray,style=dotted] (1.5000,-5.0000) -- (1.5000,5.0000);
+\draw [color=gray,style=dotted] (2.5000,-5.0000) -- (2.5000,5.0000);
+\draw [color=gray,style=dotted] (-3.0000,-4.5000) -- (3.0000,-4.5000);
+\draw [color=gray,style=dotted] (-3.0000,-3.5000) -- (3.0000,-3.5000);
+\draw [color=gray,style=dotted] (-3.0000,-2.5000) -- (3.0000,-2.5000);
+\draw [color=gray,style=dotted] (-3.0000,-1.5000) -- (3.0000,-1.5000);
+\draw [color=gray,style=dotted] (-3.0000,-0.5000) -- (3.0000,-0.5000);
+\draw [color=gray,style=dotted] (-3.0000,0.5000) -- (3.0000,0.5000);
+\draw [color=gray,style=dotted] (-3.0000,1.5000) -- (3.0000,1.5000);
+\draw [color=gray,style=dotted] (-3.0000,2.5000) -- (3.0000,2.5000);
+\draw [color=gray,style=dotted] (-3.0000,3.5000) -- (3.0000,3.5000);
+\draw [color=gray,style=dotted] (-3.0000,4.5000) -- (3.0000,4.5000);
+\draw [color=gray,style=solid] (-3.0000,-5.0000) -- (3.0000,-5.0000);
+\draw [color=gray,style=solid] (-3.0000,-4.0000) -- (3.0000,-4.0000);
+\draw [color=gray,style=solid] (-3.0000,-3.0000) -- (3.0000,-3.0000);
+\draw [color=gray,style=solid] (-3.0000,-2.0000) -- (3.0000,-2.0000);
+\draw [color=gray,style=solid] (-3.0000,-1.0000) -- (3.0000,-1.0000);
+\draw [color=gray,style=solid] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [color=gray,style=solid] (-3.0000,1.0000) -- (3.0000,1.0000);
+\draw [color=gray,style=solid] (-3.0000,2.0000) -- (3.0000,2.0000);
+\draw [color=gray,style=solid] (-3.0000,3.0000) -- (3.0000,3.0000);
+\draw [color=gray,style=solid] (-3.0000,4.0000) -- (3.0000,4.0000);
+\draw [color=gray,style=solid] (-3.0000,5.0000) -- (3.0000,5.0000);
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-5.5000) -- (0,5.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-5.5000) -- (0.0000,5.5000);
 %DEFAULT
 \draw (-3.4183,4.9138) node {\( y=cosh(x)\)};
 
-\draw [color=blue] (-2.275,4.914)--(-2.229,4.698)--(-2.183,4.492)--(-2.137,4.295)--(-2.091,4.108)--(-2.045,3.929)--(-1.999,3.758)--(-1.953,3.596)--(-1.907,3.441)--(-1.861,3.293)--(-1.815,3.152)--(-1.769,3.018)--(-1.723,2.891)--(-1.677,2.769)--(-1.631,2.653)--(-1.585,2.543)--(-1.539,2.438)--(-1.493,2.339)--(-1.448,2.244)--(-1.402,2.154)--(-1.356,2.068)--(-1.310,1.987)--(-1.264,1.911)--(-1.218,1.838)--(-1.172,1.769)--(-1.126,1.704)--(-1.080,1.642)--(-1.034,1.584)--(-0.9880,1.529)--(-0.9420,1.478)--(-0.8961,1.429)--(-0.8501,1.384)--(-0.8042,1.341)--(-0.7582,1.301)--(-0.7123,1.265)--(-0.6663,1.230)--(-0.6204,1.199)--(-0.5744,1.170)--(-0.5285,1.143)--(-0.4825,1.119)--(-0.4366,1.097)--(-0.3906,1.077)--(-0.3446,1.060)--(-0.2987,1.045)--(-0.2527,1.032)--(-0.2068,1.021)--(-0.1608,1.013)--(-0.1149,1.007)--(-0.06893,1.002)--(-0.02298,1.000)--(0.02298,1.000)--(0.06893,1.002)--(0.1149,1.007)--(0.1608,1.013)--(0.2068,1.021)--(0.2527,1.032)--(0.2987,1.045)--(0.3446,1.060)--(0.3906,1.077)--(0.4366,1.097)--(0.4825,1.119)--(0.5285,1.143)--(0.5744,1.170)--(0.6204,1.199)--(0.6663,1.230)--(0.7123,1.265)--(0.7582,1.301)--(0.8042,1.341)--(0.8501,1.384)--(0.8961,1.429)--(0.9420,1.478)--(0.9880,1.529)--(1.034,1.584)--(1.080,1.642)--(1.126,1.704)--(1.172,1.769)--(1.218,1.838)--(1.264,1.911)--(1.310,1.987)--(1.356,2.068)--(1.402,2.154)--(1.448,2.244)--(1.493,2.339)--(1.539,2.438)--(1.585,2.543)--(1.631,2.653)--(1.677,2.769)--(1.723,2.891)--(1.769,3.018)--(1.815,3.152)--(1.861,3.293)--(1.907,3.441)--(1.953,3.596)--(1.999,3.758)--(2.045,3.929)--(2.091,4.108)--(2.137,4.295)--(2.183,4.492)--(2.229,4.698)--(2.275,4.914);
-\draw (-3.4233,-4.8110) node {\( y=sinh(x)\)};
+\draw [color=blue] (-2.2746,4.9138)--(-2.2287,4.6978)--(-2.1827,4.4917)--(-2.1368,4.2952)--(-2.0908,4.1077)--(-2.0449,3.9289)--(-1.9989,3.7584)--(-1.9530,3.5958)--(-1.9070,3.4408)--(-1.8611,3.2931)--(-1.8151,3.1523)--(-1.7691,3.0182)--(-1.7232,2.8905)--(-1.6772,2.7689)--(-1.6313,2.6531)--(-1.5853,2.5430)--(-1.5394,2.4382)--(-1.4934,2.3385)--(-1.4475,2.2438)--(-1.4015,2.1538)--(-1.3556,2.0684)--(-1.3096,1.9874)--(-1.2637,1.9105)--(-1.2177,1.8377)--(-1.1718,1.7688)--(-1.1258,1.7036)--(-1.0798,1.6420)--(-1.0339,1.5838)--(-0.9879,1.5290)--(-0.9420,1.4775)--(-0.8960,1.4290)--(-0.8501,1.3836)--(-0.8041,1.3411)--(-0.7582,1.3014)--(-0.7122,1.2645)--(-0.6663,1.2303)--(-0.6203,1.1986)--(-0.5744,1.1695)--(-0.5284,1.1429)--(-0.4825,1.1186)--(-0.4365,1.0968)--(-0.3906,1.0772)--(-0.3446,1.0599)--(-0.2986,1.0449)--(-0.2527,1.0321)--(-0.2067,1.0214)--(-0.1608,1.0129)--(-0.1148,1.0066)--(-0.0689,1.0023)--(-0.0229,1.0002)--(0.0229,1.0002)--(0.0689,1.0023)--(0.1148,1.0066)--(0.1608,1.0129)--(0.2067,1.0214)--(0.2527,1.0321)--(0.2986,1.0449)--(0.3446,1.0599)--(0.3906,1.0772)--(0.4365,1.0968)--(0.4825,1.1186)--(0.5284,1.1429)--(0.5744,1.1695)--(0.6203,1.1986)--(0.6663,1.2303)--(0.7122,1.2645)--(0.7582,1.3014)--(0.8041,1.3411)--(0.8501,1.3836)--(0.8960,1.4290)--(0.9420,1.4775)--(0.9879,1.5290)--(1.0339,1.5838)--(1.0798,1.6420)--(1.1258,1.7036)--(1.1718,1.7688)--(1.2177,1.8377)--(1.2637,1.9105)--(1.3096,1.9874)--(1.3556,2.0684)--(1.4015,2.1538)--(1.4475,2.2438)--(1.4934,2.3385)--(1.5394,2.4382)--(1.5853,2.5430)--(1.6313,2.6531)--(1.6772,2.7689)--(1.7232,2.8905)--(1.7691,3.0182)--(1.8151,3.1523)--(1.8611,3.2931)--(1.9070,3.4408)--(1.9530,3.5958)--(1.9989,3.7584)--(2.0449,3.9289)--(2.0908,4.1077)--(2.1368,4.2952)--(2.1827,4.4917)--(2.2287,4.6978)--(2.2746,4.9138);
+\draw (-3.4233,-4.8109) node {\( y=sinh(x)\)};
 
-\draw [color=blue] (-2.275,-4.811)--(-2.229,-4.590)--(-2.183,-4.379)--(-2.137,-4.177)--(-2.091,-3.984)--(-2.045,-3.800)--(-1.999,-3.623)--(-1.953,-3.454)--(-1.907,-3.292)--(-1.861,-3.138)--(-1.815,-2.990)--(-1.769,-2.848)--(-1.723,-2.712)--(-1.677,-2.582)--(-1.631,-2.458)--(-1.585,-2.338)--(-1.539,-2.224)--(-1.493,-2.114)--(-1.448,-2.009)--(-1.402,-1.908)--(-1.356,-1.811)--(-1.310,-1.718)--(-1.264,-1.628)--(-1.218,-1.542)--(-1.172,-1.459)--(-1.126,-1.379)--(-1.080,-1.302)--(-1.034,-1.228)--(-0.9880,-1.157)--(-0.9420,-1.088)--(-0.8961,-1.021)--(-0.8501,-0.9563)--(-0.8042,-0.8937)--(-0.7582,-0.8330)--(-0.7123,-0.7740)--(-0.6663,-0.7167)--(-0.6204,-0.6609)--(-0.5744,-0.6065)--(-0.5285,-0.5534)--(-0.4825,-0.5014)--(-0.4366,-0.4506)--(-0.3906,-0.4006)--(-0.3446,-0.3515)--(-0.2987,-0.3032)--(-0.2527,-0.2554)--(-0.2068,-0.2083)--(-0.1608,-0.1615)--(-0.1149,-0.1151)--(-0.06893,-0.06898)--(-0.02298,-0.02298)--(0.02298,0.02298)--(0.06893,0.06898)--(0.1149,0.1151)--(0.1608,0.1615)--(0.2068,0.2083)--(0.2527,0.2554)--(0.2987,0.3032)--(0.3446,0.3515)--(0.3906,0.4006)--(0.4366,0.4506)--(0.4825,0.5014)--(0.5285,0.5534)--(0.5744,0.6065)--(0.6204,0.6609)--(0.6663,0.7167)--(0.7123,0.7740)--(0.7582,0.8330)--(0.8042,0.8937)--(0.8501,0.9563)--(0.8961,1.021)--(0.9420,1.088)--(0.9880,1.157)--(1.034,1.228)--(1.080,1.302)--(1.126,1.379)--(1.172,1.459)--(1.218,1.542)--(1.264,1.628)--(1.310,1.718)--(1.356,1.811)--(1.402,1.908)--(1.448,2.009)--(1.493,2.114)--(1.539,2.224)--(1.585,2.338)--(1.631,2.458)--(1.677,2.582)--(1.723,2.712)--(1.769,2.848)--(1.815,2.990)--(1.861,3.138)--(1.907,3.292)--(1.953,3.454)--(1.999,3.623)--(2.045,3.800)--(2.091,3.984)--(2.137,4.177)--(2.183,4.379)--(2.229,4.590)--(2.275,4.811);
+\draw [color=blue] (-2.2746,-4.8109)--(-2.2287,-4.5901)--(-2.1827,-4.3790)--(-2.1368,-4.1772)--(-2.0908,-3.9841)--(-2.0449,-3.7995)--(-1.9989,-3.6229)--(-1.9530,-3.4540)--(-1.9070,-3.2923)--(-1.8611,-3.1376)--(-1.8151,-2.9895)--(-1.7691,-2.8478)--(-1.7232,-2.7120)--(-1.6772,-2.5820)--(-1.6313,-2.4575)--(-1.5853,-2.3381)--(-1.5394,-2.2237)--(-1.4934,-2.1139)--(-1.4475,-2.0087)--(-1.4015,-1.9076)--(-1.3556,-1.8106)--(-1.3096,-1.7175)--(-1.2637,-1.6279)--(-1.2177,-1.5418)--(-1.1718,-1.4590)--(-1.1258,-1.3792)--(-1.0798,-1.3023)--(-1.0339,-1.2282)--(-0.9879,-1.1567)--(-0.9420,-1.0876)--(-0.8960,-1.0209)--(-0.8501,-0.9562)--(-0.8041,-0.8937)--(-0.7582,-0.8329)--(-0.7122,-0.7740)--(-0.6663,-0.7167)--(-0.6203,-0.6609)--(-0.5744,-0.6065)--(-0.5284,-0.5534)--(-0.4825,-0.5014)--(-0.4365,-0.4505)--(-0.3906,-0.4006)--(-0.3446,-0.3515)--(-0.2986,-0.3031)--(-0.2527,-0.2554)--(-0.2067,-0.2082)--(-0.1608,-0.1615)--(-0.1148,-0.1151)--(-0.0689,-0.0689)--(-0.0229,-0.0229)--(0.0229,0.0229)--(0.0689,0.0689)--(0.1148,0.1151)--(0.1608,0.1615)--(0.2067,0.2082)--(0.2527,0.2554)--(0.2986,0.3031)--(0.3446,0.3515)--(0.3906,0.4006)--(0.4365,0.4505)--(0.4825,0.5014)--(0.5284,0.5534)--(0.5744,0.6065)--(0.6203,0.6609)--(0.6663,0.7167)--(0.7122,0.7740)--(0.7582,0.8329)--(0.8041,0.8937)--(0.8501,0.9562)--(0.8960,1.0209)--(0.9420,1.0876)--(0.9879,1.1567)--(1.0339,1.2282)--(1.0798,1.3023)--(1.1258,1.3792)--(1.1718,1.4590)--(1.2177,1.5418)--(1.2637,1.6279)--(1.3096,1.7175)--(1.3556,1.8106)--(1.4015,1.9076)--(1.4475,2.0087)--(1.4934,2.1139)--(1.5394,2.2237)--(1.5853,2.3381)--(1.6313,2.4575)--(1.6772,2.5820)--(1.7232,2.7120)--(1.7691,2.8478)--(1.8151,2.9895)--(1.8611,3.1376)--(1.9070,3.2923)--(1.9530,3.4540)--(1.9989,3.6229)--(2.0449,3.7995)--(2.0908,3.9841)--(2.1368,4.1772)--(2.1827,4.3790)--(2.2287,4.5901)--(2.2746,4.8109);
 
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-5.0000) node {$ -5 $};
-\draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-5.0000) node {$ -5 $};
+\draw [] (-0.1000,-5.0000) -- (0.1000,-5.0000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks b/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks
index a266f3275..0a5257a4f 100644
--- a/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks
+++ b/auto/pictures_tex/Fig_UQZooGFLNEq.pstricks
@@ -108,37 +108,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-2.0708) -- (0,2.0708);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0707) -- (0.0000,2.0707);
 %DEFAULT
 
-\draw [color=blue] (-5.000,-1.373)--(-4.899,-1.369)--(-4.798,-1.365)--(-4.697,-1.361)--(-4.596,-1.357)--(-4.495,-1.352)--(-4.394,-1.347)--(-4.293,-1.342)--(-4.192,-1.337)--(-4.091,-1.331)--(-3.990,-1.325)--(-3.889,-1.319)--(-3.788,-1.313)--(-3.687,-1.306)--(-3.586,-1.299)--(-3.485,-1.291)--(-3.384,-1.283)--(-3.283,-1.275)--(-3.182,-1.266)--(-3.081,-1.257)--(-2.980,-1.247)--(-2.879,-1.236)--(-2.778,-1.225)--(-2.677,-1.213)--(-2.576,-1.200)--(-2.475,-1.187)--(-2.374,-1.172)--(-2.273,-1.156)--(-2.172,-1.139)--(-2.071,-1.121)--(-1.970,-1.101)--(-1.869,-1.079)--(-1.768,-1.056)--(-1.667,-1.030)--(-1.566,-1.002)--(-1.465,-0.9717)--(-1.364,-0.9380)--(-1.263,-0.9010)--(-1.162,-0.8600)--(-1.061,-0.8148)--(-0.9596,-0.7648)--(-0.8586,-0.7095)--(-0.7576,-0.6483)--(-0.6566,-0.5810)--(-0.5556,-0.5071)--(-0.4545,-0.4266)--(-0.3535,-0.3398)--(-0.2525,-0.2474)--(-0.1515,-0.1504)--(-0.05051,-0.05046)--(0.05051,0.05046)--(0.1515,0.1504)--(0.2525,0.2474)--(0.3535,0.3398)--(0.4545,0.4266)--(0.5556,0.5071)--(0.6566,0.5810)--(0.7576,0.6483)--(0.8586,0.7095)--(0.9596,0.7648)--(1.061,0.8148)--(1.162,0.8600)--(1.263,0.9010)--(1.364,0.9380)--(1.465,0.9717)--(1.566,1.002)--(1.667,1.030)--(1.768,1.056)--(1.869,1.079)--(1.970,1.101)--(2.071,1.121)--(2.172,1.139)--(2.273,1.156)--(2.374,1.172)--(2.475,1.187)--(2.576,1.200)--(2.677,1.213)--(2.778,1.225)--(2.879,1.236)--(2.980,1.247)--(3.081,1.257)--(3.182,1.266)--(3.283,1.275)--(3.384,1.283)--(3.485,1.291)--(3.586,1.299)--(3.687,1.306)--(3.788,1.313)--(3.889,1.319)--(3.990,1.325)--(4.091,1.331)--(4.192,1.337)--(4.293,1.342)--(4.394,1.347)--(4.495,1.352)--(4.596,1.357)--(4.697,1.361)--(4.798,1.365)--(4.899,1.369)--(5.000,1.373);
-\draw [color=red,style=dashed] (-5.00,1.57) -- (5.00,1.57);
-\draw [color=red,style=dashed] (-5.00,-1.57) -- (5.00,-1.57);
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
-\draw [] (-0.100,1.57) -- (0.100,1.57);
+\draw [color=blue] (-5.0000,-1.3734)--(-4.8989,-1.3694)--(-4.7979,-1.3653)--(-4.6969,-1.3610)--(-4.5959,-1.3565)--(-4.4949,-1.3518)--(-4.3939,-1.3470)--(-4.2929,-1.3419)--(-4.1919,-1.3366)--(-4.0909,-1.3310)--(-3.9898,-1.3252)--(-3.8888,-1.3191)--(-3.7878,-1.3126)--(-3.6868,-1.3059)--(-3.5858,-1.2988)--(-3.4848,-1.2913)--(-3.3838,-1.2834)--(-3.2828,-1.2751)--(-3.1818,-1.2662)--(-3.0808,-1.2569)--(-2.9797,-1.2470)--(-2.8787,-1.2364)--(-2.7777,-1.2252)--(-2.6767,-1.2132)--(-2.5757,-1.2004)--(-2.4747,-1.1867)--(-2.3737,-1.1720)--(-2.2727,-1.1562)--(-2.1717,-1.1392)--(-2.0707,-1.1209)--(-1.9696,-1.1010)--(-1.8686,-1.0794)--(-1.7676,-1.0559)--(-1.6666,-1.0303)--(-1.5656,-1.0023)--(-1.4646,-0.9717)--(-1.3636,-0.9380)--(-1.2626,-0.9009)--(-1.1616,-0.8600)--(-1.0606,-0.8148)--(-0.9595,-0.7647)--(-0.8585,-0.7094)--(-0.7575,-0.6483)--(-0.6565,-0.5809)--(-0.5555,-0.5070)--(-0.4545,-0.4266)--(-0.3535,-0.3398)--(-0.2525,-0.2473)--(-0.1515,-0.1503)--(-0.0505,-0.0504)--(0.0505,0.0504)--(0.1515,0.1503)--(0.2525,0.2473)--(0.3535,0.3398)--(0.4545,0.4266)--(0.5555,0.5070)--(0.6565,0.5809)--(0.7575,0.6483)--(0.8585,0.7094)--(0.9595,0.7647)--(1.0606,0.8148)--(1.1616,0.8600)--(1.2626,0.9009)--(1.3636,0.9380)--(1.4646,0.9717)--(1.5656,1.0023)--(1.6666,1.0303)--(1.7676,1.0559)--(1.8686,1.0794)--(1.9696,1.1010)--(2.0707,1.1209)--(2.1717,1.1392)--(2.2727,1.1562)--(2.3737,1.1720)--(2.4747,1.1867)--(2.5757,1.2004)--(2.6767,1.2132)--(2.7777,1.2252)--(2.8787,1.2364)--(2.9797,1.2470)--(3.0808,1.2569)--(3.1818,1.2662)--(3.2828,1.2751)--(3.3838,1.2834)--(3.4848,1.2913)--(3.5858,1.2988)--(3.6868,1.3059)--(3.7878,1.3126)--(3.8888,1.3191)--(3.9898,1.3252)--(4.0909,1.3310)--(4.1919,1.3366)--(4.2929,1.3419)--(4.3939,1.3470)--(4.4949,1.3518)--(4.5959,1.3565)--(4.6969,1.3610)--(4.7979,1.3653)--(4.8989,1.3694)--(5.0000,1.3734);
+\draw [color=red,style=dashed] (-5.0000,1.5707) -- (5.0000,1.5707);
+\draw [color=red,style=dashed] (-5.0000,-1.5707) -- (5.0000,-1.5707);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.5937,-1.5707) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-0.1000,-1.5707) -- (0.1000,-1.5707);
+\draw (-0.4518,1.5707) node {$ \frac{1}{2} \, \pi $};
+\draw [] (-0.1000,1.5707) -- (0.1000,1.5707);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks b/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks
index 71823fe52..2191d4e21 100644
--- a/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks
+++ b/auto/pictures_tex/Fig_UUNEooCNVOOs.pstricks
@@ -35,14 +35,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (0.0000,0.0000);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
@@ -75,16 +75,16 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (-1.88,1.32);
-\draw [color=red,->,>=latex] (0,0) -- (-0.11698,0.32139);
-\draw [color=green,->,>=latex] (0,0) -- (-0.88302,-0.32139);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (-1.8793,1.3159);
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (-0.1169,0.3213);
+\draw [color=green,->,>=latex] (0.0000,0.0000) -- (-0.8830,-0.3213);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks b/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks
index bfe4147ab..f8f3c74da 100644
--- a/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks
+++ b/auto/pictures_tex/Fig_UYJooCWjLgK.pstricks
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (3.46,2.00);
-\draw [] (-1.20,0) -- (13.2,0);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.19125,0.28813) node {\( 0\)};
+\draw [] (0.0000,0.0000) -- (3.4641,2.0000);
+\draw [] (-1.2000,0.0000) -- (13.200,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.1912,0.2881) node {\( 0\)};
 \draw []  (3.4641,2.0000) node [rotate=0] {$\bullet$};
-\draw (3.2681,2.2873) node {\( y\)};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.27858) node {\( x\)};
-\draw []  (12.000,0) node [rotate=0] {$\bullet$};
-\draw (12.000,-0.31406) node {\( xy\)};
+\draw (3.2680,2.2872) node {\( y\)};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.2785) node {\( x\)};
+\draw []  (12.000,0.0000) node [rotate=0] {$\bullet$};
+\draw (12.000,-0.3140) node {\( xy\)};
 \draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
 \draw (1.5408,1.2881) node {\( 1\)};
-\draw [style=dashed] (1.73,1.00) -- (6.00,0);
-\draw [style=dashed] (3.46,2.00) -- (12.0,0);
+\draw [style=dashed] (1.7320,1.0000) -- (6.0000,0.0000);
+\draw [style=dashed] (3.4641,2.0000) -- (12.000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks b/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks
index b8a022bef..265dfa4ea 100644
--- a/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks
+++ b/auto/pictures_tex/Fig_UZGooBzlYxr.pstricks
@@ -96,47 +96,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,4.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,4.5000);
 %DEFAULT
-\fill [color=lightgray] (2.83,1.00) -- (2.80,1.07) -- (2.78,1.14) -- (2.75,1.21) -- (2.72,1.27) -- (2.68,1.34) -- (2.65,1.41) -- (2.62,1.47) -- (2.58,1.53) -- (2.54,1.60) -- (2.50,1.66) -- (2.46,1.72) -- (2.41,1.78) -- (2.37,1.84) -- (2.32,1.90) -- (2.28,1.95) -- (2.23,2.01) -- (2.18,2.06) -- (2.13,2.12) -- (2.07,2.17) -- (2.02,2.22) -- (1.96,2.27) -- (1.91,2.31) -- (1.85,2.36) -- (1.79,2.41) -- (1.73,2.45) -- (1.67,2.49) -- (1.61,2.53) -- (1.55,2.57) -- (1.48,2.61) -- (1.42,2.64) -- (1.35,2.68) -- (1.29,2.71) -- (1.22,2.74) -- (1.15,2.77) -- (1.08,2.80) -- (1.01,2.82) -- (0.944,2.85) -- (0.873,2.87) -- (0.803,2.89) -- (0.731,2.91) -- (0.659,2.93) -- (0.587,2.94) -- (0.515,2.96) -- (0.442,2.97) -- (0.368,2.98) -- (0.295,2.99) -- (0.221,2.99) -- (0.148,3.00) -- (0.0740,3.00) -- (0,3.00) -- (2.00,1.00) -- (2.00,1.00) -- (2.83,1.00) --  cycle;
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 \draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.7217,0.73387) node {\( A\)};
+\draw (1.7217,0.7338) node {\( A\)};
 \draw []  (2.8284,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.1174,0.73387) node {\( B\)};
-\draw [] (-1.00,1.00) -- (4.00,1.00);
+\draw (3.1174,0.7338) node {\( B\)};
+\draw [] (-1.0000,1.0000) -- (4.0000,1.0000);
 
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-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-1.0000,4.0000)--(-0.9494,3.9494)--(-0.8989,3.8989)--(-0.8484,3.8484)--(-0.7979,3.7979)--(-0.7474,3.7474)--(-0.6969,3.6969)--(-0.6464,3.6464)--(-0.5959,3.5959)--(-0.5454,3.5454)--(-0.4949,3.4949)--(-0.4444,3.4444)--(-0.3939,3.3939)--(-0.3434,3.3434)--(-0.2929,3.2929)--(-0.2424,3.2424)--(-0.1919,3.1919)--(-0.1414,3.1414)--(-0.0909,3.0909)--(-0.0404,3.0404)--(0.0101,2.9898)--(0.0606,2.9393)--(0.1111,2.8888)--(0.1616,2.8383)--(0.2121,2.7878)--(0.2626,2.7373)--(0.3131,2.6868)--(0.3636,2.6363)--(0.4141,2.5858)--(0.4646,2.5353)--(0.5151,2.4848)--(0.5656,2.4343)--(0.6161,2.3838)--(0.6666,2.3333)--(0.7171,2.2828)--(0.7676,2.2323)--(0.8181,2.1818)--(0.8686,2.1313)--(0.9191,2.0808)--(0.9696,2.0303)--(1.0202,1.9797)--(1.0707,1.9292)--(1.1212,1.8787)--(1.1717,1.8282)--(1.2222,1.7777)--(1.2727,1.7272)--(1.3232,1.6767)--(1.3737,1.6262)--(1.4242,1.5757)--(1.4747,1.5252)--(1.5252,1.4747)--(1.5757,1.4242)--(1.6262,1.3737)--(1.6767,1.3232)--(1.7272,1.2727)--(1.7777,1.2222)--(1.8282,1.1717)--(1.8787,1.1212)--(1.9292,1.0707)--(1.9797,1.0202)--(2.0303,0.9696)--(2.0808,0.9191)--(2.1313,0.8686)--(2.1818,0.8181)--(2.2323,0.7676)--(2.2828,0.7171)--(2.3333,0.6666)--(2.3838,0.6161)--(2.4343,0.5656)--(2.4848,0.5151)--(2.5353,0.4646)--(2.5858,0.4141)--(2.6363,0.3636)--(2.6868,0.3131)--(2.7373,0.2626)--(2.7878,0.2121)--(2.8383,0.1616)--(2.8888,0.1111)--(2.9393,0.0606)--(2.9898,0.0101)--(3.0404,-0.0404)--(3.0909,-0.0909)--(3.1414,-0.1414)--(3.1919,-0.1919)--(3.2424,-0.2424)--(3.2929,-0.2929)--(3.3434,-0.3434)--(3.3939,-0.3939)--(3.4444,-0.4444)--(3.4949,-0.4949)--(3.5454,-0.5454)--(3.5959,-0.5959)--(3.6464,-0.6464)--(3.6969,-0.6969)--(3.7474,-0.7474)--(3.7979,-0.7979)--(3.8484,-0.8484)--(3.8989,-0.8989)--(3.9494,-0.9494)--(4.0000,-1.0000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UnSurxInt.pstricks b/auto/pictures_tex/Fig_UnSurxInt.pstricks
index 702ef7c25..0e671bfaf 100644
--- a/auto/pictures_tex/Fig_UnSurxInt.pstricks
+++ b/auto/pictures_tex/Fig_UnSurxInt.pstricks
@@ -87,13 +87,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.8333) -- (0,3.8333);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.8333) -- (0.0000,3.8333);
 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -101,11 +101,11 @@
 % setting the default values
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -113,35 +113,35 @@
 % setting the default values
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          hatchthickness=0.4pt}
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+\draw [color=blue] (1.0000,0.0000)--(1.0101,0.0000)--(1.0202,0.0000)--(1.0303,0.0000)--(1.0404,0.0000)--(1.0505,0.0000)--(1.0606,0.0000)--(1.0707,0.0000)--(1.0808,0.0000)--(1.0909,0.0000)--(1.1010,0.0000)--(1.1111,0.0000)--(1.1212,0.0000)--(1.1313,0.0000)--(1.1414,0.0000)--(1.1515,0.0000)--(1.1616,0.0000)--(1.1717,0.0000)--(1.1818,0.0000)--(1.1919,0.0000)--(1.2020,0.0000)--(1.2121,0.0000)--(1.2222,0.0000)--(1.2323,0.0000)--(1.2424,0.0000)--(1.2525,0.0000)--(1.2626,0.0000)--(1.2727,0.0000)--(1.2828,0.0000)--(1.2929,0.0000)--(1.3030,0.0000)--(1.3131,0.0000)--(1.3232,0.0000)--(1.3333,0.0000)--(1.3434,0.0000)--(1.3535,0.0000)--(1.3636,0.0000)--(1.3737,0.0000)--(1.3838,0.0000)--(1.3939,0.0000)--(1.4040,0.0000)--(1.4141,0.0000)--(1.4242,0.0000)--(1.4343,0.0000)--(1.4444,0.0000)--(1.4545,0.0000)--(1.4646,0.0000)--(1.4747,0.0000)--(1.4848,0.0000)--(1.4949,0.0000)--(1.5050,0.0000)--(1.5151,0.0000)--(1.5252,0.0000)--(1.5353,0.0000)--(1.5454,0.0000)--(1.5555,0.0000)--(1.5656,0.0000)--(1.5757,0.0000)--(1.5858,0.0000)--(1.5959,0.0000)--(1.6060,0.0000)--(1.6161,0.0000)--(1.6262,0.0000)--(1.6363,0.0000)--(1.6464,0.0000)--(1.6565,0.0000)--(1.6666,0.0000)--(1.6767,0.0000)--(1.6868,0.0000)--(1.6969,0.0000)--(1.7070,0.0000)--(1.7171,0.0000)--(1.7272,0.0000)--(1.7373,0.0000)--(1.7474,0.0000)--(1.7575,0.0000)--(1.7676,0.0000)--(1.7777,0.0000)--(1.7878,0.0000)--(1.7979,0.0000)--(1.8080,0.0000)--(1.8181,0.0000)--(1.8282,0.0000)--(1.8383,0.0000)--(1.8484,0.0000)--(1.8585,0.0000)--(1.8686,0.0000)--(1.8787,0.0000)--(1.8888,0.0000)--(1.8989,0.0000)--(1.9090,0.0000)--(1.9191,0.0000)--(1.9292,0.0000)--(1.9393,0.0000)--(1.9494,0.0000)--(1.9595,0.0000)--(1.9696,0.0000)--(1.9797,0.0000)--(1.9898,0.0000)--(2.0000,0.0000);
+\draw [color=brown,style=solid] (1.0000,0.0000) -- (1.0000,1.0000);
+\draw [color=brown,style=solid] (2.0000,0.5000) -- (2.0000,0.0000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_UneCellule.pstricks b/auto/pictures_tex/Fig_UneCellule.pstricks
index 5b9b823d9..9890cc16b 100644
--- a/auto/pictures_tex/Fig_UneCellule.pstricks
+++ b/auto/pictures_tex/Fig_UneCellule.pstricks
@@ -103,54 +103,54 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.41406) node {$a_1=y_{10}$};
-\draw [style=dotted] (1.00,0) -- (1.00,2.00);
-\draw [] (1.00,2.00) -- (1.00,5.00);
-\draw []  (2.2000,0) node [rotate=0] {$\bullet$};
-\draw (2.2000,-0.71406) node {$y_{11}$};
-\draw [style=dotted] (2.20,0) -- (2.20,2.00);
-\draw [] (2.20,2.00) -- (2.20,5.00);
-\draw []  (3.7000,0) node [rotate=0] {$\bullet$};
-\draw (3.7000,-0.41406) node {$y_{12}$};
-\draw [style=dotted] (3.70,0) -- (3.70,2.00);
-\draw [] (3.70,2.00) -- (3.70,5.00);
-\draw []  (4.2000,0) node [rotate=0] {$\bullet$};
-\draw (4.2000,-0.71406) node {$y_{13}$};
-\draw [style=dotted] (4.20,0) -- (4.20,2.00);
-\draw [] (4.20,2.00) -- (4.20,5.00);
-\draw []  (5.2000,0) node [rotate=0] {$\bullet$};
-\draw (5.2000,-0.41406) node {$y_{14}$};
-\draw [style=dotted] (5.20,0) -- (5.20,2.00);
-\draw [] (5.20,2.00) -- (5.20,5.00);
-\draw []  (6.2000,0) node [rotate=0] {$\bullet$};
-\draw (6.2000,-0.76222) node {$b_1=y_{15}$};
-\draw [style=dotted] (6.20,0) -- (6.20,2.00);
-\draw [] (6.20,2.00) -- (6.20,5.00);
-\draw []  (0,2.0000) node [rotate=0] {$\bullet$};
-\draw (-0.95841,2.0000) node {$a_2=y_{20}$};
-\draw [style=dotted] (0,2.00) -- (1.00,2.00);
-\draw [] (1.00,2.00) -- (6.20,2.00);
-\draw []  (0,2.5000) node [rotate=0] {$\bullet$};
-\draw (-0.53948,2.5000) node {$y_{21}$};
-\draw [style=dotted] (0,2.50) -- (1.00,2.50);
-\draw [] (1.00,2.50) -- (6.20,2.50);
-\draw []  (0,4.0000) node [rotate=0] {$\bullet$};
-\draw (-0.53948,4.0000) node {$y_{22}$};
-\draw [style=dotted] (0,4.00) -- (1.00,4.00);
-\draw [] (1.00,4.00) -- (6.20,4.00);
-\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
-\draw (-0.94026,5.0000) node {$b_2=y_{23}$};
-\draw [style=dotted] (0,5.00) -- (1.00,5.00);
-\draw [] (1.00,5.00) -- (6.20,5.00);
-\fill [color=lightgray] (4.20,4.00) -- (5.20,4.00) -- (5.20,4.00) -- (5.20,2.50) -- (5.20,2.50) -- (4.20,2.50) -- (4.20,2.50) -- (4.20,4.00) --  cycle;
-\draw [] (4.20,4.00) -- (5.20,4.00);
-\draw [] (5.20,4.00) -- (5.20,2.50);
-\draw [] (5.20,2.50) -- (4.20,2.50);
-\draw [] (4.20,2.50) -- (4.20,4.00);
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.4140) node {$a_1=y_{10}$};
+\draw [style=dotted] (1.0000,0.0000) -- (1.0000,2.0000);
+\draw [] (1.0000,2.0000) -- (1.0000,5.0000);
+\draw []  (2.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.2000,-0.7140) node {$y_{11}$};
+\draw [style=dotted] (2.2000,0.0000) -- (2.2000,2.0000);
+\draw [] (2.2000,2.0000) -- (2.2000,5.0000);
+\draw []  (3.7000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.7000,-0.4140) node {$y_{12}$};
+\draw [style=dotted] (3.7000,0.0000) -- (3.7000,2.0000);
+\draw [] (3.7000,2.0000) -- (3.7000,5.0000);
+\draw []  (4.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.2000,-0.7140) node {$y_{13}$};
+\draw [style=dotted] (4.2000,0.0000) -- (4.2000,2.0000);
+\draw [] (4.2000,2.0000) -- (4.2000,5.0000);
+\draw []  (5.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.2000,-0.4140) node {$y_{14}$};
+\draw [style=dotted] (5.2000,0.0000) -- (5.2000,2.0000);
+\draw [] (5.2000,2.0000) -- (5.2000,5.0000);
+\draw []  (6.2000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.2000,-0.7622) node {$b_1=y_{15}$};
+\draw [style=dotted] (6.2000,0.0000) -- (6.2000,2.0000);
+\draw [] (6.2000,2.0000) -- (6.2000,5.0000);
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9584,2.0000) node {$a_2=y_{20}$};
+\draw [style=dotted] (0.0000,2.0000) -- (1.0000,2.0000);
+\draw [] (1.0000,2.0000) -- (6.2000,2.0000);
+\draw []  (0.0000,2.5000) node [rotate=0] {$\bullet$};
+\draw (-0.5394,2.5000) node {$y_{21}$};
+\draw [style=dotted] (0.0000,2.5000) -- (1.0000,2.5000);
+\draw [] (1.0000,2.5000) -- (6.2000,2.5000);
+\draw []  (0.0000,4.0000) node [rotate=0] {$\bullet$};
+\draw (-0.5394,4.0000) node {$y_{22}$};
+\draw [style=dotted] (0.0000,4.0000) -- (1.0000,4.0000);
+\draw [] (1.0000,4.0000) -- (6.2000,4.0000);
+\draw []  (0.0000,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.9402,5.0000) node {$b_2=y_{23}$};
+\draw [style=dotted] (0.0000,5.0000) -- (1.0000,5.0000);
+\draw [] (1.0000,5.0000) -- (6.2000,5.0000);
+\fill [color=lightgray] (4.2000,4.0000) -- (5.2000,4.0000) -- (5.2000,4.0000) -- (5.2000,2.5000) -- (5.2000,2.5000) -- (4.2000,2.5000) -- (4.2000,2.5000) -- (4.2000,4.0000) --  cycle;
+\draw [] (4.2000,4.0000) -- (5.2000,4.0000);
+\draw [] (5.2000,4.0000) -- (5.2000,2.5000);
+\draw [] (5.2000,2.5000) -- (4.2000,2.5000);
+\draw [] (4.2000,2.5000) -- (4.2000,4.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_VANooZowSyO.pstricks b/auto/pictures_tex/Fig_VANooZowSyO.pstricks
index 653bbd4a8..88dd46939 100644
--- a/auto/pictures_tex/Fig_VANooZowSyO.pstricks
+++ b/auto/pictures_tex/Fig_VANooZowSyO.pstricks
@@ -57,23 +57,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.4995);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.4994);
 %DEFAULT
 
-\draw [color=blue] (-2.199,-1.000)--(-2.155,-0.9980)--(-2.110,-0.9920)--(-2.066,-0.9819)--(-2.021,-0.9679)--(-1.977,-0.9501)--(-1.933,-0.9284)--(-1.888,-0.9029)--(-1.844,-0.8738)--(-1.799,-0.8413)--(-1.755,-0.8053)--(-1.710,-0.7660)--(-1.666,-0.7237)--(-1.622,-0.6785)--(-1.577,-0.6306)--(-1.533,-0.5801)--(-1.488,-0.5272)--(-1.444,-0.4723)--(-1.399,-0.4154)--(-1.355,-0.3569)--(-1.311,-0.2969)--(-1.266,-0.2358)--(-1.222,-0.1736)--(-1.177,-0.1108)--(-1.133,-0.04758)--(-1.088,0.01587)--(-1.044,0.07925)--(-0.9996,0.1423)--(-0.9552,0.2048)--(-0.9107,0.2665)--(-0.8663,0.3271)--(-0.8219,0.3863)--(-0.7775,0.4441)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6442,0.6056)--(-0.5998,0.6549)--(-0.5553,0.7015)--(-0.5109,0.7453)--(-0.4665,0.7861)--(-0.4221,0.8237)--(-0.3776,0.8580)--(-0.3332,0.8888)--(-0.2888,0.9161)--(-0.2443,0.9397)--(-0.1999,0.9595)--(-0.1555,0.9754)--(-0.1111,0.9874)--(-0.06664,0.9955)--(-0.02221,0.9995)--(0.02221,0.9995)--(0.06664,0.9955)--(0.1111,0.9874)--(0.1555,0.9754)--(0.1999,0.9595)--(0.2443,0.9397)--(0.2888,0.9161)--(0.3332,0.8888)--(0.3776,0.8580)--(0.4221,0.8237)--(0.4665,0.7861)--(0.5109,0.7453)--(0.5553,0.7015)--(0.5998,0.6549)--(0.6442,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7775,0.4441)--(0.8219,0.3863)--(0.8663,0.3271)--(0.9107,0.2665)--(0.9552,0.2048)--(0.9996,0.1423)--(1.044,0.07925)--(1.088,0.01587)--(1.133,-0.04758)--(1.177,-0.1108)--(1.222,-0.1736)--(1.266,-0.2358)--(1.311,-0.2969)--(1.355,-0.3569)--(1.399,-0.4154)--(1.444,-0.4723)--(1.488,-0.5272)--(1.533,-0.5801)--(1.577,-0.6306)--(1.622,-0.6785)--(1.666,-0.7237)--(1.710,-0.7660)--(1.755,-0.8053)--(1.799,-0.8413)--(1.844,-0.8738)--(1.888,-0.9029)--(1.933,-0.9284)--(1.977,-0.9501)--(2.021,-0.9679)--(2.066,-0.9819)--(2.110,-0.9920)--(2.155,-0.9980)--(2.199,-1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,-1.0000)--(-2.1546,-0.9979)--(-2.1102,-0.9919)--(-2.0658,-0.9819)--(-2.0214,-0.9679)--(-1.9769,-0.9500)--(-1.9325,-0.9283)--(-1.8881,-0.9029)--(-1.8437,-0.8738)--(-1.7992,-0.8412)--(-1.7548,-0.8052)--(-1.7104,-0.7660)--(-1.6659,-0.7237)--(-1.6215,-0.6785)--(-1.5771,-0.6305)--(-1.5327,-0.5800)--(-1.4882,-0.5272)--(-1.4438,-0.4722)--(-1.3994,-0.4154)--(-1.3550,-0.3568)--(-1.3105,-0.2969)--(-1.2661,-0.2357)--(-1.2217,-0.1736)--(-1.1773,-0.1108)--(-1.1328,-0.0475)--(-1.0884,0.0158)--(-1.0440,0.0792)--(-0.9995,0.1423)--(-0.9551,0.2048)--(-0.9107,0.2664)--(-0.8663,0.3270)--(-0.8218,0.3863)--(-0.7774,0.4440)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6441,0.6056)--(-0.5997,0.6548)--(-0.5553,0.7014)--(-0.5109,0.7452)--(-0.4664,0.7860)--(-0.4220,0.8236)--(-0.3776,0.8579)--(-0.3331,0.8888)--(-0.2887,0.9161)--(-0.2443,0.9396)--(-0.1999,0.9594)--(-0.1554,0.9754)--(-0.1110,0.9874)--(-0.0666,0.9954)--(-0.0222,0.9994)--(0.0222,0.9994)--(0.0666,0.9954)--(0.1110,0.9874)--(0.1554,0.9754)--(0.1999,0.9594)--(0.2443,0.9396)--(0.2887,0.9161)--(0.3331,0.8888)--(0.3776,0.8579)--(0.4220,0.8236)--(0.4664,0.7860)--(0.5109,0.7452)--(0.5553,0.7014)--(0.5997,0.6548)--(0.6441,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7774,0.4440)--(0.8218,0.3863)--(0.8663,0.3270)--(0.9107,0.2664)--(0.9551,0.2048)--(0.9995,0.1423)--(1.0440,0.0792)--(1.0884,0.0158)--(1.1328,-0.0475)--(1.1773,-0.1108)--(1.2217,-0.1736)--(1.2661,-0.2357)--(1.3105,-0.2969)--(1.3550,-0.3568)--(1.3994,-0.4154)--(1.4438,-0.4722)--(1.4882,-0.5272)--(1.5327,-0.5800)--(1.5771,-0.6305)--(1.6215,-0.6785)--(1.6659,-0.7237)--(1.7104,-0.7660)--(1.7548,-0.8052)--(1.7992,-0.8412)--(1.8437,-0.8738)--(1.8881,-0.9029)--(1.9325,-0.9283)--(1.9769,-0.9500)--(2.0214,-0.9679)--(2.0658,-0.9819)--(2.1102,-0.9919)--(2.1546,-0.9979)--(2.1991,-1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -127,23 +127,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.4999) -- (0,1.4999);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4998) -- (0.0000,1.4998);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0)--(-2.155,0.06342)--(-2.110,0.1266)--(-2.066,0.1893)--(-2.021,0.2511)--(-1.977,0.3120)--(-1.933,0.3717)--(-1.888,0.4298)--(-1.844,0.4862)--(-1.799,0.5406)--(-1.755,0.5929)--(-1.710,0.6428)--(-1.666,0.6901)--(-1.622,0.7346)--(-1.577,0.7761)--(-1.533,0.8146)--(-1.488,0.8497)--(-1.444,0.8815)--(-1.399,0.9096)--(-1.355,0.9342)--(-1.311,0.9549)--(-1.266,0.9718)--(-1.222,0.9848)--(-1.177,0.9938)--(-1.133,0.9989)--(-1.088,0.9999)--(-1.044,0.9969)--(-0.9996,0.9898)--(-0.9552,0.9788)--(-0.9107,0.9638)--(-0.8663,0.9450)--(-0.8219,0.9224)--(-0.7775,0.8960)--(-0.7330,0.8660)--(-0.6886,0.8326)--(-0.6442,0.7958)--(-0.5998,0.7558)--(-0.5553,0.7127)--(-0.5109,0.6668)--(-0.4665,0.6182)--(-0.4221,0.5671)--(-0.3776,0.5137)--(-0.3332,0.4582)--(-0.2888,0.4009)--(-0.2443,0.3420)--(-0.1999,0.2817)--(-0.1555,0.2203)--(-0.1111,0.1580)--(-0.06664,0.09506)--(-0.02221,0.03173)--(0.02221,-0.03173)--(0.06664,-0.09506)--(0.1111,-0.1580)--(0.1555,-0.2203)--(0.1999,-0.2817)--(0.2443,-0.3420)--(0.2888,-0.4009)--(0.3332,-0.4582)--(0.3776,-0.5137)--(0.4221,-0.5671)--(0.4665,-0.6182)--(0.5109,-0.6668)--(0.5553,-0.7127)--(0.5998,-0.7558)--(0.6442,-0.7958)--(0.6886,-0.8326)--(0.7330,-0.8660)--(0.7775,-0.8960)--(0.8219,-0.9224)--(0.8663,-0.9450)--(0.9107,-0.9638)--(0.9552,-0.9788)--(0.9996,-0.9898)--(1.044,-0.9969)--(1.088,-0.9999)--(1.133,-0.9989)--(1.177,-0.9938)--(1.222,-0.9848)--(1.266,-0.9718)--(1.311,-0.9549)--(1.355,-0.9342)--(1.399,-0.9096)--(1.444,-0.8815)--(1.488,-0.8497)--(1.533,-0.8146)--(1.577,-0.7761)--(1.622,-0.7346)--(1.666,-0.6901)--(1.710,-0.6428)--(1.755,-0.5929)--(1.799,-0.5406)--(1.844,-0.4862)--(1.888,-0.4298)--(1.933,-0.3717)--(1.977,-0.3120)--(2.021,-0.2511)--(2.066,-0.1893)--(2.110,-0.1266)--(2.155,-0.06342)--(2.199,0);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,0.0000)--(-2.1546,0.0634)--(-2.1102,0.1265)--(-2.0658,0.1892)--(-2.0214,0.2511)--(-1.9769,0.3120)--(-1.9325,0.3716)--(-1.8881,0.4297)--(-1.8437,0.4861)--(-1.7992,0.5406)--(-1.7548,0.5929)--(-1.7104,0.6427)--(-1.6659,0.6900)--(-1.6215,0.7345)--(-1.5771,0.7761)--(-1.5327,0.8145)--(-1.4882,0.8497)--(-1.4438,0.8814)--(-1.3994,0.9096)--(-1.3550,0.9341)--(-1.3105,0.9549)--(-1.2661,0.9718)--(-1.2217,0.9848)--(-1.1773,0.9938)--(-1.1328,0.9988)--(-1.0884,0.9998)--(-1.0440,0.9968)--(-0.9995,0.9898)--(-0.9551,0.9788)--(-0.9107,0.9638)--(-0.8663,0.9450)--(-0.8218,0.9223)--(-0.7774,0.8959)--(-0.7330,0.8660)--(-0.6886,0.8325)--(-0.6441,0.7957)--(-0.5997,0.7557)--(-0.5553,0.7126)--(-0.5109,0.6667)--(-0.4664,0.6181)--(-0.4220,0.5670)--(-0.3776,0.5136)--(-0.3331,0.4582)--(-0.2887,0.4009)--(-0.2443,0.3420)--(-0.1999,0.2817)--(-0.1554,0.2203)--(-0.1110,0.1580)--(-0.0666,0.0950)--(-0.0222,0.0317)--(0.0222,-0.0317)--(0.0666,-0.0950)--(0.1110,-0.1580)--(0.1554,-0.2203)--(0.1999,-0.2817)--(0.2443,-0.3420)--(0.2887,-0.4009)--(0.3331,-0.4582)--(0.3776,-0.5136)--(0.4220,-0.5670)--(0.4664,-0.6181)--(0.5109,-0.6667)--(0.5553,-0.7126)--(0.5997,-0.7557)--(0.6441,-0.7957)--(0.6886,-0.8325)--(0.7330,-0.8660)--(0.7774,-0.8959)--(0.8218,-0.9223)--(0.8663,-0.9450)--(0.9107,-0.9638)--(0.9551,-0.9788)--(0.9995,-0.9898)--(1.0440,-0.9968)--(1.0884,-0.9998)--(1.1328,-0.9988)--(1.1773,-0.9938)--(1.2217,-0.9848)--(1.2661,-0.9718)--(1.3105,-0.9549)--(1.3550,-0.9341)--(1.3994,-0.9096)--(1.4438,-0.8814)--(1.4882,-0.8497)--(1.5327,-0.8145)--(1.5771,-0.7761)--(1.6215,-0.7345)--(1.6659,-0.6900)--(1.7104,-0.6427)--(1.7548,-0.5929)--(1.7992,-0.5406)--(1.8437,-0.4861)--(1.8881,-0.4297)--(1.9325,-0.3716)--(1.9769,-0.3120)--(2.0214,-0.2511)--(2.0658,-0.1892)--(2.1102,-0.1265)--(2.1546,-0.0634)--(2.1991,0.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -197,23 +197,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.4958) -- (0,1.4991);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4957) -- (0.0000,1.4990);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0.9991)--(-2.155,0.9989)--(-2.110,0.9988)--(-2.066,0.9986)--(-2.021,0.9985)--(-1.977,0.9982)--(-1.933,0.9980)--(-1.888,0.9977)--(-1.844,0.9974)--(-1.799,0.9971)--(-1.755,0.9967)--(-1.710,0.9962)--(-1.666,0.9957)--(-1.622,0.9951)--(-1.577,0.9945)--(-1.533,0.9937)--(-1.488,0.9929)--(-1.444,0.9919)--(-1.399,0.9908)--(-1.355,0.9896)--(-1.311,0.9882)--(-1.266,0.9866)--(-1.222,0.9848)--(-1.177,0.9827)--(-1.133,0.9804)--(-1.088,0.9778)--(-1.044,0.9748)--(-0.9996,0.9714)--(-0.9552,0.9675)--(-0.9107,0.9632)--(-0.8663,0.9582)--(-0.8219,0.9526)--(-0.7775,0.9463)--(-0.7330,0.9391)--(-0.6886,0.9309)--(-0.6442,0.9217)--(-0.5998,0.9112)--(-0.5553,0.8994)--(-0.5109,0.8861)--(-0.4665,0.8710)--(-0.4221,0.8540)--(-0.3776,0.8348)--(-0.3332,0.8131)--(-0.2888,0.7888)--(-0.2443,0.7614)--(-0.1999,0.7306)--(-0.1555,0.6961)--(-0.1111,0.6575)--(-0.06664,0.6144)--(-0.02221,0.5663)--(0.02221,0.5129)--(0.06664,0.4537)--(0.1111,0.3884)--(0.1555,0.3165)--(0.1999,0.2379)--(0.2443,0.1525)--(0.2888,0.06012)--(0.3332,-0.03882)--(0.3776,-0.1438)--(0.4221,-0.2539)--(0.4665,-0.3676)--(0.5109,-0.4829)--(0.5553,-0.5972)--(0.5998,-0.7067)--(0.6442,-0.8071)--(0.6886,-0.8928)--(0.7330,-0.9577)--(0.7775,-0.9945)--(0.8219,-0.9956)--(0.8663,-0.9536)--(0.9107,-0.8620)--(0.9552,-0.7163)--(0.9996,-0.5159)--(1.044,-0.2656)--(1.088,0.02236)--(1.133,0.3265)--(1.177,0.6156)--(1.222,0.8497)--(1.266,0.9838)--(1.311,0.9759)--(1.355,0.7986)--(1.399,0.4537)--(1.444,-0.01290)--(1.488,-0.5041)--(1.533,-0.8808)--(1.577,-0.9958)--(1.622,-0.7547)--(1.666,-0.1896)--(1.710,0.4945)--(1.755,0.9556)--(1.799,0.8754)--(1.844,0.2082)--(1.888,-0.6464)--(1.933,-0.9946)--(1.977,-0.4171)--(2.021,0.6239)--(2.066,0.9613)--(2.110,0.03819)--(2.155,-0.9628)--(2.199,-0.4089);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,0.9990)--(-2.1546,0.9989)--(-2.1102,0.9987)--(-2.0658,0.9986)--(-2.0214,0.9984)--(-1.9769,0.9982)--(-1.9325,0.9980)--(-1.8881,0.9977)--(-1.8437,0.9974)--(-1.7992,0.9970)--(-1.7548,0.9966)--(-1.7104,0.9962)--(-1.6659,0.9957)--(-1.6215,0.9951)--(-1.5771,0.9944)--(-1.5327,0.9937)--(-1.4882,0.9928)--(-1.4438,0.9919)--(-1.3994,0.9908)--(-1.3550,0.9896)--(-1.3105,0.9881)--(-1.2661,0.9866)--(-1.2217,0.9847)--(-1.1773,0.9827)--(-1.1328,0.9804)--(-1.0884,0.9777)--(-1.0440,0.9747)--(-0.9995,0.9713)--(-0.9551,0.9675)--(-0.9107,0.9631)--(-0.8663,0.9582)--(-0.8218,0.9526)--(-0.7774,0.9462)--(-0.7330,0.9390)--(-0.6886,0.9309)--(-0.6441,0.9216)--(-0.5997,0.9112)--(-0.5553,0.8994)--(-0.5109,0.8860)--(-0.4664,0.8710)--(-0.4220,0.8539)--(-0.3776,0.8347)--(-0.3331,0.8131)--(-0.2887,0.7887)--(-0.2443,0.7613)--(-0.1999,0.7306)--(-0.1554,0.6961)--(-0.1110,0.6575)--(-0.0666,0.6143)--(-0.0222,0.5663)--(0.0222,0.5128)--(0.0666,0.4537)--(0.1110,0.3883)--(0.1554,0.3165)--(0.1999,0.2379)--(0.2443,0.1524)--(0.2887,0.0601)--(0.3331,-0.0388)--(0.3776,-0.1437)--(0.4220,-0.2538)--(0.4664,-0.3675)--(0.5109,-0.4829)--(0.5553,-0.5971)--(0.5997,-0.7066)--(0.6441,-0.8070)--(0.6886,-0.8928)--(0.7330,-0.9576)--(0.7774,-0.9944)--(0.8218,-0.9956)--(0.8663,-0.9536)--(0.9107,-0.8619)--(0.9551,-0.7163)--(0.9995,-0.5158)--(1.0440,-0.2655)--(1.0884,0.0223)--(1.1328,0.3265)--(1.1773,0.6156)--(1.2217,0.8496)--(1.2661,0.9838)--(1.3105,0.9759)--(1.3550,0.7985)--(1.3994,0.4536)--(1.4438,-0.0129)--(1.4882,-0.5041)--(1.5327,-0.8808)--(1.5771,-0.9957)--(1.6215,-0.7546)--(1.6659,-0.1895)--(1.7104,0.4944)--(1.7548,0.9555)--(1.7992,0.8753)--(1.8437,0.2081)--(1.8881,-0.6463)--(1.9325,-0.9945)--(1.9769,-0.4171)--(2.0214,0.6238)--(2.0658,0.9612)--(2.1102,0.0381)--(2.1546,-0.9628)--(2.1991,-0.4089);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -271,25 +271,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4994);
 %DEFAULT
 
-\draw [color=blue] (-2.199,1.000)--(-2.155,1.002)--(-2.110,1.008)--(-2.066,1.018)--(-2.021,1.032)--(-1.977,1.050)--(-1.933,1.072)--(-1.888,1.097)--(-1.844,1.126)--(-1.799,1.159)--(-1.755,1.195)--(-1.710,1.234)--(-1.666,1.276)--(-1.622,1.321)--(-1.577,1.369)--(-1.533,1.420)--(-1.488,1.473)--(-1.444,1.528)--(-1.399,1.585)--(-1.355,1.643)--(-1.311,1.703)--(-1.266,1.764)--(-1.222,1.826)--(-1.177,1.889)--(-1.133,1.952)--(-1.088,2.016)--(-1.044,2.079)--(-0.9996,2.142)--(-0.9552,2.205)--(-0.9107,2.266)--(-0.8663,2.327)--(-0.8219,2.386)--(-0.7775,2.444)--(-0.7330,2.500)--(-0.6886,2.554)--(-0.6442,2.606)--(-0.5998,2.655)--(-0.5553,2.701)--(-0.5109,2.745)--(-0.4665,2.786)--(-0.4221,2.824)--(-0.3776,2.858)--(-0.3332,2.889)--(-0.2888,2.916)--(-0.2443,2.940)--(-0.1999,2.960)--(-0.1555,2.975)--(-0.1111,2.987)--(-0.06664,2.995)--(-0.02221,3.000)--(0.02221,3.000)--(0.06664,2.995)--(0.1111,2.987)--(0.1555,2.975)--(0.1999,2.960)--(0.2443,2.940)--(0.2888,2.916)--(0.3332,2.889)--(0.3776,2.858)--(0.4221,2.824)--(0.4665,2.786)--(0.5109,2.745)--(0.5553,2.701)--(0.5998,2.655)--(0.6442,2.606)--(0.6886,2.554)--(0.7330,2.500)--(0.7775,2.444)--(0.8219,2.386)--(0.8663,2.327)--(0.9107,2.266)--(0.9552,2.205)--(0.9996,2.142)--(1.044,2.079)--(1.088,2.016)--(1.133,1.952)--(1.177,1.889)--(1.222,1.826)--(1.266,1.764)--(1.311,1.703)--(1.355,1.643)--(1.399,1.585)--(1.444,1.528)--(1.488,1.473)--(1.533,1.420)--(1.577,1.369)--(1.622,1.321)--(1.666,1.276)--(1.710,1.234)--(1.755,1.195)--(1.799,1.159)--(1.844,1.126)--(1.888,1.097)--(1.933,1.072)--(1.977,1.050)--(2.021,1.032)--(2.066,1.018)--(2.110,1.008)--(2.155,1.002)--(2.199,1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.31058,3.1416) node {$ \pi $};
-\draw [] (-0.100,3.14) -- (0.100,3.14);
+\draw [color=blue] (-2.1991,1.0000)--(-2.1546,1.0020)--(-2.1102,1.0080)--(-2.0658,1.0180)--(-2.0214,1.0320)--(-1.9769,1.0499)--(-1.9325,1.0716)--(-1.8881,1.0970)--(-1.8437,1.1261)--(-1.7992,1.1587)--(-1.7548,1.1947)--(-1.7104,1.2339)--(-1.6659,1.2762)--(-1.6215,1.3214)--(-1.5771,1.3694)--(-1.5327,1.4199)--(-1.4882,1.4727)--(-1.4438,1.5277)--(-1.3994,1.5845)--(-1.3550,1.6431)--(-1.3105,1.7030)--(-1.2661,1.7642)--(-1.2217,1.8263)--(-1.1773,1.8891)--(-1.1328,1.9524)--(-1.0884,2.0158)--(-1.0440,2.0792)--(-0.9995,2.1423)--(-0.9551,2.2048)--(-0.9107,2.2664)--(-0.8663,2.3270)--(-0.8218,2.3863)--(-0.7774,2.4440)--(-0.7330,2.5000)--(-0.6886,2.5539)--(-0.6441,2.6056)--(-0.5997,2.6548)--(-0.5553,2.7014)--(-0.5109,2.7452)--(-0.4664,2.7860)--(-0.4220,2.8236)--(-0.3776,2.8579)--(-0.3331,2.8888)--(-0.2887,2.9161)--(-0.2443,2.9396)--(-0.1999,2.9594)--(-0.1554,2.9754)--(-0.1110,2.9874)--(-0.0666,2.9954)--(-0.0222,2.9994)--(0.0222,2.9994)--(0.0666,2.9954)--(0.1110,2.9874)--(0.1554,2.9754)--(0.1999,2.9594)--(0.2443,2.9396)--(0.2887,2.9161)--(0.3331,2.8888)--(0.3776,2.8579)--(0.4220,2.8236)--(0.4664,2.7860)--(0.5109,2.7452)--(0.5553,2.7014)--(0.5997,2.6548)--(0.6441,2.6056)--(0.6886,2.5539)--(0.7330,2.5000)--(0.7774,2.4440)--(0.8218,2.3863)--(0.8663,2.3270)--(0.9107,2.2664)--(0.9551,2.2048)--(0.9995,2.1423)--(1.0440,2.0792)--(1.0884,2.0158)--(1.1328,1.9524)--(1.1773,1.8891)--(1.2217,1.8263)--(1.2661,1.7642)--(1.3105,1.7030)--(1.3550,1.6431)--(1.3994,1.5845)--(1.4438,1.5277)--(1.4882,1.4727)--(1.5327,1.4199)--(1.5771,1.3694)--(1.6215,1.3214)--(1.6659,1.2762)--(1.7104,1.2339)--(1.7548,1.1947)--(1.7992,1.1587)--(1.8437,1.1261)--(1.8881,1.0970)--(1.9325,1.0716)--(1.9769,1.0499)--(2.0214,1.0320)--(2.0658,1.0180)--(2.1102,1.0080)--(2.1546,1.0020)--(2.1991,1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (-0.3105,3.1415) node {$ \pi $};
+\draw [] (-0.1000,3.1415) -- (0.1000,3.1415);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -343,23 +343,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.4995) -- (0,1.5000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4994) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,1.000)--(-2.155,0.9679)--(-2.110,0.8738)--(-2.066,0.7237)--(-2.021,0.5272)--(-1.977,0.2969)--(-1.933,0.04758)--(-1.888,-0.2048)--(-1.844,-0.4441)--(-1.799,-0.6549)--(-1.755,-0.8237)--(-1.710,-0.9397)--(-1.666,-0.9955)--(-1.622,-0.9874)--(-1.577,-0.9161)--(-1.533,-0.7861)--(-1.488,-0.6056)--(-1.444,-0.3863)--(-1.399,-0.1423)--(-1.355,0.1108)--(-1.311,0.3569)--(-1.266,0.5801)--(-1.222,0.7660)--(-1.177,0.9029)--(-1.133,0.9819)--(-1.088,0.9980)--(-1.044,0.9501)--(-0.9996,0.8413)--(-0.9552,0.6785)--(-0.9107,0.4723)--(-0.8663,0.2358)--(-0.8219,-0.01587)--(-0.7775,-0.2665)--(-0.7330,-0.5000)--(-0.6886,-0.7015)--(-0.6442,-0.8580)--(-0.5998,-0.9595)--(-0.5553,-0.9995)--(-0.5109,-0.9754)--(-0.4665,-0.8888)--(-0.4221,-0.7453)--(-0.3776,-0.5539)--(-0.3332,-0.3271)--(-0.2888,-0.07925)--(-0.2443,0.1736)--(-0.1999,0.4154)--(-0.1555,0.6306)--(-0.1111,0.8053)--(-0.06664,0.9284)--(-0.02221,0.9920)--(0.02221,0.9920)--(0.06664,0.9284)--(0.1111,0.8053)--(0.1555,0.6306)--(0.1999,0.4154)--(0.2443,0.1736)--(0.2888,-0.07925)--(0.3332,-0.3271)--(0.3776,-0.5539)--(0.4221,-0.7453)--(0.4665,-0.8888)--(0.5109,-0.9754)--(0.5553,-0.9995)--(0.5998,-0.9595)--(0.6442,-0.8580)--(0.6886,-0.7015)--(0.7330,-0.5000)--(0.7775,-0.2665)--(0.8219,-0.01587)--(0.8663,0.2358)--(0.9107,0.4723)--(0.9552,0.6785)--(0.9996,0.8413)--(1.044,0.9501)--(1.088,0.9980)--(1.133,0.9819)--(1.177,0.9029)--(1.222,0.7660)--(1.266,0.5801)--(1.311,0.3569)--(1.355,0.1108)--(1.399,-0.1423)--(1.444,-0.3863)--(1.488,-0.6056)--(1.533,-0.7861)--(1.577,-0.9161)--(1.622,-0.9874)--(1.666,-0.9955)--(1.710,-0.9397)--(1.755,-0.8237)--(1.799,-0.6549)--(1.844,-0.4441)--(1.888,-0.2048)--(1.933,0.04758)--(1.977,0.2969)--(2.021,0.5272)--(2.066,0.7237)--(2.110,0.8738)--(2.155,0.9679)--(2.199,1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,1.0000)--(-2.1546,0.9679)--(-2.1102,0.8738)--(-2.0658,0.7237)--(-2.0214,0.5272)--(-1.9769,0.2969)--(-1.9325,0.0475)--(-1.8881,-0.2048)--(-1.8437,-0.4440)--(-1.7992,-0.6548)--(-1.7548,-0.8236)--(-1.7104,-0.9396)--(-1.6659,-0.9954)--(-1.6215,-0.9874)--(-1.5771,-0.9161)--(-1.5327,-0.7860)--(-1.4882,-0.6056)--(-1.4438,-0.3863)--(-1.3994,-0.1423)--(-1.3550,0.1108)--(-1.3105,0.3568)--(-1.2661,0.5800)--(-1.2217,0.7660)--(-1.1773,0.9029)--(-1.1328,0.9819)--(-1.0884,0.9979)--(-1.0440,0.9500)--(-0.9995,0.8412)--(-0.9551,0.6785)--(-0.9107,0.4722)--(-0.8663,0.2357)--(-0.8218,-0.0158)--(-0.7774,-0.2664)--(-0.7330,-0.5000)--(-0.6886,-0.7014)--(-0.6441,-0.8579)--(-0.5997,-0.9594)--(-0.5553,-0.9994)--(-0.5109,-0.9754)--(-0.4664,-0.8888)--(-0.4220,-0.7452)--(-0.3776,-0.5539)--(-0.3331,-0.3270)--(-0.2887,-0.0792)--(-0.2443,0.1736)--(-0.1999,0.4154)--(-0.1554,0.6305)--(-0.1110,0.8052)--(-0.0666,0.9283)--(-0.0222,0.9919)--(0.0222,0.9919)--(0.0666,0.9283)--(0.1110,0.8052)--(0.1554,0.6305)--(0.1999,0.4154)--(0.2443,0.1736)--(0.2887,-0.0792)--(0.3331,-0.3270)--(0.3776,-0.5539)--(0.4220,-0.7452)--(0.4664,-0.8888)--(0.5109,-0.9754)--(0.5553,-0.9994)--(0.5997,-0.9594)--(0.6441,-0.8579)--(0.6886,-0.7014)--(0.7330,-0.4999)--(0.7774,-0.2664)--(0.8218,-0.0158)--(0.8663,0.2357)--(0.9107,0.4722)--(0.9551,0.6785)--(0.9995,0.8412)--(1.0440,0.9500)--(1.0884,0.9979)--(1.1328,0.9819)--(1.1773,0.9029)--(1.2217,0.7660)--(1.2661,0.5800)--(1.3105,0.3568)--(1.3550,0.1108)--(1.3994,-0.1423)--(1.4438,-0.3863)--(1.4882,-0.6056)--(1.5327,-0.7860)--(1.5771,-0.9161)--(1.6215,-0.9874)--(1.6659,-0.9954)--(1.7104,-0.9396)--(1.7548,-0.8236)--(1.7992,-0.6548)--(1.8437,-0.4440)--(1.8881,-0.2048)--(1.9325,0.0475)--(1.9769,0.2969)--(2.0214,0.5272)--(2.0658,0.7237)--(2.1102,0.8738)--(2.1546,0.9679)--(2.1991,1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -413,23 +413,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -475,19 +475,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5996,0) -- (1.5996,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4999);
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+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4999);
 %DEFAULT
 
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+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks b/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks
index 1b0b6770c..699fe6f83 100644
--- a/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks
+++ b/auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks
@@ -91,31 +91,31 @@
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 %AXES
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 %DEFAULT
 
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+\draw (-1.2000,-0.3298) node {$ -2 $};
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+\draw [] (-0.6000,-0.1000) -- (-0.6000,0.1000);
+\draw (0.6000,-0.3149) node {$ 1 $};
+\draw [] (0.6000,-0.1000) -- (0.6000,0.1000);
+\draw (1.2000,-0.3149) node {$ 2 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (-0.4331,-2.4000) node {$ -4 $};
+\draw [] (-0.1000,-2.4000) -- (0.1000,-2.4000);
+\draw (-0.4331,-1.2000) node {$ -2 $};
+\draw [] (-0.1000,-1.2000) -- (0.1000,-1.2000);
+\draw (-0.2912,1.2000) node {$ 2 $};
+\draw [] (-0.1000,1.2000) -- (0.1000,1.2000);
+\draw (-0.2912,2.4000) node {$ 4 $};
+\draw [] (-0.1000,2.4000) -- (0.1000,2.4000);
+\draw (-0.2912,3.6000) node {$ 6 $};
+\draw [] (-0.1000,3.6000) -- (0.1000,3.6000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks b/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks
index 34b8ef2f9..7c55e9baa 100644
--- a/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks
+++ b/auto/pictures_tex/Fig_VDFMooHMmFZr.pstricks
@@ -65,15 +65,15 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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+\draw [,->,>=latex] (1.2468,0.6811) -- (1.2454,0.6910);
+\draw [,->,>=latex] (0.3575,1.2187) -- (0.3480,1.2158);
+\draw [,->,>=latex] (-0.2777,0.5083) -- (-0.2808,0.4988);
+\draw [,->,>=latex] (-0.7003,0.2054) -- (-0.7094,0.2013);
+\draw [,->,>=latex] (-1.2468,-0.6811) -- (-1.2454,-0.6910);
+\draw [,->,>=latex] (-0.3575,-1.2187) -- (-0.3480,-1.2158);
+\draw [,->,>=latex] (0.2777,-0.5083) -- (0.2808,-0.4988);
+\draw [,->,>=latex] (0.7003,-0.2054) -- (0.7094,-0.2013);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks b/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks
index d8e38e460..abfb5217b 100644
--- a/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks
+++ b/auto/pictures_tex/Fig_VNBGooSqMsGU.pstricks
@@ -71,8 +71,8 @@
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 %PSTRICKS CODE
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-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,19 +81,19 @@
 % setting the default values
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+\draw [] (0.0000,0.0000) -- (0.0000,2.0000);
+\draw [color=red,style=solid] (1.0000,2.0000) -- (1.0000,0.0000);
 
-\draw [color=red] (0,2.000)--(0.01010,2.000)--(0.02020,2.000)--(0.03030,2.000)--(0.04040,2.000)--(0.05051,2.000)--(0.06061,2.000)--(0.07071,2.000)--(0.08081,2.000)--(0.09091,2.000)--(0.1010,2.000)--(0.1111,2.000)--(0.1212,2.000)--(0.1313,2.000)--(0.1414,2.000)--(0.1515,2.000)--(0.1616,2.000)--(0.1717,2.000)--(0.1818,2.000)--(0.1919,2.000)--(0.2020,2.000)--(0.2121,2.000)--(0.2222,2.000)--(0.2323,2.000)--(0.2424,2.000)--(0.2525,2.000)--(0.2626,2.000)--(0.2727,2.000)--(0.2828,2.000)--(0.2929,2.000)--(0.3030,2.000)--(0.3131,2.000)--(0.3232,2.000)--(0.3333,2.000)--(0.3434,2.000)--(0.3535,2.000)--(0.3636,2.000)--(0.3737,2.000)--(0.3838,2.000)--(0.3939,2.000)--(0.4040,2.000)--(0.4141,2.000)--(0.4242,2.000)--(0.4343,2.000)--(0.4444,2.000)--(0.4545,2.000)--(0.4646,2.000)--(0.4747,2.000)--(0.4848,2.000)--(0.4949,2.000)--(0.5051,2.000)--(0.5152,2.000)--(0.5253,2.000)--(0.5354,2.000)--(0.5455,2.000)--(0.5556,2.000)--(0.5657,2.000)--(0.5758,2.000)--(0.5859,2.000)--(0.5960,2.000)--(0.6061,2.000)--(0.6162,2.000)--(0.6263,2.000)--(0.6364,2.000)--(0.6465,2.000)--(0.6566,2.000)--(0.6667,2.000)--(0.6768,2.000)--(0.6869,2.000)--(0.6970,2.000)--(0.7071,2.000)--(0.7172,2.000)--(0.7273,2.000)--(0.7374,2.000)--(0.7475,2.000)--(0.7576,2.000)--(0.7677,2.000)--(0.7778,2.000)--(0.7879,2.000)--(0.7980,2.000)--(0.8081,2.000)--(0.8182,2.000)--(0.8283,2.000)--(0.8384,2.000)--(0.8485,2.000)--(0.8586,2.000)--(0.8687,2.000)--(0.8788,2.000)--(0.8889,2.000)--(0.8990,2.000)--(0.9091,2.000)--(0.9192,2.000)--(0.9293,2.000)--(0.9394,2.000)--(0.9495,2.000)--(0.9596,2.000)--(0.9697,2.000)--(0.9798,2.000)--(0.9899,2.000)--(1.000,2.000);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=red] (0.0000,2.0000)--(0.0101,2.0000)--(0.0202,2.0000)--(0.0303,2.0000)--(0.0404,2.0000)--(0.0505,2.0000)--(0.0606,2.0000)--(0.0707,2.0000)--(0.0808,2.0000)--(0.0909,2.0000)--(0.1010,2.0000)--(0.1111,2.0000)--(0.1212,2.0000)--(0.1313,2.0000)--(0.1414,2.0000)--(0.1515,2.0000)--(0.1616,2.0000)--(0.1717,2.0000)--(0.1818,2.0000)--(0.1919,2.0000)--(0.2020,2.0000)--(0.2121,2.0000)--(0.2222,2.0000)--(0.2323,2.0000)--(0.2424,2.0000)--(0.2525,2.0000)--(0.2626,2.0000)--(0.2727,2.0000)--(0.2828,2.0000)--(0.2929,2.0000)--(0.3030,2.0000)--(0.3131,2.0000)--(0.3232,2.0000)--(0.3333,2.0000)--(0.3434,2.0000)--(0.3535,2.0000)--(0.3636,2.0000)--(0.3737,2.0000)--(0.3838,2.0000)--(0.3939,2.0000)--(0.4040,2.0000)--(0.4141,2.0000)--(0.4242,2.0000)--(0.4343,2.0000)--(0.4444,2.0000)--(0.4545,2.0000)--(0.4646,2.0000)--(0.4747,2.0000)--(0.4848,2.0000)--(0.4949,2.0000)--(0.5050,2.0000)--(0.5151,2.0000)--(0.5252,2.0000)--(0.5353,2.0000)--(0.5454,2.0000)--(0.5555,2.0000)--(0.5656,2.0000)--(0.5757,2.0000)--(0.5858,2.0000)--(0.5959,2.0000)--(0.6060,2.0000)--(0.6161,2.0000)--(0.6262,2.0000)--(0.6363,2.0000)--(0.6464,2.0000)--(0.6565,2.0000)--(0.6666,2.0000)--(0.6767,2.0000)--(0.6868,2.0000)--(0.6969,2.0000)--(0.7070,2.0000)--(0.7171,2.0000)--(0.7272,2.0000)--(0.7373,2.0000)--(0.7474,2.0000)--(0.7575,2.0000)--(0.7676,2.0000)--(0.7777,2.0000)--(0.7878,2.0000)--(0.7979,2.0000)--(0.8080,2.0000)--(0.8181,2.0000)--(0.8282,2.0000)--(0.8383,2.0000)--(0.8484,2.0000)--(0.8585,2.0000)--(0.8686,2.0000)--(0.8787,2.0000)--(0.8888,2.0000)--(0.8989,2.0000)--(0.9090,2.0000)--(0.9191,2.0000)--(0.9292,2.0000)--(0.9393,2.0000)--(0.9494,2.0000)--(0.9595,2.0000)--(0.9696,2.0000)--(0.9797,2.0000)--(0.9898,2.0000)--(1.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks b/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks
index accdd4394..71befdfe6 100644
--- a/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks
+++ b/auto/pictures_tex/Fig_VSZRooRWgUGu.pstricks
@@ -79,11 +79,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.4000);
 %DEFAULT
 
-\draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
+\draw [color=red] (1.5000,1.5000)--(1.5606,1.6165)--(1.6212,1.7242)--(1.6818,1.8243)--(1.7424,1.9173)--(1.8030,2.0042)--(1.8636,2.0853)--(1.9242,2.1614)--(1.9848,2.2328)--(2.0454,2.3000)--(2.1060,2.3633)--(2.1666,2.4230)--(2.2272,2.4795)--(2.2878,2.5331)--(2.3484,2.5838)--(2.4090,2.6320)--(2.4696,2.6779)--(2.5303,2.7215)--(2.5909,2.7631)--(2.6515,2.8028)--(2.7121,2.8407)--(2.7727,2.8770)--(2.8333,2.9117)--(2.8939,2.9450)--(2.9545,2.9769)--(3.0151,3.0075)--(3.0757,3.0369)--(3.1363,3.0652)--(3.1969,3.0924)--(3.2575,3.1186)--(3.3181,3.1438)--(3.3787,3.1681)--(3.4393,3.1916)--(3.5000,3.2142)--(3.5606,3.2361)--(3.6212,3.2573)--(3.6818,3.2777)--(3.7424,3.2975)--(3.8030,3.3167)--(3.8636,3.3352)--(3.9242,3.3532)--(3.9848,3.3707)--(4.0454,3.3876)--(4.1060,3.4040)--(4.1666,3.4200)--(4.2272,3.4354)--(4.2878,3.4505)--(4.3484,3.4651)--(4.4090,3.4793)--(4.4696,3.4932)--(4.5303,3.5066)--(4.5909,3.5198)--(4.6515,3.5325)--(4.7121,3.5450)--(4.7727,3.5571)--(4.8333,3.5689)--(4.8939,3.5804)--(4.9545,3.5917)--(5.0151,3.6027)--(5.0757,3.6134)--(5.1363,3.6238)--(5.1969,3.6341)--(5.2575,3.6440)--(5.3181,3.6538)--(5.3787,3.6633)--(5.4393,3.6727)--(5.5000,3.6818)--(5.5606,3.6907)--(5.6212,3.6994)--(5.6818,3.7080)--(5.7424,3.7163)--(5.8030,3.7245)--(5.8636,3.7325)--(5.9242,3.7404)--(5.9848,3.7481)--(6.0454,3.7556)--(6.1060,3.7630)--(6.1666,3.7702)--(6.2272,3.7773)--(6.2878,3.7843)--(6.3484,3.7911)--(6.4090,3.7978)--(6.4696,3.8044)--(6.5303,3.8109)--(6.5909,3.8172)--(6.6515,3.8234)--(6.7121,3.8295)--(6.7727,3.8355)--(6.8333,3.8414)--(6.8939,3.8472)--(6.9545,3.8529)--(7.0151,3.8585)--(7.0757,3.8640)--(7.1363,3.8694)--(7.1969,3.8747)--(7.2575,3.8799)--(7.3181,3.8850)--(7.3787,3.8901)--(7.4393,3.8951)--(7.5000,3.9000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -91,15 +91,15 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw (6.0000,-0.3785) node {$x$};
 \draw (8.3552,3.9000) node {$f(x)$};
-\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (9.6701,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -107,11 +107,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=blue] (3.0000,3.0000)--(3.0303,3.0150)--(3.0606,3.0297)--(3.0909,3.0441)--(3.1212,3.0582)--(3.1515,3.0721)--(3.1818,3.0857)--(3.2121,3.0990)--(3.2424,3.1121)--(3.2727,3.1250)--(3.3030,3.1376)--(3.3333,3.1500)--(3.3636,3.1621)--(3.3939,3.1741)--(3.4242,3.1858)--(3.4545,3.1973)--(3.4848,3.2086)--(3.5151,3.2198)--(3.5454,3.2307)--(3.5757,3.2415)--(3.6060,3.2521)--(3.6363,3.2625)--(3.6666,3.2727)--(3.6969,3.2827)--(3.7272,3.2926)--(3.7575,3.3024)--(3.7878,3.3120)--(3.8181,3.3214)--(3.8484,3.3307)--(3.8787,3.3398)--(3.9090,3.3488)--(3.9393,3.3576)--(3.9696,3.3664)--(4.0000,3.3750)--(4.0303,3.3834)--(4.0606,3.3917)--(4.0909,3.4000)--(4.1212,3.4080)--(4.1515,3.4160)--(4.1818,3.4239)--(4.2121,3.4316)--(4.2424,3.4392)--(4.2727,3.4468)--(4.3030,3.4542)--(4.3333,3.4615)--(4.3636,3.4687)--(4.3939,3.4758)--(4.4242,3.4828)--(4.4545,3.4897)--(4.4848,3.4966)--(4.5151,3.5033)--(4.5454,3.5100)--(4.5757,3.5165)--(4.6060,3.5230)--(4.6363,3.5294)--(4.6666,3.5357)--(4.6969,3.5419)--(4.7272,3.5480)--(4.7575,3.5541)--(4.7878,3.5601)--(4.8181,3.5660)--(4.8484,3.5718)--(4.8787,3.5776)--(4.9090,3.5833)--(4.9393,3.5889)--(4.9696,3.5945)--(5.0000,3.6000)--(5.0303,3.6054)--(5.0606,3.6107)--(5.0909,3.6160)--(5.1212,3.6213)--(5.1515,3.6264)--(5.1818,3.6315)--(5.2121,3.6366)--(5.2424,3.6416)--(5.2727,3.6465)--(5.3030,3.6514)--(5.3333,3.6562)--(5.3636,3.6610)--(5.3939,3.6657)--(5.4242,3.6703)--(5.4545,3.6750)--(5.4848,3.6795)--(5.5151,3.6840)--(5.5454,3.6885)--(5.5757,3.6929)--(5.6060,3.6972)--(5.6363,3.7016)--(5.6666,3.7058)--(5.6969,3.7101)--(5.7272,3.7142)--(5.7575,3.7184)--(5.7878,3.7225)--(5.8181,3.7265)--(5.8484,3.7305)--(5.8787,3.7345)--(5.9090,3.7384)--(5.9393,3.7423)--(5.9696,3.7461)--(6.0000,3.7500);
+\draw [color=blue] (3.0000,0.0000)--(3.0303,0.0000)--(3.0606,0.0000)--(3.0909,0.0000)--(3.1212,0.0000)--(3.1515,0.0000)--(3.1818,0.0000)--(3.2121,0.0000)--(3.2424,0.0000)--(3.2727,0.0000)--(3.3030,0.0000)--(3.3333,0.0000)--(3.3636,0.0000)--(3.3939,0.0000)--(3.4242,0.0000)--(3.4545,0.0000)--(3.4848,0.0000)--(3.5151,0.0000)--(3.5454,0.0000)--(3.5757,0.0000)--(3.6060,0.0000)--(3.6363,0.0000)--(3.6666,0.0000)--(3.6969,0.0000)--(3.7272,0.0000)--(3.7575,0.0000)--(3.7878,0.0000)--(3.8181,0.0000)--(3.8484,0.0000)--(3.8787,0.0000)--(3.9090,0.0000)--(3.9393,0.0000)--(3.9696,0.0000)--(4.0000,0.0000)--(4.0303,0.0000)--(4.0606,0.0000)--(4.0909,0.0000)--(4.1212,0.0000)--(4.1515,0.0000)--(4.1818,0.0000)--(4.2121,0.0000)--(4.2424,0.0000)--(4.2727,0.0000)--(4.3030,0.0000)--(4.3333,0.0000)--(4.3636,0.0000)--(4.3939,0.0000)--(4.4242,0.0000)--(4.4545,0.0000)--(4.4848,0.0000)--(4.5151,0.0000)--(4.5454,0.0000)--(4.5757,0.0000)--(4.6060,0.0000)--(4.6363,0.0000)--(4.6666,0.0000)--(4.6969,0.0000)--(4.7272,0.0000)--(4.7575,0.0000)--(4.7878,0.0000)--(4.8181,0.0000)--(4.8484,0.0000)--(4.8787,0.0000)--(4.9090,0.0000)--(4.9393,0.0000)--(4.9696,0.0000)--(5.0000,0.0000)--(5.0303,0.0000)--(5.0606,0.0000)--(5.0909,0.0000)--(5.1212,0.0000)--(5.1515,0.0000)--(5.1818,0.0000)--(5.2121,0.0000)--(5.2424,0.0000)--(5.2727,0.0000)--(5.3030,0.0000)--(5.3333,0.0000)--(5.3636,0.0000)--(5.3939,0.0000)--(5.4242,0.0000)--(5.4545,0.0000)--(5.4848,0.0000)--(5.5151,0.0000)--(5.5454,0.0000)--(5.5757,0.0000)--(5.6060,0.0000)--(5.6363,0.0000)--(5.6666,0.0000)--(5.6969,0.0000)--(5.7272,0.0000)--(5.7575,0.0000)--(5.7878,0.0000)--(5.8181,0.0000)--(5.8484,0.0000)--(5.8787,0.0000)--(5.9090,0.0000)--(5.9393,0.0000)--(5.9696,0.0000)--(6.0000,0.0000);
+\draw [style=dashed] (3.0000,0.0000) -- (3.0000,3.0000);
+\draw [style=dashed] (6.0000,3.7500) -- (6.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks b/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks
index ba9700c1b..fdb12865a 100644
--- a/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks
+++ b/auto/pictures_tex/Fig_VWFLooPSrOqz.pstricks
@@ -83,13 +83,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6500,0) -- (3.6500,0);
-\draw [,->,>=latex] (0,-2.6500) -- (0,2.6500);
+\draw [,->,>=latex] (-1.6500,0.0000) -- (3.6500,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6500) -- (0.0000,2.6500);
 %DEFAULT
-\draw [color=red] (-0.150,2.15) -- (3.15,-1.15);
-\draw [color=red] (3.15,1.15) -- (-0.150,-2.15);
-\draw [color=red] (2.15,-2.15) -- (-1.15,1.15);
-\draw [color=red] (-1.15,-1.15) -- (2.15,2.15);
+\draw [color=red] (-0.1500,2.1500) -- (3.1500,-1.1500);
+\draw [color=red] (3.1500,1.1500) -- (-0.1500,-2.1500);
+\draw [color=red] (2.1500,-2.1500) -- (-1.1500,1.1500);
+\draw [color=red] (-1.1500,-1.1500) -- (2.1500,2.1500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,27 +97,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,1.00) -- (2.00,0) -- (2.00,0) -- (1.00,-1.00) -- (1.00,-1.00) -- (0,0) -- (0,0) -- (1.00,1.00) --  cycle;
-\draw [color=blue] (1.00,1.00) -- (2.00,0);
-\draw [color=blue] (2.00,0) -- (1.00,-1.00);
-\draw [color=blue] (1.00,-1.00) -- (0,0);
-\draw [color=blue] (0,0) -- (1.00,1.00);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,1.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (1.0000,-1.0000) -- (1.0000,-1.0000) -- (0.0000,0.0000) -- (0.0000,0.0000) -- (1.0000,1.0000) --  cycle;
+\draw [color=blue] (1.0000,1.0000) -- (2.0000,0.0000);
+\draw [color=blue] (2.0000,0.0000) -- (1.0000,-1.0000);
+\draw [color=blue] (1.0000,-1.0000) -- (0.0000,0.0000);
+\draw [color=blue] (0.0000,0.0000) -- (1.0000,1.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks b/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks
index ff1ca43f9..f4f5bb4d3 100644
--- a/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks
+++ b/auto/pictures_tex/Fig_WHCooNZAmYB.pstricks
@@ -78,22 +78,22 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.32471) node {\( O\)};
-\draw []  (1.0500,1.8187) node [rotate=0] {$\bullet$};
-\draw (1.2976,2.1166) node {\( B\)};
-\draw []  (1.8187,1.0500) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3247) node {\( O\)};
+\draw []  (1.0500,1.8186) node [rotate=0] {$\bullet$};
+\draw (1.2975,2.1165) node {\( B\)};
+\draw []  (1.8186,1.0500) node [rotate=0] {$\bullet$};
 \draw (2.1287,1.2747) node {\( A\)};
-\draw []  (2.1000,0) node [rotate=0] {$\bullet$};
-\draw (2.3360,-0.26613) node {\( I\)};
+\draw []  (2.1000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.3359,-0.2661) node {\( I\)};
 
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-\draw [] (1.05,1.82) -- (2.10,0);
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+\draw [] (0.0000,0.0000) -- (1.0500,1.8186);
+\draw [] (0.0000,0.0000) -- (2.1000,0.0000);
+\draw [] (0.0000,0.0000) -- (1.8186,1.0500);
+\draw [] (1.0500,1.8186) -- (2.1000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks b/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
index 9099168b7..d9891f8ea 100644
--- a/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
+++ b/auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks
@@ -119,41 +119,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
+\draw [,->,>=latex] (-4.0000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.7995) -- (0.0000,4.0547);
 %DEFAULT
 
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-\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
-\draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
-\draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=red] (-3.5000,3.5547)--(-3.4292,3.4996)--(-3.3585,3.4061)--(-3.2878,3.2766)--(-3.2171,3.1140)--(-3.1464,2.9213)--(-3.0757,2.7019)--(-3.0050,2.4594)--(-2.9343,2.1976)--(-2.8636,1.9206)--(-2.7929,1.6322)--(-2.7222,1.3367)--(-2.6515,1.0379)--(-2.5808,0.7400)--(-2.5101,0.4467)--(-2.4393,0.1618)--(-2.3686,-0.1113)--(-2.2979,-0.3695)--(-2.2272,-0.6098)--(-2.1565,-0.8297)--(-2.0858,-1.0268)--(-2.0151,-1.1994)--(-1.9444,-1.3460)--(-1.8737,-1.4656)--(-1.8030,-1.5575)--(-1.7323,-1.6215)--(-1.6616,-1.6577)--(-1.5909,-1.6667)--(-1.5202,-1.6496)--(-1.4494,-1.6076)--(-1.3787,-1.5424)--(-1.3080,-1.4559)--(-1.2373,-1.3503)--(-1.1666,-1.2283)--(-1.0959,-1.0923)--(-1.0252,-0.9453)--(-0.9545,-0.7901)--(-0.8838,-0.6298)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.0025)--(-0.5303,0.1441)--(-0.4595,0.2739)--(-0.3888,0.3896)--(-0.3181,0.4892)--(-0.2474,0.5710)--(-0.1767,0.6336)--(-0.1060,0.6759)--(-0.0353,0.6973)--(0.0353,0.6973)--(0.1060,0.6759)--(0.1767,0.6336)--(0.2474,0.5710)--(0.3181,0.4892)--(0.3888,0.3896)--(0.4595,0.2739)--(0.5303,0.1441)--(0.6010,0.0025)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6298)--(0.9545,-0.7901)--(1.0252,-0.9453)--(1.0959,-1.0923)--(1.1666,-1.2283)--(1.2373,-1.3503)--(1.3080,-1.4559)--(1.3787,-1.5424)--(1.4494,-1.6076)--(1.5202,-1.6496)--(1.5909,-1.6667)--(1.6616,-1.6577)--(1.7323,-1.6215)--(1.8030,-1.5575)--(1.8737,-1.4656)--(1.9444,-1.3460)--(2.0151,-1.1994)--(2.0858,-1.0268)--(2.1565,-0.8297)--(2.2272,-0.6098)--(2.2979,-0.3695)--(2.3686,-0.1113)--(2.4393,0.1618)--(2.5101,0.4467)--(2.5808,0.7400)--(2.6515,1.0379)--(2.7222,1.3367)--(2.7929,1.6322)--(2.8636,1.9206)--(2.9343,2.1976)--(3.0050,2.4594)--(3.0757,2.7019)--(3.1464,2.9213)--(3.2171,3.1140)--(3.2878,3.2766)--(3.3585,3.4061)--(3.4292,3.4996)--(3.5000,3.5547);
+\draw (-3.2986,-0.4207) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (-3.2986,-0.1000) -- (-3.2986,0.1000);
+\draw (-2.1991,-0.3210) node {$-\mathit{\pi}$};
+\draw [] (-2.1991,-0.1000) -- (-2.1991,0.1000);
+\draw (-1.0995,-0.4207) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (-1.0995,-0.1000) -- (-1.0995,0.1000);
+\draw (1.0995,-0.4207) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (1.0995,-0.1000) -- (1.0995,0.1000);
+\draw (2.1991,-0.2785) node {$\mathit{\pi}$};
+\draw [] (2.1991,-0.1000) -- (2.1991,0.1000);
+\draw (3.2986,-0.4207) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (3.2986,-0.1000) -- (3.2986,0.1000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks b/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
index ae1ef50a0..3c5bfef38 100644
--- a/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
+++ b/auto/pictures_tex/Fig_WJBooMTAhtl.pstricks
@@ -100,35 +100,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.2124,0) -- (9.9248,0);
-\draw [,->,>=latex] (0,-3.4450) -- (0,3.3170);
+\draw [,->,>=latex] (-5.2123,0.0000) -- (9.9247,0.0000);
+\draw [,->,>=latex] (0.0000,-3.4450) -- (0.0000,3.3170);
 %DEFAULT
 
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-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (9.4248,-0.31492) node {$ 3 \, \pi $};
-\draw [] (9.42,-0.100) -- (9.42,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue,style=dashed] (-1.0719,-2.9450)--(-0.9793,-2.4356)--(-0.8868,-1.9740)--(-0.7942,-1.5579)--(-0.7017,-1.1849)--(-0.6091,-0.8526)--(-0.5166,-0.5589)--(-0.4241,-0.3015)--(-0.3315,-0.0783)--(-0.2390,0.1127)--(-0.1464,0.2738)--(-0.0539,0.4068)--(0.0386,0.5136)--(0.1311,0.5962)--(0.2236,0.6564)--(0.3162,0.6959)--(0.4087,0.7166)--(0.5013,0.7200)--(0.5938,0.7078)--(0.6864,0.6816)--(0.7789,0.6429)--(0.8715,0.5932)--(0.9640,0.5338)--(1.0565,0.4662)--(1.1491,0.3917)--(1.2416,0.3115)--(1.3342,0.2268)--(1.4267,0.1389)--(1.5193,0.0487)--(1.6118,-0.0425)--(1.7044,-0.1339)--(1.7969,-0.2245)--(1.8894,-0.3134)--(1.9820,-0.3997)--(2.0745,-0.4827)--(2.1671,-0.5616)--(2.2596,-0.6356)--(2.3522,-0.7042)--(2.4447,-0.7668)--(2.5372,-0.8229)--(2.6298,-0.8718)--(2.7223,-0.9133)--(2.8149,-0.9470)--(2.9074,-0.9724)--(3.0000,-0.9894)--(3.0925,-0.9976)--(3.1851,-0.9970)--(3.2776,-0.9875)--(3.3701,-0.9688)--(3.4627,-0.9411)--(3.5552,-0.9043)--(3.6478,-0.8586)--(3.7403,-0.8040)--(3.8329,-0.7409)--(3.9254,-0.6693)--(4.0180,-0.5896)--(4.1105,-0.5022)--(4.2030,-0.4074)--(4.2956,-0.3057)--(4.3881,-0.1976)--(4.4807,-0.0836)--(4.5732,0.0355)--(4.6658,0.1593)--(4.7583,0.2869)--(4.8508,0.4178)--(4.9434,0.5509)--(5.0359,0.6855)--(5.1285,0.8206)--(5.2210,0.9553)--(5.3136,1.0885)--(5.4061,1.2192)--(5.4987,1.3463)--(5.5912,1.4685)--(5.6837,1.5847)--(5.7763,1.6936)--(5.8688,1.7938)--(5.9614,1.8840)--(6.0539,1.9627)--(6.1465,2.0284)--(6.2390,2.0797)--(6.3316,2.1148)--(6.4241,2.1323)--(6.5166,2.1303)--(6.6092,2.1072)--(6.7017,2.0610)--(6.7943,1.9901)--(6.8868,1.8925)--(6.9794,1.7662)--(7.0719,1.6092)--(7.1644,1.4195)--(7.2570,1.1950)--(7.3495,0.9334)--(7.4421,0.6327)--(7.5346,0.2905)--(7.6272,-0.0954)--(7.7197,-0.5276)--(7.8123,-1.0083)--(7.9048,-1.5401)--(7.9973,-2.1254)--(8.0899,-2.7669);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (9.4247,-0.3149) node {$ 3 \, \pi $};
+\draw [] (9.4247,-0.1000) -- (9.4247,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_WUYooCISzeB.pstricks b/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
index b138a1fb5..279ac1275 100644
--- a/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
+++ b/auto/pictures_tex/Fig_WUYooCISzeB.pstricks
@@ -61,29 +61,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.99185) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.9918) -- (0.0000,4.0275);
 %DEFAULT
 
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+\draw [color=red] (-0.8400,2.7000)--(-0.8013,2.6677)--(-0.7626,2.6355)--(-0.7239,2.6032)--(-0.6852,2.5710)--(-0.6465,2.5387)--(-0.6078,2.5065)--(-0.5691,2.4743)--(-0.5304,2.4420)--(-0.4917,2.4098)--(-0.4531,2.3775)--(-0.4144,2.3453)--(-0.3757,2.3131)--(-0.3370,2.2808)--(-0.2983,2.2486)--(-0.2596,2.2163)--(-0.2209,2.1841)--(-0.1822,2.1519)--(-0.1435,2.1196)--(-0.1049,2.0874)--(-0.0662,2.0551)--(-0.0275,2.0229)--(0.0111,1.9907)--(0.0498,1.9584)--(0.0885,1.9262)--(0.1272,1.8939)--(0.1659,1.8617)--(0.2046,1.8294)--(0.2432,1.7972)--(0.2819,1.7650)--(0.3206,1.7327)--(0.3593,1.7005)--(0.3980,1.6682)--(0.4367,1.6360)--(0.4754,1.6038)--(0.5141,1.5715)--(0.5528,1.5393)--(0.5914,1.5070)--(0.6301,1.4748)--(0.6688,1.4426)--(0.7075,1.4103)--(0.7462,1.3781)--(0.7849,1.3458)--(0.8236,1.3136)--(0.8623,1.2814)--(0.9010,1.2491)--(0.9396,1.2169)--(0.9783,1.1846)--(1.0170,1.1524)--(1.0557,1.1201)--(1.0944,1.0879)--(1.1331,1.0557)--(1.1718,1.0234)--(1.2105,0.9912)--(1.2492,0.9589)--(1.2878,0.9267)--(1.3265,0.8945)--(1.3652,0.8622)--(1.4039,0.8300)--(1.4426,0.7977)--(1.4813,0.7655)--(1.5200,0.7333)--(1.5587,0.7010)--(1.5974,0.6688)--(1.6360,0.6365)--(1.6747,0.6043)--(1.7134,0.5721)--(1.7521,0.5398)--(1.7908,0.5076)--(1.8295,0.4753)--(1.8682,0.4431)--(1.9069,0.4108)--(1.9456,0.3786)--(1.9842,0.3464)--(2.0229,0.3141)--(2.0616,0.2819)--(2.1003,0.2496)--(2.1390,0.2174)--(2.1777,0.1852)--(2.2164,0.1529)--(2.2551,0.1207)--(2.2938,0.0884)--(2.3325,0.0562)--(2.3711,0.0240)--(2.4098,-0.0082)--(2.4485,-0.0404)--(2.4872,-0.0727)--(2.5259,-0.1049)--(2.5646,-0.1371)--(2.6033,-0.1694)--(2.6420,-0.2016)--(2.6807,-0.2339)--(2.7193,-0.2661)--(2.7580,-0.2984)--(2.7967,-0.3306)--(2.8354,-0.3628)--(2.8741,-0.3951)--(2.9128,-0.4273)--(2.9515,-0.4596)--(2.9902,-0.4918);
 
-\draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -141,29 +141,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0275);
 %DEFAULT
 
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-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -221,29 +221,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.98318) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.9831) -- (0.0000,4.0275);
 %DEFAULT
 
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-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -301,29 +301,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.3400,0) -- (4.1000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0276);
+\draw [,->,>=latex] (-1.3400,0.0000) -- (4.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0275);
 %DEFAULT
 
-\draw [color=red] (-0.8400,3.499)--(-0.8086,3.401)--(-0.7773,3.307)--(-0.7459,3.218)--(-0.7145,3.133)--(-0.6831,3.051)--(-0.6517,2.974)--(-0.6204,2.900)--(-0.5890,2.830)--(-0.5576,2.763)--(-0.5263,2.699)--(-0.4949,2.639)--(-0.4635,2.581)--(-0.4321,2.527)--(-0.4008,2.474)--(-0.3694,2.425)--(-0.3380,2.378)--(-0.3066,2.333)--(-0.2753,2.291)--(-0.2439,2.251)--(-0.2125,2.212)--(-0.1811,2.176)--(-0.1498,2.142)--(-0.1184,2.109)--(-0.08701,2.078)--(-0.05563,2.049)--(-0.02426,2.021)--(0.007117,1.994)--(0.03849,1.969)--(0.06987,1.945)--(0.1012,1.922)--(0.1326,1.901)--(0.1640,1.880)--(0.1954,1.861)--(0.2267,1.843)--(0.2581,1.825)--(0.2895,1.809)--(0.3209,1.793)--(0.3522,1.778)--(0.3836,1.764)--(0.4150,1.751)--(0.4464,1.738)--(0.4777,1.727)--(0.5091,1.716)--(0.5405,1.705)--(0.5719,1.696)--(0.6032,1.687)--(0.6346,1.678)--(0.6660,1.671)--(0.6974,1.664)--(0.7287,1.658)--(0.7601,1.652)--(0.7915,1.647)--(0.8229,1.643)--(0.8542,1.640)--(0.8856,1.638)--(0.9170,1.636)--(0.9484,1.635)--(0.9797,1.635)--(1.011,1.637)--(1.042,1.639)--(1.074,1.642)--(1.105,1.646)--(1.137,1.652)--(1.168,1.658)--(1.199,1.666)--(1.231,1.676)--(1.262,1.687)--(1.293,1.699)--(1.325,1.713)--(1.356,1.729)--(1.388,1.746)--(1.419,1.765)--(1.450,1.786)--(1.482,1.810)--(1.513,1.835)--(1.544,1.863)--(1.576,1.893)--(1.607,1.925)--(1.639,1.961)--(1.670,1.998)--(1.701,2.039)--(1.733,2.083)--(1.764,2.130)--(1.795,2.180)--(1.827,2.234)--(1.858,2.291)--(1.890,2.351)--(1.921,2.416)--(1.952,2.485)--(1.984,2.557)--(2.015,2.634)--(2.046,2.716)--(2.078,2.802)--(2.109,2.893)--(2.141,2.989)--(2.172,3.090)--(2.203,3.197)--(2.235,3.309)--(2.266,3.427);
+\draw [color=red] (-0.8400,3.4987)--(-0.8086,3.4006)--(-0.7772,3.3070)--(-0.7458,3.2177)--(-0.7145,3.1325)--(-0.6831,3.0513)--(-0.6517,2.9739)--(-0.6203,2.9001)--(-0.5890,2.8299)--(-0.5576,2.7630)--(-0.5262,2.6994)--(-0.4948,2.6388)--(-0.4635,2.5812)--(-0.4321,2.5265)--(-0.4007,2.4744)--(-0.3693,2.4250)--(-0.3380,2.3780)--(-0.3066,2.3333)--(-0.2752,2.2909)--(-0.2438,2.2507)--(-0.2125,2.2124)--(-0.1811,2.1761)--(-0.1497,2.1417)--(-0.1183,2.1090)--(-0.0870,2.0780)--(-0.0556,2.0485)--(-0.0242,2.0206)--(0.0071,1.9941)--(0.0384,1.9689)--(0.0698,1.9450)--(0.1012,1.9223)--(0.1326,1.9008)--(0.1639,1.8804)--(0.1953,1.8610)--(0.2267,1.8426)--(0.2581,1.8252)--(0.2894,1.8087)--(0.3208,1.7930)--(0.3522,1.7782)--(0.3836,1.7642)--(0.4149,1.7509)--(0.4463,1.7384)--(0.4777,1.7266)--(0.5091,1.7156)--(0.5404,1.7052)--(0.5718,1.6955)--(0.6032,1.6866)--(0.6346,1.6783)--(0.6659,1.6706)--(0.6973,1.6637)--(0.7287,1.6575)--(0.7601,1.6520)--(0.7914,1.6472)--(0.8228,1.6432)--(0.8542,1.6399)--(0.8856,1.6375)--(0.9169,1.6359)--(0.9483,1.6352)--(0.9797,1.6354)--(1.0111,1.6365)--(1.0424,1.6387)--(1.0738,1.6419)--(1.1052,1.6462)--(1.1366,1.6517)--(1.1679,1.6584)--(1.1993,1.6664)--(1.2307,1.6758)--(1.2621,1.6866)--(1.2934,1.6990)--(1.3248,1.7129)--(1.3562,1.7285)--(1.3876,1.7459)--(1.4189,1.7652)--(1.4503,1.7864)--(1.4817,1.8096)--(1.5131,1.8351)--(1.5444,1.8627)--(1.5758,1.8928)--(1.6072,1.9254)--(1.6386,1.9605)--(1.6699,1.9984)--(1.7013,2.0392)--(1.7327,2.0830)--(1.7640,2.1298)--(1.7954,2.1799)--(1.8268,2.2335)--(1.8582,2.2905)--(1.8895,2.3513)--(1.9209,2.4159)--(1.9523,2.4845)--(1.9837,2.5572)--(2.0150,2.6343)--(2.0464,2.7158)--(2.0778,2.8021)--(2.1092,2.8931)--(2.1405,2.9892)--(2.1719,3.0904)--(2.2033,3.1970)--(2.2347,3.3092)--(2.2660,3.4272);
 
-\draw [color=blue] (-0.8400,3.528)--(-0.7952,3.382)--(-0.7503,3.246)--(-0.7055,3.120)--(-0.6606,3.004)--(-0.6158,2.895)--(-0.5709,2.795)--(-0.5261,2.702)--(-0.4812,2.615)--(-0.4364,2.535)--(-0.3915,2.460)--(-0.3467,2.391)--(-0.3018,2.327)--(-0.2570,2.267)--(-0.2121,2.212)--(-0.1673,2.161)--(-0.1224,2.113)--(-0.07758,2.069)--(-0.03273,2.028)--(0.01212,1.990)--(0.05697,1.955)--(0.1018,1.922)--(0.1467,1.892)--(0.1915,1.863)--(0.2364,1.837)--(0.2812,1.813)--(0.3261,1.790)--(0.3709,1.769)--(0.4158,1.750)--(0.4606,1.732)--(0.5055,1.715)--(0.5503,1.700)--(0.5952,1.685)--(0.6400,1.672)--(0.6848,1.660)--(0.7297,1.648)--(0.7745,1.638)--(0.8194,1.628)--(0.8642,1.618)--(0.9091,1.610)--(0.9539,1.602)--(0.9988,1.595)--(1.044,1.588)--(1.088,1.581)--(1.133,1.576)--(1.178,1.570)--(1.223,1.565)--(1.268,1.560)--(1.313,1.556)--(1.358,1.552)--(1.402,1.548)--(1.447,1.545)--(1.492,1.542)--(1.537,1.539)--(1.582,1.536)--(1.627,1.533)--(1.672,1.531)--(1.716,1.529)--(1.761,1.527)--(1.806,1.525)--(1.851,1.523)--(1.896,1.521)--(1.941,1.520)--(1.985,1.518)--(2.030,1.517)--(2.075,1.516)--(2.120,1.515)--(2.165,1.514)--(2.210,1.513)--(2.255,1.512)--(2.299,1.511)--(2.344,1.510)--(2.389,1.509)--(2.434,1.509)--(2.479,1.508)--(2.524,1.507)--(2.568,1.507)--(2.613,1.506)--(2.658,1.506)--(2.703,1.506)--(2.748,1.505)--(2.793,1.505)--(2.838,1.504)--(2.882,1.504)--(2.927,1.504)--(2.972,1.504)--(3.017,1.503)--(3.062,1.503)--(3.107,1.503)--(3.152,1.503)--(3.196,1.502)--(3.241,1.502)--(3.286,1.502)--(3.331,1.502)--(3.376,1.502)--(3.421,1.502)--(3.465,1.502)--(3.510,1.501)--(3.555,1.501)--(3.600,1.501);
-\draw (-1.2000,-0.32983) node {$ -1 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 1 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 2 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 3 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (-0.29125,1.0000) node {$ 2 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 4 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 6 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 8 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-0.8400,3.5275)--(-0.7951,3.3815)--(-0.7503,3.2460)--(-0.7054,3.1202)--(-0.6606,3.0036)--(-0.6157,2.8953)--(-0.5709,2.7948)--(-0.5260,2.7015)--(-0.4812,2.6150)--(-0.4363,2.5347)--(-0.3915,2.4601)--(-0.3466,2.3910)--(-0.3018,2.3268)--(-0.2569,2.2673)--(-0.2121,2.2120)--(-0.1672,2.1607)--(-0.1224,2.1131)--(-0.0775,2.0690)--(-0.0327,2.0280)--(0.0121,1.9900)--(0.0569,1.9547)--(0.1018,1.9219)--(0.1466,1.8915)--(0.1915,1.8633)--(0.2363,1.8371)--(0.2812,1.8129)--(0.3260,1.7903)--(0.3709,1.7694)--(0.4157,1.7500)--(0.4606,1.7320)--(0.5054,1.7153)--(0.5503,1.6998)--(0.5951,1.6854)--(0.6400,1.6720)--(0.6848,1.6596)--(0.7296,1.6481)--(0.7745,1.6375)--(0.8193,1.6276)--(0.8642,1.6184)--(0.9090,1.6098)--(0.9539,1.6019)--(0.9987,1.5946)--(1.0436,1.5878)--(1.0884,1.5814)--(1.1333,1.5756)--(1.1781,1.5701)--(1.2230,1.5651)--(1.2678,1.5604)--(1.3127,1.5560)--(1.3575,1.5520)--(1.4024,1.5482)--(1.4472,1.5448)--(1.4921,1.5415)--(1.5369,1.5385)--(1.5818,1.5358)--(1.6266,1.5332)--(1.6715,1.5308)--(1.7163,1.5286)--(1.7612,1.5265)--(1.8060,1.5246)--(1.8509,1.5228)--(1.8957,1.5212)--(1.9406,1.5196)--(1.9854,1.5182)--(2.0303,1.5169)--(2.0751,1.5157)--(2.1200,1.5146)--(2.1648,1.5135)--(2.2096,1.5125)--(2.2545,1.5116)--(2.2993,1.5108)--(2.3442,1.5100)--(2.3890,1.5093)--(2.4339,1.5086)--(2.4787,1.5080)--(2.5236,1.5074)--(2.5684,1.5069)--(2.6133,1.5064)--(2.6581,1.5059)--(2.7030,1.5055)--(2.7478,1.5051)--(2.7927,1.5047)--(2.8375,1.5044)--(2.8824,1.5040)--(2.9272,1.5038)--(2.9721,1.5035)--(3.0169,1.5032)--(3.0618,1.5030)--(3.1066,1.5028)--(3.1515,1.5026)--(3.1963,1.5024)--(3.2412,1.5022)--(3.2860,1.5020)--(3.3309,1.5019)--(3.3757,1.5018)--(3.4206,1.5016)--(3.4654,1.5015)--(3.5103,1.5014)--(3.5551,1.5013)--(3.6000,1.5012);
+\draw (-1.2000,-0.3298) node {$ -1 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 1 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 2 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 3 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 8 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks b/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks
index a1dcf7284..f90b8f02b 100644
--- a/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks
+++ b/auto/pictures_tex/Fig_XJMooCQTlNL.pstricks
@@ -84,35 +84,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-3.7958) -- (0,3.7958);
+\draw [,->,>=latex] (-6.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.7958) -- (0.0000,3.7958);
 %DEFAULT
 
-\draw [color=blue] (-6.000,-3.296)--(-5.879,-3.169)--(-5.758,-3.044)--(-5.636,-2.920)--(-5.515,-2.797)--(-5.394,-2.676)--(-5.273,-2.556)--(-5.151,-2.437)--(-5.030,-2.320)--(-4.909,-2.204)--(-4.788,-2.090)--(-4.667,-1.977)--(-4.545,-1.866)--(-4.424,-1.756)--(-4.303,-1.648)--(-4.182,-1.542)--(-4.061,-1.438)--(-3.939,-1.335)--(-3.818,-1.234)--(-3.697,-1.136)--(-3.576,-1.039)--(-3.455,-0.9440)--(-3.333,-0.8514)--(-3.212,-0.7609)--(-3.091,-0.6728)--(-2.970,-0.5870)--(-2.848,-0.5037)--(-2.727,-0.4229)--(-2.606,-0.3449)--(-2.485,-0.2697)--(-2.364,-0.1974)--(-2.242,-0.1283)--(-2.121,-0.06241)--(-2.000,0)--(-1.879,0.05873)--(-1.758,0.1135)--(-1.636,0.1642)--(-1.515,0.2103)--(-1.394,0.2516)--(-1.273,0.2876)--(-1.152,0.3179)--(-1.030,0.3417)--(-0.9091,0.3584)--(-0.7879,0.3670)--(-0.6667,0.3662)--(-0.5455,0.3544)--(-0.4242,0.3289)--(-0.3030,0.2859)--(-0.1818,0.2180)--(-0.06061,0.1060)--(0.06061,-0.1060)--(0.1818,-0.2180)--(0.3030,-0.2859)--(0.4242,-0.3289)--(0.5455,-0.3544)--(0.6667,-0.3662)--(0.7879,-0.3670)--(0.9091,-0.3584)--(1.030,-0.3417)--(1.152,-0.3179)--(1.273,-0.2876)--(1.394,-0.2516)--(1.515,-0.2103)--(1.636,-0.1642)--(1.758,-0.1135)--(1.879,-0.05873)--(2.000,0)--(2.121,0.06241)--(2.242,0.1283)--(2.364,0.1974)--(2.485,0.2697)--(2.606,0.3449)--(2.727,0.4229)--(2.848,0.5037)--(2.970,0.5870)--(3.091,0.6728)--(3.212,0.7609)--(3.333,0.8514)--(3.455,0.9440)--(3.576,1.039)--(3.697,1.136)--(3.818,1.234)--(3.939,1.335)--(4.061,1.438)--(4.182,1.542)--(4.303,1.648)--(4.424,1.756)--(4.545,1.866)--(4.667,1.977)--(4.788,2.090)--(4.909,2.204)--(5.030,2.320)--(5.151,2.437)--(5.273,2.556)--(5.394,2.676)--(5.515,2.797)--(5.636,2.920)--(5.758,3.044)--(5.879,3.169)--(6.000,3.296);
-\draw (-6.0000,-0.32983) node {$ -3 $};
-\draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -2 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -1 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (4.0000,-0.31492) node {$ 2 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (6.0000,-0.31492) node {$ 3 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-6.0000,-3.2958)--(-5.8787,-3.1692)--(-5.7575,-3.0439)--(-5.6363,-2.9198)--(-5.5151,-2.7971)--(-5.3939,-2.6757)--(-5.2727,-2.5556)--(-5.1515,-2.4370)--(-5.0303,-2.3198)--(-4.9090,-2.2040)--(-4.7878,-2.0897)--(-4.6666,-1.9770)--(-4.5454,-1.8658)--(-4.4242,-1.7563)--(-4.3030,-1.6484)--(-4.1818,-1.5422)--(-4.0606,-1.4378)--(-3.9393,-1.3352)--(-3.8181,-1.2344)--(-3.6969,-1.1356)--(-3.5757,-1.0388)--(-3.4545,-0.9440)--(-3.3333,-0.8513)--(-3.2121,-0.7609)--(-3.0909,-0.6727)--(-2.9696,-0.5869)--(-2.8484,-0.5036)--(-2.7272,-0.4229)--(-2.6060,-0.3449)--(-2.4848,-0.2696)--(-2.3636,-0.1974)--(-2.2424,-0.1282)--(-2.1212,-0.0624)--(-2.0000,0.0000)--(-1.8787,0.0587)--(-1.7575,0.1135)--(-1.6363,0.1641)--(-1.5151,0.2103)--(-1.3939,0.2516)--(-1.2727,0.2876)--(-1.1515,0.3178)--(-1.0303,0.3416)--(-0.9090,0.3583)--(-0.7878,0.3669)--(-0.6666,0.3662)--(-0.5454,0.3543)--(-0.4242,0.3289)--(-0.3030,0.2859)--(-0.1818,0.2179)--(-0.0606,0.1059)--(0.0606,-0.1059)--(0.1818,-0.2179)--(0.3030,-0.2859)--(0.4242,-0.3289)--(0.5454,-0.3543)--(0.6666,-0.3662)--(0.7878,-0.3669)--(0.9090,-0.3583)--(1.0303,-0.3416)--(1.1515,-0.3178)--(1.2727,-0.2876)--(1.3939,-0.2516)--(1.5151,-0.2103)--(1.6363,-0.1641)--(1.7575,-0.1135)--(1.8787,-0.0587)--(2.0000,0.0000)--(2.1212,0.0624)--(2.2424,0.1282)--(2.3636,0.1974)--(2.4848,0.2696)--(2.6060,0.3449)--(2.7272,0.4229)--(2.8484,0.5036)--(2.9696,0.5869)--(3.0909,0.6727)--(3.2121,0.7609)--(3.3333,0.8513)--(3.4545,0.9440)--(3.5757,1.0388)--(3.6969,1.1356)--(3.8181,1.2344)--(3.9393,1.3352)--(4.0606,1.4378)--(4.1818,1.5422)--(4.3030,1.6484)--(4.4242,1.7563)--(4.5454,1.8658)--(4.6666,1.9770)--(4.7878,2.0897)--(4.9090,2.2040)--(5.0303,2.3198)--(5.1515,2.4370)--(5.2727,2.5556)--(5.3939,2.6757)--(5.5151,2.7971)--(5.6363,2.9198)--(5.7575,3.0439)--(5.8787,3.1692)--(6.0000,3.2958);
+\draw (-6.0000,-0.3298) node {$ -3 $};
+\draw [] (-6.0000,-0.1000) -- (-6.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -2 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -1 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 2 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 3 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
index 75b352840..9cd54e3f6 100644
--- a/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
+++ b/auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks
@@ -120,41 +120,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-4.5252) -- (0,4.0659);
+\draw [,->,>=latex] (-6.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5251) -- (0.0000,4.0658);
 %DEFAULT
 
-\draw [color=blue] (-6.0000,3.5659)--(-5.8788,1.9708)--(-5.7576,1.0759)--(-5.6364,0.57982)--(-5.5152,0.30831)--(-5.3939,0.16165)--(-5.2727,0.083523)--(-5.1515,0.042498)--(-5.0303,0.021279)--(-4.9091,0.010476)--(-4.7879,0.0050674)--(-4.6667,0)--(-4.5455,0)--(-4.4242,0)--(-4.3030,0)--(-4.1818,0)--(-4.0606,0)--(-3.9394,0)--(-3.8182,0)--(-3.6970,0)--(-3.5758,0)--(-3.4545,0)--(-3.3333,0)--(-3.2121,0)--(-3.0909,0)--(-2.9697,0)--(-2.8485,0)--(-2.7273,0)--(-2.6061,0)--(-2.4848,0)--(-2.3636,0)--(-2.2424,0)--(-2.1212,0)--(-2.0000,0)--(-1.8788,0)--(-1.7576,0)--(-1.6364,0)--(-1.5152,0)--(-1.3939,0)--(-1.2727,0)--(-1.1515,0)--(-1.0303,0)--(-0.90909,0)--(-0.78788,0)--(-0.66667,0)--(-0.54545,0)--(-0.42424,0)--(-0.30303,0)--(-0.18182,0)--(-0.060606,0)--(0.060606,0)--(0.18182,0)--(0.30303,0)--(0.42424,0)--(0.54545,0)--(0.66667,0)--(0.78788,0)--(0.90909,0)--(1.0303,0)--(1.1515,0)--(1.2727,0)--(1.3939,0)--(1.5152,0)--(1.6364,0)--(1.7576,0)--(1.8788,0)--(2.0000,0)--(2.1212,0)--(2.2424,0)--(2.3636,0)--(2.4848,0)--(2.6061,0)--(2.7273,0)--(2.8485,0)--(2.9697,0)--(3.0909,0)--(3.2121,0)--(3.3333,0)--(3.4545,0)--(3.5758,0)--(3.6970,0)--(3.8182,0)--(3.9394,0)--(4.0606,0)--(4.1818,0)--(4.3030,0)--(4.4242,0)--(4.5455,0)--(4.6667,0)--(4.7879,-0.0054016)--(4.9091,-0.011220)--(5.0303,-0.022904)--(5.1515,-0.045980)--(5.2727,-0.090854)--(5.3939,-0.17683)--(5.5152,-0.33922)--(5.6364,-0.64184)--(5.7576,-1.1985)--(5.8788,-2.2097)--(6.0000,-4.0252);
-\draw (-6.0000,-0.32983) node {$ -10 $};
-\draw [] (-6.00,-0.100) -- (-6.00,0.100);
-\draw (-4.8000,-0.32983) node {$ -8 $};
-\draw [] (-4.80,-0.100) -- (-4.80,0.100);
-\draw (-3.6000,-0.32983) node {$ -6 $};
-\draw [] (-3.60,-0.100) -- (-3.60,0.100);
-\draw (-2.4000,-0.32983) node {$ -4 $};
-\draw [] (-2.40,-0.100) -- (-2.40,0.100);
-\draw (-1.2000,-0.32983) node {$ -2 $};
-\draw [] (-1.20,-0.100) -- (-1.20,0.100);
-\draw (1.2000,-0.31492) node {$ 2 $};
-\draw [] (1.20,-0.100) -- (1.20,0.100);
-\draw (2.4000,-0.31492) node {$ 4 $};
-\draw [] (2.40,-0.100) -- (2.40,0.100);
-\draw (3.6000,-0.31492) node {$ 6 $};
-\draw [] (3.60,-0.100) -- (3.60,0.100);
-\draw (4.8000,-0.31492) node {$ 8 $};
-\draw [] (4.80,-0.100) -- (4.80,0.100);
-\draw (6.0000,-0.31492) node {$ 10 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.59441,-4.5000) node {$ -\frac{3}{200} $};
-\draw [] (-0.100,-4.50) -- (0.100,-4.50);
-\draw (-0.59441,-3.0000) node {$ -\frac{1}{100} $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.59441,-1.5000) node {$ -\frac{1}{200} $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.45250,1.5000) node {$ \frac{1}{200} $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
-\draw (-0.45250,3.0000) node {$ \frac{1}{100} $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-6.0000,3.5658)--(-5.8787,1.9708)--(-5.7575,1.0758)--(-5.6363,0.5798)--(-5.5151,0.3083)--(-5.3939,0.1616)--(-5.2727,0.0835)--(-5.1515,0.0424)--(-5.0303,0.0212)--(-4.9090,0.0104)--(-4.7878,0.0050)--(-4.6666,0.0000)--(-4.5454,0.0000)--(-4.4242,0.0000)--(-4.3030,0.0000)--(-4.1818,0.0000)--(-4.0606,0.0000)--(-3.9393,0.0000)--(-3.8181,0.0000)--(-3.6969,0.0000)--(-3.5757,0.0000)--(-3.4545,0.0000)--(-3.3333,0.0000)--(-3.2121,0.0000)--(-3.0909,0.0000)--(-2.9696,0.0000)--(-2.8484,0.0000)--(-2.7272,0.0000)--(-2.6060,0.0000)--(-2.4848,0.0000)--(-2.3636,0.0000)--(-2.2424,0.0000)--(-2.1212,0.0000)--(-2.0000,0.0000)--(-1.8787,0.0000)--(-1.7575,0.0000)--(-1.6363,0.0000)--(-1.5151,0.0000)--(-1.3939,0.0000)--(-1.2727,0.0000)--(-1.1515,0.0000)--(-1.0303,0.0000)--(-0.9090,0.0000)--(-0.7878,0.0000)--(-0.6666,0.0000)--(-0.5454,0.0000)--(-0.4242,0.0000)--(-0.3030,0.0000)--(-0.1818,0.0000)--(-0.0606,0.0000)--(0.0606,0.0000)--(0.1818,0.0000)--(0.3030,0.0000)--(0.4242,0.0000)--(0.5454,0.0000)--(0.6666,0.0000)--(0.7878,0.0000)--(0.9090,0.0000)--(1.0303,0.0000)--(1.1515,0.0000)--(1.2727,0.0000)--(1.3939,0.0000)--(1.5151,0.0000)--(1.6363,0.0000)--(1.7575,0.0000)--(1.8787,0.0000)--(2.0000,0.0000)--(2.1212,0.0000)--(2.2424,0.0000)--(2.3636,0.0000)--(2.4848,0.0000)--(2.6060,0.0000)--(2.7272,0.0000)--(2.8484,0.0000)--(2.9696,0.0000)--(3.0909,0.0000)--(3.2121,0.0000)--(3.3333,0.0000)--(3.4545,0.0000)--(3.5757,0.0000)--(3.6969,0.0000)--(3.8181,0.0000)--(3.9393,0.0000)--(4.0606,0.0000)--(4.1818,0.0000)--(4.3030,0.0000)--(4.4242,0.0000)--(4.5454,0.0000)--(4.6666,0.0000)--(4.7878,-0.0054)--(4.9090,-0.0112)--(5.0303,-0.0229)--(5.1515,-0.0459)--(5.2727,-0.0908)--(5.3939,-0.1768)--(5.5151,-0.3392)--(5.6363,-0.6418)--(5.7575,-1.1984)--(5.8787,-2.2097)--(6.0000,-4.0251);
+\draw (-6.0000,-0.3298) node {$ -10 $};
+\draw [] (-6.0000,-0.1000) -- (-6.0000,0.1000);
+\draw (-4.8000,-0.3298) node {$ -8 $};
+\draw [] (-4.8000,-0.1000) -- (-4.8000,0.1000);
+\draw (-3.6000,-0.3298) node {$ -6 $};
+\draw [] (-3.6000,-0.1000) -- (-3.6000,0.1000);
+\draw (-2.4000,-0.3298) node {$ -4 $};
+\draw [] (-2.4000,-0.1000) -- (-2.4000,0.1000);
+\draw (-1.2000,-0.3298) node {$ -2 $};
+\draw [] (-1.2000,-0.1000) -- (-1.2000,0.1000);
+\draw (1.2000,-0.3149) node {$ 2 $};
+\draw [] (1.2000,-0.1000) -- (1.2000,0.1000);
+\draw (2.4000,-0.3149) node {$ 4 $};
+\draw [] (2.4000,-0.1000) -- (2.4000,0.1000);
+\draw (3.6000,-0.3149) node {$ 6 $};
+\draw [] (3.6000,-0.1000) -- (3.6000,0.1000);
+\draw (4.8000,-0.3149) node {$ 8 $};
+\draw [] (4.8000,-0.1000) -- (4.8000,0.1000);
+\draw (6.0000,-0.3149) node {$ 10 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.5944,-4.5000) node {$ -\frac{3}{200} $};
+\draw [] (-0.1000,-4.5000) -- (0.1000,-4.5000);
+\draw (-0.5944,-3.0000) node {$ -\frac{1}{100} $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.5944,-1.5000) node {$ -\frac{1}{200} $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.4525,1.5000) node {$ \frac{1}{200} $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
+\draw (-0.4525,3.0000) node {$ \frac{1}{100} $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks b/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks
index 766d698d6..96456fb8a 100644
--- a/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks
+++ b/auto/pictures_tex/Fig_XTGooSFFtPu.pstricks
@@ -107,93 +107,93 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-5.00,-4.00) -- (-5.00,3.00);
-\draw [color=gray,style=solid] (-4.00,-4.00) -- (-4.00,3.00);
-\draw [color=gray,style=solid] (-3.00,-4.00) -- (-3.00,3.00);
-\draw [color=gray,style=solid] (-2.00,-4.00) -- (-2.00,3.00);
-\draw [color=gray,style=solid] (-1.00,-4.00) -- (-1.00,3.00);
-\draw [color=gray,style=solid] (0,-4.00) -- (0,3.00);
-\draw [color=gray,style=solid] (1.00,-4.00) -- (1.00,3.00);
-\draw [color=gray,style=solid] (2.00,-4.00) -- (2.00,3.00);
-\draw [color=gray,style=solid] (3.00,-4.00) -- (3.00,3.00);
-\draw [color=gray,style=solid] (4.00,-4.00) -- (4.00,3.00);
-\draw [color=gray,style=solid] (5.00,-4.00) -- (5.00,3.00);
-\draw [color=gray,style=dotted] (-4.50,-4.00) -- (-4.50,3.00);
-\draw [color=gray,style=dotted] (-3.50,-4.00) -- (-3.50,3.00);
-\draw [color=gray,style=dotted] (-2.50,-4.00) -- (-2.50,3.00);
-\draw [color=gray,style=dotted] (-1.50,-4.00) -- (-1.50,3.00);
-\draw [color=gray,style=dotted] (-0.500,-4.00) -- (-0.500,3.00);
-\draw [color=gray,style=dotted] (0.500,-4.00) -- (0.500,3.00);
-\draw [color=gray,style=dotted] (1.50,-4.00) -- (1.50,3.00);
-\draw [color=gray,style=dotted] (2.50,-4.00) -- (2.50,3.00);
-\draw [color=gray,style=dotted] (3.50,-4.00) -- (3.50,3.00);
-\draw [color=gray,style=dotted] (4.50,-4.00) -- (4.50,3.00);
-\draw [color=gray,style=dotted] (-5.00,-3.50) -- (5.00,-3.50);
-\draw [color=gray,style=dotted] (-5.00,-2.50) -- (5.00,-2.50);
-\draw [color=gray,style=dotted] (-5.00,-1.50) -- (5.00,-1.50);
-\draw [color=gray,style=dotted] (-5.00,-0.500) -- (5.00,-0.500);
-\draw [color=gray,style=dotted] (-5.00,0.500) -- (5.00,0.500);
-\draw [color=gray,style=dotted] (-5.00,1.50) -- (5.00,1.50);
-\draw [color=gray,style=dotted] (-5.00,2.50) -- (5.00,2.50);
-\draw [color=gray,style=solid] (-5.00,-4.00) -- (5.00,-4.00);
-\draw [color=gray,style=solid] (-5.00,-3.00) -- (5.00,-3.00);
-\draw [color=gray,style=solid] (-5.00,-2.00) -- (5.00,-2.00);
-\draw [color=gray,style=solid] (-5.00,-1.00) -- (5.00,-1.00);
-\draw [color=gray,style=solid] (-5.00,0) -- (5.00,0);
-\draw [color=gray,style=solid] (-5.00,1.00) -- (5.00,1.00);
-\draw [color=gray,style=solid] (-5.00,2.00) -- (5.00,2.00);
-\draw [color=gray,style=solid] (-5.00,3.00) -- (5.00,3.00);
+\draw [color=gray,style=solid] (-5.0000,-4.0000) -- (-5.0000,3.0000);
+\draw [color=gray,style=solid] (-4.0000,-4.0000) -- (-4.0000,3.0000);
+\draw [color=gray,style=solid] (-3.0000,-4.0000) -- (-3.0000,3.0000);
+\draw [color=gray,style=solid] (-2.0000,-4.0000) -- (-2.0000,3.0000);
+\draw [color=gray,style=solid] (-1.0000,-4.0000) -- (-1.0000,3.0000);
+\draw [color=gray,style=solid] (0.0000,-4.0000) -- (0.0000,3.0000);
+\draw [color=gray,style=solid] (1.0000,-4.0000) -- (1.0000,3.0000);
+\draw [color=gray,style=solid] (2.0000,-4.0000) -- (2.0000,3.0000);
+\draw [color=gray,style=solid] (3.0000,-4.0000) -- (3.0000,3.0000);
+\draw [color=gray,style=solid] (4.0000,-4.0000) -- (4.0000,3.0000);
+\draw [color=gray,style=solid] (5.0000,-4.0000) -- (5.0000,3.0000);
+\draw [color=gray,style=dotted] (-4.5000,-4.0000) -- (-4.5000,3.0000);
+\draw [color=gray,style=dotted] (-3.5000,-4.0000) -- (-3.5000,3.0000);
+\draw [color=gray,style=dotted] (-2.5000,-4.0000) -- (-2.5000,3.0000);
+\draw [color=gray,style=dotted] (-1.5000,-4.0000) -- (-1.5000,3.0000);
+\draw [color=gray,style=dotted] (-0.5000,-4.0000) -- (-0.5000,3.0000);
+\draw [color=gray,style=dotted] (0.5000,-4.0000) -- (0.5000,3.0000);
+\draw [color=gray,style=dotted] (1.5000,-4.0000) -- (1.5000,3.0000);
+\draw [color=gray,style=dotted] (2.5000,-4.0000) -- (2.5000,3.0000);
+\draw [color=gray,style=dotted] (3.5000,-4.0000) -- (3.5000,3.0000);
+\draw [color=gray,style=dotted] (4.5000,-4.0000) -- (4.5000,3.0000);
+\draw [color=gray,style=dotted] (-5.0000,-3.5000) -- (5.0000,-3.5000);
+\draw [color=gray,style=dotted] (-5.0000,-2.5000) -- (5.0000,-2.5000);
+\draw [color=gray,style=dotted] (-5.0000,-1.5000) -- (5.0000,-1.5000);
+\draw [color=gray,style=dotted] (-5.0000,-0.5000) -- (5.0000,-0.5000);
+\draw [color=gray,style=dotted] (-5.0000,0.5000) -- (5.0000,0.5000);
+\draw [color=gray,style=dotted] (-5.0000,1.5000) -- (5.0000,1.5000);
+\draw [color=gray,style=dotted] (-5.0000,2.5000) -- (5.0000,2.5000);
+\draw [color=gray,style=solid] (-5.0000,-4.0000) -- (5.0000,-4.0000);
+\draw [color=gray,style=solid] (-5.0000,-3.0000) -- (5.0000,-3.0000);
+\draw [color=gray,style=solid] (-5.0000,-2.0000) -- (5.0000,-2.0000);
+\draw [color=gray,style=solid] (-5.0000,-1.0000) -- (5.0000,-1.0000);
+\draw [color=gray,style=solid] (-5.0000,0.0000) -- (5.0000,0.0000);
+\draw [color=gray,style=solid] (-5.0000,1.0000) -- (5.0000,1.0000);
+\draw [color=gray,style=solid] (-5.0000,2.0000) -- (5.0000,2.0000);
+\draw [color=gray,style=solid] (-5.0000,3.0000) -- (5.0000,3.0000);
 %AXES
-\draw [,->,>=latex] (-5.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-5.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,3.5000);
 %DEFAULT
 
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+\draw [color=blue] (1.0000,3.0000)--(1.0404,2.9595)--(1.0808,2.9191)--(1.1212,2.8787)--(1.1616,2.8383)--(1.2020,2.7979)--(1.2424,2.7575)--(1.2828,2.7171)--(1.3232,2.6767)--(1.3636,2.6363)--(1.4040,2.5959)--(1.4444,2.5555)--(1.4848,2.5151)--(1.5252,2.4747)--(1.5656,2.4343)--(1.6060,2.3939)--(1.6464,2.3535)--(1.6868,2.3131)--(1.7272,2.2727)--(1.7676,2.2323)--(1.8080,2.1919)--(1.8484,2.1515)--(1.8888,2.1111)--(1.9292,2.0707)--(1.9696,2.0303)--(2.0101,1.9898)--(2.0505,1.9494)--(2.0909,1.9090)--(2.1313,1.8686)--(2.1717,1.8282)--(2.2121,1.7878)--(2.2525,1.7474)--(2.2929,1.7070)--(2.3333,1.6666)--(2.3737,1.6262)--(2.4141,1.5858)--(2.4545,1.5454)--(2.4949,1.5050)--(2.5353,1.4646)--(2.5757,1.4242)--(2.6161,1.3838)--(2.6565,1.3434)--(2.6969,1.3030)--(2.7373,1.2626)--(2.7777,1.2222)--(2.8181,1.1818)--(2.8585,1.1414)--(2.8989,1.1010)--(2.9393,1.0606)--(2.9797,1.0202)--(3.0202,0.9797)--(3.0606,0.9393)--(3.1010,0.8989)--(3.1414,0.8585)--(3.1818,0.8181)--(3.2222,0.7777)--(3.2626,0.7373)--(3.3030,0.6969)--(3.3434,0.6565)--(3.3838,0.6161)--(3.4242,0.5757)--(3.4646,0.5353)--(3.5050,0.4949)--(3.5454,0.4545)--(3.5858,0.4141)--(3.6262,0.3737)--(3.6666,0.3333)--(3.7070,0.2929)--(3.7474,0.2525)--(3.7878,0.2121)--(3.8282,0.1717)--(3.8686,0.1313)--(3.9090,0.0909)--(3.9494,0.0505)--(3.9898,0.0101)--(4.0303,-0.0303)--(4.0707,-0.0707)--(4.1111,-0.1111)--(4.1515,-0.1515)--(4.1919,-0.1919)--(4.2323,-0.2323)--(4.2727,-0.2727)--(4.3131,-0.3131)--(4.3535,-0.3535)--(4.3939,-0.3939)--(4.4343,-0.4343)--(4.4747,-0.4747)--(4.5151,-0.5151)--(4.5555,-0.5555)--(4.5959,-0.5959)--(4.6363,-0.6363)--(4.6767,-0.6767)--(4.7171,-0.7171)--(4.7575,-0.7575)--(4.7979,-0.7979)--(4.8383,-0.8383)--(4.8787,-0.8787)--(4.9191,-0.9191)--(4.9595,-0.9595)--(5.0000,-1.0000);
 \draw [color=blue]  (-1.0000,-2.0000) node [rotate=0] {$\bullet$};
 \draw [color=blue]  (1.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw []  (0.50000,1.0000) node [rotate=0] {$\bullet$};
-\draw (0.84517,0.70650) node {\( Z_1\)};
+\draw []  (0.5000,1.0000) node [rotate=0] {$\bullet$};
+\draw (0.8451,0.7064) node {\( Z_1\)};
 \draw []  (3.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.3452,1.2935) node {\( Z_2\)};
+\draw (3.3451,1.2935) node {\( Z_2\)};
 
-\draw (-5.0000,-0.32983) node {$ -5 $};
-\draw [] (-5.00,-0.100) -- (-5.00,0.100);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (-5.0000,-0.3298) node {$ -5 $};
+\draw [] (-5.0000,-0.1000) -- (-5.0000,0.1000);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks b/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
index 824be3771..97af61aa9 100644
--- a/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
+++ b/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.02020,0.03030)--(0.04040,0.06061)--(0.06061,0.09091)--(0.08081,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5152)--(0.3636,0.5455)--(0.3838,0.5758)--(0.4040,0.6061)--(0.4242,0.6364)--(0.4444,0.6667)--(0.4646,0.6970)--(0.4848,0.7273)--(0.5051,0.7576)--(0.5253,0.7879)--(0.5455,0.8182)--(0.5657,0.8485)--(0.5859,0.8788)--(0.6061,0.9091)--(0.6263,0.9394)--(0.6465,0.9697)--(0.6667,1.000)--(0.6869,1.030)--(0.7071,1.061)--(0.7273,1.091)--(0.7475,1.121)--(0.7677,1.152)--(0.7879,1.182)--(0.8081,1.212)--(0.8283,1.242)--(0.8485,1.273)--(0.8687,1.303)--(0.8889,1.333)--(0.9091,1.364)--(0.9293,1.394)--(0.9495,1.424)--(0.9697,1.455)--(0.9899,1.485)--(1.010,1.515)--(1.030,1.545)--(1.051,1.576)--(1.071,1.606)--(1.091,1.636)--(1.111,1.667)--(1.131,1.697)--(1.152,1.727)--(1.172,1.758)--(1.192,1.788)--(1.212,1.818)--(1.232,1.848)--(1.253,1.879)--(1.273,1.909)--(1.293,1.939)--(1.313,1.970)--(1.333,2.000)--(1.354,2.030)--(1.374,2.061)--(1.394,2.091)--(1.414,2.121)--(1.434,2.152)--(1.455,2.182)--(1.475,2.212)--(1.495,2.242)--(1.515,2.273)--(1.535,2.303)--(1.556,2.333)--(1.576,2.364)--(1.596,2.394)--(1.616,2.424)--(1.636,2.455)--(1.657,2.485)--(1.677,2.515)--(1.697,2.545)--(1.717,2.576)--(1.737,2.606)--(1.758,2.636)--(1.778,2.667)--(1.798,2.697)--(1.818,2.727)--(1.838,2.758)--(1.859,2.788)--(1.879,2.818)--(1.899,2.848)--(1.919,2.879)--(1.939,2.909)--(1.960,2.939)--(1.980,2.970)--(2.000,3.000);
+\draw [color=blue] (0.0000,0.0000)--(0.0202,0.0303)--(0.0404,0.0606)--(0.0606,0.0909)--(0.0808,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5151)--(0.3636,0.5454)--(0.3838,0.5757)--(0.4040,0.6060)--(0.4242,0.6363)--(0.4444,0.6666)--(0.4646,0.6969)--(0.4848,0.7272)--(0.5050,0.7575)--(0.5252,0.7878)--(0.5454,0.8181)--(0.5656,0.8484)--(0.5858,0.8787)--(0.6060,0.9090)--(0.6262,0.9393)--(0.6464,0.9696)--(0.6666,1.0000)--(0.6868,1.0303)--(0.7070,1.0606)--(0.7272,1.0909)--(0.7474,1.1212)--(0.7676,1.1515)--(0.7878,1.1818)--(0.8080,1.2121)--(0.8282,1.2424)--(0.8484,1.2727)--(0.8686,1.3030)--(0.8888,1.3333)--(0.9090,1.3636)--(0.9292,1.3939)--(0.9494,1.4242)--(0.9696,1.4545)--(0.9898,1.4848)--(1.0101,1.5151)--(1.0303,1.5454)--(1.0505,1.5757)--(1.0707,1.6060)--(1.0909,1.6363)--(1.1111,1.6666)--(1.1313,1.6969)--(1.1515,1.7272)--(1.1717,1.7575)--(1.1919,1.7878)--(1.2121,1.8181)--(1.2323,1.8484)--(1.2525,1.8787)--(1.2727,1.9090)--(1.2929,1.9393)--(1.3131,1.9696)--(1.3333,2.0000)--(1.3535,2.0303)--(1.3737,2.0606)--(1.3939,2.0909)--(1.4141,2.1212)--(1.4343,2.1515)--(1.4545,2.1818)--(1.4747,2.2121)--(1.4949,2.2424)--(1.5151,2.2727)--(1.5353,2.3030)--(1.5555,2.3333)--(1.5757,2.3636)--(1.5959,2.3939)--(1.6161,2.4242)--(1.6363,2.4545)--(1.6565,2.4848)--(1.6767,2.5151)--(1.6969,2.5454)--(1.7171,2.5757)--(1.7373,2.6060)--(1.7575,2.6363)--(1.7777,2.6666)--(1.7979,2.6969)--(1.8181,2.7272)--(1.8383,2.7575)--(1.8585,2.7878)--(1.8787,2.8181)--(1.8989,2.8484)--(1.9191,2.8787)--(1.9393,2.9090)--(1.9595,2.9393)--(1.9797,2.9696)--(2.0000,3.0000);
 \draw []  (2.0000,3.0000) node [rotate=0] {$\bullet$};
-\draw (2.5389,3.2532) node {$(R,h)$};
-\draw (0.72367,0.33595) node {$\alpha$};
+\draw (2.5388,3.2531) node {$(R,h)$};
+\draw (0.7236,0.3359) node {$\alpha$};
 
-\draw [color=red] (0.500,0)--(0.500,0.00496)--(0.500,0.00993)--(0.500,0.0149)--(0.500,0.0198)--(0.499,0.0248)--(0.499,0.0298)--(0.499,0.0347)--(0.498,0.0397)--(0.498,0.0446)--(0.498,0.0496)--(0.497,0.0545)--(0.496,0.0594)--(0.496,0.0643)--(0.495,0.0693)--(0.494,0.0742)--(0.494,0.0791)--(0.493,0.0840)--(0.492,0.0889)--(0.491,0.0938)--(0.490,0.0986)--(0.489,0.103)--(0.488,0.108)--(0.487,0.113)--(0.486,0.118)--(0.485,0.123)--(0.483,0.128)--(0.482,0.132)--(0.481,0.137)--(0.479,0.142)--(0.478,0.147)--(0.477,0.151)--(0.475,0.156)--(0.473,0.161)--(0.472,0.166)--(0.470,0.170)--(0.468,0.175)--(0.467,0.180)--(0.465,0.184)--(0.463,0.189)--(0.461,0.193)--(0.459,0.198)--(0.457,0.202)--(0.455,0.207)--(0.453,0.212)--(0.451,0.216)--(0.449,0.220)--(0.447,0.225)--(0.444,0.229)--(0.442,0.234)--(0.440,0.238)--(0.437,0.242)--(0.435,0.247)--(0.432,0.251)--(0.430,0.255)--(0.427,0.260)--(0.425,0.264)--(0.422,0.268)--(0.419,0.272)--(0.417,0.276)--(0.414,0.281)--(0.411,0.285)--(0.408,0.289)--(0.405,0.293)--(0.402,0.297)--(0.399,0.301)--(0.396,0.305)--(0.393,0.309)--(0.390,0.312)--(0.387,0.316)--(0.384,0.320)--(0.381,0.324)--(0.378,0.328)--(0.374,0.331)--(0.371,0.335)--(0.368,0.339)--(0.364,0.342)--(0.361,0.346)--(0.357,0.350)--(0.354,0.353)--(0.350,0.357)--(0.347,0.360)--(0.343,0.364)--(0.340,0.367)--(0.336,0.370)--(0.332,0.374)--(0.329,0.377)--(0.325,0.380)--(0.321,0.383)--(0.317,0.386)--(0.313,0.390)--(0.309,0.393)--(0.306,0.396)--(0.302,0.399)--(0.298,0.402)--(0.294,0.405)--(0.290,0.408)--(0.286,0.410)--(0.281,0.413)--(0.277,0.416);
-\draw (2.0000,-0.32572) node {$\mathit{R}$};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.30273,3.0000) node {$\mathit{h}$};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=red] (0.5000,0.0000)--(0.4999,0.0049)--(0.4999,0.0099)--(0.4997,0.0148)--(0.4996,0.0198)--(0.4993,0.0248)--(0.4991,0.0297)--(0.4987,0.0347)--(0.4984,0.0396)--(0.4980,0.0446)--(0.4975,0.0495)--(0.4970,0.0544)--(0.4964,0.0594)--(0.4958,0.0643)--(0.4951,0.0692)--(0.4944,0.0741)--(0.4937,0.0790)--(0.4928,0.0839)--(0.4920,0.0888)--(0.4911,0.0937)--(0.4901,0.0986)--(0.4891,0.1034)--(0.4881,0.1083)--(0.4870,0.1131)--(0.4858,0.1180)--(0.4846,0.1228)--(0.4834,0.1276)--(0.4821,0.1324)--(0.4808,0.1371)--(0.4794,0.1419)--(0.4779,0.1467)--(0.4765,0.1514)--(0.4749,0.1561)--(0.4734,0.1608)--(0.4717,0.1655)--(0.4701,0.1702)--(0.4684,0.1749)--(0.4666,0.1795)--(0.4648,0.1841)--(0.4629,0.1887)--(0.4610,0.1933)--(0.4591,0.1979)--(0.4571,0.2024)--(0.4551,0.2070)--(0.4530,0.2115)--(0.4509,0.2160)--(0.4487,0.2204)--(0.4465,0.2249)--(0.4443,0.2293)--(0.4420,0.2337)--(0.4396,0.2381)--(0.4372,0.2424)--(0.4348,0.2467)--(0.4323,0.2511)--(0.4298,0.2553)--(0.4273,0.2596)--(0.4247,0.2638)--(0.4220,0.2680)--(0.4193,0.2722)--(0.4166,0.2763)--(0.4138,0.2805)--(0.4110,0.2846)--(0.4082,0.2886)--(0.4053,0.2927)--(0.4024,0.2967)--(0.3994,0.3007)--(0.3964,0.3046)--(0.3934,0.3085)--(0.3903,0.3124)--(0.3872,0.3163)--(0.3840,0.3201)--(0.3808,0.3239)--(0.3776,0.3277)--(0.3743,0.3314)--(0.3710,0.3351)--(0.3676,0.3388)--(0.3643,0.3424)--(0.3609,0.3460)--(0.3574,0.3496)--(0.3539,0.3531)--(0.3504,0.3566)--(0.3468,0.3601)--(0.3432,0.3635)--(0.3396,0.3669)--(0.3360,0.3702)--(0.3323,0.3735)--(0.3285,0.3768)--(0.3248,0.3801)--(0.3210,0.3833)--(0.3172,0.3864)--(0.3133,0.3896)--(0.3094,0.3927)--(0.3055,0.3957)--(0.3016,0.3987)--(0.2976,0.4017)--(0.2936,0.4046)--(0.2896,0.4075)--(0.2855,0.4104)--(0.2814,0.4132)--(0.2773,0.4160);
+\draw (2.0000,-0.3257) node {$\mathit{R}$};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.3027,3.0000) node {$\mathit{h}$};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks b/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks
index 9465f44db..b3a9e0cc9 100644
--- a/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks
+++ b/auto/pictures_tex/Fig_YQIDooBqpAdbIM.pstricks
@@ -74,24 +74,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.31799,-0.14894) node {\( A \)};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.3287,-0.15396) node {\( B \)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3179,-0.1489) node {\( A \)};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.3287,-0.1539) node {\( B \)};
 \draw []  (2.5000,3.0000) node [rotate=0] {$\bullet$};
 \draw (2.5000,3.3247) node {\( O \)};
-\draw [] (0,0) -- (5.00,0);
-\draw [] (5.00,0) -- (2.50,3.00);
-\draw [] (2.50,3.00) -- (0,0);
-\draw [color=red,->,>=latex] (0.32009,0.38411) -- (0.31241,0.39051);
+\draw [] (0.0000,0.0000) -- (5.0000,0.0000);
+\draw [] (5.0000,0.0000) -- (2.5000,3.0000);
+\draw [] (2.5000,3.0000) -- (0.0000,0.0000);
+\draw [color=red,->,>=latex] (0.3200,0.3841) -- (0.3124,0.3905);
 
-\draw [color=red] (0.500,0)--(0.500,0.00442)--(0.500,0.00885)--(0.500,0.0133)--(0.500,0.0177)--(0.500,0.0221)--(0.499,0.0265)--(0.499,0.0310)--(0.499,0.0354)--(0.498,0.0398)--(0.498,0.0442)--(0.498,0.0486)--(0.497,0.0530)--(0.497,0.0574)--(0.496,0.0618)--(0.496,0.0662)--(0.495,0.0706)--(0.494,0.0749)--(0.494,0.0793)--(0.493,0.0837)--(0.492,0.0880)--(0.491,0.0924)--(0.491,0.0967)--(0.490,0.101)--(0.489,0.105)--(0.488,0.110)--(0.487,0.114)--(0.486,0.118)--(0.485,0.123)--(0.484,0.127)--(0.482,0.131)--(0.481,0.135)--(0.480,0.140)--(0.479,0.144)--(0.478,0.148)--(0.476,0.152)--(0.475,0.157)--(0.473,0.161)--(0.472,0.165)--(0.471,0.169)--(0.469,0.173)--(0.467,0.177)--(0.466,0.182)--(0.464,0.186)--(0.463,0.190)--(0.461,0.194)--(0.459,0.198)--(0.457,0.202)--(0.456,0.206)--(0.454,0.210)--(0.452,0.214)--(0.450,0.218)--(0.448,0.222)--(0.446,0.226)--(0.444,0.230)--(0.442,0.234)--(0.440,0.238)--(0.438,0.242)--(0.436,0.245)--(0.433,0.249)--(0.431,0.253)--(0.429,0.257)--(0.427,0.261)--(0.424,0.265)--(0.422,0.268)--(0.420,0.272)--(0.417,0.276)--(0.415,0.279)--(0.412,0.283)--(0.410,0.287)--(0.407,0.290)--(0.405,0.294)--(0.402,0.297)--(0.399,0.301)--(0.397,0.305)--(0.394,0.308)--(0.391,0.311)--(0.388,0.315)--(0.386,0.318)--(0.383,0.322)--(0.380,0.325)--(0.377,0.328)--(0.374,0.332)--(0.371,0.335)--(0.368,0.338)--(0.365,0.342)--(0.362,0.345)--(0.359,0.348)--(0.356,0.351)--(0.353,0.354)--(0.350,0.357)--(0.346,0.361)--(0.343,0.364)--(0.340,0.367)--(0.337,0.370)--(0.333,0.373)--(0.330,0.375)--(0.327,0.378)--(0.323,0.381)--(0.320,0.384);
-\draw [color=red,->,>=latex] (2.8201,2.6159) -- (2.8278,2.6223);
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+\draw [color=red,->,>=latex] (2.8200,2.6158) -- (2.8277,2.6222);
 
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks b/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks
index 0ddf6c696..1fdc9e711 100644
--- a/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks
+++ b/auto/pictures_tex/Fig_YQVHooYsGLHQ.pstricks
@@ -65,230 +65,230 @@
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diff --git a/auto/pictures_tex/Fig_YWxOAkh.pstricks b/auto/pictures_tex/Fig_YWxOAkh.pstricks
index 78d8f53a2..0e493ed93 100644
--- a/auto/pictures_tex/Fig_YWxOAkh.pstricks
+++ b/auto/pictures_tex/Fig_YWxOAkh.pstricks
@@ -71,24 +71,24 @@
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+\draw [color=blue] 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+\draw [color=blue] 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+\draw [color=blue] 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+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks b/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks
index 6b17f4219..48f128d39 100644
--- a/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks
+++ b/auto/pictures_tex/Fig_YYECooQlnKtD.pstricks
@@ -104,43 +104,43 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (11.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (11.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.1682);
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw []  (1.0000,0.0013778) node [rotate=0] {$\bullet$};
-\draw []  (2.0000,0.014467) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0.090017) node [rotate=0] {$\bullet$};
-\draw []  (4.0000,0.36757) node [rotate=0] {$\bullet$};
-\draw []  (5.0000,1.0292) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.0000,0.0013) node [rotate=0] {$\bullet$};
+\draw []  (2.0000,0.0144) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,0.0900) node [rotate=0] {$\bullet$};
+\draw []  (4.0000,0.3675) node [rotate=0] {$\bullet$};
+\draw []  (5.0000,1.0291) node [rotate=0] {$\bullet$};
 \draw []  (6.0000,2.0012) node [rotate=0] {$\bullet$};
-\draw []  (7.0000,2.6683) node [rotate=0] {$\bullet$};
+\draw []  (7.0000,2.6682) node [rotate=0] {$\bullet$};
 \draw []  (8.0000,2.3347) node [rotate=0] {$\bullet$};
 \draw []  (9.0000,1.2106) node [rotate=0] {$\bullet$};
-\draw []  (10.000,0.28248) node [rotate=0] {$\bullet$};
-\draw []  (11.000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.000,-0.31492) node {$ 11 $};
-\draw [] (11.0,-0.100) -- (11.0,0.100);
+\draw []  (10.000,0.2824) node [rotate=0] {$\bullet$};
+\draw []  (11.000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (11.000,-0.3149) node {$ 11 $};
+\draw [] (11.000,-0.1000) -- (11.000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks b/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks
index 4dae1298f..ead95fd18 100644
--- a/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks
+++ b/auto/pictures_tex/Fig_ZGUDooEsqCWQ.pstricks
@@ -65,51 +65,51 @@
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+\draw [,->,>=latex] (2.1999,0.0191) -- (2.6544,0.0231);
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+\draw [,->,>=latex] (1.9681,0.9831) -- (2.3747,1.1862);
+\draw [,->,>=latex] (1.8439,1.1999) -- (2.2249,1.4478);
+\draw [,->,>=latex] (1.6959,1.4013) -- (2.0463,1.6908);
+\draw [,->,>=latex] (1.5261,1.5845) -- (1.8414,1.9119);
+\draw [,->,>=latex] (1.3366,1.7474) -- (1.6127,2.1084);
+\draw [,->,>=latex] (1.1298,1.8877) -- (1.3632,2.2777);
+\draw [,->,>=latex] (0.9085,2.0036) -- (1.0962,2.4176);
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+\draw [,->,>=latex] (-0.0634,2.1990) -- (-0.0765,2.6534);
+\draw [,->,>=latex] (-0.3123,2.1777) -- (-0.3769,2.6276);
+\draw [,->,>=latex] (-0.5573,2.1282) -- (-0.6724,2.5679);
+\draw [,->,>=latex] (-0.7950,2.0513) -- (-0.9593,2.4751);
+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2337,2.3504);
+\draw [,->,>=latex] (-1.2367,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7316,2.0119);
+\draw [,->,>=latex] (-1.6149,1.4939) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1404,1.5700);
+\draw [,->,>=latex] (-1.9100,1.0916) -- (-2.3046,1.3172);
+\draw [,->,>=latex] (-2.0215,0.8680) -- (-2.4391,1.0473);
+\draw [,->,>=latex] (-2.1069,0.6332) -- (-2.5422,0.7640);
+\draw [,->,>=latex] (-2.1651,0.3902) -- (-2.6124,0.4708);
+\draw [,->,>=latex] (-2.1953,0.1422) -- (-2.6489,0.1715);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks b/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks
index e2dc589d2..c57b3f7b3 100644
--- a/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks
+++ b/auto/pictures_tex/Fig_ZOCNoowrfvQXsr.pstricks
@@ -60,13 +60,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [] (-3.00,1.00) -- (0,1.00);
-\draw [] (3.00,2.00) -- (0.0700,2.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw []  (0,2.0000) node [rotate=0] {$o$};
+\draw [] (-3.0000,1.0000) -- (0.0000,1.0000);
+\draw [] (3.0000,2.0000) -- (0.0700,2.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,2.0000) node [rotate=0] {$o$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks b/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks
index 9a04c9614..575874fa4 100644
--- a/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks
+++ b/auto/pictures_tex/Fig_ZTTooXtHkci.pstricks
@@ -49,8 +49,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,0.50000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,0.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -59,19 +59,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.00,0) -- (2.00,0) -- (2.00,0) -- (0,-2.00) -- (0,-2.00) -- (0,-1.00) -- (0,-1.00) -- (1.00,0) --  cycle;
-\draw [] (1.00,0) -- (2.00,0);
-\draw [] (2.00,0) -- (0,-2.00);
-\draw [] (0,-2.00) -- (0,-1.00);
-\draw [] (0,-1.00) -- (1.00,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (1.0000,0.0000) -- (2.0000,0.0000) -- (2.0000,0.0000) -- (0.0000,-2.0000) -- (0.0000,-2.0000) -- (0.0000,-1.0000) -- (0.0000,-1.0000) -- (1.0000,0.0000) --  cycle;
+\draw [] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw [] (2.0000,0.0000) -- (0.0000,-2.0000);
+\draw [] (0.0000,-2.0000) -- (0.0000,-1.0000);
+\draw [] (0.0000,-1.0000) -- (1.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -117,8 +117,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -127,23 +127,23 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-2.00,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (1.00,1.00) -- (1.00,1.00) -- (-1.00,1.00) -- (-1.00,1.00) -- (-2.00,2.00) --  cycle;
-\draw [] (-2.00,2.00) -- (2.00,2.00);
-\draw [] (2.00,2.00) -- (1.00,1.00);
-\draw [] (1.00,1.00) -- (-1.00,1.00);
-\draw [] (-1.00,1.00) -- (-2.00,2.00);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (1.0000,1.0000) -- (1.0000,1.0000) -- (-1.0000,1.0000) -- (-1.0000,1.0000) -- (-2.0000,2.0000) --  cycle;
+\draw [] (-2.0000,2.0000) -- (2.0000,2.0000);
+\draw [] (2.0000,2.0000) -- (1.0000,1.0000);
+\draw [] (1.0000,1.0000) -- (-1.0000,1.0000);
+\draw [] (-1.0000,1.0000) -- (-2.0000,2.0000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_examssepti.pstricks b/auto/pictures_tex/Fig_examssepti.pstricks
index 74c463784..bfdb3d73a 100644
--- a/auto/pictures_tex/Fig_examssepti.pstricks
+++ b/auto/pictures_tex/Fig_examssepti.pstricks
@@ -75,10 +75,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
-\draw [style=dotted] (-2.36,1.85) -- (-2.36,0);
+\draw [style=dotted] (-2.3584,1.8541) -- (-2.3584,0.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -86,22 +86,22 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [color=green] (-2.36,1.85) -- (-2.36,1.85);
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-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-3.0000,0.0000)--(-2.9545,0.5202)--(-2.9090,0.7329)--(-2.8636,0.8941)--(-2.8181,1.0285)--(-2.7727,1.1454)--(-2.7272,1.2497)--(-2.6818,1.3445)--(-2.6363,1.4316)--(-2.5909,1.5123)--(-2.5454,1.5876)--(-2.5000,1.6583)--(-2.4545,1.7248)--(-2.4090,1.7878)--(-2.3636,1.8474)--(-2.3181,1.9042)--(-2.2727,1.9582)--(-2.2272,2.0097)--(-2.1818,2.0590)--(-2.1363,2.1061)--(-2.0909,2.1513)--(-2.0454,2.1945)--(-2.0000,2.2360)--(-1.9545,2.2759)--(-1.9090,2.3141)--(-1.8636,2.3509)--(-1.8181,2.3862)--(-1.7727,2.4202)--(-1.7272,2.4528)--(-1.6818,2.4842)--(-1.6363,2.5144)--(-1.5909,2.5434)--(-1.5454,2.5712)--(-1.5000,2.5980)--(-1.4545,2.6237)--(-1.4090,2.6484)--(-1.3636,2.6721)--(-1.3181,2.6948)--(-1.2727,2.7166)--(-1.2272,2.7374)--(-1.1818,2.7574)--(-1.1363,2.7764)--(-1.0909,2.7946)--(-1.0454,2.8119)--(-1.0000,2.8284)--(-0.9545,2.8440)--(-0.9090,2.8589)--(-0.8636,2.8730)--(-0.8181,2.8862)--(-0.7727,2.8987)--(-0.7272,2.9105)--(-0.6818,2.9214)--(-0.6363,2.9317)--(-0.5909,2.9412)--(-0.5454,2.9499)--(-0.5000,2.9580)--(-0.4545,2.9653)--(-0.4090,2.9719)--(-0.3636,2.9778)--(-0.3181,2.9830)--(-0.2727,2.9875)--(-0.2272,2.9913)--(-0.1818,2.9944)--(-0.1363,2.9968)--(-0.0909,2.9986)--(-0.0454,2.9996)--(0.0000,3.0000)--(0.0454,2.9996)--(0.0909,2.9986)--(0.1363,2.9968)--(0.1818,2.9944)--(0.2272,2.9913)--(0.2727,2.9875)--(0.3181,2.9830)--(0.3636,2.9778)--(0.4090,2.9719)--(0.4545,2.9653)--(0.5000,2.9580)--(0.5454,2.9499)--(0.5909,2.9412)--(0.6363,2.9317)--(0.6818,2.9214)--(0.7272,2.9105)--(0.7727,2.8987)--(0.8181,2.8862)--(0.8636,2.8730)--(0.9090,2.8589)--(0.9545,2.8440)--(1.0000,2.8284)--(1.0454,2.8119)--(1.0909,2.7946)--(1.1363,2.7764)--(1.1818,2.7574)--(1.2272,2.7374)--(1.2727,2.7166)--(1.3181,2.6948)--(1.3636,2.6721)--(1.4090,2.6484)--(1.4545,2.6237)--(1.5000,2.5980);
+\draw []  (-2.3584,1.8541) node [rotate=0] {$\bullet$};
+\draw []  (-2.3584,0.0000) node [rotate=0] {$\bullet$};
+\draw (-2.3584,-0.2335) node {\( -x_0\)};
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_examsseptii.pstricks b/auto/pictures_tex/Fig_examsseptii.pstricks
index 32328027d..df5b5e8b6 100644
--- a/auto/pictures_tex/Fig_examsseptii.pstricks
+++ b/auto/pictures_tex/Fig_examsseptii.pstricks
@@ -107,39 +107,39 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (9.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (9.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=blue] (0,-2.00) -- (0,2.00);
-\draw [color=blue] (1.00,-2.00) -- (1.00,2.00);
-\draw [color=blue] (4.00,-2.00) -- (4.00,2.00);
-\draw [color=blue] (9.00,-2.00) -- (9.00,2.00);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,-2.0000) -- (0.0000,2.0000);
+\draw [color=blue] (1.0000,-2.0000) -- (1.0000,2.0000);
+\draw [color=blue] (4.0000,-2.0000) -- (4.0000,2.0000);
+\draw [color=blue] (9.0000,-2.0000) -- (9.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks b/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks
index 644298998..e41decf28 100644
--- a/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks
+++ b/auto/pictures_tex/Fig_ooIHLPooKLIxcH.pstricks
@@ -86,17 +86,17 @@
 %PSTRICKS CODE
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-\draw (0.65195,0.18941) node {$\theta$};
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+\draw (0.6519,0.1894) node {$\theta$};
 
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+\draw [] (0.0000,0.0000) -- (1.9696,-0.3472);
+\draw [] (0.0000,0.0000) -- (1.0000,1.7320);
+\draw (0.1002,1.1407) node {$R$};
+\draw (2.4298,-0.5535) node {$\theta_0$};
+\draw (1.3148,2.1459) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/auto/pictures_tex/Fig_ratrap.pstricks b/auto/pictures_tex/Fig_ratrap.pstricks
index c0c32b68c..640d9fd7c 100644
--- a/auto/pictures_tex/Fig_ratrap.pstricks
+++ b/auto/pictures_tex/Fig_ratrap.pstricks
@@ -71,8 +71,8 @@
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 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
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 % declaring the keys in tikz
@@ -81,19 +81,19 @@
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+\draw [color=blue] (0.0000,2.0000) -- (0.0000,1.0000);
+\draw [color=blue] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.2785) node {$a$};
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.2964,1.0000) node {$a$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3267) node {$b$};
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.2783,2.0000) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/auto/pictures_tex/Fig_senotopologo.pstricks b/auto/pictures_tex/Fig_senotopologo.pstricks
index 180a3c929..c21f663cb 100644
--- a/auto/pictures_tex/Fig_senotopologo.pstricks
+++ b/auto/pictures_tex/Fig_senotopologo.pstricks
@@ -95,34 +95,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.5000,0) -- (8.5000,0);
-\draw [,->,>=latex] (0,-2.2025) -- (0,2.2025);
+\draw [,->,>=latex] (-8.5000,0.0000) -- (8.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.2024) -- (0.0000,2.2024);
 %DEFAULT
 
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+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/auto/pictures_tex/Fig_trigoWedd.pstricks b/auto/pictures_tex/Fig_trigoWedd.pstricks
index 899d83cd8..c96b2e88e 100644
--- a/auto/pictures_tex/Fig_trigoWedd.pstricks
+++ b/auto/pictures_tex/Fig_trigoWedd.pstricks
@@ -75,17 +75,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
-\draw (1.2899,0.75595) node {$z_0$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.21406) node {$q$};
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.31492) node {$1$};
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw []  (0.8660,0.5000) node [rotate=0] {$\bullet$};
+\draw (1.2898,0.7559) node {$z_0$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.2140) node {$q$};
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.3149) node {$1$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/lst_actu.py b/lst_actu.py
index a6e2b959a..fd46eb65c 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -19,8 +19,6 @@
 myRequest.ok_filenames_list.extend(["78_inversion_locale"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
-myRequest.ok_filenames_list.extend(["<++>"])
-myRequest.ok_filenames_list.extend(["<++>"])
 
 myRequest.ok_filenames_list.extend(["134_choses_finales"])
 myRequest.ok_filenames_list.extend(["157_thematique"])
diff --git a/src_phystricks/Fig_ALIzHFm.pstricks.recall b/src_phystricks/Fig_ALIzHFm.pstricks.recall
index e5623dafb..1e5c03fa2 100644
--- a/src_phystricks/Fig_ALIzHFm.pstricks.recall
+++ b/src_phystricks/Fig_ALIzHFm.pstricks.recall
@@ -74,14 +74,14 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.50000,0.86602) node [rotate=0] {$\bullet$};
-\draw (0.76404,1.1452) node {\( z_1\)};
-\draw []  (-0.50000,-0.86602) node [rotate=0] {$\bullet$};
-\draw (-0.76404,-1.1452) node {\( z_2\)};
-\draw [] (2.67,-0.384) -- (-1.67,2.12);
-\draw [] (-2.67,0.384) -- (1.67,-2.12);
-\draw [style=dotted] (2.17,-1.25) -- (-2.17,1.25);
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw []  (0.5000,0.8660) node [rotate=0] {$\bullet$};
+\draw (0.7640,1.1451) node {\( z_1\)};
+\draw []  (-0.5000,-0.8660) node [rotate=0] {$\bullet$};
+\draw (-0.7640,-1.1451) node {\( z_2\)};
+\draw [] (2.6650,-0.3839) -- (-1.6650,2.1160);
+\draw [] (-2.6650,0.3839) -- (1.6650,-2.1160);
+\draw [style=dotted] (2.1650,-1.2500) -- (-2.1650,1.2500);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall b/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall
index 6fbdb6220..4c1689b9c 100644
--- a/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall
+++ b/src_phystricks/Fig_AMDUooZZUOqa.pstricks.recall
@@ -86,17 +86,17 @@
 %PSTRICKS CODE
 %DEFAULT
 
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+\draw [] (0.0000,0.0000) -- (1.9696,-0.3472);
+\draw [] (0.0000,0.0000) -- (1.0000,1.7320);
+\draw (0.1002,1.1407) node {$R$};
+\draw (2.4298,-0.5535) node {$\theta_0$};
+\draw (1.3148,2.1459) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_AdhIntFr.pstricks.recall b/src_phystricks/Fig_AdhIntFr.pstricks.recall
index a1650bae7..857863ea3 100644
--- a/src_phystricks/Fig_AdhIntFr.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFr.pstricks.recall
@@ -83,13 +83,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
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-\draw [,->,>=latex] (0,-0.74998) -- (0,4.2500);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.7499) -- (0.0000,4.2500);
 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,32 +97,32 @@
 % setting the default values
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          hatchthickness=0.4pt}
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+\draw [] (1.0000,2.0000) -- (1.0000,2.0000);
+\draw [] (4.0000,2.0000) -- (4.0000,2.0000);
+\draw []  (3.5000,0.7500) node [rotate=0] {$\bullet$};
 
-\draw [] (3.800,0.7500)--(3.799,0.7690)--(3.798,0.7880)--(3.795,0.8068)--(3.790,0.8253)--(3.785,0.8436)--(3.779,0.8615)--(3.771,0.8789)--(3.762,0.8959)--(3.752,0.9122)--(3.742,0.9279)--(3.730,0.9428)--(3.717,0.9570)--(3.704,0.9704)--(3.689,0.9828)--(3.674,0.9944)--(3.658,1.005)--(3.642,1.014)--(3.625,1.023)--(3.607,1.030)--(3.589,1.036)--(3.571,1.042)--(3.552,1.045)--(3.533,1.048)--(3.514,1.050)--(3.495,1.050)--(3.476,1.049)--(3.457,1.047)--(3.439,1.044)--(3.420,1.039)--(3.402,1.033)--(3.384,1.027)--(3.367,1.019)--(3.350,1.010)--(3.334,0.9998)--(3.318,0.9887)--(3.304,0.9767)--(3.290,0.9638)--(3.276,0.9500)--(3.264,0.9354)--(3.253,0.9201)--(3.243,0.9041)--(3.233,0.8875)--(3.225,0.8703)--(3.218,0.8526)--(3.212,0.8345)--(3.207,0.8161)--(3.204,0.7974)--(3.201,0.7785)--(3.200,0.7595)--(3.200,0.7405)--(3.201,0.7215)--(3.204,0.7026)--(3.207,0.6839)--(3.212,0.6655)--(3.218,0.6474)--(3.225,0.6297)--(3.233,0.6125)--(3.243,0.5959)--(3.253,0.5799)--(3.264,0.5646)--(3.276,0.5500)--(3.290,0.5362)--(3.304,0.5233)--(3.318,0.5113)--(3.334,0.5002)--(3.350,0.4902)--(3.367,0.4812)--(3.384,0.4733)--(3.402,0.4665)--(3.420,0.4608)--(3.439,0.4564)--(3.457,0.4531)--(3.476,0.4509)--(3.495,0.4500)--(3.514,0.4503)--(3.533,0.4518)--(3.552,0.4546)--(3.571,0.4585)--(3.589,0.4635)--(3.607,0.4698)--(3.625,0.4771)--(3.642,0.4856)--(3.658,0.4951)--(3.674,0.5056)--(3.689,0.5172)--(3.704,0.5296)--(3.717,0.5430)--(3.730,0.5572)--(3.742,0.5721)--(3.752,0.5878)--(3.762,0.6041)--(3.771,0.6211)--(3.779,0.6385)--(3.785,0.6564)--(3.790,0.6747)--(3.795,0.6932)--(3.798,0.7120)--(3.799,0.7310)--(3.800,0.7500);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
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+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall b/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall
index ff57c58d7..272312662 100644
--- a/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFrDeux.pstricks.recall
@@ -91,13 +91,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,6.5000);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,6.5000);
 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -105,36 +105,36 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (1.0000,2.0000) -- (1.0000,2.0000);
+\draw [] (3.0000,4.0000) -- (3.0000,6.0000);
 \draw []  (1.0000,2.0000) node [rotate=0] {$\bullet$};
 
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-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
-\draw (-0.29125,6.0000) node {$ 6 $};
-\draw [] (-0.100,6.00) -- (0.100,6.00);
+\draw [] (1.7000,2.0000)--(1.6985,2.0443)--(1.6943,2.0886)--(1.6873,2.1324)--(1.6775,2.1758)--(1.6650,2.2184)--(1.6498,2.2601)--(1.6320,2.3008)--(1.6116,2.3403)--(1.5888,2.3784)--(1.5636,2.4150)--(1.5362,2.4499)--(1.5066,2.4830)--(1.4749,2.5142)--(1.4413,2.5433)--(1.4060,2.5702)--(1.3690,2.5948)--(1.3305,2.6170)--(1.2907,2.6367)--(1.2498,2.6539)--(1.2078,2.6684)--(1.1650,2.6802)--(1.1215,2.6893)--(1.0775,2.6956)--(1.0333,2.6992)--(0.9888,2.6999)--(0.9445,2.6977)--(0.9003,2.6928)--(0.8566,2.6851)--(0.8134,2.6746)--(0.7710,2.6615)--(0.7295,2.6456)--(0.6891,2.6271)--(0.6500,2.6062)--(0.6122,2.5827)--(0.5760,2.5570)--(0.5415,2.5290)--(0.5089,2.4988)--(0.4783,2.4667)--(0.4497,2.4327)--(0.4234,2.3969)--(0.3994,2.3595)--(0.3778,2.3207)--(0.3587,2.2806)--(0.3422,2.2394)--(0.3283,2.1972)--(0.3171,2.1542)--(0.3087,2.1106)--(0.3031,2.0665)--(0.3003,2.0222)--(0.3003,1.9777)--(0.3031,1.9334)--(0.3087,1.8893)--(0.3171,1.8457)--(0.3283,1.8027)--(0.3422,1.7605)--(0.3587,1.7193)--(0.3778,1.6792)--(0.3994,1.6404)--(0.4234,1.6030)--(0.4497,1.5672)--(0.4783,1.5332)--(0.5089,1.5011)--(0.5415,1.4709)--(0.5760,1.4429)--(0.6122,1.4172)--(0.6500,1.3937)--(0.6891,1.3728)--(0.7295,1.3543)--(0.7710,1.3384)--(0.8134,1.3253)--(0.8566,1.3148)--(0.9003,1.3071)--(0.9445,1.3022)--(0.9888,1.3000)--(1.0333,1.3007)--(1.0775,1.3043)--(1.1215,1.3106)--(1.1650,1.3197)--(1.2078,1.3315)--(1.2498,1.3460)--(1.2907,1.3632)--(1.3305,1.3829)--(1.3690,1.4051)--(1.4060,1.4297)--(1.4413,1.4566)--(1.4749,1.4857)--(1.5066,1.5169)--(1.5362,1.5500)--(1.5636,1.5849)--(1.5888,1.6215)--(1.6116,1.6596)--(1.6320,1.6991)--(1.6498,1.7398)--(1.6650,1.7815)--(1.6775,1.8241)--(1.6873,1.8675)--(1.6943,1.9113)--(1.6985,1.9556)--(1.7000,2.0000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
+\draw (-0.2912,6.0000) node {$ 6 $};
+\draw [] (-0.1000,6.0000) -- (0.1000,6.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_AdhIntFrSix.pstricks.recall b/src_phystricks/Fig_AdhIntFrSix.pstricks.recall
index 855693c3c..c04f51d28 100644
--- a/src_phystricks/Fig_AdhIntFrSix.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFrSix.pstricks.recall
@@ -67,116 +67,116 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.1000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.5000);
+\draw [,->,>=latex] (-1.1000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.5000);
 %DEFAULT
-\draw [color=blue] (4.00,0) -- (4.00,4.00);
-\draw [color=blue] (2.00,0) -- (2.00,4.00);
-\draw [color=blue] (1.33,0) -- (1.33,4.00);
-\draw [color=blue] (1.00,0) -- (1.00,4.00);
-\draw [color=blue] (0.800,0) -- (0.800,4.00);
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+\draw [] (0.0000,0.0000) -- (0.0000,4.0000);
+\draw []  (0.0000,3.2000) node [rotate=0] {$\bullet$};
 
-\draw [] (0.600,3.20)--(0.599,3.24)--(0.595,3.28)--(0.589,3.31)--(0.581,3.35)--(0.570,3.39)--(0.557,3.42)--(0.542,3.46)--(0.524,3.49)--(0.505,3.52)--(0.483,3.56)--(0.460,3.59)--(0.434,3.61)--(0.407,3.64)--(0.378,3.67)--(0.348,3.69)--(0.316,3.71)--(0.283,3.73)--(0.249,3.75)--(0.214,3.76)--(0.178,3.77)--(0.141,3.78)--(0.104,3.79)--(0.0665,3.80)--(0.0285,3.80)--(-0.00952,3.80)--(-0.0476,3.80)--(-0.0854,3.79)--(-0.123,3.79)--(-0.160,3.78)--(-0.196,3.77)--(-0.232,3.75)--(-0.266,3.74)--(-0.300,3.72)--(-0.332,3.70)--(-0.363,3.68)--(-0.393,3.65)--(-0.421,3.63)--(-0.447,3.60)--(-0.472,3.57)--(-0.494,3.54)--(-0.515,3.51)--(-0.533,3.47)--(-0.550,3.44)--(-0.564,3.41)--(-0.576,3.37)--(-0.585,3.33)--(-0.592,3.29)--(-0.597,3.26)--(-0.600,3.22)--(-0.600,3.18)--(-0.597,3.14)--(-0.592,3.11)--(-0.585,3.07)--(-0.576,3.03)--(-0.564,2.99)--(-0.550,2.96)--(-0.533,2.93)--(-0.515,2.89)--(-0.494,2.86)--(-0.472,2.83)--(-0.447,2.80)--(-0.421,2.77)--(-0.393,2.75)--(-0.363,2.72)--(-0.332,2.70)--(-0.300,2.68)--(-0.266,2.66)--(-0.232,2.65)--(-0.196,2.63)--(-0.160,2.62)--(-0.123,2.61)--(-0.0854,2.61)--(-0.0476,2.60)--(-0.00952,2.60)--(0.0285,2.60)--(0.0665,2.60)--(0.104,2.61)--(0.141,2.62)--(0.178,2.63)--(0.214,2.64)--(0.249,2.65)--(0.283,2.67)--(0.316,2.69)--(0.348,2.71)--(0.378,2.73)--(0.407,2.76)--(0.434,2.79)--(0.460,2.81)--(0.483,2.84)--(0.505,2.88)--(0.524,2.91)--(0.542,2.94)--(0.557,2.98)--(0.570,3.01)--(0.581,3.05)--(0.589,3.09)--(0.595,3.12)--(0.599,3.16)--(0.600,3.20);
-\draw (4.0000,-0.31492) node {$ 1 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.29125,4.0000) node {$ 1 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [] (0.6000,3.2000)--(0.5987,3.2380)--(0.5951,3.2759)--(0.5891,3.3135)--(0.5807,3.3506)--(0.5700,3.3872)--(0.5570,3.4229)--(0.5417,3.4578)--(0.5243,3.4917)--(0.5047,3.5243)--(0.4831,3.5557)--(0.4596,3.5856)--(0.4342,3.6140)--(0.4071,3.6407)--(0.3783,3.6656)--(0.3480,3.6887)--(0.3163,3.7098)--(0.2833,3.7288)--(0.2492,3.7457)--(0.2141,3.7604)--(0.1781,3.7729)--(0.1414,3.7830)--(0.1041,3.7908)--(0.0665,3.7963)--(0.0285,3.7993)--(-0.0095,3.7999)--(-0.0475,3.7981)--(-0.0853,3.7938)--(-0.1228,3.7872)--(-0.1598,3.7783)--(-0.1962,3.7670)--(-0.2318,3.7534)--(-0.2664,3.7375)--(-0.3000,3.7196)--(-0.3323,3.6995)--(-0.3633,3.6774)--(-0.3929,3.6534)--(-0.4208,3.6276)--(-0.4471,3.6000)--(-0.4716,3.5708)--(-0.4942,3.5402)--(-0.5147,3.5082)--(-0.5333,3.4749)--(-0.5496,3.4405)--(-0.5638,3.4052)--(-0.5756,3.3690)--(-0.5852,3.3321)--(-0.5924,3.2948)--(-0.5972,3.2570)--(-0.5996,3.2190)--(-0.5996,3.1809)--(-0.5972,3.1429)--(-0.5924,3.1051)--(-0.5852,3.0678)--(-0.5756,3.0309)--(-0.5638,2.9947)--(-0.5496,2.9594)--(-0.5333,2.9250)--(-0.5147,2.8917)--(-0.4942,2.8597)--(-0.4716,2.8291)--(-0.4471,2.7999)--(-0.4208,2.7723)--(-0.3929,2.7465)--(-0.3633,2.7225)--(-0.3323,2.7004)--(-0.3000,2.6803)--(-0.2664,2.6624)--(-0.2318,2.6465)--(-0.1962,2.6329)--(-0.1598,2.6216)--(-0.1228,2.6127)--(-0.0853,2.6061)--(-0.0475,2.6018)--(-0.0095,2.6000)--(0.0285,2.6006)--(0.0665,2.6036)--(0.1041,2.6091)--(0.1414,2.6169)--(0.1781,2.6270)--(0.2141,2.6395)--(0.2492,2.6542)--(0.2833,2.6711)--(0.3163,2.6901)--(0.3480,2.7112)--(0.3783,2.7343)--(0.4071,2.7592)--(0.4342,2.7859)--(0.4596,2.8143)--(0.4831,2.8442)--(0.5047,2.8756)--(0.5243,2.9082)--(0.5417,2.9421)--(0.5570,2.9770)--(0.5700,3.0127)--(0.5807,3.0493)--(0.5891,3.0864)--(0.5951,3.1240)--(0.5987,3.1619)--(0.6000,3.2000);
+\draw (4.0000,-0.3149) node {$ 1 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.2912,4.0000) node {$ 1 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall b/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall
index 0ccc5fe2d..cc82200c7 100644
--- a/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall
+++ b/src_phystricks/Fig_AdhIntFrTrois.pstricks.recall
@@ -79,35 +79,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (8.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (8.5000,0.0000);
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+\draw [color=gray,style=dashed] 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+\draw [color=gray,style=dashed] 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+\draw [color=gray,style=dashed] 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+\draw [color=gray,style=dashed] 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
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -115,37 +115,37 @@
 % setting the default values
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+\draw [color=blue,style=solid] (4.0000,3.0000) -- (4.0000,0.8414);
+\draw []  (0.0000,0.7000) node [rotate=0] {$\bullet$};
 
-\draw [] (0.400,0.700)--(0.399,0.706)--(0.397,0.713)--(0.393,0.719)--(0.387,0.725)--(0.380,0.731)--(0.371,0.737)--(0.361,0.743)--(0.350,0.749)--(0.336,0.754)--(0.322,0.759)--(0.306,0.764)--(0.289,0.769)--(0.271,0.773)--(0.252,0.778)--(0.232,0.781)--(0.211,0.785)--(0.189,0.788)--(0.166,0.791)--(0.143,0.793)--(0.119,0.795)--(0.0943,0.797)--(0.0695,0.798)--(0.0443,0.799)--(0.0190,0.800)--(-0.00635,0.800)--(-0.0317,0.800)--(-0.0569,0.799)--(-0.0819,0.798)--(-0.107,0.796)--(-0.131,0.794)--(-0.155,0.792)--(-0.178,0.790)--(-0.200,0.787)--(-0.222,0.783)--(-0.242,0.780)--(-0.262,0.776)--(-0.281,0.771)--(-0.298,0.767)--(-0.314,0.762)--(-0.329,0.757)--(-0.343,0.751)--(-0.356,0.746)--(-0.366,0.740)--(-0.376,0.734)--(-0.384,0.728)--(-0.390,0.722)--(-0.395,0.716)--(-0.398,0.710)--(-0.400,0.703)--(-0.400,0.697)--(-0.398,0.690)--(-0.395,0.684)--(-0.390,0.678)--(-0.384,0.672)--(-0.376,0.666)--(-0.366,0.660)--(-0.356,0.654)--(-0.343,0.649)--(-0.329,0.643)--(-0.314,0.638)--(-0.298,0.633)--(-0.281,0.629)--(-0.262,0.624)--(-0.242,0.620)--(-0.222,0.617)--(-0.200,0.613)--(-0.178,0.610)--(-0.155,0.608)--(-0.131,0.606)--(-0.107,0.604)--(-0.0819,0.602)--(-0.0569,0.601)--(-0.0317,0.600)--(-0.00635,0.600)--(0.0190,0.600)--(0.0443,0.601)--(0.0695,0.602)--(0.0943,0.603)--(0.119,0.604)--(0.143,0.607)--(0.166,0.609)--(0.189,0.612)--(0.211,0.615)--(0.232,0.619)--(0.252,0.622)--(0.271,0.627)--(0.289,0.631)--(0.306,0.636)--(0.322,0.641)--(0.336,0.646)--(0.350,0.651)--(0.361,0.657)--(0.371,0.663)--(0.380,0.669)--(0.387,0.675)--(0.393,0.681)--(0.397,0.687)--(0.399,0.694)--(0.400,0.700);
-\draw (-4.0000,-0.32983) node {$ -1 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (4.0000,-0.31492) node {$ 1 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (8.0000,-0.31492) node {$ 2 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [] (0.4000,0.7000)--(0.3991,0.7063)--(0.3967,0.7126)--(0.3927,0.7189)--(0.3871,0.7251)--(0.3800,0.7312)--(0.3713,0.7371)--(0.3611,0.7429)--(0.3495,0.7486)--(0.3365,0.7540)--(0.3221,0.7592)--(0.3064,0.7642)--(0.2894,0.7690)--(0.2714,0.7734)--(0.2522,0.7776)--(0.2320,0.7814)--(0.2108,0.7849)--(0.1889,0.7881)--(0.1661,0.7909)--(0.1427,0.7934)--(0.1187,0.7954)--(0.0943,0.7971)--(0.0694,0.7984)--(0.0443,0.7993)--(0.0190,0.7998)--(-0.0063,0.7999)--(-0.0316,0.7996)--(-0.0569,0.7989)--(-0.0819,0.7978)--(-0.1065,0.7963)--(-0.1308,0.7945)--(-0.1545,0.7922)--(-0.1776,0.7895)--(-0.2000,0.7866)--(-0.2215,0.7832)--(-0.2422,0.7795)--(-0.2619,0.7755)--(-0.2805,0.7712)--(-0.2981,0.7666)--(-0.3144,0.7618)--(-0.3294,0.7567)--(-0.3431,0.7513)--(-0.3555,0.7458)--(-0.3664,0.7400)--(-0.3758,0.7342)--(-0.3837,0.7281)--(-0.3901,0.7220)--(-0.3949,0.7158)--(-0.3981,0.7095)--(-0.3997,0.7031)--(-0.3997,0.6968)--(-0.3981,0.6904)--(-0.3949,0.6841)--(-0.3901,0.6779)--(-0.3837,0.6718)--(-0.3758,0.6657)--(-0.3664,0.6599)--(-0.3555,0.6541)--(-0.3431,0.6486)--(-0.3294,0.6432)--(-0.3144,0.6381)--(-0.2981,0.6333)--(-0.2805,0.6287)--(-0.2619,0.6244)--(-0.2422,0.6204)--(-0.2215,0.6167)--(-0.2000,0.6133)--(-0.1776,0.6104)--(-0.1545,0.6077)--(-0.1308,0.6054)--(-0.1065,0.6036)--(-0.0819,0.6021)--(-0.0569,0.6010)--(-0.0316,0.6003)--(-0.0063,0.6000)--(0.0190,0.6001)--(0.0443,0.6006)--(0.0694,0.6015)--(0.0943,0.6028)--(0.1187,0.6045)--(0.1427,0.6065)--(0.1661,0.6090)--(0.1889,0.6118)--(0.2108,0.6150)--(0.2320,0.6185)--(0.2522,0.6223)--(0.2714,0.6265)--(0.2894,0.6309)--(0.3064,0.6357)--(0.3221,0.6407)--(0.3365,0.6459)--(0.3495,0.6513)--(0.3611,0.6570)--(0.3713,0.6628)--(0.3800,0.6687)--(0.3871,0.6748)--(0.3927,0.6810)--(0.3967,0.6873)--(0.3991,0.6936)--(0.4000,0.7000);
+\draw (-4.0000,-0.3298) node {$ -1 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 1 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 2 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ArcLongueurFinesse.pstricks.recall b/src_phystricks/Fig_ArcLongueurFinesse.pstricks.recall
index 3ce96b5bd..8ba02ddc0 100644
--- a/src_phystricks/Fig_ArcLongueurFinesse.pstricks.recall
+++ b/src_phystricks/Fig_ArcLongueurFinesse.pstricks.recall
@@ -65,23 +65,23 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=blue,style=dashed] (0,0.70000)--(0.13328,0.68735)--(0.26656,0.64986)--(0.39984,0.58888)--(0.53312,0.50661)--(0.66640,0.40604)--(0.79968,0.29079)--(0.93296,0.16503)--(1.0662,0.033307)--(1.1995,-0.099620)--(1.3328,-0.22895)--(1.4661,-0.35000)--(1.5994,-0.45840)--(1.7326,-0.55024)--(1.8659,-0.62218)--(1.9992,-0.67165)--(2.1325,-0.69683)--(2.2658,-0.69683)--(2.3990,-0.67165)--(2.5323,-0.62218)--(2.6656,-0.55024)--(2.7989,-0.45840)--(2.9322,-0.35000)--(3.0654,-0.22895)--(3.1987,-0.099620)--(3.3320,0.033307)--(3.4653,0.16503)--(3.5986,0.29079)--(3.7318,0.40604)--(3.8651,0.50661)--(3.9984,0.58888)--(4.1317,0.64986)--(4.2650,0.68735)--(4.3982,0.70000)--(4.5315,0.68735)--(4.6648,0.64986)--(4.7981,0.58888)--(4.9314,0.50661)--(5.0646,0.40604)--(5.1979,0.29079)--(5.3312,0.16503)--(5.4645,0.033307)--(5.5977,-0.099620)--(5.7310,-0.22895)--(5.8643,-0.35000)--(5.9976,-0.45840)--(6.1309,-0.55024)--(6.2641,-0.62218)--(6.3974,-0.67165)--(6.5307,-0.69683)--(6.6640,-0.69683)--(6.7973,-0.67165)--(6.9305,-0.62218)--(7.0638,-0.55024)--(7.1971,-0.45840)--(7.3304,-0.35000)--(7.4637,-0.22895)--(7.5969,-0.099620)--(7.7302,0.033307)--(7.8635,0.16503)--(7.9968,0.29079)--(8.1301,0.40604)--(8.2633,0.50661)--(8.3966,0.58888)--(8.5299,0.64986)--(8.6632,0.68735)--(8.7965,0.70000)--(8.9297,0.68735)--(9.0630,0.64986)--(9.1963,0.58888)--(9.3296,0.50661)--(9.4629,0.40604)--(9.5961,0.29079)--(9.7294,0.16503)--(9.8627,0.033307)--(9.9960,-0.099620)--(10.129,-0.22895)--(10.263,-0.35000)--(10.396,-0.45840)--(10.529,-0.55024)--(10.662,-0.62218)--(10.796,-0.67165)--(10.929,-0.69683)--(11.062,-0.69683)--(11.195,-0.67165)--(11.329,-0.62218)--(11.462,-0.55024)--(11.595,-0.45840)--(11.729,-0.35000)--(11.862,-0.22895)--(11.995,-0.099620)--(12.128,0.033307)--(12.262,0.16503)--(12.395,0.29079)--(12.528,0.40604)--(12.662,0.50661)--(12.795,0.58888)--(12.928,0.64986)--(13.061,0.68735)--(13.195,0.70000);
-\draw [color=red] (0,0.700) -- (2.64,-0.566);
-\draw [color=red] (2.64,-0.566) -- (5.28,0.216);
-\draw [color=red] (5.28,0.216) -- (7.92,0.217);
-\draw [color=red] (7.92,0.217) -- (10.6,-0.566);
-\draw [color=red] (10.6,-0.566) -- (13.2,0.700);
-\draw [color=green] (0,0.700) -- (1.20,-0.0996);
-\draw [color=green] (1.20,-0.0996) -- (2.40,-0.672);
-\draw [color=green] (2.40,-0.672) -- (3.60,0.291);
-\draw [color=green] (3.60,0.291) -- (4.80,0.589);
-\draw [color=green] (4.80,0.589) -- (6.00,-0.459);
-\draw [color=green] (6.00,-0.459) -- (7.20,-0.459);
-\draw [color=green] (7.20,-0.459) -- (8.40,0.589);
-\draw [color=green] (8.40,0.589) -- (9.60,0.291);
-\draw [color=green] (9.60,0.291) -- (10.8,-0.672);
-\draw [color=green] (10.8,-0.672) -- (12.0,-0.0991);
-\draw [color=green] (12.0,-0.0991) -- (13.2,0.700);
+\draw [color=blue,style=dashed] (0.0000,0.7000)--(0.1332,0.6873)--(0.2665,0.6498)--(0.3998,0.5888)--(0.5331,0.5066)--(0.6663,0.4060)--(0.7996,0.2907)--(0.9329,0.1650)--(1.0662,0.0333)--(1.1995,-0.0996)--(1.3327,-0.2289)--(1.4660,-0.3500)--(1.5993,-0.4584)--(1.7326,-0.5502)--(1.8659,-0.6221)--(1.9991,-0.6716)--(2.1324,-0.6968)--(2.2657,-0.6968)--(2.3990,-0.6716)--(2.5323,-0.6221)--(2.6655,-0.5502)--(2.7988,-0.4584)--(2.9321,-0.3500)--(3.0654,-0.2289)--(3.1987,-0.0996)--(3.3319,0.0333)--(3.4652,0.1650)--(3.5985,0.2907)--(3.7318,0.4060)--(3.8651,0.5066)--(3.9983,0.5888)--(4.1316,0.6498)--(4.2649,0.6873)--(4.3982,0.7000)--(4.5315,0.6873)--(4.6647,0.6498)--(4.7980,0.5888)--(4.9313,0.5066)--(5.0646,0.4060)--(5.1979,0.2907)--(5.3311,0.1650)--(5.4644,0.0333)--(5.5977,-0.0996)--(5.7310,-0.2289)--(5.8643,-0.3499)--(5.9975,-0.4584)--(6.1308,-0.5502)--(6.2641,-0.6221)--(6.3974,-0.6716)--(6.5307,-0.6968)--(6.6639,-0.6968)--(6.7972,-0.6716)--(6.9305,-0.6221)--(7.0638,-0.5502)--(7.1971,-0.4584)--(7.3303,-0.3500)--(7.4636,-0.2289)--(7.5969,-0.0996)--(7.7302,0.0333)--(7.8635,0.1650)--(7.9967,0.2907)--(8.1300,0.4060)--(8.2633,0.5066)--(8.3966,0.5888)--(8.5299,0.6498)--(8.6631,0.6873)--(8.7964,0.7000)--(8.9297,0.6873)--(9.0630,0.6498)--(9.1962,0.5888)--(9.3295,0.5066)--(9.4628,0.4060)--(9.5961,0.2907)--(9.7294,0.1650)--(9.8626,0.0333)--(9.9959,-0.0996)--(10.129,-0.2289)--(10.262,-0.3499)--(10.395,-0.4584)--(10.529,-0.5502)--(10.662,-0.6221)--(10.795,-0.6716)--(10.928,-0.6968)--(11.062,-0.6968)--(11.195,-0.6716)--(11.328,-0.6221)--(11.462,-0.5502)--(11.595,-0.4584)--(11.728,-0.3500)--(11.861,-0.2289)--(11.995,-0.0996)--(12.128,0.0333)--(12.261,0.1650)--(12.395,0.2907)--(12.528,0.4060)--(12.661,0.5066)--(12.794,0.5888)--(12.928,0.6498)--(13.061,0.6873)--(13.194,0.7000);
+\draw [color=red] (0.0000,0.7000) -- (2.6389,-0.5663);
+\draw [color=red] (2.6389,-0.5663) -- (5.2778,0.2163);
+\draw [color=red] (5.2778,0.2163) -- (7.9168,0.2163);
+\draw [color=red] (7.9168,0.2163) -- (10.555,-0.5663);
+\draw [color=red] (10.555,-0.5663) -- (13.194,0.7000);
+\draw [color=green] (0.0000,0.7000) -- (1.1995,-0.0996);
+\draw [color=green] (1.1995,-0.0996) -- (2.3990,-0.6716);
+\draw [color=green] (2.3990,-0.6716) -- (3.5985,0.2907);
+\draw [color=green] (3.5985,0.2907) -- (4.7980,0.5888);
+\draw [color=green] (4.7980,0.5888) -- (5.9975,-0.4584);
+\draw [color=green] (5.9975,-0.4584) -- (7.1971,-0.4584);
+\draw [color=green] (7.1971,-0.4584) -- (8.3966,0.5888);
+\draw [color=green] (8.3966,0.5888) -- (9.5961,0.2907);
+\draw [color=green] (9.5961,0.2907) -- (10.795,-0.6716);
+\draw [color=green] (10.795,-0.6716) -- (11.995,-0.0996);
+\draw [color=green] (11.995,-0.0996) -- (13.194,0.7000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall b/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall
index a64d6dad8..5fddf7ead 100644
--- a/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall
+++ b/src_phystricks/Fig_BEHTooWsdrys.pstricks.recall
@@ -65,58 +65,58 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
+\fill [color=cyan] (1.6000,-1.2000) -- (3.0666,-1.2000) -- (3.0666,-1.2000) -- (3.0666,1.2000) -- (3.0666,1.2000) -- (1.6000,1.2000) -- (1.6000,1.2000) -- (1.6000,-1.2000) --  cycle;
 
 
 \draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
-\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
-\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
-\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
-\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
-\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,-0.6666) -- (2.2250,-0.6666);
+\draw [,->,>=latex] (1.6000,-0.3333) -- (2.2250,-0.3333);
+\draw [,->,>=latex] (1.6000,0.0000) -- (2.2250,0.0000);
+\draw [,->,>=latex] (1.6000,0.3333) -- (2.2250,0.3333);
+\draw [,->,>=latex] (1.6000,0.6666) -- (2.2250,0.6666);
 \draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
 \draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
-\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
-\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
-\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
-\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
-\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,-0.6666) -- (2.7619,-0.6666);
+\draw [,->,>=latex] (2.3333,-0.3333) -- (2.7619,-0.3333);
+\draw [,->,>=latex] (2.3333,0.0000) -- (2.7619,0.0000);
+\draw [,->,>=latex] (2.3333,0.3333) -- (2.7619,0.3333);
+\draw [,->,>=latex] (2.3333,0.6666) -- (2.7619,0.6666);
 \draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
-\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
-\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
-\draw [,->,>=latex] (3.0667,-0.33333) -- (3.3928,-0.33333);
-\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
-\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
-\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
-\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
-\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
-\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
-\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
-\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
-\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
-\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
-\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (3.0666,-1.0000) -- (3.3927,-1.0000);
+\draw [,->,>=latex] (3.0666,-0.6666) -- (3.3927,-0.6666);
+\draw [,->,>=latex] (3.0666,-0.3333) -- (3.3927,-0.3333);
+\draw [,->,>=latex] (3.0666,0.0000) -- (3.3927,0.0000);
+\draw [,->,>=latex] (3.0666,0.3333) -- (3.3927,0.3333);
+\draw [,->,>=latex] (3.0666,0.6666) -- (3.3927,0.6666);
+\draw [,->,>=latex] (3.0666,1.0000) -- (3.3927,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0631,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.6666) -- (4.0631,-0.6666);
+\draw [,->,>=latex] (3.8000,-0.3333) -- (4.0631,-0.3333);
+\draw [,->,>=latex] (3.8000,0.0000) -- (4.0631,0.0000);
+\draw [,->,>=latex] (3.8000,0.3333) -- (4.0631,0.3333);
+\draw [,->,>=latex] (3.8000,0.6666) -- (4.0631,0.6666);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0631,1.0000);
 \draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
-\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
-\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
-\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
-\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
-\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,-0.6666) -- (4.7539,-0.6666);
+\draw [,->,>=latex] (4.5333,-0.3333) -- (4.7539,-0.3333);
+\draw [,->,>=latex] (4.5333,0.0000) -- (4.7539,0.0000);
+\draw [,->,>=latex] (4.5333,0.3333) -- (4.7539,0.3333);
+\draw [,->,>=latex] (4.5333,0.6666) -- (4.7539,0.6666);
 \draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
-\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
-\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
-\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
-\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
-\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
-\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
-\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
-\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
-\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
-\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
-\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
-\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
-\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
-\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
+\draw [,->,>=latex] (5.2666,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2666,-0.6666) -- (5.4565,-0.6666);
+\draw [,->,>=latex] (5.2666,-0.3333) -- (5.4565,-0.3333);
+\draw [,->,>=latex] (5.2666,0.0000) -- (5.4565,0.0000);
+\draw [,->,>=latex] (5.2666,0.3333) -- (5.4565,0.3333);
+\draw [,->,>=latex] (5.2666,0.6666) -- (5.4565,0.6666);
+\draw [,->,>=latex] (5.2666,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1666,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.6666) -- (6.1666,-0.6666);
+\draw [,->,>=latex] (6.0000,-0.3333) -- (6.1666,-0.3333);
+\draw [,->,>=latex] (6.0000,0.0000) -- (6.1666,0.0000);
+\draw [,->,>=latex] (6.0000,0.3333) -- (6.1666,0.3333);
+\draw [,->,>=latex] (6.0000,0.6666) -- (6.1666,0.6666);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1666,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall b/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall
index 7ac338be2..5d8af556c 100644
--- a/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall
+++ b/src_phystricks/Fig_BIFooDsvVHb.pstricks.recall
@@ -72,21 +72,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-2.0000) -- (0,2.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [] (1.50,0)--(1.50,0.0951)--(1.49,0.190)--(1.47,0.284)--(1.45,0.377)--(1.43,0.468)--(1.39,0.557)--(1.35,0.645)--(1.31,0.729)--(1.26,0.811)--(1.21,0.889)--(1.15,0.964)--(1.09,1.04)--(1.02,1.10)--(0.946,1.16)--(0.870,1.22)--(0.791,1.27)--(0.708,1.32)--(0.623,1.36)--(0.535,1.40)--(0.445,1.43)--(0.354,1.46)--(0.260,1.48)--(0.166,1.49)--(0.0714,1.50)--(-0.0238,1.50)--(-0.119,1.50)--(-0.213,1.48)--(-0.307,1.47)--(-0.400,1.45)--(-0.491,1.42)--(-0.580,1.38)--(-0.666,1.34)--(-0.750,1.30)--(-0.831,1.25)--(-0.908,1.19)--(-0.982,1.13)--(-1.05,1.07)--(-1.12,1.00)--(-1.18,0.927)--(-1.24,0.851)--(-1.29,0.771)--(-1.33,0.687)--(-1.37,0.601)--(-1.41,0.513)--(-1.44,0.423)--(-1.46,0.330)--(-1.48,0.237)--(-1.49,0.143)--(-1.50,0.0476)--(-1.50,-0.0476)--(-1.49,-0.143)--(-1.48,-0.237)--(-1.46,-0.330)--(-1.44,-0.423)--(-1.41,-0.513)--(-1.37,-0.601)--(-1.33,-0.687)--(-1.29,-0.771)--(-1.24,-0.851)--(-1.18,-0.927)--(-1.12,-1.00)--(-1.05,-1.07)--(-0.982,-1.13)--(-0.908,-1.19)--(-0.831,-1.25)--(-0.750,-1.30)--(-0.666,-1.34)--(-0.580,-1.38)--(-0.491,-1.42)--(-0.400,-1.45)--(-0.307,-1.47)--(-0.213,-1.48)--(-0.119,-1.50)--(-0.0238,-1.50)--(0.0714,-1.50)--(0.166,-1.49)--(0.260,-1.48)--(0.354,-1.46)--(0.445,-1.43)--(0.535,-1.40)--(0.623,-1.36)--(0.708,-1.32)--(0.791,-1.27)--(0.870,-1.22)--(0.946,-1.16)--(1.02,-1.10)--(1.09,-1.04)--(1.15,-0.964)--(1.21,-0.889)--(1.26,-0.811)--(1.31,-0.729)--(1.35,-0.645)--(1.39,-0.557)--(1.43,-0.468)--(1.45,-0.377)--(1.47,-0.284)--(1.49,-0.190)--(1.50,-0.0951)--(1.50,0);
-\draw (1.0127,0.25615) node {\( \theta\)};
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw (0.7147,0.3590) node {\( \theta\)};
 
-\draw [] (0.500,0)--(0.500,0.00264)--(0.500,0.00529)--(0.500,0.00793)--(0.500,0.0106)--(0.500,0.0132)--(0.500,0.0159)--(0.500,0.0185)--(0.500,0.0211)--(0.499,0.0238)--(0.499,0.0264)--(0.499,0.0291)--(0.499,0.0317)--(0.499,0.0344)--(0.499,0.0370)--(0.498,0.0396)--(0.498,0.0423)--(0.498,0.0449)--(0.498,0.0475)--(0.497,0.0502)--(0.497,0.0528)--(0.497,0.0554)--(0.497,0.0580)--(0.496,0.0607)--(0.496,0.0633)--(0.496,0.0659)--(0.495,0.0685)--(0.495,0.0712)--(0.495,0.0738)--(0.494,0.0764)--(0.494,0.0790)--(0.493,0.0816)--(0.493,0.0842)--(0.492,0.0868)--(0.492,0.0894)--(0.491,0.0920)--(0.491,0.0946)--(0.490,0.0972)--(0.490,0.0998)--(0.489,0.102)--(0.489,0.105)--(0.488,0.108)--(0.488,0.110)--(0.487,0.113)--(0.487,0.115)--(0.486,0.118)--(0.485,0.120)--(0.485,0.123)--(0.484,0.126)--(0.483,0.128)--(0.483,0.131)--(0.482,0.133)--(0.481,0.136)--(0.480,0.138)--(0.480,0.141)--(0.479,0.143)--(0.478,0.146)--(0.477,0.148)--(0.477,0.151)--(0.476,0.154)--(0.475,0.156)--(0.474,0.159)--(0.473,0.161)--(0.473,0.164)--(0.472,0.166)--(0.471,0.169)--(0.470,0.171)--(0.469,0.173)--(0.468,0.176)--(0.467,0.178)--(0.466,0.181)--(0.465,0.183)--(0.464,0.186)--(0.463,0.188)--(0.462,0.191)--(0.461,0.193)--(0.460,0.196)--(0.459,0.198)--(0.458,0.200)--(0.457,0.203)--(0.456,0.205)--(0.455,0.208)--(0.454,0.210)--(0.453,0.213)--(0.451,0.215)--(0.450,0.217)--(0.449,0.220)--(0.448,0.222)--(0.447,0.224)--(0.446,0.227)--(0.444,0.229)--(0.443,0.231)--(0.442,0.234)--(0.441,0.236)--(0.439,0.238)--(0.438,0.241)--(0.437,0.243)--(0.436,0.245)--(0.434,0.248)--(0.433,0.250);
-\draw []  (1.2990,0) node [rotate=0] {$\bullet$};
-\draw (1.2990,-0.27858) node {\( x\)};
-\draw []  (0,0.75000) node [rotate=0] {$\bullet$};
-\draw (-0.29602,0.75000) node {\( y\)};
-\draw [] (0,0) -- (1.30,0.750);
-\draw [style=dashed] (1.30,0.750) -- (1.30,0);
-\draw [style=dashed] (1.30,0.750) -- (0,0.750);
+\draw [] (0.5000,0.0000)--(0.4999,0.0048)--(0.4999,0.0096)--(0.4997,0.0145)--(0.4996,0.0193)--(0.4994,0.0242)--(0.4991,0.0290)--(0.4988,0.0339)--(0.4984,0.0387)--(0.4980,0.0435)--(0.4976,0.0484)--(0.4971,0.0532)--(0.4966,0.0580)--(0.4960,0.0628)--(0.4954,0.0676)--(0.4947,0.0724)--(0.4939,0.0772)--(0.4932,0.0820)--(0.4924,0.0868)--(0.4915,0.0915)--(0.4906,0.0963)--(0.4896,0.1011)--(0.4886,0.1058)--(0.4876,0.1105)--(0.4865,0.1153)--(0.4853,0.1200)--(0.4841,0.1247)--(0.4829,0.1294)--(0.4816,0.1340)--(0.4803,0.1387)--(0.4789,0.1434)--(0.4775,0.1480)--(0.4761,0.1526)--(0.4746,0.1572)--(0.4730,0.1618)--(0.4714,0.1664)--(0.4698,0.1710)--(0.4681,0.1755)--(0.4664,0.1800)--(0.4646,0.1846)--(0.4628,0.1890)--(0.4610,0.1935)--(0.4591,0.1980)--(0.4571,0.2024)--(0.4551,0.2069)--(0.4531,0.2113)--(0.4510,0.2156)--(0.4489,0.2200)--(0.4468,0.2243)--(0.4446,0.2287)--(0.4423,0.2330)--(0.4401,0.2373)--(0.4377,0.2415)--(0.4354,0.2457)--(0.4330,0.2500)--(0.4305,0.2541)--(0.4280,0.2583)--(0.4255,0.2624)--(0.4229,0.2666)--(0.4203,0.2706)--(0.4177,0.2747)--(0.4150,0.2787)--(0.4123,0.2828)--(0.4095,0.2867)--(0.4067,0.2907)--(0.4039,0.2946)--(0.4010,0.2985)--(0.3981,0.3024)--(0.3951,0.3063)--(0.3922,0.3101)--(0.3891,0.3139)--(0.3861,0.3176)--(0.3830,0.3213)--(0.3798,0.3250)--(0.3767,0.3287)--(0.3735,0.3323)--(0.3702,0.3360)--(0.3669,0.3395)--(0.3636,0.3431)--(0.3603,0.3466)--(0.3569,0.3501)--(0.3535,0.3535)--(0.3501,0.3569)--(0.3466,0.3603)--(0.3431,0.3636)--(0.3395,0.3669)--(0.3360,0.3702)--(0.3323,0.3735)--(0.3287,0.3767)--(0.3250,0.3798)--(0.3213,0.3830)--(0.3176,0.3861)--(0.3139,0.3891)--(0.3101,0.3922)--(0.3063,0.3951)--(0.3024,0.3981)--(0.2985,0.4010)--(0.2946,0.4039)--(0.2907,0.4067)--(0.2867,0.4095);
+\draw []  (1.1471,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.1471,-0.2785) node {\( x\)};
+\draw []  (0.0000,1.6383) node [rotate=0] {$\bullet$};
+\draw (-0.2960,1.6383) node {\( y\)};
+\draw [] (0.0000,0.0000) -- (1.1471,1.6383);
+\draw [style=dashed] (1.1471,1.6383) -- (1.1471,0.0000);
+\draw [style=dashed] (1.1471,1.6383) -- (0.0000,1.6383);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall b/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall
index 3f5418461..39102d176 100644
--- a/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall
+++ b/src_phystricks/Fig_BQXKooPqSEMN.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.2499);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (0.0000,0.0000) -- (0.0000,0.9722);
+\draw [] (5.0000,2.6388) -- (5.0000,0.0000);
 
-\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3485,0.6490)--(-0.2727,0.7215)--(-0.1970,0.7928)--(-0.1212,0.8628)--(-0.04545,0.9316)--(0.03030,0.9991)--(0.1061,1.065)--(0.1818,1.130)--(0.2576,1.194)--(0.3333,1.256)--(0.4091,1.317)--(0.4848,1.377)--(0.5606,1.436)--(0.6364,1.493)--(0.7121,1.549)--(0.7879,1.604)--(0.8636,1.657)--(0.9394,1.709)--(1.015,1.760)--(1.091,1.810)--(1.167,1.858)--(1.242,1.905)--(1.318,1.951)--(1.394,1.995)--(1.470,2.039)--(1.545,2.081)--(1.621,2.121)--(1.697,2.161)--(1.773,2.199)--(1.848,2.236)--(1.924,2.271)--(2.000,2.306)--(2.076,2.339)--(2.152,2.370)--(2.227,2.401)--(2.303,2.430)--(2.379,2.458)--(2.455,2.485)--(2.530,2.510)--(2.606,2.534)--(2.682,2.557)--(2.758,2.578)--(2.833,2.599)--(2.909,2.618)--(2.985,2.635)--(3.061,2.652)--(3.136,2.667)--(3.212,2.681)--(3.288,2.694)--(3.364,2.705)--(3.439,2.715)--(3.515,2.724)--(3.591,2.731)--(3.667,2.738)--(3.742,2.743)--(3.818,2.746)--(3.894,2.749)--(3.970,2.750)--(4.045,2.750)--(4.121,2.748)--(4.197,2.746)--(4.273,2.742)--(4.349,2.737)--(4.424,2.730)--(4.500,2.722)--(4.576,2.713)--(4.651,2.703)--(4.727,2.691)--(4.803,2.678)--(4.879,2.664)--(4.955,2.649)--(5.030,2.632)--(5.106,2.614)--(5.182,2.595)--(5.258,2.574)--(5.333,2.552)--(5.409,2.529)--(5.485,2.505)--(5.561,2.479)--(5.636,2.452)--(5.712,2.424)--(5.788,2.395)--(5.864,2.364)--(5.939,2.332)--(6.015,2.299)--(6.091,2.264)--(6.167,2.228)--(6.242,2.191)--(6.318,2.153)--(6.394,2.113)--(6.470,2.072)--(6.545,2.030)--(6.621,1.987)--(6.697,1.942)--(6.773,1.896)--(6.849,1.848)--(6.924,1.800)--(7.000,1.750);
+\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3484,0.6489)--(-0.2727,0.7215)--(-0.1969,0.7928)--(-0.1212,0.8628)--(-0.0454,0.9315)--(0.0303,0.9990)--(0.1060,1.0652)--(0.1818,1.1301)--(0.2575,1.1938)--(0.3333,1.2561)--(0.4090,1.3172)--(0.4848,1.3770)--(0.5606,1.4356)--(0.6363,1.4928)--(0.7121,1.5488)--(0.7878,1.6035)--(0.8636,1.6570)--(0.9393,1.7091)--(1.0151,1.7600)--(1.0909,1.8096)--(1.1666,1.8580)--(1.2424,1.9050)--(1.3181,1.9508)--(1.3939,1.9953)--(1.4696,2.0386)--(1.5454,2.0805)--(1.6212,2.1212)--(1.6969,2.1606)--(1.7727,2.1988)--(1.8484,2.2356)--(1.9242,2.2712)--(2.0000,2.3055)--(2.0757,2.3385)--(2.1515,2.3703)--(2.2272,2.4008)--(2.3030,2.4300)--(2.3787,2.4579)--(2.4545,2.4846)--(2.5303,2.5099)--(2.6060,2.5341)--(2.6818,2.5569)--(2.7575,2.5784)--(2.8333,2.5987)--(2.9090,2.6177)--(2.9848,2.6354)--(3.0606,2.6519)--(3.1363,2.6671)--(3.2121,2.6810)--(3.2878,2.6936)--(3.3636,2.7050)--(3.4393,2.7150)--(3.5151,2.7238)--(3.5909,2.7314)--(3.6666,2.7376)--(3.7424,2.7426)--(3.8181,2.7463)--(3.8939,2.7487)--(3.9696,2.7498)--(4.0454,2.7497)--(4.1212,2.7483)--(4.1969,2.7456)--(4.2727,2.7417)--(4.3484,2.7365)--(4.4242,2.7300)--(4.5000,2.7222)--(4.5757,2.7131)--(4.6515,2.7028)--(4.7272,2.6912)--(4.8030,2.6783)--(4.8787,2.6641)--(4.9545,2.6487)--(5.0303,2.6320)--(5.1060,2.6140)--(5.1818,2.5948)--(5.2575,2.5742)--(5.3333,2.5524)--(5.4090,2.5293)--(5.4848,2.5050)--(5.5606,2.4793)--(5.6363,2.4524)--(5.7121,2.4242)--(5.7878,2.3948)--(5.8636,2.3640)--(5.9393,2.3320)--(6.0151,2.2987)--(6.0909,2.2642)--(6.1666,2.2283)--(6.2424,2.1912)--(6.3181,2.1528)--(6.3939,2.1132)--(6.4696,2.0722)--(6.5454,2.0300)--(6.6212,1.9865)--(6.6969,1.9418)--(6.7727,1.8957)--(6.8484,1.8484)--(6.9242,1.7998)--(7.0000,1.7500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
-\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
-\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
-\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
-\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
-\draw (5.4420,3.2118) node {$f(x)$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.27858) node {$x$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.34557) node {$x+\Delta x$};
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,2.6388) -- (6.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,0.0000) --  cycle;
+\draw [color=red,style=dashed] (5.0000,0.0000) -- (6.0000,0.0000);
+\draw [color=red,style=dashed] (6.0000,0.0000) -- (6.0000,2.6388);
+\draw [color=red,style=dashed] (6.0000,2.6388) -- (5.0000,2.6388);
+\draw [color=red,style=dashed] (5.0000,2.6388) -- (5.0000,0.0000);
+\draw []  (5.0000,2.6388) node [rotate=0] {$\bullet$};
+\draw (5.4419,3.2118) node {$f(x)$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.2785) node {$x$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.3455) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_BoulePtLoin.pstricks.recall b/src_phystricks/Fig_BoulePtLoin.pstricks.recall
index 7e8a51e8e..c0058deeb 100644
--- a/src_phystricks/Fig_BoulePtLoin.pstricks.recall
+++ b/src_phystricks/Fig_BoulePtLoin.pstricks.recall
@@ -77,14 +77,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [style=dashed] (1.41,1.41) -- (1.94,1.94);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.30860,0.29071) node {$a$};
+\draw [style=dashed] (1.4142,1.4142) -- (1.9445,1.9445);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3085,0.2907) node {$a$};
 \draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.0501,1.6428) node {$x$};
-\draw [,->,>=latex] (0,0) -- (1.4142,1.4142);
+\draw (1.0501,1.6427) node {$x$};
+\draw [,->,>=latex] (0.0000,0.0000) -- (1.4142,1.4142);
 
-\draw [style=dotted] (2.414,1.414)--(2.412,1.478)--(2.406,1.541)--(2.396,1.603)--(2.382,1.665)--(2.364,1.726)--(2.343,1.786)--(2.317,1.844)--(2.288,1.900)--(2.255,1.955)--(2.219,2.007)--(2.180,2.057)--(2.138,2.104)--(2.093,2.149)--(2.045,2.190)--(1.994,2.229)--(1.941,2.264)--(1.886,2.296)--(1.830,2.324)--(1.771,2.348)--(1.711,2.369)--(1.650,2.386)--(1.588,2.399)--(1.525,2.408)--(1.462,2.413)--(1.398,2.414)--(1.335,2.411)--(1.272,2.404)--(1.209,2.393)--(1.148,2.378)--(1.087,2.359)--(1.028,2.337)--(0.9701,2.310)--(0.9142,2.280)--(0.8603,2.247)--(0.8086,2.210)--(0.7594,2.170)--(0.7127,2.127)--(0.6689,2.081)--(0.6282,2.032)--(0.5905,1.981)--(0.5562,1.928)--(0.5254,1.872)--(0.4981,1.815)--(0.4745,1.756)--(0.4547,1.696)--(0.4388,1.635)--(0.4268,1.572)--(0.4187,1.509)--(0.4147,1.446)--(0.4147,1.382)--(0.4187,1.319)--(0.4268,1.256)--(0.4388,1.194)--(0.4547,1.132)--(0.4745,1.072)--(0.4981,1.013)--(0.5254,0.9560)--(0.5562,0.9005)--(0.5905,0.8472)--(0.6282,0.7961)--(0.6689,0.7474)--(0.7127,0.7015)--(0.7594,0.6585)--(0.8086,0.6185)--(0.8603,0.5816)--(0.9142,0.5482)--(0.9701,0.5182)--(1.028,0.4919)--(1.087,0.4692)--(1.148,0.4504)--(1.209,0.4354)--(1.272,0.4244)--(1.335,0.4174)--(1.398,0.4143)--(1.462,0.4153)--(1.525,0.4204)--(1.588,0.4294)--(1.650,0.4424)--(1.711,0.4593)--(1.771,0.4801)--(1.830,0.5046)--(1.886,0.5328)--(1.941,0.5645)--(1.994,0.5996)--(2.045,0.6381)--(2.093,0.6796)--(2.138,0.7241)--(2.180,0.7714)--(2.219,0.8213)--(2.255,0.8736)--(2.288,0.9280)--(2.317,0.9844)--(2.343,1.043)--(2.364,1.102)--(2.382,1.163)--(2.396,1.225)--(2.406,1.288)--(2.412,1.351)--(2.414,1.414);
+\draw [style=dotted] (2.4142,1.4142)--(2.4122,1.4776)--(2.4061,1.5408)--(2.3961,1.6034)--(2.3821,1.6653)--(2.3642,1.7262)--(2.3425,1.7858)--(2.3171,1.8440)--(2.2880,1.9004)--(2.2554,1.9548)--(2.2194,2.0071)--(2.1802,2.0570)--(2.1379,2.1042)--(2.0927,2.1488)--(2.0447,2.1903)--(1.9942,2.2287)--(1.9414,2.2639)--(1.8864,2.2956)--(1.8296,2.3238)--(1.7710,2.3483)--(1.7111,2.3691)--(1.6499,2.3860)--(1.5878,2.3990)--(1.5250,2.4080)--(1.4617,2.4130)--(1.3983,2.4140)--(1.3349,2.4110)--(1.2718,2.4040)--(1.2094,2.3930)--(1.1477,2.3780)--(1.0871,2.3592)--(1.0278,2.3365)--(0.9701,2.3102)--(0.9142,2.2802)--(0.8602,2.2467)--(0.8086,2.2099)--(0.7593,2.1699)--(0.7127,2.1269)--(0.6689,2.0809)--(0.6281,2.0323)--(0.5905,1.9812)--(0.5562,1.9278)--(0.5253,1.8724)--(0.4981,1.8151)--(0.4745,1.7562)--(0.4547,1.6959)--(0.4387,1.6345)--(0.4267,1.5722)--(0.4187,1.5092)--(0.4147,1.4459)--(0.4147,1.3824)--(0.4187,1.3191)--(0.4267,1.2562)--(0.4387,1.1939)--(0.4547,1.1324)--(0.4745,1.0721)--(0.4981,1.0132)--(0.5253,0.9559)--(0.5562,0.9005)--(0.5905,0.8471)--(0.6281,0.7960)--(0.6689,0.7474)--(0.7127,0.7015)--(0.7593,0.6584)--(0.8086,0.6184)--(0.8602,0.5816)--(0.9142,0.5481)--(0.9701,0.5182)--(1.0278,0.4918)--(1.0871,0.4692)--(1.1477,0.4503)--(1.2094,0.4354)--(1.2718,0.4243)--(1.3349,0.4173)--(1.3983,0.4143)--(1.4617,0.4153)--(1.5250,0.4203)--(1.5878,0.4294)--(1.6499,0.4424)--(1.7111,0.4593)--(1.7710,0.4800)--(1.8296,0.5045)--(1.8864,0.5327)--(1.9414,0.5644)--(1.9942,0.5996)--(2.0447,0.6380)--(2.0927,0.6796)--(2.1379,0.7241)--(2.1802,0.7714)--(2.2194,0.8213)--(2.2554,0.8735)--(2.2880,0.9280)--(2.3171,0.9844)--(2.3425,1.0425)--(2.3642,1.1021)--(2.3821,1.1630)--(2.3961,1.2249)--(2.4061,1.2876)--(2.4122,1.3507)--(2.4142,1.4142);
 \draw []  (1.9445,1.9445) node [rotate=0] {$\bullet$};
 \draw (1.9445,1.5198) node {$P$};
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CELooGVvzMc.pstricks.recall b/src_phystricks/Fig_CELooGVvzMc.pstricks.recall
index 3ceb434c7..1f9995268 100644
--- a/src_phystricks/Fig_CELooGVvzMc.pstricks.recall
+++ b/src_phystricks/Fig_CELooGVvzMc.pstricks.recall
@@ -80,34 +80,34 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.7900,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.7900);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.7900,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.7900);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.02323,0)--(0.04646,0.002159)--(0.06970,0.004858)--(0.09293,0.008636)--(0.1162,0.01349)--(0.1394,0.01943)--(0.1626,0.02645)--(0.1859,0.03454)--(0.2091,0.04372)--(0.2323,0.05397)--(0.2556,0.06531)--(0.2788,0.07772)--(0.3020,0.09122)--(0.3253,0.1058)--(0.3485,0.1214)--(0.3717,0.1382)--(0.3950,0.1560)--(0.4182,0.1749)--(0.4414,0.1948)--(0.4646,0.2159)--(0.4879,0.2380)--(0.5111,0.2612)--(0.5343,0.2855)--(0.5576,0.3109)--(0.5808,0.3373)--(0.6040,0.3649)--(0.6273,0.3935)--(0.6505,0.4232)--(0.6737,0.4539)--(0.6970,0.4858)--(0.7202,0.5187)--(0.7434,0.5527)--(0.7667,0.5878)--(0.7899,0.6239)--(0.8131,0.6612)--(0.8364,0.6995)--(0.8596,0.7389)--(0.8828,0.7794)--(0.9061,0.8209)--(0.9293,0.8636)--(0.9525,0.9073)--(0.9758,0.9521)--(0.9990,0.9980)--(1.022,1.045)--(1.045,1.093)--(1.069,1.142)--(1.092,1.192)--(1.115,1.244)--(1.138,1.296)--(1.162,1.349)--(1.185,1.404)--(1.208,1.459)--(1.231,1.516)--(1.255,1.574)--(1.278,1.633)--(1.301,1.693)--(1.324,1.754)--(1.347,1.816)--(1.371,1.879)--(1.394,1.943)--(1.417,2.008)--(1.440,2.075)--(1.464,2.142)--(1.487,2.211)--(1.510,2.280)--(1.533,2.351)--(1.557,2.423)--(1.580,2.496)--(1.603,2.570)--(1.626,2.645)--(1.649,2.721)--(1.673,2.798)--(1.696,2.876)--(1.719,2.956)--(1.742,3.036)--(1.766,3.118)--(1.789,3.200)--(1.812,3.284)--(1.835,3.369)--(1.859,3.454)--(1.882,3.541)--(1.905,3.629)--(1.928,3.718)--(1.952,3.808)--(1.975,3.900)--(1.998,3.992)--(2.021,4.085)--(2.044,4.180)--(2.068,4.275)--(2.091,4.372)--(2.114,4.470)--(2.137,4.568)--(2.161,4.668)--(2.184,4.769)--(2.207,4.871)--(2.230,4.974)--(2.254,5.078)--(2.277,5.184)--(2.300,5.290);
+\draw [color=blue] (0.0000,0.0000)--(0.0232,0.0000)--(0.0464,0.0021)--(0.0696,0.0048)--(0.0929,0.0086)--(0.1161,0.0134)--(0.1393,0.0194)--(0.1626,0.0264)--(0.1858,0.0345)--(0.2090,0.0437)--(0.2323,0.0539)--(0.2555,0.0653)--(0.2787,0.0777)--(0.3020,0.0912)--(0.3252,0.1057)--(0.3484,0.1214)--(0.3717,0.1381)--(0.3949,0.1559)--(0.4181,0.1748)--(0.4414,0.1948)--(0.4646,0.2158)--(0.4878,0.2380)--(0.5111,0.2612)--(0.5343,0.2855)--(0.5575,0.3108)--(0.5808,0.3373)--(0.6040,0.3648)--(0.6272,0.3934)--(0.6505,0.4231)--(0.6737,0.4539)--(0.6969,0.4857)--(0.7202,0.5186)--(0.7434,0.5526)--(0.7666,0.5877)--(0.7898,0.6239)--(0.8131,0.6611)--(0.8363,0.6995)--(0.8595,0.7389)--(0.8828,0.7793)--(0.9060,0.8209)--(0.9292,0.8635)--(0.9525,0.9073)--(0.9757,0.9521)--(0.9989,0.9979)--(1.0222,1.0449)--(1.0454,1.0929)--(1.0686,1.1420)--(1.0919,1.1922)--(1.1151,1.2435)--(1.1383,1.2959)--(1.1616,1.3493)--(1.1848,1.4038)--(1.2080,1.4594)--(1.2313,1.5161)--(1.2545,1.5738)--(1.2777,1.6327)--(1.3010,1.6926)--(1.3242,1.7536)--(1.3474,1.8156)--(1.3707,1.8788)--(1.3939,1.9430)--(1.4171,2.0083)--(1.4404,2.0747)--(1.4636,2.1422)--(1.4868,2.2107)--(1.5101,2.2804)--(1.5333,2.3511)--(1.5565,2.4228)--(1.5797,2.4957)--(1.6030,2.5697)--(1.6262,2.6447)--(1.6494,2.7208)--(1.6727,2.7980)--(1.6959,2.8762)--(1.7191,2.9556)--(1.7424,3.0360)--(1.7656,3.1175)--(1.7888,3.2001)--(1.8121,3.2837)--(1.8353,3.3685)--(1.8585,3.4543)--(1.8818,3.5412)--(1.9050,3.6292)--(1.9282,3.7182)--(1.9515,3.8084)--(1.9747,3.8996)--(1.9979,3.9919)--(2.0212,4.0852)--(2.0444,4.1797)--(2.0676,4.2752)--(2.0909,4.3719)--(2.1141,4.4695)--(2.1373,4.5683)--(2.1606,4.6682)--(2.1838,4.7691)--(2.2070,4.8711)--(2.2303,4.9742)--(2.2535,5.0784)--(2.2767,5.1836)--(2.3000,5.2900);
 
-\draw [color=blue] (0,0)--(0.05343,0.2312)--(0.1069,0.3269)--(0.1603,0.4004)--(0.2137,0.4623)--(0.2672,0.5169)--(0.3206,0.5662)--(0.3740,0.6116)--(0.4275,0.6538)--(0.4809,0.6935)--(0.5343,0.7310)--(0.5878,0.7667)--(0.6412,0.8008)--(0.6946,0.8335)--(0.7481,0.8649)--(0.8015,0.8953)--(0.8549,0.9246)--(0.9084,0.9531)--(0.9618,0.9807)--(1.015,1.008)--(1.069,1.034)--(1.122,1.059)--(1.176,1.084)--(1.229,1.109)--(1.282,1.132)--(1.336,1.156)--(1.389,1.179)--(1.443,1.201)--(1.496,1.223)--(1.550,1.245)--(1.603,1.266)--(1.656,1.287)--(1.710,1.308)--(1.763,1.328)--(1.817,1.348)--(1.870,1.368)--(1.924,1.387)--(1.977,1.406)--(2.031,1.425)--(2.084,1.444)--(2.137,1.462)--(2.191,1.480)--(2.244,1.498)--(2.298,1.516)--(2.351,1.533)--(2.405,1.551)--(2.458,1.568)--(2.511,1.585)--(2.565,1.602)--(2.618,1.618)--(2.672,1.635)--(2.725,1.651)--(2.779,1.667)--(2.832,1.683)--(2.885,1.699)--(2.939,1.714)--(2.992,1.730)--(3.046,1.745)--(3.099,1.760)--(3.153,1.776)--(3.206,1.791)--(3.259,1.805)--(3.313,1.820)--(3.366,1.835)--(3.420,1.849)--(3.473,1.864)--(3.527,1.878)--(3.580,1.892)--(3.634,1.906)--(3.687,1.920)--(3.740,1.934)--(3.794,1.948)--(3.847,1.961)--(3.901,1.975)--(3.954,1.988)--(4.008,2.002)--(4.061,2.015)--(4.114,2.028)--(4.168,2.042)--(4.221,2.055)--(4.275,2.068)--(4.328,2.080)--(4.382,2.093)--(4.435,2.106)--(4.488,2.119)--(4.542,2.131)--(4.595,2.144)--(4.649,2.156)--(4.702,2.168)--(4.756,2.181)--(4.809,2.193)--(4.863,2.205)--(4.916,2.217)--(4.969,2.229)--(5.023,2.241)--(5.076,2.253)--(5.130,2.265)--(5.183,2.277)--(5.237,2.288)--(5.290,2.300);
-\draw [style=dashed] (0,0) -- (3.79,3.79);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0534,0.2311)--(0.1068,0.3269)--(0.1603,0.4003)--(0.2137,0.4623)--(0.2671,0.5168)--(0.3206,0.5662)--(0.3740,0.6115)--(0.4274,0.6538)--(0.4809,0.6934)--(0.5343,0.7309)--(0.5877,0.7666)--(0.6412,0.8007)--(0.6946,0.8334)--(0.7480,0.8649)--(0.8015,0.8952)--(0.8549,0.9246)--(0.9083,0.9530)--(0.9618,0.9807)--(1.0152,1.0075)--(1.0686,1.0337)--(1.1221,1.0593)--(1.1755,1.0842)--(1.2289,1.1085)--(1.2824,1.1324)--(1.3358,1.1557)--(1.3892,1.1786)--(1.4427,1.2011)--(1.4961,1.2231)--(1.5495,1.2448)--(1.6030,1.2661)--(1.6564,1.2870)--(1.7098,1.3076)--(1.7633,1.3279)--(1.8167,1.3478)--(1.8702,1.3675)--(1.9236,1.3869)--(1.9770,1.4060)--(2.0305,1.4249)--(2.0839,1.4435)--(2.1373,1.4619)--(2.1908,1.4801)--(2.2442,1.4980)--(2.2976,1.5158)--(2.3511,1.5333)--(2.4045,1.5506)--(2.4579,1.5677)--(2.5114,1.5847)--(2.5648,1.6015)--(2.6182,1.6181)--(2.6717,1.6345)--(2.7251,1.6508)--(2.7785,1.6669)--(2.8320,1.6828)--(2.8854,1.6986)--(2.9388,1.7143)--(2.9923,1.7298)--(3.0457,1.7452)--(3.0991,1.7604)--(3.1526,1.7755)--(3.2060,1.7905)--(3.2594,1.8054)--(3.3129,1.8201)--(3.3663,1.8347)--(3.4197,1.8492)--(3.4732,1.8636)--(3.5266,1.8779)--(3.5801,1.8921)--(3.6335,1.9061)--(3.6869,1.9201)--(3.7404,1.9340)--(3.7938,1.9477)--(3.8472,1.9614)--(3.9007,1.9750)--(3.9541,1.9885)--(4.0075,2.0018)--(4.0610,2.0151)--(4.1144,2.0284)--(4.1678,2.0415)--(4.2213,2.0545)--(4.2747,2.0675)--(4.3281,2.0804)--(4.3816,2.0932)--(4.4350,2.1059)--(4.4884,2.1186)--(4.5419,2.1311)--(4.5953,2.1436)--(4.6487,2.1561)--(4.7022,2.1684)--(4.7556,2.1807)--(4.8090,2.1929)--(4.8625,2.2051)--(4.9159,2.2171)--(4.9693,2.2292)--(5.0228,2.2411)--(5.0762,2.2530)--(5.1296,2.2648)--(5.1831,2.2766)--(5.2365,2.2883)--(5.2900,2.3000);
+\draw [style=dashed] (0.0000,0.0000) -- (3.7950,3.7950);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall b/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall
index 0eb3ad650..31f37cf34 100644
--- a/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall
+++ b/src_phystricks/Fig_CFMooGzvfRP.pstricks.recall
@@ -64,21 +64,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [] (2.00,0)--(2.00,0.127)--(1.98,0.253)--(1.96,0.379)--(1.94,0.502)--(1.90,0.624)--(1.86,0.743)--(1.81,0.860)--(1.75,0.972)--(1.68,1.08)--(1.61,1.19)--(1.53,1.29)--(1.45,1.38)--(1.36,1.47)--(1.26,1.55)--(1.16,1.63)--(1.05,1.70)--(0.945,1.76)--(0.831,1.82)--(0.714,1.87)--(0.594,1.91)--(0.472,1.94)--(0.347,1.97)--(0.222,1.99)--(0.0952,2.00)--(-0.0317,2.00)--(-0.158,1.99)--(-0.285,1.98)--(-0.410,1.96)--(-0.533,1.93)--(-0.654,1.89)--(-0.773,1.84)--(-0.888,1.79)--(-1.00,1.73)--(-1.11,1.67)--(-1.21,1.59)--(-1.31,1.51)--(-1.40,1.43)--(-1.49,1.33)--(-1.57,1.24)--(-1.65,1.13)--(-1.72,1.03)--(-1.78,0.916)--(-1.83,0.802)--(-1.88,0.684)--(-1.92,0.563)--(-1.95,0.441)--(-1.97,0.316)--(-1.99,0.190)--(-2.00,0.0635)--(-2.00,-0.0635)--(-1.99,-0.190)--(-1.97,-0.316)--(-1.95,-0.441)--(-1.92,-0.563)--(-1.88,-0.684)--(-1.83,-0.802)--(-1.78,-0.916)--(-1.72,-1.03)--(-1.65,-1.13)--(-1.57,-1.24)--(-1.49,-1.33)--(-1.40,-1.43)--(-1.31,-1.51)--(-1.21,-1.59)--(-1.11,-1.67)--(-1.00,-1.73)--(-0.888,-1.79)--(-0.773,-1.84)--(-0.654,-1.89)--(-0.533,-1.93)--(-0.410,-1.96)--(-0.285,-1.98)--(-0.158,-1.99)--(-0.0317,-2.00)--(0.0952,-2.00)--(0.222,-1.99)--(0.347,-1.97)--(0.472,-1.94)--(0.594,-1.91)--(0.714,-1.87)--(0.831,-1.82)--(0.945,-1.76)--(1.05,-1.70)--(1.16,-1.63)--(1.26,-1.55)--(1.36,-1.47)--(1.45,-1.38)--(1.53,-1.29)--(1.61,-1.19)--(1.68,-1.08)--(1.75,-0.972)--(1.81,-0.860)--(1.86,-0.743)--(1.90,-0.624)--(1.94,-0.502)--(1.96,-0.379)--(1.98,-0.253)--(2.00,-0.127)--(2.00,0);
-\draw [] (0,0) -- (1.73,1.00);
-\draw [] (0,0) -- (-1.73,1.00);
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (1.7320,1.0000);
+\draw [] (0.0000,0.0000) -- (-1.7320,1.0000);
 \draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (-1.7320,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3949,0.31186) node {\( \pi/6\)};
+\draw (1.3948,0.3118) node {\( \pi/6\)};
 
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-\draw (-1.3949,0.31186) node {\( \pi/6\)};
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+\draw (-1.3948,0.3118) node {\( \pi/6\)};
 
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+\draw [] (-0.4330,0.2500)--(-0.4343,0.2477)--(-0.4356,0.2454)--(-0.4369,0.2430)--(-0.4382,0.2407)--(-0.4394,0.2384)--(-0.4407,0.2361)--(-0.4419,0.2338)--(-0.4431,0.2314)--(-0.4444,0.2291)--(-0.4456,0.2267)--(-0.4468,0.2243)--(-0.4479,0.2220)--(-0.4491,0.2196)--(-0.4503,0.2172)--(-0.4514,0.2148)--(-0.4525,0.2125)--(-0.4537,0.2101)--(-0.4548,0.2077)--(-0.4559,0.2052)--(-0.4569,0.2028)--(-0.4580,0.2004)--(-0.4591,0.1980)--(-0.4601,0.1956)--(-0.4611,0.1931)--(-0.4621,0.1907)--(-0.4631,0.1882)--(-0.4641,0.1858)--(-0.4651,0.1833)--(-0.4661,0.1809)--(-0.4670,0.1784)--(-0.4680,0.1759)--(-0.4689,0.1734)--(-0.4698,0.1710)--(-0.4707,0.1685)--(-0.4716,0.1660)--(-0.4725,0.1635)--(-0.4733,0.1610)--(-0.4742,0.1585)--(-0.4750,0.1560)--(-0.4758,0.1535)--(-0.4766,0.1509)--(-0.4774,0.1484)--(-0.4782,0.1459)--(-0.4789,0.1434)--(-0.4797,0.1408)--(-0.4804,0.1383)--(-0.4812,0.1357)--(-0.4819,0.1332)--(-0.4826,0.1306)--(-0.4833,0.1281)--(-0.4839,0.1255)--(-0.4846,0.1230)--(-0.4852,0.1204)--(-0.4859,0.1178)--(-0.4865,0.1153)--(-0.4871,0.1127)--(-0.4877,0.1101)--(-0.4882,0.1075)--(-0.4888,0.1049)--(-0.4894,0.1024)--(-0.4899,0.0998)--(-0.4904,0.0972)--(-0.4909,0.0946)--(-0.4914,0.0920)--(-0.4919,0.0894)--(-0.4924,0.0868)--(-0.4928,0.0842)--(-0.4932,0.0816)--(-0.4937,0.0790)--(-0.4941,0.0763)--(-0.4945,0.0737)--(-0.4949,0.0711)--(-0.4952,0.0685)--(-0.4956,0.0659)--(-0.4959,0.0632)--(-0.4963,0.0606)--(-0.4966,0.0580)--(-0.4969,0.0554)--(-0.4972,0.0527)--(-0.4974,0.0501)--(-0.4977,0.0475)--(-0.4979,0.0448)--(-0.4982,0.0422)--(-0.4984,0.0396)--(-0.4986,0.0369)--(-0.4988,0.0343)--(-0.4989,0.0317)--(-0.4991,0.0290)--(-0.4993,0.0264)--(-0.4994,0.0237)--(-0.4995,0.0211)--(-0.4996,0.0185)--(-0.4997,0.0158)--(-0.4998,0.0132)--(-0.4998,0.0105)--(-0.4999,0.0079)--(-0.4999,0.0052)--(-0.4999,0.0026)--(-0.5000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CSCiii.pstricks.recall b/src_phystricks/Fig_CSCiii.pstricks.recall
index ebdb1078f..a027cc720 100644
--- a/src_phystricks/Fig_CSCiii.pstricks.recall
+++ b/src_phystricks/Fig_CSCiii.pstricks.recall
@@ -41,16 +41,16 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.6734);
+\draw [,->,>=latex] (-1.1515,0.0000) -- (3.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.6734);
 %DEFAULT
-\draw [color=blue] (3.00,0.0300)--(3.00,0.125)--(2.99,0.219)--(2.98,0.314)--(2.96,0.407)--(2.94,0.500)--(2.92,0.591)--(2.89,0.682)--(2.86,0.771)--(2.83,0.859)--(2.79,0.945)--(2.75,1.03)--(2.71,1.11)--(2.66,1.19)--(2.61,1.27)--(2.55,1.34)--(2.49,1.42)--(2.44,1.49)--(2.37,1.55)--(2.31,1.62)--(2.24,1.68)--(2.17,1.73)--(2.10,1.79)--(2.02,1.84)--(1.95,1.89)--(1.87,1.93)--(1.79,1.97)--(1.71,2.01)--(1.63,2.04)--(1.55,2.07)--(1.47,2.10)--(1.39,2.12)--(1.31,2.14)--(1.22,2.15)--(1.14,2.16)--(1.06,2.17)--(0.976,2.17)--(0.894,2.17)--(0.814,2.17)--(0.734,2.16)--(0.655,2.15)--(0.578,2.14)--(0.502,2.12)--(0.428,2.10)--(0.355,2.08)--(0.284,2.05)--(0.215,2.02)--(0.148,1.99)--(0.0832,1.96)--(0.0208,1.92)--(-0.0392,1.88)--(-0.0968,1.84)--(-0.152,1.80)--(-0.204,1.76)--(-0.254,1.71)--(-0.300,1.66)--(-0.344,1.61)--(-0.385,1.56)--(-0.423,1.51)--(-0.459,1.45)--(-0.491,1.40)--(-0.520,1.35)--(-0.546,1.29)--(-0.570,1.24)--(-0.590,1.18)--(-0.607,1.12)--(-0.622,1.07)--(-0.633,1.01)--(-0.642,0.958)--(-0.648,0.904)--(-0.651,0.850)--(-0.652,0.797)--(-0.650,0.745)--(-0.645,0.695)--(-0.638,0.645)--(-0.629,0.597)--(-0.617,0.550)--(-0.604,0.504)--(-0.588,0.460)--(-0.570,0.418)--(-0.551,0.378)--(-0.530,0.339)--(-0.507,0.302)--(-0.483,0.268)--(-0.457,0.235)--(-0.430,0.204)--(-0.403,0.176)--(-0.374,0.149)--(-0.344,0.125)--(-0.314,0.103)--(-0.283,0.0828)--(-0.252,0.0651)--(-0.220,0.0496)--(-0.188,0.0362)--(-0.156,0.0249)--(-0.125,0.0158)--(-0.0929,0.00884)--(-0.0615,0.00390)--(-0.0305,0)--(0,0);
-\draw (1.5000,-0.42071) node {$ \frac{1}{2} $};
-\draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.31083,1.5000) node {$ \frac{1}{2} $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
+\draw [color=blue] (2.9998,0.0299)--(2.9965,0.1248)--(2.9892,0.2194)--(2.9780,0.3135)--(2.9628,0.4070)--(2.9437,0.4997)--(2.9208,0.5914)--(2.8940,0.6819)--(2.8635,0.7711)--(2.8293,0.8587)--(2.7915,0.9447)--(2.7502,1.0287)--(2.7055,1.1108)--(2.6575,1.1906)--(2.6063,1.2681)--(2.5521,1.3432)--(2.4949,1.4156)--(2.4349,1.4853)--(2.3722,1.5521)--(2.3071,1.6159)--(2.2396,1.6767)--(2.1698,1.7342)--(2.0981,1.7885)--(2.0244,1.8394)--(1.9491,1.8868)--(1.8722,1.9308)--(1.7939,1.9712)--(1.7145,2.0079)--(1.6340,2.0411)--(1.5528,2.0705)--(1.4708,2.0963)--(1.3884,2.1183)--(1.3058,2.1367)--(1.2230,2.1513)--(1.1403,2.1623)--(1.0578,2.1696)--(0.9758,2.1733)--(0.8944,2.1734)--(0.8137,2.1700)--(0.7340,2.1631)--(0.6553,2.1528)--(0.5779,2.1393)--(0.5019,2.1224)--(0.4275,2.1025)--(0.3547,2.0795)--(0.2838,2.0536)--(0.2148,2.0248)--(0.1479,1.9934)--(0.0832,1.9593)--(0.0207,1.9228)--(-0.0392,1.8840)--(-0.0967,1.8430)--(-0.1517,1.8000)--(-0.2040,1.7551)--(-0.2535,1.7084)--(-0.3003,1.6601)--(-0.3442,1.6104)--(-0.3853,1.5594)--(-0.4234,1.5073)--(-0.4586,1.4542)--(-0.4908,1.4003)--(-0.5200,1.3457)--(-0.5463,1.2906)--(-0.5695,1.2352)--(-0.5898,1.1796)--(-0.6071,1.1239)--(-0.6216,1.0684)--(-0.6331,1.0131)--(-0.6418,0.9582)--(-0.6478,0.9039)--(-0.6510,0.8502)--(-0.6515,0.7974)--(-0.6495,0.7455)--(-0.6450,0.6946)--(-0.6380,0.6449)--(-0.6287,0.5966)--(-0.6172,0.5496)--(-0.6035,0.5041)--(-0.5878,0.4602)--(-0.5701,0.4180)--(-0.5507,0.3775)--(-0.5295,0.3389)--(-0.5068,0.3022)--(-0.4826,0.2675)--(-0.4571,0.2347)--(-0.4304,0.2041)--(-0.4025,0.1755)--(-0.3738,0.1491)--(-0.3442,0.1248)--(-0.3139,0.1027)--(-0.2830,0.0828)--(-0.2517,0.0651)--(-0.2201,0.0495)--(-0.1882,0.0361)--(-0.1564,0.0249)--(-0.1245,0.0158)--(-0.0928,0.0088)--(-0.0614,0.0038)--(-0.0304,0.0000)--(0.0000,0.0000);
+\draw (1.5000,-0.4207) node {$ \frac{1}{2} $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.3108,1.5000) node {$ \frac{1}{2} $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -92,19 +92,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.2388,0) -- (2.6268,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.3345);
+\draw [,->,>=latex] (-2.2387,0.0000) -- (2.6268,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.3344);
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+\draw (-1.5000,-0.4207) node {$ -\frac{1}{20} $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (1.5000,-0.4207) node {$ \frac{1}{20} $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (-0.3816,3.0000) node {$ \frac{1}{10} $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -146,27 +146,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.1516,0) -- (3.4998,0);
-\draw [,->,>=latex] (0,-2.6734) -- (0,2.6734);
+\draw [,->,>=latex] (-1.1515,0.0000) -- (3.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-2.6734) -- (0.0000,2.6734);
 %DEFAULT
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+\draw [color=red] (0.0000,0.0000)--(-0.0033,0.0000)--(-0.0067,0.0000)--(-0.0100,0.0000)--(-0.0133,-0.0017)--(-0.0166,-0.0026)--(-0.0198,-0.0038)--(-0.0229,-0.0051)--(-0.0260,-0.0067)--(-0.0289,-0.0085)--(-0.0318,-0.0104)--(-0.0345,-0.0125)--(-0.0371,-0.0148)--(-0.0395,-0.0173)--(-0.0418,-0.0199)--(-0.0439,-0.0226)--(-0.0459,-0.0255)--(-0.0476,-0.0285)--(-0.0492,-0.0316)--(-0.0506,-0.0348)--(-0.0518,-0.0381)--(-0.0528,-0.0415)--(-0.0535,-0.0449)--(-0.0541,-0.0484)--(-0.0544,-0.0519)--(-0.0545,-0.0554)--(-0.0544,-0.0589)--(-0.0541,-0.0624)--(-0.0536,-0.0659)--(-0.0528,-0.0694)--(-0.0518,-0.0728)--(-0.0506,-0.0761)--(-0.0493,-0.0794)--(-0.0477,-0.0826)--(-0.0459,-0.0857)--(-0.0439,-0.0886)--(-0.0417,-0.0914)--(-0.0394,-0.0941)--(-0.0369,-0.0967)--(-0.0342,-0.0990)--(-0.0314,-0.1012)--(-0.0285,-0.1033)--(-0.0255,-0.1051)--(-0.0223,-0.1068)--(-0.0190,-0.1082)--(-0.0157,-0.1094)--(-0.0123,-0.1105)--(-0.0088,-0.1113)--(-0.0053,-0.1118)--(-0.0017,-0.1122)--(0.0017,-0.1123)--(0.0053,-0.1122)--(0.0089,-0.1119)--(0.0124,-0.1114)--(0.0159,-0.1106)--(0.0193,-0.1096)--(0.0226,-0.1084)--(0.0259,-0.1070)--(0.0291,-0.1054)--(0.0322,-0.1036)--(0.0351,-0.1015)--(0.0379,-0.0993)--(0.0406,-0.0970)--(0.0431,-0.0944)--(0.0454,-0.0917)--(0.0476,-0.0889)--(0.0496,-0.0859)--(0.0513,-0.0828)--(0.0529,-0.0796)--(0.0543,-0.0763)--(0.0554,-0.0729)--(0.0564,-0.0694)--(0.0571,-0.0659)--(0.0576,-0.0623)--(0.0578,-0.0587)--(0.0578,-0.0551)--(0.0576,-0.0515)--(0.0571,-0.0479)--(0.0565,-0.0444)--(0.0555,-0.0409)--(0.0544,-0.0374)--(0.0530,-0.0341)--(0.0515,-0.0308)--(0.0497,-0.0276)--(0.0477,-0.0245)--(0.0455,-0.0216)--(0.0431,-0.0188)--(0.0405,-0.0162)--(0.0378,-0.0137)--(0.0349,-0.0114)--(0.0319,-0.0093)--(0.0287,-0.0074)--(0.0254,-0.0057)--(0.0220,-0.0042)--(0.0185,-0.0029)--(0.0149,-0.0019)--(0.0112,-0.0010)--(0.0075,0.0000)--(0.0037,0.0000)--(0.0000,0.0000);
+\draw (1.5000,-0.4207) node {$ \frac{1}{2} $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4527,-1.5000) node {$ -\frac{1}{2} $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.3108,1.5000) node {$ \frac{1}{2} $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CSCvi.pstricks.recall b/src_phystricks/Fig_CSCvi.pstricks.recall
index 8a0a3f7b3..8dc9d59f3 100644
--- a/src_phystricks/Fig_CSCvi.pstricks.recall
+++ b/src_phystricks/Fig_CSCvi.pstricks.recall
@@ -61,30 +61,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (2.0708,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4332);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (2.0707,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4331);
 %DEFAULT
 
-\draw [color=blue] (-1.171,4.933)--(-1.143,4.605)--(-1.115,4.316)--(-1.088,4.059)--(-1.060,3.830)--(-1.032,3.624)--(-1.005,3.438)--(-0.9769,3.268)--(-0.9493,3.114)--(-0.9216,2.972)--(-0.8939,2.841)--(-0.8662,2.720)--(-0.8385,2.608)--(-0.8108,2.504)--(-0.7831,2.406)--(-0.7554,2.315)--(-0.7277,2.230)--(-0.7000,2.150)--(-0.6723,2.074)--(-0.6446,2.003)--(-0.6169,1.935)--(-0.5892,1.871)--(-0.5616,1.811)--(-0.5339,1.753)--(-0.5062,1.698)--(-0.4785,1.645)--(-0.4508,1.595)--(-0.4231,1.547)--(-0.3954,1.501)--(-0.3677,1.457)--(-0.3400,1.414)--(-0.3123,1.374)--(-0.2846,1.334)--(-0.2569,1.297)--(-0.2292,1.260)--(-0.2015,1.225)--(-0.1739,1.191)--(-0.1462,1.158)--(-0.1185,1.126)--(-0.09077,1.095)--(-0.06308,1.065)--(-0.03539,1.036)--(-0.007696,1.008)--(0.02000,0.9802)--(0.04769,0.9534)--(0.07538,0.9273)--(0.1031,0.9019)--(0.1308,0.8771)--(0.1585,0.8529)--(0.1862,0.8292)--(0.2138,0.8061)--(0.2415,0.7835)--(0.2692,0.7614)--(0.2969,0.7398)--(0.3246,0.7186)--(0.3523,0.6978)--(0.3800,0.6774)--(0.4077,0.6574)--(0.4354,0.6377)--(0.4631,0.6184)--(0.4908,0.5994)--(0.5185,0.5808)--(0.5462,0.5624)--(0.5739,0.5443)--(0.6015,0.5265)--(0.6292,0.5089)--(0.6569,0.4916)--(0.6846,0.4746)--(0.7123,0.4577)--(0.7400,0.4411)--(0.7677,0.4246)--(0.7954,0.4084)--(0.8231,0.3923)--(0.8508,0.3764)--(0.8785,0.3607)--(0.9062,0.3451)--(0.9339,0.3297)--(0.9616,0.3144)--(0.9893,0.2993)--(1.017,0.2842)--(1.045,0.2693)--(1.072,0.2545)--(1.100,0.2398)--(1.128,0.2252)--(1.155,0.2107)--(1.183,0.1963)--(1.211,0.1820)--(1.238,0.1677)--(1.266,0.1535)--(1.294,0.1394)--(1.322,0.1253)--(1.349,0.1112)--(1.377,0.09723)--(1.405,0.08327)--(1.432,0.06934)--(1.460,0.05544)--(1.488,0.04156)--(1.515,0.02770)--(1.543,0.01385)--(1.571,0);
-\draw [color=lightgray,style=dashed] (-1.57,0) -- (-1.57,4.93);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue] (-1.1707,4.9331)--(-1.1431,4.6047)--(-1.1154,4.3157)--(-1.0877,4.0592)--(-1.0600,3.8301)--(-1.0323,3.6240)--(-1.0046,3.4377)--(-0.9769,3.2682)--(-0.9492,3.1135)--(-0.9215,2.9715)--(-0.8938,2.8408)--(-0.8661,2.7199)--(-0.8384,2.6079)--(-0.8107,2.5036)--(-0.7830,2.4063)--(-0.7554,2.3153)--(-0.7277,2.2300)--(-0.7000,2.1497)--(-0.6723,2.0742)--(-0.6446,2.0028)--(-0.6169,1.9353)--(-0.5892,1.8713)--(-0.5615,1.8105)--(-0.5338,1.7527)--(-0.5061,1.6977)--(-0.4784,1.6451)--(-0.4507,1.5949)--(-0.4230,1.5469)--(-0.3953,1.5009)--(-0.3677,1.4568)--(-0.3400,1.4144)--(-0.3123,1.3737)--(-0.2846,1.3344)--(-0.2569,1.2966)--(-0.2292,1.2602)--(-0.2015,1.2249)--(-0.1738,1.1909)--(-0.1461,1.1579)--(-0.1184,1.1260)--(-0.0907,1.0951)--(-0.0630,1.0651)--(-0.0353,1.0360)--(-0.0076,1.0077)--(0.0199,0.9802)--(0.0476,0.9534)--(0.0753,0.9273)--(0.1030,0.9018)--(0.1307,0.8770)--(0.1584,0.8528)--(0.1861,0.8292)--(0.2138,0.8061)--(0.2415,0.7835)--(0.2692,0.7614)--(0.2969,0.7397)--(0.3246,0.7185)--(0.3523,0.6977)--(0.3800,0.6773)--(0.4076,0.6573)--(0.4353,0.6377)--(0.4630,0.6184)--(0.4907,0.5994)--(0.5184,0.5807)--(0.5461,0.5624)--(0.5738,0.5443)--(0.6015,0.5265)--(0.6292,0.5089)--(0.6569,0.4916)--(0.6846,0.4745)--(0.7123,0.4577)--(0.7400,0.4410)--(0.7677,0.4246)--(0.7953,0.4083)--(0.8230,0.3923)--(0.8507,0.3764)--(0.8784,0.3606)--(0.9061,0.3451)--(0.9338,0.3296)--(0.9615,0.3144)--(0.9892,0.2992)--(1.0169,0.2842)--(1.0446,0.2693)--(1.0723,0.2545)--(1.1000,0.2398)--(1.1277,0.2252)--(1.1554,0.2107)--(1.1830,0.1963)--(1.2107,0.1819)--(1.2384,0.1677)--(1.2661,0.1534)--(1.2938,0.1393)--(1.3215,0.1252)--(1.3492,0.1112)--(1.3769,0.0972)--(1.4046,0.0832)--(1.4323,0.0693)--(1.4600,0.0554)--(1.4877,0.0415)--(1.5154,0.0276)--(1.5431,0.0138)--(1.5707,0.0000);
+\draw [color=lightgray,style=dashed] (-1.5707,0.0000) -- (-1.5707,4.9331);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -142,24 +142,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.4211,0);
-\draw [,->,>=latex] (0,-5.0437) -- (0,0.80024);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.4210,0.0000);
+\draw [,->,>=latex] (0.0000,-5.0437) -- (0.0000,0.8002);
 %DEFAULT
-\draw [color=blue] (1.921,-4.544)--(1.910,-4.190)--(1.898,-3.876)--(1.886,-3.595)--(1.872,-3.341)--(1.858,-3.111)--(1.844,-2.901)--(1.829,-2.709)--(1.813,-2.531)--(1.797,-2.367)--(1.779,-2.214)--(1.762,-2.072)--(1.744,-1.939)--(1.725,-1.815)--(1.705,-1.698)--(1.686,-1.587)--(1.665,-1.483)--(1.644,-1.385)--(1.623,-1.292)--(1.601,-1.204)--(1.579,-1.120)--(1.556,-1.040)--(1.533,-0.9641)--(1.509,-0.8919)--(1.485,-0.8231)--(1.460,-0.7575)--(1.436,-0.6949)--(1.411,-0.6352)--(1.385,-0.5781)--(1.359,-0.5237)--(1.333,-0.4717)--(1.307,-0.4221)--(1.281,-0.3747)--(1.254,-0.3295)--(1.227,-0.2864)--(1.200,-0.2452)--(1.173,-0.2060)--(1.146,-0.1686)--(1.118,-0.1331)--(1.091,-0.09928)--(1.063,-0.06715)--(1.035,-0.03666)--(1.008,-0.007756)--(0.9800,0.01960)--(0.9523,0.04545)--(0.9247,0.06984)--(0.8971,0.09280)--(0.8696,0.1144)--(0.8422,0.1346)--(0.8149,0.1535)--(0.7878,0.1711)--(0.7608,0.1874)--(0.7340,0.2025)--(0.7074,0.2164)--(0.6811,0.2292)--(0.6549,0.2408)--(0.6291,0.2513)--(0.6035,0.2607)--(0.5782,0.2690)--(0.5533,0.2763)--(0.5287,0.2825)--(0.5044,0.2878)--(0.4806,0.2921)--(0.4571,0.2955)--(0.4341,0.2980)--(0.4115,0.2995)--(0.3893,0.3002)--(0.3676,0.3001)--(0.3464,0.2991)--(0.3257,0.2974)--(0.3055,0.2949)--(0.2859,0.2916)--(0.2668,0.2877)--(0.2482,0.2830)--(0.2302,0.2776)--(0.2129,0.2717)--(0.1961,0.2650)--(0.1799,0.2578)--(0.1644,0.2501)--(0.1495,0.2417)--(0.1353,0.2329)--(0.1217,0.2236)--(0.1088,0.2137)--(0.09657,0.2035)--(0.08504,0.1928)--(0.07422,0.1817)--(0.06411,0.1703)--(0.05471,0.1585)--(0.04604,0.1464)--(0.03810,0.1340)--(0.03090,0.1214)--(0.02444,0.1085)--(0.01873,0.09541)--(0.01377,0.08212)--(0.009571,0.06868)--(0.006129,0.05510)--(0.003449,0.04142)--(0.001533,0.02766)--(0,0.01384)--(0,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-5.0000) node {$ -5 $};
-\draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
+\draw [color=blue] (1.9210,-4.5437)--(1.9099,-4.1899)--(1.8980,-3.8759)--(1.8855,-3.5947)--(1.8723,-3.3412)--(1.8584,-3.1112)--(1.8439,-2.9013)--(1.8287,-2.7087)--(1.8129,-2.5312)--(1.7965,-2.3669)--(1.7795,-2.2144)--(1.7618,-2.0722)--(1.7436,-1.9393)--(1.7248,-1.8147)--(1.7054,-1.6976)--(1.6855,-1.5873)--(1.6651,-1.4833)--(1.6442,-1.3849)--(1.6228,-1.2918)--(1.6009,-1.2035)--(1.5785,-1.1196)--(1.5557,-1.0399)--(1.5325,-0.9641)--(1.5088,-0.8919)--(1.4848,-0.8230)--(1.4604,-0.7574)--(1.4356,-0.6948)--(1.4105,-0.6351)--(1.3851,-0.5781)--(1.3594,-0.5237)--(1.3334,-0.4717)--(1.3072,-0.4220)--(1.2807,-0.3747)--(1.2541,-0.3295)--(1.2272,-0.2863)--(1.2001,-0.2452)--(1.1729,-0.2060)--(1.1456,-0.1686)--(1.1181,-0.1330)--(1.0906,-0.0992)--(1.0630,-0.0671)--(1.0353,-0.0366)--(1.0076,-0.0077)--(0.9800,0.0195)--(0.9523,0.0454)--(0.9246,0.0698)--(0.8971,0.0927)--(0.8696,0.1143)--(0.8422,0.1345)--(0.8149,0.1534)--(0.7877,0.1710)--(0.7608,0.1874)--(0.7340,0.2025)--(0.7074,0.2164)--(0.6810,0.2291)--(0.6549,0.2407)--(0.6290,0.2512)--(0.6035,0.2606)--(0.5782,0.2689)--(0.5532,0.2762)--(0.5286,0.2825)--(0.5044,0.2878)--(0.4805,0.2921)--(0.4571,0.2954)--(0.4340,0.2979)--(0.4114,0.2995)--(0.3893,0.3002)--(0.3676,0.3001)--(0.3464,0.2991)--(0.3257,0.2974)--(0.3055,0.2948)--(0.2858,0.2916)--(0.2667,0.2876)--(0.2482,0.2829)--(0.2302,0.2776)--(0.2128,0.2716)--(0.1960,0.2650)--(0.1799,0.2578)--(0.1643,0.2500)--(0.1494,0.2417)--(0.1352,0.2328)--(0.1216,0.2235)--(0.1087,0.2137)--(0.0965,0.2034)--(0.0850,0.1928)--(0.0742,0.1817)--(0.0641,0.1703)--(0.0547,0.1585)--(0.0460,0.1464)--(0.0381,0.1340)--(0.0308,0.1213)--(0.0244,0.1085)--(0.0187,0.0954)--(0.0137,0.0821)--(0.0095,0.0686)--(0.0061,0.0551)--(0.0034,0.0414)--(0.0015,0.0276)--(0.0000,0.0138)--(0.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-5.0000) node {$ -5 $};
+\draw [] (-0.1000,-5.0000) -- (0.1000,-5.0000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CURGooXvruWV.pstricks.recall b/src_phystricks/Fig_CURGooXvruWV.pstricks.recall
index 8d44d1bc7..4389d953c 100644
--- a/src_phystricks/Fig_CURGooXvruWV.pstricks.recall
+++ b/src_phystricks/Fig_CURGooXvruWV.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,23 +81,23 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=green] (0.0000,0.0000)--(0.0202,0.0202)--(0.0404,0.0404)--(0.0606,0.0606)--(0.0808,0.0808)--(0.1010,0.1010)--(0.1212,0.1212)--(0.1414,0.1414)--(0.1616,0.1616)--(0.1818,0.1818)--(0.2020,0.2020)--(0.2222,0.2222)--(0.2424,0.2424)--(0.2626,0.2626)--(0.2828,0.2828)--(0.3030,0.3030)--(0.3232,0.3232)--(0.3434,0.3434)--(0.3636,0.3636)--(0.3838,0.3838)--(0.4040,0.4040)--(0.4242,0.4242)--(0.4444,0.4444)--(0.4646,0.4646)--(0.4848,0.4848)--(0.5050,0.5050)--(0.5252,0.5252)--(0.5454,0.5454)--(0.5656,0.5656)--(0.5858,0.5858)--(0.6060,0.6060)--(0.6262,0.6262)--(0.6464,0.6464)--(0.6666,0.6666)--(0.6868,0.6868)--(0.7070,0.7070)--(0.7272,0.7272)--(0.7474,0.7474)--(0.7676,0.7676)--(0.7878,0.7878)--(0.8080,0.8080)--(0.8282,0.8282)--(0.8484,0.8484)--(0.8686,0.8686)--(0.8888,0.8888)--(0.9090,0.9090)--(0.9292,0.9292)--(0.9494,0.9494)--(0.9696,0.9696)--(0.9898,0.9898)--(1.0101,1.0101)--(1.0303,1.0303)--(1.0505,1.0505)--(1.0707,1.0707)--(1.0909,1.0909)--(1.1111,1.1111)--(1.1313,1.1313)--(1.1515,1.1515)--(1.1717,1.1717)--(1.1919,1.1919)--(1.2121,1.2121)--(1.2323,1.2323)--(1.2525,1.2525)--(1.2727,1.2727)--(1.2929,1.2929)--(1.3131,1.3131)--(1.3333,1.3333)--(1.3535,1.3535)--(1.3737,1.3737)--(1.3939,1.3939)--(1.4141,1.4141)--(1.4343,1.4343)--(1.4545,1.4545)--(1.4747,1.4747)--(1.4949,1.4949)--(1.5151,1.5151)--(1.5353,1.5353)--(1.5555,1.5555)--(1.5757,1.5757)--(1.5959,1.5959)--(1.6161,1.6161)--(1.6363,1.6363)--(1.6565,1.6565)--(1.6767,1.6767)--(1.6969,1.6969)--(1.7171,1.7171)--(1.7373,1.7373)--(1.7575,1.7575)--(1.7777,1.7777)--(1.7979,1.7979)--(1.8181,1.8181)--(1.8383,1.8383)--(1.8585,1.8585)--(1.8787,1.8787)--(1.8989,1.8989)--(1.9191,1.9191)--(1.9393,1.9393)--(1.9595,1.9595)--(1.9797,1.9797)--(2.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_Cardioideexo.pstricks.recall b/src_phystricks/Fig_Cardioideexo.pstricks.recall
index 1b2d525b7..87bc7a7ef 100644
--- a/src_phystricks/Fig_Cardioideexo.pstricks.recall
+++ b/src_phystricks/Fig_Cardioideexo.pstricks.recall
@@ -79,37 +79,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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-\draw [color=lightgray] (0,1.00) -- (1.00,1.00);
-\draw [color=lightgray] (1.00,1.00) -- (1.00,0);
-\draw [color=lightgray] (1.00,0) -- (0,0);
-\draw [color=lightgray] (0,0) -- (0,1.00);
-\draw [color=lightgray,style=dashed] (0,0) -- (1.21,1.21);
+\draw [color=lightgray] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw [color=blue] (2.0000,0.0000)--(1.9939,0.1267)--(1.9759,0.2521)--(1.9461,0.3750)--(1.9048,0.4942)--(1.8527,0.6084)--(1.7902,0.7167)--(1.7182,0.8178)--(1.6374,0.9110)--(1.5489,0.9954)--(1.4537,1.0703)--(1.3528,1.1351)--(1.2475,1.1895)--(1.1388,1.2330)--(1.0281,1.2655)--(0.9165,1.2870)--(0.8051,1.2977)--(0.6953,1.2977)--(0.5879,1.2875)--(0.4842,1.2675)--(0.3850,1.2384)--(0.2913,1.2009)--(0.2038,1.1558)--(0.1231,1.1039)--(0.0498,1.0463)--(-0.0156,0.9840)--(-0.0729,0.9178)--(-0.1220,0.8489)--(-0.1628,0.7783)--(-0.1954,0.7070)--(-0.2200,0.6359)--(-0.2370,0.5660)--(-0.2468,0.4981)--(-0.2500,0.4330)--(-0.2470,0.3713)--(-0.2388,0.3138)--(-0.2260,0.2608)--(-0.2094,0.2127)--(-0.1898,0.1698)--(-0.1681,0.1322)--(-0.1452,0.0999)--(-0.1218,0.0729)--(-0.0988,0.0509)--(-0.0768,0.0336)--(-0.0566,0.0206)--(-0.0388,0.0114)--(-0.0239,0.0054)--(-0.0124,0.0019)--(-0.0045,0.0000)--(0.0000,0.0000)--(0.0000,0.0000)--(-0.0045,0.0000)--(-0.0124,-0.0019)--(-0.0239,-0.0054)--(-0.0388,-0.0114)--(-0.0566,-0.0206)--(-0.0768,-0.0336)--(-0.0988,-0.0509)--(-0.1218,-0.0729)--(-0.1452,-0.0999)--(-0.1681,-0.1322)--(-0.1898,-0.1698)--(-0.2094,-0.2127)--(-0.2260,-0.2608)--(-0.2388,-0.3138)--(-0.2470,-0.3713)--(-0.2500,-0.4330)--(-0.2468,-0.4981)--(-0.2370,-0.5660)--(-0.2200,-0.6359)--(-0.1954,-0.7070)--(-0.1628,-0.7783)--(-0.1220,-0.8489)--(-0.0729,-0.9178)--(-0.0156,-0.9840)--(0.0498,-1.0463)--(0.1231,-1.1039)--(0.2038,-1.1558)--(0.2913,-1.2009)--(0.3850,-1.2384)--(0.4842,-1.2675)--(0.5879,-1.2875)--(0.6953,-1.2977)--(0.8051,-1.2977)--(0.9165,-1.2870)--(1.0281,-1.2655)--(1.1388,-1.2330)--(1.2475,-1.1895)--(1.3528,-1.1351)--(1.4537,-1.0703)--(1.5489,-0.9954)--(1.6374,-0.9110)--(1.7182,-0.8178)--(1.7902,-0.7167)--(1.8527,-0.6084)--(1.9048,-0.4942)--(1.9461,-0.3750)--(1.9759,-0.2521)--(1.9939,-0.1267)--(2.0000,0.0000);
+\draw [color=lightgray] (0.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=lightgray] (1.0000,1.0000) -- (1.0000,0.0000);
+\draw [color=lightgray] (1.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=lightgray] (0.0000,0.0000) -- (0.0000,1.0000);
+\draw [color=lightgray,style=dashed] (0.0000,0.0000) -- (1.2071,1.2071);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.2071,1.2071) node [rotate=0] {$\bullet$};
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CbCartTuii.pstricks.recall b/src_phystricks/Fig_CbCartTuii.pstricks.recall
index f3a5e95ac..912d80e0f 100644
--- a/src_phystricks/Fig_CbCartTuii.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTuii.pstricks.recall
@@ -67,12 +67,12 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.1489) -- (0,1.1489);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.1488) -- (0.0000,1.1488);
 %DEFAULT
-\draw [color=blue] (2.00,0)--(1.99,0.126)--(1.97,0.247)--(1.93,0.358)--(1.87,0.456)--(1.81,0.535)--(1.72,0.595)--(1.63,0.633)--(1.53,0.649)--(1.42,0.644)--(1.30,0.619)--(1.17,0.578)--(1.05,0.523)--(0.921,0.459)--(0.795,0.389)--(0.673,0.318)--(0.556,0.249)--(0.446,0.186)--(0.345,0.130)--(0.255,0.0849)--(0.176,0.0500)--(0.111,0.0255)--(0.0603,0.0103)--(0.0246,0.00271)--(0.00453,0)--(0,0)--(0.0126,0)--(0.0405,-0.00571)--(0.0839,-0.0168)--(0.142,-0.0365)--(0.214,-0.0661)--(0.299,-0.106)--(0.394,-0.157)--(0.500,-0.217)--(0.614,-0.283)--(0.734,-0.354)--(0.858,-0.424)--(0.984,-0.492)--(1.11,-0.552)--(1.24,-0.600)--(1.36,-0.634)--(1.47,-0.649)--(1.58,-0.644)--(1.68,-0.617)--(1.77,-0.568)--(1.84,-0.498)--(1.90,-0.409)--(1.95,-0.304)--(1.98,-0.188)--(2.00,-0.0634)--(2.00,0.0634)--(1.98,0.188)--(1.95,0.304)--(1.90,0.409)--(1.84,0.498)--(1.77,0.568)--(1.68,0.617)--(1.58,0.644)--(1.47,0.649)--(1.36,0.634)--(1.24,0.600)--(1.11,0.552)--(0.984,0.492)--(0.858,0.424)--(0.734,0.354)--(0.614,0.283)--(0.500,0.217)--(0.394,0.157)--(0.299,0.106)--(0.214,0.0661)--(0.142,0.0365)--(0.0839,0.0168)--(0.0405,0.00571)--(0.0126,0)--(0,0)--(0.00453,0)--(0.0246,-0.00271)--(0.0603,-0.0103)--(0.111,-0.0255)--(0.176,-0.0500)--(0.255,-0.0849)--(0.345,-0.130)--(0.446,-0.186)--(0.556,-0.249)--(0.673,-0.318)--(0.795,-0.389)--(0.921,-0.459)--(1.05,-0.523)--(1.17,-0.578)--(1.30,-0.619)--(1.42,-0.644)--(1.53,-0.649)--(1.63,-0.633)--(1.72,-0.595)--(1.81,-0.535)--(1.87,-0.456)--(1.93,-0.358)--(1.97,-0.247)--(1.99,-0.126)--(2.00,0);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
+\draw [color=blue] (2.0000,0.0000)--(1.9919,0.1260)--(1.9679,0.2471)--(1.9283,0.3583)--(1.8738,0.4555)--(1.8052,0.5351)--(1.7237,0.5947)--(1.6305,0.6327)--(1.5272,0.6488)--(1.4154,0.6437)--(1.2969,0.6192)--(1.1736,0.5779)--(1.0475,0.5231)--(0.9207,0.4589)--(0.7951,0.3891)--(0.6729,0.3179)--(0.5559,0.2490)--(0.4460,0.1856)--(0.3451,0.1304)--(0.2547,0.0849)--(0.1763,0.0499)--(0.1111,0.0254)--(0.0603,0.0103)--(0.0245,0.0027)--(0.0045,0.0000)--(0.0000,0.0000)--(0.0125,0.0000)--(0.0405,-0.0057)--(0.0838,-0.0168)--(0.1420,-0.0364)--(0.2139,-0.0661)--(0.2985,-0.1063)--(0.3943,-0.1569)--(0.5000,-0.2165)--(0.6136,-0.2830)--(0.7335,-0.3535)--(0.8576,-0.4244)--(0.9841,-0.4920)--(1.1108,-0.5519)--(1.2357,-0.6004)--(1.3568,-0.6337)--(1.4722,-0.6488)--(1.5800,-0.6435)--(1.6785,-0.6165)--(1.7660,-0.5675)--(1.8412,-0.4977)--(1.9029,-0.4089)--(1.9500,-0.3042)--(1.9819,-0.1875)--(1.9979,-0.0633)--(1.9979,0.0633)--(1.9819,0.1875)--(1.9500,0.3042)--(1.9029,0.4089)--(1.8412,0.4977)--(1.7660,0.5675)--(1.6785,0.6165)--(1.5800,0.6435)--(1.4722,0.6488)--(1.3568,0.6337)--(1.2357,0.6004)--(1.1108,0.5519)--(0.9841,0.4920)--(0.8576,0.4244)--(0.7335,0.3535)--(0.6136,0.2830)--(0.4999,0.2165)--(0.3943,0.1569)--(0.2985,0.1063)--(0.2139,0.0661)--(0.1420,0.0364)--(0.0838,0.0168)--(0.0405,0.0057)--(0.0125,0.0000)--(0.0000,0.0000)--(0.0045,0.0000)--(0.0245,-0.0027)--(0.0603,-0.0103)--(0.1111,-0.0254)--(0.1763,-0.0499)--(0.2547,-0.0849)--(0.3451,-0.1304)--(0.4460,-0.1856)--(0.5559,-0.2490)--(0.6729,-0.3179)--(0.7951,-0.3891)--(0.9207,-0.4589)--(1.0475,-0.5231)--(1.1736,-0.5779)--(1.2969,-0.6192)--(1.4154,-0.6437)--(1.5272,-0.6488)--(1.6305,-0.6327)--(1.7237,-0.5947)--(1.8052,-0.5351)--(1.8738,-0.4555)--(1.9283,-0.3583)--(1.9679,-0.2471)--(1.9919,-0.1260)--(2.0000,0.0000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CbCartTuiii.pstricks.recall b/src_phystricks/Fig_CbCartTuiii.pstricks.recall
index 8b557fe13..c83988743 100644
--- a/src_phystricks/Fig_CbCartTuiii.pstricks.recall
+++ b/src_phystricks/Fig_CbCartTuiii.pstricks.recall
@@ -71,18 +71,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4997,0) -- (2.4997,0);
-\draw [,->,>=latex] (0,-2.4977) -- (0,2.4977);
+\draw [,->,>=latex] (-2.4997,0.0000) -- (2.4997,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4977) -- (0.0000,2.4977);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.253,0.379)--(0.502,0.743)--(0.743,1.08)--(0.972,1.38)--(1.19,1.63)--(1.38,1.82)--(1.55,1.94)--(1.70,2.00)--(1.82,1.98)--(1.91,1.89)--(1.97,1.73)--(2.00,1.51)--(1.99,1.24)--(1.96,0.916)--(1.89,0.563)--(1.79,0.190)--(1.67,-0.190)--(1.51,-0.563)--(1.33,-0.916)--(1.13,-1.24)--(0.916,-1.51)--(0.684,-1.73)--(0.441,-1.89)--(0.190,-1.98)--(-0.0635,-2.00)--(-0.316,-1.94)--(-0.563,-1.82)--(-0.802,-1.63)--(-1.03,-1.38)--(-1.24,-1.08)--(-1.43,-0.743)--(-1.59,-0.379)--(-1.73,0)--(-1.84,0.379)--(-1.93,0.743)--(-1.98,1.08)--(-2.00,1.38)--(-1.99,1.63)--(-1.94,1.82)--(-1.87,1.94)--(-1.76,2.00)--(-1.63,1.98)--(-1.47,1.89)--(-1.29,1.73)--(-1.08,1.51)--(-0.860,1.24)--(-0.624,0.916)--(-0.379,0.563)--(-0.127,0.190)--(0.127,-0.190)--(0.379,-0.563)--(0.624,-0.916)--(0.860,-1.24)--(1.08,-1.51)--(1.29,-1.73)--(1.47,-1.89)--(1.63,-1.98)--(1.76,-2.00)--(1.87,-1.94)--(1.94,-1.82)--(1.99,-1.63)--(2.00,-1.38)--(1.98,-1.08)--(1.93,-0.743)--(1.84,-0.379)--(1.73,0)--(1.59,0.379)--(1.43,0.743)--(1.24,1.08)--(1.03,1.38)--(0.802,1.63)--(0.563,1.82)--(0.316,1.94)--(0.0635,2.00)--(-0.190,1.98)--(-0.441,1.89)--(-0.684,1.73)--(-0.916,1.51)--(-1.13,1.24)--(-1.33,0.916)--(-1.51,0.563)--(-1.67,0.190)--(-1.79,-0.190)--(-1.89,-0.563)--(-1.96,-0.916)--(-1.99,-1.24)--(-2.00,-1.51)--(-1.97,-1.73)--(-1.91,-1.89)--(-1.82,-1.98)--(-1.70,-2.00)--(-1.55,-1.94)--(-1.38,-1.82)--(-1.19,-1.63)--(-0.972,-1.38)--(-0.743,-1.08)--(-0.502,-0.743)--(-0.253,-0.379)--(0,0);
-\draw (-2.0000,-0.32983) node {$ -1 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.43316,-2.0000) node {$ -1 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.2531,0.3785)--(0.5022,0.7433)--(0.7433,1.0812)--(0.9723,1.3801)--(1.1858,1.6291)--(1.3801,1.8192)--(1.5522,1.9436)--(1.6994,1.9977)--(1.8192,1.9796)--(1.9098,1.8900)--(1.9696,1.7320)--(1.9977,1.5114)--(1.9937,1.2363)--(1.9576,0.9164)--(1.8900,0.5634)--(1.7919,0.1901)--(1.6651,-0.1901)--(1.5114,-0.5634)--(1.3335,-0.9164)--(1.1341,-1.2363)--(0.9164,-1.5114)--(0.6840,-1.7320)--(0.4406,-1.8900)--(0.1901,-1.9796)--(-0.0634,-1.9977)--(-0.3160,-1.9436)--(-0.5634,-1.8192)--(-0.8018,-1.6291)--(-1.0273,-1.3801)--(-1.2363,-1.0812)--(-1.4253,-0.7433)--(-1.5915,-0.3785)--(-1.7320,0.0000)--(-1.8447,0.3785)--(-1.9276,0.7433)--(-1.9796,1.0812)--(-1.9997,1.3801)--(-1.9876,1.6291)--(-1.9436,1.8192)--(-1.8682,1.9436)--(-1.7629,1.9977)--(-1.6291,1.9796)--(-1.4691,1.8900)--(-1.2855,1.7320)--(-1.0812,1.5114)--(-0.8595,1.2363)--(-0.6240,0.9164)--(-0.3785,0.5634)--(-0.1268,0.1901)--(0.1268,-0.1901)--(0.3785,-0.5634)--(0.6240,-0.9164)--(0.8595,-1.2363)--(1.0812,-1.5114)--(1.2855,-1.7320)--(1.4691,-1.8900)--(1.6291,-1.9796)--(1.7629,-1.9977)--(1.8682,-1.9436)--(1.9436,-1.8192)--(1.9876,-1.6291)--(1.9997,-1.3801)--(1.9796,-1.0812)--(1.9276,-0.7433)--(1.8447,-0.3785)--(1.7320,0.0000)--(1.5915,0.3785)--(1.4253,0.7433)--(1.2363,1.0812)--(1.0273,1.3801)--(0.8018,1.6291)--(0.5634,1.8192)--(0.3160,1.9436)--(0.0634,1.9977)--(-0.1901,1.9796)--(-0.4406,1.8900)--(-0.6840,1.7320)--(-0.9164,1.5114)--(-1.1341,1.2363)--(-1.3335,0.9164)--(-1.5114,0.5634)--(-1.6651,0.1901)--(-1.7919,-0.1901)--(-1.8900,-0.5634)--(-1.9576,-0.9164)--(-1.9937,-1.2363)--(-1.9977,-1.5114)--(-1.9696,-1.7320)--(-1.9098,-1.8900)--(-1.8192,-1.9796)--(-1.6994,-1.9977)--(-1.5522,-1.9436)--(-1.3801,-1.8192)--(-1.1858,-1.6291)--(-0.9723,-1.3801)--(-0.7433,-1.0812)--(-0.5022,-0.7433)--(-0.2531,-0.3785)--(0.0000,0.0000);
+\draw (-2.0000,-0.3298) node {$ -1 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CercleImplicite.pstricks.recall b/src_phystricks/Fig_CercleImplicite.pstricks.recall
index 65090a7cf..570ddd92b 100644
--- a/src_phystricks/Fig_CercleImplicite.pstricks.recall
+++ b/src_phystricks/Fig_CercleImplicite.pstricks.recall
@@ -79,20 +79,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 \draw []  (1.4142,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.7689,1.7511) node {$P$};
+\draw (1.7688,1.7510) node {$P$};
 \draw []  (1.4142,-1.4142) node [rotate=0] {$\bullet$};
 \draw (1.8158,-1.7671) node {\( P'\)};
-\draw []  (-2.0000,0) node [rotate=0] {$\bullet$};
-\draw (-2.3564,0.37233) node {\( Q\)};
-\draw []  (1.4142,0) node [rotate=0] {$\bullet$};
-\draw (1.0978,-0.29071) node {\( x\)};
-\draw [color=red,style=dotted] (1.41,1.41) -- (1.41,-1.41);
+\draw []  (-2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-2.3564,0.3723) node {\( Q\)};
+\draw []  (1.4142,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0977,-0.2907) node {\( x\)};
+\draw [color=red,style=dotted] (1.4142,1.4142) -- (1.4142,-1.4142);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_CercleTnu.pstricks.recall b/src_phystricks/Fig_CercleTnu.pstricks.recall
index 3feef34ae..4b6a8499f 100644
--- a/src_phystricks/Fig_CercleTnu.pstricks.recall
+++ b/src_phystricks/Fig_CercleTnu.pstricks.recall
@@ -73,14 +73,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=green,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
-\draw (2.3385,1.8384) node {\( n\)};
+\draw [color=green,->,>=latex] (1.7320,1.0000) -- (2.5980,1.5000);
+\draw (2.3385,1.8383) node {\( n\)};
 \draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
-\draw (1.6552,2.1243) node {\( e_{\theta}\)};
-\draw [color=green,->,>=latex] (-0.68404,1.8794) -- (-1.0261,2.8191);
-\draw (-1.4175,2.6379) node {\( n\)};
-\draw [color=red,->,>=latex] (-0.68404,1.8794) -- (-1.6237,1.5374);
-\draw (-1.8897,1.9276) node {\( e_{\theta}\)};
+\draw (1.6551,2.1243) node {\( e_{\theta}\)};
+\draw [color=green,->,>=latex] (-0.6840,1.8793) -- (-1.0260,2.8190);
+\draw (-1.4175,2.6378) node {\( n\)};
+\draw [color=red,->,>=latex] (-0.6840,1.8793) -- (-1.6237,1.5373);
+\draw (-1.8896,1.9275) node {\( e_{\theta}\)};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -88,8 +88,8 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.00,0) -- (2.00,0.127) -- (1.98,0.253) -- (1.96,0.379) -- (1.94,0.502) -- (1.90,0.624) -- (1.86,0.743) -- (1.81,0.860) -- (1.75,0.972) -- (1.68,1.08) -- (1.61,1.19) -- (1.53,1.29) -- (1.45,1.38) -- (1.36,1.47) -- (1.26,1.55) -- (1.16,1.63) -- (1.05,1.70) -- (0.945,1.76) -- (0.831,1.82) -- (0.714,1.87) -- (0.594,1.91) -- (0.472,1.94) -- (0.347,1.97) -- (0.222,1.99) -- (0.0952,2.00) -- (-0.0317,2.00) -- (-0.158,1.99) -- (-0.285,1.98) -- (-0.410,1.96) -- (-0.533,1.93) -- (-0.654,1.89) -- (-0.773,1.84) -- (-0.888,1.79) -- (-1.00,1.73) -- (-1.11,1.67) -- (-1.21,1.59) -- (-1.31,1.51) -- (-1.40,1.43) -- (-1.49,1.33) -- (-1.57,1.24) -- (-1.65,1.13) -- (-1.72,1.03) -- (-1.78,0.916) -- (-1.83,0.802) -- (-1.88,0.684) -- (-1.92,0.563) -- (-1.95,0.441) -- (-1.97,0.316) -- (-1.99,0.190) -- (-2.00,0.0635) -- (-2.00,-0.0635) -- (-1.99,-0.190) -- (-1.97,-0.316) -- (-1.95,-0.441) -- (-1.92,-0.563) -- (-1.88,-0.684) -- (-1.83,-0.802) -- (-1.78,-0.916) -- (-1.72,-1.03) -- (-1.65,-1.13) -- (-1.57,-1.24) -- (-1.49,-1.33) -- (-1.40,-1.43) -- (-1.31,-1.51) -- (-1.21,-1.59) -- (-1.11,-1.67) -- (-1.00,-1.73) -- (-0.888,-1.79) -- (-0.773,-1.84) -- (-0.654,-1.89) -- (-0.533,-1.93) -- (-0.410,-1.96) -- (-0.285,-1.98) -- (-0.158,-1.99) -- (-0.0317,-2.00) -- (0.0952,-2.00) -- (0.222,-1.99) -- (0.347,-1.97) -- (0.472,-1.94) -- (0.594,-1.91) -- (0.714,-1.87) -- (0.831,-1.82) -- (0.945,-1.76) -- (1.05,-1.70) -- (1.16,-1.63) -- (1.26,-1.55) -- (1.36,-1.47) -- (1.45,-1.38) -- (1.53,-1.29) -- (1.61,-1.19) -- (1.68,-1.08) -- (1.75,-0.972) -- (1.81,-0.860) -- (1.86,-0.743) -- (1.90,-0.624) -- (1.94,-0.502) -- (1.96,-0.379) -- (1.98,-0.253) -- (2.00,-0.127) -- (2.00,0) --  cycle;
-\draw [color=brown] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.0000,0.0000) -- (1.9959,0.1268) -- (1.9839,0.2531) -- (1.9638,0.3785) -- (1.9358,0.5022) -- (1.9001,0.6240) -- (1.8567,0.7433) -- (1.8058,0.8595) -- (1.7476,0.9723) -- (1.6825,1.0812) -- (1.6105,1.1858) -- (1.5320,1.2855) -- (1.4474,1.3801) -- (1.3570,1.4691) -- (1.2611,1.5522) -- (1.1601,1.6291) -- (1.0544,1.6994) -- (0.9445,1.7629) -- (0.8308,1.8192) -- (0.7137,1.8682) -- (0.5938,1.9098) -- (0.4715,1.9436) -- (0.3472,1.9696) -- (0.2216,1.9876) -- (0.0951,1.9977) -- (-0.0317,1.9997) -- (-0.1584,1.9937) -- (-0.2846,1.9796) -- (-0.4096,1.9576) -- (-0.5329,1.9276) -- (-0.6541,1.8900) -- (-0.7726,1.8447) -- (-0.8881,1.7919) -- (-1.0000,1.7320) -- (-1.1078,1.6651) -- (-1.2112,1.5915) -- (-1.3097,1.5114) -- (-1.4029,1.4253) -- (-1.4905,1.3335) -- (-1.5721,1.2363) -- (-1.6473,1.1341) -- (-1.7159,1.0273) -- (-1.7776,0.9164) -- (-1.8322,0.8018) -- (-1.8793,0.6840) -- (-1.9189,0.5634) -- (-1.9508,0.4406) -- (-1.9748,0.3160) -- (-1.9909,0.1901) -- (-1.9989,0.0634) -- (-1.9989,-0.0634) -- (-1.9909,-0.1901) -- (-1.9748,-0.3160) -- (-1.9508,-0.4406) -- (-1.9189,-0.5634) -- (-1.8793,-0.6840) -- (-1.8322,-0.8018) -- (-1.7776,-0.9164) -- (-1.7159,-1.0273) -- (-1.6473,-1.1341) -- (-1.5721,-1.2363) -- (-1.4905,-1.3335) -- (-1.4029,-1.4253) -- (-1.3097,-1.5114) -- (-1.2112,-1.5915) -- (-1.1078,-1.6651) -- (-0.9999,-1.7320) -- (-0.8881,-1.7919) -- (-0.7726,-1.8447) -- (-0.6541,-1.8900) -- (-0.5329,-1.9276) -- (-0.4096,-1.9576) -- (-0.2846,-1.9796) -- (-0.1584,-1.9937) -- (-0.0317,-1.9997) -- (0.0951,-1.9977) -- (0.2216,-1.9876) -- (0.3472,-1.9696) -- (0.4715,-1.9436) -- (0.5938,-1.9098) -- (0.7137,-1.8682) -- (0.8308,-1.8192) -- (0.9445,-1.7629) -- (1.0544,-1.6994) -- (1.1601,-1.6291) -- (1.2611,-1.5522) -- (1.3570,-1.4691) -- (1.4474,-1.3801) -- (1.5320,-1.2855) -- (1.6105,-1.1858) -- (1.6825,-1.0812) -- (1.7476,-0.9723) -- (1.8058,-0.8595) -- (1.8567,-0.7433) -- (1.9001,-0.6240) -- (1.9358,-0.5022) -- (1.9638,-0.3785) -- (1.9839,-0.2531) -- (1.9959,-0.1268) -- (2.0000,0.0000) --  cycle;
+\draw [color=brown] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CercleTrigono.pstricks.recall b/src_phystricks/Fig_CercleTrigono.pstricks.recall
index 391fc1759..bb88ff553 100644
--- a/src_phystricks/Fig_CercleTrigono.pstricks.recall
+++ b/src_phystricks/Fig_CercleTrigono.pstricks.recall
@@ -79,26 +79,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
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-\draw [color=brown,style=dotted] (1.73,1.00) -- (1.73,0);
-\draw [color=brown,style=dotted] (1.73,1.00) -- (0,1.00);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
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-\draw [,->,>=latex] (-0.20000,0.50000) -- (-0.20000,1.0000);
-\draw (-0.75804,0.50000) node {$\sin(\theta)$};
-\draw [,->,>=latex] (0.86602,-0.20000) -- (0,-0.20000);
-\draw [,->,>=latex] (0.86602,-0.20000) -- (1.7320,-0.20000);
-\draw (0.86602,-0.48246) node {$\cos(\theta)$};
-\draw [color=brown,->,>=latex] (0,0) -- (1.7320,1.0000);
-\draw (0.91614,0.23026) node {$\theta$};
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw [color=brown,style=dotted] (1.7320,1.0000) -- (1.7320,0.0000);
+\draw [color=brown,style=dotted] (1.7320,1.0000) -- (0.0000,1.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (1.7320,0.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (-0.2000,0.5000) -- (-0.2000,0.0000);
+\draw [,->,>=latex] (-0.2000,0.5000) -- (-0.2000,1.0000);
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+\draw [,->,>=latex] (0.8660,-0.2000) -- (0.0000,-0.2000);
+\draw [,->,>=latex] (0.8660,-0.2000) -- (1.7320,-0.2000);
+\draw (0.8660,-0.4824) node {$\cos(\theta)$};
+\draw [color=brown,->,>=latex] (0.0000,0.0000) -- (1.7320,1.0000);
+\draw (0.9161,0.2302) node {$\theta$};
 
-\draw [] (0.400,0)--(0.400,0.00212)--(0.400,0.00423)--(0.400,0.00635)--(0.400,0.00846)--(0.400,0.0106)--(0.400,0.0127)--(0.400,0.0148)--(0.400,0.0169)--(0.400,0.0190)--(0.399,0.0211)--(0.399,0.0233)--(0.399,0.0254)--(0.399,0.0275)--(0.399,0.0296)--(0.399,0.0317)--(0.399,0.0338)--(0.398,0.0359)--(0.398,0.0380)--(0.398,0.0401)--(0.398,0.0422)--(0.398,0.0443)--(0.397,0.0464)--(0.397,0.0485)--(0.397,0.0506)--(0.397,0.0527)--(0.396,0.0548)--(0.396,0.0569)--(0.396,0.0590)--(0.395,0.0611)--(0.395,0.0632)--(0.395,0.0653)--(0.394,0.0674)--(0.394,0.0695)--(0.394,0.0715)--(0.393,0.0736)--(0.393,0.0757)--(0.392,0.0778)--(0.392,0.0798)--(0.392,0.0819)--(0.391,0.0840)--(0.391,0.0861)--(0.390,0.0881)--(0.390,0.0902)--(0.389,0.0922)--(0.389,0.0943)--(0.388,0.0964)--(0.388,0.0984)--(0.387,0.100)--(0.387,0.103)--(0.386,0.105)--(0.386,0.107)--(0.385,0.109)--(0.384,0.111)--(0.384,0.113)--(0.383,0.115)--(0.383,0.117)--(0.382,0.119)--(0.381,0.121)--(0.381,0.123)--(0.380,0.125)--(0.379,0.127)--(0.379,0.129)--(0.378,0.131)--(0.377,0.133)--(0.377,0.135)--(0.376,0.137)--(0.375,0.139)--(0.374,0.141)--(0.374,0.143)--(0.373,0.145)--(0.372,0.147)--(0.371,0.149)--(0.371,0.151)--(0.370,0.153)--(0.369,0.155)--(0.368,0.156)--(0.367,0.158)--(0.366,0.160)--(0.366,0.162)--(0.365,0.164)--(0.364,0.166)--(0.363,0.168)--(0.362,0.170)--(0.361,0.172)--(0.360,0.174)--(0.359,0.176)--(0.358,0.178)--(0.357,0.180)--(0.357,0.181)--(0.356,0.183)--(0.355,0.185)--(0.354,0.187)--(0.353,0.189)--(0.352,0.191)--(0.351,0.193)--(0.350,0.194)--(0.349,0.196)--(0.347,0.198)--(0.346,0.200);
-\draw (2.1344,1.2747) node {$P$};
+\draw [] (0.4000,0.0000)--(0.3999,0.0021)--(0.3999,0.0042)--(0.3999,0.0063)--(0.3999,0.0084)--(0.3998,0.0105)--(0.3997,0.0126)--(0.3997,0.0148)--(0.3996,0.0169)--(0.3995,0.0190)--(0.3994,0.0211)--(0.3993,0.0232)--(0.3991,0.0253)--(0.3990,0.0274)--(0.3989,0.0295)--(0.3987,0.0316)--(0.3985,0.0338)--(0.3983,0.0359)--(0.3981,0.0380)--(0.3979,0.0401)--(0.3977,0.0422)--(0.3975,0.0443)--(0.3972,0.0464)--(0.3970,0.0485)--(0.3967,0.0506)--(0.3965,0.0527)--(0.3962,0.0548)--(0.3959,0.0569)--(0.3956,0.0590)--(0.3953,0.0611)--(0.3949,0.0632)--(0.3946,0.0652)--(0.3942,0.0673)--(0.3939,0.0694)--(0.3935,0.0715)--(0.3931,0.0736)--(0.3927,0.0757)--(0.3923,0.0777)--(0.3919,0.0798)--(0.3915,0.0819)--(0.3910,0.0839)--(0.3906,0.0860)--(0.3901,0.0881)--(0.3897,0.0901)--(0.3892,0.0922)--(0.3887,0.0943)--(0.3882,0.0963)--(0.3877,0.0984)--(0.3871,0.1004)--(0.3866,0.1025)--(0.3860,0.1045)--(0.3855,0.1065)--(0.3849,0.1086)--(0.3843,0.1106)--(0.3837,0.1126)--(0.3831,0.1147)--(0.3825,0.1167)--(0.3819,0.1187)--(0.3813,0.1207)--(0.3806,0.1228)--(0.3800,0.1248)--(0.3793,0.1268)--(0.3786,0.1288)--(0.3780,0.1308)--(0.3773,0.1328)--(0.3765,0.1348)--(0.3758,0.1368)--(0.3751,0.1387)--(0.3744,0.1407)--(0.3736,0.1427)--(0.3728,0.1447)--(0.3721,0.1466)--(0.3713,0.1486)--(0.3705,0.1506)--(0.3697,0.1525)--(0.3689,0.1545)--(0.3681,0.1564)--(0.3672,0.1584)--(0.3664,0.1603)--(0.3655,0.1623)--(0.3647,0.1642)--(0.3638,0.1661)--(0.3629,0.1680)--(0.3620,0.1700)--(0.3611,0.1719)--(0.3602,0.1738)--(0.3593,0.1757)--(0.3583,0.1776)--(0.3574,0.1795)--(0.3564,0.1814)--(0.3555,0.1832)--(0.3545,0.1851)--(0.3535,0.1870)--(0.3525,0.1889)--(0.3515,0.1907)--(0.3505,0.1926)--(0.3495,0.1944)--(0.3485,0.1963)--(0.3474,0.1981)--(0.3464,0.2000);
+\draw (2.1343,1.2747) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_ChiSquared.pstricks.recall b/src_phystricks/Fig_ChiSquared.pstricks.recall
index 77a4f4ecd..50b7f9efb 100644
--- a/src_phystricks/Fig_ChiSquared.pstricks.recall
+++ b/src_phystricks/Fig_ChiSquared.pstricks.recall
@@ -95,27 +95,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (15.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.3819);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (15.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.3819);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.15152,0)--(0.30303,0.0064874)--(0.45455,0.028225)--(0.60606,0.076663)--(0.75758,0.16085)--(0.90909,0.28665)--(1.0606,0.45638)--(1.2121,0.66911)--(1.3636,0.92109)--(1.5152,1.2065)--(1.6667,1.5181)--(1.8182,1.8478)--(1.9697,2.1872)--(2.1212,2.5283)--(2.2727,2.8634)--(2.4242,3.1856)--(2.5758,3.4891)--(2.7273,3.7688)--(2.8788,4.0209)--(3.0303,4.2426)--(3.1818,4.4318)--(3.3333,4.5877)--(3.4848,4.7099)--(3.6364,4.7989)--(3.7879,4.8558)--(3.9394,4.8819)--(4.0909,4.8792)--(4.2424,4.8498)--(4.3939,4.7960)--(4.5455,4.7203)--(4.6970,4.6252)--(4.8485,4.5132)--(5.0000,4.3867)--(5.1515,4.2481)--(5.3030,4.0997)--(5.4545,3.9435)--(5.6061,3.7816)--(5.7576,3.6158)--(5.9091,3.4477)--(6.0606,3.2787)--(6.2121,3.1103)--(6.3636,2.9435)--(6.5152,2.7793)--(6.6667,2.6186)--(6.8182,2.4621)--(6.9697,2.3104)--(7.1212,2.1639)--(7.2727,2.0231)--(7.4242,1.8881)--(7.5758,1.7592)--(7.7273,1.6365)--(7.8788,1.5200)--(8.0303,1.4097)--(8.1818,1.3056)--(8.3333,1.2075)--(8.4848,1.1153)--(8.6364,1.0288)--(8.7879,0.94785)--(8.9394,0.87223)--(9.0909,0.80173)--(9.2424,0.73610)--(9.3939,0.67512)--(9.5455,0.61855)--(9.6970,0.56615)--(9.8485,0.51768)--(10.000,0.47292)--(10.152,0.43162)--(10.303,0.39359)--(10.455,0.35859)--(10.606,0.32643)--(10.758,0.29691)--(10.909,0.26985)--(11.061,0.24507)--(11.212,0.22239)--(11.364,0.20167)--(11.515,0.18274)--(11.667,0.16548)--(11.818,0.14975)--(11.970,0.13542)--(12.121,0.12239)--(12.273,0.11054)--(12.424,0.099777)--(12.576,0.090008)--(12.727,0.081149)--(12.879,0.073121)--(13.030,0.065850)--(13.182,0.059271)--(13.333,0.053320)--(13.485,0.047942)--(13.636,0.043085)--(13.788,0.038701)--(13.939,0.034746)--(14.091,0.031180)--(14.242,0.027968)--(14.394,0.025075)--(14.545,0.022471)--(14.697,0.020129)--(14.848,0.018024)--(15.000,0.016132);
-\draw (2.5000,-0.31492) node {$ 5 $};
-\draw [] (2.50,-0.100) -- (2.50,0.100);
-\draw (5.0000,-0.31492) node {$ 10 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (7.5000,-0.31492) node {$ 15 $};
-\draw [] (7.50,-0.100) -- (7.50,0.100);
-\draw (10.000,-0.31492) node {$ 20 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (12.500,-0.31492) node {$ 25 $};
-\draw [] (12.5,-0.100) -- (12.5,0.100);
-\draw (15.000,-0.31492) node {$ 30 $};
-\draw [] (15.0,-0.100) -- (15.0,0.100);
-\draw (-0.38167,2.5000) node {$ \frac{1}{20} $};
-\draw [] (-0.100,2.50) -- (0.100,2.50);
-\draw (-0.38167,5.0000) node {$ \frac{1}{10} $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue] (0.0000,0.0000)--(0.1515,0.0000)--(0.3030,0.0064)--(0.4545,0.0282)--(0.6060,0.0766)--(0.7575,0.1608)--(0.9090,0.2866)--(1.0606,0.4563)--(1.2121,0.6691)--(1.3636,0.9210)--(1.5151,1.2065)--(1.6666,1.5180)--(1.8181,1.8477)--(1.9696,2.1872)--(2.1212,2.5283)--(2.2727,2.8634)--(2.4242,3.1856)--(2.5757,3.4890)--(2.7272,3.7688)--(2.8787,4.0209)--(3.0303,4.2425)--(3.1818,4.4318)--(3.3333,4.5877)--(3.4848,4.7099)--(3.6363,4.7989)--(3.7878,4.8557)--(3.9393,4.8819)--(4.0909,4.8792)--(4.2424,4.8497)--(4.3939,4.7960)--(4.5454,4.7203)--(4.6969,4.6252)--(4.8484,4.5131)--(5.0000,4.3866)--(5.1515,4.2480)--(5.3030,4.0996)--(5.4545,3.9435)--(5.6060,3.7816)--(5.7575,3.6157)--(5.9090,3.4476)--(6.0606,3.2787)--(6.2121,3.1102)--(6.3636,2.9434)--(6.5151,2.7792)--(6.6666,2.6185)--(6.8181,2.4620)--(6.9696,2.3103)--(7.1212,2.1639)--(7.2727,2.0230)--(7.4242,1.8881)--(7.5757,1.7592)--(7.7272,1.6365)--(7.8787,1.5200)--(8.0303,1.4097)--(8.1818,1.3055)--(8.3333,1.2074)--(8.4848,1.1152)--(8.6363,1.0287)--(8.7878,0.9478)--(8.9393,0.8722)--(9.0909,0.8017)--(9.2424,0.7361)--(9.3939,0.6751)--(9.5454,0.6185)--(9.6969,0.5661)--(9.8484,0.5176)--(10.000,0.4729)--(10.151,0.4316)--(10.303,0.3935)--(10.454,0.3585)--(10.606,0.3264)--(10.757,0.2969)--(10.909,0.2698)--(11.060,0.2450)--(11.212,0.2223)--(11.363,0.2016)--(11.515,0.1827)--(11.666,0.1654)--(11.818,0.1497)--(11.969,0.1354)--(12.121,0.1223)--(12.272,0.1105)--(12.424,0.0997)--(12.575,0.0900)--(12.727,0.0811)--(12.878,0.0731)--(13.030,0.0658)--(13.181,0.0592)--(13.333,0.0533)--(13.484,0.0479)--(13.636,0.0430)--(13.787,0.0387)--(13.939,0.0347)--(14.090,0.0311)--(14.242,0.0279)--(14.393,0.0250)--(14.545,0.0224)--(14.696,0.0201)--(14.848,0.0180)--(15.000,0.0161);
+\draw (2.5000,-0.3149) node {$ 5 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (5.0000,-0.3149) node {$ 10 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (7.5000,-0.3149) node {$ 15 $};
+\draw [] (7.5000,-0.1000) -- (7.5000,0.1000);
+\draw (10.000,-0.3149) node {$ 20 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (12.500,-0.3149) node {$ 25 $};
+\draw [] (12.500,-0.1000) -- (12.500,0.1000);
+\draw (15.000,-0.3149) node {$ 30 $};
+\draw [] (15.000,-0.1000) -- (15.000,0.1000);
+\draw (-0.3816,2.5000) node {$ \frac{1}{20} $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.3816,5.0000) node {$ \frac{1}{10} $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall b/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
index 8e28a13e5..aac167aaf 100644
--- a/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
+++ b/src_phystricks/Fig_ChiSquaresQuantile.pstricks.recall
@@ -95,11 +95,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (0,0) -- (9.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.3842);
+\draw [,->,>=latex] (0.0000,0.0000) -- (9.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.3841);
 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -107,11 +107,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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@@ -119,27 +119,27 @@
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+\draw [color=blue] (4.8000,0.0000)--(4.8424,0.0000)--(4.8848,0.0000)--(4.9272,0.0000)--(4.9696,0.0000)--(5.0121,0.0000)--(5.0545,0.0000)--(5.0969,0.0000)--(5.1393,0.0000)--(5.1818,0.0000)--(5.2242,0.0000)--(5.2666,0.0000)--(5.3090,0.0000)--(5.3515,0.0000)--(5.3939,0.0000)--(5.4363,0.0000)--(5.4787,0.0000)--(5.5212,0.0000)--(5.5636,0.0000)--(5.6060,0.0000)--(5.6484,0.0000)--(5.6909,0.0000)--(5.7333,0.0000)--(5.7757,0.0000)--(5.8181,0.0000)--(5.8606,0.0000)--(5.9030,0.0000)--(5.9454,0.0000)--(5.9878,0.0000)--(6.0303,0.0000)--(6.0727,0.0000)--(6.1151,0.0000)--(6.1575,0.0000)--(6.2000,0.0000)--(6.2424,0.0000)--(6.2848,0.0000)--(6.3272,0.0000)--(6.3696,0.0000)--(6.4121,0.0000)--(6.4545,0.0000)--(6.4969,0.0000)--(6.5393,0.0000)--(6.5818,0.0000)--(6.6242,0.0000)--(6.6666,0.0000)--(6.7090,0.0000)--(6.7515,0.0000)--(6.7939,0.0000)--(6.8363,0.0000)--(6.8787,0.0000)--(6.9212,0.0000)--(6.9636,0.0000)--(7.0060,0.0000)--(7.0484,0.0000)--(7.0909,0.0000)--(7.1333,0.0000)--(7.1757,0.0000)--(7.2181,0.0000)--(7.2606,0.0000)--(7.3030,0.0000)--(7.3454,0.0000)--(7.3878,0.0000)--(7.4303,0.0000)--(7.4727,0.0000)--(7.5151,0.0000)--(7.5575,0.0000)--(7.6000,0.0000)--(7.6424,0.0000)--(7.6848,0.0000)--(7.7272,0.0000)--(7.7696,0.0000)--(7.8121,0.0000)--(7.8545,0.0000)--(7.8969,0.0000)--(7.9393,0.0000)--(7.9818,0.0000)--(8.0242,0.0000)--(8.0666,0.0000)--(8.1090,0.0000)--(8.1515,0.0000)--(8.1939,0.0000)--(8.2363,0.0000)--(8.2787,0.0000)--(8.3212,0.0000)--(8.3636,0.0000)--(8.4060,0.0000)--(8.4484,0.0000)--(8.4909,0.0000)--(8.5333,0.0000)--(8.5757,0.0000)--(8.6181,0.0000)--(8.6606,0.0000)--(8.7030,0.0000)--(8.7454,0.0000)--(8.7878,0.0000)--(8.8303,0.0000)--(8.8727,0.0000)--(8.9151,0.0000)--(8.9575,0.0000)--(9.0000,0.0000);
+\draw [] (4.8000,0.0000) -- (4.8000,1.4313);
+\draw [] (9.0000,0.0161) -- (9.0000,0.0000);
+\draw (1.5000,-0.3149) node {$ 5 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 10 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.5000,-0.3149) node {$ 15 $};
+\draw [] (4.5000,-0.1000) -- (4.5000,0.1000);
+\draw (6.0000,-0.3149) node {$ 20 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.5000,-0.3149) node {$ 25 $};
+\draw [] (7.5000,-0.1000) -- (7.5000,0.1000);
+\draw (9.0000,-0.3149) node {$ 30 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (-0.3816,2.5000) node {$ \frac{1}{20} $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.3816,5.0000) node {$ \frac{1}{10} $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ContourSqL.pstricks.recall b/src_phystricks/Fig_ContourSqL.pstricks.recall
index 766a75511..4aaf950f6 100644
--- a/src_phystricks/Fig_ContourSqL.pstricks.recall
+++ b/src_phystricks/Fig_ContourSqL.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -77,17 +77,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=blue] (0.0000,0.0000)--(0.0303,0.0303)--(0.0606,0.0606)--(0.0909,0.0909)--(0.1212,0.1212)--(0.1515,0.1515)--(0.1818,0.1818)--(0.2121,0.2121)--(0.2424,0.2424)--(0.2727,0.2727)--(0.3030,0.3030)--(0.3333,0.3333)--(0.3636,0.3636)--(0.3939,0.3939)--(0.4242,0.4242)--(0.4545,0.4545)--(0.4848,0.4848)--(0.5151,0.5151)--(0.5454,0.5454)--(0.5757,0.5757)--(0.6060,0.6060)--(0.6363,0.6363)--(0.6666,0.6666)--(0.6969,0.6969)--(0.7272,0.7272)--(0.7575,0.7575)--(0.7878,0.7878)--(0.8181,0.8181)--(0.8484,0.8484)--(0.8787,0.8787)--(0.9090,0.9090)--(0.9393,0.9393)--(0.9696,0.9696)--(1.0000,1.0000)--(1.0303,1.0303)--(1.0606,1.0606)--(1.0909,1.0909)--(1.1212,1.1212)--(1.1515,1.1515)--(1.1818,1.1818)--(1.2121,1.2121)--(1.2424,1.2424)--(1.2727,1.2727)--(1.3030,1.3030)--(1.3333,1.3333)--(1.3636,1.3636)--(1.3939,1.3939)--(1.4242,1.4242)--(1.4545,1.4545)--(1.4848,1.4848)--(1.5151,1.5151)--(1.5454,1.5454)--(1.5757,1.5757)--(1.6060,1.6060)--(1.6363,1.6363)--(1.6666,1.6666)--(1.6969,1.6969)--(1.7272,1.7272)--(1.7575,1.7575)--(1.7878,1.7878)--(1.8181,1.8181)--(1.8484,1.8484)--(1.8787,1.8787)--(1.9090,1.9090)--(1.9393,1.9393)--(1.9696,1.9696)--(2.0000,2.0000)--(2.0303,2.0303)--(2.0606,2.0606)--(2.0909,2.0909)--(2.1212,2.1212)--(2.1515,2.1515)--(2.1818,2.1818)--(2.2121,2.2121)--(2.2424,2.2424)--(2.2727,2.2727)--(2.3030,2.3030)--(2.3333,2.3333)--(2.3636,2.3636)--(2.3939,2.3939)--(2.4242,2.4242)--(2.4545,2.4545)--(2.4848,2.4848)--(2.5151,2.5151)--(2.5454,2.5454)--(2.5757,2.5757)--(2.6060,2.6060)--(2.6363,2.6363)--(2.6666,2.6666)--(2.6969,2.6969)--(2.7272,2.7272)--(2.7575,2.7575)--(2.7878,2.7878)--(2.8181,2.8181)--(2.8484,2.8484)--(2.8787,2.8787)--(2.9090,2.9090)--(2.9393,2.9393)--(2.9696,2.9696)--(3.0000,3.0000);
+\draw [,->,>=latex] (1.5000,1.5000) -- (1.4787,1.4787);
+\draw [color=blue] (0.0000,0.0000)--(0.0303,0.0000)--(0.0606,0.0012)--(0.0909,0.0027)--(0.1212,0.0048)--(0.1515,0.0076)--(0.1818,0.0110)--(0.2121,0.0149)--(0.2424,0.0195)--(0.2727,0.0247)--(0.3030,0.0306)--(0.3333,0.0370)--(0.3636,0.0440)--(0.3939,0.0517)--(0.4242,0.0599)--(0.4545,0.0688)--(0.4848,0.0783)--(0.5151,0.0884)--(0.5454,0.0991)--(0.5757,0.1104)--(0.6060,0.1224)--(0.6363,0.1349)--(0.6666,0.1481)--(0.6969,0.1619)--(0.7272,0.1763)--(0.7575,0.1913)--(0.7878,0.2069)--(0.8181,0.2231)--(0.8484,0.2399)--(0.8787,0.2574)--(0.9090,0.2754)--(0.9393,0.2941)--(0.9696,0.3134)--(1.0000,0.3333)--(1.0303,0.3538)--(1.0606,0.3749)--(1.0909,0.3966)--(1.1212,0.4190)--(1.1515,0.4419)--(1.1818,0.4655)--(1.2121,0.4897)--(1.2424,0.5145)--(1.2727,0.5399)--(1.3030,0.5659)--(1.3333,0.5925)--(1.3636,0.6198)--(1.3939,0.6476)--(1.4242,0.6761)--(1.4545,0.7052)--(1.4848,0.7349)--(1.5151,0.7652)--(1.5454,0.7961)--(1.5757,0.8276)--(1.6060,0.8598)--(1.6363,0.8925)--(1.6666,0.9259)--(1.6969,0.9599)--(1.7272,0.9944)--(1.7575,1.0296)--(1.7878,1.0655)--(1.8181,1.1019)--(1.8484,1.1389)--(1.8787,1.1766)--(1.9090,1.2148)--(1.9393,1.2537)--(1.9696,1.2932)--(2.0000,1.3333)--(2.0303,1.3740)--(2.0606,1.4153)--(2.0909,1.4573)--(2.1212,1.4998)--(2.1515,1.5430)--(2.1818,1.5867)--(2.2121,1.6311)--(2.2424,1.6761)--(2.2727,1.7217)--(2.3030,1.7679)--(2.3333,1.8148)--(2.3636,1.8622)--(2.3939,1.9103)--(2.4242,1.9589)--(2.4545,2.0082)--(2.4848,2.0581)--(2.5151,2.1086)--(2.5454,2.1597)--(2.5757,2.2115)--(2.6060,2.2638)--(2.6363,2.3168)--(2.6666,2.3703)--(2.6969,2.4245)--(2.7272,2.4793)--(2.7575,2.5347)--(2.7878,2.5907)--(2.8181,2.6473)--(2.8484,2.7046)--(2.8787,2.7624)--(2.9090,2.8209)--(2.9393,2.8800)--(2.9696,2.9397)--(3.0000,3.0000);
+\draw [,->,>=latex] (1.5000,0.7500) -- (1.5212,0.7712);
+\draw [] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (3.0000,3.0000) -- (3.0000,3.0000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CourbeRectifiable.pstricks.recall b/src_phystricks/Fig_CourbeRectifiable.pstricks.recall
index c87a9cf28..06e3bd8f9 100644
--- a/src_phystricks/Fig_CourbeRectifiable.pstricks.recall
+++ b/src_phystricks/Fig_CourbeRectifiable.pstricks.recall
@@ -85,21 +85,21 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [color=red] (-14.0,0) -- (-13.6,1.40);
-\draw [color=red] (-13.6,1.40) -- (-12.3,0);
-\draw [color=red] (-12.3,0) -- (-10.2,-1.40);
-\draw [color=red] (-10.2,-1.40) -- (-7.56,0);
-\draw []  (-14.000,0) node [rotate=0] {$\bullet$};
-\draw (-14.592,0) node {$\gamma(t_{0})$};
-\draw []  (-13.565,1.4000) node [rotate=0] {$\bullet$};
-\draw (-13.565,1.7825) node {$\gamma(t_{1})$};
-\draw []  (-12.286,0) node [rotate=0] {$\bullet$};
-\draw (-12.837,-0.30378) node {$\gamma(t_{2})$};
-\draw []  (-10.244,-1.4000) node [rotate=0] {$\bullet$};
-\draw (-10.244,-1.7825) node {$\gamma(t_{3})$};
-\draw []  (-7.5642,0) node [rotate=0] {$\bullet$};
-\draw (-8.0755,0.34271) node {$\gamma(t_{4})$};
-\draw [color=blue] (-14.000,0)--(-13.999,0.088794)--(-13.997,0.17723)--(-13.994,0.26495)--(-13.989,0.35161)--(-13.982,0.43685)--(-13.974,0.52033)--(-13.965,0.60171)--(-13.954,0.68068)--(-13.942,0.75690)--(-13.929,0.83007)--(-13.914,0.89990)--(-13.897,0.96611)--(-13.879,1.0284)--(-13.860,1.0866)--(-13.840,1.1404)--(-13.818,1.1896)--(-13.794,1.2340)--(-13.769,1.2735)--(-13.743,1.3078)--(-13.715,1.3369)--(-13.686,1.3605)--(-13.656,1.3787)--(-13.624,1.3914)--(-13.591,1.3984)--(-13.556,1.3998)--(-13.520,1.3956)--(-13.483,1.3857)--(-13.444,1.3703)--(-13.404,1.3494)--(-13.362,1.3230)--(-13.319,1.2913)--(-13.275,1.2544)--(-13.229,1.2124)--(-13.182,1.1656)--(-13.134,1.1141)--(-13.085,1.0581)--(-13.034,0.99777)--(-12.981,0.93348)--(-12.928,0.86542)--(-12.873,0.79388)--(-12.816,0.71915)--(-12.759,0.64152)--(-12.700,0.56130)--(-12.640,0.47883)--(-12.578,0.39443)--(-12.516,0.30843)--(-12.452,0.22120)--(-12.386,0.13308)--(-12.320,0.044419)--(-12.252,-0.044419)--(-12.183,-0.13308)--(-12.113,-0.22120)--(-12.041,-0.30843)--(-11.968,-0.39443)--(-11.895,-0.47883)--(-11.819,-0.56130)--(-11.743,-0.64152)--(-11.665,-0.71915)--(-11.587,-0.79388)--(-11.507,-0.86542)--(-11.425,-0.93348)--(-11.343,-0.99777)--(-11.260,-1.0581)--(-11.175,-1.1141)--(-11.089,-1.1656)--(-11.002,-1.2124)--(-10.914,-1.2544)--(-10.825,-1.2913)--(-10.735,-1.3230)--(-10.644,-1.3494)--(-10.551,-1.3703)--(-10.458,-1.3857)--(-10.363,-1.3956)--(-10.268,-1.3998)--(-10.171,-1.3984)--(-10.073,-1.3914)--(-9.9746,-1.3787)--(-9.8749,-1.3605)--(-9.7742,-1.3369)--(-9.6724,-1.3078)--(-9.5697,-1.2735)--(-9.4660,-1.2340)--(-9.3613,-1.1896)--(-9.2557,-1.1404)--(-9.1491,-1.0866)--(-9.0416,-1.0284)--(-8.9332,-0.96611)--(-8.8239,-0.89990)--(-8.7136,-0.83007)--(-8.6025,-0.75690)--(-8.4905,-0.68068)--(-8.3776,-0.60171)--(-8.2639,-0.52033)--(-8.1493,-0.43685)--(-8.0339,-0.35161)--(-7.9177,-0.26495)--(-7.8007,-0.17723)--(-7.6828,-0.088794)--(-7.5642,0);
+\draw [color=red] (-14.000,0.0000) -- (-13.564,1.4000);
+\draw [color=red] (-13.564,1.4000) -- (-12.286,0.0000);
+\draw [color=red] (-12.286,0.0000) -- (-10.243,-1.4000);
+\draw [color=red] (-10.243,-1.4000) -- (-7.5642,0.0000);
+\draw []  (-14.000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-14.591,0.0000) node {$\gamma(t_{0})$};
+\draw []  (-13.564,1.4000) node [rotate=0] {$\bullet$};
+\draw (-13.564,1.7824) node {$\gamma(t_{1})$};
+\draw []  (-12.286,0.0000) node [rotate=0] {$\bullet$};
+\draw (-12.836,-0.3037) node {$\gamma(t_{2})$};
+\draw []  (-10.243,-1.4000) node [rotate=0] {$\bullet$};
+\draw (-10.243,-1.7824) node {$\gamma(t_{3})$};
+\draw []  (-7.5642,0.0000) node [rotate=0] {$\bullet$};
+\draw (-8.0754,0.3427) node {$\gamma(t_{4})$};
+\draw [color=blue] (-14.000,0.0000)--(-13.999,0.0887)--(-13.997,0.1772)--(-13.993,0.2649)--(-13.988,0.3516)--(-13.982,0.4368)--(-13.974,0.5203)--(-13.965,0.6017)--(-13.954,0.6806)--(-13.942,0.7568)--(-13.928,0.8300)--(-13.913,0.8999)--(-13.897,0.9661)--(-13.879,1.0284)--(-13.860,1.0866)--(-13.839,1.1404)--(-13.817,1.1896)--(-13.794,1.2340)--(-13.769,1.2734)--(-13.742,1.3078)--(-13.715,1.3368)--(-13.686,1.3605)--(-13.655,1.3787)--(-13.623,1.3913)--(-13.590,1.3984)--(-13.555,1.3998)--(-13.519,1.3955)--(-13.482,1.3857)--(-13.443,1.3703)--(-13.403,1.3493)--(-13.362,1.3230)--(-13.319,1.2912)--(-13.274,1.2543)--(-13.229,1.2124)--(-13.182,1.1655)--(-13.134,1.1140)--(-13.084,1.0580)--(-13.033,0.9977)--(-12.981,0.9334)--(-12.927,0.8654)--(-12.872,0.7938)--(-12.816,0.7191)--(-12.758,0.6415)--(-12.700,0.5613)--(-12.639,0.4788)--(-12.578,0.3944)--(-12.515,0.3084)--(-12.451,0.2212)--(-12.386,0.1330)--(-12.319,0.0444)--(-12.252,-0.0444)--(-12.183,-0.1330)--(-12.112,-0.2212)--(-12.041,-0.3084)--(-11.968,-0.3944)--(-11.894,-0.4788)--(-11.819,-0.5613)--(-11.742,-0.6415)--(-11.665,-0.7191)--(-11.586,-0.7938)--(-11.506,-0.8654)--(-11.425,-0.9334)--(-11.343,-0.9977)--(-11.259,-1.0580)--(-11.175,-1.1140)--(-11.089,-1.1655)--(-11.002,-1.2124)--(-10.914,-1.2543)--(-10.825,-1.2912)--(-10.735,-1.3230)--(-10.643,-1.3493)--(-10.551,-1.3703)--(-10.457,-1.3857)--(-10.363,-1.3955)--(-10.267,-1.3998)--(-10.171,-1.3984)--(-10.073,-1.3913)--(-9.9746,-1.3787)--(-9.8749,-1.3605)--(-9.7741,-1.3368)--(-9.6724,-1.3078)--(-9.5696,-1.2734)--(-9.4659,-1.2340)--(-9.3613,-1.1896)--(-9.2556,-1.1404)--(-9.1491,-1.0866)--(-9.0416,-1.0284)--(-8.9331,-0.9661)--(-8.8238,-0.8999)--(-8.7136,-0.8300)--(-8.6024,-0.7568)--(-8.4904,-0.6806)--(-8.3776,-0.6017)--(-8.2638,-0.5203)--(-8.1493,-0.4368)--(-8.0339,-0.3516)--(-7.9176,-0.2649)--(-7.8006,-0.1772)--(-7.6828,-0.0887)--(-7.5642,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CouroneExam.pstricks.recall b/src_phystricks/Fig_CouroneExam.pstricks.recall
index 33009d868..6c2bbbb83 100644
--- a/src_phystricks/Fig_CouroneExam.pstricks.recall
+++ b/src_phystricks/Fig_CouroneExam.pstricks.recall
@@ -71,22 +71,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
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+\draw [color=blue] (2.0000,0.0000)--(1.9997,0.0317)--(1.9989,0.0634)--(1.9977,0.0951)--(1.9959,0.1268)--(1.9937,0.1584)--(1.9909,0.1901)--(1.9876,0.2216)--(1.9839,0.2531)--(1.9796,0.2846)--(1.9748,0.3160)--(1.9696,0.3472)--(1.9638,0.3785)--(1.9576,0.4096)--(1.9508,0.4406)--(1.9436,0.4715)--(1.9358,0.5022)--(1.9276,0.5329)--(1.9189,0.5634)--(1.9098,0.5938)--(1.9001,0.6240)--(1.8900,0.6541)--(1.8793,0.6840)--(1.8682,0.7137)--(1.8567,0.7433)--(1.8447,0.7726)--(1.8322,0.8018)--(1.8192,0.8308)--(1.8058,0.8595)--(1.7919,0.8881)--(1.7776,0.9164)--(1.7629,0.9445)--(1.7476,0.9723)--(1.7320,1.0000)--(1.7159,1.0273)--(1.6994,1.0544)--(1.6825,1.0812)--(1.6651,1.1078)--(1.6473,1.1341)--(1.6291,1.1601)--(1.6105,1.1858)--(1.5915,1.2112)--(1.5721,1.2363)--(1.5522,1.2611)--(1.5320,1.2855)--(1.5114,1.3097)--(1.4905,1.3335)--(1.4691,1.3570)--(1.4474,1.3801)--(1.4253,1.4029)--(1.4029,1.4253)--(1.3801,1.4474)--(1.3570,1.4691)--(1.3335,1.4905)--(1.3097,1.5114)--(1.2855,1.5320)--(1.2611,1.5522)--(1.2363,1.5721)--(1.2112,1.5915)--(1.1858,1.6105)--(1.1601,1.6291)--(1.1341,1.6473)--(1.1078,1.6651)--(1.0812,1.6825)--(1.0544,1.6994)--(1.0273,1.7159)--(1.0000,1.7320)--(0.9723,1.7476)--(0.9445,1.7629)--(0.9164,1.7776)--(0.8881,1.7919)--(0.8595,1.8058)--(0.8308,1.8192)--(0.8018,1.8322)--(0.7726,1.8447)--(0.7433,1.8567)--(0.7137,1.8682)--(0.6840,1.8793)--(0.6541,1.8900)--(0.6240,1.9001)--(0.5938,1.9098)--(0.5634,1.9189)--(0.5329,1.9276)--(0.5022,1.9358)--(0.4715,1.9436)--(0.4406,1.9508)--(0.4096,1.9576)--(0.3785,1.9638)--(0.3472,1.9696)--(0.3160,1.9748)--(0.2846,1.9796)--(0.2531,1.9839)--(0.2216,1.9876)--(0.1901,1.9909)--(0.1584,1.9937)--(0.1268,1.9959)--(0.0951,1.9977)--(0.0634,1.9989)--(0.0317,1.9997)--(0.0000,2.0000);
+\draw [color=blue] (0.0000,2.0000) -- (0.0000,1.0000);
+\draw [color=blue] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall b/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall
index 401de7a4e..2c06f37c0 100644
--- a/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall
+++ b/src_phystricks/Fig_CurvilignesPolaires.pstricks.recall
@@ -44,12 +44,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [color=brown,style=dashed] (2.000,0)--(2.000,0.02116)--(2.000,0.04231)--(1.999,0.06346)--(1.998,0.08460)--(1.997,0.1057)--(1.996,0.1268)--(1.995,0.1480)--(1.993,0.1690)--(1.991,0.1901)--(1.989,0.2112)--(1.986,0.2322)--(1.984,0.2532)--(1.981,0.2742)--(1.978,0.2951)--(1.975,0.3160)--(1.971,0.3369)--(1.968,0.3577)--(1.964,0.3785)--(1.960,0.3993)--(1.955,0.4200)--(1.951,0.4406)--(1.946,0.4612)--(1.941,0.4818)--(1.936,0.5023)--(1.930,0.5227)--(1.925,0.5431)--(1.919,0.5635)--(1.913,0.5837)--(1.907,0.6039)--(1.900,0.6241)--(1.893,0.6441)--(1.887,0.6641)--(1.879,0.6840)--(1.872,0.7039)--(1.865,0.7236)--(1.857,0.7433)--(1.849,0.7629)--(1.841,0.7824)--(1.832,0.8019)--(1.824,0.8212)--(1.815,0.8404)--(1.806,0.8596)--(1.797,0.8786)--(1.787,0.8976)--(1.778,0.9165)--(1.768,0.9352)--(1.758,0.9538)--(1.748,0.9724)--(1.737,0.9908)--(1.727,1.009)--(1.716,1.027)--(1.705,1.045)--(1.694,1.063)--(1.683,1.081)--(1.671,1.099)--(1.659,1.117)--(1.647,1.134)--(1.635,1.151)--(1.623,1.169)--(1.611,1.186)--(1.598,1.203)--(1.585,1.220)--(1.572,1.236)--(1.559,1.253)--(1.546,1.269)--(1.532,1.286)--(1.518,1.302)--(1.505,1.318)--(1.491,1.334)--(1.476,1.349)--(1.462,1.365)--(1.447,1.380)--(1.433,1.395)--(1.418,1.410)--(1.403,1.425)--(1.388,1.440)--(1.372,1.455)--(1.357,1.469)--(1.341,1.483)--(1.326,1.498)--(1.310,1.512)--(1.294,1.525)--(1.277,1.539)--(1.261,1.552)--(1.245,1.566)--(1.228,1.579)--(1.211,1.592)--(1.194,1.604)--(1.177,1.617)--(1.160,1.629)--(1.143,1.641)--(1.125,1.653)--(1.108,1.665)--(1.090,1.677)--(1.072,1.688)--(1.054,1.699)--(1.036,1.711)--(1.018,1.721)--(1.000,1.732);
-\draw [color=brown,style=dashed] (1.30,0.750) -- (3.03,1.75);
-\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
-\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=brown,style=dashed] (2.0000,0.0000)--(1.9998,0.0211)--(1.9995,0.0423)--(1.9989,0.0634)--(1.9982,0.0845)--(1.9972,0.1057)--(1.9959,0.1268)--(1.9945,0.1479)--(1.9928,0.1690)--(1.9909,0.1901)--(1.9888,0.2111)--(1.9864,0.2321)--(1.9839,0.2531)--(1.9811,0.2741)--(1.9781,0.2950)--(1.9748,0.3160)--(1.9714,0.3368)--(1.9677,0.3577)--(1.9638,0.3785)--(1.9597,0.3992)--(1.9554,0.4199)--(1.9508,0.4406)--(1.9460,0.4612)--(1.9411,0.4817)--(1.9358,0.5022)--(1.9304,0.5227)--(1.9248,0.5431)--(1.9189,0.5634)--(1.9129,0.5837)--(1.9066,0.6039)--(1.9001,0.6240)--(1.8934,0.6441)--(1.8865,0.6641)--(1.8793,0.6840)--(1.8720,0.7038)--(1.8644,0.7236)--(1.8567,0.7433)--(1.8487,0.7629)--(1.8405,0.7824)--(1.8322,0.8018)--(1.8236,0.8211)--(1.8148,0.8404)--(1.8058,0.8595)--(1.7966,0.8786)--(1.7872,0.8975)--(1.7776,0.9164)--(1.7678,0.9352)--(1.7578,0.9538)--(1.7476,0.9723)--(1.7373,0.9908)--(1.7267,1.0091)--(1.7159,1.0273)--(1.7050,1.0454)--(1.6938,1.0634)--(1.6825,1.0812)--(1.6709,1.0990)--(1.6592,1.1166)--(1.6473,1.1341)--(1.6352,1.1514)--(1.6229,1.1687)--(1.6105,1.1858)--(1.5979,1.2027)--(1.5850,1.2196)--(1.5721,1.2363)--(1.5589,1.2528)--(1.5456,1.2692)--(1.5320,1.2855)--(1.5184,1.3017)--(1.5045,1.3176)--(1.4905,1.3335)--(1.4763,1.3492)--(1.4619,1.3647)--(1.4474,1.3801)--(1.4327,1.3953)--(1.4179,1.4104)--(1.4029,1.4253)--(1.3877,1.4401)--(1.3724,1.4547)--(1.3570,1.4691)--(1.3414,1.4834)--(1.3256,1.4975)--(1.3097,1.5114)--(1.2936,1.5252)--(1.2774,1.5388)--(1.2611,1.5522)--(1.2446,1.5655)--(1.2279,1.5786)--(1.2112,1.5915)--(1.1943,1.6042)--(1.1772,1.6167)--(1.1601,1.6291)--(1.1428,1.6413)--(1.1253,1.6533)--(1.1078,1.6651)--(1.0901,1.6767)--(1.0723,1.6882)--(1.0544,1.6994)--(1.0364,1.7105)--(1.0182,1.7213)--(1.0000,1.7320);
+\draw [color=brown,style=dashed] (1.2990,0.7500) -- (3.0310,1.7500);
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5980,1.5000);
+\draw (2.2869,1.8657) node {$e_{r}$};
 \draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
-\draw (1.6075,2.1865) node {$e_{\theta}$};
+\draw (1.6075,2.1864) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -89,12 +89,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [color=brown,style=dashed] (-1.477,0.2605)--(-1.480,0.2448)--(-1.482,0.2292)--(-1.485,0.2135)--(-1.487,0.1978)--(-1.489,0.1820)--(-1.491,0.1663)--(-1.492,0.1505)--(-1.494,0.1347)--(-1.495,0.1189)--(-1.496,0.1031)--(-1.497,0.08722)--(-1.498,0.07137)--(-1.499,0.05552)--(-1.499,0.03966)--(-1.500,0.02380)--(-1.500,0.007933)--(-1.500,-0.007933)--(-1.500,-0.02380)--(-1.499,-0.03966)--(-1.499,-0.05552)--(-1.498,-0.07137)--(-1.497,-0.08722)--(-1.496,-0.1031)--(-1.495,-0.1189)--(-1.494,-0.1347)--(-1.492,-0.1505)--(-1.491,-0.1663)--(-1.489,-0.1820)--(-1.487,-0.1978)--(-1.485,-0.2135)--(-1.482,-0.2292)--(-1.480,-0.2448)--(-1.477,-0.2605)--(-1.474,-0.2761)--(-1.471,-0.2917)--(-1.468,-0.3072)--(-1.465,-0.3227)--(-1.461,-0.3382)--(-1.458,-0.3536)--(-1.454,-0.3690)--(-1.450,-0.3844)--(-1.446,-0.3997)--(-1.441,-0.4150)--(-1.437,-0.4302)--(-1.432,-0.4454)--(-1.428,-0.4605)--(-1.423,-0.4756)--(-1.417,-0.4906)--(-1.412,-0.5056)--(-1.407,-0.5205)--(-1.401,-0.5353)--(-1.395,-0.5501)--(-1.390,-0.5648)--(-1.384,-0.5795)--(-1.377,-0.5941)--(-1.371,-0.6087)--(-1.364,-0.6231)--(-1.358,-0.6375)--(-1.351,-0.6518)--(-1.344,-0.6661)--(-1.337,-0.6803)--(-1.330,-0.6944)--(-1.322,-0.7084)--(-1.315,-0.7224)--(-1.307,-0.7362)--(-1.299,-0.7500)--(-1.291,-0.7637)--(-1.283,-0.7773)--(-1.275,-0.7908)--(-1.266,-0.8043)--(-1.258,-0.8176)--(-1.249,-0.8309)--(-1.240,-0.8440)--(-1.231,-0.8571)--(-1.222,-0.8701)--(-1.213,-0.8830)--(-1.203,-0.8957)--(-1.194,-0.9084)--(-1.184,-0.9210)--(-1.174,-0.9335)--(-1.164,-0.9458)--(-1.154,-0.9581)--(-1.144,-0.9702)--(-1.134,-0.9823)--(-1.123,-0.9942)--(-1.113,-1.006)--(-1.102,-1.018)--(-1.091,-1.029)--(-1.080,-1.041)--(-1.069,-1.052)--(-1.058,-1.063)--(-1.047,-1.075)--(-1.035,-1.086)--(-1.024,-1.096)--(-1.012,-1.107)--(-1.000,-1.118)--(-0.9883,-1.128)--(-0.9763,-1.139)--(-0.9642,-1.149);
-\draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
-\draw (-2.0855,-1.2429) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
-\draw (-0.69205,-1.1323) node {$e_{\theta}$};
+\draw [color=brown,style=dashed] (-1.4772,0.2604)--(-1.4798,0.2448)--(-1.4823,0.2291)--(-1.4847,0.2134)--(-1.4869,0.1977)--(-1.4889,0.1820)--(-1.4907,0.1662)--(-1.4924,0.1504)--(-1.4939,0.1346)--(-1.4952,0.1188)--(-1.4964,0.1030)--(-1.4974,0.0872)--(-1.4983,0.0713)--(-1.4989,0.0555)--(-1.4994,0.0396)--(-1.4998,0.0237)--(-1.4999,0.0079)--(-1.4999,-0.0079)--(-1.4998,-0.0237)--(-1.4994,-0.0396)--(-1.4989,-0.0555)--(-1.4983,-0.0713)--(-1.4974,-0.0872)--(-1.4964,-0.1030)--(-1.4952,-0.1188)--(-1.4939,-0.1346)--(-1.4924,-0.1504)--(-1.4907,-0.1662)--(-1.4889,-0.1820)--(-1.4869,-0.1977)--(-1.4847,-0.2134)--(-1.4823,-0.2291)--(-1.4798,-0.2448)--(-1.4772,-0.2604)--(-1.4743,-0.2760)--(-1.4713,-0.2916)--(-1.4682,-0.3072)--(-1.4648,-0.3227)--(-1.4613,-0.3381)--(-1.4577,-0.3536)--(-1.4538,-0.3690)--(-1.4499,-0.3843)--(-1.4457,-0.3997)--(-1.4414,-0.4149)--(-1.4369,-0.4302)--(-1.4323,-0.4453)--(-1.4275,-0.4605)--(-1.4226,-0.4755)--(-1.4175,-0.4906)--(-1.4122,-0.5055)--(-1.4068,-0.5204)--(-1.4012,-0.5353)--(-1.3954,-0.5501)--(-1.3895,-0.5648)--(-1.3835,-0.5795)--(-1.3773,-0.5941)--(-1.3709,-0.6086)--(-1.3644,-0.6231)--(-1.3577,-0.6375)--(-1.3509,-0.6518)--(-1.3439,-0.6660)--(-1.3368,-0.6802)--(-1.3295,-0.6943)--(-1.3221,-0.7084)--(-1.3146,-0.7223)--(-1.3068,-0.7362)--(-1.2990,-0.7500)--(-1.2910,-0.7636)--(-1.2828,-0.7773)--(-1.2745,-0.7908)--(-1.2661,-0.8042)--(-1.2575,-0.8176)--(-1.2488,-0.8308)--(-1.2399,-0.8440)--(-1.2309,-0.8571)--(-1.2218,-0.8700)--(-1.2125,-0.8829)--(-1.2031,-0.8957)--(-1.1936,-0.9084)--(-1.1839,-0.9209)--(-1.1741,-0.9334)--(-1.1642,-0.9458)--(-1.1541,-0.9580)--(-1.1439,-0.9702)--(-1.1336,-0.9822)--(-1.1231,-0.9942)--(-1.1125,-1.0060)--(-1.1018,-1.0177)--(-1.0910,-1.0293)--(-1.0801,-1.0408)--(-1.0690,-1.0522)--(-1.0578,-1.0634)--(-1.0465,-1.0745)--(-1.0351,-1.0856)--(-1.0235,-1.0964)--(-1.0119,-1.1072)--(-1.0001,-1.1178)--(-0.9882,-1.1284)--(-0.9762,-1.1388)--(-0.9641,-1.1490);
+\draw [color=brown,style=dashed] (-0.9396,-0.3420) -- (-2.8190,-1.0260);
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-2.3492,-0.8550);
+\draw (-2.0854,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-1.0675,-1.4527);
+\draw (-0.6920,-1.1322) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_CycloideA.pstricks.recall b/src_phystricks/Fig_CycloideA.pstricks.recall
index 22616691c..9fe6d18e5 100644
--- a/src_phystricks/Fig_CycloideA.pstricks.recall
+++ b/src_phystricks/Fig_CycloideA.pstricks.recall
@@ -115,40 +115,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (13.066,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.4995);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (13.066,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.4994);
 %DEFAULT
-\draw [color=blue] (0,0)--(0,0.0080452)--(0.0027181,0.032051)--(0.0091366,0.071632)--(0.021535,0.12615)--(0.041757,0.19473)--(0.071519,0.27627)--(0.11238,0.36945)--(0.16574,0.47277)--(0.23277,0.58459)--(0.31443,0.70308)--(0.41146,0.82635)--(0.52433,0.95242)--(0.65327,1.0793)--(0.79826,1.2048)--(0.95899,1.3271)--(1.1349,1.4441)--(1.3253,1.5539)--(1.5290,1.6549)--(1.7450,1.7453)--(1.9716,1.8237)--(2.2074,1.8888)--(2.4505,1.9397)--(2.6992,1.9754)--(2.9513,1.9955)--(3.2051,1.9995)--(3.4583,1.9874)--(3.7089,1.9595)--(3.9551,1.9161)--(4.1947,1.8580)--(4.4261,1.7861)--(4.6476,1.7015)--(4.8576,1.6056)--(5.0548,1.5000)--(5.2381,1.3863)--(5.4065,1.2665)--(5.5594,1.1423)--(5.6964,1.0159)--(5.8173,0.88916)--(5.9222,0.76424)--(6.0115,0.64311)--(6.0857,0.52773)--(6.1458,0.41994)--(6.1927,0.32149)--(6.2278,0.23396)--(6.2526,0.15875)--(6.2687,0.097073)--(6.2779,0.049929)--(6.2820,0.018071)--(6.2831,0.0020133)--(6.2832,0.0020133)--(6.2843,0.018071)--(6.2885,0.049929)--(6.2977,0.097073)--(6.3137,0.15875)--(6.3385,0.23396)--(6.3737,0.32149)--(6.4206,0.41994)--(6.4807,0.52773)--(6.5549,0.64311)--(6.6442,0.76424)--(6.7491,0.88916)--(6.8700,1.0159)--(7.0070,1.1423)--(7.1599,1.2665)--(7.3283,1.3863)--(7.5116,1.5000)--(7.7087,1.6056)--(7.9188,1.7015)--(8.1402,1.7861)--(8.3716,1.8580)--(8.6113,1.9161)--(8.8575,1.9595)--(9.1081,1.9874)--(9.3613,1.9995)--(9.6150,1.9955)--(9.8672,1.9754)--(10.116,1.9397)--(10.359,1.8888)--(10.595,1.8237)--(10.821,1.7453)--(11.037,1.6549)--(11.241,1.5539)--(11.431,1.4441)--(11.607,1.3271)--(11.768,1.2048)--(11.913,1.0793)--(12.042,0.95242)--(12.155,0.82635)--(12.252,0.70308)--(12.334,0.58459)--(12.401,0.47277)--(12.454,0.36945)--(12.495,0.27627)--(12.525,0.19473)--(12.545,0.12615)--(12.557,0.071632)--(12.564,0.032051)--(12.566,0.0080452)--(12.566,0);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (9.0000,-0.31492) node {$ 9 $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (10.000,-0.31492) node {$ 10 $};
-\draw [] (10.0,-0.100) -- (10.0,0.100);
-\draw (11.000,-0.31492) node {$ 11 $};
-\draw [] (11.0,-0.100) -- (11.0,0.100);
-\draw (12.000,-0.31492) node {$ 12 $};
-\draw [] (12.0,-0.100) -- (12.0,0.100);
-\draw (13.000,-0.31492) node {$ 13 $};
-\draw [] (13.0,-0.100) -- (13.0,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0000,0.0080)--(0.0027,0.0320)--(0.0091,0.0716)--(0.0215,0.1261)--(0.0417,0.1947)--(0.0715,0.2762)--(0.1123,0.3694)--(0.1657,0.4727)--(0.2327,0.5845)--(0.3144,0.7030)--(0.4114,0.8263)--(0.5243,0.9524)--(0.6532,1.0792)--(0.7982,1.2048)--(0.9589,1.3270)--(1.1349,1.4440)--(1.3252,1.5539)--(1.5290,1.6548)--(1.7449,1.7452)--(1.9716,1.8236)--(2.2073,1.8888)--(2.4505,1.9396)--(2.6991,1.9754)--(2.9513,1.9954)--(3.2050,1.9994)--(3.4582,1.9874)--(3.7089,1.9594)--(3.9550,1.9161)--(4.1947,1.8579)--(4.4261,1.7860)--(4.6476,1.7014)--(4.8576,1.6056)--(5.0548,1.5000)--(5.2380,1.3863)--(5.4064,1.2664)--(5.5594,1.1423)--(5.6963,1.0158)--(5.8172,0.8891)--(5.9221,0.7642)--(6.0114,0.6431)--(6.0857,0.5277)--(6.1457,0.4199)--(6.1927,0.3214)--(6.2278,0.2339)--(6.2526,0.1587)--(6.2687,0.0970)--(6.2778,0.0499)--(6.2820,0.0180)--(6.2831,0.0020)--(6.2832,0.0020)--(6.2843,0.0180)--(6.2884,0.0499)--(6.2976,0.0970)--(6.3137,0.1587)--(6.3385,0.2339)--(6.3736,0.3214)--(6.4206,0.4199)--(6.4806,0.5277)--(6.5549,0.6431)--(6.6441,0.7642)--(6.7490,0.8891)--(6.8699,1.0158)--(7.0069,1.1423)--(7.1598,1.2664)--(7.3282,1.3863)--(7.5115,1.5000)--(7.7087,1.6056)--(7.9187,1.7014)--(8.1402,1.7860)--(8.3716,1.8579)--(8.6113,1.9161)--(8.8574,1.9594)--(9.1081,1.9874)--(9.3613,1.9994)--(9.6150,1.9954)--(9.8672,1.9754)--(10.115,1.9396)--(10.359,1.8888)--(10.594,1.8236)--(10.821,1.7452)--(11.037,1.6548)--(11.241,1.5539)--(11.431,1.4440)--(11.607,1.3270)--(11.768,1.2048)--(11.913,1.0792)--(12.042,0.9524)--(12.154,0.8263)--(12.251,0.7030)--(12.333,0.5845)--(12.400,0.4727)--(12.453,0.3694)--(12.494,0.2762)--(12.524,0.1947)--(12.544,0.1261)--(12.557,0.0716)--(12.563,0.0320)--(12.566,0.0080)--(12.566,0.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (9.0000,-0.3149) node {$ 9 $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (10.000,-0.3149) node {$ 10 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (11.000,-0.3149) node {$ 11 $};
+\draw [] (11.000,-0.1000) -- (11.000,0.1000);
+\draw (12.000,-0.3149) node {$ 12 $};
+\draw [] (12.000,-0.1000) -- (12.000,0.1000);
+\draw (13.000,-0.3149) node {$ 13 $};
+\draw [] (13.000,-0.1000) -- (13.000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall b/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall
index 28b9aef27..dc2f78bc4 100644
--- a/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall
+++ b/src_phystricks/Fig_DGFSooWgbuuMoB.pstricks.recall
@@ -60,21 +60,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.2500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.2500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue,style=dashed] (2.5000,1.0000)--(2.5126,0.9898)--(2.5252,0.9797)--(2.5378,0.9696)--(2.5505,0.9595)--(2.5631,0.9494)--(2.5757,0.9393)--(2.5883,0.9292)--(2.6010,0.9191)--(2.6136,0.9090)--(2.6262,0.8989)--(2.6388,0.8888)--(2.6515,0.8787)--(2.6641,0.8686)--(2.6767,0.8585)--(2.6893,0.8484)--(2.7020,0.8383)--(2.7146,0.8282)--(2.7272,0.8181)--(2.7398,0.8080)--(2.7525,0.7979)--(2.7651,0.7878)--(2.7777,0.7777)--(2.7904,0.7676)--(2.8030,0.7575)--(2.8156,0.7474)--(2.8282,0.7373)--(2.8409,0.7272)--(2.8535,0.7171)--(2.8661,0.7070)--(2.8787,0.6969)--(2.8914,0.6868)--(2.9040,0.6767)--(2.9166,0.6666)--(2.9292,0.6565)--(2.9419,0.6464)--(2.9545,0.6363)--(2.9671,0.6262)--(2.9797,0.6161)--(2.9924,0.6060)--(3.0050,0.5959)--(3.0176,0.5858)--(3.0303,0.5757)--(3.0429,0.5656)--(3.0555,0.5555)--(3.0681,0.5454)--(3.0808,0.5353)--(3.0934,0.5252)--(3.1060,0.5151)--(3.1186,0.5050)--(3.1313,0.4949)--(3.1439,0.4848)--(3.1565,0.4747)--(3.1691,0.4646)--(3.1818,0.4545)--(3.1944,0.4444)--(3.2070,0.4343)--(3.2196,0.4242)--(3.2323,0.4141)--(3.2449,0.4040)--(3.2575,0.3939)--(3.2702,0.3838)--(3.2828,0.3737)--(3.2954,0.3636)--(3.3080,0.3535)--(3.3207,0.3434)--(3.3333,0.3333)--(3.3459,0.3232)--(3.3585,0.3131)--(3.3712,0.3030)--(3.3838,0.2929)--(3.3964,0.2828)--(3.4090,0.2727)--(3.4217,0.2626)--(3.4343,0.2525)--(3.4469,0.2424)--(3.4595,0.2323)--(3.4722,0.2222)--(3.4848,0.2121)--(3.4974,0.2020)--(3.5101,0.1919)--(3.5227,0.1818)--(3.5353,0.1717)--(3.5479,0.1616)--(3.5606,0.1515)--(3.5732,0.1414)--(3.5858,0.1313)--(3.5984,0.1212)--(3.6111,0.1111)--(3.6237,0.1010)--(3.6363,0.0909)--(3.6489,0.0808)--(3.6616,0.0707)--(3.6742,0.0606)--(3.6868,0.0505)--(3.6994,0.0404)--(3.7121,0.0303)--(3.7247,0.0202)--(3.7373,0.0101)--(3.7500,0.0000);
+\draw [] (1.2500,-0.1000) -- (1.2500,0.1000);
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw [] (3.7500,-0.1000) -- (3.7500,0.1000);
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DerivTangenteOM.pstricks.recall b/src_phystricks/Fig_DerivTangenteOM.pstricks.recall
index a0c5b4ece..606108c8a 100644
--- a/src_phystricks/Fig_DerivTangenteOM.pstricks.recall
+++ b/src_phystricks/Fig_DerivTangenteOM.pstricks.recall
@@ -87,33 +87,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.8750,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,7.8518);
 %DEFAULT
-\draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
-\draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
-\draw [color=green,style=dashed] (3.00,1.50) -- (3.00,0);
-\draw [color=green,style=dashed] (6.50,6.09) -- (0,6.09);
-\draw [color=green,style=dashed] (3.00,1.50) -- (0,1.50);
-\draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
+\draw [color=cyan] (2.1250,0.3535) -- (7.3750,7.2320);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (6.5000,0.0000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (3.0000,0.0000);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (0.0000,6.0856);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (0.0000,1.5000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (6.5000,1.5000);
 
-\draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
+\draw [color=blue] (1.0000,1.0185)--(1.0606,1.0220)--(1.1212,1.0261)--(1.1818,1.0305)--(1.2424,1.0355)--(1.3030,1.0409)--(1.3636,1.0469)--(1.4242,1.0535)--(1.4848,1.0606)--(1.5454,1.0683)--(1.6060,1.0767)--(1.6666,1.0857)--(1.7272,1.0954)--(1.7878,1.1058)--(1.8484,1.1169)--(1.9090,1.1288)--(1.9696,1.1415)--(2.0303,1.1549)--(2.0909,1.1692)--(2.1515,1.1844)--(2.2121,1.2004)--(2.2727,1.2173)--(2.3333,1.2352)--(2.3939,1.2540)--(2.4545,1.2738)--(2.5151,1.2946)--(2.5757,1.3164)--(2.6363,1.3393)--(2.6969,1.3632)--(2.7575,1.3883)--(2.8181,1.4144)--(2.8787,1.4418)--(2.9393,1.4703)--(3.0000,1.5000)--(3.0606,1.5309)--(3.1212,1.5630)--(3.1818,1.5965)--(3.2424,1.6312)--(3.3030,1.6673)--(3.3636,1.7047)--(3.4242,1.7435)--(3.4848,1.7837)--(3.5454,1.8253)--(3.6060,1.8683)--(3.6666,1.9128)--(3.7272,1.9589)--(3.7878,2.0064)--(3.8484,2.0555)--(3.9090,2.1061)--(3.9696,2.1584)--(4.0303,2.2123)--(4.0909,2.2678)--(4.1515,2.3250)--(4.2121,2.3839)--(4.2727,2.4445)--(4.3333,2.5068)--(4.3939,2.5709)--(4.4545,2.6368)--(4.5151,2.7046)--(4.5757,2.7741)--(4.6363,2.8456)--(4.6969,2.9189)--(4.7575,2.9941)--(4.8181,3.0713)--(4.8787,3.1505)--(4.9393,3.2316)--(5.0000,3.3148)--(5.0606,3.4000)--(5.1212,3.4872)--(5.1818,3.5766)--(5.2424,3.6681)--(5.3030,3.7617)--(5.3636,3.8574)--(5.4242,3.9554)--(5.4848,4.0556)--(5.5454,4.1580)--(5.6060,4.2627)--(5.6666,4.3696)--(5.7272,4.4789)--(5.7878,4.5905)--(5.8484,4.7045)--(5.9090,4.8209)--(5.9696,4.9396)--(6.0303,5.0609)--(6.0909,5.1845)--(6.1515,5.3107)--(6.2121,5.4394)--(6.2727,5.5706)--(6.3333,5.7043)--(6.3939,5.8407)--(6.4545,5.9797)--(6.5151,6.1212)--(6.5757,6.2655)--(6.6363,6.4124)--(6.6969,6.5621)--(6.7575,6.7145)--(6.8181,6.8696)--(6.8787,7.0275)--(6.9393,7.1882)--(7.0000,7.3518);
 \draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,-0.27858) node {$a$};
-\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
-\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.2785) node {$a$};
+\draw []  (0.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.4473,1.5000) node {$f(a)$};
 \draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
-\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
-\draw (6.5000,-0.27858) node {$x$};
-\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
-\draw (-0.45521,6.0856) node {$f(x)$};
+\draw []  (6.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.2785) node {$x$};
+\draw []  (0.0000,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.4552,6.0856) node {$f(x)$};
 \draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
 \draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
-\draw (4.7500,0.97897) node {$x-a$};
+\draw (4.7500,0.9789) node {$x-a$};
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
-\draw (7.8256,3.7928) node {$f(x)-f(a)$};
+\draw (7.8255,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DessinLim.pstricks.recall b/src_phystricks/Fig_DessinLim.pstricks.recall
index 59d0386bb..509b71d62 100644
--- a/src_phystricks/Fig_DessinLim.pstricks.recall
+++ b/src_phystricks/Fig_DessinLim.pstricks.recall
@@ -87,25 +87,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.8000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.8000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.8000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8000);
 %DEFAULT
 
-\draw [color=blue] (2.300,0)--(2.300,0.03649)--(2.299,0.07297)--(2.297,0.1094)--(2.295,0.1459)--(2.293,0.1823)--(2.290,0.2186)--(2.286,0.2549)--(2.281,0.2912)--(2.277,0.3273)--(2.271,0.3634)--(2.265,0.3994)--(2.258,0.4353)--(2.251,0.4711)--(2.243,0.5067)--(2.235,0.5422)--(2.226,0.5776)--(2.217,0.6129)--(2.207,0.6480)--(2.196,0.6829)--(2.185,0.7177)--(2.173,0.7523)--(2.161,0.7866)--(2.149,0.8208)--(2.135,0.8548)--(2.121,0.8886)--(2.107,0.9221)--(2.092,0.9555)--(2.077,0.9885)--(2.061,1.021)--(2.044,1.054)--(2.027,1.086)--(2.010,1.118)--(1.992,1.150)--(1.973,1.181)--(1.954,1.213)--(1.935,1.243)--(1.915,1.274)--(1.894,1.304)--(1.874,1.334)--(1.852,1.364)--(1.830,1.393)--(1.808,1.422)--(1.785,1.450)--(1.762,1.478)--(1.738,1.506)--(1.714,1.534)--(1.690,1.561)--(1.665,1.587)--(1.639,1.613)--(1.613,1.639)--(1.587,1.665)--(1.561,1.690)--(1.534,1.714)--(1.506,1.738)--(1.478,1.762)--(1.450,1.785)--(1.422,1.808)--(1.393,1.830)--(1.364,1.852)--(1.334,1.874)--(1.304,1.894)--(1.274,1.915)--(1.243,1.935)--(1.213,1.954)--(1.181,1.973)--(1.150,1.992)--(1.118,2.010)--(1.086,2.027)--(1.054,2.044)--(1.021,2.061)--(0.9885,2.077)--(0.9555,2.092)--(0.9221,2.107)--(0.8886,2.121)--(0.8548,2.135)--(0.8208,2.149)--(0.7866,2.161)--(0.7523,2.173)--(0.7177,2.185)--(0.6829,2.196)--(0.6480,2.207)--(0.6129,2.217)--(0.5776,2.226)--(0.5422,2.235)--(0.5067,2.243)--(0.4711,2.251)--(0.4353,2.258)--(0.3994,2.265)--(0.3634,2.271)--(0.3273,2.277)--(0.2912,2.281)--(0.2549,2.286)--(0.2186,2.290)--(0.1823,2.293)--(0.1459,2.295)--(0.1094,2.297)--(0.07297,2.299)--(0.03649,2.300)--(0,2.300);
-\draw [] (0,0) -- (2.30,2.30);
-\draw [style=dashed] (0,1.63) -- (1.63,1.63);
-\draw [style=dashed] (1.63,0) -- (1.63,1.63);
-\draw [] (2.30,2.30) -- (2.30,0);
-\draw []  (0,1.6263) node [rotate=0] {$\bullet$};
-\draw (-0.57160,1.6263) node {\( \sin(x)\)};
-\draw []  (1.6263,0) node [rotate=0] {$\bullet$};
-\draw (1.6263,-0.28245) node {\( \cos(x)\)};
-\draw []  (2.3000,0) node [rotate=0] {$\bullet$};
-\draw (2.4869,-0.21131) node {\( A\)};
+\draw [color=blue] (2.3000,0.0000)--(2.2997,0.0364)--(2.2988,0.0729)--(2.2973,0.1094)--(2.2953,0.1458)--(2.2927,0.1822)--(2.2895,0.2186)--(2.2858,0.2549)--(2.2814,0.2911)--(2.2765,0.3273)--(2.2711,0.3634)--(2.2650,0.3993)--(2.2584,0.4352)--(2.2512,0.4710)--(2.2434,0.5067)--(2.2351,0.5422)--(2.2262,0.5776)--(2.2168,0.6128)--(2.2068,0.6479)--(2.1962,0.6829)--(2.1851,0.7176)--(2.1735,0.7522)--(2.1612,0.7866)--(2.1485,0.8208)--(2.1352,0.8548)--(2.1214,0.8885)--(2.1070,0.9221)--(2.0921,0.9554)--(2.0767,0.9885)--(2.0607,1.0213)--(2.0443,1.0539)--(2.0273,1.0862)--(2.0098,1.1182)--(1.9918,1.1500)--(1.9733,1.1814)--(1.9543,1.2126)--(1.9348,1.2434)--(1.9149,1.2740)--(1.8944,1.3042)--(1.8735,1.3341)--(1.8521,1.3636)--(1.8302,1.3929)--(1.8079,1.4217)--(1.7851,1.4502)--(1.7619,1.4784)--(1.7382,1.5061)--(1.7141,1.5335)--(1.6895,1.5605)--(1.6645,1.5871)--(1.6391,1.6133)--(1.6133,1.6391)--(1.5871,1.6645)--(1.5605,1.6895)--(1.5335,1.7141)--(1.5061,1.7382)--(1.4784,1.7619)--(1.4502,1.7851)--(1.4217,1.8079)--(1.3929,1.8302)--(1.3636,1.8521)--(1.3341,1.8735)--(1.3042,1.8944)--(1.2740,1.9149)--(1.2434,1.9348)--(1.2126,1.9543)--(1.1814,1.9733)--(1.1500,1.9918)--(1.1182,2.0098)--(1.0862,2.0273)--(1.0539,2.0443)--(1.0213,2.0607)--(0.9885,2.0767)--(0.9554,2.0921)--(0.9221,2.1070)--(0.8885,2.1214)--(0.8548,2.1352)--(0.8208,2.1485)--(0.7866,2.1612)--(0.7522,2.1735)--(0.7176,2.1851)--(0.6829,2.1962)--(0.6479,2.2068)--(0.6128,2.2168)--(0.5776,2.2262)--(0.5422,2.2351)--(0.5067,2.2434)--(0.4710,2.2512)--(0.4352,2.2584)--(0.3993,2.2650)--(0.3634,2.2711)--(0.3273,2.2765)--(0.2911,2.2814)--(0.2549,2.2858)--(0.2186,2.2895)--(0.1822,2.2927)--(0.1458,2.2953)--(0.1094,2.2973)--(0.0729,2.2988)--(0.0364,2.2997)--(0.0000,2.3000);
+\draw [] (0.0000,0.0000) -- (2.3000,2.3000);
+\draw [style=dashed] (0.0000,1.6263) -- (1.6263,1.6263);
+\draw [style=dashed] (1.6263,0.0000) -- (1.6263,1.6263);
+\draw [] (2.3000,2.3000) -- (2.3000,0.0000);
+\draw []  (0.0000,1.6263) node [rotate=0] {$\bullet$};
+\draw (-0.5715,1.6263) node {\( \sin(x)\)};
+\draw []  (1.6263,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.6263,-0.2824) node {\( \cos(x)\)};
+\draw []  (2.3000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.4868,-0.2113) node {\( A\)};
 \draw []  (2.3000,2.3000) node [rotate=0] {$\bullet$};
-\draw (2.5320,2.3000) node {\( T\)};
+\draw (2.5319,2.3000) node {\( T\)};
 \draw []  (1.6263,1.6263) node [rotate=0] {$\bullet$};
-\draw (2.0689,1.6263) node {\( P\)};
+\draw (2.0688,1.6263) node {\( P\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DeuxCercles.pstricks.recall b/src_phystricks/Fig_DeuxCercles.pstricks.recall
index b1136c0f4..c6ffd32e5 100644
--- a/src_phystricks/Fig_DeuxCercles.pstricks.recall
+++ b/src_phystricks/Fig_DeuxCercles.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -76,17 +76,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_DisqueConv.pstricks.recall b/src_phystricks/Fig_DisqueConv.pstricks.recall
index f268bbcad..3daf3e818 100644
--- a/src_phystricks/Fig_DisqueConv.pstricks.recall
+++ b/src_phystricks/Fig_DisqueConv.pstricks.recall
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
-\draw (1.7653,1.8233) node {$z_0$};
+\draw (1.7652,1.8233) node {$z_0$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -79,8 +79,8 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\fill [color=green,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (3.0000,2.0000) -- (2.9979,2.0634) -- (2.9919,2.1265) -- (2.9819,2.1892) -- (2.9679,2.2511) -- (2.9500,2.3120) -- (2.9283,2.3716) -- (2.9029,2.4297) -- (2.8738,2.4861) -- (2.8412,2.5406) -- (2.8052,2.5929) -- (2.7660,2.6427) -- (2.7237,2.6900) -- (2.6785,2.7345) -- (2.6305,2.7761) -- (2.5800,2.8145) -- (2.5272,2.8497) -- (2.4722,2.8814) -- (2.4154,2.9096) -- (2.3568,2.9341) -- (2.2969,2.9549) -- (2.2357,2.9718) -- (2.1736,2.9848) -- (2.1108,2.9938) -- (2.0475,2.9988) -- (1.9841,2.9998) -- (1.9207,2.9968) -- (1.8576,2.9898) -- (1.7951,2.9788) -- (1.7335,2.9638) -- (1.6729,2.9450) -- (1.6136,2.9223) -- (1.5559,2.8959) -- (1.5000,2.8660) -- (1.4460,2.8325) -- (1.3943,2.7957) -- (1.3451,2.7557) -- (1.2985,2.7126) -- (1.2547,2.6667) -- (1.2139,2.6181) -- (1.1763,2.5670) -- (1.1420,2.5136) -- (1.1111,2.4582) -- (1.0838,2.4009) -- (1.0603,2.3420) -- (1.0405,2.2817) -- (1.0245,2.2203) -- (1.0125,2.1580) -- (1.0045,2.0950) -- (1.0005,2.0317) -- (1.0005,1.9682) -- (1.0045,1.9049) -- (1.0125,1.8419) -- (1.0245,1.7796) -- (1.0405,1.7182) -- (1.0603,1.6579) -- (1.0838,1.5990) -- (1.1111,1.5417) -- (1.1420,1.4863) -- (1.1763,1.4329) -- (1.2139,1.3818) -- (1.2547,1.3332) -- (1.2985,1.2873) -- (1.3451,1.2442) -- (1.3943,1.2042) -- (1.4460,1.1674) -- (1.5000,1.1339) -- (1.5559,1.1040) -- (1.6136,1.0776) -- (1.6729,1.0549) -- (1.7335,1.0361) -- (1.7951,1.0211) -- (1.8576,1.0101) -- (1.9207,1.0031) -- (1.9841,1.0001) -- (2.0475,1.0011) -- (2.1108,1.0061) -- (2.1736,1.0151) -- (2.2357,1.0281) -- (2.2969,1.0450) -- (2.3568,1.0658) -- (2.4154,1.0903) -- (2.4722,1.1185) -- (2.5272,1.1502) -- (2.5800,1.1854) -- (2.6305,1.2238) -- (2.6785,1.2654) -- (2.7237,1.3099) -- (2.7660,1.3572) -- (2.8052,1.4070) -- (2.8412,1.4593) -- (2.8738,1.5138) -- (2.9029,1.5702) -- (2.9283,1.6283) -- (2.9500,1.6879) -- (2.9679,1.7488) -- (2.9819,1.8107) -- (2.9919,1.8734) -- (2.9979,1.9365) -- (3.0000,2.0000) --  cycle;
+\draw [color=red] (3.0000,2.0000)--(2.9979,2.0634)--(2.9919,2.1265)--(2.9819,2.1892)--(2.9679,2.2511)--(2.9500,2.3120)--(2.9283,2.3716)--(2.9029,2.4297)--(2.8738,2.4861)--(2.8412,2.5406)--(2.8052,2.5929)--(2.7660,2.6427)--(2.7237,2.6900)--(2.6785,2.7345)--(2.6305,2.7761)--(2.5800,2.8145)--(2.5272,2.8497)--(2.4722,2.8814)--(2.4154,2.9096)--(2.3568,2.9341)--(2.2969,2.9549)--(2.2357,2.9718)--(2.1736,2.9848)--(2.1108,2.9938)--(2.0475,2.9988)--(1.9841,2.9998)--(1.9207,2.9968)--(1.8576,2.9898)--(1.7951,2.9788)--(1.7335,2.9638)--(1.6729,2.9450)--(1.6136,2.9223)--(1.5559,2.8959)--(1.5000,2.8660)--(1.4460,2.8325)--(1.3943,2.7957)--(1.3451,2.7557)--(1.2985,2.7126)--(1.2547,2.6667)--(1.2139,2.6181)--(1.1763,2.5670)--(1.1420,2.5136)--(1.1111,2.4582)--(1.0838,2.4009)--(1.0603,2.3420)--(1.0405,2.2817)--(1.0245,2.2203)--(1.0125,2.1580)--(1.0045,2.0950)--(1.0005,2.0317)--(1.0005,1.9682)--(1.0045,1.9049)--(1.0125,1.8419)--(1.0245,1.7796)--(1.0405,1.7182)--(1.0603,1.6579)--(1.0838,1.5990)--(1.1111,1.5417)--(1.1420,1.4863)--(1.1763,1.4329)--(1.2139,1.3818)--(1.2547,1.3332)--(1.2985,1.2873)--(1.3451,1.2442)--(1.3943,1.2042)--(1.4460,1.1674)--(1.5000,1.1339)--(1.5559,1.1040)--(1.6136,1.0776)--(1.6729,1.0549)--(1.7335,1.0361)--(1.7951,1.0211)--(1.8576,1.0101)--(1.9207,1.0031)--(1.9841,1.0001)--(2.0475,1.0011)--(2.1108,1.0061)--(2.1736,1.0151)--(2.2357,1.0281)--(2.2969,1.0450)--(2.3568,1.0658)--(2.4154,1.0903)--(2.4722,1.1185)--(2.5272,1.1502)--(2.5800,1.1854)--(2.6305,1.2238)--(2.6785,1.2654)--(2.7237,1.3099)--(2.7660,1.3572)--(2.8052,1.4070)--(2.8412,1.4593)--(2.8738,1.5138)--(2.9029,1.5702)--(2.9283,1.6283)--(2.9500,1.6879)--(2.9679,1.7488)--(2.9819,1.8107)--(2.9919,1.8734)--(2.9979,1.9365)--(3.0000,2.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DistanceEnsemble.pstricks.recall b/src_phystricks/Fig_DistanceEnsemble.pstricks.recall
index fb287b1fd..9b810198e 100644
--- a/src_phystricks/Fig_DistanceEnsemble.pstricks.recall
+++ b/src_phystricks/Fig_DistanceEnsemble.pstricks.recall
@@ -77,16 +77,16 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw (1.7632,1.7511) node {$A$};
-\draw [] (-3.98,-2.30) -- (-1.73,-1.00);
-\draw [style=dotted] (-3.98,-2.30) -- (-0.347,1.97);
-\draw [style=dotted] (-3.98,-2.30) -- (0,-1.40);
+\draw (1.7632,1.7510) node {$A$};
+\draw [] (-3.9837,-2.3000) -- (-1.7320,-1.0000);
+\draw [style=dotted] (-3.9837,-2.3000) -- (-0.3472,1.9696);
+\draw [style=dotted] (-3.9837,-2.3000) -- (0.0000,-1.4000);
 
-\draw [] (2.00,0)--(2.00,0.127)--(1.98,0.253)--(1.96,0.379)--(1.94,0.502)--(1.90,0.624)--(1.86,0.743)--(1.81,0.860)--(1.75,0.972)--(1.68,1.08)--(1.61,1.19)--(1.53,1.29)--(1.45,1.38)--(1.36,1.47)--(1.26,1.55)--(1.16,1.63)--(1.05,1.70)--(0.945,1.76)--(0.831,1.82)--(0.714,1.87)--(0.594,1.91)--(0.472,1.94)--(0.347,1.97)--(0.222,1.99)--(0.0952,2.00)--(-0.0317,2.00)--(-0.158,1.99)--(-0.285,1.98)--(-0.410,1.96)--(-0.533,1.93)--(-0.654,1.89)--(-0.773,1.84)--(-0.888,1.79)--(-1.00,1.73)--(-1.11,1.67)--(-1.21,1.59)--(-1.31,1.51)--(-1.40,1.43)--(-1.49,1.33)--(-1.57,1.24)--(-1.65,1.13)--(-1.72,1.03)--(-1.78,0.916)--(-1.83,0.802)--(-1.88,0.684)--(-1.92,0.563)--(-1.95,0.441)--(-1.97,0.316)--(-1.99,0.190)--(-2.00,0.0635)--(-2.00,-0.0635)--(-1.99,-0.190)--(-1.97,-0.316)--(-1.95,-0.441)--(-1.92,-0.563)--(-1.88,-0.684)--(-1.83,-0.802)--(-1.78,-0.916)--(-1.72,-1.03)--(-1.65,-1.13)--(-1.57,-1.24)--(-1.49,-1.33)--(-1.40,-1.43)--(-1.31,-1.51)--(-1.21,-1.59)--(-1.11,-1.67)--(-1.00,-1.73)--(-0.888,-1.79)--(-0.773,-1.84)--(-0.654,-1.89)--(-0.533,-1.93)--(-0.410,-1.96)--(-0.285,-1.98)--(-0.158,-1.99)--(-0.0317,-2.00)--(0.0952,-2.00)--(0.222,-1.99)--(0.347,-1.97)--(0.472,-1.94)--(0.594,-1.91)--(0.714,-1.87)--(0.831,-1.82)--(0.945,-1.76)--(1.05,-1.70)--(1.16,-1.63)--(1.26,-1.55)--(1.36,-1.47)--(1.45,-1.38)--(1.53,-1.29)--(1.61,-1.19)--(1.68,-1.08)--(1.75,-0.972)--(1.81,-0.860)--(1.86,-0.743)--(1.90,-0.624)--(1.94,-0.502)--(1.96,-0.379)--(1.98,-0.253)--(2.00,-0.127)--(2.00,0);
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 \draw []  (-1.7320,-1.0000) node [rotate=0] {$\bullet$};
-\draw (-1.7320,-1.6141) node {$p$};
-\draw []  (0,-1.4000) node [rotate=0] {$\bullet$};
-\draw []  (-0.34729,1.9696) node [rotate=0] {$\bullet$};
+\draw (-1.7320,-1.6140) node {$p$};
+\draw []  (0.0000,-1.4000) node [rotate=0] {$\bullet$};
+\draw []  (-0.3472,1.9696) node [rotate=0] {$\bullet$};
 \draw []  (-3.9837,-2.3000) node [rotate=0] {$\bullet$};
 \draw (-4.3880,-2.3000) node {$x$};
 %END PSPICTURE
diff --git a/src_phystricks/Fig_DivergenceTrois.pstricks.recall b/src_phystricks/Fig_DivergenceTrois.pstricks.recall
index 8a25fb77c..1c983862e 100644
--- a/src_phystricks/Fig_DivergenceTrois.pstricks.recall
+++ b/src_phystricks/Fig_DivergenceTrois.pstricks.recall
@@ -65,51 +65,51 @@
 %OTHER STUFF
 %PSTRICKS CODE
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+\draw [,->,>=latex] (0.9999,0.0087) -- (1.9999,0.0174);
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+\draw [,->,>=latex] (-0.9275,0.3735) -- (-1.8551,0.7471);
+\draw [,->,>=latex] (-0.9911,0.1324) -- (-1.9823,0.2649);
+\draw [,->,>=latex] (2.1999,0.0191) -- (2.6544,0.0231);
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+\draw [,->,>=latex] (2.0668,0.7536) -- (2.4939,0.9093);
+\draw [,->,>=latex] (1.9681,0.9831) -- (2.3747,1.1862);
+\draw [,->,>=latex] (1.8439,1.1999) -- (2.2249,1.4478);
+\draw [,->,>=latex] (1.6959,1.4013) -- (2.0463,1.6908);
+\draw [,->,>=latex] (1.5261,1.5845) -- (1.8414,1.9119);
+\draw [,->,>=latex] (1.3366,1.7474) -- (1.6127,2.1084);
+\draw [,->,>=latex] (1.1298,1.8877) -- (1.3632,2.2777);
+\draw [,->,>=latex] (0.9085,2.0036) -- (1.0962,2.4176);
+\draw [,->,>=latex] (0.6754,2.0937) -- (0.8150,2.5263);
+\draw [,->,>=latex] (0.4336,2.1568) -- (0.5233,2.6024);
+\draw [,->,>=latex] (0.1863,2.1920) -- (0.2248,2.6450);
+\draw [,->,>=latex] (-0.0634,2.1990) -- (-0.0765,2.6534);
+\draw [,->,>=latex] (-0.3123,2.1777) -- (-0.3769,2.6276);
+\draw [,->,>=latex] (-0.5573,2.1282) -- (-0.6724,2.5679);
+\draw [,->,>=latex] (-0.7950,2.0513) -- (-0.9593,2.4751);
+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2337,2.3504);
+\draw [,->,>=latex] (-1.2367,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7316,2.0119);
+\draw [,->,>=latex] (-1.6149,1.4939) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1404,1.5700);
+\draw [,->,>=latex] (-1.9100,1.0916) -- (-2.3046,1.3172);
+\draw [,->,>=latex] (-2.0215,0.8680) -- (-2.4391,1.0473);
+\draw [,->,>=latex] (-2.1069,0.6332) -- (-2.5422,0.7640);
+\draw [,->,>=latex] (-2.1651,0.3902) -- (-2.6124,0.4708);
+\draw [,->,>=latex] (-2.1953,0.1422) -- (-2.6489,0.1715);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_DivergenceUn.pstricks.recall b/src_phystricks/Fig_DivergenceUn.pstricks.recall
index 8f1c5afca..eb4722e04 100644
--- a/src_phystricks/Fig_DivergenceUn.pstricks.recall
+++ b/src_phystricks/Fig_DivergenceUn.pstricks.recall
@@ -65,58 +65,58 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
+\fill [color=cyan] (1.6000,-1.2000) -- (3.0666,-1.2000) -- (3.0666,-1.2000) -- (3.0666,1.2000) -- (3.0666,1.2000) -- (1.6000,1.2000) -- (1.6000,1.2000) -- (1.6000,-1.2000) --  cycle;
 
 
 \draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
-\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
-\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
-\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
-\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
-\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,-0.6666) -- (2.2250,-0.6666);
+\draw [,->,>=latex] (1.6000,-0.3333) -- (2.2250,-0.3333);
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 \draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
 \draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
-\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
-\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
-\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
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-\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
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-\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
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-\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
-\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
-\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
-\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
-\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,-0.6666) -- (4.7539,-0.6666);
+\draw [,->,>=latex] (4.5333,-0.3333) -- (4.7539,-0.3333);
+\draw [,->,>=latex] (4.5333,0.0000) -- (4.7539,0.0000);
+\draw [,->,>=latex] (4.5333,0.3333) -- (4.7539,0.3333);
+\draw [,->,>=latex] (4.5333,0.6666) -- (4.7539,0.6666);
 \draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
-\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
-\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
-\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
-\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
-\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
-\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
-\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
-\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
-\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
-\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
-\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
-\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
-\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
-\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
+\draw [,->,>=latex] (5.2666,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2666,-0.6666) -- (5.4565,-0.6666);
+\draw [,->,>=latex] (5.2666,-0.3333) -- (5.4565,-0.3333);
+\draw [,->,>=latex] (5.2666,0.0000) -- (5.4565,0.0000);
+\draw [,->,>=latex] (5.2666,0.3333) -- (5.4565,0.3333);
+\draw [,->,>=latex] (5.2666,0.6666) -- (5.4565,0.6666);
+\draw [,->,>=latex] (5.2666,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1666,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.6666) -- (6.1666,-0.6666);
+\draw [,->,>=latex] (6.0000,-0.3333) -- (6.1666,-0.3333);
+\draw [,->,>=latex] (6.0000,0.0000) -- (6.1666,0.0000);
+\draw [,->,>=latex] (6.0000,0.3333) -- (6.1666,0.3333);
+\draw [,->,>=latex] (6.0000,0.6666) -- (6.1666,0.6666);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1666,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall b/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall
index ad2371f2a..129d6ad96 100644
--- a/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall
+++ b/src_phystricks/Fig_EELKooMwkockxB.pstricks.recall
@@ -94,29 +94,29 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (8.00,2.00);
-\draw []  (1.6000,0.40000) node [rotate=0] {$\bullet$};
-\draw (1.7450,0.12740) node {\( a\)};
+\draw [] (0.0000,0.0000) -- (8.0000,2.0000);
+\draw []  (1.6000,0.4000) node [rotate=0] {$\bullet$};
+\draw (1.7449,0.1273) node {\( a\)};
 \draw []  (4.8000,1.2000) node [rotate=0] {$\bullet$};
-\draw (4.9268,0.87924) node {\( b\)};
+\draw (4.9268,0.8792) node {\( b\)};
 \draw []  (6.4000,1.6000) node [rotate=0] {$\bullet$};
-\draw (6.5275,1.3274) node {\( c\)};
+\draw (6.5274,1.3273) node {\( c\)};
 \draw []  (4.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (4.1528,0.72740) node {\( x\)};
-\draw [] (1.60,0.400) -- (2.60,3.40);
-\draw [] (4.80,1.20) -- (2.60,3.40);
+\draw (4.1528,0.7273) node {\( x\)};
+\draw [] (1.6000,0.4000) -- (2.6000,3.4000);
+\draw [] (4.8000,1.2000) -- (2.6000,3.4000);
 \draw []  (2.2000,2.2000) node [rotate=0] {$\bullet$};
 \draw (1.9184,2.3773) node {\( p\)};
-\draw []  (4.0667,1.9333) node [rotate=0] {$\bullet$};
-\draw (4.3353,2.1333) node {\( q\)};
-\draw [] (6.40,1.60) -- (0.940,2.38);
+\draw []  (4.0666,1.9333) node [rotate=0] {$\bullet$};
+\draw (4.3353,2.1332) node {\( q\)};
+\draw [] (6.4000,1.6000) -- (0.9400,2.3800);
 \draw []  (2.6000,3.4000) node [rotate=0] {$\bullet$};
 \draw (2.8817,3.6374) node {\( m\)};
-\draw [] (4.07,1.93) -- (1.60,0.400);
-\draw [] (2.20,2.20) -- (4.80,1.20);
-\draw []  (3.6182,1.6545) node [rotate=0] {$\bullet$};
-\draw (3.3268,1.4928) node {\( n\)};
-\draw [] (2.60,3.40) -- (4.00,1.00);
+\draw [] (4.0666,1.9333) -- (1.6000,0.4000);
+\draw [] (2.2000,2.2000) -- (4.8000,1.2000);
+\draw []  (3.6181,1.6545) node [rotate=0] {$\bullet$};
+\draw (3.3267,1.4927) node {\( n\)};
+\draw [] (2.6000,3.4000) -- (4.0000,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall b/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall
index 61e7d4c9d..9925ad92e 100644
--- a/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall
+++ b/src_phystricks/Fig_EHDooGDwfjC.pstricks.recall
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,0.32471) node {\( A\)};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.2890,0.26613) node {\( B\)};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,0.3247) node {\( A\)};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.2890,0.2661) node {\( B\)};
 \draw []  (12.000,-3.0000) node [rotate=0] {$\bullet$};
-\draw (12.285,-3.2661) node {\( C\)};
+\draw (12.284,-3.2661) node {\( C\)};
 \draw []  (4.0000,-1.0000) node [rotate=0] {$\bullet$};
 \draw (3.6905,-1.2661) node {\( K\)};
-\draw []  (1.3333,0) node [rotate=0] {$\bullet$};
-\draw (1.3333,0.32471) node {\( L\)};
-\draw [] (0,0) -- (4.00,0);
-\draw [] (0,0) -- (12.0,-3.00);
-\draw [style=dashed] (12.0,-3.00) -- (4.00,0);
-\draw [style=dashed] (4.00,-1.00) -- (1.33,0);
+\draw []  (1.3333,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.3333,0.3247) node {\( L\)};
+\draw [] (0.0000,0.0000) -- (4.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (12.000,-3.0000);
+\draw [style=dashed] (12.000,-3.0000) -- (4.0000,0.0000);
+\draw [style=dashed] (4.0000,-1.0000) -- (1.3333,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall b/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall
index 31c8bb72e..662c87d49 100644
--- a/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall
+++ b/src_phystricks/Fig_ERPMooZibfNOiU.pstricks.recall
@@ -78,33 +78,33 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw []  (2.0868,1.7595) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.4102,2.9353);
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (3.4102,2.9353);
 \draw []  (1.7350,1.4643) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.34725,0.36521);
-\draw []  (2.6575,2.4377) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.2478,3.0982);
-\draw []  (0.96811,1.0201) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.21509,0.55343);
-\draw []  (2.8537,2.7936) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (3.1558,3.1756);
-\draw []  (0.58356,0.88873) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.15485,0.65749);
-\draw []  (3.0136,3.3088) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.9997,3.2873);
-\draw []  (0.048375,0.82073) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (0.071918,0.83064);
-\draw []  (2.8961,4.0071) node [rotate=0] {$\bullet$};
-\draw [color=red,->,>=latex] (1.9151,1.6070) -- (2.6718,3.4583);
-\draw []  (-0.61890,1.0577) node [rotate=0] {$\bullet$};
-\draw [color=blue,->,>=latex] (1.9151,1.6070) -- (-0.039497,1.1833);
-\draw [] (-3.00,0) -- (3.00,0);
-\draw [] (0,0) -- (3.83,3.21);
-\draw (0.77455,0.24959) node {\( \alpha\)};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.3472,0.3652);
+\draw []  (2.6575,2.4376) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (3.2478,3.0982);
+\draw []  (0.9681,1.0200) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.2150,0.5534);
+\draw []  (2.8536,2.7935) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (3.1558,3.1756);
+\draw []  (0.5835,0.8887) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.1548,0.6574);
+\draw []  (3.0135,3.3088) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (2.9997,3.2873);
+\draw []  (0.0483,0.8207) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (0.0719,0.8306);
+\draw []  (2.8960,4.0070) node [rotate=0] {$\bullet$};
+\draw [color=red,->,>=latex] (1.9151,1.6069) -- (2.6717,3.4583);
+\draw []  (-0.6188,1.0576) node [rotate=0] {$\bullet$};
+\draw [color=blue,->,>=latex] (1.9151,1.6069) -- (-0.0394,1.1832);
+\draw [] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (3.8302,3.2139);
+\draw (0.7745,0.2495) node {\( \alpha\)};
 
-\draw [] (0.500,0)--(0.500,0.00353)--(0.500,0.00705)--(0.500,0.0106)--(0.500,0.0141)--(0.500,0.0176)--(0.500,0.0211)--(0.499,0.0247)--(0.499,0.0282)--(0.499,0.0317)--(0.499,0.0352)--(0.499,0.0387)--(0.498,0.0423)--(0.498,0.0458)--(0.498,0.0493)--(0.497,0.0528)--(0.497,0.0563)--(0.496,0.0598)--(0.496,0.0633)--(0.496,0.0668)--(0.495,0.0703)--(0.495,0.0738)--(0.494,0.0773)--(0.493,0.0807)--(0.493,0.0842)--(0.492,0.0877)--(0.492,0.0912)--(0.491,0.0946)--(0.490,0.0981)--(0.490,0.102)--(0.489,0.105)--(0.488,0.108)--(0.487,0.112)--(0.487,0.115)--(0.486,0.119)--(0.485,0.122)--(0.484,0.126)--(0.483,0.129)--(0.482,0.132)--(0.481,0.136)--(0.480,0.139)--(0.479,0.143)--(0.478,0.146)--(0.477,0.149)--(0.476,0.153)--(0.475,0.156)--(0.474,0.159)--(0.473,0.163)--(0.472,0.166)--(0.470,0.169)--(0.469,0.173)--(0.468,0.176)--(0.467,0.179)--(0.465,0.183)--(0.464,0.186)--(0.463,0.189)--(0.462,0.192)--(0.460,0.196)--(0.459,0.199)--(0.457,0.202)--(0.456,0.205)--(0.454,0.209)--(0.453,0.212)--(0.451,0.215)--(0.450,0.218)--(0.448,0.221)--(0.447,0.224)--(0.445,0.228)--(0.444,0.231)--(0.442,0.234)--(0.440,0.237)--(0.439,0.240)--(0.437,0.243)--(0.435,0.246)--(0.433,0.249)--(0.432,0.252)--(0.430,0.255)--(0.428,0.258)--(0.426,0.261)--(0.424,0.264)--(0.423,0.267)--(0.421,0.270)--(0.419,0.273)--(0.417,0.276)--(0.415,0.279)--(0.413,0.282)--(0.411,0.285)--(0.409,0.288)--(0.407,0.291)--(0.405,0.294)--(0.403,0.296)--(0.401,0.299)--(0.398,0.302)--(0.396,0.305)--(0.394,0.308)--(0.392,0.310)--(0.390,0.313)--(0.388,0.316)--(0.385,0.319)--(0.383,0.321);
-\draw []  (1.9151,1.6070) node [rotate=0] {$\bullet$};
-\draw (2.1862,1.3291) node {\( P\)};
-\draw [] plot [smooth,tension=1] coordinates {(2.84,4.08)(2.78,4.14)(2.72,4.20)(2.65,4.24)(2.57,4.28)(2.49,4.32)(2.40,4.35)(2.31,4.37)(2.21,4.38)(2.11,4.39)(2.00,4.38)(1.89,4.38)(1.78,4.36)(1.66,4.34)(1.54,4.31)(1.42,4.27)(1.30,4.23)(1.17,4.18)(1.05,4.12)(0.927,4.06)(0.804,3.99)(0.683,3.92)(0.562,3.84)(0.444,3.76)(0.329,3.67)(0.216,3.58)(0.107,3.48)(0.00146,3.38)(-0.0994,3.27)(-0.196,3.17)(-0.287,3.06)(-0.372,2.95)(-0.452,2.84)(-0.525,2.73)(-0.592,2.61)(-0.653,2.50)(-0.706,2.39)(-0.752,2.28)(-0.791,2.17)(-0.822,2.06)(-0.846,1.95)(-0.862,1.85)(-0.870,1.75)(-0.870,1.65)(-0.862,1.55)(-0.847,1.47)(-0.823,1.38)(-0.792,1.30)(-0.754,1.23)(-0.708,1.16)(-0.656,1.09)(-0.595,1.04)(-0.529,0.986)(-0.455,0.942)(-0.376,0.904)(-0.291,0.873)(-0.200,0.849)(-0.104,0.833)(-0.00305,0.823)(0.102,0.820)(0.211,0.825)(0.323,0.837)(0.439,0.856)(0.557,0.882)(0.677,0.915)(0.799,0.955)(0.922,1.00)(1.05,1.05)(1.17,1.11)(1.29,1.18)(1.42,1.25)(1.54,1.33)(1.66,1.41)(1.77,1.49)(1.89,1.58)(2.00,1.68)(2.10,1.78)(2.21,1.88)(2.31,1.98)(2.40,2.09)(2.49,2.20)(2.57,2.31)(2.65,2.42)(2.72,2.54)(2.78,2.65)(2.84,2.76)(2.89,2.87)(2.93,2.99)(2.97,3.09)(2.99,3.20)(3.01,3.31)(3.03,3.41)(3.03,3.51)(3.03,3.60)(3.01,3.70)(2.99,3.78)(2.97,3.87)(2.93,3.94)(2.89,4.01)(2.84,4.08)};
+\draw [] (0.5000,0.0000)--(0.4999,0.0035)--(0.4999,0.0070)--(0.4998,0.0105)--(0.4998,0.0141)--(0.4996,0.0176)--(0.4995,0.0211)--(0.4993,0.0246)--(0.4992,0.0281)--(0.4989,0.0317)--(0.4987,0.0352)--(0.4984,0.0387)--(0.4982,0.0422)--(0.4979,0.0457)--(0.4975,0.0492)--(0.4972,0.0527)--(0.4968,0.0562)--(0.4964,0.0597)--(0.4959,0.0632)--(0.4955,0.0667)--(0.4950,0.0702)--(0.4945,0.0737)--(0.4939,0.0772)--(0.4934,0.0807)--(0.4928,0.0842)--(0.4922,0.0876)--(0.4916,0.0911)--(0.4909,0.0946)--(0.4902,0.0980)--(0.4895,0.1015)--(0.4888,0.1049)--(0.4881,0.1084)--(0.4873,0.1118)--(0.4865,0.1153)--(0.4856,0.1187)--(0.4848,0.1221)--(0.4839,0.1255)--(0.4830,0.1289)--(0.4821,0.1323)--(0.4812,0.1357)--(0.4802,0.1391)--(0.4792,0.1425)--(0.4782,0.1459)--(0.4771,0.1493)--(0.4761,0.1526)--(0.4750,0.1560)--(0.4739,0.1593)--(0.4727,0.1627)--(0.4716,0.1660)--(0.4704,0.1693)--(0.4692,0.1726)--(0.4680,0.1759)--(0.4667,0.1792)--(0.4654,0.1825)--(0.4641,0.1858)--(0.4628,0.1890)--(0.4615,0.1923)--(0.4601,0.1956)--(0.4587,0.1988)--(0.4573,0.2020)--(0.4559,0.2052)--(0.4544,0.2085)--(0.4529,0.2117)--(0.4514,0.2148)--(0.4499,0.2180)--(0.4483,0.2212)--(0.4468,0.2243)--(0.4452,0.2275)--(0.4436,0.2306)--(0.4419,0.2338)--(0.4403,0.2369)--(0.4386,0.2400)--(0.4369,0.2430)--(0.4351,0.2461)--(0.4334,0.2492)--(0.4316,0.2522)--(0.4298,0.2553)--(0.4280,0.2583)--(0.4262,0.2613)--(0.4243,0.2643)--(0.4225,0.2673)--(0.4206,0.2703)--(0.4187,0.2732)--(0.4167,0.2762)--(0.4148,0.2791)--(0.4128,0.2820)--(0.4108,0.2849)--(0.4088,0.2878)--(0.4067,0.2907)--(0.4047,0.2936)--(0.4026,0.2964)--(0.4005,0.2992)--(0.3984,0.3021)--(0.3962,0.3049)--(0.3941,0.3076)--(0.3919,0.3104)--(0.3897,0.3132)--(0.3875,0.3159)--(0.3852,0.3186)--(0.3830,0.3213);
+\draw []  (1.9151,1.6069) node [rotate=0] {$\bullet$};
+\draw (2.1861,1.3290) node {\( P\)};
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 \draw (4.1391,3.5225) node {\( \ell_p\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExSinLarge.pstricks.recall b/src_phystricks/Fig_ExSinLarge.pstricks.recall
index 306d962f8..c2542d003 100644
--- a/src_phystricks/Fig_ExSinLarge.pstricks.recall
+++ b/src_phystricks/Fig_ExSinLarge.pstricks.recall
@@ -75,26 +75,26 @@
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 %PSTRICKS CODE
 %AXES
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+\draw [color=blue,style=solid] (0.0000,1.0000)--(0.0317,1.0317)--(0.0634,1.0634)--(0.0951,1.0950)--(0.1269,1.1265)--(0.1586,1.1580)--(0.1903,1.1892)--(0.2221,1.2203)--(0.2538,1.2511)--(0.2855,1.2817)--(0.3173,1.3120)--(0.3490,1.3420)--(0.3807,1.3716)--(0.4125,1.4009)--(0.4442,1.4297)--(0.4759,1.4582)--(0.5077,1.4861)--(0.5394,1.5136)--(0.5711,1.5406)--(0.6029,1.5670)--(0.6346,1.5929)--(0.6663,1.6181)--(0.6981,1.6427)--(0.7298,1.6667)--(0.7615,1.6900)--(0.7933,1.7126)--(0.8250,1.7345)--(0.8567,1.7557)--(0.8885,1.7761)--(0.9202,1.7957)--(0.9519,1.8145)--(0.9837,1.8325)--(1.0154,1.8497)--(1.0471,1.8660)--(1.0789,1.8814)--(1.1106,1.8959)--(1.1423,1.9096)--(1.1741,1.9223)--(1.2058,1.9341)--(1.2375,1.9450)--(1.2693,1.9549)--(1.3010,1.9638)--(1.3327,1.9718)--(1.3645,1.9788)--(1.3962,1.9848)--(1.4279,1.9898)--(1.4597,1.9938)--(1.4914,1.9968)--(1.5231,1.9988)--(1.5549,1.9998)--(1.5866,1.9998)--(1.6183,1.9988)--(1.6501,1.9968)--(1.6818,1.9938)--(1.7135,1.9898)--(1.7453,1.9848)--(1.7770,1.9788)--(1.8087,1.9718)--(1.8405,1.9638)--(1.8722,1.9549)--(1.9039,1.9450)--(1.9357,1.9341)--(1.9674,1.9223)--(1.9991,1.9096)--(2.0309,1.8959)--(2.0626,1.8814)--(2.0943,1.8660)--(2.1261,1.8497)--(2.1578,1.8325)--(2.1895,1.8145)--(2.2213,1.7957)--(2.2530,1.7761)--(2.2847,1.7557)--(2.3165,1.7345)--(2.3482,1.7126)--(2.3799,1.6900)--(2.4117,1.6667)--(2.4434,1.6427)--(2.4751,1.6181)--(2.5069,1.5929)--(2.5386,1.5670)--(2.5703,1.5406)--(2.6021,1.5136)--(2.6338,1.4861)--(2.6655,1.4582)--(2.6973,1.4297)--(2.7290,1.4009)--(2.7607,1.3716)--(2.7925,1.3420)--(2.8242,1.3120)--(2.8559,1.2817)--(2.8877,1.2511)--(2.9194,1.2203)--(2.9511,1.1892)--(2.9829,1.1580)--(3.0146,1.1265)--(3.0463,1.0950)--(3.0781,1.0634)--(3.1098,1.0317)--(3.1415,1.0000);
+\draw [color=blue,style=solid] (0.0000,2.0000)--(0.0317,2.0317)--(0.0634,2.0634)--(0.0951,2.0950)--(0.1269,2.1265)--(0.1586,2.1580)--(0.1903,2.1892)--(0.2221,2.2203)--(0.2538,2.2511)--(0.2855,2.2817)--(0.3173,2.3120)--(0.3490,2.3420)--(0.3807,2.3716)--(0.4125,2.4009)--(0.4442,2.4297)--(0.4759,2.4582)--(0.5077,2.4861)--(0.5394,2.5136)--(0.5711,2.5406)--(0.6029,2.5670)--(0.6346,2.5929)--(0.6663,2.6181)--(0.6981,2.6427)--(0.7298,2.6667)--(0.7615,2.6900)--(0.7933,2.7126)--(0.8250,2.7345)--(0.8567,2.7557)--(0.8885,2.7761)--(0.9202,2.7957)--(0.9519,2.8145)--(0.9837,2.8325)--(1.0154,2.8497)--(1.0471,2.8660)--(1.0789,2.8814)--(1.1106,2.8959)--(1.1423,2.9096)--(1.1741,2.9223)--(1.2058,2.9341)--(1.2375,2.9450)--(1.2693,2.9549)--(1.3010,2.9638)--(1.3327,2.9718)--(1.3645,2.9788)--(1.3962,2.9848)--(1.4279,2.9898)--(1.4597,2.9938)--(1.4914,2.9968)--(1.5231,2.9988)--(1.5549,2.9998)--(1.5866,2.9998)--(1.6183,2.9988)--(1.6501,2.9968)--(1.6818,2.9938)--(1.7135,2.9898)--(1.7453,2.9848)--(1.7770,2.9788)--(1.8087,2.9718)--(1.8405,2.9638)--(1.8722,2.9549)--(1.9039,2.9450)--(1.9357,2.9341)--(1.9674,2.9223)--(1.9991,2.9096)--(2.0309,2.8959)--(2.0626,2.8814)--(2.0943,2.8660)--(2.1261,2.8497)--(2.1578,2.8325)--(2.1895,2.8145)--(2.2213,2.7957)--(2.2530,2.7761)--(2.2847,2.7557)--(2.3165,2.7345)--(2.3482,2.7126)--(2.3799,2.6900)--(2.4117,2.6667)--(2.4434,2.6427)--(2.4751,2.6181)--(2.5069,2.5929)--(2.5386,2.5670)--(2.5703,2.5406)--(2.6021,2.5136)--(2.6338,2.4861)--(2.6655,2.4582)--(2.6973,2.4297)--(2.7290,2.4009)--(2.7607,2.3716)--(2.7925,2.3420)--(2.8242,2.3120)--(2.8559,2.2817)--(2.8877,2.2511)--(2.9194,2.2203)--(2.9511,2.1892)--(2.9829,2.1580)--(3.0146,2.1265)--(3.0463,2.0950)--(3.0781,2.0634)--(3.1098,2.0317)--(3.1415,2.0000);
+\draw [] (0.0000,2.0000) -- (0.0000,1.0000);
+\draw [] (3.1415,1.0000) -- (3.1415,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExampleIntegration.pstricks.recall b/src_phystricks/Fig_ExampleIntegration.pstricks.recall
index 63012a8e2..82c7f4343 100644
--- a/src_phystricks/Fig_ExampleIntegration.pstricks.recall
+++ b/src_phystricks/Fig_ExampleIntegration.pstricks.recall
@@ -79,8 +79,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -89,27 +89,27 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
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+\draw [color=blue,style=solid] (0.0000,-1.0000)--(0.0202,-0.9995)--(0.0404,-0.9983)--(0.0606,-0.9963)--(0.0808,-0.9934)--(0.1010,-0.9897)--(0.1212,-0.9853)--(0.1414,-0.9800)--(0.1616,-0.9738)--(0.1818,-0.9669)--(0.2020,-0.9591)--(0.2222,-0.9506)--(0.2424,-0.9412)--(0.2626,-0.9310)--(0.2828,-0.9200)--(0.3030,-0.9081)--(0.3232,-0.8955)--(0.3434,-0.8820)--(0.3636,-0.8677)--(0.3838,-0.8526)--(0.4040,-0.8367)--(0.4242,-0.8200)--(0.4444,-0.8024)--(0.4646,-0.7841)--(0.4848,-0.7649)--(0.5050,-0.7449)--(0.5252,-0.7241)--(0.5454,-0.7024)--(0.5656,-0.6800)--(0.5858,-0.6567)--(0.6060,-0.6326)--(0.6262,-0.6077)--(0.6464,-0.5820)--(0.6666,-0.5555)--(0.6868,-0.5282)--(0.7070,-0.5000)--(0.7272,-0.4710)--(0.7474,-0.4412)--(0.7676,-0.4106)--(0.7878,-0.3792)--(0.8080,-0.3470)--(0.8282,-0.3139)--(0.8484,-0.2800)--(0.8686,-0.2453)--(0.8888,-0.2098)--(0.9090,-0.1735)--(0.9292,-0.1364)--(0.9494,-0.0984)--(0.9696,-0.0596)--(0.9898,-0.0200)--(1.0101,0.0203)--(1.0303,0.0615)--(1.0505,0.1035)--(1.0707,0.1464)--(1.0909,0.1900)--(1.1111,0.2345)--(1.1313,0.2798)--(1.1515,0.3259)--(1.1717,0.3729)--(1.1919,0.4206)--(1.2121,0.4692)--(1.2323,0.5186)--(1.2525,0.5688)--(1.2727,0.6198)--(1.2929,0.6716)--(1.3131,0.7243)--(1.3333,0.7777)--(1.3535,0.8320)--(1.3737,0.8871)--(1.3939,0.9430)--(1.4141,0.9997)--(1.4343,1.0573)--(1.4545,1.1157)--(1.4747,1.1748)--(1.4949,1.2348)--(1.5151,1.2956)--(1.5353,1.3573)--(1.5555,1.4197)--(1.5757,1.4830)--(1.5959,1.5470)--(1.6161,1.6119)--(1.6363,1.6776)--(1.6565,1.7442)--(1.6767,1.8115)--(1.6969,1.8797)--(1.7171,1.9486)--(1.7373,2.0184)--(1.7575,2.0890)--(1.7777,2.1604)--(1.7979,2.2327)--(1.8181,2.3057)--(1.8383,2.3796)--(1.8585,2.4543)--(1.8787,2.5298)--(1.8989,2.6061)--(1.9191,2.6832)--(1.9393,2.7612)--(1.9595,2.8400)--(1.9797,2.9196)--(2.0000,3.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall b/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall
index 8e4adfdc0..cec7cc8f0 100644
--- a/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall
+++ b/src_phystricks/Fig_ExampleIntegrationdeux.pstricks.recall
@@ -103,8 +103,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-4.5000) -- (0,5.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-4.5000) -- (0.0000,5.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -113,11 +113,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (-1.0000,2.0000) -- (-1.0000,-2.0000);
+\draw [] (5.0000,4.0000) -- (5.0000,4.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -125,53 +125,53 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
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-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue,style=solid] (-3.0000,0.0000)--(-2.9696,-0.2461)--(-2.9393,-0.3481)--(-2.9090,-0.4264)--(-2.8787,-0.4923)--(-2.8484,-0.5504)--(-2.8181,-0.6030)--(-2.7878,-0.6513)--(-2.7575,-0.6963)--(-2.7272,-0.7385)--(-2.6969,-0.7784)--(-2.6666,-0.8164)--(-2.6363,-0.8528)--(-2.6060,-0.8876)--(-2.5757,-0.9211)--(-2.5454,-0.9534)--(-2.5151,-0.9847)--(-2.4848,-1.0150)--(-2.4545,-1.0444)--(-2.4242,-1.0730)--(-2.3939,-1.1009)--(-2.3636,-1.1281)--(-2.3333,-1.1547)--(-2.3030,-1.1806)--(-2.2727,-1.2060)--(-2.2424,-1.2309)--(-2.2121,-1.2552)--(-2.1818,-1.2792)--(-2.1515,-1.3026)--(-2.1212,-1.3257)--(-2.0909,-1.3483)--(-2.0606,-1.3706)--(-2.0303,-1.3926)--(-2.0000,-1.4142)--(-1.9696,-1.4354)--(-1.9393,-1.4564)--(-1.9090,-1.4770)--(-1.8787,-1.4974)--(-1.8484,-1.5175)--(-1.8181,-1.5374)--(-1.7878,-1.5569)--(-1.7575,-1.5763)--(-1.7272,-1.5954)--(-1.6969,-1.6143)--(-1.6666,-1.6329)--(-1.6363,-1.6514)--(-1.6060,-1.6696)--(-1.5757,-1.6877)--(-1.5454,-1.7056)--(-1.5151,-1.7232)--(-1.4848,-1.7407)--(-1.4545,-1.7580)--(-1.4242,-1.7752)--(-1.3939,-1.7922)--(-1.3636,-1.8090)--(-1.3333,-1.8257)--(-1.3030,-1.8422)--(-1.2727,-1.8586)--(-1.2424,-1.8748)--(-1.2121,-1.8909)--(-1.1818,-1.9069)--(-1.1515,-1.9227)--(-1.1212,-1.9384)--(-1.0909,-1.9540)--(-1.0606,-1.9694)--(-1.0303,-1.9847)--(-1.0000,-2.0000)--(-0.9696,-2.0150)--(-0.9393,-2.0300)--(-0.9090,-2.0449)--(-0.8787,-2.0597)--(-0.8484,-2.0743)--(-0.8181,-2.0889)--(-0.7878,-2.1033)--(-0.7575,-2.1177)--(-0.7272,-2.1320)--(-0.6969,-2.1461)--(-0.6666,-2.1602)--(-0.6363,-2.1742)--(-0.6060,-2.1881)--(-0.5757,-2.2019)--(-0.5454,-2.2156)--(-0.5151,-2.2292)--(-0.4848,-2.2428)--(-0.4545,-2.2563)--(-0.4242,-2.2696)--(-0.3939,-2.2830)--(-0.3636,-2.2962)--(-0.3333,-2.3094)--(-0.3030,-2.3224)--(-0.2727,-2.3354)--(-0.2424,-2.3484)--(-0.2121,-2.3613)--(-0.1818,-2.3741)--(-0.1515,-2.3868)--(-0.1212,-2.3994)--(-0.0909,-2.4120)--(-0.0606,-2.4246)--(-0.0303,-2.4370)--(0.0000,-2.4494);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExempleArcParam.pstricks.recall b/src_phystricks/Fig_ExempleArcParam.pstricks.recall
index 3f874e1c9..9be72d754 100644
--- a/src_phystricks/Fig_ExempleArcParam.pstricks.recall
+++ b/src_phystricks/Fig_ExempleArcParam.pstricks.recall
@@ -65,30 +65,30 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.8998,0) -- (1.8998,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.8982);
+\draw [,->,>=latex] (-1.8998,0.0000) -- (1.8998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.8982);
 %DEFAULT
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-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
-\draw (-0.29125,4.2000) node {$ 6 $};
-\draw [] (-0.100,4.20) -- (0.100,4.20);
+\draw [color=blue] (0.0000,0.0000)--(0.0887,0.0444)--(0.1772,0.0888)--(0.2649,0.1332)--(0.3516,0.1777)--(0.4368,0.2221)--(0.5203,0.2665)--(0.6017,0.3109)--(0.6806,0.3554)--(0.7568,0.3998)--(0.8300,0.4442)--(0.8999,0.4886)--(0.9661,0.5331)--(1.0284,0.5775)--(1.0866,0.6219)--(1.1404,0.6663)--(1.1896,0.7108)--(1.2340,0.7552)--(1.2734,0.7996)--(1.3078,0.8441)--(1.3368,0.8885)--(1.3605,0.9329)--(1.3787,0.9773)--(1.3913,1.0218)--(1.3984,1.0662)--(1.3998,1.1106)--(1.3955,1.1550)--(1.3857,1.1995)--(1.3703,1.2439)--(1.3493,1.2883)--(1.3230,1.3327)--(1.2912,1.3772)--(1.2543,1.4216)--(1.2124,1.4660)--(1.1655,1.5105)--(1.1140,1.5549)--(1.0580,1.5993)--(0.9977,1.6437)--(0.9334,1.6882)--(0.8654,1.7326)--(0.7938,1.7770)--(0.7191,1.8214)--(0.6415,1.8659)--(0.5613,1.9103)--(0.4788,1.9547)--(0.3944,1.9991)--(0.3084,2.0436)--(0.2212,2.0880)--(0.1330,2.1324)--(0.0444,2.1769)--(-0.0444,2.2213)--(-0.1330,2.2657)--(-0.2212,2.3101)--(-0.3084,2.3546)--(-0.3944,2.3990)--(-0.4788,2.4434)--(-0.5613,2.4878)--(-0.6415,2.5323)--(-0.7191,2.5767)--(-0.7938,2.6211)--(-0.8654,2.6655)--(-0.9334,2.7100)--(-0.9977,2.7544)--(-1.0580,2.7988)--(-1.1140,2.8433)--(-1.1655,2.8877)--(-1.2124,2.9321)--(-1.2543,2.9765)--(-1.2912,3.0210)--(-1.3230,3.0654)--(-1.3493,3.1098)--(-1.3703,3.1542)--(-1.3857,3.1987)--(-1.3955,3.2431)--(-1.3998,3.2875)--(-1.3984,3.3319)--(-1.3913,3.3764)--(-1.3787,3.4208)--(-1.3605,3.4652)--(-1.3368,3.5096)--(-1.3078,3.5541)--(-1.2734,3.5985)--(-1.2340,3.6429)--(-1.1896,3.6874)--(-1.1404,3.7318)--(-1.0866,3.7762)--(-1.0284,3.8206)--(-0.9661,3.8651)--(-0.8999,3.9095)--(-0.8300,3.9539)--(-0.7568,3.9983)--(-0.6806,4.0428)--(-0.6017,4.0872)--(-0.5203,4.1316)--(-0.4368,4.1760)--(-0.3516,4.2205)--(-0.2649,4.2649)--(-0.1772,4.3093)--(-0.0887,4.3538)--(0.0000,4.3982);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
+\draw (-0.2912,4.2000) node {$ 6 $};
+\draw [] (-0.1000,4.2000) -- (0.1000,4.2000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -126,18 +126,18 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.9998,0) -- (1.9998,0);
-\draw [,->,>=latex] (0,-1.9998) -- (0,1.9998);
+\draw [,->,>=latex] (-1.9998,0.0000) -- (1.9998,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9998) -- (0.0000,1.9998);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.190,-0.0951)--(0.377,-0.190)--(0.557,-0.284)--(0.729,-0.377)--(0.889,-0.468)--(1.04,-0.557)--(1.16,-0.645)--(1.27,-0.729)--(1.36,-0.811)--(1.43,-0.889)--(1.48,-0.964)--(1.50,-1.04)--(1.50,-1.10)--(1.47,-1.16)--(1.42,-1.22)--(1.34,-1.27)--(1.25,-1.32)--(1.13,-1.36)--(1.00,-1.40)--(0.851,-1.43)--(0.687,-1.46)--(0.513,-1.48)--(0.330,-1.49)--(0.143,-1.50)--(-0.0476,-1.50)--(-0.237,-1.50)--(-0.423,-1.48)--(-0.601,-1.47)--(-0.771,-1.45)--(-0.927,-1.42)--(-1.07,-1.38)--(-1.19,-1.34)--(-1.30,-1.30)--(-1.38,-1.25)--(-1.45,-1.19)--(-1.48,-1.13)--(-1.50,-1.07)--(-1.49,-1.00)--(-1.46,-0.927)--(-1.40,-0.851)--(-1.32,-0.771)--(-1.22,-0.687)--(-1.10,-0.601)--(-0.964,-0.513)--(-0.811,-0.423)--(-0.645,-0.330)--(-0.468,-0.237)--(-0.284,-0.143)--(-0.0951,-0.0476)--(0.0951,0.0476)--(0.284,0.143)--(0.468,0.237)--(0.645,0.330)--(0.811,0.423)--(0.964,0.513)--(1.10,0.601)--(1.22,0.687)--(1.32,0.771)--(1.40,0.851)--(1.46,0.927)--(1.49,1.00)--(1.50,1.07)--(1.48,1.13)--(1.45,1.19)--(1.38,1.25)--(1.30,1.30)--(1.19,1.34)--(1.07,1.38)--(0.927,1.42)--(0.771,1.45)--(0.601,1.47)--(0.423,1.48)--(0.237,1.50)--(0.0476,1.50)--(-0.143,1.50)--(-0.330,1.49)--(-0.513,1.48)--(-0.687,1.46)--(-0.851,1.43)--(-1.00,1.40)--(-1.13,1.36)--(-1.25,1.32)--(-1.34,1.27)--(-1.42,1.22)--(-1.47,1.16)--(-1.50,1.10)--(-1.50,1.04)--(-1.48,0.964)--(-1.43,0.889)--(-1.36,0.811)--(-1.27,0.729)--(-1.16,0.645)--(-1.04,0.557)--(-0.889,0.468)--(-0.729,0.377)--(-0.557,0.284)--(-0.377,0.190)--(-0.190,0.0951)--(0,0);
-\draw (-1.5000,-0.32983) node {$ -1 $};
-\draw [] (-1.50,-0.100) -- (-1.50,0.100);
-\draw (1.5000,-0.31492) node {$ 1 $};
-\draw [] (1.50,-0.100) -- (1.50,0.100);
-\draw (-0.43316,-1.5000) node {$ -1 $};
-\draw [] (-0.100,-1.50) -- (0.100,-1.50);
-\draw (-0.29125,1.5000) node {$ 1 $};
-\draw [] (-0.100,1.50) -- (0.100,1.50);
+\draw [color=blue] (0.0000,0.0000)--(0.1898,-0.0951)--(0.3767,-0.1898)--(0.5574,-0.2838)--(0.7292,-0.3767)--(0.8893,-0.4680)--(1.0351,-0.5574)--(1.1642,-0.6446)--(1.2745,-0.7292)--(1.3644,-0.8109)--(1.4323,-0.8893)--(1.4772,-0.9641)--(1.4983,-1.0351)--(1.4952,-1.1018)--(1.4682,-1.1642)--(1.4175,-1.2218)--(1.3439,-1.2745)--(1.2488,-1.3221)--(1.1336,-1.3644)--(1.0001,-1.4012)--(0.8505,-1.4323)--(0.6873,-1.4577)--(0.5130,-1.4772)--(0.3304,-1.4907)--(0.1425,-1.4983)--(-0.0475,-1.4998)--(-0.2370,-1.4952)--(-0.4225,-1.4847)--(-0.6013,-1.4682)--(-0.7705,-1.4457)--(-0.9272,-1.4175)--(-1.0690,-1.3835)--(-1.1936,-1.3439)--(-1.2990,-1.2990)--(-1.3835,-1.2488)--(-1.4457,-1.1936)--(-1.4847,-1.1336)--(-1.4998,-1.0690)--(-1.4907,-1.0001)--(-1.4577,-0.9272)--(-1.4012,-0.8505)--(-1.3221,-0.7705)--(-1.2218,-0.6873)--(-1.1018,-0.6013)--(-0.9641,-0.5130)--(-0.8109,-0.4225)--(-0.6446,-0.3304)--(-0.4680,-0.2370)--(-0.2838,-0.1425)--(-0.0951,-0.0475)--(0.0951,0.0475)--(0.2838,0.1425)--(0.4680,0.2370)--(0.6446,0.3304)--(0.8109,0.4225)--(0.9641,0.5130)--(1.1018,0.6013)--(1.2218,0.6873)--(1.3221,0.7705)--(1.4012,0.8505)--(1.4577,0.9272)--(1.4907,1.0001)--(1.4998,1.0690)--(1.4847,1.1336)--(1.4457,1.1936)--(1.3835,1.2488)--(1.2990,1.2990)--(1.1936,1.3439)--(1.0690,1.3835)--(0.9272,1.4175)--(0.7705,1.4457)--(0.6013,1.4682)--(0.4225,1.4847)--(0.2370,1.4952)--(0.0475,1.4998)--(-0.1425,1.4983)--(-0.3304,1.4907)--(-0.5130,1.4772)--(-0.6873,1.4577)--(-0.8505,1.4323)--(-1.0001,1.4012)--(-1.1336,1.3644)--(-1.2488,1.3221)--(-1.3439,1.2745)--(-1.4175,1.2218)--(-1.4682,1.1642)--(-1.4952,1.1018)--(-1.4983,1.0351)--(-1.4772,0.9641)--(-1.4323,0.8893)--(-1.3644,0.8109)--(-1.2745,0.7292)--(-1.1642,0.6446)--(-1.0351,0.5574)--(-0.8893,0.4680)--(-0.7292,0.3767)--(-0.5574,0.2838)--(-0.3767,0.1898)--(-0.1898,0.0951)--(0.0000,0.0000);
+\draw (-1.5000,-0.3298) node {$ -1 $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (1.5000,-0.3149) node {$ 1 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (-0.4331,-1.5000) node {$ -1 $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.2912,1.5000) node {$ 1 $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExoParamCD.pstricks.recall b/src_phystricks/Fig_ExoParamCD.pstricks.recall
index 126669a8f..9e60d2dd7 100644
--- a/src_phystricks/Fig_ExoParamCD.pstricks.recall
+++ b/src_phystricks/Fig_ExoParamCD.pstricks.recall
@@ -71,26 +71,26 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.190,0.285)--(0.380,0.568)--(0.568,0.845)--(0.753,1.11)--(0.936,1.37)--(1.11,1.62)--(1.29,1.85)--(1.46,2.07)--(1.62,2.27)--(1.78,2.44)--(1.93,2.60)--(2.07,2.73)--(2.20,2.83)--(2.33,2.92)--(2.44,2.97)--(2.55,3.00)--(2.64,3.00)--(2.73,2.97)--(2.80,2.92)--(2.86,2.83)--(2.92,2.73)--(2.95,2.60)--(2.98,2.44)--(3.00,2.27)--(3.00,2.07)--(2.99,1.85)--(2.97,1.62)--(2.94,1.37)--(2.89,1.11)--(2.83,0.845)--(2.77,0.568)--(2.69,0.285)--(2.60,0)--(2.50,-0.285)--(2.39,-0.568)--(2.27,-0.845)--(2.14,-1.11)--(2.00,-1.37)--(1.85,-1.62)--(1.70,-1.85)--(1.54,-2.07)--(1.37,-2.27)--(1.20,-2.44)--(1.03,-2.60)--(0.845,-2.73)--(0.661,-2.83)--(0.474,-2.92)--(0.285,-2.97)--(0.0952,-3.00)--(-0.0952,-3.00)--(-0.285,-2.97)--(-0.474,-2.92)--(-0.661,-2.83)--(-0.845,-2.73)--(-1.03,-2.60)--(-1.20,-2.44)--(-1.37,-2.27)--(-1.54,-2.07)--(-1.70,-1.85)--(-1.85,-1.62)--(-2.00,-1.37)--(-2.14,-1.11)--(-2.27,-0.845)--(-2.39,-0.568)--(-2.50,-0.285)--(-2.60,0)--(-2.69,0.285)--(-2.77,0.568)--(-2.83,0.845)--(-2.89,1.11)--(-2.94,1.37)--(-2.97,1.62)--(-2.99,1.85)--(-3.00,2.07)--(-3.00,2.27)--(-2.98,2.44)--(-2.95,2.60)--(-2.92,2.73)--(-2.86,2.83)--(-2.80,2.92)--(-2.73,2.97)--(-2.64,3.00)--(-2.55,3.00)--(-2.44,2.97)--(-2.33,2.92)--(-2.20,2.83)--(-2.07,2.73)--(-1.93,2.60)--(-1.78,2.44)--(-1.62,2.27)--(-1.46,2.07)--(-1.29,1.85)--(-1.11,1.62)--(-0.936,1.37)--(-0.753,1.11)--(-0.568,0.845)--(-0.380,0.568)--(-0.190,0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.016641,0.024962);
+\draw [color=blue] (0.0000,0.0000)--(0.1902,0.2851)--(0.3797,0.5677)--(0.5677,0.8451)--(0.7534,1.1149)--(0.9361,1.3746)--(1.1149,1.6219)--(1.2893,1.8544)--(1.4585,2.0702)--(1.6219,2.2672)--(1.7787,2.4437)--(1.9283,2.5980)--(2.0702,2.7288)--(2.2037,2.8350)--(2.3284,2.9154)--(2.4437,2.9694)--(2.5491,2.9966)--(2.6443,2.9966)--(2.7288,2.9694)--(2.8024,2.9154)--(2.8647,2.8350)--(2.9154,2.7288)--(2.9544,2.5980)--(2.9815,2.4437)--(2.9966,2.2672)--(2.9996,2.0702)--(2.9905,1.8544)--(2.9694,1.6219)--(2.9364,1.3746)--(2.8915,1.1149)--(2.8350,0.8451)--(2.7670,0.5677)--(2.6879,0.2851)--(2.5980,0.0000)--(2.4977,-0.2851)--(2.3872,-0.5677)--(2.2672,-0.8451)--(2.1380,-1.1149)--(2.0003,-1.3746)--(1.8544,-1.6219)--(1.7011,-1.8544)--(1.5410,-2.0702)--(1.3746,-2.2672)--(1.2027,-2.4437)--(1.0260,-2.5980)--(0.8451,-2.7288)--(0.6609,-2.8350)--(0.4740,-2.9154)--(0.2851,-2.9694)--(0.0951,-2.9966)--(-0.0951,-2.9966)--(-0.2851,-2.9694)--(-0.4740,-2.9154)--(-0.6609,-2.8350)--(-0.8451,-2.7288)--(-1.0260,-2.5980)--(-1.2027,-2.4437)--(-1.3746,-2.2672)--(-1.5410,-2.0702)--(-1.7011,-1.8544)--(-1.8544,-1.6219)--(-2.0003,-1.3746)--(-2.1380,-1.1149)--(-2.2672,-0.8451)--(-2.3872,-0.5677)--(-2.4977,-0.2851)--(-2.5980,0.0000)--(-2.6879,0.2851)--(-2.7670,0.5677)--(-2.8350,0.8451)--(-2.8915,1.1149)--(-2.9364,1.3746)--(-2.9694,1.6219)--(-2.9905,1.8544)--(-2.9996,2.0702)--(-2.9966,2.2672)--(-2.9815,2.4437)--(-2.9544,2.5980)--(-2.9154,2.7288)--(-2.8647,2.8350)--(-2.8024,2.9154)--(-2.7288,2.9694)--(-2.6443,2.9966)--(-2.5491,2.9966)--(-2.4437,2.9694)--(-2.3284,2.9154)--(-2.2037,2.8350)--(-2.0702,2.7288)--(-1.9283,2.5980)--(-1.7787,2.4437)--(-1.6219,2.2672)--(-1.4585,2.0702)--(-1.2893,1.8544)--(-1.1149,1.6219)--(-0.9361,1.3746)--(-0.7534,1.1149)--(-0.5677,0.8451)--(-0.3797,0.5677)--(-0.1902,0.2851)--(0.0000,0.0000);
+\draw [,->,>=latex] (0.0000,0.0000) -- (0.0166,0.0249);
 \draw [,->,>=latex] (3.0000,2.1213) -- (3.0000,2.0913);
-\draw [,->,>=latex] (0,-3.0000) -- (-0.030000,-3.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (-0.0300,-3.0000);
 \draw [,->,>=latex] (-3.0000,2.1213) -- (-3.0000,2.1513);
-\draw [color=red] (0,0)--(0.190,-0.285)--(0.380,-0.568)--(0.568,-0.845)--(0.753,-1.11)--(0.936,-1.37)--(1.11,-1.62)--(1.29,-1.85)--(1.46,-2.07)--(1.62,-2.27)--(1.78,-2.44)--(1.93,-2.60)--(2.07,-2.73)--(2.20,-2.83)--(2.33,-2.92)--(2.44,-2.97)--(2.55,-3.00)--(2.64,-3.00)--(2.73,-2.97)--(2.80,-2.92)--(2.86,-2.83)--(2.92,-2.73)--(2.95,-2.60)--(2.98,-2.44)--(3.00,-2.27)--(3.00,-2.07)--(2.99,-1.85)--(2.97,-1.62)--(2.94,-1.37)--(2.89,-1.11)--(2.83,-0.845)--(2.77,-0.568)--(2.69,-0.285)--(2.60,0)--(2.50,0.285)--(2.39,0.568)--(2.27,0.845)--(2.14,1.11)--(2.00,1.37)--(1.85,1.62)--(1.70,1.85)--(1.54,2.07)--(1.37,2.27)--(1.20,2.44)--(1.03,2.60)--(0.845,2.73)--(0.661,2.83)--(0.474,2.92)--(0.285,2.97)--(0.0952,3.00)--(-0.0952,3.00)--(-0.285,2.97)--(-0.474,2.92)--(-0.661,2.83)--(-0.845,2.73)--(-1.03,2.60)--(-1.20,2.44)--(-1.37,2.27)--(-1.54,2.07)--(-1.70,1.85)--(-1.85,1.62)--(-2.00,1.37)--(-2.14,1.11)--(-2.27,0.845)--(-2.39,0.568)--(-2.50,0.285)--(-2.60,0)--(-2.69,-0.285)--(-2.77,-0.568)--(-2.83,-0.845)--(-2.89,-1.11)--(-2.94,-1.37)--(-2.97,-1.62)--(-2.99,-1.85)--(-3.00,-2.07)--(-3.00,-2.27)--(-2.98,-2.44)--(-2.95,-2.60)--(-2.92,-2.73)--(-2.86,-2.83)--(-2.80,-2.92)--(-2.73,-2.97)--(-2.64,-3.00)--(-2.55,-3.00)--(-2.44,-2.97)--(-2.33,-2.92)--(-2.20,-2.83)--(-2.07,-2.73)--(-1.93,-2.60)--(-1.78,-2.44)--(-1.62,-2.27)--(-1.46,-2.07)--(-1.29,-1.85)--(-1.11,-1.62)--(-0.936,-1.37)--(-0.753,-1.11)--(-0.568,-0.845)--(-0.380,-0.568)--(-0.190,-0.285)--(0,0);
-\draw [,->,>=latex] (0,0) -- (0.016641,-0.024962);
+\draw [color=red] (0.0000,0.0000)--(0.1902,-0.2851)--(0.3797,-0.5677)--(0.5677,-0.8451)--(0.7534,-1.1149)--(0.9361,-1.3746)--(1.1149,-1.6219)--(1.2893,-1.8544)--(1.4585,-2.0702)--(1.6219,-2.2672)--(1.7787,-2.4437)--(1.9283,-2.5980)--(2.0702,-2.7288)--(2.2037,-2.8350)--(2.3284,-2.9154)--(2.4437,-2.9694)--(2.5491,-2.9966)--(2.6443,-2.9966)--(2.7288,-2.9694)--(2.8024,-2.9154)--(2.8647,-2.8350)--(2.9154,-2.7288)--(2.9544,-2.5980)--(2.9815,-2.4437)--(2.9966,-2.2672)--(2.9996,-2.0702)--(2.9905,-1.8544)--(2.9694,-1.6219)--(2.9364,-1.3746)--(2.8915,-1.1149)--(2.8350,-0.8451)--(2.7670,-0.5677)--(2.6879,-0.2851)--(2.5980,0.0000)--(2.4977,0.2851)--(2.3872,0.5677)--(2.2672,0.8451)--(2.1380,1.1149)--(2.0003,1.3746)--(1.8544,1.6219)--(1.7011,1.8544)--(1.5410,2.0702)--(1.3746,2.2672)--(1.2027,2.4437)--(1.0260,2.5980)--(0.8451,2.7288)--(0.6609,2.8350)--(0.4740,2.9154)--(0.2851,2.9694)--(0.0951,2.9966)--(-0.0951,2.9966)--(-0.2851,2.9694)--(-0.4740,2.9154)--(-0.6609,2.8350)--(-0.8451,2.7288)--(-1.0260,2.5980)--(-1.2027,2.4437)--(-1.3746,2.2672)--(-1.5410,2.0702)--(-1.7011,1.8544)--(-1.8544,1.6219)--(-2.0003,1.3746)--(-2.1380,1.1149)--(-2.2672,0.8451)--(-2.3872,0.5677)--(-2.4977,0.2851)--(-2.5980,0.0000)--(-2.6879,-0.2851)--(-2.7670,-0.5677)--(-2.8350,-0.8451)--(-2.8915,-1.1149)--(-2.9364,-1.3746)--(-2.9694,-1.6219)--(-2.9905,-1.8544)--(-2.9996,-2.0702)--(-2.9966,-2.2672)--(-2.9815,-2.4437)--(-2.9544,-2.5980)--(-2.9154,-2.7288)--(-2.8647,-2.8350)--(-2.8024,-2.9154)--(-2.7288,-2.9694)--(-2.6443,-2.9966)--(-2.5491,-2.9966)--(-2.4437,-2.9694)--(-2.3284,-2.9154)--(-2.2037,-2.8350)--(-2.0702,-2.7288)--(-1.9283,-2.5980)--(-1.7787,-2.4437)--(-1.6219,-2.2672)--(-1.4585,-2.0702)--(-1.2893,-1.8544)--(-1.1149,-1.6219)--(-0.9361,-1.3746)--(-0.7534,-1.1149)--(-0.5677,-0.8451)--(-0.3797,-0.5677)--(-0.1902,-0.2851)--(0.0000,0.0000);
+\draw [,->,>=latex] (0.0000,0.0000) -- (0.0166,-0.0249);
 \draw [,->,>=latex] (3.0000,-2.1213) -- (3.0000,-2.0913);
-\draw [,->,>=latex] (0,3.0000) -- (-0.030000,3.0000);
-\draw (-3.0000,-0.32983) node {$ -1 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -1 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [,->,>=latex] (0.0000,3.0000) -- (-0.0300,3.0000);
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -1 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ExoVarj.pstricks.recall b/src_phystricks/Fig_ExoVarj.pstricks.recall
index 9f183c7db..0bde7f009 100644
--- a/src_phystricks/Fig_ExoVarj.pstricks.recall
+++ b/src_phystricks/Fig_ExoVarj.pstricks.recall
@@ -71,17 +71,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-1.9999) -- (0,1.9999);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9998) -- (0.0000,1.9998);
 %DEFAULT
 
-\draw [color=blue] (-3.00,1.46)--(-2.94,1.47)--(-2.88,1.48)--(-2.82,1.49)--(-2.76,1.49)--(-2.70,1.50)--(-2.64,1.50)--(-2.58,1.50)--(-2.52,1.50)--(-2.45,1.49)--(-2.39,1.49)--(-2.33,1.48)--(-2.27,1.47)--(-2.21,1.46)--(-2.15,1.45)--(-2.09,1.44)--(-2.03,1.42)--(-1.97,1.41)--(-1.91,1.39)--(-1.85,1.37)--(-1.79,1.34)--(-1.73,1.32)--(-1.67,1.29)--(-1.61,1.27)--(-1.55,1.24)--(-1.48,1.21)--(-1.42,1.18)--(-1.36,1.14)--(-1.30,1.11)--(-1.24,1.07)--(-1.18,1.03)--(-1.12,0.990)--(-1.06,0.947)--(-1.00,0.904)--(-0.939,0.858)--(-0.879,0.812)--(-0.818,0.763)--(-0.758,0.713)--(-0.697,0.662)--(-0.636,0.609)--(-0.576,0.556)--(-0.515,0.501)--(-0.455,0.444)--(-0.394,0.387)--(-0.333,0.329)--(-0.273,0.271)--(-0.212,0.211)--(-0.152,0.151)--(-0.0909,0.0908)--(-0.0303,0.0303)--(0.0303,0.0303)--(0.0909,0.0908)--(0.152,0.151)--(0.212,0.211)--(0.273,0.271)--(0.333,0.329)--(0.394,0.387)--(0.455,0.444)--(0.515,0.501)--(0.576,0.556)--(0.636,0.609)--(0.697,0.662)--(0.758,0.713)--(0.818,0.763)--(0.879,0.812)--(0.939,0.858)--(1.00,0.904)--(1.06,0.947)--(1.12,0.990)--(1.18,1.03)--(1.24,1.07)--(1.30,1.11)--(1.36,1.14)--(1.42,1.18)--(1.48,1.21)--(1.55,1.24)--(1.61,1.27)--(1.67,1.29)--(1.73,1.32)--(1.79,1.34)--(1.85,1.37)--(1.91,1.39)--(1.97,1.41)--(2.03,1.42)--(2.09,1.44)--(2.15,1.45)--(2.21,1.46)--(2.27,1.47)--(2.33,1.48)--(2.39,1.49)--(2.45,1.49)--(2.52,1.50)--(2.58,1.50)--(2.64,1.50)--(2.70,1.50)--(2.76,1.49)--(2.82,1.49)--(2.88,1.48)--(2.94,1.47)--(3.00,1.46);
+\draw [color=blue] (-3.0000,1.4576)--(-2.9393,1.4696)--(-2.8787,1.4796)--(-2.8181,1.4875)--(-2.7575,1.4935)--(-2.6969,1.4975)--(-2.6363,1.4996)--(-2.5757,1.4998)--(-2.5151,1.4982)--(-2.4545,1.4949)--(-2.3939,1.4897)--(-2.3333,1.4828)--(-2.2727,1.4742)--(-2.2121,1.4638)--(-2.1515,1.4517)--(-2.0909,1.4380)--(-2.0303,1.4225)--(-1.9696,1.4054)--(-1.9090,1.3866)--(-1.8484,1.3661)--(-1.7878,1.3439)--(-1.7272,1.3200)--(-1.6666,1.2945)--(-1.6060,1.2673)--(-1.5454,1.2384)--(-1.4848,1.2079)--(-1.4242,1.1756)--(-1.3636,1.1417)--(-1.3030,1.1062)--(-1.2424,1.0689)--(-1.1818,1.0300)--(-1.1212,0.9895)--(-1.0606,0.9474)--(-1.0000,0.9036)--(-0.9393,0.8583)--(-0.8787,0.8115)--(-0.8181,0.7631)--(-0.7575,0.7132)--(-0.6969,0.6620)--(-0.6363,0.6094)--(-0.5757,0.5556)--(-0.5151,0.5005)--(-0.4545,0.4444)--(-0.3939,0.3873)--(-0.3333,0.3292)--(-0.2727,0.2705)--(-0.2121,0.2110)--(-0.1515,0.1511)--(-0.0909,0.0908)--(-0.0303,0.0302)--(0.0303,0.0302)--(0.0909,0.0908)--(0.1515,0.1511)--(0.2121,0.2110)--(0.2727,0.2705)--(0.3333,0.3292)--(0.3939,0.3873)--(0.4545,0.4444)--(0.5151,0.5005)--(0.5757,0.5556)--(0.6363,0.6094)--(0.6969,0.6620)--(0.7575,0.7132)--(0.8181,0.7631)--(0.8787,0.8115)--(0.9393,0.8583)--(1.0000,0.9036)--(1.0606,0.9474)--(1.1212,0.9895)--(1.1818,1.0300)--(1.2424,1.0689)--(1.3030,1.1062)--(1.3636,1.1417)--(1.4242,1.1756)--(1.4848,1.2079)--(1.5454,1.2384)--(1.6060,1.2673)--(1.6666,1.2945)--(1.7272,1.3200)--(1.7878,1.3439)--(1.8484,1.3661)--(1.9090,1.3866)--(1.9696,1.4054)--(2.0303,1.4225)--(2.0909,1.4380)--(2.1515,1.4517)--(2.2121,1.4638)--(2.2727,1.4742)--(2.3333,1.4828)--(2.3939,1.4897)--(2.4545,1.4949)--(2.5151,1.4982)--(2.5757,1.4998)--(2.6363,1.4996)--(2.6969,1.4975)--(2.7575,1.4935)--(2.8181,1.4875)--(2.8787,1.4796)--(2.9393,1.4696)--(3.0000,1.4576);
 
-\draw [color=red] (-3.00,-1.46)--(-2.94,-1.47)--(-2.88,-1.48)--(-2.82,-1.49)--(-2.76,-1.49)--(-2.70,-1.50)--(-2.64,-1.50)--(-2.58,-1.50)--(-2.52,-1.50)--(-2.45,-1.49)--(-2.39,-1.49)--(-2.33,-1.48)--(-2.27,-1.47)--(-2.21,-1.46)--(-2.15,-1.45)--(-2.09,-1.44)--(-2.03,-1.42)--(-1.97,-1.41)--(-1.91,-1.39)--(-1.85,-1.37)--(-1.79,-1.34)--(-1.73,-1.32)--(-1.67,-1.29)--(-1.61,-1.27)--(-1.55,-1.24)--(-1.48,-1.21)--(-1.42,-1.18)--(-1.36,-1.14)--(-1.30,-1.11)--(-1.24,-1.07)--(-1.18,-1.03)--(-1.12,-0.990)--(-1.06,-0.947)--(-1.00,-0.904)--(-0.939,-0.858)--(-0.879,-0.812)--(-0.818,-0.763)--(-0.758,-0.713)--(-0.697,-0.662)--(-0.636,-0.609)--(-0.576,-0.556)--(-0.515,-0.501)--(-0.455,-0.444)--(-0.394,-0.387)--(-0.333,-0.329)--(-0.273,-0.271)--(-0.212,-0.211)--(-0.152,-0.151)--(-0.0909,-0.0908)--(-0.0303,-0.0303)--(0.0303,-0.0303)--(0.0909,-0.0908)--(0.152,-0.151)--(0.212,-0.211)--(0.273,-0.271)--(0.333,-0.329)--(0.394,-0.387)--(0.455,-0.444)--(0.515,-0.501)--(0.576,-0.556)--(0.636,-0.609)--(0.697,-0.662)--(0.758,-0.713)--(0.818,-0.763)--(0.879,-0.812)--(0.939,-0.858)--(1.00,-0.904)--(1.06,-0.947)--(1.12,-0.990)--(1.18,-1.03)--(1.24,-1.07)--(1.30,-1.11)--(1.36,-1.14)--(1.42,-1.18)--(1.48,-1.21)--(1.55,-1.24)--(1.61,-1.27)--(1.67,-1.29)--(1.73,-1.32)--(1.79,-1.34)--(1.85,-1.37)--(1.91,-1.39)--(1.97,-1.41)--(2.03,-1.42)--(2.09,-1.44)--(2.15,-1.45)--(2.21,-1.46)--(2.27,-1.47)--(2.33,-1.48)--(2.39,-1.49)--(2.45,-1.49)--(2.52,-1.50)--(2.58,-1.50)--(2.64,-1.50)--(2.70,-1.50)--(2.76,-1.49)--(2.82,-1.49)--(2.88,-1.48)--(2.94,-1.47)--(3.00,-1.46);
-\draw (-3.0000,-0.32983) node {$ -1 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
+\draw [color=red] (-3.0000,-1.4576)--(-2.9393,-1.4696)--(-2.8787,-1.4796)--(-2.8181,-1.4875)--(-2.7575,-1.4935)--(-2.6969,-1.4975)--(-2.6363,-1.4996)--(-2.5757,-1.4998)--(-2.5151,-1.4982)--(-2.4545,-1.4949)--(-2.3939,-1.4897)--(-2.3333,-1.4828)--(-2.2727,-1.4742)--(-2.2121,-1.4638)--(-2.1515,-1.4517)--(-2.0909,-1.4380)--(-2.0303,-1.4225)--(-1.9696,-1.4054)--(-1.9090,-1.3866)--(-1.8484,-1.3661)--(-1.7878,-1.3439)--(-1.7272,-1.3200)--(-1.6666,-1.2945)--(-1.6060,-1.2673)--(-1.5454,-1.2384)--(-1.4848,-1.2079)--(-1.4242,-1.1756)--(-1.3636,-1.1417)--(-1.3030,-1.1062)--(-1.2424,-1.0689)--(-1.1818,-1.0300)--(-1.1212,-0.9895)--(-1.0606,-0.9474)--(-1.0000,-0.9036)--(-0.9393,-0.8583)--(-0.8787,-0.8115)--(-0.8181,-0.7631)--(-0.7575,-0.7132)--(-0.6969,-0.6620)--(-0.6363,-0.6094)--(-0.5757,-0.5556)--(-0.5151,-0.5005)--(-0.4545,-0.4444)--(-0.3939,-0.3873)--(-0.3333,-0.3292)--(-0.2727,-0.2705)--(-0.2121,-0.2110)--(-0.1515,-0.1511)--(-0.0909,-0.0908)--(-0.0303,-0.0302)--(0.0303,-0.0302)--(0.0909,-0.0908)--(0.1515,-0.1511)--(0.2121,-0.2110)--(0.2727,-0.2705)--(0.3333,-0.3292)--(0.3939,-0.3873)--(0.4545,-0.4444)--(0.5151,-0.5005)--(0.5757,-0.5556)--(0.6363,-0.6094)--(0.6969,-0.6620)--(0.7575,-0.7132)--(0.8181,-0.7631)--(0.8787,-0.8115)--(0.9393,-0.8583)--(1.0000,-0.9036)--(1.0606,-0.9474)--(1.1212,-0.9895)--(1.1818,-1.0300)--(1.2424,-1.0689)--(1.3030,-1.1062)--(1.3636,-1.1417)--(1.4242,-1.1756)--(1.4848,-1.2079)--(1.5454,-1.2384)--(1.6060,-1.2673)--(1.6666,-1.2945)--(1.7272,-1.3200)--(1.7878,-1.3439)--(1.8484,-1.3661)--(1.9090,-1.3866)--(1.9696,-1.4054)--(2.0303,-1.4225)--(2.0909,-1.4380)--(2.1515,-1.4517)--(2.2121,-1.4638)--(2.2727,-1.4742)--(2.3333,-1.4828)--(2.3939,-1.4897)--(2.4545,-1.4949)--(2.5151,-1.4982)--(2.5757,-1.4998)--(2.6363,-1.4996)--(2.6969,-1.4975)--(2.7575,-1.4935)--(2.8181,-1.4875)--(2.8787,-1.4796)--(2.9393,-1.4696)--(3.0000,-1.4576);
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall b/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall
index aacb17a58..7d8c5c8c6 100644
--- a/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall
+++ b/src_phystricks/Fig_FCUEooTpEPFoeQ.pstricks.recall
@@ -99,7 +99,7 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (1.4562,-1.1844) node {$
+\draw (1.4562,-1.1843) node {$
         \begin{pmatrix}
         \phantom{
         \begin{matrix}
@@ -108,40 +108,40 @@
         }
         \end{pmatrix}$
         };
-\draw (0.29125,-0.23688) node {*};
-\draw (0.87375,-0.23688) node {*};
-\draw (1.4562,-0.23688) node {*};
-\draw (2.0387,-0.23688) node {*};
-\draw (2.6213,-0.23688) node {*};
-\draw (0.29125,-0.71062) node {0};
-\draw (0.87375,-0.71062) node {*};
-\draw (1.4562,-0.71062) node {*};
-\draw (2.0387,-0.71062) node {*};
-\draw (2.6213,-0.71062) node {*};
-\draw (0.29125,-1.1844) node {0};
-\draw (0.87375,-1.1844) node {0};
-\draw (1.4562,-1.1844) node {*};
-\draw (2.0387,-1.1844) node {*};
-\draw (2.6213,-1.1844) node {*};
-\draw (0.29125,-1.6581) node {0};
-\draw (0.87375,-1.6581) node {0};
+\draw (0.2912,-0.2368) node {*};
+\draw (0.8737,-0.2368) node {*};
+\draw (1.4562,-0.2368) node {*};
+\draw (2.0387,-0.2368) node {*};
+\draw (2.6212,-0.2368) node {*};
+\draw (0.2912,-0.7106) node {0};
+\draw (0.8737,-0.7106) node {*};
+\draw (1.4562,-0.7106) node {*};
+\draw (2.0387,-0.7106) node {*};
+\draw (2.6212,-0.7106) node {*};
+\draw (0.2912,-1.1843) node {0};
+\draw (0.8737,-1.1843) node {0};
+\draw (1.4562,-1.1843) node {*};
+\draw (2.0387,-1.1843) node {*};
+\draw (2.6212,-1.1843) node {*};
+\draw (0.2912,-1.6581) node {0};
+\draw (0.8737,-1.6581) node {0};
 \draw (1.4562,-1.6581) node {*};
 \draw (2.0387,-1.6581) node {*};
-\draw (2.6213,-1.6581) node {*};
-\draw (0.29125,-2.1319) node {0};
-\draw (0.87375,-2.1319) node {0};
-\draw (1.4562,-2.1319) node {*};
-\draw (2.0387,-2.1319) node {*};
-\draw (2.6213,-2.1319) node {*};
-\draw [color=red] (0.100,-0.0500) -- (1.06,-0.0500);
-\draw [color=red] (1.06,-0.0500) -- (1.06,-0.898);
-\draw [color=red] (1.06,-0.898) -- (0.100,-0.898);
-\draw [color=red] (0.100,-0.898) -- (0.100,-0.0500);
-\draw [color=blue] (1.27,-0.997) -- (2.81,-0.997);
-\draw [color=blue] (2.81,-0.997) -- (2.81,-2.32);
-\draw [color=blue] (2.81,-2.32) -- (1.27,-2.32);
-\draw [color=blue] (1.27,-2.32) -- (1.27,-0.997);
-\draw (1.2124,0.22642) node {\( \Delta_k(A_2)\)};
+\draw (2.6212,-1.6581) node {*};
+\draw (0.2912,-2.1318) node {0};
+\draw (0.8737,-2.1318) node {0};
+\draw (1.4562,-2.1318) node {*};
+\draw (2.0387,-2.1318) node {*};
+\draw (2.6212,-2.1318) node {*};
+\draw [color=red] (0.1000,-0.0500) -- (1.0649,-0.0500);
+\draw [color=red] (1.0649,-0.0500) -- (1.0649,-0.8975);
+\draw [color=red] (1.0649,-0.8975) -- (0.1000,-0.8975);
+\draw [color=red] (0.1000,-0.8975) -- (0.1000,-0.0500);
+\draw [color=blue] (1.2649,-0.9975) -- (2.8124,-0.9975);
+\draw [color=blue] (2.8124,-0.9975) -- (2.8124,-2.3187);
+\draw [color=blue] (2.8124,-2.3187) -- (1.2649,-2.3187);
+\draw [color=blue] (1.2649,-2.3187) -- (1.2649,-0.9975);
+\draw (1.2123,0.2264) node {\( \Delta_k(A_2)\)};
 \draw (2.0387,-2.6012) node {\( \Omega_{k+1}(A_2)\)};
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_FGRooDhFkch.pstricks.recall b/src_phystricks/Fig_FGRooDhFkch.pstricks.recall
index f4993c771..935fc87f8 100644
--- a/src_phystricks/Fig_FGRooDhFkch.pstricks.recall
+++ b/src_phystricks/Fig_FGRooDhFkch.pstricks.recall
@@ -86,39 +86,39 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-3.00,-3.14) -- (-3.00,3.14);
-\draw [color=gray,style=solid] (0,-3.14) -- (0,3.14);
-\draw [color=gray,style=solid] (3.00,-3.14) -- (3.00,3.14);
-\draw [color=gray,style=dotted] (-1.50,-3.14) -- (-1.50,3.14);
-\draw [color=gray,style=dotted] (1.50,-3.14) -- (1.50,3.14);
-\draw [color=gray,style=dotted] (-3.00,-2.36) -- (3.00,-2.36);
-\draw [color=gray,style=dotted] (-3.00,-0.785) -- (3.00,-0.785);
-\draw [color=gray,style=dotted] (-3.00,0.785) -- (3.00,0.785);
-\draw [color=gray,style=dotted] (-3.00,2.36) -- (3.00,2.36);
-\draw [color=gray,style=solid] (-3.00,-3.14) -- (3.00,-3.14);
-\draw [color=gray,style=solid] (-3.00,-1.57) -- (3.00,-1.57);
-\draw [color=gray,style=solid] (-3.00,0) -- (3.00,0);
-\draw [color=gray,style=solid] (-3.00,1.57) -- (3.00,1.57);
-\draw [color=gray,style=solid] (-3.00,3.14) -- (3.00,3.14);
+\draw [color=gray,style=solid] (-3.0000,-3.1415) -- (-3.0000,3.1415);
+\draw [color=gray,style=solid] (0.0000,-3.1415) -- (0.0000,3.1415);
+\draw [color=gray,style=solid] (3.0000,-3.1415) -- (3.0000,3.1415);
+\draw [color=gray,style=dotted] (-1.5000,-3.1415) -- (-1.5000,3.1415);
+\draw [color=gray,style=dotted] (1.5000,-3.1415) -- (1.5000,3.1415);
+\draw [color=gray,style=dotted] (-3.0000,-2.3561) -- (3.0000,-2.3561);
+\draw [color=gray,style=dotted] (-3.0000,-0.7853) -- (3.0000,-0.7853);
+\draw [color=gray,style=dotted] (-3.0000,0.7853) -- (3.0000,0.7853);
+\draw [color=gray,style=dotted] (-3.0000,2.3561) -- (3.0000,2.3561);
+\draw [color=gray,style=solid] (-3.0000,-3.1415) -- (3.0000,-3.1415);
+\draw [color=gray,style=solid] (-3.0000,-1.5707) -- (3.0000,-1.5707);
+\draw [color=gray,style=solid] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [color=gray,style=solid] (-3.0000,1.5707) -- (3.0000,1.5707);
+\draw [color=gray,style=solid] (-3.0000,3.1415) -- (3.0000,3.1415);
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.6416) -- (0,3.6416);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.6415) -- (0.0000,3.6415);
 %DEFAULT
 
-\draw [color=blue] (-3.000,-1.571)--(-2.939,-1.369)--(-2.879,-1.286)--(-2.818,-1.221)--(-2.758,-1.166)--(-2.697,-1.117)--(-2.636,-1.073)--(-2.576,-1.033)--(-2.515,-0.9943)--(-2.455,-0.9582)--(-2.394,-0.9239)--(-2.333,-0.8911)--(-2.273,-0.8596)--(-2.212,-0.8292)--(-2.152,-0.7997)--(-2.091,-0.7712)--(-2.030,-0.7434)--(-1.970,-0.7163)--(-1.909,-0.6898)--(-1.848,-0.6639)--(-1.788,-0.6385)--(-1.727,-0.6135)--(-1.667,-0.5890)--(-1.606,-0.5649)--(-1.545,-0.5412)--(-1.485,-0.5178)--(-1.424,-0.4947)--(-1.364,-0.4719)--(-1.303,-0.4493)--(-1.242,-0.4270)--(-1.182,-0.4049)--(-1.121,-0.3830)--(-1.061,-0.3613)--(-1.000,-0.3398)--(-0.9394,-0.3185)--(-0.8788,-0.2973)--(-0.8182,-0.2762)--(-0.7576,-0.2553)--(-0.6970,-0.2345)--(-0.6364,-0.2137)--(-0.5758,-0.1931)--(-0.5152,-0.1726)--(-0.4545,-0.1521)--(-0.3939,-0.1317)--(-0.3333,-0.1113)--(-0.2727,-0.09103)--(-0.2121,-0.07077)--(-0.1515,-0.05053)--(-0.09091,-0.03031)--(-0.03030,-0.01010)--(0.03030,0.01010)--(0.09091,0.03031)--(0.1515,0.05053)--(0.2121,0.07077)--(0.2727,0.09103)--(0.3333,0.1113)--(0.3939,0.1317)--(0.4545,0.1521)--(0.5152,0.1726)--(0.5758,0.1931)--(0.6364,0.2137)--(0.6970,0.2345)--(0.7576,0.2553)--(0.8182,0.2762)--(0.8788,0.2973)--(0.9394,0.3185)--(1.000,0.3398)--(1.061,0.3613)--(1.121,0.3830)--(1.182,0.4049)--(1.242,0.4270)--(1.303,0.4493)--(1.364,0.4719)--(1.424,0.4947)--(1.485,0.5178)--(1.545,0.5412)--(1.606,0.5649)--(1.667,0.5890)--(1.727,0.6135)--(1.788,0.6385)--(1.848,0.6639)--(1.909,0.6898)--(1.970,0.7163)--(2.030,0.7434)--(2.091,0.7712)--(2.152,0.7997)--(2.212,0.8292)--(2.273,0.8596)--(2.333,0.8911)--(2.394,0.9239)--(2.455,0.9582)--(2.515,0.9943)--(2.576,1.033)--(2.636,1.073)--(2.697,1.117)--(2.758,1.166)--(2.818,1.221)--(2.879,1.286)--(2.939,1.369)--(3.000,1.571);
+\draw [color=blue] (-3.0000,-1.5707)--(-2.9393,-1.3694)--(-2.8787,-1.2855)--(-2.8181,-1.2208)--(-2.7575,-1.1660)--(-2.6969,-1.1174)--(-2.6363,-1.0733)--(-2.5757,-1.0325)--(-2.5151,-0.9943)--(-2.4545,-0.9582)--(-2.3939,-0.9239)--(-2.3333,-0.8911)--(-2.2727,-0.8595)--(-2.2121,-0.8291)--(-2.1515,-0.7997)--(-2.0909,-0.7711)--(-2.0303,-0.7433)--(-1.9696,-0.7162)--(-1.9090,-0.6897)--(-1.8484,-0.6638)--(-1.7878,-0.6384)--(-1.7272,-0.6135)--(-1.6666,-0.5890)--(-1.6060,-0.5649)--(-1.5454,-0.5411)--(-1.4848,-0.5177)--(-1.4242,-0.4946)--(-1.3636,-0.4718)--(-1.3030,-0.4493)--(-1.2424,-0.4269)--(-1.1818,-0.4049)--(-1.1212,-0.3830)--(-1.0606,-0.3613)--(-1.0000,-0.3398)--(-0.9393,-0.3184)--(-0.8787,-0.2972)--(-0.8181,-0.2762)--(-0.7575,-0.2552)--(-0.6969,-0.2344)--(-0.6363,-0.2137)--(-0.5757,-0.1931)--(-0.5151,-0.1725)--(-0.4545,-0.1521)--(-0.3939,-0.1316)--(-0.3333,-0.1113)--(-0.2727,-0.0910)--(-0.2121,-0.0707)--(-0.1515,-0.0505)--(-0.0909,-0.0303)--(-0.0303,-0.0101)--(0.0303,0.0101)--(0.0909,0.0303)--(0.1515,0.0505)--(0.2121,0.0707)--(0.2727,0.0910)--(0.3333,0.1113)--(0.3939,0.1316)--(0.4545,0.1521)--(0.5151,0.1725)--(0.5757,0.1931)--(0.6363,0.2137)--(0.6969,0.2344)--(0.7575,0.2552)--(0.8181,0.2762)--(0.8787,0.2972)--(0.9393,0.3184)--(1.0000,0.3398)--(1.0606,0.3613)--(1.1212,0.3830)--(1.1818,0.4049)--(1.2424,0.4269)--(1.3030,0.4493)--(1.3636,0.4718)--(1.4242,0.4946)--(1.4848,0.5177)--(1.5454,0.5411)--(1.6060,0.5649)--(1.6666,0.5890)--(1.7272,0.6135)--(1.7878,0.6384)--(1.8484,0.6638)--(1.9090,0.6897)--(1.9696,0.7162)--(2.0303,0.7433)--(2.0909,0.7711)--(2.1515,0.7997)--(2.2121,0.8291)--(2.2727,0.8595)--(2.3333,0.8911)--(2.3939,0.9239)--(2.4545,0.9582)--(2.5151,0.9943)--(2.5757,1.0325)--(2.6363,1.0733)--(2.6969,1.1174)--(2.7575,1.1660)--(2.8181,1.2208)--(2.8787,1.2855)--(2.9393,1.3694)--(3.0000,1.5707);
 
-\draw (-3.0000,-0.32983) node {$ -1 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.45249,-3.1416) node {$ -\pi $};
-\draw [] (-0.100,-3.14) -- (0.100,-3.14);
-\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
-\draw [] (-0.100,1.57) -- (0.100,1.57);
-\draw (-0.31058,3.1416) node {$ \pi $};
-\draw [] (-0.100,3.14) -- (0.100,3.14);
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4524,-3.1415) node {$ -\pi $};
+\draw [] (-0.1000,-3.1415) -- (0.1000,-3.1415);
+\draw (-0.5937,-1.5707) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-0.1000,-1.5707) -- (0.1000,-1.5707);
+\draw (-0.4518,1.5707) node {$ \frac{1}{2} \, \pi $};
+\draw [] (-0.1000,1.5707) -- (0.1000,1.5707);
+\draw (-0.3105,3.1415) node {$ \pi $};
+\draw [] (-0.1000,3.1415) -- (0.1000,3.1415);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_FnCosApprox.pstricks.recall b/src_phystricks/Fig_FnCosApprox.pstricks.recall
index 890159f3e..6f99255f5 100644
--- a/src_phystricks/Fig_FnCosApprox.pstricks.recall
+++ b/src_phystricks/Fig_FnCosApprox.pstricks.recall
@@ -91,25 +91,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (0,2.000)--(0.06347,1.999)--(0.1269,1.996)--(0.1904,1.991)--(0.2539,1.984)--(0.3173,1.975)--(0.3808,1.964)--(0.4443,1.951)--(0.5077,1.936)--(0.5712,1.919)--(0.6347,1.900)--(0.6981,1.879)--(0.7616,1.857)--(0.8251,1.832)--(0.8885,1.806)--(0.9520,1.778)--(1.015,1.748)--(1.079,1.716)--(1.142,1.683)--(1.206,1.647)--(1.269,1.611)--(1.333,1.572)--(1.396,1.532)--(1.460,1.491)--(1.523,1.447)--(1.587,1.403)--(1.650,1.357)--(1.714,1.310)--(1.777,1.261)--(1.841,1.211)--(1.904,1.160)--(1.967,1.108)--(2.031,1.054)--(2.094,1.000)--(2.158,0.9445)--(2.221,0.8881)--(2.285,0.8308)--(2.348,0.7727)--(2.412,0.7138)--(2.475,0.6541)--(2.539,0.5938)--(2.602,0.5330)--(2.666,0.4715)--(2.729,0.4096)--(2.793,0.3473)--(2.856,0.2846)--(2.919,0.2217)--(2.983,0.1585)--(3.046,0.09516)--(3.110,0.03173)--(3.173,-0.03173)--(3.237,-0.09516)--(3.300,-0.1585)--(3.364,-0.2217)--(3.427,-0.2846)--(3.491,-0.3473)--(3.554,-0.4096)--(3.618,-0.4715)--(3.681,-0.5330)--(3.745,-0.5938)--(3.808,-0.6541)--(3.871,-0.7138)--(3.935,-0.7727)--(3.998,-0.8308)--(4.062,-0.8881)--(4.125,-0.9445)--(4.189,-1.000)--(4.252,-1.054)--(4.316,-1.108)--(4.379,-1.160)--(4.443,-1.211)--(4.506,-1.261)--(4.570,-1.310)--(4.633,-1.357)--(4.697,-1.403)--(4.760,-1.447)--(4.823,-1.491)--(4.887,-1.532)--(4.950,-1.572)--(5.014,-1.611)--(5.077,-1.647)--(5.141,-1.683)--(5.204,-1.716)--(5.268,-1.748)--(5.331,-1.778)--(5.395,-1.806)--(5.458,-1.832)--(5.522,-1.857)--(5.585,-1.879)--(5.648,-1.900)--(5.712,-1.919)--(5.775,-1.936)--(5.839,-1.951)--(5.902,-1.964)--(5.966,-1.975)--(6.029,-1.984)--(6.093,-1.991)--(6.156,-1.996)--(6.220,-1.999)--(6.283,-2.000);
-\draw []  (1.5708,1.4142) node [rotate=0] {$\bullet$};
-\draw (1.7999,1.6614) node {$P$};
-\draw (1.5708,-0.42071) node {$ \frac{1}{4} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{4} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.27858) node {$ \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.43316,-2.0000) node {$ -1 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,2.0000)--(0.0634,1.9989)--(0.1269,1.9959)--(0.1903,1.9909)--(0.2538,1.9839)--(0.3173,1.9748)--(0.3807,1.9638)--(0.4442,1.9508)--(0.5077,1.9358)--(0.5711,1.9189)--(0.6346,1.9001)--(0.6981,1.8793)--(0.7615,1.8567)--(0.8250,1.8322)--(0.8885,1.8058)--(0.9519,1.7776)--(1.0154,1.7476)--(1.0789,1.7159)--(1.1423,1.6825)--(1.2058,1.6473)--(1.2693,1.6105)--(1.3327,1.5721)--(1.3962,1.5320)--(1.4597,1.4905)--(1.5231,1.4474)--(1.5866,1.4029)--(1.6501,1.3570)--(1.7135,1.3097)--(1.7770,1.2611)--(1.8405,1.2112)--(1.9039,1.1601)--(1.9674,1.1078)--(2.0309,1.0544)--(2.0943,1.0000)--(2.1578,0.9445)--(2.2213,0.8881)--(2.2847,0.8308)--(2.3482,0.7726)--(2.4117,0.7137)--(2.4751,0.6541)--(2.5386,0.5938)--(2.6021,0.5329)--(2.6655,0.4715)--(2.7290,0.4096)--(2.7925,0.3472)--(2.8559,0.2846)--(2.9194,0.2216)--(2.9829,0.1584)--(3.0463,0.0951)--(3.1098,0.0317)--(3.1733,-0.0317)--(3.2367,-0.0951)--(3.3002,-0.1584)--(3.3637,-0.2216)--(3.4271,-0.2846)--(3.4906,-0.3472)--(3.5541,-0.4096)--(3.6175,-0.4715)--(3.6810,-0.5329)--(3.7445,-0.5938)--(3.8079,-0.6541)--(3.8714,-0.7137)--(3.9349,-0.7726)--(3.9983,-0.8308)--(4.0618,-0.8881)--(4.1253,-0.9445)--(4.1887,-1.0000)--(4.2522,-1.0544)--(4.3157,-1.1078)--(4.3791,-1.1601)--(4.4426,-1.2112)--(4.5061,-1.2611)--(4.5695,-1.3097)--(4.6330,-1.3570)--(4.6965,-1.4029)--(4.7599,-1.4474)--(4.8234,-1.4905)--(4.8869,-1.5320)--(4.9503,-1.5721)--(5.0138,-1.6105)--(5.0773,-1.6473)--(5.1407,-1.6825)--(5.2042,-1.7159)--(5.2677,-1.7476)--(5.3311,-1.7776)--(5.3946,-1.8058)--(5.4581,-1.8322)--(5.5215,-1.8567)--(5.5850,-1.8793)--(5.6485,-1.9001)--(5.7119,-1.9189)--(5.7754,-1.9358)--(5.8389,-1.9508)--(5.9023,-1.9638)--(5.9658,-1.9748)--(6.0293,-1.9839)--(6.0927,-1.9909)--(6.1562,-1.9959)--(6.2197,-1.9989)--(6.2831,-2.0000);
+\draw []  (1.5707,1.4142) node [rotate=0] {$\bullet$};
+\draw (1.7999,1.6613) node {$P$};
+\draw (1.5707,-0.4207) node {$ \frac{1}{4} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{4} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.2785) node {$ \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall b/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
index 5fcac2505..3571a0d1c 100644
--- a/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
+++ b/src_phystricks/Fig_FonctionEtDeriveOM.pstricks.recall
@@ -119,41 +119,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
+\draw [,->,>=latex] (-4.0000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.7995) -- (0.0000,4.0547);
 %DEFAULT
 
-\draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
+\draw [color=blue] (-3.5000,-0.9928)--(-3.4292,-0.6362)--(-3.3585,-0.2871)--(-3.2878,0.0506)--(-3.2171,0.3737)--(-3.1464,0.6787)--(-3.0757,0.9630)--(-3.0050,1.2238)--(-2.9343,1.4592)--(-2.8636,1.6673)--(-2.7929,1.8468)--(-2.7222,1.9968)--(-2.6515,2.1167)--(-2.5808,2.2065)--(-2.5101,2.2664)--(-2.4393,2.2970)--(-2.3686,2.2995)--(-2.2979,2.2750)--(-2.2272,2.2254)--(-2.1565,2.1525)--(-2.0858,2.0586)--(-2.0151,1.9459)--(-1.9444,1.8171)--(-1.8737,1.6749)--(-1.8030,1.5220)--(-1.7323,1.3612)--(-1.6616,1.1953)--(-1.5909,1.0272)--(-1.5202,0.8595)--(-1.4494,0.6948)--(-1.3787,0.5355)--(-1.3080,0.3839)--(-1.2373,0.2420)--(-1.1666,0.1116)--(-1.0959,-0.0056)--(-1.0252,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2680)--(-0.8131,-0.3235)--(-0.7424,-0.3625)--(-0.6717,-0.3854)--(-0.6010,-0.3927)--(-0.5303,-0.3852)--(-0.4595,-0.3640)--(-0.3888,-0.3304)--(-0.3181,-0.2858)--(-0.2474,-0.2321)--(-0.1767,-0.1711)--(-0.1060,-0.1048)--(-0.0353,-0.0353)--(0.0353,0.0353)--(0.1060,0.1048)--(0.1767,0.1711)--(0.2474,0.2321)--(0.3181,0.2858)--(0.3888,0.3304)--(0.4595,0.3640)--(0.5303,0.3852)--(0.6010,0.3927)--(0.6717,0.3854)--(0.7424,0.3625)--(0.8131,0.3235)--(0.8838,0.2680)--(0.9545,0.1963)--(1.0252,0.1086)--(1.0959,0.0056)--(1.1666,-0.1116)--(1.2373,-0.2420)--(1.3080,-0.3839)--(1.3787,-0.5355)--(1.4494,-0.6948)--(1.5202,-0.8595)--(1.5909,-1.0272)--(1.6616,-1.1953)--(1.7323,-1.3612)--(1.8030,-1.5220)--(1.8737,-1.6749)--(1.9444,-1.8171)--(2.0151,-1.9459)--(2.0858,-2.0586)--(2.1565,-2.1525)--(2.2272,-2.2254)--(2.2979,-2.2750)--(2.3686,-2.2995)--(2.4393,-2.2970)--(2.5101,-2.2664)--(2.5808,-2.2065)--(2.6515,-2.1167)--(2.7222,-1.9968)--(2.7929,-1.8468)--(2.8636,-1.6673)--(2.9343,-1.4592)--(3.0050,-1.2238)--(3.0757,-0.9630)--(3.1464,-0.6787)--(3.2171,-0.3737)--(3.2878,-0.0506)--(3.3585,0.2871)--(3.4292,0.6362)--(3.5000,0.9928);
 
-\draw [color=red] (-3.500,3.555)--(-3.429,3.500)--(-3.359,3.406)--(-3.288,3.277)--(-3.217,3.114)--(-3.146,2.921)--(-3.076,2.702)--(-3.005,2.459)--(-2.934,2.198)--(-2.864,1.921)--(-2.793,1.632)--(-2.722,1.337)--(-2.652,1.038)--(-2.581,0.7401)--(-2.510,0.4468)--(-2.439,0.1618)--(-2.369,-0.1114)--(-2.298,-0.3696)--(-2.227,-0.6099)--(-2.157,-0.8297)--(-2.086,-1.027)--(-2.015,-1.199)--(-1.944,-1.346)--(-1.874,-1.466)--(-1.803,-1.558)--(-1.732,-1.622)--(-1.662,-1.658)--(-1.591,-1.667)--(-1.520,-1.650)--(-1.449,-1.608)--(-1.379,-1.542)--(-1.308,-1.456)--(-1.237,-1.350)--(-1.167,-1.228)--(-1.096,-1.092)--(-1.025,-0.9453)--(-0.9545,-0.7902)--(-0.8838,-0.6299)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.002540)--(-0.5303,0.1441)--(-0.4596,0.2739)--(-0.3889,0.3896)--(-0.3182,0.4892)--(-0.2475,0.5710)--(-0.1768,0.6336)--(-0.1061,0.6760)--(-0.03535,0.6973)--(0.03535,0.6973)--(0.1061,0.6760)--(0.1768,0.6336)--(0.2475,0.5710)--(0.3182,0.4892)--(0.3889,0.3896)--(0.4596,0.2739)--(0.5303,0.1441)--(0.6010,0.002540)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6299)--(0.9545,-0.7902)--(1.025,-0.9453)--(1.096,-1.092)--(1.167,-1.228)--(1.237,-1.350)--(1.308,-1.456)--(1.379,-1.542)--(1.449,-1.608)--(1.520,-1.650)--(1.591,-1.667)--(1.662,-1.658)--(1.732,-1.622)--(1.803,-1.558)--(1.874,-1.466)--(1.944,-1.346)--(2.015,-1.199)--(2.086,-1.027)--(2.157,-0.8297)--(2.227,-0.6099)--(2.298,-0.3696)--(2.369,-0.1114)--(2.439,0.1618)--(2.510,0.4468)--(2.581,0.7401)--(2.652,1.038)--(2.722,1.337)--(2.793,1.632)--(2.864,1.921)--(2.934,2.198)--(3.005,2.459)--(3.076,2.702)--(3.146,2.921)--(3.217,3.114)--(3.288,3.277)--(3.359,3.406)--(3.429,3.500)--(3.500,3.555);
-\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
-\draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
-\draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=red] (-3.5000,3.5547)--(-3.4292,3.4996)--(-3.3585,3.4061)--(-3.2878,3.2766)--(-3.2171,3.1140)--(-3.1464,2.9213)--(-3.0757,2.7019)--(-3.0050,2.4594)--(-2.9343,2.1976)--(-2.8636,1.9206)--(-2.7929,1.6322)--(-2.7222,1.3367)--(-2.6515,1.0379)--(-2.5808,0.7400)--(-2.5101,0.4467)--(-2.4393,0.1618)--(-2.3686,-0.1113)--(-2.2979,-0.3695)--(-2.2272,-0.6098)--(-2.1565,-0.8297)--(-2.0858,-1.0268)--(-2.0151,-1.1994)--(-1.9444,-1.3460)--(-1.8737,-1.4656)--(-1.8030,-1.5575)--(-1.7323,-1.6215)--(-1.6616,-1.6577)--(-1.5909,-1.6667)--(-1.5202,-1.6496)--(-1.4494,-1.6076)--(-1.3787,-1.5424)--(-1.3080,-1.4559)--(-1.2373,-1.3503)--(-1.1666,-1.2283)--(-1.0959,-1.0923)--(-1.0252,-0.9453)--(-0.9545,-0.7901)--(-0.8838,-0.6298)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.0025)--(-0.5303,0.1441)--(-0.4595,0.2739)--(-0.3888,0.3896)--(-0.3181,0.4892)--(-0.2474,0.5710)--(-0.1767,0.6336)--(-0.1060,0.6759)--(-0.0353,0.6973)--(0.0353,0.6973)--(0.1060,0.6759)--(0.1767,0.6336)--(0.2474,0.5710)--(0.3181,0.4892)--(0.3888,0.3896)--(0.4595,0.2739)--(0.5303,0.1441)--(0.6010,0.0025)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6298)--(0.9545,-0.7901)--(1.0252,-0.9453)--(1.0959,-1.0923)--(1.1666,-1.2283)--(1.2373,-1.3503)--(1.3080,-1.4559)--(1.3787,-1.5424)--(1.4494,-1.6076)--(1.5202,-1.6496)--(1.5909,-1.6667)--(1.6616,-1.6577)--(1.7323,-1.6215)--(1.8030,-1.5575)--(1.8737,-1.4656)--(1.9444,-1.3460)--(2.0151,-1.1994)--(2.0858,-1.0268)--(2.1565,-0.8297)--(2.2272,-0.6098)--(2.2979,-0.3695)--(2.3686,-0.1113)--(2.4393,0.1618)--(2.5101,0.4467)--(2.5808,0.7400)--(2.6515,1.0379)--(2.7222,1.3367)--(2.7929,1.6322)--(2.8636,1.9206)--(2.9343,2.1976)--(3.0050,2.4594)--(3.0757,2.7019)--(3.1464,2.9213)--(3.2171,3.1140)--(3.2878,3.2766)--(3.3585,3.4061)--(3.4292,3.4996)--(3.5000,3.5547);
+\draw (-3.2986,-0.4207) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (-3.2986,-0.1000) -- (-3.2986,0.1000);
+\draw (-2.1991,-0.3210) node {$-\mathit{\pi}$};
+\draw [] (-2.1991,-0.1000) -- (-2.1991,0.1000);
+\draw (-1.0995,-0.4207) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (-1.0995,-0.1000) -- (-1.0995,0.1000);
+\draw (1.0995,-0.4207) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (1.0995,-0.1000) -- (1.0995,0.1000);
+\draw (2.1991,-0.2785) node {$\mathit{\pi}$};
+\draw [] (2.1991,-0.1000) -- (2.1991,0.1000);
+\draw (3.2986,-0.4207) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (3.2986,-0.1000) -- (3.2986,0.1000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall b/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall
index 115bccffb..bec0dac16 100644
--- a/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall
+++ b/src_phystricks/Fig_GCNooKEbjWB.pstricks.recall
@@ -63,30 +63,30 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (0,0) -- (0,3.00);
-\draw [color=gray,style=solid] (3.00,0) -- (3.00,3.00);
-\draw [color=gray,style=dotted] (1.50,0) -- (1.50,3.00);
-\draw [color=gray,style=dotted] (0,1.50) -- (3.00,1.50);
-\draw [color=gray,style=solid] (0,0) -- (3.00,0);
-\draw [color=gray,style=solid] (0,3.00) -- (3.00,3.00);
+\draw [color=gray,style=solid] (0.0000,0.0000) -- (0.0000,3.0000);
+\draw [color=gray,style=solid] (3.0000,0.0000) -- (3.0000,3.0000);
+\draw [color=gray,style=dotted] (1.5000,0.0000) -- (1.5000,3.0000);
+\draw [color=gray,style=dotted] (0.0000,1.5000) -- (3.0000,1.5000);
+\draw [color=gray,style=solid] (0.0000,0.0000) -- (3.0000,0.0000);
+\draw [color=gray,style=solid] (0.0000,3.0000) -- (3.0000,3.0000);
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
-\draw [color=blue] (0,0)--(0.0152,0.0152)--(0.0303,0.0303)--(0.0455,0.0455)--(0.0606,0.0606)--(0.0758,0.0758)--(0.0909,0.0909)--(0.106,0.106)--(0.121,0.121)--(0.136,0.136)--(0.152,0.152)--(0.167,0.167)--(0.182,0.182)--(0.197,0.197)--(0.212,0.212)--(0.227,0.227)--(0.242,0.242)--(0.258,0.258)--(0.273,0.273)--(0.288,0.288)--(0.303,0.303)--(0.318,0.318)--(0.333,0.333)--(0.348,0.348)--(0.364,0.364)--(0.379,0.379)--(0.394,0.394)--(0.409,0.409)--(0.424,0.424)--(0.439,0.439)--(0.455,0.455)--(0.470,0.470)--(0.485,0.485)--(0.500,0.500)--(0.515,0.515)--(0.530,0.530)--(0.545,0.545)--(0.561,0.561)--(0.576,0.576)--(0.591,0.591)--(0.606,0.606)--(0.621,0.621)--(0.636,0.636)--(0.651,0.651)--(0.667,0.667)--(0.682,0.682)--(0.697,0.697)--(0.712,0.712)--(0.727,0.727)--(0.742,0.742)--(0.758,0.758)--(0.773,0.773)--(0.788,0.788)--(0.803,0.803)--(0.818,0.818)--(0.833,0.833)--(0.849,0.849)--(0.864,0.864)--(0.879,0.879)--(0.894,0.894)--(0.909,0.909)--(0.924,0.924)--(0.939,0.939)--(0.955,0.955)--(0.970,0.970)--(0.985,0.985)--(1.00,1.00)--(1.02,1.02)--(1.03,1.03)--(1.05,1.05)--(1.06,1.06)--(1.08,1.08)--(1.09,1.09)--(1.11,1.11)--(1.12,1.12)--(1.14,1.14)--(1.15,1.15)--(1.17,1.17)--(1.18,1.18)--(1.20,1.20)--(1.21,1.21)--(1.23,1.23)--(1.24,1.24)--(1.26,1.26)--(1.27,1.27)--(1.29,1.29)--(1.30,1.30)--(1.32,1.32)--(1.33,1.33)--(1.35,1.35)--(1.36,1.36)--(1.38,1.38)--(1.39,1.39)--(1.41,1.41)--(1.42,1.42)--(1.44,1.44)--(1.45,1.45)--(1.47,1.47)--(1.48,1.48)--(1.50,1.50);
+\draw [color=blue] (0.0000,0.0000)--(0.0151,0.0151)--(0.0303,0.0303)--(0.0454,0.0454)--(0.0606,0.0606)--(0.0757,0.0757)--(0.0909,0.0909)--(0.1060,0.1060)--(0.1212,0.1212)--(0.1363,0.1363)--(0.1515,0.1515)--(0.1666,0.1666)--(0.1818,0.1818)--(0.1969,0.1969)--(0.2121,0.2121)--(0.2272,0.2272)--(0.2424,0.2424)--(0.2575,0.2575)--(0.2727,0.2727)--(0.2878,0.2878)--(0.3030,0.3030)--(0.3181,0.3181)--(0.3333,0.3333)--(0.3484,0.3484)--(0.3636,0.3636)--(0.3787,0.3787)--(0.3939,0.3939)--(0.4090,0.4090)--(0.4242,0.4242)--(0.4393,0.4393)--(0.4545,0.4545)--(0.4696,0.4696)--(0.4848,0.4848)--(0.5000,0.5000)--(0.5151,0.5151)--(0.5303,0.5303)--(0.5454,0.5454)--(0.5606,0.5606)--(0.5757,0.5757)--(0.5909,0.5909)--(0.6060,0.6060)--(0.6212,0.6212)--(0.6363,0.6363)--(0.6515,0.6515)--(0.6666,0.6666)--(0.6818,0.6818)--(0.6969,0.6969)--(0.7121,0.7121)--(0.7272,0.7272)--(0.7424,0.7424)--(0.7575,0.7575)--(0.7727,0.7727)--(0.7878,0.7878)--(0.8030,0.8030)--(0.8181,0.8181)--(0.8333,0.8333)--(0.8484,0.8484)--(0.8636,0.8636)--(0.8787,0.8787)--(0.8939,0.8939)--(0.9090,0.9090)--(0.9242,0.9242)--(0.9393,0.9393)--(0.9545,0.9545)--(0.9696,0.9696)--(0.9848,0.9848)--(1.0000,1.0000)--(1.0151,1.0151)--(1.0303,1.0303)--(1.0454,1.0454)--(1.0606,1.0606)--(1.0757,1.0757)--(1.0909,1.0909)--(1.1060,1.1060)--(1.1212,1.1212)--(1.1363,1.1363)--(1.1515,1.1515)--(1.1666,1.1666)--(1.1818,1.1818)--(1.1969,1.1969)--(1.2121,1.2121)--(1.2272,1.2272)--(1.2424,1.2424)--(1.2575,1.2575)--(1.2727,1.2727)--(1.2878,1.2878)--(1.3030,1.3030)--(1.3181,1.3181)--(1.3333,1.3333)--(1.3484,1.3484)--(1.3636,1.3636)--(1.3787,1.3787)--(1.3939,1.3939)--(1.4090,1.4090)--(1.4242,1.4242)--(1.4393,1.4393)--(1.4545,1.4545)--(1.4696,1.4696)--(1.4848,1.4848)--(1.5000,1.5000);
 
-\draw [color=blue] (1.50,3.00)--(1.52,2.98)--(1.53,2.97)--(1.55,2.95)--(1.56,2.94)--(1.58,2.92)--(1.59,2.91)--(1.61,2.89)--(1.62,2.88)--(1.64,2.86)--(1.65,2.85)--(1.67,2.83)--(1.68,2.82)--(1.70,2.80)--(1.71,2.79)--(1.73,2.77)--(1.74,2.76)--(1.76,2.74)--(1.77,2.73)--(1.79,2.71)--(1.80,2.70)--(1.82,2.68)--(1.83,2.67)--(1.85,2.65)--(1.86,2.64)--(1.88,2.62)--(1.89,2.61)--(1.91,2.59)--(1.92,2.58)--(1.94,2.56)--(1.95,2.55)--(1.97,2.53)--(1.98,2.52)--(2.00,2.50)--(2.02,2.48)--(2.03,2.47)--(2.05,2.45)--(2.06,2.44)--(2.08,2.42)--(2.09,2.41)--(2.11,2.39)--(2.12,2.38)--(2.14,2.36)--(2.15,2.35)--(2.17,2.33)--(2.18,2.32)--(2.20,2.30)--(2.21,2.29)--(2.23,2.27)--(2.24,2.26)--(2.26,2.24)--(2.27,2.23)--(2.29,2.21)--(2.30,2.20)--(2.32,2.18)--(2.33,2.17)--(2.35,2.15)--(2.36,2.14)--(2.38,2.12)--(2.39,2.11)--(2.41,2.09)--(2.42,2.08)--(2.44,2.06)--(2.45,2.05)--(2.47,2.03)--(2.48,2.02)--(2.50,2.00)--(2.52,1.98)--(2.53,1.97)--(2.55,1.95)--(2.56,1.94)--(2.58,1.92)--(2.59,1.91)--(2.61,1.89)--(2.62,1.88)--(2.64,1.86)--(2.65,1.85)--(2.67,1.83)--(2.68,1.82)--(2.70,1.80)--(2.71,1.79)--(2.73,1.77)--(2.74,1.76)--(2.76,1.74)--(2.77,1.73)--(2.79,1.71)--(2.80,1.70)--(2.82,1.68)--(2.83,1.67)--(2.85,1.65)--(2.86,1.64)--(2.88,1.62)--(2.89,1.61)--(2.91,1.59)--(2.92,1.58)--(2.94,1.56)--(2.95,1.55)--(2.97,1.53)--(2.98,1.52)--(3.00,1.50);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
+\draw [color=blue] (1.5000,3.0000)--(1.5151,2.9848)--(1.5303,2.9696)--(1.5454,2.9545)--(1.5606,2.9393)--(1.5757,2.9242)--(1.5909,2.9090)--(1.6060,2.8939)--(1.6212,2.8787)--(1.6363,2.8636)--(1.6515,2.8484)--(1.6666,2.8333)--(1.6818,2.8181)--(1.6969,2.8030)--(1.7121,2.7878)--(1.7272,2.7727)--(1.7424,2.7575)--(1.7575,2.7424)--(1.7727,2.7272)--(1.7878,2.7121)--(1.8030,2.6969)--(1.8181,2.6818)--(1.8333,2.6666)--(1.8484,2.6515)--(1.8636,2.6363)--(1.8787,2.6212)--(1.8939,2.6060)--(1.9090,2.5909)--(1.9242,2.5757)--(1.9393,2.5606)--(1.9545,2.5454)--(1.9696,2.5303)--(1.9848,2.5151)--(2.0000,2.5000)--(2.0151,2.4848)--(2.0303,2.4696)--(2.0454,2.4545)--(2.0606,2.4393)--(2.0757,2.4242)--(2.0909,2.4090)--(2.1060,2.3939)--(2.1212,2.3787)--(2.1363,2.3636)--(2.1515,2.3484)--(2.1666,2.3333)--(2.1818,2.3181)--(2.1969,2.3030)--(2.2121,2.2878)--(2.2272,2.2727)--(2.2424,2.2575)--(2.2575,2.2424)--(2.2727,2.2272)--(2.2878,2.2121)--(2.3030,2.1969)--(2.3181,2.1818)--(2.3333,2.1666)--(2.3484,2.1515)--(2.3636,2.1363)--(2.3787,2.1212)--(2.3939,2.1060)--(2.4090,2.0909)--(2.4242,2.0757)--(2.4393,2.0606)--(2.4545,2.0454)--(2.4696,2.0303)--(2.4848,2.0151)--(2.5000,2.0000)--(2.5151,1.9848)--(2.5303,1.9696)--(2.5454,1.9545)--(2.5606,1.9393)--(2.5757,1.9242)--(2.5909,1.9090)--(2.6060,1.8939)--(2.6212,1.8787)--(2.6363,1.8636)--(2.6515,1.8484)--(2.6666,1.8333)--(2.6818,1.8181)--(2.6969,1.8030)--(2.7121,1.7878)--(2.7272,1.7727)--(2.7424,1.7575)--(2.7575,1.7424)--(2.7727,1.7272)--(2.7878,1.7121)--(2.8030,1.6969)--(2.8181,1.6818)--(2.8333,1.6666)--(2.8484,1.6515)--(2.8636,1.6363)--(2.8787,1.6212)--(2.8939,1.6060)--(2.9090,1.5909)--(2.9242,1.5757)--(2.9393,1.5606)--(2.9545,1.5454)--(2.9696,1.5303)--(2.9848,1.5151)--(3.0000,1.5000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.5000,1.5000) node [rotate=0] {$o$};
 \draw []  (1.5000,3.0000) node [rotate=0] {$\bullet$};
 \draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw [style=dashed] (1.50,1.50) -- (1.50,3.00);
+\draw [style=dashed] (1.5000,1.5000) -- (1.5000,3.0000);
 
-\draw (3.0000,-0.31492) node {$ 1 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall b/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall
index 83e1fe1b2..379aaa659 100644
--- a/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall
+++ b/src_phystricks/Fig_GMIooJvcCXg.pstricks.recall
@@ -95,53 +95,53 @@
 \begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
 %PSTRICKS CODE
 %GRID
-\draw [color=gray,style=solid] (-1.00,-1.57) -- (-1.00,4.71);
-\draw [color=gray,style=solid] (0,-1.57) -- (0,4.71);
-\draw [color=gray,style=solid] (1.00,-1.57) -- (1.00,4.71);
-\draw [color=gray,style=solid] (2.00,-1.57) -- (2.00,4.71);
-\draw [color=gray,style=solid] (3.00,-1.57) -- (3.00,4.71);
-\draw [color=gray,style=solid] (4.00,-1.57) -- (4.00,4.71);
-\draw [color=gray,style=dotted] (-0.500,-1.57) -- (-0.500,4.71);
-\draw [color=gray,style=dotted] (0.500,-1.57) -- (0.500,4.71);
-\draw [color=gray,style=dotted] (1.50,-1.57) -- (1.50,4.71);
-\draw [color=gray,style=dotted] (2.50,-1.57) -- (2.50,4.71);
-\draw [color=gray,style=dotted] (3.50,-1.57) -- (3.50,4.71);
-\draw [color=gray,style=dotted] (-1.00,-0.785) -- (4.00,-0.785);
-\draw [color=gray,style=dotted] (-1.00,0.785) -- (4.00,0.785);
-\draw [color=gray,style=dotted] (-1.00,2.36) -- (4.00,2.36);
-\draw [color=gray,style=dotted] (-1.00,3.93) -- (4.00,3.93);
-\draw [color=gray,style=solid] (-1.00,-1.57) -- (4.00,-1.57);
-\draw [color=gray,style=solid] (-1.00,0) -- (4.00,0);
-\draw [color=gray,style=solid] (-1.00,1.57) -- (4.00,1.57);
-\draw [color=gray,style=solid] (-1.00,3.14) -- (4.00,3.14);
-\draw [color=gray,style=solid] (-1.00,4.71) -- (4.00,4.71);
+\draw [color=gray,style=solid] (-1.0000,-1.5707) -- (-1.0000,4.7123);
+\draw [color=gray,style=solid] (0.0000,-1.5707) -- (0.0000,4.7123);
+\draw [color=gray,style=solid] (1.0000,-1.5707) -- (1.0000,4.7123);
+\draw [color=gray,style=solid] (2.0000,-1.5707) -- (2.0000,4.7123);
+\draw [color=gray,style=solid] (3.0000,-1.5707) -- (3.0000,4.7123);
+\draw [color=gray,style=solid] (4.0000,-1.5707) -- (4.0000,4.7123);
+\draw [color=gray,style=dotted] (-0.5000,-1.5707) -- (-0.5000,4.7123);
+\draw [color=gray,style=dotted] (0.5000,-1.5707) -- (0.5000,4.7123);
+\draw [color=gray,style=dotted] (1.5000,-1.5707) -- (1.5000,4.7123);
+\draw [color=gray,style=dotted] (2.5000,-1.5707) -- (2.5000,4.7123);
+\draw [color=gray,style=dotted] (3.5000,-1.5707) -- (3.5000,4.7123);
+\draw [color=gray,style=dotted] (-1.0000,-0.7853) -- (4.0000,-0.7853);
+\draw [color=gray,style=dotted] (-1.0000,0.7853) -- (4.0000,0.7853);
+\draw [color=gray,style=dotted] (-1.0000,2.3561) -- (4.0000,2.3561);
+\draw [color=gray,style=dotted] (-1.0000,3.9269) -- (4.0000,3.9269);
+\draw [color=gray,style=solid] (-1.0000,-1.5707) -- (4.0000,-1.5707);
+\draw [color=gray,style=solid] (-1.0000,0.0000) -- (4.0000,0.0000);
+\draw [color=gray,style=solid] (-1.0000,1.5707) -- (4.0000,1.5707);
+\draw [color=gray,style=solid] (-1.0000,3.1415) -- (4.0000,3.1415);
+\draw [color=gray,style=solid] (-1.0000,4.7123) -- (4.0000,4.7123);
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-2.0708) -- (0,5.2124);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.0707) -- (0.0000,5.2123);
 %DEFAULT
 
-\draw [color=blue] (0,1.000)--(0.03173,0.9995)--(0.06347,0.9980)--(0.09520,0.9955)--(0.1269,0.9920)--(0.1587,0.9874)--(0.1904,0.9819)--(0.2221,0.9754)--(0.2539,0.9679)--(0.2856,0.9595)--(0.3173,0.9501)--(0.3491,0.9397)--(0.3808,0.9284)--(0.4125,0.9161)--(0.4443,0.9029)--(0.4760,0.8888)--(0.5077,0.8738)--(0.5395,0.8580)--(0.5712,0.8413)--(0.6029,0.8237)--(0.6347,0.8053)--(0.6664,0.7861)--(0.6981,0.7660)--(0.7299,0.7453)--(0.7616,0.7237)--(0.7933,0.7015)--(0.8251,0.6785)--(0.8568,0.6549)--(0.8885,0.6306)--(0.9203,0.6056)--(0.9520,0.5801)--(0.9837,0.5539)--(1.015,0.5272)--(1.047,0.5000)--(1.079,0.4723)--(1.111,0.4441)--(1.142,0.4154)--(1.174,0.3863)--(1.206,0.3569)--(1.238,0.3271)--(1.269,0.2969)--(1.301,0.2665)--(1.333,0.2358)--(1.365,0.2048)--(1.396,0.1736)--(1.428,0.1423)--(1.460,0.1108)--(1.491,0.07925)--(1.523,0.04758)--(1.555,0.01587)--(1.587,-0.01587)--(1.618,-0.04758)--(1.650,-0.07925)--(1.682,-0.1108)--(1.714,-0.1423)--(1.745,-0.1736)--(1.777,-0.2048)--(1.809,-0.2358)--(1.841,-0.2665)--(1.872,-0.2969)--(1.904,-0.3271)--(1.936,-0.3569)--(1.967,-0.3863)--(1.999,-0.4154)--(2.031,-0.4441)--(2.063,-0.4723)--(2.094,-0.5000)--(2.126,-0.5272)--(2.158,-0.5539)--(2.190,-0.5801)--(2.221,-0.6056)--(2.253,-0.6306)--(2.285,-0.6549)--(2.317,-0.6785)--(2.348,-0.7015)--(2.380,-0.7237)--(2.412,-0.7453)--(2.443,-0.7660)--(2.475,-0.7861)--(2.507,-0.8053)--(2.539,-0.8237)--(2.570,-0.8413)--(2.602,-0.8580)--(2.634,-0.8738)--(2.666,-0.8888)--(2.697,-0.9029)--(2.729,-0.9161)--(2.761,-0.9284)--(2.793,-0.9397)--(2.824,-0.9501)--(2.856,-0.9595)--(2.888,-0.9679)--(2.919,-0.9754)--(2.951,-0.9819)--(2.983,-0.9874)--(3.015,-0.9920)--(3.046,-0.9955)--(3.078,-0.9980)--(3.110,-0.9995)--(3.142,-1.000);
+\draw [color=blue] (0.0000,1.0000)--(0.0317,0.9994)--(0.0634,0.9979)--(0.0951,0.9954)--(0.1269,0.9919)--(0.1586,0.9874)--(0.1903,0.9819)--(0.2221,0.9754)--(0.2538,0.9679)--(0.2855,0.9594)--(0.3173,0.9500)--(0.3490,0.9396)--(0.3807,0.9283)--(0.4125,0.9161)--(0.4442,0.9029)--(0.4759,0.8888)--(0.5077,0.8738)--(0.5394,0.8579)--(0.5711,0.8412)--(0.6029,0.8236)--(0.6346,0.8052)--(0.6663,0.7860)--(0.6981,0.7660)--(0.7298,0.7452)--(0.7615,0.7237)--(0.7933,0.7014)--(0.8250,0.6785)--(0.8567,0.6548)--(0.8885,0.6305)--(0.9202,0.6056)--(0.9519,0.5800)--(0.9837,0.5539)--(1.0154,0.5272)--(1.0471,0.5000)--(1.0789,0.4722)--(1.1106,0.4440)--(1.1423,0.4154)--(1.1741,0.3863)--(1.2058,0.3568)--(1.2375,0.3270)--(1.2693,0.2969)--(1.3010,0.2664)--(1.3327,0.2357)--(1.3645,0.2048)--(1.3962,0.1736)--(1.4279,0.1423)--(1.4597,0.1108)--(1.4914,0.0792)--(1.5231,0.0475)--(1.5549,0.0158)--(1.5866,-0.0158)--(1.6183,-0.0475)--(1.6501,-0.0792)--(1.6818,-0.1108)--(1.7135,-0.1423)--(1.7453,-0.1736)--(1.7770,-0.2048)--(1.8087,-0.2357)--(1.8405,-0.2664)--(1.8722,-0.2969)--(1.9039,-0.3270)--(1.9357,-0.3568)--(1.9674,-0.3863)--(1.9991,-0.4154)--(2.0309,-0.4440)--(2.0626,-0.4722)--(2.0943,-0.5000)--(2.1261,-0.5272)--(2.1578,-0.5539)--(2.1895,-0.5800)--(2.2213,-0.6056)--(2.2530,-0.6305)--(2.2847,-0.6548)--(2.3165,-0.6785)--(2.3482,-0.7014)--(2.3799,-0.7237)--(2.4117,-0.7452)--(2.4434,-0.7660)--(2.4751,-0.7860)--(2.5069,-0.8052)--(2.5386,-0.8236)--(2.5703,-0.8412)--(2.6021,-0.8579)--(2.6338,-0.8738)--(2.6655,-0.8888)--(2.6973,-0.9029)--(2.7290,-0.9161)--(2.7607,-0.9283)--(2.7925,-0.9396)--(2.8242,-0.9500)--(2.8559,-0.9594)--(2.8877,-0.9679)--(2.9194,-0.9754)--(2.9511,-0.9819)--(2.9829,-0.9874)--(3.0146,-0.9919)--(3.0463,-0.9954)--(3.0781,-0.9979)--(3.1098,-0.9994)--(3.1415,-1.0000);
 
-\draw [color=blue] (-1.000,3.142)--(-0.9798,2.940)--(-0.9596,2.856)--(-0.9394,2.792)--(-0.9192,2.737)--(-0.8990,2.688)--(-0.8788,2.644)--(-0.8586,2.603)--(-0.8384,2.565)--(-0.8182,2.529)--(-0.7980,2.495)--(-0.7778,2.462)--(-0.7576,2.430)--(-0.7374,2.400)--(-0.7172,2.371)--(-0.6970,2.342)--(-0.6768,2.314)--(-0.6566,2.287)--(-0.6364,2.261)--(-0.6162,2.235)--(-0.5960,2.209)--(-0.5758,2.184)--(-0.5556,2.160)--(-0.5354,2.136)--(-0.5152,2.112)--(-0.4949,2.089)--(-0.4747,2.065)--(-0.4545,2.043)--(-0.4343,2.020)--(-0.4141,1.998)--(-0.3939,1.976)--(-0.3737,1.954)--(-0.3535,1.932)--(-0.3333,1.911)--(-0.3131,1.889)--(-0.2929,1.868)--(-0.2727,1.847)--(-0.2525,1.826)--(-0.2323,1.805)--(-0.2121,1.785)--(-0.1919,1.764)--(-0.1717,1.743)--(-0.1515,1.723)--(-0.1313,1.702)--(-0.1111,1.682)--(-0.09091,1.662)--(-0.07071,1.642)--(-0.05051,1.621)--(-0.03030,1.601)--(-0.01010,1.581)--(0.01010,1.561)--(0.03030,1.540)--(0.05051,1.520)--(0.07071,1.500)--(0.09091,1.480)--(0.1111,1.459)--(0.1313,1.439)--(0.1515,1.419)--(0.1717,1.398)--(0.1919,1.378)--(0.2121,1.357)--(0.2323,1.336)--(0.2525,1.316)--(0.2727,1.295)--(0.2929,1.274)--(0.3131,1.252)--(0.3333,1.231)--(0.3535,1.209)--(0.3737,1.188)--(0.3939,1.166)--(0.4141,1.144)--(0.4343,1.121)--(0.4545,1.099)--(0.4747,1.076)--(0.4949,1.053)--(0.5152,1.030)--(0.5354,1.006)--(0.5556,0.9818)--(0.5758,0.9573)--(0.5960,0.9323)--(0.6162,0.9069)--(0.6364,0.8810)--(0.6566,0.8545)--(0.6768,0.8274)--(0.6970,0.7996)--(0.7172,0.7711)--(0.7374,0.7416)--(0.7576,0.7112)--(0.7778,0.6797)--(0.7980,0.6469)--(0.8182,0.6126)--(0.8384,0.5765)--(0.8586,0.5383)--(0.8788,0.4975)--(0.8990,0.4533)--(0.9192,0.4048)--(0.9394,0.3499)--(0.9596,0.2852)--(0.9798,0.2013)--(1.000,0);
+\draw [color=blue] (-1.0000,3.1415)--(-0.9797,2.9402)--(-0.9595,2.8563)--(-0.9393,2.7916)--(-0.9191,2.7368)--(-0.8989,2.6882)--(-0.8787,2.6441)--(-0.8585,2.6033)--(-0.8383,2.5651)--(-0.8181,2.5290)--(-0.7979,2.4947)--(-0.7777,2.4619)--(-0.7575,2.4303)--(-0.7373,2.3999)--(-0.7171,2.3705)--(-0.6969,2.3419)--(-0.6767,2.3141)--(-0.6565,2.2870)--(-0.6363,2.2605)--(-0.6161,2.2346)--(-0.5959,2.2092)--(-0.5757,2.1843)--(-0.5555,2.1598)--(-0.5353,2.1357)--(-0.5151,2.1119)--(-0.4949,2.0885)--(-0.4747,2.0654)--(-0.4545,2.0426)--(-0.4343,2.0201)--(-0.4141,1.9977)--(-0.3939,1.9757)--(-0.3737,1.9538)--(-0.3535,1.9321)--(-0.3333,1.9106)--(-0.3131,1.8892)--(-0.2929,1.8680)--(-0.2727,1.8470)--(-0.2525,1.8260)--(-0.2323,1.8052)--(-0.2121,1.7845)--(-0.1919,1.7639)--(-0.1717,1.7433)--(-0.1515,1.7228)--(-0.1313,1.7024)--(-0.1111,1.6821)--(-0.0909,1.6618)--(-0.0707,1.6415)--(-0.0505,1.6213)--(-0.0303,1.6011)--(-0.0101,1.5808)--(0.0101,1.5606)--(0.0303,1.5404)--(0.0505,1.5202)--(0.0707,1.5000)--(0.0909,1.4797)--(0.1111,1.4594)--(0.1313,1.4391)--(0.1515,1.4186)--(0.1717,1.3982)--(0.1919,1.3776)--(0.2121,1.3570)--(0.2323,1.3363)--(0.2525,1.3155)--(0.2727,1.2945)--(0.2929,1.2735)--(0.3131,1.2523)--(0.3333,1.2309)--(0.3535,1.2094)--(0.3737,1.1877)--(0.3939,1.1658)--(0.4141,1.1437)--(0.4343,1.1214)--(0.4545,1.0989)--(0.4747,1.0761)--(0.4949,1.0530)--(0.5151,1.0296)--(0.5353,1.0058)--(0.5555,0.9817)--(0.5757,0.9572)--(0.5959,0.9323)--(0.6161,0.9069)--(0.6363,0.8810)--(0.6565,0.8545)--(0.6767,0.8274)--(0.6969,0.7996)--(0.7171,0.7710)--(0.7373,0.7416)--(0.7575,0.7112)--(0.7777,0.6796)--(0.7979,0.6468)--(0.8181,0.6125)--(0.8383,0.5764)--(0.8585,0.5382)--(0.8787,0.4974)--(0.8989,0.4533)--(0.9191,0.4047)--(0.9393,0.3499)--(0.9595,0.2852)--(0.9797,0.2013)--(1.0000,0.0000);
 
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.59374,-1.5708) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-0.100,-1.57) -- (0.100,-1.57);
-\draw (-0.45183,1.5708) node {$ \frac{1}{2} \, \pi $};
-\draw [] (-0.100,1.57) -- (0.100,1.57);
-\draw (-0.31058,3.1416) node {$ \pi $};
-\draw [] (-0.100,3.14) -- (0.100,3.14);
-\draw (-0.45183,4.7124) node {$ \frac{3}{2} \, \pi $};
-\draw [] (-0.100,4.71) -- (0.100,4.71);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.5937,-1.5707) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-0.1000,-1.5707) -- (0.1000,-1.5707);
+\draw (-0.4518,1.5707) node {$ \frac{1}{2} \, \pi $};
+\draw [] (-0.1000,1.5707) -- (0.1000,1.5707);
+\draw (-0.3105,3.1415) node {$ \pi $};
+\draw [] (-0.1000,3.1415) -- (0.1000,3.1415);
+\draw (-0.4518,4.7123) node {$ \frac{3}{2} \, \pi $};
+\draw [] (-0.1000,4.7123) -- (0.1000,4.7123);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall b/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall
index 02a0a4b23..63d8b55e5 100644
--- a/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall
+++ b/src_phystricks/Fig_GVDJooYzMxLW.pstricks.recall
@@ -89,24 +89,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (2.00,3.46) -- (0,0);
-\draw [] (2.00,3.46) -- (4.00,0);
-\draw [] (4.00,0) -- (0,0);
-\draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.2518,0.36492) node {$60$};
+\draw [] (2.0000,3.4641) -- (0.0000,0.0000);
+\draw [] (2.0000,3.4641) -- (4.0000,0.0000);
+\draw [] (4.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=blue,style=dotted] (2.0000,3.4641) -- (2.0000,0.0000);
+\draw (3.2517,0.3649) node {$60$};
 
-\draw [color=red] (3.75,0.433)--(3.75,0.430)--(3.74,0.428)--(3.74,0.425)--(3.73,0.422)--(3.73,0.419)--(3.72,0.416)--(3.72,0.413)--(3.71,0.410)--(3.71,0.407)--(3.71,0.404)--(3.70,0.401)--(3.70,0.398)--(3.69,0.395)--(3.69,0.391)--(3.68,0.388)--(3.68,0.385)--(3.68,0.381)--(3.67,0.378)--(3.67,0.374)--(3.66,0.371)--(3.66,0.367)--(3.66,0.364)--(3.65,0.360)--(3.65,0.356)--(3.65,0.353)--(3.64,0.349)--(3.64,0.345)--(3.63,0.341)--(3.63,0.337)--(3.63,0.333)--(3.62,0.329)--(3.62,0.325)--(3.62,0.321)--(3.61,0.317)--(3.61,0.313)--(3.61,0.309)--(3.60,0.305)--(3.60,0.301)--(3.60,0.296)--(3.59,0.292)--(3.59,0.288)--(3.59,0.284)--(3.59,0.279)--(3.58,0.275)--(3.58,0.270)--(3.58,0.266)--(3.57,0.261)--(3.57,0.257)--(3.57,0.252)--(3.57,0.248)--(3.56,0.243)--(3.56,0.238)--(3.56,0.234)--(3.56,0.229)--(3.55,0.224)--(3.55,0.220)--(3.55,0.215)--(3.55,0.210)--(3.54,0.205)--(3.54,0.200)--(3.54,0.196)--(3.54,0.191)--(3.54,0.186)--(3.53,0.181)--(3.53,0.176)--(3.53,0.171)--(3.53,0.166)--(3.53,0.161)--(3.52,0.156)--(3.52,0.151)--(3.52,0.146)--(3.52,0.141)--(3.52,0.136)--(3.52,0.131)--(3.52,0.126)--(3.51,0.120)--(3.51,0.115)--(3.51,0.110)--(3.51,0.105)--(3.51,0.0998)--(3.51,0.0946)--(3.51,0.0894)--(3.51,0.0842)--(3.51,0.0790)--(3.51,0.0738)--(3.50,0.0685)--(3.50,0.0633)--(3.50,0.0580)--(3.50,0.0528)--(3.50,0.0475)--(3.50,0.0423)--(3.50,0.0370)--(3.50,0.0317)--(3.50,0.0264)--(3.50,0.0211)--(3.50,0.0159)--(3.50,0.0106)--(3.50,0.00529)--(3.50,0);
+\draw [color=red] (3.7500,0.4330)--(3.7454,0.4303)--(3.7408,0.4276)--(3.7363,0.4248)--(3.7319,0.4220)--(3.7274,0.4191)--(3.7230,0.4162)--(3.7186,0.4133)--(3.7142,0.4103)--(3.7099,0.4072)--(3.7056,0.4041)--(3.7014,0.4010)--(3.6971,0.3978)--(3.6930,0.3946)--(3.6888,0.3913)--(3.6847,0.3880)--(3.6806,0.3847)--(3.6765,0.3813)--(3.6725,0.3778)--(3.6685,0.3743)--(3.6646,0.3708)--(3.6607,0.3672)--(3.6568,0.3636)--(3.6530,0.3600)--(3.6492,0.3563)--(3.6455,0.3526)--(3.6418,0.3488)--(3.6381,0.3450)--(3.6345,0.3411)--(3.6309,0.3373)--(3.6273,0.3333)--(3.6238,0.3294)--(3.6203,0.3254)--(3.6169,0.3213)--(3.6135,0.3173)--(3.6102,0.3132)--(3.6069,0.3090)--(3.6037,0.3049)--(3.6005,0.3006)--(3.5973,0.2964)--(3.5942,0.2921)--(3.5911,0.2878)--(3.5881,0.2835)--(3.5851,0.2791)--(3.5822,0.2747)--(3.5793,0.2703)--(3.5765,0.2658)--(3.5737,0.2613)--(3.5710,0.2568)--(3.5683,0.2522)--(3.5656,0.2477)--(3.5630,0.2430)--(3.5605,0.2384)--(3.5580,0.2338)--(3.5555,0.2291)--(3.5531,0.2243)--(3.5508,0.2196)--(3.5485,0.2148)--(3.5462,0.2101)--(3.5440,0.2052)--(3.5419,0.2004)--(3.5398,0.1956)--(3.5378,0.1907)--(3.5358,0.1858)--(3.5338,0.1809)--(3.5319,0.1759)--(3.5301,0.1710)--(3.5283,0.1660)--(3.5266,0.1610)--(3.5249,0.1560)--(3.5233,0.1509)--(3.5217,0.1459)--(3.5202,0.1408)--(3.5187,0.1357)--(3.5173,0.1306)--(3.5160,0.1255)--(3.5147,0.1204)--(3.5134,0.1153)--(3.5122,0.1101)--(3.5111,0.1049)--(3.5100,0.0998)--(3.5090,0.0946)--(3.5080,0.0894)--(3.5071,0.0842)--(3.5062,0.0790)--(3.5054,0.0737)--(3.5047,0.0685)--(3.5040,0.0632)--(3.5033,0.0580)--(3.5027,0.0527)--(3.5022,0.0475)--(3.5017,0.0422)--(3.5013,0.0369)--(3.5010,0.0317)--(3.5006,0.0264)--(3.5004,0.0211)--(3.5002,0.0158)--(3.5001,0.0105)--(3.5000,0.0052)--(3.5000,0.0000);
 \draw (2.3119,2.2340) node {$30$};
 
-\draw [color=cyan] (2.00,2.96)--(2.00,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.01,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.02,2.96)--(2.03,2.96)--(2.03,2.96)--(2.03,2.97)--(2.03,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.04,2.97)--(2.05,2.97)--(2.05,2.97)--(2.05,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.06,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.07,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.08,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.09,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.97)--(2.10,2.98)--(2.11,2.98)--(2.11,2.98)--(2.11,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.12,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.13,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.98)--(2.14,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.15,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.16,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,2.99)--(2.17,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.18,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.19,3.00)--(2.20,3.00)--(2.20,3.00)--(2.20,3.01)--(2.20,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.21,3.01)--(2.22,3.01)--(2.22,3.01)--(2.22,3.02)--(2.22,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.23,3.02)--(2.24,3.02)--(2.24,3.02)--(2.24,3.03)--(2.24,3.03)--(2.25,3.03)--(2.25,3.03)--(2.25,3.03);
+\draw [color=cyan] (2.0000,2.9641)--(2.0026,2.9641)--(2.0052,2.9641)--(2.0079,2.9641)--(2.0105,2.9642)--(2.0132,2.9642)--(2.0158,2.9643)--(2.0185,2.9644)--(2.0211,2.9645)--(2.0237,2.9646)--(2.0264,2.9648)--(2.0290,2.9649)--(2.0317,2.9651)--(2.0343,2.9652)--(2.0369,2.9654)--(2.0396,2.9656)--(2.0422,2.9658)--(2.0448,2.9661)--(2.0475,2.9663)--(2.0501,2.9666)--(2.0527,2.9668)--(2.0554,2.9671)--(2.0580,2.9674)--(2.0606,2.9677)--(2.0632,2.9681)--(2.0659,2.9684)--(2.0685,2.9688)--(2.0711,2.9691)--(2.0737,2.9695)--(2.0763,2.9699)--(2.0790,2.9703)--(2.0816,2.9708)--(2.0842,2.9712)--(2.0868,2.9716)--(2.0894,2.9721)--(2.0920,2.9726)--(2.0946,2.9731)--(2.0972,2.9736)--(2.0998,2.9741)--(2.1024,2.9747)--(2.1049,2.9752)--(2.1075,2.9758)--(2.1101,2.9763)--(2.1127,2.9769)--(2.1153,2.9775)--(2.1178,2.9781)--(2.1204,2.9788)--(2.1230,2.9794)--(2.1255,2.9801)--(2.1281,2.9807)--(2.1306,2.9814)--(2.1332,2.9821)--(2.1357,2.9828)--(2.1383,2.9836)--(2.1408,2.9843)--(2.1434,2.9851)--(2.1459,2.9858)--(2.1484,2.9866)--(2.1509,2.9874)--(2.1535,2.9882)--(2.1560,2.9890)--(2.1585,2.9898)--(2.1610,2.9907)--(2.1635,2.9916)--(2.1660,2.9924)--(2.1685,2.9933)--(2.1710,2.9942)--(2.1734,2.9951)--(2.1759,2.9960)--(2.1784,2.9970)--(2.1809,2.9979)--(2.1833,2.9989)--(2.1858,2.9999)--(2.1882,3.0009)--(2.1907,3.0019)--(2.1931,3.0029)--(2.1956,3.0039)--(2.1980,3.0049)--(2.2004,3.0060)--(2.2028,3.0071)--(2.2052,3.0081)--(2.2077,3.0092)--(2.2101,3.0103)--(2.2125,3.0115)--(2.2148,3.0126)--(2.2172,3.0137)--(2.2196,3.0149)--(2.2220,3.0161)--(2.2243,3.0172)--(2.2267,3.0184)--(2.2291,3.0196)--(2.2314,3.0209)--(2.2338,3.0221)--(2.2361,3.0233)--(2.2384,3.0246)--(2.2407,3.0258)--(2.2430,3.0271)--(2.2454,3.0284)--(2.2477,3.0297)--(2.2500,3.0310);
 \draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
 \draw (2.0000,3.8888) node {$A$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.44758,0) node {$B$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.4435,0) node {$C$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.42471) node {$H$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.4475,0.0000) node {$B$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.4434,0.0000) node {$C$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.4247) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall b/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall
index bbb7404a6..1fcfa15f5 100644
--- a/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall
+++ b/src_phystricks/Fig_GWOYooRxHKSm.pstricks.recall
@@ -87,33 +87,33 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.8750,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,7.8519);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.8750,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,7.8518);
 %DEFAULT
-\draw [color=cyan] (2.12,0.354) -- (7.38,7.23);
-\draw [color=green,style=dashed] (6.50,6.09) -- (6.50,0);
-\draw [color=green,style=dashed] (3.00,1.50) -- (3.00,0);
-\draw [color=green,style=dashed] (6.50,6.09) -- (0,6.09);
-\draw [color=green,style=dashed] (3.00,1.50) -- (0,1.50);
-\draw [color=green,style=dashed] (3.00,1.50) -- (6.50,1.50);
+\draw [color=cyan] (2.1250,0.3535) -- (7.3750,7.2320);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (6.5000,0.0000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (3.0000,0.0000);
+\draw [color=green,style=dashed] (6.5000,6.0856) -- (0.0000,6.0856);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (0.0000,1.5000);
+\draw [color=green,style=dashed] (3.0000,1.5000) -- (6.5000,1.5000);
 
-\draw [color=blue] (1.000,1.019)--(1.061,1.022)--(1.121,1.026)--(1.182,1.031)--(1.242,1.036)--(1.303,1.041)--(1.364,1.047)--(1.424,1.053)--(1.485,1.061)--(1.545,1.068)--(1.606,1.077)--(1.667,1.086)--(1.727,1.095)--(1.788,1.106)--(1.848,1.117)--(1.909,1.129)--(1.970,1.142)--(2.030,1.155)--(2.091,1.169)--(2.152,1.184)--(2.212,1.200)--(2.273,1.217)--(2.333,1.235)--(2.394,1.254)--(2.455,1.274)--(2.515,1.295)--(2.576,1.316)--(2.636,1.339)--(2.697,1.363)--(2.758,1.388)--(2.818,1.414)--(2.879,1.442)--(2.939,1.470)--(3.000,1.500)--(3.061,1.531)--(3.121,1.563)--(3.182,1.597)--(3.242,1.631)--(3.303,1.667)--(3.364,1.705)--(3.424,1.744)--(3.485,1.784)--(3.545,1.825)--(3.606,1.868)--(3.667,1.913)--(3.727,1.959)--(3.788,2.006)--(3.848,2.056)--(3.909,2.106)--(3.970,2.158)--(4.030,2.212)--(4.091,2.268)--(4.151,2.325)--(4.212,2.384)--(4.273,2.445)--(4.333,2.507)--(4.394,2.571)--(4.455,2.637)--(4.515,2.705)--(4.576,2.774)--(4.636,2.846)--(4.697,2.919)--(4.758,2.994)--(4.818,3.071)--(4.879,3.151)--(4.939,3.232)--(5.000,3.315)--(5.061,3.400)--(5.121,3.487)--(5.182,3.577)--(5.242,3.668)--(5.303,3.762)--(5.364,3.857)--(5.424,3.955)--(5.485,4.056)--(5.545,4.158)--(5.606,4.263)--(5.667,4.370)--(5.727,4.479)--(5.788,4.591)--(5.849,4.705)--(5.909,4.821)--(5.970,4.940)--(6.030,5.061)--(6.091,5.185)--(6.151,5.311)--(6.212,5.439)--(6.273,5.571)--(6.333,5.704)--(6.394,5.841)--(6.455,5.980)--(6.515,6.121)--(6.576,6.266)--(6.636,6.412)--(6.697,6.562)--(6.758,6.715)--(6.818,6.870)--(6.879,7.028)--(6.939,7.188)--(7.000,7.352);
+\draw [color=blue] (1.0000,1.0185)--(1.0606,1.0220)--(1.1212,1.0261)--(1.1818,1.0305)--(1.2424,1.0355)--(1.3030,1.0409)--(1.3636,1.0469)--(1.4242,1.0535)--(1.4848,1.0606)--(1.5454,1.0683)--(1.6060,1.0767)--(1.6666,1.0857)--(1.7272,1.0954)--(1.7878,1.1058)--(1.8484,1.1169)--(1.9090,1.1288)--(1.9696,1.1415)--(2.0303,1.1549)--(2.0909,1.1692)--(2.1515,1.1844)--(2.2121,1.2004)--(2.2727,1.2173)--(2.3333,1.2352)--(2.3939,1.2540)--(2.4545,1.2738)--(2.5151,1.2946)--(2.5757,1.3164)--(2.6363,1.3393)--(2.6969,1.3632)--(2.7575,1.3883)--(2.8181,1.4144)--(2.8787,1.4418)--(2.9393,1.4703)--(3.0000,1.5000)--(3.0606,1.5309)--(3.1212,1.5630)--(3.1818,1.5965)--(3.2424,1.6312)--(3.3030,1.6673)--(3.3636,1.7047)--(3.4242,1.7435)--(3.4848,1.7837)--(3.5454,1.8253)--(3.6060,1.8683)--(3.6666,1.9128)--(3.7272,1.9589)--(3.7878,2.0064)--(3.8484,2.0555)--(3.9090,2.1061)--(3.9696,2.1584)--(4.0303,2.2123)--(4.0909,2.2678)--(4.1515,2.3250)--(4.2121,2.3839)--(4.2727,2.4445)--(4.3333,2.5068)--(4.3939,2.5709)--(4.4545,2.6368)--(4.5151,2.7046)--(4.5757,2.7741)--(4.6363,2.8456)--(4.6969,2.9189)--(4.7575,2.9941)--(4.8181,3.0713)--(4.8787,3.1505)--(4.9393,3.2316)--(5.0000,3.3148)--(5.0606,3.4000)--(5.1212,3.4872)--(5.1818,3.5766)--(5.2424,3.6681)--(5.3030,3.7617)--(5.3636,3.8574)--(5.4242,3.9554)--(5.4848,4.0556)--(5.5454,4.1580)--(5.6060,4.2627)--(5.6666,4.3696)--(5.7272,4.4789)--(5.7878,4.5905)--(5.8484,4.7045)--(5.9090,4.8209)--(5.9696,4.9396)--(6.0303,5.0609)--(6.0909,5.1845)--(6.1515,5.3107)--(6.2121,5.4394)--(6.2727,5.5706)--(6.3333,5.7043)--(6.3939,5.8407)--(6.4545,5.9797)--(6.5151,6.1212)--(6.5757,6.2655)--(6.6363,6.4124)--(6.6969,6.5621)--(6.7575,6.7145)--(6.8181,6.8696)--(6.8787,7.0275)--(6.9393,7.1882)--(7.0000,7.3518);
 \draw []  (3.0000,1.5000) node [rotate=0] {$\bullet$};
-\draw []  (3.0000,0) node [rotate=0] {$\bullet$};
-\draw (3.0000,-0.27858) node {$a$};
-\draw []  (0,1.5000) node [rotate=0] {$\bullet$};
-\draw (-0.44737,1.5000) node {$f(a)$};
+\draw []  (3.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.0000,-0.2785) node {$a$};
+\draw []  (0.0000,1.5000) node [rotate=0] {$\bullet$};
+\draw (-0.4473,1.5000) node {$f(a)$};
 \draw []  (6.5000,6.0856) node [rotate=0] {$\bullet$};
-\draw []  (6.5000,0) node [rotate=0] {$\bullet$};
-\draw (6.5000,-0.27858) node {$x$};
-\draw []  (0,6.0856) node [rotate=0] {$\bullet$};
-\draw (-0.45521,6.0856) node {$f(x)$};
+\draw []  (6.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.2785) node {$x$};
+\draw []  (0.0000,6.0856) node [rotate=0] {$\bullet$};
+\draw (-0.4552,6.0856) node {$f(x)$};
 \draw [,->,>=latex] (4.7500,1.3000) -- (3.0000,1.3000);
 \draw [,->,>=latex] (4.7500,1.3000) -- (6.5000,1.3000);
-\draw (4.7500,0.97897) node {$x-a$};
+\draw (4.7500,0.9789) node {$x-a$};
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,6.0856);
 \draw [,->,>=latex] (6.7000,3.7928) -- (6.7000,1.5000);
-\draw (7.8256,3.7928) node {$f(x)-f(a)$};
+\draw (7.8255,3.7928) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall b/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall
index 61cae4865..51d91f6b5 100644
--- a/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall
+++ b/src_phystricks/Fig_Grapheunsurunmoinsx.pstricks.recall
@@ -107,54 +107,54 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-5.2619) -- (0,5.2619);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-5.2619) -- (0.0000,5.2619);
 %DEFAULT
 
-\draw [color=red] (-4.000,0.2000)--(-3.952,0.2020)--(-3.903,0.2039)--(-3.855,0.2060)--(-3.806,0.2081)--(-3.758,0.2102)--(-3.710,0.2123)--(-3.661,0.2145)--(-3.613,0.2168)--(-3.565,0.2191)--(-3.516,0.2214)--(-3.468,0.2238)--(-3.419,0.2263)--(-3.371,0.2288)--(-3.323,0.2313)--(-3.274,0.2340)--(-3.226,0.2366)--(-3.177,0.2394)--(-3.129,0.2422)--(-3.081,0.2451)--(-3.032,0.2480)--(-2.984,0.2510)--(-2.936,0.2541)--(-2.887,0.2573)--(-2.839,0.2605)--(-2.790,0.2638)--(-2.742,0.2672)--(-2.694,0.2707)--(-2.645,0.2743)--(-2.597,0.2780)--(-2.548,0.2818)--(-2.500,0.2857)--(-2.452,0.2897)--(-2.403,0.2938)--(-2.355,0.2981)--(-2.307,0.3024)--(-2.258,0.3069)--(-2.210,0.3115)--(-2.161,0.3163)--(-2.113,0.3212)--(-2.065,0.3263)--(-2.016,0.3315)--(-1.968,0.3369)--(-1.919,0.3425)--(-1.871,0.3483)--(-1.823,0.3543)--(-1.774,0.3604)--(-1.726,0.3668)--(-1.678,0.3735)--(-1.629,0.3803)--(-1.581,0.3875)--(-1.532,0.3949)--(-1.484,0.4026)--(-1.436,0.4106)--(-1.387,0.4189)--(-1.339,0.4276)--(-1.291,0.4366)--(-1.242,0.4460)--(-1.194,0.4558)--(-1.145,0.4661)--(-1.097,0.4769)--(-1.049,0.4881)--(-1.000,0.5000)--(-0.9518,0.5123)--(-0.9034,0.5254)--(-0.8550,0.5391)--(-0.8067,0.5535)--(-0.7583,0.5687)--(-0.7099,0.5848)--(-0.6615,0.6019)--(-0.6131,0.6199)--(-0.5648,0.6391)--(-0.5164,0.6595)--(-0.4680,0.6812)--(-0.4196,0.7044)--(-0.3712,0.7293)--(-0.3228,0.7560)--(-0.2744,0.7847)--(-0.2261,0.8156)--(-0.1777,0.8491)--(-0.1293,0.8855)--(-0.08091,0.9251)--(-0.03253,0.9685)--(0.01586,1.016)--(0.06424,1.069)--(0.1126,1.127)--(0.1610,1.192)--(0.2094,1.265)--(0.2578,1.347)--(0.3062,1.441)--(0.3545,1.549)--(0.4029,1.675)--(0.4513,1.823)--(0.4997,1.999)--(0.5481,2.213)--(0.5965,2.478)--(0.6449,2.816)--(0.6932,3.260)--(0.7416,3.870)--(0.7900,4.762);
+\draw [color=red] (-4.0000,0.2000)--(-3.9516,0.2019)--(-3.9032,0.2039)--(-3.8548,0.2059)--(-3.8064,0.2080)--(-3.7580,0.2101)--(-3.7096,0.2123)--(-3.6613,0.2145)--(-3.6129,0.2167)--(-3.5645,0.2190)--(-3.5161,0.2214)--(-3.4677,0.2238)--(-3.4193,0.2262)--(-3.3710,0.2287)--(-3.3226,0.2313)--(-3.2742,0.2339)--(-3.2258,0.2366)--(-3.1774,0.2393)--(-3.1290,0.2421)--(-3.0807,0.2450)--(-3.0323,0.2479)--(-2.9839,0.2510)--(-2.9355,0.2540)--(-2.8871,0.2572)--(-2.8387,0.2604)--(-2.7904,0.2638)--(-2.7420,0.2672)--(-2.6936,0.2707)--(-2.6452,0.2743)--(-2.5968,0.2780)--(-2.5484,0.2818)--(-2.5001,0.2857)--(-2.4517,0.2897)--(-2.4033,0.2938)--(-2.3549,0.2980)--(-2.3065,0.3024)--(-2.2581,0.3069)--(-2.2097,0.3115)--(-2.1614,0.3163)--(-2.1130,0.3212)--(-2.0646,0.3263)--(-2.0162,0.3315)--(-1.9678,0.3369)--(-1.9194,0.3425)--(-1.8711,0.3482)--(-1.8227,0.3542)--(-1.7743,0.3604)--(-1.7259,0.3668)--(-1.6775,0.3734)--(-1.6291,0.3803)--(-1.5808,0.3874)--(-1.5324,0.3948)--(-1.4840,0.4025)--(-1.4356,0.4105)--(-1.3872,0.4188)--(-1.3388,0.4275)--(-1.2905,0.4365)--(-1.2421,0.4460)--(-1.1937,0.4558)--(-1.1453,0.4661)--(-1.0969,0.4768)--(-1.0485,0.4881)--(-1.0002,0.4999)--(-0.9518,0.5123)--(-0.9034,0.5253)--(-0.8550,0.5390)--(-0.8066,0.5535)--(-0.7582,0.5687)--(-0.7098,0.5848)--(-0.6615,0.6018)--(-0.6131,0.6199)--(-0.5647,0.6390)--(-0.5163,0.6594)--(-0.4679,0.6812)--(-0.4195,0.7044)--(-0.3712,0.7292)--(-0.3228,0.7559)--(-0.2744,0.7846)--(-0.2260,0.8156)--(-0.1776,0.8491)--(-0.1292,0.8855)--(-0.0809,0.9251)--(-0.0325,0.9684)--(0.0158,1.0161)--(0.0642,1.0686)--(0.1126,1.1269)--(0.1610,1.1919)--(0.2093,1.2648)--(0.2577,1.3473)--(0.3061,1.4412)--(0.3545,1.5492)--(0.4029,1.6748)--(0.4513,1.8225)--(0.4996,1.9987)--(0.5480,2.2127)--(0.5964,2.4780)--(0.6448,2.8156)--(0.6932,3.2597)--(0.7416,3.8702)--(0.7900,4.7619);
 
-\draw [color=blue] (1.210,-4.762)--(1.258,-3.870)--(1.307,-3.260)--(1.355,-2.816)--(1.404,-2.478)--(1.452,-2.213)--(1.500,-1.999)--(1.549,-1.823)--(1.597,-1.675)--(1.645,-1.549)--(1.694,-1.441)--(1.742,-1.347)--(1.791,-1.265)--(1.839,-1.192)--(1.887,-1.127)--(1.936,-1.069)--(1.984,-1.016)--(2.033,-0.9685)--(2.081,-0.9251)--(2.129,-0.8855)--(2.178,-0.8491)--(2.226,-0.8156)--(2.274,-0.7847)--(2.323,-0.7560)--(2.371,-0.7293)--(2.420,-0.7044)--(2.468,-0.6812)--(2.516,-0.6595)--(2.565,-0.6391)--(2.613,-0.6199)--(2.662,-0.6019)--(2.710,-0.5848)--(2.758,-0.5687)--(2.807,-0.5535)--(2.855,-0.5391)--(2.903,-0.5254)--(2.952,-0.5123)--(3.000,-0.5000)--(3.049,-0.4881)--(3.097,-0.4769)--(3.145,-0.4661)--(3.194,-0.4558)--(3.242,-0.4460)--(3.290,-0.4366)--(3.339,-0.4276)--(3.387,-0.4189)--(3.436,-0.4106)--(3.484,-0.4026)--(3.532,-0.3949)--(3.581,-0.3875)--(3.629,-0.3803)--(3.678,-0.3735)--(3.726,-0.3668)--(3.774,-0.3604)--(3.823,-0.3543)--(3.871,-0.3483)--(3.919,-0.3425)--(3.968,-0.3369)--(4.016,-0.3315)--(4.065,-0.3263)--(4.113,-0.3212)--(4.161,-0.3163)--(4.210,-0.3115)--(4.258,-0.3069)--(4.307,-0.3024)--(4.355,-0.2981)--(4.403,-0.2938)--(4.452,-0.2897)--(4.500,-0.2857)--(4.548,-0.2818)--(4.597,-0.2780)--(4.645,-0.2743)--(4.694,-0.2707)--(4.742,-0.2672)--(4.790,-0.2638)--(4.839,-0.2605)--(4.887,-0.2573)--(4.936,-0.2541)--(4.984,-0.2510)--(5.032,-0.2480)--(5.081,-0.2451)--(5.129,-0.2422)--(5.177,-0.2394)--(5.226,-0.2366)--(5.274,-0.2340)--(5.323,-0.2313)--(5.371,-0.2288)--(5.419,-0.2263)--(5.468,-0.2238)--(5.516,-0.2214)--(5.565,-0.2191)--(5.613,-0.2168)--(5.661,-0.2145)--(5.710,-0.2123)--(5.758,-0.2102)--(5.806,-0.2081)--(5.855,-0.2060)--(5.903,-0.2039)--(5.952,-0.2020)--(6.000,-0.2000);
-\draw [style=dotted] (1.00,-4.76) -- (1.00,4.76);
-\draw (-4.0000,-0.32983) node {$ -4 $};
-\draw [] (-4.00,-0.100) -- (-4.00,0.100);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (-0.43316,-5.0000) node {$ -5 $};
-\draw [] (-0.100,-5.00) -- (0.100,-5.00);
-\draw (-0.43316,-4.0000) node {$ -4 $};
-\draw [] (-0.100,-4.00) -- (0.100,-4.00);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw (-0.29125,5.0000) node {$ 5 $};
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw [color=blue] (1.2100,-4.7619)--(1.2583,-3.8702)--(1.3067,-3.2597)--(1.3551,-2.8156)--(1.4035,-2.4780)--(1.4519,-2.2127)--(1.5003,-1.9987)--(1.5486,-1.8225)--(1.5970,-1.6748)--(1.6454,-1.5492)--(1.6938,-1.4412)--(1.7422,-1.3473)--(1.7906,-1.2648)--(1.8389,-1.1919)--(1.8873,-1.1269)--(1.9357,-1.0686)--(1.9841,-1.0161)--(2.0325,-0.9684)--(2.0809,-0.9251)--(2.1292,-0.8855)--(2.1776,-0.8491)--(2.2260,-0.8156)--(2.2744,-0.7846)--(2.3228,-0.7559)--(2.3712,-0.7292)--(2.4195,-0.7044)--(2.4679,-0.6812)--(2.5163,-0.6594)--(2.5647,-0.6390)--(2.6131,-0.6199)--(2.6615,-0.6018)--(2.7098,-0.5848)--(2.7582,-0.5687)--(2.8066,-0.5535)--(2.8550,-0.5390)--(2.9034,-0.5253)--(2.9518,-0.5123)--(3.0002,-0.4999)--(3.0485,-0.4881)--(3.0969,-0.4768)--(3.1453,-0.4661)--(3.1937,-0.4558)--(3.2421,-0.4460)--(3.2905,-0.4365)--(3.3388,-0.4275)--(3.3872,-0.4188)--(3.4356,-0.4105)--(3.4840,-0.4025)--(3.5324,-0.3948)--(3.5808,-0.3874)--(3.6291,-0.3803)--(3.6775,-0.3734)--(3.7259,-0.3668)--(3.7743,-0.3604)--(3.8227,-0.3542)--(3.8711,-0.3482)--(3.9194,-0.3425)--(3.9678,-0.3369)--(4.0162,-0.3315)--(4.0646,-0.3263)--(4.1130,-0.3212)--(4.1614,-0.3163)--(4.2097,-0.3115)--(4.2581,-0.3069)--(4.3065,-0.3024)--(4.3549,-0.2980)--(4.4033,-0.2938)--(4.4517,-0.2897)--(4.5001,-0.2857)--(4.5484,-0.2818)--(4.5968,-0.2780)--(4.6452,-0.2743)--(4.6936,-0.2707)--(4.7420,-0.2672)--(4.7904,-0.2638)--(4.8387,-0.2604)--(4.8871,-0.2572)--(4.9355,-0.2540)--(4.9839,-0.2510)--(5.0323,-0.2479)--(5.0807,-0.2450)--(5.1290,-0.2421)--(5.1774,-0.2393)--(5.2258,-0.2366)--(5.2742,-0.2339)--(5.3226,-0.2313)--(5.3710,-0.2287)--(5.4193,-0.2262)--(5.4677,-0.2238)--(5.5161,-0.2214)--(5.5645,-0.2190)--(5.6129,-0.2167)--(5.6613,-0.2145)--(5.7096,-0.2123)--(5.7580,-0.2101)--(5.8064,-0.2080)--(5.8548,-0.2059)--(5.9032,-0.2039)--(5.9516,-0.2019)--(6.0000,-0.2000);
+\draw [style=dotted] (1.0000,-4.7619) -- (1.0000,4.7619);
+\draw (-4.0000,-0.3298) node {$ -4 $};
+\draw [] (-4.0000,-0.1000) -- (-4.0000,0.1000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (-0.4331,-5.0000) node {$ -5 $};
+\draw [] (-0.1000,-5.0000) -- (0.1000,-5.0000);
+\draw (-0.4331,-4.0000) node {$ -4 $};
+\draw [] (-0.1000,-4.0000) -- (0.1000,-4.0000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
+\draw (-0.2912,5.0000) node {$ 5 $};
+\draw [] (-0.1000,5.0000) -- (0.1000,5.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall b/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall
index 1d7103b76..24c9873b6 100644
--- a/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall
+++ b/src_phystricks/Fig_HCJPooHsaTgI.pstricks.recall
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4998);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -51,25 +51,25 @@
 % setting the default values
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+\draw [color=blue] (1.0000,1.6347)--(1.0505,1.5624)--(1.1010,1.4970)--(1.1515,1.4392)--(1.2020,1.3895)--(1.2525,1.3484)--(1.3030,1.3161)--(1.3535,1.2928)--(1.4040,1.2785)--(1.4545,1.2731)--(1.5050,1.2763)--(1.5555,1.2878)--(1.6060,1.3071)--(1.6565,1.3337)--(1.7070,1.3669)--(1.7575,1.4059)--(1.8080,1.4499)--(1.8585,1.4980)--(1.9090,1.5493)--(1.9595,1.6029)--(2.0101,1.6578)--(2.0606,1.7132)--(2.1111,1.7680)--(2.1616,1.8214)--(2.2121,1.8726)--(2.2626,1.9208)--(2.3131,1.9653)--(2.3636,2.0056)--(2.4141,2.0411)--(2.4646,2.0714)--(2.5151,2.0963)--(2.5656,2.1155)--(2.6161,2.1290)--(2.6666,2.1368)--(2.7171,2.1391)--(2.7676,2.1362)--(2.8181,2.1284)--(2.8686,2.1162)--(2.9191,2.1001)--(2.9696,2.0808)--(3.0202,2.0588)--(3.0707,2.0350)--(3.1212,2.0101)--(3.1717,1.9849)--(3.2222,1.9602)--(3.2727,1.9367)--(3.3232,1.9152)--(3.3737,1.8964)--(3.4242,1.8810)--(3.4747,1.8696)--(3.5252,1.8627)--(3.5757,1.8608)--(3.6262,1.8642)--(3.6767,1.8731)--(3.7272,1.8877)--(3.7777,1.9080)--(3.8282,1.9339)--(3.8787,1.9653)--(3.9292,2.0017)--(3.9797,2.0428)--(4.0303,2.0881)--(4.0808,2.1370)--(4.1313,2.1887)--(4.1818,2.2424)--(4.2323,2.2974)--(4.2828,2.3527)--(4.3333,2.4075)--(4.3838,2.4607)--(4.4343,2.5115)--(4.4848,2.5589)--(4.5353,2.6020)--(4.5858,2.6399)--(4.6363,2.6718)--(4.6868,2.6971)--(4.7373,2.7149)--(4.7878,2.7249)--(4.8383,2.7265)--(4.8888,2.7194)--(4.9393,2.7033)--(4.9898,2.6783)--(5.0404,2.6442)--(5.0909,2.6014)--(5.1414,2.5502)--(5.1919,2.4909)--(5.2424,2.4241)--(5.2929,2.3505)--(5.3434,2.2708)--(5.3939,2.1861)--(5.4444,2.0972)--(5.4949,2.0052)--(5.5454,1.9112)--(5.5959,1.8163)--(5.6464,1.7217)--(5.6969,1.6286)--(5.7474,1.5381)--(5.7979,1.4515)--(5.8484,1.3698)--(5.8989,1.2941)--(5.9494,1.2254)--(6.0000,1.1645);
 
-\draw [color=cyan] (1.00,5.00) -- (6.00,5.00);
-\draw [color=cyan] (6.00,5.00) -- (6.00,1.16);
-\draw [color=cyan] (6.00,1.16) -- (1.00,1.16);
-\draw [color=cyan] (1.00,1.16) -- (1.00,5.00);
+\draw [color=cyan] (1.0000,4.9998) -- (6.0000,4.9998);
+\draw [color=cyan] (6.0000,4.9998) -- (6.0000,1.1645);
+\draw [color=cyan] (6.0000,1.1645) -- (1.0000,1.1645);
+\draw [color=cyan] (1.0000,1.1645) -- (1.0000,4.9998);
 
 %OTHER STUFF
 %END PSPICTURE
@@ -108,8 +108,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -118,24 +118,24 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-0.27898,1.0000) node {$c$};
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-\draw [color=cyan] (1.16,6.00) -- (5.00,6.00);
-\draw [color=cyan] (5.00,6.00) -- (5.00,1.00);
-\draw [color=cyan] (5.00,1.00) -- (1.16,1.00);
-\draw [color=cyan] (1.16,1.00) -- (1.16,6.00);
+\draw [color=cyan] (1.1645,6.0000) -- (4.9998,6.0000);
+\draw [color=cyan] (4.9998,6.0000) -- (4.9998,1.0000);
+\draw [color=cyan] (4.9998,1.0000) -- (1.1645,1.0000);
+\draw [color=cyan] (1.1645,1.0000) -- (1.1645,6.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall b/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall
index bf237b1bd..f19c097bb 100644
--- a/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall
+++ b/src_phystricks/Fig_HFAYooOrfMAA.pstricks.recall
@@ -67,21 +67,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.0000) -- (0.0000,2.5000);
 %DEFAULT
 
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+\draw [color=red,style=solid] (0.0000,0.0000)--(0.0158,0.0000)--(0.0317,0.0000)--(0.0475,0.0011)--(0.0634,0.0020)--(0.0792,0.0031)--(0.0950,0.0045)--(0.1108,0.0061)--(0.1265,0.0080)--(0.1423,0.0101)--(0.1580,0.0125)--(0.1736,0.0151)--(0.1892,0.0180)--(0.2048,0.0211)--(0.2203,0.0245)--(0.2357,0.0281)--(0.2511,0.0320)--(0.2664,0.0361)--(0.2817,0.0405)--(0.2969,0.0450)--(0.3120,0.0499)--(0.3270,0.0549)--(0.3420,0.0603)--(0.3568,0.0658)--(0.3716,0.0716)--(0.3863,0.0776)--(0.4009,0.0838)--(0.4154,0.0903)--(0.4297,0.0970)--(0.4440,0.1040)--(0.4582,0.1111)--(0.4722,0.1185)--(0.4861,0.1261)--(0.5000,0.1339)--(0.5136,0.1420)--(0.5272,0.1502)--(0.5406,0.1587)--(0.5539,0.1674)--(0.5670,0.1763)--(0.5800,0.1854)--(0.5929,0.1947)--(0.6056,0.2042)--(0.6181,0.2139)--(0.6305,0.2238)--(0.6427,0.2339)--(0.6548,0.2442)--(0.6667,0.2547)--(0.6785,0.2654)--(0.6900,0.2762)--(0.7014,0.2873)--(0.7126,0.2985)--(0.7237,0.3099)--(0.7345,0.3214)--(0.7452,0.3332)--(0.7557,0.3451)--(0.7660,0.3572)--(0.7761,0.3694)--(0.7860,0.3818)--(0.7957,0.3943)--(0.8052,0.4070)--(0.8145,0.4199)--(0.8236,0.4329)--(0.8325,0.4460)--(0.8412,0.4593)--(0.8497,0.4727)--(0.8579,0.4863)--(0.8660,0.5000)--(0.8738,0.5138)--(0.8814,0.5277)--(0.8888,0.5417)--(0.8959,0.5559)--(0.9029,0.5702)--(0.9096,0.5845)--(0.9161,0.5990)--(0.9223,0.6136)--(0.9283,0.6283)--(0.9341,0.6431)--(0.9396,0.6579)--(0.9450,0.6729)--(0.9500,0.6879)--(0.9549,0.7030)--(0.9594,0.7182)--(0.9638,0.7335)--(0.9679,0.7488)--(0.9718,0.7642)--(0.9754,0.7796)--(0.9788,0.7951)--(0.9819,0.8107)--(0.9848,0.8263)--(0.9874,0.8419)--(0.9898,0.8576)--(0.9919,0.8734)--(0.9938,0.8891)--(0.9954,0.9049)--(0.9968,0.9207)--(0.9979,0.9365)--(0.9988,0.9524)--(0.9994,0.9682)--(0.9998,0.9841)--(1.0000,1.0000);
+\draw [] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (1.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=red] (-0.5000,-0.5000) -- (1.5000,1.5000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3546,0.66316) node {$P$};
+\draw (1.3546,0.6631) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall b/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall
index feb15aa45..07f9cdb05 100644
--- a/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall
+++ b/src_phystricks/Fig_HGQPooKrRtAN.pstricks.recall
@@ -44,12 +44,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
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-\draw [color=brown,style=dashed] (1.30,0.750) -- (3.03,1.75);
-\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5981,1.5000);
-\draw (2.2869,1.8658) node {$e_{r}$};
+\draw [color=brown,style=dashed] (2.0000,0.0000)--(1.9998,0.0211)--(1.9995,0.0423)--(1.9989,0.0634)--(1.9982,0.0845)--(1.9972,0.1057)--(1.9959,0.1268)--(1.9945,0.1479)--(1.9928,0.1690)--(1.9909,0.1901)--(1.9888,0.2111)--(1.9864,0.2321)--(1.9839,0.2531)--(1.9811,0.2741)--(1.9781,0.2950)--(1.9748,0.3160)--(1.9714,0.3368)--(1.9677,0.3577)--(1.9638,0.3785)--(1.9597,0.3992)--(1.9554,0.4199)--(1.9508,0.4406)--(1.9460,0.4612)--(1.9411,0.4817)--(1.9358,0.5022)--(1.9304,0.5227)--(1.9248,0.5431)--(1.9189,0.5634)--(1.9129,0.5837)--(1.9066,0.6039)--(1.9001,0.6240)--(1.8934,0.6441)--(1.8865,0.6641)--(1.8793,0.6840)--(1.8720,0.7038)--(1.8644,0.7236)--(1.8567,0.7433)--(1.8487,0.7629)--(1.8405,0.7824)--(1.8322,0.8018)--(1.8236,0.8211)--(1.8148,0.8404)--(1.8058,0.8595)--(1.7966,0.8786)--(1.7872,0.8975)--(1.7776,0.9164)--(1.7678,0.9352)--(1.7578,0.9538)--(1.7476,0.9723)--(1.7373,0.9908)--(1.7267,1.0091)--(1.7159,1.0273)--(1.7050,1.0454)--(1.6938,1.0634)--(1.6825,1.0812)--(1.6709,1.0990)--(1.6592,1.1166)--(1.6473,1.1341)--(1.6352,1.1514)--(1.6229,1.1687)--(1.6105,1.1858)--(1.5979,1.2027)--(1.5850,1.2196)--(1.5721,1.2363)--(1.5589,1.2528)--(1.5456,1.2692)--(1.5320,1.2855)--(1.5184,1.3017)--(1.5045,1.3176)--(1.4905,1.3335)--(1.4763,1.3492)--(1.4619,1.3647)--(1.4474,1.3801)--(1.4327,1.3953)--(1.4179,1.4104)--(1.4029,1.4253)--(1.3877,1.4401)--(1.3724,1.4547)--(1.3570,1.4691)--(1.3414,1.4834)--(1.3256,1.4975)--(1.3097,1.5114)--(1.2936,1.5252)--(1.2774,1.5388)--(1.2611,1.5522)--(1.2446,1.5655)--(1.2279,1.5786)--(1.2112,1.5915)--(1.1943,1.6042)--(1.1772,1.6167)--(1.1601,1.6291)--(1.1428,1.6413)--(1.1253,1.6533)--(1.1078,1.6651)--(1.0901,1.6767)--(1.0723,1.6882)--(1.0544,1.6994)--(1.0364,1.7105)--(1.0182,1.7213)--(1.0000,1.7320);
+\draw [color=brown,style=dashed] (1.2990,0.7500) -- (3.0310,1.7500);
+\draw [color=red,->,>=latex] (1.7320,1.0000) -- (2.5980,1.5000);
+\draw (2.2869,1.8657) node {$e_{r}$};
 \draw [color=red,->,>=latex] (1.7320,1.0000) -- (1.2320,1.8660);
-\draw (1.6075,2.1865) node {$e_{\theta}$};
+\draw (1.6075,2.1864) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
@@ -89,12 +89,12 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [color=brown,style=dashed] (-1.477,0.2605)--(-1.480,0.2448)--(-1.482,0.2292)--(-1.485,0.2135)--(-1.487,0.1978)--(-1.489,0.1820)--(-1.491,0.1663)--(-1.492,0.1505)--(-1.494,0.1347)--(-1.495,0.1189)--(-1.496,0.1031)--(-1.497,0.08722)--(-1.498,0.07137)--(-1.499,0.05552)--(-1.499,0.03966)--(-1.500,0.02380)--(-1.500,0.007933)--(-1.500,-0.007933)--(-1.500,-0.02380)--(-1.499,-0.03966)--(-1.499,-0.05552)--(-1.498,-0.07137)--(-1.497,-0.08722)--(-1.496,-0.1031)--(-1.495,-0.1189)--(-1.494,-0.1347)--(-1.492,-0.1505)--(-1.491,-0.1663)--(-1.489,-0.1820)--(-1.487,-0.1978)--(-1.485,-0.2135)--(-1.482,-0.2292)--(-1.480,-0.2448)--(-1.477,-0.2605)--(-1.474,-0.2761)--(-1.471,-0.2917)--(-1.468,-0.3072)--(-1.465,-0.3227)--(-1.461,-0.3382)--(-1.458,-0.3536)--(-1.454,-0.3690)--(-1.450,-0.3844)--(-1.446,-0.3997)--(-1.441,-0.4150)--(-1.437,-0.4302)--(-1.432,-0.4454)--(-1.428,-0.4605)--(-1.423,-0.4756)--(-1.417,-0.4906)--(-1.412,-0.5056)--(-1.407,-0.5205)--(-1.401,-0.5353)--(-1.395,-0.5501)--(-1.390,-0.5648)--(-1.384,-0.5795)--(-1.377,-0.5941)--(-1.371,-0.6087)--(-1.364,-0.6231)--(-1.358,-0.6375)--(-1.351,-0.6518)--(-1.344,-0.6661)--(-1.337,-0.6803)--(-1.330,-0.6944)--(-1.322,-0.7084)--(-1.315,-0.7224)--(-1.307,-0.7362)--(-1.299,-0.7500)--(-1.291,-0.7637)--(-1.283,-0.7773)--(-1.275,-0.7908)--(-1.266,-0.8043)--(-1.258,-0.8176)--(-1.249,-0.8309)--(-1.240,-0.8440)--(-1.231,-0.8571)--(-1.222,-0.8701)--(-1.213,-0.8830)--(-1.203,-0.8957)--(-1.194,-0.9084)--(-1.184,-0.9210)--(-1.174,-0.9335)--(-1.164,-0.9458)--(-1.154,-0.9581)--(-1.144,-0.9702)--(-1.134,-0.9823)--(-1.123,-0.9942)--(-1.113,-1.006)--(-1.102,-1.018)--(-1.091,-1.029)--(-1.080,-1.041)--(-1.069,-1.052)--(-1.058,-1.063)--(-1.047,-1.075)--(-1.035,-1.086)--(-1.024,-1.096)--(-1.012,-1.107)--(-1.000,-1.118)--(-0.9883,-1.128)--(-0.9763,-1.139)--(-0.9642,-1.149);
-\draw [color=brown,style=dashed] (-0.940,-0.342) -- (-2.82,-1.03);
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-2.3492,-0.85505);
-\draw (-2.0855,-1.2429) node {$e_{r}$};
-\draw [color=red,->,>=latex] (-1.4095,-0.51303) -- (-1.0675,-1.4527);
-\draw (-0.69205,-1.1323) node {$e_{\theta}$};
+\draw [color=brown,style=dashed] (-1.4772,0.2604)--(-1.4798,0.2448)--(-1.4823,0.2291)--(-1.4847,0.2134)--(-1.4869,0.1977)--(-1.4889,0.1820)--(-1.4907,0.1662)--(-1.4924,0.1504)--(-1.4939,0.1346)--(-1.4952,0.1188)--(-1.4964,0.1030)--(-1.4974,0.0872)--(-1.4983,0.0713)--(-1.4989,0.0555)--(-1.4994,0.0396)--(-1.4998,0.0237)--(-1.4999,0.0079)--(-1.4999,-0.0079)--(-1.4998,-0.0237)--(-1.4994,-0.0396)--(-1.4989,-0.0555)--(-1.4983,-0.0713)--(-1.4974,-0.0872)--(-1.4964,-0.1030)--(-1.4952,-0.1188)--(-1.4939,-0.1346)--(-1.4924,-0.1504)--(-1.4907,-0.1662)--(-1.4889,-0.1820)--(-1.4869,-0.1977)--(-1.4847,-0.2134)--(-1.4823,-0.2291)--(-1.4798,-0.2448)--(-1.4772,-0.2604)--(-1.4743,-0.2760)--(-1.4713,-0.2916)--(-1.4682,-0.3072)--(-1.4648,-0.3227)--(-1.4613,-0.3381)--(-1.4577,-0.3536)--(-1.4538,-0.3690)--(-1.4499,-0.3843)--(-1.4457,-0.3997)--(-1.4414,-0.4149)--(-1.4369,-0.4302)--(-1.4323,-0.4453)--(-1.4275,-0.4605)--(-1.4226,-0.4755)--(-1.4175,-0.4906)--(-1.4122,-0.5055)--(-1.4068,-0.5204)--(-1.4012,-0.5353)--(-1.3954,-0.5501)--(-1.3895,-0.5648)--(-1.3835,-0.5795)--(-1.3773,-0.5941)--(-1.3709,-0.6086)--(-1.3644,-0.6231)--(-1.3577,-0.6375)--(-1.3509,-0.6518)--(-1.3439,-0.6660)--(-1.3368,-0.6802)--(-1.3295,-0.6943)--(-1.3221,-0.7084)--(-1.3146,-0.7223)--(-1.3068,-0.7362)--(-1.2990,-0.7500)--(-1.2910,-0.7636)--(-1.2828,-0.7773)--(-1.2745,-0.7908)--(-1.2661,-0.8042)--(-1.2575,-0.8176)--(-1.2488,-0.8308)--(-1.2399,-0.8440)--(-1.2309,-0.8571)--(-1.2218,-0.8700)--(-1.2125,-0.8829)--(-1.2031,-0.8957)--(-1.1936,-0.9084)--(-1.1839,-0.9209)--(-1.1741,-0.9334)--(-1.1642,-0.9458)--(-1.1541,-0.9580)--(-1.1439,-0.9702)--(-1.1336,-0.9822)--(-1.1231,-0.9942)--(-1.1125,-1.0060)--(-1.1018,-1.0177)--(-1.0910,-1.0293)--(-1.0801,-1.0408)--(-1.0690,-1.0522)--(-1.0578,-1.0634)--(-1.0465,-1.0745)--(-1.0351,-1.0856)--(-1.0235,-1.0964)--(-1.0119,-1.1072)--(-1.0001,-1.1178)--(-0.9882,-1.1284)--(-0.9762,-1.1388)--(-0.9641,-1.1490);
+\draw [color=brown,style=dashed] (-0.9396,-0.3420) -- (-2.8190,-1.0260);
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-2.3492,-0.8550);
+\draw (-2.0854,-1.2429) node {$e_{r}$};
+\draw [color=red,->,>=latex] (-1.4095,-0.5130) -- (-1.0675,-1.4527);
+\draw (-0.6920,-1.1322) node {$e_{\theta}$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall b/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall
index b7cae7f35..b7851a4a4 100644
--- a/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall
+++ b/src_phystricks/Fig_IOCTooePeHGCXH.pstricks.recall
@@ -95,24 +95,24 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [] (2.000,2.000)--(1.996,2.127)--(1.984,2.253)--(1.964,2.379)--(1.936,2.502)--(1.900,2.624)--(1.857,2.743)--(1.806,2.860)--(1.748,2.972)--(1.683,3.081)--(1.611,3.186)--(1.532,3.286)--(1.447,3.380)--(1.357,3.469)--(1.261,3.552)--(1.160,3.629)--(1.054,3.699)--(0.9445,3.763)--(0.8308,3.819)--(0.7138,3.868)--(0.5938,3.910)--(0.4715,3.944)--(0.3473,3.970)--(0.2217,3.988)--(0.09516,3.998)--(-0.03173,4.000)--(-0.1585,3.994)--(-0.2846,3.980)--(-0.4096,3.958)--(-0.5330,3.928)--(-0.6541,3.890)--(-0.7727,3.845)--(-0.8881,3.792)--(-1.000,3.732)--(-1.108,3.665)--(-1.211,3.592)--(-1.310,3.512)--(-1.403,3.425)--(-1.491,3.334)--(-1.572,3.236)--(-1.647,3.134)--(-1.716,3.027)--(-1.778,2.916)--(-1.832,2.802)--(-1.879,2.684)--(-1.919,2.563)--(-1.951,2.441)--(-1.975,2.316)--(-1.991,2.190)--(-1.999,2.063)--(-1.999,1.937)--(-1.991,1.810)--(-1.975,1.684)--(-1.951,1.559)--(-1.919,1.437)--(-1.879,1.316)--(-1.832,1.198)--(-1.778,1.084)--(-1.716,0.9726)--(-1.647,0.8659)--(-1.572,0.7637)--(-1.491,0.6665)--(-1.403,0.5746)--(-1.310,0.4885)--(-1.211,0.4085)--(-1.108,0.3349)--(-1.000,0.2679)--(-0.8881,0.2080)--(-0.7727,0.1553)--(-0.6541,0.1100)--(-0.5330,0.07232)--(-0.4096,0.04240)--(-0.2846,0.02036)--(-0.1585,0.006290)--(-0.03173,0)--(0.09516,0.002265)--(0.2217,0.01232)--(0.3473,0.03038)--(0.4715,0.05638)--(0.5938,0.09020)--(0.7138,0.1317)--(0.8308,0.1807)--(0.9445,0.2371)--(1.054,0.3005)--(1.160,0.3708)--(1.261,0.4477)--(1.357,0.5308)--(1.447,0.6198)--(1.532,0.7144)--(1.611,0.8142)--(1.683,0.9187)--(1.748,1.028)--(1.806,1.140)--(1.857,1.257)--(1.900,1.376)--(1.936,1.498)--(1.964,1.621)--(1.984,1.747)--(1.996,1.873)--(2.000,2.000);
+\draw [] (2.0000,2.0000)--(1.9959,2.1268)--(1.9839,2.2531)--(1.9638,2.3785)--(1.9358,2.5022)--(1.9001,2.6240)--(1.8567,2.7433)--(1.8058,2.8595)--(1.7476,2.9723)--(1.6825,3.0812)--(1.6105,3.1858)--(1.5320,3.2855)--(1.4474,3.3801)--(1.3570,3.4691)--(1.2611,3.5522)--(1.1601,3.6291)--(1.0544,3.6994)--(0.9445,3.7629)--(0.8308,3.8192)--(0.7137,3.8682)--(0.5938,3.9098)--(0.4715,3.9436)--(0.3472,3.9696)--(0.2216,3.9876)--(0.0951,3.9977)--(-0.0317,3.9997)--(-0.1584,3.9937)--(-0.2846,3.9796)--(-0.4096,3.9576)--(-0.5329,3.9276)--(-0.6541,3.8900)--(-0.7726,3.8447)--(-0.8881,3.7919)--(-1.0000,3.7320)--(-1.1078,3.6651)--(-1.2112,3.5915)--(-1.3097,3.5114)--(-1.4029,3.4253)--(-1.4905,3.3335)--(-1.5721,3.2363)--(-1.6473,3.1341)--(-1.7159,3.0273)--(-1.7776,2.9164)--(-1.8322,2.8018)--(-1.8793,2.6840)--(-1.9189,2.5634)--(-1.9508,2.4406)--(-1.9748,2.3160)--(-1.9909,2.1901)--(-1.9989,2.0634)--(-1.9989,1.9365)--(-1.9909,1.8098)--(-1.9748,1.6839)--(-1.9508,1.5593)--(-1.9189,1.4365)--(-1.8793,1.3159)--(-1.8322,1.1981)--(-1.7776,1.0835)--(-1.7159,0.9726)--(-1.6473,0.8658)--(-1.5721,0.7636)--(-1.4905,0.6664)--(-1.4029,0.5746)--(-1.3097,0.4885)--(-1.2112,0.4084)--(-1.1078,0.3348)--(-0.9999,0.2679)--(-0.8881,0.2080)--(-0.7726,0.1552)--(-0.6541,0.1099)--(-0.5329,0.0723)--(-0.4096,0.0423)--(-0.2846,0.0203)--(-0.1584,0.0062)--(-0.0317,0.0000)--(0.0951,0.0022)--(0.2216,0.0123)--(0.3472,0.0303)--(0.4715,0.0563)--(0.5938,0.0901)--(0.7137,0.1317)--(0.8308,0.1807)--(0.9445,0.2370)--(1.0544,0.3005)--(1.1601,0.3708)--(1.2611,0.4477)--(1.3570,0.5308)--(1.4474,0.6198)--(1.5320,0.7144)--(1.6105,0.8141)--(1.6825,0.9187)--(1.7476,1.0276)--(1.8058,1.1404)--(1.8567,1.2566)--(1.9001,1.3759)--(1.9358,1.4977)--(1.9638,1.6214)--(1.9839,1.7468)--(1.9959,1.8731)--(2.0000,2.0000);
 
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-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.50870) node {\( \tilde \phi(a)\)};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.5172,-0.33726) node {\( \tilde\phi(c)\)};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.50870) node {\( \tilde \phi(b)\)};
-\draw [] (-4.00,0) -- (8.00,0);
-\draw []  (1.0790,3.6840) node [rotate=0] {$\bullet$};
-\draw (1.6899,3.6840) node {\( \tilde\phi(m)\)};
-\draw (-2.2928,3.3087) node {\( \tilde \phi(A)\)};
-\draw (6.6610,5.4714) node {\( \tilde \phi(B)\)};
+\draw [] (7.0000,3.0000)--(6.9939,3.1902)--(6.9758,3.3797)--(6.9457,3.5677)--(6.9038,3.7534)--(6.8502,3.9361)--(6.7851,4.1149)--(6.7087,4.2893)--(6.6215,4.4585)--(6.5237,4.6219)--(6.4158,4.7787)--(6.2981,4.9283)--(6.1712,5.0702)--(6.0355,5.2037)--(5.8916,5.3284)--(5.7401,5.4437)--(5.5816,5.5491)--(5.4168,5.6443)--(5.2462,5.7288)--(5.0706,5.8024)--(4.8907,5.8647)--(4.7072,5.9154)--(4.5209,5.9544)--(4.3325,5.9815)--(4.1427,5.9966)--(3.9524,5.9996)--(3.7622,5.9905)--(3.5730,5.9694)--(3.3855,5.9364)--(3.2005,5.8915)--(3.0187,5.8350)--(2.8409,5.7670)--(2.6678,5.6879)--(2.5000,5.5980)--(2.3382,5.4977)--(2.1831,5.3872)--(2.0354,5.2672)--(1.8955,5.1380)--(1.7642,5.0003)--(1.6418,4.8544)--(1.5289,4.7011)--(1.4260,4.5410)--(1.3334,4.3746)--(1.2516,4.2027)--(1.1809,4.0260)--(1.1215,3.8451)--(1.0737,3.6609)--(1.0376,3.4740)--(1.0135,3.2851)--(1.0015,3.0951)--(1.0015,2.9048)--(1.0135,2.7148)--(1.0376,2.5259)--(1.0737,2.3390)--(1.1215,2.1548)--(1.1809,1.9739)--(1.2516,1.7972)--(1.3334,1.6253)--(1.4260,1.4589)--(1.5289,1.2988)--(1.6418,1.1455)--(1.7642,0.9996)--(1.8955,0.8619)--(2.0354,0.7327)--(2.1831,0.6127)--(2.3382,0.5022)--(2.5000,0.4019)--(2.6678,0.3120)--(2.8409,0.2329)--(3.0187,0.1649)--(3.2005,0.1084)--(3.3855,0.0635)--(3.5730,0.0305)--(3.7622,0.0094)--(3.9524,0.0000)--(4.1427,0.0033)--(4.3325,0.0184)--(4.5209,0.0455)--(4.7072,0.0845)--(4.8907,0.1352)--(5.0706,0.1975)--(5.2462,0.2711)--(5.4168,0.3556)--(5.5816,0.4508)--(5.7401,0.5562)--(5.8916,0.6715)--(6.0355,0.7962)--(6.1712,0.9297)--(6.2981,1.0716)--(6.4158,1.2212)--(6.5237,1.3780)--(6.6215,1.5414)--(6.7087,1.7106)--(6.7851,1.8850)--(6.8502,2.0638)--(6.9038,2.2465)--(6.9457,2.4322)--(6.9758,2.6202)--(6.9939,2.8097)--(7.0000,3.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.5086) node {\( \tilde \phi(a)\)};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.5171,-0.3372) node {\( \tilde\phi(c)\)};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.5086) node {\( \tilde \phi(b)\)};
+\draw [] (-4.0000,0.0000) -- (8.0000,0.0000);
+\draw []  (1.0790,3.6839) node [rotate=0] {$\bullet$};
+\draw (1.6898,3.6839) node {\( \tilde\phi(m)\)};
+\draw (-2.2927,3.3086) node {\( \tilde \phi(A)\)};
+\draw (6.6609,5.4714) node {\( \tilde \phi(B)\)};
 \draw []  (1.7445,1.0219) node [rotate=0] {$\bullet$};
-\draw (2.0358,1.0219) node {\( 0\)};
-\draw [] (0.746,5.02) -- (2.08,-0.309);
-\draw (8.5574,0) node {\( \tilde\phi(\mC)\)};
+\draw (2.0357,1.0219) node {\( 0\)};
+\draw [] (0.7462,5.0149) -- (2.0772,-0.3091);
+\draw (8.5573,0.0000) node {\( \tilde\phi(\mC)\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_IWuPxFc.pstricks.recall b/src_phystricks/Fig_IWuPxFc.pstricks.recall
index a40e3d90c..ed922425f 100644
--- a/src_phystricks/Fig_IWuPxFc.pstricks.recall
+++ b/src_phystricks/Fig_IWuPxFc.pstricks.recall
@@ -95,40 +95,40 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.4989,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-3.1457) -- (0,3.1457);
+\draw [,->,>=latex] (-2.4988,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.1456) -- (0.0000,3.1456);
 %DEFAULT
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+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_IntBoutCercle.pstricks.recall b/src_phystricks/Fig_IntBoutCercle.pstricks.recall
index 01492c84e..b55e125c4 100644
--- a/src_phystricks/Fig_IntBoutCercle.pstricks.recall
+++ b/src_phystricks/Fig_IntBoutCercle.pstricks.recall
@@ -67,21 +67,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.0000) -- (0.0000,2.5000);
 %DEFAULT
 
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+\draw [color=red,style=solid] (0.0000,0.0000)--(0.0158,0.0000)--(0.0317,0.0000)--(0.0475,0.0011)--(0.0634,0.0020)--(0.0792,0.0031)--(0.0950,0.0045)--(0.1108,0.0061)--(0.1265,0.0080)--(0.1423,0.0101)--(0.1580,0.0125)--(0.1736,0.0151)--(0.1892,0.0180)--(0.2048,0.0211)--(0.2203,0.0245)--(0.2357,0.0281)--(0.2511,0.0320)--(0.2664,0.0361)--(0.2817,0.0405)--(0.2969,0.0450)--(0.3120,0.0499)--(0.3270,0.0549)--(0.3420,0.0603)--(0.3568,0.0658)--(0.3716,0.0716)--(0.3863,0.0776)--(0.4009,0.0838)--(0.4154,0.0903)--(0.4297,0.0970)--(0.4440,0.1040)--(0.4582,0.1111)--(0.4722,0.1185)--(0.4861,0.1261)--(0.5000,0.1339)--(0.5136,0.1420)--(0.5272,0.1502)--(0.5406,0.1587)--(0.5539,0.1674)--(0.5670,0.1763)--(0.5800,0.1854)--(0.5929,0.1947)--(0.6056,0.2042)--(0.6181,0.2139)--(0.6305,0.2238)--(0.6427,0.2339)--(0.6548,0.2442)--(0.6667,0.2547)--(0.6785,0.2654)--(0.6900,0.2762)--(0.7014,0.2873)--(0.7126,0.2985)--(0.7237,0.3099)--(0.7345,0.3214)--(0.7452,0.3332)--(0.7557,0.3451)--(0.7660,0.3572)--(0.7761,0.3694)--(0.7860,0.3818)--(0.7957,0.3943)--(0.8052,0.4070)--(0.8145,0.4199)--(0.8236,0.4329)--(0.8325,0.4460)--(0.8412,0.4593)--(0.8497,0.4727)--(0.8579,0.4863)--(0.8660,0.5000)--(0.8738,0.5138)--(0.8814,0.5277)--(0.8888,0.5417)--(0.8959,0.5559)--(0.9029,0.5702)--(0.9096,0.5845)--(0.9161,0.5990)--(0.9223,0.6136)--(0.9283,0.6283)--(0.9341,0.6431)--(0.9396,0.6579)--(0.9450,0.6729)--(0.9500,0.6879)--(0.9549,0.7030)--(0.9594,0.7182)--(0.9638,0.7335)--(0.9679,0.7488)--(0.9718,0.7642)--(0.9754,0.7796)--(0.9788,0.7951)--(0.9819,0.8107)--(0.9848,0.8263)--(0.9874,0.8419)--(0.9898,0.8576)--(0.9919,0.8734)--(0.9938,0.8891)--(0.9954,0.9049)--(0.9968,0.9207)--(0.9979,0.9365)--(0.9988,0.9524)--(0.9994,0.9682)--(0.9998,0.9841)--(1.0000,1.0000);
+\draw [] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (1.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=red] (-0.5000,-0.5000) -- (1.5000,1.5000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3546,0.66316) node {$P$};
+\draw (1.3546,0.6631) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_IntEcourbe.pstricks.recall b/src_phystricks/Fig_IntEcourbe.pstricks.recall
index f9fb48682..ee8fb43b6 100644
--- a/src_phystricks/Fig_IntEcourbe.pstricks.recall
+++ b/src_phystricks/Fig_IntEcourbe.pstricks.recall
@@ -79,13 +79,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.0000);
 %DEFAULT
 
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-\draw [color=blue] (1.000,1.341)--(1.020,1.352)--(1.040,1.363)--(1.061,1.373)--(1.081,1.382)--(1.101,1.392)--(1.121,1.401)--(1.141,1.409)--(1.162,1.417)--(1.182,1.425)--(1.202,1.433)--(1.222,1.440)--(1.242,1.447)--(1.263,1.453)--(1.283,1.459)--(1.303,1.464)--(1.323,1.470)--(1.343,1.474)--(1.364,1.479)--(1.384,1.483)--(1.404,1.486)--(1.424,1.489)--(1.444,1.492)--(1.465,1.494)--(1.485,1.496)--(1.505,1.498)--(1.525,1.499)--(1.545,1.500)--(1.566,1.500)--(1.586,1.500)--(1.606,1.499)--(1.626,1.498)--(1.646,1.497)--(1.667,1.495)--(1.687,1.493)--(1.707,1.491)--(1.727,1.488)--(1.747,1.484)--(1.768,1.481)--(1.788,1.477)--(1.808,1.472)--(1.828,1.467)--(1.848,1.462)--(1.869,1.456)--(1.889,1.450)--(1.909,1.443)--(1.929,1.436)--(1.949,1.429)--(1.970,1.421)--(1.990,1.413)--(2.010,1.405)--(2.030,1.396)--(2.051,1.387)--(2.071,1.378)--(2.091,1.368)--(2.111,1.358)--(2.131,1.347)--(2.152,1.336)--(2.172,1.325)--(2.192,1.313)--(2.212,1.301)--(2.232,1.289)--(2.253,1.276)--(2.273,1.264)--(2.293,1.250)--(2.313,1.237)--(2.333,1.223)--(2.354,1.209)--(2.374,1.195)--(2.394,1.180)--(2.414,1.165)--(2.434,1.150)--(2.455,1.134)--(2.475,1.119)--(2.495,1.103)--(2.515,1.086)--(2.535,1.070)--(2.556,1.053)--(2.576,1.036)--(2.596,1.019)--(2.616,1.002)--(2.636,0.9840)--(2.657,0.9662)--(2.677,0.9483)--(2.697,0.9301)--(2.717,0.9118)--(2.737,0.8933)--(2.758,0.8746)--(2.778,0.8558)--(2.798,0.8369)--(2.818,0.8178)--(2.838,0.7986)--(2.859,0.7792)--(2.879,0.7598)--(2.899,0.7402)--(2.919,0.7206)--(2.939,0.7008)--(2.960,0.6810)--(2.980,0.6611)--(3.000,0.6411);
+\draw [color=blue] (1.0000,1.3414)--(1.0202,1.3522)--(1.0404,1.3626)--(1.0606,1.3726)--(1.0808,1.3823)--(1.1010,1.3916)--(1.1212,1.4006)--(1.1414,1.4092)--(1.1616,1.4174)--(1.1818,1.4252)--(1.2020,1.4327)--(1.2222,1.4398)--(1.2424,1.4465)--(1.2626,1.4528)--(1.2828,1.4588)--(1.3030,1.4643)--(1.3232,1.4695)--(1.3434,1.4742)--(1.3636,1.4786)--(1.3838,1.4825)--(1.4040,1.4861)--(1.4242,1.4892)--(1.4444,1.4920)--(1.4646,1.4943)--(1.4848,1.4963)--(1.5050,1.4978)--(1.5252,1.4989)--(1.5454,1.4996)--(1.5656,1.4999)--(1.5858,1.4998)--(1.6060,1.4993)--(1.6262,1.4984)--(1.6464,1.4971)--(1.6666,1.4954)--(1.6868,1.4932)--(1.7070,1.4907)--(1.7272,1.4877)--(1.7474,1.4844)--(1.7676,1.4806)--(1.7878,1.4765)--(1.8080,1.4719)--(1.8282,1.4670)--(1.8484,1.4616)--(1.8686,1.4559)--(1.8888,1.4498)--(1.9090,1.4433)--(1.9292,1.4364)--(1.9494,1.4291)--(1.9696,1.4214)--(1.9898,1.4134)--(2.0101,1.4050)--(2.0303,1.3962)--(2.0505,1.3871)--(2.0707,1.3776)--(2.0909,1.3677)--(2.1111,1.3575)--(2.1313,1.3469)--(2.1515,1.3360)--(2.1717,1.3248)--(2.1919,1.3132)--(2.2121,1.3013)--(2.2323,1.2890)--(2.2525,1.2764)--(2.2727,1.2635)--(2.2929,1.2503)--(2.3131,1.2368)--(2.3333,1.2230)--(2.3535,1.2089)--(2.3737,1.1945)--(2.3939,1.1799)--(2.4141,1.1649)--(2.4343,1.1497)--(2.4545,1.1342)--(2.4747,1.1185)--(2.4949,1.1025)--(2.5151,1.0862)--(2.5353,1.0697)--(2.5555,1.0530)--(2.5757,1.0361)--(2.5959,1.0189)--(2.6161,1.0015)--(2.6363,0.9840)--(2.6565,0.9662)--(2.6767,0.9482)--(2.6969,0.9301)--(2.7171,0.9117)--(2.7373,0.8933)--(2.7575,0.8746)--(2.7777,0.8558)--(2.7979,0.8368)--(2.8181,0.8178)--(2.8383,0.7985)--(2.8585,0.7792)--(2.8787,0.7597)--(2.8989,0.7402)--(2.9191,0.7205)--(2.9393,0.7008)--(2.9595,0.6809)--(2.9797,0.6610)--(3.0000,0.6411);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -93,30 +93,30 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [] (3.00,3.50) -- (3.00,0.641);
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+\draw [color=red,style=solid] (1.0000,3.5000)--(1.0202,3.4600)--(1.0404,3.4208)--(1.0606,3.3824)--(1.0808,3.3449)--(1.1010,3.3081)--(1.1212,3.2722)--(1.1414,3.2371)--(1.1616,3.2028)--(1.1818,3.1694)--(1.2020,3.1367)--(1.2222,3.1049)--(1.2424,3.0739)--(1.2626,3.0437)--(1.2828,3.0143)--(1.3030,2.9857)--(1.3232,2.9580)--(1.3434,2.9310)--(1.3636,2.9049)--(1.3838,2.8796)--(1.4040,2.8551)--(1.4242,2.8314)--(1.4444,2.8086)--(1.4646,2.7866)--(1.4848,2.7653)--(1.5050,2.7449)--(1.5252,2.7253)--(1.5454,2.7066)--(1.5656,2.6886)--(1.5858,2.6715)--(1.6060,2.6551)--(1.6262,2.6396)--(1.6464,2.6249)--(1.6666,2.6111)--(1.6868,2.5980)--(1.7070,2.5858)--(1.7272,2.5743)--(1.7474,2.5637)--(1.7676,2.5539)--(1.7878,2.5449)--(1.8080,2.5368)--(1.8282,2.5294)--(1.8484,2.5229)--(1.8686,2.5172)--(1.8888,2.5123)--(1.9090,2.5082)--(1.9292,2.5049)--(1.9494,2.5025)--(1.9696,2.5009)--(1.9898,2.5001)--(2.0101,2.5001)--(2.0303,2.5009)--(2.0505,2.5025)--(2.0707,2.5049)--(2.0909,2.5082)--(2.1111,2.5123)--(2.1313,2.5172)--(2.1515,2.5229)--(2.1717,2.5294)--(2.1919,2.5368)--(2.2121,2.5449)--(2.2323,2.5539)--(2.2525,2.5637)--(2.2727,2.5743)--(2.2929,2.5858)--(2.3131,2.5980)--(2.3333,2.6111)--(2.3535,2.6249)--(2.3737,2.6396)--(2.3939,2.6551)--(2.4141,2.6715)--(2.4343,2.6886)--(2.4545,2.7066)--(2.4747,2.7253)--(2.4949,2.7449)--(2.5151,2.7653)--(2.5353,2.7866)--(2.5555,2.8086)--(2.5757,2.8314)--(2.5959,2.8551)--(2.6161,2.8796)--(2.6363,2.9049)--(2.6565,2.9310)--(2.6767,2.9580)--(2.6969,2.9857)--(2.7171,3.0143)--(2.7373,3.0437)--(2.7575,3.0739)--(2.7777,3.1049)--(2.7979,3.1367)--(2.8181,3.1694)--(2.8383,3.2028)--(2.8585,3.2371)--(2.8787,3.2722)--(2.8989,3.3081)--(2.9191,3.3449)--(2.9393,3.3824)--(2.9595,3.4208)--(2.9797,3.4600)--(3.0000,3.5000);
+\draw [color=red,style=solid] (1.0000,1.3414)--(1.0202,1.3522)--(1.0404,1.3626)--(1.0606,1.3726)--(1.0808,1.3823)--(1.1010,1.3916)--(1.1212,1.4006)--(1.1414,1.4092)--(1.1616,1.4174)--(1.1818,1.4252)--(1.2020,1.4327)--(1.2222,1.4398)--(1.2424,1.4465)--(1.2626,1.4528)--(1.2828,1.4588)--(1.3030,1.4643)--(1.3232,1.4695)--(1.3434,1.4742)--(1.3636,1.4786)--(1.3838,1.4825)--(1.4040,1.4861)--(1.4242,1.4892)--(1.4444,1.4920)--(1.4646,1.4943)--(1.4848,1.4963)--(1.5050,1.4978)--(1.5252,1.4989)--(1.5454,1.4996)--(1.5656,1.4999)--(1.5858,1.4998)--(1.6060,1.4993)--(1.6262,1.4984)--(1.6464,1.4971)--(1.6666,1.4954)--(1.6868,1.4932)--(1.7070,1.4907)--(1.7272,1.4877)--(1.7474,1.4844)--(1.7676,1.4806)--(1.7878,1.4765)--(1.8080,1.4719)--(1.8282,1.4670)--(1.8484,1.4616)--(1.8686,1.4559)--(1.8888,1.4498)--(1.9090,1.4433)--(1.9292,1.4364)--(1.9494,1.4291)--(1.9696,1.4214)--(1.9898,1.4134)--(2.0101,1.4050)--(2.0303,1.3962)--(2.0505,1.3871)--(2.0707,1.3776)--(2.0909,1.3677)--(2.1111,1.3575)--(2.1313,1.3469)--(2.1515,1.3360)--(2.1717,1.3248)--(2.1919,1.3132)--(2.2121,1.3013)--(2.2323,1.2890)--(2.2525,1.2764)--(2.2727,1.2635)--(2.2929,1.2503)--(2.3131,1.2368)--(2.3333,1.2230)--(2.3535,1.2089)--(2.3737,1.1945)--(2.3939,1.1799)--(2.4141,1.1649)--(2.4343,1.1497)--(2.4545,1.1342)--(2.4747,1.1185)--(2.4949,1.1025)--(2.5151,1.0862)--(2.5353,1.0697)--(2.5555,1.0530)--(2.5757,1.0361)--(2.5959,1.0189)--(2.6161,1.0015)--(2.6363,0.9840)--(2.6565,0.9662)--(2.6767,0.9482)--(2.6969,0.9301)--(2.7171,0.9117)--(2.7373,0.8933)--(2.7575,0.8746)--(2.7777,0.8558)--(2.7979,0.8368)--(2.8181,0.8178)--(2.8383,0.7985)--(2.8585,0.7792)--(2.8787,0.7597)--(2.8989,0.7402)--(2.9191,0.7205)--(2.9393,0.7008)--(2.9595,0.6809)--(2.9797,0.6610)--(3.0000,0.6411);
+\draw [] (1.0000,1.3414) -- (1.0000,3.5000);
+\draw [] (3.0000,3.5000) -- (3.0000,0.6411);
 
-\draw [color=cyan] (1.00,3.50) -- (3.00,3.50);
-\draw [color=cyan] (3.00,3.50) -- (3.00,0.641);
-\draw [color=cyan] (3.00,0.641) -- (1.00,0.641);
-\draw [color=cyan] (1.00,0.641) -- (1.00,3.50);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=cyan] (1.0000,3.5000) -- (3.0000,3.5000);
+\draw [color=cyan] (3.0000,3.5000) -- (3.0000,0.6411);
+\draw [color=cyan] (3.0000,0.6411) -- (1.0000,0.6411);
+\draw [color=cyan] (1.0000,0.6411) -- (1.0000,3.5000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_IntTriangle.pstricks.recall b/src_phystricks/Fig_IntTriangle.pstricks.recall
index 5831092a4..7c812e549 100644
--- a/src_phystricks/Fig_IntTriangle.pstricks.recall
+++ b/src_phystricks/Fig_IntTriangle.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -81,23 +81,23 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
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-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=green] (0.0000,0.0000)--(0.0202,0.0202)--(0.0404,0.0404)--(0.0606,0.0606)--(0.0808,0.0808)--(0.1010,0.1010)--(0.1212,0.1212)--(0.1414,0.1414)--(0.1616,0.1616)--(0.1818,0.1818)--(0.2020,0.2020)--(0.2222,0.2222)--(0.2424,0.2424)--(0.2626,0.2626)--(0.2828,0.2828)--(0.3030,0.3030)--(0.3232,0.3232)--(0.3434,0.3434)--(0.3636,0.3636)--(0.3838,0.3838)--(0.4040,0.4040)--(0.4242,0.4242)--(0.4444,0.4444)--(0.4646,0.4646)--(0.4848,0.4848)--(0.5050,0.5050)--(0.5252,0.5252)--(0.5454,0.5454)--(0.5656,0.5656)--(0.5858,0.5858)--(0.6060,0.6060)--(0.6262,0.6262)--(0.6464,0.6464)--(0.6666,0.6666)--(0.6868,0.6868)--(0.7070,0.7070)--(0.7272,0.7272)--(0.7474,0.7474)--(0.7676,0.7676)--(0.7878,0.7878)--(0.8080,0.8080)--(0.8282,0.8282)--(0.8484,0.8484)--(0.8686,0.8686)--(0.8888,0.8888)--(0.9090,0.9090)--(0.9292,0.9292)--(0.9494,0.9494)--(0.9696,0.9696)--(0.9898,0.9898)--(1.0101,1.0101)--(1.0303,1.0303)--(1.0505,1.0505)--(1.0707,1.0707)--(1.0909,1.0909)--(1.1111,1.1111)--(1.1313,1.1313)--(1.1515,1.1515)--(1.1717,1.1717)--(1.1919,1.1919)--(1.2121,1.2121)--(1.2323,1.2323)--(1.2525,1.2525)--(1.2727,1.2727)--(1.2929,1.2929)--(1.3131,1.3131)--(1.3333,1.3333)--(1.3535,1.3535)--(1.3737,1.3737)--(1.3939,1.3939)--(1.4141,1.4141)--(1.4343,1.4343)--(1.4545,1.4545)--(1.4747,1.4747)--(1.4949,1.4949)--(1.5151,1.5151)--(1.5353,1.5353)--(1.5555,1.5555)--(1.5757,1.5757)--(1.5959,1.5959)--(1.6161,1.6161)--(1.6363,1.6363)--(1.6565,1.6565)--(1.6767,1.6767)--(1.6969,1.6969)--(1.7171,1.7171)--(1.7373,1.7373)--(1.7575,1.7575)--(1.7777,1.7777)--(1.7979,1.7979)--(1.8181,1.8181)--(1.8383,1.8383)--(1.8585,1.8585)--(1.8787,1.8787)--(1.8989,1.8989)--(1.9191,1.9191)--(1.9393,1.9393)--(1.9595,1.9595)--(1.9797,1.9797)--(2.0000,2.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_IntTrois.pstricks.recall b/src_phystricks/Fig_IntTrois.pstricks.recall
index 654f2a3b0..5afdcb38a 100644
--- a/src_phystricks/Fig_IntTrois.pstricks.recall
+++ b/src_phystricks/Fig_IntTrois.pstricks.recall
@@ -37,20 +37,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\fill [color=lightgray] (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,0) --  cycle;
-\draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
-\draw [] (1.73,1.00) -- (1.73,2.00);
-\draw [color=red] (0,2.00) -- (2.00,2.00);
-\draw [color=red] (2.00,2.00) -- (2.00,0);
+\fill [color=lightgray] (2.0000,0.0000) -- (1.9997,0.0317) -- (1.9989,0.0634) -- (1.9977,0.0951) -- (1.9959,0.1268) -- (1.9937,0.1584) -- (1.9909,0.1901) -- (1.9876,0.2216) -- (1.9839,0.2531) -- (1.9796,0.2846) -- (1.9748,0.3160) -- (1.9696,0.3472) -- (1.9638,0.3785) -- (1.9576,0.4096) -- (1.9508,0.4406) -- (1.9436,0.4715) -- (1.9358,0.5022) -- (1.9276,0.5329) -- (1.9189,0.5634) -- (1.9098,0.5938) -- (1.9001,0.6240) -- (1.8900,0.6541) -- (1.8793,0.6840) -- (1.8682,0.7137) -- (1.8567,0.7433) -- (1.8447,0.7726) -- (1.8322,0.8018) -- (1.8192,0.8308) -- (1.8058,0.8595) -- (1.7919,0.8881) -- (1.7776,0.9164) -- (1.7629,0.9445) -- (1.7476,0.9723) -- (1.7320,1.0000) -- (1.7159,1.0273) -- (1.6994,1.0544) -- (1.6825,1.0812) -- (1.6651,1.1078) -- (1.6473,1.1341) -- (1.6291,1.1601) -- (1.6105,1.1858) -- (1.5915,1.2112) -- (1.5721,1.2363) -- (1.5522,1.2611) -- (1.5320,1.2855) -- (1.5114,1.3097) -- (1.4905,1.3335) -- (1.4691,1.3570) -- (1.4474,1.3801) -- (1.4253,1.4029) -- (1.4029,1.4253) -- (1.3801,1.4474) -- (1.3570,1.4691) -- (1.3335,1.4905) -- (1.3097,1.5114) -- (1.2855,1.5320) -- (1.2611,1.5522) -- (1.2363,1.5721) -- (1.2112,1.5915) -- (1.1858,1.6105) -- (1.1601,1.6291) -- (1.1341,1.6473) -- (1.1078,1.6651) -- (1.0812,1.6825) -- (1.0544,1.6994) -- (1.0273,1.7159) -- (1.0000,1.7320) -- (0.9723,1.7476) -- (0.9445,1.7629) -- (0.9164,1.7776) -- (0.8881,1.7919) -- (0.8595,1.8058) -- (0.8308,1.8192) -- (0.8018,1.8322) -- (0.7726,1.8447) -- (0.7433,1.8567) -- (0.7137,1.8682) -- (0.6840,1.8793) -- (0.6541,1.8900) -- (0.6240,1.9001) -- (0.5938,1.9098) -- (0.5634,1.9189) -- (0.5329,1.9276) -- (0.5022,1.9358) -- (0.4715,1.9436) -- (0.4406,1.9508) -- (0.4096,1.9576) -- (0.3785,1.9638) -- (0.3472,1.9696) -- (0.3160,1.9748) -- (0.2846,1.9796) -- (0.2531,1.9839) -- (0.2216,1.9876) -- (0.1901,1.9909) -- (0.1584,1.9937) -- (0.1268,1.9959) -- (0.0951,1.9977) -- (0.0634,1.9989) -- (0.0317,1.9997) -- (0.0000,2.0000) -- (0.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,0.0000) --  cycle;
+\draw [color=red] (2.0000,0.0000)--(1.9997,0.0317)--(1.9989,0.0634)--(1.9977,0.0951)--(1.9959,0.1268)--(1.9937,0.1584)--(1.9909,0.1901)--(1.9876,0.2216)--(1.9839,0.2531)--(1.9796,0.2846)--(1.9748,0.3160)--(1.9696,0.3472)--(1.9638,0.3785)--(1.9576,0.4096)--(1.9508,0.4406)--(1.9436,0.4715)--(1.9358,0.5022)--(1.9276,0.5329)--(1.9189,0.5634)--(1.9098,0.5938)--(1.9001,0.6240)--(1.8900,0.6541)--(1.8793,0.6840)--(1.8682,0.7137)--(1.8567,0.7433)--(1.8447,0.7726)--(1.8322,0.8018)--(1.8192,0.8308)--(1.8058,0.8595)--(1.7919,0.8881)--(1.7776,0.9164)--(1.7629,0.9445)--(1.7476,0.9723)--(1.7320,1.0000)--(1.7159,1.0273)--(1.6994,1.0544)--(1.6825,1.0812)--(1.6651,1.1078)--(1.6473,1.1341)--(1.6291,1.1601)--(1.6105,1.1858)--(1.5915,1.2112)--(1.5721,1.2363)--(1.5522,1.2611)--(1.5320,1.2855)--(1.5114,1.3097)--(1.4905,1.3335)--(1.4691,1.3570)--(1.4474,1.3801)--(1.4253,1.4029)--(1.4029,1.4253)--(1.3801,1.4474)--(1.3570,1.4691)--(1.3335,1.4905)--(1.3097,1.5114)--(1.2855,1.5320)--(1.2611,1.5522)--(1.2363,1.5721)--(1.2112,1.5915)--(1.1858,1.6105)--(1.1601,1.6291)--(1.1341,1.6473)--(1.1078,1.6651)--(1.0812,1.6825)--(1.0544,1.6994)--(1.0273,1.7159)--(1.0000,1.7320)--(0.9723,1.7476)--(0.9445,1.7629)--(0.9164,1.7776)--(0.8881,1.7919)--(0.8595,1.8058)--(0.8308,1.8192)--(0.8018,1.8322)--(0.7726,1.8447)--(0.7433,1.8567)--(0.7137,1.8682)--(0.6840,1.8793)--(0.6541,1.8900)--(0.6240,1.9001)--(0.5938,1.9098)--(0.5634,1.9189)--(0.5329,1.9276)--(0.5022,1.9358)--(0.4715,1.9436)--(0.4406,1.9508)--(0.4096,1.9576)--(0.3785,1.9638)--(0.3472,1.9696)--(0.3160,1.9748)--(0.2846,1.9796)--(0.2531,1.9839)--(0.2216,1.9876)--(0.1901,1.9909)--(0.1584,1.9937)--(0.1268,1.9959)--(0.0951,1.9977)--(0.0634,1.9989)--(0.0317,1.9997)--(0.0000,2.0000);
+\draw [] (1.7320,1.0000) -- (1.7320,2.0000);
+\draw [color=red] (0.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=red] (2.0000,2.0000) -- (2.0000,0.0000);
 \draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
 \draw []  (1.7320,2.0000) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -84,25 +84,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [style=dotted] (0,0) -- (1.64,1.15);
-\draw [style=dotted] (0,0) -- (1.15,1.64);
-\fill [color=lightgray] (2.00,0) -- (2.00,0.0317) -- (2.00,0.0635) -- (2.00,0.0952) -- (2.00,0.127) -- (1.99,0.158) -- (1.99,0.190) -- (1.99,0.222) -- (1.98,0.253) -- (1.98,0.285) -- (1.97,0.316) -- (1.97,0.347) -- (1.96,0.379) -- (1.96,0.410) -- (1.95,0.441) -- (1.94,0.472) -- (1.94,0.502) -- (1.93,0.533) -- (1.92,0.563) -- (1.91,0.594) -- (1.90,0.624) -- (1.89,0.654) -- (1.88,0.684) -- (1.87,0.714) -- (1.86,0.743) -- (1.84,0.773) -- (1.83,0.802) -- (1.82,0.831) -- (1.81,0.860) -- (1.79,0.888) -- (1.78,0.916) -- (1.76,0.945) -- (1.75,0.972) -- (1.73,1.00) -- (1.72,1.03) -- (1.70,1.05) -- (1.68,1.08) -- (1.67,1.11) -- (1.65,1.13) -- (1.63,1.16) -- (1.61,1.19) -- (1.59,1.21) -- (1.57,1.24) -- (1.55,1.26) -- (1.53,1.29) -- (1.51,1.31) -- (1.49,1.33) -- (1.47,1.36) -- (1.45,1.38) -- (1.43,1.40) -- (1.40,1.43) -- (1.38,1.45) -- (1.36,1.47) -- (1.33,1.49) -- (1.31,1.51) -- (1.29,1.53) -- (1.26,1.55) -- (1.24,1.57) -- (1.21,1.59) -- (1.19,1.61) -- (1.16,1.63) -- (1.13,1.65) -- (1.11,1.67) -- (1.08,1.68) -- (1.05,1.70) -- (1.03,1.72) -- (1.00,1.73) -- (0.972,1.75) -- (0.945,1.76) -- (0.916,1.78) -- (0.888,1.79) -- (0.860,1.81) -- (0.831,1.82) -- (0.802,1.83) -- (0.773,1.84) -- (0.743,1.86) -- (0.714,1.87) -- (0.684,1.88) -- (0.654,1.89) -- (0.624,1.90) -- (0.594,1.91) -- (0.563,1.92) -- (0.533,1.93) -- (0.502,1.94) -- (0.472,1.94) -- (0.441,1.95) -- (0.410,1.96) -- (0.379,1.96) -- (0.347,1.97) -- (0.316,1.97) -- (0.285,1.98) -- (0.253,1.98) -- (0.222,1.99) -- (0.190,1.99) -- (0.158,1.99) -- (0.127,2.00) -- (0.0952,2.00) -- (0.0635,2.00) -- (0.0317,2.00) -- (0,2.00) -- (0,2.00) -- (2.00,2.00) -- (2.00,2.00) -- (2.00,0) --  cycle;
-\draw [color=red] (0,2.00) -- (2.00,2.00);
-\draw [color=red] (2.00,2.00) -- (2.00,0);
-\draw [color=red] (2.00,0)--(2.00,0.0317)--(2.00,0.0635)--(2.00,0.0952)--(2.00,0.127)--(1.99,0.158)--(1.99,0.190)--(1.99,0.222)--(1.98,0.253)--(1.98,0.285)--(1.97,0.316)--(1.97,0.347)--(1.96,0.379)--(1.96,0.410)--(1.95,0.441)--(1.94,0.472)--(1.94,0.502)--(1.93,0.533)--(1.92,0.563)--(1.91,0.594)--(1.90,0.624)--(1.89,0.654)--(1.88,0.684)--(1.87,0.714)--(1.86,0.743)--(1.84,0.773)--(1.83,0.802)--(1.82,0.831)--(1.81,0.860)--(1.79,0.888)--(1.78,0.916)--(1.76,0.945)--(1.75,0.972)--(1.73,1.00)--(1.72,1.03)--(1.70,1.05)--(1.68,1.08)--(1.67,1.11)--(1.65,1.13)--(1.63,1.16)--(1.61,1.19)--(1.59,1.21)--(1.57,1.24)--(1.55,1.26)--(1.53,1.29)--(1.51,1.31)--(1.49,1.33)--(1.47,1.36)--(1.45,1.38)--(1.43,1.40)--(1.40,1.43)--(1.38,1.45)--(1.36,1.47)--(1.33,1.49)--(1.31,1.51)--(1.29,1.53)--(1.26,1.55)--(1.24,1.57)--(1.21,1.59)--(1.19,1.61)--(1.16,1.63)--(1.13,1.65)--(1.11,1.67)--(1.08,1.68)--(1.05,1.70)--(1.03,1.72)--(1.00,1.73)--(0.972,1.75)--(0.945,1.76)--(0.916,1.78)--(0.888,1.79)--(0.860,1.81)--(0.831,1.82)--(0.802,1.83)--(0.773,1.84)--(0.743,1.86)--(0.714,1.87)--(0.684,1.88)--(0.654,1.89)--(0.624,1.90)--(0.594,1.91)--(0.563,1.92)--(0.533,1.93)--(0.502,1.94)--(0.472,1.94)--(0.441,1.95)--(0.410,1.96)--(0.379,1.96)--(0.347,1.97)--(0.316,1.97)--(0.285,1.98)--(0.253,1.98)--(0.222,1.99)--(0.190,1.99)--(0.158,1.99)--(0.127,2.00)--(0.0952,2.00)--(0.0635,2.00)--(0.0317,2.00)--(0,2.00);
-\draw [] (1.64,1.15) -- (2.00,1.40);
-\draw []  (1.6383,1.1472) node [rotate=0] {$\bullet$};
+\draw [style=dotted] (0.0000,0.0000) -- (1.6383,1.1471);
+\draw [style=dotted] (0.0000,0.0000) -- (1.1471,1.6383);
+\fill [color=lightgray] (2.0000,0.0000) -- (1.9997,0.0317) -- (1.9989,0.0634) -- (1.9977,0.0951) -- (1.9959,0.1268) -- (1.9937,0.1584) -- (1.9909,0.1901) -- (1.9876,0.2216) -- (1.9839,0.2531) -- (1.9796,0.2846) -- (1.9748,0.3160) -- (1.9696,0.3472) -- (1.9638,0.3785) -- (1.9576,0.4096) -- (1.9508,0.4406) -- (1.9436,0.4715) -- (1.9358,0.5022) -- (1.9276,0.5329) -- (1.9189,0.5634) -- (1.9098,0.5938) -- (1.9001,0.6240) -- (1.8900,0.6541) -- (1.8793,0.6840) -- (1.8682,0.7137) -- (1.8567,0.7433) -- (1.8447,0.7726) -- (1.8322,0.8018) -- (1.8192,0.8308) -- (1.8058,0.8595) -- (1.7919,0.8881) -- (1.7776,0.9164) -- (1.7629,0.9445) -- (1.7476,0.9723) -- (1.7320,1.0000) -- (1.7159,1.0273) -- (1.6994,1.0544) -- (1.6825,1.0812) -- (1.6651,1.1078) -- (1.6473,1.1341) -- (1.6291,1.1601) -- (1.6105,1.1858) -- (1.5915,1.2112) -- (1.5721,1.2363) -- (1.5522,1.2611) -- (1.5320,1.2855) -- (1.5114,1.3097) -- (1.4905,1.3335) -- (1.4691,1.3570) -- (1.4474,1.3801) -- (1.4253,1.4029) -- (1.4029,1.4253) -- (1.3801,1.4474) -- (1.3570,1.4691) -- (1.3335,1.4905) -- (1.3097,1.5114) -- (1.2855,1.5320) -- (1.2611,1.5522) -- (1.2363,1.5721) -- (1.2112,1.5915) -- (1.1858,1.6105) -- (1.1601,1.6291) -- (1.1341,1.6473) -- (1.1078,1.6651) -- (1.0812,1.6825) -- (1.0544,1.6994) -- (1.0273,1.7159) -- (1.0000,1.7320) -- (0.9723,1.7476) -- (0.9445,1.7629) -- (0.9164,1.7776) -- (0.8881,1.7919) -- (0.8595,1.8058) -- (0.8308,1.8192) -- (0.8018,1.8322) -- (0.7726,1.8447) -- (0.7433,1.8567) -- (0.7137,1.8682) -- (0.6840,1.8793) -- (0.6541,1.8900) -- (0.6240,1.9001) -- (0.5938,1.9098) -- (0.5634,1.9189) -- (0.5329,1.9276) -- (0.5022,1.9358) -- (0.4715,1.9436) -- (0.4406,1.9508) -- (0.4096,1.9576) -- (0.3785,1.9638) -- (0.3472,1.9696) -- (0.3160,1.9748) -- (0.2846,1.9796) -- (0.2531,1.9839) -- (0.2216,1.9876) -- (0.1901,1.9909) -- (0.1584,1.9937) -- (0.1268,1.9959) -- (0.0951,1.9977) -- (0.0634,1.9989) -- (0.0317,1.9997) -- (0.0000,2.0000) -- (0.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,2.0000) -- (2.0000,0.0000) --  cycle;
+\draw [color=red] (0.0000,2.0000) -- (2.0000,2.0000);
+\draw [color=red] (2.0000,2.0000) -- (2.0000,0.0000);
+\draw [color=red] (2.0000,0.0000)--(1.9997,0.0317)--(1.9989,0.0634)--(1.9977,0.0951)--(1.9959,0.1268)--(1.9937,0.1584)--(1.9909,0.1901)--(1.9876,0.2216)--(1.9839,0.2531)--(1.9796,0.2846)--(1.9748,0.3160)--(1.9696,0.3472)--(1.9638,0.3785)--(1.9576,0.4096)--(1.9508,0.4406)--(1.9436,0.4715)--(1.9358,0.5022)--(1.9276,0.5329)--(1.9189,0.5634)--(1.9098,0.5938)--(1.9001,0.6240)--(1.8900,0.6541)--(1.8793,0.6840)--(1.8682,0.7137)--(1.8567,0.7433)--(1.8447,0.7726)--(1.8322,0.8018)--(1.8192,0.8308)--(1.8058,0.8595)--(1.7919,0.8881)--(1.7776,0.9164)--(1.7629,0.9445)--(1.7476,0.9723)--(1.7320,1.0000)--(1.7159,1.0273)--(1.6994,1.0544)--(1.6825,1.0812)--(1.6651,1.1078)--(1.6473,1.1341)--(1.6291,1.1601)--(1.6105,1.1858)--(1.5915,1.2112)--(1.5721,1.2363)--(1.5522,1.2611)--(1.5320,1.2855)--(1.5114,1.3097)--(1.4905,1.3335)--(1.4691,1.3570)--(1.4474,1.3801)--(1.4253,1.4029)--(1.4029,1.4253)--(1.3801,1.4474)--(1.3570,1.4691)--(1.3335,1.4905)--(1.3097,1.5114)--(1.2855,1.5320)--(1.2611,1.5522)--(1.2363,1.5721)--(1.2112,1.5915)--(1.1858,1.6105)--(1.1601,1.6291)--(1.1341,1.6473)--(1.1078,1.6651)--(1.0812,1.6825)--(1.0544,1.6994)--(1.0273,1.7159)--(1.0000,1.7320)--(0.9723,1.7476)--(0.9445,1.7629)--(0.9164,1.7776)--(0.8881,1.7919)--(0.8595,1.8058)--(0.8308,1.8192)--(0.8018,1.8322)--(0.7726,1.8447)--(0.7433,1.8567)--(0.7137,1.8682)--(0.6840,1.8793)--(0.6541,1.8900)--(0.6240,1.9001)--(0.5938,1.9098)--(0.5634,1.9189)--(0.5329,1.9276)--(0.5022,1.9358)--(0.4715,1.9436)--(0.4406,1.9508)--(0.4096,1.9576)--(0.3785,1.9638)--(0.3472,1.9696)--(0.3160,1.9748)--(0.2846,1.9796)--(0.2531,1.9839)--(0.2216,1.9876)--(0.1901,1.9909)--(0.1584,1.9937)--(0.1268,1.9959)--(0.0951,1.9977)--(0.0634,1.9989)--(0.0317,1.9997)--(0.0000,2.0000);
+\draw [] (1.6383,1.1471) -- (2.0000,1.4004);
+\draw []  (1.6383,1.1471) node [rotate=0] {$\bullet$};
 \draw []  (2.0000,1.4004) node [rotate=0] {$\bullet$};
-\draw []  (1.1472,1.6383) node [rotate=0] {$\bullet$};
+\draw []  (1.1471,1.6383) node [rotate=0] {$\bullet$};
 \draw []  (1.4004,2.0000) node [rotate=0] {$\bullet$};
-\draw [] (1.15,1.64) -- (1.40,2.00);
-\draw (2.0000,-0.31492) node {$ 1 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (-0.29125,2.0000) node {$ 1 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [] (1.1471,1.6383) -- (1.4004,2.0000);
+\draw (2.0000,-0.3149) node {$ 1 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_IntegraleSimple.pstricks.recall b/src_phystricks/Fig_IntegraleSimple.pstricks.recall
index 50af4c229..0d91ab0cb 100644
--- a/src_phystricks/Fig_IntegraleSimple.pstricks.recall
+++ b/src_phystricks/Fig_IntegraleSimple.pstricks.recall
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4994);
 %DEFAULT
-\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
-\draw (-1.5708,-0.27858) node {$a$};
-\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
-\draw (6.2832,-0.32674) node {$b$};
+\draw []  (-1.5707,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5707,-0.2785) node {$a$};
+\draw []  (6.2831,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.2831,-0.3267) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [color=blue,style=solid] (-1.571,1.000)--(-1.491,1.003)--(-1.412,1.013)--(-1.333,1.028)--(-1.253,1.050)--(-1.174,1.078)--(-1.095,1.111)--(-1.015,1.150)--(-0.9361,1.195)--(-0.8568,1.244)--(-0.7775,1.299)--(-0.6981,1.357)--(-0.6188,1.420)--(-0.5395,1.486)--(-0.4601,1.556)--(-0.3808,1.628)--(-0.3015,1.703)--(-0.2221,1.780)--(-0.1428,1.858)--(-0.06347,1.937)--(0.01587,2.016)--(0.09520,2.095)--(0.1745,2.174)--(0.2539,2.251)--(0.3332,2.327)--(0.4125,2.401)--(0.4919,2.472)--(0.5712,2.541)--(0.6505,2.606)--(0.7299,2.667)--(0.8092,2.724)--(0.8885,2.776)--(0.9679,2.824)--(1.047,2.866)--(1.127,2.903)--(1.206,2.934)--(1.285,2.960)--(1.365,2.979)--(1.444,2.992)--(1.523,2.999)--(1.603,3.000)--(1.682,2.994)--(1.761,2.982)--(1.841,2.964)--(1.920,2.940)--(1.999,2.910)--(2.079,2.874)--(2.158,2.833)--(2.237,2.786)--(2.317,2.735)--(2.396,2.678)--(2.475,2.618)--(2.555,2.554)--(2.634,2.486)--(2.713,2.415)--(2.793,2.342)--(2.872,2.266)--(2.951,2.189)--(3.031,2.111)--(3.110,2.032)--(3.189,1.952)--(3.269,1.873)--(3.348,1.795)--(3.427,1.718)--(3.507,1.643)--(3.586,1.570)--(3.665,1.500)--(3.745,1.433)--(3.824,1.369)--(3.903,1.310)--(3.983,1.255)--(4.062,1.204)--(4.141,1.159)--(4.221,1.119)--(4.300,1.084)--(4.379,1.055)--(4.458,1.032)--(4.538,1.015)--(4.617,1.005)--(4.697,1.000)--(4.776,1.002)--(4.855,1.010)--(4.935,1.025)--(5.014,1.045)--(5.093,1.072)--(5.173,1.104)--(5.252,1.142)--(5.331,1.185)--(5.411,1.234)--(5.490,1.287)--(5.569,1.345)--(5.648,1.407)--(5.728,1.473)--(5.807,1.542)--(5.887,1.614)--(5.966,1.688)--(6.045,1.764)--(6.125,1.842)--(6.204,1.921)--(6.283,2.000);
-\draw [color=blue] (-1.571,0)--(-1.491,0)--(-1.412,0)--(-1.333,0)--(-1.253,0)--(-1.174,0)--(-1.095,0)--(-1.015,0)--(-0.9361,0)--(-0.8568,0)--(-0.7775,0)--(-0.6981,0)--(-0.6188,0)--(-0.5395,0)--(-0.4601,0)--(-0.3808,0)--(-0.3015,0)--(-0.2221,0)--(-0.1428,0)--(-0.06347,0)--(0.01587,0)--(0.09520,0)--(0.1745,0)--(0.2539,0)--(0.3332,0)--(0.4125,0)--(0.4919,0)--(0.5712,0)--(0.6505,0)--(0.7299,0)--(0.8092,0)--(0.8885,0)--(0.9679,0)--(1.047,0)--(1.127,0)--(1.206,0)--(1.285,0)--(1.365,0)--(1.444,0)--(1.523,0)--(1.603,0)--(1.682,0)--(1.761,0)--(1.841,0)--(1.920,0)--(1.999,0)--(2.079,0)--(2.158,0)--(2.237,0)--(2.317,0)--(2.396,0)--(2.475,0)--(2.555,0)--(2.634,0)--(2.713,0)--(2.793,0)--(2.872,0)--(2.951,0)--(3.031,0)--(3.110,0)--(3.189,0)--(3.269,0)--(3.348,0)--(3.427,0)--(3.507,0)--(3.586,0)--(3.665,0)--(3.745,0)--(3.824,0)--(3.903,0)--(3.983,0)--(4.062,0)--(4.141,0)--(4.221,0)--(4.300,0)--(4.379,0)--(4.458,0)--(4.538,0)--(4.617,0)--(4.697,0)--(4.776,0)--(4.855,0)--(4.935,0)--(5.014,0)--(5.093,0)--(5.173,0)--(5.252,0)--(5.331,0)--(5.411,0)--(5.490,0)--(5.569,0)--(5.648,0)--(5.728,0)--(5.807,0)--(5.887,0)--(5.966,0)--(6.045,0)--(6.125,0)--(6.204,0)--(6.283,0);
-\draw [style=dashed] (-1.57,0) -- (-1.57,1.00);
-\draw [style=dashed] (6.28,2.00) -- (6.28,0);
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+\draw [color=blue,style=solid] (-1.5707,1.0000)--(-1.4914,1.0031)--(-1.4121,1.0125)--(-1.3327,1.0281)--(-1.2534,1.0499)--(-1.1741,1.0776)--(-1.0947,1.1111)--(-1.0154,1.1502)--(-0.9361,1.1947)--(-0.8567,1.2442)--(-0.7774,1.2985)--(-0.6981,1.3572)--(-0.6187,1.4199)--(-0.5394,1.4863)--(-0.4601,1.5559)--(-0.3807,1.6283)--(-0.3014,1.7030)--(-0.2221,1.7796)--(-0.1427,1.8576)--(-0.0634,1.9365)--(0.0158,2.0158)--(0.0951,2.0950)--(0.1745,2.1736)--(0.2538,2.2511)--(0.3331,2.3270)--(0.4125,2.4009)--(0.4918,2.4722)--(0.5711,2.5406)--(0.6505,2.6056)--(0.7298,2.6667)--(0.8091,2.7237)--(0.8885,2.7761)--(0.9678,2.8236)--(1.0471,2.8660)--(1.1265,2.9029)--(1.2058,2.9341)--(1.2851,2.9594)--(1.3645,2.9788)--(1.4438,2.9919)--(1.5231,2.9988)--(1.6025,2.9994)--(1.6818,2.9938)--(1.7611,2.9819)--(1.8405,2.9638)--(1.9198,2.9396)--(1.9991,2.9096)--(2.0785,2.8738)--(2.1578,2.8325)--(2.2371,2.7860)--(2.3165,2.7345)--(2.3958,2.6785)--(2.4751,2.6181)--(2.5545,2.5539)--(2.6338,2.4861)--(2.7131,2.4154)--(2.7925,2.3420)--(2.8718,2.2664)--(2.9511,2.1892)--(3.0305,2.1108)--(3.1098,2.0317)--(3.1891,1.9524)--(3.2685,1.8734)--(3.3478,1.7951)--(3.4271,1.7182)--(3.5065,1.6431)--(3.5858,1.5702)--(3.6651,1.5000)--(3.7445,1.4329)--(3.8238,1.3694)--(3.9031,1.3099)--(3.9825,1.2547)--(4.0618,1.2042)--(4.1411,1.1587)--(4.2205,1.1185)--(4.2998,1.0838)--(4.3791,1.0549)--(4.4585,1.0320)--(4.5378,1.0151)--(4.6171,1.0045)--(4.6965,1.0001)--(4.7758,1.0020)--(4.8551,1.0101)--(4.9345,1.0245)--(5.0138,1.0450)--(5.0931,1.0716)--(5.1725,1.1040)--(5.2518,1.1420)--(5.3311,1.1854)--(5.4105,1.2339)--(5.4898,1.2873)--(5.5691,1.3451)--(5.6485,1.4070)--(5.7278,1.4727)--(5.8071,1.5417)--(5.8865,1.6136)--(5.9658,1.6879)--(6.0451,1.7642)--(6.1245,1.8419)--(6.2038,1.9207)--(6.2831,2.0000);
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+\draw [style=dashed] (-1.5707,0.0000) -- (-1.5707,1.0000);
+\draw [style=dashed] (6.2831,2.0000) -- (6.2831,0.0000);
 
-\draw [color=blue] (-1.571,1.000)--(-1.491,1.003)--(-1.412,1.013)--(-1.333,1.028)--(-1.253,1.050)--(-1.174,1.078)--(-1.095,1.111)--(-1.015,1.150)--(-0.9361,1.195)--(-0.8568,1.244)--(-0.7775,1.299)--(-0.6981,1.357)--(-0.6188,1.420)--(-0.5395,1.486)--(-0.4601,1.556)--(-0.3808,1.628)--(-0.3015,1.703)--(-0.2221,1.780)--(-0.1428,1.858)--(-0.06347,1.937)--(0.01587,2.016)--(0.09520,2.095)--(0.1745,2.174)--(0.2539,2.251)--(0.3332,2.327)--(0.4125,2.401)--(0.4919,2.472)--(0.5712,2.541)--(0.6505,2.606)--(0.7299,2.667)--(0.8092,2.724)--(0.8885,2.776)--(0.9679,2.824)--(1.047,2.866)--(1.127,2.903)--(1.206,2.934)--(1.285,2.960)--(1.365,2.979)--(1.444,2.992)--(1.523,2.999)--(1.603,3.000)--(1.682,2.994)--(1.761,2.982)--(1.841,2.964)--(1.920,2.940)--(1.999,2.910)--(2.079,2.874)--(2.158,2.833)--(2.237,2.786)--(2.317,2.735)--(2.396,2.678)--(2.475,2.618)--(2.555,2.554)--(2.634,2.486)--(2.713,2.415)--(2.793,2.342)--(2.872,2.266)--(2.951,2.189)--(3.031,2.111)--(3.110,2.032)--(3.189,1.952)--(3.269,1.873)--(3.348,1.795)--(3.427,1.718)--(3.507,1.643)--(3.586,1.570)--(3.665,1.500)--(3.745,1.433)--(3.824,1.369)--(3.903,1.310)--(3.983,1.255)--(4.062,1.204)--(4.141,1.159)--(4.221,1.119)--(4.300,1.084)--(4.379,1.055)--(4.458,1.032)--(4.538,1.015)--(4.617,1.005)--(4.697,1.000)--(4.776,1.002)--(4.855,1.010)--(4.935,1.025)--(5.014,1.045)--(5.093,1.072)--(5.173,1.104)--(5.252,1.142)--(5.331,1.185)--(5.411,1.234)--(5.490,1.287)--(5.569,1.345)--(5.648,1.407)--(5.728,1.473)--(5.807,1.542)--(5.887,1.614)--(5.966,1.688)--(6.045,1.764)--(6.125,1.842)--(6.204,1.921)--(6.283,2.000);
+\draw [color=blue] (-1.5707,1.0000)--(-1.4914,1.0031)--(-1.4121,1.0125)--(-1.3327,1.0281)--(-1.2534,1.0499)--(-1.1741,1.0776)--(-1.0947,1.1111)--(-1.0154,1.1502)--(-0.9361,1.1947)--(-0.8567,1.2442)--(-0.7774,1.2985)--(-0.6981,1.3572)--(-0.6187,1.4199)--(-0.5394,1.4863)--(-0.4601,1.5559)--(-0.3807,1.6283)--(-0.3014,1.7030)--(-0.2221,1.7796)--(-0.1427,1.8576)--(-0.0634,1.9365)--(0.0158,2.0158)--(0.0951,2.0950)--(0.1745,2.1736)--(0.2538,2.2511)--(0.3331,2.3270)--(0.4125,2.4009)--(0.4918,2.4722)--(0.5711,2.5406)--(0.6505,2.6056)--(0.7298,2.6667)--(0.8091,2.7237)--(0.8885,2.7761)--(0.9678,2.8236)--(1.0471,2.8660)--(1.1265,2.9029)--(1.2058,2.9341)--(1.2851,2.9594)--(1.3645,2.9788)--(1.4438,2.9919)--(1.5231,2.9988)--(1.6025,2.9994)--(1.6818,2.9938)--(1.7611,2.9819)--(1.8405,2.9638)--(1.9198,2.9396)--(1.9991,2.9096)--(2.0785,2.8738)--(2.1578,2.8325)--(2.2371,2.7860)--(2.3165,2.7345)--(2.3958,2.6785)--(2.4751,2.6181)--(2.5545,2.5539)--(2.6338,2.4861)--(2.7131,2.4154)--(2.7925,2.3420)--(2.8718,2.2664)--(2.9511,2.1892)--(3.0305,2.1108)--(3.1098,2.0317)--(3.1891,1.9524)--(3.2685,1.8734)--(3.3478,1.7951)--(3.4271,1.7182)--(3.5065,1.6431)--(3.5858,1.5702)--(3.6651,1.5000)--(3.7445,1.4329)--(3.8238,1.3694)--(3.9031,1.3099)--(3.9825,1.2547)--(4.0618,1.2042)--(4.1411,1.1587)--(4.2205,1.1185)--(4.2998,1.0838)--(4.3791,1.0549)--(4.4585,1.0320)--(4.5378,1.0151)--(4.6171,1.0045)--(4.6965,1.0001)--(4.7758,1.0020)--(4.8551,1.0101)--(4.9345,1.0245)--(5.0138,1.0450)--(5.0931,1.0716)--(5.1725,1.1040)--(5.2518,1.1420)--(5.3311,1.1854)--(5.4105,1.2339)--(5.4898,1.2873)--(5.5691,1.3451)--(5.6485,1.4070)--(5.7278,1.4727)--(5.8071,1.5417)--(5.8865,1.6136)--(5.9658,1.6879)--(6.0451,1.7642)--(6.1245,1.8419)--(6.2038,1.9207)--(6.2831,2.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_JJAooWpimYW.pstricks.recall b/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
index 9f0dd8da9..79ade3f77 100644
--- a/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
+++ b/src_phystricks/Fig_JJAooWpimYW.pstricks.recall
@@ -108,35 +108,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
-\draw [,->,>=latex] (0,-1.4989) -- (0,1.4999);
+\draw [,->,>=latex] (-8.3539,0.0000) -- (8.3539,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4988) -- (0.0000,1.4998);
 %DEFAULT
 
-\draw [color=blue] (-7.854,0)--(-7.695,0.1580)--(-7.537,0.3120)--(-7.378,0.4582)--(-7.219,0.5929)--(-7.061,0.7127)--(-6.902,0.8146)--(-6.743,0.8960)--(-6.585,0.9549)--(-6.426,0.9898)--(-6.267,0.9999)--(-6.109,0.9848)--(-5.950,0.9450)--(-5.791,0.8815)--(-5.633,0.7958)--(-5.474,0.6901)--(-5.315,0.5671)--(-5.157,0.4298)--(-4.998,0.2817)--(-4.839,0.1266)--(-4.681,-0.03173)--(-4.522,-0.1893)--(-4.363,-0.3420)--(-4.205,-0.4862)--(-4.046,-0.6182)--(-3.887,-0.7346)--(-3.729,-0.8326)--(-3.570,-0.9096)--(-3.411,-0.9638)--(-3.253,-0.9938)--(-3.094,-0.9989)--(-2.935,-0.9788)--(-2.777,-0.9342)--(-2.618,-0.8660)--(-2.459,-0.7761)--(-2.301,-0.6668)--(-2.142,-0.5406)--(-1.983,-0.4009)--(-1.825,-0.2511)--(-1.666,-0.09506)--(-1.507,0.06342)--(-1.349,0.2203)--(-1.190,0.3717)--(-1.031,0.5137)--(-0.8727,0.6428)--(-0.7140,0.7558)--(-0.5553,0.8497)--(-0.3967,0.9224)--(-0.2380,0.9718)--(-0.07933,0.9969)--(0.07933,0.9969)--(0.2380,0.9718)--(0.3967,0.9224)--(0.5553,0.8497)--(0.7140,0.7558)--(0.8727,0.6428)--(1.031,0.5137)--(1.190,0.3717)--(1.349,0.2203)--(1.507,0.06342)--(1.666,-0.09506)--(1.825,-0.2511)--(1.983,-0.4009)--(2.142,-0.5406)--(2.301,-0.6668)--(2.459,-0.7761)--(2.618,-0.8660)--(2.777,-0.9342)--(2.935,-0.9788)--(3.094,-0.9989)--(3.253,-0.9938)--(3.411,-0.9638)--(3.570,-0.9096)--(3.729,-0.8326)--(3.887,-0.7346)--(4.046,-0.6182)--(4.205,-0.4862)--(4.363,-0.3420)--(4.522,-0.1893)--(4.681,-0.03173)--(4.839,0.1266)--(4.998,0.2817)--(5.157,0.4298)--(5.315,0.5671)--(5.474,0.6901)--(5.633,0.7958)--(5.791,0.8815)--(5.950,0.9450)--(6.109,0.9848)--(6.267,0.9999)--(6.426,0.9898)--(6.585,0.9549)--(6.743,0.8960)--(6.902,0.8146)--(7.061,0.7127)--(7.219,0.5929)--(7.378,0.4582)--(7.537,0.3120)--(7.695,0.1580)--(7.854,0);
-\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
-\draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
-\draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-7.8539,0.0000)--(-7.6953,0.1580)--(-7.5366,0.3120)--(-7.3779,0.4582)--(-7.2193,0.5929)--(-7.0606,0.7126)--(-6.9019,0.8145)--(-6.7433,0.8959)--(-6.5846,0.9549)--(-6.4259,0.9898)--(-6.2673,0.9998)--(-6.1086,0.9848)--(-5.9499,0.9450)--(-5.7913,0.8814)--(-5.6326,0.7957)--(-5.4739,0.6900)--(-5.3153,0.5670)--(-5.1566,0.4297)--(-4.9979,0.2817)--(-4.8393,0.1265)--(-4.6806,-0.0317)--(-4.5219,-0.1892)--(-4.3633,-0.3420)--(-4.2046,-0.4861)--(-4.0459,-0.6181)--(-3.8873,-0.7345)--(-3.7286,-0.8325)--(-3.5699,-0.9096)--(-3.4113,-0.9638)--(-3.2526,-0.9938)--(-3.0939,-0.9988)--(-2.9353,-0.9788)--(-2.7766,-0.9341)--(-2.6179,-0.8660)--(-2.4593,-0.7761)--(-2.3006,-0.6667)--(-2.1419,-0.5406)--(-1.9833,-0.4009)--(-1.8246,-0.2511)--(-1.6659,-0.0950)--(-1.5073,0.0634)--(-1.3486,0.2203)--(-1.1899,0.3716)--(-1.0313,0.5136)--(-0.8726,0.6427)--(-0.7139,0.7557)--(-0.5553,0.8497)--(-0.3966,0.9223)--(-0.2379,0.9718)--(-0.0793,0.9968)--(0.0793,0.9968)--(0.2379,0.9718)--(0.3966,0.9223)--(0.5553,0.8497)--(0.7139,0.7557)--(0.8726,0.6427)--(1.0313,0.5136)--(1.1899,0.3716)--(1.3486,0.2203)--(1.5073,0.0634)--(1.6659,-0.0950)--(1.8246,-0.2511)--(1.9833,-0.4009)--(2.1419,-0.5406)--(2.3006,-0.6667)--(2.4593,-0.7761)--(2.6179,-0.8660)--(2.7766,-0.9341)--(2.9353,-0.9788)--(3.0939,-0.9988)--(3.2526,-0.9938)--(3.4113,-0.9638)--(3.5699,-0.9096)--(3.7286,-0.8325)--(3.8873,-0.7345)--(4.0459,-0.6181)--(4.2046,-0.4861)--(4.3633,-0.3420)--(4.5219,-0.1892)--(4.6806,-0.0317)--(4.8393,0.1265)--(4.9979,0.2817)--(5.1566,0.4297)--(5.3153,0.5670)--(5.4739,0.6900)--(5.6326,0.7957)--(5.7913,0.8814)--(5.9499,0.9450)--(6.1086,0.9848)--(6.2673,0.9998)--(6.4259,0.9898)--(6.5846,0.9549)--(6.7433,0.8959)--(6.9019,0.8145)--(7.0606,0.7126)--(7.2193,0.5929)--(7.3779,0.4582)--(7.5366,0.3120)--(7.6953,0.1580)--(7.8539,0.0000);
+\draw (-7.8539,-0.4207) node {$ -\frac{5}{2} \, \pi $};
+\draw [] (-7.8539,-0.1000) -- (-7.8539,0.1000);
+\draw (-6.2831,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-6.2831,-0.1000) -- (-6.2831,0.1000);
+\draw (-4.7123,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-4.7123,-0.1000) -- (-4.7123,0.1000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (7.8539,-0.4207) node {$ \frac{5}{2} \, \pi $};
+\draw [] (7.8539,-0.1000) -- (7.8539,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall b/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
index ab72b83d5..e2b94607d 100644
--- a/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
+++ b/src_phystricks/Fig_JSLooFJWXtB.pstricks.recall
@@ -92,8 +92,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.2124,0) -- (5.2124,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-5.2123,0.0000) -- (5.2123,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -102,11 +102,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [color=blue] (-4.712,1.000)--(-4.697,0.9999)--(-4.681,0.9995)--(-4.665,0.9989)--(-4.649,0.9980)--(-4.633,0.9969)--(-4.617,0.9955)--(-4.601,0.9938)--(-4.585,0.9920)--(-4.570,0.9898)--(-4.554,0.9874)--(-4.538,0.9848)--(-4.522,0.9819)--(-4.506,0.9788)--(-4.490,0.9754)--(-4.474,0.9718)--(-4.458,0.9679)--(-4.443,0.9638)--(-4.427,0.9595)--(-4.411,0.9549)--(-4.395,0.9501)--(-4.379,0.9450)--(-4.363,0.9397)--(-4.347,0.9342)--(-4.332,0.9284)--(-4.316,0.9224)--(-4.300,0.9161)--(-4.284,0.9096)--(-4.268,0.9029)--(-4.252,0.8960)--(-4.236,0.8888)--(-4.221,0.8815)--(-4.205,0.8738)--(-4.189,0.8660)--(-4.173,0.8580)--(-4.157,0.8497)--(-4.141,0.8413)--(-4.125,0.8326)--(-4.109,0.8237)--(-4.094,0.8146)--(-4.078,0.8053)--(-4.062,0.7958)--(-4.046,0.7861)--(-4.030,0.7761)--(-4.014,0.7660)--(-3.998,0.7558)--(-3.983,0.7453)--(-3.967,0.7346)--(-3.951,0.7237)--(-3.935,0.7127)--(-3.919,0.7015)--(-3.903,0.6901)--(-3.887,0.6785)--(-3.871,0.6668)--(-3.856,0.6549)--(-3.840,0.6428)--(-3.824,0.6306)--(-3.808,0.6182)--(-3.792,0.6056)--(-3.776,0.5929)--(-3.760,0.5801)--(-3.745,0.5671)--(-3.729,0.5539)--(-3.713,0.5406)--(-3.697,0.5272)--(-3.681,0.5137)--(-3.665,0.5000)--(-3.649,0.4862)--(-3.633,0.4723)--(-3.618,0.4582)--(-3.602,0.4441)--(-3.586,0.4298)--(-3.570,0.4154)--(-3.554,0.4009)--(-3.538,0.3863)--(-3.522,0.3717)--(-3.507,0.3569)--(-3.491,0.3420)--(-3.475,0.3271)--(-3.459,0.3120)--(-3.443,0.2969)--(-3.427,0.2817)--(-3.411,0.2665)--(-3.395,0.2511)--(-3.380,0.2358)--(-3.364,0.2203)--(-3.348,0.2048)--(-3.332,0.1893)--(-3.316,0.1736)--(-3.300,0.1580)--(-3.284,0.1423)--(-3.269,0.1266)--(-3.253,0.1108)--(-3.237,0.09506)--(-3.221,0.07925)--(-3.205,0.06342)--(-3.189,0.04758)--(-3.173,0.03173)--(-3.157,0.01587)--(-3.142,0);
-\draw [color=blue] (-4.712,0)--(-4.697,0)--(-4.681,0)--(-4.665,0)--(-4.649,0)--(-4.633,0)--(-4.617,0)--(-4.601,0)--(-4.585,0)--(-4.570,0)--(-4.554,0)--(-4.538,0)--(-4.522,0)--(-4.506,0)--(-4.490,0)--(-4.474,0)--(-4.458,0)--(-4.443,0)--(-4.427,0)--(-4.411,0)--(-4.395,0)--(-4.379,0)--(-4.363,0)--(-4.347,0)--(-4.332,0)--(-4.316,0)--(-4.300,0)--(-4.284,0)--(-4.268,0)--(-4.252,0)--(-4.236,0)--(-4.221,0)--(-4.205,0)--(-4.189,0)--(-4.173,0)--(-4.157,0)--(-4.141,0)--(-4.125,0)--(-4.109,0)--(-4.094,0)--(-4.078,0)--(-4.062,0)--(-4.046,0)--(-4.030,0)--(-4.014,0)--(-3.998,0)--(-3.983,0)--(-3.967,0)--(-3.951,0)--(-3.935,0)--(-3.919,0)--(-3.903,0)--(-3.887,0)--(-3.871,0)--(-3.856,0)--(-3.840,0)--(-3.824,0)--(-3.808,0)--(-3.792,0)--(-3.776,0)--(-3.760,0)--(-3.745,0)--(-3.729,0)--(-3.713,0)--(-3.697,0)--(-3.681,0)--(-3.665,0)--(-3.649,0)--(-3.633,0)--(-3.618,0)--(-3.602,0)--(-3.586,0)--(-3.570,0)--(-3.554,0)--(-3.538,0)--(-3.522,0)--(-3.507,0)--(-3.491,0)--(-3.475,0)--(-3.459,0)--(-3.443,0)--(-3.427,0)--(-3.411,0)--(-3.395,0)--(-3.380,0)--(-3.364,0)--(-3.348,0)--(-3.332,0)--(-3.316,0)--(-3.300,0)--(-3.284,0)--(-3.269,0)--(-3.253,0)--(-3.237,0)--(-3.221,0)--(-3.205,0)--(-3.189,0)--(-3.173,0)--(-3.157,0)--(-3.142,0);
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-\draw [] (-3.14,0) -- (-3.14,0);
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+\draw [] (-3.1415,0.0000) -- (-3.1415,0.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -114,11 +114,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -126,11 +126,11 @@
 % setting the default values
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          hatchthickness=0.4pt}
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -138,29 +138,29 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (3.1415,0.0000) -- (3.1415,0.0000);
+\draw [] (4.7123,-1.0000) -- (4.7123,0.0000);
 
-\draw [color=blue] (-4.712,1.000)--(-4.617,0.9955)--(-4.522,0.9819)--(-4.427,0.9595)--(-4.332,0.9284)--(-4.236,0.8888)--(-4.141,0.8413)--(-4.046,0.7861)--(-3.951,0.7237)--(-3.856,0.6549)--(-3.760,0.5801)--(-3.665,0.5000)--(-3.570,0.4154)--(-3.475,0.3271)--(-3.380,0.2358)--(-3.284,0.1423)--(-3.189,0.04758)--(-3.094,-0.04758)--(-2.999,-0.1423)--(-2.904,-0.2358)--(-2.808,-0.3271)--(-2.713,-0.4154)--(-2.618,-0.5000)--(-2.523,-0.5801)--(-2.428,-0.6549)--(-2.332,-0.7237)--(-2.237,-0.7861)--(-2.142,-0.8413)--(-2.047,-0.8888)--(-1.952,-0.9284)--(-1.856,-0.9595)--(-1.761,-0.9819)--(-1.666,-0.9955)--(-1.571,-1.000)--(-1.476,-0.9955)--(-1.380,-0.9819)--(-1.285,-0.9595)--(-1.190,-0.9284)--(-1.095,-0.8888)--(-0.9996,-0.8413)--(-0.9044,-0.7861)--(-0.8092,-0.7237)--(-0.7140,-0.6549)--(-0.6188,-0.5801)--(-0.5236,-0.5000)--(-0.4284,-0.4154)--(-0.3332,-0.3271)--(-0.2380,-0.2358)--(-0.1428,-0.1423)--(-0.04760,-0.04758)--(0.04760,0.04758)--(0.1428,0.1423)--(0.2380,0.2358)--(0.3332,0.3271)--(0.4284,0.4154)--(0.5236,0.5000)--(0.6188,0.5801)--(0.7140,0.6549)--(0.8092,0.7237)--(0.9044,0.7861)--(0.9996,0.8413)--(1.095,0.8888)--(1.190,0.9284)--(1.285,0.9595)--(1.380,0.9819)--(1.476,0.9955)--(1.571,1.000)--(1.666,0.9955)--(1.761,0.9819)--(1.856,0.9595)--(1.952,0.9284)--(2.047,0.8888)--(2.142,0.8413)--(2.237,0.7861)--(2.332,0.7237)--(2.428,0.6549)--(2.523,0.5801)--(2.618,0.5000)--(2.713,0.4154)--(2.808,0.3271)--(2.904,0.2358)--(2.999,0.1423)--(3.094,0.04758)--(3.189,-0.04758)--(3.284,-0.1423)--(3.380,-0.2358)--(3.475,-0.3271)--(3.570,-0.4154)--(3.665,-0.5000)--(3.760,-0.5801)--(3.856,-0.6549)--(3.951,-0.7237)--(4.046,-0.7861)--(4.141,-0.8413)--(4.236,-0.8888)--(4.332,-0.9284)--(4.427,-0.9595)--(4.522,-0.9819)--(4.617,-0.9955)--(4.712,-1.000);
-\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-4.7123,1.0000)--(-4.6171,0.9954)--(-4.5219,0.9819)--(-4.4267,0.9594)--(-4.3315,0.9283)--(-4.2363,0.8888)--(-4.1411,0.8412)--(-4.0459,0.7860)--(-3.9507,0.7237)--(-3.8555,0.6548)--(-3.7603,0.5800)--(-3.6651,0.5000)--(-3.5699,0.4154)--(-3.4747,0.3270)--(-3.3795,0.2357)--(-3.2843,0.1423)--(-3.1891,0.0475)--(-3.0939,-0.0475)--(-2.9987,-0.1423)--(-2.9035,-0.2357)--(-2.8083,-0.3270)--(-2.7131,-0.4154)--(-2.6179,-0.5000)--(-2.5227,-0.5800)--(-2.4275,-0.6548)--(-2.3323,-0.7237)--(-2.2371,-0.7860)--(-2.1419,-0.8412)--(-2.0467,-0.8888)--(-1.9515,-0.9283)--(-1.8563,-0.9594)--(-1.7611,-0.9819)--(-1.6659,-0.9954)--(-1.5707,-1.0000)--(-1.4755,-0.9954)--(-1.3803,-0.9819)--(-1.2851,-0.9594)--(-1.1899,-0.9283)--(-1.0947,-0.8888)--(-0.9995,-0.8412)--(-0.9043,-0.7860)--(-0.8091,-0.7237)--(-0.7139,-0.6548)--(-0.6187,-0.5800)--(-0.5235,-0.5000)--(-0.4283,-0.4154)--(-0.3331,-0.3270)--(-0.2379,-0.2357)--(-0.1427,-0.1423)--(-0.0475,-0.0475)--(0.0475,0.0475)--(0.1427,0.1423)--(0.2379,0.2357)--(0.3331,0.3270)--(0.4283,0.4154)--(0.5235,0.4999)--(0.6187,0.5800)--(0.7139,0.6548)--(0.8091,0.7237)--(0.9043,0.7860)--(0.9995,0.8412)--(1.0947,0.8888)--(1.1899,0.9283)--(1.2851,0.9594)--(1.3803,0.9819)--(1.4755,0.9954)--(1.5707,1.0000)--(1.6659,0.9954)--(1.7611,0.9819)--(1.8563,0.9594)--(1.9515,0.9283)--(2.0467,0.8888)--(2.1419,0.8412)--(2.2371,0.7860)--(2.3323,0.7237)--(2.4275,0.6548)--(2.5227,0.5800)--(2.6179,0.5000)--(2.7131,0.4154)--(2.8083,0.3270)--(2.9035,0.2357)--(2.9987,0.1423)--(3.0939,0.0475)--(3.1891,-0.0475)--(3.2843,-0.1423)--(3.3795,-0.2357)--(3.4747,-0.3270)--(3.5699,-0.4154)--(3.6651,-0.4999)--(3.7603,-0.5800)--(3.8555,-0.6548)--(3.9507,-0.7237)--(4.0459,-0.7860)--(4.1411,-0.8412)--(4.2363,-0.8888)--(4.3315,-0.9283)--(4.4267,-0.9594)--(4.5219,-0.9819)--(4.6171,-0.9954)--(4.7123,-1.0000);
+\draw (-4.7123,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-4.7123,-0.1000) -- (-4.7123,0.1000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall b/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall
index 92685399c..f9cdf1b0f 100644
--- a/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall
+++ b/src_phystricks/Fig_KKLooMbjxdI.pstricks.recall
@@ -71,13 +71,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.0708,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
+\draw [,->,>=latex] (-2.0707,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4994);
 %DEFAULT
-\draw []  (-1.5708,0) node [rotate=0] {$\bullet$};
-\draw (-1.5708,-0.27858) node {$a$};
-\draw []  (6.2832,0) node [rotate=0] {$\bullet$};
-\draw (6.2832,-0.32674) node {$b$};
+\draw []  (-1.5707,0.0000) node [rotate=0] {$\bullet$};
+\draw (-1.5707,-0.2785) node {$a$};
+\draw []  (6.2831,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.2831,-0.3267) node {$b$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [color=blue,style=solid] (-1.571,1.000)--(-1.491,1.003)--(-1.412,1.013)--(-1.333,1.028)--(-1.253,1.050)--(-1.174,1.078)--(-1.095,1.111)--(-1.015,1.150)--(-0.9361,1.195)--(-0.8568,1.244)--(-0.7775,1.299)--(-0.6981,1.357)--(-0.6188,1.420)--(-0.5395,1.486)--(-0.4601,1.556)--(-0.3808,1.628)--(-0.3015,1.703)--(-0.2221,1.780)--(-0.1428,1.858)--(-0.06347,1.937)--(0.01587,2.016)--(0.09520,2.095)--(0.1745,2.174)--(0.2539,2.251)--(0.3332,2.327)--(0.4125,2.401)--(0.4919,2.472)--(0.5712,2.541)--(0.6505,2.606)--(0.7299,2.667)--(0.8092,2.724)--(0.8885,2.776)--(0.9679,2.824)--(1.047,2.866)--(1.127,2.903)--(1.206,2.934)--(1.285,2.960)--(1.365,2.979)--(1.444,2.992)--(1.523,2.999)--(1.603,3.000)--(1.682,2.994)--(1.761,2.982)--(1.841,2.964)--(1.920,2.940)--(1.999,2.910)--(2.079,2.874)--(2.158,2.833)--(2.237,2.786)--(2.317,2.735)--(2.396,2.678)--(2.475,2.618)--(2.555,2.554)--(2.634,2.486)--(2.713,2.415)--(2.793,2.342)--(2.872,2.266)--(2.951,2.189)--(3.031,2.111)--(3.110,2.032)--(3.189,1.952)--(3.269,1.873)--(3.348,1.795)--(3.427,1.718)--(3.507,1.643)--(3.586,1.570)--(3.665,1.500)--(3.745,1.433)--(3.824,1.369)--(3.903,1.310)--(3.983,1.255)--(4.062,1.204)--(4.141,1.159)--(4.221,1.119)--(4.300,1.084)--(4.379,1.055)--(4.458,1.032)--(4.538,1.015)--(4.617,1.005)--(4.697,1.000)--(4.776,1.002)--(4.855,1.010)--(4.935,1.025)--(5.014,1.045)--(5.093,1.072)--(5.173,1.104)--(5.252,1.142)--(5.331,1.185)--(5.411,1.234)--(5.490,1.287)--(5.569,1.345)--(5.648,1.407)--(5.728,1.473)--(5.807,1.542)--(5.887,1.614)--(5.966,1.688)--(6.045,1.764)--(6.125,1.842)--(6.204,1.921)--(6.283,2.000);
-\draw [color=blue] (-1.571,0)--(-1.491,0)--(-1.412,0)--(-1.333,0)--(-1.253,0)--(-1.174,0)--(-1.095,0)--(-1.015,0)--(-0.9361,0)--(-0.8568,0)--(-0.7775,0)--(-0.6981,0)--(-0.6188,0)--(-0.5395,0)--(-0.4601,0)--(-0.3808,0)--(-0.3015,0)--(-0.2221,0)--(-0.1428,0)--(-0.06347,0)--(0.01587,0)--(0.09520,0)--(0.1745,0)--(0.2539,0)--(0.3332,0)--(0.4125,0)--(0.4919,0)--(0.5712,0)--(0.6505,0)--(0.7299,0)--(0.8092,0)--(0.8885,0)--(0.9679,0)--(1.047,0)--(1.127,0)--(1.206,0)--(1.285,0)--(1.365,0)--(1.444,0)--(1.523,0)--(1.603,0)--(1.682,0)--(1.761,0)--(1.841,0)--(1.920,0)--(1.999,0)--(2.079,0)--(2.158,0)--(2.237,0)--(2.317,0)--(2.396,0)--(2.475,0)--(2.555,0)--(2.634,0)--(2.713,0)--(2.793,0)--(2.872,0)--(2.951,0)--(3.031,0)--(3.110,0)--(3.189,0)--(3.269,0)--(3.348,0)--(3.427,0)--(3.507,0)--(3.586,0)--(3.665,0)--(3.745,0)--(3.824,0)--(3.903,0)--(3.983,0)--(4.062,0)--(4.141,0)--(4.221,0)--(4.300,0)--(4.379,0)--(4.458,0)--(4.538,0)--(4.617,0)--(4.697,0)--(4.776,0)--(4.855,0)--(4.935,0)--(5.014,0)--(5.093,0)--(5.173,0)--(5.252,0)--(5.331,0)--(5.411,0)--(5.490,0)--(5.569,0)--(5.648,0)--(5.728,0)--(5.807,0)--(5.887,0)--(5.966,0)--(6.045,0)--(6.125,0)--(6.204,0)--(6.283,0);
-\draw [style=dashed] (-1.57,0) -- (-1.57,1.00);
-\draw [style=dashed] (6.28,2.00) -- (6.28,0);
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+\draw [color=blue] (-1.5707,0.0000)--(-1.4914,0.0000)--(-1.4121,0.0000)--(-1.3327,0.0000)--(-1.2534,0.0000)--(-1.1741,0.0000)--(-1.0947,0.0000)--(-1.0154,0.0000)--(-0.9361,0.0000)--(-0.8567,0.0000)--(-0.7774,0.0000)--(-0.6981,0.0000)--(-0.6187,0.0000)--(-0.5394,0.0000)--(-0.4601,0.0000)--(-0.3807,0.0000)--(-0.3014,0.0000)--(-0.2221,0.0000)--(-0.1427,0.0000)--(-0.0634,0.0000)--(0.0158,0.0000)--(0.0951,0.0000)--(0.1745,0.0000)--(0.2538,0.0000)--(0.3331,0.0000)--(0.4125,0.0000)--(0.4918,0.0000)--(0.5711,0.0000)--(0.6505,0.0000)--(0.7298,0.0000)--(0.8091,0.0000)--(0.8885,0.0000)--(0.9678,0.0000)--(1.0471,0.0000)--(1.1265,0.0000)--(1.2058,0.0000)--(1.2851,0.0000)--(1.3645,0.0000)--(1.4438,0.0000)--(1.5231,0.0000)--(1.6025,0.0000)--(1.6818,0.0000)--(1.7611,0.0000)--(1.8405,0.0000)--(1.9198,0.0000)--(1.9991,0.0000)--(2.0785,0.0000)--(2.1578,0.0000)--(2.2371,0.0000)--(2.3165,0.0000)--(2.3958,0.0000)--(2.4751,0.0000)--(2.5545,0.0000)--(2.6338,0.0000)--(2.7131,0.0000)--(2.7925,0.0000)--(2.8718,0.0000)--(2.9511,0.0000)--(3.0305,0.0000)--(3.1098,0.0000)--(3.1891,0.0000)--(3.2685,0.0000)--(3.3478,0.0000)--(3.4271,0.0000)--(3.5065,0.0000)--(3.5858,0.0000)--(3.6651,0.0000)--(3.7445,0.0000)--(3.8238,0.0000)--(3.9031,0.0000)--(3.9825,0.0000)--(4.0618,0.0000)--(4.1411,0.0000)--(4.2205,0.0000)--(4.2998,0.0000)--(4.3791,0.0000)--(4.4585,0.0000)--(4.5378,0.0000)--(4.6171,0.0000)--(4.6965,0.0000)--(4.7758,0.0000)--(4.8551,0.0000)--(4.9345,0.0000)--(5.0138,0.0000)--(5.0931,0.0000)--(5.1725,0.0000)--(5.2518,0.0000)--(5.3311,0.0000)--(5.4105,0.0000)--(5.4898,0.0000)--(5.5691,0.0000)--(5.6485,0.0000)--(5.7278,0.0000)--(5.8071,0.0000)--(5.8865,0.0000)--(5.9658,0.0000)--(6.0451,0.0000)--(6.1245,0.0000)--(6.2038,0.0000)--(6.2831,0.0000);
+\draw [style=dashed] (-1.5707,0.0000) -- (-1.5707,1.0000);
+\draw [style=dashed] (6.2831,2.0000) -- (6.2831,0.0000);
 
-\draw [color=blue] (-1.571,1.000)--(-1.491,1.003)--(-1.412,1.013)--(-1.333,1.028)--(-1.253,1.050)--(-1.174,1.078)--(-1.095,1.111)--(-1.015,1.150)--(-0.9361,1.195)--(-0.8568,1.244)--(-0.7775,1.299)--(-0.6981,1.357)--(-0.6188,1.420)--(-0.5395,1.486)--(-0.4601,1.556)--(-0.3808,1.628)--(-0.3015,1.703)--(-0.2221,1.780)--(-0.1428,1.858)--(-0.06347,1.937)--(0.01587,2.016)--(0.09520,2.095)--(0.1745,2.174)--(0.2539,2.251)--(0.3332,2.327)--(0.4125,2.401)--(0.4919,2.472)--(0.5712,2.541)--(0.6505,2.606)--(0.7299,2.667)--(0.8092,2.724)--(0.8885,2.776)--(0.9679,2.824)--(1.047,2.866)--(1.127,2.903)--(1.206,2.934)--(1.285,2.960)--(1.365,2.979)--(1.444,2.992)--(1.523,2.999)--(1.603,3.000)--(1.682,2.994)--(1.761,2.982)--(1.841,2.964)--(1.920,2.940)--(1.999,2.910)--(2.079,2.874)--(2.158,2.833)--(2.237,2.786)--(2.317,2.735)--(2.396,2.678)--(2.475,2.618)--(2.555,2.554)--(2.634,2.486)--(2.713,2.415)--(2.793,2.342)--(2.872,2.266)--(2.951,2.189)--(3.031,2.111)--(3.110,2.032)--(3.189,1.952)--(3.269,1.873)--(3.348,1.795)--(3.427,1.718)--(3.507,1.643)--(3.586,1.570)--(3.665,1.500)--(3.745,1.433)--(3.824,1.369)--(3.903,1.310)--(3.983,1.255)--(4.062,1.204)--(4.141,1.159)--(4.221,1.119)--(4.300,1.084)--(4.379,1.055)--(4.458,1.032)--(4.538,1.015)--(4.617,1.005)--(4.697,1.000)--(4.776,1.002)--(4.855,1.010)--(4.935,1.025)--(5.014,1.045)--(5.093,1.072)--(5.173,1.104)--(5.252,1.142)--(5.331,1.185)--(5.411,1.234)--(5.490,1.287)--(5.569,1.345)--(5.648,1.407)--(5.728,1.473)--(5.807,1.542)--(5.887,1.614)--(5.966,1.688)--(6.045,1.764)--(6.125,1.842)--(6.204,1.921)--(6.283,2.000);
+\draw [color=blue] (-1.5707,1.0000)--(-1.4914,1.0031)--(-1.4121,1.0125)--(-1.3327,1.0281)--(-1.2534,1.0499)--(-1.1741,1.0776)--(-1.0947,1.1111)--(-1.0154,1.1502)--(-0.9361,1.1947)--(-0.8567,1.2442)--(-0.7774,1.2985)--(-0.6981,1.3572)--(-0.6187,1.4199)--(-0.5394,1.4863)--(-0.4601,1.5559)--(-0.3807,1.6283)--(-0.3014,1.7030)--(-0.2221,1.7796)--(-0.1427,1.8576)--(-0.0634,1.9365)--(0.0158,2.0158)--(0.0951,2.0950)--(0.1745,2.1736)--(0.2538,2.2511)--(0.3331,2.3270)--(0.4125,2.4009)--(0.4918,2.4722)--(0.5711,2.5406)--(0.6505,2.6056)--(0.7298,2.6667)--(0.8091,2.7237)--(0.8885,2.7761)--(0.9678,2.8236)--(1.0471,2.8660)--(1.1265,2.9029)--(1.2058,2.9341)--(1.2851,2.9594)--(1.3645,2.9788)--(1.4438,2.9919)--(1.5231,2.9988)--(1.6025,2.9994)--(1.6818,2.9938)--(1.7611,2.9819)--(1.8405,2.9638)--(1.9198,2.9396)--(1.9991,2.9096)--(2.0785,2.8738)--(2.1578,2.8325)--(2.2371,2.7860)--(2.3165,2.7345)--(2.3958,2.6785)--(2.4751,2.6181)--(2.5545,2.5539)--(2.6338,2.4861)--(2.7131,2.4154)--(2.7925,2.3420)--(2.8718,2.2664)--(2.9511,2.1892)--(3.0305,2.1108)--(3.1098,2.0317)--(3.1891,1.9524)--(3.2685,1.8734)--(3.3478,1.7951)--(3.4271,1.7182)--(3.5065,1.6431)--(3.5858,1.5702)--(3.6651,1.5000)--(3.7445,1.4329)--(3.8238,1.3694)--(3.9031,1.3099)--(3.9825,1.2547)--(4.0618,1.2042)--(4.1411,1.1587)--(4.2205,1.1185)--(4.2998,1.0838)--(4.3791,1.0549)--(4.4585,1.0320)--(4.5378,1.0151)--(4.6171,1.0045)--(4.6965,1.0001)--(4.7758,1.0020)--(4.8551,1.0101)--(4.9345,1.0245)--(5.0138,1.0450)--(5.0931,1.0716)--(5.1725,1.1040)--(5.2518,1.1420)--(5.3311,1.1854)--(5.4105,1.2339)--(5.4898,1.2873)--(5.5691,1.3451)--(5.6485,1.4070)--(5.7278,1.4727)--(5.8071,1.5417)--(5.8865,1.6136)--(5.9658,1.6879)--(6.0451,1.7642)--(6.1245,1.8419)--(6.2038,1.9207)--(6.2831,2.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_KKRooHseDzC.pstricks.recall b/src_phystricks/Fig_KKRooHseDzC.pstricks.recall
index 62e8d054f..0537c7bdf 100644
--- a/src_phystricks/Fig_KKRooHseDzC.pstricks.recall
+++ b/src_phystricks/Fig_KKRooHseDzC.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.3750);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.3749);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (0.0000,0.0000) -- (0.0000,1.9861);
+\draw [] (5.0000,2.8194) -- (5.0000,0.0000);
 
-\draw [color=black] (-0.5000,1.750)--(-0.4242,1.788)--(-0.3485,1.824)--(-0.2727,1.861)--(-0.1970,1.896)--(-0.1212,1.931)--(-0.04545,1.966)--(0.03030,2.000)--(0.1061,2.033)--(0.1818,2.065)--(0.2576,2.097)--(0.3333,2.128)--(0.4091,2.159)--(0.4848,2.189)--(0.5606,2.218)--(0.6364,2.246)--(0.7121,2.274)--(0.7879,2.302)--(0.8636,2.329)--(0.9394,2.355)--(1.015,2.380)--(1.091,2.405)--(1.167,2.429)--(1.242,2.453)--(1.318,2.475)--(1.394,2.498)--(1.470,2.519)--(1.545,2.540)--(1.621,2.561)--(1.697,2.580)--(1.773,2.599)--(1.848,2.618)--(1.924,2.636)--(2.000,2.653)--(2.076,2.669)--(2.152,2.685)--(2.227,2.700)--(2.303,2.715)--(2.379,2.729)--(2.455,2.742)--(2.530,2.755)--(2.606,2.767)--(2.682,2.778)--(2.758,2.789)--(2.833,2.799)--(2.909,2.809)--(2.985,2.818)--(3.061,2.826)--(3.136,2.834)--(3.212,2.841)--(3.288,2.847)--(3.364,2.853)--(3.439,2.858)--(3.515,2.862)--(3.591,2.866)--(3.667,2.869)--(3.742,2.871)--(3.818,2.873)--(3.894,2.874)--(3.970,2.875)--(4.045,2.875)--(4.121,2.874)--(4.197,2.873)--(4.273,2.871)--(4.349,2.868)--(4.424,2.865)--(4.500,2.861)--(4.576,2.857)--(4.651,2.851)--(4.727,2.846)--(4.803,2.839)--(4.879,2.832)--(4.955,2.824)--(5.030,2.816)--(5.106,2.807)--(5.182,2.797)--(5.258,2.787)--(5.333,2.776)--(5.409,2.765)--(5.485,2.753)--(5.561,2.740)--(5.636,2.726)--(5.712,2.712)--(5.788,2.697)--(5.864,2.682)--(5.939,2.666)--(6.015,2.649)--(6.091,2.632)--(6.167,2.614)--(6.242,2.596)--(6.318,2.576)--(6.394,2.557)--(6.470,2.536)--(6.545,2.515)--(6.621,2.493)--(6.697,2.471)--(6.773,2.448)--(6.849,2.424)--(6.924,2.400)--(7.000,2.375);
+\draw [color=black] (-0.5000,1.7500)--(-0.4242,1.7875)--(-0.3484,1.8244)--(-0.2727,1.8607)--(-0.1969,1.8964)--(-0.1212,1.9314)--(-0.0454,1.9657)--(0.0303,1.9995)--(0.1060,2.0326)--(0.1818,2.0650)--(0.2575,2.0969)--(0.3333,2.1280)--(0.4090,2.1586)--(0.4848,2.1885)--(0.5606,2.2178)--(0.6363,2.2464)--(0.7121,2.2744)--(0.7878,2.3017)--(0.8636,2.3285)--(0.9393,2.3545)--(1.0151,2.3800)--(1.0909,2.4048)--(1.1666,2.4290)--(1.2424,2.4525)--(1.3181,2.4754)--(1.3939,2.4976)--(1.4696,2.5193)--(1.5454,2.5402)--(1.6212,2.5606)--(1.6969,2.5803)--(1.7727,2.5994)--(1.8484,2.6178)--(1.9242,2.6356)--(2.0000,2.6527)--(2.0757,2.6692)--(2.1515,2.6851)--(2.2272,2.7004)--(2.3030,2.7150)--(2.3787,2.7289)--(2.4545,2.7423)--(2.5303,2.7549)--(2.6060,2.7670)--(2.6818,2.7784)--(2.7575,2.7892)--(2.8333,2.7993)--(2.9090,2.8088)--(2.9848,2.8177)--(3.0606,2.8259)--(3.1363,2.8335)--(3.2121,2.8405)--(3.2878,2.8468)--(3.3636,2.8525)--(3.4393,2.8575)--(3.5151,2.8619)--(3.5909,2.8657)--(3.6666,2.8688)--(3.7424,2.8713)--(3.8181,2.8731)--(3.8939,2.8743)--(3.9696,2.8749)--(4.0454,2.8748)--(4.1212,2.8741)--(4.1969,2.8728)--(4.2727,2.8708)--(4.3484,2.8682)--(4.4242,2.8650)--(4.5000,2.8611)--(4.5757,2.8565)--(4.6515,2.8514)--(4.7272,2.8456)--(4.8030,2.8391)--(4.8787,2.8320)--(4.9545,2.8243)--(5.0303,2.8160)--(5.1060,2.8070)--(5.1818,2.7974)--(5.2575,2.7871)--(5.3333,2.7762)--(5.4090,2.7646)--(5.4848,2.7525)--(5.5606,2.7396)--(5.6363,2.7262)--(5.7121,2.7121)--(5.7878,2.6974)--(5.8636,2.6820)--(5.9393,2.6660)--(6.0151,2.6493)--(6.0909,2.6321)--(6.1666,2.6141)--(6.2424,2.5956)--(6.3181,2.5764)--(6.3939,2.5566)--(6.4696,2.5361)--(6.5454,2.5150)--(6.6212,2.4932)--(6.6969,2.4709)--(6.7727,2.4478)--(6.8484,2.4242)--(6.9242,2.3999)--(7.0000,2.3750);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.82) -- (6.00,2.82) -- (5.00,2.82) -- (5.00,2.82) -- (5.00,0) --  cycle;
-\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.82);
-\draw [color=red,style=dashed] (6.00,2.82) -- (5.00,2.82);
-\draw [color=red,style=dashed] (5.00,2.82) -- (5.00,0);
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,2.8194) -- (6.0000,2.8194) -- (5.0000,2.8194) -- (5.0000,2.8194) -- (5.0000,0.0000) --  cycle;
+\draw [color=red,style=dashed] (5.0000,0.0000) -- (6.0000,0.0000);
+\draw [color=red,style=dashed] (6.0000,0.0000) -- (6.0000,2.8194);
+\draw [color=red,style=dashed] (6.0000,2.8194) -- (5.0000,2.8194);
+\draw [color=red,style=dashed] (5.0000,2.8194) -- (5.0000,0.0000);
 \draw []  (5.0000,2.8194) node [rotate=0] {$\bullet$};
-\draw (5.4420,3.3924) node {$f(x)$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.27858) node {$x$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.34557) node {$x+\Delta x$};
+\draw (5.4419,3.3923) node {$f(x)$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.2785) node {$x$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.3455) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_KScolorD.pstricks.recall b/src_phystricks/Fig_KScolorD.pstricks.recall
index 091738bd0..92eee38ae 100644
--- a/src_phystricks/Fig_KScolorD.pstricks.recall
+++ b/src_phystricks/Fig_KScolorD.pstricks.recall
@@ -63,17 +63,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6969,0) -- (1.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.6940);
+\draw [,->,>=latex] (-1.6968,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.6940);
 %DEFAULT
 
-\draw [color=blue] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
+\draw [color=blue] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
 
-\draw [color=black] plot [smooth,tension=1] coordinates {(0,1.00)(-0.119,1.19)(-0.159,0.784)(-0.354,1.15)(-0.311,0.737)(-0.575,1.05)(-0.452,0.660)(-0.773,0.918)(-0.574,0.557)(-0.940,0.746)(-0.673,0.433)(-1.07,0.544)(-0.745,0.290)(-1.16,0.322)(-0.788,0.137)(-1.20,0.0869)(-1.00,0)};
+\draw [color=black] plot [smooth,tension=1] coordinates {(0.0000,1.0000)(-0.1194,1.1940)(-0.1591,0.7840)(-0.3538,1.1466)(-0.3105,0.7372)(-0.5746,1.0534)(-0.4515,0.6603)(-0.7733,0.9175)(-0.5742,0.5569)(-0.9399,0.7460)(-0.6729,0.4325)(-1.0694,0.5443)(-0.7454,0.2904)(-1.1560,0.3216)(-0.7882,0.1367)(-1.1968,0.0868)(-1.0000,0.0000)};
 
-\draw [color=black] (0,-1.00)--(0.0159,-1.00)--(0.0317,-1.00)--(0.0476,-0.999)--(0.0634,-0.998)--(0.0792,-0.997)--(0.0951,-0.995)--(0.111,-0.994)--(0.127,-0.992)--(0.142,-0.990)--(0.158,-0.987)--(0.174,-0.985)--(0.189,-0.982)--(0.205,-0.979)--(0.220,-0.975)--(0.236,-0.972)--(0.251,-0.968)--(0.266,-0.964)--(0.282,-0.959)--(0.297,-0.955)--(0.312,-0.950)--(0.327,-0.945)--(0.342,-0.940)--(0.357,-0.934)--(0.372,-0.928)--(0.386,-0.922)--(0.401,-0.916)--(0.415,-0.910)--(0.430,-0.903)--(0.444,-0.896)--(0.458,-0.889)--(0.472,-0.881)--(0.486,-0.874)--(0.500,-0.866)--(0.514,-0.858)--(0.527,-0.850)--(0.541,-0.841)--(0.554,-0.833)--(0.567,-0.824)--(0.580,-0.815)--(0.593,-0.805)--(0.606,-0.796)--(0.618,-0.786)--(0.631,-0.776)--(0.643,-0.766)--(0.655,-0.756)--(0.667,-0.745)--(0.679,-0.735)--(0.690,-0.724)--(0.701,-0.713)--(0.713,-0.701)--(0.724,-0.690)--(0.735,-0.679)--(0.745,-0.667)--(0.756,-0.655)--(0.766,-0.643)--(0.776,-0.631)--(0.786,-0.618)--(0.796,-0.606)--(0.805,-0.593)--(0.815,-0.580)--(0.824,-0.567)--(0.833,-0.554)--(0.841,-0.541)--(0.850,-0.527)--(0.858,-0.514)--(0.866,-0.500)--(0.874,-0.486)--(0.881,-0.472)--(0.889,-0.458)--(0.896,-0.444)--(0.903,-0.430)--(0.910,-0.415)--(0.916,-0.401)--(0.922,-0.386)--(0.928,-0.372)--(0.934,-0.357)--(0.940,-0.342)--(0.945,-0.327)--(0.950,-0.312)--(0.955,-0.297)--(0.959,-0.282)--(0.964,-0.266)--(0.968,-0.251)--(0.972,-0.236)--(0.975,-0.220)--(0.979,-0.205)--(0.982,-0.189)--(0.985,-0.174)--(0.987,-0.158)--(0.990,-0.142)--(0.992,-0.127)--(0.994,-0.111)--(0.995,-0.0951)--(0.997,-0.0792)--(0.998,-0.0634)--(0.999,-0.0476)--(1.00,-0.0317)--(1.00,-0.0159)--(1.00,0);
-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw []  (0,-1.0000) node [rotate=0] {$\bullet$};
+\draw [color=black] (0.0000,-1.0000)--(0.0158,-0.9998)--(0.0317,-0.9994)--(0.0475,-0.9988)--(0.0634,-0.9979)--(0.0792,-0.9968)--(0.0950,-0.9954)--(0.1108,-0.9938)--(0.1265,-0.9919)--(0.1423,-0.9898)--(0.1580,-0.9874)--(0.1736,-0.9848)--(0.1892,-0.9819)--(0.2048,-0.9788)--(0.2203,-0.9754)--(0.2357,-0.9718)--(0.2511,-0.9679)--(0.2664,-0.9638)--(0.2817,-0.9594)--(0.2969,-0.9549)--(0.3120,-0.9500)--(0.3270,-0.9450)--(0.3420,-0.9396)--(0.3568,-0.9341)--(0.3716,-0.9283)--(0.3863,-0.9223)--(0.4009,-0.9161)--(0.4154,-0.9096)--(0.4297,-0.9029)--(0.4440,-0.8959)--(0.4582,-0.8888)--(0.4722,-0.8814)--(0.4861,-0.8738)--(0.5000,-0.8660)--(0.5136,-0.8579)--(0.5272,-0.8497)--(0.5406,-0.8412)--(0.5539,-0.8325)--(0.5670,-0.8236)--(0.5800,-0.8145)--(0.5929,-0.8052)--(0.6056,-0.7957)--(0.6181,-0.7860)--(0.6305,-0.7761)--(0.6427,-0.7660)--(0.6548,-0.7557)--(0.6667,-0.7452)--(0.6785,-0.7345)--(0.6900,-0.7237)--(0.7014,-0.7126)--(0.7126,-0.7014)--(0.7237,-0.6900)--(0.7345,-0.6785)--(0.7452,-0.6667)--(0.7557,-0.6548)--(0.7660,-0.6427)--(0.7761,-0.6305)--(0.7860,-0.6181)--(0.7957,-0.6056)--(0.8052,-0.5929)--(0.8145,-0.5800)--(0.8236,-0.5670)--(0.8325,-0.5539)--(0.8412,-0.5406)--(0.8497,-0.5272)--(0.8579,-0.5136)--(0.8660,-0.5000)--(0.8738,-0.4861)--(0.8814,-0.4722)--(0.8888,-0.4582)--(0.8959,-0.4440)--(0.9029,-0.4297)--(0.9096,-0.4154)--(0.9161,-0.4009)--(0.9223,-0.3863)--(0.9283,-0.3716)--(0.9341,-0.3568)--(0.9396,-0.3420)--(0.9450,-0.3270)--(0.9500,-0.3120)--(0.9549,-0.2969)--(0.9594,-0.2817)--(0.9638,-0.2664)--(0.9679,-0.2511)--(0.9718,-0.2357)--(0.9754,-0.2203)--(0.9788,-0.2048)--(0.9819,-0.1892)--(0.9848,-0.1736)--(0.9874,-0.1580)--(0.9898,-0.1423)--(0.9919,-0.1265)--(0.9938,-0.1108)--(0.9954,-0.0950)--(0.9968,-0.0792)--(0.9979,-0.0634)--(0.9988,-0.0475)--(0.9994,-0.0317)--(0.9998,-0.0158)--(1.0000,0.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall b/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall
index 03400f4b5..2a7bc9ce4 100644
--- a/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall
+++ b/src_phystricks/Fig_LLVMooWOkvAB.pstricks.recall
@@ -67,8 +67,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -77,17 +77,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [,->,>=latex] (1.5000,1.5000) -- (1.4787,1.4787);
+\draw [color=green] (0.0000,0.0000)--(0.0303,0.0000)--(0.0606,0.0012)--(0.0909,0.0027)--(0.1212,0.0048)--(0.1515,0.0076)--(0.1818,0.0110)--(0.2121,0.0149)--(0.2424,0.0195)--(0.2727,0.0247)--(0.3030,0.0306)--(0.3333,0.0370)--(0.3636,0.0440)--(0.3939,0.0517)--(0.4242,0.0599)--(0.4545,0.0688)--(0.4848,0.0783)--(0.5151,0.0884)--(0.5454,0.0991)--(0.5757,0.1104)--(0.6060,0.1224)--(0.6363,0.1349)--(0.6666,0.1481)--(0.6969,0.1619)--(0.7272,0.1763)--(0.7575,0.1913)--(0.7878,0.2069)--(0.8181,0.2231)--(0.8484,0.2399)--(0.8787,0.2574)--(0.9090,0.2754)--(0.9393,0.2941)--(0.9696,0.3134)--(1.0000,0.3333)--(1.0303,0.3538)--(1.0606,0.3749)--(1.0909,0.3966)--(1.1212,0.4190)--(1.1515,0.4419)--(1.1818,0.4655)--(1.2121,0.4897)--(1.2424,0.5145)--(1.2727,0.5399)--(1.3030,0.5659)--(1.3333,0.5925)--(1.3636,0.6198)--(1.3939,0.6476)--(1.4242,0.6761)--(1.4545,0.7052)--(1.4848,0.7349)--(1.5151,0.7652)--(1.5454,0.7961)--(1.5757,0.8276)--(1.6060,0.8598)--(1.6363,0.8925)--(1.6666,0.9259)--(1.6969,0.9599)--(1.7272,0.9944)--(1.7575,1.0296)--(1.7878,1.0655)--(1.8181,1.1019)--(1.8484,1.1389)--(1.8787,1.1766)--(1.9090,1.2148)--(1.9393,1.2537)--(1.9696,1.2932)--(2.0000,1.3333)--(2.0303,1.3740)--(2.0606,1.4153)--(2.0909,1.4573)--(2.1212,1.4998)--(2.1515,1.5430)--(2.1818,1.5867)--(2.2121,1.6311)--(2.2424,1.6761)--(2.2727,1.7217)--(2.3030,1.7679)--(2.3333,1.8148)--(2.3636,1.8622)--(2.3939,1.9103)--(2.4242,1.9589)--(2.4545,2.0082)--(2.4848,2.0581)--(2.5151,2.1086)--(2.5454,2.1597)--(2.5757,2.2115)--(2.6060,2.2638)--(2.6363,2.3168)--(2.6666,2.3703)--(2.6969,2.4245)--(2.7272,2.4793)--(2.7575,2.5347)--(2.7878,2.5907)--(2.8181,2.6473)--(2.8484,2.7046)--(2.8787,2.7624)--(2.9090,2.8209)--(2.9393,2.8800)--(2.9696,2.9397)--(3.0000,3.0000);
+\draw [,->,>=latex] (1.5000,0.7500) -- (1.5212,0.7712);
+\draw [color=green] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=green] (3.0000,3.0000) -- (3.0000,3.0000);
+\draw (3.0000,-0.3149) node {$ 1 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_LesSubFigures.pstricks.recall b/src_phystricks/Fig_LesSubFigures.pstricks.recall
index 29f58eaae..6757fb5ec 100644
--- a/src_phystricks/Fig_LesSubFigures.pstricks.recall
+++ b/src_phystricks/Fig_LesSubFigures.pstricks.recall
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.7141,2.0547) -- (3.2758,3.1849);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
 \draw (3.0787,2.5199) node {$Q_{0}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.5492,1.9345) -- (3.0382,3.2170);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
-\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (2.3975,2.8868) node [rotate=0] {$\bullet$};
+\draw (2.6952,2.4360) node {$Q_{1}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.4022,1.7769) -- (2.7827,3.2509);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
-\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (1.9950,2.7631) node [rotate=0] {$\bullet$};
+\draw (2.3223,2.3215) node {$Q_{2}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7788);
 %DEFAULT
-\draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.2850,1.5627) -- (2.4974,3.2788);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
-\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw (1.9663,2.1571) node {$Q_{3}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall b/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall
index 27a5ac04b..3b29a8a32 100644
--- a/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall
+++ b/src_phystricks/Fig_LesSubFiguresOM.pstricks.recall
@@ -81,17 +81,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.714,2.05) -- (3.28,3.19);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.7141,2.0547) -- (3.2758,3.1849);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (2.8000,2.9750) node [rotate=0] {$\bullet$};
 \draw (3.0787,2.5199) node {$Q_{0}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -170,17 +170,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.549,1.93) -- (3.04,3.22);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.5492,1.9345) -- (3.0382,3.2170);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (2.3975,2.8869) node [rotate=0] {$\bullet$};
-\draw (2.6953,2.4360) node {$Q_{1}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (2.3975,2.8868) node [rotate=0] {$\bullet$};
+\draw (2.6952,2.4360) node {$Q_{1}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -259,17 +259,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7730);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7729);
 %DEFAULT
-\draw [color=cyan] (0.402,1.78) -- (2.78,3.25);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.4022,1.7769) -- (2.7827,3.2509);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
-\draw []  (1.9950,2.7632) node [rotate=0] {$\bullet$};
-\draw (2.3224,2.3215) node {$Q_{2}$};
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
+\draw []  (1.9950,2.7631) node [rotate=0] {$\bullet$};
+\draw (2.3223,2.3215) node {$Q_{2}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -348,17 +348,17 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.7789);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.7788);
 %DEFAULT
-\draw [color=cyan] (0.285,1.56) -- (2.50,3.28);
-\draw [color=red] (0.219,1.26) -- (2.16,3.27);
+\draw [color=cyan] (0.2850,1.5627) -- (2.4974,3.2788);
+\draw [color=red] (0.2187,1.2564) -- (2.1612,3.2729);
 
-\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6753)--(0.5508,0.8312)--(0.5812,0.9708)--(0.6116,1.097)--(0.6420,1.210)--(0.6724,1.314)--(0.7028,1.408)--(0.7332,1.495)--(0.7636,1.575)--(0.7940,1.649)--(0.8244,1.717)--(0.8549,1.780)--(0.8853,1.839)--(0.9157,1.895)--(0.9461,1.946)--(0.9765,1.995)--(1.007,2.040)--(1.037,2.083)--(1.068,2.123)--(1.098,2.161)--(1.128,2.197)--(1.159,2.232)--(1.189,2.264)--(1.220,2.295)--(1.250,2.324)--(1.281,2.352)--(1.311,2.379)--(1.341,2.404)--(1.372,2.428)--(1.402,2.452)--(1.433,2.474)--(1.463,2.495)--(1.493,2.516)--(1.524,2.535)--(1.554,2.554)--(1.585,2.572)--(1.615,2.590)--(1.645,2.607)--(1.676,2.623)--(1.706,2.638)--(1.737,2.654)--(1.767,2.668)--(1.797,2.682)--(1.828,2.696)--(1.858,2.709)--(1.889,2.722)--(1.919,2.734)--(1.949,2.746)--(1.980,2.758)--(2.010,2.769)--(2.041,2.780)--(2.071,2.790)--(2.101,2.800)--(2.132,2.810)--(2.162,2.820)--(2.193,2.830)--(2.223,2.839)--(2.253,2.848)--(2.284,2.856)--(2.314,2.865)--(2.345,2.873)--(2.375,2.881)--(2.405,2.889)--(2.436,2.897)--(2.466,2.904)--(2.497,2.911)--(2.527,2.918)--(2.557,2.925)--(2.588,2.932)--(2.618,2.939)--(2.649,2.945)--(2.679,2.951)--(2.710,2.957)--(2.740,2.963)--(2.770,2.969)--(2.801,2.975)--(2.831,2.981)--(2.862,2.986)--(2.892,2.992)--(2.922,2.997)--(2.953,3.002)--(2.983,3.007)--(3.014,3.012)--(3.044,3.017)--(3.074,3.022)--(3.105,3.027)--(3.135,3.031)--(3.166,3.036)--(3.196,3.040)--(3.226,3.044)--(3.257,3.049)--(3.287,3.053)--(3.318,3.057)--(3.348,3.061)--(3.378,3.065)--(3.409,3.069)--(3.439,3.073)--(3.470,3.076)--(3.500,3.080);
+\draw [color=blue] (0.4900,0.5000)--(0.5204,0.6752)--(0.5508,0.8311)--(0.5812,0.9708)--(0.6116,1.0965)--(0.6420,1.2103)--(0.6724,1.3138)--(0.7028,1.4084)--(0.7332,1.4951)--(0.7636,1.5750)--(0.7940,1.6487)--(0.8244,1.7169)--(0.8548,1.7803)--(0.8852,1.8394)--(0.9156,1.8945)--(0.9460,1.9461)--(0.9764,1.9945)--(1.0068,2.0400)--(1.0372,2.0828)--(1.0676,2.1231)--(1.0980,2.1613)--(1.1284,2.1973)--(1.1588,2.2315)--(1.1892,2.2639)--(1.2196,2.2947)--(1.2501,2.3240)--(1.2805,2.3520)--(1.3109,2.3786)--(1.3413,2.4040)--(1.3717,2.4283)--(1.4021,2.4515)--(1.4325,2.4738)--(1.4629,2.4951)--(1.4933,2.5156)--(1.5237,2.5352)--(1.5541,2.5541)--(1.5845,2.5722)--(1.6149,2.5897)--(1.6453,2.6065)--(1.6757,2.6227)--(1.7061,2.6384)--(1.7365,2.6535)--(1.7669,2.6680)--(1.7973,2.6821)--(1.8277,2.6957)--(1.8581,2.7089)--(1.8885,2.7216)--(1.9189,2.7339)--(1.9493,2.7459)--(1.9797,2.7575)--(2.0102,2.7687)--(2.0406,2.7796)--(2.0710,2.7902)--(2.1014,2.8004)--(2.1318,2.8104)--(2.1622,2.8201)--(2.1926,2.8295)--(2.2230,2.8387)--(2.2534,2.8476)--(2.2838,2.8563)--(2.3142,2.8648)--(2.3446,2.8730)--(2.3750,2.8810)--(2.4054,2.8888)--(2.4358,2.8965)--(2.4662,2.9039)--(2.4966,2.9112)--(2.5270,2.9182)--(2.5574,2.9252)--(2.5878,2.9319)--(2.6182,2.9385)--(2.6486,2.9450)--(2.6790,2.9513)--(2.7094,2.9574)--(2.7398,2.9634)--(2.7703,2.9693)--(2.8007,2.9751)--(2.8311,2.9807)--(2.8615,2.9862)--(2.8919,2.9916)--(2.9223,2.9969)--(2.9527,3.0021)--(2.9831,3.0072)--(3.0135,3.0122)--(3.0439,3.0170)--(3.0743,3.0218)--(3.1047,3.0265)--(3.1351,3.0311)--(3.1655,3.0356)--(3.1959,3.0400)--(3.2263,3.0443)--(3.2567,3.0486)--(3.2871,3.0528)--(3.3175,3.0569)--(3.3479,3.0609)--(3.3783,3.0648)--(3.4087,3.0687)--(3.4391,3.0725)--(3.4695,3.0763)--(3.5000,3.0800);
 \draw []  (1.5925,2.5769) node [rotate=0] {$\bullet$};
-\draw (1.9664,2.1572) node {$Q_{3}$};
+\draw (1.9663,2.1571) node {$Q_{3}$};
 \draw []  (1.1900,2.2647) node [rotate=0] {$\bullet$};
-\draw (0.83143,2.5975) node {$P$};
+\draw (0.8314,2.5975) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_MCKyvdk.pstricks.recall b/src_phystricks/Fig_MCKyvdk.pstricks.recall
index 8e7561c6c..0cb581bbd 100644
--- a/src_phystricks/Fig_MCKyvdk.pstricks.recall
+++ b/src_phystricks/Fig_MCKyvdk.pstricks.recall
@@ -95,26 +95,26 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw (-0.27830,3.2661) node {\( A\)};
+\draw (-0.2782,3.2661) node {\( A\)};
 \draw (3.0000,3.3247) node {\( B\)};
-\draw (3.2849,-0.26613) node {\( C\)};
-\draw (-0.35616,0) node {\( D\)};
-\draw (1.0124,4.0161) node {\( E\)};
+\draw (3.2849,-0.2661) node {\( C\)};
+\draw (-0.3561,0.0000) node {\( D\)};
+\draw (1.0123,4.0161) node {\( E\)};
 \draw (4.6417,3.7500) node {\( F\)};
-\draw (4.6425,0.75000) node {\( G\)};
-\draw (0.99109,1.0161) node {\( H\)};
-\draw [] (0,3.00) -- (1.30,3.75);
-\draw [] (3.00,3.00) -- (4.30,3.75);
-\draw [] (3.00,0) -- (4.30,0.750);
-\draw [style=dashed] (0,0) -- (1.30,0.750);
-\draw [] (1.30,3.75) -- (4.30,3.75);
-\draw [] (4.30,3.75) -- (4.30,0.750);
-\draw [style=dashed] (4.30,0.750) -- (1.30,0.750);
-\draw [style=dashed] (1.30,0.750) -- (1.30,3.75);
-\draw [] (0,3.00) -- (3.00,3.00);
-\draw [] (3.00,3.00) -- (3.00,0);
-\draw [] (3.00,0) -- (0,0);
-\draw [] (0,0) -- (0,3.00);
+\draw (4.6425,0.7500) node {\( G\)};
+\draw (0.9910,1.0161) node {\( H\)};
+\draw [] (0.0000,3.0000) -- (1.2990,3.7500);
+\draw [] (3.0000,3.0000) -- (4.2990,3.7500);
+\draw [] (3.0000,0.0000) -- (4.2990,0.7500);
+\draw [style=dashed] (0.0000,0.0000) -- (1.2990,0.7500);
+\draw [] (1.2990,3.7500) -- (4.2990,3.7500);
+\draw [] (4.2990,3.7500) -- (4.2990,0.7500);
+\draw [style=dashed] (4.2990,0.7500) -- (1.2990,0.7500);
+\draw [style=dashed] (1.2990,0.7500) -- (1.2990,3.7500);
+\draw [] (0.0000,3.0000) -- (3.0000,3.0000);
+\draw [] (3.0000,3.0000) -- (3.0000,0.0000);
+\draw [] (3.0000,0.0000) -- (0.0000,0.0000);
+\draw [] (0.0000,0.0000) -- (0.0000,3.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_MCQueGF.pstricks.recall b/src_phystricks/Fig_MCQueGF.pstricks.recall
index 8594e6364..89057aa14 100644
--- a/src_phystricks/Fig_MCQueGF.pstricks.recall
+++ b/src_phystricks/Fig_MCQueGF.pstricks.recall
@@ -87,37 +87,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
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+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_MaxVraissLp.pstricks.recall b/src_phystricks/Fig_MaxVraissLp.pstricks.recall
index 4d9809e18..2acb70e7e 100644
--- a/src_phystricks/Fig_MaxVraissLp.pstricks.recall
+++ b/src_phystricks/Fig_MaxVraissLp.pstricks.recall
@@ -87,25 +87,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.1683);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.1682);
 %DEFAULT
-\draw []  (3.0000,2.6683) node [rotate=0] {$\bullet$};
+\draw []  (3.0000,2.6682) node [rotate=0] {$\bullet$};
 
-\draw [color=blue] (0,0)--(0.101,0.00115)--(0.202,0.00858)--(0.303,0.0269)--(0.404,0.0593)--(0.505,0.108)--(0.606,0.172)--(0.707,0.254)--(0.808,0.351)--(0.909,0.463)--(1.01,0.587)--(1.11,0.722)--(1.21,0.865)--(1.31,1.01)--(1.41,1.17)--(1.52,1.32)--(1.62,1.47)--(1.72,1.63)--(1.82,1.77)--(1.92,1.91)--(2.02,2.04)--(2.12,2.16)--(2.22,2.27)--(2.32,2.36)--(2.42,2.45)--(2.53,2.52)--(2.63,2.58)--(2.73,2.62)--(2.83,2.65)--(2.93,2.67)--(3.03,2.67)--(3.13,2.66)--(3.23,2.64)--(3.33,2.60)--(3.43,2.56)--(3.54,2.50)--(3.64,2.44)--(3.74,2.37)--(3.84,2.29)--(3.94,2.20)--(4.04,2.11)--(4.14,2.02)--(4.24,1.92)--(4.34,1.82)--(4.44,1.72)--(4.55,1.62)--(4.65,1.52)--(4.75,1.42)--(4.85,1.32)--(4.95,1.22)--(5.05,1.12)--(5.15,1.03)--(5.25,0.945)--(5.35,0.861)--(5.45,0.781)--(5.56,0.705)--(5.66,0.633)--(5.76,0.567)--(5.86,0.504)--(5.96,0.446)--(6.06,0.393)--(6.16,0.345)--(6.26,0.300)--(6.36,0.260)--(6.46,0.224)--(6.57,0.191)--(6.67,0.163)--(6.77,0.137)--(6.87,0.115)--(6.97,0.0953)--(7.07,0.0785)--(7.17,0.0641)--(7.27,0.0518)--(7.37,0.0415)--(7.47,0.0328)--(7.58,0.0257)--(7.68,0.0198)--(7.78,0.0151)--(7.88,0.0113)--(7.98,0.00837)--(8.08,0.00607)--(8.18,0.00432)--(8.28,0.00300)--(8.38,0.00204)--(8.48,0.00134)--(8.59,0)--(8.69,0)--(8.79,0)--(8.89,0)--(8.99,0)--(9.09,0)--(9.19,0)--(9.29,0)--(9.39,0)--(9.50,0)--(9.60,0)--(9.70,0)--(9.80,0)--(9.90,0)--(10.0,0);
-\draw [style=dotted] (3.00,2.67) -- (3.00,0);
-\draw (10.500,-0.41406) node {$p$};
-\draw (10.500,-0.41406) node {$p$};
-\draw (3.0000,-0.42071) node {$ \frac{3}{10} $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (6.0000,-0.42071) node {$ \frac{3}{5} $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (9.0000,-0.42071) node {$ \frac{9}{10} $};
-\draw [] (9.00,-0.100) -- (9.00,0.100);
-\draw (-0.65793,3.1683) node {$L(p)$};
-\draw (-0.65793,3.1683) node {$L(p)$};
-\draw (-0.31083,2.0000) node {$ \frac{1}{5} $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.1010,0.0011)--(0.2020,0.0085)--(0.3030,0.0269)--(0.4040,0.0593)--(0.5050,0.1075)--(0.6060,0.1724)--(0.7070,0.2538)--(0.8080,0.3510)--(0.9090,0.4626)--(1.0101,0.5868)--(1.1111,0.7217)--(1.2121,0.8649)--(1.3131,1.0142)--(1.4141,1.1672)--(1.5151,1.3214)--(1.6161,1.4748)--(1.7171,1.6250)--(1.8181,1.7702)--(1.9191,1.9086)--(2.0202,2.0384)--(2.1212,2.1584)--(2.2222,2.2674)--(2.3232,2.3643)--(2.4242,2.4484)--(2.5252,2.5192)--(2.6262,2.5763)--(2.7272,2.6196)--(2.8282,2.6491)--(2.9292,2.6650)--(3.0303,2.6676)--(3.1313,2.6575)--(3.2323,2.6351)--(3.3333,2.6012)--(3.4343,2.5565)--(3.5353,2.5020)--(3.6363,2.4384)--(3.7373,2.3668)--(3.8383,2.2881)--(3.9393,2.2033)--(4.0404,2.1133)--(4.1414,2.0191)--(4.2424,1.9217)--(4.3434,1.8220)--(4.4444,1.7207)--(4.5454,1.6189)--(4.6464,1.5171)--(4.7474,1.4162)--(4.8484,1.3168)--(4.9494,1.2195)--(5.0505,1.1248)--(5.1515,1.0333)--(5.2525,0.9452)--(5.3535,0.8609)--(5.4545,0.7807)--(5.5555,0.7048)--(5.6565,0.6333)--(5.7575,0.5664)--(5.8585,0.5041)--(5.9595,0.4464)--(6.0606,0.3933)--(6.1616,0.3445)--(6.2626,0.3002)--(6.3636,0.2599)--(6.4646,0.2237)--(6.5656,0.1913)--(6.6666,0.1625)--(6.7676,0.1371)--(6.8686,0.1147)--(6.9696,0.0953)--(7.0707,0.0785)--(7.1717,0.0640)--(7.2727,0.0518)--(7.3737,0.0414)--(7.4747,0.0328)--(7.5757,0.0256)--(7.6767,0.0198)--(7.7777,0.0151)--(7.8787,0.0113)--(7.9797,0.0083)--(8.0808,0.0060)--(8.1818,0.0043)--(8.2828,0.0030)--(8.3838,0.0020)--(8.4848,0.0013)--(8.5858,0.0000)--(8.6868,0.0000)--(8.7878,0.0000)--(8.8888,0.0000)--(8.9898,0.0000)--(9.0909,0.0000)--(9.1919,0.0000)--(9.2929,0.0000)--(9.3939,0.0000)--(9.4949,0.0000)--(9.5959,0.0000)--(9.6969,0.0000)--(9.7979,0.0000)--(9.8989,0.0000)--(10.000,0.0000);
+\draw [style=dotted] (3.0000,2.6682) -- (3.0000,0.0000);
+\draw (10.500,-0.4140) node {$p$};
+\draw (10.500,-0.4140) node {$p$};
+\draw (3.0000,-0.4207) node {$ \frac{3}{10} $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (6.0000,-0.4207) node {$ \frac{3}{5} $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (9.0000,-0.4207) node {$ \frac{9}{10} $};
+\draw [] (9.0000,-0.1000) -- (9.0000,0.1000);
+\draw (-0.6579,3.1682) node {$L(p)$};
+\draw (-0.6579,3.1682) node {$L(p)$};
+\draw (-0.3108,2.0000) node {$ \frac{1}{5} $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_MoulinEau.pstricks.recall b/src_phystricks/Fig_MoulinEau.pstricks.recall
index a61cb53e6..cc264ec62 100644
--- a/src_phystricks/Fig_MoulinEau.pstricks.recall
+++ b/src_phystricks/Fig_MoulinEau.pstricks.recall
@@ -35,14 +35,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (0.0000,0.0000);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
@@ -75,16 +75,16 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (-1.29,0.468);
-\draw [color=red,->,>=latex] (-0.64279,1.2340) -- (-1.2296,1.7264);
-\draw [color=green,->,>=latex] (-0.64279,1.2340) -- (-1.0560,0.74155);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (-1.2855,0.4679);
+\draw [color=red,->,>=latex] (-0.6427,1.2339) -- (-1.2296,1.7263);
+\draw [color=green,->,>=latex] (-0.6427,1.2339) -- (-1.0559,0.7415);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall b/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall
index fc28bdea2..307497c31 100644
--- a/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall
+++ b/src_phystricks/Fig_NOCGooYRHLCn.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.0000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.2500);
+\draw [,->,>=latex] (-1.0000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.2499);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,13 +85,13 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [] (0.0000,0.0000) -- (0.0000,0.9722);
+\draw [] (5.0000,2.6388) -- (5.0000,0.0000);
 
-\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3485,0.6490)--(-0.2727,0.7215)--(-0.1970,0.7928)--(-0.1212,0.8628)--(-0.04545,0.9316)--(0.03030,0.9991)--(0.1061,1.065)--(0.1818,1.130)--(0.2576,1.194)--(0.3333,1.256)--(0.4091,1.317)--(0.4848,1.377)--(0.5606,1.436)--(0.6364,1.493)--(0.7121,1.549)--(0.7879,1.604)--(0.8636,1.657)--(0.9394,1.709)--(1.015,1.760)--(1.091,1.810)--(1.167,1.858)--(1.242,1.905)--(1.318,1.951)--(1.394,1.995)--(1.470,2.039)--(1.545,2.081)--(1.621,2.121)--(1.697,2.161)--(1.773,2.199)--(1.848,2.236)--(1.924,2.271)--(2.000,2.306)--(2.076,2.339)--(2.152,2.370)--(2.227,2.401)--(2.303,2.430)--(2.379,2.458)--(2.455,2.485)--(2.530,2.510)--(2.606,2.534)--(2.682,2.557)--(2.758,2.578)--(2.833,2.599)--(2.909,2.618)--(2.985,2.635)--(3.061,2.652)--(3.136,2.667)--(3.212,2.681)--(3.288,2.694)--(3.364,2.705)--(3.439,2.715)--(3.515,2.724)--(3.591,2.731)--(3.667,2.738)--(3.742,2.743)--(3.818,2.746)--(3.894,2.749)--(3.970,2.750)--(4.045,2.750)--(4.121,2.748)--(4.197,2.746)--(4.273,2.742)--(4.349,2.737)--(4.424,2.730)--(4.500,2.722)--(4.576,2.713)--(4.651,2.703)--(4.727,2.691)--(4.803,2.678)--(4.879,2.664)--(4.955,2.649)--(5.030,2.632)--(5.106,2.614)--(5.182,2.595)--(5.258,2.574)--(5.333,2.552)--(5.409,2.529)--(5.485,2.505)--(5.561,2.479)--(5.636,2.452)--(5.712,2.424)--(5.788,2.395)--(5.864,2.364)--(5.939,2.332)--(6.015,2.299)--(6.091,2.264)--(6.167,2.228)--(6.242,2.191)--(6.318,2.153)--(6.394,2.113)--(6.470,2.072)--(6.545,2.030)--(6.621,1.987)--(6.697,1.942)--(6.773,1.896)--(6.849,1.848)--(6.924,1.800)--(7.000,1.750);
+\draw [color=brown] (-0.5000,0.5000)--(-0.4242,0.5751)--(-0.3484,0.6489)--(-0.2727,0.7215)--(-0.1969,0.7928)--(-0.1212,0.8628)--(-0.0454,0.9315)--(0.0303,0.9990)--(0.1060,1.0652)--(0.1818,1.1301)--(0.2575,1.1938)--(0.3333,1.2561)--(0.4090,1.3172)--(0.4848,1.3770)--(0.5606,1.4356)--(0.6363,1.4928)--(0.7121,1.5488)--(0.7878,1.6035)--(0.8636,1.6570)--(0.9393,1.7091)--(1.0151,1.7600)--(1.0909,1.8096)--(1.1666,1.8580)--(1.2424,1.9050)--(1.3181,1.9508)--(1.3939,1.9953)--(1.4696,2.0386)--(1.5454,2.0805)--(1.6212,2.1212)--(1.6969,2.1606)--(1.7727,2.1988)--(1.8484,2.2356)--(1.9242,2.2712)--(2.0000,2.3055)--(2.0757,2.3385)--(2.1515,2.3703)--(2.2272,2.4008)--(2.3030,2.4300)--(2.3787,2.4579)--(2.4545,2.4846)--(2.5303,2.5099)--(2.6060,2.5341)--(2.6818,2.5569)--(2.7575,2.5784)--(2.8333,2.5987)--(2.9090,2.6177)--(2.9848,2.6354)--(3.0606,2.6519)--(3.1363,2.6671)--(3.2121,2.6810)--(3.2878,2.6936)--(3.3636,2.7050)--(3.4393,2.7150)--(3.5151,2.7238)--(3.5909,2.7314)--(3.6666,2.7376)--(3.7424,2.7426)--(3.8181,2.7463)--(3.8939,2.7487)--(3.9696,2.7498)--(4.0454,2.7497)--(4.1212,2.7483)--(4.1969,2.7456)--(4.2727,2.7417)--(4.3484,2.7365)--(4.4242,2.7300)--(4.5000,2.7222)--(4.5757,2.7131)--(4.6515,2.7028)--(4.7272,2.6912)--(4.8030,2.6783)--(4.8787,2.6641)--(4.9545,2.6487)--(5.0303,2.6320)--(5.1060,2.6140)--(5.1818,2.5948)--(5.2575,2.5742)--(5.3333,2.5524)--(5.4090,2.5293)--(5.4848,2.5050)--(5.5606,2.4793)--(5.6363,2.4524)--(5.7121,2.4242)--(5.7878,2.3948)--(5.8636,2.3640)--(5.9393,2.3320)--(6.0151,2.2987)--(6.0909,2.2642)--(6.1666,2.2283)--(6.2424,2.1912)--(6.3181,2.1528)--(6.3939,2.1132)--(6.4696,2.0722)--(6.5454,2.0300)--(6.6212,1.9865)--(6.6969,1.9418)--(6.7727,1.8957)--(6.8484,1.8484)--(6.9242,1.7998)--(7.0000,1.7500);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -99,17 +99,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.00,0) -- (6.00,0) -- (6.00,0) -- (6.00,2.64) -- (6.00,2.64) -- (5.00,2.64) -- (5.00,2.64) -- (5.00,0) --  cycle;
-\draw [color=red,style=dashed] (5.00,0) -- (6.00,0);
-\draw [color=red,style=dashed] (6.00,0) -- (6.00,2.64);
-\draw [color=red,style=dashed] (6.00,2.64) -- (5.00,2.64);
-\draw [color=red,style=dashed] (5.00,2.64) -- (5.00,0);
-\draw []  (5.0000,2.6389) node [rotate=0] {$\bullet$};
-\draw (5.4420,3.2118) node {$f(x)$};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.27858) node {$x$};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.34557) node {$x+\Delta x$};
+\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (5.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (6.0000,2.6388) -- (6.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,2.6388) -- (5.0000,0.0000) --  cycle;
+\draw [color=red,style=dashed] (5.0000,0.0000) -- (6.0000,0.0000);
+\draw [color=red,style=dashed] (6.0000,0.0000) -- (6.0000,2.6388);
+\draw [color=red,style=dashed] (6.0000,2.6388) -- (5.0000,2.6388);
+\draw [color=red,style=dashed] (5.0000,2.6388) -- (5.0000,0.0000);
+\draw []  (5.0000,2.6388) node [rotate=0] {$\bullet$};
+\draw (5.4419,3.2118) node {$f(x)$};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.2785) node {$x$};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.3455) node {$x+\Delta x$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_NWDooOObSHB.pstricks.recall b/src_phystricks/Fig_NWDooOObSHB.pstricks.recall
index 0769eba3a..ddce63f29 100644
--- a/src_phystricks/Fig_NWDooOObSHB.pstricks.recall
+++ b/src_phystricks/Fig_NWDooOObSHB.pstricks.recall
@@ -75,8 +75,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.3000,0) -- (7.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.7812);
+\draw [,->,>=latex] (-8.3000,0.0000) -- (7.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.7812);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -85,11 +85,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
-\fill [color=red,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (-7.80,1.12) -- (-7.75,1.13) -- (-7.69,1.14) -- (-7.64,1.14) -- (-7.59,1.15) -- (-7.54,1.15) -- (-7.48,1.16) -- (-7.43,1.16) -- (-7.38,1.17) -- (-7.33,1.17) -- (-7.27,1.18) -- (-7.22,1.19) -- (-7.17,1.19) -- (-7.12,1.20) -- (-7.06,1.21) -- (-7.01,1.21) -- (-6.96,1.22) -- (-6.91,1.23) -- (-6.85,1.23) -- (-6.80,1.24) -- (-6.75,1.25) -- (-6.70,1.25) -- (-6.64,1.26) -- (-6.59,1.27) -- (-6.54,1.28) -- (-6.49,1.28) -- (-6.43,1.29) -- (-6.38,1.30) -- (-6.33,1.31) -- (-6.28,1.31) -- (-6.22,1.32) -- (-6.17,1.33) -- (-6.12,1.34) -- (-6.07,1.35) -- (-6.01,1.36) -- (-5.96,1.36) -- (-5.91,1.37) -- (-5.86,1.38) -- (-5.80,1.39) -- (-5.75,1.40) -- (-5.70,1.41) -- (-5.65,1.42) -- (-5.59,1.43) -- (-5.54,1.44) -- (-5.49,1.45) -- (-5.44,1.46) -- (-5.38,1.47) -- (-5.33,1.48) -- (-5.28,1.48) -- (-5.23,1.49) -- (-5.17,1.51) -- (-5.12,1.52) -- (-5.07,1.53) -- (-5.02,1.54) -- (-4.96,1.55) -- (-4.91,1.56) -- (-4.86,1.57) -- (-4.81,1.58) -- (-4.75,1.59) -- (-4.70,1.60) -- (-4.65,1.61) -- (-4.60,1.62) -- (-4.54,1.63) -- (-4.49,1.65) -- (-4.44,1.66) -- (-4.39,1.67) -- (-4.33,1.68) -- (-4.28,1.69) -- (-4.23,1.70) -- (-4.18,1.72) -- (-4.12,1.73) -- (-4.07,1.74) -- (-4.02,1.75) -- (-3.97,1.77) -- (-3.91,1.78) -- (-3.86,1.79) -- (-3.81,1.80) -- (-3.76,1.82) -- (-3.70,1.83) -- (-3.65,1.84) -- (-3.60,1.86) -- (-3.55,1.87) -- (-3.49,1.88) -- (-3.44,1.90) -- (-3.39,1.91) -- (-3.34,1.92) -- (-3.28,1.94) -- (-3.23,1.95) -- (-3.18,1.96) -- (-3.13,1.98) -- (-3.07,1.99) -- (-3.02,2.01) -- (-2.97,2.02) -- (-2.92,2.04) -- (-2.86,2.05) -- (-2.81,2.07) -- (-2.76,2.08) -- (-2.71,2.09) -- (-2.65,2.11) -- (-2.60,2.12) -- (-2.60,2.12) -- (-2.60,0) -- (-2.60,0) -- (-2.65,0) -- (-2.71,0) -- (-2.76,0) -- (-2.81,0) -- (-2.86,0) -- (-2.92,0) -- (-2.97,0) -- (-3.02,0) -- (-3.07,0) -- (-3.13,0) -- (-3.18,0) -- (-3.23,0) -- (-3.28,0) -- (-3.34,0) -- (-3.39,0) -- (-3.44,0) -- (-3.49,0) -- (-3.55,0) -- (-3.60,0) -- (-3.65,0) -- (-3.70,0) -- (-3.76,0) -- (-3.81,0) -- (-3.86,0) -- (-3.91,0) -- (-3.97,0) -- (-4.02,0) -- (-4.07,0) -- (-4.12,0) -- (-4.18,0) -- (-4.23,0) -- (-4.28,0) -- (-4.33,0) -- (-4.39,0) -- (-4.44,0) -- (-4.49,0) -- (-4.54,0) -- (-4.60,0) -- (-4.65,0) -- (-4.70,0) -- (-4.75,0) -- (-4.81,0) -- (-4.86,0) -- (-4.91,0) -- (-4.96,0) -- (-5.02,0) -- (-5.07,0) -- (-5.12,0) -- (-5.17,0) -- (-5.23,0) -- (-5.28,0) -- (-5.33,0) -- (-5.38,0) -- (-5.44,0) -- (-5.49,0) -- (-5.54,0) -- (-5.59,0) -- (-5.65,0) -- (-5.70,0) -- (-5.75,0) -- (-5.80,0) -- (-5.86,0) -- (-5.91,0) -- (-5.96,0) -- (-6.01,0) -- (-6.07,0) -- (-6.12,0) -- (-6.17,0) -- (-6.22,0) -- (-6.28,0) -- (-6.33,0) -- (-6.38,0) -- (-6.43,0) -- (-6.49,0) -- (-6.54,0) -- (-6.59,0) -- (-6.64,0) -- (-6.70,0) -- (-6.75,0) -- (-6.80,0) -- (-6.85,0) -- (-6.91,0) -- (-6.96,0) -- (-7.01,0) -- (-7.06,0) -- (-7.12,0) -- (-7.17,0) -- (-7.22,0) -- (-7.27,0) -- (-7.33,0) -- (-7.38,0) -- (-7.43,0) -- (-7.48,0) -- (-7.54,0) -- (-7.59,0) -- (-7.64,0) -- (-7.69,0) -- (-7.75,0) -- (-7.80,0) -- (-7.80,0) -- (-7.80,1.12) --  cycle;
-\draw [color=blue] (-7.800,1.125)--(-7.747,1.130)--(-7.695,1.135)--(-7.642,1.141)--(-7.590,1.146)--(-7.537,1.152)--(-7.485,1.157)--(-7.432,1.163)--(-7.380,1.169)--(-7.327,1.175)--(-7.275,1.181)--(-7.222,1.187)--(-7.170,1.193)--(-7.117,1.199)--(-7.065,1.206)--(-7.012,1.212)--(-6.960,1.219)--(-6.907,1.226)--(-6.855,1.232)--(-6.802,1.239)--(-6.750,1.246)--(-6.697,1.254)--(-6.644,1.261)--(-6.592,1.268)--(-6.539,1.276)--(-6.487,1.283)--(-6.434,1.291)--(-6.382,1.299)--(-6.329,1.306)--(-6.277,1.314)--(-6.224,1.322)--(-6.172,1.331)--(-6.119,1.339)--(-6.067,1.347)--(-6.014,1.356)--(-5.962,1.364)--(-5.909,1.373)--(-5.857,1.382)--(-5.804,1.391)--(-5.752,1.400)--(-5.699,1.409)--(-5.646,1.418)--(-5.594,1.427)--(-5.541,1.436)--(-5.489,1.446)--(-5.436,1.456)--(-5.384,1.465)--(-5.331,1.475)--(-5.279,1.485)--(-5.226,1.495)--(-5.174,1.505)--(-5.121,1.515)--(-5.069,1.526)--(-5.016,1.536)--(-4.964,1.546)--(-4.911,1.557)--(-4.859,1.568)--(-4.806,1.579)--(-4.754,1.590)--(-4.701,1.601)--(-4.648,1.612)--(-4.596,1.623)--(-4.543,1.634)--(-4.491,1.646)--(-4.438,1.657)--(-4.386,1.669)--(-4.333,1.681)--(-4.281,1.692)--(-4.228,1.704)--(-4.176,1.716)--(-4.123,1.729)--(-4.071,1.741)--(-4.018,1.753)--(-3.966,1.766)--(-3.913,1.778)--(-3.861,1.791)--(-3.808,1.803)--(-3.756,1.816)--(-3.703,1.829)--(-3.651,1.842)--(-3.598,1.856)--(-3.545,1.869)--(-3.493,1.882)--(-3.440,1.896)--(-3.388,1.909)--(-3.335,1.923)--(-3.283,1.937)--(-3.230,1.951)--(-3.178,1.965)--(-3.125,1.979)--(-3.073,1.993)--(-3.020,2.007)--(-2.968,2.021)--(-2.915,2.036)--(-2.863,2.051)--(-2.810,2.065)--(-2.758,2.080)--(-2.705,2.095)--(-2.653,2.110)--(-2.600,2.125);
-\draw [color=blue] (-7.800,0)--(-7.747,0)--(-7.695,0)--(-7.642,0)--(-7.590,0)--(-7.537,0)--(-7.485,0)--(-7.432,0)--(-7.380,0)--(-7.327,0)--(-7.275,0)--(-7.222,0)--(-7.170,0)--(-7.117,0)--(-7.065,0)--(-7.012,0)--(-6.960,0)--(-6.907,0)--(-6.855,0)--(-6.802,0)--(-6.750,0)--(-6.697,0)--(-6.644,0)--(-6.592,0)--(-6.539,0)--(-6.487,0)--(-6.434,0)--(-6.382,0)--(-6.329,0)--(-6.277,0)--(-6.224,0)--(-6.172,0)--(-6.119,0)--(-6.067,0)--(-6.014,0)--(-5.962,0)--(-5.909,0)--(-5.857,0)--(-5.804,0)--(-5.752,0)--(-5.699,0)--(-5.646,0)--(-5.594,0)--(-5.541,0)--(-5.489,0)--(-5.436,0)--(-5.384,0)--(-5.331,0)--(-5.279,0)--(-5.226,0)--(-5.174,0)--(-5.121,0)--(-5.069,0)--(-5.016,0)--(-4.964,0)--(-4.911,0)--(-4.859,0)--(-4.806,0)--(-4.754,0)--(-4.701,0)--(-4.648,0)--(-4.596,0)--(-4.543,0)--(-4.491,0)--(-4.438,0)--(-4.386,0)--(-4.333,0)--(-4.281,0)--(-4.228,0)--(-4.176,0)--(-4.123,0)--(-4.071,0)--(-4.018,0)--(-3.966,0)--(-3.913,0)--(-3.861,0)--(-3.808,0)--(-3.756,0)--(-3.703,0)--(-3.651,0)--(-3.598,0)--(-3.545,0)--(-3.493,0)--(-3.440,0)--(-3.388,0)--(-3.335,0)--(-3.283,0)--(-3.230,0)--(-3.178,0)--(-3.125,0)--(-3.073,0)--(-3.020,0)--(-2.968,0)--(-2.915,0)--(-2.863,0)--(-2.810,0)--(-2.758,0)--(-2.705,0)--(-2.653,0)--(-2.600,0);
-\draw [] (-7.80,0) -- (-7.80,1.12);
-\draw [] (-2.60,2.12) -- (-2.60,0);
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+\draw [] (-7.8000,0.0000) -- (-7.8000,1.1250);
+\draw [] (-2.6000,2.1250) -- (-2.6000,0.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -97,17 +97,17 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=blue] (-2.6000,0.0000)--(-2.5080,0.0000)--(-2.4161,0.0000)--(-2.3242,0.0000)--(-2.2323,0.0000)--(-2.1404,0.0000)--(-2.0484,0.0000)--(-1.9565,0.0000)--(-1.8646,0.0000)--(-1.7727,0.0000)--(-1.6808,0.0000)--(-1.5888,0.0000)--(-1.4969,0.0000)--(-1.4050,0.0000)--(-1.3131,0.0000)--(-1.2212,0.0000)--(-1.1292,0.0000)--(-1.0373,0.0000)--(-0.9454,0.0000)--(-0.8535,0.0000)--(-0.7616,0.0000)--(-0.6696,0.0000)--(-0.5777,0.0000)--(-0.4858,0.0000)--(-0.3939,0.0000)--(-0.3020,0.0000)--(-0.2101,0.0000)--(-0.1181,0.0000)--(-0.0262,0.0000)--(0.0656,0.0000)--(0.1575,0.0000)--(0.2494,0.0000)--(0.3414,0.0000)--(0.4333,0.0000)--(0.5252,0.0000)--(0.6171,0.0000)--(0.7090,0.0000)--(0.8010,0.0000)--(0.8929,0.0000)--(0.9848,0.0000)--(1.0767,0.0000)--(1.1686,0.0000)--(1.2606,0.0000)--(1.3525,0.0000)--(1.4444,0.0000)--(1.5363,0.0000)--(1.6282,0.0000)--(1.7202,0.0000)--(1.8121,0.0000)--(1.9040,0.0000)--(1.9959,0.0000)--(2.0878,0.0000)--(2.1797,0.0000)--(2.2717,0.0000)--(2.3636,0.0000)--(2.4555,0.0000)--(2.5474,0.0000)--(2.6393,0.0000)--(2.7313,0.0000)--(2.8232,0.0000)--(2.9151,0.0000)--(3.0070,0.0000)--(3.0989,0.0000)--(3.1909,0.0000)--(3.2828,0.0000)--(3.3747,0.0000)--(3.4666,0.0000)--(3.5585,0.0000)--(3.6505,0.0000)--(3.7424,0.0000)--(3.8343,0.0000)--(3.9262,0.0000)--(4.0181,0.0000)--(4.1101,0.0000)--(4.2020,0.0000)--(4.2939,0.0000)--(4.3858,0.0000)--(4.4777,0.0000)--(4.5696,0.0000)--(4.6616,0.0000)--(4.7535,0.0000)--(4.8454,0.0000)--(4.9373,0.0000)--(5.0292,0.0000)--(5.1212,0.0000)--(5.2131,0.0000)--(5.3050,0.0000)--(5.3969,0.0000)--(5.4888,0.0000)--(5.5808,0.0000)--(5.6727,0.0000)--(5.7646,0.0000)--(5.8565,0.0000)--(5.9484,0.0000)--(6.0404,0.0000)--(6.1323,0.0000)--(6.2242,0.0000)--(6.3161,0.0000)--(6.4080,0.0000)--(6.5000,0.0000);
+\draw [] (-2.6000,0.0000) -- (-2.6000,2.1250);
+\draw [] (6.5000,6.2812) -- (6.5000,0.0000);
+\draw []  (-7.8000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-7.8000,-0.2785) node {\( a\)};
+\draw []  (-2.6000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-2.6000,-0.3267) node {\( b\)};
+\draw []  (6.5000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.5000,-0.2785) node {\( c\)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_Osculateur.pstricks.recall b/src_phystricks/Fig_Osculateur.pstricks.recall
index 3136c04e9..c76dc0bf1 100644
--- a/src_phystricks/Fig_Osculateur.pstricks.recall
+++ b/src_phystricks/Fig_Osculateur.pstricks.recall
@@ -67,17 +67,17 @@
 %DEFAULT
 \draw []  (-19.378,2.0000) node [rotate=0] {$\bullet$};
 
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 \draw []  (-19.378,2.0000) node [rotate=0] {$\bullet$};
-\draw []  (-19.107,1.9021) node [rotate=0] {$\bullet$};
+\draw []  (-19.106,1.9021) node [rotate=0] {$\bullet$};
 
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-\draw []  (-15.297,-1.9021) node [rotate=0] {$\bullet$};
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+\draw []  (-15.296,-1.9021) node [rotate=0] {$\bullet$};
 
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 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_PHTVjfk.pstricks.recall b/src_phystricks/Fig_PHTVjfk.pstricks.recall
index 833669663..e9326c40f 100644
--- a/src_phystricks/Fig_PHTVjfk.pstricks.recall
+++ b/src_phystricks/Fig_PHTVjfk.pstricks.recall
@@ -79,29 +79,29 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.2315,0) -- (2.2315,0);
-\draw [,->,>=latex] (0,-2.2315) -- (0,2.2315);
+\draw [,->,>=latex] (-2.2314,0.0000) -- (2.2314,0.0000);
+\draw [,->,>=latex] (0.0000,-2.2314) -- (0.0000,2.2314);
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+\draw [] plot [smooth,tension=1] coordinates {(-0.3020,-1.3814)(-0.2348,-1.3942)(-0.1677,-1.4038)(-0.1006,-1.4102)(-0.0335,-1.4134)(0.0335,-1.4134)(0.1006,-1.4102)(0.1677,-1.4038)(0.2348,-1.3942)(0.3020,-1.3814)(0.3256,-1.3758)(0.3691,-1.3649)(0.4362,-1.3448)(0.5033,-1.3213)(0.5348,-1.3087)(0.5704,-1.2937)(0.6375,-1.2619)(0.6762,-1.2416)(0.7046,-1.2258)(0.7718,-1.1847)(0.7872,-1.1744)(0.8389,-1.1380)(0.8789,-1.1073)(0.9060,-1.0854)(0.9576,-1.0402)(0.9731,-1.0257)(1.0257,-0.9731)(1.0402,-0.9576)(1.0854,-0.9060)(1.1073,-0.8789)(1.1380,-0.8389)(1.1744,-0.7872)(1.1847,-0.7718)(1.2258,-0.7046)(1.2416,-0.6762)(1.2619,-0.6375)(1.2937,-0.5704)(1.3087,-0.5348)(1.3213,-0.5033)(1.3448,-0.4362)(1.3649,-0.3691)(1.3758,-0.3256)(1.3814,-0.3020)(1.3942,-0.2348)(1.4038,-0.1677)(1.4102,-0.1006)(1.4134,-0.0335)(1.4134,0.0335)(1.4102,0.1006)(1.4038,0.1677)(1.3942,0.2348)(1.3814,0.3020)(1.3758,0.3256)(1.3649,0.3691)(1.3448,0.4362)(1.3213,0.5033)(1.3087,0.5348)(1.2937,0.5704)(1.2619,0.6375)(1.2416,0.6762)(1.2258,0.7046)(1.1847,0.7718)(1.1744,0.7872)(1.1380,0.8389)(1.1073,0.8789)(1.0854,0.9060)(1.0402,0.9576)(1.0257,0.9731)(0.9731,1.0257)(0.9576,1.0402)(0.9060,1.0854)(0.8789,1.1073)(0.8389,1.1380)(0.7872,1.1744)(0.7718,1.1847)(0.7046,1.2258)(0.6762,1.2416)(0.6375,1.2619)(0.5704,1.2937)(0.5348,1.3087)(0.5033,1.3213)(0.4362,1.3448)(0.3691,1.3649)(0.3256,1.3758)(0.3020,1.3814)(0.2348,1.3942)(0.1677,1.4038)(0.1006,1.4102)(0.0335,1.4134)(-0.0335,1.4134)(-0.1006,1.4102)(-0.1677,1.4038)(-0.2348,1.3942)(-0.3020,1.3814)(-0.3256,1.3758)(-0.3691,1.3649)(-0.4362,1.3448)(-0.5033,1.3213)(-0.5348,1.3087)(-0.5704,1.2937)(-0.6375,1.2619)(-0.6762,1.2416)(-0.7046,1.2258)(-0.7718,1.1847)(-0.7872,1.1744)(-0.8389,1.1380)(-0.8789,1.1073)(-0.9060,1.0854)(-0.9576,1.0402)(-0.9731,1.0257)(-1.0257,0.9731)(-1.0402,0.9576)(-1.0854,0.9060)(-1.1073,0.8789)(-1.1380,0.8389)(-1.1744,0.7872)(-1.1847,0.7718)(-1.2258,0.7046)(-1.2416,0.6762)(-1.2619,0.6375)(-1.2937,0.5704)(-1.3087,0.5348)(-1.3213,0.5033)(-1.3448,0.4362)(-1.3649,0.3691)(-1.3758,0.3256)(-1.3814,0.3020)(-1.3942,0.2348)(-1.4038,0.1677)(-1.4102,0.1006)(-1.4134,0.0335)(-1.4134,-0.0335)(-1.4102,-0.1006)(-1.4038,-0.1677)(-1.3942,-0.2348)(-1.3814,-0.3020)(-1.3758,-0.3256)(-1.3649,-0.3691)(-1.3448,-0.4362)(-1.3213,-0.5033)(-1.3087,-0.5348)(-1.2937,-0.5704)(-1.2619,-0.6375)(-1.2416,-0.6762)(-1.2258,-0.7046)(-1.1847,-0.7718)(-1.1744,-0.7872)(-1.1380,-0.8389)(-1.1073,-0.8789)(-1.0854,-0.9060)(-1.0402,-0.9576)(-1.0257,-0.9731)(-0.9731,-1.0257)(-0.9576,-1.0402)(-0.9060,-1.0854)(-0.8789,-1.1073)(-0.8389,-1.1380)(-0.7872,-1.1744)(-0.7718,-1.1847)(-0.7046,-1.2258)(-0.6762,-1.2416)(-0.6375,-1.2619)(-0.5704,-1.2937)(-0.5348,-1.3087)(-0.5033,-1.3213)(-0.4362,-1.3448)(-0.3691,-1.3649)(-0.3256,-1.3758)(-0.3020,-1.3814)};
+\draw [] plot [smooth,tension=1] coordinates {(-0.1677,-0.9854)(-0.1006,-0.9944)(-0.0335,-0.9989)(0.0335,-0.9989)(0.1006,-0.9944)(0.1677,-0.9854)(0.2294,-0.9731)(0.2348,-0.9719)(0.3020,-0.9528)(0.3691,-0.9288)(0.4223,-0.9060)(0.4362,-0.8996)(0.5033,-0.8634)(0.5432,-0.8389)(0.5704,-0.8207)(0.6357,-0.7718)(0.6375,-0.7703)(0.7046,-0.7093)(0.7093,-0.7046)(0.7703,-0.6375)(0.7718,-0.6357)(0.8207,-0.5704)(0.8389,-0.5432)(0.8634,-0.5033)(0.8996,-0.4362)(0.9060,-0.4223)(0.9288,-0.3691)(0.9528,-0.3020)(0.9719,-0.2348)(0.9731,-0.2294)(0.9854,-0.1677)(0.9944,-0.1006)(0.9989,-0.0335)(0.9989,0.0335)(0.9944,0.1006)(0.9854,0.1677)(0.9731,0.2294)(0.9719,0.2348)(0.9528,0.3020)(0.9288,0.3691)(0.9060,0.4223)(0.8996,0.4362)(0.8634,0.5033)(0.8389,0.5432)(0.8207,0.5704)(0.7718,0.6357)(0.7703,0.6375)(0.7093,0.7046)(0.7046,0.7093)(0.6375,0.7703)(0.6357,0.7718)(0.5704,0.8207)(0.5432,0.8389)(0.5033,0.8634)(0.4362,0.8996)(0.4223,0.9060)(0.3691,0.9288)(0.3020,0.9528)(0.2348,0.9719)(0.2294,0.9731)(0.1677,0.9854)(0.1006,0.9944)(0.0335,0.9989)(-0.0335,0.9989)(-0.1006,0.9944)(-0.1677,0.9854)(-0.2294,0.9731)(-0.2348,0.9719)(-0.3020,0.9528)(-0.3691,0.9288)(-0.4223,0.9060)(-0.4362,0.8996)(-0.5033,0.8634)(-0.5432,0.8389)(-0.5704,0.8207)(-0.6357,0.7718)(-0.6375,0.7703)(-0.7046,0.7093)(-0.7093,0.7046)(-0.7703,0.6375)(-0.7718,0.6357)(-0.8207,0.5704)(-0.8389,0.5432)(-0.8634,0.5033)(-0.8996,0.4362)(-0.9060,0.4223)(-0.9288,0.3691)(-0.9528,0.3020)(-0.9719,0.2348)(-0.9731,0.2294)(-0.9854,0.1677)(-0.9944,0.1006)(-0.9989,0.0335)(-0.9989,-0.0335)(-0.9944,-0.1006)(-0.9854,-0.1677)(-0.9731,-0.2294)(-0.9719,-0.2348)(-0.9528,-0.3020)(-0.9288,-0.3691)(-0.9060,-0.4223)(-0.8996,-0.4362)(-0.8634,-0.5033)(-0.8389,-0.5432)(-0.8207,-0.5704)(-0.7718,-0.6357)(-0.7703,-0.6375)(-0.7093,-0.7046)(-0.7046,-0.7093)(-0.6375,-0.7703)(-0.6357,-0.7718)(-0.5704,-0.8207)(-0.5432,-0.8389)(-0.5033,-0.8634)(-0.4362,-0.8996)(-0.4223,-0.9060)(-0.3691,-0.9288)(-0.3020,-0.9528)(-0.2348,-0.9719)(-0.2294,-0.9731)(-0.1677,-0.9854)};
+\draw [] plot [smooth,tension=1] coordinates {(-0.0335,-0.7062)(0.0335,-0.7062)(0.0505,-0.7046)(0.1006,-0.6996)(0.1677,-0.6862)(0.2348,-0.6661)(0.3020,-0.6392)(0.3054,-0.6375)(0.3691,-0.6021)(0.4166,-0.5704)(0.4362,-0.5558)(0.4961,-0.5033)(0.5033,-0.4961)(0.5558,-0.4362)(0.5704,-0.4166)(0.6021,-0.3691)(0.6375,-0.3054)(0.6392,-0.3020)(0.6661,-0.2348)(0.6862,-0.1677)(0.6996,-0.1006)(0.7046,-0.0505)(0.7062,-0.0335)(0.7062,0.0335)(0.7046,0.0505)(0.6996,0.1006)(0.6862,0.1677)(0.6661,0.2348)(0.6392,0.3020)(0.6375,0.3054)(0.6021,0.3691)(0.5704,0.4166)(0.5558,0.4362)(0.5033,0.4961)(0.4961,0.5033)(0.4362,0.5558)(0.4166,0.5704)(0.3691,0.6021)(0.3054,0.6375)(0.3020,0.6392)(0.2348,0.6661)(0.1677,0.6862)(0.1006,0.6996)(0.0505,0.7046)(0.0335,0.7062)(-0.0335,0.7062)(-0.0505,0.7046)(-0.1006,0.6996)(-0.1677,0.6862)(-0.2348,0.6661)(-0.3020,0.6392)(-0.3054,0.6375)(-0.3691,0.6021)(-0.4166,0.5704)(-0.4362,0.5558)(-0.4961,0.5033)(-0.5033,0.4961)(-0.5558,0.4362)(-0.5704,0.4166)(-0.6021,0.3691)(-0.6375,0.3054)(-0.6392,0.3020)(-0.6661,0.2348)(-0.6862,0.1677)(-0.6996,0.1006)(-0.7046,0.0505)(-0.7062,0.0335)(-0.7062,-0.0335)(-0.7046,-0.0505)(-0.6996,-0.1006)(-0.6862,-0.1677)(-0.6661,-0.2348)(-0.6392,-0.3020)(-0.6375,-0.3054)(-0.6021,-0.3691)(-0.5704,-0.4166)(-0.5558,-0.4362)(-0.5033,-0.4961)(-0.4961,-0.5033)(-0.4362,-0.5558)(-0.4166,-0.5704)(-0.3691,-0.6021)(-0.3054,-0.6375)(-0.3020,-0.6392)(-0.2348,-0.6661)(-0.1677,-0.6862)(-0.1006,-0.6996)(-0.0505,-0.7046)(-0.0335,-0.7062)};
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall b/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall
index ab49695de..9cb321f63 100644
--- a/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall
+++ b/src_phystricks/Fig_PVJooJDyNAg.pstricks.recall
@@ -143,65 +143,65 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.9978,0) -- (5.9978,0);
-\draw [,->,>=latex] (0,-4.0000) -- (0,4.0000);
+\draw [,->,>=latex] (-5.9977,0.0000) -- (5.9977,0.0000);
+\draw [,->,>=latex] (0.0000,-4.0000) -- (0.0000,4.0000);
 %DEFAULT
 
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-\draw [color=blue] (3.469,-2.827)--(3.488,-2.530)--(3.507,-2.286)--(3.526,-2.082)--(3.545,-1.908)--(3.564,-1.758)--(3.583,-1.628)--(3.602,-1.513)--(3.621,-1.411)--(3.640,-1.319)--(3.659,-1.237)--(3.678,-1.162)--(3.697,-1.093)--(3.716,-1.030)--(3.736,-0.9722)--(3.755,-0.9184)--(3.774,-0.8683)--(3.793,-0.8214)--(3.812,-0.7775)--(3.831,-0.7361)--(3.850,-0.6971)--(3.869,-0.6601)--(3.888,-0.6250)--(3.907,-0.5915)--(3.926,-0.5596)--(3.945,-0.5290)--(3.964,-0.4996)--(3.983,-0.4714)--(4.002,-0.4442)--(4.021,-0.4179)--(4.041,-0.3924)--(4.060,-0.3677)--(4.079,-0.3437)--(4.098,-0.3204)--(4.117,-0.2976)--(4.136,-0.2753)--(4.155,-0.2535)--(4.174,-0.2322)--(4.193,-0.2112)--(4.212,-0.1906)--(4.231,-0.1702)--(4.250,-0.1502)--(4.269,-0.1303)--(4.288,-0.1107)--(4.307,-0.09122)--(4.327,-0.07190)--(4.346,-0.05268)--(4.365,-0.03354)--(4.384,-0.01445)--(4.403,0.004624)--(4.422,0.02370)--(4.441,0.04281)--(4.460,0.06199)--(4.479,0.08125)--(4.498,0.1006)--(4.517,0.1202)--(4.536,0.1399)--(4.555,0.1599)--(4.574,0.1800)--(4.594,0.2005)--(4.613,0.2213)--(4.632,0.2425)--(4.651,0.2640)--(4.670,0.2861)--(4.689,0.3086)--(4.708,0.3316)--(4.727,0.3553)--(4.746,0.3796)--(4.765,0.4047)--(4.784,0.4305)--(4.803,0.4572)--(4.822,0.4849)--(4.841,0.5137)--(4.860,0.5437)--(4.880,0.5749)--(4.899,0.6076)--(4.918,0.6418)--(4.937,0.6778)--(4.956,0.7158)--(4.975,0.7559)--(4.994,0.7985)--(5.013,0.8438)--(5.032,0.8921)--(5.051,0.9440)--(5.070,0.9998)--(5.089,1.060)--(5.108,1.126)--(5.127,1.197)--(5.146,1.276)--(5.166,1.362)--(5.185,1.459)--(5.204,1.567)--(5.223,1.689)--(5.242,1.828)--(5.261,1.989)--(5.280,2.176)--(5.299,2.398)--(5.318,2.666)--(5.337,2.997)--(5.356,3.414);
-\draw [style=dashed] (-5.50,-3.50) -- (-5.50,3.50);
-\draw [style=dashed] (-3.30,-3.50) -- (-3.30,3.50);
-\draw [style=dashed] (-1.10,-3.50) -- (-1.10,3.50);
-\draw [style=dashed] (1.10,-3.50) -- (1.10,3.50);
-\draw [style=dashed] (3.30,-3.50) -- (3.30,3.50);
-\draw [style=dashed] (5.50,-3.50) -- (5.50,3.50);
-\draw (-5.4978,-0.42071) node {$ -\frac{5}{2} \, \pi $};
-\draw [] (-5.50,-0.100) -- (-5.50,0.100);
-\draw (-4.3982,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-4.40,-0.100) -- (-4.40,0.100);
-\draw (-3.2987,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.1991,-0.32103) node {$ -\pi $};
-\draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.0996,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.0996,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.1991,-0.27858) node {$ \pi $};
-\draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.2987,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (4.3982,-0.31492) node {$ 2 \, \pi $};
-\draw [] (4.40,-0.100) -- (4.40,0.100);
-\draw (5.4978,-0.42071) node {$ \frac{5}{2} \, \pi $};
-\draw [] (5.50,-0.100) -- (5.50,0.100);
-\draw (-0.43316,-3.5000) node {$ -5 $};
-\draw [] (-0.100,-3.50) -- (0.100,-3.50);
-\draw (-0.43316,-2.8000) node {$ -4 $};
-\draw [] (-0.100,-2.80) -- (0.100,-2.80);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=blue] (3.4685,-2.8273)--(3.4876,-2.5299)--(3.5066,-2.2857)--(3.5257,-2.0814)--(3.5448,-1.9078)--(3.5638,-1.7582)--(3.5829,-1.6277)--(3.6020,-1.5128)--(3.6210,-1.4107)--(3.6401,-1.3191)--(3.6592,-1.2366)--(3.6783,-1.1616)--(3.6973,-1.0931)--(3.7164,-1.0302)--(3.7355,-0.9721)--(3.7545,-0.9183)--(3.7736,-0.8682)--(3.7927,-0.8214)--(3.8117,-0.7774)--(3.8308,-0.7361)--(3.8499,-0.6970)--(3.8689,-0.6601)--(3.8880,-0.6249)--(3.9071,-0.5915)--(3.9261,-0.5595)--(3.9452,-0.5289)--(3.9643,-0.4996)--(3.9833,-0.4713)--(4.0024,-0.4441)--(4.0215,-0.4178)--(4.0405,-0.3924)--(4.0596,-0.3677)--(4.0787,-0.3437)--(4.0977,-0.3203)--(4.1168,-0.2975)--(4.1359,-0.2753)--(4.1549,-0.2535)--(4.1740,-0.2321)--(4.1931,-0.2111)--(4.2121,-0.1905)--(4.2312,-0.1702)--(4.2503,-0.1501)--(4.2693,-0.1303)--(4.2884,-0.1106)--(4.3075,-0.0912)--(4.3265,-0.0718)--(4.3456,-0.0526)--(4.3647,-0.0335)--(4.3837,-0.0144)--(4.4028,0.0046)--(4.4219,0.0237)--(4.4409,0.0428)--(4.4600,0.0619)--(4.4791,0.0812)--(4.4981,0.1006)--(4.5172,0.1201)--(4.5363,0.1399)--(4.5553,0.1598)--(4.5744,0.1800)--(4.5935,0.2005)--(4.6125,0.2213)--(4.6316,0.2424)--(4.6507,0.2640)--(4.6697,0.2860)--(4.6888,0.3085)--(4.7079,0.3316)--(4.7269,0.3552)--(4.7460,0.3796)--(4.7651,0.4046)--(4.7841,0.4305)--(4.8032,0.4572)--(4.8223,0.4849)--(4.8413,0.5137)--(4.8604,0.5436)--(4.8795,0.5748)--(4.8986,0.6075)--(4.9176,0.6418)--(4.9367,0.6778)--(4.9558,0.7157)--(4.9748,0.7558)--(4.9939,0.7984)--(5.0130,0.8437)--(5.0320,0.8921)--(5.0511,0.9439)--(5.0702,0.9997)--(5.0892,1.0601)--(5.1083,1.1256)--(5.1274,1.1971)--(5.1464,1.2756)--(5.1655,1.3623)--(5.1846,1.4587)--(5.2036,1.5668)--(5.2227,1.6888)--(5.2418,1.8281)--(5.2608,1.9887)--(5.2799,2.1762)--(5.2990,2.3984)--(5.3180,2.6664)--(5.3371,2.9965)--(5.3562,3.4137);
+\draw [style=dashed] (-5.4977,-3.5000) -- (-5.4977,3.5000);
+\draw [style=dashed] (-3.2986,-3.5000) -- (-3.2986,3.5000);
+\draw [style=dashed] (-1.0995,-3.5000) -- (-1.0995,3.5000);
+\draw [style=dashed] (1.0995,-3.5000) -- (1.0995,3.5000);
+\draw [style=dashed] (3.2986,-3.5000) -- (3.2986,3.5000);
+\draw [style=dashed] (5.4977,-3.5000) -- (5.4977,3.5000);
+\draw (-5.4977,-0.4207) node {$ -\frac{5}{2} \, \pi $};
+\draw [] (-5.4977,-0.1000) -- (-5.4977,0.1000);
+\draw (-4.3982,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-4.3982,-0.1000) -- (-4.3982,0.1000);
+\draw (-3.2986,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-3.2986,-0.1000) -- (-3.2986,0.1000);
+\draw (-2.1991,-0.3210) node {$ -\pi $};
+\draw [] (-2.1991,-0.1000) -- (-2.1991,0.1000);
+\draw (-1.0995,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.0995,-0.1000) -- (-1.0995,0.1000);
+\draw (1.0995,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.0995,-0.1000) -- (1.0995,0.1000);
+\draw (2.1991,-0.2785) node {$ \pi $};
+\draw [] (2.1991,-0.1000) -- (2.1991,0.1000);
+\draw (3.2986,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (3.2986,-0.1000) -- (3.2986,0.1000);
+\draw (4.3982,-0.3149) node {$ 2 \, \pi $};
+\draw [] (4.3982,-0.1000) -- (4.3982,0.1000);
+\draw (5.4977,-0.4207) node {$ \frac{5}{2} \, \pi $};
+\draw [] (5.4977,-0.1000) -- (5.4977,0.1000);
+\draw (-0.4331,-3.5000) node {$ -5 $};
+\draw [] (-0.1000,-3.5000) -- (0.1000,-3.5000);
+\draw (-0.4331,-2.8000) node {$ -4 $};
+\draw [] (-0.1000,-2.8000) -- (0.1000,-2.8000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ProjPoly.pstricks.recall b/src_phystricks/Fig_ProjPoly.pstricks.recall
index 2b8360aeb..4abb9a4fc 100644
--- a/src_phystricks/Fig_ProjPoly.pstricks.recall
+++ b/src_phystricks/Fig_ProjPoly.pstricks.recall
@@ -111,57 +111,57 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.8117,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-3.0000) -- (0,3.0000);
+\draw [,->,>=latex] (-2.8116,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0000) -- (0.0000,3.0000);
 %DEFAULT
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+\draw [] plot [smooth,tension=1] coordinates {(2.5000,2.1793)(2.4853,2.1644)(2.4664,2.1447)(2.4529,2.1308)(2.4328,2.1100)(2.4204,2.0973)(2.3993,2.0752)(2.3880,2.0637)(2.3657,2.0405)(2.3557,2.0302)(2.3322,2.0057)(2.3233,1.9966)(2.2986,1.9708)(2.2910,1.9630)(2.2651,1.9359)(2.2588,1.9295)(2.2315,1.9010)(2.2266,1.8959)(2.1979,1.8660)(2.1944,1.8624)(2.1644,1.8309)(2.1623,1.8288)(2.1308,1.7958)(2.1303,1.7953)(2.0983,1.7617)(2.0973,1.7606)(2.0663,1.7281)(2.0637,1.7254)(2.0344,1.6946)(2.0302,1.6901)(2.0025,1.6610)(1.9966,1.6547)(1.9707,1.6275)(1.9630,1.6192)(1.9390,1.5939)(1.9295,1.5836)(1.9074,1.5604)(1.8959,1.5480)(1.8758,1.5268)(1.8624,1.5123)(1.8444,1.4932)(1.8288,1.4764)(1.8130,1.4597)(1.7953,1.4405)(1.7817,1.4261)(1.7617,1.4044)(1.7505,1.3926)(1.7281,1.3683)(1.7194,1.3590)(1.6946,1.3320)(1.6885,1.3255)(1.6610,1.2955)(1.6576,1.2919)(1.6275,1.2589)(1.6269,1.2583)(1.5963,1.2248)(1.5939,1.2221)(1.5658,1.1912)(1.5604,1.1851)(1.5355,1.1577)(1.5268,1.1478)(1.5053,1.1241)(1.4932,1.1104)(1.4754,1.0906)(1.4597,1.0726)(1.4456,1.0570)(1.4261,1.0347)(1.4161,1.0234)(1.3926,0.9964)(1.3867,0.9899)(1.3590,0.9578)(1.3577,0.9563)(1.3288,0.9228)(1.3255,0.9187)(1.3002,0.8892)(1.2919,0.8791)(1.2719,0.8557)(1.2583,0.8390)(1.2440,0.8221)(1.2248,0.7984)(1.2165,0.7885)(1.1912,0.7572)(1.1894,0.7550)(1.1626,0.7214)(1.1577,0.7150)(1.1362,0.6879)(1.1241,0.6718)(1.1104,0.6543)(1.0906,0.6276)(1.0853,0.6208)(1.0607,0.5872)(1.0570,0.5819)(1.0366,0.5536)(1.0234,0.5343)(1.0134,0.5201)(0.9910,0.4865)(0.9899,0.4847)(0.9693,0.4530)(0.9563,0.4315)(0.9487,0.4194)(0.9291,0.3859)(0.9228,0.3740)(0.9105,0.3523)(0.8934,0.3187)(0.8892,0.3097)(0.8773,0.2852)(0.8629,0.2516)(0.8557,0.2325)(0.8499,0.2181)(0.8384,0.1845)(0.8288,0.1510)(0.8221,0.1213)(0.8211,0.1174)(0.8151,0.0838)(0.8110,0.0503)(0.8090,0.0167)(0.8090,-0.0167)(0.8110,-0.0503)(0.8151,-0.0838)(0.8211,-0.1174)(0.8221,-0.1213)(0.8288,-0.1510)(0.8384,-0.1845)(0.8499,-0.2181)(0.8557,-0.2325)(0.8629,-0.2516)(0.8773,-0.2852)(0.8892,-0.3097)(0.8934,-0.3187)(0.9105,-0.3523)(0.9228,-0.3740)(0.9291,-0.3859)(0.9487,-0.4194)(0.9563,-0.4315)(0.9693,-0.4530)(0.9899,-0.4847)(0.9910,-0.4865)(1.0134,-0.5201)(1.0234,-0.5343)(1.0366,-0.5536)(1.0570,-0.5819)(1.0607,-0.5872)(1.0853,-0.6208)(1.0906,-0.6276)(1.1104,-0.6543)(1.1241,-0.6718)(1.1362,-0.6879)(1.1577,-0.7150)(1.1626,-0.7214)(1.1894,-0.7550)(1.1912,-0.7572)(1.2165,-0.7885)(1.2248,-0.7984)(1.2440,-0.8221)(1.2583,-0.8390)(1.2719,-0.8557)(1.2919,-0.8791)(1.3002,-0.8892)(1.3255,-0.9187)(1.3288,-0.9228)(1.3577,-0.9563)(1.3590,-0.9578)(1.3867,-0.9899)(1.3926,-0.9964)(1.4161,-1.0234)(1.4261,-1.0347)(1.4456,-1.0570)(1.4597,-1.0726)(1.4754,-1.0906)(1.4932,-1.1104)(1.5053,-1.1241)(1.5268,-1.1478)(1.5355,-1.1577)(1.5604,-1.1851)(1.5658,-1.1912)(1.5939,-1.2221)(1.5963,-1.2248)(1.6269,-1.2583)(1.6275,-1.2589)(1.6576,-1.2919)(1.6610,-1.2955)(1.6885,-1.3255)(1.6946,-1.3320)(1.7194,-1.3590)(1.7281,-1.3683)(1.7505,-1.3926)(1.7617,-1.4044)(1.7817,-1.4261)(1.7953,-1.4405)(1.8130,-1.4597)(1.8288,-1.4764)(1.8444,-1.4932)(1.8624,-1.5123)(1.8758,-1.5268)(1.8959,-1.5480)(1.9074,-1.5604)(1.9295,-1.5836)(1.9390,-1.5939)(1.9630,-1.6192)(1.9707,-1.6275)(1.9966,-1.6547)(2.0025,-1.6610)(2.0302,-1.6901)(2.0344,-1.6946)(2.0637,-1.7254)(2.0663,-1.7281)(2.0973,-1.7606)(2.0983,-1.7617)(2.1303,-1.7953)(2.1308,-1.7958)(2.1623,-1.8288)(2.1644,-1.8309)(2.1944,-1.8624)(2.1979,-1.8660)(2.2266,-1.8959)(2.2315,-1.9010)(2.2588,-1.9295)(2.2651,-1.9359)(2.2910,-1.9630)(2.2986,-1.9708)(2.3233,-1.9966)(2.3322,-2.0057)(2.3557,-2.0302)(2.3657,-2.0405)(2.3880,-2.0637)(2.3993,-2.0752)(2.4204,-2.0973)(2.4328,-2.1100)(2.4529,-2.1308)(2.4664,-2.1447)(2.4853,-2.1644)(2.5000,-2.1793)};
+\draw (-2.5000,-0.3298) node {$ -5 $};
+\draw [] (-2.5000,-0.1000) -- (-2.5000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -4 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.5000,-0.3298) node {$ -3 $};
+\draw [] (-1.5000,-0.1000) -- (-1.5000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -2 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (-0.5000,-0.3298) node {$ -1 $};
+\draw [] (-0.5000,-0.1000) -- (-0.5000,0.1000);
+\draw (0.5000,-0.3149) node {$ 1 $};
+\draw [] (0.5000,-0.1000) -- (0.5000,0.1000);
+\draw (1.0000,-0.3149) node {$ 2 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (1.5000,-0.3149) node {$ 3 $};
+\draw [] (1.5000,-0.1000) -- (1.5000,0.1000);
+\draw (2.0000,-0.3149) node {$ 4 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (2.5000,-0.3149) node {$ 5 $};
+\draw [] (2.5000,-0.1000) -- (2.5000,0.1000);
+\draw (3.0000,-0.3149) node {$ 6 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -6 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.5000) node {$ -5 $};
+\draw [] (-0.1000,-2.5000) -- (0.1000,-2.5000);
+\draw (-0.4331,-2.0000) node {$ -4 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.5000) node {$ -3 $};
+\draw [] (-0.1000,-1.5000) -- (0.1000,-1.5000);
+\draw (-0.4331,-1.0000) node {$ -2 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.4331,-0.5000) node {$ -1 $};
+\draw [] (-0.1000,-0.5000) -- (0.1000,-0.5000);
+\draw (-0.2912,0.5000) node {$ 1 $};
+\draw [] (-0.1000,0.5000) -- (0.1000,0.5000);
+\draw (-0.2912,1.0000) node {$ 2 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,1.5000) node {$ 3 $};
+\draw [] (-0.1000,1.5000) -- (0.1000,1.5000);
+\draw (-0.2912,2.0000) node {$ 4 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,2.5000) node {$ 5 $};
+\draw [] (-0.1000,2.5000) -- (0.1000,2.5000);
+\draw (-0.2912,3.0000) node {$ 6 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall b/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall
index 3b06e3eda..7a7dd07e9 100644
--- a/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall
+++ b/src_phystricks/Fig_QIZooQNQSJj.pstricks.recall
@@ -79,17 +79,17 @@
 %PSTRICKS CODE
 %DEFAULT
 \draw (-1.4116,1.5804) node {\( A\)};
-\draw (-1.9800,-0.79165) node {\( B\)};
-\draw (1.9759,0.79165) node {\( C\)};
+\draw (-1.9799,-0.7916) node {\( B\)};
+\draw (1.9758,0.7916) node {\( C\)};
 
-\draw [] (1.750,0)--(1.746,0.1110)--(1.736,0.2215)--(1.718,0.3312)--(1.694,0.4395)--(1.663,0.5461)--(1.625,0.6504)--(1.580,0.7521)--(1.529,0.8508)--(1.472,0.9461)--(1.409,1.038)--(1.341,1.125)--(1.267,1.208)--(1.187,1.286)--(1.103,1.358)--(1.015,1.426)--(0.9226,1.487)--(0.8265,1.543)--(0.7270,1.592)--(0.6245,1.635)--(0.5196,1.671)--(0.4126,1.701)--(0.3039,1.723)--(0.1940,1.739)--(0.08327,1.748)--(-0.02777,1.750)--(-0.1387,1.744)--(-0.2491,1.732)--(-0.3584,1.713)--(-0.4663,1.687)--(-0.5724,1.654)--(-0.6761,1.614)--(-0.7771,1.568)--(-0.8750,1.516)--(-0.9694,1.457)--(-1.060,1.393)--(-1.146,1.323)--(-1.228,1.247)--(-1.304,1.167)--(-1.376,1.082)--(-1.441,0.9924)--(-1.501,0.8989)--(-1.555,0.8019)--(-1.603,0.7016)--(-1.644,0.5985)--(-1.679,0.4930)--(-1.707,0.3855)--(-1.728,0.2765)--(-1.742,0.1663)--(-1.749,0.05552)--(-1.749,-0.05552)--(-1.742,-0.1663)--(-1.728,-0.2765)--(-1.707,-0.3855)--(-1.679,-0.4930)--(-1.644,-0.5985)--(-1.603,-0.7016)--(-1.555,-0.8019)--(-1.501,-0.8989)--(-1.441,-0.9924)--(-1.376,-1.082)--(-1.304,-1.167)--(-1.228,-1.247)--(-1.146,-1.323)--(-1.060,-1.393)--(-0.9694,-1.457)--(-0.8750,-1.516)--(-0.7771,-1.568)--(-0.6761,-1.614)--(-0.5724,-1.654)--(-0.4663,-1.687)--(-0.3584,-1.713)--(-0.2491,-1.732)--(-0.1387,-1.744)--(-0.02777,-1.750)--(0.08327,-1.748)--(0.1940,-1.739)--(0.3039,-1.723)--(0.4126,-1.701)--(0.5196,-1.671)--(0.6245,-1.635)--(0.7270,-1.592)--(0.8265,-1.543)--(0.9226,-1.487)--(1.015,-1.426)--(1.103,-1.358)--(1.187,-1.286)--(1.267,-1.208)--(1.341,-1.125)--(1.409,-1.038)--(1.472,-0.9461)--(1.529,-0.8508)--(1.580,-0.7521)--(1.625,-0.6504)--(1.663,-0.5461)--(1.694,-0.4395)--(1.718,-0.3312)--(1.736,-0.2215)--(1.746,-0.1110)--(1.750,0);
-\draw (-0.42285,-0.55206) node {\( H\)};
-\draw [] (-1.21,1.27) -- (-0.658,-0.239);
-\draw [] (-1.21,1.27) -- (-1.64,-0.599);
-\draw [] (-1.64,-0.599) -- (1.64,0.599);
-\draw [] (1.64,0.599) -- (-1.21,1.27);
-\draw [] (-0.478,0.145) -- (-0.760,0.0425);
-\draw [] (-0.478,0.145) -- (-0.376,-0.137);
+\draw [] (1.7500,0.0000)--(1.7464,0.1109)--(1.7359,0.2215)--(1.7183,0.3311)--(1.6939,0.4395)--(1.6626,0.5460)--(1.6246,0.6504)--(1.5801,0.7521)--(1.5292,0.8508)--(1.4721,0.9461)--(1.4092,1.0375)--(1.3405,1.1248)--(1.2665,1.2076)--(1.1873,1.2855)--(1.1034,1.3582)--(1.0150,1.4255)--(0.9226,1.4870)--(0.8264,1.5425)--(0.7269,1.5918)--(0.6245,1.6347)--(0.5196,1.6710)--(0.4125,1.7006)--(0.3038,1.7234)--(0.1939,1.7392)--(0.0832,1.7480)--(-0.0277,1.7497)--(-0.1386,1.7444)--(-0.2490,1.7321)--(-0.3584,1.7129)--(-0.4663,1.6867)--(-0.5723,1.6537)--(-0.6761,1.6141)--(-0.7771,1.5679)--(-0.8750,1.5155)--(-0.9693,1.4569)--(-1.0598,1.3925)--(-1.1460,1.3225)--(-1.2275,1.2472)--(-1.3042,1.1668)--(-1.3755,1.0817)--(-1.4414,0.9923)--(-1.5014,0.8989)--(-1.5554,0.8018)--(-1.6031,0.7016)--(-1.6444,0.5985)--(-1.6791,0.4930)--(-1.7070,0.3855)--(-1.7280,0.2765)--(-1.7420,0.1663)--(-1.7491,0.0555)--(-1.7491,-0.0555)--(-1.7420,-0.1663)--(-1.7280,-0.2765)--(-1.7070,-0.3855)--(-1.6791,-0.4930)--(-1.6444,-0.5985)--(-1.6031,-0.7016)--(-1.5554,-0.8018)--(-1.5014,-0.8989)--(-1.4414,-0.9923)--(-1.3755,-1.0817)--(-1.3042,-1.1668)--(-1.2275,-1.2472)--(-1.1460,-1.3225)--(-1.0598,-1.3925)--(-0.9693,-1.4569)--(-0.8749,-1.5155)--(-0.7771,-1.5679)--(-0.6761,-1.6141)--(-0.5723,-1.6537)--(-0.4663,-1.6867)--(-0.3584,-1.7129)--(-0.2490,-1.7321)--(-0.1386,-1.7444)--(-0.0277,-1.7497)--(0.0832,-1.7480)--(0.1939,-1.7392)--(0.3038,-1.7234)--(0.4125,-1.7006)--(0.5196,-1.6710)--(0.6245,-1.6347)--(0.7269,-1.5918)--(0.8264,-1.5425)--(0.9226,-1.4870)--(1.0150,-1.4255)--(1.1034,-1.3582)--(1.1873,-1.2855)--(1.2665,-1.2076)--(1.3405,-1.1248)--(1.4092,-1.0375)--(1.4721,-0.9461)--(1.5292,-0.8508)--(1.5801,-0.7521)--(1.6246,-0.6504)--(1.6626,-0.5460)--(1.6939,-0.4395)--(1.7183,-0.3311)--(1.7359,-0.2215)--(1.7464,-0.1109)--(1.7500,0.0000);
+\draw (-0.4228,-0.5520) node {\( H\)};
+\draw [] (-1.2063,1.2677) -- (-0.6577,-0.2394);
+\draw [] (-1.2063,1.2677) -- (-1.6444,-0.5985);
+\draw [] (-1.6444,-0.5985) -- (1.6444,0.5985);
+\draw [] (1.6444,0.5985) -- (-1.2063,1.2677);
+\draw [] (-0.4784,0.1450) -- (-0.7603,0.0424);
+\draw [] (-0.4784,0.1450) -- (-0.3758,-0.1368);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall b/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall
index 21bc4eded..bfe913e1b 100644
--- a/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall
+++ b/src_phystricks/Fig_QOBAooZZZOrl.pstricks.recall
@@ -41,22 +41,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4250);
 %DEFAULT
 
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+\draw [] (2.1742,0.0000) -- (2.1742,1.6500);
+\draw [] (5.8257,1.6500) -- (5.8257,0.0000);
+\draw []  (2.1742,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.1742,-0.3785) node {$a$};
+\draw []  (5.8257,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,22 +95,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4250);
 %DEFAULT
 
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+\draw [] (2.1742,0.0000) -- (2.1742,1.6500);
+\draw [] (5.8257,1.6500) -- (5.8257,0.0000);
+\draw []  (2.1742,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.1742,-0.3785) node {$a$};
+\draw []  (5.8257,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,22 +149,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (7.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (7.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4250);
 %DEFAULT
 
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+\draw [] (2.1742,1.6500) -- (2.1742,1.6500);
+\draw [] (5.8257,1.6500) -- (5.8257,1.6500);
+\draw []  (2.1742,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.1742,-0.3785) node {$a$};
+\draw []  (5.8257,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.8257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QQa.pstricks.recall b/src_phystricks/Fig_QQa.pstricks.recall
index b9cb733d7..d4ad1e802 100644
--- a/src_phystricks/Fig_QQa.pstricks.recall
+++ b/src_phystricks/Fig_QQa.pstricks.recall
@@ -33,10 +33,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (0.50000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (0.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,-2.0000)(0.0700,-1.9000)(-0.0700,-1.8000)(0.0700,-1.7000)(-0.0700,-1.6000)(0.0700,-1.5000)(-0.0700,-1.4000)(0.0700,-1.3000)(-0.0700,-1.2000)(0.0700,-1.1000)(-0.0700,-1.0000)(0.0700,-0.9000)(-0.0700,-0.8000)(0.0700,-0.7000)(-0.0700,-0.6000)(0.0700,-0.5000)(-0.0700,-0.4000)(0.0700,-0.3000)(-0.0700,-0.2000)(0.0700,-0.1000)(-0.0700,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -67,11 +67,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,0)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,0.0000)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -102,10 +102,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,-2.00)(-1.98,-1.88)(-1.81,-1.91)(-1.84,-1.74)(-1.66,-1.76)(-1.69,-1.59)(-1.52,-1.62)(-1.55,-1.45)(-1.38,-1.48)(-1.41,-1.31)(-1.24,-1.34)(-1.26,-1.16)(-1.09,-1.19)(-1.12,-1.02)(-0.951,-1.05)(-0.978,-0.879)(-0.808,-0.907)(-0.835,-0.736)(-0.665,-0.764)(-0.692,-0.593)(-0.522,-0.621)(-0.549,-0.451)(-0.379,-0.478)(-0.407,-0.308)(-0.236,-0.335)(-0.264,-0.165)(-0.0934,-0.192)(-0.121,-0.0219)(0.0495,-0.0495)(0.0219,0.121)(0.192,0.0934)(0.165,0.264)(0.335,0.236)(0.308,0.407)(0.478,0.379)(0.451,0.549)(0.621,0.522)(0.593,0.692)(0.764,0.665)(0.736,0.835)(0.907,0.808)(0.879,0.978)(1.05,0.951)(1.02,1.12)(1.19,1.09)(1.16,1.26)(1.34,1.24)(1.31,1.41)(1.48,1.38)(1.45,1.55)(1.62,1.52)(1.59,1.69)(1.76,1.66)(1.74,1.84)(1.91,1.81)(1.88,1.98)(2.00,2.00)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(-2.0000,-2.0000)(-1.9780,-1.8790)(-1.8076,-1.9066)(-1.8352,-1.7362)(-1.6647,-1.7637)(-1.6923,-1.5933)(-1.5219,-1.6209)(-1.5494,-1.4505)(-1.3790,-1.4780)(-1.4066,-1.3076)(-1.2362,-1.3352)(-1.2637,-1.1647)(-1.0933,-1.1923)(-1.1209,-1.0219)(-0.9505,-1.0494)(-0.9780,-0.8790)(-0.8076,-0.9066)(-0.8352,-0.7362)(-0.6647,-0.7637)(-0.6923,-0.5933)(-0.5219,-0.6209)(-0.5494,-0.4505)(-0.3790,-0.4780)(-0.4066,-0.3076)(-0.2362,-0.3352)(-0.2637,-0.1647)(-0.0933,-0.1923)(-0.1209,-0.0219)(0.0494,-0.0494)(0.0219,0.1209)(0.1923,0.0933)(0.1647,0.2637)(0.3352,0.2362)(0.3076,0.4066)(0.4780,0.3790)(0.4505,0.5494)(0.6209,0.5219)(0.5933,0.6923)(0.7637,0.6647)(0.7362,0.8352)(0.9066,0.8076)(0.8790,0.9780)(1.0494,0.9505)(1.0219,1.1209)(1.1923,1.0933)(1.1647,1.2637)(1.3352,1.2362)(1.3076,1.4066)(1.4780,1.3790)(1.4505,1.5494)(1.6209,1.5219)(1.5933,1.6923)(1.7637,1.6647)(1.7362,1.8352)(1.9066,1.8076)(1.8790,1.9780)(2.0000,2.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -136,11 +136,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.8026) -- (0,1.2352);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.8025) -- (0.0000,1.2351);
 %DEFAULT
 
-\draw [color=red] plot [smooth,tension=1] coordinates {(0.100,-2.30)(0.0409,-2.20)(0.191,-2.11)(0.0654,-1.99)(0.218,-1.91)(0.0952,-1.79)(0.250,-1.72)(0.131,-1.60)(0.289,-1.53)(0.175,-1.40)(0.336,-1.33)(0.228,-1.20)(0.391,-1.15)(0.291,-1.01)(0.457,-0.962)(0.365,-0.817)(0.534,-0.783)(0.453,-0.631)(0.624,-0.609)(0.554,-0.452)(0.726,-0.443)(0.668,-0.281)(0.840,-0.286)(0.794,-0.120)(0.966,-0.137)(0.933,0.0318)(1.10,0.00217)(1.08,0.173)(1.25,0.133)(1.24,0.304)(1.40,0.253)(1.40,0.425)(1.57,0.365)(1.58,0.537)(1.73,0.469)(1.75,0.640)(1.90,0.565)(1.93,0.735)(2.00,0.693)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.1000,-2.3025)(0.0408,-2.1956)(0.1914,-2.1124)(0.0654,-1.9945)(0.2181,-1.9146)(0.0952,-1.7947)(0.2501,-1.7199)(0.1313,-1.5958)(0.2889,-1.5263)(0.1751,-1.3972)(0.3357,-1.3345)(0.2280,-1.2004)(0.3912,-1.1464)(0.2906,-1.0076)(0.4570,-0.9622)(0.3652,-0.8172)(0.5343,-0.7827)(0.4528,-0.6314)(0.6239,-0.6089)(0.5537,-0.4520)(0.7258,-0.4429)(0.6681,-0.2807)(0.8396,-0.2857)(0.7941,-0.1201)(0.9656,-0.1372)(0.9329,0.0317)(1.1025,0.0021)(1.0818,0.1730)(1.2495,0.1325)(1.2394,0.3040)(1.4038,0.2528)(1.4047,0.4253)(1.5653,0.3647)(1.5751,0.5369)(1.7324,0.4684)(1.7490,0.6395)(1.9043,0.5649)(1.9279,0.7351)(2.0000,0.6931)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -171,11 +171,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,-2.0000)(0.0700,-1.9000)(-0.0700,-1.8000)(0.0700,-1.7000)(-0.0700,-1.6000)(0.0700,-1.5000)(-0.0700,-1.4000)(0.0700,-1.3000)(-0.0700,-1.2000)(0.0700,-1.1000)(-0.0700,-1.0000)(0.0700,-0.9000)(-0.0700,-0.8000)(0.0700,-0.7000)(-0.0700,-0.6000)(0.0700,-0.5000)(-0.0700,-0.4000)(0.0700,-0.3000)(-0.0700,-0.2000)(0.0700,-0.1000)(-0.0700,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(-2.0000,0.0000)(-1.9000,0.0700)(-1.8000,-0.0700)(-1.7000,0.0700)(-1.6000,-0.0700)(-1.5000,0.0700)(-1.4000,-0.0700)(-1.3000,0.0700)(-1.2000,-0.0700)(-1.1000,0.0700)(-1.0000,-0.0700)(-0.9000,0.0700)(-0.8000,-0.0700)(-0.7000,0.0700)(-0.6000,-0.0700)(-0.5000,0.0700)(-0.4000,-0.0700)(-0.3000,0.0700)(-0.2000,-0.0700)(-0.1000,0.0700)(0.0000,-0.0700)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -206,11 +206,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
-\draw [color=red] plot [smooth,tension=1] coordinates {(0,-2.00)(0.0700,-1.90)(-0.0700,-1.80)(0.0700,-1.70)(-0.0700,-1.60)(0.0700,-1.50)(-0.0700,-1.40)(0.0700,-1.30)(-0.0700,-1.20)(0.0700,-1.10)(-0.0700,-1.00)(0.0700,-0.900)(-0.0700,-0.800)(0.0700,-0.700)(-0.0700,-0.600)(0.0700,-0.500)(-0.0700,-0.400)(0.0700,-0.300)(-0.0700,-0.200)(0.0700,-0.100)(-0.0700,0)(0.0700,0.100)(-0.0700,0.200)(0.0700,0.300)(-0.0700,0.400)(0.0700,0.500)(-0.0700,0.600)(0.0700,0.700)(-0.0700,0.800)(0.0700,0.900)(-0.0700,1.00)(0.0700,1.10)(-0.0700,1.20)(0.0700,1.30)(-0.0700,1.40)(0.0700,1.50)(-0.0700,1.60)(0.0700,1.70)(-0.0700,1.80)(0.0700,1.90)(0,2.00)};
-\draw [color=red] plot [smooth,tension=1] coordinates {(-2.00,0)(-1.90,0.0700)(-1.80,-0.0700)(-1.70,0.0700)(-1.60,-0.0700)(-1.50,0.0700)(-1.40,-0.0700)(-1.30,0.0700)(-1.20,-0.0700)(-1.10,0.0700)(-1.00,-0.0700)(-0.900,0.0700)(-0.800,-0.0700)(-0.700,0.0700)(-0.600,-0.0700)(-0.500,0.0700)(-0.400,-0.0700)(-0.300,0.0700)(-0.200,-0.0700)(-0.100,0.0700)(0,-0.0700)(0.100,0.0700)(0.200,-0.0700)(0.300,0.0700)(0.400,-0.0700)(0.500,0.0700)(0.600,-0.0700)(0.700,0.0700)(0.800,-0.0700)(0.900,0.0700)(1.00,-0.0700)(1.10,0.0700)(1.20,-0.0700)(1.30,0.0700)(1.40,-0.0700)(1.50,0.0700)(1.60,-0.0700)(1.70,0.0700)(1.80,-0.0700)(1.90,0.0700)(2.00,0)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(0.0000,-2.0000)(0.0700,-1.9000)(-0.0700,-1.8000)(0.0700,-1.7000)(-0.0700,-1.6000)(0.0700,-1.5000)(-0.0700,-1.4000)(0.0700,-1.3000)(-0.0700,-1.2000)(0.0700,-1.1000)(-0.0700,-1.0000)(0.0700,-0.9000)(-0.0700,-0.8000)(0.0700,-0.7000)(-0.0700,-0.6000)(0.0700,-0.5000)(-0.0700,-0.4000)(0.0700,-0.3000)(-0.0700,-0.2000)(0.0700,-0.1000)(-0.0700,0.0000)(0.0700,0.1000)(-0.0700,0.2000)(0.0700,0.3000)(-0.0700,0.4000)(0.0700,0.5000)(-0.0700,0.6000)(0.0700,0.7000)(-0.0700,0.8000)(0.0700,0.9000)(-0.0700,1.0000)(0.0700,1.1000)(-0.0700,1.2000)(0.0700,1.3000)(-0.0700,1.4000)(0.0700,1.5000)(-0.0700,1.6000)(0.0700,1.7000)(-0.0700,1.8000)(0.0700,1.9000)(0.0000,2.0000)};
+\draw [color=red] plot [smooth,tension=1] coordinates {(-2.0000,0.0000)(-1.9000,0.0700)(-1.8000,-0.0700)(-1.7000,0.0700)(-1.6000,-0.0700)(-1.5000,0.0700)(-1.4000,-0.0700)(-1.3000,0.0700)(-1.2000,-0.0700)(-1.1000,0.0700)(-1.0000,-0.0700)(-0.9000,0.0700)(-0.8000,-0.0700)(-0.7000,0.0700)(-0.6000,-0.0700)(-0.5000,0.0700)(-0.4000,-0.0700)(-0.3000,0.0700)(-0.2000,-0.0700)(-0.1000,0.0700)(0.0000,-0.0700)(0.1000,0.0700)(0.2000,-0.0700)(0.3000,0.0700)(0.4000,-0.0700)(0.5000,0.0700)(0.6000,-0.0700)(0.7000,0.0700)(0.8000,-0.0700)(0.9000,0.0700)(1.0000,-0.0700)(1.1000,0.0700)(1.2000,-0.0700)(1.3000,0.0700)(1.4000,-0.0700)(1.5000,0.0700)(1.6000,-0.0700)(1.7000,0.0700)(1.8000,-0.0700)(1.9000,0.0700)(2.0000,0.0000)};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QXyVaKD.pstricks.recall b/src_phystricks/Fig_QXyVaKD.pstricks.recall
index c8f85cd29..5d96b472f 100644
--- a/src_phystricks/Fig_QXyVaKD.pstricks.recall
+++ b/src_phystricks/Fig_QXyVaKD.pstricks.recall
@@ -67,21 +67,21 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-1.0000) -- (0,2.5000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.0000) -- (0.0000,2.5000);
 %DEFAULT
 
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+\draw [color=cyan] (0.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=cyan] (1.0000,1.0000) -- (1.0000,1.0000);
+\draw [color=red] (-0.5000,-0.5000) -- (1.5000,1.5000);
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.3546,0.66316) node {$P$};
+\draw (1.3546,0.6631) node {$P$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_QuelCote.pstricks.recall b/src_phystricks/Fig_QuelCote.pstricks.recall
index 314ffc1de..62f4d980b 100644
--- a/src_phystricks/Fig_QuelCote.pstricks.recall
+++ b/src_phystricks/Fig_QuelCote.pstricks.recall
@@ -85,18 +85,18 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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-\draw [color=brown,->,>=latex] (-2.0667,0.29444) -- (-1.6866,2.2580);
-\draw (-0.82720,2.2580) node {$\gamma'(t)$};
-\draw [,->,>=latex] (-2.0667,0.29444) -- (-3.3161,1.8562);
-\draw (-2.7118,2.2547) node {$\gamma''(t)$};
-\draw [color=green,->,>=latex] (-2.0667,0.29444) -- (-0.10311,-0.085599);
-\draw (-0.10311,0.39686) node {$n(t)$};
-\draw [color=green,style=dashed,->,>=latex] (-2.0667,0.29444) -- (-4.0302,0.67449);
+\draw [color=blue] (-3.0000,-1.7500)--(-2.9803,-1.7204)--(-2.9607,-1.6907)--(-2.9413,-1.6610)--(-2.9219,-1.6312)--(-2.9026,-1.6014)--(-2.8834,-1.5714)--(-2.8643,-1.5414)--(-2.8453,-1.5114)--(-2.8264,-1.4812)--(-2.8076,-1.4510)--(-2.7889,-1.4206)--(-2.7703,-1.3902)--(-2.7519,-1.3597)--(-2.7335,-1.3291)--(-2.7153,-1.2984)--(-2.6971,-1.2676)--(-2.6791,-1.2367)--(-2.6612,-1.2057)--(-2.6435,-1.1746)--(-2.6259,-1.1434)--(-2.6084,-1.1121)--(-2.5910,-1.0806)--(-2.5738,-1.0491)--(-2.5567,-1.0174)--(-2.5398,-0.9856)--(-2.5230,-0.9536)--(-2.5063,-0.9215)--(-2.4898,-0.8893)--(-2.4735,-0.8569)--(-2.4574,-0.8244)--(-2.4414,-0.7917)--(-2.4255,-0.7589)--(-2.4099,-0.7259)--(-2.3944,-0.6927)--(-2.3791,-0.6593)--(-2.3640,-0.6258)--(-2.3491,-0.5921)--(-2.3344,-0.5581)--(-2.3199,-0.5240)--(-2.3056,-0.4896)--(-2.2915,-0.4551)--(-2.2777,-0.4203)--(-2.2640,-0.3853)--(-2.2506,-0.3500)--(-2.2375,-0.3145)--(-2.2246,-0.2787)--(-2.2119,-0.2427)--(-2.1995,-0.2063)--(-2.1874,-0.1697)--(-2.1755,-0.1327)--(-2.1640,-0.0955)--(-2.1527,-0.0579)--(-2.1417,-0.0200)--(-2.1311,0.0182)--(-2.1207,0.0569)--(-2.1107,0.0959)--(-2.1010,0.1354)--(-2.0917,0.1752)--(-2.0828,0.2156)--(-2.0742,0.2563)--(-2.0660,0.2976)--(-2.0582,0.3393)--(-2.0508,0.3816)--(-2.0439,0.4244)--(-2.0374,0.4678)--(-2.0313,0.5118)--(-2.0257,0.5565)--(-2.0207,0.6017)--(-2.0161,0.6477)--(-2.0120,0.6944)--(-2.0085,0.7418)--(-2.0056,0.7901)--(-2.0032,0.8392)--(-2.0015,0.8891)--(-2.0004,0.9400)--(-2.0000,0.9919)--(-2.0002,1.0448)--(-2.0012,1.0987)--(-2.0029,1.1539)--(-2.0053,1.2102)--(-2.0086,1.2678)--(-2.0127,1.3267)--(-2.0177,1.3871)--(-2.0236,1.4489)--(-2.0305,1.5124)--(-2.0383,1.5776)--(-2.0473,1.6445)--(-2.0573,1.7134)--(-2.0684,1.7843)--(-2.0808,1.8574)--(-2.0944,1.9328)--(-2.1093,2.0107)--(-2.1256,2.0912)--(-2.1434,2.1745)--(-2.1627,2.2608)--(-2.1837,2.3503)--(-2.2063,2.4433)--(-2.2307,2.5400)--(-2.2571,2.6408);
+\draw [color=brown,style=dashed] (-2.4467,-1.6691) -- (-1.6866,2.2580);
+\draw [color=brown,->,>=latex] (-2.0666,0.2944) -- (-1.6866,2.2580);
+\draw (-0.8272,2.2580) node {$\gamma'(t)$};
+\draw [,->,>=latex] (-2.0666,0.2944) -- (-3.3160,1.8561);
+\draw (-2.7117,2.2547) node {$\gamma''(t)$};
+\draw [color=green,->,>=latex] (-2.0666,0.2944) -- (-0.1031,-0.0855);
+\draw (-0.1031,0.3968) node {$n(t)$};
+\draw [color=green,style=dashed,->,>=latex] (-2.0666,0.2944) -- (-4.0302,0.6744);
 \draw (-4.0302,1.1569) node {$-n(t)$};
-\draw []  (-2.0667,0.29444) node [rotate=0] {$\bullet$};
-\draw (-1.7120,-0.042396) node {$P$};
+\draw []  (-2.0666,0.2944) node [rotate=0] {$\bullet$};
+\draw (-1.7120,-0.0423) node {$P$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_RLuqsrr.pstricks.recall b/src_phystricks/Fig_RLuqsrr.pstricks.recall
index f50e8f6d6..4afde3c45 100644
--- a/src_phystricks/Fig_RLuqsrr.pstricks.recall
+++ b/src_phystricks/Fig_RLuqsrr.pstricks.recall
@@ -87,25 +87,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.7832,0);
-\draw [,->,>=latex] (0,-0.91381) -- (0,2.9142);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-0.9138) -- (0.0000,2.9141);
 %DEFAULT
 
-\draw [color=red] (0,0)--(0.06347,0.1346)--(0.1269,0.2832)--(0.1904,0.4433)--(0.2539,0.6124)--(0.3173,0.7876)--(0.3808,0.9663)--(0.4443,1.146)--(0.5077,1.322)--(0.5712,1.494)--(0.6347,1.658)--(0.6981,1.811)--(0.7616,1.951)--(0.8251,2.076)--(0.8885,2.184)--(0.9520,2.272)--(1.015,2.340)--(1.079,2.387)--(1.142,2.411)--(1.206,2.412)--(1.269,2.391)--(1.333,2.347)--(1.396,2.282)--(1.460,2.196)--(1.523,2.091)--(1.587,1.968)--(1.650,1.829)--(1.714,1.678)--(1.777,1.515)--(1.841,1.344)--(1.904,1.168)--(1.967,0.9888)--(2.031,0.8098)--(2.094,0.6340)--(2.158,0.4640)--(2.221,0.3026)--(2.285,0.1525)--(2.348,0.01599)--(2.412,-0.1047)--(2.475,-0.2076)--(2.539,-0.2910)--(2.602,-0.3537)--(2.666,-0.3946)--(2.729,-0.4131)--(2.793,-0.4088)--(2.856,-0.3819)--(2.919,-0.3327)--(2.983,-0.2621)--(3.046,-0.1712)--(3.110,-0.06141)--(3.173,0.06544)--(3.237,0.2073)--(3.300,0.3620)--(3.364,0.5269)--(3.427,0.6994)--(3.491,0.8767)--(3.554,1.056)--(3.618,1.235)--(3.681,1.409)--(3.745,1.577)--(3.808,1.736)--(3.871,1.883)--(3.935,2.016)--(3.998,2.132)--(4.062,2.230)--(4.125,2.309)--(4.189,2.366)--(4.252,2.401)--(4.316,2.414)--(4.379,2.404)--(4.443,2.372)--(4.506,2.317)--(4.570,2.241)--(4.633,2.145)--(4.697,2.031)--(4.760,1.900)--(4.823,1.755)--(4.887,1.598)--(4.950,1.431)--(5.014,1.257)--(5.077,1.078)--(5.141,0.8991)--(5.204,0.7214)--(5.268,0.5481)--(5.331,0.3821)--(5.395,0.2260)--(5.458,0.08240)--(5.522,-0.04645)--(5.585,-0.1585)--(5.648,-0.2518)--(5.712,-0.3250)--(5.775,-0.3769)--(5.839,-0.4067)--(5.902,-0.4138)--(5.966,-0.3982)--(6.029,-0.3600)--(6.093,-0.3000)--(6.156,-0.2191)--(6.220,-0.1185)--(6.283,0);
+\draw [color=red] (0.0000,0.0000)--(0.0634,0.1346)--(0.1269,0.2831)--(0.1903,0.4432)--(0.2538,0.6123)--(0.3173,0.7876)--(0.3807,0.9663)--(0.4442,1.1455)--(0.5077,1.3224)--(0.5711,1.4942)--(0.6346,1.6579)--(0.6981,1.8111)--(0.7615,1.9512)--(0.8250,2.0761)--(0.8885,2.1836)--(0.9519,2.2720)--(1.0154,2.3400)--(1.0789,2.3864)--(1.1423,2.4106)--(1.2058,2.4120)--(1.2693,2.3907)--(1.3327,2.3470)--(1.3962,2.2817)--(1.4597,2.1957)--(1.5231,2.0905)--(1.5866,1.9677)--(1.6501,1.8294)--(1.7135,1.6777)--(1.7770,1.5151)--(1.8405,1.3443)--(1.9039,1.1678)--(1.9674,0.9887)--(2.0309,0.8098)--(2.0943,0.6339)--(2.1578,0.4639)--(2.2213,0.3026)--(2.2847,0.1524)--(2.3482,0.0159)--(2.4117,-0.1046)--(2.4751,-0.2075)--(2.5386,-0.2910)--(2.6021,-0.3537)--(2.6655,-0.3946)--(2.7290,-0.4131)--(2.7925,-0.4088)--(2.8559,-0.3818)--(2.9194,-0.3327)--(2.9829,-0.2621)--(3.0463,-0.1711)--(3.1098,-0.0614)--(3.1733,0.0654)--(3.2367,0.2073)--(3.3002,0.3619)--(3.3637,0.5268)--(3.4271,0.6993)--(3.4906,0.8767)--(3.5541,1.0560)--(3.6175,1.2345)--(3.6810,1.4091)--(3.7445,1.5772)--(3.8079,1.7360)--(3.8714,1.8830)--(3.9349,2.0157)--(3.9983,2.1321)--(4.0618,2.2303)--(4.1253,2.3086)--(4.1887,2.3660)--(4.2522,2.4013)--(4.3157,2.4141)--(4.3791,2.4042)--(4.4426,2.3716)--(4.5061,2.3170)--(4.5695,2.2412)--(4.6330,2.1454)--(4.6965,2.0312)--(4.7599,1.9004)--(4.8234,1.7551)--(4.8869,1.5976)--(4.9503,1.4306)--(5.0138,1.2566)--(5.0773,1.0784)--(5.1407,0.8991)--(5.2042,0.7213)--(5.2677,0.5480)--(5.3311,0.3820)--(5.3946,0.2260)--(5.4581,0.0823)--(5.5215,-0.0464)--(5.5850,-0.1584)--(5.6485,-0.2518)--(5.7119,-0.3250)--(5.7754,-0.3769)--(5.8389,-0.4066)--(5.9023,-0.4138)--(5.9658,-0.3981)--(6.0293,-0.3600)--(6.0927,-0.3000)--(6.1562,-0.2190)--(6.2197,-0.1185)--(6.2831,0.0000);
 
-\draw [color=blue] (0,0)--(0.06347,0.008045)--(0.1269,0.03205)--(0.1904,0.07163)--(0.2539,0.1262)--(0.3173,0.1947)--(0.3808,0.2763)--(0.4443,0.3694)--(0.5077,0.4728)--(0.5712,0.5846)--(0.6347,0.7031)--(0.6981,0.8264)--(0.7616,0.9524)--(0.8251,1.079)--(0.8885,1.205)--(0.9520,1.327)--(1.015,1.444)--(1.079,1.554)--(1.142,1.655)--(1.206,1.745)--(1.269,1.824)--(1.333,1.889)--(1.396,1.940)--(1.460,1.975)--(1.523,1.995)--(1.587,1.999)--(1.650,1.987)--(1.714,1.959)--(1.777,1.916)--(1.841,1.858)--(1.904,1.786)--(1.967,1.701)--(2.031,1.606)--(2.094,1.500)--(2.158,1.386)--(2.221,1.266)--(2.285,1.142)--(2.348,1.016)--(2.412,0.8892)--(2.475,0.7642)--(2.539,0.6431)--(2.602,0.5277)--(2.666,0.4199)--(2.729,0.3215)--(2.793,0.2340)--(2.856,0.1587)--(2.919,0.09707)--(2.983,0.04993)--(3.046,0.01807)--(3.110,0.002013)--(3.173,0.002013)--(3.237,0.01807)--(3.300,0.04993)--(3.364,0.09707)--(3.427,0.1587)--(3.491,0.2340)--(3.554,0.3215)--(3.618,0.4199)--(3.681,0.5277)--(3.745,0.6431)--(3.808,0.7642)--(3.871,0.8892)--(3.935,1.016)--(3.998,1.142)--(4.062,1.266)--(4.125,1.386)--(4.189,1.500)--(4.252,1.606)--(4.316,1.701)--(4.379,1.786)--(4.443,1.858)--(4.506,1.916)--(4.570,1.959)--(4.633,1.987)--(4.697,1.999)--(4.760,1.995)--(4.823,1.975)--(4.887,1.940)--(4.950,1.889)--(5.014,1.824)--(5.077,1.745)--(5.141,1.655)--(5.204,1.554)--(5.268,1.444)--(5.331,1.327)--(5.395,1.205)--(5.458,1.079)--(5.522,0.9524)--(5.585,0.8264)--(5.648,0.7031)--(5.712,0.5846)--(5.775,0.4728)--(5.839,0.3694)--(5.902,0.2763)--(5.966,0.1947)--(6.029,0.1262)--(6.093,0.07163)--(6.156,0.03205)--(6.220,0.008045)--(6.283,0);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (0.0000,0.0000)--(0.0634,0.0080)--(0.1269,0.0320)--(0.1903,0.0716)--(0.2538,0.1261)--(0.3173,0.1947)--(0.3807,0.2762)--(0.4442,0.3694)--(0.5077,0.4727)--(0.5711,0.5845)--(0.6346,0.7030)--(0.6981,0.8263)--(0.7615,0.9524)--(0.8250,1.0792)--(0.8885,1.2048)--(0.9519,1.3270)--(1.0154,1.4440)--(1.0789,1.5539)--(1.1423,1.6548)--(1.2058,1.7452)--(1.2693,1.8236)--(1.3327,1.8888)--(1.3962,1.9396)--(1.4597,1.9754)--(1.5231,1.9954)--(1.5866,1.9994)--(1.6501,1.9874)--(1.7135,1.9594)--(1.7770,1.9161)--(1.8405,1.8579)--(1.9039,1.7860)--(1.9674,1.7014)--(2.0309,1.6056)--(2.0943,1.5000)--(2.1578,1.3863)--(2.2213,1.2664)--(2.2847,1.1423)--(2.3482,1.0158)--(2.4117,0.8891)--(2.4751,0.7642)--(2.5386,0.6431)--(2.6021,0.5277)--(2.6655,0.4199)--(2.7290,0.3214)--(2.7925,0.2339)--(2.8559,0.1587)--(2.9194,0.0970)--(2.9829,0.0499)--(3.0463,0.0180)--(3.1098,0.0020)--(3.1733,0.0020)--(3.2367,0.0180)--(3.3002,0.0499)--(3.3637,0.0970)--(3.4271,0.1587)--(3.4906,0.2339)--(3.5541,0.3214)--(3.6175,0.4199)--(3.6810,0.5277)--(3.7445,0.6431)--(3.8079,0.7642)--(3.8714,0.8891)--(3.9349,1.0158)--(3.9983,1.1423)--(4.0618,1.2664)--(4.1253,1.3863)--(4.1887,1.5000)--(4.2522,1.6056)--(4.3157,1.7014)--(4.3791,1.7860)--(4.4426,1.8579)--(4.5061,1.9161)--(4.5695,1.9594)--(4.6330,1.9874)--(4.6965,1.9994)--(4.7599,1.9954)--(4.8234,1.9754)--(4.8869,1.9396)--(4.9503,1.8888)--(5.0138,1.8236)--(5.0773,1.7452)--(5.1407,1.6548)--(5.2042,1.5539)--(5.2677,1.4440)--(5.3311,1.3270)--(5.3946,1.2048)--(5.4581,1.0792)--(5.5215,0.9524)--(5.5850,0.8263)--(5.6485,0.7030)--(5.7119,0.5845)--(5.7754,0.4727)--(5.8389,0.3694)--(5.9023,0.2762)--(5.9658,0.1947)--(6.0293,0.1261)--(6.0927,0.0716)--(6.1562,0.0320)--(6.2197,0.0080)--(6.2831,0.0000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall b/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall
index 9b902f047..733a2337b 100644
--- a/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall
+++ b/src_phystricks/Fig_RPNooQXxpZZ.pstricks.recall
@@ -92,27 +92,27 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-6.5000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-1.9457) -- (0,1.9457);
+\draw [,->,>=latex] (-6.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.9456) -- (0.0000,1.9456);
 %DEFAULT
 
-\draw [color=blue] (-6.0000,0.059192)--(-5.8788,0.027135)--(-5.7576,0.0060480)--(-5.6364,0)--(-5.5152,0.011586)--(-5.3939,0.039487)--(-5.2727,0.080491)--(-5.1515,0.12891)--(-5.0303,0.17762)--(-4.9091,0.21910)--(-4.7879,0.24669)--(-4.6667,0.25565)--(-4.5455,0.24409)--(-4.4242,0.21334)--(-4.3030,0.16799)--(-4.1818,0.11529)--(-4.0606,0.064108)--(-3.9394,0.023666)--(-3.8182,0.0020298)--(-3.6970,0.0047735)--(-3.5758,0.033930)--(-3.4545,0.087448)--(-3.3333,0.15925)--(-3.2121,0.23994)--(-3.0909,0.31806)--(-2.9697,0.38179)--(-2.8485,0.42076)--(-2.7273,0.42785)--(-2.6061,0.40058)--(-2.4848,0.34187)--(-2.3636,0.26014)--(-2.2424,0.16851)--(-2.1212,0.083272)--(-2.0000,0.021790)--(-1.8788,0)--(-1.7576,0.030313)--(-1.6364,0.11884)--(-1.5152,0.26464)--(-1.3939,0.45879)--(-1.2727,0.68492)--(-1.1515,0.92065)--(-1.0303,1.1399)--(-0.90909,1.3159)--(-0.78788,1.4242)--(-0.66667,1.4457)--(-0.54545,1.3694)--(-0.42424,1.1936)--(-0.30303,0.92708)--(-0.18182,0.58774)--(-0.060606,0.20133)--(0.060606,-0.20133)--(0.18182,-0.58774)--(0.30303,-0.92708)--(0.42424,-1.1936)--(0.54545,-1.3694)--(0.66667,-1.4457)--(0.78788,-1.4242)--(0.90909,-1.3159)--(1.0303,-1.1399)--(1.1515,-0.92065)--(1.2727,-0.68492)--(1.3939,-0.45879)--(1.5152,-0.26464)--(1.6364,-0.11884)--(1.7576,-0.030313)--(1.8788,0)--(2.0000,-0.021790)--(2.1212,-0.083272)--(2.2424,-0.16851)--(2.3636,-0.26014)--(2.4848,-0.34187)--(2.6061,-0.40058)--(2.7273,-0.42785)--(2.8485,-0.42076)--(2.9697,-0.38179)--(3.0909,-0.31806)--(3.2121,-0.23994)--(3.3333,-0.15925)--(3.4545,-0.087448)--(3.5758,-0.033930)--(3.6970,-0.0047735)--(3.8182,-0.0020298)--(3.9394,-0.023666)--(4.0606,-0.064108)--(4.1818,-0.11529)--(4.3030,-0.16799)--(4.4242,-0.21334)--(4.5455,-0.24409)--(4.6667,-0.25565)--(4.7879,-0.24669)--(4.9091,-0.21910)--(5.0303,-0.17762)--(5.1515,-0.12891)--(5.2727,-0.080491)--(5.3939,-0.039487)--(5.5152,-0.011586)--(5.6364,0)--(5.7576,-0.0060480)--(5.8788,-0.027135)--(6.0000,-0.059192);
-\draw (-5.6549,-0.32983) node {$ -6 \, \pi $};
-\draw [] (-5.66,-0.100) -- (-5.66,0.100);
-\draw (-3.7699,-0.32983) node {$ -4 \, \pi $};
-\draw [] (-3.77,-0.100) -- (-3.77,0.100);
-\draw (-1.8850,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-1.88,-0.100) -- (-1.88,0.100);
-\draw (1.8850,-0.31492) node {$ 2 \, \pi $};
-\draw [] (1.88,-0.100) -- (1.88,0.100);
-\draw (3.7699,-0.31492) node {$ 4 \, \pi $};
-\draw [] (3.77,-0.100) -- (3.77,0.100);
-\draw (5.6549,-0.31492) node {$ 6 \, \pi $};
-\draw [] (5.66,-0.100) -- (5.66,0.100);
-\draw (-0.45274,-1.0000) node {$ -\frac{1}{2} $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.31083,1.0000) node {$ \frac{1}{2} $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-6.0000,0.0591)--(-5.8787,0.0271)--(-5.7575,0.0060)--(-5.6363,0.0000)--(-5.5151,0.0115)--(-5.3939,0.0394)--(-5.2727,0.0804)--(-5.1515,0.1289)--(-5.0303,0.1776)--(-4.9090,0.2191)--(-4.7878,0.2466)--(-4.6666,0.2556)--(-4.5454,0.2440)--(-4.4242,0.2133)--(-4.3030,0.1679)--(-4.1818,0.1152)--(-4.0606,0.0641)--(-3.9393,0.0236)--(-3.8181,0.0020)--(-3.6969,0.0047)--(-3.5757,0.0339)--(-3.4545,0.0874)--(-3.3333,0.1592)--(-3.2121,0.2399)--(-3.0909,0.3180)--(-2.9696,0.3817)--(-2.8484,0.4207)--(-2.7272,0.4278)--(-2.6060,0.4005)--(-2.4848,0.3418)--(-2.3636,0.2601)--(-2.2424,0.1685)--(-2.1212,0.0832)--(-2.0000,0.0217)--(-1.8787,0.0000)--(-1.7575,0.0303)--(-1.6363,0.1188)--(-1.5151,0.2646)--(-1.3939,0.4587)--(-1.2727,0.6849)--(-1.1515,0.9206)--(-1.0303,1.1399)--(-0.9090,1.3159)--(-0.7878,1.4241)--(-0.6666,1.4456)--(-0.5454,1.3693)--(-0.4242,1.1936)--(-0.3030,0.9270)--(-0.1818,0.5877)--(-0.0606,0.2013)--(0.0606,-0.2013)--(0.1818,-0.5877)--(0.3030,-0.9270)--(0.4242,-1.1936)--(0.5454,-1.3693)--(0.6666,-1.4456)--(0.7878,-1.4241)--(0.9090,-1.3159)--(1.0303,-1.1399)--(1.1515,-0.9206)--(1.2727,-0.6849)--(1.3939,-0.4587)--(1.5151,-0.2646)--(1.6363,-0.1188)--(1.7575,-0.0303)--(1.8787,0.0000)--(2.0000,-0.0217)--(2.1212,-0.0832)--(2.2424,-0.1685)--(2.3636,-0.2601)--(2.4848,-0.3418)--(2.6060,-0.4005)--(2.7272,-0.4278)--(2.8484,-0.4207)--(2.9696,-0.3817)--(3.0909,-0.3180)--(3.2121,-0.2399)--(3.3333,-0.1592)--(3.4545,-0.0874)--(3.5757,-0.0339)--(3.6969,-0.0047)--(3.8181,-0.0020)--(3.9393,-0.0236)--(4.0606,-0.0641)--(4.1818,-0.1152)--(4.3030,-0.1679)--(4.4242,-0.2133)--(4.5454,-0.2440)--(4.6666,-0.2556)--(4.7878,-0.2466)--(4.9090,-0.2191)--(5.0303,-0.1776)--(5.1515,-0.1289)--(5.2727,-0.0804)--(5.3939,-0.0394)--(5.5151,-0.0115)--(5.6363,0.0000)--(5.7575,-0.0060)--(5.8787,-0.0271)--(6.0000,-0.0591);
+\draw (-5.6548,-0.3298) node {$ -6 \, \pi $};
+\draw [] (-5.6548,-0.1000) -- (-5.6548,0.1000);
+\draw (-3.7699,-0.3298) node {$ -4 \, \pi $};
+\draw [] (-3.7699,-0.1000) -- (-3.7699,0.1000);
+\draw (-1.8849,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-1.8849,-0.1000) -- (-1.8849,0.1000);
+\draw (1.8849,-0.3149) node {$ 2 \, \pi $};
+\draw [] (1.8849,-0.1000) -- (1.8849,0.1000);
+\draw (3.7699,-0.3149) node {$ 4 \, \pi $};
+\draw [] (3.7699,-0.1000) -- (3.7699,0.1000);
+\draw (5.6548,-0.3149) node {$ 6 \, \pi $};
+\draw [] (5.6548,-0.1000) -- (5.6548,0.1000);
+\draw (-0.4527,-1.0000) node {$ -\frac{1}{2} $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.3108,1.0000) node {$ \frac{1}{2} $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall b/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall
index 27241d0e2..5c4449b10 100644
--- a/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall
+++ b/src_phystricks/Fig_RegioniPrimoeSecondoTipo.pstricks.recall
@@ -65,8 +65,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -75,23 +75,23 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw [style=dashed] (4.00,1.04) -- (4.00,2.00);
-\draw []  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw (1.0000,-0.37858) node {$a$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.42674) node {$b$};
-\draw (2.1287,1.3162) node {$g_1$};
-\draw (2.8783,6.0762) node {$g_2$};
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+\draw [color=red] (1.0000,5.0000)--(1.0303,5.0596)--(1.0606,5.1175)--(1.0909,5.1735)--(1.1212,5.2277)--(1.1515,5.2800)--(1.1818,5.3305)--(1.2121,5.3792)--(1.2424,5.4260)--(1.2727,5.4710)--(1.3030,5.5142)--(1.3333,5.5555)--(1.3636,5.5950)--(1.3939,5.6326)--(1.4242,5.6685)--(1.4545,5.7024)--(1.4848,5.7346)--(1.5151,5.7649)--(1.5454,5.7933)--(1.5757,5.8200)--(1.6060,5.8448)--(1.6363,5.8677)--(1.6666,5.8888)--(1.6969,5.9081)--(1.7272,5.9256)--(1.7575,5.9412)--(1.7878,5.9550)--(1.8181,5.9669)--(1.8484,5.9770)--(1.8787,5.9853)--(1.9090,5.9917)--(1.9393,5.9963)--(1.9696,5.9990)--(2.0000,6.0000)--(2.0303,5.9990)--(2.0606,5.9963)--(2.0909,5.9917)--(2.1212,5.9853)--(2.1515,5.9770)--(2.1818,5.9669)--(2.2121,5.9550)--(2.2424,5.9412)--(2.2727,5.9256)--(2.3030,5.9081)--(2.3333,5.8888)--(2.3636,5.8677)--(2.3939,5.8448)--(2.4242,5.8200)--(2.4545,5.7933)--(2.4848,5.7649)--(2.5151,5.7346)--(2.5454,5.7024)--(2.5757,5.6685)--(2.6060,5.6326)--(2.6363,5.5950)--(2.6666,5.5555)--(2.6969,5.5142)--(2.7272,5.4710)--(2.7575,5.4260)--(2.7878,5.3792)--(2.8181,5.3305)--(2.8484,5.2800)--(2.8787,5.2277)--(2.9090,5.1735)--(2.9393,5.1175)--(2.9696,5.0596)--(3.0000,5.0000)--(3.0303,4.9384)--(3.0606,4.8751)--(3.0909,4.8099)--(3.1212,4.7428)--(3.1515,4.6740)--(3.1818,4.6033)--(3.2121,4.5307)--(3.2424,4.4563)--(3.2727,4.3801)--(3.3030,4.3021)--(3.3333,4.2222)--(3.3636,4.1404)--(3.3939,4.0569)--(3.4242,3.9715)--(3.4545,3.8842)--(3.4848,3.7952)--(3.5151,3.7043)--(3.5454,3.6115)--(3.5757,3.5169)--(3.6060,3.4205)--(3.6363,3.3223)--(3.6666,3.2222)--(3.6969,3.1202)--(3.7272,3.0165)--(3.7575,2.9109)--(3.7878,2.8034)--(3.8181,2.6942)--(3.8484,2.5831)--(3.8787,2.4701)--(3.9090,2.3553)--(3.9393,2.2387)--(3.9696,2.1202)--(4.0000,2.0000);
 
 %OTHER STUFF
 %END PSPICTURE
@@ -154,8 +154,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -163,19 +163,19 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw []  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw (-0.37898,1.0000) node {$c$};
-\draw []  (0,5.0000) node [rotate=0] {$\bullet$};
-\draw (-0.39499,5.0000) node {$d$};
+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (2.0000,1.0000) -- (2.0403,1.0404) -- (2.0807,1.0808) -- (2.1209,1.1212) -- (2.1609,1.1616) -- (2.2006,1.2020) -- (2.2400,1.2424) -- (2.2790,1.2828) -- (2.3176,1.3232) -- (2.3556,1.3636) -- (2.3931,1.4040) -- (2.4299,1.4444) -- (2.4660,1.4848) -- (2.5014,1.5252) -- (2.5359,1.5656) -- (2.5696,1.6060) -- (2.6023,1.6464) -- (2.6341,1.6868) -- (2.6648,1.7272) -- (2.6944,1.7676) -- (2.7229,1.8080) -- (2.7502,1.8484) -- (2.7763,1.8888) -- (2.8011,1.9292) -- (2.8247,1.9696) -- (2.8468,2.0101) -- (2.8676,2.0505) -- (2.8870,2.0909) -- (2.9049,2.1313) -- (2.9214,2.1717) -- (2.9363,2.2121) -- (2.9497,2.2525) -- (2.9616,2.2929) -- (2.9719,2.3333) -- (2.9806,2.3737) -- (2.9877,2.4141) -- (2.9932,2.4545) -- (2.9971,2.4949) -- (2.9993,2.5353) -- (2.9999,2.5757) -- (2.9989,2.6161) -- (2.9963,2.6565) -- (2.9920,2.6969) -- (2.9861,2.7373) -- (2.9786,2.7777) -- (2.9695,2.8181) -- (2.9588,2.8585) -- (2.9466,2.8989) -- (2.9328,2.9393) -- (2.9175,2.9797) -- (2.9007,3.0202) -- (2.8824,3.0606) -- (2.8626,3.1010) -- (2.8415,3.1414) -- (2.8190,3.1818) -- (2.7952,3.2222) -- (2.7700,3.2626) -- (2.7436,3.3030) -- (2.7160,3.3434) -- (2.6872,3.3838) -- (2.6573,3.4242) -- (2.6264,3.4646) -- (2.5944,3.5050) -- (2.5614,3.5454) -- (2.5275,3.5858) -- (2.4928,3.6262) -- (2.4572,3.6666) -- (2.4209,3.7070) -- (2.3839,3.7474) -- (2.3463,3.7878) -- (2.3082,3.8282) -- (2.2695,3.8686) -- (2.2304,3.9090) -- (2.1909,3.9494) -- (2.1511,3.9898) -- (2.1110,4.0303) -- (2.0708,4.0707) -- (2.0304,4.1111) -- (1.9900,4.1515) -- (1.9496,4.1919) -- (1.9093,4.2323) -- (1.8692,4.2727) -- (1.8293,4.3131) -- (1.7896,4.3535) -- (1.7503,4.3939) -- (1.7114,4.4343) -- (1.6729,4.4747) -- (1.6350,4.5151) -- (1.5977,4.5555) -- (1.5611,4.5959) -- (1.5251,4.6363) -- (1.4900,4.6767) -- (1.4556,4.7171) -- (1.4222,4.7575) -- (1.3897,4.7979) -- (1.3582,4.8383) -- (1.3277,4.8787) -- (1.2984,4.9191) -- (1.2702,4.9595) -- (1.2431,5.0000) -- (1.2431,5.0000) -- (6.0000,5.0000) -- (2.0000,1.0000) -- (5.1111,1.0000) -- (5.1111,1.0000) -- (5.1023,1.0404) -- (5.0938,1.0808) -- (5.0858,1.1212) -- (5.0780,1.1616) -- (5.0707,1.2020) -- (5.0637,1.2424) -- (5.0571,1.2828) -- (5.0508,1.3232) -- (5.0449,1.3636) -- (5.0394,1.4040) -- (5.0342,1.4444) -- (5.0294,1.4848) -- (5.0250,1.5252) -- (5.0209,1.5656) -- (5.0172,1.6060) -- (5.0138,1.6464) -- (5.0108,1.6868) -- (5.0082,1.7272) -- (5.0059,1.7676) -- (5.0040,1.8080) -- (5.0025,1.8484) -- (5.0013,1.8888) -- (5.0005,1.9292) -- (5.0001,1.9696) -- (5.0000,2.0101) -- (5.0002,2.0505) -- (5.0009,2.0909) -- (5.0019,2.1313) -- (5.0032,2.1717) -- (5.0049,2.2121) -- (5.0070,2.2525) -- (5.0095,2.2929) -- (5.0123,2.3333) -- (5.0155,2.3737) -- (5.0190,2.4141) -- (5.0229,2.4545) -- (5.0272,2.4949) -- (5.0318,2.5353) -- (5.0368,2.5757) -- (5.0421,2.6161) -- (5.0478,2.6565) -- (5.0539,2.6969) -- (5.0604,2.7373) -- (5.0672,2.7777) -- (5.0743,2.8181) -- (5.0819,2.8585) -- (5.0897,2.8989) -- (5.0980,2.9393) -- (5.1066,2.9797) -- (5.1156,3.0202) -- (5.1249,3.0606) -- (5.1346,3.1010) -- (5.1447,3.1414) -- (5.1551,3.1818) -- (5.1659,3.2222) -- (5.1771,3.2626) -- (5.1886,3.3030) -- (5.2005,3.3434) -- (5.2127,3.3838) -- (5.2253,3.4242) -- (5.2383,3.4646) -- (5.2516,3.5050) -- (5.2653,3.5454) -- (5.2794,3.5858) -- (5.2938,3.6262) -- (5.3086,3.6666) -- (5.3237,3.7070) -- (5.3392,3.7474) -- (5.3551,3.7878) -- (5.3714,3.8282) -- (5.3879,3.8686) -- (5.4049,3.9090) -- (5.4222,3.9494) -- (5.4399,3.9898) -- (5.4580,4.0303) -- (5.4764,4.0707) -- (5.4951,4.1111) -- (5.5143,4.1515) -- (5.5338,4.1919) -- (5.5536,4.2323) -- (5.5739,4.2727) -- (5.5945,4.3131) -- (5.6154,4.3535) -- (5.6367,4.3939) -- (5.6584,4.4343) -- (5.6804,4.4747) -- (5.7028,4.5151) -- (5.7256,4.5555) -- (5.7487,4.5959) -- (5.7722,4.6363) -- (5.7961,4.6767) -- (5.8203,4.7171) -- (5.8449,4.7575) -- (5.8698,4.7979) -- (5.8951,4.8383) -- (5.9208,4.8787) -- (5.9468,4.9191) -- (5.9732,4.9595) -- (6.0000,5.0000) --  cycle;
+\draw [color=blue] (2.0000,1.0000)--(2.0403,1.0404)--(2.0807,1.0808)--(2.1209,1.1212)--(2.1609,1.1616)--(2.2006,1.2020)--(2.2400,1.2424)--(2.2790,1.2828)--(2.3176,1.3232)--(2.3556,1.3636)--(2.3931,1.4040)--(2.4299,1.4444)--(2.4660,1.4848)--(2.5014,1.5252)--(2.5359,1.5656)--(2.5696,1.6060)--(2.6023,1.6464)--(2.6341,1.6868)--(2.6648,1.7272)--(2.6944,1.7676)--(2.7229,1.8080)--(2.7502,1.8484)--(2.7763,1.8888)--(2.8011,1.9292)--(2.8247,1.9696)--(2.8468,2.0101)--(2.8676,2.0505)--(2.8870,2.0909)--(2.9049,2.1313)--(2.9214,2.1717)--(2.9363,2.2121)--(2.9497,2.2525)--(2.9616,2.2929)--(2.9719,2.3333)--(2.9806,2.3737)--(2.9877,2.4141)--(2.9932,2.4545)--(2.9971,2.4949)--(2.9993,2.5353)--(2.9999,2.5757)--(2.9989,2.6161)--(2.9963,2.6565)--(2.9920,2.6969)--(2.9861,2.7373)--(2.9786,2.7777)--(2.9695,2.8181)--(2.9588,2.8585)--(2.9466,2.8989)--(2.9328,2.9393)--(2.9175,2.9797)--(2.9007,3.0202)--(2.8824,3.0606)--(2.8626,3.1010)--(2.8415,3.1414)--(2.8190,3.1818)--(2.7952,3.2222)--(2.7700,3.2626)--(2.7436,3.3030)--(2.7160,3.3434)--(2.6872,3.3838)--(2.6573,3.4242)--(2.6264,3.4646)--(2.5944,3.5050)--(2.5614,3.5454)--(2.5275,3.5858)--(2.4928,3.6262)--(2.4572,3.6666)--(2.4209,3.7070)--(2.3839,3.7474)--(2.3463,3.7878)--(2.3082,3.8282)--(2.2695,3.8686)--(2.2304,3.9090)--(2.1909,3.9494)--(2.1511,3.9898)--(2.1110,4.0303)--(2.0708,4.0707)--(2.0304,4.1111)--(1.9900,4.1515)--(1.9496,4.1919)--(1.9093,4.2323)--(1.8692,4.2727)--(1.8293,4.3131)--(1.7896,4.3535)--(1.7503,4.3939)--(1.7114,4.4343)--(1.6729,4.4747)--(1.6350,4.5151)--(1.5977,4.5555)--(1.5611,4.5959)--(1.5251,4.6363)--(1.4900,4.6767)--(1.4556,4.7171)--(1.4222,4.7575)--(1.3897,4.7979)--(1.3582,4.8383)--(1.3277,4.8787)--(1.2984,4.9191)--(1.2702,4.9595)--(1.2431,5.0000);
+\draw [style=dashed] (1.2431,5.0000) -- (6.0000,5.0000);
+\draw [style=dashed] (2.0000,1.0000) -- (5.1111,1.0000);
+\draw [color=blue] (5.1111,1.0000)--(5.1023,1.0404)--(5.0938,1.0808)--(5.0858,1.1212)--(5.0780,1.1616)--(5.0707,1.2020)--(5.0637,1.2424)--(5.0571,1.2828)--(5.0508,1.3232)--(5.0449,1.3636)--(5.0394,1.4040)--(5.0342,1.4444)--(5.0294,1.4848)--(5.0250,1.5252)--(5.0209,1.5656)--(5.0172,1.6060)--(5.0138,1.6464)--(5.0108,1.6868)--(5.0082,1.7272)--(5.0059,1.7676)--(5.0040,1.8080)--(5.0025,1.8484)--(5.0013,1.8888)--(5.0005,1.9292)--(5.0001,1.9696)--(5.0000,2.0101)--(5.0002,2.0505)--(5.0009,2.0909)--(5.0019,2.1313)--(5.0032,2.1717)--(5.0049,2.2121)--(5.0070,2.2525)--(5.0095,2.2929)--(5.0123,2.3333)--(5.0155,2.3737)--(5.0190,2.4141)--(5.0229,2.4545)--(5.0272,2.4949)--(5.0318,2.5353)--(5.0368,2.5757)--(5.0421,2.6161)--(5.0478,2.6565)--(5.0539,2.6969)--(5.0604,2.7373)--(5.0672,2.7777)--(5.0743,2.8181)--(5.0819,2.8585)--(5.0897,2.8989)--(5.0980,2.9393)--(5.1066,2.9797)--(5.1156,3.0202)--(5.1249,3.0606)--(5.1346,3.1010)--(5.1447,3.1414)--(5.1551,3.1818)--(5.1659,3.2222)--(5.1771,3.2626)--(5.1886,3.3030)--(5.2005,3.3434)--(5.2127,3.3838)--(5.2253,3.4242)--(5.2383,3.4646)--(5.2516,3.5050)--(5.2653,3.5454)--(5.2794,3.5858)--(5.2938,3.6262)--(5.3086,3.6666)--(5.3237,3.7070)--(5.3392,3.7474)--(5.3551,3.7878)--(5.3714,3.8282)--(5.3879,3.8686)--(5.4049,3.9090)--(5.4222,3.9494)--(5.4399,3.9898)--(5.4580,4.0303)--(5.4764,4.0707)--(5.4951,4.1111)--(5.5143,4.1515)--(5.5338,4.1919)--(5.5536,4.2323)--(5.5739,4.2727)--(5.5945,4.3131)--(5.6154,4.3535)--(5.6367,4.3939)--(5.6584,4.4343)--(5.6804,4.4747)--(5.7028,4.5151)--(5.7256,4.5555)--(5.7487,4.5959)--(5.7722,4.6363)--(5.7961,4.6767)--(5.8203,4.7171)--(5.8449,4.7575)--(5.8698,4.7979)--(5.8951,4.8383)--(5.9208,4.8787)--(5.9468,4.9191)--(5.9732,4.9595)--(6.0000,5.0000);
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3789,1.0000) node {$c$};
+\draw []  (0.0000,5.0000) node [rotate=0] {$\bullet$};
+\draw (-0.3949,5.0000) node {$d$};
 \draw (2.4480,2.7306) node {$h_1$};
 \draw (5.5954,3.0000) node {$h_2$};
-\draw [style=dotted] (2.00,1.00) -- (0,1.00);
-\draw [style=dotted] (1.24,5.00) -- (0,5.00);
+\draw [style=dotted] (2.0000,1.0000) -- (0.0000,1.0000);
+\draw [style=dotted] (1.2431,5.0000) -- (0.0000,5.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SFdgHdO.pstricks.recall b/src_phystricks/Fig_SFdgHdO.pstricks.recall
index aa92dc4e9..3afa9a990 100644
--- a/src_phystricks/Fig_SFdgHdO.pstricks.recall
+++ b/src_phystricks/Fig_SFdgHdO.pstricks.recall
@@ -67,23 +67,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5390,0) -- (1.5474,0);
-\draw [,->,>=latex] (0,-1.5497) -- (0,1.5497);
+\draw [,->,>=latex] (-1.5389,0.0000) -- (1.5473,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5496) -- (0.0000,1.5496);
 %DEFAULT
 
-\draw [color=black] plot [smooth,tension=1] coordinates {(0.00100,1.00)(0.105,1.04)(0.189,0.931)(0.310,1.00)(0.370,0.875)(0.504,0.921)(0.537,0.784)(0.677,0.803)(0.682,0.661)(0.823,0.652)(0.800,0.513)(0.936,0.476)(0.886,0.344)(1.01,0.281)(0.936,0.162)(1.05,0.0740)(0.999,0.0447)};
+\draw [color=black] plot [smooth,tension=1] coordinates {(0.0010,0.9999)(0.1054,1.0446)(0.1887,0.9310)(0.3103,1.0030)(0.3695,0.8751)(0.5035,0.9213)(0.5365,0.7839)(0.6767,0.8028)(0.6820,0.6613)(0.8229,0.6520)(0.7996,0.5128)(0.9361,0.4756)(0.8856,0.3437)(1.0117,0.2809)(0.9361,0.1616)(1.0473,0.0739)(0.9990,0.0447)};
 
-\draw [color=green] plot [smooth,tension=1] coordinates {(-0.999,0.0447)(-1.04,0.152)(-0.922,0.230)(-0.988,0.356)(-0.857,0.410)(-0.898,0.545)(-0.759,0.571)(-0.772,0.712)(-0.630,0.711)(-0.615,0.851)(-0.476,0.822)(-0.433,0.957)(-0.303,0.900)(-0.233,1.02)(-0.118,0.943)(-0.0252,1.05)(-0.00100,1.00)};
+\draw [color=green] plot [smooth,tension=1] coordinates {(-0.9990,0.0447)(-1.0389,0.1517)(-0.9216,0.2302)(-0.9878,0.3557)(-0.8571,0.4095)(-0.8975,0.5449)(-0.7589,0.5713)(-0.7717,0.7119)(-0.6301,0.7109)(-0.6150,0.8510)(-0.4761,0.8220)(-0.4329,0.9565)(-0.3031,0.9003)(-0.2333,1.0237)(-0.1175,0.9427)(-0.0252,1.0496)(-0.0010,0.9999)};
 
-\draw [color=green] plot [smooth,tension=1] coordinates {(0.00100,-1.00)(0.105,-1.04)(0.189,-0.931)(0.310,-1.00)(0.370,-0.875)(0.504,-0.921)(0.537,-0.784)(0.677,-0.803)(0.682,-0.661)(0.823,-0.652)(0.800,-0.513)(0.936,-0.476)(0.886,-0.344)(1.01,-0.281)(0.936,-0.162)(1.05,-0.0740)(0.999,-0.0447)};
+\draw [color=green] plot [smooth,tension=1] coordinates {(0.0010,-0.9999)(0.1054,-1.0446)(0.1887,-0.9310)(0.3103,-1.0030)(0.3695,-0.8751)(0.5035,-0.9213)(0.5365,-0.7839)(0.6767,-0.8028)(0.6820,-0.6613)(0.8229,-0.6520)(0.7996,-0.5128)(0.9361,-0.4756)(0.8856,-0.3437)(1.0117,-0.2809)(0.9361,-0.1616)(1.0473,-0.0739)(0.9990,-0.0447)};
 
-\draw [color=black] plot [smooth,tension=1] coordinates {(-0.999,-0.0447)(-1.04,-0.152)(-0.922,-0.230)(-0.988,-0.356)(-0.857,-0.410)(-0.898,-0.545)(-0.759,-0.571)(-0.772,-0.712)(-0.630,-0.711)(-0.615,-0.851)(-0.476,-0.822)(-0.433,-0.957)(-0.303,-0.900)(-0.233,-1.02)(-0.118,-0.943)(-0.0252,-1.05)(-0.00100,-1.00)};
-\draw [color=red]  (1.0000,0) node [rotate=0] {$\bullet$};
-\draw [color=red]  (-1.0000,0) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown]  (0,-1.0000) node [rotate=0] {$\bullet$};
-\draw (2.0495,-0.36509) node {$K_H$};
-\draw (2.0495,-0.36509) node {$K_H$};
+\draw [color=black] plot [smooth,tension=1] coordinates {(-0.9990,-0.0447)(-1.0389,-0.1517)(-0.9216,-0.2302)(-0.9878,-0.3557)(-0.8571,-0.4095)(-0.8975,-0.5449)(-0.7589,-0.5713)(-0.7717,-0.7119)(-0.6301,-0.7109)(-0.6150,-0.8510)(-0.4761,-0.8220)(-0.4329,-0.9565)(-0.3031,-0.9003)(-0.2333,-1.0237)(-0.1175,-0.9427)(-0.0252,-1.0496)(-0.0010,-0.9999)};
+\draw [color=red]  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=red]  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown]  (0.0000,-1.0000) node [rotate=0] {$\bullet$};
+\draw (2.0494,-0.3650) node {$K_H$};
+\draw (2.0494,-0.3650) node {$K_H$};
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall b/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall
index d7811ff6e..657a0615e 100644
--- a/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall
+++ b/src_phystricks/Fig_SJAWooRDGzIkrj.pstricks.recall
@@ -87,21 +87,21 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [] (1.00,0)--(0.998,0.0634)--(0.992,0.127)--(0.982,0.189)--(0.968,0.251)--(0.950,0.312)--(0.928,0.372)--(0.903,0.430)--(0.874,0.486)--(0.841,0.541)--(0.805,0.593)--(0.766,0.643)--(0.724,0.690)--(0.679,0.735)--(0.631,0.776)--(0.580,0.815)--(0.527,0.850)--(0.472,0.881)--(0.415,0.910)--(0.357,0.934)--(0.297,0.955)--(0.236,0.972)--(0.174,0.985)--(0.111,0.994)--(0.0476,0.999)--(-0.0159,1.00)--(-0.0792,0.997)--(-0.142,0.990)--(-0.205,0.979)--(-0.266,0.964)--(-0.327,0.945)--(-0.386,0.922)--(-0.444,0.896)--(-0.500,0.866)--(-0.554,0.833)--(-0.606,0.796)--(-0.655,0.756)--(-0.701,0.713)--(-0.745,0.667)--(-0.786,0.618)--(-0.824,0.567)--(-0.858,0.514)--(-0.889,0.458)--(-0.916,0.401)--(-0.940,0.342)--(-0.959,0.282)--(-0.975,0.220)--(-0.987,0.158)--(-0.995,0.0951)--(-1.00,0.0317)--(-1.00,-0.0317)--(-0.995,-0.0951)--(-0.987,-0.158)--(-0.975,-0.220)--(-0.959,-0.282)--(-0.940,-0.342)--(-0.916,-0.401)--(-0.889,-0.458)--(-0.858,-0.514)--(-0.824,-0.567)--(-0.786,-0.618)--(-0.745,-0.667)--(-0.701,-0.713)--(-0.655,-0.756)--(-0.606,-0.796)--(-0.554,-0.833)--(-0.500,-0.866)--(-0.444,-0.896)--(-0.386,-0.922)--(-0.327,-0.945)--(-0.266,-0.964)--(-0.205,-0.979)--(-0.142,-0.990)--(-0.0792,-0.997)--(-0.0159,-1.00)--(0.0476,-0.999)--(0.111,-0.994)--(0.174,-0.985)--(0.236,-0.972)--(0.297,-0.955)--(0.357,-0.934)--(0.415,-0.910)--(0.472,-0.881)--(0.527,-0.850)--(0.580,-0.815)--(0.631,-0.776)--(0.679,-0.735)--(0.724,-0.690)--(0.766,-0.643)--(0.805,-0.593)--(0.841,-0.541)--(0.874,-0.486)--(0.903,-0.430)--(0.928,-0.372)--(0.950,-0.312)--(0.968,-0.251)--(0.982,-0.189)--(0.992,-0.127)--(0.998,-0.0634)--(1.00,0);
-\draw []  (0.34202,0.93969) node [rotate=0] {$\bullet$};
-\draw (0.50689,1.2062) node {\( a\)};
-\draw []  (0.34202,-0.93969) node [rotate=0] {$\bullet$};
-\draw (0.48875,-1.2544) node {\( b\)};
-\draw []  (2.9238,0) node [rotate=0] {$\bullet$};
-\draw (3.4840,0) node {\( m\)};
-\draw [] (-0.303,1.17) -- (3.57,-0.235);
-\draw [] (-0.303,-1.17) -- (3.57,0.235);
-\draw []  (0.86602,0.50000) node [rotate=0] {$\bullet$};
-\draw (0.58852,0.32142) node {\( x\)};
-\draw []  (-0.54003,0.84164) node [rotate=0] {$\bullet$};
-\draw (-0.72702,1.0885) node {\( c\)};
-\draw [] (3.44,-0.126) -- (-1.06,0.968);
-\draw (-1.2344,-0.53513) node {\( \mC\)};
+\draw [] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
+\draw []  (0.3420,0.9396) node [rotate=0] {$\bullet$};
+\draw (0.5068,1.2062) node {\( a\)};
+\draw []  (0.3420,-0.9396) node [rotate=0] {$\bullet$};
+\draw (0.4887,-1.2543) node {\( b\)};
+\draw []  (2.9238,0.0000) node [rotate=0] {$\bullet$};
+\draw (3.4840,0.0000) node {\( m\)};
+\draw [] (-0.3034,1.1746) -- (3.5692,-0.2349);
+\draw [] (-0.3034,-1.1746) -- (3.5692,0.2349);
+\draw []  (0.8660,0.5000) node [rotate=0] {$\bullet$};
+\draw (0.5885,0.3214) node {\( x\)};
+\draw []  (-0.5400,0.8416) node [rotate=0] {$\bullet$};
+\draw (-0.7270,1.0885) node {\( c\)};
+\draw [] (3.4433,-0.1262) -- (-1.0596,0.9678);
+\draw (-1.2343,-0.5351) node {\( \mC\)};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_SenoTopologo.pstricks.recall b/src_phystricks/Fig_SenoTopologo.pstricks.recall
index 18e3d2a7e..a1ade4ed6 100644
--- a/src_phystricks/Fig_SenoTopologo.pstricks.recall
+++ b/src_phystricks/Fig_SenoTopologo.pstricks.recall
@@ -63,14 +63,14 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.0000,0) -- (3.0000,0);
-\draw [,->,>=latex] (0,-1.5862) -- (0,2.7732);
+\draw [,->,>=latex] (-3.0000,0.0000) -- (3.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5861) -- (0.0000,2.7732);
 %DEFAULT
 
-\draw [color=blue] 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+\draw [color=blue] 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+\draw [color=blue] (1.2518,-0.9425)--(1.2543,-0.9377)--(1.2568,-0.9329)--(1.2593,-0.9281)--(1.2618,-0.9232)--(1.2643,-0.9183)--(1.2668,-0.9132)--(1.2693,-0.9082)--(1.2718,-0.9031)--(1.2743,-0.8979)--(1.2768,-0.8927)--(1.2793,-0.8874)--(1.2818,-0.8820)--(1.2843,-0.8767)--(1.2868,-0.8712)--(1.2893,-0.8657)--(1.2918,-0.8602)--(1.2943,-0.8546)--(1.2968,-0.8490)--(1.2993,-0.8433)--(1.3019,-0.8376)--(1.3044,-0.8318)--(1.3069,-0.8260)--(1.3094,-0.8202)--(1.3119,-0.8143)--(1.3144,-0.8083)--(1.3169,-0.8023)--(1.3194,-0.7963)--(1.3219,-0.7902)--(1.3244,-0.7841)--(1.3269,-0.7780)--(1.3294,-0.7718)--(1.3319,-0.7656)--(1.3344,-0.7593)--(1.3369,-0.7530)--(1.3394,-0.7467)--(1.3419,-0.7403)--(1.3444,-0.7339)--(1.3469,-0.7274)--(1.3494,-0.7210)--(1.3519,-0.7144)--(1.3544,-0.7079)--(1.3569,-0.7013)--(1.3594,-0.6947)--(1.3619,-0.6881)--(1.3644,-0.6814)--(1.3669,-0.6747)--(1.3694,-0.6679)--(1.3719,-0.6612)--(1.3744,-0.6544)--(1.3769,-0.6475)--(1.3794,-0.6407)--(1.3819,-0.6338)--(1.3844,-0.6269)--(1.3869,-0.6200)--(1.3894,-0.6130)--(1.3919,-0.6060)--(1.3944,-0.5990)--(1.3969,-0.5920)--(1.3994,-0.5849)--(1.4019,-0.5778)--(1.4044,-0.5707)--(1.4069,-0.5636)--(1.4094,-0.5564)--(1.4119,-0.5493)--(1.4144,-0.5421)--(1.4169,-0.5348)--(1.4194,-0.5276)--(1.4219,-0.5203)--(1.4244,-0.5131)--(1.4269,-0.5058)--(1.4294,-0.4985)--(1.4319,-0.4911)--(1.4344,-0.4838)--(1.4369,-0.4764)--(1.4394,-0.4690)--(1.4419,-0.4616)--(1.4444,-0.4542)--(1.4469,-0.4467)--(1.4494,-0.4393)--(1.4519,-0.4318)--(1.4544,-0.4243)--(1.4569,-0.4168)--(1.4594,-0.4093)--(1.4619,-0.4018)--(1.4644,-0.3942)--(1.4669,-0.3867)--(1.4694,-0.3791)--(1.4719,-0.3715)--(1.4744,-0.3639)--(1.4769,-0.3563)--(1.4794,-0.3487)--(1.4819,-0.3411)--(1.4844,-0.3334)--(1.4869,-0.3258)--(1.4894,-0.3181)--(1.4919,-0.3104)--(1.4944,-0.3027)--(1.4969,-0.2950)--(1.4994,-0.2873)--(1.5020,-0.2796)--(1.5045,-0.2719)--(1.5070,-0.2642)--(1.5095,-0.2565)--(1.5120,-0.2487)--(1.5145,-0.2410)--(1.5170,-0.2332)--(1.5195,-0.2254)--(1.5220,-0.2177)--(1.5245,-0.2099)--(1.5270,-0.2021)--(1.5295,-0.1943)--(1.5320,-0.1865)--(1.5345,-0.1787)--(1.5370,-0.1709)--(1.5395,-0.1631)--(1.5420,-0.1553)--(1.5445,-0.1475)--(1.5470,-0.1396)--(1.5495,-0.1318)--(1.5520,-0.1240)--(1.5545,-0.1162)--(1.5570,-0.1083)--(1.5595,-0.1005)--(1.5620,-0.0926)--(1.5645,-0.0848)--(1.5670,-0.0769)--(1.5695,-0.0691)--(1.5720,-0.0612)--(1.5745,-0.0534)--(1.5770,-0.0455)--(1.5795,-0.0377)--(1.5820,-0.0298)--(1.5845,-0.0220)--(1.5870,-0.0141)--(1.5895,-0.0062)--(1.5920,0.0015)--(1.5945,0.0094)--(1.5970,0.0172)--(1.5995,0.0251)--(1.6020,0.0329)--(1.6045,0.0408)--(1.6070,0.0487)--(1.6095,0.0565)--(1.6120,0.0644)--(1.6145,0.0722)--(1.6170,0.0801)--(1.6195,0.0879)--(1.6220,0.0957)--(1.6245,0.1036)--(1.6270,0.1114)--(1.6295,0.1193)--(1.6320,0.1271)--(1.6345,0.1349)--(1.6370,0.1428)--(1.6395,0.1506)--(1.6420,0.1584)--(1.6445,0.1662)--(1.6470,0.1741)--(1.6495,0.1819)--(1.6520,0.1897)--(1.6545,0.1975)--(1.6570,0.2053)--(1.6595,0.2131)--(1.6620,0.2209)--(1.6645,0.2287)--(1.6670,0.2364)--(1.6695,0.2442)--(1.6720,0.2520)--(1.6745,0.2598)--(1.6770,0.2675)--(1.6795,0.2753)--(1.6820,0.2830)--(1.6845,0.2908)--(1.6870,0.2985)--(1.6895,0.3063)--(1.6920,0.3140)--(1.6945,0.3217)--(1.6970,0.3294)--(1.6995,0.3371)--(1.7021,0.3449)--(1.7046,0.3526)--(1.7071,0.3602)--(1.7096,0.3679)--(1.7121,0.3756)--(1.7146,0.3833)--(1.7171,0.3909)--(1.7196,0.3986)--(1.7221,0.4063)--(1.7246,0.4139)--(1.7271,0.4215)--(1.7296,0.4292)--(1.7321,0.4368)--(1.7346,0.4444)--(1.7371,0.4520)--(1.7396,0.4596)--(1.7421,0.4672)--(1.7446,0.4748)--(1.7471,0.4824)--(1.7496,0.4899)--(1.7521,0.4975)--(1.7546,0.5050)--(1.7571,0.5126)--(1.7596,0.5201)--(1.7621,0.5276)--(1.7646,0.5351)--(1.7671,0.5426)--(1.7696,0.5501)--(1.7721,0.5576)--(1.7746,0.5651)--(1.7771,0.5726)--(1.7796,0.5801)--(1.7821,0.5875)--(1.7846,0.5950)--(1.7871,0.6024)--(1.7896,0.6098)--(1.7921,0.6172)--(1.7946,0.6246)--(1.7971,0.6320)--(1.7996,0.6394)--(1.8021,0.6468)--(1.8046,0.6542)--(1.8071,0.6615)--(1.8096,0.6689)--(1.8121,0.6762)--(1.8146,0.6836)--(1.8171,0.6909)--(1.8196,0.6982)--(1.8221,0.7055)--(1.8246,0.7128)--(1.8271,0.7201)--(1.8296,0.7273)--(1.8321,0.7346)--(1.8346,0.7419)--(1.8371,0.7491)--(1.8396,0.7563)--(1.8421,0.7635)--(1.8446,0.7708)--(1.8471,0.7780)--(1.8496,0.7851)--(1.8521,0.7923)--(1.8546,0.7995)--(1.8571,0.8066)--(1.8596,0.8138)--(1.8621,0.8209)--(1.8646,0.8281)--(1.8671,0.8352)--(1.8696,0.8423)--(1.8721,0.8494)--(1.8746,0.8565)--(1.8771,0.8635)--(1.8796,0.8706)--(1.8821,0.8776)--(1.8846,0.8847)--(1.8871,0.8917)--(1.8896,0.8987)--(1.8921,0.9057)--(1.8946,0.9127)--(1.8971,0.9197)--(1.8996,0.9267)--(1.9022,0.9336)--(1.9047,0.9406)--(1.9072,0.9475)--(1.9097,0.9545)--(1.9122,0.9614)--(1.9147,0.9683)--(1.9172,0.9752)--(1.9197,0.9821)--(1.9222,0.9889)--(1.9247,0.9958)--(1.9272,1.0026)--(1.9297,1.0095)--(1.9322,1.0163)--(1.9347,1.0231)--(1.9372,1.0299)--(1.9397,1.0367)--(1.9422,1.0435)--(1.9447,1.0503)--(1.9472,1.0570)--(1.9497,1.0638)--(1.9522,1.0705)--(1.9547,1.0772)--(1.9572,1.0839)--(1.9597,1.0906)--(1.9622,1.0973)--(1.9647,1.1040)--(1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
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SolsSinpA.pstricks.recall b/src_phystricks/Fig_SolsSinpA.pstricks.recall
index b23b7aff3..50f508c52 100644
--- a/src_phystricks/Fig_SolsSinpA.pstricks.recall
+++ b/src_phystricks/Fig_SolsSinpA.pstricks.recall
@@ -87,31 +87,31 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.6416,0) -- (3.6416,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-3.6415,0.0000) -- (3.6415,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
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+\draw [color=cyan] (-2.3561,-1.0000)--(-2.3244,-0.9994)--(-2.2927,-0.9979)--(-2.2609,-0.9954)--(-2.2292,-0.9919)--(-2.1975,-0.9874)--(-2.1657,-0.9819)--(-2.1340,-0.9754)--(-2.1023,-0.9679)--(-2.0705,-0.9594)--(-2.0388,-0.9500)--(-2.0071,-0.9396)--(-1.9753,-0.9283)--(-1.9436,-0.9161)--(-1.9119,-0.9029)--(-1.8801,-0.8888)--(-1.8484,-0.8738)--(-1.8167,-0.8579)--(-1.7849,-0.8412)--(-1.7532,-0.8236)--(-1.7215,-0.8052)--(-1.6897,-0.7860)--(-1.6580,-0.7660)--(-1.6263,-0.7452)--(-1.5945,-0.7237)--(-1.5628,-0.7014)--(-1.5311,-0.6785)--(-1.4993,-0.6548)--(-1.4676,-0.6305)--(-1.4359,-0.6056)--(-1.4041,-0.5800)--(-1.3724,-0.5539)--(-1.3407,-0.5272)--(-1.3089,-0.5000)--(-1.2772,-0.4722)--(-1.2455,-0.4440)--(-1.2137,-0.4154)--(-1.1820,-0.3863)--(-1.1503,-0.3568)--(-1.1185,-0.3270)--(-1.0868,-0.2969)--(-1.0551,-0.2664)--(-1.0233,-0.2357)--(-0.9916,-0.2048)--(-0.9599,-0.1736)--(-0.9281,-0.1423)--(-0.8964,-0.1108)--(-0.8647,-0.0792)--(-0.8329,-0.0475)--(-0.8012,-0.0158)--(-0.7695,0.0158)--(-0.7377,0.0475)--(-0.7060,0.0792)--(-0.6743,0.1108)--(-0.6425,0.1423)--(-0.6108,0.1736)--(-0.5791,0.2048)--(-0.5473,0.2357)--(-0.5156,0.2664)--(-0.4839,0.2969)--(-0.4521,0.3270)--(-0.4204,0.3568)--(-0.3887,0.3863)--(-0.3569,0.4154)--(-0.3252,0.4440)--(-0.2935,0.4722)--(-0.2617,0.5000)--(-0.2300,0.5272)--(-0.1983,0.5539)--(-0.1665,0.5800)--(-0.1348,0.6056)--(-0.1031,0.6305)--(-0.0713,0.6548)--(-0.0396,0.6785)--(-0.0079,0.7014)--(0.0237,0.7237)--(0.0555,0.7452)--(0.0872,0.7660)--(0.1189,0.7860)--(0.1507,0.8052)--(0.1824,0.8236)--(0.2141,0.8412)--(0.2459,0.8579)--(0.2776,0.8738)--(0.3093,0.8888)--(0.3411,0.9029)--(0.3728,0.9161)--(0.4045,0.9283)--(0.4363,0.9396)--(0.4680,0.9500)--(0.4997,0.9594)--(0.5315,0.9679)--(0.5632,0.9754)--(0.5949,0.9819)--(0.6267,0.9874)--(0.6584,0.9919)--(0.6901,0.9954)--(0.7219,0.9979)--(0.7536,0.9994)--(0.7853,1.0000);
 
-\draw [color=brown] (-3.142,-1.000)--(-3.110,-0.9995)--(-3.078,-0.9980)--(-3.046,-0.9955)--(-3.015,-0.9920)--(-2.983,-0.9874)--(-2.951,-0.9819)--(-2.919,-0.9754)--(-2.888,-0.9679)--(-2.856,-0.9595)--(-2.824,-0.9501)--(-2.793,-0.9397)--(-2.761,-0.9284)--(-2.729,-0.9161)--(-2.697,-0.9029)--(-2.666,-0.8888)--(-2.634,-0.8738)--(-2.602,-0.8580)--(-2.570,-0.8413)--(-2.539,-0.8237)--(-2.507,-0.8053)--(-2.475,-0.7861)--(-2.443,-0.7660)--(-2.412,-0.7453)--(-2.380,-0.7237)--(-2.348,-0.7015)--(-2.317,-0.6785)--(-2.285,-0.6549)--(-2.253,-0.6306)--(-2.221,-0.6056)--(-2.190,-0.5801)--(-2.158,-0.5539)--(-2.126,-0.5272)--(-2.094,-0.5000)--(-2.063,-0.4723)--(-2.031,-0.4441)--(-1.999,-0.4154)--(-1.967,-0.3863)--(-1.936,-0.3569)--(-1.904,-0.3271)--(-1.872,-0.2969)--(-1.841,-0.2665)--(-1.809,-0.2358)--(-1.777,-0.2048)--(-1.745,-0.1736)--(-1.714,-0.1423)--(-1.682,-0.1108)--(-1.650,-0.07925)--(-1.618,-0.04758)--(-1.587,-0.01587)--(-1.555,0.01587)--(-1.523,0.04758)--(-1.491,0.07925)--(-1.460,0.1108)--(-1.428,0.1423)--(-1.396,0.1736)--(-1.365,0.2048)--(-1.333,0.2358)--(-1.301,0.2665)--(-1.269,0.2969)--(-1.238,0.3271)--(-1.206,0.3569)--(-1.174,0.3863)--(-1.142,0.4154)--(-1.111,0.4441)--(-1.079,0.4723)--(-1.047,0.5000)--(-1.015,0.5272)--(-0.9837,0.5539)--(-0.9520,0.5801)--(-0.9203,0.6056)--(-0.8885,0.6306)--(-0.8568,0.6549)--(-0.8251,0.6785)--(-0.7933,0.7015)--(-0.7616,0.7237)--(-0.7299,0.7453)--(-0.6981,0.7660)--(-0.6664,0.7861)--(-0.6347,0.8053)--(-0.6029,0.8237)--(-0.5712,0.8413)--(-0.5395,0.8580)--(-0.5077,0.8738)--(-0.4760,0.8888)--(-0.4443,0.9029)--(-0.4125,0.9161)--(-0.3808,0.9284)--(-0.3491,0.9397)--(-0.3173,0.9501)--(-0.2856,0.9595)--(-0.2539,0.9679)--(-0.2221,0.9754)--(-0.1904,0.9819)--(-0.1587,0.9874)--(-0.1269,0.9920)--(-0.09520,0.9955)--(-0.06347,0.9980)--(-0.03173,0.9995)--(0,1.000);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=brown] (-3.1415,-1.0000)--(-3.1098,-0.9994)--(-3.0781,-0.9979)--(-3.0463,-0.9954)--(-3.0146,-0.9919)--(-2.9829,-0.9874)--(-2.9511,-0.9819)--(-2.9194,-0.9754)--(-2.8877,-0.9679)--(-2.8559,-0.9594)--(-2.8242,-0.9500)--(-2.7925,-0.9396)--(-2.7607,-0.9283)--(-2.7290,-0.9161)--(-2.6973,-0.9029)--(-2.6655,-0.8888)--(-2.6338,-0.8738)--(-2.6021,-0.8579)--(-2.5703,-0.8412)--(-2.5386,-0.8236)--(-2.5069,-0.8052)--(-2.4751,-0.7860)--(-2.4434,-0.7660)--(-2.4117,-0.7452)--(-2.3799,-0.7237)--(-2.3482,-0.7014)--(-2.3165,-0.6785)--(-2.2847,-0.6548)--(-2.2530,-0.6305)--(-2.2213,-0.6056)--(-2.1895,-0.5800)--(-2.1578,-0.5539)--(-2.1261,-0.5272)--(-2.0943,-0.5000)--(-2.0626,-0.4722)--(-2.0309,-0.4440)--(-1.9991,-0.4154)--(-1.9674,-0.3863)--(-1.9357,-0.3568)--(-1.9039,-0.3270)--(-1.8722,-0.2969)--(-1.8405,-0.2664)--(-1.8087,-0.2357)--(-1.7770,-0.2048)--(-1.7453,-0.1736)--(-1.7135,-0.1423)--(-1.6818,-0.1108)--(-1.6501,-0.0792)--(-1.6183,-0.0475)--(-1.5866,-0.0158)--(-1.5549,0.0158)--(-1.5231,0.0475)--(-1.4914,0.0792)--(-1.4597,0.1108)--(-1.4279,0.1423)--(-1.3962,0.1736)--(-1.3645,0.2048)--(-1.3327,0.2357)--(-1.3010,0.2664)--(-1.2693,0.2969)--(-1.2375,0.3270)--(-1.2058,0.3568)--(-1.1741,0.3863)--(-1.1423,0.4154)--(-1.1106,0.4440)--(-1.0789,0.4722)--(-1.0471,0.5000)--(-1.0154,0.5272)--(-0.9837,0.5539)--(-0.9519,0.5800)--(-0.9202,0.6056)--(-0.8885,0.6305)--(-0.8567,0.6548)--(-0.8250,0.6785)--(-0.7933,0.7014)--(-0.7615,0.7237)--(-0.7298,0.7452)--(-0.6981,0.7660)--(-0.6663,0.7860)--(-0.6346,0.8052)--(-0.6029,0.8236)--(-0.5711,0.8412)--(-0.5394,0.8579)--(-0.5077,0.8738)--(-0.4759,0.8888)--(-0.4442,0.9029)--(-0.4125,0.9161)--(-0.3807,0.9283)--(-0.3490,0.9396)--(-0.3173,0.9500)--(-0.2855,0.9594)--(-0.2538,0.9679)--(-0.2221,0.9754)--(-0.1903,0.9819)--(-0.1586,0.9874)--(-0.1269,0.9919)--(-0.0951,0.9954)--(-0.0634,0.9979)--(-0.0317,0.9994)--(0.0000,1.0000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SpiraleLimite.pstricks.recall b/src_phystricks/Fig_SpiraleLimite.pstricks.recall
index 3908c2a7d..89c0ff412 100644
--- a/src_phystricks/Fig_SpiraleLimite.pstricks.recall
+++ b/src_phystricks/Fig_SpiraleLimite.pstricks.recall
@@ -63,11 +63,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.7107);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.7106);
 %DEFAULT
-\draw [color=blue] (0,0)--(0.0214,0.0214)--(0.0429,0.0429)--(0.0643,0.0643)--(0.0857,0.0857)--(0.107,0.107)--(0.129,0.129)--(0.150,0.150)--(0.171,0.171)--(0.193,0.193)--(0.214,0.214)--(0.236,0.236)--(0.257,0.257)--(0.279,0.278)--(0.300,0.300)--(0.322,0.321)--(0.344,0.342)--(0.365,0.363)--(0.387,0.385)--(0.409,0.406)--(0.430,0.427)--(0.452,0.448)--(0.474,0.469)--(0.496,0.490)--(0.518,0.511)--(0.540,0.531)--(0.562,0.552)--(0.584,0.573)--(0.607,0.593)--(0.629,0.614)--(0.652,0.634)--(0.674,0.654)--(0.697,0.674)--(0.720,0.694)--(0.743,0.714)--(0.766,0.733)--(0.790,0.753)--(0.813,0.772)--(0.837,0.791)--(0.861,0.810)--(0.885,0.828)--(0.909,0.847)--(0.934,0.865)--(0.958,0.883)--(0.983,0.900)--(1.01,0.918)--(1.03,0.935)--(1.06,0.952)--(1.09,0.968)--(1.11,0.984)--(1.14,1.00)--(1.17,1.02)--(1.19,1.03)--(1.22,1.04)--(1.25,1.06)--(1.28,1.07)--(1.30,1.09)--(1.33,1.10)--(1.36,1.11)--(1.39,1.12)--(1.42,1.13)--(1.45,1.14)--(1.48,1.15)--(1.51,1.16)--(1.55,1.17)--(1.58,1.18)--(1.61,1.19)--(1.64,1.19)--(1.68,1.20)--(1.71,1.20)--(1.74,1.21)--(1.78,1.21)--(1.82,1.21)--(1.85,1.21)--(1.89,1.21)--(1.92,1.21)--(1.96,1.21)--(2.00,1.20)--(2.04,1.19)--(2.08,1.19)--(2.12,1.18)--(2.16,1.17)--(2.20,1.15)--(2.24,1.14)--(2.28,1.12)--(2.33,1.10)--(2.37,1.08)--(2.42,1.06)--(2.46,1.03)--(2.51,0.997)--(2.55,0.962)--(2.60,0.922)--(2.65,0.876)--(2.70,0.824)--(2.74,0.764)--(2.79,0.694)--(2.84,0.611)--(2.90,0.507)--(2.95,0.364)--(3.00,0);
-\draw [] (3.00,-0.100) -- (3.00,0.100);
+\draw [color=blue] (0.0000,0.0000)--(0.0214,0.0214)--(0.0428,0.0428)--(0.0642,0.0642)--(0.0857,0.0857)--(0.1071,0.1071)--(0.1285,0.1285)--(0.1500,0.1499)--(0.1714,0.1713)--(0.1929,0.1927)--(0.2143,0.2141)--(0.2358,0.2355)--(0.2573,0.2569)--(0.2788,0.2782)--(0.3004,0.2995)--(0.3219,0.3208)--(0.3435,0.3421)--(0.3651,0.3633)--(0.3868,0.3845)--(0.4085,0.4056)--(0.4303,0.4267)--(0.4521,0.4478)--(0.4739,0.4688)--(0.4959,0.4897)--(0.5179,0.5105)--(0.5399,0.5313)--(0.5621,0.5520)--(0.5843,0.5726)--(0.6067,0.5931)--(0.6291,0.6135)--(0.6517,0.6338)--(0.6743,0.6539)--(0.6971,0.6740)--(0.7200,0.6938)--(0.7431,0.7136)--(0.7663,0.7332)--(0.7897,0.7526)--(0.8132,0.7718)--(0.8369,0.7908)--(0.8608,0.8097)--(0.8849,0.8283)--(0.9091,0.8467)--(0.9336,0.8649)--(0.9583,0.8828)--(0.9833,0.9004)--(1.0084,0.9178)--(1.0339,0.9349)--(1.0596,0.9516)--(1.0855,0.9681)--(1.1117,0.9842)--(1.1382,0.9999)--(1.1651,1.0153)--(1.1922,1.0303)--(1.2196,1.0449)--(1.2474,1.0590)--(1.2755,1.0727)--(1.3040,1.0859)--(1.3328,1.0986)--(1.3620,1.1108)--(1.3916,1.1224)--(1.4215,1.1335)--(1.4519,1.1440)--(1.4827,1.1538)--(1.5139,1.1630)--(1.5455,1.1715)--(1.5776,1.1793)--(1.6101,1.1863)--(1.6431,1.1925)--(1.6766,1.1979)--(1.7105,1.2024)--(1.7450,1.2060)--(1.7799,1.2086)--(1.8154,1.2101)--(1.8514,1.2106)--(1.8879,1.2100)--(1.9249,1.2081)--(1.9625,1.2050)--(2.0007,1.2005)--(2.0395,1.1946)--(2.0788,1.1871)--(2.1187,1.1781)--(2.1592,1.1672)--(2.2003,1.1545)--(2.2420,1.1397)--(2.2844,1.1228)--(2.3274,1.1034)--(2.3710,1.0815)--(2.4153,1.0567)--(2.4602,1.0287)--(2.5058,0.9972)--(2.5520,0.9616)--(2.5990,0.9215)--(2.6466,0.8760)--(2.6949,0.8240)--(2.7440,0.7643)--(2.7937,0.6944)--(2.8442,0.6108)--(2.8954,0.5065)--(2.9473,0.3636)--(3.0000,0.0000);
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall b/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall
index 19a5c0484..c5e7152c3 100644
--- a/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall
+++ b/src_phystricks/Fig_SubfiguresCDUTraceCycloide.pstricks.recall
@@ -69,65 +69,65 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,3.1310);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.4902,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,3.1309);
 %DEFAULT
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.0000,1.0000) -- (0,1.0000);
-\draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw [color=green,->,>=latex] (1.0000,1.0000) -- (1.7071,0.29289);
-\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (2.1416,1.0000) -- (3.1416,1.0000);
-\draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw [color=green,->,>=latex] (2.1416,1.0000) -- (1.4345,0.29289);
-\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (4.7124,0) -- (4.7124,1.0000);
-\draw [color=brown] (3.71,0) -- (5.71,0);
-\draw [color=green,->,>=latex] (4.7124,0) -- (4.7124,-1.0000);
-\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (7.2832,1.0000) -- (6.2832,1.0000);
-\draw [color=brown] (6.58,0.293) -- (7.99,1.71);
-\draw [color=green,->,>=latex] (7.2832,1.0000) -- (7.9903,0.29289);
+\draw [,->,>=latex] (1.0000,1.0000) -- (0.0000,1.0000);
+\draw [color=brown] (0.2928,0.2928) -- (1.7071,1.7071);
+\draw [color=green,->,>=latex] (1.0000,1.0000) -- (1.7071,0.2928);
+\draw []  (2.1415,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (2.1415,1.0000) -- (3.1415,1.0000);
+\draw [color=brown] (1.4344,1.7071) -- (2.8486,0.2928);
+\draw [color=green,->,>=latex] (2.1415,1.0000) -- (1.4344,0.2928);
+\draw []  (4.7123,0.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (4.7123,0.0000) -- (4.7123,1.0000);
+\draw [color=brown] (3.7123,0.0000) -- (5.7123,0.0000);
+\draw [color=green,->,>=latex] (4.7123,0.0000) -- (4.7123,-1.0000);
+\draw []  (7.2831,1.0000) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (7.2831,1.0000) -- (6.2831,1.0000);
+\draw [color=brown] (6.5760,0.2928) -- (7.9902,1.7071);
+\draw [color=green,->,>=latex] (7.2831,1.0000) -- (7.9902,0.2928);
 \draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.4925,1.7071) -- (0.78540,1.0000);
-\draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw [color=green,->,>=latex] (1.4925,1.7071) -- (2.4164,1.3244);
-\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (1.6491,1.7071) -- (2.3562,1.0000);
-\draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw [color=green,->,>=latex] (1.6491,1.7071) -- (0.72521,1.3244);
-\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (3.2199,0.29289) -- (3.9270,1.0000);
-\draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw [color=green,->,>=latex] (3.2199,0.29289) -- (2.8372,-0.63099);
-\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
-\draw [,->,>=latex] (6.2049,0.29289) -- (5.4978,1.0000);
-\draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
-\draw [color=green,->,>=latex] (6.2049,0.29289) -- (6.5876,-0.63099);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [,->,>=latex] (1.4925,1.7071) -- (0.7853,1.0000);
+\draw [color=brown] (1.1098,0.7832) -- (1.8751,2.6309);
+\draw [color=green,->,>=latex] (1.4925,1.7071) -- (2.4163,1.3244);
+\draw []  (1.6490,1.7071) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (1.6490,1.7071) -- (2.3561,1.0000);
+\draw [color=brown] (1.2664,2.6309) -- (2.0317,0.7832);
+\draw [color=green,->,>=latex] (1.6490,1.7071) -- (0.7252,1.3244);
+\draw []  (3.2198,0.2928) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (3.2198,0.2928) -- (3.9269,1.0000);
+\draw [color=brown] (2.2960,0.6755) -- (4.1437,-0.0897);
+\draw [color=green,->,>=latex] (3.2198,0.2928) -- (2.8372,-0.6309);
+\draw []  (6.2048,0.2928) node [rotate=0] {$\bullet$};
+\draw [,->,>=latex] (6.2048,0.2928) -- (5.4977,1.0000);
+\draw [color=brown] (5.2810,-0.0897) -- (7.1287,0.6755);
+\draw [color=green,->,>=latex] (6.2048,0.2928) -- (6.5875,-0.6309);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -189,48 +189,48 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.4903,0);
-\draw [,->,>=latex] (0,-0.58979) -- (0,3.1310);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.4902,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5897) -- (0.0000,3.1309);
 %DEFAULT
 \draw []  (1.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown] (0.293,0.293) -- (1.71,1.71);
-\draw []  (2.1416,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown] (1.43,1.71) -- (2.85,0.293);
-\draw []  (4.7124,0) node [rotate=0] {$\bullet$};
-\draw [color=brown] (3.71,0) -- (5.71,0);
-\draw []  (7.2832,1.0000) node [rotate=0] {$\bullet$};
-\draw [color=brown] (6.58,0.293) -- (7.99,1.71);
+\draw [color=brown] (0.2928,0.2928) -- (1.7071,1.7071);
+\draw []  (2.1415,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown] (1.4344,1.7071) -- (2.8486,0.2928);
+\draw []  (4.7123,0.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown] (3.7123,0.0000) -- (5.7123,0.0000);
+\draw []  (7.2831,1.0000) node [rotate=0] {$\bullet$};
+\draw [color=brown] (6.5760,0.2928) -- (7.9902,1.7071);
 \draw []  (1.4925,1.7071) node [rotate=0] {$\bullet$};
-\draw [color=brown] (1.11,0.783) -- (1.88,2.63);
-\draw []  (1.6491,1.7071) node [rotate=0] {$\bullet$};
-\draw [color=brown] (1.27,2.63) -- (2.03,0.783);
-\draw []  (3.2199,0.29289) node [rotate=0] {$\bullet$};
-\draw [color=brown] (2.30,0.676) -- (4.14,-0.0897);
-\draw []  (6.2049,0.29289) node [rotate=0] {$\bullet$};
-\draw [color=brown] (5.28,-0.0897) -- (7.13,0.676);
-\draw [color=blue,style=dashed] (1.000,1.000)--(1.061,1.063)--(1.119,1.127)--(1.172,1.189)--(1.222,1.251)--(1.267,1.312)--(1.309,1.372)--(1.347,1.430)--(1.382,1.486)--(1.412,1.541)--(1.440,1.593)--(1.464,1.643)--(1.485,1.690)--(1.504,1.735)--(1.519,1.776)--(1.532,1.815)--(1.543,1.850)--(1.551,1.881)--(1.558,1.910)--(1.563,1.934)--(1.566,1.955)--(1.569,1.972)--(1.570,1.985)--(1.571,1.994)--(1.571,1.999)--(1.571,2.000)--(1.571,1.997)--(1.571,1.990)--(1.572,1.979)--(1.574,1.964)--(1.577,1.945)--(1.581,1.922)--(1.587,1.896)--(1.594,1.866)--(1.604,1.833)--(1.616,1.796)--(1.630,1.756)--(1.647,1.713)--(1.666,1.667)--(1.689,1.618)--(1.715,1.567)--(1.744,1.514)--(1.777,1.458)--(1.813,1.401)--(1.853,1.342)--(1.896,1.282)--(1.944,1.220)--(1.995,1.158)--(2.051,1.095)--(2.110,1.032)--(2.174,0.9683)--(2.241,0.9049)--(2.313,0.8420)--(2.388,0.7797)--(2.468,0.7183)--(2.551,0.6580)--(2.638,0.5991)--(2.729,0.5418)--(2.823,0.4863)--(2.921,0.4329)--(3.022,0.3818)--(3.126,0.3332)--(3.233,0.2873)--(3.344,0.2443)--(3.456,0.2042)--(3.571,0.1674)--(3.689,0.1340)--(3.808,0.1040)--(3.929,0.07765)--(4.052,0.05500)--(4.176,0.03616)--(4.301,0.02120)--(4.427,0.01018)--(4.554,0.003145)--(4.681,0)--(4.808,0.001133)--(4.934,0.006162)--(5.061,0.01519)--(5.186,0.02819)--(5.311,0.04510)--(5.434,0.06585)--(5.556,0.09037)--(5.677,0.1185)--(5.795,0.1503)--(5.911,0.1854)--(6.025,0.2239)--(6.137,0.2654)--(6.245,0.3099)--(6.351,0.3572)--(6.454,0.4071)--(6.553,0.4594)--(6.649,0.5138)--(6.742,0.5702)--(6.831,0.6283)--(6.916,0.6880)--(6.997,0.7489)--(7.075,0.8107)--(7.148,0.8734)--(7.218,0.9366)--(7.283,1.000);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (8.0000,-0.31492) node {$ 8 $};
-\draw [] (8.00,-0.100) -- (8.00,0.100);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=brown] (1.1098,0.7832) -- (1.8751,2.6309);
+\draw []  (1.6490,1.7071) node [rotate=0] {$\bullet$};
+\draw [color=brown] (1.2664,2.6309) -- (2.0317,0.7832);
+\draw []  (3.2198,0.2928) node [rotate=0] {$\bullet$};
+\draw [color=brown] (2.2960,0.6755) -- (4.1437,-0.0897);
+\draw []  (6.2048,0.2928) node [rotate=0] {$\bullet$};
+\draw [color=brown] (5.2810,-0.0897) -- (7.1287,0.6755);
+\draw [color=blue,style=dashed] (1.0000,1.0000)--(1.0614,1.0634)--(1.1188,1.1265)--(1.1723,1.1892)--(1.2218,1.2511)--(1.2674,1.3120)--(1.3091,1.3716)--(1.3471,1.4297)--(1.3815,1.4861)--(1.4124,1.5406)--(1.4399,1.5929)--(1.4641,1.6427)--(1.4853,1.6900)--(1.5035,1.7345)--(1.5190,1.7761)--(1.5320,1.8145)--(1.5426,1.8497)--(1.5512,1.8814)--(1.5578,1.9096)--(1.5627,1.9341)--(1.5662,1.9549)--(1.5685,1.9718)--(1.5699,1.9848)--(1.5705,1.9938)--(1.5707,1.9988)--(1.5707,1.9998)--(1.5708,1.9968)--(1.5712,1.9898)--(1.5722,1.9788)--(1.5740,1.9638)--(1.5769,1.9450)--(1.5811,1.9223)--(1.5868,1.8959)--(1.5943,1.8660)--(1.6039,1.8325)--(1.6157,1.7957)--(1.6299,1.7557)--(1.6467,1.7126)--(1.6664,1.6667)--(1.6891,1.6181)--(1.7149,1.5670)--(1.7441,1.5136)--(1.7767,1.4582)--(1.8129,1.4009)--(1.8528,1.3420)--(1.8965,1.2817)--(1.9440,1.2203)--(1.9954,1.1580)--(2.0509,1.0950)--(2.1103,1.0317)--(2.1738,0.9682)--(2.2413,0.9049)--(2.3128,0.8419)--(2.3882,0.7796)--(2.4676,0.7182)--(2.5509,0.6579)--(2.6380,0.5990)--(2.7287,0.5417)--(2.8230,0.4863)--(2.9208,0.4329)--(3.0219,0.3818)--(3.1261,0.3332)--(3.2334,0.2873)--(3.3435,0.2442)--(3.4562,0.2042)--(3.5714,0.1674)--(3.6887,0.1339)--(3.8081,0.1040)--(3.9293,0.0776)--(4.0521,0.0549)--(4.1761,0.0361)--(4.3013,0.0211)--(4.4272,0.0101)--(4.5538,0.0031)--(4.6806,0.0000)--(4.8075,0.0011)--(4.9342,0.0061)--(5.0605,0.0151)--(5.1861,0.0281)--(5.3107,0.0450)--(5.4342,0.0658)--(5.5562,0.0903)--(5.6765,0.1185)--(5.7949,0.1502)--(5.9112,0.1854)--(6.0252,0.2238)--(6.1366,0.2654)--(6.2453,0.3099)--(6.3510,0.3572)--(6.4537,0.4070)--(6.5532,0.4593)--(6.6493,0.5138)--(6.7418,0.5702)--(6.8307,0.6283)--(6.9159,0.6879)--(6.9972,0.7488)--(7.0747,0.8107)--(7.1482,0.8734)--(7.2177,0.9365)--(7.2831,1.0000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (8.0000,-0.3149) node {$ 8 $};
+\draw [] (8.0000,-0.1000) -- (8.0000,0.1000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall b/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall
index 3a69fd3a0..f9d0646be 100644
--- a/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall
+++ b/src_phystricks/Fig_SurfaceEntreCourbes.pstricks.recall
@@ -41,22 +41,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8999);
 %DEFAULT
+\fill [color=brown] (0.6742,1.6500) -- (0.7111,1.6799) -- (0.7480,1.7093) -- (0.7849,1.7381) -- (0.8217,1.7663) -- (0.8586,1.7938) -- (0.8955,1.8207) -- (0.9324,1.8471) -- (0.9693,1.8728) -- (1.0062,1.8979) -- (1.0430,1.9224) -- (1.0799,1.9462) -- (1.1168,1.9695) -- (1.1537,1.9922) -- (1.1906,2.0142) -- (1.2275,2.0356) -- (1.2643,2.0564) -- (1.3012,2.0766) -- (1.3381,2.0962) -- (1.3750,2.1152) -- (1.4119,2.1336) -- (1.4488,2.1513) -- (1.4856,2.1685) -- (1.5225,2.1850) -- (1.5594,2.2009) -- (1.5963,2.2162) -- (1.6332,2.2309) -- (1.6701,2.2450) -- (1.7070,2.2585) -- (1.7438,2.2713) -- (1.7807,2.2836) -- (1.8176,2.2952) -- (1.8545,2.3062) -- (1.8914,2.3166) -- (1.9283,2.3264) -- (1.9651,2.3356) -- (2.0020,2.3442) -- (2.0389,2.3521) -- (2.0758,2.3595) -- (2.1127,2.3662) -- (2.1496,2.3723) -- (2.1864,2.3778) -- (2.2233,2.3827) -- (2.2602,2.3870) -- (2.2971,2.3907) -- (2.3340,2.3938) -- (2.3709,2.3962) -- (2.4077,2.3980) -- (2.4446,2.3993) -- (2.4815,2.3999) -- (2.5184,2.3999) -- (2.5553,2.3993) -- (2.5922,2.3980) -- (2.6290,2.3962) -- (2.6659,2.3938) -- (2.7028,2.3907) -- (2.7397,2.3870) -- (2.7766,2.3827) -- (2.8135,2.3778) -- (2.8503,2.3723) -- (2.8872,2.3662) -- (2.9241,2.3595) -- (2.9610,2.3521) -- (2.9979,2.3442) -- (3.0348,2.3356) -- (3.0716,2.3264) -- (3.1085,2.3166) -- (3.1454,2.3062) -- (3.1823,2.2952) -- (3.2192,2.2836) -- (3.2561,2.2713) -- (3.2929,2.2585) -- (3.3298,2.2450) -- (3.3667,2.2309) -- (3.4036,2.2162) -- (3.4405,2.2009) -- (3.4774,2.1850) -- (3.5143,2.1685) -- (3.5511,2.1513) -- (3.5880,2.1336) -- (3.6249,2.1152) -- (3.6618,2.0962) -- (3.6987,2.0766) -- (3.7356,2.0564) -- (3.7724,2.0356) -- (3.8093,2.0142) -- (3.8462,1.9922) -- (3.8831,1.9695) -- (3.9200,1.9462) -- (3.9569,1.9224) -- (3.9937,1.8979) -- (4.0306,1.8728) -- (4.0675,1.8471) -- (4.1044,1.8207) -- (4.1413,1.7938) -- (4.1782,1.7663) -- (4.2150,1.7381) -- (4.2519,1.7093) -- (4.2888,1.6799) -- (4.3257,1.6500) -- (4.3257,1.6500) -- (4.3257,0.0000) -- (4.3257,0.0000) -- (4.2888,0.0000) -- (4.2519,0.0000) -- (4.2150,0.0000) -- (4.1782,0.0000) -- (4.1413,0.0000) -- (4.1044,0.0000) -- (4.0675,0.0000) -- (4.0306,0.0000) -- (3.9937,0.0000) -- (3.9569,0.0000) -- (3.9200,0.0000) -- (3.8831,0.0000) -- (3.8462,0.0000) -- (3.8093,0.0000) -- (3.7724,0.0000) -- (3.7356,0.0000) -- (3.6987,0.0000) -- (3.6618,0.0000) -- (3.6249,0.0000) -- (3.5880,0.0000) -- (3.5511,0.0000) -- (3.5143,0.0000) -- (3.4774,0.0000) -- (3.4405,0.0000) -- (3.4036,0.0000) -- (3.3667,0.0000) -- (3.3298,0.0000) -- (3.2929,0.0000) -- (3.2561,0.0000) -- (3.2192,0.0000) -- (3.1823,0.0000) -- (3.1454,0.0000) -- (3.1085,0.0000) -- (3.0716,0.0000) -- (3.0348,0.0000) -- (2.9979,0.0000) -- (2.9610,0.0000) -- (2.9241,0.0000) -- (2.8872,0.0000) -- (2.8503,0.0000) -- (2.8135,0.0000) -- (2.7766,0.0000) -- (2.7397,0.0000) -- (2.7028,0.0000) -- (2.6659,0.0000) -- (2.6290,0.0000) -- (2.5922,0.0000) -- (2.5553,0.0000) -- (2.5184,0.0000) -- (2.4815,0.0000) -- (2.4446,0.0000) -- (2.4077,0.0000) -- (2.3709,0.0000) -- (2.3340,0.0000) -- (2.2971,0.0000) -- (2.2602,0.0000) -- (2.2233,0.0000) -- (2.1864,0.0000) -- (2.1496,0.0000) -- (2.1127,0.0000) -- (2.0758,0.0000) -- (2.0389,0.0000) -- (2.0020,0.0000) -- (1.9651,0.0000) -- (1.9283,0.0000) -- (1.8914,0.0000) -- (1.8545,0.0000) -- (1.8176,0.0000) -- (1.7807,0.0000) -- (1.7438,0.0000) -- (1.7070,0.0000) -- (1.6701,0.0000) -- (1.6332,0.0000) -- (1.5963,0.0000) -- (1.5594,0.0000) -- (1.5225,0.0000) -- (1.4856,0.0000) -- (1.4488,0.0000) -- (1.4119,0.0000) -- (1.3750,0.0000) -- (1.3381,0.0000) -- (1.3012,0.0000) -- (1.2643,0.0000) -- (1.2275,0.0000) -- (1.1906,0.0000) -- (1.1537,0.0000) -- (1.1168,0.0000) -- (1.0799,0.0000) -- (1.0430,0.0000) -- (1.0062,0.0000) -- (0.9693,0.0000) -- (0.9324,0.0000) -- (0.8955,0.0000) -- (0.8586,0.0000) -- (0.8217,0.0000) -- (0.7849,0.0000) -- (0.7480,0.0000) -- (0.7111,0.0000) -- (0.6742,0.0000) -- (0.6742,0.0000) -- (0.6742,1.6500) --  cycle;
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+\draw [] (0.6742,0.0000) -- (0.6742,1.6500);
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+\draw []  (0.6742,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.6742,-0.3785) node {$a$};
+\draw []  (4.3257,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.3257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -95,22 +95,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.1000,0.0000);
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+\draw [] (0.6742,0.0000) -- (0.6742,1.6500);
+\draw [] (4.3257,1.6500) -- (4.3257,0.0000);
+\draw []  (0.6742,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.6742,-0.3785) node {$a$};
+\draw []  (4.3257,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.3257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
@@ -149,22 +149,22 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4250);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8999);
 %DEFAULT
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-\draw (3.8257,-0.42674) node {$b$};
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+\draw []  (0.6742,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.6742,-0.3785) node {$a$};
+\draw []  (4.3257,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.3257,-0.4267) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall b/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall
index a3b7cc8a7..e40581b9a 100644
--- a/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall
+++ b/src_phystricks/Fig_SurfaceHorizVerti.pstricks.recall
@@ -41,8 +41,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (6.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.4999);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (6.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.4998);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -51,26 +51,26 @@
 % setting the default values
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+\draw [color=blue] (1.0000,1.6347)--(1.0505,1.5624)--(1.1010,1.4970)--(1.1515,1.4392)--(1.2020,1.3895)--(1.2525,1.3484)--(1.3030,1.3161)--(1.3535,1.2928)--(1.4040,1.2785)--(1.4545,1.2731)--(1.5050,1.2763)--(1.5555,1.2878)--(1.6060,1.3071)--(1.6565,1.3337)--(1.7070,1.3669)--(1.7575,1.4059)--(1.8080,1.4499)--(1.8585,1.4980)--(1.9090,1.5493)--(1.9595,1.6029)--(2.0101,1.6578)--(2.0606,1.7132)--(2.1111,1.7680)--(2.1616,1.8214)--(2.2121,1.8726)--(2.2626,1.9208)--(2.3131,1.9653)--(2.3636,2.0056)--(2.4141,2.0411)--(2.4646,2.0714)--(2.5151,2.0963)--(2.5656,2.1155)--(2.6161,2.1290)--(2.6666,2.1368)--(2.7171,2.1391)--(2.7676,2.1362)--(2.8181,2.1284)--(2.8686,2.1162)--(2.9191,2.1001)--(2.9696,2.0808)--(3.0202,2.0588)--(3.0707,2.0350)--(3.1212,2.0101)--(3.1717,1.9849)--(3.2222,1.9602)--(3.2727,1.9367)--(3.3232,1.9152)--(3.3737,1.8964)--(3.4242,1.8810)--(3.4747,1.8696)--(3.5252,1.8627)--(3.5757,1.8608)--(3.6262,1.8642)--(3.6767,1.8731)--(3.7272,1.8877)--(3.7777,1.9080)--(3.8282,1.9339)--(3.8787,1.9653)--(3.9292,2.0017)--(3.9797,2.0428)--(4.0303,2.0881)--(4.0808,2.1370)--(4.1313,2.1887)--(4.1818,2.2424)--(4.2323,2.2974)--(4.2828,2.3527)--(4.3333,2.4075)--(4.3838,2.4607)--(4.4343,2.5115)--(4.4848,2.5589)--(4.5353,2.6020)--(4.5858,2.6399)--(4.6363,2.6718)--(4.6868,2.6971)--(4.7373,2.7149)--(4.7878,2.7249)--(4.8383,2.7265)--(4.8888,2.7194)--(4.9393,2.7033)--(4.9898,2.6783)--(5.0404,2.6442)--(5.0909,2.6014)--(5.1414,2.5502)--(5.1919,2.4909)--(5.2424,2.4241)--(5.2929,2.3505)--(5.3434,2.2708)--(5.3939,2.1861)--(5.4444,2.0972)--(5.4949,2.0052)--(5.5454,1.9112)--(5.5959,1.8163)--(5.6464,1.7217)--(5.6969,1.6286)--(5.7474,1.5381)--(5.7979,1.4515)--(5.8484,1.3698)--(5.8989,1.2941)--(5.9494,1.2254)--(6.0000,1.1645);
 
-\draw [color=cyan] (1.00,5.00) -- (6.00,5.00);
-\draw [color=cyan] (6.00,5.00) -- (6.00,1.16);
-\draw [color=cyan] (6.00,1.16) -- (1.00,1.16);
-\draw [color=cyan] (1.00,1.16) -- (1.00,5.00);
+\draw [color=cyan] (1.0000,4.9998) -- (6.0000,4.9998);
+\draw [color=cyan] (6.0000,4.9998) -- (6.0000,1.1645);
+\draw [color=cyan] (6.0000,1.1645) -- (1.0000,1.1645);
+\draw [color=cyan] (1.0000,1.1645) -- (1.0000,4.9998);
 
 %OTHER STUFF
 %END PSPICTURE
@@ -109,8 +109,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.4999,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,6.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.4998,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,6.5000);
 %DEFAULT
 
 % declaring the keys in tikz
@@ -119,24 +119,24 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-0.27898,1.0000) node {$c$};
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+\draw [color=blue,style=solid] (1.6347,1.0000)--(1.5624,1.0505)--(1.4970,1.1010)--(1.4392,1.1515)--(1.3895,1.2020)--(1.3484,1.2525)--(1.3161,1.3030)--(1.2928,1.3535)--(1.2785,1.4040)--(1.2731,1.4545)--(1.2763,1.5050)--(1.2878,1.5555)--(1.3071,1.6060)--(1.3337,1.6565)--(1.3669,1.7070)--(1.4059,1.7575)--(1.4499,1.8080)--(1.4980,1.8585)--(1.5493,1.9090)--(1.6029,1.9595)--(1.6578,2.0101)--(1.7132,2.0606)--(1.7680,2.1111)--(1.8214,2.1616)--(1.8726,2.2121)--(1.9208,2.2626)--(1.9653,2.3131)--(2.0056,2.3636)--(2.0411,2.4141)--(2.0714,2.4646)--(2.0963,2.5151)--(2.1155,2.5656)--(2.1290,2.6161)--(2.1368,2.6666)--(2.1391,2.7171)--(2.1362,2.7676)--(2.1284,2.8181)--(2.1162,2.8686)--(2.1001,2.9191)--(2.0808,2.9696)--(2.0588,3.0202)--(2.0350,3.0707)--(2.0101,3.1212)--(1.9849,3.1717)--(1.9602,3.2222)--(1.9367,3.2727)--(1.9152,3.3232)--(1.8964,3.3737)--(1.8810,3.4242)--(1.8696,3.4747)--(1.8627,3.5252)--(1.8608,3.5757)--(1.8642,3.6262)--(1.8731,3.6767)--(1.8877,3.7272)--(1.9080,3.7777)--(1.9339,3.8282)--(1.9653,3.8787)--(2.0017,3.9292)--(2.0428,3.9797)--(2.0881,4.0303)--(2.1370,4.0808)--(2.1887,4.1313)--(2.2424,4.1818)--(2.2974,4.2323)--(2.3527,4.2828)--(2.4075,4.3333)--(2.4607,4.3838)--(2.5115,4.4343)--(2.5589,4.4848)--(2.6020,4.5353)--(2.6399,4.5858)--(2.6718,4.6363)--(2.6971,4.6868)--(2.7149,4.7373)--(2.7249,4.7878)--(2.7265,4.8383)--(2.7194,4.8888)--(2.7033,4.9393)--(2.6783,4.9898)--(2.6442,5.0404)--(2.6014,5.0909)--(2.5502,5.1414)--(2.4909,5.1919)--(2.4241,5.2424)--(2.3505,5.2929)--(2.2708,5.3434)--(2.1861,5.3939)--(2.0972,5.4444)--(2.0052,5.4949)--(1.9112,5.5454)--(1.8163,5.5959)--(1.7217,5.6464)--(1.6286,5.6969)--(1.5381,5.7474)--(1.4515,5.7979)--(1.3698,5.8484)--(1.2941,5.8989)--(1.2254,5.9494)--(1.1645,6.0000);
 
-\draw [color=cyan] (1.16,6.00) -- (5.00,6.00);
-\draw [color=cyan] (5.00,6.00) -- (5.00,1.00);
-\draw [color=cyan] (5.00,1.00) -- (1.16,1.00);
-\draw [color=cyan] (1.16,1.00) -- (1.16,6.00);
+\draw [color=cyan] (1.1645,6.0000) -- (4.9998,6.0000);
+\draw [color=cyan] (4.9998,6.0000) -- (4.9998,1.0000);
+\draw [color=cyan] (4.9998,1.0000) -- (1.1645,1.0000);
+\draw [color=cyan] (1.1645,1.0000) -- (1.1645,6.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall b/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall
index fd9c3b21d..0b50e3485 100644
--- a/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall
+++ b/src_phystricks/Fig_SurfacePrimiteGeog.pstricks.recall
@@ -79,11 +79,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.4000);
 %DEFAULT
 
-\draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
+\draw [color=red] (1.5000,1.5000)--(1.5606,1.6165)--(1.6212,1.7242)--(1.6818,1.8243)--(1.7424,1.9173)--(1.8030,2.0042)--(1.8636,2.0853)--(1.9242,2.1614)--(1.9848,2.2328)--(2.0454,2.3000)--(2.1060,2.3633)--(2.1666,2.4230)--(2.2272,2.4795)--(2.2878,2.5331)--(2.3484,2.5838)--(2.4090,2.6320)--(2.4696,2.6779)--(2.5303,2.7215)--(2.5909,2.7631)--(2.6515,2.8028)--(2.7121,2.8407)--(2.7727,2.8770)--(2.8333,2.9117)--(2.8939,2.9450)--(2.9545,2.9769)--(3.0151,3.0075)--(3.0757,3.0369)--(3.1363,3.0652)--(3.1969,3.0924)--(3.2575,3.1186)--(3.3181,3.1438)--(3.3787,3.1681)--(3.4393,3.1916)--(3.5000,3.2142)--(3.5606,3.2361)--(3.6212,3.2573)--(3.6818,3.2777)--(3.7424,3.2975)--(3.8030,3.3167)--(3.8636,3.3352)--(3.9242,3.3532)--(3.9848,3.3707)--(4.0454,3.3876)--(4.1060,3.4040)--(4.1666,3.4200)--(4.2272,3.4354)--(4.2878,3.4505)--(4.3484,3.4651)--(4.4090,3.4793)--(4.4696,3.4932)--(4.5303,3.5066)--(4.5909,3.5198)--(4.6515,3.5325)--(4.7121,3.5450)--(4.7727,3.5571)--(4.8333,3.5689)--(4.8939,3.5804)--(4.9545,3.5917)--(5.0151,3.6027)--(5.0757,3.6134)--(5.1363,3.6238)--(5.1969,3.6341)--(5.2575,3.6440)--(5.3181,3.6538)--(5.3787,3.6633)--(5.4393,3.6727)--(5.5000,3.6818)--(5.5606,3.6907)--(5.6212,3.6994)--(5.6818,3.7080)--(5.7424,3.7163)--(5.8030,3.7245)--(5.8636,3.7325)--(5.9242,3.7404)--(5.9848,3.7481)--(6.0454,3.7556)--(6.1060,3.7630)--(6.1666,3.7702)--(6.2272,3.7773)--(6.2878,3.7843)--(6.3484,3.7911)--(6.4090,3.7978)--(6.4696,3.8044)--(6.5303,3.8109)--(6.5909,3.8172)--(6.6515,3.8234)--(6.7121,3.8295)--(6.7727,3.8355)--(6.8333,3.8414)--(6.8939,3.8472)--(6.9545,3.8529)--(7.0151,3.8585)--(7.0757,3.8640)--(7.1363,3.8694)--(7.1969,3.8747)--(7.2575,3.8799)--(7.3181,3.8850)--(7.3787,3.8901)--(7.4393,3.8951)--(7.5000,3.9000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -91,15 +91,15 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw (6.0000,-0.3785) node {$x$};
 \draw (8.3552,3.9000) node {$f(x)$};
-\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (9.6701,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -107,11 +107,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\fill [color=blue,  pattern=custom north west lines,hatchspread=10pt,hatchthickness=1pt ] (3.0000,3.0000) -- (3.0303,3.0150) -- (3.0606,3.0297) -- (3.0909,3.0441) -- (3.1212,3.0582) -- (3.1515,3.0721) -- (3.1818,3.0857) -- (3.2121,3.0990) -- (3.2424,3.1121) -- (3.2727,3.1250) -- (3.3030,3.1376) -- (3.3333,3.1500) -- (3.3636,3.1621) -- (3.3939,3.1741) -- (3.4242,3.1858) -- (3.4545,3.1973) -- (3.4848,3.2086) -- (3.5151,3.2198) -- (3.5454,3.2307) -- (3.5757,3.2415) -- (3.6060,3.2521) -- (3.6363,3.2625) -- (3.6666,3.2727) -- (3.6969,3.2827) -- (3.7272,3.2926) -- (3.7575,3.3024) -- (3.7878,3.3120) -- (3.8181,3.3214) -- (3.8484,3.3307) -- (3.8787,3.3398) -- (3.9090,3.3488) -- (3.9393,3.3576) -- (3.9696,3.3664) -- (4.0000,3.3750) -- (4.0303,3.3834) -- (4.0606,3.3917) -- (4.0909,3.4000) -- (4.1212,3.4080) -- (4.1515,3.4160) -- (4.1818,3.4239) -- (4.2121,3.4316) -- (4.2424,3.4392) -- (4.2727,3.4468) -- (4.3030,3.4542) -- (4.3333,3.4615) -- (4.3636,3.4687) -- (4.3939,3.4758) -- (4.4242,3.4828) -- (4.4545,3.4897) -- (4.4848,3.4966) -- (4.5151,3.5033) -- (4.5454,3.5100) -- (4.5757,3.5165) -- (4.6060,3.5230) -- (4.6363,3.5294) -- (4.6666,3.5357) -- (4.6969,3.5419) -- (4.7272,3.5480) -- (4.7575,3.5541) -- (4.7878,3.5601) -- (4.8181,3.5660) -- (4.8484,3.5718) -- (4.8787,3.5776) -- (4.9090,3.5833) -- (4.9393,3.5889) -- (4.9696,3.5945) -- (5.0000,3.6000) -- (5.0303,3.6054) -- (5.0606,3.6107) -- (5.0909,3.6160) -- (5.1212,3.6213) -- (5.1515,3.6264) -- (5.1818,3.6315) -- (5.2121,3.6366) -- (5.2424,3.6416) -- (5.2727,3.6465) -- (5.3030,3.6514) -- (5.3333,3.6562) -- (5.3636,3.6610) -- (5.3939,3.6657) -- (5.4242,3.6703) -- (5.4545,3.6750) -- (5.4848,3.6795) -- (5.5151,3.6840) -- (5.5454,3.6885) -- (5.5757,3.6929) -- (5.6060,3.6972) -- (5.6363,3.7016) -- (5.6666,3.7058) -- (5.6969,3.7101) -- (5.7272,3.7142) -- (5.7575,3.7184) -- (5.7878,3.7225) -- (5.8181,3.7265) -- (5.8484,3.7305) -- (5.8787,3.7345) -- (5.9090,3.7384) -- (5.9393,3.7423) -- (5.9696,3.7461) -- (6.0000,3.7500) -- (6.0000,3.7500) -- (6.0000,0.0000) -- (6.0000,0.0000) -- (5.9696,0.0000) -- (5.9393,0.0000) -- (5.9090,0.0000) -- (5.8787,0.0000) -- (5.8484,0.0000) -- (5.8181,0.0000) -- (5.7878,0.0000) -- (5.7575,0.0000) -- (5.7272,0.0000) -- (5.6969,0.0000) -- (5.6666,0.0000) -- (5.6363,0.0000) -- (5.6060,0.0000) -- (5.5757,0.0000) -- (5.5454,0.0000) -- (5.5151,0.0000) -- (5.4848,0.0000) -- (5.4545,0.0000) -- (5.4242,0.0000) -- (5.3939,0.0000) -- (5.3636,0.0000) -- (5.3333,0.0000) -- (5.3030,0.0000) -- (5.2727,0.0000) -- (5.2424,0.0000) -- (5.2121,0.0000) -- (5.1818,0.0000) -- (5.1515,0.0000) -- (5.1212,0.0000) -- (5.0909,0.0000) -- (5.0606,0.0000) -- (5.0303,0.0000) -- (5.0000,0.0000) -- (4.9696,0.0000) -- (4.9393,0.0000) -- (4.9090,0.0000) -- (4.8787,0.0000) -- (4.8484,0.0000) -- (4.8181,0.0000) -- (4.7878,0.0000) -- (4.7575,0.0000) -- (4.7272,0.0000) -- (4.6969,0.0000) -- (4.6666,0.0000) -- (4.6363,0.0000) -- (4.6060,0.0000) -- (4.5757,0.0000) -- (4.5454,0.0000) -- (4.5151,0.0000) -- (4.4848,0.0000) -- (4.4545,0.0000) -- (4.4242,0.0000) -- (4.3939,0.0000) -- (4.3636,0.0000) -- (4.3333,0.0000) -- (4.3030,0.0000) -- (4.2727,0.0000) -- (4.2424,0.0000) -- (4.2121,0.0000) -- (4.1818,0.0000) -- (4.1515,0.0000) -- (4.1212,0.0000) -- (4.0909,0.0000) -- (4.0606,0.0000) -- (4.0303,0.0000) -- (4.0000,0.0000) -- (3.9696,0.0000) -- (3.9393,0.0000) -- (3.9090,0.0000) -- (3.8787,0.0000) -- (3.8484,0.0000) -- (3.8181,0.0000) -- (3.7878,0.0000) -- (3.7575,0.0000) -- (3.7272,0.0000) -- (3.6969,0.0000) -- (3.6666,0.0000) -- (3.6363,0.0000) -- (3.6060,0.0000) -- (3.5757,0.0000) -- (3.5454,0.0000) -- (3.5151,0.0000) -- (3.4848,0.0000) -- (3.4545,0.0000) -- (3.4242,0.0000) -- (3.3939,0.0000) -- (3.3636,0.0000) -- (3.3333,0.0000) -- (3.3030,0.0000) -- (3.2727,0.0000) -- (3.2424,0.0000) -- (3.2121,0.0000) -- (3.1818,0.0000) -- (3.1515,0.0000) -- (3.1212,0.0000) -- (3.0909,0.0000) -- (3.0606,0.0000) -- (3.0303,0.0000) -- (3.0000,0.0000) -- (3.0000,0.0000) -- (3.0000,3.0000) --  cycle;
+\draw [color=blue] (3.0000,3.0000)--(3.0303,3.0150)--(3.0606,3.0297)--(3.0909,3.0441)--(3.1212,3.0582)--(3.1515,3.0721)--(3.1818,3.0857)--(3.2121,3.0990)--(3.2424,3.1121)--(3.2727,3.1250)--(3.3030,3.1376)--(3.3333,3.1500)--(3.3636,3.1621)--(3.3939,3.1741)--(3.4242,3.1858)--(3.4545,3.1973)--(3.4848,3.2086)--(3.5151,3.2198)--(3.5454,3.2307)--(3.5757,3.2415)--(3.6060,3.2521)--(3.6363,3.2625)--(3.6666,3.2727)--(3.6969,3.2827)--(3.7272,3.2926)--(3.7575,3.3024)--(3.7878,3.3120)--(3.8181,3.3214)--(3.8484,3.3307)--(3.8787,3.3398)--(3.9090,3.3488)--(3.9393,3.3576)--(3.9696,3.3664)--(4.0000,3.3750)--(4.0303,3.3834)--(4.0606,3.3917)--(4.0909,3.4000)--(4.1212,3.4080)--(4.1515,3.4160)--(4.1818,3.4239)--(4.2121,3.4316)--(4.2424,3.4392)--(4.2727,3.4468)--(4.3030,3.4542)--(4.3333,3.4615)--(4.3636,3.4687)--(4.3939,3.4758)--(4.4242,3.4828)--(4.4545,3.4897)--(4.4848,3.4966)--(4.5151,3.5033)--(4.5454,3.5100)--(4.5757,3.5165)--(4.6060,3.5230)--(4.6363,3.5294)--(4.6666,3.5357)--(4.6969,3.5419)--(4.7272,3.5480)--(4.7575,3.5541)--(4.7878,3.5601)--(4.8181,3.5660)--(4.8484,3.5718)--(4.8787,3.5776)--(4.9090,3.5833)--(4.9393,3.5889)--(4.9696,3.5945)--(5.0000,3.6000)--(5.0303,3.6054)--(5.0606,3.6107)--(5.0909,3.6160)--(5.1212,3.6213)--(5.1515,3.6264)--(5.1818,3.6315)--(5.2121,3.6366)--(5.2424,3.6416)--(5.2727,3.6465)--(5.3030,3.6514)--(5.3333,3.6562)--(5.3636,3.6610)--(5.3939,3.6657)--(5.4242,3.6703)--(5.4545,3.6750)--(5.4848,3.6795)--(5.5151,3.6840)--(5.5454,3.6885)--(5.5757,3.6929)--(5.6060,3.6972)--(5.6363,3.7016)--(5.6666,3.7058)--(5.6969,3.7101)--(5.7272,3.7142)--(5.7575,3.7184)--(5.7878,3.7225)--(5.8181,3.7265)--(5.8484,3.7305)--(5.8787,3.7345)--(5.9090,3.7384)--(5.9393,3.7423)--(5.9696,3.7461)--(6.0000,3.7500);
+\draw [color=blue] (3.0000,0.0000)--(3.0303,0.0000)--(3.0606,0.0000)--(3.0909,0.0000)--(3.1212,0.0000)--(3.1515,0.0000)--(3.1818,0.0000)--(3.2121,0.0000)--(3.2424,0.0000)--(3.2727,0.0000)--(3.3030,0.0000)--(3.3333,0.0000)--(3.3636,0.0000)--(3.3939,0.0000)--(3.4242,0.0000)--(3.4545,0.0000)--(3.4848,0.0000)--(3.5151,0.0000)--(3.5454,0.0000)--(3.5757,0.0000)--(3.6060,0.0000)--(3.6363,0.0000)--(3.6666,0.0000)--(3.6969,0.0000)--(3.7272,0.0000)--(3.7575,0.0000)--(3.7878,0.0000)--(3.8181,0.0000)--(3.8484,0.0000)--(3.8787,0.0000)--(3.9090,0.0000)--(3.9393,0.0000)--(3.9696,0.0000)--(4.0000,0.0000)--(4.0303,0.0000)--(4.0606,0.0000)--(4.0909,0.0000)--(4.1212,0.0000)--(4.1515,0.0000)--(4.1818,0.0000)--(4.2121,0.0000)--(4.2424,0.0000)--(4.2727,0.0000)--(4.3030,0.0000)--(4.3333,0.0000)--(4.3636,0.0000)--(4.3939,0.0000)--(4.4242,0.0000)--(4.4545,0.0000)--(4.4848,0.0000)--(4.5151,0.0000)--(4.5454,0.0000)--(4.5757,0.0000)--(4.6060,0.0000)--(4.6363,0.0000)--(4.6666,0.0000)--(4.6969,0.0000)--(4.7272,0.0000)--(4.7575,0.0000)--(4.7878,0.0000)--(4.8181,0.0000)--(4.8484,0.0000)--(4.8787,0.0000)--(4.9090,0.0000)--(4.9393,0.0000)--(4.9696,0.0000)--(5.0000,0.0000)--(5.0303,0.0000)--(5.0606,0.0000)--(5.0909,0.0000)--(5.1212,0.0000)--(5.1515,0.0000)--(5.1818,0.0000)--(5.2121,0.0000)--(5.2424,0.0000)--(5.2727,0.0000)--(5.3030,0.0000)--(5.3333,0.0000)--(5.3636,0.0000)--(5.3939,0.0000)--(5.4242,0.0000)--(5.4545,0.0000)--(5.4848,0.0000)--(5.5151,0.0000)--(5.5454,0.0000)--(5.5757,0.0000)--(5.6060,0.0000)--(5.6363,0.0000)--(5.6666,0.0000)--(5.6969,0.0000)--(5.7272,0.0000)--(5.7575,0.0000)--(5.7878,0.0000)--(5.8181,0.0000)--(5.8484,0.0000)--(5.8787,0.0000)--(5.9090,0.0000)--(5.9393,0.0000)--(5.9696,0.0000)--(6.0000,0.0000);
+\draw [style=dashed] (3.0000,0.0000) -- (3.0000,3.0000);
+\draw [style=dashed] (6.0000,3.7500) -- (6.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall b/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall
index 4c1dc4011..22ee5927c 100644
--- a/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall
+++ b/src_phystricks/Fig_TKXZooLwXzjS.pstricks.recall
@@ -65,58 +65,58 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\fill [color=cyan] (1.60,-1.20) -- (3.07,-1.20) -- (3.07,-1.20) -- (3.07,1.20) -- (3.07,1.20) -- (1.60,1.20) -- (1.60,1.20) -- (1.60,-1.20) --  cycle;
+\fill [color=cyan] (1.6000,-1.2000) -- (3.0666,-1.2000) -- (3.0666,-1.2000) -- (3.0666,1.2000) -- (3.0666,1.2000) -- (1.6000,1.2000) -- (1.6000,1.2000) -- (1.6000,-1.2000) --  cycle;
 
 
 \draw [,->,>=latex] (1.6000,-1.0000) -- (2.2250,-1.0000);
-\draw [,->,>=latex] (1.6000,-0.66667) -- (2.2250,-0.66667);
-\draw [,->,>=latex] (1.6000,-0.33333) -- (2.2250,-0.33333);
-\draw [,->,>=latex] (1.6000,0) -- (2.2250,0);
-\draw [,->,>=latex] (1.6000,0.33333) -- (2.2250,0.33333);
-\draw [,->,>=latex] (1.6000,0.66667) -- (2.2250,0.66667);
+\draw [,->,>=latex] (1.6000,-0.6666) -- (2.2250,-0.6666);
+\draw [,->,>=latex] (1.6000,-0.3333) -- (2.2250,-0.3333);
+\draw [,->,>=latex] (1.6000,0.0000) -- (2.2250,0.0000);
+\draw [,->,>=latex] (1.6000,0.3333) -- (2.2250,0.3333);
+\draw [,->,>=latex] (1.6000,0.6666) -- (2.2250,0.6666);
 \draw [,->,>=latex] (1.6000,1.0000) -- (2.2250,1.0000);
 \draw [,->,>=latex] (2.3333,-1.0000) -- (2.7619,-1.0000);
-\draw [,->,>=latex] (2.3333,-0.66667) -- (2.7619,-0.66667);
-\draw [,->,>=latex] (2.3333,-0.33333) -- (2.7619,-0.33333);
-\draw [,->,>=latex] (2.3333,0) -- (2.7619,0);
-\draw [,->,>=latex] (2.3333,0.33333) -- (2.7619,0.33333);
-\draw [,->,>=latex] (2.3333,0.66667) -- (2.7619,0.66667);
+\draw [,->,>=latex] (2.3333,-0.6666) -- (2.7619,-0.6666);
+\draw [,->,>=latex] (2.3333,-0.3333) -- (2.7619,-0.3333);
+\draw [,->,>=latex] (2.3333,0.0000) -- (2.7619,0.0000);
+\draw [,->,>=latex] (2.3333,0.3333) -- (2.7619,0.3333);
+\draw [,->,>=latex] (2.3333,0.6666) -- (2.7619,0.6666);
 \draw [,->,>=latex] (2.3333,1.0000) -- (2.7619,1.0000);
-\draw [,->,>=latex] (3.0667,-1.0000) -- (3.3928,-1.0000);
-\draw [,->,>=latex] (3.0667,-0.66667) -- (3.3928,-0.66667);
-\draw [,->,>=latex] (3.0667,-0.33333) -- (3.3928,-0.33333);
-\draw [,->,>=latex] (3.0667,0) -- (3.3928,0);
-\draw [,->,>=latex] (3.0667,0.33333) -- (3.3928,0.33333);
-\draw [,->,>=latex] (3.0667,0.66667) -- (3.3928,0.66667);
-\draw [,->,>=latex] (3.0667,1.0000) -- (3.3928,1.0000);
-\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0632,-1.0000);
-\draw [,->,>=latex] (3.8000,-0.66667) -- (4.0632,-0.66667);
-\draw [,->,>=latex] (3.8000,-0.33333) -- (4.0632,-0.33333);
-\draw [,->,>=latex] (3.8000,0) -- (4.0632,0);
-\draw [,->,>=latex] (3.8000,0.33333) -- (4.0632,0.33333);
-\draw [,->,>=latex] (3.8000,0.66667) -- (4.0632,0.66667);
-\draw [,->,>=latex] (3.8000,1.0000) -- (4.0632,1.0000);
+\draw [,->,>=latex] (3.0666,-1.0000) -- (3.3927,-1.0000);
+\draw [,->,>=latex] (3.0666,-0.6666) -- (3.3927,-0.6666);
+\draw [,->,>=latex] (3.0666,-0.3333) -- (3.3927,-0.3333);
+\draw [,->,>=latex] (3.0666,0.0000) -- (3.3927,0.0000);
+\draw [,->,>=latex] (3.0666,0.3333) -- (3.3927,0.3333);
+\draw [,->,>=latex] (3.0666,0.6666) -- (3.3927,0.6666);
+\draw [,->,>=latex] (3.0666,1.0000) -- (3.3927,1.0000);
+\draw [,->,>=latex] (3.8000,-1.0000) -- (4.0631,-1.0000);
+\draw [,->,>=latex] (3.8000,-0.6666) -- (4.0631,-0.6666);
+\draw [,->,>=latex] (3.8000,-0.3333) -- (4.0631,-0.3333);
+\draw [,->,>=latex] (3.8000,0.0000) -- (4.0631,0.0000);
+\draw [,->,>=latex] (3.8000,0.3333) -- (4.0631,0.3333);
+\draw [,->,>=latex] (3.8000,0.6666) -- (4.0631,0.6666);
+\draw [,->,>=latex] (3.8000,1.0000) -- (4.0631,1.0000);
 \draw [,->,>=latex] (4.5333,-1.0000) -- (4.7539,-1.0000);
-\draw [,->,>=latex] (4.5333,-0.66667) -- (4.7539,-0.66667);
-\draw [,->,>=latex] (4.5333,-0.33333) -- (4.7539,-0.33333);
-\draw [,->,>=latex] (4.5333,0) -- (4.7539,0);
-\draw [,->,>=latex] (4.5333,0.33333) -- (4.7539,0.33333);
-\draw [,->,>=latex] (4.5333,0.66667) -- (4.7539,0.66667);
+\draw [,->,>=latex] (4.5333,-0.6666) -- (4.7539,-0.6666);
+\draw [,->,>=latex] (4.5333,-0.3333) -- (4.7539,-0.3333);
+\draw [,->,>=latex] (4.5333,0.0000) -- (4.7539,0.0000);
+\draw [,->,>=latex] (4.5333,0.3333) -- (4.7539,0.3333);
+\draw [,->,>=latex] (4.5333,0.6666) -- (4.7539,0.6666);
 \draw [,->,>=latex] (4.5333,1.0000) -- (4.7539,1.0000);
-\draw [,->,>=latex] (5.2667,-1.0000) -- (5.4565,-1.0000);
-\draw [,->,>=latex] (5.2667,-0.66667) -- (5.4565,-0.66667);
-\draw [,->,>=latex] (5.2667,-0.33333) -- (5.4565,-0.33333);
-\draw [,->,>=latex] (5.2667,0) -- (5.4565,0);
-\draw [,->,>=latex] (5.2667,0.33333) -- (5.4565,0.33333);
-\draw [,->,>=latex] (5.2667,0.66667) -- (5.4565,0.66667);
-\draw [,->,>=latex] (5.2667,1.0000) -- (5.4565,1.0000);
-\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1667,-1.0000);
-\draw [,->,>=latex] (6.0000,-0.66667) -- (6.1667,-0.66667);
-\draw [,->,>=latex] (6.0000,-0.33333) -- (6.1667,-0.33333);
-\draw [,->,>=latex] (6.0000,0) -- (6.1667,0);
-\draw [,->,>=latex] (6.0000,0.33333) -- (6.1667,0.33333);
-\draw [,->,>=latex] (6.0000,0.66667) -- (6.1667,0.66667);
-\draw [,->,>=latex] (6.0000,1.0000) -- (6.1667,1.0000);
+\draw [,->,>=latex] (5.2666,-1.0000) -- (5.4565,-1.0000);
+\draw [,->,>=latex] (5.2666,-0.6666) -- (5.4565,-0.6666);
+\draw [,->,>=latex] (5.2666,-0.3333) -- (5.4565,-0.3333);
+\draw [,->,>=latex] (5.2666,0.0000) -- (5.4565,0.0000);
+\draw [,->,>=latex] (5.2666,0.3333) -- (5.4565,0.3333);
+\draw [,->,>=latex] (5.2666,0.6666) -- (5.4565,0.6666);
+\draw [,->,>=latex] (5.2666,1.0000) -- (5.4565,1.0000);
+\draw [,->,>=latex] (6.0000,-1.0000) -- (6.1666,-1.0000);
+\draw [,->,>=latex] (6.0000,-0.6666) -- (6.1666,-0.6666);
+\draw [,->,>=latex] (6.0000,-0.3333) -- (6.1666,-0.3333);
+\draw [,->,>=latex] (6.0000,0.0000) -- (6.1666,0.0000);
+\draw [,->,>=latex] (6.0000,0.3333) -- (6.1666,0.3333);
+\draw [,->,>=latex] (6.0000,0.6666) -- (6.1666,0.6666);
+\draw [,->,>=latex] (6.0000,1.0000) -- (6.1666,1.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall b/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
index 66e9b1259..f15627b9e 100644
--- a/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
+++ b/src_phystricks/Fig_TWHooJjXEtS.pstricks.recall
@@ -108,35 +108,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-8.3540,0) -- (8.3540,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-8.3539,0.0000) -- (8.3539,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [color=blue] (-7.854,-1.000)--(-7.695,-0.9874)--(-7.537,-0.9501)--(-7.378,-0.8888)--(-7.219,-0.8053)--(-7.061,-0.7015)--(-6.902,-0.5801)--(-6.743,-0.4441)--(-6.585,-0.2969)--(-6.426,-0.1423)--(-6.267,0.01587)--(-6.109,0.1736)--(-5.950,0.3271)--(-5.791,0.4723)--(-5.633,0.6056)--(-5.474,0.7237)--(-5.315,0.8237)--(-5.157,0.9029)--(-4.998,0.9595)--(-4.839,0.9920)--(-4.681,0.9995)--(-4.522,0.9819)--(-4.363,0.9397)--(-4.205,0.8738)--(-4.046,0.7861)--(-3.887,0.6785)--(-3.729,0.5539)--(-3.570,0.4154)--(-3.411,0.2665)--(-3.253,0.1108)--(-3.094,-0.04758)--(-2.935,-0.2048)--(-2.777,-0.3569)--(-2.618,-0.5000)--(-2.459,-0.6306)--(-2.301,-0.7453)--(-2.142,-0.8413)--(-1.983,-0.9161)--(-1.825,-0.9679)--(-1.666,-0.9955)--(-1.507,-0.9980)--(-1.349,-0.9754)--(-1.190,-0.9284)--(-1.031,-0.8580)--(-0.8727,-0.7660)--(-0.7140,-0.6549)--(-0.5553,-0.5272)--(-0.3967,-0.3863)--(-0.2380,-0.2358)--(-0.07933,-0.07925)--(0.07933,0.07925)--(0.2380,0.2358)--(0.3967,0.3863)--(0.5553,0.5272)--(0.7140,0.6549)--(0.8727,0.7660)--(1.031,0.8580)--(1.190,0.9284)--(1.349,0.9754)--(1.507,0.9980)--(1.666,0.9955)--(1.825,0.9679)--(1.983,0.9161)--(2.142,0.8413)--(2.301,0.7453)--(2.459,0.6306)--(2.618,0.5000)--(2.777,0.3569)--(2.935,0.2048)--(3.094,0.04758)--(3.253,-0.1108)--(3.411,-0.2665)--(3.570,-0.4154)--(3.729,-0.5539)--(3.887,-0.6785)--(4.046,-0.7861)--(4.205,-0.8738)--(4.363,-0.9397)--(4.522,-0.9819)--(4.681,-0.9995)--(4.839,-0.9920)--(4.998,-0.9595)--(5.157,-0.9029)--(5.315,-0.8237)--(5.474,-0.7237)--(5.633,-0.6056)--(5.791,-0.4723)--(5.950,-0.3271)--(6.109,-0.1736)--(6.267,-0.01587)--(6.426,0.1423)--(6.585,0.2969)--(6.743,0.4441)--(6.902,0.5801)--(7.061,0.7015)--(7.219,0.8053)--(7.378,0.8888)--(7.537,0.9501)--(7.695,0.9874)--(7.854,1.000);
-\draw (-7.8540,-0.42071) node {$ -\frac{5}{2} \, \pi $};
-\draw [] (-7.85,-0.100) -- (-7.85,0.100);
-\draw (-6.2832,-0.32983) node {$ -2 \, \pi $};
-\draw [] (-6.28,-0.100) -- (-6.28,0.100);
-\draw (-4.7124,-0.42071) node {$ -\frac{3}{2} \, \pi $};
-\draw [] (-4.71,-0.100) -- (-4.71,0.100);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (-1.5708,-0.42071) node {$ -\frac{1}{2} \, \pi $};
-\draw [] (-1.57,-0.100) -- (-1.57,0.100);
-\draw (1.5708,-0.42071) node {$ \frac{1}{2} \, \pi $};
-\draw [] (1.57,-0.100) -- (1.57,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (4.7124,-0.42071) node {$ \frac{3}{2} \, \pi $};
-\draw [] (4.71,-0.100) -- (4.71,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (7.8540,-0.42071) node {$ \frac{5}{2} \, \pi $};
-\draw [] (7.85,-0.100) -- (7.85,0.100);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
+\draw [color=blue] (-7.8539,-1.0000)--(-7.6953,-0.9874)--(-7.5366,-0.9500)--(-7.3779,-0.8888)--(-7.2193,-0.8052)--(-7.0606,-0.7014)--(-6.9019,-0.5800)--(-6.7433,-0.4440)--(-6.5846,-0.2969)--(-6.4259,-0.1423)--(-6.2673,0.0158)--(-6.1086,0.1736)--(-5.9499,0.3270)--(-5.7913,0.4722)--(-5.6326,0.6056)--(-5.4739,0.7237)--(-5.3153,0.8236)--(-5.1566,0.9029)--(-4.9979,0.9594)--(-4.8393,0.9919)--(-4.6806,0.9994)--(-4.5219,0.9819)--(-4.3633,0.9396)--(-4.2046,0.8738)--(-4.0459,0.7860)--(-3.8873,0.6785)--(-3.7286,0.5539)--(-3.5699,0.4154)--(-3.4113,0.2664)--(-3.2526,0.1108)--(-3.0939,-0.0475)--(-2.9353,-0.2048)--(-2.7766,-0.3568)--(-2.6179,-0.5000)--(-2.4593,-0.6305)--(-2.3006,-0.7452)--(-2.1419,-0.8412)--(-1.9833,-0.9161)--(-1.8246,-0.9679)--(-1.6659,-0.9954)--(-1.5073,-0.9979)--(-1.3486,-0.9754)--(-1.1899,-0.9283)--(-1.0313,-0.8579)--(-0.8726,-0.7660)--(-0.7139,-0.6548)--(-0.5553,-0.5272)--(-0.3966,-0.3863)--(-0.2379,-0.2357)--(-0.0793,-0.0792)--(0.0793,0.0792)--(0.2379,0.2357)--(0.3966,0.3863)--(0.5553,0.5272)--(0.7139,0.6548)--(0.8726,0.7660)--(1.0313,0.8579)--(1.1899,0.9283)--(1.3486,0.9754)--(1.5073,0.9979)--(1.6659,0.9954)--(1.8246,0.9679)--(1.9833,0.9161)--(2.1419,0.8412)--(2.3006,0.7452)--(2.4593,0.6305)--(2.6179,0.5000)--(2.7766,0.3568)--(2.9353,0.2048)--(3.0939,0.0475)--(3.2526,-0.1108)--(3.4113,-0.2664)--(3.5699,-0.4154)--(3.7286,-0.5539)--(3.8873,-0.6785)--(4.0459,-0.7860)--(4.2046,-0.8738)--(4.3633,-0.9396)--(4.5219,-0.9819)--(4.6806,-0.9994)--(4.8393,-0.9919)--(4.9979,-0.9594)--(5.1566,-0.9029)--(5.3153,-0.8236)--(5.4739,-0.7237)--(5.6326,-0.6056)--(5.7913,-0.4722)--(5.9499,-0.3270)--(6.1086,-0.1736)--(6.2673,-0.0158)--(6.4259,0.1423)--(6.5846,0.2969)--(6.7433,0.4440)--(6.9019,0.5800)--(7.0606,0.7014)--(7.2193,0.8052)--(7.3779,0.8888)--(7.5366,0.9500)--(7.6953,0.9874)--(7.8539,1.0000);
+\draw (-7.8539,-0.4207) node {$ -\frac{5}{2} \, \pi $};
+\draw [] (-7.8539,-0.1000) -- (-7.8539,0.1000);
+\draw (-6.2831,-0.3298) node {$ -2 \, \pi $};
+\draw [] (-6.2831,-0.1000) -- (-6.2831,0.1000);
+\draw (-4.7123,-0.4207) node {$ -\frac{3}{2} \, \pi $};
+\draw [] (-4.7123,-0.1000) -- (-4.7123,0.1000);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (-1.5707,-0.4207) node {$ -\frac{1}{2} \, \pi $};
+\draw [] (-1.5707,-0.1000) -- (-1.5707,0.1000);
+\draw (1.5707,-0.4207) node {$ \frac{1}{2} \, \pi $};
+\draw [] (1.5707,-0.1000) -- (1.5707,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (4.7123,-0.4207) node {$ \frac{3}{2} \, \pi $};
+\draw [] (4.7123,-0.1000) -- (4.7123,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (7.8539,-0.4207) node {$ \frac{5}{2} \, \pi $};
+\draw [] (7.8539,-0.1000) -- (7.8539,0.1000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_TZCISko.pstricks.recall b/src_phystricks/Fig_TZCISko.pstricks.recall
index 068402609..fcf282bc4 100644
--- a/src_phystricks/Fig_TZCISko.pstricks.recall
+++ b/src_phystricks/Fig_TZCISko.pstricks.recall
@@ -71,20 +71,20 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (10.500,0);
-\draw [,->,>=latex] (0,-2.5000) -- (0,2.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (10.500,0.0000);
+\draw [,->,>=latex] (0.0000,-2.4999) -- (0.0000,2.4999);
 %DEFAULT
 
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+\draw [color=blue] 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+\draw [color=blue] 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+\draw [color=blue] 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+\draw (10.000,-0.3149) node {$ 1 $};
+\draw [] (10.000,-0.1000) -- (10.000,0.1000);
+\draw (-0.4331,-2.0000) node {$ -1 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.2912,2.0000) node {$ 1 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_TangentSegment.pstricks.recall b/src_phystricks/Fig_TangentSegment.pstricks.recall
index 7ff8d9539..d4da27183 100644
--- a/src_phystricks/Fig_TangentSegment.pstricks.recall
+++ b/src_phystricks/Fig_TangentSegment.pstricks.recall
@@ -103,45 +103,45 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.7038,0) -- (7.7832,0);
-\draw [,->,>=latex] (0,-3.0689) -- (0,2.5000);
+\draw [,->,>=latex] (-3.7037,0.0000) -- (7.7831,0.0000);
+\draw [,->,>=latex] (0.0000,-3.0689) -- (0.0000,2.5000);
 %DEFAULT
 
-\draw [color=blue] (-3.142,-2.000)--(-3.036,-1.997)--(-2.931,-1.989)--(-2.826,-1.975)--(-2.720,-1.956)--(-2.615,-1.931)--(-2.510,-1.901)--(-2.404,-1.866)--(-2.299,-1.825)--(-2.194,-1.780)--(-2.089,-1.729)--(-1.983,-1.674)--(-1.878,-1.614)--(-1.773,-1.550)--(-1.667,-1.481)--(-1.562,-1.408)--(-1.457,-1.331)--(-1.351,-1.251)--(-1.246,-1.167)--(-1.141,-1.080)--(-1.036,-0.9899)--(-0.9303,-0.8971)--(-0.8250,-0.8018)--(-0.7197,-0.7042)--(-0.6144,-0.6048)--(-0.5091,-0.5036)--(-0.4038,-0.4010)--(-0.2985,-0.2974)--(-0.1932,-0.1929)--(-0.08787,-0.08784)--(0.01743,0.01743)--(0.1227,0.1227)--(0.2280,0.2275)--(0.3333,0.3318)--(0.4386,0.4351)--(0.5439,0.5373)--(0.6492,0.6379)--(0.7545,0.7368)--(0.8598,0.8336)--(0.9651,0.9281)--(1.070,1.020)--(1.176,1.109)--(1.281,1.195)--(1.386,1.278)--(1.492,1.357)--(1.597,1.433)--(1.702,1.504)--(1.808,1.571)--(1.913,1.634)--(2.018,1.693)--(2.123,1.746)--(2.229,1.795)--(2.334,1.839)--(2.439,1.878)--(2.545,1.912)--(2.650,1.940)--(2.755,1.963)--(2.861,1.980)--(2.966,1.992)--(3.071,1.999)--(3.176,2.000)--(3.282,1.995)--(3.387,1.985)--(3.492,1.969)--(3.598,1.948)--(3.703,1.922)--(3.808,1.890)--(3.914,1.853)--(4.019,1.811)--(4.124,1.763)--(4.229,1.711)--(4.335,1.655)--(4.440,1.593)--(4.545,1.527)--(4.651,1.457)--(4.756,1.383)--(4.861,1.305)--(4.967,1.224)--(5.072,1.139)--(5.177,1.050)--(5.282,0.9595)--(5.388,0.8658)--(5.493,0.7697)--(5.598,0.6715)--(5.704,0.5714)--(5.809,0.4698)--(5.914,0.3668)--(6.020,0.2628)--(6.125,0.1581)--(6.230,0.05300)--(6.335,-0.05229)--(6.441,-0.1574)--(6.546,-0.2621)--(6.651,-0.3661)--(6.757,-0.4691)--(6.862,-0.5708)--(6.967,-0.6708)--(7.073,-0.7691)--(7.178,-0.8652)--(7.283,-0.9589);
-\draw [] (-3.20,-2.57) -- (0.0622,-0.259);
-\draw []  (-1.5708,-1.4142) node [rotate=0] {$\bullet$};
-\draw [] (1.14,2.00) -- (5.14,2.00);
-\draw []  (3.1416,2.0000) node [rotate=0] {$\bullet$};
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (5.0000,-0.31492) node {$ 5 $};
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw (6.0000,-0.31492) node {$ 6 $};
-\draw [] (6.00,-0.100) -- (6.00,0.100);
-\draw (7.0000,-0.31492) node {$ 7 $};
-\draw [] (7.00,-0.100) -- (7.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
+\draw [color=blue] (-3.1415,-2.0000)--(-3.0362,-1.9972)--(-2.9309,-1.9889)--(-2.8256,-1.9751)--(-2.7203,-1.9558)--(-2.6150,-1.9310)--(-2.5097,-1.9010)--(-2.4044,-1.8656)--(-2.2991,-1.8251)--(-2.1938,-1.7796)--(-2.0885,-1.7291)--(-1.9832,-1.6738)--(-1.8779,-1.6139)--(-1.7726,-1.5495)--(-1.6673,-1.4808)--(-1.5620,-1.4080)--(-1.4567,-1.3313)--(-1.3514,-1.2509)--(-1.2461,-1.1670)--(-1.1408,-1.0800)--(-1.0355,-0.9899)--(-0.9302,-0.8970)--(-0.8249,-0.8017)--(-0.7196,-0.7042)--(-0.6143,-0.6047)--(-0.5090,-0.5035)--(-0.4037,-0.4010)--(-0.2984,-0.2973)--(-0.1931,-0.1928)--(-0.0878,-0.0878)--(0.0174,0.0174)--(0.1227,0.1226)--(0.2280,0.2275)--(0.3333,0.3317)--(0.4386,0.4351)--(0.5439,0.5372)--(0.6492,0.6378)--(0.7545,0.7367)--(0.8598,0.8335)--(0.9651,0.9281)--(1.0704,1.0200)--(1.1757,1.1091)--(1.2810,1.1952)--(1.3863,1.2779)--(1.4916,1.3571)--(1.5969,1.4325)--(1.7022,1.5040)--(1.8075,1.5713)--(1.9128,1.6342)--(2.0181,1.6926)--(2.1234,1.7463)--(2.2287,1.7952)--(2.3340,1.8391)--(2.4393,1.8779)--(2.5446,1.9115)--(2.6499,1.9398)--(2.7552,1.9628)--(2.8605,1.9802)--(2.9658,1.9922)--(3.0711,1.9987)--(3.1764,1.9996)--(3.2817,1.9950)--(3.3870,1.9849)--(3.4923,1.9693)--(3.5976,1.9482)--(3.7029,1.9217)--(3.8082,1.8899)--(3.9135,1.8528)--(4.0188,1.8106)--(4.1241,1.7634)--(4.2294,1.7113)--(4.3347,1.6545)--(4.4400,1.5930)--(4.5453,1.5272)--(4.6506,1.4571)--(4.7559,1.3830)--(4.8612,1.3051)--(4.9665,1.2235)--(5.0718,1.1386)--(5.1771,1.0504)--(5.2824,0.9594)--(5.3877,0.8657)--(5.4930,0.7697)--(5.5983,0.6715)--(5.7036,0.5714)--(5.8089,0.4697)--(5.9142,0.3668)--(6.0195,0.2628)--(6.1248,0.1581)--(6.2301,0.0530)--(6.3354,-0.0522)--(6.4407,-0.1574)--(6.5460,-0.2621)--(6.6513,-0.3661)--(6.7566,-0.4690)--(6.8619,-0.5707)--(6.9672,-0.6708)--(7.0725,-0.7690)--(7.1778,-0.8651)--(7.2831,-0.9588);
+\draw [] (-3.2037,-2.5689) -- (0.0621,-0.2595);
+\draw []  (-1.5707,-1.4142) node [rotate=0] {$\bullet$};
+\draw [] (1.1415,2.0000) -- (5.1415,2.0000);
+\draw []  (3.1415,2.0000) node [rotate=0] {$\bullet$};
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (5.0000,-0.3149) node {$ 5 $};
+\draw [] (5.0000,-0.1000) -- (5.0000,0.1000);
+\draw (6.0000,-0.3149) node {$ 6 $};
+\draw [] (6.0000,-0.1000) -- (6.0000,0.1000);
+\draw (7.0000,-0.3149) node {$ 7 $};
+\draw [] (7.0000,-0.1000) -- (7.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_TangenteDetail.pstricks.recall b/src_phystricks/Fig_TangenteDetail.pstricks.recall
index bfa327892..cbca04407 100644
--- a/src_phystricks/Fig_TangenteDetail.pstricks.recall
+++ b/src_phystricks/Fig_TangenteDetail.pstricks.recall
@@ -111,35 +111,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.1051);
 %DEFAULT
-\draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
-\draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
-\draw [color=green,style=dashed] (1.70,3.24) -- (1.70,0);
-\draw [color=green,style=dashed] (4.00,4.25) -- (0,4.25);
-\draw [color=green,style=dashed] (1.70,3.24) -- (0,3.24);
-\draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
+\draw [color=cyan] (0.8950,2.8801) -- (4.8050,4.6051);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (4.0000,0.0000);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (1.7000,0.0000);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (0.0000,4.2500);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (0.0000,3.2352);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (4.0000,3.2352);
 
-\draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw [color=blue] (0.7000,0.7142)--(0.7434,0.9646)--(0.7868,1.1874)--(0.8303,1.3868)--(0.8737,1.5664)--(0.9171,1.7290)--(0.9606,1.8769)--(1.0040,2.0120)--(1.0474,2.1359)--(1.0909,2.2500)--(1.1343,2.3552)--(1.1777,2.4528)--(1.2212,2.5434)--(1.2646,2.6277)--(1.3080,2.7065)--(1.3515,2.7802)--(1.3949,2.8493)--(1.4383,2.9143)--(1.4818,2.9754)--(1.5252,3.0331)--(1.5686,3.0875)--(1.6121,3.1390)--(1.6555,3.1879)--(1.6989,3.2342)--(1.7424,3.2782)--(1.7858,3.3201)--(1.8292,3.3600)--(1.8727,3.3980)--(1.9161,3.4343)--(1.9595,3.4690)--(2.0030,3.5022)--(2.0464,3.5340)--(2.0898,3.5645)--(2.1333,3.5937)--(2.1767,3.6218)--(2.2202,3.6487)--(2.2636,3.6746)--(2.3070,3.6996)--(2.3505,3.7236)--(2.3939,3.7468)--(2.4373,3.7691)--(2.4808,3.7907)--(2.5242,3.8115)--(2.5676,3.8316)--(2.6111,3.8510)--(2.6545,3.8698)--(2.6979,3.8880)--(2.7414,3.9056)--(2.7848,3.9227)--(2.8282,3.9392)--(2.8717,3.9553)--(2.9151,3.9708)--(2.9585,3.9860)--(3.0020,4.0006)--(3.0454,4.0149)--(3.0888,4.0287)--(3.1323,4.0422)--(3.1757,4.0553)--(3.2191,4.0680)--(3.2626,4.0804)--(3.3060,4.0925)--(3.3494,4.1043)--(3.3929,4.1158)--(3.4363,4.1269)--(3.4797,4.1378)--(3.5232,4.1485)--(3.5666,4.1588)--(3.6101,4.1689)--(3.6535,4.1788)--(3.6969,4.1885)--(3.7404,4.1979)--(3.7838,4.2071)--(3.8272,4.2161)--(3.8707,4.2249)--(3.9141,4.2335)--(3.9575,4.2419)--(4.0010,4.2501)--(4.0444,4.2582)--(4.0878,4.2661)--(4.1313,4.2738)--(4.1747,4.2813)--(4.2181,4.2887)--(4.2616,4.2960)--(4.3050,4.3031)--(4.3484,4.3101)--(4.3919,4.3169)--(4.4353,4.3236)--(4.4787,4.3301)--(4.5222,4.3366)--(4.5656,4.3429)--(4.6090,4.3491)--(4.6525,4.3551)--(4.6959,4.3611)--(4.7393,4.3670)--(4.7828,4.3727)--(4.8262,4.3784)--(4.8696,4.3839)--(4.9131,4.3893)--(4.9565,4.3947)--(5.0000,4.4000);
+\draw []  (1.7000,3.2352) node [rotate=0] {$\bullet$};
 \draw (1.3414,3.5681) node {$P$};
-\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
-\draw (1.7000,-0.27858) node {$a$};
-\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
-\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (1.7000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.2785) node {$a$};
+\draw []  (0.0000,3.2352) node [rotate=0] {$\bullet$};
+\draw (-0.4473,3.2352) node {$f(a)$};
 \draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
-\draw (3.8004,4.7051) node {$Q$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.27858) node {$x$};
-\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
-\draw (-0.45521,4.2500) node {$f(x)$};
-\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
-\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
-\draw (2.8500,2.7143) node {$x-a$};
+\draw (3.8004,4.7050) node {$Q$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.2785) node {$x$};
+\draw []  (0.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.4552,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0352) -- (1.7000,3.0352);
+\draw [,->,>=latex] (2.8500,3.0352) -- (4.0000,3.0352);
+\draw (2.8500,2.7142) node {$x-a$};
 \draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
-\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
-\draw (5.3256,3.7426) node {$f(x)-f(a)$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2352);
+\draw (5.3255,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TangenteDetailOM.pstricks.recall b/src_phystricks/Fig_TangenteDetailOM.pstricks.recall
index 4771ca36f..53f08374e 100644
--- a/src_phystricks/Fig_TangenteDetailOM.pstricks.recall
+++ b/src_phystricks/Fig_TangenteDetailOM.pstricks.recall
@@ -111,35 +111,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.1051);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.1051);
 %DEFAULT
-\draw [color=cyan] (0.895,2.88) -- (4.81,4.60);
-\draw [color=green,style=dashed] (4.00,4.25) -- (4.00,0);
-\draw [color=green,style=dashed] (1.70,3.24) -- (1.70,0);
-\draw [color=green,style=dashed] (4.00,4.25) -- (0,4.25);
-\draw [color=green,style=dashed] (1.70,3.24) -- (0,3.24);
-\draw [color=green,style=dashed] (1.70,3.24) -- (4.00,3.24);
+\draw [color=cyan] (0.8950,2.8801) -- (4.8050,4.6051);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (4.0000,0.0000);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (1.7000,0.0000);
+\draw [color=green,style=dashed] (4.0000,4.2500) -- (0.0000,4.2500);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (0.0000,3.2352);
+\draw [color=green,style=dashed] (1.7000,3.2352) -- (4.0000,3.2352);
 
-\draw [color=blue] (0.7000,0.7143)--(0.7434,0.9647)--(0.7869,1.187)--(0.8303,1.387)--(0.8737,1.566)--(0.9172,1.729)--(0.9606,1.877)--(1.004,2.012)--(1.047,2.136)--(1.091,2.250)--(1.134,2.355)--(1.178,2.453)--(1.221,2.543)--(1.265,2.628)--(1.308,2.707)--(1.352,2.780)--(1.395,2.849)--(1.438,2.914)--(1.482,2.975)--(1.525,3.033)--(1.569,3.088)--(1.612,3.139)--(1.656,3.188)--(1.699,3.234)--(1.742,3.278)--(1.786,3.320)--(1.829,3.360)--(1.873,3.398)--(1.916,3.434)--(1.960,3.469)--(2.003,3.502)--(2.046,3.534)--(2.090,3.565)--(2.133,3.594)--(2.177,3.622)--(2.220,3.649)--(2.264,3.675)--(2.307,3.700)--(2.350,3.724)--(2.394,3.747)--(2.437,3.769)--(2.481,3.791)--(2.524,3.812)--(2.568,3.832)--(2.611,3.851)--(2.655,3.870)--(2.698,3.888)--(2.741,3.906)--(2.785,3.923)--(2.828,3.939)--(2.872,3.955)--(2.915,3.971)--(2.959,3.986)--(3.002,4.001)--(3.045,4.015)--(3.089,4.029)--(3.132,4.042)--(3.176,4.055)--(3.219,4.068)--(3.263,4.081)--(3.306,4.093)--(3.349,4.104)--(3.393,4.116)--(3.436,4.127)--(3.480,4.138)--(3.523,4.148)--(3.567,4.159)--(3.610,4.169)--(3.654,4.179)--(3.697,4.189)--(3.740,4.198)--(3.784,4.207)--(3.827,4.216)--(3.871,4.225)--(3.914,4.234)--(3.958,4.242)--(4.001,4.250)--(4.044,4.258)--(4.088,4.266)--(4.131,4.274)--(4.175,4.281)--(4.218,4.289)--(4.262,4.296)--(4.305,4.303)--(4.349,4.310)--(4.392,4.317)--(4.435,4.324)--(4.479,4.330)--(4.522,4.337)--(4.566,4.343)--(4.609,4.349)--(4.653,4.355)--(4.696,4.361)--(4.739,4.367)--(4.783,4.373)--(4.826,4.378)--(4.870,4.384)--(4.913,4.389)--(4.957,4.395)--(5.000,4.400);
-\draw []  (1.7000,3.2353) node [rotate=0] {$\bullet$};
+\draw [color=blue] (0.7000,0.7142)--(0.7434,0.9646)--(0.7868,1.1874)--(0.8303,1.3868)--(0.8737,1.5664)--(0.9171,1.7290)--(0.9606,1.8769)--(1.0040,2.0120)--(1.0474,2.1359)--(1.0909,2.2500)--(1.1343,2.3552)--(1.1777,2.4528)--(1.2212,2.5434)--(1.2646,2.6277)--(1.3080,2.7065)--(1.3515,2.7802)--(1.3949,2.8493)--(1.4383,2.9143)--(1.4818,2.9754)--(1.5252,3.0331)--(1.5686,3.0875)--(1.6121,3.1390)--(1.6555,3.1879)--(1.6989,3.2342)--(1.7424,3.2782)--(1.7858,3.3201)--(1.8292,3.3600)--(1.8727,3.3980)--(1.9161,3.4343)--(1.9595,3.4690)--(2.0030,3.5022)--(2.0464,3.5340)--(2.0898,3.5645)--(2.1333,3.5937)--(2.1767,3.6218)--(2.2202,3.6487)--(2.2636,3.6746)--(2.3070,3.6996)--(2.3505,3.7236)--(2.3939,3.7468)--(2.4373,3.7691)--(2.4808,3.7907)--(2.5242,3.8115)--(2.5676,3.8316)--(2.6111,3.8510)--(2.6545,3.8698)--(2.6979,3.8880)--(2.7414,3.9056)--(2.7848,3.9227)--(2.8282,3.9392)--(2.8717,3.9553)--(2.9151,3.9708)--(2.9585,3.9860)--(3.0020,4.0006)--(3.0454,4.0149)--(3.0888,4.0287)--(3.1323,4.0422)--(3.1757,4.0553)--(3.2191,4.0680)--(3.2626,4.0804)--(3.3060,4.0925)--(3.3494,4.1043)--(3.3929,4.1158)--(3.4363,4.1269)--(3.4797,4.1378)--(3.5232,4.1485)--(3.5666,4.1588)--(3.6101,4.1689)--(3.6535,4.1788)--(3.6969,4.1885)--(3.7404,4.1979)--(3.7838,4.2071)--(3.8272,4.2161)--(3.8707,4.2249)--(3.9141,4.2335)--(3.9575,4.2419)--(4.0010,4.2501)--(4.0444,4.2582)--(4.0878,4.2661)--(4.1313,4.2738)--(4.1747,4.2813)--(4.2181,4.2887)--(4.2616,4.2960)--(4.3050,4.3031)--(4.3484,4.3101)--(4.3919,4.3169)--(4.4353,4.3236)--(4.4787,4.3301)--(4.5222,4.3366)--(4.5656,4.3429)--(4.6090,4.3491)--(4.6525,4.3551)--(4.6959,4.3611)--(4.7393,4.3670)--(4.7828,4.3727)--(4.8262,4.3784)--(4.8696,4.3839)--(4.9131,4.3893)--(4.9565,4.3947)--(5.0000,4.4000);
+\draw []  (1.7000,3.2352) node [rotate=0] {$\bullet$};
 \draw (1.3414,3.5681) node {$P$};
-\draw []  (1.7000,0) node [rotate=0] {$\bullet$};
-\draw (1.7000,-0.27858) node {$a$};
-\draw []  (0,3.2353) node [rotate=0] {$\bullet$};
-\draw (-0.44737,3.2353) node {$f(a)$};
+\draw []  (1.7000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.7000,-0.2785) node {$a$};
+\draw []  (0.0000,3.2352) node [rotate=0] {$\bullet$};
+\draw (-0.4473,3.2352) node {$f(a)$};
 \draw []  (4.0000,4.2500) node [rotate=0] {$\bullet$};
-\draw (3.8004,4.7051) node {$Q$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.0000,-0.27858) node {$x$};
-\draw []  (0,4.2500) node [rotate=0] {$\bullet$};
-\draw (-0.45521,4.2500) node {$f(x)$};
-\draw [,->,>=latex] (2.8500,3.0353) -- (1.7000,3.0353);
-\draw [,->,>=latex] (2.8500,3.0353) -- (4.0000,3.0353);
-\draw (2.8500,2.7143) node {$x-a$};
+\draw (3.8004,4.7050) node {$Q$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.0000,-0.2785) node {$x$};
+\draw []  (0.0000,4.2500) node [rotate=0] {$\bullet$};
+\draw (-0.4552,4.2500) node {$f(x)$};
+\draw [,->,>=latex] (2.8500,3.0352) -- (1.7000,3.0352);
+\draw [,->,>=latex] (2.8500,3.0352) -- (4.0000,3.0352);
+\draw (2.8500,2.7142) node {$x-a$};
 \draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,4.2500);
-\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2353);
-\draw (5.3256,3.7426) node {$f(x)-f(a)$};
+\draw [,->,>=latex] (4.2000,3.7426) -- (4.2000,3.2352);
+\draw (5.3255,3.7426) node {$f(x)-f(a)$};
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_TgCercleTrigono.pstricks.recall b/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
index 4b392a788..312cf3840 100644
--- a/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
+++ b/src_phystricks/Fig_TgCercleTrigono.pstricks.recall
@@ -79,24 +79,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-2.5393) -- (0,3.7681);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.5392) -- (0.0000,3.7680);
 %DEFAULT
-\draw [color=red,style=dashed] (0,0) -- (2.00,2.38);
-\draw [color=cyan,style=dashed] (0,0) -- (2.00,-1.15);
+\draw [color=red,style=dashed] (0.0000,0.0000) -- (2.0000,2.3835);
+\draw [color=cyan,style=dashed] (0.0000,0.0000) -- (2.0000,-1.1547);
 
-\draw [] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
-\draw (0.75659,0.33805) node {$\theta$};
+\draw [] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
+\draw (0.7565,0.3380) node {$\theta$};
 
-\draw [color=red] (0.500,0)--(0.500,0.00441)--(0.500,0.00881)--(0.500,0.0132)--(0.500,0.0176)--(0.500,0.0220)--(0.499,0.0264)--(0.499,0.0308)--(0.499,0.0352)--(0.498,0.0396)--(0.498,0.0440)--(0.498,0.0484)--(0.497,0.0528)--(0.497,0.0572)--(0.496,0.0615)--(0.496,0.0659)--(0.495,0.0703)--(0.494,0.0746)--(0.494,0.0790)--(0.493,0.0834)--(0.492,0.0877)--(0.491,0.0920)--(0.491,0.0964)--(0.490,0.101)--(0.489,0.105)--(0.488,0.109)--(0.487,0.114)--(0.486,0.118)--(0.485,0.122)--(0.484,0.126)--(0.483,0.131)--(0.481,0.135)--(0.480,0.139)--(0.479,0.143)--(0.478,0.148)--(0.476,0.152)--(0.475,0.156)--(0.474,0.160)--(0.472,0.164)--(0.471,0.169)--(0.469,0.173)--(0.468,0.177)--(0.466,0.181)--(0.465,0.185)--(0.463,0.189)--(0.461,0.193)--(0.459,0.197)--(0.458,0.201)--(0.456,0.205)--(0.454,0.209)--(0.452,0.213)--(0.450,0.217)--(0.448,0.221)--(0.446,0.225)--(0.444,0.229)--(0.442,0.233)--(0.440,0.237)--(0.438,0.241)--(0.436,0.245)--(0.434,0.248)--(0.432,0.252)--(0.429,0.256)--(0.427,0.260)--(0.425,0.264)--(0.423,0.267)--(0.420,0.271)--(0.418,0.275)--(0.415,0.278)--(0.413,0.282)--(0.410,0.286)--(0.408,0.289)--(0.405,0.293)--(0.403,0.296)--(0.400,0.300)--(0.397,0.303)--(0.395,0.307)--(0.392,0.310)--(0.389,0.314)--(0.386,0.317)--(0.384,0.321)--(0.381,0.324)--(0.378,0.327)--(0.375,0.331)--(0.372,0.334)--(0.369,0.337)--(0.366,0.341)--(0.363,0.344)--(0.360,0.347)--(0.357,0.350)--(0.354,0.353)--(0.351,0.356)--(0.348,0.359)--(0.344,0.362)--(0.341,0.366)--(0.338,0.368)--(0.335,0.371)--(0.331,0.374)--(0.328,0.377)--(0.325,0.380)--(0.321,0.383);
-\draw (0.34879,0.69703) node {$\varphi$};
+\draw [color=red] (0.5000,0.0000)--(0.4999,0.0044)--(0.4999,0.0088)--(0.4998,0.0132)--(0.4996,0.0176)--(0.4995,0.0220)--(0.4993,0.0264)--(0.4990,0.0308)--(0.4987,0.0352)--(0.4984,0.0396)--(0.4980,0.0440)--(0.4976,0.0484)--(0.4972,0.0527)--(0.4967,0.0571)--(0.4961,0.0615)--(0.4956,0.0659)--(0.4950,0.0702)--(0.4943,0.0746)--(0.4937,0.0790)--(0.4930,0.0833)--(0.4922,0.0876)--(0.4914,0.0920)--(0.4906,0.0963)--(0.4897,0.1006)--(0.4888,0.1049)--(0.4879,0.1092)--(0.4869,0.1135)--(0.4859,0.1178)--(0.4848,0.1221)--(0.4837,0.1264)--(0.4826,0.1306)--(0.4814,0.1349)--(0.4802,0.1391)--(0.4789,0.1434)--(0.4777,0.1476)--(0.4763,0.1518)--(0.4750,0.1560)--(0.4736,0.1601)--(0.4722,0.1643)--(0.4707,0.1685)--(0.4692,0.1726)--(0.4677,0.1767)--(0.4661,0.1809)--(0.4645,0.1850)--(0.4628,0.1890)--(0.4611,0.1931)--(0.4594,0.1972)--(0.4577,0.2012)--(0.4559,0.2052)--(0.4540,0.2093)--(0.4522,0.2133)--(0.4503,0.2172)--(0.4483,0.2212)--(0.4464,0.2251)--(0.4444,0.2291)--(0.4423,0.2330)--(0.4403,0.2369)--(0.4382,0.2407)--(0.4360,0.2446)--(0.4338,0.2484)--(0.4316,0.2522)--(0.4294,0.2560)--(0.4271,0.2598)--(0.4248,0.2636)--(0.4225,0.2673)--(0.4201,0.2710)--(0.4177,0.2747)--(0.4153,0.2784)--(0.4128,0.2820)--(0.4103,0.2857)--(0.4077,0.2893)--(0.4052,0.2928)--(0.4026,0.2964)--(0.4000,0.2999)--(0.3973,0.3035)--(0.3946,0.3069)--(0.3919,0.3104)--(0.3891,0.3139)--(0.3864,0.3173)--(0.3835,0.3207)--(0.3807,0.3240)--(0.3778,0.3274)--(0.3749,0.3307)--(0.3720,0.3340)--(0.3690,0.3373)--(0.3660,0.3405)--(0.3630,0.3437)--(0.3600,0.3469)--(0.3569,0.3501)--(0.3538,0.3532)--(0.3507,0.3563)--(0.3475,0.3594)--(0.3444,0.3624)--(0.3411,0.3654)--(0.3379,0.3684)--(0.3346,0.3714)--(0.3314,0.3743)--(0.3280,0.3772)--(0.3247,0.3801)--(0.3213,0.3830);
+\draw (0.3487,0.6970) node {$\varphi$};
 
-\draw [color=cyan] (0.500,0)--(0.500,0.0132)--(0.499,0.0264)--(0.498,0.0396)--(0.497,0.0528)--(0.496,0.0659)--(0.494,0.0790)--(0.491,0.0920)--(0.489,0.105)--(0.486,0.118)--(0.483,0.131)--(0.479,0.143)--(0.475,0.156)--(0.471,0.169)--(0.466,0.181)--(0.461,0.193)--(0.456,0.205)--(0.450,0.217)--(0.444,0.229)--(0.438,0.241)--(0.432,0.252)--(0.425,0.264)--(0.418,0.275)--(0.410,0.286)--(0.403,0.296)--(0.395,0.307)--(0.386,0.317)--(0.378,0.327)--(0.369,0.337)--(0.360,0.347)--(0.351,0.356)--(0.341,0.366)--(0.331,0.374)--(0.321,0.383)--(0.311,0.391)--(0.301,0.399)--(0.290,0.407)--(0.279,0.415)--(0.268,0.422)--(0.257,0.429)--(0.245,0.436)--(0.234,0.442)--(0.222,0.448)--(0.210,0.454)--(0.198,0.459)--(0.186,0.464)--(0.173,0.469)--(0.161,0.473)--(0.148,0.477)--(0.136,0.481)--(0.123,0.485)--(0.110,0.488)--(0.0972,0.490)--(0.0842,0.493)--(0.0712,0.495)--(0.0580,0.497)--(0.0449,0.498)--(0.0317,0.499)--(0.0185,0.500)--(0.00529,0.500)--(-0.00793,0.500)--(-0.0211,0.500)--(-0.0344,0.499)--(-0.0475,0.498)--(-0.0607,0.496)--(-0.0738,0.495)--(-0.0868,0.492)--(-0.0998,0.490)--(-0.113,0.487)--(-0.126,0.484)--(-0.138,0.480)--(-0.151,0.477)--(-0.164,0.473)--(-0.176,0.468)--(-0.188,0.463)--(-0.200,0.458)--(-0.213,0.453)--(-0.224,0.447)--(-0.236,0.441)--(-0.248,0.434)--(-0.259,0.428)--(-0.270,0.421)--(-0.281,0.413)--(-0.292,0.406)--(-0.303,0.398)--(-0.313,0.390)--(-0.323,0.381)--(-0.333,0.373)--(-0.343,0.364)--(-0.353,0.354)--(-0.362,0.345)--(-0.371,0.335)--(-0.380,0.325)--(-0.388,0.315)--(-0.396,0.305)--(-0.404,0.294)--(-0.412,0.284)--(-0.419,0.273)--(-0.426,0.261)--(-0.433,0.250);
-\draw [color=red,->,>=latex] (0,0) -- (1.2856,1.5321);
-\draw [color=cyan,->,>=latex] (0,0) -- (-1.7320,1.0000);
-\draw [] (2.00,3.27) -- (2.00,-2.04);
+\draw [color=cyan] (0.5000,0.0000)--(0.4998,0.0132)--(0.4993,0.0264)--(0.4984,0.0396)--(0.4972,0.0527)--(0.4956,0.0659)--(0.4937,0.0790)--(0.4914,0.0920)--(0.4888,0.1049)--(0.4859,0.1178)--(0.4826,0.1306)--(0.4789,0.1434)--(0.4750,0.1560)--(0.4707,0.1685)--(0.4661,0.1809)--(0.4611,0.1931)--(0.4559,0.2052)--(0.4503,0.2172)--(0.4444,0.2291)--(0.4382,0.2407)--(0.4316,0.2522)--(0.4248,0.2636)--(0.4177,0.2747)--(0.4103,0.2857)--(0.4026,0.2964)--(0.3946,0.3069)--(0.3864,0.3173)--(0.3778,0.3274)--(0.3690,0.3373)--(0.3600,0.3469)--(0.3507,0.3563)--(0.3411,0.3654)--(0.3314,0.3743)--(0.3213,0.3830)--(0.3111,0.3913)--(0.3006,0.3994)--(0.2900,0.4072)--(0.2791,0.4148)--(0.2680,0.4220)--(0.2568,0.4289)--(0.2454,0.4356)--(0.2338,0.4419)--(0.2220,0.4479)--(0.2101,0.4537)--(0.1980,0.4591)--(0.1858,0.4641)--(0.1734,0.4689)--(0.1610,0.4733)--(0.1484,0.4774)--(0.1357,0.4812)--(0.1230,0.4846)--(0.1101,0.4877)--(0.0972,0.4904)--(0.0842,0.4928)--(0.0711,0.4949)--(0.0580,0.4966)--(0.0448,0.4979)--(0.0317,0.4989)--(0.0185,0.4996)--(0.0052,0.4999)--(-0.0079,0.4999)--(-0.0211,0.4995)--(-0.0343,0.4988)--(-0.0475,0.4977)--(-0.0606,0.4963)--(-0.0737,0.4945)--(-0.0868,0.4924)--(-0.0998,0.4899)--(-0.1127,0.4871)--(-0.1255,0.4839)--(-0.1383,0.4804)--(-0.1509,0.4766)--(-0.1635,0.4725)--(-0.1759,0.4680)--(-0.1882,0.4631)--(-0.2004,0.4580)--(-0.2125,0.4525)--(-0.2243,0.4468)--(-0.2361,0.4407)--(-0.2477,0.4343)--(-0.2591,0.4276)--(-0.2703,0.4206)--(-0.2813,0.4133)--(-0.2921,0.4057)--(-0.3028,0.3978)--(-0.3132,0.3897)--(-0.3234,0.3813)--(-0.3333,0.3726)--(-0.3431,0.3636)--(-0.3526,0.3544)--(-0.3618,0.3450)--(-0.3708,0.3353)--(-0.3796,0.3254)--(-0.3880,0.3152)--(-0.3962,0.3049)--(-0.4041,0.2943)--(-0.4118,0.2835)--(-0.4191,0.2725)--(-0.4262,0.2613)--(-0.4330,0.2500);
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (1.2855,1.5320);
+\draw [color=cyan,->,>=latex] (0.0000,0.0000) -- (-1.7320,1.0000);
+\draw [] (2.0000,3.2680) -- (2.0000,-2.0392);
 \draw [color=red]  (2.0000,2.3835) node [rotate=0] {$\bullet$};
-\draw (2.6976,2.3835) node {$\tan(\theta)$};
+\draw (2.6975,2.3835) node {$\tan(\theta)$};
 \draw [color=cyan]  (2.0000,-1.1547) node [rotate=0] {$\bullet$};
 \draw (2.7262,-1.1547) node {$\tan(\varphi)$};
 
diff --git a/src_phystricks/Fig_TriangleRectangle.pstricks.recall b/src_phystricks/Fig_TriangleRectangle.pstricks.recall
index 1807898e1..fb1989307 100644
--- a/src_phystricks/Fig_TriangleRectangle.pstricks.recall
+++ b/src_phystricks/Fig_TriangleRectangle.pstricks.recall
@@ -89,24 +89,24 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (2.00,3.46) -- (0,0);
-\draw [] (2.00,3.46) -- (4.00,0);
-\draw [] (4.00,0) -- (0,0);
-\draw [color=blue,style=dotted] (2.00,3.46) -- (2.00,0);
-\draw (3.2518,0.36492) node {$60$};
+\draw [] (2.0000,3.4641) -- (0.0000,0.0000);
+\draw [] (2.0000,3.4641) -- (4.0000,0.0000);
+\draw [] (4.0000,0.0000) -- (0.0000,0.0000);
+\draw [color=blue,style=dotted] (2.0000,3.4641) -- (2.0000,0.0000);
+\draw (3.2517,0.3649) node {$60$};
 
-\draw [color=red] (3.75,0.433)--(3.75,0.430)--(3.74,0.428)--(3.74,0.425)--(3.73,0.422)--(3.73,0.419)--(3.72,0.416)--(3.72,0.413)--(3.71,0.410)--(3.71,0.407)--(3.71,0.404)--(3.70,0.401)--(3.70,0.398)--(3.69,0.395)--(3.69,0.391)--(3.68,0.388)--(3.68,0.385)--(3.68,0.381)--(3.67,0.378)--(3.67,0.374)--(3.66,0.371)--(3.66,0.367)--(3.66,0.364)--(3.65,0.360)--(3.65,0.356)--(3.65,0.353)--(3.64,0.349)--(3.64,0.345)--(3.63,0.341)--(3.63,0.337)--(3.63,0.333)--(3.62,0.329)--(3.62,0.325)--(3.62,0.321)--(3.61,0.317)--(3.61,0.313)--(3.61,0.309)--(3.60,0.305)--(3.60,0.301)--(3.60,0.296)--(3.59,0.292)--(3.59,0.288)--(3.59,0.284)--(3.59,0.279)--(3.58,0.275)--(3.58,0.270)--(3.58,0.266)--(3.57,0.261)--(3.57,0.257)--(3.57,0.252)--(3.57,0.248)--(3.56,0.243)--(3.56,0.238)--(3.56,0.234)--(3.56,0.229)--(3.55,0.224)--(3.55,0.220)--(3.55,0.215)--(3.55,0.210)--(3.54,0.205)--(3.54,0.200)--(3.54,0.196)--(3.54,0.191)--(3.54,0.186)--(3.53,0.181)--(3.53,0.176)--(3.53,0.171)--(3.53,0.166)--(3.53,0.161)--(3.52,0.156)--(3.52,0.151)--(3.52,0.146)--(3.52,0.141)--(3.52,0.136)--(3.52,0.131)--(3.52,0.126)--(3.51,0.120)--(3.51,0.115)--(3.51,0.110)--(3.51,0.105)--(3.51,0.0998)--(3.51,0.0946)--(3.51,0.0894)--(3.51,0.0842)--(3.51,0.0790)--(3.51,0.0738)--(3.50,0.0685)--(3.50,0.0633)--(3.50,0.0580)--(3.50,0.0528)--(3.50,0.0475)--(3.50,0.0423)--(3.50,0.0370)--(3.50,0.0317)--(3.50,0.0264)--(3.50,0.0211)--(3.50,0.0159)--(3.50,0.0106)--(3.50,0.00529)--(3.50,0);
+\draw [color=red] (3.7500,0.4330)--(3.7454,0.4303)--(3.7408,0.4276)--(3.7363,0.4248)--(3.7319,0.4220)--(3.7274,0.4191)--(3.7230,0.4162)--(3.7186,0.4133)--(3.7142,0.4103)--(3.7099,0.4072)--(3.7056,0.4041)--(3.7014,0.4010)--(3.6971,0.3978)--(3.6930,0.3946)--(3.6888,0.3913)--(3.6847,0.3880)--(3.6806,0.3847)--(3.6765,0.3813)--(3.6725,0.3778)--(3.6685,0.3743)--(3.6646,0.3708)--(3.6607,0.3672)--(3.6568,0.3636)--(3.6530,0.3600)--(3.6492,0.3563)--(3.6455,0.3526)--(3.6418,0.3488)--(3.6381,0.3450)--(3.6345,0.3411)--(3.6309,0.3373)--(3.6273,0.3333)--(3.6238,0.3294)--(3.6203,0.3254)--(3.6169,0.3213)--(3.6135,0.3173)--(3.6102,0.3132)--(3.6069,0.3090)--(3.6037,0.3049)--(3.6005,0.3006)--(3.5973,0.2964)--(3.5942,0.2921)--(3.5911,0.2878)--(3.5881,0.2835)--(3.5851,0.2791)--(3.5822,0.2747)--(3.5793,0.2703)--(3.5765,0.2658)--(3.5737,0.2613)--(3.5710,0.2568)--(3.5683,0.2522)--(3.5656,0.2477)--(3.5630,0.2430)--(3.5605,0.2384)--(3.5580,0.2338)--(3.5555,0.2291)--(3.5531,0.2243)--(3.5508,0.2196)--(3.5485,0.2148)--(3.5462,0.2101)--(3.5440,0.2052)--(3.5419,0.2004)--(3.5398,0.1956)--(3.5378,0.1907)--(3.5358,0.1858)--(3.5338,0.1809)--(3.5319,0.1759)--(3.5301,0.1710)--(3.5283,0.1660)--(3.5266,0.1610)--(3.5249,0.1560)--(3.5233,0.1509)--(3.5217,0.1459)--(3.5202,0.1408)--(3.5187,0.1357)--(3.5173,0.1306)--(3.5160,0.1255)--(3.5147,0.1204)--(3.5134,0.1153)--(3.5122,0.1101)--(3.5111,0.1049)--(3.5100,0.0998)--(3.5090,0.0946)--(3.5080,0.0894)--(3.5071,0.0842)--(3.5062,0.0790)--(3.5054,0.0737)--(3.5047,0.0685)--(3.5040,0.0632)--(3.5033,0.0580)--(3.5027,0.0527)--(3.5022,0.0475)--(3.5017,0.0422)--(3.5013,0.0369)--(3.5010,0.0317)--(3.5006,0.0264)--(3.5004,0.0211)--(3.5002,0.0158)--(3.5001,0.0105)--(3.5000,0.0052)--(3.5000,0.0000);
 \draw (2.3119,2.2340) node {$30$};
 
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+\draw [color=cyan] (2.0000,2.9641)--(2.0026,2.9641)--(2.0052,2.9641)--(2.0079,2.9641)--(2.0105,2.9642)--(2.0132,2.9642)--(2.0158,2.9643)--(2.0185,2.9644)--(2.0211,2.9645)--(2.0237,2.9646)--(2.0264,2.9648)--(2.0290,2.9649)--(2.0317,2.9651)--(2.0343,2.9652)--(2.0369,2.9654)--(2.0396,2.9656)--(2.0422,2.9658)--(2.0448,2.9661)--(2.0475,2.9663)--(2.0501,2.9666)--(2.0527,2.9668)--(2.0554,2.9671)--(2.0580,2.9674)--(2.0606,2.9677)--(2.0632,2.9681)--(2.0659,2.9684)--(2.0685,2.9688)--(2.0711,2.9691)--(2.0737,2.9695)--(2.0763,2.9699)--(2.0790,2.9703)--(2.0816,2.9708)--(2.0842,2.9712)--(2.0868,2.9716)--(2.0894,2.9721)--(2.0920,2.9726)--(2.0946,2.9731)--(2.0972,2.9736)--(2.0998,2.9741)--(2.1024,2.9747)--(2.1049,2.9752)--(2.1075,2.9758)--(2.1101,2.9763)--(2.1127,2.9769)--(2.1153,2.9775)--(2.1178,2.9781)--(2.1204,2.9788)--(2.1230,2.9794)--(2.1255,2.9801)--(2.1281,2.9807)--(2.1306,2.9814)--(2.1332,2.9821)--(2.1357,2.9828)--(2.1383,2.9836)--(2.1408,2.9843)--(2.1434,2.9851)--(2.1459,2.9858)--(2.1484,2.9866)--(2.1509,2.9874)--(2.1535,2.9882)--(2.1560,2.9890)--(2.1585,2.9898)--(2.1610,2.9907)--(2.1635,2.9916)--(2.1660,2.9924)--(2.1685,2.9933)--(2.1710,2.9942)--(2.1734,2.9951)--(2.1759,2.9960)--(2.1784,2.9970)--(2.1809,2.9979)--(2.1833,2.9989)--(2.1858,2.9999)--(2.1882,3.0009)--(2.1907,3.0019)--(2.1931,3.0029)--(2.1956,3.0039)--(2.1980,3.0049)--(2.2004,3.0060)--(2.2028,3.0071)--(2.2052,3.0081)--(2.2077,3.0092)--(2.2101,3.0103)--(2.2125,3.0115)--(2.2148,3.0126)--(2.2172,3.0137)--(2.2196,3.0149)--(2.2220,3.0161)--(2.2243,3.0172)--(2.2267,3.0184)--(2.2291,3.0196)--(2.2314,3.0209)--(2.2338,3.0221)--(2.2361,3.0233)--(2.2384,3.0246)--(2.2407,3.0258)--(2.2430,3.0271)--(2.2454,3.0284)--(2.2477,3.0297)--(2.2500,3.0310);
 \draw []  (2.0000,3.4641) node [rotate=0] {$\bullet$};
 \draw (2.0000,3.8888) node {$A$};
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.44758,0) node {$B$};
-\draw []  (4.0000,0) node [rotate=0] {$\bullet$};
-\draw (4.4435,0) node {$C$};
-\draw []  (2.0000,0) node [rotate=0] {$\bullet$};
-\draw (2.0000,-0.42471) node {$H$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.4475,0.0000) node {$B$};
+\draw []  (4.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (4.4434,0.0000) node {$C$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.4247) node {$H$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall b/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall
index 23fd0b1bb..b1653b7ec 100644
--- a/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall
+++ b/src_phystricks/Fig_UEGEooHEDIJVPn.pstricks.recall
@@ -96,46 +96,37 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (5.5000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,5.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,5.5000);
 %DEFAULT
-\draw [] (0,0) -- (5.00,5.00);
+\draw [] (0.0000,0.0000) -- (5.0000,5.0000);
 
-\draw [color=blue] (0.1000,4.996)--(0.1495,4.594)--(0.1990,4.308)--(0.2485,4.086)--(0.2980,3.904)--(0.3475,3.750)--(0.3970,3.617)--(0.4465,3.500)--(0.4960,3.394)--(0.5455,3.299)--(0.5949,3.212)--(0.6444,3.133)--(0.6939,3.059)--(0.7434,2.990)--(0.7929,2.925)--(0.8424,2.865)--(0.8919,2.808)--(0.9414,2.754)--(0.9909,2.702)--(1.040,2.654)--(1.090,2.607)--(1.139,2.563)--(1.189,2.520)--(1.238,2.479)--(1.288,2.440)--(1.337,2.402)--(1.387,2.366)--(1.436,2.331)--(1.486,2.297)--(1.535,2.264)--(1.585,2.233)--(1.634,2.202)--(1.684,2.172)--(1.733,2.143)--(1.783,2.115)--(1.832,2.088)--(1.882,2.061)--(1.931,2.035)--(1.981,2.010)--(2.030,1.985)--(2.080,1.961)--(2.129,1.937)--(2.179,1.914)--(2.228,1.892)--(2.278,1.870)--(2.327,1.848)--(2.377,1.827)--(2.426,1.807)--(2.476,1.787)--(2.525,1.767)--(2.575,1.747)--(2.624,1.728)--(2.674,1.710)--(2.723,1.691)--(2.773,1.673)--(2.822,1.656)--(2.872,1.638)--(2.921,1.621)--(2.971,1.604)--(3.020,1.588)--(3.070,1.572)--(3.119,1.556)--(3.169,1.540)--(3.218,1.524)--(3.268,1.509)--(3.317,1.494)--(3.367,1.479)--(3.416,1.465)--(3.466,1.450)--(3.515,1.436)--(3.565,1.422)--(3.614,1.408)--(3.664,1.395)--(3.713,1.381)--(3.763,1.368)--(3.812,1.355)--(3.862,1.342)--(3.911,1.329)--(3.961,1.317)--(4.010,1.304)--(4.060,1.292)--(4.109,1.280)--(4.159,1.268)--(4.208,1.256)--(4.258,1.244)--(4.307,1.233)--(4.357,1.221)--(4.406,1.210)--(4.456,1.199)--(4.505,1.188)--(4.555,1.177)--(4.604,1.166)--(4.654,1.156)--(4.703,1.145)--(4.753,1.134)--(4.802,1.124)--(4.852,1.114)--(4.901,1.104)--(4.951,1.094)--(5.000,1.084);
+\draw [color=blue] (0.1000,4.9957)--(0.1494,4.5936)--(0.1989,4.3076)--(0.2484,4.0855)--(0.2979,3.9038)--(0.3474,3.7502)--(0.3969,3.6170)--(0.4464,3.4995)--(0.4959,3.3944)--(0.5454,3.2992)--(0.5949,3.2124)--(0.6444,3.1325)--(0.6939,3.0585)--(0.7434,2.9896)--(0.7929,2.9251)--(0.8424,2.8646)--(0.8919,2.8075)--(0.9414,2.7535)--(0.9909,2.7022)--(1.0404,2.6535)--(1.0898,2.6070)--(1.1393,2.5626)--(1.1888,2.5201)--(1.2383,2.4793)--(1.2878,2.4401)--(1.3373,2.4024)--(1.3868,2.3660)--(1.4363,2.3310)--(1.4858,2.2971)--(1.5353,2.2643)--(1.5848,2.2326)--(1.6343,2.2019)--(1.6838,2.1720)--(1.7333,2.1431)--(1.7828,2.1149)--(1.8323,2.0875)--(1.8818,2.0609)--(1.9313,2.0349)--(1.9808,2.0096)--(2.0303,1.9849)--(2.0797,1.9608)--(2.1292,1.9373)--(2.1787,1.9143)--(2.2282,1.8919)--(2.2777,1.8699)--(2.3272,1.8484)--(2.3767,1.8274)--(2.4262,1.8067)--(2.4757,1.7866)--(2.5252,1.7668)--(2.5747,1.7473)--(2.6242,1.7283)--(2.6737,1.7096)--(2.7232,1.6913)--(2.7727,1.6733)--(2.8222,1.6556)--(2.8717,1.6382)--(2.9212,1.6211)--(2.9707,1.6043)--(3.0202,1.5878)--(3.0696,1.5715)--(3.1191,1.5555)--(3.1686,1.5398)--(3.2181,1.5243)--(3.2676,1.5090)--(3.3171,1.4940)--(3.3666,1.4792)--(3.4161,1.4646)--(3.4656,1.4502)--(3.5151,1.4360)--(3.5646,1.4220)--(3.6141,1.4082)--(3.6636,1.3946)--(3.7131,1.3812)--(3.7626,1.3680)--(3.8121,1.3549)--(3.8616,1.3420)--(3.9111,1.3293)--(3.9606,1.3167)--(4.0101,1.3043)--(4.0595,1.2920)--(4.1090,1.2799)--(4.1585,1.2679)--(4.2080,1.2561)--(4.2575,1.2444)--(4.3070,1.2328)--(4.3565,1.2214)--(4.4060,1.2101)--(4.4555,1.1989)--(4.5050,1.1879)--(4.5545,1.1770)--(4.6040,1.1662)--(4.6535,1.1555)--(4.7030,1.1449)--(4.7525,1.1344)--(4.8020,1.1241)--(4.8515,1.1138)--(4.9010,1.1037)--(4.9505,1.0936)--(5.0000,1.0837);
 \draw []  (5.0000,1.0837) node [rotate=0] {$\bullet$};
-\draw (5.3918,1.2782) node {\( P_{ 0  }\)};
+\draw (5.3917,1.2781) node {\( P_{ 0  }\)};
 \draw []  (1.0837,1.0837) node [rotate=0] {$\bullet$};
-\draw (0.71885,1.3853) node {\( Q_{0}\)};
-\draw []  (5.0000,0) node [rotate=0] {$\bullet$};
-\draw (5.0000,-0.30595) node {\( x_{ 0  }\)};
-\draw [color=cyan,style=dashed] (5.00,1.08) -- (1.08,1.08);
-\draw [color=red,style=dashed] (5.00,1.08) -- (5.00,0);
-\draw []  (1.0837,2.6128) node [rotate=0] {$\bullet$};
-\draw (1.3567,2.9496) node {\( P_{ 1  }\)};
-\draw []  (2.6128,2.6128) node [rotate=0] {$\bullet$};
-\draw (2.2479,2.9144) node {\( Q_{1}\)};
-\draw []  (1.0837,0) node [rotate=0] {$\bullet$};
-\draw (1.0837,-0.30595) node {\( x_{ 1  }\)};
-\draw [color=cyan,style=dashed] (1.08,2.61) -- (2.61,2.61);
-\draw [color=red,style=dashed] (1.08,2.61) -- (1.08,0);
-\draw []  (2.6128,1.7327) node [rotate=0] {$\bullet$};
-\draw (2.9758,1.9954) node {\( P_{ 2  }\)};
+\draw (0.7188,1.3853) node {\( Q_{0}\)};
+\draw []  (5.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (5.0000,-0.3059) node {\( x_{ 0  }\)};
+\draw [color=cyan,style=dashed] (5.0000,1.0837) -- (1.0837,1.0837);
+\draw [color=red,style=dashed] (5.0000,1.0837) -- (5.0000,0.0000);
+\draw []  (1.0837,2.6127) node [rotate=0] {$\bullet$};
+\draw (1.3566,2.9495) node {\( P_{ 1  }\)};
+\draw []  (2.6127,2.6127) node [rotate=0] {$\bullet$};
+\draw (2.2478,2.9143) node {\( Q_{1}\)};
+\draw []  (1.0837,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0837,-0.3059) node {\( x_{ 1  }\)};
+\draw [color=cyan,style=dashed] (1.0837,2.6127) -- (2.6127,2.6127);
+\draw [color=red,style=dashed] (1.0837,2.6127) -- (1.0837,0.0000);
+\draw []  (2.6127,1.7327) node [rotate=0] {$\bullet$};
+\draw (2.9757,1.9953) node {\( P_{ 2  }\)};
 \draw []  (1.7327,1.7327) node [rotate=0] {$\bullet$};
-\draw (1.3679,2.0344) node {\( Q_{2}\)};
-\draw []  (2.6128,0) node [rotate=0] {$\bullet$};
-\draw (2.6128,-0.30595) node {\( x_{ 2  }\)};
-\draw [color=cyan,style=dashed] (2.61,1.73) -- (1.73,1.73);
-\draw [color=red,style=dashed] (2.61,1.73) -- (2.61,0);
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw [] (5.00,-0.100) -- (5.00,0.100);
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw [] (-0.100,4.00) -- (0.100,4.00);
-\draw [] (-0.100,5.00) -- (0.100,5.00);
+\draw (1.3678,2.0343) node {\( Q_{2}\)};
+\draw []  (2.6127,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.6127,-0.3059) node {\( x_{ 2  }\)};
+\draw [color=cyan,style=dashed] (2.6127,1.7327) -- (1.7327,1.7327);
+\draw [color=red,style=dashed] (2.6127,1.7327) -- (2.6127,0.0000);
+
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall b/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall
index 71823fe52..2191d4e21 100644
--- a/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall
+++ b/src_phystricks/Fig_UUNEooCNVOOs.pstricks.recall
@@ -35,14 +35,14 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (0,0);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (0.0000,0.0000);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
@@ -75,16 +75,16 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,2.00) -- (-1.88,1.32);
-\draw [color=red,->,>=latex] (0,0) -- (-0.11698,0.32139);
-\draw [color=green,->,>=latex] (0,0) -- (-0.88302,-0.32139);
-\draw [color=blue,->,>=latex] (-2.0000,0) -- (-3.0000,0);
+\draw [] (0.0000,2.0000) -- (-1.8793,1.3159);
+\draw [color=red,->,>=latex] (0.0000,0.0000) -- (-0.1169,0.3213);
+\draw [color=green,->,>=latex] (0.0000,0.0000) -- (-0.8830,-0.3213);
+\draw [color=blue,->,>=latex] (-2.0000,0.0000) -- (-3.0000,0.0000);
 \draw [color=blue,->,>=latex] (-2.0000,1.0000) -- (-3.0000,1.0000);
 \draw [color=blue,->,>=latex] (-2.0000,2.0000) -- (-3.0000,2.0000);
-\draw [color=blue,->,>=latex] (0,0) -- (-1.0000,0);
-\draw [color=blue,->,>=latex] (0,1.0000) -- (-1.0000,1.0000);
-\draw [color=blue,->,>=latex] (0,2.0000) -- (-1.0000,2.0000);
-\draw [color=blue,->,>=latex] (2.0000,0) -- (1.0000,0);
+\draw [color=blue,->,>=latex] (0.0000,0.0000) -- (-1.0000,0.0000);
+\draw [color=blue,->,>=latex] (0.0000,1.0000) -- (-1.0000,1.0000);
+\draw [color=blue,->,>=latex] (0.0000,2.0000) -- (-1.0000,2.0000);
+\draw [color=blue,->,>=latex] (2.0000,0.0000) -- (1.0000,0.0000);
 \draw [color=blue,->,>=latex] (2.0000,1.0000) -- (1.0000,1.0000);
 \draw [color=blue,->,>=latex] (2.0000,2.0000) -- (1.0000,2.0000);
 %END PSPICTURE
diff --git a/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall b/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall
index bfe4147ab..f8f3c74da 100644
--- a/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall
+++ b/src_phystricks/Fig_UYJooCWjLgK.pstricks.recall
@@ -82,20 +82,20 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw [] (0,0) -- (3.46,2.00);
-\draw [] (-1.20,0) -- (13.2,0);
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (-0.19125,0.28813) node {\( 0\)};
+\draw [] (0.0000,0.0000) -- (3.4641,2.0000);
+\draw [] (-1.2000,0.0000) -- (13.200,0.0000);
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (-0.1912,0.2881) node {\( 0\)};
 \draw []  (3.4641,2.0000) node [rotate=0] {$\bullet$};
-\draw (3.2681,2.2873) node {\( y\)};
-\draw []  (6.0000,0) node [rotate=0] {$\bullet$};
-\draw (6.0000,-0.27858) node {\( x\)};
-\draw []  (12.000,0) node [rotate=0] {$\bullet$};
-\draw (12.000,-0.31406) node {\( xy\)};
+\draw (3.2680,2.2872) node {\( y\)};
+\draw []  (6.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (6.0000,-0.2785) node {\( x\)};
+\draw []  (12.000,0.0000) node [rotate=0] {$\bullet$};
+\draw (12.000,-0.3140) node {\( xy\)};
 \draw []  (1.7320,1.0000) node [rotate=0] {$\bullet$};
 \draw (1.5408,1.2881) node {\( 1\)};
-\draw [style=dashed] (1.73,1.00) -- (6.00,0);
-\draw [style=dashed] (3.46,2.00) -- (12.0,0);
+\draw [style=dashed] (1.7320,1.0000) -- (6.0000,0.0000);
+\draw [style=dashed] (3.4641,2.0000) -- (12.000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall b/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall
index b8a022bef..265dfa4ea 100644
--- a/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall
+++ b/src_phystricks/Fig_UZGooBzlYxr.pstricks.recall
@@ -96,47 +96,47 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (4.5000,0);
-\draw [,->,>=latex] (0,-3.5000) -- (0,4.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,4.5000);
 %DEFAULT
-\fill [color=lightgray] (2.83,1.00) -- (2.80,1.07) -- (2.78,1.14) -- (2.75,1.21) -- (2.72,1.27) -- (2.68,1.34) -- (2.65,1.41) -- (2.62,1.47) -- (2.58,1.53) -- (2.54,1.60) -- (2.50,1.66) -- (2.46,1.72) -- (2.41,1.78) -- (2.37,1.84) -- (2.32,1.90) -- (2.28,1.95) -- (2.23,2.01) -- (2.18,2.06) -- (2.13,2.12) -- (2.07,2.17) -- (2.02,2.22) -- (1.96,2.27) -- (1.91,2.31) -- (1.85,2.36) -- (1.79,2.41) -- (1.73,2.45) -- (1.67,2.49) -- (1.61,2.53) -- (1.55,2.57) -- (1.48,2.61) -- (1.42,2.64) -- (1.35,2.68) -- (1.29,2.71) -- (1.22,2.74) -- (1.15,2.77) -- (1.08,2.80) -- (1.01,2.82) -- (0.944,2.85) -- (0.873,2.87) -- (0.803,2.89) -- (0.731,2.91) -- (0.659,2.93) -- (0.587,2.94) -- (0.515,2.96) -- (0.442,2.97) -- (0.368,2.98) -- (0.295,2.99) -- (0.221,2.99) -- (0.148,3.00) -- (0.0740,3.00) -- (0,3.00) -- (2.00,1.00) -- (2.00,1.00) -- (2.83,1.00) --  cycle;
+\fill [color=lightgray] (2.8284,1.0000) -- (2.8029,1.0693) -- (2.7757,1.1379) -- (2.7469,1.2059) -- (2.7164,1.2732) -- (2.6842,1.3397) -- (2.6504,1.4053) -- (2.6150,1.4702) -- (2.5780,1.5341) -- (2.5395,1.5971) -- (2.4994,1.6591) -- (2.4578,1.7201) -- (2.4147,1.7801) -- (2.3701,1.8390) -- (2.3241,1.8968) -- (2.2767,1.9535) -- (2.2280,2.0089) -- (2.1778,2.0632) -- (2.1264,2.1161) -- (2.0737,2.1678) -- (2.0197,2.2182) -- (1.9644,2.2673) -- (1.9080,2.3150) -- (1.8505,2.3612) -- (1.7918,2.4061) -- (1.7320,2.4494) -- (1.6712,2.4913) -- (1.6093,2.5317) -- (1.5465,2.5706) -- (1.4828,2.6079) -- (1.4181,2.6436) -- (1.3526,2.6777) -- (1.2863,2.7102) -- (1.2192,2.7410) -- (1.1513,2.7702) -- (1.0828,2.7977) -- (1.0136,2.8235) -- (0.9438,2.8476) -- (0.8734,2.8700) -- (0.8025,2.8906) -- (0.7311,2.9095) -- (0.6592,2.9266) -- (0.5870,2.9420) -- (0.5144,2.9555) -- (0.4415,2.9673) -- (0.3683,2.9772) -- (0.2949,2.9854) -- (0.2213,2.9918) -- (0.1476,2.9963) -- (0.0738,2.9990) -- (0.0000,3.0000) -- (2.0000,1.0000) -- (2.0000,1.0000) -- (2.8284,1.0000) --  cycle;
 
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 \draw []  (2.0000,1.0000) node [rotate=0] {$\bullet$};
-\draw (1.7217,0.73387) node {\( A\)};
+\draw (1.7217,0.7338) node {\( A\)};
 \draw []  (2.8284,1.0000) node [rotate=0] {$\bullet$};
-\draw (3.1174,0.73387) node {\( B\)};
-\draw [] (-1.00,1.00) -- (4.00,1.00);
+\draw (3.1174,0.7338) node {\( B\)};
+\draw [] (-1.0000,1.0000) -- (4.0000,1.0000);
 
-\draw [color=blue] (-1.000,4.000)--(-0.9495,3.949)--(-0.8990,3.899)--(-0.8485,3.848)--(-0.7980,3.798)--(-0.7475,3.747)--(-0.6970,3.697)--(-0.6465,3.646)--(-0.5960,3.596)--(-0.5455,3.545)--(-0.4949,3.495)--(-0.4444,3.444)--(-0.3939,3.394)--(-0.3434,3.343)--(-0.2929,3.293)--(-0.2424,3.242)--(-0.1919,3.192)--(-0.1414,3.141)--(-0.09091,3.091)--(-0.04040,3.040)--(0.01010,2.990)--(0.06061,2.939)--(0.1111,2.889)--(0.1616,2.838)--(0.2121,2.788)--(0.2626,2.737)--(0.3131,2.687)--(0.3636,2.636)--(0.4141,2.586)--(0.4646,2.535)--(0.5152,2.485)--(0.5657,2.434)--(0.6162,2.384)--(0.6667,2.333)--(0.7172,2.283)--(0.7677,2.232)--(0.8182,2.182)--(0.8687,2.131)--(0.9192,2.081)--(0.9697,2.030)--(1.020,1.980)--(1.071,1.929)--(1.121,1.879)--(1.172,1.828)--(1.222,1.778)--(1.273,1.727)--(1.323,1.677)--(1.374,1.626)--(1.424,1.576)--(1.475,1.525)--(1.525,1.475)--(1.576,1.424)--(1.626,1.374)--(1.677,1.323)--(1.727,1.273)--(1.778,1.222)--(1.828,1.172)--(1.879,1.121)--(1.929,1.071)--(1.980,1.020)--(2.030,0.9697)--(2.081,0.9192)--(2.131,0.8687)--(2.182,0.8182)--(2.232,0.7677)--(2.283,0.7172)--(2.333,0.6667)--(2.384,0.6162)--(2.434,0.5657)--(2.485,0.5152)--(2.535,0.4646)--(2.586,0.4141)--(2.636,0.3636)--(2.687,0.3131)--(2.737,0.2626)--(2.788,0.2121)--(2.838,0.1616)--(2.889,0.1111)--(2.939,0.06061)--(2.990,0.01010)--(3.040,-0.04040)--(3.091,-0.09091)--(3.141,-0.1414)--(3.192,-0.1919)--(3.242,-0.2424)--(3.293,-0.2929)--(3.343,-0.3434)--(3.394,-0.3939)--(3.444,-0.4444)--(3.495,-0.4949)--(3.545,-0.5455)--(3.596,-0.5960)--(3.646,-0.6465)--(3.697,-0.6970)--(3.747,-0.7475)--(3.798,-0.7980)--(3.848,-0.8485)--(3.899,-0.8990)--(3.949,-0.9495)--(4.000,-1.000);
-\draw (-3.0000,-0.32983) node {$ -3 $};
-\draw [] (-3.00,-0.100) -- (-3.00,0.100);
-\draw (-2.0000,-0.32983) node {$ -2 $};
-\draw [] (-2.00,-0.100) -- (-2.00,0.100);
-\draw (-1.0000,-0.32983) node {$ -1 $};
-\draw [] (-1.00,-0.100) -- (-1.00,0.100);
-\draw (1.0000,-0.31492) node {$ 1 $};
-\draw [] (1.00,-0.100) -- (1.00,0.100);
-\draw (2.0000,-0.31492) node {$ 2 $};
-\draw [] (2.00,-0.100) -- (2.00,0.100);
-\draw (3.0000,-0.31492) node {$ 3 $};
-\draw [] (3.00,-0.100) -- (3.00,0.100);
-\draw (4.0000,-0.31492) node {$ 4 $};
-\draw [] (4.00,-0.100) -- (4.00,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
-\draw (-0.29125,4.0000) node {$ 4 $};
-\draw [] (-0.100,4.00) -- (0.100,4.00);
+\draw [color=blue] (-1.0000,4.0000)--(-0.9494,3.9494)--(-0.8989,3.8989)--(-0.8484,3.8484)--(-0.7979,3.7979)--(-0.7474,3.7474)--(-0.6969,3.6969)--(-0.6464,3.6464)--(-0.5959,3.5959)--(-0.5454,3.5454)--(-0.4949,3.4949)--(-0.4444,3.4444)--(-0.3939,3.3939)--(-0.3434,3.3434)--(-0.2929,3.2929)--(-0.2424,3.2424)--(-0.1919,3.1919)--(-0.1414,3.1414)--(-0.0909,3.0909)--(-0.0404,3.0404)--(0.0101,2.9898)--(0.0606,2.9393)--(0.1111,2.8888)--(0.1616,2.8383)--(0.2121,2.7878)--(0.2626,2.7373)--(0.3131,2.6868)--(0.3636,2.6363)--(0.4141,2.5858)--(0.4646,2.5353)--(0.5151,2.4848)--(0.5656,2.4343)--(0.6161,2.3838)--(0.6666,2.3333)--(0.7171,2.2828)--(0.7676,2.2323)--(0.8181,2.1818)--(0.8686,2.1313)--(0.9191,2.0808)--(0.9696,2.0303)--(1.0202,1.9797)--(1.0707,1.9292)--(1.1212,1.8787)--(1.1717,1.8282)--(1.2222,1.7777)--(1.2727,1.7272)--(1.3232,1.6767)--(1.3737,1.6262)--(1.4242,1.5757)--(1.4747,1.5252)--(1.5252,1.4747)--(1.5757,1.4242)--(1.6262,1.3737)--(1.6767,1.3232)--(1.7272,1.2727)--(1.7777,1.2222)--(1.8282,1.1717)--(1.8787,1.1212)--(1.9292,1.0707)--(1.9797,1.0202)--(2.0303,0.9696)--(2.0808,0.9191)--(2.1313,0.8686)--(2.1818,0.8181)--(2.2323,0.7676)--(2.2828,0.7171)--(2.3333,0.6666)--(2.3838,0.6161)--(2.4343,0.5656)--(2.4848,0.5151)--(2.5353,0.4646)--(2.5858,0.4141)--(2.6363,0.3636)--(2.6868,0.3131)--(2.7373,0.2626)--(2.7878,0.2121)--(2.8383,0.1616)--(2.8888,0.1111)--(2.9393,0.0606)--(2.9898,0.0101)--(3.0404,-0.0404)--(3.0909,-0.0909)--(3.1414,-0.1414)--(3.1919,-0.1919)--(3.2424,-0.2424)--(3.2929,-0.2929)--(3.3434,-0.3434)--(3.3939,-0.3939)--(3.4444,-0.4444)--(3.4949,-0.4949)--(3.5454,-0.5454)--(3.5959,-0.5959)--(3.6464,-0.6464)--(3.6969,-0.6969)--(3.7474,-0.7474)--(3.7979,-0.7979)--(3.8484,-0.8484)--(3.8989,-0.8989)--(3.9494,-0.9494)--(4.0000,-1.0000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (4.0000,-0.3149) node {$ 4 $};
+\draw [] (4.0000,-0.1000) -- (4.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
+\draw (-0.2912,4.0000) node {$ 4 $};
+\draw [] (-0.1000,4.0000) -- (0.1000,4.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_UnSurxInt.pstricks.recall b/src_phystricks/Fig_UnSurxInt.pstricks.recall
index 702ef7c25..0e671bfaf 100644
--- a/src_phystricks/Fig_UnSurxInt.pstricks.recall
+++ b/src_phystricks/Fig_UnSurxInt.pstricks.recall
@@ -87,13 +87,13 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (3.5000,0);
-\draw [,->,>=latex] (0,-3.8333) -- (0,3.8333);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-3.8333) -- (0.0000,3.8333);
 %DEFAULT
 
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -101,11 +101,11 @@
 % setting the default values
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 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -113,35 +113,35 @@
 % setting the default values
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          hatchthickness=0.4pt}
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+\draw [color=blue] (1.0000,0.0000)--(1.0101,0.0000)--(1.0202,0.0000)--(1.0303,0.0000)--(1.0404,0.0000)--(1.0505,0.0000)--(1.0606,0.0000)--(1.0707,0.0000)--(1.0808,0.0000)--(1.0909,0.0000)--(1.1010,0.0000)--(1.1111,0.0000)--(1.1212,0.0000)--(1.1313,0.0000)--(1.1414,0.0000)--(1.1515,0.0000)--(1.1616,0.0000)--(1.1717,0.0000)--(1.1818,0.0000)--(1.1919,0.0000)--(1.2020,0.0000)--(1.2121,0.0000)--(1.2222,0.0000)--(1.2323,0.0000)--(1.2424,0.0000)--(1.2525,0.0000)--(1.2626,0.0000)--(1.2727,0.0000)--(1.2828,0.0000)--(1.2929,0.0000)--(1.3030,0.0000)--(1.3131,0.0000)--(1.3232,0.0000)--(1.3333,0.0000)--(1.3434,0.0000)--(1.3535,0.0000)--(1.3636,0.0000)--(1.3737,0.0000)--(1.3838,0.0000)--(1.3939,0.0000)--(1.4040,0.0000)--(1.4141,0.0000)--(1.4242,0.0000)--(1.4343,0.0000)--(1.4444,0.0000)--(1.4545,0.0000)--(1.4646,0.0000)--(1.4747,0.0000)--(1.4848,0.0000)--(1.4949,0.0000)--(1.5050,0.0000)--(1.5151,0.0000)--(1.5252,0.0000)--(1.5353,0.0000)--(1.5454,0.0000)--(1.5555,0.0000)--(1.5656,0.0000)--(1.5757,0.0000)--(1.5858,0.0000)--(1.5959,0.0000)--(1.6060,0.0000)--(1.6161,0.0000)--(1.6262,0.0000)--(1.6363,0.0000)--(1.6464,0.0000)--(1.6565,0.0000)--(1.6666,0.0000)--(1.6767,0.0000)--(1.6868,0.0000)--(1.6969,0.0000)--(1.7070,0.0000)--(1.7171,0.0000)--(1.7272,0.0000)--(1.7373,0.0000)--(1.7474,0.0000)--(1.7575,0.0000)--(1.7676,0.0000)--(1.7777,0.0000)--(1.7878,0.0000)--(1.7979,0.0000)--(1.8080,0.0000)--(1.8181,0.0000)--(1.8282,0.0000)--(1.8383,0.0000)--(1.8484,0.0000)--(1.8585,0.0000)--(1.8686,0.0000)--(1.8787,0.0000)--(1.8888,0.0000)--(1.8989,0.0000)--(1.9090,0.0000)--(1.9191,0.0000)--(1.9292,0.0000)--(1.9393,0.0000)--(1.9494,0.0000)--(1.9595,0.0000)--(1.9696,0.0000)--(1.9797,0.0000)--(1.9898,0.0000)--(2.0000,0.0000);
+\draw [color=brown,style=solid] (1.0000,0.0000) -- (1.0000,1.0000);
+\draw [color=brown,style=solid] (2.0000,0.5000) -- (2.0000,0.0000);
+\draw (-3.0000,-0.3298) node {$ -3 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-2.0000,-0.3298) node {$ -2 $};
+\draw [] (-2.0000,-0.1000) -- (-2.0000,0.1000);
+\draw (-1.0000,-0.3298) node {$ -1 $};
+\draw [] (-1.0000,-0.1000) -- (-1.0000,0.1000);
+\draw (1.0000,-0.3149) node {$ 1 $};
+\draw [] (1.0000,-0.1000) -- (1.0000,0.1000);
+\draw (2.0000,-0.3149) node {$ 2 $};
+\draw [] (2.0000,-0.1000) -- (2.0000,0.1000);
+\draw (3.0000,-0.3149) node {$ 3 $};
+\draw [] (3.0000,-0.1000) -- (3.0000,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_VANooZowSyO.pstricks.recall b/src_phystricks/Fig_VANooZowSyO.pstricks.recall
index 653bbd4a8..88dd46939 100644
--- a/src_phystricks/Fig_VANooZowSyO.pstricks.recall
+++ b/src_phystricks/Fig_VANooZowSyO.pstricks.recall
@@ -57,23 +57,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.4995);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.4994);
 %DEFAULT
 
-\draw [color=blue] (-2.199,-1.000)--(-2.155,-0.9980)--(-2.110,-0.9920)--(-2.066,-0.9819)--(-2.021,-0.9679)--(-1.977,-0.9501)--(-1.933,-0.9284)--(-1.888,-0.9029)--(-1.844,-0.8738)--(-1.799,-0.8413)--(-1.755,-0.8053)--(-1.710,-0.7660)--(-1.666,-0.7237)--(-1.622,-0.6785)--(-1.577,-0.6306)--(-1.533,-0.5801)--(-1.488,-0.5272)--(-1.444,-0.4723)--(-1.399,-0.4154)--(-1.355,-0.3569)--(-1.311,-0.2969)--(-1.266,-0.2358)--(-1.222,-0.1736)--(-1.177,-0.1108)--(-1.133,-0.04758)--(-1.088,0.01587)--(-1.044,0.07925)--(-0.9996,0.1423)--(-0.9552,0.2048)--(-0.9107,0.2665)--(-0.8663,0.3271)--(-0.8219,0.3863)--(-0.7775,0.4441)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6442,0.6056)--(-0.5998,0.6549)--(-0.5553,0.7015)--(-0.5109,0.7453)--(-0.4665,0.7861)--(-0.4221,0.8237)--(-0.3776,0.8580)--(-0.3332,0.8888)--(-0.2888,0.9161)--(-0.2443,0.9397)--(-0.1999,0.9595)--(-0.1555,0.9754)--(-0.1111,0.9874)--(-0.06664,0.9955)--(-0.02221,0.9995)--(0.02221,0.9995)--(0.06664,0.9955)--(0.1111,0.9874)--(0.1555,0.9754)--(0.1999,0.9595)--(0.2443,0.9397)--(0.2888,0.9161)--(0.3332,0.8888)--(0.3776,0.8580)--(0.4221,0.8237)--(0.4665,0.7861)--(0.5109,0.7453)--(0.5553,0.7015)--(0.5998,0.6549)--(0.6442,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7775,0.4441)--(0.8219,0.3863)--(0.8663,0.3271)--(0.9107,0.2665)--(0.9552,0.2048)--(0.9996,0.1423)--(1.044,0.07925)--(1.088,0.01587)--(1.133,-0.04758)--(1.177,-0.1108)--(1.222,-0.1736)--(1.266,-0.2358)--(1.311,-0.2969)--(1.355,-0.3569)--(1.399,-0.4154)--(1.444,-0.4723)--(1.488,-0.5272)--(1.533,-0.5801)--(1.577,-0.6306)--(1.622,-0.6785)--(1.666,-0.7237)--(1.710,-0.7660)--(1.755,-0.8053)--(1.799,-0.8413)--(1.844,-0.8738)--(1.888,-0.9029)--(1.933,-0.9284)--(1.977,-0.9501)--(2.021,-0.9679)--(2.066,-0.9819)--(2.110,-0.9920)--(2.155,-0.9980)--(2.199,-1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,-1.0000)--(-2.1546,-0.9979)--(-2.1102,-0.9919)--(-2.0658,-0.9819)--(-2.0214,-0.9679)--(-1.9769,-0.9500)--(-1.9325,-0.9283)--(-1.8881,-0.9029)--(-1.8437,-0.8738)--(-1.7992,-0.8412)--(-1.7548,-0.8052)--(-1.7104,-0.7660)--(-1.6659,-0.7237)--(-1.6215,-0.6785)--(-1.5771,-0.6305)--(-1.5327,-0.5800)--(-1.4882,-0.5272)--(-1.4438,-0.4722)--(-1.3994,-0.4154)--(-1.3550,-0.3568)--(-1.3105,-0.2969)--(-1.2661,-0.2357)--(-1.2217,-0.1736)--(-1.1773,-0.1108)--(-1.1328,-0.0475)--(-1.0884,0.0158)--(-1.0440,0.0792)--(-0.9995,0.1423)--(-0.9551,0.2048)--(-0.9107,0.2664)--(-0.8663,0.3270)--(-0.8218,0.3863)--(-0.7774,0.4440)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6441,0.6056)--(-0.5997,0.6548)--(-0.5553,0.7014)--(-0.5109,0.7452)--(-0.4664,0.7860)--(-0.4220,0.8236)--(-0.3776,0.8579)--(-0.3331,0.8888)--(-0.2887,0.9161)--(-0.2443,0.9396)--(-0.1999,0.9594)--(-0.1554,0.9754)--(-0.1110,0.9874)--(-0.0666,0.9954)--(-0.0222,0.9994)--(0.0222,0.9994)--(0.0666,0.9954)--(0.1110,0.9874)--(0.1554,0.9754)--(0.1999,0.9594)--(0.2443,0.9396)--(0.2887,0.9161)--(0.3331,0.8888)--(0.3776,0.8579)--(0.4220,0.8236)--(0.4664,0.7860)--(0.5109,0.7452)--(0.5553,0.7014)--(0.5997,0.6548)--(0.6441,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7774,0.4440)--(0.8218,0.3863)--(0.8663,0.3270)--(0.9107,0.2664)--(0.9551,0.2048)--(0.9995,0.1423)--(1.0440,0.0792)--(1.0884,0.0158)--(1.1328,-0.0475)--(1.1773,-0.1108)--(1.2217,-0.1736)--(1.2661,-0.2357)--(1.3105,-0.2969)--(1.3550,-0.3568)--(1.3994,-0.4154)--(1.4438,-0.4722)--(1.4882,-0.5272)--(1.5327,-0.5800)--(1.5771,-0.6305)--(1.6215,-0.6785)--(1.6659,-0.7237)--(1.7104,-0.7660)--(1.7548,-0.8052)--(1.7992,-0.8412)--(1.8437,-0.8738)--(1.8881,-0.9029)--(1.9325,-0.9283)--(1.9769,-0.9500)--(2.0214,-0.9679)--(2.0658,-0.9819)--(2.1102,-0.9919)--(2.1546,-0.9979)--(2.1991,-1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -127,23 +127,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.4999) -- (0,1.4999);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4998) -- (0.0000,1.4998);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0)--(-2.155,0.06342)--(-2.110,0.1266)--(-2.066,0.1893)--(-2.021,0.2511)--(-1.977,0.3120)--(-1.933,0.3717)--(-1.888,0.4298)--(-1.844,0.4862)--(-1.799,0.5406)--(-1.755,0.5929)--(-1.710,0.6428)--(-1.666,0.6901)--(-1.622,0.7346)--(-1.577,0.7761)--(-1.533,0.8146)--(-1.488,0.8497)--(-1.444,0.8815)--(-1.399,0.9096)--(-1.355,0.9342)--(-1.311,0.9549)--(-1.266,0.9718)--(-1.222,0.9848)--(-1.177,0.9938)--(-1.133,0.9989)--(-1.088,0.9999)--(-1.044,0.9969)--(-0.9996,0.9898)--(-0.9552,0.9788)--(-0.9107,0.9638)--(-0.8663,0.9450)--(-0.8219,0.9224)--(-0.7775,0.8960)--(-0.7330,0.8660)--(-0.6886,0.8326)--(-0.6442,0.7958)--(-0.5998,0.7558)--(-0.5553,0.7127)--(-0.5109,0.6668)--(-0.4665,0.6182)--(-0.4221,0.5671)--(-0.3776,0.5137)--(-0.3332,0.4582)--(-0.2888,0.4009)--(-0.2443,0.3420)--(-0.1999,0.2817)--(-0.1555,0.2203)--(-0.1111,0.1580)--(-0.06664,0.09506)--(-0.02221,0.03173)--(0.02221,-0.03173)--(0.06664,-0.09506)--(0.1111,-0.1580)--(0.1555,-0.2203)--(0.1999,-0.2817)--(0.2443,-0.3420)--(0.2888,-0.4009)--(0.3332,-0.4582)--(0.3776,-0.5137)--(0.4221,-0.5671)--(0.4665,-0.6182)--(0.5109,-0.6668)--(0.5553,-0.7127)--(0.5998,-0.7558)--(0.6442,-0.7958)--(0.6886,-0.8326)--(0.7330,-0.8660)--(0.7775,-0.8960)--(0.8219,-0.9224)--(0.8663,-0.9450)--(0.9107,-0.9638)--(0.9552,-0.9788)--(0.9996,-0.9898)--(1.044,-0.9969)--(1.088,-0.9999)--(1.133,-0.9989)--(1.177,-0.9938)--(1.222,-0.9848)--(1.266,-0.9718)--(1.311,-0.9549)--(1.355,-0.9342)--(1.399,-0.9096)--(1.444,-0.8815)--(1.488,-0.8497)--(1.533,-0.8146)--(1.577,-0.7761)--(1.622,-0.7346)--(1.666,-0.6901)--(1.710,-0.6428)--(1.755,-0.5929)--(1.799,-0.5406)--(1.844,-0.4862)--(1.888,-0.4298)--(1.933,-0.3717)--(1.977,-0.3120)--(2.021,-0.2511)--(2.066,-0.1893)--(2.110,-0.1266)--(2.155,-0.06342)--(2.199,0);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,0.0000)--(-2.1546,0.0634)--(-2.1102,0.1265)--(-2.0658,0.1892)--(-2.0214,0.2511)--(-1.9769,0.3120)--(-1.9325,0.3716)--(-1.8881,0.4297)--(-1.8437,0.4861)--(-1.7992,0.5406)--(-1.7548,0.5929)--(-1.7104,0.6427)--(-1.6659,0.6900)--(-1.6215,0.7345)--(-1.5771,0.7761)--(-1.5327,0.8145)--(-1.4882,0.8497)--(-1.4438,0.8814)--(-1.3994,0.9096)--(-1.3550,0.9341)--(-1.3105,0.9549)--(-1.2661,0.9718)--(-1.2217,0.9848)--(-1.1773,0.9938)--(-1.1328,0.9988)--(-1.0884,0.9998)--(-1.0440,0.9968)--(-0.9995,0.9898)--(-0.9551,0.9788)--(-0.9107,0.9638)--(-0.8663,0.9450)--(-0.8218,0.9223)--(-0.7774,0.8959)--(-0.7330,0.8660)--(-0.6886,0.8325)--(-0.6441,0.7957)--(-0.5997,0.7557)--(-0.5553,0.7126)--(-0.5109,0.6667)--(-0.4664,0.6181)--(-0.4220,0.5670)--(-0.3776,0.5136)--(-0.3331,0.4582)--(-0.2887,0.4009)--(-0.2443,0.3420)--(-0.1999,0.2817)--(-0.1554,0.2203)--(-0.1110,0.1580)--(-0.0666,0.0950)--(-0.0222,0.0317)--(0.0222,-0.0317)--(0.0666,-0.0950)--(0.1110,-0.1580)--(0.1554,-0.2203)--(0.1999,-0.2817)--(0.2443,-0.3420)--(0.2887,-0.4009)--(0.3331,-0.4582)--(0.3776,-0.5136)--(0.4220,-0.5670)--(0.4664,-0.6181)--(0.5109,-0.6667)--(0.5553,-0.7126)--(0.5997,-0.7557)--(0.6441,-0.7957)--(0.6886,-0.8325)--(0.7330,-0.8660)--(0.7774,-0.8959)--(0.8218,-0.9223)--(0.8663,-0.9450)--(0.9107,-0.9638)--(0.9551,-0.9788)--(0.9995,-0.9898)--(1.0440,-0.9968)--(1.0884,-0.9998)--(1.1328,-0.9988)--(1.1773,-0.9938)--(1.2217,-0.9848)--(1.2661,-0.9718)--(1.3105,-0.9549)--(1.3550,-0.9341)--(1.3994,-0.9096)--(1.4438,-0.8814)--(1.4882,-0.8497)--(1.5327,-0.8145)--(1.5771,-0.7761)--(1.6215,-0.7345)--(1.6659,-0.6900)--(1.7104,-0.6427)--(1.7548,-0.5929)--(1.7992,-0.5406)--(1.8437,-0.4861)--(1.8881,-0.4297)--(1.9325,-0.3716)--(1.9769,-0.3120)--(2.0214,-0.2511)--(2.0658,-0.1892)--(2.1102,-0.1265)--(2.1546,-0.0634)--(2.1991,0.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -197,23 +197,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.4958) -- (0,1.4991);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4957) -- (0.0000,1.4990);
 %DEFAULT
 
-\draw [color=blue] (-2.199,0.9991)--(-2.155,0.9989)--(-2.110,0.9988)--(-2.066,0.9986)--(-2.021,0.9985)--(-1.977,0.9982)--(-1.933,0.9980)--(-1.888,0.9977)--(-1.844,0.9974)--(-1.799,0.9971)--(-1.755,0.9967)--(-1.710,0.9962)--(-1.666,0.9957)--(-1.622,0.9951)--(-1.577,0.9945)--(-1.533,0.9937)--(-1.488,0.9929)--(-1.444,0.9919)--(-1.399,0.9908)--(-1.355,0.9896)--(-1.311,0.9882)--(-1.266,0.9866)--(-1.222,0.9848)--(-1.177,0.9827)--(-1.133,0.9804)--(-1.088,0.9778)--(-1.044,0.9748)--(-0.9996,0.9714)--(-0.9552,0.9675)--(-0.9107,0.9632)--(-0.8663,0.9582)--(-0.8219,0.9526)--(-0.7775,0.9463)--(-0.7330,0.9391)--(-0.6886,0.9309)--(-0.6442,0.9217)--(-0.5998,0.9112)--(-0.5553,0.8994)--(-0.5109,0.8861)--(-0.4665,0.8710)--(-0.4221,0.8540)--(-0.3776,0.8348)--(-0.3332,0.8131)--(-0.2888,0.7888)--(-0.2443,0.7614)--(-0.1999,0.7306)--(-0.1555,0.6961)--(-0.1111,0.6575)--(-0.06664,0.6144)--(-0.02221,0.5663)--(0.02221,0.5129)--(0.06664,0.4537)--(0.1111,0.3884)--(0.1555,0.3165)--(0.1999,0.2379)--(0.2443,0.1525)--(0.2888,0.06012)--(0.3332,-0.03882)--(0.3776,-0.1438)--(0.4221,-0.2539)--(0.4665,-0.3676)--(0.5109,-0.4829)--(0.5553,-0.5972)--(0.5998,-0.7067)--(0.6442,-0.8071)--(0.6886,-0.8928)--(0.7330,-0.9577)--(0.7775,-0.9945)--(0.8219,-0.9956)--(0.8663,-0.9536)--(0.9107,-0.8620)--(0.9552,-0.7163)--(0.9996,-0.5159)--(1.044,-0.2656)--(1.088,0.02236)--(1.133,0.3265)--(1.177,0.6156)--(1.222,0.8497)--(1.266,0.9838)--(1.311,0.9759)--(1.355,0.7986)--(1.399,0.4537)--(1.444,-0.01290)--(1.488,-0.5041)--(1.533,-0.8808)--(1.577,-0.9958)--(1.622,-0.7547)--(1.666,-0.1896)--(1.710,0.4945)--(1.755,0.9556)--(1.799,0.8754)--(1.844,0.2082)--(1.888,-0.6464)--(1.933,-0.9946)--(1.977,-0.4171)--(2.021,0.6239)--(2.066,0.9613)--(2.110,0.03819)--(2.155,-0.9628)--(2.199,-0.4089);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,0.9990)--(-2.1546,0.9989)--(-2.1102,0.9987)--(-2.0658,0.9986)--(-2.0214,0.9984)--(-1.9769,0.9982)--(-1.9325,0.9980)--(-1.8881,0.9977)--(-1.8437,0.9974)--(-1.7992,0.9970)--(-1.7548,0.9966)--(-1.7104,0.9962)--(-1.6659,0.9957)--(-1.6215,0.9951)--(-1.5771,0.9944)--(-1.5327,0.9937)--(-1.4882,0.9928)--(-1.4438,0.9919)--(-1.3994,0.9908)--(-1.3550,0.9896)--(-1.3105,0.9881)--(-1.2661,0.9866)--(-1.2217,0.9847)--(-1.1773,0.9827)--(-1.1328,0.9804)--(-1.0884,0.9777)--(-1.0440,0.9747)--(-0.9995,0.9713)--(-0.9551,0.9675)--(-0.9107,0.9631)--(-0.8663,0.9582)--(-0.8218,0.9526)--(-0.7774,0.9462)--(-0.7330,0.9390)--(-0.6886,0.9309)--(-0.6441,0.9216)--(-0.5997,0.9112)--(-0.5553,0.8994)--(-0.5109,0.8860)--(-0.4664,0.8710)--(-0.4220,0.8539)--(-0.3776,0.8347)--(-0.3331,0.8131)--(-0.2887,0.7887)--(-0.2443,0.7613)--(-0.1999,0.7306)--(-0.1554,0.6961)--(-0.1110,0.6575)--(-0.0666,0.6143)--(-0.0222,0.5663)--(0.0222,0.5128)--(0.0666,0.4537)--(0.1110,0.3883)--(0.1554,0.3165)--(0.1999,0.2379)--(0.2443,0.1524)--(0.2887,0.0601)--(0.3331,-0.0388)--(0.3776,-0.1437)--(0.4220,-0.2538)--(0.4664,-0.3675)--(0.5109,-0.4829)--(0.5553,-0.5971)--(0.5997,-0.7066)--(0.6441,-0.8070)--(0.6886,-0.8928)--(0.7330,-0.9576)--(0.7774,-0.9944)--(0.8218,-0.9956)--(0.8663,-0.9536)--(0.9107,-0.8619)--(0.9551,-0.7163)--(0.9995,-0.5158)--(1.0440,-0.2655)--(1.0884,0.0223)--(1.1328,0.3265)--(1.1773,0.6156)--(1.2217,0.8496)--(1.2661,0.9838)--(1.3105,0.9759)--(1.3550,0.7985)--(1.3994,0.4536)--(1.4438,-0.0129)--(1.4882,-0.5041)--(1.5327,-0.8808)--(1.5771,-0.9957)--(1.6215,-0.7546)--(1.6659,-0.1895)--(1.7104,0.4944)--(1.7548,0.9555)--(1.7992,0.8753)--(1.8437,0.2081)--(1.8881,-0.6463)--(1.9325,-0.9945)--(1.9769,-0.4171)--(2.0214,0.6238)--(2.0658,0.9612)--(2.1102,0.0381)--(2.1546,-0.9628)--(2.1991,-0.4089);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -271,25 +271,25 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.4995);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.4994);
 %DEFAULT
 
-\draw [color=blue] (-2.199,1.000)--(-2.155,1.002)--(-2.110,1.008)--(-2.066,1.018)--(-2.021,1.032)--(-1.977,1.050)--(-1.933,1.072)--(-1.888,1.097)--(-1.844,1.126)--(-1.799,1.159)--(-1.755,1.195)--(-1.710,1.234)--(-1.666,1.276)--(-1.622,1.321)--(-1.577,1.369)--(-1.533,1.420)--(-1.488,1.473)--(-1.444,1.528)--(-1.399,1.585)--(-1.355,1.643)--(-1.311,1.703)--(-1.266,1.764)--(-1.222,1.826)--(-1.177,1.889)--(-1.133,1.952)--(-1.088,2.016)--(-1.044,2.079)--(-0.9996,2.142)--(-0.9552,2.205)--(-0.9107,2.266)--(-0.8663,2.327)--(-0.8219,2.386)--(-0.7775,2.444)--(-0.7330,2.500)--(-0.6886,2.554)--(-0.6442,2.606)--(-0.5998,2.655)--(-0.5553,2.701)--(-0.5109,2.745)--(-0.4665,2.786)--(-0.4221,2.824)--(-0.3776,2.858)--(-0.3332,2.889)--(-0.2888,2.916)--(-0.2443,2.940)--(-0.1999,2.960)--(-0.1555,2.975)--(-0.1111,2.987)--(-0.06664,2.995)--(-0.02221,3.000)--(0.02221,3.000)--(0.06664,2.995)--(0.1111,2.987)--(0.1555,2.975)--(0.1999,2.960)--(0.2443,2.940)--(0.2888,2.916)--(0.3332,2.889)--(0.3776,2.858)--(0.4221,2.824)--(0.4665,2.786)--(0.5109,2.745)--(0.5553,2.701)--(0.5998,2.655)--(0.6442,2.606)--(0.6886,2.554)--(0.7330,2.500)--(0.7775,2.444)--(0.8219,2.386)--(0.8663,2.327)--(0.9107,2.266)--(0.9552,2.205)--(0.9996,2.142)--(1.044,2.079)--(1.088,2.016)--(1.133,1.952)--(1.177,1.889)--(1.222,1.826)--(1.266,1.764)--(1.311,1.703)--(1.355,1.643)--(1.399,1.585)--(1.444,1.528)--(1.488,1.473)--(1.533,1.420)--(1.577,1.369)--(1.622,1.321)--(1.666,1.276)--(1.710,1.234)--(1.755,1.195)--(1.799,1.159)--(1.844,1.126)--(1.888,1.097)--(1.933,1.072)--(1.977,1.050)--(2.021,1.032)--(2.066,1.018)--(2.110,1.008)--(2.155,1.002)--(2.199,1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
-\draw (-0.31058,3.1416) node {$ \pi $};
-\draw [] (-0.100,3.14) -- (0.100,3.14);
+\draw [color=blue] (-2.1991,1.0000)--(-2.1546,1.0020)--(-2.1102,1.0080)--(-2.0658,1.0180)--(-2.0214,1.0320)--(-1.9769,1.0499)--(-1.9325,1.0716)--(-1.8881,1.0970)--(-1.8437,1.1261)--(-1.7992,1.1587)--(-1.7548,1.1947)--(-1.7104,1.2339)--(-1.6659,1.2762)--(-1.6215,1.3214)--(-1.5771,1.3694)--(-1.5327,1.4199)--(-1.4882,1.4727)--(-1.4438,1.5277)--(-1.3994,1.5845)--(-1.3550,1.6431)--(-1.3105,1.7030)--(-1.2661,1.7642)--(-1.2217,1.8263)--(-1.1773,1.8891)--(-1.1328,1.9524)--(-1.0884,2.0158)--(-1.0440,2.0792)--(-0.9995,2.1423)--(-0.9551,2.2048)--(-0.9107,2.2664)--(-0.8663,2.3270)--(-0.8218,2.3863)--(-0.7774,2.4440)--(-0.7330,2.5000)--(-0.6886,2.5539)--(-0.6441,2.6056)--(-0.5997,2.6548)--(-0.5553,2.7014)--(-0.5109,2.7452)--(-0.4664,2.7860)--(-0.4220,2.8236)--(-0.3776,2.8579)--(-0.3331,2.8888)--(-0.2887,2.9161)--(-0.2443,2.9396)--(-0.1999,2.9594)--(-0.1554,2.9754)--(-0.1110,2.9874)--(-0.0666,2.9954)--(-0.0222,2.9994)--(0.0222,2.9994)--(0.0666,2.9954)--(0.1110,2.9874)--(0.1554,2.9754)--(0.1999,2.9594)--(0.2443,2.9396)--(0.2887,2.9161)--(0.3331,2.8888)--(0.3776,2.8579)--(0.4220,2.8236)--(0.4664,2.7860)--(0.5109,2.7452)--(0.5553,2.7014)--(0.5997,2.6548)--(0.6441,2.6056)--(0.6886,2.5539)--(0.7330,2.5000)--(0.7774,2.4440)--(0.8218,2.3863)--(0.8663,2.3270)--(0.9107,2.2664)--(0.9551,2.2048)--(0.9995,2.1423)--(1.0440,2.0792)--(1.0884,2.0158)--(1.1328,1.9524)--(1.1773,1.8891)--(1.2217,1.8263)--(1.2661,1.7642)--(1.3105,1.7030)--(1.3550,1.6431)--(1.3994,1.5845)--(1.4438,1.5277)--(1.4882,1.4727)--(1.5327,1.4199)--(1.5771,1.3694)--(1.6215,1.3214)--(1.6659,1.2762)--(1.7104,1.2339)--(1.7548,1.1947)--(1.7992,1.1587)--(1.8437,1.1261)--(1.8881,1.0970)--(1.9325,1.0716)--(1.9769,1.0499)--(2.0214,1.0320)--(2.0658,1.0180)--(2.1102,1.0080)--(2.1546,1.0020)--(2.1991,1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
+\draw (-0.3105,3.1415) node {$ \pi $};
+\draw [] (-0.1000,3.1415) -- (0.1000,3.1415);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -343,23 +343,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-1.4995) -- (0,1.5000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4994) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,1.000)--(-2.155,0.9679)--(-2.110,0.8738)--(-2.066,0.7237)--(-2.021,0.5272)--(-1.977,0.2969)--(-1.933,0.04758)--(-1.888,-0.2048)--(-1.844,-0.4441)--(-1.799,-0.6549)--(-1.755,-0.8237)--(-1.710,-0.9397)--(-1.666,-0.9955)--(-1.622,-0.9874)--(-1.577,-0.9161)--(-1.533,-0.7861)--(-1.488,-0.6056)--(-1.444,-0.3863)--(-1.399,-0.1423)--(-1.355,0.1108)--(-1.311,0.3569)--(-1.266,0.5801)--(-1.222,0.7660)--(-1.177,0.9029)--(-1.133,0.9819)--(-1.088,0.9980)--(-1.044,0.9501)--(-0.9996,0.8413)--(-0.9552,0.6785)--(-0.9107,0.4723)--(-0.8663,0.2358)--(-0.8219,-0.01587)--(-0.7775,-0.2665)--(-0.7330,-0.5000)--(-0.6886,-0.7015)--(-0.6442,-0.8580)--(-0.5998,-0.9595)--(-0.5553,-0.9995)--(-0.5109,-0.9754)--(-0.4665,-0.8888)--(-0.4221,-0.7453)--(-0.3776,-0.5539)--(-0.3332,-0.3271)--(-0.2888,-0.07925)--(-0.2443,0.1736)--(-0.1999,0.4154)--(-0.1555,0.6306)--(-0.1111,0.8053)--(-0.06664,0.9284)--(-0.02221,0.9920)--(0.02221,0.9920)--(0.06664,0.9284)--(0.1111,0.8053)--(0.1555,0.6306)--(0.1999,0.4154)--(0.2443,0.1736)--(0.2888,-0.07925)--(0.3332,-0.3271)--(0.3776,-0.5539)--(0.4221,-0.7453)--(0.4665,-0.8888)--(0.5109,-0.9754)--(0.5553,-0.9995)--(0.5998,-0.9595)--(0.6442,-0.8580)--(0.6886,-0.7015)--(0.7330,-0.5000)--(0.7775,-0.2665)--(0.8219,-0.01587)--(0.8663,0.2358)--(0.9107,0.4723)--(0.9552,0.6785)--(0.9996,0.8413)--(1.044,0.9501)--(1.088,0.9980)--(1.133,0.9819)--(1.177,0.9029)--(1.222,0.7660)--(1.266,0.5801)--(1.311,0.3569)--(1.355,0.1108)--(1.399,-0.1423)--(1.444,-0.3863)--(1.488,-0.6056)--(1.533,-0.7861)--(1.577,-0.9161)--(1.622,-0.9874)--(1.666,-0.9955)--(1.710,-0.9397)--(1.755,-0.8237)--(1.799,-0.6549)--(1.844,-0.4441)--(1.888,-0.2048)--(1.933,0.04758)--(1.977,0.2969)--(2.021,0.5272)--(2.066,0.7237)--(2.110,0.8738)--(2.155,0.9679)--(2.199,1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,1.0000)--(-2.1546,0.9679)--(-2.1102,0.8738)--(-2.0658,0.7237)--(-2.0214,0.5272)--(-1.9769,0.2969)--(-1.9325,0.0475)--(-1.8881,-0.2048)--(-1.8437,-0.4440)--(-1.7992,-0.6548)--(-1.7548,-0.8236)--(-1.7104,-0.9396)--(-1.6659,-0.9954)--(-1.6215,-0.9874)--(-1.5771,-0.9161)--(-1.5327,-0.7860)--(-1.4882,-0.6056)--(-1.4438,-0.3863)--(-1.3994,-0.1423)--(-1.3550,0.1108)--(-1.3105,0.3568)--(-1.2661,0.5800)--(-1.2217,0.7660)--(-1.1773,0.9029)--(-1.1328,0.9819)--(-1.0884,0.9979)--(-1.0440,0.9500)--(-0.9995,0.8412)--(-0.9551,0.6785)--(-0.9107,0.4722)--(-0.8663,0.2357)--(-0.8218,-0.0158)--(-0.7774,-0.2664)--(-0.7330,-0.5000)--(-0.6886,-0.7014)--(-0.6441,-0.8579)--(-0.5997,-0.9594)--(-0.5553,-0.9994)--(-0.5109,-0.9754)--(-0.4664,-0.8888)--(-0.4220,-0.7452)--(-0.3776,-0.5539)--(-0.3331,-0.3270)--(-0.2887,-0.0792)--(-0.2443,0.1736)--(-0.1999,0.4154)--(-0.1554,0.6305)--(-0.1110,0.8052)--(-0.0666,0.9283)--(-0.0222,0.9919)--(0.0222,0.9919)--(0.0666,0.9283)--(0.1110,0.8052)--(0.1554,0.6305)--(0.1999,0.4154)--(0.2443,0.1736)--(0.2887,-0.0792)--(0.3331,-0.3270)--(0.3776,-0.5539)--(0.4220,-0.7452)--(0.4664,-0.8888)--(0.5109,-0.9754)--(0.5553,-0.9994)--(0.5997,-0.9594)--(0.6441,-0.8579)--(0.6886,-0.7014)--(0.7330,-0.4999)--(0.7774,-0.2664)--(0.8218,-0.0158)--(0.8663,0.2357)--(0.9107,0.4722)--(0.9551,0.6785)--(0.9995,0.8412)--(1.0440,0.9500)--(1.0884,0.9979)--(1.1328,0.9819)--(1.1773,0.9029)--(1.2217,0.7660)--(1.2661,0.5800)--(1.3105,0.3568)--(1.3550,0.1108)--(1.3994,-0.1423)--(1.4438,-0.3863)--(1.4882,-0.6056)--(1.5327,-0.7860)--(1.5771,-0.9161)--(1.6215,-0.9874)--(1.6659,-0.9954)--(1.7104,-0.9396)--(1.7548,-0.8236)--(1.7992,-0.6548)--(1.8437,-0.4440)--(1.8881,-0.2048)--(1.9325,0.0475)--(1.9769,0.2969)--(2.0214,0.5272)--(2.0658,0.7237)--(2.1102,0.8738)--(2.1546,0.9679)--(2.1991,1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -413,23 +413,23 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.6991,0) -- (2.6991,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.5000);
+\draw [,->,>=latex] (-2.6991,0.0000) -- (2.6991,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
 %DEFAULT
 
-\draw [color=blue] (-2.199,1.000)--(-2.155,0.9980)--(-2.110,0.9920)--(-2.066,0.9819)--(-2.021,0.9679)--(-1.977,0.9501)--(-1.933,0.9284)--(-1.888,0.9029)--(-1.844,0.8738)--(-1.799,0.8413)--(-1.755,0.8053)--(-1.710,0.7660)--(-1.666,0.7237)--(-1.622,0.6785)--(-1.577,0.6306)--(-1.533,0.5801)--(-1.488,0.5272)--(-1.444,0.4723)--(-1.399,0.4154)--(-1.355,0.3569)--(-1.311,0.2969)--(-1.266,0.2358)--(-1.222,0.1736)--(-1.177,0.1108)--(-1.133,0.04758)--(-1.088,0.01587)--(-1.044,0.07925)--(-0.9996,0.1423)--(-0.9552,0.2048)--(-0.9107,0.2665)--(-0.8663,0.3271)--(-0.8219,0.3863)--(-0.7775,0.4441)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6442,0.6056)--(-0.5998,0.6549)--(-0.5553,0.7015)--(-0.5109,0.7453)--(-0.4665,0.7861)--(-0.4221,0.8237)--(-0.3776,0.8580)--(-0.3332,0.8888)--(-0.2888,0.9161)--(-0.2443,0.9397)--(-0.1999,0.9595)--(-0.1555,0.9754)--(-0.1111,0.9874)--(-0.06664,0.9955)--(-0.02221,0.9995)--(0.02221,0.9995)--(0.06664,0.9955)--(0.1111,0.9874)--(0.1555,0.9754)--(0.1999,0.9595)--(0.2443,0.9397)--(0.2888,0.9161)--(0.3332,0.8888)--(0.3776,0.8580)--(0.4221,0.8237)--(0.4665,0.7861)--(0.5109,0.7453)--(0.5553,0.7015)--(0.5998,0.6549)--(0.6442,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7775,0.4441)--(0.8219,0.3863)--(0.8663,0.3271)--(0.9107,0.2665)--(0.9552,0.2048)--(0.9996,0.1423)--(1.044,0.07925)--(1.088,0.01587)--(1.133,0.04758)--(1.177,0.1108)--(1.222,0.1736)--(1.266,0.2358)--(1.311,0.2969)--(1.355,0.3569)--(1.399,0.4154)--(1.444,0.4723)--(1.488,0.5272)--(1.533,0.5801)--(1.577,0.6306)--(1.622,0.6785)--(1.666,0.7237)--(1.710,0.7660)--(1.755,0.8053)--(1.799,0.8413)--(1.844,0.8738)--(1.888,0.9029)--(1.933,0.9284)--(1.977,0.9501)--(2.021,0.9679)--(2.066,0.9819)--(2.110,0.9920)--(2.155,0.9980)--(2.199,1.000);
-\draw (-2.1000,-0.32983) node {$ -3 $};
-\draw [] (-2.10,-0.100) -- (-2.10,0.100);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
-\draw (2.1000,-0.31492) node {$ 3 $};
-\draw [] (2.10,-0.100) -- (2.10,0.100);
+\draw [color=blue] (-2.1991,1.0000)--(-2.1546,0.9979)--(-2.1102,0.9919)--(-2.0658,0.9819)--(-2.0214,0.9679)--(-1.9769,0.9500)--(-1.9325,0.9283)--(-1.8881,0.9029)--(-1.8437,0.8738)--(-1.7992,0.8412)--(-1.7548,0.8052)--(-1.7104,0.7660)--(-1.6659,0.7237)--(-1.6215,0.6785)--(-1.5771,0.6305)--(-1.5327,0.5800)--(-1.4882,0.5272)--(-1.4438,0.4722)--(-1.3994,0.4154)--(-1.3550,0.3568)--(-1.3105,0.2969)--(-1.2661,0.2357)--(-1.2217,0.1736)--(-1.1773,0.1108)--(-1.1328,0.0475)--(-1.0884,0.0158)--(-1.0440,0.0792)--(-0.9995,0.1423)--(-0.9551,0.2048)--(-0.9107,0.2664)--(-0.8663,0.3270)--(-0.8218,0.3863)--(-0.7774,0.4440)--(-0.7330,0.5000)--(-0.6886,0.5539)--(-0.6441,0.6056)--(-0.5997,0.6548)--(-0.5553,0.7014)--(-0.5109,0.7452)--(-0.4664,0.7860)--(-0.4220,0.8236)--(-0.3776,0.8579)--(-0.3331,0.8888)--(-0.2887,0.9161)--(-0.2443,0.9396)--(-0.1999,0.9594)--(-0.1554,0.9754)--(-0.1110,0.9874)--(-0.0666,0.9954)--(-0.0222,0.9994)--(0.0222,0.9994)--(0.0666,0.9954)--(0.1110,0.9874)--(0.1554,0.9754)--(0.1999,0.9594)--(0.2443,0.9396)--(0.2887,0.9161)--(0.3331,0.8888)--(0.3776,0.8579)--(0.4220,0.8236)--(0.4664,0.7860)--(0.5109,0.7452)--(0.5553,0.7014)--(0.5997,0.6548)--(0.6441,0.6056)--(0.6886,0.5539)--(0.7330,0.5000)--(0.7774,0.4440)--(0.8218,0.3863)--(0.8663,0.3270)--(0.9107,0.2664)--(0.9551,0.2048)--(0.9995,0.1423)--(1.0440,0.0792)--(1.0884,0.0158)--(1.1328,0.0475)--(1.1773,0.1108)--(1.2217,0.1736)--(1.2661,0.2357)--(1.3105,0.2969)--(1.3550,0.3568)--(1.3994,0.4154)--(1.4438,0.4722)--(1.4882,0.5272)--(1.5327,0.5800)--(1.5771,0.6305)--(1.6215,0.6785)--(1.6659,0.7237)--(1.7104,0.7660)--(1.7548,0.8052)--(1.7992,0.8412)--(1.8437,0.8738)--(1.8881,0.9029)--(1.9325,0.9283)--(1.9769,0.9500)--(2.0214,0.9679)--(2.0658,0.9819)--(2.1102,0.9919)--(2.1546,0.9979)--(2.1991,1.0000);
+\draw (-2.1000,-0.3298) node {$ -3 $};
+\draw [] (-2.1000,-0.1000) -- (-2.1000,0.1000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
+\draw (2.1000,-0.3149) node {$ 3 $};
+\draw [] (2.1000,-0.1000) -- (2.1000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
@@ -475,19 +475,19 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5996,0) -- (1.5996,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,1.4999);
+\draw [,->,>=latex] (-1.5995,0.0000) -- (1.5995,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.4999);
 %DEFAULT
 
-\draw [color=blue] (-1.100,0)--(-1.077,0.1781)--(-1.055,0.2518)--(-1.033,0.3083)--(-1.011,0.3558)--(-0.9885,0.3975)--(-0.9663,0.4350)--(-0.9441,0.4694)--(-0.9219,0.5011)--(-0.8996,0.5308)--(-0.8774,0.5586)--(-0.8552,0.5848)--(-0.8330,0.6096)--(-0.8108,0.6332)--(-0.7886,0.6556)--(-0.7664,0.6769)--(-0.7441,0.6973)--(-0.7219,0.7167)--(-0.6997,0.7353)--(-0.6775,0.7530)--(-0.6553,0.7700)--(-0.6331,0.7862)--(-0.6109,0.8017)--(-0.5887,0.8166)--(-0.5664,0.8307)--(-0.5442,0.8442)--(-0.5220,0.8571)--(-0.4998,0.8693)--(-0.4776,0.8810)--(-0.4554,0.8921)--(-0.4332,0.9025)--(-0.4109,0.9125)--(-0.3887,0.9218)--(-0.3665,0.9306)--(-0.3443,0.9389)--(-0.3221,0.9466)--(-0.2999,0.9537)--(-0.2777,0.9604)--(-0.2555,0.9665)--(-0.2332,0.9721)--(-0.2110,0.9772)--(-0.1888,0.9818)--(-0.1666,0.9858)--(-0.1444,0.9893)--(-0.1222,0.9924)--(-0.09996,0.9949)--(-0.07775,0.9969)--(-0.05553,0.9984)--(-0.03332,0.9994)--(-0.01111,0.9999)--(0.01111,0.9999)--(0.03332,0.9994)--(0.05553,0.9984)--(0.07775,0.9969)--(0.09996,0.9949)--(0.1222,0.9924)--(0.1444,0.9893)--(0.1666,0.9858)--(0.1888,0.9818)--(0.2110,0.9772)--(0.2332,0.9721)--(0.2555,0.9665)--(0.2777,0.9604)--(0.2999,0.9537)--(0.3221,0.9466)--(0.3443,0.9389)--(0.3665,0.9306)--(0.3887,0.9218)--(0.4109,0.9125)--(0.4332,0.9025)--(0.4554,0.8921)--(0.4776,0.8810)--(0.4998,0.8693)--(0.5220,0.8571)--(0.5442,0.8442)--(0.5664,0.8307)--(0.5887,0.8166)--(0.6109,0.8017)--(0.6331,0.7862)--(0.6553,0.7700)--(0.6775,0.7530)--(0.6997,0.7353)--(0.7219,0.7167)--(0.7441,0.6973)--(0.7664,0.6769)--(0.7886,0.6556)--(0.8108,0.6332)--(0.8330,0.6096)--(0.8552,0.5848)--(0.8774,0.5586)--(0.8996,0.5308)--(0.9219,0.5011)--(0.9441,0.4694)--(0.9663,0.4350)--(0.9885,0.3975)--(1.011,0.3558)--(1.033,0.3083)--(1.055,0.2518)--(1.077,0.1781)--(1.100,0);
-\draw (-1.4000,-0.32983) node {$ -2 $};
-\draw [] (-1.40,-0.100) -- (-1.40,0.100);
-\draw (-0.70000,-0.32983) node {$ -1 $};
-\draw [] (-0.700,-0.100) -- (-0.700,0.100);
-\draw (0.70000,-0.31492) node {$ 1 $};
-\draw [] (0.700,-0.100) -- (0.700,0.100);
-\draw (1.4000,-0.31492) node {$ 2 $};
-\draw [] (1.40,-0.100) -- (1.40,0.100);
+\draw [color=blue] (-1.0995,0.0000)--(-1.0773,0.1781)--(-1.0551,0.2518)--(-1.0329,0.3083)--(-1.0107,0.3557)--(-0.9884,0.3974)--(-0.9662,0.4350)--(-0.9440,0.4693)--(-0.9218,0.5011)--(-0.8996,0.5307)--(-0.8774,0.5585)--(-0.8552,0.5848)--(-0.8329,0.6096)--(-0.8107,0.6331)--(-0.7885,0.6555)--(-0.7663,0.6769)--(-0.7441,0.6972)--(-0.7219,0.7167)--(-0.6997,0.7352)--(-0.6775,0.7530)--(-0.6552,0.7700)--(-0.6330,0.7862)--(-0.6108,0.8017)--(-0.5886,0.8165)--(-0.5664,0.8307)--(-0.5442,0.8442)--(-0.5220,0.8570)--(-0.4997,0.8693)--(-0.4775,0.8809)--(-0.4553,0.8920)--(-0.4331,0.9025)--(-0.4109,0.9124)--(-0.3887,0.9218)--(-0.3665,0.9306)--(-0.3443,0.9388)--(-0.3220,0.9465)--(-0.2998,0.9537)--(-0.2776,0.9603)--(-0.2554,0.9665)--(-0.2332,0.9721)--(-0.2110,0.9771)--(-0.1888,0.9817)--(-0.1665,0.9858)--(-0.1443,0.9893)--(-0.1221,0.9923)--(-0.0999,0.9948)--(-0.0777,0.9969)--(-0.0555,0.9984)--(-0.0333,0.9994)--(-0.0111,0.9999)--(0.0111,0.9999)--(0.0333,0.9994)--(0.0555,0.9984)--(0.0777,0.9969)--(0.0999,0.9948)--(0.1221,0.9923)--(0.1443,0.9893)--(0.1665,0.9858)--(0.1888,0.9817)--(0.2110,0.9771)--(0.2332,0.9721)--(0.2554,0.9665)--(0.2776,0.9603)--(0.2998,0.9537)--(0.3220,0.9465)--(0.3443,0.9388)--(0.3665,0.9306)--(0.3887,0.9218)--(0.4109,0.9124)--(0.4331,0.9025)--(0.4553,0.8920)--(0.4775,0.8809)--(0.4997,0.8693)--(0.5220,0.8570)--(0.5442,0.8442)--(0.5664,0.8307)--(0.5886,0.8165)--(0.6108,0.8017)--(0.6330,0.7862)--(0.6552,0.7700)--(0.6775,0.7530)--(0.6997,0.7352)--(0.7219,0.7167)--(0.7441,0.6972)--(0.7663,0.6769)--(0.7885,0.6555)--(0.8107,0.6331)--(0.8329,0.6096)--(0.8552,0.5848)--(0.8774,0.5585)--(0.8996,0.5307)--(0.9218,0.5011)--(0.9440,0.4693)--(0.9662,0.4350)--(0.9884,0.3974)--(1.0107,0.3557)--(1.0329,0.3083)--(1.0551,0.2518)--(1.0773,0.1781)--(1.0995,0.0000);
+\draw (-1.4000,-0.3298) node {$ -2 $};
+\draw [] (-1.4000,-0.1000) -- (-1.4000,0.1000);
+\draw (-0.7000,-0.3298) node {$ -1 $};
+\draw [] (-0.7000,-0.1000) -- (-0.7000,0.1000);
+\draw (0.7000,-0.3149) node {$ 1 $};
+\draw [] (0.7000,-0.1000) -- (0.7000,0.1000);
+\draw (1.4000,-0.3149) node {$ 2 $};
+\draw [] (1.4000,-0.1000) -- (1.4000,0.1000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall b/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall
index accdd4394..71befdfe6 100644
--- a/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall
+++ b/src_phystricks/Fig_VSZRooRWgUGu.pstricks.recall
@@ -79,11 +79,11 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (8.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,4.4000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,4.4000);
 %DEFAULT
 
-\draw [color=red] (1.500,1.500)--(1.561,1.617)--(1.621,1.724)--(1.682,1.824)--(1.742,1.917)--(1.803,2.004)--(1.864,2.085)--(1.924,2.161)--(1.985,2.233)--(2.045,2.300)--(2.106,2.363)--(2.167,2.423)--(2.227,2.480)--(2.288,2.533)--(2.348,2.584)--(2.409,2.632)--(2.470,2.678)--(2.530,2.722)--(2.591,2.763)--(2.652,2.803)--(2.712,2.841)--(2.773,2.877)--(2.833,2.912)--(2.894,2.945)--(2.955,2.977)--(3.015,3.008)--(3.076,3.037)--(3.136,3.065)--(3.197,3.092)--(3.258,3.119)--(3.318,3.144)--(3.379,3.168)--(3.439,3.192)--(3.500,3.214)--(3.561,3.236)--(3.621,3.257)--(3.682,3.278)--(3.742,3.298)--(3.803,3.317)--(3.864,3.335)--(3.924,3.353)--(3.985,3.371)--(4.045,3.388)--(4.106,3.404)--(4.167,3.420)--(4.227,3.435)--(4.288,3.451)--(4.349,3.465)--(4.409,3.479)--(4.470,3.493)--(4.530,3.507)--(4.591,3.520)--(4.651,3.533)--(4.712,3.545)--(4.773,3.557)--(4.833,3.569)--(4.894,3.581)--(4.955,3.592)--(5.015,3.603)--(5.076,3.613)--(5.136,3.624)--(5.197,3.634)--(5.258,3.644)--(5.318,3.654)--(5.379,3.663)--(5.439,3.673)--(5.500,3.682)--(5.561,3.691)--(5.621,3.699)--(5.682,3.708)--(5.742,3.716)--(5.803,3.725)--(5.864,3.733)--(5.924,3.740)--(5.985,3.748)--(6.045,3.756)--(6.106,3.763)--(6.167,3.770)--(6.227,3.777)--(6.288,3.784)--(6.349,3.791)--(6.409,3.798)--(6.470,3.804)--(6.530,3.811)--(6.591,3.817)--(6.651,3.823)--(6.712,3.830)--(6.773,3.836)--(6.833,3.841)--(6.894,3.847)--(6.955,3.853)--(7.015,3.859)--(7.076,3.864)--(7.136,3.869)--(7.197,3.875)--(7.258,3.880)--(7.318,3.885)--(7.379,3.890)--(7.439,3.895)--(7.500,3.900);
+\draw [color=red] (1.5000,1.5000)--(1.5606,1.6165)--(1.6212,1.7242)--(1.6818,1.8243)--(1.7424,1.9173)--(1.8030,2.0042)--(1.8636,2.0853)--(1.9242,2.1614)--(1.9848,2.2328)--(2.0454,2.3000)--(2.1060,2.3633)--(2.1666,2.4230)--(2.2272,2.4795)--(2.2878,2.5331)--(2.3484,2.5838)--(2.4090,2.6320)--(2.4696,2.6779)--(2.5303,2.7215)--(2.5909,2.7631)--(2.6515,2.8028)--(2.7121,2.8407)--(2.7727,2.8770)--(2.8333,2.9117)--(2.8939,2.9450)--(2.9545,2.9769)--(3.0151,3.0075)--(3.0757,3.0369)--(3.1363,3.0652)--(3.1969,3.0924)--(3.2575,3.1186)--(3.3181,3.1438)--(3.3787,3.1681)--(3.4393,3.1916)--(3.5000,3.2142)--(3.5606,3.2361)--(3.6212,3.2573)--(3.6818,3.2777)--(3.7424,3.2975)--(3.8030,3.3167)--(3.8636,3.3352)--(3.9242,3.3532)--(3.9848,3.3707)--(4.0454,3.3876)--(4.1060,3.4040)--(4.1666,3.4200)--(4.2272,3.4354)--(4.2878,3.4505)--(4.3484,3.4651)--(4.4090,3.4793)--(4.4696,3.4932)--(4.5303,3.5066)--(4.5909,3.5198)--(4.6515,3.5325)--(4.7121,3.5450)--(4.7727,3.5571)--(4.8333,3.5689)--(4.8939,3.5804)--(4.9545,3.5917)--(5.0151,3.6027)--(5.0757,3.6134)--(5.1363,3.6238)--(5.1969,3.6341)--(5.2575,3.6440)--(5.3181,3.6538)--(5.3787,3.6633)--(5.4393,3.6727)--(5.5000,3.6818)--(5.5606,3.6907)--(5.6212,3.6994)--(5.6818,3.7080)--(5.7424,3.7163)--(5.8030,3.7245)--(5.8636,3.7325)--(5.9242,3.7404)--(5.9848,3.7481)--(6.0454,3.7556)--(6.1060,3.7630)--(6.1666,3.7702)--(6.2272,3.7773)--(6.2878,3.7843)--(6.3484,3.7911)--(6.4090,3.7978)--(6.4696,3.8044)--(6.5303,3.8109)--(6.5909,3.8172)--(6.6515,3.8234)--(6.7121,3.8295)--(6.7727,3.8355)--(6.8333,3.8414)--(6.8939,3.8472)--(6.9545,3.8529)--(7.0151,3.8585)--(7.0757,3.8640)--(7.1363,3.8694)--(7.1969,3.8747)--(7.2575,3.8799)--(7.3181,3.8850)--(7.3787,3.8901)--(7.4393,3.8951)--(7.5000,3.9000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -91,15 +91,15 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw (6.0000,-0.3785) node {$x$};
 \draw (8.3552,3.9000) node {$f(x)$};
-\draw (9.6702,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
+\draw (9.6701,1.8750) node {$S=F(x)=\int_a^xf(t)dt$};
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -107,11 +107,11 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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+\draw [color=blue] (3.0000,0.0000)--(3.0303,0.0000)--(3.0606,0.0000)--(3.0909,0.0000)--(3.1212,0.0000)--(3.1515,0.0000)--(3.1818,0.0000)--(3.2121,0.0000)--(3.2424,0.0000)--(3.2727,0.0000)--(3.3030,0.0000)--(3.3333,0.0000)--(3.3636,0.0000)--(3.3939,0.0000)--(3.4242,0.0000)--(3.4545,0.0000)--(3.4848,0.0000)--(3.5151,0.0000)--(3.5454,0.0000)--(3.5757,0.0000)--(3.6060,0.0000)--(3.6363,0.0000)--(3.6666,0.0000)--(3.6969,0.0000)--(3.7272,0.0000)--(3.7575,0.0000)--(3.7878,0.0000)--(3.8181,0.0000)--(3.8484,0.0000)--(3.8787,0.0000)--(3.9090,0.0000)--(3.9393,0.0000)--(3.9696,0.0000)--(4.0000,0.0000)--(4.0303,0.0000)--(4.0606,0.0000)--(4.0909,0.0000)--(4.1212,0.0000)--(4.1515,0.0000)--(4.1818,0.0000)--(4.2121,0.0000)--(4.2424,0.0000)--(4.2727,0.0000)--(4.3030,0.0000)--(4.3333,0.0000)--(4.3636,0.0000)--(4.3939,0.0000)--(4.4242,0.0000)--(4.4545,0.0000)--(4.4848,0.0000)--(4.5151,0.0000)--(4.5454,0.0000)--(4.5757,0.0000)--(4.6060,0.0000)--(4.6363,0.0000)--(4.6666,0.0000)--(4.6969,0.0000)--(4.7272,0.0000)--(4.7575,0.0000)--(4.7878,0.0000)--(4.8181,0.0000)--(4.8484,0.0000)--(4.8787,0.0000)--(4.9090,0.0000)--(4.9393,0.0000)--(4.9696,0.0000)--(5.0000,0.0000)--(5.0303,0.0000)--(5.0606,0.0000)--(5.0909,0.0000)--(5.1212,0.0000)--(5.1515,0.0000)--(5.1818,0.0000)--(5.2121,0.0000)--(5.2424,0.0000)--(5.2727,0.0000)--(5.3030,0.0000)--(5.3333,0.0000)--(5.3636,0.0000)--(5.3939,0.0000)--(5.4242,0.0000)--(5.4545,0.0000)--(5.4848,0.0000)--(5.5151,0.0000)--(5.5454,0.0000)--(5.5757,0.0000)--(5.6060,0.0000)--(5.6363,0.0000)--(5.6666,0.0000)--(5.6969,0.0000)--(5.7272,0.0000)--(5.7575,0.0000)--(5.7878,0.0000)--(5.8181,0.0000)--(5.8484,0.0000)--(5.8787,0.0000)--(5.9090,0.0000)--(5.9393,0.0000)--(5.9696,0.0000)--(6.0000,0.0000);
+\draw [style=dashed] (3.0000,0.0000) -- (3.0000,3.0000);
+\draw [style=dashed] (6.0000,3.7500) -- (6.0000,0.0000);
 
 %OTHER STUFF
 %END PSPICTURE
diff --git a/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall b/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall
index ff1ca43f9..f4f5bb4d3 100644
--- a/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall
+++ b/src_phystricks/Fig_WHCooNZAmYB.pstricks.recall
@@ -78,22 +78,22 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
-\draw []  (0,0) node [rotate=0] {$\bullet$};
-\draw (0,-0.32471) node {\( O\)};
-\draw []  (1.0500,1.8187) node [rotate=0] {$\bullet$};
-\draw (1.2976,2.1166) node {\( B\)};
-\draw []  (1.8187,1.0500) node [rotate=0] {$\bullet$};
+\draw []  (0.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (0.0000,-0.3247) node {\( O\)};
+\draw []  (1.0500,1.8186) node [rotate=0] {$\bullet$};
+\draw (1.2975,2.1165) node {\( B\)};
+\draw []  (1.8186,1.0500) node [rotate=0] {$\bullet$};
 \draw (2.1287,1.2747) node {\( A\)};
-\draw []  (2.1000,0) node [rotate=0] {$\bullet$};
-\draw (2.3360,-0.26613) node {\( I\)};
+\draw []  (2.1000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.3359,-0.2661) node {\( I\)};
 
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+\draw [] (2.1000,0.0000)--(2.0957,0.1331)--(2.0831,0.2658)--(2.0620,0.3974)--(2.0326,0.5274)--(1.9951,0.6552)--(1.9495,0.7804)--(1.8961,0.9025)--(1.8350,1.0210)--(1.7666,1.1353)--(1.6910,1.2451)--(1.6086,1.3498)--(1.5198,1.4491)--(1.4248,1.5426)--(1.3241,1.6299)--(1.2181,1.7106)--(1.1071,1.7844)--(0.9917,1.8510)--(0.8723,1.9102)--(0.7494,1.9617)--(0.6235,2.0052)--(0.4950,2.0408)--(0.3646,2.0680)--(0.2327,2.0870)--(0.0999,2.0976)--(-0.0333,2.0997)--(-0.1664,2.0933)--(-0.2988,2.0786)--(-0.4300,2.0554)--(-0.5595,2.0240)--(-0.6868,1.9845)--(-0.8113,1.9369)--(-0.9325,1.8815)--(-1.0500,1.8186)--(-1.1632,1.7483)--(-1.2717,1.6710)--(-1.3752,1.5870)--(-1.4730,1.4966)--(-1.5650,1.4002)--(-1.6507,1.2981)--(-1.7297,1.1908)--(-1.8017,1.0787)--(-1.8665,0.9622)--(-1.9238,0.8419)--(-1.9733,0.7182)--(-2.0149,0.5916)--(-2.0484,0.4626)--(-2.0736,0.3318)--(-2.0904,0.1996)--(-2.0989,0.0666)--(-2.0989,-0.0666)--(-2.0904,-0.1996)--(-2.0736,-0.3318)--(-2.0484,-0.4626)--(-2.0149,-0.5916)--(-1.9733,-0.7182)--(-1.9238,-0.8419)--(-1.8665,-0.9622)--(-1.8017,-1.0787)--(-1.7297,-1.1908)--(-1.6507,-1.2981)--(-1.5650,-1.4002)--(-1.4730,-1.4966)--(-1.3752,-1.5870)--(-1.2717,-1.6710)--(-1.1632,-1.7483)--(-1.0500,-1.8186)--(-0.9325,-1.8815)--(-0.8113,-1.9369)--(-0.6868,-1.9845)--(-0.5595,-2.0240)--(-0.4300,-2.0554)--(-0.2988,-2.0786)--(-0.1664,-2.0933)--(-0.0333,-2.0997)--(0.0999,-2.0976)--(0.2327,-2.0870)--(0.3646,-2.0680)--(0.4950,-2.0408)--(0.6235,-2.0052)--(0.7494,-1.9617)--(0.8723,-1.9102)--(0.9917,-1.8510)--(1.1071,-1.7844)--(1.2181,-1.7106)--(1.3241,-1.6299)--(1.4248,-1.5426)--(1.5198,-1.4491)--(1.6086,-1.3498)--(1.6910,-1.2451)--(1.7666,-1.1353)--(1.8350,-1.0210)--(1.8961,-0.9025)--(1.9495,-0.7804)--(1.9951,-0.6552)--(2.0326,-0.5274)--(2.0620,-0.3974)--(2.0831,-0.2658)--(2.0957,-0.1331)--(2.1000,0.0000);
 
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-\draw [] (0,0) -- (1.05,1.82);
-\draw [] (0,0) -- (2.10,0);
-\draw [] (0,0) -- (1.82,1.05);
-\draw [] (1.05,1.82) -- (2.10,0);
+\draw [color=blue,style=dashed] (2.9056,1.0500)--(2.9035,1.1189)--(2.8969,1.1876)--(2.8860,1.2557)--(2.8708,1.3230)--(2.8514,1.3891)--(2.8278,1.4540)--(2.8001,1.5172)--(2.7685,1.5785)--(2.7331,1.6376)--(2.6940,1.6945)--(2.6513,1.7487)--(2.6053,1.8001)--(2.5562,1.8485)--(2.5040,1.8937)--(2.4491,1.9354)--(2.3917,1.9736)--(2.3320,2.0081)--(2.2702,2.0388)--(2.2066,2.0654)--(2.1414,2.0880)--(2.0749,2.1063)--(2.0074,2.1205)--(1.9391,2.1303)--(1.8703,2.1358)--(1.8014,2.1369)--(1.7325,2.1336)--(1.6639,2.1259)--(1.5960,2.1139)--(1.5289,2.0977)--(1.4631,2.0772)--(1.3986,2.0526)--(1.3359,2.0239)--(1.2751,1.9914)--(1.2165,1.9550)--(1.1603,1.9150)--(1.1067,1.8715)--(1.0561,1.8247)--(1.0085,1.7748)--(0.9641,1.7219)--(0.9232,1.6664)--(0.8859,1.6083)--(0.8524,1.5481)--(0.8228,1.4858)--(0.7971,1.4217)--(0.7756,1.3562)--(0.7583,1.2894)--(0.7452,1.2217)--(0.7365,1.1533)--(0.7321,1.0844)--(0.7321,1.0155)--(0.7365,0.9466)--(0.7452,0.8782)--(0.7583,0.8105)--(0.7756,0.7437)--(0.7971,0.6782)--(0.8228,0.6141)--(0.8524,0.5518)--(0.8859,0.4916)--(0.9232,0.4335)--(0.9641,0.3780)--(1.0085,0.3251)--(1.0561,0.2752)--(1.1067,0.2284)--(1.1603,0.1849)--(1.2165,0.1449)--(1.2751,0.1085)--(1.3359,0.0760)--(1.3986,0.0473)--(1.4631,0.0227)--(1.5289,0.0022)--(1.5960,-0.0139)--(1.6639,-0.0259)--(1.7325,-0.0336)--(1.8014,-0.0369)--(1.8703,-0.0358)--(1.9391,-0.0303)--(2.0074,-0.0205)--(2.0749,-0.0063)--(2.1414,0.0119)--(2.2066,0.0345)--(2.2702,0.0611)--(2.3320,0.0918)--(2.3917,0.1263)--(2.4491,0.1645)--(2.5040,0.2062)--(2.5562,0.2514)--(2.6053,0.2998)--(2.6513,0.3512)--(2.6940,0.4054)--(2.7331,0.4623)--(2.7685,0.5214)--(2.8001,0.5827)--(2.8278,0.6459)--(2.8514,0.7108)--(2.8708,0.7769)--(2.8860,0.8442)--(2.8969,0.9123)--(2.9035,0.9810)--(2.9056,1.0500);
+\draw [] (0.0000,0.0000) -- (1.0500,1.8186);
+\draw [] (0.0000,0.0000) -- (2.1000,0.0000);
+\draw [] (0.0000,0.0000) -- (1.8186,1.0500);
+\draw [] (1.0500,1.8186) -- (2.1000,0.0000);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall b/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
index 9099168b7..d9891f8ea 100644
--- a/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
+++ b/src_phystricks/Fig_WIRAooTCcpOV.pstricks.recall
@@ -119,41 +119,41 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.0000,0) -- (4.0000,0);
-\draw [,->,>=latex] (0,-2.7995) -- (0,4.0548);
+\draw [,->,>=latex] (-4.0000,0.0000) -- (4.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-2.7995) -- (0.0000,4.0547);
 %DEFAULT
 
-\draw [color=blue] (-3.500,-0.9928)--(-3.429,-0.6362)--(-3.359,-0.2871)--(-3.288,0.05069)--(-3.217,0.3737)--(-3.146,0.6788)--(-3.076,0.9630)--(-3.005,1.224)--(-2.934,1.459)--(-2.864,1.667)--(-2.793,1.847)--(-2.722,1.997)--(-2.652,2.117)--(-2.581,2.207)--(-2.510,2.266)--(-2.439,2.297)--(-2.369,2.300)--(-2.298,2.275)--(-2.227,2.225)--(-2.157,2.153)--(-2.086,2.059)--(-2.015,1.946)--(-1.944,1.817)--(-1.874,1.675)--(-1.803,1.522)--(-1.732,1.361)--(-1.662,1.195)--(-1.591,1.027)--(-1.520,0.8595)--(-1.449,0.6948)--(-1.379,0.5355)--(-1.308,0.3839)--(-1.237,0.2420)--(-1.167,0.1117)--(-1.096,-0.005633)--(-1.025,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2681)--(-0.8131,-0.3235)--(-0.7424,-0.3626)--(-0.6717,-0.3855)--(-0.6010,-0.3928)--(-0.5303,-0.3853)--(-0.4596,-0.3640)--(-0.3889,-0.3304)--(-0.3182,-0.2859)--(-0.2475,-0.2322)--(-0.1768,-0.1712)--(-0.1061,-0.1048)--(-0.03535,-0.03531)--(0.03535,0.03531)--(0.1061,0.1048)--(0.1768,0.1712)--(0.2475,0.2322)--(0.3182,0.2859)--(0.3889,0.3304)--(0.4596,0.3640)--(0.5303,0.3853)--(0.6010,0.3928)--(0.6717,0.3855)--(0.7424,0.3626)--(0.8131,0.3235)--(0.8838,0.2681)--(0.9545,0.1963)--(1.025,0.1086)--(1.096,0.005633)--(1.167,-0.1117)--(1.237,-0.2420)--(1.308,-0.3839)--(1.379,-0.5355)--(1.449,-0.6948)--(1.520,-0.8595)--(1.591,-1.027)--(1.662,-1.195)--(1.732,-1.361)--(1.803,-1.522)--(1.874,-1.675)--(1.944,-1.817)--(2.015,-1.946)--(2.086,-2.059)--(2.157,-2.153)--(2.227,-2.225)--(2.298,-2.275)--(2.369,-2.300)--(2.439,-2.297)--(2.510,-2.266)--(2.581,-2.207)--(2.652,-2.117)--(2.722,-1.997)--(2.793,-1.847)--(2.864,-1.667)--(2.934,-1.459)--(3.005,-1.224)--(3.076,-0.9630)--(3.146,-0.6788)--(3.217,-0.3737)--(3.288,-0.05069)--(3.359,0.2871)--(3.429,0.6362)--(3.500,0.9928);
+\draw [color=blue] (-3.5000,-0.9928)--(-3.4292,-0.6362)--(-3.3585,-0.2871)--(-3.2878,0.0506)--(-3.2171,0.3737)--(-3.1464,0.6787)--(-3.0757,0.9630)--(-3.0050,1.2238)--(-2.9343,1.4592)--(-2.8636,1.6673)--(-2.7929,1.8468)--(-2.7222,1.9968)--(-2.6515,2.1167)--(-2.5808,2.2065)--(-2.5101,2.2664)--(-2.4393,2.2970)--(-2.3686,2.2995)--(-2.2979,2.2750)--(-2.2272,2.2254)--(-2.1565,2.1525)--(-2.0858,2.0586)--(-2.0151,1.9459)--(-1.9444,1.8171)--(-1.8737,1.6749)--(-1.8030,1.5220)--(-1.7323,1.3612)--(-1.6616,1.1953)--(-1.5909,1.0272)--(-1.5202,0.8595)--(-1.4494,0.6948)--(-1.3787,0.5355)--(-1.3080,0.3839)--(-1.2373,0.2420)--(-1.1666,0.1116)--(-1.0959,-0.0056)--(-1.0252,-0.1086)--(-0.9545,-0.1963)--(-0.8838,-0.2680)--(-0.8131,-0.3235)--(-0.7424,-0.3625)--(-0.6717,-0.3854)--(-0.6010,-0.3927)--(-0.5303,-0.3852)--(-0.4595,-0.3640)--(-0.3888,-0.3304)--(-0.3181,-0.2858)--(-0.2474,-0.2321)--(-0.1767,-0.1711)--(-0.1060,-0.1048)--(-0.0353,-0.0353)--(0.0353,0.0353)--(0.1060,0.1048)--(0.1767,0.1711)--(0.2474,0.2321)--(0.3181,0.2858)--(0.3888,0.3304)--(0.4595,0.3640)--(0.5303,0.3852)--(0.6010,0.3927)--(0.6717,0.3854)--(0.7424,0.3625)--(0.8131,0.3235)--(0.8838,0.2680)--(0.9545,0.1963)--(1.0252,0.1086)--(1.0959,0.0056)--(1.1666,-0.1116)--(1.2373,-0.2420)--(1.3080,-0.3839)--(1.3787,-0.5355)--(1.4494,-0.6948)--(1.5202,-0.8595)--(1.5909,-1.0272)--(1.6616,-1.1953)--(1.7323,-1.3612)--(1.8030,-1.5220)--(1.8737,-1.6749)--(1.9444,-1.8171)--(2.0151,-1.9459)--(2.0858,-2.0586)--(2.1565,-2.1525)--(2.2272,-2.2254)--(2.2979,-2.2750)--(2.3686,-2.2995)--(2.4393,-2.2970)--(2.5101,-2.2664)--(2.5808,-2.2065)--(2.6515,-2.1167)--(2.7222,-1.9968)--(2.7929,-1.8468)--(2.8636,-1.6673)--(2.9343,-1.4592)--(3.0050,-1.2238)--(3.0757,-0.9630)--(3.1464,-0.6787)--(3.2171,-0.3737)--(3.2878,-0.0506)--(3.3585,0.2871)--(3.4292,0.6362)--(3.5000,0.9928);
 
-\draw [color=red] (-3.500,3.555)--(-3.429,3.500)--(-3.359,3.406)--(-3.288,3.277)--(-3.217,3.114)--(-3.146,2.921)--(-3.076,2.702)--(-3.005,2.459)--(-2.934,2.198)--(-2.864,1.921)--(-2.793,1.632)--(-2.722,1.337)--(-2.652,1.038)--(-2.581,0.7401)--(-2.510,0.4468)--(-2.439,0.1618)--(-2.369,-0.1114)--(-2.298,-0.3696)--(-2.227,-0.6099)--(-2.157,-0.8297)--(-2.086,-1.027)--(-2.015,-1.199)--(-1.944,-1.346)--(-1.874,-1.466)--(-1.803,-1.558)--(-1.732,-1.622)--(-1.662,-1.658)--(-1.591,-1.667)--(-1.520,-1.650)--(-1.449,-1.608)--(-1.379,-1.542)--(-1.308,-1.456)--(-1.237,-1.350)--(-1.167,-1.228)--(-1.096,-1.092)--(-1.025,-0.9453)--(-0.9545,-0.7902)--(-0.8838,-0.6299)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.002540)--(-0.5303,0.1441)--(-0.4596,0.2739)--(-0.3889,0.3896)--(-0.3182,0.4892)--(-0.2475,0.5710)--(-0.1768,0.6336)--(-0.1061,0.6760)--(-0.03535,0.6973)--(0.03535,0.6973)--(0.1061,0.6760)--(0.1768,0.6336)--(0.2475,0.5710)--(0.3182,0.4892)--(0.3889,0.3896)--(0.4596,0.2739)--(0.5303,0.1441)--(0.6010,0.002540)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6299)--(0.9545,-0.7902)--(1.025,-0.9453)--(1.096,-1.092)--(1.167,-1.228)--(1.237,-1.350)--(1.308,-1.456)--(1.379,-1.542)--(1.449,-1.608)--(1.520,-1.650)--(1.591,-1.667)--(1.662,-1.658)--(1.732,-1.622)--(1.803,-1.558)--(1.874,-1.466)--(1.944,-1.346)--(2.015,-1.199)--(2.086,-1.027)--(2.157,-0.8297)--(2.227,-0.6099)--(2.298,-0.3696)--(2.369,-0.1114)--(2.439,0.1618)--(2.510,0.4468)--(2.581,0.7401)--(2.652,1.038)--(2.722,1.337)--(2.793,1.632)--(2.864,1.921)--(2.934,2.198)--(3.005,2.459)--(3.076,2.702)--(3.146,2.921)--(3.217,3.114)--(3.288,3.277)--(3.359,3.406)--(3.429,3.500)--(3.500,3.555);
-\draw (-3.2987,-0.42071) node {$-\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (-3.30,-0.100) -- (-3.30,0.100);
-\draw (-2.1991,-0.32103) node {$-\mathit{\pi}$};
-\draw [] (-2.20,-0.100) -- (-2.20,0.100);
-\draw (-1.0996,-0.42071) node {$-\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (-1.10,-0.100) -- (-1.10,0.100);
-\draw (1.0996,-0.42071) node {$\frac{1}{2} \, \mathit{\pi}$};
-\draw [] (1.10,-0.100) -- (1.10,0.100);
-\draw (2.1991,-0.27858) node {$\mathit{\pi}$};
-\draw [] (2.20,-0.100) -- (2.20,0.100);
-\draw (3.2987,-0.42071) node {$\frac{3}{2} \, \mathit{\pi}$};
-\draw [] (3.30,-0.100) -- (3.30,0.100);
-\draw (-0.43316,-2.1000) node {$ -3 $};
-\draw [] (-0.100,-2.10) -- (0.100,-2.10);
-\draw (-0.43316,-1.4000) node {$ -2 $};
-\draw [] (-0.100,-1.40) -- (0.100,-1.40);
-\draw (-0.43316,-0.70000) node {$ -1 $};
-\draw [] (-0.100,-0.700) -- (0.100,-0.700);
-\draw (-0.29125,0.70000) node {$ 1 $};
-\draw [] (-0.100,0.700) -- (0.100,0.700);
-\draw (-0.29125,1.4000) node {$ 2 $};
-\draw [] (-0.100,1.40) -- (0.100,1.40);
-\draw (-0.29125,2.1000) node {$ 3 $};
-\draw [] (-0.100,2.10) -- (0.100,2.10);
-\draw (-0.29125,2.8000) node {$ 4 $};
-\draw [] (-0.100,2.80) -- (0.100,2.80);
-\draw (-0.29125,3.5000) node {$ 5 $};
-\draw [] (-0.100,3.50) -- (0.100,3.50);
+\draw [color=red] (-3.5000,3.5547)--(-3.4292,3.4996)--(-3.3585,3.4061)--(-3.2878,3.2766)--(-3.2171,3.1140)--(-3.1464,2.9213)--(-3.0757,2.7019)--(-3.0050,2.4594)--(-2.9343,2.1976)--(-2.8636,1.9206)--(-2.7929,1.6322)--(-2.7222,1.3367)--(-2.6515,1.0379)--(-2.5808,0.7400)--(-2.5101,0.4467)--(-2.4393,0.1618)--(-2.3686,-0.1113)--(-2.2979,-0.3695)--(-2.2272,-0.6098)--(-2.1565,-0.8297)--(-2.0858,-1.0268)--(-2.0151,-1.1994)--(-1.9444,-1.3460)--(-1.8737,-1.4656)--(-1.8030,-1.5575)--(-1.7323,-1.6215)--(-1.6616,-1.6577)--(-1.5909,-1.6667)--(-1.5202,-1.6496)--(-1.4494,-1.6076)--(-1.3787,-1.5424)--(-1.3080,-1.4559)--(-1.2373,-1.3503)--(-1.1666,-1.2283)--(-1.0959,-1.0923)--(-1.0252,-0.9453)--(-0.9545,-0.7901)--(-0.8838,-0.6298)--(-0.8131,-0.4675)--(-0.7424,-0.3060)--(-0.6717,-0.1484)--(-0.6010,0.0025)--(-0.5303,0.1441)--(-0.4595,0.2739)--(-0.3888,0.3896)--(-0.3181,0.4892)--(-0.2474,0.5710)--(-0.1767,0.6336)--(-0.1060,0.6759)--(-0.0353,0.6973)--(0.0353,0.6973)--(0.1060,0.6759)--(0.1767,0.6336)--(0.2474,0.5710)--(0.3181,0.4892)--(0.3888,0.3896)--(0.4595,0.2739)--(0.5303,0.1441)--(0.6010,0.0025)--(0.6717,-0.1484)--(0.7424,-0.3060)--(0.8131,-0.4675)--(0.8838,-0.6298)--(0.9545,-0.7901)--(1.0252,-0.9453)--(1.0959,-1.0923)--(1.1666,-1.2283)--(1.2373,-1.3503)--(1.3080,-1.4559)--(1.3787,-1.5424)--(1.4494,-1.6076)--(1.5202,-1.6496)--(1.5909,-1.6667)--(1.6616,-1.6577)--(1.7323,-1.6215)--(1.8030,-1.5575)--(1.8737,-1.4656)--(1.9444,-1.3460)--(2.0151,-1.1994)--(2.0858,-1.0268)--(2.1565,-0.8297)--(2.2272,-0.6098)--(2.2979,-0.3695)--(2.3686,-0.1113)--(2.4393,0.1618)--(2.5101,0.4467)--(2.5808,0.7400)--(2.6515,1.0379)--(2.7222,1.3367)--(2.7929,1.6322)--(2.8636,1.9206)--(2.9343,2.1976)--(3.0050,2.4594)--(3.0757,2.7019)--(3.1464,2.9213)--(3.2171,3.1140)--(3.2878,3.2766)--(3.3585,3.4061)--(3.4292,3.4996)--(3.5000,3.5547);
+\draw (-3.2986,-0.4207) node {$-\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (-3.2986,-0.1000) -- (-3.2986,0.1000);
+\draw (-2.1991,-0.3210) node {$-\mathit{\pi}$};
+\draw [] (-2.1991,-0.1000) -- (-2.1991,0.1000);
+\draw (-1.0995,-0.4207) node {$-\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (-1.0995,-0.1000) -- (-1.0995,0.1000);
+\draw (1.0995,-0.4207) node {$\frac{1}{2} \, \mathit{\pi}$};
+\draw [] (1.0995,-0.1000) -- (1.0995,0.1000);
+\draw (2.1991,-0.2785) node {$\mathit{\pi}$};
+\draw [] (2.1991,-0.1000) -- (2.1991,0.1000);
+\draw (3.2986,-0.4207) node {$\frac{3}{2} \, \mathit{\pi}$};
+\draw [] (3.2986,-0.1000) -- (3.2986,0.1000);
+\draw (-0.4331,-2.1000) node {$ -3 $};
+\draw [] (-0.1000,-2.1000) -- (0.1000,-2.1000);
+\draw (-0.4331,-1.4000) node {$ -2 $};
+\draw [] (-0.1000,-1.4000) -- (0.1000,-1.4000);
+\draw (-0.4331,-0.7000) node {$ -1 $};
+\draw [] (-0.1000,-0.7000) -- (0.1000,-0.7000);
+\draw (-0.2912,0.7000) node {$ 1 $};
+\draw [] (-0.1000,0.7000) -- (0.1000,0.7000);
+\draw (-0.2912,1.4000) node {$ 2 $};
+\draw [] (-0.1000,1.4000) -- (0.1000,1.4000);
+\draw (-0.2912,2.1000) node {$ 3 $};
+\draw [] (-0.1000,2.1000) -- (0.1000,2.1000);
+\draw (-0.2912,2.8000) node {$ 4 $};
+\draw [] (-0.1000,2.8000) -- (0.1000,2.8000);
+\draw (-0.2912,3.5000) node {$ 5 $};
+\draw [] (-0.1000,3.5000) -- (0.1000,3.5000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall b/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
index ae1ef50a0..3c5bfef38 100644
--- a/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
+++ b/src_phystricks/Fig_WJBooMTAhtl.pstricks.recall
@@ -100,35 +100,35 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-5.2124,0) -- (9.9248,0);
-\draw [,->,>=latex] (0,-3.4450) -- (0,3.3170);
+\draw [,->,>=latex] (-5.2123,0.0000) -- (9.9247,0.0000);
+\draw [,->,>=latex] (0.0000,-3.4450) -- (0.0000,3.3170);
 %DEFAULT
 
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+\draw [color=red,style=dashed] (-3.8629,2.8170)--(-3.7851,2.3891)--(-3.7072,1.9986)--(-3.6294,1.6437)--(-3.5516,1.3226)--(-3.4737,1.0337)--(-3.3959,0.7752)--(-3.3181,0.5457)--(-3.2402,0.3434)--(-3.1624,0.1669)--(-3.0845,0.0147)--(-3.0067,-0.1147)--(-2.9289,-0.2229)--(-2.8510,-0.3111)--(-2.7732,-0.3808)--(-2.6954,-0.4332)--(-2.6175,-0.4697)--(-2.5397,-0.4915)--(-2.4619,-0.4998)--(-2.3840,-0.4958)--(-2.3062,-0.4806)--(-2.2284,-0.4554)--(-2.1505,-0.4212)--(-2.0727,-0.3790)--(-1.9949,-0.3299)--(-1.9170,-0.2747)--(-1.8392,-0.2145)--(-1.7614,-0.1501)--(-1.6835,-0.0824)--(-1.6057,-0.0121)--(-1.5278,0.0598)--(-1.4500,0.1328)--(-1.3722,0.2062)--(-1.2943,0.2792)--(-1.2165,0.3512)--(-1.1387,0.4217)--(-1.0608,0.4900)--(-0.9830,0.5557)--(-0.9052,0.6182)--(-0.8273,0.6772)--(-0.7495,0.7322)--(-0.6717,0.7828)--(-0.5938,0.8288)--(-0.5160,0.8698)--(-0.4382,0.9055)--(-0.3603,0.9357)--(-0.2825,0.9603)--(-0.2046,0.9791)--(-0.1268,0.9919)--(-0.0490,0.9987)--(0.0288,0.9995)--(0.1066,0.9943)--(0.1844,0.9830)--(0.2623,0.9657)--(0.3401,0.9427)--(0.4179,0.9139)--(0.4958,0.8796)--(0.5736,0.8399)--(0.6514,0.7952)--(0.7293,0.7458)--(0.8071,0.6919)--(0.8849,0.6339)--(0.9628,0.5722)--(1.0406,0.5073)--(1.1184,0.4396)--(1.1963,0.3697)--(1.2741,0.2980)--(1.3520,0.2252)--(1.4298,0.1519)--(1.5076,0.0787)--(1.5855,0.0063)--(1.6633,-0.0644)--(1.7411,-0.1328)--(1.8190,-0.1982)--(1.8968,-0.2596)--(1.9746,-0.3161)--(2.0525,-0.3669)--(2.1303,-0.4109)--(2.2081,-0.4473)--(2.2860,-0.4750)--(2.3638,-0.4929)--(2.4416,-0.4999)--(2.5195,-0.4949)--(2.5973,-0.4767)--(2.6751,-0.4442)--(2.7530,-0.3960)--(2.8308,-0.3310)--(2.9087,-0.2477)--(2.9865,-0.1448)--(3.0643,-0.0210)--(3.1422,0.1251)--(3.2200,0.2952)--(3.2978,0.4906)--(3.3757,0.7129)--(3.4535,0.9637)--(3.5313,1.2445)--(3.6092,1.5571)--(3.6870,1.9030)--(3.7648,2.2841)--(3.8427,2.7021);
 
-\draw [color=blue,style=dashed] (-1.072,-2.945)--(-0.9794,-2.436)--(-0.8868,-1.974)--(-0.7943,-1.558)--(-0.7017,-1.185)--(-0.6092,-0.8526)--(-0.5167,-0.5589)--(-0.4241,-0.3015)--(-0.3316,-0.07832)--(-0.2390,0.1128)--(-0.1465,0.2738)--(-0.05393,0.4068)--(0.03861,0.5136)--(0.1312,0.5962)--(0.2237,0.6564)--(0.3162,0.6960)--(0.4088,0.7166)--(0.5013,0.7200)--(0.5939,0.7079)--(0.6864,0.6817)--(0.7790,0.6430)--(0.8715,0.5932)--(0.9641,0.5339)--(1.057,0.4663)--(1.149,0.3917)--(1.242,0.3115)--(1.334,0.2269)--(1.427,0.1389)--(1.519,0.04875)--(1.612,-0.04255)--(1.704,-0.1340)--(1.797,-0.2246)--(1.889,-0.3135)--(1.982,-0.3998)--(2.075,-0.4828)--(2.167,-0.5616)--(2.260,-0.6357)--(2.352,-0.7043)--(2.445,-0.7669)--(2.537,-0.8229)--(2.630,-0.8719)--(2.722,-0.9134)--(2.815,-0.9470)--(2.907,-0.9725)--(3.000,-0.9894)--(3.093,-0.9977)--(3.185,-0.9971)--(3.278,-0.9875)--(3.370,-0.9689)--(3.463,-0.9411)--(3.555,-0.9044)--(3.648,-0.8586)--(3.740,-0.8041)--(3.833,-0.7409)--(3.925,-0.6693)--(4.018,-0.5897)--(4.111,-0.5022)--(4.203,-0.4074)--(4.296,-0.3058)--(4.388,-0.1977)--(4.481,-0.08370)--(4.573,0.03551)--(4.666,0.1593)--(4.758,0.2870)--(4.851,0.4178)--(4.943,0.5509)--(5.036,0.6855)--(5.129,0.8206)--(5.221,0.9553)--(5.314,1.089)--(5.406,1.219)--(5.499,1.346)--(5.591,1.469)--(5.684,1.585)--(5.776,1.694)--(5.869,1.794)--(5.961,1.884)--(6.054,1.963)--(6.146,2.028)--(6.239,2.080)--(6.332,2.115)--(6.424,2.132)--(6.517,2.130)--(6.609,2.107)--(6.702,2.061)--(6.794,1.990)--(6.887,1.893)--(6.979,1.766)--(7.072,1.609)--(7.164,1.420)--(7.257,1.195)--(7.350,0.9335)--(7.442,0.6328)--(7.535,0.2905)--(7.627,-0.09545)--(7.720,-0.5276)--(7.812,-1.008)--(7.905,-1.540)--(7.997,-2.125)--(8.090,-2.767);
-\draw (-3.1416,-0.32103) node {$ -\pi $};
-\draw [] (-3.14,-0.100) -- (-3.14,0.100);
-\draw (3.1416,-0.27858) node {$ \pi $};
-\draw [] (3.14,-0.100) -- (3.14,0.100);
-\draw (6.2832,-0.31492) node {$ 2 \, \pi $};
-\draw [] (6.28,-0.100) -- (6.28,0.100);
-\draw (9.4248,-0.31492) node {$ 3 \, \pi $};
-\draw [] (9.42,-0.100) -- (9.42,0.100);
-\draw (-0.43316,-3.0000) node {$ -3 $};
-\draw [] (-0.100,-3.00) -- (0.100,-3.00);
-\draw (-0.43316,-2.0000) node {$ -2 $};
-\draw [] (-0.100,-2.00) -- (0.100,-2.00);
-\draw (-0.43316,-1.0000) node {$ -1 $};
-\draw [] (-0.100,-1.00) -- (0.100,-1.00);
-\draw (-0.29125,1.0000) node {$ 1 $};
-\draw [] (-0.100,1.00) -- (0.100,1.00);
-\draw (-0.29125,2.0000) node {$ 2 $};
-\draw [] (-0.100,2.00) -- (0.100,2.00);
-\draw (-0.29125,3.0000) node {$ 3 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue,style=dashed] (-1.0719,-2.9450)--(-0.9793,-2.4356)--(-0.8868,-1.9740)--(-0.7942,-1.5579)--(-0.7017,-1.1849)--(-0.6091,-0.8526)--(-0.5166,-0.5589)--(-0.4241,-0.3015)--(-0.3315,-0.0783)--(-0.2390,0.1127)--(-0.1464,0.2738)--(-0.0539,0.4068)--(0.0386,0.5136)--(0.1311,0.5962)--(0.2236,0.6564)--(0.3162,0.6959)--(0.4087,0.7166)--(0.5013,0.7200)--(0.5938,0.7078)--(0.6864,0.6816)--(0.7789,0.6429)--(0.8715,0.5932)--(0.9640,0.5338)--(1.0565,0.4662)--(1.1491,0.3917)--(1.2416,0.3115)--(1.3342,0.2268)--(1.4267,0.1389)--(1.5193,0.0487)--(1.6118,-0.0425)--(1.7044,-0.1339)--(1.7969,-0.2245)--(1.8894,-0.3134)--(1.9820,-0.3997)--(2.0745,-0.4827)--(2.1671,-0.5616)--(2.2596,-0.6356)--(2.3522,-0.7042)--(2.4447,-0.7668)--(2.5372,-0.8229)--(2.6298,-0.8718)--(2.7223,-0.9133)--(2.8149,-0.9470)--(2.9074,-0.9724)--(3.0000,-0.9894)--(3.0925,-0.9976)--(3.1851,-0.9970)--(3.2776,-0.9875)--(3.3701,-0.9688)--(3.4627,-0.9411)--(3.5552,-0.9043)--(3.6478,-0.8586)--(3.7403,-0.8040)--(3.8329,-0.7409)--(3.9254,-0.6693)--(4.0180,-0.5896)--(4.1105,-0.5022)--(4.2030,-0.4074)--(4.2956,-0.3057)--(4.3881,-0.1976)--(4.4807,-0.0836)--(4.5732,0.0355)--(4.6658,0.1593)--(4.7583,0.2869)--(4.8508,0.4178)--(4.9434,0.5509)--(5.0359,0.6855)--(5.1285,0.8206)--(5.2210,0.9553)--(5.3136,1.0885)--(5.4061,1.2192)--(5.4987,1.3463)--(5.5912,1.4685)--(5.6837,1.5847)--(5.7763,1.6936)--(5.8688,1.7938)--(5.9614,1.8840)--(6.0539,1.9627)--(6.1465,2.0284)--(6.2390,2.0797)--(6.3316,2.1148)--(6.4241,2.1323)--(6.5166,2.1303)--(6.6092,2.1072)--(6.7017,2.0610)--(6.7943,1.9901)--(6.8868,1.8925)--(6.9794,1.7662)--(7.0719,1.6092)--(7.1644,1.4195)--(7.2570,1.1950)--(7.3495,0.9334)--(7.4421,0.6327)--(7.5346,0.2905)--(7.6272,-0.0954)--(7.7197,-0.5276)--(7.8123,-1.0083)--(7.9048,-1.5401)--(7.9973,-2.1254)--(8.0899,-2.7669);
+\draw (-3.1415,-0.3210) node {$ -\pi $};
+\draw [] (-3.1415,-0.1000) -- (-3.1415,0.1000);
+\draw (3.1415,-0.2785) node {$ \pi $};
+\draw [] (3.1415,-0.1000) -- (3.1415,0.1000);
+\draw (6.2831,-0.3149) node {$ 2 \, \pi $};
+\draw [] (6.2831,-0.1000) -- (6.2831,0.1000);
+\draw (9.4247,-0.3149) node {$ 3 \, \pi $};
+\draw [] (9.4247,-0.1000) -- (9.4247,0.1000);
+\draw (-0.4331,-3.0000) node {$ -3 $};
+\draw [] (-0.1000,-3.0000) -- (0.1000,-3.0000);
+\draw (-0.4331,-2.0000) node {$ -2 $};
+\draw [] (-0.1000,-2.0000) -- (0.1000,-2.0000);
+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
+\draw (-0.2912,2.0000) node {$ 2 $};
+\draw [] (-0.1000,2.0000) -- (0.1000,2.0000);
+\draw (-0.2912,3.0000) node {$ 3 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_YWxOAkh.pstricks.recall b/src_phystricks/Fig_YWxOAkh.pstricks.recall
index 78d8f53a2..0e493ed93 100644
--- a/src_phystricks/Fig_YWxOAkh.pstricks.recall
+++ b/src_phystricks/Fig_YWxOAkh.pstricks.recall
@@ -71,24 +71,24 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
-\draw [,->,>=latex] (0,-1.5000) -- (0,1.5000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.4999) -- (0.0000,1.4999);
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+\draw [color=blue] 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+\draw [color=blue] 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+\draw [color=blue] 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+\draw (-0.4331,-1.0000) node {$ -1 $};
+\draw [] (-0.1000,-1.0000) -- (0.1000,-1.0000);
+\draw (-0.2912,1.0000) node {$ 1 $};
+\draw [] (-0.1000,1.0000) -- (0.1000,1.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall b/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall
index 4dae1298f..ead95fd18 100644
--- a/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall
+++ b/src_phystricks/Fig_ZGUDooEsqCWQ.pstricks.recall
@@ -65,51 +65,51 @@
 %OTHER STUFF
 %PSTRICKS CODE
 %DEFAULT
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+\draw [] (0.9999,0.0087)--(0.9991,0.0403)--(0.9974,0.0719)--(0.9946,0.1034)--(0.9908,0.1348)--(0.9860,0.1661)--(0.9803,0.1972)--(0.9736,0.2282)--(0.9659,0.2589)--(0.9572,0.2893)--(0.9475,0.3194)--(0.9370,0.3492)--(0.9254,0.3787)--(0.9130,0.4078)--(0.8996,0.4365)--(0.8854,0.4647)--(0.8702,0.4925)--(0.8542,0.5198)--(0.8373,0.5466)--(0.8196,0.5728)--(0.8011,0.5985)--(0.7817,0.6235)--(0.7616,0.6479)--(0.7407,0.6717)--(0.7191,0.6948)--(0.6968,0.7172)--(0.6737,0.7389)--(0.6500,0.7598)--(0.6256,0.7800)--(0.6006,0.7994)--(0.5750,0.8180)--(0.5489,0.8358)--(0.5221,0.8528)--(0.4949,0.8689)--(0.4672,0.8841)--(0.4390,0.8984)--(0.4103,0.9119)--(0.3812,0.9244)--(0.3518,0.9360)--(0.3220,0.9467)--(0.2919,0.9564)--(0.2615,0.9651)--(0.2308,0.9729)--(0.1999,0.9798)--(0.1688,0.9856)--(0.1376,0.9904)--(0.1061,0.9943)--(0.0746,0.9972)--(0.0430,0.9990)--(0.0114,0.9999)--(-0.0201,0.9997)--(-0.0518,0.9986)--(-0.0833,0.9965)--(-0.1148,0.9933)--(-0.1462,0.9892)--(-0.1774,0.9841)--(-0.2085,0.9780)--(-0.2393,0.9709)--(-0.2699,0.9628)--(-0.3002,0.9538)--(-0.3303,0.9438)--(-0.3600,0.9329)--(-0.3893,0.9210)--(-0.4182,0.9083)--(-0.4468,0.8946)--(-0.4749,0.8800)--(-0.5025,0.8645)--(-0.5296,0.8482)--(-0.5561,0.8310)--(-0.5822,0.8130)--(-0.6076,0.7942)--(-0.6324,0.7745)--(-0.6566,0.7541)--(-0.6801,0.7330)--(-0.7030,0.7111)--(-0.7251,0.6885)--(-0.7466,0.6652)--(-0.7672,0.6413)--(-0.7871,0.6167)--(-0.8063,0.5914)--(-0.8246,0.5656)--(-0.8421,0.5393)--(-0.8587,0.5123)--(-0.8745,0.4849)--(-0.8894,0.4570)--(-0.9034,0.4286)--(-0.9165,0.3998)--(-0.9287,0.3706)--(-0.9400,0.3411)--(-0.9503,0.3111)--(-0.9597,0.2809)--(-0.9681,0.2504)--(-0.9755,0.2197)--(-0.9820,0.1887)--(-0.9875,0.1575)--(-0.9919,0.1262)--(-0.9954,0.0947)--(-0.9979,0.0632)--(-0.9994,0.0316)--(-1.0000,0.0000);
+\draw [,->,>=latex] (0.9999,0.0087) -- (1.9999,0.0174);
+\draw [,->,>=latex] (0.9667,0.2558) -- (1.9334,0.5116);
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+\draw [,->,>=latex] (-0.8063,0.5914) -- (-1.6126,1.1829);
+\draw [,->,>=latex] (-0.9275,0.3735) -- (-1.8551,0.7471);
+\draw [,->,>=latex] (-0.9911,0.1324) -- (-1.9823,0.2649);
+\draw [,->,>=latex] (2.1999,0.0191) -- (2.6544,0.0231);
+\draw [,->,>=latex] (2.1835,0.2685) -- (2.6346,0.3240);
+\draw [,->,>=latex] (2.1390,0.5143) -- (2.5809,0.6206);
+\draw [,->,>=latex] (2.0668,0.7536) -- (2.4939,0.9093);
+\draw [,->,>=latex] (1.9681,0.9831) -- (2.3747,1.1862);
+\draw [,->,>=latex] (1.8439,1.1999) -- (2.2249,1.4478);
+\draw [,->,>=latex] (1.6959,1.4013) -- (2.0463,1.6908);
+\draw [,->,>=latex] (1.5261,1.5845) -- (1.8414,1.9119);
+\draw [,->,>=latex] (1.3366,1.7474) -- (1.6127,2.1084);
+\draw [,->,>=latex] (1.1298,1.8877) -- (1.3632,2.2777);
+\draw [,->,>=latex] (0.9085,2.0036) -- (1.0962,2.4176);
+\draw [,->,>=latex] (0.6754,2.0937) -- (0.8150,2.5263);
+\draw [,->,>=latex] (0.4336,2.1568) -- (0.5233,2.6024);
+\draw [,->,>=latex] (0.1863,2.1920) -- (0.2248,2.6450);
+\draw [,->,>=latex] (-0.0634,2.1990) -- (-0.0765,2.6534);
+\draw [,->,>=latex] (-0.3123,2.1777) -- (-0.3769,2.6276);
+\draw [,->,>=latex] (-0.5573,2.1282) -- (-0.6724,2.5679);
+\draw [,->,>=latex] (-0.7950,2.0513) -- (-0.9593,2.4751);
+\draw [,->,>=latex] (-1.0225,1.9479) -- (-1.2337,2.3504);
+\draw [,->,>=latex] (-1.2367,1.8194) -- (-1.4923,2.1953);
+\draw [,->,>=latex] (-1.4351,1.6674) -- (-1.7316,2.0119);
+\draw [,->,>=latex] (-1.6149,1.4939) -- (-1.9486,1.8026);
+\draw [,->,>=latex] (-1.7739,1.3012) -- (-2.1404,1.5700);
+\draw [,->,>=latex] (-1.9100,1.0916) -- (-2.3046,1.3172);
+\draw [,->,>=latex] (-2.0215,0.8680) -- (-2.4391,1.0473);
+\draw [,->,>=latex] (-2.1069,0.6332) -- (-2.5422,0.7640);
+\draw [,->,>=latex] (-2.1651,0.3902) -- (-2.6124,0.4708);
+\draw [,->,>=latex] (-2.1953,0.1422) -- (-2.6489,0.1715);
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_examssepti.pstricks.recall b/src_phystricks/Fig_examssepti.pstricks.recall
index 74c463784..bfdb3d73a 100644
--- a/src_phystricks/Fig_examssepti.pstricks.recall
+++ b/src_phystricks/Fig_examssepti.pstricks.recall
@@ -75,10 +75,10 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0) -- (2.0000,0);
-\draw [,->,>=latex] (0,-0.50000) -- (0,3.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (2.0000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
-\draw [style=dotted] (-2.36,1.85) -- (-2.36,0);
+\draw [style=dotted] (-2.3584,1.8541) -- (-2.3584,0.0000);
 
 % declaring the keys in tikz
 \tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
@@ -86,22 +86,22 @@
 % setting the default values
 \tikzset{hatchspread=3pt,
          hatchthickness=0.4pt}
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-\draw (-0.29125,3.0000) node {$ 1 $};
-\draw [] (-0.100,3.00) -- (0.100,3.00);
+\draw [color=blue] (-3.0000,0.0000)--(-2.9545,0.5202)--(-2.9090,0.7329)--(-2.8636,0.8941)--(-2.8181,1.0285)--(-2.7727,1.1454)--(-2.7272,1.2497)--(-2.6818,1.3445)--(-2.6363,1.4316)--(-2.5909,1.5123)--(-2.5454,1.5876)--(-2.5000,1.6583)--(-2.4545,1.7248)--(-2.4090,1.7878)--(-2.3636,1.8474)--(-2.3181,1.9042)--(-2.2727,1.9582)--(-2.2272,2.0097)--(-2.1818,2.0590)--(-2.1363,2.1061)--(-2.0909,2.1513)--(-2.0454,2.1945)--(-2.0000,2.2360)--(-1.9545,2.2759)--(-1.9090,2.3141)--(-1.8636,2.3509)--(-1.8181,2.3862)--(-1.7727,2.4202)--(-1.7272,2.4528)--(-1.6818,2.4842)--(-1.6363,2.5144)--(-1.5909,2.5434)--(-1.5454,2.5712)--(-1.5000,2.5980)--(-1.4545,2.6237)--(-1.4090,2.6484)--(-1.3636,2.6721)--(-1.3181,2.6948)--(-1.2727,2.7166)--(-1.2272,2.7374)--(-1.1818,2.7574)--(-1.1363,2.7764)--(-1.0909,2.7946)--(-1.0454,2.8119)--(-1.0000,2.8284)--(-0.9545,2.8440)--(-0.9090,2.8589)--(-0.8636,2.8730)--(-0.8181,2.8862)--(-0.7727,2.8987)--(-0.7272,2.9105)--(-0.6818,2.9214)--(-0.6363,2.9317)--(-0.5909,2.9412)--(-0.5454,2.9499)--(-0.5000,2.9580)--(-0.4545,2.9653)--(-0.4090,2.9719)--(-0.3636,2.9778)--(-0.3181,2.9830)--(-0.2727,2.9875)--(-0.2272,2.9913)--(-0.1818,2.9944)--(-0.1363,2.9968)--(-0.0909,2.9986)--(-0.0454,2.9996)--(0.0000,3.0000)--(0.0454,2.9996)--(0.0909,2.9986)--(0.1363,2.9968)--(0.1818,2.9944)--(0.2272,2.9913)--(0.2727,2.9875)--(0.3181,2.9830)--(0.3636,2.9778)--(0.4090,2.9719)--(0.4545,2.9653)--(0.5000,2.9580)--(0.5454,2.9499)--(0.5909,2.9412)--(0.6363,2.9317)--(0.6818,2.9214)--(0.7272,2.9105)--(0.7727,2.8987)--(0.8181,2.8862)--(0.8636,2.8730)--(0.9090,2.8589)--(0.9545,2.8440)--(1.0000,2.8284)--(1.0454,2.8119)--(1.0909,2.7946)--(1.1363,2.7764)--(1.1818,2.7574)--(1.2272,2.7374)--(1.2727,2.7166)--(1.3181,2.6948)--(1.3636,2.6721)--(1.4090,2.6484)--(1.4545,2.6237)--(1.5000,2.5980);
+\draw []  (-2.3584,1.8541) node [rotate=0] {$\bullet$};
+\draw []  (-2.3584,0.0000) node [rotate=0] {$\bullet$};
+\draw (-2.3584,-0.2335) node {\( -x_0\)};
+\draw (-3.0000,-0.3298) node {$ -1 $};
+\draw [] (-3.0000,-0.1000) -- (-3.0000,0.1000);
+\draw (-0.2912,3.0000) node {$ 1 $};
+\draw [] (-0.1000,3.0000) -- (0.1000,3.0000);
 %OTHER STUFF
 %END PSPICTURE
 \end{tikzpicture}
diff --git a/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall b/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall
index 644298998..e41decf28 100644
--- a/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall
+++ b/src_phystricks/Fig_ooIHLPooKLIxcH.pstricks.recall
@@ -86,17 +86,17 @@
 %PSTRICKS CODE
 %DEFAULT
 
-\draw [style=dashed] (2.000,0)--(1.996,0.1268)--(1.984,0.2532)--(1.964,0.3785)--(1.936,0.5023)--(1.900,0.6241)--(1.857,0.7433)--(1.806,0.8596)--(1.748,0.9724)--(1.683,1.081)--(1.611,1.186)--(1.532,1.286)--(1.447,1.380)--(1.357,1.469)--(1.261,1.552)--(1.160,1.629)--(1.054,1.699)--(0.9445,1.763)--(0.8308,1.819)--(0.7138,1.868)--(0.5938,1.910)--(0.4715,1.944)--(0.3473,1.970)--(0.2217,1.988)--(0.09516,1.998)--(-0.03173,2.000)--(-0.1585,1.994)--(-0.2846,1.980)--(-0.4096,1.958)--(-0.5330,1.928)--(-0.6541,1.890)--(-0.7727,1.845)--(-0.8881,1.792)--(-1.000,1.732)--(-1.108,1.665)--(-1.211,1.592)--(-1.310,1.512)--(-1.403,1.425)--(-1.491,1.334)--(-1.572,1.236)--(-1.647,1.134)--(-1.716,1.027)--(-1.778,0.9165)--(-1.832,0.8019)--(-1.879,0.6840)--(-1.919,0.5635)--(-1.951,0.4406)--(-1.975,0.3160)--(-1.991,0.1901)--(-1.999,0.06346)--(-1.999,-0.06346)--(-1.991,-0.1901)--(-1.975,-0.3160)--(-1.951,-0.4406)--(-1.919,-0.5635)--(-1.879,-0.6840)--(-1.832,-0.8019)--(-1.778,-0.9165)--(-1.716,-1.027)--(-1.647,-1.134)--(-1.572,-1.236)--(-1.491,-1.334)--(-1.403,-1.425)--(-1.310,-1.512)--(-1.211,-1.592)--(-1.108,-1.665)--(-1.000,-1.732)--(-0.8881,-1.792)--(-0.7727,-1.845)--(-0.6541,-1.890)--(-0.5330,-1.928)--(-0.4096,-1.958)--(-0.2846,-1.980)--(-0.1585,-1.994)--(-0.03173,-2.000)--(0.09516,-1.998)--(0.2217,-1.988)--(0.3473,-1.970)--(0.4715,-1.944)--(0.5938,-1.910)--(0.7138,-1.868)--(0.8308,-1.819)--(0.9445,-1.763)--(1.054,-1.699)--(1.160,-1.629)--(1.261,-1.552)--(1.357,-1.469)--(1.447,-1.380)--(1.532,-1.286)--(1.611,-1.186)--(1.683,-1.081)--(1.748,-0.9724)--(1.806,-0.8596)--(1.857,-0.7433)--(1.900,-0.6241)--(1.936,-0.5023)--(1.964,-0.3785)--(1.984,-0.2532)--(1.996,-0.1268)--(2.000,0);
+\draw [style=dashed] (2.0000,0.0000)--(1.9959,0.1268)--(1.9839,0.2531)--(1.9638,0.3785)--(1.9358,0.5022)--(1.9001,0.6240)--(1.8567,0.7433)--(1.8058,0.8595)--(1.7476,0.9723)--(1.6825,1.0812)--(1.6105,1.1858)--(1.5320,1.2855)--(1.4474,1.3801)--(1.3570,1.4691)--(1.2611,1.5522)--(1.1601,1.6291)--(1.0544,1.6994)--(0.9445,1.7629)--(0.8308,1.8192)--(0.7137,1.8682)--(0.5938,1.9098)--(0.4715,1.9436)--(0.3472,1.9696)--(0.2216,1.9876)--(0.0951,1.9977)--(-0.0317,1.9997)--(-0.1584,1.9937)--(-0.2846,1.9796)--(-0.4096,1.9576)--(-0.5329,1.9276)--(-0.6541,1.8900)--(-0.7726,1.8447)--(-0.8881,1.7919)--(-1.0000,1.7320)--(-1.1078,1.6651)--(-1.2112,1.5915)--(-1.3097,1.5114)--(-1.4029,1.4253)--(-1.4905,1.3335)--(-1.5721,1.2363)--(-1.6473,1.1341)--(-1.7159,1.0273)--(-1.7776,0.9164)--(-1.8322,0.8018)--(-1.8793,0.6840)--(-1.9189,0.5634)--(-1.9508,0.4406)--(-1.9748,0.3160)--(-1.9909,0.1901)--(-1.9989,0.0634)--(-1.9989,-0.0634)--(-1.9909,-0.1901)--(-1.9748,-0.3160)--(-1.9508,-0.4406)--(-1.9189,-0.5634)--(-1.8793,-0.6840)--(-1.8322,-0.8018)--(-1.7776,-0.9164)--(-1.7159,-1.0273)--(-1.6473,-1.1341)--(-1.5721,-1.2363)--(-1.4905,-1.3335)--(-1.4029,-1.4253)--(-1.3097,-1.5114)--(-1.2112,-1.5915)--(-1.1078,-1.6651)--(-0.9999,-1.7320)--(-0.8881,-1.7919)--(-0.7726,-1.8447)--(-0.6541,-1.8900)--(-0.5329,-1.9276)--(-0.4096,-1.9576)--(-0.2846,-1.9796)--(-0.1584,-1.9937)--(-0.0317,-1.9997)--(0.0951,-1.9977)--(0.2216,-1.9876)--(0.3472,-1.9696)--(0.4715,-1.9436)--(0.5938,-1.9098)--(0.7137,-1.8682)--(0.8308,-1.8192)--(0.9445,-1.7629)--(1.0544,-1.6994)--(1.1601,-1.6291)--(1.2611,-1.5522)--(1.3570,-1.4691)--(1.4474,-1.3801)--(1.5320,-1.2855)--(1.6105,-1.1858)--(1.6825,-1.0812)--(1.7476,-0.9723)--(1.8058,-0.8595)--(1.8567,-0.7433)--(1.9001,-0.6240)--(1.9358,-0.5022)--(1.9638,-0.3785)--(1.9839,-0.2531)--(1.9959,-0.1268)--(2.0000,0.0000);
 
-\draw [color=blue,style=dashed] (1.970,-0.3473)--(1.974,-0.3230)--(1.978,-0.2986)--(1.981,-0.2742)--(1.984,-0.2497)--(1.987,-0.2252)--(1.990,-0.2006)--(1.992,-0.1761)--(1.994,-0.1515)--(1.996,-0.1268)--(1.997,-0.1022)--(1.998,-0.07755)--(1.999,-0.05288)--(2.000,-0.02821)--(2.000,-0.003526)--(2.000,0.02116)--(1.999,0.04583)--(1.999,0.07050)--(1.998,0.09516)--(1.996,0.1198)--(1.995,0.1444)--(1.993,0.1690)--(1.991,0.1936)--(1.988,0.2182)--(1.985,0.2427)--(1.982,0.2672)--(1.979,0.2916)--(1.975,0.3160)--(1.971,0.3404)--(1.966,0.3646)--(1.962,0.3889)--(1.957,0.4131)--(1.952,0.4372)--(1.946,0.4612)--(1.940,0.4852)--(1.934,0.5091)--(1.928,0.5330)--(1.921,0.5567)--(1.914,0.5804)--(1.907,0.6039)--(1.899,0.6274)--(1.891,0.6508)--(1.883,0.6741)--(1.875,0.6973)--(1.866,0.7204)--(1.857,0.7433)--(1.847,0.7662)--(1.838,0.7889)--(1.828,0.8115)--(1.818,0.8340)--(1.807,0.8564)--(1.797,0.8786)--(1.786,0.9007)--(1.774,0.9227)--(1.763,0.9445)--(1.751,0.9662)--(1.739,0.9878)--(1.727,1.009)--(1.714,1.030)--(1.701,1.051)--(1.688,1.072)--(1.675,1.093)--(1.661,1.114)--(1.647,1.134)--(1.633,1.154)--(1.619,1.174)--(1.604,1.194)--(1.589,1.214)--(1.574,1.234)--(1.559,1.253)--(1.543,1.272)--(1.528,1.291)--(1.512,1.310)--(1.495,1.328)--(1.479,1.347)--(1.462,1.365)--(1.445,1.383)--(1.428,1.400)--(1.410,1.418)--(1.393,1.435)--(1.375,1.452)--(1.357,1.469)--(1.339,1.486)--(1.320,1.502)--(1.302,1.518)--(1.283,1.534)--(1.264,1.550)--(1.245,1.566)--(1.225,1.581)--(1.206,1.596)--(1.186,1.611)--(1.166,1.625)--(1.146,1.639)--(1.125,1.653)--(1.105,1.667)--(1.084,1.681)--(1.063,1.694)--(1.042,1.707)--(1.021,1.720)--(1.000,1.732);
-\draw (0.65195,0.18941) node {$\theta$};
+\draw [color=blue,style=dashed] (1.9696,-0.3472)--(1.9737,-0.3229)--(1.9775,-0.2985)--(1.9811,-0.2741)--(1.9843,-0.2496)--(1.9872,-0.2251)--(1.9899,-0.2006)--(1.9922,-0.1760)--(1.9942,-0.1514)--(1.9959,-0.1268)--(1.9973,-0.1022)--(1.9984,-0.0775)--(1.9993,-0.0528)--(1.9998,-0.0282)--(1.9999,-0.0035)--(1.9998,0.0211)--(1.9994,0.0458)--(1.9987,0.0705)--(1.9977,0.0951)--(1.9964,0.1198)--(1.9947,0.1444)--(1.9928,0.1690)--(1.9906,0.1936)--(1.9880,0.2181)--(1.9852,0.2426)--(1.9820,0.2671)--(1.9786,0.2916)--(1.9748,0.3160)--(1.9708,0.3403)--(1.9664,0.3646)--(1.9618,0.3888)--(1.9568,0.4130)--(1.9516,0.4371)--(1.9460,0.4612)--(1.9402,0.4852)--(1.9341,0.5091)--(1.9276,0.5329)--(1.9209,0.5566)--(1.9139,0.5803)--(1.9066,0.6039)--(1.8990,0.6274)--(1.8911,0.6508)--(1.8829,0.6740)--(1.8745,0.6972)--(1.8657,0.7203)--(1.8567,0.7433)--(1.8474,0.7661)--(1.8378,0.7889)--(1.8279,0.8115)--(1.8177,0.8340)--(1.8073,0.8564)--(1.7966,0.8786)--(1.7856,0.9007)--(1.7744,0.9227)--(1.7629,0.9445)--(1.7511,0.9662)--(1.7390,0.9877)--(1.7267,1.0091)--(1.7141,1.0303)--(1.7013,1.0514)--(1.6882,1.0723)--(1.6748,1.0931)--(1.6612,1.1137)--(1.6473,1.1341)--(1.6332,1.1543)--(1.6188,1.1744)--(1.6042,1.1943)--(1.5893,1.2140)--(1.5742,1.2335)--(1.5589,1.2528)--(1.5433,1.2720)--(1.5275,1.2909)--(1.5114,1.3097)--(1.4952,1.3282)--(1.4787,1.3466)--(1.4619,1.3647)--(1.4450,1.3827)--(1.4278,1.4004)--(1.4104,1.4179)--(1.3928,1.4352)--(1.3750,1.4523)--(1.3570,1.4691)--(1.3387,1.4858)--(1.3203,1.5022)--(1.3017,1.5184)--(1.2828,1.5343)--(1.2638,1.5500)--(1.2446,1.5655)--(1.2252,1.5807)--(1.2056,1.5957)--(1.1858,1.6105)--(1.1658,1.6250)--(1.1457,1.6393)--(1.1253,1.6533)--(1.1049,1.6670)--(1.0842,1.6805)--(1.0634,1.6938)--(1.0424,1.7068)--(1.0212,1.7195)--(1.0000,1.7320);
+\draw (0.6519,0.1894) node {$\theta$};
 
-\draw [] (0.492,-0.0868)--(0.493,-0.0807)--(0.494,-0.0746)--(0.495,-0.0685)--(0.496,-0.0624)--(0.497,-0.0563)--(0.497,-0.0502)--(0.498,-0.0440)--(0.499,-0.0379)--(0.499,-0.0317)--(0.499,-0.0256)--(0.500,-0.0194)--(0.500,-0.0132)--(0.500,-0.00705)--(0.500,0)--(0.500,0.00529)--(0.500,0.0115)--(0.500,0.0176)--(0.499,0.0238)--(0.499,0.0300)--(0.499,0.0361)--(0.498,0.0423)--(0.498,0.0484)--(0.497,0.0545)--(0.496,0.0607)--(0.496,0.0668)--(0.495,0.0729)--(0.494,0.0790)--(0.493,0.0851)--(0.492,0.0912)--(0.490,0.0972)--(0.489,0.103)--(0.488,0.109)--(0.487,0.115)--(0.485,0.121)--(0.484,0.127)--(0.482,0.133)--(0.480,0.139)--(0.478,0.145)--(0.477,0.151)--(0.475,0.157)--(0.473,0.163)--(0.471,0.169)--(0.469,0.174)--(0.466,0.180)--(0.464,0.186)--(0.462,0.192)--(0.459,0.197)--(0.457,0.203)--(0.454,0.209)--(0.452,0.214)--(0.449,0.220)--(0.446,0.225)--(0.444,0.231)--(0.441,0.236)--(0.438,0.242)--(0.435,0.247)--(0.432,0.252)--(0.429,0.258)--(0.425,0.263)--(0.422,0.268)--(0.419,0.273)--(0.415,0.278)--(0.412,0.284)--(0.408,0.289)--(0.405,0.294)--(0.401,0.299)--(0.397,0.303)--(0.394,0.308)--(0.390,0.313)--(0.386,0.318)--(0.382,0.323)--(0.378,0.327)--(0.374,0.332)--(0.370,0.337)--(0.366,0.341)--(0.361,0.346)--(0.357,0.350)--(0.353,0.354)--(0.348,0.359)--(0.344,0.363)--(0.339,0.367)--(0.335,0.371)--(0.330,0.376)--(0.325,0.380)--(0.321,0.384)--(0.316,0.388)--(0.311,0.391)--(0.306,0.395)--(0.301,0.399)--(0.296,0.403)--(0.291,0.406)--(0.286,0.410)--(0.281,0.413)--(0.276,0.417)--(0.271,0.420)--(0.266,0.423)--(0.261,0.427)--(0.255,0.430)--(0.250,0.433);
-\draw [] (0,0) -- (1.97,-0.347);
-\draw [] (0,0) -- (1.00,1.73);
-\draw (0.10021,1.1407) node {$R$};
-\draw (2.4299,-0.55350) node {$\theta_0$};
-\draw (1.3148,2.1460) node {$\theta_1$};
+\draw [] (0.4924,-0.0868)--(0.4934,-0.0807)--(0.4943,-0.0746)--(0.4952,-0.0685)--(0.4960,-0.0624)--(0.4968,-0.0562)--(0.4974,-0.0501)--(0.4980,-0.0440)--(0.4985,-0.0378)--(0.4989,-0.0317)--(0.4993,-0.0255)--(0.4996,-0.0193)--(0.4998,-0.0132)--(0.4999,-0.0070)--(0.4999,0.0000)--(0.4999,0.0052)--(0.4998,0.0114)--(0.4996,0.0176)--(0.4994,0.0237)--(0.4991,0.0299)--(0.4986,0.0361)--(0.4982,0.0422)--(0.4976,0.0484)--(0.4970,0.0545)--(0.4963,0.0606)--(0.4955,0.0667)--(0.4946,0.0729)--(0.4937,0.0790)--(0.4927,0.0850)--(0.4916,0.0911)--(0.4904,0.0972)--(0.4892,0.1032)--(0.4879,0.1092)--(0.4865,0.1153)--(0.4850,0.1213)--(0.4835,0.1272)--(0.4819,0.1332)--(0.4802,0.1391)--(0.4784,0.1450)--(0.4766,0.1509)--(0.4747,0.1568)--(0.4727,0.1627)--(0.4707,0.1685)--(0.4686,0.1743)--(0.4664,0.1800)--(0.4641,0.1858)--(0.4618,0.1915)--(0.4594,0.1972)--(0.4569,0.2028)--(0.4544,0.2085)--(0.4518,0.2141)--(0.4491,0.2196)--(0.4464,0.2251)--(0.4436,0.2306)--(0.4407,0.2361)--(0.4377,0.2415)--(0.4347,0.2469)--(0.4316,0.2522)--(0.4285,0.2575)--(0.4253,0.2628)--(0.4220,0.2680)--(0.4187,0.2732)--(0.4153,0.2784)--(0.4118,0.2835)--(0.4083,0.2885)--(0.4047,0.2936)--(0.4010,0.2985)--(0.3973,0.3035)--(0.3935,0.3083)--(0.3897,0.3132)--(0.3858,0.3180)--(0.3818,0.3227)--(0.3778,0.3274)--(0.3738,0.3320)--(0.3696,0.3366)--(0.3654,0.3411)--(0.3612,0.3456)--(0.3569,0.3501)--(0.3526,0.3544)--(0.3482,0.3588)--(0.3437,0.3630)--(0.3392,0.3672)--(0.3346,0.3714)--(0.3300,0.3755)--(0.3254,0.3796)--(0.3207,0.3835)--(0.3159,0.3875)--(0.3111,0.3913)--(0.3063,0.3951)--(0.3014,0.3989)--(0.2964,0.4026)--(0.2914,0.4062)--(0.2864,0.4098)--(0.2813,0.4133)--(0.2762,0.4167)--(0.2710,0.4201)--(0.2658,0.4234)--(0.2606,0.4267)--(0.2553,0.4298)--(0.2500,0.4330);
+\draw [] (0.0000,0.0000) -- (1.9696,-0.3472);
+\draw [] (0.0000,0.0000) -- (1.0000,1.7320);
+\draw (0.1002,1.1407) node {$R$};
+\draw (2.4298,-0.5535) node {$\theta_0$};
+\draw (1.3148,2.1459) node {$\theta_1$};
 %END PSPICTURE
 \end{tikzpicture}
 %AFTER PSPICTURE
diff --git a/src_phystricks/Fig_ratrap.pstricks.recall b/src_phystricks/Fig_ratrap.pstricks.recall
index c0c32b68c..640d9fd7c 100644
--- a/src_phystricks/Fig_ratrap.pstricks.recall
+++ b/src_phystricks/Fig_ratrap.pstricks.recall
@@ -71,8 +71,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.50000,0) -- (2.5000,0);
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 % declaring the keys in tikz
@@ -81,19 +81,19 @@
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+\draw [color=blue] (1.0000,0.0000)--(0.9998,0.0158)--(0.9994,0.0317)--(0.9988,0.0475)--(0.9979,0.0634)--(0.9968,0.0792)--(0.9954,0.0950)--(0.9938,0.1108)--(0.9919,0.1265)--(0.9898,0.1423)--(0.9874,0.1580)--(0.9848,0.1736)--(0.9819,0.1892)--(0.9788,0.2048)--(0.9754,0.2203)--(0.9718,0.2357)--(0.9679,0.2511)--(0.9638,0.2664)--(0.9594,0.2817)--(0.9549,0.2969)--(0.9500,0.3120)--(0.9450,0.3270)--(0.9396,0.3420)--(0.9341,0.3568)--(0.9283,0.3716)--(0.9223,0.3863)--(0.9161,0.4009)--(0.9096,0.4154)--(0.9029,0.4297)--(0.8959,0.4440)--(0.8888,0.4582)--(0.8814,0.4722)--(0.8738,0.4861)--(0.8660,0.5000)--(0.8579,0.5136)--(0.8497,0.5272)--(0.8412,0.5406)--(0.8325,0.5539)--(0.8236,0.5670)--(0.8145,0.5800)--(0.8052,0.5929)--(0.7957,0.6056)--(0.7860,0.6181)--(0.7761,0.6305)--(0.7660,0.6427)--(0.7557,0.6548)--(0.7452,0.6667)--(0.7345,0.6785)--(0.7237,0.6900)--(0.7126,0.7014)--(0.7014,0.7126)--(0.6900,0.7237)--(0.6785,0.7345)--(0.6667,0.7452)--(0.6548,0.7557)--(0.6427,0.7660)--(0.6305,0.7761)--(0.6181,0.7860)--(0.6056,0.7957)--(0.5929,0.8052)--(0.5800,0.8145)--(0.5670,0.8236)--(0.5539,0.8325)--(0.5406,0.8412)--(0.5272,0.8497)--(0.5136,0.8579)--(0.5000,0.8660)--(0.4861,0.8738)--(0.4722,0.8814)--(0.4582,0.8888)--(0.4440,0.8959)--(0.4297,0.9029)--(0.4154,0.9096)--(0.4009,0.9161)--(0.3863,0.9223)--(0.3716,0.9283)--(0.3568,0.9341)--(0.3420,0.9396)--(0.3270,0.9450)--(0.3120,0.9500)--(0.2969,0.9549)--(0.2817,0.9594)--(0.2664,0.9638)--(0.2511,0.9679)--(0.2357,0.9718)--(0.2203,0.9754)--(0.2048,0.9788)--(0.1892,0.9819)--(0.1736,0.9848)--(0.1580,0.9874)--(0.1423,0.9898)--(0.1265,0.9919)--(0.1108,0.9938)--(0.0950,0.9954)--(0.0792,0.9968)--(0.0634,0.9979)--(0.0475,0.9988)--(0.0317,0.9994)--(0.0158,0.9998)--(0.0000,1.0000);
+\draw [color=blue] (2.0000,0.0000)--(1.9997,0.0317)--(1.9989,0.0634)--(1.9977,0.0951)--(1.9959,0.1268)--(1.9937,0.1584)--(1.9909,0.1901)--(1.9876,0.2216)--(1.9839,0.2531)--(1.9796,0.2846)--(1.9748,0.3160)--(1.9696,0.3472)--(1.9638,0.3785)--(1.9576,0.4096)--(1.9508,0.4406)--(1.9436,0.4715)--(1.9358,0.5022)--(1.9276,0.5329)--(1.9189,0.5634)--(1.9098,0.5938)--(1.9001,0.6240)--(1.8900,0.6541)--(1.8793,0.6840)--(1.8682,0.7137)--(1.8567,0.7433)--(1.8447,0.7726)--(1.8322,0.8018)--(1.8192,0.8308)--(1.8058,0.8595)--(1.7919,0.8881)--(1.7776,0.9164)--(1.7629,0.9445)--(1.7476,0.9723)--(1.7320,1.0000)--(1.7159,1.0273)--(1.6994,1.0544)--(1.6825,1.0812)--(1.6651,1.1078)--(1.6473,1.1341)--(1.6291,1.1601)--(1.6105,1.1858)--(1.5915,1.2112)--(1.5721,1.2363)--(1.5522,1.2611)--(1.5320,1.2855)--(1.5114,1.3097)--(1.4905,1.3335)--(1.4691,1.3570)--(1.4474,1.3801)--(1.4253,1.4029)--(1.4029,1.4253)--(1.3801,1.4474)--(1.3570,1.4691)--(1.3335,1.4905)--(1.3097,1.5114)--(1.2855,1.5320)--(1.2611,1.5522)--(1.2363,1.5721)--(1.2112,1.5915)--(1.1858,1.6105)--(1.1601,1.6291)--(1.1341,1.6473)--(1.1078,1.6651)--(1.0812,1.6825)--(1.0544,1.6994)--(1.0273,1.7159)--(1.0000,1.7320)--(0.9723,1.7476)--(0.9445,1.7629)--(0.9164,1.7776)--(0.8881,1.7919)--(0.8595,1.8058)--(0.8308,1.8192)--(0.8018,1.8322)--(0.7726,1.8447)--(0.7433,1.8567)--(0.7137,1.8682)--(0.6840,1.8793)--(0.6541,1.8900)--(0.6240,1.9001)--(0.5938,1.9098)--(0.5634,1.9189)--(0.5329,1.9276)--(0.5022,1.9358)--(0.4715,1.9436)--(0.4406,1.9508)--(0.4096,1.9576)--(0.3785,1.9638)--(0.3472,1.9696)--(0.3160,1.9748)--(0.2846,1.9796)--(0.2531,1.9839)--(0.2216,1.9876)--(0.1901,1.9909)--(0.1584,1.9937)--(0.1268,1.9959)--(0.0951,1.9977)--(0.0634,1.9989)--(0.0317,1.9997)--(0.0000,2.0000);
+\draw [color=blue] (0.0000,2.0000) -- (0.0000,1.0000);
+\draw [color=blue] (1.0000,0.0000) -- (2.0000,0.0000);
+\draw []  (1.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (1.0000,-0.2785) node {$a$};
+\draw []  (0.0000,1.0000) node [rotate=0] {$\bullet$};
+\draw (-0.2964,1.0000) node {$a$};
+\draw []  (2.0000,0.0000) node [rotate=0] {$\bullet$};
+\draw (2.0000,-0.3267) node {$b$};
+\draw []  (0.0000,2.0000) node [rotate=0] {$\bullet$};
+\draw (-0.2783,2.0000) node {$b$};
 
 %OTHER STUFF
 %END PSPICTURE

From 1c41beb275a4f2921317289f09953e171c67a91f Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 13:45:14 +0200
Subject: [PATCH 35/64] (pictures) Add the requested import from sage.all

The reason is that ''phystricks' does not import 'sage.all *' anymore.
---
 auto/pictures_tex/Fig_BIFooDsvVHb.pstricks     | 2 +-
 auto/pictures_tex/Fig_CFMooGzvfRP.pstricks     | 2 +-
 auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks  | 4 ++--
 auto/pictures_tex/Fig_Cardioideexo.pstricks    | 2 +-
 auto/pictures_tex/Fig_CercleImplicite.pstricks | 2 +-
 auto/pictures_tex/Fig_CercleTrigono.pstricks   | 2 +-
 auto/pictures_tex/Fig_ConeRevolution.pstricks  | 2 +-
 auto/pictures_tex/Fig_CoordPolaires.pstricks   | 2 +-
 auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks  | 2 +-
 auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks  | 2 +-
 auto/pictures_tex/Fig_DessinLim.pstricks       | 4 ++--
 auto/pictures_tex/Fig_DeuxCercles.pstricks     | 2 +-
 auto/pictures_tex/Fig_DisqueConv.pstricks      | 2 +-
 auto/pictures_tex/Fig_ExoPolaire.pstricks      | 2 +-
 auto/pictures_tex/Fig_JGuKEjH.pstricks         | 2 +-
 auto/pictures_tex/Fig_JWINooSfKCeA.pstricks    | 2 +-
 auto/pictures_tex/Fig_KScolorD.pstricks        | 4 ++--
 auto/pictures_tex/Fig_LesSpheres.pstricks      | 2 +-
 auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks  | 2 +-
 auto/pictures_tex/Fig_TgCercleTrigono.pstricks | 2 +-
 auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks    | 2 +-
 src_phystricks/phystricksChiSquared.py         | 1 +
 src_phystricks/phystricksChiSquaresQuantile.py | 1 +
 src_phystricks/phystricksFGRooDhFkch.py        | 1 +
 src_phystricks/phystricksGMIooJvcCXg.py        | 1 +
 src_phystricks/phystricksMantisse.py           | 1 +
 src_phystricks/phystricksMaxVraissLp.py        | 1 +
 src_phystricks/phystricksPartieEntiere.py      | 1 +
 src_phystricks/phystricksRLuqsrr.py            | 4 +---
 src_phystricks/phystricksSYNKooZBuEWsWw.py     | 1 +
 src_phystricks/phystricksSpiraleLimite.py      | 1 +
 src_phystricks/phystricksUGCFooQoCihh.py       | 4 +---
 src_phystricks/phystricksUNVooMsXxHa.py        | 1 +
 src_phystricks/phystricksYYECooQlnKtD.py       | 1 +
 34 files changed, 37 insertions(+), 30 deletions(-)

diff --git a/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks b/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
index 5d8af556c..4b2b8960b 100644
--- a/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
+++ b/auto/pictures_tex/Fig_BIFooDsvVHb.pstricks
@@ -72,7 +72,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks b/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
index 31f37cf34..92ab0e718 100644
--- a/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
+++ b/auto/pictures_tex/Fig_CFMooGzvfRP.pstricks
@@ -64,7 +64,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
index 53a4f6010..ac874c7ad 100644
--- a/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
+++ b/auto/pictures_tex/Fig_CWKJooppMsZXjw.pstricks
@@ -68,8 +68,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-3.5000,0.0000) -- (1.0000,0.0000);
-\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,0.5000);
+\draw [,->,>=latex] (-3.5000,0.0000) -- (1.1000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,0.6000);
 %DEFAULT
 \draw [,->,>=latex] (-3.0000,-1.0000) -- (-3.0000,0.0000);
 \draw [] (0.0000,0.0000) -- (-3.0000,-1.0000);
diff --git a/auto/pictures_tex/Fig_Cardioideexo.pstricks b/auto/pictures_tex/Fig_Cardioideexo.pstricks
index 87bc7a7ef..569b76e57 100644
--- a/auto/pictures_tex/Fig_Cardioideexo.pstricks
+++ b/auto/pictures_tex/Fig_Cardioideexo.pstricks
@@ -79,7 +79,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_CercleImplicite.pstricks b/auto/pictures_tex/Fig_CercleImplicite.pstricks
index 570ddd92b..f73d234ae 100644
--- a/auto/pictures_tex/Fig_CercleImplicite.pstricks
+++ b/auto/pictures_tex/Fig_CercleImplicite.pstricks
@@ -79,7 +79,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_CercleTrigono.pstricks b/auto/pictures_tex/Fig_CercleTrigono.pstricks
index bb88ff553..fc8423387 100644
--- a/auto/pictures_tex/Fig_CercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_CercleTrigono.pstricks
@@ -79,7 +79,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_ConeRevolution.pstricks b/auto/pictures_tex/Fig_ConeRevolution.pstricks
index 8f2baeb65..44b78d6a4 100644
--- a/auto/pictures_tex/Fig_ConeRevolution.pstricks
+++ b/auto/pictures_tex/Fig_ConeRevolution.pstricks
@@ -80,7 +80,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
-\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
+\draw [,->,>=latex] (0.0000,-0.6000) -- (0.0000,3.5000);
 %DEFAULT
 
 \draw [color=blue] (0.0000,0.0000)--(0.0202,0.0303)--(0.0404,0.0606)--(0.0606,0.0909)--(0.0808,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5151)--(0.3636,0.5454)--(0.3838,0.5757)--(0.4040,0.6060)--(0.4242,0.6363)--(0.4444,0.6666)--(0.4646,0.6969)--(0.4848,0.7272)--(0.5050,0.7575)--(0.5252,0.7878)--(0.5454,0.8181)--(0.5656,0.8484)--(0.5858,0.8787)--(0.6060,0.9090)--(0.6262,0.9393)--(0.6464,0.9696)--(0.6666,1.0000)--(0.6868,1.0303)--(0.7070,1.0606)--(0.7272,1.0909)--(0.7474,1.1212)--(0.7676,1.1515)--(0.7878,1.1818)--(0.8080,1.2121)--(0.8282,1.2424)--(0.8484,1.2727)--(0.8686,1.3030)--(0.8888,1.3333)--(0.9090,1.3636)--(0.9292,1.3939)--(0.9494,1.4242)--(0.9696,1.4545)--(0.9898,1.4848)--(1.0101,1.5151)--(1.0303,1.5454)--(1.0505,1.5757)--(1.0707,1.6060)--(1.0909,1.6363)--(1.1111,1.6666)--(1.1313,1.6969)--(1.1515,1.7272)--(1.1717,1.7575)--(1.1919,1.7878)--(1.2121,1.8181)--(1.2323,1.8484)--(1.2525,1.8787)--(1.2727,1.9090)--(1.2929,1.9393)--(1.3131,1.9696)--(1.3333,2.0000)--(1.3535,2.0303)--(1.3737,2.0606)--(1.3939,2.0909)--(1.4141,2.1212)--(1.4343,2.1515)--(1.4545,2.1818)--(1.4747,2.2121)--(1.4949,2.2424)--(1.5151,2.2727)--(1.5353,2.3030)--(1.5555,2.3333)--(1.5757,2.3636)--(1.5959,2.3939)--(1.6161,2.4242)--(1.6363,2.4545)--(1.6565,2.4848)--(1.6767,2.5151)--(1.6969,2.5454)--(1.7171,2.5757)--(1.7373,2.6060)--(1.7575,2.6363)--(1.7777,2.6666)--(1.7979,2.6969)--(1.8181,2.7272)--(1.8383,2.7575)--(1.8585,2.7878)--(1.8787,2.8181)--(1.8989,2.8484)--(1.9191,2.8787)--(1.9393,2.9090)--(1.9595,2.9393)--(1.9797,2.9696)--(2.0000,3.0000);
diff --git a/auto/pictures_tex/Fig_CoordPolaires.pstricks b/auto/pictures_tex/Fig_CoordPolaires.pstricks
index e35320430..beadbe2bd 100644
--- a/auto/pictures_tex/Fig_CoordPolaires.pstricks
+++ b/auto/pictures_tex/Fig_CoordPolaires.pstricks
@@ -84,7 +84,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
-\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
+\draw [,->,>=latex] (0.0000,-0.6000) -- (0.0000,2.5000);
 %DEFAULT
 \draw (1.5233,2.0000) node {$(x,y)$};
 \draw [,->,>=latex] (0.0000,0.0000) -- (1.0000,2.0000);
diff --git a/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks b/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
index 2e7cbdaeb..4c99b1669 100644
--- a/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
+++ b/auto/pictures_tex/Fig_DNHRooqGtffLkd.pstricks
@@ -107,7 +107,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0.0000) -- (5.5000,0.0000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (5.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks b/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
index 83dad0aaf..f9e5d5f84 100644
--- a/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
+++ b/auto/pictures_tex/Fig_DNRRooJWRHgOCw.pstricks
@@ -95,7 +95,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000,0.0000) -- (8.5000,0.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (8.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5000) -- (0.0000,2.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_DessinLim.pstricks b/auto/pictures_tex/Fig_DessinLim.pstricks
index 509b71d62..d7d9d041a 100644
--- a/auto/pictures_tex/Fig_DessinLim.pstricks
+++ b/auto/pictures_tex/Fig_DessinLim.pstricks
@@ -87,8 +87,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000,0.0000) -- (2.8000,0.0000);
-\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.8000);
+\draw [,->,>=latex] (-0.6000,0.0000) -- (2.9000,0.0000);
+\draw [,->,>=latex] (0.0000,-0.6000) -- (0.0000,2.9000);
 %DEFAULT
 
 \draw [color=blue] (2.3000,0.0000)--(2.2997,0.0364)--(2.2988,0.0729)--(2.2973,0.1094)--(2.2953,0.1458)--(2.2927,0.1822)--(2.2895,0.2186)--(2.2858,0.2549)--(2.2814,0.2911)--(2.2765,0.3273)--(2.2711,0.3634)--(2.2650,0.3993)--(2.2584,0.4352)--(2.2512,0.4710)--(2.2434,0.5067)--(2.2351,0.5422)--(2.2262,0.5776)--(2.2168,0.6128)--(2.2068,0.6479)--(2.1962,0.6829)--(2.1851,0.7176)--(2.1735,0.7522)--(2.1612,0.7866)--(2.1485,0.8208)--(2.1352,0.8548)--(2.1214,0.8885)--(2.1070,0.9221)--(2.0921,0.9554)--(2.0767,0.9885)--(2.0607,1.0213)--(2.0443,1.0539)--(2.0273,1.0862)--(2.0098,1.1182)--(1.9918,1.1500)--(1.9733,1.1814)--(1.9543,1.2126)--(1.9348,1.2434)--(1.9149,1.2740)--(1.8944,1.3042)--(1.8735,1.3341)--(1.8521,1.3636)--(1.8302,1.3929)--(1.8079,1.4217)--(1.7851,1.4502)--(1.7619,1.4784)--(1.7382,1.5061)--(1.7141,1.5335)--(1.6895,1.5605)--(1.6645,1.5871)--(1.6391,1.6133)--(1.6133,1.6391)--(1.5871,1.6645)--(1.5605,1.6895)--(1.5335,1.7141)--(1.5061,1.7382)--(1.4784,1.7619)--(1.4502,1.7851)--(1.4217,1.8079)--(1.3929,1.8302)--(1.3636,1.8521)--(1.3341,1.8735)--(1.3042,1.8944)--(1.2740,1.9149)--(1.2434,1.9348)--(1.2126,1.9543)--(1.1814,1.9733)--(1.1500,1.9918)--(1.1182,2.0098)--(1.0862,2.0273)--(1.0539,2.0443)--(1.0213,2.0607)--(0.9885,2.0767)--(0.9554,2.0921)--(0.9221,2.1070)--(0.8885,2.1214)--(0.8548,2.1352)--(0.8208,2.1485)--(0.7866,2.1612)--(0.7522,2.1735)--(0.7176,2.1851)--(0.6829,2.1962)--(0.6479,2.2068)--(0.6128,2.2168)--(0.5776,2.2262)--(0.5422,2.2351)--(0.5067,2.2434)--(0.4710,2.2512)--(0.4352,2.2584)--(0.3993,2.2650)--(0.3634,2.2711)--(0.3273,2.2765)--(0.2911,2.2814)--(0.2549,2.2858)--(0.2186,2.2895)--(0.1822,2.2927)--(0.1458,2.2953)--(0.1094,2.2973)--(0.0729,2.2988)--(0.0364,2.2997)--(0.0000,2.3000);
diff --git a/auto/pictures_tex/Fig_DeuxCercles.pstricks b/auto/pictures_tex/Fig_DeuxCercles.pstricks
index c6ffd32e5..d48c24f59 100644
--- a/auto/pictures_tex/Fig_DeuxCercles.pstricks
+++ b/auto/pictures_tex/Fig_DeuxCercles.pstricks
@@ -67,7 +67,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,2.5000);
 %DEFAULT
 % declaring the keys in tikz
diff --git a/auto/pictures_tex/Fig_DisqueConv.pstricks b/auto/pictures_tex/Fig_DisqueConv.pstricks
index 3daf3e818..cdcbc987d 100644
--- a/auto/pictures_tex/Fig_DisqueConv.pstricks
+++ b/auto/pictures_tex/Fig_DisqueConv.pstricks
@@ -67,7 +67,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-0.5000,0.0000) -- (3.5000,0.0000);
+\draw [,->,>=latex] (-0.5000,0.0000) -- (3.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
 %DEFAULT
 \draw []  (2.0000,2.0000) node [rotate=0] {$\bullet$};
diff --git a/auto/pictures_tex/Fig_ExoPolaire.pstricks b/auto/pictures_tex/Fig_ExoPolaire.pstricks
index f17182b99..b7f1abe90 100644
--- a/auto/pictures_tex/Fig_ExoPolaire.pstricks
+++ b/auto/pictures_tex/Fig_ExoPolaire.pstricks
@@ -76,7 +76,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000,0.0000) -- (2.2320,0.0000);
-\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,1.5000);
+\draw [,->,>=latex] (0.0000,-0.6000) -- (0.0000,1.5000);
 %DEFAULT
 \draw (2.3896,1.0000) node {$(\sqrt{3},1)$};
 \draw (0.6579,0.8865) node {$l$};
diff --git a/auto/pictures_tex/Fig_JGuKEjH.pstricks b/auto/pictures_tex/Fig_JGuKEjH.pstricks
index 3f97fb0e7..875ee8bc1 100644
--- a/auto/pictures_tex/Fig_JGuKEjH.pstricks
+++ b/auto/pictures_tex/Fig_JGuKEjH.pstricks
@@ -63,7 +63,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks b/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
index 66eb5ab83..60c0807e9 100644
--- a/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
+++ b/auto/pictures_tex/Fig_JWINooSfKCeA.pstricks
@@ -84,7 +84,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000,0.0000) -- (1.5000,0.0000);
-\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,2.5000);
+\draw [,->,>=latex] (0.0000,-0.6000) -- (0.0000,2.5000);
 %DEFAULT
 \draw (1.5233,2.0000) node {$(x,y)$};
 \draw [,->,>=latex] (0.0000,0.0000) -- (1.0000,2.0000);
diff --git a/auto/pictures_tex/Fig_KScolorD.pstricks b/auto/pictures_tex/Fig_KScolorD.pstricks
index 92eee38ae..43372e3a2 100644
--- a/auto/pictures_tex/Fig_KScolorD.pstricks
+++ b/auto/pictures_tex/Fig_KScolorD.pstricks
@@ -63,8 +63,8 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.6968,0.0000) -- (1.5000,0.0000);
-\draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.6940);
+\draw [,->,>=latex] (-1.6968,0.0000) -- (1.6000,0.0000);
+\draw [,->,>=latex] (0.0000,-1.6000) -- (0.0000,1.6940);
 %DEFAULT
 
 \draw [color=blue] (1.0000,0.0000)--(0.9979,0.0634)--(0.9919,0.1265)--(0.9819,0.1892)--(0.9679,0.2511)--(0.9500,0.3120)--(0.9283,0.3716)--(0.9029,0.4297)--(0.8738,0.4861)--(0.8412,0.5406)--(0.8052,0.5929)--(0.7660,0.6427)--(0.7237,0.6900)--(0.6785,0.7345)--(0.6305,0.7761)--(0.5800,0.8145)--(0.5272,0.8497)--(0.4722,0.8814)--(0.4154,0.9096)--(0.3568,0.9341)--(0.2969,0.9549)--(0.2357,0.9718)--(0.1736,0.9848)--(0.1108,0.9938)--(0.0475,0.9988)--(-0.0158,0.9998)--(-0.0792,0.9968)--(-0.1423,0.9898)--(-0.2048,0.9788)--(-0.2664,0.9638)--(-0.3270,0.9450)--(-0.3863,0.9223)--(-0.4440,0.8959)--(-0.5000,0.8660)--(-0.5539,0.8325)--(-0.6056,0.7957)--(-0.6548,0.7557)--(-0.7014,0.7126)--(-0.7452,0.6667)--(-0.7860,0.6181)--(-0.8236,0.5670)--(-0.8579,0.5136)--(-0.8888,0.4582)--(-0.9161,0.4009)--(-0.9396,0.3420)--(-0.9594,0.2817)--(-0.9754,0.2203)--(-0.9874,0.1580)--(-0.9954,0.0950)--(-0.9994,0.0317)--(-0.9994,-0.0317)--(-0.9954,-0.0950)--(-0.9874,-0.1580)--(-0.9754,-0.2203)--(-0.9594,-0.2817)--(-0.9396,-0.3420)--(-0.9161,-0.4009)--(-0.8888,-0.4582)--(-0.8579,-0.5136)--(-0.8236,-0.5670)--(-0.7860,-0.6181)--(-0.7452,-0.6667)--(-0.7014,-0.7126)--(-0.6548,-0.7557)--(-0.6056,-0.7957)--(-0.5539,-0.8325)--(-0.5000,-0.8660)--(-0.4440,-0.8959)--(-0.3863,-0.9223)--(-0.3270,-0.9450)--(-0.2664,-0.9638)--(-0.2048,-0.9788)--(-0.1423,-0.9898)--(-0.0792,-0.9968)--(-0.0158,-0.9998)--(0.0475,-0.9988)--(0.1108,-0.9938)--(0.1736,-0.9848)--(0.2357,-0.9718)--(0.2969,-0.9549)--(0.3568,-0.9341)--(0.4154,-0.9096)--(0.4722,-0.8814)--(0.5272,-0.8497)--(0.5800,-0.8145)--(0.6305,-0.7761)--(0.6785,-0.7345)--(0.7237,-0.6900)--(0.7660,-0.6427)--(0.8052,-0.5929)--(0.8412,-0.5406)--(0.8738,-0.4861)--(0.9029,-0.4297)--(0.9283,-0.3716)--(0.9500,-0.3120)--(0.9679,-0.2511)--(0.9819,-0.1892)--(0.9919,-0.1265)--(0.9979,-0.0634)--(1.0000,0.0000);
diff --git a/auto/pictures_tex/Fig_LesSpheres.pstricks b/auto/pictures_tex/Fig_LesSpheres.pstricks
index cd673c46c..9914101e0 100644
--- a/auto/pictures_tex/Fig_LesSpheres.pstricks
+++ b/auto/pictures_tex/Fig_LesSpheres.pstricks
@@ -93,7 +93,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-1.5000,0.0000) -- (1.5000,0.0000);
+\draw [,->,>=latex] (-1.5000,0.0000) -- (1.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-1.5000) -- (0.0000,1.5000);
 %DEFAULT
 
diff --git a/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks b/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
index f6e6c8bba..b6d68fb3f 100644
--- a/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
+++ b/auto/pictures_tex/Fig_QMWKooRRulrgcH.pstricks
@@ -80,7 +80,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-4.5000,0.0000) -- (4.5000,0.0000);
+\draw [,->,>=latex] (-4.5000,0.0000) -- (4.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-3.5000) -- (0.0000,3.5000);
 %DEFAULT
 \draw []  (-1.0000,0.0000) node [rotate=0] {$\bullet$};
diff --git a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
index 312cf3840..23be84fe7 100644
--- a/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
+++ b/auto/pictures_tex/Fig_TgCercleTrigono.pstricks
@@ -79,7 +79,7 @@
 %GRID
 %PSTRICKS CODE
 %AXES
-\draw [,->,>=latex] (-2.5000,0.0000) -- (2.5000,0.0000);
+\draw [,->,>=latex] (-2.5000,0.0000) -- (2.6000,0.0000);
 \draw [,->,>=latex] (0.0000,-2.5392) -- (0.0000,3.7680);
 %DEFAULT
 \draw [color=red,style=dashed] (0.0000,0.0000) -- (2.0000,2.3835);
diff --git a/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks b/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
index 97af61aa9..2cefcf8c4 100644
--- a/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
+++ b/auto/pictures_tex/Fig_YHJYooTEXLLn.pstricks
@@ -80,7 +80,7 @@
 %PSTRICKS CODE
 %AXES
 \draw [,->,>=latex] (-0.5000,0.0000) -- (2.5000,0.0000);
-\draw [,->,>=latex] (0.0000,-0.5000) -- (0.0000,3.5000);
+\draw [,->,>=latex] (0.0000,-0.6000) -- (0.0000,3.5000);
 %DEFAULT
 
 \draw [color=blue] (0.0000,0.0000)--(0.0202,0.0303)--(0.0404,0.0606)--(0.0606,0.0909)--(0.0808,0.1212)--(0.1010,0.1515)--(0.1212,0.1818)--(0.1414,0.2121)--(0.1616,0.2424)--(0.1818,0.2727)--(0.2020,0.3030)--(0.2222,0.3333)--(0.2424,0.3636)--(0.2626,0.3939)--(0.2828,0.4242)--(0.3030,0.4545)--(0.3232,0.4848)--(0.3434,0.5151)--(0.3636,0.5454)--(0.3838,0.5757)--(0.4040,0.6060)--(0.4242,0.6363)--(0.4444,0.6666)--(0.4646,0.6969)--(0.4848,0.7272)--(0.5050,0.7575)--(0.5252,0.7878)--(0.5454,0.8181)--(0.5656,0.8484)--(0.5858,0.8787)--(0.6060,0.9090)--(0.6262,0.9393)--(0.6464,0.9696)--(0.6666,1.0000)--(0.6868,1.0303)--(0.7070,1.0606)--(0.7272,1.0909)--(0.7474,1.1212)--(0.7676,1.1515)--(0.7878,1.1818)--(0.8080,1.2121)--(0.8282,1.2424)--(0.8484,1.2727)--(0.8686,1.3030)--(0.8888,1.3333)--(0.9090,1.3636)--(0.9292,1.3939)--(0.9494,1.4242)--(0.9696,1.4545)--(0.9898,1.4848)--(1.0101,1.5151)--(1.0303,1.5454)--(1.0505,1.5757)--(1.0707,1.6060)--(1.0909,1.6363)--(1.1111,1.6666)--(1.1313,1.6969)--(1.1515,1.7272)--(1.1717,1.7575)--(1.1919,1.7878)--(1.2121,1.8181)--(1.2323,1.8484)--(1.2525,1.8787)--(1.2727,1.9090)--(1.2929,1.9393)--(1.3131,1.9696)--(1.3333,2.0000)--(1.3535,2.0303)--(1.3737,2.0606)--(1.3939,2.0909)--(1.4141,2.1212)--(1.4343,2.1515)--(1.4545,2.1818)--(1.4747,2.2121)--(1.4949,2.2424)--(1.5151,2.2727)--(1.5353,2.3030)--(1.5555,2.3333)--(1.5757,2.3636)--(1.5959,2.3939)--(1.6161,2.4242)--(1.6363,2.4545)--(1.6565,2.4848)--(1.6767,2.5151)--(1.6969,2.5454)--(1.7171,2.5757)--(1.7373,2.6060)--(1.7575,2.6363)--(1.7777,2.6666)--(1.7979,2.6969)--(1.8181,2.7272)--(1.8383,2.7575)--(1.8585,2.7878)--(1.8787,2.8181)--(1.8989,2.8484)--(1.9191,2.8787)--(1.9393,2.9090)--(1.9595,2.9393)--(1.9797,2.9696)--(2.0000,3.0000);
diff --git a/src_phystricks/phystricksChiSquared.py b/src_phystricks/phystricksChiSquared.py
index 2735d05cc..86a341c1c 100644
--- a/src_phystricks/phystricksChiSquared.py
+++ b/src_phystricks/phystricksChiSquared.py
@@ -1,3 +1,4 @@
+from sage.all import RealDistribution
 from phystricks import *
 def ChiSquared():
     pspict,fig = SinglePicture("ChiSquared")
diff --git a/src_phystricks/phystricksChiSquaresQuantile.py b/src_phystricks/phystricksChiSquaresQuantile.py
index a5108ab20..26f2d83e6 100644
--- a/src_phystricks/phystricksChiSquaresQuantile.py
+++ b/src_phystricks/phystricksChiSquaresQuantile.py
@@ -1,3 +1,4 @@
+from sage.all import RealDistribution
 from phystricks import *
 def ChiSquaresQuantile():
     pspict,fig = SinglePicture("ChiSquaresQuantile")
diff --git a/src_phystricks/phystricksFGRooDhFkch.py b/src_phystricks/phystricksFGRooDhFkch.py
index 3e01b9db3..bc071ef22 100644
--- a/src_phystricks/phystricksFGRooDhFkch.py
+++ b/src_phystricks/phystricksFGRooDhFkch.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import arcsin
 from phystricks import *
 def FGRooDhFkch():
     pspict,fig = SinglePicture("FGRooDhFkch")
diff --git a/src_phystricks/phystricksGMIooJvcCXg.py b/src_phystricks/phystricksGMIooJvcCXg.py
index 401ead8a1..7e454c481 100644
--- a/src_phystricks/phystricksGMIooJvcCXg.py
+++ b/src_phystricks/phystricksGMIooJvcCXg.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import arccos
 from phystricks import *
 def GMIooJvcCXg():
     pspict,fig = SinglePicture("GMIooJvcCXg")
diff --git a/src_phystricks/phystricksMantisse.py b/src_phystricks/phystricksMantisse.py
index 6ca87d49b..844b5f9c6 100644
--- a/src_phystricks/phystricksMantisse.py
+++ b/src_phystricks/phystricksMantisse.py
@@ -1,3 +1,4 @@
+from sage.all import floor
 from phystricks import *
 def Mantisse():
     pspict,fig = SinglePicture("Mantisse")
diff --git a/src_phystricks/phystricksMaxVraissLp.py b/src_phystricks/phystricksMaxVraissLp.py
index d630275d9..590fe5920 100644
--- a/src_phystricks/phystricksMaxVraissLp.py
+++ b/src_phystricks/phystricksMaxVraissLp.py
@@ -1,3 +1,4 @@
+from sage.all import binomial
 from phystricks import *
 def MaxVraissLp():
     pspict,fig = SinglePicture("MaxVraissLp")
diff --git a/src_phystricks/phystricksPartieEntiere.py b/src_phystricks/phystricksPartieEntiere.py
index b80268b02..e400c1859 100644
--- a/src_phystricks/phystricksPartieEntiere.py
+++ b/src_phystricks/phystricksPartieEntiere.py
@@ -1,3 +1,4 @@
+from sage.all import floor
 from phystricks import *
 def PartieEntiere():
 	pspict,fig = SinglePicture("PartieEntiere")
diff --git a/src_phystricks/phystricksRLuqsrr.py b/src_phystricks/phystricksRLuqsrr.py
index 153173083..42a9221a9 100644
--- a/src_phystricks/phystricksRLuqsrr.py
+++ b/src_phystricks/phystricksRLuqsrr.py
@@ -16,7 +16,7 @@ def n(u,v):
     f2 = phyFunction( n(0,1) )
 
     mx = 0
-    Mx = mx+2*math.pi
+    Mx = mx+2*pi
 
     F1 = f1.graph(mx,Mx)
     F2 = f2.graph(mx,Mx)
@@ -30,7 +30,5 @@ def n(u,v):
     pspict.axes.single_axeX.Dx=0.5
     pspict.DrawDefaultAxes()
 
-
     fig.conclude()
     fig.write_the_file()
-
diff --git a/src_phystricks/phystricksSYNKooZBuEWsWw.py b/src_phystricks/phystricksSYNKooZBuEWsWw.py
index fec6f89af..ee1cb26f3 100644
--- a/src_phystricks/phystricksSYNKooZBuEWsWw.py
+++ b/src_phystricks/phystricksSYNKooZBuEWsWw.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import sinh,cosh
 from phystricks import *
 def SYNKooZBuEWsWw():
     pspict,fig = SinglePicture("SYNKooZBuEWsWw")
diff --git a/src_phystricks/phystricksSpiraleLimite.py b/src_phystricks/phystricksSpiraleLimite.py
index 095e0417c..82dbfa306 100644
--- a/src_phystricks/phystricksSpiraleLimite.py
+++ b/src_phystricks/phystricksSpiraleLimite.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import arccos
 from phystricks import *
 def SpiraleLimite():
 	pspict,fig = SinglePicture("SpiraleLimite")
diff --git a/src_phystricks/phystricksUGCFooQoCihh.py b/src_phystricks/phystricksUGCFooQoCihh.py
index fecfa8209..8f9dbfa16 100644
--- a/src_phystricks/phystricksUGCFooQoCihh.py
+++ b/src_phystricks/phystricksUGCFooQoCihh.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import factorial
 from phystricks import *
 
 def proba(l,k):
@@ -6,7 +7,6 @@ def proba(l,k):
 
 def UGCFooQoCihh():
     pspict,fig = SinglePicture("UGCFooQoCihh")
-    #pspict.dilatation_X(1)
     pspict.dilatation_Y(10)
 
     l=2
@@ -19,5 +19,3 @@ def UGCFooQoCihh():
     fig.no_figure()
     fig.conclude()
     fig.write_the_file()
-
-
diff --git a/src_phystricks/phystricksUNVooMsXxHa.py b/src_phystricks/phystricksUNVooMsXxHa.py
index 61a0c8258..aa26d08ed 100644
--- a/src_phystricks/phystricksUNVooMsXxHa.py
+++ b/src_phystricks/phystricksUNVooMsXxHa.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import cosh,sinh
 from phystricks import *
 def UNVooMsXxHa():
     pspict,fig = SinglePicture("UNVooMsXxHa")
diff --git a/src_phystricks/phystricksYYECooQlnKtD.py b/src_phystricks/phystricksYYECooQlnKtD.py
index f4f5af667..7d5cc910d 100644
--- a/src_phystricks/phystricksYYECooQlnKtD.py
+++ b/src_phystricks/phystricksYYECooQlnKtD.py
@@ -1,4 +1,5 @@
 # -*- coding: utf8 -*-
+from sage.all import binomial
 from phystricks import *
 
 def proba(n,p,k):

From a3ceb501239bc907d7eb1d2fca59203490a3b96e Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 13:50:22 +0200
Subject: [PATCH 36/64] =?UTF-8?q?(num=C3=A9rique)=20Ajoute=20une=20figure?=
 =?UTF-8?q?=20donnant=20une=20image=20du=20maillage=20=C3=A0=205=20points?=
 =?UTF-8?q?=20dans=20la=20m=C3=A9thode=20des=20diff=C3=A9rences=20finies.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 auto/pictures_tex/Fig_GMRNooCNBpIl.pstricks | 185 ++++++++++++++++++++
 src_phystricks/figures_mazhe.py             |   6 +-
 2 files changed, 190 insertions(+), 1 deletion(-)
 create mode 100644 auto/pictures_tex/Fig_GMRNooCNBpIl.pstricks

diff --git a/auto/pictures_tex/Fig_GMRNooCNBpIl.pstricks b/auto/pictures_tex/Fig_GMRNooCNBpIl.pstricks
new file mode 100644
index 000000000..af43e05d4
--- /dev/null
+++ b/auto/pictures_tex/Fig_GMRNooCNBpIl.pstricks
@@ -0,0 +1,185 @@
+%ENTETE FIGURE
+% This file is automatically generated by phystricks
+% See the documentation 
+% http://student.ulb.ac.be/~lclaesse/phystricks-doc.pdf 
+% http://student.ulb.ac.be/~lclaesse/phystricks-documentation/_build/html/index.html 
+% and the projects phystricks and phystricks-doc at 
+% https://github.com/LaurentClaessens/phystricks
+%SPECIFIC_NEEDS
+%HATCHING_COMMANDS
+                 \makeatletter
+% If hatchspread is not defined, we define it
+\ifthenelse{\value{defHatch}=0}{
+\setcounter{defHatch}{1}
+\newlength{\hatchspread}%
+\newlength{\hatchthickness}%
+}{}
+               \makeatother               
+               \makeatletter
+\ifthenelse{\value{defPattern}=0}{
+\setcounter{defPattern}{1}
+\pgfdeclarepatternformonly[\hatchspread,\hatchthickness]% variables
+   {custom north west lines}% name
+   {\pgfqpoint{-2\hatchthickness}{-2\hatchthickness}}% lower left corner
+   {\pgfqpoint{\dimexpr\hatchspread+2\hatchthickness}{\dimexpr\hatchspread+2\hatchthickness}}% upper right corner
+   {\pgfqpoint{\hatchspread}{\hatchspread}}% tile size
+   {% shape description
+    \pgfsetlinewidth{\hatchthickness}
+    \pgfpathmoveto{\pgfqpoint{0pt}{\hatchspread}}
+    \pgfpathlineto{\pgfqpoint{\dimexpr\hatchspread+0.15pt}{-0.15pt}}
+        \pgfusepath{stroke}
+   }
+   }{}
+   \makeatother               
+%SUBFIGURES
+%AFTER SUBFIGURES
+%DEFAULT
+%BEFORE PSPICTURE
+%PSPICTURE
+%ENTETE PSPICTURE
+% This file is automatically generated by phystricks
+% See the documentation 
+% http://student.ulb.ac.be/~lclaesse/phystricks-doc.pdf 
+% http://student.ulb.ac.be/~lclaesse/phystricks-documentation/_build/html/index.html 
+% and the projects phystricks and phystricks-doc at 
+% https://github.com/LaurentClaessens/phystricks
+%OPEN_WRITE_AND_LABEL
+\ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
+\ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
+\immediate\openout\writeOfphystricks=GMRNooCNBpIl.phystricks.aux%
+\ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
+\ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
+\immediate\openout\writeOfphystricks=GMRNooCNBpIl.phystricks.aux%
+%WRITE_AND_LABEL
+\setlength{\lengthOfforphystricks}{\totalheightof{\cdot}}%
+\immediate\write\writeOfphystricks{totalheightof9204fd4be60aaa10dec0929315049a65155d8ffc:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\cdot}}%
+\immediate\write\writeOfphystricks{widthof9204fd4be60aaa10dec0929315049a65155d8ffc:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{*}}%
+\immediate\write\writeOfphystricks{totalheightofdf58248c414f342c81e056b40bee12d17a08bf61:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{*}}%
+\immediate\write\writeOfphystricks{widthofdf58248c414f342c81e056b40bee12d17a08bf61:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{\( N_y\)}}%
+\immediate\write\writeOfphystricks{totalheightof37cf4223835560070b1d8c85dd7a877627cf7518:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\( N_y\)}}%
+\immediate\write\writeOfphystricks{widthof37cf4223835560070b1d8c85dd7a877627cf7518:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{\( 0\)}}%
+\immediate\write\writeOfphystricks{totalheightofbeb9fe9f7236f6b0aa74bd7fd2526542efcfdb9b:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\( 0\)}}%
+\immediate\write\writeOfphystricks{widthofbeb9fe9f7236f6b0aa74bd7fd2526542efcfdb9b:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{\( N_x\)}}%
+\immediate\write\writeOfphystricks{totalheightofdcde614ad9b53cf0597a74425858664a51fe5d23:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\( N_x\)}}%
+\immediate\write\writeOfphystricks{widthofdcde614ad9b53cf0597a74425858664a51fe5d23:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$
+        \begin{matrix}
+        \phantom{
+        \begin{matrix}
+        \rule{ 4.83353000000000cm  }{2.58334933333333cm}
+        \end{matrix}
+        }
+        \end{matrix}$
+        }}%
+\immediate\write\writeOfphystricks{totalheightof1109a030fa0011251dfed13b79aef75f00f00634:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$
+        \begin{matrix}
+        \phantom{
+        \begin{matrix}
+        \rule{ 4.83353000000000cm  }{2.58334933333333cm}
+        \end{matrix}
+        }
+        \end{matrix}$
+        }}%
+\immediate\write\writeOfphystricks{widthof1109a030fa0011251dfed13b79aef75f00f00634:\the\lengthOfforphystricks-}
+%CLOSE_WRITE_AND_LABEL
+\immediate\closeout\writeOfphystricks%
+%BEFORE PSPICTURE
+%BEGIN PSPICTURE
+\tikzsetnextfilename{tikzGMRNooCNBpIl}
+\begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
+%GRID
+%AXES
+%OTHER STUFF
+%PSTRICKS CODE
+%DEFAULT
+
+\draw (2.4168,-1.2917) node {$
+        \begin{matrix}
+        \phantom{
+        \begin{matrix}
+        \rule{ 4.83353000000000cm  }{2.58334933333333cm}
+        \end{matrix}
+        }
+        \end{matrix}$
+        };
+\draw (0.031598,0.14734) node {\( N_y\)};
+\draw (0.40279,-0.17209) node {\cdot};
+\draw (1.2084,-0.17209) node {\cdot};
+\draw (2.0140,-0.17209) node {\cdot};
+\draw (2.8196,-0.17209) node {\cdot};
+\draw (3.6251,-0.17209) node {\cdot};
+\draw (4.4307,-0.17209) node {\cdot};
+\draw (0.40279,-0.58105) node {\cdot};
+\draw (1.2084,-0.58105) node {*};
+\draw (2.0140,-0.58105) node {*};
+\draw (2.8196,-0.58105) node {*};
+\draw (3.6251,-0.58105) node {*};
+\draw (4.4307,-0.58105) node {\cdot};
+\draw (0.40279,-1.0548) node {\cdot};
+\draw (1.2084,-1.0548) node {*};
+\draw (2.0140,-1.0548) node {*};
+\draw (2.8196,-1.0548) node {*};
+\draw (3.6251,-1.0548) node {*};
+\draw (4.4307,-1.0548) node {\cdot};
+\draw (0.40279,-1.5285) node {\cdot};
+\draw (1.2084,-1.5285) node {*};
+\draw (2.0140,-1.5285) node {*};
+\draw (2.8196,-1.5285) node {*};
+\draw (3.6251,-1.5285) node {*};
+\draw (4.4307,-1.5285) node {\cdot};
+\draw (0.40279,-2.0023) node {\cdot};
+\draw (1.2084,-2.0023) node {*};
+\draw (2.0140,-2.0023) node {*};
+\draw (2.8196,-2.0023) node {*};
+\draw (3.6251,-2.0023) node {*};
+\draw (4.4307,-2.0023) node {\cdot};
+\draw (0.11154,-2.4113) node {\( 0\)};
+\draw (0.40279,-2.7262) node {\( 0\)};
+\draw (0.40279,-2.4113) node {\cdot};
+\draw (1.2084,-2.4113) node {\cdot};
+\draw (2.0140,-2.4113) node {\cdot};
+\draw (2.8196,-2.4113) node {\cdot};
+\draw (3.6251,-2.4113) node {\cdot};
+\draw (4.8069,-2.7048) node {\( N_x\)};
+\draw (4.4307,-2.4113) node {\cdot};
+\draw [] (0.906,-0.394) -- (1.51,-0.394);
+\draw [] (1.51,-0.394) -- (1.51,-2.19);
+\draw [] (1.51,-2.19) -- (0.906,-2.19);
+\draw [] (0.906,-2.19) -- (0.906,-0.394);
+\draw [] (3.32,-0.394) -- (3.93,-0.394);
+\draw [] (3.93,-0.394) -- (3.93,-2.19);
+\draw [] (3.93,-2.19) -- (3.32,-2.19);
+\draw [] (3.32,-2.19) -- (3.32,-0.394);
+\draw [] (0.906,-1.82) -- (3.93,-1.82);
+\draw [] (3.93,-1.82) -- (3.93,-2.19);
+\draw [] (3.93,-2.19) -- (0.906,-2.19);
+\draw [] (0.906,-2.19) -- (0.906,-1.82);
+\draw [] (0.906,-0.394) -- (3.93,-0.394);
+\draw [] (3.93,-0.394) -- (3.93,-0.768);
+\draw [] (3.93,-0.768) -- (0.906,-0.768);
+\draw [] (0.906,-0.768) -- (0.906,-0.394);
+\draw [style=dashed] (1.71,-0.868) -- (3.12,-0.868);
+\draw [style=dashed] (3.12,-0.868) -- (3.12,-1.72);
+\draw [style=dashed] (3.12,-1.72) -- (1.71,-1.72);
+\draw [style=dashed] (1.71,-1.72) -- (1.71,-0.868);
+\draw (0.031598,0.14734) node {\( N_y\)};
+\draw (0.40279,-0.17209) node {\cdot};
+\draw (0.11154,-2.4113) node {\( 0\)};
+\draw (0.40279,-2.7262) node {\( 0\)};
+\draw (0.40279,-2.4113) node {\cdot};
+\draw (4.8069,-2.7048) node {\( N_x\)};
+\draw (4.4307,-2.4113) node {\cdot};
+%END PSPICTURE
+\end{tikzpicture}
+%AFTER PSPICTURE
+%AFTER PSPICTURE
diff --git a/src_phystricks/figures_mazhe.py b/src_phystricks/figures_mazhe.py
index 8fd2c3c0b..6560b8ba0 100755
--- a/src_phystricks/figures_mazhe.py
+++ b/src_phystricks/figures_mazhe.py
@@ -7,6 +7,9 @@
 from phystricks import *
 import sys
 
+from phystricksGMRNooCNBpIl import GMRNooCNBpIl 
+
+
 from phystricksCWKJooppMsZXjw import CWKJooppMsZXjw
 from phystricksDNRRooJWRHgOCw import DNRRooJWRHgOCw
 from phystricksDNHRooqGtffLkd import DNHRooqGtffLkd
@@ -568,10 +571,11 @@ def append_picture(fun,number):
 append_picture(ZOCNoowrfvQXsr,1)
 append_picture(UCDQooMCxpDszQ,1)
 
+append_picture(GMRNooCNBpIll,2)
+
 """
 append_picture(<++>,1)
 append_picture(<++>,1)
-append_picture(<++>,1)
 """
 
 def AllFigures():

From b35f43f4db166e0a08b9b7e78784d552a8d2c06a Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 13:51:42 +0200
Subject: [PATCH 37/64] =?UTF-8?q?(testing)=20Ajout=20d'une=20r=C3=A9f?=
 =?UTF-8?q?=C3=A9rence=20ok=20vers=20le=20futur.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 commons.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/commons.py b/commons.py
index 9e94441bf..c08cb7e95 100644
--- a/commons.py
+++ b/commons.py
@@ -22,7 +22,7 @@
 ok_hash=[]
 ok_hash.append("59080de1fd3e3d70e33ae6b8abd5d5bf4fce79b0")
 ok_hash.append("9f89a9fe96b2ef03f28f3f978e53b1058cf9b606")
-ok_hash.append("<++>")
+ok_hash.append("cd26df5e364185f46fd763ecc01e39bf7be2029a")
 ok_hash.append("fe793b1330cf5ee38543842fe28f30b51196c145")
 ok_hash.append("<++>")
 ok_hash.append("<++>")

From f7e60c05ebf0aff6275795960708e4003203e376 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 13:53:08 +0200
Subject: [PATCH 38/64] (pictures) Ajout de quelque md5 et pdf

---
 auto/pictures_tikz/tikzAMDUooZZUOqa.md5    |   2 +-
 auto/pictures_tikz/tikzAMDUooZZUOqa.pdf    | Bin 24866 -> 24994 bytes
 auto/pictures_tikz/tikzCWKJooppMsZXjw.md5  |   2 +-
 auto/pictures_tikz/tikzCWKJooppMsZXjw.pdf  | Bin 32165 -> 32186 bytes
 auto/pictures_tikz/tikzCercleTrigono.md5   |   2 +-
 auto/pictures_tikz/tikzCercleTrigono.pdf   | Bin 31860 -> 31882 bytes
 auto/pictures_tikz/tikzGMRNooCNBpIl.md5    |   1 +
 auto/pictures_tikz/tikzGMRNooCNBpIl.pdf    | Bin 0 -> 30438 bytes
 auto/pictures_tikz/tikzTgCercleTrigono.md5 |   2 +-
 auto/pictures_tikz/tikzTgCercleTrigono.pdf | Bin 32501 -> 32524 bytes
 10 files changed, 5 insertions(+), 4 deletions(-)
 create mode 100644 auto/pictures_tikz/tikzGMRNooCNBpIl.md5
 create mode 100644 auto/pictures_tikz/tikzGMRNooCNBpIl.pdf

diff --git a/auto/pictures_tikz/tikzAMDUooZZUOqa.md5 b/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
index 653cc956d..a8896e52f 100644
--- a/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
+++ b/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {85A99FF3936E7506B736DF05A3966190}%
+\def \tikzexternallastkey {3345F266A94F0440D55ED2B0B5D88AD8}%
diff --git a/auto/pictures_tikz/tikzAMDUooZZUOqa.pdf b/auto/pictures_tikz/tikzAMDUooZZUOqa.pdf
index d09ff7249c287851f0bc53440a5908f20628cb95..2f7b65fca38675d89823ce1bb47870947bb55c2a 100644
GIT binary patch
delta 4693
zcmbW4Wl+?Q*2e`!U}<D25lIp0U07ggq(eFcDPbvTSYnBvh;+kBN`rJSwJOpfD!HV9
zNG>3vOXvOHnfvOQXXbh4zB)0VbH1<6%=s4g5Eb_l@$u@bs0oP)OMrOCrut_=65=op
zQ4Y`Nt{_=ikcbY#!^zK?10pG%o<hN&st*N*QF(A&++Zd*yTS)@g;h@VN~ijuE9k_Z
zk@E$GWXaC9`!*-u+gs}=$9AGX7<@t+^&540PM{kNyg+?ue0<Mh^5Qb|${M9}Io6y|
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diff --git a/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5 b/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
index a514d5041..f46935921 100644
--- a/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
+++ b/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6FAF8888BC7B4AEAEFA57F6D91268658}%
+\def \tikzexternallastkey {6D0C8FB875137B88ECCD08E37B34FF12}%
diff --git a/auto/pictures_tikz/tikzCWKJooppMsZXjw.pdf b/auto/pictures_tikz/tikzCWKJooppMsZXjw.pdf
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diff --git a/auto/pictures_tikz/tikzCercleTrigono.md5 b/auto/pictures_tikz/tikzCercleTrigono.md5
index ecdf3ddd0..1dfeabd2a 100644
--- a/auto/pictures_tikz/tikzCercleTrigono.md5
+++ b/auto/pictures_tikz/tikzCercleTrigono.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6B12D470F848F82D122FDEF013DA0F81}%
+\def \tikzexternallastkey {C70CB22CF717378D3DB2778DF83DC686}%
diff --git a/auto/pictures_tikz/tikzCercleTrigono.pdf b/auto/pictures_tikz/tikzCercleTrigono.pdf
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diff --git a/auto/pictures_tikz/tikzGMRNooCNBpIl.md5 b/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
new file mode 100644
index 000000000..3f5c13b0d
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diff --git a/auto/pictures_tikz/tikzTgCercleTrigono.md5 b/auto/pictures_tikz/tikzTgCercleTrigono.md5
index ea87b4672..0f527aef0 100644
--- a/auto/pictures_tikz/tikzTgCercleTrigono.md5
+++ b/auto/pictures_tikz/tikzTgCercleTrigono.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {9098D410EECADE035E8EC84DD59942EF}%
+\def \tikzexternallastkey {6123A04DDC9811FB76E71C6DB97C148A}%
diff --git a/auto/pictures_tikz/tikzTgCercleTrigono.pdf b/auto/pictures_tikz/tikzTgCercleTrigono.pdf
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From e84a63edcba70e9cd8fd3a27366ed96aedccba0a Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 13:53:49 +0200
Subject: [PATCH 39/64] Une image de maillage.

---
 src_phystricks/phystricksGMRNooCNBpIl.py | 39 ++++++++++++++++++++++++
 1 file changed, 39 insertions(+)
 create mode 100644 src_phystricks/phystricksGMRNooCNBpIl.py

diff --git a/src_phystricks/phystricksGMRNooCNBpIl.py b/src_phystricks/phystricksGMRNooCNBpIl.py
new file mode 100644
index 000000000..0df4342fc
--- /dev/null
+++ b/src_phystricks/phystricksGMRNooCNBpIl.py
@@ -0,0 +1,39 @@
+# -*- coding: utf8 -*-
+from phystricks import *
+def GMRNooCNBpIl():
+    pspict,fig = SinglePicture("GMRNooCNBpIl")
+    pspict.dilatation(1)
+
+    matrix=phyMatrix(6,6)
+    matrix.set_matrix_environment("matrix")
+    for el in matrix.getElements() :
+        el.text="\cdot"
+
+    for i in range(2,6):
+        for j in range(2,6):
+            matrix.elements[i,j].text="*"
+
+
+    I4=matrix.square(   (2,2) , (2,5),pspict )
+    I1=matrix.square(   (2,2) , (5,2),pspict )
+    I2=matrix.square(   (2,5) , (5,5),pspict )
+    I3=matrix.square(   (5,2) , (5,5),pspict )
+
+    S=matrix.square( (3,3),(4,4),pspict  )
+    S.edges_parameters.style="dashed"
+
+    Ny=matrix.elements[1,1].central_point
+    Ny.put_mark(0.2,angle=90+45,added_angle=0,text="\( N_y\)",pspict=pspict)
+
+    Z=matrix.elements[6,1].central_point
+    Z.put_mark(0.2,angle=180,added_angle=0,text="\( 0\)",pspict=pspict)
+    Z.put_mark(0.2,angle=-90,added_angle=0,text="\( 0\)",pspict=pspict)
+
+    Nx=matrix.elements[6,6].central_point
+    Nx.put_mark(0.2,angle=-45,added_angle=0,text="\( N_x\)",pspict=pspict)
+
+    pspict.DrawGraphs(matrix,I1,I2,I3,I4,S,Ny,Z,Nx)
+    
+    fig.no_figure()
+    fig.conclude()
+    fig.write_the_file()

From 2cab5e95475ca1342f7c076aac16e5237068909b Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 14:22:14 +0200
Subject: [PATCH 40/64] (pictures) Fix a typo in a picture name.

---
 src_phystricks/figures_mazhe.py | 4 +---
 1 file changed, 1 insertion(+), 3 deletions(-)

diff --git a/src_phystricks/figures_mazhe.py b/src_phystricks/figures_mazhe.py
index 17de03ad1..b549e0a7a 100755
--- a/src_phystricks/figures_mazhe.py
+++ b/src_phystricks/figures_mazhe.py
@@ -8,8 +8,6 @@
 import sys
 
 from phystricksGMRNooCNBpIl import GMRNooCNBpIl 
-
-
 from phystricksCWKJooppMsZXjw import CWKJooppMsZXjw
 from phystricksDNRRooJWRHgOCw import DNRRooJWRHgOCw
 from phystricksDNHRooqGtffLkd import DNHRooqGtffLkd
@@ -551,7 +549,7 @@ def append_picture(fun,number):
 append_picture(ZOCNoowrfvQXsr,1)
 append_picture(UCDQooMCxpDszQ,1)
 
-append_picture(GMRNooCNBpIll,2)
+append_picture(GMRNooCNBpIl,2)
 
 """
 append_picture(<++>,1)

From 8f6e7039f7ac3d3c416f733af5ae5437c461c3e6 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 14:47:47 +0200
Subject: [PATCH 41/64] (picture) Add the recall file for 'GMRN'

---
 .../Fig_GMRNooCNBpIl.pstricks.recall          | 185 ++++++++++++++++++
 1 file changed, 185 insertions(+)
 create mode 100644 src_phystricks/Fig_GMRNooCNBpIl.pstricks.recall

diff --git a/src_phystricks/Fig_GMRNooCNBpIl.pstricks.recall b/src_phystricks/Fig_GMRNooCNBpIl.pstricks.recall
new file mode 100644
index 000000000..c75aeef9f
--- /dev/null
+++ b/src_phystricks/Fig_GMRNooCNBpIl.pstricks.recall
@@ -0,0 +1,185 @@
+%ENTETE FIGURE
+% This file is automatically generated by phystricks
+% See the documentation 
+% http://student.ulb.ac.be/~lclaesse/phystricks-doc.pdf 
+% http://student.ulb.ac.be/~lclaesse/phystricks-documentation/_build/html/index.html 
+% and the projects phystricks and phystricks-doc at 
+% https://github.com/LaurentClaessens/phystricks
+%SPECIFIC_NEEDS
+%HATCHING_COMMANDS
+                 \makeatletter
+% If hatchspread is not defined, we define it
+\ifthenelse{\value{defHatch}=0}{
+\setcounter{defHatch}{1}
+\newlength{\hatchspread}%
+\newlength{\hatchthickness}%
+}{}
+               \makeatother               
+               \makeatletter
+\ifthenelse{\value{defPattern}=0}{
+\setcounter{defPattern}{1}
+\pgfdeclarepatternformonly[\hatchspread,\hatchthickness]% variables
+   {custom north west lines}% name
+   {\pgfqpoint{-2\hatchthickness}{-2\hatchthickness}}% lower left corner
+   {\pgfqpoint{\dimexpr\hatchspread+2\hatchthickness}{\dimexpr\hatchspread+2\hatchthickness}}% upper right corner
+   {\pgfqpoint{\hatchspread}{\hatchspread}}% tile size
+   {% shape description
+    \pgfsetlinewidth{\hatchthickness}
+    \pgfpathmoveto{\pgfqpoint{0pt}{\hatchspread}}
+    \pgfpathlineto{\pgfqpoint{\dimexpr\hatchspread+0.15pt}{-0.15pt}}
+        \pgfusepath{stroke}
+   }
+   }{}
+   \makeatother               
+%SUBFIGURES
+%AFTER SUBFIGURES
+%DEFAULT
+%BEFORE PSPICTURE
+%PSPICTURE
+%ENTETE PSPICTURE
+% This file is automatically generated by phystricks
+% See the documentation 
+% http://student.ulb.ac.be/~lclaesse/phystricks-doc.pdf 
+% http://student.ulb.ac.be/~lclaesse/phystricks-documentation/_build/html/index.html 
+% and the projects phystricks and phystricks-doc at 
+% https://github.com/LaurentClaessens/phystricks
+%OPEN_WRITE_AND_LABEL
+\ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
+\ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
+\immediate\openout\writeOfphystricks=GMRNooCNBpIl.phystricks.aux%
+\ifthenelse{\isundefined{\writeOfphystricks}}{\newwrite{\writeOfphystricks}}{}
+\ifthenelse{\isundefined{\lengthOfforphystricks}}{\newlength{\lengthOfforphystricks}}{}
+\immediate\openout\writeOfphystricks=GMRNooCNBpIl.phystricks.aux%
+%WRITE_AND_LABEL
+\setlength{\lengthOfforphystricks}{\totalheightof{\cdot}}%
+\immediate\write\writeOfphystricks{totalheightof9204fd4be60aaa10dec0929315049a65155d8ffc:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\cdot}}%
+\immediate\write\writeOfphystricks{widthof9204fd4be60aaa10dec0929315049a65155d8ffc:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{*}}%
+\immediate\write\writeOfphystricks{totalheightofdf58248c414f342c81e056b40bee12d17a08bf61:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{*}}%
+\immediate\write\writeOfphystricks{widthofdf58248c414f342c81e056b40bee12d17a08bf61:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{\( N_y\)}}%
+\immediate\write\writeOfphystricks{totalheightof37cf4223835560070b1d8c85dd7a877627cf7518:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\( N_y\)}}%
+\immediate\write\writeOfphystricks{widthof37cf4223835560070b1d8c85dd7a877627cf7518:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{\( 0\)}}%
+\immediate\write\writeOfphystricks{totalheightofbeb9fe9f7236f6b0aa74bd7fd2526542efcfdb9b:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\( 0\)}}%
+\immediate\write\writeOfphystricks{widthofbeb9fe9f7236f6b0aa74bd7fd2526542efcfdb9b:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{\( N_x\)}}%
+\immediate\write\writeOfphystricks{totalheightofdcde614ad9b53cf0597a74425858664a51fe5d23:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{\( N_x\)}}%
+\immediate\write\writeOfphystricks{widthofdcde614ad9b53cf0597a74425858664a51fe5d23:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\totalheightof{$
+        \begin{matrix}
+        \phantom{
+        \begin{matrix}
+        \rule{ 4.83353000000000cm  }{2.58334933333333cm}
+        \end{matrix}
+        }
+        \end{matrix}$
+        }}%
+\immediate\write\writeOfphystricks{totalheightof1109a030fa0011251dfed13b79aef75f00f00634:\the\lengthOfforphystricks-}
+\setlength{\lengthOfforphystricks}{\widthof{$
+        \begin{matrix}
+        \phantom{
+        \begin{matrix}
+        \rule{ 4.83353000000000cm  }{2.58334933333333cm}
+        \end{matrix}
+        }
+        \end{matrix}$
+        }}%
+\immediate\write\writeOfphystricks{widthof1109a030fa0011251dfed13b79aef75f00f00634:\the\lengthOfforphystricks-}
+%CLOSE_WRITE_AND_LABEL
+\immediate\closeout\writeOfphystricks%
+%BEFORE PSPICTURE
+%BEGIN PSPICTURE
+\tikzsetnextfilename{tikzGMRNooCNBpIl}
+\begin{tikzpicture}[xscale=1,yscale=1,inner sep=2.25pt,outer sep=0pt]
+%GRID
+%AXES
+%OTHER STUFF
+%PSTRICKS CODE
+%DEFAULT
+
+\draw (2.4167,-1.2916) node {$
+        \begin{matrix}
+        \phantom{
+        \begin{matrix}
+        \rule{ 4.83353000000000cm  }{2.58334933333333cm}
+        \end{matrix}
+        }
+        \end{matrix}$
+        };
+\draw (0.0315,0.1473) node {\( N_y\)};
+\draw (0.4027,-0.1720) node {\cdot};
+\draw (1.2083,-0.1720) node {\cdot};
+\draw (2.0139,-0.1720) node {\cdot};
+\draw (2.8195,-0.1720) node {\cdot};
+\draw (3.6251,-0.1720) node {\cdot};
+\draw (4.4307,-0.1720) node {\cdot};
+\draw (0.4027,-0.5810) node {\cdot};
+\draw (1.2083,-0.5810) node {*};
+\draw (2.0139,-0.5810) node {*};
+\draw (2.8195,-0.5810) node {*};
+\draw (3.6251,-0.5810) node {*};
+\draw (4.4307,-0.5810) node {\cdot};
+\draw (0.4027,-1.0547) node {\cdot};
+\draw (1.2083,-1.0547) node {*};
+\draw (2.0139,-1.0547) node {*};
+\draw (2.8195,-1.0547) node {*};
+\draw (3.6251,-1.0547) node {*};
+\draw (4.4307,-1.0547) node {\cdot};
+\draw (0.4027,-1.5285) node {\cdot};
+\draw (1.2083,-1.5285) node {*};
+\draw (2.0139,-1.5285) node {*};
+\draw (2.8195,-1.5285) node {*};
+\draw (3.6251,-1.5285) node {*};
+\draw (4.4307,-1.5285) node {\cdot};
+\draw (0.4027,-2.0022) node {\cdot};
+\draw (1.2083,-2.0022) node {*};
+\draw (2.0139,-2.0022) node {*};
+\draw (2.8195,-2.0022) node {*};
+\draw (3.6251,-2.0022) node {*};
+\draw (4.4307,-2.0022) node {\cdot};
+\draw (0.1115,-2.4112) node {\( 0\)};
+\draw (0.4027,-2.7261) node {\( 0\)};
+\draw (0.4027,-2.4112) node {\cdot};
+\draw (1.2083,-2.4112) node {\cdot};
+\draw (2.0139,-2.4112) node {\cdot};
+\draw (2.8195,-2.4112) node {\cdot};
+\draw (3.6251,-2.4112) node {\cdot};
+\draw (4.8068,-2.7047) node {\( N_x\)};
+\draw (4.4307,-2.4112) node {\cdot};
+\draw [] (0.9055,-0.3941) -- (1.5111,-0.3941);
+\draw [] (1.5111,-0.3941) -- (1.5111,-2.1891);
+\draw [] (1.5111,-2.1891) -- (0.9055,-2.1891);
+\draw [] (0.9055,-2.1891) -- (0.9055,-0.3941);
+\draw [] (3.3223,-0.3941) -- (3.9279,-0.3941);
+\draw [] (3.9279,-0.3941) -- (3.9279,-2.1891);
+\draw [] (3.9279,-2.1891) -- (3.3223,-2.1891);
+\draw [] (3.3223,-2.1891) -- (3.3223,-0.3941);
+\draw [] (0.9055,-1.8154) -- (3.9279,-1.8154);
+\draw [] (3.9279,-1.8154) -- (3.9279,-2.1891);
+\draw [] (3.9279,-2.1891) -- (0.9055,-2.1891);
+\draw [] (0.9055,-2.1891) -- (0.9055,-1.8154);
+\draw [] (0.9055,-0.3941) -- (3.9279,-0.3941);
+\draw [] (3.9279,-0.3941) -- (3.9279,-0.7679);
+\draw [] (3.9279,-0.7679) -- (0.9055,-0.7679);
+\draw [] (0.9055,-0.7679) -- (0.9055,-0.3941);
+\draw [style=dashed] (1.7111,-0.8679) -- (3.1223,-0.8679);
+\draw [style=dashed] (3.1223,-0.8679) -- (3.1223,-1.7154);
+\draw [style=dashed] (3.1223,-1.7154) -- (1.7111,-1.7154);
+\draw [style=dashed] (1.7111,-1.7154) -- (1.7111,-0.8679);
+\draw (0.0315,0.1473) node {\( N_y\)};
+\draw (0.4027,-0.1720) node {\cdot};
+\draw (0.1115,-2.4112) node {\( 0\)};
+\draw (0.4027,-2.7261) node {\( 0\)};
+\draw (0.4027,-2.4112) node {\cdot};
+\draw (4.8068,-2.7047) node {\( N_x\)};
+\draw (4.4307,-2.4112) node {\cdot};
+%END PSPICTURE
+\end{tikzpicture}
+%AFTER PSPICTURE
+%AFTER PSPICTURE

From 62443909a08e861b358c76ea4208498eb3e1cb4f Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 15:23:10 +0200
Subject: [PATCH 42/64] (pytex) Remove the 'test_recall' plugin.

---
 lst_frido.py     |  1 -
 plugins_agreg.py | 22 ----------------------
 2 files changed, 23 deletions(-)

diff --git a/lst_frido.py b/lst_frido.py
index 37f3443f8..34e584d89 100644
--- a/lst_frido.py
+++ b/lst_frido.py
@@ -18,7 +18,6 @@
 myRequest.add_plugin(plugins_agreg.set_boolean("isFrido","true"),"before_pytex")
 myRequest.add_plugin(plugins_agreg.set_commit_hexsha,"after_pytex")
 myRequest.add_plugin(plugins_agreg.assert_MonCerveau_first,"after_compilation")
-myRequest.add_plugin(plugins_agreg.check_recall,"before_compilation")
 
 myRequest.new_output_filename="0-lefrido.pdf"
 
diff --git a/plugins_agreg.py b/plugins_agreg.py
index 6e5e57125..05985b0d9 100644
--- a/plugins_agreg.py
+++ b/plugins_agreg.py
@@ -172,25 +172,3 @@ def assert_MonCerveau_first():
                 
                 """.format(filename))
         raise
-
-def check_recall():
-    # import from the path name:
-# https://stackoverflow.com/questions/27381264/python-3-4-how-to-import-a-module-given-the-full-path
-    import sys,os
-    import importlib
-
-    module_path="testing/TestRecall.py"
-    sys.path.append(os.path.dirname(module_path))
-    module_name = os.path.splitext(os.path.basename(module_path))[0]
-    TestRecall = importlib.import_module(module_name)                       
-    sys.path.pop()
-
-    mfl,wfl=TestRecall.wrong_file_list(os.getcwd())
-    if mfl != []:
-        print("There are missing recall files :")
-        for f in mfl:
-            print(f)
-    if wfl != []:
-        print("There are wrong recall/pstricks files :")
-        for f in wfl:
-            print(f)

From c2a16550ad8757f542ed4977397859dda24cd42b Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Thu, 22 Jun 2017 16:41:19 +0200
Subject: [PATCH 43/64] typo

---
 tex/frido/173_differentielle.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/tex/frido/173_differentielle.tex b/tex/frido/173_differentielle.tex
index 37cd50db6..dcb92bacf 100644
--- a/tex/frido/173_differentielle.tex
+++ b/tex/frido/173_differentielle.tex
@@ -194,7 +194,7 @@ \subsection{Définition de la différentielle}
 \begin{equation}        \label{EqDiffPartRap}
     \begin{aligned}
         df_a\colon \eR^n&\to \eR^m \\
-        u&\mapsto df_a(u)=\frac{ \partial f }{ \partial u }(a)=\sum_i \frac{ \partial f }{ \partial x_i }u^i,
+        u&\mapsto df_a(u)=\frac{ \partial f }{ \partial u }(a)=\sum_i \frac{ \partial f }{ \partial x_i }u_i,
     \end{aligned}
 \end{equation}
 si les $u^i$ sont les composantes de $u$ dans la base canonique de $\eR^n$.

From 282d9bf836fe45b864d617776b9f8f417d4210d2 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Fri, 23 Jun 2017 08:01:16 +0200
Subject: [PATCH 44/64] (pictures) Update the 'pdf' and 'md5' files of the
 pictures.

---
 auto/pictures_tikz/tikzADUGmRRA.md5           |   2 +-
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 auto/pictures_tikz/tikzADUGmRRB.md5           |   2 +-
 auto/pictures_tikz/tikzADUGmRRB.pdf           | Bin 26349 -> 26349 bytes
 auto/pictures_tikz/tikzADUGmRRC.md5           |   2 +-
 auto/pictures_tikz/tikzADUGmRRC.pdf           | Bin 26743 -> 26747 bytes
 auto/pictures_tikz/tikzAIFsOQO.md5            |   2 +-
 auto/pictures_tikz/tikzAIFsOQO.pdf            | Bin 1991 -> 1994 bytes
 auto/pictures_tikz/tikzALIzHFm.md5            |   2 +-
 auto/pictures_tikz/tikzALIzHFm.pdf            | Bin 28521 -> 28520 bytes
 auto/pictures_tikz/tikzAMDUooZZUOqa.md5       |   2 +-
 auto/pictures_tikz/tikzAMDUooZZUOqa.pdf       | Bin 0 -> 25073 bytes
 auto/pictures_tikz/tikzASHYooUVHkak.md5       |   2 +-
 auto/pictures_tikz/tikzASHYooUVHkak.pdf       | Bin 29435 -> 29472 bytes
 auto/pictures_tikz/tikzAccumulationIsole.md5  |   2 +-
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 auto/pictures_tikz/tikzAdhIntFr.md5           |   2 +-
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diff --git a/auto/pictures_tikz/tikzADUGmRRA.md5 b/auto/pictures_tikz/tikzADUGmRRA.md5
index d3d63378f..2080ed089 100644
--- a/auto/pictures_tikz/tikzADUGmRRA.md5
+++ b/auto/pictures_tikz/tikzADUGmRRA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {DE648265946A3EB2B502C7AF26D81CE7}%
+\def \tikzexternallastkey {D5B289E062124C50D8C76063A6A249A8}%
diff --git a/auto/pictures_tikz/tikzADUGmRRA.pdf b/auto/pictures_tikz/tikzADUGmRRA.pdf
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diff --git a/auto/pictures_tikz/tikzADUGmRRB.md5 b/auto/pictures_tikz/tikzADUGmRRB.md5
index 21e834365..a9a1d3894 100644
--- a/auto/pictures_tikz/tikzADUGmRRB.md5
+++ b/auto/pictures_tikz/tikzADUGmRRB.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7272448D54EB3947265EDBBA7924386E}%
+\def \tikzexternallastkey {BB663E65943A83AB658AFC07A17BDFC1}%
diff --git a/auto/pictures_tikz/tikzADUGmRRB.pdf b/auto/pictures_tikz/tikzADUGmRRB.pdf
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diff --git a/auto/pictures_tikz/tikzAIFsOQO.md5 b/auto/pictures_tikz/tikzAIFsOQO.md5
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+++ b/auto/pictures_tikz/tikzAIFsOQO.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {017F8C4594019A29B95F27775178075E}%
+\def \tikzexternallastkey {D869F62E37304BB5A1997314A7771F47}%
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diff --git a/auto/pictures_tikz/tikzALIzHFm.md5 b/auto/pictures_tikz/tikzALIzHFm.md5
index a6c285af2..17566703a 100644
--- a/auto/pictures_tikz/tikzALIzHFm.md5
+++ b/auto/pictures_tikz/tikzALIzHFm.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {AA73CBE6819F3CECF47301E145232B8C}%
+\def \tikzexternallastkey {65ECF5F391EDCAF8F64F82532A334F0C}%
diff --git a/auto/pictures_tikz/tikzALIzHFm.pdf b/auto/pictures_tikz/tikzALIzHFm.pdf
index 4f1122784ca9647cab15094a77e993137023b0a3..fd6368f56c3d439ca4b955dfbe08df870053c465 100644
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diff --git a/auto/pictures_tikz/tikzAMDUooZZUOqa.md5 b/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
index a8896e52f..813409499 100644
--- a/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
+++ b/auto/pictures_tikz/tikzAMDUooZZUOqa.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {43A1F1A0DD087E6539EA05582748B841}%
diff --git a/auto/pictures_tikz/tikzAMDUooZZUOqa.pdf b/auto/pictures_tikz/tikzAMDUooZZUOqa.pdf
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diff --git a/auto/pictures_tikz/tikzASHYooUVHkak.md5 b/auto/pictures_tikz/tikzASHYooUVHkak.md5
index cdee77570..f891c1ba1 100644
--- a/auto/pictures_tikz/tikzASHYooUVHkak.md5
+++ b/auto/pictures_tikz/tikzASHYooUVHkak.md5
@@ -1 +1 @@
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diff --git a/auto/pictures_tikz/tikzAccumulationIsole.md5 b/auto/pictures_tikz/tikzAccumulationIsole.md5
index 5e3a2dd12..dd6238cdf 100644
--- a/auto/pictures_tikz/tikzAccumulationIsole.md5
+++ b/auto/pictures_tikz/tikzAccumulationIsole.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B36C1E06728EC11FFA9929B2CF038087}%
+\def \tikzexternallastkey {04C0806DC1F6A27B37F5063AA44D26E8}%
diff --git a/auto/pictures_tikz/tikzAccumulationIsole.pdf b/auto/pictures_tikz/tikzAccumulationIsole.pdf
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diff --git a/auto/pictures_tikz/tikzAdhIntFr.md5 b/auto/pictures_tikz/tikzAdhIntFr.md5
index 10fec0df4..b89a71ece 100644
--- a/auto/pictures_tikz/tikzAdhIntFr.md5
+++ b/auto/pictures_tikz/tikzAdhIntFr.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7832CB8DDE645E4B672C8216A883D5FE}%
+\def \tikzexternallastkey {B98B92B109A819FB26EBF21BD9205692}%
diff --git a/auto/pictures_tikz/tikzAdhIntFr.pdf b/auto/pictures_tikz/tikzAdhIntFr.pdf
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diff --git a/auto/pictures_tikz/tikzAdhIntFrDeux.md5 b/auto/pictures_tikz/tikzAdhIntFrDeux.md5
index 5d7fca98b..24f362688 100644
--- a/auto/pictures_tikz/tikzAdhIntFrDeux.md5
+++ b/auto/pictures_tikz/tikzAdhIntFrDeux.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B37186EC04D59BC716C24ED2C6034F79}%
+\def \tikzexternallastkey {C5F8D762BAFF37F3A1040725ACAA5317}%
diff --git a/auto/pictures_tikz/tikzAdhIntFrDeux.pdf b/auto/pictures_tikz/tikzAdhIntFrDeux.pdf
index 32fcf507b7c17ea6e97fc8647f34ec1a5d265b49..411eff01dac3f57bf11d712128d83a06694c4289 100644
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diff --git a/auto/pictures_tikz/tikzAdhIntFrSix.md5 b/auto/pictures_tikz/tikzAdhIntFrSix.md5
index 4eacf9f43..585cf0f40 100644
--- a/auto/pictures_tikz/tikzAdhIntFrSix.md5
+++ b/auto/pictures_tikz/tikzAdhIntFrSix.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6097FFFEEF1CA1555017AAF2C5DBA46F}%
+\def \tikzexternallastkey {3BAA09BF83A6B0ECB90A2469435C62FF}%
diff --git a/auto/pictures_tikz/tikzAdhIntFrSix.pdf b/auto/pictures_tikz/tikzAdhIntFrSix.pdf
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diff --git a/auto/pictures_tikz/tikzAdhIntFrTrois.md5 b/auto/pictures_tikz/tikzAdhIntFrTrois.md5
index 3bc5bc4ba..2610cc233 100644
--- a/auto/pictures_tikz/tikzAdhIntFrTrois.md5
+++ b/auto/pictures_tikz/tikzAdhIntFrTrois.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {994DE06D00A22906DE5AF2A9F5D867A6}%
+\def \tikzexternallastkey {4D6855E6B80A83BEAAAC8F2AC261787C}%
diff --git a/auto/pictures_tikz/tikzAdhIntFrTrois.pdf b/auto/pictures_tikz/tikzAdhIntFrTrois.pdf
index 4cb9e620114693a96c894c2809b582754bef428e..c7e72fac5c910bb43999a6052546c8297885b55c 100644
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diff --git a/auto/pictures_tikz/tikzAireParabole.md5 b/auto/pictures_tikz/tikzAireParabole.md5
index d45a839e7..4380fdcef 100644
--- a/auto/pictures_tikz/tikzAireParabole.md5
+++ b/auto/pictures_tikz/tikzAireParabole.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {1D7A8555D9BD0C4157DA8901EE26241E}%
+\def \tikzexternallastkey {A3EE859B0C2990D04DC25423D47FF282}%
diff --git a/auto/pictures_tikz/tikzAireParabole.pdf b/auto/pictures_tikz/tikzAireParabole.pdf
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+++ b/auto/pictures_tikz/tikzBEHTooWsdrys.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {A0A851811A1B2A68D5C08342B8A31AA2}%
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diff --git a/auto/pictures_tikz/tikzBIFooDsvVHb.md5 b/auto/pictures_tikz/tikzBIFooDsvVHb.md5
index 61193bbd4..37800c7b1 100644
--- a/auto/pictures_tikz/tikzBIFooDsvVHb.md5
+++ b/auto/pictures_tikz/tikzBIFooDsvVHb.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {DD67C0EE577EF8AB908963183635BA16}%
+\def \tikzexternallastkey {D7AB8BE93FEE7582634CE22FFBF25AA0}%
diff --git a/auto/pictures_tikz/tikzBIFooDsvVHb.pdf b/auto/pictures_tikz/tikzBIFooDsvVHb.pdf
index 75df380f1e1d34826908eaf87d0a361a4b184393..88f3fa39e9a3e02386387b1b7d93224e90648449 100644
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diff --git a/auto/pictures_tikz/tikzBNHLooLDxdPA.md5 b/auto/pictures_tikz/tikzBNHLooLDxdPA.md5
index 7425c1393..350160d73 100644
--- a/auto/pictures_tikz/tikzBNHLooLDxdPA.md5
+++ b/auto/pictures_tikz/tikzBNHLooLDxdPA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C4659604B9EAD0FDAA8DF65F4B251D31}%
+\def \tikzexternallastkey {1D843C5F03A88080783548470AB9B01F}%
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diff --git a/auto/pictures_tikz/tikzBQXKooPqSEMN.md5 b/auto/pictures_tikz/tikzBQXKooPqSEMN.md5
index d59108f0c..808537c54 100644
--- a/auto/pictures_tikz/tikzBQXKooPqSEMN.md5
+++ b/auto/pictures_tikz/tikzBQXKooPqSEMN.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6DC5C6D56D86E876BDC7DCFA08AAB33B}%
+\def \tikzexternallastkey {3BE3BA2D9FA9A50544B690672DA578D0}%
diff --git a/auto/pictures_tikz/tikzBQXKooPqSEMN.pdf b/auto/pictures_tikz/tikzBQXKooPqSEMN.pdf
index 258b0d8b358880dfa4ba4d79212ecf360d8062c0..6139db034149f5c96823f244f619a6d86467d563 100644
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diff --git a/auto/pictures_tikz/tikzBateau.md5 b/auto/pictures_tikz/tikzBateau.md5
index 1e3e655d8..bfdaff257 100644
--- a/auto/pictures_tikz/tikzBateau.md5
+++ b/auto/pictures_tikz/tikzBateau.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D5AB16985616278680CDB4A3E4051EDB}%
+\def \tikzexternallastkey {0EB9A65E2BACA098DB0380E54E93B7CC}%
diff --git a/auto/pictures_tikz/tikzBateau.pdf b/auto/pictures_tikz/tikzBateau.pdf
index ec6569a0603d5be18577855c4cfda1549a63828c..4f5cbbf34adc5ed4a4f7e3d09746c5413223843b 100644
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diff --git a/auto/pictures_tikz/tikzBiaisOuPas.md5 b/auto/pictures_tikz/tikzBiaisOuPas.md5
index be5f6ae0d..351a6ec3f 100644
--- a/auto/pictures_tikz/tikzBiaisOuPas.md5
+++ b/auto/pictures_tikz/tikzBiaisOuPas.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {41F7CDB48054E0728DDB02C3473F18A2}%
+\def \tikzexternallastkey {5477336FEF74664ED85FBE3EEF105B79}%
diff --git a/auto/pictures_tikz/tikzBiaisOuPas.pdf b/auto/pictures_tikz/tikzBiaisOuPas.pdf
index 9f7c6892ec5d56313bfd870655c90e17072ba8b3..16aa01e33652bfb3bf44e62c039d204d427441b3 100644
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diff --git a/auto/pictures_tikz/tikzBoulePtLoin.md5 b/auto/pictures_tikz/tikzBoulePtLoin.md5
index e44164ee5..695164908 100644
--- a/auto/pictures_tikz/tikzBoulePtLoin.md5
+++ b/auto/pictures_tikz/tikzBoulePtLoin.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E0CD6F87211430C41FA9A7CCCDA1807C}%
+\def \tikzexternallastkey {18E389E01135F999BE3A2C68E133BCA5}%
diff --git a/auto/pictures_tikz/tikzBoulePtLoin.pdf b/auto/pictures_tikz/tikzBoulePtLoin.pdf
index 0e3a4b6375ede86585c68354ff67666e16faaf43..fa30f35e6e5bcac75a7ed1f5f842ab625864f4c6 100644
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diff --git a/auto/pictures_tikz/tikzCELooGVvzMc.md5 b/auto/pictures_tikz/tikzCELooGVvzMc.md5
index 25e11d213..2de40e47a 100644
--- a/auto/pictures_tikz/tikzCELooGVvzMc.md5
+++ b/auto/pictures_tikz/tikzCELooGVvzMc.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {51CA4B251DE244AA8CFEC206D899C934}%
+\def \tikzexternallastkey {E2A9B51D89FF4F4DC46914EEA1E86CB8}%
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diff --git a/auto/pictures_tikz/tikzCFMooGzvfRP.md5 b/auto/pictures_tikz/tikzCFMooGzvfRP.md5
index 9fe2a5fe4..131f6dae1 100644
--- a/auto/pictures_tikz/tikzCFMooGzvfRP.md5
+++ b/auto/pictures_tikz/tikzCFMooGzvfRP.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {33F05A613E3BD4AF5983B72A85664BD8}%
+\def \tikzexternallastkey {A9E26A3254F5BA814481B2CF0BE5E2A3}%
diff --git a/auto/pictures_tikz/tikzCFMooGzvfRP.pdf b/auto/pictures_tikz/tikzCFMooGzvfRP.pdf
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diff --git a/auto/pictures_tikz/tikzCMMAooQegASg.md5 b/auto/pictures_tikz/tikzCMMAooQegASg.md5
index df7469f28..637443521 100644
--- a/auto/pictures_tikz/tikzCMMAooQegASg.md5
+++ b/auto/pictures_tikz/tikzCMMAooQegASg.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A6EF2FE79D6E3FB223266FAA24B5D3CA}%
+\def \tikzexternallastkey {FA1A75F84A76B028CF3FBCE294A54858}%
diff --git a/auto/pictures_tikz/tikzCMMAooQegASg.pdf b/auto/pictures_tikz/tikzCMMAooQegASg.pdf
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diff --git a/auto/pictures_tikz/tikzCQIXooBEDnfK.md5 b/auto/pictures_tikz/tikzCQIXooBEDnfK.md5
index 21152876a..bd30b5cc5 100644
--- a/auto/pictures_tikz/tikzCQIXooBEDnfK.md5
+++ b/auto/pictures_tikz/tikzCQIXooBEDnfK.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CA90AAF8AB80E321AE73315278C9A3B5}%
+\def \tikzexternallastkey {7833820658FC4257E1271B33091C44F4}%
diff --git a/auto/pictures_tikz/tikzCQIXooBEDnfK.pdf b/auto/pictures_tikz/tikzCQIXooBEDnfK.pdf
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diff --git a/auto/pictures_tikz/tikzCSCii.md5 b/auto/pictures_tikz/tikzCSCii.md5
index 5fd1e147e..559e71838 100644
--- a/auto/pictures_tikz/tikzCSCii.md5
+++ b/auto/pictures_tikz/tikzCSCii.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {F92BEF1EC1D21248DADEA6B1EBCD80A8}%
+\def \tikzexternallastkey {4F84F6F351404210088019064A58289E}%
diff --git a/auto/pictures_tikz/tikzCSCii.pdf b/auto/pictures_tikz/tikzCSCii.pdf
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diff --git a/auto/pictures_tikz/tikzCSCiv.md5 b/auto/pictures_tikz/tikzCSCiv.md5
index 313df147a..f47369a14 100644
--- a/auto/pictures_tikz/tikzCSCiv.md5
+++ b/auto/pictures_tikz/tikzCSCiv.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0305934C84081BE649F598B3AEB77795}%
+\def \tikzexternallastkey {F29AA3ECB2E42CB7EB489D58FBDF69B1}%
diff --git a/auto/pictures_tikz/tikzCSCiv.pdf b/auto/pictures_tikz/tikzCSCiv.pdf
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diff --git a/auto/pictures_tikz/tikzCURGooXvruWV.md5 b/auto/pictures_tikz/tikzCURGooXvruWV.md5
index 918566458..f072cabe6 100644
--- a/auto/pictures_tikz/tikzCURGooXvruWV.md5
+++ b/auto/pictures_tikz/tikzCURGooXvruWV.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7AA7EF02F8FF899DC02A3464EA502A2A}%
+\def \tikzexternallastkey {410218383FBC62D2089EDE3F072B6788}%
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diff --git a/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5 b/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
index f46935921..262d6b4db 100644
--- a/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
+++ b/auto/pictures_tikz/tikzCWKJooppMsZXjw.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6D0C8FB875137B88ECCD08E37B34FF12}%
+\def \tikzexternallastkey {122FDAF5740AFE1E211EEBBFC87F98C1}%
diff --git a/auto/pictures_tikz/tikzCWKJooppMsZXjw.pdf b/auto/pictures_tikz/tikzCWKJooppMsZXjw.pdf
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index e2c6a048f..4e3e7aff7 100644
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diff --git a/auto/pictures_tikz/tikzCbCartTui.md5 b/auto/pictures_tikz/tikzCbCartTui.md5
index 2e25f4199..9287bd853 100644
--- a/auto/pictures_tikz/tikzCbCartTui.md5
+++ b/auto/pictures_tikz/tikzCbCartTui.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0CA9C29D28452112F22FDD0F88A742E2}%
+\def \tikzexternallastkey {10DB130EDC30C0E8A5ADB1501074FF3C}%
diff --git a/auto/pictures_tikz/tikzCbCartTui.pdf b/auto/pictures_tikz/tikzCbCartTui.pdf
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diff --git a/auto/pictures_tikz/tikzCbCartTuii.md5 b/auto/pictures_tikz/tikzCbCartTuii.md5
index 7da0631aa..889c06941 100644
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diff --git a/auto/pictures_tikz/tikzCercleImplicite.md5 b/auto/pictures_tikz/tikzCercleImplicite.md5
index f4bda332e..31b10ecb9 100644
--- a/auto/pictures_tikz/tikzCercleImplicite.md5
+++ b/auto/pictures_tikz/tikzCercleImplicite.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {32C55B039224F89BCAE9B1A8F623F966}%
+\def \tikzexternallastkey {9F3F2099C4FED368CF508CD5B8761F57}%
diff --git a/auto/pictures_tikz/tikzCercleImplicite.pdf b/auto/pictures_tikz/tikzCercleImplicite.pdf
index 3d9d90106a65b0f8937e00a93640fd33bfc95d45..b57c7f0d01e800078c5aae0c94c70f8cbf49538b 100644
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diff --git a/auto/pictures_tikz/tikzCercleTnu.md5 b/auto/pictures_tikz/tikzCercleTnu.md5
index 1ff146ae3..6c5d3a09a 100644
--- a/auto/pictures_tikz/tikzCercleTnu.md5
+++ b/auto/pictures_tikz/tikzCercleTnu.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D2F25749A4C45F055DC0FF262F0584FC}%
+\def \tikzexternallastkey {BD8226141503E1E306F325123E55045B}%
diff --git a/auto/pictures_tikz/tikzCercleTnu.pdf b/auto/pictures_tikz/tikzCercleTnu.pdf
index 611f712690367792e2290297b99452e198464c09..fa23d4f95dcce1645ef2992ed49cd1360d3a0996 100644
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diff --git a/auto/pictures_tikz/tikzCercleTrigono.md5 b/auto/pictures_tikz/tikzCercleTrigono.md5
index 1dfeabd2a..7937ce6ff 100644
--- a/auto/pictures_tikz/tikzCercleTrigono.md5
+++ b/auto/pictures_tikz/tikzCercleTrigono.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C70CB22CF717378D3DB2778DF83DC686}%
+\def \tikzexternallastkey {FA3FA2FF953240D23AA7D479AAC3DA76}%
diff --git a/auto/pictures_tikz/tikzCercleTrigono.pdf b/auto/pictures_tikz/tikzCercleTrigono.pdf
new file mode 100644
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diff --git a/auto/pictures_tikz/tikzChampGraviation.md5 b/auto/pictures_tikz/tikzChampGraviation.md5
index 0c5544456..bdaf2fcd2 100644
--- a/auto/pictures_tikz/tikzChampGraviation.md5
+++ b/auto/pictures_tikz/tikzChampGraviation.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {96AB15FD51F7A08F29785301FFA82E36}%
+\def \tikzexternallastkey {5EC94F2D472138637101095DE117E352}%
diff --git a/auto/pictures_tikz/tikzChampGraviation.pdf b/auto/pictures_tikz/tikzChampGraviation.pdf
index 025ae86b59d2b0da57cf47ae7676d4f8cf534717..31a1feb04060e4eb92651eaebd1449924beef436 100644
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diff --git a/auto/pictures_tikz/tikzCheminFresnel.md5 b/auto/pictures_tikz/tikzCheminFresnel.md5
index 04edb2a56..865fe8720 100644
--- a/auto/pictures_tikz/tikzCheminFresnel.md5
+++ b/auto/pictures_tikz/tikzCheminFresnel.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B6575B62245C4AA710B16D162A976354}%
+\def \tikzexternallastkey {26B7F35BDF6483039412F694951F7B99}%
diff --git a/auto/pictures_tikz/tikzCheminFresnel.pdf b/auto/pictures_tikz/tikzCheminFresnel.pdf
index 21a1af55e674a44ec061a9c99157d610d5efd014..530428d1bc74502a73e52b39749ffd577d52eb97 100644
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diff --git a/auto/pictures_tikz/tikzChiSquared.md5 b/auto/pictures_tikz/tikzChiSquared.md5
index 431fa2202..f82721592 100644
--- a/auto/pictures_tikz/tikzChiSquared.md5
+++ b/auto/pictures_tikz/tikzChiSquared.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {2B9D8611EA7E50B1367E5F720050C5CD}%
+\def \tikzexternallastkey {4227CF2DE0D5755A70403519A16235A8}%
diff --git a/auto/pictures_tikz/tikzChiSquared.pdf b/auto/pictures_tikz/tikzChiSquared.pdf
index c22e6481c0edf55a3ed8c07b614f2852f34730f3..505a4d8522a444a115805b7092ae831100af28bb 100644
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diff --git a/auto/pictures_tikz/tikzChiSquaresQuantile.md5 b/auto/pictures_tikz/tikzChiSquaresQuantile.md5
index 82f38de15..7d6d844f9 100644
--- a/auto/pictures_tikz/tikzChiSquaresQuantile.md5
+++ b/auto/pictures_tikz/tikzChiSquaresQuantile.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {166C54795087BEBF2693DF8612E8CFAB}%
+\def \tikzexternallastkey {05AA336649EF360906CEC957C696FB44}%
diff --git a/auto/pictures_tikz/tikzChiSquaresQuantile.pdf b/auto/pictures_tikz/tikzChiSquaresQuantile.pdf
index ee3615fe8893d61c1e2445dfca65263592a60aeb..130ab140e7ac2871e8f999e6737b24cb2711e007 100644
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diff --git a/auto/pictures_tikz/tikzCoinPasVar.md5 b/auto/pictures_tikz/tikzCoinPasVar.md5
index 2c10d8056..71a720de1 100644
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diff --git a/auto/pictures_tikz/tikzContourGreen.md5 b/auto/pictures_tikz/tikzContourGreen.md5
index 71e0fda7d..609630174 100644
--- a/auto/pictures_tikz/tikzContourGreen.md5
+++ b/auto/pictures_tikz/tikzContourGreen.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {3C589F8246B297B66318C2B15AA532F1}%
+\def \tikzexternallastkey {67A7437D3D31C0A3BC16C96B88F2F53A}%
diff --git a/auto/pictures_tikz/tikzContourGreen.pdf b/auto/pictures_tikz/tikzContourGreen.pdf
index a5cc37333b19c17ab0d2e4ff2ccee4fb0c2b60f9..3c61a25c430996ea338036d5118c296ff0b32602 100644
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diff --git a/auto/pictures_tikz/tikzContourSqL.md5 b/auto/pictures_tikz/tikzContourSqL.md5
index fc4717328..e9ceda8f1 100644
--- a/auto/pictures_tikz/tikzContourSqL.md5
+++ b/auto/pictures_tikz/tikzContourSqL.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CE60B1B7C8DD9F794F5404D345C5F5CA}%
+\def \tikzexternallastkey {53BE081EE54A3ABDF816EF7EDC45F246}%
diff --git a/auto/pictures_tikz/tikzContourSqL.pdf b/auto/pictures_tikz/tikzContourSqL.pdf
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diff --git a/auto/pictures_tikz/tikzContourTgNDivergence.md5 b/auto/pictures_tikz/tikzContourTgNDivergence.md5
index fed6d6d3f..6be4d673b 100644
--- a/auto/pictures_tikz/tikzContourTgNDivergence.md5
+++ b/auto/pictures_tikz/tikzContourTgNDivergence.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D65C5CC685E5D98F074F0C608BCA5820}%
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diff --git a/auto/pictures_tikz/tikzCornetGlace.md5 b/auto/pictures_tikz/tikzCornetGlace.md5
index d60e90f6d..2d22aa70a 100644
--- a/auto/pictures_tikz/tikzCornetGlace.md5
+++ b/auto/pictures_tikz/tikzCornetGlace.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {200A56F252CEC0503FC8D9AB39611789}%
+\def \tikzexternallastkey {967990A707D845E65ACA9175C9AC53D2}%
diff --git a/auto/pictures_tikz/tikzCornetGlace.pdf b/auto/pictures_tikz/tikzCornetGlace.pdf
index 3ff8a5836e0e27b7e636fc216666ce04b560fd43..69838f8f448b1e5f4eccc43dde4bdb0c8342a5d2 100644
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diff --git a/auto/pictures_tikz/tikzCourbeRectifiable.md5 b/auto/pictures_tikz/tikzCourbeRectifiable.md5
index 77869a412..c6542ce23 100644
--- a/auto/pictures_tikz/tikzCourbeRectifiable.md5
+++ b/auto/pictures_tikz/tikzCourbeRectifiable.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {479E8E3D51E431E5F90F8F646943A0EE}%
+\def \tikzexternallastkey {25F51D44499C3DDB7E1861B834D7B3E9}%
diff --git a/auto/pictures_tikz/tikzCourbeRectifiable.pdf b/auto/pictures_tikz/tikzCourbeRectifiable.pdf
index f6da3d71668b73f2ad36f3350ef299759a5a5b6c..c80491e8d2467bf71f3667d4832d1d48203605b8 100644
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diff --git a/auto/pictures_tikz/tikzCouroneExam.md5 b/auto/pictures_tikz/tikzCouroneExam.md5
index 47826baf9..f4582eb0c 100644
--- a/auto/pictures_tikz/tikzCouroneExam.md5
+++ b/auto/pictures_tikz/tikzCouroneExam.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CFC133F42C68AEFC13F9F7D787182369}%
+\def \tikzexternallastkey {C3A9BBB019E5CCB2F7F98BF55656FF3E}%
diff --git a/auto/pictures_tikz/tikzCouroneExam.pdf b/auto/pictures_tikz/tikzCouroneExam.pdf
index 117fbb29978bd9e819e99b5ec6bfd5a6e3664865..578cd1e31e41518c6424efd8fae6e8dfa966c166 100644
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diff --git a/auto/pictures_tikz/tikzCycloideA.md5 b/auto/pictures_tikz/tikzCycloideA.md5
index da6d1413e..325f9e440 100644
--- a/auto/pictures_tikz/tikzCycloideA.md5
+++ b/auto/pictures_tikz/tikzCycloideA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C5246F4A5900C196442D8862CB78B8F4}%
+\def \tikzexternallastkey {96E0EFB5D6866C07A5A91E9355CAAD53}%
diff --git a/auto/pictures_tikz/tikzCycloideA.pdf b/auto/pictures_tikz/tikzCycloideA.pdf
index 5169d050da11c0405f488969c124e6c31ded4fdf..2e3c3db9e166cff064cb528d0771153c444f97c4 100644
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diff --git a/auto/pictures_tikz/tikzDDCTooYscVzA.md5 b/auto/pictures_tikz/tikzDDCTooYscVzA.md5
index fed6d6d3f..6be4d673b 100644
--- a/auto/pictures_tikz/tikzDDCTooYscVzA.md5
+++ b/auto/pictures_tikz/tikzDDCTooYscVzA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D65C5CC685E5D98F074F0C608BCA5820}%
+\def \tikzexternallastkey {41EBD0A36E382DE0DF83F7EB8ECF9B04}%
diff --git a/auto/pictures_tikz/tikzDDCTooYscVzA.pdf b/auto/pictures_tikz/tikzDDCTooYscVzA.pdf
index 740ddc3d1b0b307e5e363165c080840c057e4ccd..a343b7458e0873afb8c1a533bad84676cd614791 100644
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diff --git a/auto/pictures_tikz/tikzDGFSooWgbuuMoB.md5 b/auto/pictures_tikz/tikzDGFSooWgbuuMoB.md5
index 71efcf7ef..5316f018d 100644
--- a/auto/pictures_tikz/tikzDGFSooWgbuuMoB.md5
+++ b/auto/pictures_tikz/tikzDGFSooWgbuuMoB.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {423D56D0D705518398094C60AB4C992F}%
+\def \tikzexternallastkey {25F2B740290673AD548F35B2641BF60F}%
diff --git a/auto/pictures_tikz/tikzDGFSooWgbuuMoB.pdf b/auto/pictures_tikz/tikzDGFSooWgbuuMoB.pdf
index 5cf88c9a95522efdb337f2c5b07213d297d0de3e..6c93afddc057a26ff23adff5bf87cbc4ba125e18 100644
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diff --git a/auto/pictures_tikz/tikzDNHRooqGtffLkd.md5 b/auto/pictures_tikz/tikzDNHRooqGtffLkd.md5
index 85ecb3453..519370aed 100644
--- a/auto/pictures_tikz/tikzDNHRooqGtffLkd.md5
+++ b/auto/pictures_tikz/tikzDNHRooqGtffLkd.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {967AF4EA290203D051C3ABDFD0C507B4}%
+\def \tikzexternallastkey {57ACB11B96ED4D67F3ABAE6B1A7E09F2}%
diff --git a/auto/pictures_tikz/tikzDNHRooqGtffLkd.pdf b/auto/pictures_tikz/tikzDNHRooqGtffLkd.pdf
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diff --git a/auto/pictures_tikz/tikzDNRRooJWRHgOCw.md5 b/auto/pictures_tikz/tikzDNRRooJWRHgOCw.md5
index 33b51fd04..086eb4561 100644
--- a/auto/pictures_tikz/tikzDNRRooJWRHgOCw.md5
+++ b/auto/pictures_tikz/tikzDNRRooJWRHgOCw.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A65655F0637335B0944FA837129CE1C9}%
+\def \tikzexternallastkey {CC5A4C40D75AEEFD4609160957C93C7F}%
diff --git a/auto/pictures_tikz/tikzDNRRooJWRHgOCw.pdf b/auto/pictures_tikz/tikzDNRRooJWRHgOCw.pdf
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diff --git a/auto/pictures_tikz/tikzDTIYKkP.md5 b/auto/pictures_tikz/tikzDTIYKkP.md5
index 0fa306707..c896d8892 100644
--- a/auto/pictures_tikz/tikzDTIYKkP.md5
+++ b/auto/pictures_tikz/tikzDTIYKkP.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {FC3B237D575F8366E8850CA5E312C892}%
+\def \tikzexternallastkey {A296FCB47A715FC4448F61FEBA7E9118}%
diff --git a/auto/pictures_tikz/tikzDTIYKkP.pdf b/auto/pictures_tikz/tikzDTIYKkP.pdf
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diff --git a/auto/pictures_tikz/tikzDefinitionCartesiennes.md5 b/auto/pictures_tikz/tikzDefinitionCartesiennes.md5
index 9f2fd926b..1833cfa44 100644
--- a/auto/pictures_tikz/tikzDefinitionCartesiennes.md5
+++ b/auto/pictures_tikz/tikzDefinitionCartesiennes.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {FDCB15FD9F9837FB894F2D20A74E97AE}%
+\def \tikzexternallastkey {268E1C0C1396ACEA154FD926B35E7F68}%
diff --git a/auto/pictures_tikz/tikzDefinitionCartesiennes.pdf b/auto/pictures_tikz/tikzDefinitionCartesiennes.pdf
index 736f931099875b9bf4dbccdc11ebe29f76dd5783..cbc5822beeaf823ca49860ba660f12f4c5d6ebf0 100644
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diff --git a/auto/pictures_tikz/tikzDerivTangenteOM.md5 b/auto/pictures_tikz/tikzDerivTangenteOM.md5
index 2c8a82fcc..b2e60de06 100644
--- a/auto/pictures_tikz/tikzDerivTangenteOM.md5
+++ b/auto/pictures_tikz/tikzDerivTangenteOM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6DB76C3DFE1327B569DAED52AD66AF02}%
+\def \tikzexternallastkey {15E89CEB0409F169A8F8DA9512463C60}%
diff --git a/auto/pictures_tikz/tikzDerivTangenteOM.pdf b/auto/pictures_tikz/tikzDerivTangenteOM.pdf
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diff --git a/auto/pictures_tikz/tikzDessinLim.md5 b/auto/pictures_tikz/tikzDessinLim.md5
index ba751db90..b6bd0b1b1 100644
--- a/auto/pictures_tikz/tikzDessinLim.md5
+++ b/auto/pictures_tikz/tikzDessinLim.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {79EE9B46A3EC0235580CC4AD5875599C}%
+\def \tikzexternallastkey {F10082AA2AF9765E8A8D3B4F67537F8C}%
diff --git a/auto/pictures_tikz/tikzDessinLim.pdf b/auto/pictures_tikz/tikzDessinLim.pdf
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diff --git a/auto/pictures_tikz/tikzDeuxCercles.md5 b/auto/pictures_tikz/tikzDeuxCercles.md5
index 5b2ec442b..5427e3061 100644
--- a/auto/pictures_tikz/tikzDeuxCercles.md5
+++ b/auto/pictures_tikz/tikzDeuxCercles.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {87A4C4E2C4BAC0159C6898AFA47EB2D7}%
+\def \tikzexternallastkey {138EE3636AAFEC27568830B6A395F53A}%
diff --git a/auto/pictures_tikz/tikzDeuxCercles.pdf b/auto/pictures_tikz/tikzDeuxCercles.pdf
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diff --git a/auto/pictures_tikz/tikzDifferentielle.md5 b/auto/pictures_tikz/tikzDifferentielle.md5
index 029348c50..71244d967 100644
--- a/auto/pictures_tikz/tikzDifferentielle.md5
+++ b/auto/pictures_tikz/tikzDifferentielle.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A7553211A22D688CEE5DEDE410198A34}%
+\def \tikzexternallastkey {ADFF7BD751DA7E73E903FE1D5513E8ED}%
diff --git a/auto/pictures_tikz/tikzDifferentielle.pdf b/auto/pictures_tikz/tikzDifferentielle.pdf
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diff --git a/auto/pictures_tikz/tikzDisqueConv.md5 b/auto/pictures_tikz/tikzDisqueConv.md5
index 79d12297f..ce380ddeb 100644
--- a/auto/pictures_tikz/tikzDisqueConv.md5
+++ b/auto/pictures_tikz/tikzDisqueConv.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E8CDF0EABB0AB612E4D22EB006577C07}%
+\def \tikzexternallastkey {5CEB94189F2FEF301C8ABB24E1CFEAF6}%
diff --git a/auto/pictures_tikz/tikzDisqueConv.pdf b/auto/pictures_tikz/tikzDisqueConv.pdf
index 126139a6332ee417fa1b33ec9424dd551fc908f3..8e3183876e9550651221be07776e4a4aadb13c5a 100644
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diff --git a/auto/pictures_tikz/tikzDistanceEnsemble.md5 b/auto/pictures_tikz/tikzDistanceEnsemble.md5
index 9d9c6b70c..51f1ac034 100644
--- a/auto/pictures_tikz/tikzDistanceEnsemble.md5
+++ b/auto/pictures_tikz/tikzDistanceEnsemble.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {1DB08103BDEB686C7AC1B05988D3911F}%
+\def \tikzexternallastkey {13152CB2DE5F08A07E8C832439A910CA}%
diff --git a/auto/pictures_tikz/tikzDistanceEnsemble.pdf b/auto/pictures_tikz/tikzDistanceEnsemble.pdf
index 42aa29b5ca6a920ad2d5ad595ec985cd42b3023c..58e09e32bf4b00472f34d91eef702d89f06b5eb4 100644
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diff --git a/auto/pictures_tikz/tikzDistanceEuclide.md5 b/auto/pictures_tikz/tikzDistanceEuclide.md5
index f1b46d7ba..40af61f5a 100644
--- a/auto/pictures_tikz/tikzDistanceEuclide.md5
+++ b/auto/pictures_tikz/tikzDistanceEuclide.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {24743AD4623383D9E485BD6D12D66680}%
+\def \tikzexternallastkey {ED96346285E5F4FD70A09F9731527F25}%
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diff --git a/auto/pictures_tikz/tikzDivergenceDeux.md5 b/auto/pictures_tikz/tikzDivergenceDeux.md5
index 15538fc16..24393d32a 100644
--- a/auto/pictures_tikz/tikzDivergenceDeux.md5
+++ b/auto/pictures_tikz/tikzDivergenceDeux.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C1E76FBDBEC88ED3FD93DA9B1D935535}%
+\def \tikzexternallastkey {B9179F47BE4732594D65B1E02D560F51}%
diff --git a/auto/pictures_tikz/tikzDivergenceDeux.pdf b/auto/pictures_tikz/tikzDivergenceDeux.pdf
index 1649fda3485fa3ac42184bb91e1a00ecaccf9473..c6f941ed654b35c25fbe23ba61d3423c53a9e615 100644
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diff --git a/auto/pictures_tikz/tikzDivergenceTrois.md5 b/auto/pictures_tikz/tikzDivergenceTrois.md5
index 7842d83b3..dd234bb5c 100644
--- a/auto/pictures_tikz/tikzDivergenceTrois.md5
+++ b/auto/pictures_tikz/tikzDivergenceTrois.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {42A98F39F4F00B9A41387B04B2355EAE}%
+\def \tikzexternallastkey {72228E51743D5ABE69CF833BF3E714B6}%
diff --git a/auto/pictures_tikz/tikzDivergenceTrois.pdf b/auto/pictures_tikz/tikzDivergenceTrois.pdf
index 7948d3284fe325288b2c31ad6301c1e03e2a3608..472ea33dec8e0801125cd05b80c8a1982367dabc 100644
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diff --git a/auto/pictures_tikz/tikzDynkinpWjUbE.md5 b/auto/pictures_tikz/tikzDynkinpWjUbE.md5
index 80b112fff..7a42cd4b9 100644
--- a/auto/pictures_tikz/tikzDynkinpWjUbE.md5
+++ b/auto/pictures_tikz/tikzDynkinpWjUbE.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C33318BAB24044E46878BC9726CDA755}%
+\def \tikzexternallastkey {5A153D5F02EFE6D7F5471D2E06EA8F42}%
diff --git a/auto/pictures_tikz/tikzDynkinpWjUbE.pdf b/auto/pictures_tikz/tikzDynkinpWjUbE.pdf
index 0d6e375d2e687de6b4ca3ae86b1d1d55bd95522f..4e632750c69722fe8d1219e69bac6edaf6145898 100644
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diff --git a/auto/pictures_tikz/tikzDynkinqlgIQl.md5 b/auto/pictures_tikz/tikzDynkinqlgIQl.md5
index ee35dc9ef..0a089387f 100644
--- a/auto/pictures_tikz/tikzDynkinqlgIQl.md5
+++ b/auto/pictures_tikz/tikzDynkinqlgIQl.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {33D26589A963D31B1A5AE905B997C613}%
+\def \tikzexternallastkey {3DE9544C93574AA8F0F33210969D9B20}%
diff --git a/auto/pictures_tikz/tikzDynkinqlgIQl.pdf b/auto/pictures_tikz/tikzDynkinqlgIQl.pdf
index 21128121598f104953126a0265f81f11b1214abb..a9fc12f8aaa0a7c3d37be5104fe43168b5d55c58 100644
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diff --git a/auto/pictures_tikz/tikzDynkinrjbHIu.md5 b/auto/pictures_tikz/tikzDynkinrjbHIu.md5
index 794150d26..e52e866ed 100644
--- a/auto/pictures_tikz/tikzDynkinrjbHIu.md5
+++ b/auto/pictures_tikz/tikzDynkinrjbHIu.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {460C0D0AA59F76E589BE6C40F3625C46}%
+\def \tikzexternallastkey {6E78C06A3D0FBC12B7500A5925E4EF43}%
diff --git a/auto/pictures_tikz/tikzDynkinrjbHIu.pdf b/auto/pictures_tikz/tikzDynkinrjbHIu.pdf
index c92f4d0f9e399fdf65f0d3a704d0c8a5a60f28f4..f79322fd8931d827aa264f4c8d98111448bd0186 100644
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diff --git a/auto/pictures_tikz/tikzEELKooMwkockxB.md5 b/auto/pictures_tikz/tikzEELKooMwkockxB.md5
index abb1cdf94..cf5c92791 100644
--- a/auto/pictures_tikz/tikzEELKooMwkockxB.md5
+++ b/auto/pictures_tikz/tikzEELKooMwkockxB.md5
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diff --git a/auto/pictures_tikz/tikzEHDooGDwfjC.md5 b/auto/pictures_tikz/tikzEHDooGDwfjC.md5
index 6605accb7..ec931791a 100644
--- a/auto/pictures_tikz/tikzEHDooGDwfjC.md5
+++ b/auto/pictures_tikz/tikzEHDooGDwfjC.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {A3F2FC9F0F9C5890318841CB6FAE5755}%
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diff --git a/auto/pictures_tikz/tikzEJRsWXw.md5 b/auto/pictures_tikz/tikzEJRsWXw.md5
index 7286983fd..89742f5ba 100644
--- a/auto/pictures_tikz/tikzEJRsWXw.md5
+++ b/auto/pictures_tikz/tikzEJRsWXw.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {ADAE7228E005A732677E9E4AF23CB55E}%
+\def \tikzexternallastkey {388FF0F62E71A90AB3A329A809DEB81F}%
diff --git a/auto/pictures_tikz/tikzEJRsWXw.pdf b/auto/pictures_tikz/tikzEJRsWXw.pdf
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diff --git a/auto/pictures_tikz/tikzERPMooZibfNOiU.md5 b/auto/pictures_tikz/tikzERPMooZibfNOiU.md5
index 9232d7324..341269040 100644
--- a/auto/pictures_tikz/tikzERPMooZibfNOiU.md5
+++ b/auto/pictures_tikz/tikzERPMooZibfNOiU.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8C100F6B8209FE6AAE821558A5CA1E23}%
+\def \tikzexternallastkey {20F949A5A33622E6D9C117E3EE0828D7}%
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diff --git a/auto/pictures_tikz/tikzExPolygone.md5 b/auto/pictures_tikz/tikzExPolygone.md5
index a1aa29658..12ecdc808 100644
--- a/auto/pictures_tikz/tikzExPolygone.md5
+++ b/auto/pictures_tikz/tikzExPolygone.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {51E520617D86BA1994EC20957DD54DC4}%
+\def \tikzexternallastkey {F343A3225200B91911DD3EBDEB6206AC}%
diff --git a/auto/pictures_tikz/tikzExPolygone.pdf b/auto/pictures_tikz/tikzExPolygone.pdf
index 03c243766c0e90a9ec212f55b827a1d55bd66296..0616fccb584f52d8b260313ad255ade2e3c01dcc 100644
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diff --git a/auto/pictures_tikz/tikzExSinLarge.md5 b/auto/pictures_tikz/tikzExSinLarge.md5
index 69b62281e..60c1771c7 100644
--- a/auto/pictures_tikz/tikzExSinLarge.md5
+++ b/auto/pictures_tikz/tikzExSinLarge.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {9D303FDA6411EFA8A76511BFDE28A5E3}%
+\def \tikzexternallastkey {0A8E1A1CC25D6E7F0C0ECE6691B93077}%
diff --git a/auto/pictures_tikz/tikzExSinLarge.pdf b/auto/pictures_tikz/tikzExSinLarge.pdf
index 5c182fa208b0527cbdb080270b5e0e7267779f6c..0e0fab910bf51f693651b0309ccf7861972ce42e 100644
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diff --git a/auto/pictures_tikz/tikzExampleIntegration.md5 b/auto/pictures_tikz/tikzExampleIntegration.md5
index d232e20d4..1e40e954b 100644
--- a/auto/pictures_tikz/tikzExampleIntegration.md5
+++ b/auto/pictures_tikz/tikzExampleIntegration.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6FB2FEFBC3E9C072F3A627307D9739E5}%
+\def \tikzexternallastkey {3712612AC8E97CDD2BC333BCDD10EFD9}%
diff --git a/auto/pictures_tikz/tikzExampleIntegration.pdf b/auto/pictures_tikz/tikzExampleIntegration.pdf
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diff --git a/auto/pictures_tikz/tikzExampleIntegrationdeux.md5 b/auto/pictures_tikz/tikzExampleIntegrationdeux.md5
index 41f88923a..2d6d9dae6 100644
--- a/auto/pictures_tikz/tikzExampleIntegrationdeux.md5
+++ b/auto/pictures_tikz/tikzExampleIntegrationdeux.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B24881098A33F689F03E37DE6098AAF9}%
+\def \tikzexternallastkey {F5FEDB4F9F00601982D05DEF9A25797E}%
diff --git a/auto/pictures_tikz/tikzExampleIntegrationdeux.pdf b/auto/pictures_tikz/tikzExampleIntegrationdeux.pdf
index cd2245365ed9da71872d4f92a0d3e10d66280abd..b6476d1be54a02b290544a0ab582caa454eced92 100644
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diff --git a/auto/pictures_tikz/tikzExempleNonRang.md5 b/auto/pictures_tikz/tikzExempleNonRang.md5
index 424ade9fd..7fd8dae84 100644
--- a/auto/pictures_tikz/tikzExempleNonRang.md5
+++ b/auto/pictures_tikz/tikzExempleNonRang.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {F5B9D16B553165030B72202026B9E3C1}%
+\def \tikzexternallastkey {0FD186E2240B583E9A3E14195EAF3CD3}%
diff --git a/auto/pictures_tikz/tikzExempleNonRang.pdf b/auto/pictures_tikz/tikzExempleNonRang.pdf
index 2c92cef2fbbf7862be385747c61b11b846a1b8c0..29229f8269fba7e182f74fd6b512388b56ef7b54 100644
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diff --git a/auto/pictures_tikz/tikzExoCUd.md5 b/auto/pictures_tikz/tikzExoCUd.md5
index 4b33e2103..241a8277e 100644
--- a/auto/pictures_tikz/tikzExoCUd.md5
+++ b/auto/pictures_tikz/tikzExoCUd.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {1CD82CF2BFBEF9BF91A4F0CAE77304A8}%
+\def \tikzexternallastkey {FF89EA3127A42E84A45B02328BA9A7C3}%
diff --git a/auto/pictures_tikz/tikzExoCUd.pdf b/auto/pictures_tikz/tikzExoCUd.pdf
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diff --git a/auto/pictures_tikz/tikzExoMagnetique.md5 b/auto/pictures_tikz/tikzExoMagnetique.md5
index 17839462c..9fed71a54 100644
--- a/auto/pictures_tikz/tikzExoMagnetique.md5
+++ b/auto/pictures_tikz/tikzExoMagnetique.md5
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diff --git a/auto/pictures_tikz/tikzExoMagnetique.pdf b/auto/pictures_tikz/tikzExoMagnetique.pdf
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index cc772d6dd..ca5e513cd 100644
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+++ b/auto/pictures_tikz/tikzExoParamCD.md5
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diff --git a/auto/pictures_tikz/tikzExoPolaire.md5 b/auto/pictures_tikz/tikzExoPolaire.md5
index 08af2c533..8800ec14c 100644
--- a/auto/pictures_tikz/tikzExoPolaire.md5
+++ b/auto/pictures_tikz/tikzExoPolaire.md5
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diff --git a/auto/pictures_tikz/tikzExoProjection.md5 b/auto/pictures_tikz/tikzExoProjection.md5
index 26a21af4a..41601f418 100644
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@@ -1 +1 @@
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+\def \tikzexternallastkey {553863C96770B9DC400928E9E51CBE5F}%
diff --git a/auto/pictures_tikz/tikzExoProjection.pdf b/auto/pictures_tikz/tikzExoProjection.pdf
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diff --git a/auto/pictures_tikz/tikzExoUnSurxPolaire.md5 b/auto/pictures_tikz/tikzExoUnSurxPolaire.md5
index 8a5844987..fd93ca067 100644
--- a/auto/pictures_tikz/tikzExoUnSurxPolaire.md5
+++ b/auto/pictures_tikz/tikzExoUnSurxPolaire.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A49E2A1B20E9235BE9BDA1C280FA7679}%
+\def \tikzexternallastkey {D05D2AA0B46B8011359B99CE3CBFE4B1}%
diff --git a/auto/pictures_tikz/tikzExoUnSurxPolaire.pdf b/auto/pictures_tikz/tikzExoUnSurxPolaire.pdf
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index 3ecabd914..fce3505bc 100644
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+++ b/auto/pictures_tikz/tikzExoVarj.md5
@@ -1 +1 @@
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index 80ce9138d..35995bd3d 100644
--- a/auto/pictures_tikz/tikzExoXLVL.md5
+++ b/auto/pictures_tikz/tikzExoXLVL.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {69CA1D00E2C0EBCA74673B9DFD1FE2CF}%
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diff --git a/auto/pictures_tikz/tikzFCUEooTpEPFoeQ.md5 b/auto/pictures_tikz/tikzFCUEooTpEPFoeQ.md5
index 613288e6c..0a5a25258 100644
--- a/auto/pictures_tikz/tikzFCUEooTpEPFoeQ.md5
+++ b/auto/pictures_tikz/tikzFCUEooTpEPFoeQ.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D4102DFAA21C489BC7844F118E5A9552}%
+\def \tikzexternallastkey {0298D67F252D3BD13BF021B1148E8444}%
diff --git a/auto/pictures_tikz/tikzFCUEooTpEPFoeQ.pdf b/auto/pictures_tikz/tikzFCUEooTpEPFoeQ.pdf
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diff --git a/auto/pictures_tikz/tikzFGRooDhFkch.md5 b/auto/pictures_tikz/tikzFGRooDhFkch.md5
index 0a3663322..3083cc11d 100644
--- a/auto/pictures_tikz/tikzFGRooDhFkch.md5
+++ b/auto/pictures_tikz/tikzFGRooDhFkch.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A31CD6206E8D01D378424A2CB466A690}%
+\def \tikzexternallastkey {D711DD12E517F75727DA60AD8AB70762}%
diff --git a/auto/pictures_tikz/tikzFGRooDhFkch.pdf b/auto/pictures_tikz/tikzFGRooDhFkch.pdf
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diff --git a/auto/pictures_tikz/tikzFGWjJBX.md5 b/auto/pictures_tikz/tikzFGWjJBX.md5
index 55ab6d3f3..bcb1972fc 100644
--- a/auto/pictures_tikz/tikzFGWjJBX.md5
+++ b/auto/pictures_tikz/tikzFGWjJBX.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {13C2E041B21568A2A04DE5E3205B4A01}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ0PICTACUooQwcDMZpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ0PICTACUooQwcDMZpspict0.md5
index 3f7da20e9..66fb71cf7 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ0PICTACUooQwcDMZpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ0PICTACUooQwcDMZpspict0.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {6D58C4EE906BF521EBBDDBDA15693E86}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ1PICTACUooQwcDMZpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ1PICTACUooQwcDMZpspict1.md5
index 14dada0e4..653d1a331 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ1PICTACUooQwcDMZpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ1PICTACUooQwcDMZpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B2F7EBA168B7F42F26578CF9D4B4299A}%
+\def \tikzexternallastkey {2AD7EDADC73A665670D8747C800CF21F}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ1PICTACUooQwcDMZpspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ1PICTACUooQwcDMZpspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ2PICTACUooQwcDMZpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ2PICTACUooQwcDMZpspict2.md5
index 7d1db4173..02dcf080a 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ3PICTACUooQwcDMZpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ3PICTACUooQwcDMZpspict3.md5
index 3b554b859..39fb35f97 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ3PICTACUooQwcDMZpspict3.md5
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+\def \tikzexternallastkey {379627D73C4A1F3331CE38DA3533E8D0}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ4PICTACUooQwcDMZpspict4.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ4PICTACUooQwcDMZpspict4.md5
index cf870841f..bb484e8cb 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ4PICTACUooQwcDMZpspict4.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ4PICTACUooQwcDMZpspict4.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {121761AA7681C0F9F2BBE2A119ACD474}%
+\def \tikzexternallastkey {A884E5A31C8B4FF5E2DCBCB824154743}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ4PICTACUooQwcDMZpspict4.pdf b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ4PICTACUooQwcDMZpspict4.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ5PICTACUooQwcDMZpspict5.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ5PICTACUooQwcDMZpspict5.md5
index 7b050e6c9..57ff09129 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ5PICTACUooQwcDMZpspict5.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ5PICTACUooQwcDMZpspict5.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {351E3475BFD8047ECFB9C18CE228E3F9}%
+\def \tikzexternallastkey {598B30D8242CD7247E88815FDDBE8FB3}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ5PICTACUooQwcDMZpspict5.pdf b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ5PICTACUooQwcDMZpspict5.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ6PICTACUooQwcDMZpspict6.md5 b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ6PICTACUooQwcDMZpspict6.md5
index cef0e17f7..d7696bb7e 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ6PICTACUooQwcDMZpspict6.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ6PICTACUooQwcDMZpspict6.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {492C5A13D96A559FDA60587C625DB79B}%
+\def \tikzexternallastkey {A47FD3BBA6DA9D08980375CF24ED75F3}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ6PICTACUooQwcDMZpspict6.pdf b/auto/pictures_tikz/tikzFIGLabelFigACUooQwcDMZssLabelSubFigACUooQwcDMZ6PICTACUooQwcDMZpspict6.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii0PICTCSCiiipspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii0PICTCSCiiipspict0.md5
index 53976765c..c4782b784 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii0PICTCSCiiipspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii0PICTCSCiiipspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6C1B6F1A1F82C681FC362F05DF516148}%
+\def \tikzexternallastkey {74809F57116AD3FEB7294CCA72EEE94E}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii0PICTCSCiiipspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii0PICTCSCiiipspict0.pdf
index fa360502b339bc5608e5264f4eac86f32ff0301b..5867cce63d902b392358b6e59eb1c6a0314a30f4 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii1PICTCSCiiipspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii1PICTCSCiiipspict1.md5
index d3fe9b903..e739cffe9 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii1PICTCSCiiipspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii1PICTCSCiiipspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B5F143B4866E03FFCCCB93152A57B98C}%
+\def \tikzexternallastkey {610276ED53789872F5A09E95D03AF610}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii1PICTCSCiiipspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii1PICTCSCiiipspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii2PICTCSCiiipspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii2PICTCSCiiipspict2.md5
index 4627c6f29..88d05177b 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii2PICTCSCiiipspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii2PICTCSCiiipspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {514A96211FF33E5C53DD3E733B75442D}%
+\def \tikzexternallastkey {7AD57D4D8B5A9A480D5411475C2F7EE7}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii2PICTCSCiiipspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigCSCiiissLabelSubFigCSCiii2PICTCSCiiipspict2.pdf
index 62de2ec102dccb2e4405d8c3684d6dde1e864a26..0c19e0159fefdcd647b545797c52a5464085e5df 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCvissLabelSubFigCSCvi0PICTCSCvipspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCvissLabelSubFigCSCvi0PICTCSCvipspict0.md5
index 25d5b2971..8bca2d727 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCvissLabelSubFigCSCvi1PICTCSCvipspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCvissLabelSubFigCSCvi1PICTCSCvipspict1.md5
index 4c7adde12..6dfe70edf 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCSCvissLabelSubFigCSCvi1PICTCSCvipspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCSCvissLabelSubFigCSCvi1PICTCSCvipspict1.md5
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv0PICTCSCvpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv0PICTCSCvpspict0.md5
index ebf324770..2ed4c95dd 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv0PICTCSCvpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv0PICTCSCvpspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A7741ADBECEF1E88D87BFE030C7EC686}%
+\def \tikzexternallastkey {A19D174E6FE2C2B8C54BDD4813DCAA97}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv0PICTCSCvpspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv0PICTCSCvpspict0.pdf
index 5d78a367c65d1f6a8354fb0630ab14eef7b09c38..c4a57a125241f6334c6db54a9f3f95900f82d744 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv1PICTCSCvpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv1PICTCSCvpspict1.md5
index 7442674e8..1a3100f73 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv1PICTCSCvpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCSCvssLabelSubFigCSCv1PICTCSCvpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {4659AC920AE8DACB75DEA2D07DB18350}%
+\def \tikzexternallastkey {B2FED15D66570C7CFB385EE85B272546}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini0PICTChoixInfinipspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini0PICTChoixInfinipspict0.md5
index 2d808c466..22e18185c 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini0PICTChoixInfinipspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini0PICTChoixInfinipspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {31B70603BEE2ECA0CBA525AC86537A1C}%
+\def \tikzexternallastkey {51C6C64BE5672253233130C55BD1D2A6}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini1PICTChoixInfinipspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini1PICTChoixInfinipspict1.md5
index 3c1d3d5c8..8d0e3475f 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini1PICTChoixInfinipspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigChoixInfinissLabelSubFigChoixInfini1PICTChoixInfinipspict1.md5
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+\def \tikzexternallastkey {92CA71756AD474F4F22173AA8A9B7144}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubONEPICT.md5 b/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubONEPICT.md5
index 458a28da1..58756b481 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubONEPICT.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubONEPICT.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {5DF4E2597B5DF8672CF47B33D0E4C4B2}%
+\def \tikzexternallastkey {55ECB4AF2115A9A87906C676F7309484}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubZEROPICT.md5 b/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubZEROPICT.md5
index 84642062e..ba09b3181 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubZEROPICT.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigCurvilignesPolairessssubZEROPICT.md5
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf0PICTDZVooQZLUtfpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf0PICTDZVooQZLUtfpspict0.md5
index fe0388707..5cd0f4bd1 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf0PICTDZVooQZLUtfpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf0PICTDZVooQZLUtfpspict0.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {F7F43837ED6C7FC9D29449DD6BB4933A}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf0PICTDZVooQZLUtfpspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf0PICTDZVooQZLUtfpspict0.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf1PICTDZVooQZLUtfpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf1PICTDZVooQZLUtfpspict1.md5
index e94455485..1d44a17a3 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf1PICTDZVooQZLUtfpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf1PICTDZVooQZLUtfpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {325410A9A567037356EF7938EC82E324}%
+\def \tikzexternallastkey {C177BC424C6E72D7B3CAE9D11CC1AB1D}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf1PICTDZVooQZLUtfpspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf1PICTDZVooQZLUtfpspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf2PICTDZVooQZLUtfpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf2PICTDZVooQZLUtfpspict2.md5
index 1fd17b54b..747156176 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf2PICTDZVooQZLUtfpspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf2PICTDZVooQZLUtfpspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {51C5DB7E11C51622A7B750E3DEFFDEB2}%
+\def \tikzexternallastkey {813F50E10231A444814AE1638E0C788E}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf2PICTDZVooQZLUtfpspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf2PICTDZVooQZLUtfpspict2.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf3PICTDZVooQZLUtfpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf3PICTDZVooQZLUtfpspict3.md5
index 26c25a6d1..070fbd00a 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf3PICTDZVooQZLUtfpspict3.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf3PICTDZVooQZLUtfpspict3.md5
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+\def \tikzexternallastkey {435FA10FDD489ECEC6D6999DBDF6C9D2}%
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index 0e6cfe5af..e7551e91a 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf4PICTDZVooQZLUtfpspict4.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf4PICTDZVooQZLUtfpspict4.md5
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+\def \tikzexternallastkey {C8BAA7A0CB1D305CE432BCD47F334A00}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf5PICTDZVooQZLUtfpspict5.md5 b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf5PICTDZVooQZLUtfpspict5.md5
index 716d45bdd..4b4381951 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf5PICTDZVooQZLUtfpspict5.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf5PICTDZVooQZLUtfpspict5.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {9BA627005D5997B6B6FE51B5B0E1762C}%
+\def \tikzexternallastkey {7DB7F223F06EEE23941DC574FA7B1DF7}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf6PICTDZVooQZLUtfpspict6.md5 b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf6PICTDZVooQZLUtfpspict6.md5
index 6719c0f42..ea7dc14f6 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf6PICTDZVooQZLUtfpspict6.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf6PICTDZVooQZLUtfpspict6.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {BA2547F2E68CCF6A7FCB89291C557282}%
+\def \tikzexternallastkey {99B42F0E95A59BC2CCDCC757C8C5655F}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf6PICTDZVooQZLUtfpspict6.pdf b/auto/pictures_tikz/tikzFIGLabelFigDZVooQZLUtfssLabelSubFigDZVooQZLUtf6PICTDZVooQZLUtfpspict6.pdf
index 3a989e9b0b716fda4e8ea29845cf0ec233f40194..2499bf5971f9ab60dfc825dbbc67261c3977ef73 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx0PICTDynkinNUtPJxpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx0PICTDynkinNUtPJxpspict0.md5
index f9151bc26..536f3eecc 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx0PICTDynkinNUtPJxpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx0PICTDynkinNUtPJxpspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {2BD908ED74DE86D8EC170F90A633CC0E}%
+\def \tikzexternallastkey {A6D9412EE6B1FCA5117A8EB2846894C4}%
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+++ b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx1PICTDynkinNUtPJxpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {96792490DE249A7B8A70C145D2975AB2}%
+\def \tikzexternallastkey {9BA990158667440B1BB8DE97E941021F}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx2PICTDynkinNUtPJxpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx2PICTDynkinNUtPJxpspict2.md5
index e5074eb0b..72bb48bee 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx2PICTDynkinNUtPJxpspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx2PICTDynkinNUtPJxpspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C798FBDE0AFF7860F38347942166916D}%
+\def \tikzexternallastkey {9C6A6063D1BCB9DAC1D6A2810A7C32B2}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx2PICTDynkinNUtPJxpspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx2PICTDynkinNUtPJxpspict2.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx3PICTDynkinNUtPJxpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx3PICTDynkinNUtPJxpspict3.md5
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+\def \tikzexternallastkey {0FABA7FD9D7CF3E3B3E179A0100125EC}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx4PICTDynkinNUtPJxpspict4.md5 b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx4PICTDynkinNUtPJxpspict4.md5
index 527da6ffc..9bd270a30 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx4PICTDynkinNUtPJxpspict4.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigDynkinNUtPJxssLabelSubFigDynkinNUtPJx4PICTDynkinNUtPJxpspict4.md5
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+\def \tikzexternallastkey {AAE95AFEA9EC9F31A63B412B4A0BE780}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExempleArcParamssLabelArcUnPICTgammasintti.md5 b/auto/pictures_tikz/tikzFIGLabelFigExempleArcParamssLabelArcUnPICTgammasintti.md5
index 580fad946..5888a75ed 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigExempleArcParamssLabelArcUnPICTgammasintti.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigExempleArcParamssLabelArcUnPICTgammasintti.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0B99F1857FE5608B327B87514180AE4F}%
+\def \tikzexternallastkey {ADB510D16B6DAF16CF64F9230D4A0D13}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigExempleArcParamssLabelArcUnPICTgammasintti.pdf b/auto/pictures_tikz/tikzFIGLabelFigExempleArcParamssLabelArcUnPICTgammasintti.pdf
index 8db8ce2267bc014894c671871a76f2f96ab7c096..4d5fac4545d77b92561a4abcbe4839dc0db201bb 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosexPICTcosex.md5 b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosexPICTcosex.md5
new file mode 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosinusPICTcosinus.md5 b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosinusPICTcosinus.md5
new file mode 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox.md5 b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox.md5
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox.pdf b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosquattroxPICTcosquattrox.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosxplusunPICTcosxplusun.md5 b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissscosxplusunPICTcosxplusun.md5
new file mode 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissssinusPICTsinus.md5 b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbissssinusPICTsinus.md5
new file mode 100644
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new file mode 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.md5 b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.md5
new file mode 100644
index 000000000..6917acda3
--- /dev/null
+++ b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.md5
@@ -0,0 +1 @@
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.pdf b/auto/pictures_tikz/tikzFIGLabelFigExerciceGraphesbisssvalabsoluecosinusPICTvalabsoluecosinus.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI0PICTHCJPooHsaTgIpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI0PICTHCJPooHsaTgIpspict0.md5
index d9f29cc92..5589a5305 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI0PICTHCJPooHsaTgIpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI0PICTHCJPooHsaTgIpspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D37A75A0B3E7850DFF071133F50E8E03}%
+\def \tikzexternallastkey {751F8CC181109168913D849D0F046170}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI0PICTHCJPooHsaTgIpspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI0PICTHCJPooHsaTgIpspict0.pdf
index b5c7508fcf0f7be4f5fdecf6458af8fc5f38900c..354324fcbbc55b2f772cc0116d50e68f2fb846fb 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI1PICTHCJPooHsaTgIpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI1PICTHCJPooHsaTgIpspict1.md5
index 29725df78..13d188809 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI1PICTHCJPooHsaTgIpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI1PICTHCJPooHsaTgIpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {361BF836D23F47999D121DBC7543F540}%
+\def \tikzexternallastkey {A812A8400BE3FBE77384D358B629FCFA}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI1PICTHCJPooHsaTgIpspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigHCJPooHsaTgIssLabelSubFigHCJPooHsaTgI1PICTHCJPooHsaTgIpspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubONEPICT.md5 b/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubONEPICT.md5
index 458a28da1..58756b481 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubONEPICT.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubONEPICT.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {5DF4E2597B5DF8672CF47B33D0E4C4B2}%
+\def \tikzexternallastkey {55ECB4AF2115A9A87906C676F7309484}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubONEPICT.pdf b/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubONEPICT.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubZEROPICT.md5 b/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubZEROPICT.md5
index 84642062e..ba09b3181 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubZEROPICT.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubZEROPICT.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {504462D747BAC4BD5116BC1B7554A615}%
+\def \tikzexternallastkey {181908206F88D702B70E687F93B111AF}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubZEROPICT.pdf b/auto/pictures_tikz/tikzFIGLabelFigHGQPooKrRtANsssubZEROPICT.pdf
index 020b3b4618a50de92f7e8e7d28599fbbff0aef3b..32fc94c85df44ffdd94e3e784bd868f12200c01f 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelBonPICTBon.md5 b/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelBonPICTBon.md5
index 09fbc500b..918e156f0 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelBonPICTBon.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelBonPICTBon.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {66CC0D781BA58932DFA03ECD38AA216F}%
+\def \tikzexternallastkey {071284F4E1E6E0D6BC49D0AAE35B4FE8}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelMauvaisPICTmauvais.md5 b/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelMauvaisPICTmauvais.md5
index c3e38b14f..6489f657a 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelMauvaisPICTmauvais.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelMauvaisPICTmauvais.md5
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+\def \tikzexternallastkey {97B355C98E723C5655FF3F675E470972}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelMauvaisPICTmauvais.pdf b/auto/pictures_tikz/tikzFIGLabelFigIntTroisssLabelMauvaisPICTmauvais.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSpheresssLabelSubFigLesSpheres1PICTLesSpherespspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSpheresssLabelSubFigLesSpheres1PICTLesSpherespspict1.md5
index 4c26e783d..9b3e8e6cb 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSpheresssLabelSubFigLesSpheres2PICTLesSpherespspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSpheresssLabelSubFigLesSpheres2PICTLesSpherespspict2.md5
index abec689ec..9fd8472f4 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM0PICTLesSubFiguresOMpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM0PICTLesSubFiguresOMpspict0.md5
index 2e39d9f4f..cbafe9870 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM0PICTLesSubFiguresOMpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM0PICTLesSubFiguresOMpspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8B5E141803DA788AD6A48A45C64660F0}%
+\def \tikzexternallastkey {D67BD3AE1757C3B19BFADAE542B5F127}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM0PICTLesSubFiguresOMpspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM0PICTLesSubFiguresOMpspict0.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM1PICTLesSubFiguresOMpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM1PICTLesSubFiguresOMpspict1.md5
index e168d0f67..5d1a618f6 100644
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+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM1PICTLesSubFiguresOMpspict1.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {A9558C1C291B176C935B194C4E64B9F7}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM2PICTLesSubFiguresOMpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM2PICTLesSubFiguresOMpspict2.md5
index cab83fd6b..49205b10d 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM2PICTLesSubFiguresOMpspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM2PICTLesSubFiguresOMpspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {72614A23E68E171A3F74B68FC956C691}%
+\def \tikzexternallastkey {0DCBF5BCCEE74E5734A5A73755F84D9E}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM2PICTLesSubFiguresOMpspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM2PICTLesSubFiguresOMpspict2.pdf
index 49b7d4980a43b59cb615928266e9aa4fd17cc18e..dc09e2f3f54f69324a47ef58daca399898ab8c5c 100644
GIT binary patch
delta 2589
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zSIFJY5g<c|$by^uCARA8GO_!|Ws0snezMmFTYvSjD=L$}ZyyOaHVP#rMNdWwcWkhK

delta 2570
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM3PICTLesSubFiguresOMpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM3PICTLesSubFiguresOMpspict3.md5
index 63e8dce85..d75be480e 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM3PICTLesSubFiguresOMpspict3.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM3PICTLesSubFiguresOMpspict3.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {DD21606884FDE32A7ADE044D715F5CB8}%
+\def \tikzexternallastkey {35E230B866B348983548FF206A7A0533}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM3PICTLesSubFiguresOMpspict3.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresOMssLabelSubFigLesSubFiguresOM3PICTLesSubFiguresOMpspict3.pdf
index 6f5dcda79d427f35074e3532e075376b2235c709..53642a15dff922c5ec3f23207659955e34e12929 100644
GIT binary patch
delta 2638
zcmai!X&@7f1BYqRT2vCba%U5<vDxIluL$L8<R%lA+}CnOMoP$;Hw<CkA=k3G&om)N
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures0PICTLesSubFigurespspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures0PICTLesSubFigurespspict0.md5
index 2e39d9f4f..cbafe9870 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures0PICTLesSubFigurespspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures0PICTLesSubFigurespspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8B5E141803DA788AD6A48A45C64660F0}%
+\def \tikzexternallastkey {D67BD3AE1757C3B19BFADAE542B5F127}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures0PICTLesSubFigurespspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures0PICTLesSubFigurespspict0.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures1PICTLesSubFigurespspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures1PICTLesSubFigurespspict1.md5
index e168d0f67..5d1a618f6 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures1PICTLesSubFigurespspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures1PICTLesSubFigurespspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {98D4CC08C0CF992874112C24C4641158}%
+\def \tikzexternallastkey {A9558C1C291B176C935B194C4E64B9F7}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures1PICTLesSubFigurespspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures1PICTLesSubFigurespspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures2PICTLesSubFigurespspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures2PICTLesSubFigurespspict2.md5
index cab83fd6b..49205b10d 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures2PICTLesSubFigurespspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures2PICTLesSubFigurespspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {72614A23E68E171A3F74B68FC956C691}%
+\def \tikzexternallastkey {0DCBF5BCCEE74E5734A5A73755F84D9E}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures2PICTLesSubFigurespspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures2PICTLesSubFigurespspict2.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures3PICTLesSubFigurespspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures3PICTLesSubFigurespspict3.md5
index 63e8dce85..d75be480e 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures3PICTLesSubFigurespspict3.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures3PICTLesSubFigurespspict3.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {DD21606884FDE32A7ADE044D715F5CB8}%
+\def \tikzexternallastkey {35E230B866B348983548FF206A7A0533}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures3PICTLesSubFigurespspict3.pdf b/auto/pictures_tikz/tikzFIGLabelFigLesSubFiguresssLabelSubFigLesSubFigures3PICTLesSubFigurespspict3.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigMoulinEaussLabelSubFigMoulinEau0PICTMoulinEaupspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigMoulinEaussLabelSubFigMoulinEau0PICTMoulinEaupspict0.md5
index d2b94dfb3..f74ba097a 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigMoulinEaussLabelSubFigMoulinEau1PICTMoulinEaupspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigMoulinEaussLabelSubFigMoulinEau1PICTMoulinEaupspict1.md5
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl1PICTQOBAooZZZOrlpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl1PICTQOBAooZZZOrlpspict1.md5
index d40d4b200..31beb4b3d 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl1PICTQOBAooZZZOrlpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl1PICTQOBAooZZZOrlpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D0809B0385D6E56588277164F6D4CA42}%
+\def \tikzexternallastkey {464929C3A1FADAC91B2E19AAA1F54A3D}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl1PICTQOBAooZZZOrlpspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl1PICTQOBAooZZZOrlpspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl2PICTQOBAooZZZOrlpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl2PICTQOBAooZZZOrlpspict2.md5
index a94093aec..5d2c595cc 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl2PICTQOBAooZZZOrlpspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigQOBAooZZZOrlssLabelSubFigQOBAooZZZOrl2PICTQOBAooZZZOrlpspict2.md5
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+\def \tikzexternallastkey {6BE83B9A1F10B093749E6C03BFC0E08B}%
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index b1a0e36dd..2f1014e4f 100644
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index 7518b2e30..85f74fd69 100644
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+++ b/auto/pictures_tikz/tikzFIGLabelFigQQassLabelSubFigQQa3PICTQQapspict3.md5
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigRegioniPrimoeSecondoTiposssecondotipoPICTregdeux.md5 b/auto/pictures_tikz/tikzFIGLabelFigRegioniPrimoeSecondoTiposssecondotipoPICTregdeux.md5
index 034329b5c..09b88ed03 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigRegioniPrimoeSecondoTiposssecondotipoPICTregdeux.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigRegioniPrimoeSecondoTiposssecondotipoPICTregdeux.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {F3EC871D79B29164CE1EE9E192F181F4}%
+\def \tikzexternallastkey {5C1FE84F597B951B89499B0E762C6784}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigRegioniPrimoeSecondoTiposssecondotipoPICTregdeux.pdf b/auto/pictures_tikz/tikzFIGLabelFigRegioniPrimoeSecondoTiposssecondotipoPICTregdeux.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS1TraceCycloidePICTTraceCycloidepsp1.md5 b/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS1TraceCycloidePICTTraceCycloidepsp1.md5
index 40589bf33..e4765b4ff 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS1TraceCycloidePICTTraceCycloidepsp1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS1TraceCycloidePICTTraceCycloidepsp1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {33E3C6078F5DEDB95FB87577383ACD77}%
+\def \tikzexternallastkey {9411C6F89ACBD6BAA7EB5AFF61173EB7}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS1TraceCycloidePICTTraceCycloidepsp1.pdf b/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS1TraceCycloidePICTTraceCycloidepsp1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS2TraceCycloidePICTTraceCycloidepsp2.md5 b/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS2TraceCycloidePICTTraceCycloidepsp2.md5
index e1d790917..4422cb45f 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS2TraceCycloidePICTTraceCycloidepsp2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS2TraceCycloidePICTTraceCycloidepsp2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {01BA78443CDE2CFB775CDDD3B1DEC5CD}%
+\def \tikzexternallastkey {06347780B08CC6C61FA21E259EF80925}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS2TraceCycloidePICTTraceCycloidepsp2.pdf b/auto/pictures_tikz/tikzFIGLabelFigSubfiguresCDUTraceCycloidessSS2TraceCycloidePICTTraceCycloidepsp2.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes0PICTSurfaceEntreCourbespspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes0PICTSurfaceEntreCourbespspict0.md5
index 4118803d3..2b3b7b4bc 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes0PICTSurfaceEntreCourbespspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes0PICTSurfaceEntreCourbespspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {4A7A1ABAF5FFD59C799BB20AE57F265F}%
+\def \tikzexternallastkey {93313D6E62A94877FB05E80B44C3C0B8}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes0PICTSurfaceEntreCourbespspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes0PICTSurfaceEntreCourbespspict0.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes1PICTSurfaceEntreCourbespspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes1PICTSurfaceEntreCourbespspict1.md5
index d0723a682..9e6af8512 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes1PICTSurfaceEntreCourbespspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes1PICTSurfaceEntreCourbespspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {50AE3B60401A3E37DBCF9BED47E1C594}%
+\def \tikzexternallastkey {E283546080FC3C765FF410F68D5D4A9E}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes1PICTSurfaceEntreCourbespspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes1PICTSurfaceEntreCourbespspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes2PICTSurfaceEntreCourbespspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes2PICTSurfaceEntreCourbespspict2.md5
index 73382b6aa..7839e355e 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes2PICTSurfaceEntreCourbespspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes2PICTSurfaceEntreCourbespspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B40A43B2D059707E03B5F1A1D50AE48B}%
+\def \tikzexternallastkey {F443CD13E58E6D2F7E74E6BF53A3C8E2}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes2PICTSurfaceEntreCourbespspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigSurfaceEntreCourbesssLabelSubFigSurfaceEntreCourbes2PICTSurfaceEntreCourbespspict2.pdf
index 9fe79d82d14f58d498f6babc2ac54c256e388b62..dbdcbf8ca49c41240a2de1ef947010d628362591 100644
GIT binary patch
delta 4822
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti0PICTSurfaceHorizVertipspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti0PICTSurfaceHorizVertipspict0.md5
index bf104a5f1..44198c342 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti0PICTSurfaceHorizVertipspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti0PICTSurfaceHorizVertipspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {01EED7739677F51DDBAAD5EFA8D9EE6F}%
+\def \tikzexternallastkey {8CECD281F84A4DD846DEB219D056F627}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti0PICTSurfaceHorizVertipspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti0PICTSurfaceHorizVertipspict0.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti1PICTSurfaceHorizVertipspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti1PICTSurfaceHorizVertipspict1.md5
index 29725df78..13d188809 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti1PICTSurfaceHorizVertipspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti1PICTSurfaceHorizVertipspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {361BF836D23F47999D121DBC7543F540}%
+\def \tikzexternallastkey {A812A8400BE3FBE77384D358B629FCFA}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti1PICTSurfaceHorizVertipspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigSurfaceHorizVertissLabelSubFigSurfaceHorizVerti1PICTSurfaceHorizVertipspict1.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn0PICTTracerUnpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn0PICTTracerUnpspict0.md5
index f6623e86c..9fa0dcb32 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn0PICTTracerUnpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn0PICTTracerUnpspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B300147BD07F178BF53776D53DFE6D50}%
+\def \tikzexternallastkey {895AA374B66C3980F9187D4A081ECA4D}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn0PICTTracerUnpspict0.pdf b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn0PICTTracerUnpspict0.pdf
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index adb7badde..7bfa7ff11 100644
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+++ b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn1PICTTracerUnpspict1.md5
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+\def \tikzexternallastkey {A158632C5F9F3270607F8CDCA7DEDA4D}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn3PICTTracerUnpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn3PICTTracerUnpspict3.md5
index 835f8c9b5..4e396733a 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn3PICTTracerUnpspict3.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn3PICTTracerUnpspict3.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {E35956AD6A19813357B7B4596F864F00}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn3PICTTracerUnpspict3.pdf b/auto/pictures_tikz/tikzFIGLabelFigTracerUnssLabelSubFigTracerUn3PICTTracerUnpspict3.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigUUNEooCNVOOsssLabelSubFigUUNEooCNVOOs0PICTUUNEooCNVOOspspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigUUNEooCNVOOsssLabelSubFigUUNEooCNVOOs0PICTUUNEooCNVOOspspict0.md5
index d2b94dfb3..f74ba097a 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigUUNEooCNVOOsssLabelSubFigUUNEooCNVOOs0PICTUUNEooCNVOOspspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigUUNEooCNVOOsssLabelSubFigUUNEooCNVOOs0PICTUUNEooCNVOOspspict0.md5
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index 44499974b..f60899867 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigUUNEooCNVOOsssLabelSubFigUUNEooCNVOOs1PICTUUNEooCNVOOspspict1.md5
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO0PICTVANooZowSyOpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO0PICTVANooZowSyOpspict0.md5
index f587627e5..ae196d014 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO1PICTVANooZowSyOpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO1PICTVANooZowSyOpspict1.md5
index d2cfc61fc..e35d7af99 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO1PICTVANooZowSyOpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO1PICTVANooZowSyOpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0FE45CC7AB5941680AFC43F11E77E176}%
+\def \tikzexternallastkey {EB7B869AFEB63F6832A3E12E612F237B}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO1PICTVANooZowSyOpspict1.pdf b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO1PICTVANooZowSyOpspict1.pdf
index c9fcd5ce6aea0dd7f6034478397b33f75c93985e..b501edd430a86a0553b9cce7b27ed07e0e149664 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO2PICTVANooZowSyOpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO2PICTVANooZowSyOpspict2.md5
index 9cca96a7e..5287b0174 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO2PICTVANooZowSyOpspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO2PICTVANooZowSyOpspict2.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {1D912D7D4C6EA7DA6F3FE712298A825B}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO2PICTVANooZowSyOpspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO2PICTVANooZowSyOpspict2.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO3PICTVANooZowSyOpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO3PICTVANooZowSyOpspict3.md5
index d46266d9b..cf4b462fd 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO3PICTVANooZowSyOpspict3.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO3PICTVANooZowSyOpspict3.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {ABDC8068A1A492F98196E1BE855BC6C2}%
+\def \tikzexternallastkey {90301D5641096D7D4A76FF058BB29781}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO3PICTVANooZowSyOpspict3.pdf b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO3PICTVANooZowSyOpspict3.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO4PICTVANooZowSyOpspict4.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO4PICTVANooZowSyOpspict4.md5
index e760c890a..c2e30aae4 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO4PICTVANooZowSyOpspict4.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO4PICTVANooZowSyOpspict4.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8AEE872360EB13528D18AC5191FA2D40}%
+\def \tikzexternallastkey {D33FDE3C7D2A29D3D84887AC1963F6CC}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO4PICTVANooZowSyOpspict4.pdf b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO4PICTVANooZowSyOpspict4.pdf
index f7eccd0b62bbfbacc98afc50d160237137e66717..550c0a8a3330a6e96cfbc77477987b21c5224b10 100644
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO5PICTVANooZowSyOpspict5.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO5PICTVANooZowSyOpspict5.md5
index 8f0588b62..8dcc83786 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO5PICTVANooZowSyOpspict5.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO5PICTVANooZowSyOpspict5.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6E91D3FF341070FFC9C8421644CB6D0C}%
+\def \tikzexternallastkey {FAEB4235148325D0042B39B8C62627FC}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO5PICTVANooZowSyOpspict5.pdf b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO5PICTVANooZowSyOpspict5.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO6PICTVANooZowSyOpspict6.md5 b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO6PICTVANooZowSyOpspict6.md5
index afb58ad5a..c2b249f9d 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO6PICTVANooZowSyOpspict6.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO6PICTVANooZowSyOpspict6.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {41529B3CBE27688F885F553C9F3C5A82}%
+\def \tikzexternallastkey {2330EF6CF77F5EC5539E95E5A6EDCE8B}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO6PICTVANooZowSyOpspict6.pdf b/auto/pictures_tikz/tikzFIGLabelFigVANooZowSyOssLabelSubFigVANooZowSyO6PICTVANooZowSyOpspict6.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB0PICTWUYooCISzeBpspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB0PICTWUYooCISzeBpspict0.md5
index f98a1f671..54cec4c4c 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB0PICTWUYooCISzeBpspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB0PICTWUYooCISzeBpspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7D8C6FC9F54D48187CBAA36B4724EB19}%
+\def \tikzexternallastkey {AC6F502047A8384482133C976D332AF8}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB1PICTWUYooCISzeBpspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB1PICTWUYooCISzeBpspict1.md5
index b8fe08676..8cbbf6374 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB1PICTWUYooCISzeBpspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB1PICTWUYooCISzeBpspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {32C9B97040840F8C45FCB15921B29A2B}%
+\def \tikzexternallastkey {86128F8EF911B0779A9939C31C31D734}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB2PICTWUYooCISzeBpspict2.md5 b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB2PICTWUYooCISzeBpspict2.md5
index 69e670cb8..07745602b 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB2PICTWUYooCISzeBpspict2.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB2PICTWUYooCISzeBpspict2.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7280320C776E4B60047B4755C6F7109D}%
+\def \tikzexternallastkey {9BCA14C130FC3D67C216FE06D72FC3E3}%
diff --git a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB2PICTWUYooCISzeBpspict2.pdf b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB2PICTWUYooCISzeBpspict2.pdf
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB3PICTWUYooCISzeBpspict3.md5 b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB3PICTWUYooCISzeBpspict3.md5
index 7701a25e2..ff6200a27 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB3PICTWUYooCISzeBpspict3.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigWUYooCISzeBssLabelSubFigWUYooCISzeB3PICTWUYooCISzeBpspict3.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {11B733830DB336C0B8A7AE082D2D5382}%
+\def \tikzexternallastkey {FFE5EA23640099FE384C12D439EC3CD6}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci0PICTZTTooXtHkcipspict0.md5 b/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci0PICTZTTooXtHkcipspict0.md5
index b24e97bfe..5f7cd9038 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci0PICTZTTooXtHkcipspict0.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci0PICTZTTooXtHkcipspict0.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EDB5DF7A295670756CCA0D1C442D75EC}%
+\def \tikzexternallastkey {01ADC9A685D88AE04029A65D4E886F5D}%
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diff --git a/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci1PICTZTTooXtHkcipspict1.md5 b/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci1PICTZTTooXtHkcipspict1.md5
index d23cc640a..b76e9dab4 100644
--- a/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci1PICTZTTooXtHkcipspict1.md5
+++ b/auto/pictures_tikz/tikzFIGLabelFigZTTooXtHkcissLabelSubFigZTTooXtHkci1PICTZTTooXtHkcipspict1.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {61C3AB37EABFAFC7D9286E73D257579B}%
+\def \tikzexternallastkey {2635F5447E597444F884C64EFA896A4B}%
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diff --git a/auto/pictures_tikz/tikzFNBQooYgkAmS.md5 b/auto/pictures_tikz/tikzFNBQooYgkAmS.md5
index 0ef612337..6cc68e4a9 100644
--- a/auto/pictures_tikz/tikzFNBQooYgkAmS.md5
+++ b/auto/pictures_tikz/tikzFNBQooYgkAmS.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {2DB6073AE1C4C2526465F90A6FC73B96}%
+\def \tikzexternallastkey {E77FC0BFB9FACE93A7382B6D970FB41B}%
diff --git a/auto/pictures_tikz/tikzFNBQooYgkAmS.pdf b/auto/pictures_tikz/tikzFNBQooYgkAmS.pdf
index 4eeab1ada232b7eb7f39e7a051dc468474416852..c3a40c9c747c9a7b8382440172c693e7a0635e09 100644
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diff --git a/auto/pictures_tikz/tikzFWJuNhU.md5 b/auto/pictures_tikz/tikzFWJuNhU.md5
index 80ce9138d..7238b5d4f 100644
--- a/auto/pictures_tikz/tikzFWJuNhU.md5
+++ b/auto/pictures_tikz/tikzFWJuNhU.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {903C57A8DEF696986247EF37A6C2FD86}%
+\def \tikzexternallastkey {27EB9FAB99004161B34AA3F583B8E929}%
diff --git a/auto/pictures_tikz/tikzFWJuNhU.pdf b/auto/pictures_tikz/tikzFWJuNhU.pdf
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diff --git a/auto/pictures_tikz/tikzFXVooJYAfif.md5 b/auto/pictures_tikz/tikzFXVooJYAfif.md5
index 381b4bfa1..1f1b921be 100644
--- a/auto/pictures_tikz/tikzFXVooJYAfif.md5
+++ b/auto/pictures_tikz/tikzFXVooJYAfif.md5
@@ -1 +1 @@
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diff --git a/auto/pictures_tikz/tikzFonctionEtDeriveOM.md5 b/auto/pictures_tikz/tikzFonctionEtDeriveOM.md5
index 98cc099b7..fe4e3480c 100644
--- a/auto/pictures_tikz/tikzFonctionEtDeriveOM.md5
+++ b/auto/pictures_tikz/tikzFonctionEtDeriveOM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {792A4F586A9858BD981AC2996A41F698}%
+\def \tikzexternallastkey {0CB5BC2E2D41D21B82D7DCB2435B2AA4}%
diff --git a/auto/pictures_tikz/tikzFonctionEtDeriveOM.pdf b/auto/pictures_tikz/tikzFonctionEtDeriveOM.pdf
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diff --git a/auto/pictures_tikz/tikzFonctionXtroisOM.md5 b/auto/pictures_tikz/tikzFonctionXtroisOM.md5
index edcea4c49..3f6303605 100644
--- a/auto/pictures_tikz/tikzFonctionXtroisOM.md5
+++ b/auto/pictures_tikz/tikzFonctionXtroisOM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {71C7C0C0F017C273D6C718354B3A7FF2}%
+\def \tikzexternallastkey {0B8C1BE31000F3606A32C42B23694923}%
diff --git a/auto/pictures_tikz/tikzFonctionXtroisOM.pdf b/auto/pictures_tikz/tikzFonctionXtroisOM.pdf
index 1bf130d507e96cb4c63232c6825c3801a94c216a..0f2229ed1d975ec0a2f14554233fcac3dfc2ad90 100644
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diff --git a/auto/pictures_tikz/tikzGBnUivi.md5 b/auto/pictures_tikz/tikzGBnUivi.md5
index a5fc2f817..fcfb3b7d5 100644
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+++ b/auto/pictures_tikz/tikzGBnUivi.md5
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index d3ba9da22..f0699d7b7 100644
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diff --git a/auto/pictures_tikz/tikzGMIooJvcCXg.md5 b/auto/pictures_tikz/tikzGMIooJvcCXg.md5
index 4827c0993..6cf987cd0 100644
--- a/auto/pictures_tikz/tikzGMIooJvcCXg.md5
+++ b/auto/pictures_tikz/tikzGMIooJvcCXg.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {2E8ECF54C677D0031E21E2AA158AB659}%
+\def \tikzexternallastkey {AAD81FF779673016A775989C0B5C7F62}%
diff --git a/auto/pictures_tikz/tikzGMIooJvcCXg.pdf b/auto/pictures_tikz/tikzGMIooJvcCXg.pdf
index 14dd654454fb8c350eeec8570ebd3d3a4fe68059..6598806b58534fb7e63877f4324a648265406bde 100644
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diff --git a/auto/pictures_tikz/tikzGMRNooCNBpIl.md5 b/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
index 3f5c13b0d..86dec2867 100644
--- a/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
+++ b/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D8B8D925942BA98161F2FF2F13A860D7}%
+\def \tikzexternallastkey {EBBB58E0F976B0E8ACCCAFC417AA4D36}%
diff --git a/auto/pictures_tikz/tikzGMRNooCNBpIl.pdf b/auto/pictures_tikz/tikzGMRNooCNBpIl.pdf
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diff --git a/auto/pictures_tikz/tikzGVDJooYzMxLW.md5 b/auto/pictures_tikz/tikzGVDJooYzMxLW.md5
index 18ae9d838..6210e3970 100644
--- a/auto/pictures_tikz/tikzGVDJooYzMxLW.md5
+++ b/auto/pictures_tikz/tikzGVDJooYzMxLW.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A75E882129B3FA5D0B74ACEB446BA35E}%
+\def \tikzexternallastkey {8CC2A69023968DACEF3BFDA0B947A164}%
diff --git a/auto/pictures_tikz/tikzGVDJooYzMxLW.pdf b/auto/pictures_tikz/tikzGVDJooYzMxLW.pdf
index 111ea83afb59f853313bc405cfecaf9e6c0480b2..be4ee14a7a0f295a669466f8374f30c705b7a649 100644
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diff --git a/auto/pictures_tikz/tikzGWOYooRxHKSm.md5 b/auto/pictures_tikz/tikzGWOYooRxHKSm.md5
index 2c8a82fcc..b2e60de06 100644
--- a/auto/pictures_tikz/tikzGWOYooRxHKSm.md5
+++ b/auto/pictures_tikz/tikzGWOYooRxHKSm.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6DB76C3DFE1327B569DAED52AD66AF02}%
+\def \tikzexternallastkey {15E89CEB0409F169A8F8DA9512463C60}%
diff --git a/auto/pictures_tikz/tikzGWOYooRxHKSm.pdf b/auto/pictures_tikz/tikzGWOYooRxHKSm.pdf
index 0a8dc406b70723d4cb3f79c51b69cfd1347e8e53..2a2a6083b70e27e9b819d06803b238716691b716 100644
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diff --git a/auto/pictures_tikz/tikzGYODoojTiGZSkJ.md5 b/auto/pictures_tikz/tikzGYODoojTiGZSkJ.md5
index f6624be99..8c7d9d23c 100644
--- a/auto/pictures_tikz/tikzGYODoojTiGZSkJ.md5
+++ b/auto/pictures_tikz/tikzGYODoojTiGZSkJ.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {F9AA2B0C9ACA466AB95CB3DD0AA92B98}%
+\def \tikzexternallastkey {025698556C916D74C13006435063E3DD}%
diff --git a/auto/pictures_tikz/tikzGYODoojTiGZSkJ.pdf b/auto/pictures_tikz/tikzGYODoojTiGZSkJ.pdf
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diff --git a/auto/pictures_tikz/tikzGrapheunsurunmoinsx.md5 b/auto/pictures_tikz/tikzGrapheunsurunmoinsx.md5
index 3bfb39d0b..b6b0c439f 100644
--- a/auto/pictures_tikz/tikzGrapheunsurunmoinsx.md5
+++ b/auto/pictures_tikz/tikzGrapheunsurunmoinsx.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7BA61F36C8806B9AFF7B816E03A6AF1C}%
+\def \tikzexternallastkey {59524DCD56144CF2A35CD1BB13405953}%
diff --git a/auto/pictures_tikz/tikzGrapheunsurunmoinsx.pdf b/auto/pictures_tikz/tikzGrapheunsurunmoinsx.pdf
index b3aae53bab3cc8659e1852e3a277f53e18b4b0ba..276cc969046e91530cd56a7882bfff4cda0a35ee 100644
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diff --git a/auto/pictures_tikz/tikzHFAYooOrfMAA.md5 b/auto/pictures_tikz/tikzHFAYooOrfMAA.md5
index 806e0226e..9a6ee6279 100644
--- a/auto/pictures_tikz/tikzHFAYooOrfMAA.md5
+++ b/auto/pictures_tikz/tikzHFAYooOrfMAA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B5D3C7310CB4AC552544EC77A662DA06}%
+\def \tikzexternallastkey {B468421C787A3981947F59725736A98E}%
diff --git a/auto/pictures_tikz/tikzHFAYooOrfMAA.pdf b/auto/pictures_tikz/tikzHFAYooOrfMAA.pdf
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diff --git a/auto/pictures_tikz/tikzHLJooGDZnqF.md5 b/auto/pictures_tikz/tikzHLJooGDZnqF.md5
index 121b3ea75..a1a832d66 100644
--- a/auto/pictures_tikz/tikzHLJooGDZnqF.md5
+++ b/auto/pictures_tikz/tikzHLJooGDZnqF.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0BB818A01FE573CAC9DB0C3E614ADA52}%
+\def \tikzexternallastkey {157779B17717D83A291221BEB7938595}%
diff --git a/auto/pictures_tikz/tikzHLJooGDZnqF.pdf b/auto/pictures_tikz/tikzHLJooGDZnqF.pdf
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diff --git a/auto/pictures_tikz/tikzHNxitLj.md5 b/auto/pictures_tikz/tikzHNxitLj.md5
index d41315166..78abcc657 100644
--- a/auto/pictures_tikz/tikzHNxitLj.md5
+++ b/auto/pictures_tikz/tikzHNxitLj.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {581462227314AC5A89A28BAF7EAD6F68}%
+\def \tikzexternallastkey {88003510F18B8FEFAD033D1BC06A24BC}%
diff --git a/auto/pictures_tikz/tikzHNxitLj.pdf b/auto/pictures_tikz/tikzHNxitLj.pdf
index 4e11538c96332443f20e0021877da89aaba0eb52..44c395147de3a4de7f08974650dd0b691fa650cd 100644
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diff --git a/auto/pictures_tikz/tikzHasseAGdfdy.md5 b/auto/pictures_tikz/tikzHasseAGdfdy.md5
index 76244c447..5b9775893 100644
--- a/auto/pictures_tikz/tikzHasseAGdfdy.md5
+++ b/auto/pictures_tikz/tikzHasseAGdfdy.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8BDD9D3E2ACEA79718F04BA6756A942A}%
+\def \tikzexternallastkey {77CF693C20A6AD28CAA80B03CA705E42}%
diff --git a/auto/pictures_tikz/tikzHasseAGdfdy.pdf b/auto/pictures_tikz/tikzHasseAGdfdy.pdf
index bed16bb22b9d334f04f64cd5f1ec664b4f81487f..e18fff8081be289ae597c2d5f27ff47bb3adc0f5 100644
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diff --git a/auto/pictures_tikz/tikzIOCTooePeHGCXH.md5 b/auto/pictures_tikz/tikzIOCTooePeHGCXH.md5
index f513537c4..6b3b711d4 100644
--- a/auto/pictures_tikz/tikzIOCTooePeHGCXH.md5
+++ b/auto/pictures_tikz/tikzIOCTooePeHGCXH.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {3528B3849B337BEA708C297BBB73E40E}%
+\def \tikzexternallastkey {65EF3FE573FD73B38258881306915A50}%
diff --git a/auto/pictures_tikz/tikzIOCTooePeHGCXH.pdf b/auto/pictures_tikz/tikzIOCTooePeHGCXH.pdf
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diff --git a/auto/pictures_tikz/tikzIWuPxFc.md5 b/auto/pictures_tikz/tikzIWuPxFc.md5
index 0203ad33f..8404c5c60 100644
--- a/auto/pictures_tikz/tikzIWuPxFc.md5
+++ b/auto/pictures_tikz/tikzIWuPxFc.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8E45D43A49F94E6CB6CE1E694C375A89}%
+\def \tikzexternallastkey {B9940B041BAC221CC9DA112C066B6D82}%
diff --git a/auto/pictures_tikz/tikzIWuPxFc.pdf b/auto/pictures_tikz/tikzIWuPxFc.pdf
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diff --git a/auto/pictures_tikz/tikzIYAvSvI.md5 b/auto/pictures_tikz/tikzIYAvSvI.md5
index 74d30cf17..787b86371 100644
--- a/auto/pictures_tikz/tikzIYAvSvI.md5
+++ b/auto/pictures_tikz/tikzIYAvSvI.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7D9570945F7551CAB3432AF8134C5357}%
+\def \tikzexternallastkey {BFD9A5835133003636B8063977FF4C04}%
diff --git a/auto/pictures_tikz/tikzIYAvSvI.pdf b/auto/pictures_tikz/tikzIYAvSvI.pdf
index 11ee866adf4c956072a2751a9f7f395e540cf7de..26946af997c792bbdc11837f8741064469f737e5 100644
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index 806e0226e..9a6ee6279 100644
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@@ -1 +1 @@
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+\def \tikzexternallastkey {CA228E5822D8CE51008A65466AA874D3}%
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index 32819a72d..4cae228f0 100644
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+++ b/auto/pictures_tikz/tikzIntRectangle.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {7071B6B63EAA322A4F46FA4B4BAE0896}%
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index 918566458..f072cabe6 100644
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diff --git a/auto/pictures_tikz/tikzIntegraleSimple.md5 b/auto/pictures_tikz/tikzIntegraleSimple.md5
index b7fae6095..f7e82f031 100644
--- a/auto/pictures_tikz/tikzIntegraleSimple.md5
+++ b/auto/pictures_tikz/tikzIntegraleSimple.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E5AB361436E342712CF526A50E26B271}%
+\def \tikzexternallastkey {E178B14D73A0157D3E8470657EA7DAC8}%
diff --git a/auto/pictures_tikz/tikzIntegraleSimple.pdf b/auto/pictures_tikz/tikzIntegraleSimple.pdf
index d4fd01a1385eea93708dd694c04fab82a52b1d78..34297812b1edba25322022aa9b8c8972eade1f9d 100644
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diff --git a/auto/pictures_tikz/tikzIntervalleUn.md5 b/auto/pictures_tikz/tikzIntervalleUn.md5
index d8815ffaf..fa0bc79a5 100644
--- a/auto/pictures_tikz/tikzIntervalleUn.md5
+++ b/auto/pictures_tikz/tikzIntervalleUn.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {FD2410086D36F840A2E7DECA23659F99}%
+\def \tikzexternallastkey {57DEC5A9366A6AE7AAAFDAF06F45E79E}%
diff --git a/auto/pictures_tikz/tikzIntervalleUn.pdf b/auto/pictures_tikz/tikzIntervalleUn.pdf
index b2b9d5ef3803d8771f5c4bad4ac6aa6a7814b036..a7e201e31f7783bb5062260df913951676ccaece 100644
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index 1b4d44b73..6e438cca1 100644
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diff --git a/auto/pictures_tikz/tikzJJAooWpimYW.md5 b/auto/pictures_tikz/tikzJJAooWpimYW.md5
index 0f776a2f6..365798c4b 100644
--- a/auto/pictures_tikz/tikzJJAooWpimYW.md5
+++ b/auto/pictures_tikz/tikzJJAooWpimYW.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {5843CBA01231E15DB19356BCF273E16F}%
+\def \tikzexternallastkey {BE0F028108E7817EAAE02ADCEC3795B0}%
diff --git a/auto/pictures_tikz/tikzJJAooWpimYW.pdf b/auto/pictures_tikz/tikzJJAooWpimYW.pdf
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diff --git a/auto/pictures_tikz/tikzJSLooFJWXtB.md5 b/auto/pictures_tikz/tikzJSLooFJWXtB.md5
index 4b28aa456..2da9aa266 100644
--- a/auto/pictures_tikz/tikzJSLooFJWXtB.md5
+++ b/auto/pictures_tikz/tikzJSLooFJWXtB.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CCA233E2A8EC082CCAAD6007C88285D1}%
+\def \tikzexternallastkey {BCF92AD69074D949B5D1F71324ABCE8D}%
diff --git a/auto/pictures_tikz/tikzJSLooFJWXtB.pdf b/auto/pictures_tikz/tikzJSLooFJWXtB.pdf
index 3314ac1d77d26b9fb7044f0c4e8dfe890b3a73b2..0e7d7e108c831b1b51c8a3796dc975607c1e4c38 100644
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diff --git a/auto/pictures_tikz/tikzJWINooSfKCeA.md5 b/auto/pictures_tikz/tikzJWINooSfKCeA.md5
index 67dad41da..635b76005 100644
--- a/auto/pictures_tikz/tikzJWINooSfKCeA.md5
+++ b/auto/pictures_tikz/tikzJWINooSfKCeA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {91616F6C3276CF2641936ABE53B329AC}%
+\def \tikzexternallastkey {666AA64833B2BB4DB635B792CEA4AC21}%
diff --git a/auto/pictures_tikz/tikzJWINooSfKCeA.pdf b/auto/pictures_tikz/tikzJWINooSfKCeA.pdf
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diff --git a/auto/pictures_tikz/tikzKGQXooZFNVnW.md5 b/auto/pictures_tikz/tikzKGQXooZFNVnW.md5
index 8d014ddea..103c94577 100644
--- a/auto/pictures_tikz/tikzKGQXooZFNVnW.md5
+++ b/auto/pictures_tikz/tikzKGQXooZFNVnW.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0B901939ED4148D4DEA1D7469B0066E5}%
+\def \tikzexternallastkey {4A361B5FABA09C8473C023959F119644}%
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diff --git a/auto/pictures_tikz/tikzKKJAooubQzgBgP.md5 b/auto/pictures_tikz/tikzKKJAooubQzgBgP.md5
index 5f9a66a51..9e76e3ffb 100644
--- a/auto/pictures_tikz/tikzKKJAooubQzgBgP.md5
+++ b/auto/pictures_tikz/tikzKKJAooubQzgBgP.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E526CED2508323DA532F64F78CA562E1}%
+\def \tikzexternallastkey {69182FA84C8FAF3846269F657F9FF722}%
diff --git a/auto/pictures_tikz/tikzKKJAooubQzgBgP.pdf b/auto/pictures_tikz/tikzKKJAooubQzgBgP.pdf
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diff --git a/auto/pictures_tikz/tikzKKLooMbjxdI.md5 b/auto/pictures_tikz/tikzKKLooMbjxdI.md5
index b7fae6095..f7e82f031 100644
--- a/auto/pictures_tikz/tikzKKLooMbjxdI.md5
+++ b/auto/pictures_tikz/tikzKKLooMbjxdI.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E5AB361436E342712CF526A50E26B271}%
+\def \tikzexternallastkey {E178B14D73A0157D3E8470657EA7DAC8}%
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diff --git a/auto/pictures_tikz/tikzKKRooHseDzC.md5 b/auto/pictures_tikz/tikzKKRooHseDzC.md5
index 05c6b4273..b527b4007 100644
--- a/auto/pictures_tikz/tikzKKRooHseDzC.md5
+++ b/auto/pictures_tikz/tikzKKRooHseDzC.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EF67D1865FCDCF2B1AB0F1B9C57F2D83}%
+\def \tikzexternallastkey {06BEE2641B2385B2D5B246892C6642F1}%
diff --git a/auto/pictures_tikz/tikzKKRooHseDzC.pdf b/auto/pictures_tikz/tikzKKRooHseDzC.pdf
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diff --git a/auto/pictures_tikz/tikzLAfWmaN.md5 b/auto/pictures_tikz/tikzLAfWmaN.md5
index a8d9584b1..480442334 100644
--- a/auto/pictures_tikz/tikzLAfWmaN.md5
+++ b/auto/pictures_tikz/tikzLAfWmaN.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {3AB839236FC8AA1D0AB52F4ED4392A59}%
+\def \tikzexternallastkey {E5CC95BE797EC80DC3AD7EDBAD7F325F}%
diff --git a/auto/pictures_tikz/tikzLAfWmaN.pdf b/auto/pictures_tikz/tikzLAfWmaN.pdf
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diff --git a/auto/pictures_tikz/tikzLBGooAdteCt.md5 b/auto/pictures_tikz/tikzLBGooAdteCt.md5
index 0c10bda80..68d508d3c 100644
--- a/auto/pictures_tikz/tikzLBGooAdteCt.md5
+++ b/auto/pictures_tikz/tikzLBGooAdteCt.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6C394FC61BD35763203BACF6BD3BC136}%
+\def \tikzexternallastkey {A7F1C680549E8DE72FE8B28D54FDEFBC}%
diff --git a/auto/pictures_tikz/tikzLBGooAdteCt.pdf b/auto/pictures_tikz/tikzLBGooAdteCt.pdf
index c200d1f68a6b2ca6b193f391f5d07947a56de4c5..a1e277177e70682866d1e2cd029dafe19dfd3d29 100644
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diff --git a/auto/pictures_tikz/tikzLEJNDxI.md5 b/auto/pictures_tikz/tikzLEJNDxI.md5
index e44bf1d03..29805a09a 100644
--- a/auto/pictures_tikz/tikzLEJNDxI.md5
+++ b/auto/pictures_tikz/tikzLEJNDxI.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {138ADC6331ADB33D3BDBA1BF140413B1}%
+\def \tikzexternallastkey {9CBFF8C99E623ECFAEE9D46CF6F7ED70}%
diff --git a/auto/pictures_tikz/tikzLEJNDxI.pdf b/auto/pictures_tikz/tikzLEJNDxI.pdf
index b124552a30aab8f7e20c6e76af41c013cbaa616d..18d50bf27f78a07255305700e0cf9721f30ab207 100644
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diff --git a/auto/pictures_tikz/tikzLLVMooWOkvAB.md5 b/auto/pictures_tikz/tikzLLVMooWOkvAB.md5
index bb043406f..83c1cab61 100644
--- a/auto/pictures_tikz/tikzLLVMooWOkvAB.md5
+++ b/auto/pictures_tikz/tikzLLVMooWOkvAB.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {898922BAC49866174A0DCD68E0782ACF}%
+\def \tikzexternallastkey {DCF40E8C3D5BE0C6E0C68077A24D5C6F}%
diff --git a/auto/pictures_tikz/tikzLLVMooWOkvAB.pdf b/auto/pictures_tikz/tikzLLVMooWOkvAB.pdf
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diff --git a/auto/pictures_tikz/tikzLMHMooCscXNNdU.md5 b/auto/pictures_tikz/tikzLMHMooCscXNNdU.md5
index 41e575e63..61eadf29d 100644
--- a/auto/pictures_tikz/tikzLMHMooCscXNNdU.md5
+++ b/auto/pictures_tikz/tikzLMHMooCscXNNdU.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {F5B3984C7E95A4F7AC5CF74634CE26C0}%
+\def \tikzexternallastkey {ADE8A835A1804E62BBF4A015A9DD93A5}%
diff --git a/auto/pictures_tikz/tikzLMHMooCscXNNdU.pdf b/auto/pictures_tikz/tikzLMHMooCscXNNdU.pdf
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diff --git a/auto/pictures_tikz/tikzLaurin.md5 b/auto/pictures_tikz/tikzLaurin.md5
index 4236e950e..e03ca1cef 100644
--- a/auto/pictures_tikz/tikzLaurin.md5
+++ b/auto/pictures_tikz/tikzLaurin.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {1783A868820F4BFAF9F4324483D9AC24}%
+\def \tikzexternallastkey {F9DD8ECEE69CD7C345D57ABE9EB0A33F}%
diff --git a/auto/pictures_tikz/tikzLaurin.pdf b/auto/pictures_tikz/tikzLaurin.pdf
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index de32250ab..a0dbb6dbc 100644
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+++ b/auto/pictures_tikz/tikzMCQueGF.md5
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+\def \tikzexternallastkey {826D7FA50FF63A554F4F048092478BEE}%
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diff --git a/auto/pictures_tikz/tikzMNICGhR.md5 b/auto/pictures_tikz/tikzMNICGhR.md5
index ee028f6c7..0821e118c 100644
--- a/auto/pictures_tikz/tikzMNICGhR.md5
+++ b/auto/pictures_tikz/tikzMNICGhR.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {4888AA984F0A313B72DBA196034179DF}%
+\def \tikzexternallastkey {8BA1D154B2DC40C62E4EAF8730D74F38}%
diff --git a/auto/pictures_tikz/tikzMNICGhR.pdf b/auto/pictures_tikz/tikzMNICGhR.pdf
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diff --git a/auto/pictures_tikz/tikzMantisse.md5 b/auto/pictures_tikz/tikzMantisse.md5
index 9ebb86bc7..cd6ba6207 100644
--- a/auto/pictures_tikz/tikzMantisse.md5
+++ b/auto/pictures_tikz/tikzMantisse.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {84F2E6436C6BB5D41E04BF2FB49F53DC}%
+\def \tikzexternallastkey {7BB73FA56F934C4F548E22C4D547B6A0}%
diff --git a/auto/pictures_tikz/tikzMantisse.pdf b/auto/pictures_tikz/tikzMantisse.pdf
index 798532fd32be74e3e81cdaf598ab96c3a0112988..3cec93de4862cf8b5fe4638ce1fcc2d737612da7 100644
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diff --git a/auto/pictures_tikz/tikzMaxVraissLp.md5 b/auto/pictures_tikz/tikzMaxVraissLp.md5
index e1aff0cea..46987331c 100644
--- a/auto/pictures_tikz/tikzMaxVraissLp.md5
+++ b/auto/pictures_tikz/tikzMaxVraissLp.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {DDCA726E8636198BFA5D805BC9E9BA29}%
+\def \tikzexternallastkey {3B3EDD733332AB758F259EDCFC2EB79E}%
diff --git a/auto/pictures_tikz/tikzMaxVraissLp.pdf b/auto/pictures_tikz/tikzMaxVraissLp.pdf
index 2a8e75eeb012773a128af0a227e206a263d7c8a8..b5819fd4f7fc13335134b3dd78501db52a6be5aa 100644
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diff --git a/auto/pictures_tikz/tikzMethodeChemin.md5 b/auto/pictures_tikz/tikzMethodeChemin.md5
index cd450833d..11e01b862 100644
--- a/auto/pictures_tikz/tikzMethodeChemin.md5
+++ b/auto/pictures_tikz/tikzMethodeChemin.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {3A373E4CE60D3678A36C4E7347ED7696}%
+\def \tikzexternallastkey {D3ECC297399B6B21948B29694A6E5F35}%
diff --git a/auto/pictures_tikz/tikzMethodeChemin.pdf b/auto/pictures_tikz/tikzMethodeChemin.pdf
index 5d4688424ce0dd21105d5620988e557e6da63b8d..382a9af4977b52ed105bc7f43d6ce09a63f67af6 100644
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diff --git a/auto/pictures_tikz/tikzMethodeNewton.md5 b/auto/pictures_tikz/tikzMethodeNewton.md5
index 661b71183..770bffc84 100644
--- a/auto/pictures_tikz/tikzMethodeNewton.md5
+++ b/auto/pictures_tikz/tikzMethodeNewton.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {37CC86A0ED8D01012C37CF48DF99243A}%
+\def \tikzexternallastkey {63DF905292EC0B44053ECDCB51F67A89}%
diff --git a/auto/pictures_tikz/tikzMethodeNewton.pdf b/auto/pictures_tikz/tikzMethodeNewton.pdf
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index 0a952eef7..e5f769bea 100644
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diff --git a/auto/pictures_tikz/tikzNOCGooYRHLCn.md5 b/auto/pictures_tikz/tikzNOCGooYRHLCn.md5
index 9dc57a749..808537c54 100644
--- a/auto/pictures_tikz/tikzNOCGooYRHLCn.md5
+++ b/auto/pictures_tikz/tikzNOCGooYRHLCn.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {172F20B3822B043D0ABEB816E9DCC81E}%
+\def \tikzexternallastkey {3BE3BA2D9FA9A50544B690672DA578D0}%
diff --git a/auto/pictures_tikz/tikzNOCGooYRHLCn.pdf b/auto/pictures_tikz/tikzNOCGooYRHLCn.pdf
index 4056139cdc293eaa22e3469672d9b21c0276fe81..13192c1d3cda629d842b771689a13ad50bd3c18d 100644
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diff --git a/auto/pictures_tikz/tikzNWDooOObSHB.md5 b/auto/pictures_tikz/tikzNWDooOObSHB.md5
index 022d0dd9c..3b66ff2e3 100644
--- a/auto/pictures_tikz/tikzNWDooOObSHB.md5
+++ b/auto/pictures_tikz/tikzNWDooOObSHB.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {FBAB1AA9F8B459844EECE817AE2E926E}%
+\def \tikzexternallastkey {287D4F42AD84D3C950000D0494F486B3}%
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diff --git a/auto/pictures_tikz/tikzNiveauHyperbole.md5 b/auto/pictures_tikz/tikzNiveauHyperbole.md5
index 21152876a..bd30b5cc5 100644
--- a/auto/pictures_tikz/tikzNiveauHyperbole.md5
+++ b/auto/pictures_tikz/tikzNiveauHyperbole.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CA90AAF8AB80E321AE73315278C9A3B5}%
+\def \tikzexternallastkey {7833820658FC4257E1271B33091C44F4}%
diff --git a/auto/pictures_tikz/tikzNiveauHyperbole.pdf b/auto/pictures_tikz/tikzNiveauHyperbole.pdf
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diff --git a/auto/pictures_tikz/tikzNiveauHyperboleDeux.md5 b/auto/pictures_tikz/tikzNiveauHyperboleDeux.md5
index 8d014ddea..103c94577 100644
--- a/auto/pictures_tikz/tikzNiveauHyperboleDeux.md5
+++ b/auto/pictures_tikz/tikzNiveauHyperboleDeux.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0B901939ED4148D4DEA1D7469B0066E5}%
+\def \tikzexternallastkey {4A361B5FABA09C8473C023959F119644}%
diff --git a/auto/pictures_tikz/tikzNiveauHyperboleDeux.pdf b/auto/pictures_tikz/tikzNiveauHyperboleDeux.pdf
index e10a888fc08979e5ee838f6ab82a240f0df7d459..976071217920c2a973c790fc12f62b913aa90a39 100644
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diff --git a/auto/pictures_tikz/tikzOQTEoodIwAPfZE.md5 b/auto/pictures_tikz/tikzOQTEoodIwAPfZE.md5
index 131259d24..bdd7d37d1 100644
--- a/auto/pictures_tikz/tikzOQTEoodIwAPfZE.md5
+++ b/auto/pictures_tikz/tikzOQTEoodIwAPfZE.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E50932C23ACF25EC634B0C897FB9EF6A}%
+\def \tikzexternallastkey {03832FA75779C09FEDB2FFC82C3743B8}%
diff --git a/auto/pictures_tikz/tikzOQTEoodIwAPfZE.pdf b/auto/pictures_tikz/tikzOQTEoodIwAPfZE.pdf
index 183dd264c5aec72bfcce81bbf0b0f9081525d3e9..8a076a44127b4f7c3f1c01bafc63baa285648308 100644
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diff --git a/auto/pictures_tikz/tikzOsculateur.md5 b/auto/pictures_tikz/tikzOsculateur.md5
index ea0e51f5e..b0dbb30a2 100644
--- a/auto/pictures_tikz/tikzOsculateur.md5
+++ b/auto/pictures_tikz/tikzOsculateur.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {03C94579D3531DAF812690865BEE9BA2}%
+\def \tikzexternallastkey {0ED926580928D8448AAF4E7553F6A705}%
diff --git a/auto/pictures_tikz/tikzOsculateur.pdf b/auto/pictures_tikz/tikzOsculateur.pdf
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diff --git a/auto/pictures_tikz/tikzPHTVjfk.md5 b/auto/pictures_tikz/tikzPHTVjfk.md5
index d45afea02..0b87b93f2 100644
--- a/auto/pictures_tikz/tikzPHTVjfk.md5
+++ b/auto/pictures_tikz/tikzPHTVjfk.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {06826F1A235B448B59BF37DFCF7012AA}%
+\def \tikzexternallastkey {666FADDBA0CB9FD862AFFF72A2815F8D}%
diff --git a/auto/pictures_tikz/tikzPHTVjfk.pdf b/auto/pictures_tikz/tikzPHTVjfk.pdf
index cd534f048901a978f30ae560c2384d01485e2bec..e3dc60afae450e6f9db62e3e06e173f800c5f002 100644
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diff --git a/auto/pictures_tikz/tikzPLTWoocPNeiZir.md5 b/auto/pictures_tikz/tikzPLTWoocPNeiZir.md5
index 2d1c5309c..a6da26a43 100644
--- a/auto/pictures_tikz/tikzPLTWoocPNeiZir.md5
+++ b/auto/pictures_tikz/tikzPLTWoocPNeiZir.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EA2AA486941B36F0720FAD2BE5897CC4}%
+\def \tikzexternallastkey {BB5CE726933BA71687F19A5AF55E2D27}%
diff --git a/auto/pictures_tikz/tikzPLTWoocPNeiZir.pdf b/auto/pictures_tikz/tikzPLTWoocPNeiZir.pdf
index 8aa32352f45fc036956dfae735b425aceeaf2203..377feb328d1bfc726b5abfed69384de3890ee063 100644
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diff --git a/auto/pictures_tikz/tikzPONXooXYjEot.md5 b/auto/pictures_tikz/tikzPONXooXYjEot.md5
index 61f1d7a8b..25bd3acf7 100644
--- a/auto/pictures_tikz/tikzPONXooXYjEot.md5
+++ b/auto/pictures_tikz/tikzPONXooXYjEot.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {25E5D10E4BBF0131A11000A127B1475D}%
+\def \tikzexternallastkey {E3F4441BCBA05FA6DE2E679FE58F35A5}%
diff --git a/auto/pictures_tikz/tikzPONXooXYjEot.pdf b/auto/pictures_tikz/tikzPONXooXYjEot.pdf
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diff --git a/auto/pictures_tikz/tikzPVJooJDyNAg.md5 b/auto/pictures_tikz/tikzPVJooJDyNAg.md5
index dc926fd9c..b1aed2661 100644
--- a/auto/pictures_tikz/tikzPVJooJDyNAg.md5
+++ b/auto/pictures_tikz/tikzPVJooJDyNAg.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {DAB0B16788BC1540192C57C5C6FC7AB7}%
+\def \tikzexternallastkey {949EAFF49DD237148F2DF6F7F47C601E}%
diff --git a/auto/pictures_tikz/tikzPVJooJDyNAg.pdf b/auto/pictures_tikz/tikzPVJooJDyNAg.pdf
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diff --git a/auto/pictures_tikz/tikzParallelogrammeOM.md5 b/auto/pictures_tikz/tikzParallelogrammeOM.md5
index 9c18f5316..8bc9d9622 100644
--- a/auto/pictures_tikz/tikzParallelogrammeOM.md5
+++ b/auto/pictures_tikz/tikzParallelogrammeOM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CA8B9F5A34D246CF08B92CEB51CC41C4}%
+\def \tikzexternallastkey {1BCFE4B98F363C2A0F168EA437F34839}%
diff --git a/auto/pictures_tikz/tikzParallelogrammeOM.pdf b/auto/pictures_tikz/tikzParallelogrammeOM.pdf
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diff --git a/auto/pictures_tikz/tikzParamTangente.md5 b/auto/pictures_tikz/tikzParamTangente.md5
index 7194a8d67..36889cfb5 100644
--- a/auto/pictures_tikz/tikzParamTangente.md5
+++ b/auto/pictures_tikz/tikzParamTangente.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {CC87D06EEE65EA8B002F4682A62155D9}%
+\def \tikzexternallastkey {AE057392A3444253BBC705E1D770942F}%
diff --git a/auto/pictures_tikz/tikzParamTangente.pdf b/auto/pictures_tikz/tikzParamTangente.pdf
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diff --git a/auto/pictures_tikz/tikzPartieEntiere.md5 b/auto/pictures_tikz/tikzPartieEntiere.md5
index 76f125b45..eeeb0e4fe 100644
--- a/auto/pictures_tikz/tikzPartieEntiere.md5
+++ b/auto/pictures_tikz/tikzPartieEntiere.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7412102A1C500C0C25A0E26FD8198E83}%
+\def \tikzexternallastkey {93D8E93D4F1AF9273667A72E47B2D020}%
diff --git a/auto/pictures_tikz/tikzPartieEntiere.pdf b/auto/pictures_tikz/tikzPartieEntiere.pdf
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diff --git a/auto/pictures_tikz/tikzPolirettangolo.md5 b/auto/pictures_tikz/tikzPolirettangolo.md5
index c4515b007..43597e4ec 100644
--- a/auto/pictures_tikz/tikzPolirettangolo.md5
+++ b/auto/pictures_tikz/tikzPolirettangolo.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EEC20E90EE9F7BF486AD5AD3CF93D071}%
+\def \tikzexternallastkey {F518B4DC3D807832581E269038798A9D}%
diff --git a/auto/pictures_tikz/tikzPolirettangolo.pdf b/auto/pictures_tikz/tikzPolirettangolo.pdf
index c1e1b26e53e32f81ba590e44a804d0797f2e982a..e7bb5dab40273ebcb7ddf071672cec993ac2149a 100644
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diff --git a/auto/pictures_tikz/tikzProjPoly.md5 b/auto/pictures_tikz/tikzProjPoly.md5
index 87eadaf74..a70be43ef 100644
--- a/auto/pictures_tikz/tikzProjPoly.md5
+++ b/auto/pictures_tikz/tikzProjPoly.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {ED375D4E63ADB328065DAEC2A5DA2027}%
+\def \tikzexternallastkey {25202CA4EFD8B21FE10F2547E1DDA7BC}%
diff --git a/auto/pictures_tikz/tikzProjPoly.pdf b/auto/pictures_tikz/tikzProjPoly.pdf
index 86cd848ccc82559bffa90dcbfcdcf09e4754742e..19d4b045c77fe9dfaa304d966fa95a7566f87dc1 100644
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diff --git a/auto/pictures_tikz/tikzProjectionScalaire.md5 b/auto/pictures_tikz/tikzProjectionScalaire.md5
index eb607adce..76e6451b3 100644
--- a/auto/pictures_tikz/tikzProjectionScalaire.md5
+++ b/auto/pictures_tikz/tikzProjectionScalaire.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {98D77D2FE7E711BF3DA403E7F7CDD513}%
+\def \tikzexternallastkey {A8B2BC20883E52BD172FD2921B2435E3}%
diff --git a/auto/pictures_tikz/tikzProjectionScalaire.pdf b/auto/pictures_tikz/tikzProjectionScalaire.pdf
index 7c60bec392fc60e92bfdf1ea62c593e9aa1640f9..cc555ee987b42af7b52234ef6a6511dc32a35c99 100644
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diff --git a/auto/pictures_tikz/tikzQCb.md5 b/auto/pictures_tikz/tikzQCb.md5
index 227250062..3334d4482 100644
--- a/auto/pictures_tikz/tikzQCb.md5
+++ b/auto/pictures_tikz/tikzQCb.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {ECEEBADEA23BD3BCC75100140AD3E76E}%
+\def \tikzexternallastkey {2EE65816E0919CA86890E6FF8EDDA7E7}%
diff --git a/auto/pictures_tikz/tikzQCb.pdf b/auto/pictures_tikz/tikzQCb.pdf
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--- a/auto/pictures_tikz/tikzQIZooQNQSJj.md5
+++ b/auto/pictures_tikz/tikzQIZooQNQSJj.md5
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diff --git a/auto/pictures_tikz/tikzQMWKooRRulrgcH.md5 b/auto/pictures_tikz/tikzQMWKooRRulrgcH.md5
index c6bde59c7..4e5121f24 100644
--- a/auto/pictures_tikz/tikzQMWKooRRulrgcH.md5
+++ b/auto/pictures_tikz/tikzQMWKooRRulrgcH.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {E2795A33EB11BDEB9C15270877498DAA}%
+\def \tikzexternallastkey {0B0E03D2D1C76BAA30C6004FA58AD87B}%
diff --git a/auto/pictures_tikz/tikzQMWKooRRulrgcH.pdf b/auto/pictures_tikz/tikzQMWKooRRulrgcH.pdf
index 313f50f3653b2bf16066272d8bb7c3e157ab8de7..a1ba282b761325aa349ab14900d54b6849ceb31d 100644
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diff --git a/auto/pictures_tikz/tikzQPcdHwP.md5 b/auto/pictures_tikz/tikzQPcdHwP.md5
index e0676ba60..bb60d40f8 100644
--- a/auto/pictures_tikz/tikzQPcdHwP.md5
+++ b/auto/pictures_tikz/tikzQPcdHwP.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {07E854C091C997AA5DB05FCE854A10ED}%
+\def \tikzexternallastkey {76AFEC15ACB80535D13BF79BF2DC8AC2}%
diff --git a/auto/pictures_tikz/tikzQPcdHwP.pdf b/auto/pictures_tikz/tikzQPcdHwP.pdf
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diff --git a/auto/pictures_tikz/tikzQSKDooujUbDCsu.md5 b/auto/pictures_tikz/tikzQSKDooujUbDCsu.md5
index 24955daee..8fd3bb1af 100644
--- a/auto/pictures_tikz/tikzQSKDooujUbDCsu.md5
+++ b/auto/pictures_tikz/tikzQSKDooujUbDCsu.md5
@@ -1 +1 @@
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index 863d3a104..ce5cf0457 100644
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+++ b/auto/pictures_tikz/tikzQXyVaKD.md5
@@ -1 +1 @@
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diff --git a/auto/pictures_tikz/tikzQuelCote.md5 b/auto/pictures_tikz/tikzQuelCote.md5
index 417c13d06..2b8a83b5e 100644
--- a/auto/pictures_tikz/tikzQuelCote.md5
+++ b/auto/pictures_tikz/tikzQuelCote.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D2178A60DB02D5C52268DE37B55CB82A}%
+\def \tikzexternallastkey {A4AF236089882D43D3BEDC30198A823B}%
diff --git a/auto/pictures_tikz/tikzQuelCote.pdf b/auto/pictures_tikz/tikzQuelCote.pdf
index 6d0f399d8752cfa76a066e14652f4ad59a5d8173..145c58170cb9c505e994ca7993c09aad24440047 100644
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diff --git a/auto/pictures_tikz/tikzRGjjpwF.md5 b/auto/pictures_tikz/tikzRGjjpwF.md5
index c353d3814..b805d2ded 100644
--- a/auto/pictures_tikz/tikzRGjjpwF.md5
+++ b/auto/pictures_tikz/tikzRGjjpwF.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {819B7CF0EFD7955868BD2745891F2A68}%
+\def \tikzexternallastkey {82EFA673EEED1C71C3BF0E7E847938A5}%
diff --git a/auto/pictures_tikz/tikzRGjjpwF.pdf b/auto/pictures_tikz/tikzRGjjpwF.pdf
index e887f509729f078dfe8277826864781280efa1d6..6e591ba784c682464409d28a7c12bfb13e6b47f5 100644
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diff --git a/auto/pictures_tikz/tikzRLuqsrr.md5 b/auto/pictures_tikz/tikzRLuqsrr.md5
index 32bf38142..386330541 100644
--- a/auto/pictures_tikz/tikzRLuqsrr.md5
+++ b/auto/pictures_tikz/tikzRLuqsrr.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {61F17DED83F1494E984A0DEBCFFB475F}%
+\def \tikzexternallastkey {0BEBFE3092DEB38195D6BBB3CEA89811}%
diff --git a/auto/pictures_tikz/tikzRLuqsrr.pdf b/auto/pictures_tikz/tikzRLuqsrr.pdf
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diff --git a/auto/pictures_tikz/tikzROAOooPgUZIt.md5 b/auto/pictures_tikz/tikzROAOooPgUZIt.md5
index 815e53a53..47c438964 100644
--- a/auto/pictures_tikz/tikzROAOooPgUZIt.md5
+++ b/auto/pictures_tikz/tikzROAOooPgUZIt.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C806DEBD38C28D388779CA7C04B2014C}%
+\def \tikzexternallastkey {954F48525994546FDA6F3E1A6B989C35}%
diff --git a/auto/pictures_tikz/tikzROAOooPgUZIt.pdf b/auto/pictures_tikz/tikzROAOooPgUZIt.pdf
index eba064aa3415c7fcc5766ba0df7f63954e7b6030..4af47f53e22332a6e779769dc6471960c64c9a19 100644
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diff --git a/auto/pictures_tikz/tikzRPNooQXxpZZ.md5 b/auto/pictures_tikz/tikzRPNooQXxpZZ.md5
index 54a0440fc..83234eacc 100644
--- a/auto/pictures_tikz/tikzRPNooQXxpZZ.md5
+++ b/auto/pictures_tikz/tikzRPNooQXxpZZ.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EBA8451174AE4814ADA74FCDECB07E56}%
+\def \tikzexternallastkey {FFCCA2B5E2EB009581E41CF96AF0C314}%
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index 7993e4971..507906508 100644
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diff --git a/auto/pictures_tikz/tikzSBTooEasQsT.md5 b/auto/pictures_tikz/tikzSBTooEasQsT.md5
index 59a68c36c..2ef413ed8 100644
--- a/auto/pictures_tikz/tikzSBTooEasQsT.md5
+++ b/auto/pictures_tikz/tikzSBTooEasQsT.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {5BE106EE6DA175DBD7DEBA88AFCBF332}%
+\def \tikzexternallastkey {D4FF70EB14D5C735CB83E3460D1F1738}%
diff --git a/auto/pictures_tikz/tikzSBTooEasQsT.pdf b/auto/pictures_tikz/tikzSBTooEasQsT.pdf
index 8a6fb8c0e54595fbbc801006bb3e41442eba970d..abbc1a8e3f2ac34fca0cdf092b3c7cbfd2a61f39 100644
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diff --git a/auto/pictures_tikz/tikzSFdgHdO.md5 b/auto/pictures_tikz/tikzSFdgHdO.md5
index 3ead1f8f0..863800549 100644
--- a/auto/pictures_tikz/tikzSFdgHdO.md5
+++ b/auto/pictures_tikz/tikzSFdgHdO.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D2A1590AB65EE935D84AF88AD996A7CB}%
+\def \tikzexternallastkey {C7D4FF88958F6D5F6CCB38D6D93CFFDD}%
diff --git a/auto/pictures_tikz/tikzSFdgHdO.pdf b/auto/pictures_tikz/tikzSFdgHdO.pdf
index d0eab7b453f88d81afa65912b23d44f65cce656a..c283bb9b35afe1394066fbad069a8f8f347afb45 100644
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diff --git a/auto/pictures_tikz/tikzSJAWooRDGzIkrj.md5 b/auto/pictures_tikz/tikzSJAWooRDGzIkrj.md5
index 9cc1599f2..1cfec12c4 100644
--- a/auto/pictures_tikz/tikzSJAWooRDGzIkrj.md5
+++ b/auto/pictures_tikz/tikzSJAWooRDGzIkrj.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {AE48546DCEC02EB705E36986BCF2398C}%
+\def \tikzexternallastkey {85CA1176FF692FBA838A3DD6C078A27A}%
diff --git a/auto/pictures_tikz/tikzSJAWooRDGzIkrj.pdf b/auto/pictures_tikz/tikzSJAWooRDGzIkrj.pdf
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diff --git a/auto/pictures_tikz/tikzSQNPooPTrLRQ.md5 b/auto/pictures_tikz/tikzSQNPooPTrLRQ.md5
index 0c5544456..bdaf2fcd2 100644
--- a/auto/pictures_tikz/tikzSQNPooPTrLRQ.md5
+++ b/auto/pictures_tikz/tikzSQNPooPTrLRQ.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {96AB15FD51F7A08F29785301FFA82E36}%
+\def \tikzexternallastkey {5EC94F2D472138637101095DE117E352}%
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diff --git a/auto/pictures_tikz/tikzSTdyNTH.md5 b/auto/pictures_tikz/tikzSTdyNTH.md5
index b57f85baf..8c48436e7 100644
--- a/auto/pictures_tikz/tikzSTdyNTH.md5
+++ b/auto/pictures_tikz/tikzSTdyNTH.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8ACC9F336608A3FFD7E65E4C7E3F5FA5}%
+\def \tikzexternallastkey {28BA7CF53180DF3E3D49ACE4DB26D9D8}%
diff --git a/auto/pictures_tikz/tikzSTdyNTH.pdf b/auto/pictures_tikz/tikzSTdyNTH.pdf
index 06c1a50ad46d19d192fcb2791d27cda650902096..95ba2d4791ee24b5e2ef4dcf69dc5180519cce56 100644
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diff --git a/auto/pictures_tikz/tikzSYNKooZBuEWsWw.md5 b/auto/pictures_tikz/tikzSYNKooZBuEWsWw.md5
index 29f40d056..7114b1096 100644
--- a/auto/pictures_tikz/tikzSYNKooZBuEWsWw.md5
+++ b/auto/pictures_tikz/tikzSYNKooZBuEWsWw.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7E0D03286E1B691E498D09AC2ED8AD8C}%
+\def \tikzexternallastkey {FE5E6E9997DB4CD9781C6FB91B5309C2}%
diff --git a/auto/pictures_tikz/tikzSYNKooZBuEWsWw.pdf b/auto/pictures_tikz/tikzSYNKooZBuEWsWw.pdf
index 3d9fefeba356257cc6c4e9ebc33b7d1eb808c9e1..38a7ea155e20c2ab13b8367a56f5dc7f5f791b63 100644
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diff --git a/auto/pictures_tikz/tikzSolsEqDiffSin.md5 b/auto/pictures_tikz/tikzSolsEqDiffSin.md5
index 41d926e71..435a1fc17 100644
--- a/auto/pictures_tikz/tikzSolsEqDiffSin.md5
+++ b/auto/pictures_tikz/tikzSolsEqDiffSin.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {7B5CA19D7CBEC100B9D28CC65D39F71F}%
+\def \tikzexternallastkey {45DC66EBA6D01A5B77EF6D6BCF2DC8AC}%
diff --git a/auto/pictures_tikz/tikzSolsEqDiffSin.pdf b/auto/pictures_tikz/tikzSolsEqDiffSin.pdf
index 7d6c5a836cd36a5200130795c25e18ee98940afa..2c2d500c880aa239ff5dff2bdc2e892c84dadbce 100644
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diff --git a/auto/pictures_tikz/tikzSolsSinpA.md5 b/auto/pictures_tikz/tikzSolsSinpA.md5
index a28ebd95f..f0cf2bf7e 100644
--- a/auto/pictures_tikz/tikzSolsSinpA.md5
+++ b/auto/pictures_tikz/tikzSolsSinpA.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A79E1512901924A14585C9D4F6DD6017}%
+\def \tikzexternallastkey {A8862D3193F076A69F3D6E9AF132F596}%
diff --git a/auto/pictures_tikz/tikzSolsSinpA.pdf b/auto/pictures_tikz/tikzSolsSinpA.pdf
index baa6b1ff0d802469614ccf0f0f8caa6448db1058..37f65d1662f0ae1491c0407425574b10608db74d 100644
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diff --git a/auto/pictures_tikz/tikzSpiraleLimite.md5 b/auto/pictures_tikz/tikzSpiraleLimite.md5
index 98e63a5d3..be1ceaece 100644
--- a/auto/pictures_tikz/tikzSpiraleLimite.md5
+++ b/auto/pictures_tikz/tikzSpiraleLimite.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {9B4E5A3254642B2B686E203ACB0A788F}%
+\def \tikzexternallastkey {84A9F089A7A8DBD7B27F2DA2B15DB105}%
diff --git a/auto/pictures_tikz/tikzSpiraleLimite.pdf b/auto/pictures_tikz/tikzSpiraleLimite.pdf
index 262d23b8a09f6112715ca4410f8af2de2a5ddd57..475d7c8b1f54851de8a80851deaee5b97e525c38 100644
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diff --git a/auto/pictures_tikz/tikzSuiteInverseAlterne.md5 b/auto/pictures_tikz/tikzSuiteInverseAlterne.md5
index 57c217193..d71213b64 100644
--- a/auto/pictures_tikz/tikzSuiteInverseAlterne.md5
+++ b/auto/pictures_tikz/tikzSuiteInverseAlterne.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {DD7EDE3B2E2775EA3070BE641361066E}%
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diff --git a/auto/pictures_tikz/tikzSuiteUnSurn.md5 b/auto/pictures_tikz/tikzSuiteUnSurn.md5
index 8f827cc91..e5d0d9657 100644
--- a/auto/pictures_tikz/tikzSuiteUnSurn.md5
+++ b/auto/pictures_tikz/tikzSuiteUnSurn.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B7A7EE08068072EF0664F0B4B12E72D8}%
+\def \tikzexternallastkey {50C1AD86687D2A335196991899BD3E36}%
diff --git a/auto/pictures_tikz/tikzSuiteUnSurn.pdf b/auto/pictures_tikz/tikzSuiteUnSurn.pdf
index 1dcfbfb8391e93dad640ebc2728e3bba053fd639..56c020f5ccab44ff6127b5b3bdccf08d5c437ceb 100644
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diff --git a/auto/pictures_tikz/tikzSurfaceCercle.md5 b/auto/pictures_tikz/tikzSurfaceCercle.md5
index df7469f28..637443521 100644
--- a/auto/pictures_tikz/tikzSurfaceCercle.md5
+++ b/auto/pictures_tikz/tikzSurfaceCercle.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A6EF2FE79D6E3FB223266FAA24B5D3CA}%
+\def \tikzexternallastkey {FA1A75F84A76B028CF3FBCE294A54858}%
diff --git a/auto/pictures_tikz/tikzSurfaceCercle.pdf b/auto/pictures_tikz/tikzSurfaceCercle.pdf
index 094bdb043c30f12f1d5fab77a75c6fb5d52873f7..c7aff1087f1e25f547c5699af860f28efa318f62 100644
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diff --git a/auto/pictures_tikz/tikzSurfacePrimiteGeog.md5 b/auto/pictures_tikz/tikzSurfacePrimiteGeog.md5
index 51788152d..c401b25fc 100644
--- a/auto/pictures_tikz/tikzSurfacePrimiteGeog.md5
+++ b/auto/pictures_tikz/tikzSurfacePrimiteGeog.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B8FFF17330D6D19DE86A28287A8DDD0B}%
+\def \tikzexternallastkey {0A2700085CD24AC2EF9ADBEB2BD5FF2F}%
diff --git a/auto/pictures_tikz/tikzSurfacePrimiteGeog.pdf b/auto/pictures_tikz/tikzSurfacePrimiteGeog.pdf
index 17d7f38af20bb285c3842fc9de7b730a72e8d01f..040da609e499f12d6a808af4408699dfce58bb53 100644
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diff --git a/auto/pictures_tikz/tikzTKXZooLwXzjS.md5 b/auto/pictures_tikz/tikzTKXZooLwXzjS.md5
index 50efe39b3..3c39a23c3 100644
--- a/auto/pictures_tikz/tikzTKXZooLwXzjS.md5
+++ b/auto/pictures_tikz/tikzTKXZooLwXzjS.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {297D83DA86C0FA08EDEBB57EEF8C7740}%
+\def \tikzexternallastkey {A0A851811A1B2A68D5C08342B8A31AA2}%
diff --git a/auto/pictures_tikz/tikzTKXZooLwXzjS.pdf b/auto/pictures_tikz/tikzTKXZooLwXzjS.pdf
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diff --git a/auto/pictures_tikz/tikzTVXooWoKkqV.md5 b/auto/pictures_tikz/tikzTVXooWoKkqV.md5
index dc24cbd7e..b27749abf 100644
--- a/auto/pictures_tikz/tikzTVXooWoKkqV.md5
+++ b/auto/pictures_tikz/tikzTVXooWoKkqV.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {9A68CA21A1B5D83A689D65E961E4E4A9}%
+\def \tikzexternallastkey {28C076E644023F6FEA2795A81F26BE5F}%
diff --git a/auto/pictures_tikz/tikzTVXooWoKkqV.pdf b/auto/pictures_tikz/tikzTVXooWoKkqV.pdf
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diff --git a/auto/pictures_tikz/tikzTWHooJjXEtS.md5 b/auto/pictures_tikz/tikzTWHooJjXEtS.md5
index 787348311..2aa19b971 100644
--- a/auto/pictures_tikz/tikzTWHooJjXEtS.md5
+++ b/auto/pictures_tikz/tikzTWHooJjXEtS.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {BE7363C918108F3C49DE1A9177D5EEDD}%
+\def \tikzexternallastkey {1E9324EE2F8C7B69657F630B484CDACB}%
diff --git a/auto/pictures_tikz/tikzTWHooJjXEtS.pdf b/auto/pictures_tikz/tikzTWHooJjXEtS.pdf
index 093c41cbb185fcbcdd8b286a6b4ebfcc988e81a5..d689adf42c9e36137d40f88a43ad113255400e96 100644
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diff --git a/auto/pictures_tikz/tikzTZCISko.md5 b/auto/pictures_tikz/tikzTZCISko.md5
index a161af102..7517c4369 100644
--- a/auto/pictures_tikz/tikzTZCISko.md5
+++ b/auto/pictures_tikz/tikzTZCISko.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {3CE04227B1CEB84A14A8FD2BE44D929F}%
+\def \tikzexternallastkey {7FEB32A6E3117C0212C38A4C81598B4A}%
diff --git a/auto/pictures_tikz/tikzTZCISko.pdf b/auto/pictures_tikz/tikzTZCISko.pdf
index 22f78442f1fa7195f166fbe281424cf8fa5b7c54..5a5bb7c711942e16e1a5e4d0b7276bf95b97334e 100644
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diff --git a/auto/pictures_tikz/tikzTangentSegment.md5 b/auto/pictures_tikz/tikzTangentSegment.md5
index 882ef0af2..2c4202bb9 100644
--- a/auto/pictures_tikz/tikzTangentSegment.md5
+++ b/auto/pictures_tikz/tikzTangentSegment.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {BF224A29D7E18E66133E8403893EDCFE}%
+\def \tikzexternallastkey {B8AB036C685D028F1199999F3A6845E4}%
diff --git a/auto/pictures_tikz/tikzTangentSegment.pdf b/auto/pictures_tikz/tikzTangentSegment.pdf
index ac3b5ffba4caa9ae1a42b19d6246dde9dba78a3c..86427b881f954b9bc0eb6b57f2898b67c77f4f18 100644
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diff --git a/auto/pictures_tikz/tikzTangenteDetail.md5 b/auto/pictures_tikz/tikzTangenteDetail.md5
index 80604b947..aee218cfa 100644
--- a/auto/pictures_tikz/tikzTangenteDetail.md5
+++ b/auto/pictures_tikz/tikzTangenteDetail.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {2303176568AF740ABEA5432A4C57A2F3}%
+\def \tikzexternallastkey {125EA76C25F48B08F0E7E3C6A6982E8C}%
diff --git a/auto/pictures_tikz/tikzTangenteDetail.pdf b/auto/pictures_tikz/tikzTangenteDetail.pdf
index cc547b98d407256fa29a1a24ebd3124794abce4d..39e370394320db4df81f2e8d5cc1a3472019f746 100644
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diff --git a/auto/pictures_tikz/tikzTangenteDetailOM.md5 b/auto/pictures_tikz/tikzTangenteDetailOM.md5
index 80604b947..aee218cfa 100644
--- a/auto/pictures_tikz/tikzTangenteDetailOM.md5
+++ b/auto/pictures_tikz/tikzTangenteDetailOM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {2303176568AF740ABEA5432A4C57A2F3}%
+\def \tikzexternallastkey {125EA76C25F48B08F0E7E3C6A6982E8C}%
diff --git a/auto/pictures_tikz/tikzTangenteDetailOM.pdf b/auto/pictures_tikz/tikzTangenteDetailOM.pdf
index 027e8d39a7ba884a65995146eee2224d5979dab6..c3e862351edea9342011cf4c0e495d5f0d691f9e 100644
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diff --git a/auto/pictures_tikz/tikzTangenteQuestion.md5 b/auto/pictures_tikz/tikzTangenteQuestion.md5
index 940c66d3f..aa87321f6 100644
--- a/auto/pictures_tikz/tikzTangenteQuestion.md5
+++ b/auto/pictures_tikz/tikzTangenteQuestion.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EFB842B45B0906090D40C0569EED2BCC}%
+\def \tikzexternallastkey {C9FE2537E83D8B9B6F7C679CDCE4F5D5}%
diff --git a/auto/pictures_tikz/tikzTangenteQuestion.pdf b/auto/pictures_tikz/tikzTangenteQuestion.pdf
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diff --git a/auto/pictures_tikz/tikzTangenteQuestionOM.md5 b/auto/pictures_tikz/tikzTangenteQuestionOM.md5
index 940c66d3f..aa87321f6 100644
--- a/auto/pictures_tikz/tikzTangenteQuestionOM.md5
+++ b/auto/pictures_tikz/tikzTangenteQuestionOM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EFB842B45B0906090D40C0569EED2BCC}%
+\def \tikzexternallastkey {C9FE2537E83D8B9B6F7C679CDCE4F5D5}%
diff --git a/auto/pictures_tikz/tikzTangenteQuestionOM.pdf b/auto/pictures_tikz/tikzTangenteQuestionOM.pdf
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diff --git a/auto/pictures_tikz/tikzTgCercleTrigono.md5 b/auto/pictures_tikz/tikzTgCercleTrigono.md5
index 0f527aef0..7379eb653 100644
--- a/auto/pictures_tikz/tikzTgCercleTrigono.md5
+++ b/auto/pictures_tikz/tikzTgCercleTrigono.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {6123A04DDC9811FB76E71C6DB97C148A}%
+\def \tikzexternallastkey {77C37DD499BC8A6189C962A2410522E9}%
diff --git a/auto/pictures_tikz/tikzTgCercleTrigono.pdf b/auto/pictures_tikz/tikzTgCercleTrigono.pdf
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diff --git a/auto/pictures_tikz/tikzToreRevolution.md5 b/auto/pictures_tikz/tikzToreRevolution.md5
index 815e53a53..47c438964 100644
--- a/auto/pictures_tikz/tikzToreRevolution.md5
+++ b/auto/pictures_tikz/tikzToreRevolution.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C806DEBD38C28D388779CA7C04B2014C}%
+\def \tikzexternallastkey {954F48525994546FDA6F3E1A6B989C35}%
diff --git a/auto/pictures_tikz/tikzToreRevolution.pdf b/auto/pictures_tikz/tikzToreRevolution.pdf
index d6ace4c9720e48bc47fe51c89dbe39c4d581f12b..95233e98d57bb7bf259a1d7160acfd5e16dc560f 100644
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diff --git a/auto/pictures_tikz/tikzTrajs.md5 b/auto/pictures_tikz/tikzTrajs.md5
index 8a5c92ef0..aa9cd17f7 100644
--- a/auto/pictures_tikz/tikzTrajs.md5
+++ b/auto/pictures_tikz/tikzTrajs.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {D7FB2876CCC76DA8FC1FAB7672D2D095}%
+\def \tikzexternallastkey {6E5864CF32FF59933C624708BF8CFEE7}%
diff --git a/auto/pictures_tikz/tikzTrajs.pdf b/auto/pictures_tikz/tikzTrajs.pdf
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diff --git a/auto/pictures_tikz/tikzTriangleRectangle.md5 b/auto/pictures_tikz/tikzTriangleRectangle.md5
index 18ae9d838..6210e3970 100644
--- a/auto/pictures_tikz/tikzTriangleRectangle.md5
+++ b/auto/pictures_tikz/tikzTriangleRectangle.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A75E882129B3FA5D0B74ACEB446BA35E}%
+\def \tikzexternallastkey {8CC2A69023968DACEF3BFDA0B947A164}%
diff --git a/auto/pictures_tikz/tikzTriangleRectangle.pdf b/auto/pictures_tikz/tikzTriangleRectangle.pdf
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index d152e3fad..e4afa45a3 100644
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diff --git a/auto/pictures_tikz/tikzTriangleUV.pdf b/auto/pictures_tikz/tikzTriangleUV.pdf
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index 61589c97d..e4f379cbb 100644
--- a/auto/pictures_tikz/tikzUEGEooHEDIJVPn.md5
+++ b/auto/pictures_tikz/tikzUEGEooHEDIJVPn.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {7FAAA5DA87D678F779CFBC4E5AC80B38}%
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GIT binary patch
delta 4130
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delta 4240
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diff --git a/auto/pictures_tikz/tikzUMEBooVTMyfD.md5 b/auto/pictures_tikz/tikzUMEBooVTMyfD.md5
index dc4feec31..5edba78e1 100644
--- a/auto/pictures_tikz/tikzUMEBooVTMyfD.md5
+++ b/auto/pictures_tikz/tikzUMEBooVTMyfD.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {034D066813FB34DD1CFDB1E884DC58B0}%
+\def \tikzexternallastkey {4290828C9A9814BED9DC84611C4AEE6E}%
diff --git a/auto/pictures_tikz/tikzUMEBooVTMyfD.pdf b/auto/pictures_tikz/tikzUMEBooVTMyfD.pdf
index 0ce845f28a7857b7a7dc37b9c5e05ede5501e692..eeab473815705d757852bfdbd86f9293892ece17 100644
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diff --git a/auto/pictures_tikz/tikzUNVooMsXxHa.md5 b/auto/pictures_tikz/tikzUNVooMsXxHa.md5
index 46c00ff6d..d8180afae 100644
--- a/auto/pictures_tikz/tikzUNVooMsXxHa.md5
+++ b/auto/pictures_tikz/tikzUNVooMsXxHa.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {63CD621F5C1DBA5C70190B939DDE4A54}%
+\def \tikzexternallastkey {240E36C9ECE67E08DBE14A142CEA3946}%
diff --git a/auto/pictures_tikz/tikzUNVooMsXxHa.pdf b/auto/pictures_tikz/tikzUNVooMsXxHa.pdf
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diff --git a/auto/pictures_tikz/tikzUQZooGFLNEq.md5 b/auto/pictures_tikz/tikzUQZooGFLNEq.md5
index daa2c5b3a..4d645d4a4 100644
--- a/auto/pictures_tikz/tikzUQZooGFLNEq.md5
+++ b/auto/pictures_tikz/tikzUQZooGFLNEq.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A329943B872036083FCEE5960CA06339}%
+\def \tikzexternallastkey {452A404CFE45F9A6056FBAAC52383E59}%
diff --git a/auto/pictures_tikz/tikzUQZooGFLNEq.pdf b/auto/pictures_tikz/tikzUQZooGFLNEq.pdf
index 643c872312d52e74a50d2048088f8711b3638ae9..0c0b1cfd23d59c87a780015049189fcbdbfc7a38 100644
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diff --git a/auto/pictures_tikz/tikzUYJooCWjLgK.md5 b/auto/pictures_tikz/tikzUYJooCWjLgK.md5
index 8864651d6..f51f917d2 100644
--- a/auto/pictures_tikz/tikzUYJooCWjLgK.md5
+++ b/auto/pictures_tikz/tikzUYJooCWjLgK.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {FCFEAE3179405B25F532AB97937460BA}%
+\def \tikzexternallastkey {54E43393635FD18072B3E4ED5C02E13D}%
diff --git a/auto/pictures_tikz/tikzUYJooCWjLgK.pdf b/auto/pictures_tikz/tikzUYJooCWjLgK.pdf
index ad07d53f312998ebe5e3e429cc17527572a25df2..95ddb8172f413887fe396fb41fd8ea41c0892818 100644
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diff --git a/auto/pictures_tikz/tikzUZGooBzlYxr.md5 b/auto/pictures_tikz/tikzUZGooBzlYxr.md5
index 46cdd4ee8..ced306d0d 100644
--- a/auto/pictures_tikz/tikzUZGooBzlYxr.md5
+++ b/auto/pictures_tikz/tikzUZGooBzlYxr.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {ACFF014CE5ED6BC58788673145DF566B}%
+\def \tikzexternallastkey {7A2FE9EAC6F4EDA69C1A32FA12E2899A}%
diff --git a/auto/pictures_tikz/tikzUZGooBzlYxr.pdf b/auto/pictures_tikz/tikzUZGooBzlYxr.pdf
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diff --git a/auto/pictures_tikz/tikzUnSurxInt.md5 b/auto/pictures_tikz/tikzUnSurxInt.md5
index 761da852e..6d8bd0f55 100644
--- a/auto/pictures_tikz/tikzUnSurxInt.md5
+++ b/auto/pictures_tikz/tikzUnSurxInt.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {1F907523E2084AA365A90F0B41A720E8}%
+\def \tikzexternallastkey {1F301355F545AD14D37DF15CE54FE8CD}%
diff --git a/auto/pictures_tikz/tikzUnSurxInt.pdf b/auto/pictures_tikz/tikzUnSurxInt.pdf
index a9810b85b67e88ff1f718c7b0a40e27c9c11dc6e..8dc134fad89fc8698f31df868f2f7d7d87403d7f 100644
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diff --git a/auto/pictures_tikz/tikzUneCellule.md5 b/auto/pictures_tikz/tikzUneCellule.md5
index bfcefb8c1..6699e4b60 100644
--- a/auto/pictures_tikz/tikzUneCellule.md5
+++ b/auto/pictures_tikz/tikzUneCellule.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EB90DE934FDB74321E23799F143A5E9D}%
+\def \tikzexternallastkey {FC12D2E8156918501617078DFCD2DED7}%
diff --git a/auto/pictures_tikz/tikzUneCellule.pdf b/auto/pictures_tikz/tikzUneCellule.pdf
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diff --git a/auto/pictures_tikz/tikzVBOIooRHhKOH.md5 b/auto/pictures_tikz/tikzVBOIooRHhKOH.md5
index edcea4c49..3f6303605 100644
--- a/auto/pictures_tikz/tikzVBOIooRHhKOH.md5
+++ b/auto/pictures_tikz/tikzVBOIooRHhKOH.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {71C7C0C0F017C273D6C718354B3A7FF2}%
+\def \tikzexternallastkey {0B8C1BE31000F3606A32C42B23694923}%
diff --git a/auto/pictures_tikz/tikzVBOIooRHhKOH.pdf b/auto/pictures_tikz/tikzVBOIooRHhKOH.pdf
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diff --git a/auto/pictures_tikz/tikzVSZRooRWgUGu.md5 b/auto/pictures_tikz/tikzVSZRooRWgUGu.md5
index 51788152d..c401b25fc 100644
--- a/auto/pictures_tikz/tikzVSZRooRWgUGu.md5
+++ b/auto/pictures_tikz/tikzVSZRooRWgUGu.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {B8FFF17330D6D19DE86A28287A8DDD0B}%
+\def \tikzexternallastkey {0A2700085CD24AC2EF9ADBEB2BD5FF2F}%
diff --git a/auto/pictures_tikz/tikzVSZRooRWgUGu.pdf b/auto/pictures_tikz/tikzVSZRooRWgUGu.pdf
index 63afce94a0106a10bc3d22648877dcef3f235751..e502138cfcb89c2a253cb510f9ba2059857ae856 100644
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diff --git a/auto/pictures_tikz/tikzVWFLooPSrOqz.md5 b/auto/pictures_tikz/tikzVWFLooPSrOqz.md5
index a1aa29658..12ecdc808 100644
--- a/auto/pictures_tikz/tikzVWFLooPSrOqz.md5
+++ b/auto/pictures_tikz/tikzVWFLooPSrOqz.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {51E520617D86BA1994EC20957DD54DC4}%
+\def \tikzexternallastkey {F343A3225200B91911DD3EBDEB6206AC}%
diff --git a/auto/pictures_tikz/tikzVWFLooPSrOqz.pdf b/auto/pictures_tikz/tikzVWFLooPSrOqz.pdf
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diff --git a/auto/pictures_tikz/tikzWHCooNZAmYB.md5 b/auto/pictures_tikz/tikzWHCooNZAmYB.md5
index 5158fc1da..664b74c9c 100644
--- a/auto/pictures_tikz/tikzWHCooNZAmYB.md5
+++ b/auto/pictures_tikz/tikzWHCooNZAmYB.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {382AD32006651ECD0B044902CABC734B}%
+\def \tikzexternallastkey {2D708735399A35BB1351A5A9C4868A93}%
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diff --git a/auto/pictures_tikz/tikzWIRAooTCcpOV.md5 b/auto/pictures_tikz/tikzWIRAooTCcpOV.md5
index 98cc099b7..fe4e3480c 100644
--- a/auto/pictures_tikz/tikzWIRAooTCcpOV.md5
+++ b/auto/pictures_tikz/tikzWIRAooTCcpOV.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {792A4F586A9858BD981AC2996A41F698}%
+\def \tikzexternallastkey {0CB5BC2E2D41D21B82D7DCB2435B2AA4}%
diff --git a/auto/pictures_tikz/tikzWIRAooTCcpOV.pdf b/auto/pictures_tikz/tikzWIRAooTCcpOV.pdf
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diff --git a/auto/pictures_tikz/tikzWJBooMTAhtl.md5 b/auto/pictures_tikz/tikzWJBooMTAhtl.md5
index f41ee08d7..536743af6 100644
--- a/auto/pictures_tikz/tikzWJBooMTAhtl.md5
+++ b/auto/pictures_tikz/tikzWJBooMTAhtl.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {41D44098881210689159C79A230FCE1C}%
+\def \tikzexternallastkey {DBF7EF45A1DAE4CF7C218B866C075C5E}%
diff --git a/auto/pictures_tikz/tikzWJBooMTAhtl.pdf b/auto/pictures_tikz/tikzWJBooMTAhtl.pdf
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diff --git a/auto/pictures_tikz/tikzXJMooCQTlNL.md5 b/auto/pictures_tikz/tikzXJMooCQTlNL.md5
index acc491749..ee47b9af7 100644
--- a/auto/pictures_tikz/tikzXJMooCQTlNL.md5
+++ b/auto/pictures_tikz/tikzXJMooCQTlNL.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {4CFEF314D52B27D7F3D9B7ED7B3BA618}%
+\def \tikzexternallastkey {B847DB6EFCC291537D4D804BF48C6395}%
diff --git a/auto/pictures_tikz/tikzXJMooCQTlNL.pdf b/auto/pictures_tikz/tikzXJMooCQTlNL.pdf
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diff --git a/auto/pictures_tikz/tikzXOLBooGcrjiwoU.md5 b/auto/pictures_tikz/tikzXOLBooGcrjiwoU.md5
index 6fd66fec9..4f815175a 100644
--- a/auto/pictures_tikz/tikzXOLBooGcrjiwoU.md5
+++ b/auto/pictures_tikz/tikzXOLBooGcrjiwoU.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {922AC9A8A71A40013D284E82FC08AA01}%
+\def \tikzexternallastkey {F1899328C78498BEDAD7EBFD2D79DB58}%
diff --git a/auto/pictures_tikz/tikzXOLBooGcrjiwoU.pdf b/auto/pictures_tikz/tikzXOLBooGcrjiwoU.pdf
index 376d440f4f1223a4c935278be89348b5692635de..5f0b40d0e4a5c3431c7ca21182ee47e19f2f0965 100644
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diff --git a/auto/pictures_tikz/tikzXTGooSFFtPu.md5 b/auto/pictures_tikz/tikzXTGooSFFtPu.md5
index 2521ae51b..c08bd36e4 100644
--- a/auto/pictures_tikz/tikzXTGooSFFtPu.md5
+++ b/auto/pictures_tikz/tikzXTGooSFFtPu.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {25DE3DF65721E5F1856DA8CE85945B66}%
+\def \tikzexternallastkey {FE9C75E6F3EA55F7547DFCC1C06232FD}%
diff --git a/auto/pictures_tikz/tikzXTGooSFFtPu.pdf b/auto/pictures_tikz/tikzXTGooSFFtPu.pdf
index 2b38acd3aa4c117a5f2e045db8004e807069aa86..ed320f18f098d0adcffa5ab5fd2404de6c3d80d8 100644
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diff --git a/auto/pictures_tikz/tikzYHJYooTEXLLn.md5 b/auto/pictures_tikz/tikzYHJYooTEXLLn.md5
index e97f98c4e..c4136a548 100644
--- a/auto/pictures_tikz/tikzYHJYooTEXLLn.md5
+++ b/auto/pictures_tikz/tikzYHJYooTEXLLn.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {A570E1DA36BFEC42B0147BCDF045FF97}%
+\def \tikzexternallastkey {DC11305CB47954537BF59E8AC0974ECE}%
diff --git a/auto/pictures_tikz/tikzYHJYooTEXLLn.pdf b/auto/pictures_tikz/tikzYHJYooTEXLLn.pdf
index 4666af7c8834e67090f86bc8602b68796b57b78e..26045dd0b1629a2b40af08d09405dedfa4d95f31 100644
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diff --git a/auto/pictures_tikz/tikzYQIDooBqpAdbIM.md5 b/auto/pictures_tikz/tikzYQIDooBqpAdbIM.md5
index 63359ebef..5a2c02c76 100644
--- a/auto/pictures_tikz/tikzYQIDooBqpAdbIM.md5
+++ b/auto/pictures_tikz/tikzYQIDooBqpAdbIM.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {39F730BEAEF3F9476D5D5036AF98C383}%
+\def \tikzexternallastkey {5CE5111FE11EBC241D6ACDEF01ACFFD2}%
diff --git a/auto/pictures_tikz/tikzYQIDooBqpAdbIM.pdf b/auto/pictures_tikz/tikzYQIDooBqpAdbIM.pdf
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diff --git a/auto/pictures_tikz/tikzYQVHooYsGLHQ.md5 b/auto/pictures_tikz/tikzYQVHooYsGLHQ.md5
index 15538fc16..24393d32a 100644
--- a/auto/pictures_tikz/tikzYQVHooYsGLHQ.md5
+++ b/auto/pictures_tikz/tikzYQVHooYsGLHQ.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {C1E76FBDBEC88ED3FD93DA9B1D935535}%
+\def \tikzexternallastkey {B9179F47BE4732594D65B1E02D560F51}%
diff --git a/auto/pictures_tikz/tikzYQVHooYsGLHQ.pdf b/auto/pictures_tikz/tikzYQVHooYsGLHQ.pdf
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diff --git a/auto/pictures_tikz/tikzYWxOAkh.md5 b/auto/pictures_tikz/tikzYWxOAkh.md5
index c738a3b52..9170f0dc1 100644
--- a/auto/pictures_tikz/tikzYWxOAkh.md5
+++ b/auto/pictures_tikz/tikzYWxOAkh.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {83AF4E620AD42408EB7A60794DE435AA}%
+\def \tikzexternallastkey {E2149AFC135877A967F933737FC48231}%
diff --git a/auto/pictures_tikz/tikzYWxOAkh.pdf b/auto/pictures_tikz/tikzYWxOAkh.pdf
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diff --git a/auto/pictures_tikz/tikzYYECooQlnKtD.md5 b/auto/pictures_tikz/tikzYYECooQlnKtD.md5
index 305d3a0fc..4cb6e78cc 100644
--- a/auto/pictures_tikz/tikzYYECooQlnKtD.md5
+++ b/auto/pictures_tikz/tikzYYECooQlnKtD.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {1E95624176E34967C3BD8B088CA465D4}%
+\def \tikzexternallastkey {F81E06C40FF174E724942AB6789433C7}%
diff --git a/auto/pictures_tikz/tikzYYECooQlnKtD.pdf b/auto/pictures_tikz/tikzYYECooQlnKtD.pdf
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diff --git a/auto/pictures_tikz/tikzZGUDooEsqCWQ.md5 b/auto/pictures_tikz/tikzZGUDooEsqCWQ.md5
index 7842d83b3..dd234bb5c 100644
--- a/auto/pictures_tikz/tikzZGUDooEsqCWQ.md5
+++ b/auto/pictures_tikz/tikzZGUDooEsqCWQ.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {42A98F39F4F00B9A41387B04B2355EAE}%
+\def \tikzexternallastkey {72228E51743D5ABE69CF833BF3E714B6}%
diff --git a/auto/pictures_tikz/tikzZGUDooEsqCWQ.pdf b/auto/pictures_tikz/tikzZGUDooEsqCWQ.pdf
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diff --git a/auto/pictures_tikz/tikzZOCNoowrfvQXsr.md5 b/auto/pictures_tikz/tikzZOCNoowrfvQXsr.md5
index 709d41fad..8887f2f3b 100644
--- a/auto/pictures_tikz/tikzZOCNoowrfvQXsr.md5
+++ b/auto/pictures_tikz/tikzZOCNoowrfvQXsr.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {ABEBCF003D002B6C696BBD37F60BCA4C}%
+\def \tikzexternallastkey {43E77186571E16569883773A3EF8EA11}%
diff --git a/auto/pictures_tikz/tikzZOCNoowrfvQXsr.pdf b/auto/pictures_tikz/tikzZOCNoowrfvQXsr.pdf
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diff --git a/auto/pictures_tikz/tikzexamssepti.md5 b/auto/pictures_tikz/tikzexamssepti.md5
index 5260c3041..151776d76 100644
--- a/auto/pictures_tikz/tikzexamssepti.md5
+++ b/auto/pictures_tikz/tikzexamssepti.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {434FBB9961312D9054D313A534DB8291}%
+\def \tikzexternallastkey {8E59057DF38BB6285C0BA1D0DF2D7A4B}%
diff --git a/auto/pictures_tikz/tikzexamssepti.pdf b/auto/pictures_tikz/tikzexamssepti.pdf
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diff --git a/auto/pictures_tikz/tikzexamsseptii.md5 b/auto/pictures_tikz/tikzexamsseptii.md5
index 67672e7a6..6ec85630c 100644
--- a/auto/pictures_tikz/tikzexamsseptii.md5
+++ b/auto/pictures_tikz/tikzexamsseptii.md5
@@ -1 +1 @@
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+\def \tikzexternallastkey {E0048E0AFC11B32AB187D29F901C71C7}%
diff --git a/auto/pictures_tikz/tikzexamsseptii.pdf b/auto/pictures_tikz/tikzexamsseptii.pdf
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diff --git a/auto/pictures_tikz/tikzooIHLPooKLIxcH.md5 b/auto/pictures_tikz/tikzooIHLPooKLIxcH.md5
new file mode 100644
index 000000000..7596b9cda
--- /dev/null
+++ b/auto/pictures_tikz/tikzooIHLPooKLIxcH.md5
@@ -0,0 +1 @@
+\def \tikzexternallastkey {F6CE50E63BF3CB5919960B5718549775}%
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diff --git a/auto/pictures_tikz/tikzratrap.md5 b/auto/pictures_tikz/tikzratrap.md5
index 7034b9a9c..58df00419 100644
--- a/auto/pictures_tikz/tikzratrap.md5
+++ b/auto/pictures_tikz/tikzratrap.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {248CAA0C930E2B033B1D12907D4F6EE3}%
+\def \tikzexternallastkey {2247F7AB3C43BEA6BDD0C1FF69430C5C}%
diff --git a/auto/pictures_tikz/tikzratrap.pdf b/auto/pictures_tikz/tikzratrap.pdf
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diff --git a/auto/pictures_tikz/tikzsenotopologo.md5 b/auto/pictures_tikz/tikzsenotopologo.md5
index 25ea1af1b..2a9c9d212 100644
--- a/auto/pictures_tikz/tikzsenotopologo.md5
+++ b/auto/pictures_tikz/tikzsenotopologo.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {8F9C16EB8B4BC7985E03E6714E9EBB27}%
+\def \tikzexternallastkey {EB74FAA89ECC284DC3E0BB95B47C1B06}%
diff --git a/auto/pictures_tikz/tikzsenotopologo.pdf b/auto/pictures_tikz/tikzsenotopologo.pdf
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diff --git a/auto/pictures_tikz/tikztrigoWedd.md5 b/auto/pictures_tikz/tikztrigoWedd.md5
index d38c973af..da7160e95 100644
--- a/auto/pictures_tikz/tikztrigoWedd.md5
+++ b/auto/pictures_tikz/tikztrigoWedd.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {0A88DA1985E3B7A7A8BC55A8EDC626B5}%
+\def \tikzexternallastkey {46481C299D7B8EE18A6AA337DA88B753}%
diff --git a/auto/pictures_tikz/tikztrigoWedd.pdf b/auto/pictures_tikz/tikztrigoWedd.pdf
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From f882ff721fad457e659629b2b63a0c0448e9f8e0 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Fri, 23 Jun 2017 08:50:59 +0200
Subject: [PATCH 45/64] Preparing a merge

---
 lst_actu.py | 8 +++-----
 1 file changed, 3 insertions(+), 5 deletions(-)

diff --git a/lst_actu.py b/lst_actu.py
index 0f3130704..b08c7396c 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -16,12 +16,10 @@
 myRequest.original_filename="mazhe.tex"
 
 myRequest.ok_filenames_list=["e_mazhe"]
-myRequest.ok_filenames_list.extend(["182_numerique"])
-myRequest.ok_filenames_list.extend(["<++>"])
-myRequest.ok_filenames_list.extend(["<++>"])
-myRequest.ok_filenames_list.extend(["<++>"])
-myRequest.ok_filenames_list.extend(["<++>"])
+myRequest.ok_filenames_list.extend(["79_inversion_locale"])
+myRequest.ok_filenames_list.extend(["68_Chap_calcul_differentiel"])
 myRequest.ok_filenames_list.extend(["<++>"])
+myRequest.ok_filenames_list.extend(["68_Chap_calcul_differentiel"])
 
 myRequest.ok_filenames_list.extend(["134_choses_finales"])
 myRequest.ok_filenames_list.extend(["157_thematique"])

From 05e0a699bc6ec3234672528e3be91b1098eaacbe Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Fri, 23 Jun 2017 11:31:16 +0200
Subject: [PATCH 46/64] =?UTF-8?q?(index=20th=C3=A9matique)=20Ajout=20des?=
 =?UTF-8?q?=20formules=20de=20la=20diff=C3=A9rentielle.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 commons.py                         | 14 ++++++++++----
 tex/front_back_matter/65_theme.tex |  5 ++---
 2 files changed, 12 insertions(+), 7 deletions(-)

diff --git a/commons.py b/commons.py
index c08cb7e95..9a082958b 100644
--- a/commons.py
+++ b/commons.py
@@ -20,16 +20,22 @@
 
 
 ok_hash=[]
-ok_hash.append("59080de1fd3e3d70e33ae6b8abd5d5bf4fce79b0")
-ok_hash.append("9f89a9fe96b2ef03f28f3f978e53b1058cf9b606")
-ok_hash.append("cd26df5e364185f46fd763ecc01e39bf7be2029a")
-ok_hash.append("fe793b1330cf5ee38543842fe28f30b51196c145")
+ok_hash.append("")
+ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
+ok_hash.append("77a4d1ada815b1388561bd7b44b18ebc93c38af6")
+ok_hash.append("246a74658264789058fbfbd84086f5ef1016df1b")
+ok_hash.append("3c76e8da908a59108dacf311f2ceeb8849192007")
+ok_hash.append("faeadafa6e012a3f8fdee0df8c3e1d2414870feb")
 ok_hash.append("9598573994f5293fb255383df00c0cfb8fd413e4")
 ok_hash.append("ee8ddea7a3769f303e5cc70cd72a5f6d2c1979fa")
+ok_hash.append("fe793b1330cf5ee38543842fe28f30b51196c145")
+ok_hash.append("59080de1fd3e3d70e33ae6b8abd5d5bf4fce79b0")
+ok_hash.append("9f89a9fe96b2ef03f28f3f978e53b1058cf9b606")
+ok_hash.append("cd26df5e364185f46fd763ecc01e39bf7be2029a")
 ok_hash.append("7094ee4918e6f22b77ead4b006507e7ab6f81c13")
 ok_hash.append("4e4e3a8c2c2387a9796d82c155035a14783f225d")
 ok_hash.append("2c7d4dba8f8f23f29b53b1491aa6b7c7e8564b1e")
diff --git a/tex/front_back_matter/65_theme.tex b/tex/front_back_matter/65_theme.tex
index e8eb42d39..4410e7713 100644
--- a/tex/front_back_matter/65_theme.tex
+++ b/tex/front_back_matter/65_theme.tex
@@ -3,9 +3,8 @@
     \item
         La recherche d'extrema d'une fonction sur \( \eR^n\) passe par la seconde différentielle, proposition \ref{PropoExtreRn}.
     \item
-        La différentielle est liée aux dérivées partielles par les formules
+        La différentielle est liée aux dérivées partielles par les formules données au lemme \ref{LemdfaSurLesPartielles}
 	\begin{equation}
-        df_a(u)=\frac{ \partial f }{ \partial u }(a)=\Dsdd{ f(a+tu) }{t}{0}=\sum_{i=1}^mu_i\frac{ \partial f }{ \partial x_i }(a)=\nabla f(a)\cdot u
+        df_a(u)=\frac{ \partial f }{ \partial u }(a)=\Dsdd{ f(a+tu) }{t}{0}=\sum_{i=1}^mu_i\frac{ \partial f }{ \partial x_i }(a)=\nabla f(a)\cdot u.
 	\end{equation}
-    du lemme \ref{LemdfaSurLesPartielles}.
 \end{enumerate}

From 144e8df3c8b5fda23508d056e7fcae447a0efdfc Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 24 Jun 2017 23:27:33 +0200
Subject: [PATCH 47/64] (testing) Re-activate the 'normal' tesing

---
 testing/testing.sh | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/testing/testing.sh b/testing/testing.sh
index 9d081f2ca..dbb725cf8 100755
--- a/testing/testing.sh
+++ b/testing/testing.sh
@@ -107,8 +107,8 @@ then
     test_death_links&
 fi
 
-#compile_everything&
-#compile_frido
+compile_everything&
+compile_frido
 
 
 cd $MAIN_DIR

From 3cd8790af4bff7de68634b838aa2fb91a69703be Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 24 Jun 2017 23:28:33 +0200
Subject: [PATCH 48/64] (pictures) The 'md5' and 'pdf' of two pictures.

---
 auto/pictures_tikz/tikzGMRNooCNBpIl.md5 |   2 +-
 auto/pictures_tikz/tikzGMRNooCNBpIl.pdf | Bin 31903 -> 31922 bytes
 2 files changed, 1 insertion(+), 1 deletion(-)

diff --git a/auto/pictures_tikz/tikzGMRNooCNBpIl.md5 b/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
index 86dec2867..ab91b7149 100644
--- a/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
+++ b/auto/pictures_tikz/tikzGMRNooCNBpIl.md5
@@ -1 +1 @@
-\def \tikzexternallastkey {EBBB58E0F976B0E8ACCCAFC417AA4D36}%
+\def \tikzexternallastkey {46B87348CD4051022E49D2A5395220DC}%
diff --git a/auto/pictures_tikz/tikzGMRNooCNBpIl.pdf b/auto/pictures_tikz/tikzGMRNooCNBpIl.pdf
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From acdd46574655e76215f9aafb35ec5ab9fa1e223a Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 24 Jun 2017 23:29:35 +0200
Subject: [PATCH 49/64] =?UTF-8?q?(fonctions=20convexes)=20Convexit=C3=A9?=
 =?UTF-8?q?=20et=20gradient/hessienne=20:=20d=C3=A9but=20du=20travail.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 commons.py                                |   4 +-
 lst_actu.py                               |   1 -
 tex/frido/68_Chap_calcul_differentiel.tex | 161 +++++++++++++++++++++-
 3 files changed, 159 insertions(+), 7 deletions(-)

diff --git a/commons.py b/commons.py
index 9a082958b..d29d3f94a 100644
--- a/commons.py
+++ b/commons.py
@@ -20,12 +20,14 @@
 
 
 ok_hash=[]
-ok_hash.append("")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("ccced7c771d6b95a06cf3339fc4ae85a7624e8ca")
 ok_hash.append("77a4d1ada815b1388561bd7b44b18ebc93c38af6")
 ok_hash.append("246a74658264789058fbfbd84086f5ef1016df1b")
 ok_hash.append("3c76e8da908a59108dacf311f2ceeb8849192007")
diff --git a/lst_actu.py b/lst_actu.py
index 29de2264a..6a3793fc0 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -21,7 +21,6 @@
 myRequest.ok_filenames_list.extend(["78_inversion_locale"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
-myRequest.ok_filenames_list.extend(["68_Chap_calcul_differentiel"])
 
 myRequest.ok_filenames_list.extend(["134_choses_finales"])
 myRequest.ok_filenames_list.extend(["157_thematique"])
diff --git a/tex/frido/68_Chap_calcul_differentiel.tex b/tex/frido/68_Chap_calcul_differentiel.tex
index ddf18071e..8b221c051 100644
--- a/tex/frido/68_Chap_calcul_differentiel.tex
+++ b/tex/frido/68_Chap_calcul_differentiel.tex
@@ -759,7 +759,7 @@ \subsection{Différentielle seconde, fonction de classe \( C^2\)}
 \end{example}
 
 Tout ceci est un peu résumé dans la proposition suivante.
-\begin{proposition}
+\begin{proposition}     \label{PROPooFWZYooUQwzjW}
     Soit une fonction \( f\colon \eR^n\to \eR\) de classe \( C^2\). Alors en désignant par \( H_af\) sa matrice hessienne au point \( a\) nous avons
     \begin{equation}
         (d^2f)_a(u,v)=\frac{ \partial^2f }{ \partial u\partial v }(a)=\langle (H_af)u, v\rangle 
@@ -979,7 +979,7 @@ \subsection{Dérivées d'une fonction convexe}
     La fonction est \( C^2\), donc \( f''\) est positive si et seulement si \( f'\) est croissante (proposition \ref{PropGFkZMwD}) alors que la proposition \ref{PropYKwTDPX} nous jure que \( f\) sera convexe si et seulement si \( f'\) est croissante.
 \end{proof}
 
-\begin{remark}
+\begin{remark}      \label{REMooVRPQooIybxmp}
     Une fonction peut être strictement convexe sans que sa dérivée seconde ne soit toujours strictement positive. En exemple : \( x\mapsto x^4\) est strictement convexe alors que sa dérivée seconde s'annule en zéro.
 \end{remark}
 
@@ -996,9 +996,6 @@ \subsection{Dérivées d'une fonction convexe}
     \end{enumerate}
 \end{example}
 
-\begin{proposition}[\cite{CLTooTlwZoz}] \label{PropHRLooTqIJPS}
-    Si \( f\colon \eR^n\to \eR\) est de classe \( C^2\), elle est convexe si et seulement si sa matrice hessienne est définie positive en tout point.
-\end{proposition}
 
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Graphe d'une fonction convexe}
@@ -1279,6 +1276,160 @@ \subsection{Graphe d'une fonction convexe}
     ce qui est bien \eqref{EqGFLooOElciS}.
 \end{proof}
 
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{En dimension supérieure}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit une partie convexe \( U\) de \( \eR^n\) et une fonction \( f\colon U\to \eR\). 
+    \begin{enumerate}
+        \item
+        La fonction \( f\) est \defe{convexe}{convexe!fonction sur \( \eR^n\)} si pour tout \( x,y\in U\) avec \( x\neq y\) et pour tout \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous avons
+            \begin{equation}
+                f\big( \theta x+(1-\theta)y \big)\leq \theta f(x)+(1-\theta)f(y).
+            \end{equation}
+        \item
+            Elle est \defe{strictement convexe}{strictement!convexe!sur \( \eR^n\)} si nous avons l'inégalité stricte.
+    \end{enumerate}
+\end{definition}
+
+\begin{proposition}[\cite{ooLJMHooMSBWki}]
+    Soit \( \Omega\) ouvert dans \( \eR^n\) et \( U\) convexe dans \( \Omega\), et une fonction différentiable \( f\colon U\to \eR\).
+    \begin{enumerate}
+        \item       \label{ITEMooRVIVooIayuPS}
+            La fonction \( f\) est convexe sur \( U\) si et seulement si pour tout \( x,y\in U\),
+            \begin{equation}
+                f(y)\geq f(x)+df_x(y-x).
+            \end{equation}
+        \item       \label{ITEMooCWEWooFtNnKl}
+            La fonction \( f\) est strictement convexe sur \( U\) si et seulement si pour tout \( x,y\in U\) avec \( x\neq y\),
+            \begin{equation}
+                f(y)>f(x)+df_x(y-x).
+            \end{equation}
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Nous avons quatre petites choses à démontrer.
+    \begin{subproof}
+    \item[\ref{ITEMooRVIVooIayuPS}, sens direct]
+        Soit une fonction convexe \( f\). Nous avons :
+        \begin{equation}
+            f\big( (1-\theta)x+\theta y \big)\leq (1-\theta)f(x)+\theta f(y),
+        \end{equation}
+        donc
+        \begin{equation}
+            f\big( x+\theta(y-x) \big)-f(x)\leq \theta\big( f(y)-f(x) \big)
+        \end{equation}
+        Vu que \( \theta>0\) nous pouvons diviser par \( \theta\) sans changer le sens de l'inégalité :
+        \begin{equation}        \label{EQooAXXFooHWtiJh}
+            \frac{ f\big( x+\theta(y-x) \big)-f(x) }{ \theta }\leq f(y)-f(x).
+        \end{equation}
+        Nous prenons la limite \( \theta\to 0^+\). Cette limite est égale à a limite simple \( \theta\to 0\) et vaut (parce que \( f\) est différentiable) :
+        \begin{equation}
+            \frac{ \partial f }{ \partial (y-x) }(x)\leq f(y)-f(x),
+        \end{equation}
+        et aussi
+        \begin{equation}
+            df_x(y-x)\leq f(y)-f(x)
+        \end{equation}
+        par le lemme \ref{LemdfaSurLesPartielles}.
+    \item[\ref{ITEMooRVIVooIayuPS}, sens inverse]
+        Pour tout \( a\neq b\) dans \( U\) nous avons
+        \begin{equation}        \label{EQooEALSooJOszWr}
+            f(b)\geq f(a)+df_a(b-a).
+        \end{equation}
+    Pour \( x\neq y\) dans \( U\) et pour \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous écrivons \eqref{EQooEALSooJOszWr} pour les couples \( \big( \theta x+(1-\theta)y,y \big)\) et \( \big( \theta x+(1-\theta)y,x \big)\). Ça donne :
+        \begin{equation}
+            f(y)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( \theta(y-x) \big),
+        \end{equation}
+        et
+        \begin{equation}
+            f(x)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( (1-\theta)(x-y) \big).
+        \end{equation}
+        La différentielle est linéaire; en multipliant la première par \( (1-\theta)\) et la seconde par \( \theta\) et en la somme, les termes en \( df\) se simplifient et nous trouvons
+        \begin{equation}
+            \theta f(x)+(1-\theta)f(y)\geq f\big( \theta x+(1-\theta)y \big).
+        \end{equation}
+    \item[\ref{ITEMooCWEWooFtNnKl}, sens direct]
+        Nous avons encore l'équation \eqref{EQooAXXFooHWtiJh}, avec une inégalité stricte. Par contre, ça ne va pas être suffisant parce que le passage à la limite ne conserve pas les inégalités strictes. Nous devons donc être plus malins. 
+
+        Soient \( 0<\theta<\omega<1\). Nous avons \( (1-\theta)x+\theta y\in \mathopen[ x , (1-\omega)x+\omega y \mathclose]\), donc nous pouvons écrire \( (1-\theta)x+\theta y\) sous la forme \( (1-s)x+s\big( (1-\omega)x+\omega y \big)\). Il se fait que c'est bon pour \( s=\theta/\omega\) (et aussi que nous avons \( \theta/\omega<1\)). Donc nous avons
+        \begin{subequations}
+            \begin{align}
+            f\big( (1-\theta)x+\theta y \big)&=f\Big( (1-\frac{ \theta }{ \omega })x+\frac{ \theta }{ \omega }\big( (1-\omega)x+\omega y \big) \Big)\\
+            &<(1-\frac{ \theta }{ \omega })f(x)+\frac{ \theta }{ \omega }f\big( (1-\omega)x+\omega y \big).
+            \end{align}
+        \end{subequations}
+        Cela nous permet d'écrire
+        \begin{equation}
+            \frac{ f\big( (1-\theta)x+\theta y \big)-f(x) }{ \theta }<\frac{ f\big( (1-\omega)x+\omega y \big) }{ \omega }<f(y)-f(x).
+        \end{equation}
+        Le seconde inégalité est le pendant de \eqref{EQooAXXFooHWtiJh}. Maintenant en passant à la limite pour \( \theta\) nous conservons une inégalité stricte par rapport à \( f(y)-f(x)\) :
+        \begin{equation}
+            df_x(y-x)<f(y)-f(x).
+        \end{equation}
+    \end{subproof}
+\end{proof}
+
+% Il faut laisser les sauts de lignes suivants, pour rechercher efficacement les références vers le futur.
+Avant de lire la proposition suivante, il faut relire la proposition \ref{PROPooFWZYooUQwzjW} et ce qui s'y rapporte. 
+Lire aussi la remarque \ref{REMooVRPQooIybxmp} qui indique
+qu'il n'y a pas de réciproque dans l'énoncé \ref{ITEMooHAGQooYZyhQk}.     
+\begin{proposition}
+    Soit une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) et un convexe \( U\subset \Omega\).
+    \begin{enumerate}
+        \item       \label{ITEMooZQCAooIFjHOn}
+            La fonction \( f\) est convexe sur \( U\) si et seulement si
+            \begin{equation}        \label{EQooIBDCooJYdiBb}
+                (d^2f)_x(y-x,y-x)\geq 0
+            \end{equation}
+            pour tout \( x,y\in U\).
+        \item       \label{ITEMooHAGQooYZyhQk}
+            Si pour tout \( x\neq y\) dans \( U\) nous avons
+            \begin{equation}
+                (d^2f)_x(y-x,y-x)>0
+            \end{equation}
+            alors la fonction \( f\) est strictement convexe sur \( U\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{remark}
+    Notons que la condition \eqref{EQooIBDCooJYdiBb} n'est pas équivalente à demander \( (d^2f)_x(h,h)\geq 0\) pour tout \( h\). En effet nous ne demandons la positivité que dans les directions atteignables comme différence de deux éléments de \( U\). La partie \( U\) n'est pas spécialement ouverte; elle pourrait n'être qu'une droite dans \( \eR^3\). Dans ce cas, demander que \( f\) (qui est \( C^2\) sur l'ouvert \( \Omega\)) soit convexe sur \( U\) ne demande que la positivité de \( (d^2f)_x\) appliqué à des vecteurs situés sur la droite \( U\).
+\end{remark}
+
+\begin{proof}
+
+    Il y a trois parties à démontrer.
+    \begin{subproof}
+    \item[\ref{ITEMooZQCAooIFjHOn} sens direct]
+
+        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci à un jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
+        \begin{equation}
+            f(x+th)=f(x)+tdf_x(h)+\frac{ t^2 }{2}(d^2_x)(h,h)+t^2\| h \|^2\alpha(th)
+        \end{equation}
+        avec \( \lim_{s\to 0}\alpha(s)=0\). Le fait que \( f\) soit convexe donne
+        \begin{equation}
+            0\leq f(x+th)-f(x)-tdf_x(h),
+        \end{equation}
+        et donc
+        \begin{equation}
+            0\leq \frac{ t^2 }{2}(d^2f)_x(h,h)+f^2\| h \|^2\alpha(th).
+        \end{equation}
+        En multipliant par \( 2\) et en divisant par \( t^2\),
+        \begin{equation}
+            0\leq (d^2f)_x(h,h)+2\| h \|^2\alpha(th).
+        \end{equation}
+        En prenant \( t\to 0\) nous avons bien pour tout \( h\) : \( (d^2f)_x(h,h)\geq 0\).
+    \end{subproof}
+    <++>
+\end{proof}
+<++>
+
+\begin{proposition}[\cite{CLTooTlwZoz}] %\label{PropHRLooTqIJPS}
+    Si \( f\colon \eR^n\to \eR\) est de classe \( C^2\), elle est convexe si et seulement si sa matrice hessienne est définie positive en tout point.
+\end{proposition}
+
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Quelque inégalités}
 %---------------------------------------------------------------------------------------------------------------------------

From a2f02dc52917afa4ddd336d185c65b49253aa58b Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 24 Jun 2017 23:42:48 +0200
Subject: [PATCH 50/64] =?UTF-8?q?(organisation)=20Renomme=20quelque=20fich?=
 =?UTF-8?q?iers=20pour=20avoir=20de=20la=20coh=C3=A9rence=20entre=20les=20?=
 =?UTF-8?q?noms=20de=20chapitres=20et=20noms=20de=20fichiers.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/frido/138_EspacesVectos.tex |  980 +++++++++++++++++
 tex/frido/141_EspacesVectos.tex |  843 +++++++++++++++
 tex/frido/144_EspacesVectos.tex | 1648 ++++++++++++++++++++++++++++
 tex/frido/55_EspacesVectos.tex  | 1035 ++++++++++++++++++
 tex/frido/57_EspacesVectos.tex  | 1791 +++++++++++++++++++++++++++++++
 tex/frido/59_EspacesVectos.tex  |  266 +++++
 tex/frido/60_EspacesVectos.tex  |  431 ++++++++
 7 files changed, 6994 insertions(+)
 create mode 100644 tex/frido/138_EspacesVectos.tex
 create mode 100644 tex/frido/141_EspacesVectos.tex
 create mode 100644 tex/frido/144_EspacesVectos.tex
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 create mode 100644 tex/frido/59_EspacesVectos.tex
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diff --git a/tex/frido/138_EspacesVectos.tex b/tex/frido/138_EspacesVectos.tex
new file mode 100644
index 000000000..3faadbb73
--- /dev/null
+++ b/tex/frido/138_EspacesVectos.tex
@@ -0,0 +1,980 @@
+% This is part of Mes notes de mathématique
+% Copyright (c) 2011-2016
+%   Laurent Claessens, Carlotta Donadello
+% See the file fdl-1.3.txt for copying conditions.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Extension du corps de base}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\label{SECooAUOWooNdYTZf}
+
+Nous avons discuté dans la section \ref{SECooLQVJooTGeqiR} de ce qui arrive au corps lorsqu'on l'étend. Dans cette sections nous allons étudier ce qui arrive aux applications linéaires entre deux \( \eK\)-espaces vectoriels lorsque nous étendons le corps \( \eK\) en un corps \( \eL\).
+
+Soit donc un corps \( \eK\) et deux \( \eK\)-espaces vectoriels \( E\) et \( F\), et entrons dans le vif du sujet\footnote{Le sujet étant le corps étendu.}. Soit \( \eK\) un corps (commutatif) et une extension \( \eL\) de \( \eK\). Soient \( E\) et \( F\), des \( \eK\)-espaces vectoriels de dimension finie. 
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Extension des applications linéaires}
+%---------------------------------------------------------------------------------------------------------------------------
+
+
+\begin{definition}[\cite{ooAFBYooYvTCCN}]
+    L'espace vectoriel obtenu par \defe{extension du corps de base}{extension!corps de base} de \( E\) est l'espace vectoriel
+    \begin{equation}
+        E_{\eL}=\eL\otimes_{\eK}E.
+    \end{equation}
+    Ce dernier est le quotient \( \eL\otimes_{\eK}E=(\eL\times E)/\sim\) par la relation d'équivalence
+    \begin{equation}
+        (\lambda,v)\sim\big( a\lambda,\frac{1}{ a }v \big)
+    \end{equation}
+    pour tout \( a\in \eK\). Nous noterons \( [\lambda,v]\) ou \( \lambda\otimes v\) ou encore \( \lambda\otimes_{\eK}v\) la classe de \( (\lambda,v)\).
+\end{definition}
+Un élément de \( E_{\eL}\) est de la forme \( \sum_k[\lambda_k,v_k]\) avec \( \lambda_k\in \eL\) et \( v_k\in E\). Si \( f\colon E\to F\) est une applications linéaire nous définissons
+\begin{equation}
+    \begin{aligned}
+        f_{\eL}\colon E_{\eL}&\to F_{\eL} \\
+        [\lambda,v]&\mapsto [\lambda,f(v)]. 
+    \end{aligned}
+\end{equation}
+
+\begin{remark}
+    Si deux vecteurs de \( E_{\eL}\) sont linéairement indépendants pour \( \eK\), ils ne le sont pas spécialement pour \( \eL\). Par exemple si \( \eC\) est vu comme \( \eR\)-espace vectoriel, alors \( \{ 1,i \}\) est une partie libre. Mais dans \( \eC\) vu comme \( \eC\)-espace vectoriel, la partie \( \{ 1,i \}\) n'est pas libre.
+\end{remark}
+
+Nous définissons aussi l'injection canonique
+\begin{equation}
+    \begin{aligned}
+        \iota\colon E&\to E_{\eL} \\
+        v&\mapsto [1,v]. 
+    \end{aligned}
+\end{equation}
+
+\begin{proposition}[\cite{ooEPEFooQiPESf}]      \label{PropooWECLooHPzIHw}
+    Injectivité et surjectivité respectées.
+    \begin{enumerate}
+        \item
+            L'application \( f_{\eL}\) est injective si et seulement si \( f\) est injective.    
+        \item
+            L'application \( f_{\eL}\) est surjective si et seulement si \( f\) est surjective.    
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Supposons pour commencer que \( f_{\eL}\) est injective.
+    Le diagramme
+    \begin{equation}
+        \xymatrix{%
+            E \ar[r]^-{f}\ar[d]_-{\tau}      &   F\ar[d]^{\tau}\\
+            E_{\eL} \ar[r]_{f_{\eL}}  &   F_{\eL}
+           }
+    \end{equation}
+    est un diagramme commutatif. En effet 
+    \begin{equation}
+        (\tau\circ f)(v)=[1,f(v)]
+    \end{equation}
+    tandis que 
+    \begin{equation}
+        (f_{\eL\circ\tau})(v)=f_{\eL}[1,v]=[1,f(v)].
+    \end{equation}
+    Donc si \( f(v)=0\) avec \( v\neq 0\) nous aurions \( (\tau\circ f)(v)=0\) et donc aussi \( (f_{\eL}\circ \tau)(v)=0\), alors que \( \tau(v)\neq 0\) dans \( E_{\eL}\).
+
+    Réciproquement, nous supposons que \( f\) est injective et nous prouvons que \( f_{\eL}\) est injective. Par le lemme \ref{LEMooDAACooElDsYb}\ref{ITEMooEZEWooZGoqsZ}, nous savons qu'il existe \( g\colon F\to E\) telle que \( f\circ g=\id|_F\). Nous en déduisons que \( f_{\eL}\circ g_{\eL}=\id|_{F_{\eL}}\) parce que si \( [\lambda,v]\in F_{\eL}\) alors
+    \begin{equation}
+        (f_{\eL}\circ g_{\eL})[\lambda,v]=f_{\eL}[\lambda,g(v)]=[\lambda,(f\circ g)(v)]=[\lambda,v].
+    \end{equation}
+    Notons que \( g\) est injective, donc \( g_{\eL}\) est injective et l'égalité \( f_{\eL}\circ g_{\eL}=\id|_{F_{\eL}} \) implique que \( f_{\eL}\) est également injective.
+\end{proof}
+
+\begin{proposition}[\cite{MonCerveau}] \label{PROPooMHARooUycAts}
+    Soit \( \{ e_i \}_{i=1,\ldots, p}\) une base de \( E\). Alors \( \{ 1\otimes e_i \}_i\) est une base de \( E_{\eL}=\eL\otimes_{\eK}E\).
+\end{proposition}
+
+\begin{proof}
+    L'espace vectoriel \( E\) peut être écrit comme somme directe \( E=\bigoplus_i\eK e_i\). Si \( \lambda\in \eL\) et \( k\in \eK\) nous avons
+    \begin{equation}
+        \lambda\otimes ke_i=\frac{ \lambda }{ k }\otimes e_i=\frac{ \lambda }{ k }(1\otimes e_i).
+    \end{equation}
+    Cela pour introduire que l'application
+    \begin{equation}
+        \begin{aligned}
+            \psi\colon \eL\otimes_{\eK}E&\to \bigoplus_i\eL(1\otimes e_i) \\
+            \sum_k \lambda_k\otimes v_k&\mapsto \oplus_i \sum_k(\lambda_k v_{ik})(1\otimes e_i)
+        \end{aligned}
+    \end{equation}
+    où \( v_k=\sum_i v_{ik}e_i\) avec \( v_{ik}\in \eK\) est un isomorphisme de \( \eL\)-espaces vectoriels. La surjectivité est facile. En ce qui concerne l'injectivité, si
+    \begin{equation}
+        \sum_i\sum_k(\lambda_kv_{ik})(1\otimes e_i)=0
+    \end{equation}
+    alors les choses suivantes sont nulles également :
+    \begin{equation}
+        \sum_i\sum_k(\lambda_kv_{ik})(1\otimes e_i)=\sum_{ik}(\lambda_k\otimes v_{ik}e_i)=\sum_k(\lambda_k\otimes \sum_iv_{ik}e_i)=\sum_k(\lambda_k\otimes v_k).
+    \end{equation}
+    Le dernier est l'argument de \( \psi\). Le fait que ce soit nul implique que \( \psi\) est injective.
+\end{proof}
+
+\begin{remark}
+    Nous n'avons pas dû prouver que chacun des \( \lambda_k\otimes v_k\) était nul. Et encore heureux, parce que cela pouvait très bien être faux, vu qu'il y a plusieurs façons de noter un élément de \( E_{\eL}\) sous la forme de tels termes.
+\end{remark}
+
+\begin{corollary}       \label{CORooTQGHooIKhNtr}
+    La \( \eL\)-dimension de \( E_{\eL}\) est égale à la \( \eK\)-dimension de \( E\).
+\end{corollary}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Projections}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{probleme}
+    Nous allons définir \( \pr\colon \aL(E_{\eL},F_{\eL})\to \aL(E,F)\) en faisant appel à des bases et en prouvant que les choses définies ne dépendent pas des bases choisies. Il y a sûrement une façon plus «intrinsèque» de faire.
+\end{probleme}
+
+
+Nous savons que \( \eL\) est un \( \eK\)-espace vectoriel dans lequel nous pouvons voir \( \eK\) comme un sous-espace (lemme \ref{LemooOLIIooXzdppM}). Dans cette optique nous choisissons dans \( \eL\) un supplémentaire de \( \eK\), c'est à dire un sous-espace vectoriel de \( \eL\) tel que
+\begin{equation}
+    \eL=\eK\oplus V.
+\end{equation}
+Nous avons alors naturellement une projection \( \pr\colon \eL\to \eK\).
+
+Soit \( \{ e_i \}\) une base de \( E \) et \(\{ e_a \}\) une  de\( F\). Nous noterons également \( e_i\) et \( e_a\) les éléments \( \tau e_i\) et \( \tau e_a\) correspondants. Grâce à la proposition \ref{PROPooMHARooUycAts}, ce sont des bases de \( E_{\eL}\) et \( F_{\eL}\). Si la fonction \( f\colon E_{\eL}\to F_{\eL}\) s'écrit dans ce ces bases comme
+\begin{equation}
+    f(e_i)=\sum_af_{ai}e_a
+\end{equation}
+alors nous définissons \( \pr(f)\) par 
+\begin{equation}        \label{EQooSAFRooJnfkLO}
+    (\pr f)e_i=\sum_a\pr(f_{ai})e_a.
+\end{equation}
+
+\begin{proposition}[\cite{MonCerveau}]      \label{PROPooOEHTooHyjuZQ}
+    L'application \( \pr\) définie en \eqref{EQooSAFRooJnfkLO} est indépendante du choix des bases.
+\end{proposition}
+
+\begin{proof}
+    Notons que dans ce qui suit, les sommes sur \( a\) ou \( b\) et celles sur \( i\) ou \( j\) ne vont pas jusqu'au même indice (dimensions de \( E\) et \( F\)). De plus nous manipulons deux choses qui se notent \( \pr\). La première est la projection \( \pr\colon \eL\to \eK\) qui ne dépend que d'un choix de supplémentaire et que nous supposons fixée ici. D'autre part il y a \( \pr\colon E_{\eL}\to E\) qui dépend a priori des bases choisies.
+
+    Nous choisissons de nouvelles bases qui sont liées aux anciennes bases par
+    \begin{subequations}
+        \begin{numcases}{}
+            e'_b=\sum_aB_{ab}e_a\\
+            e'_i=\sum_jA_{ji}e_j.
+        \end{numcases}
+    \end{subequations}
+    Les matrices \( A\) et \( B\) sont dans \( \GL(\eK)\). Nous allons écrire l'opérateur \( \pr'\) qui correspond à ces bases et montrer que pour toute application linéaire \( f\colon E_{\eL}\to F_{\eL} \) nous avons \( \pr(f)=\pr'(f)\). Nous avons :
+    \begin{subequations}
+        \begin{align}
+            f(e'_j)&=\sum_iA_{ji}f(e_i)\\
+            &=\sum_a\sum_b\sum_iA_{ji}f_{ai}(B^{-1})_{ba}e'b\\
+            &=\sum_b\Big( \sum_{ai}A_{ji}f_{ai}(B^{-1})_{ba} \Big)e'b,
+        \end{align}
+    \end{subequations}
+    ce qui fait que
+    \begin{equation}        \label{EQooUQNBooMWHRbD}
+        (\pr'f)e'_j=\sum_b\Big( \pr\big( A_{ji}f_{ai}(B^{-1})_{ba} \big) \Big)e'_b.
+    \end{equation}
+    Nous calculons maintenant \( (\pr'f)e_j\) en substituant \( e_j=\sum_l(A^{-1})_{lj}e'_l\) et en utilisant \eqref{EQooUQNBooMWHRbD} et la linéarité de \( \pr'\) et la \( \eK\)-linéarité de \( \pr\colon \eL\to \eK\) :
+    \begin{subequations}
+        \begin{align}
+            (\pr'f)\Big( \sum_l(A^{-1})_{lj}e'_l \Big)
+            &=\sum_l(A^{-1})_{lj}\sum_b\sum_{ai}\pr\big(A_{li}f_{ai}(B^{-1})_{ba}\big)e_b\\
+            &=\sum_a\pr(f_{aj})e_a\\
+            &=(\pr f)e_j.
+        \end{align}
+    \end{subequations}
+    Donc \( \pr=\pr'\).
+\end{proof}
+
+Note au passage comme toujours : il y a un abus systématique de notation entre \( e_i\in E\) et \( \tau(e_i)=1\otimes e_i\in E_{\eL}\).
+
+\begin{remark}[\cite{MonCerveau}]       \label{REMooBEXGooLgpHzg}
+    L'opération \( \pr\colon \aL(E_{\eL},F_{\eL})\to \aL(E,F)\) ne dépend pas des bases choisies un peu partout. Mais elle dépend de l'application \( pr\colon \eL\to \eK\) déjà construite. Et celle-là dépend du choix d'un supplémentaire $V$ qui fournit \( \eL=\eK\oplus V\).
+
+    Si \( \pr(\lambda)=0\) pour un de ces choix, cela n'implique nullement que \( \lambda=0\). Penser à \( i\in \eC\) si la projection \( \pr\colon \eC\to \eR\) est l'application \( (x+iy)\mapsto x\) parallèle à l'axe des imaginaires.
+
+    Par contre si \( \pr(\lambda)=0\) pour tout choix de \( V\), alors nous avons bien \( \lambda=0\). Dans la suit nous «fixons» un choix de \( V\) générique, et lorsque nous rencontrerons l'égalité \( \pr(\lambda)=0\) nous en déduirons \( \lambda=0\).
+\end{remark}
+
+\begin{proposition} \label{PROPooPWDKooFNFWRI}
+    Si \( f\colon E\to F\) et si \( f_{\eL}e_j=\sum_a(f_{\eL})_{aj}e_a\) et si \( f(e_j)=\sum_af_{aj}e_a\) alors
+    \begin{enumerate}
+        \item
+            \( \pr f_{\eL}=f\),
+        \item       \label{ITEMooNMPYooXosGhI}
+            \( (f_{\eL})_{ja}=f_{ja} \in \eK\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Nous avons
+    \begin{equation}
+        f_{\eL}(e_i)=\sum_a f_{ai}(1\otimes e_a)=\sum_a f_{ai}\tau(e_a),
+    \end{equation}
+    donc
+    \begin{equation}
+        (\pr f_{\eL})e_i=\sum_a\pr(f_{ai})e_a=\sum_af_{ai}e_a=f(e_i).
+    \end{equation}
+    Cela prouve que \( \pr f_{\eL}=f\).
+
+    Par ailleurs, 
+    \begin{equation}        \label{EQooIOTFooNAdkit}
+        f_{\eL}(\tau e_i)=f_{\eL}(1\otimes e_i)=1\otimes f(e_i)=\tau\big( f(e_i) \big)=\sum_af_{ai}\tau(e_a)
+    \end{equation}
+    alors que par définition,
+    \begin{equation}        \label{EQooMYSCooPFWATG}
+        f_{\eL}(\tau e_i)=\sum_a(f_{\eL})_{ai}\tau(e_a).
+    \end{equation}
+    Les éléments \( \tau(e_a)\) formant une base\footnote{Encore la proposition \ref{PROPooMHARooUycAts}.}, la comparaison de \eqref{EQooIOTFooNAdkit} avec \eqref{EQooMYSCooPFWATG} donne \( (f_{\eL})_{ai}=f_{ai}\in \eK\).
+\end{proof}
+
+\begin{lemma}       \label{LEMooWZGSooONEnjZ}
+    Soient
+    \begin{enumerate}
+        \item
+            Une base \( \{ e_i \}\) de \( E\) et une application linéaire \( f\colon E\to F\);
+        \item
+            une base \( \{ e_a \}\) de \( F\) et une application linéaire \( g\colon G\to F\);
+        \item
+            une base \( \{ e_{\alpha} \} \) de \( G\) et une application linéaire \( \tilde h\colon G_{\eL}\to E_{\eL}\).
+    \end{enumerate}
+    Alors nous avons
+    \begin{equation}
+        \pr(f_{\eL}\circ \tilde h)=\pr(f_{\eL})\circ\pr(\tilde h).
+    \end{equation}
+\end{lemma}
+
+\begin{proof}
+    Pour écrire \( \pr(f_{\eL}\circ \tilde h)\) à partir de la définition \eqref{EQooSAFRooJnfkLO} nous commençons par écrire
+    \begin{equation}
+        (f_{\eL}\circ \tilde h)e_{\alpha}=\sum_a(f_{\eL}\circ \tilde h)_{a\alpha}e_a=\sum_{ai}(f_{\eL})_{ai}(\tilde h)_{i\alpha}e_a=\sum_a\Big( \sum_{i}f_{ai}(\tilde h)_{i\alpha} \Big)e_a
+    \end{equation}
+    où nous avons utilisé le fait que \( (f_{\eL})_{ai}=f_{ai}\). Donc, en utilisant la \( \eK\)-linéarité de \( \pr\),
+    \begin{equation}        \label{EQooZGCGooQsCBQH}
+        \pr(f_{\eL}\circ \tilde h)e_{\alpha}=\sum_a\sum_i\pr\Big( f_{ai}(\tilde h)_{i\alpha} \Big)e_a=\sum_a\sum_if_{ai}\pr\Big( (\tilde h)_{i\alpha} \Big)e_a.
+    \end{equation}
+    D'autre part,
+    \begin{equation}
+        \begin{aligned}[]
+            \pr(f_{\eL})\circ \pr(\tilde h)e_{\alpha}&=\pr(f_{\eL})\sum_i\pr\Big( (\tilde h)_{i\alpha} \Big)e_i\\
+            &=\sum_i\pr\Big( (\tilde h)_{i\alpha} \Big)\sum_af_{ai}e_a\\
+            &=\sum_{ai}\pr\Big( (\tilde h)_{i\alpha} \Big)f_{ai}e_a,
+        \end{aligned}
+    \end{equation}
+    et c'est égal à \eqref{EQooZGCGooQsCBQH}.
+\end{proof}
+
+\begin{remark}
+    Nous n'avons en général pas \( \pr(xy)=\pr(x)\pr(y)\) pour tout \( x,y\in \eL\). Par exemple si \( \eK=\eR\) et \( \eL=\eC\) avec la projection canonique,
+    \begin{equation}
+        \pr(i\cdot i)=\pr(-1)=-1
+    \end{equation}
+    alors que \( \pr(i)=0\).
+\end{remark}
+
+\begin{proposition}
+    Soit \( f\in\aL(E,F)\) et \( g\in\aL(F,E)\). Alors il existe \( h\colon G\to E\) tel que \( f\circ h=g\) si et seulement s'il existe \( \tilde g\colon G_{\eL}\to E_{\eL}\) tel que \( f_{\eL}\circ \tilde g=g_{\eL}\).
+\end{proposition}
+
+\begin{proof}
+    Dans le sens direct, il suffit de poser \( \tilde h=h_{\eL}\).
+
+    Dans le sens inverse, si nous avons \( \tilde h\colon G_{\eL}\to E_{\eL}\) tel que \( f_{\eL}\circ\tilde h=g_{\eL}\) alors en appliquant \( \pr\) des deux côtés et en utilisant le lemme \ref{LEMooWZGSooONEnjZ},
+    \begin{equation}
+        \pr(f_{\eL})\circ\pr(\tilde h)=\pr(g_{\eL})
+    \end{equation}
+    c'est à dire
+    \begin{equation}
+        f\circ\pr(\tilde h)=g,
+    \end{equation}
+    c'est à dire que l'application \( \pr\tilde h\colon G\to E\) est la réponse à la proposition.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Rang, polynôme minimal, polynôme caractéristique}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[Stabilité du rang par extension des scalaires\cite{ooEPEFooQiPESf}]     \label{PROPooJFQDooZSsxMf}
+    Si \( f\colon E\to F\) est linéaire alors nous avons
+    \begin{equation}
+        \rang(f)=\rang(f_{\eL}).
+    \end{equation}
+    où à droite nous considérons le rang de l'application \( \eL\)-linéaire \( f_{\eL}\colon E_{\eL}\to F_{\eL}\).
+\end{proposition}
+
+\begin{proof}
+    Il existe un supplémentaire \( V\) tel que \( E=\ker(f)\oplus V\) avec \( \dim(V)=\rang(f)\). Nous pouvons factoriser \( f\) en 
+    \begin{equation}
+        f=f_2\circ f_1
+    \end{equation}
+    avec \( f_1\colon E\to V\) est la projection parallèle à \( \ker(f)\) et est surjective (vers \( V\)) parce que \( \dim(V)=\rang(f)=\dim\big( \Image(f) \big)\). De plus \( f_2\colon V\to F\) est injective parce que si \( v\in V\) est tel que \( f_2(v)=0\) alors on aurait
+    \begin{equation}
+        f(v)=(f_2\circ f_1)(v)=f_2(v)=0.
+    \end{equation}
+    Cela donne \( v\in\ker(f)\cap V=\{ 0 \}\). Par la proposition \ref{PropooWECLooHPzIHw}, les applications \( (f_1)_{\eL}\) et \( (f_2)_{\eL}\) sont respectivement surjective et injective.
+
+    L'application \( (f_2)_{\eL}\colon V_{\eL}\to F_{\eL}\) est forcément surjective sur son image, donc
+    \begin{equation}
+        (f_2)_{\eL}\colon V_{\eL}\to \Image(f_{\eL}) 
+    \end{equation}
+    est un isomorphisme de \( \eL\)-espaces vectoriels. Nous avons alors les égalités
+    \begin{equation}        \label{EQooWLOIooKlYWTL}
+        \dim_{\eL}(V_{\eL})=\dim_{\eL}\big( \Image(f_{\eL}) \big)=\rang(f_{\eL}).
+    \end{equation}
+    Mais aussi, par les définitions posées plus haut,
+    \begin{equation}        \label{EQooEVCGooAGjmoU}
+        \dim(V)=\rang(f)=\dim\big( \Image(f) \big).
+    \end{equation}
+    Mais le corollaire \ref{CORooTQGHooIKhNtr} nous dit que \( \dim_{\eL}(V_{\eL})=\dim_{\eK}(V)\). Donc il y a égalité des deux lignes \eqref{EQooWLOIooKlYWTL} et \eqref{EQooEVCGooAGjmoU} donne \( \rang(f)=\rang(f_{\eL})\).
+\end{proof}
+
+\begin{proposition}     \label{PROPooZAZFooUFdCUv}
+    Nous avons
+    \begin{enumerate}
+        \item
+            \( \det(f)=\det(f_{\eL})\)
+        \item
+            \( \chi_f=\chi_{f_{\eL}}\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Dès que l'on a des bases nous avons \( (f_{\eL})_{ai}=f_{ai}\) par la proposition \ref{PROPooPWDKooFNFWRI}\ref{ITEMooNMPYooXosGhI}. Le nombre \( \det(f)\in \eK\) est un polynôme en les \( f_{ai}\). Entendons nous : il existe un polynôme indépendant de \( f\) et de \( \eK\) et de \( \eL\) donnant le déterminant de n'importe quelle matrice. Donc \( \det(f)=\det(f_{\eL})\).
+
+    Même chose pour le polynôme caractéristique (définition \ref{DefOWQooXbybYD}) : les coefficient de ce polynôme sont des polynôme en les \( f_{ai}\) qui sont indépendants de \( \eL\), de \( \eK\) et de \( f\).
+
+    Notons que \( \chi_{f_{\eL}}\) est un polynôme à coefficients dans \( \eK\).
+\end{proof}
+
+La situation est très différente avec le polynôme minimal\footnote{Définition \ref{DefCVMooFGSAgL}.}. Autant il existe une «recette» pour créer le polynôme caractéristique, il n'en n'existe pas pour le polynôme minimal (ou en tout cas, il ne suffit pas d'appliquer des polynôme en les coefficients de la matrice). La proposition suivante montre que le polynôme minimal est conservé par extension de corps, mais que pour le voir, il faut travailler plus.
+
+\begin{proposition}[\cite{ooEPEFooQiPESf,MonCerveau}]      \label{PROPooXVZMooXcJrsJ}
+    Soit \( \eL\) une extension du corps \( \eK\) et une application linéaire \( f\colon E\to F\) entre deux \( \eK\)-espaces vectoriels. Alors \( \mu_f=\mu_{f_{\eL}}\).
+\end{proposition}
+
+\begin{proof}
+    Nous allons montrer que l'application
+    \begin{equation}
+        \begin{aligned}
+            \tilde g\colon \frac{ \eL[X] }{ (\mu) }&\to \End(E_{\eL}) \\
+            \bar P&\mapsto P(f_{\eL}) 
+        \end{aligned}
+    \end{equation}
+    est bien définie et injective. La proposition \ref{PROPooVUJPooMzxzjE} nous dira alors que \( \mu\) est le polynôme minimal de \( f_{\eL}\).
+
+    Pour prouver que l'application \( \tilde g\) est bien définie, nous commençons par prouver que  \( P(f_{\eL})=P(f)_{\eL}\) :
+    \begin{subequations}
+        \begin{align}
+            P(f_{\eL})\lambda\otimes v&=\sum_ka_kf_{\eL}^k\lambda\otimes v\\
+            &=\lambda\otimes \sum_ka_kf^k(v)\\
+            &=\lambda\otimes P(f)v\\
+            &=P(f)_{\eL}\lambda\otimes v.
+        \end{align}
+    \end{subequations}
+    Par conséquent \( \mu(f_{\eL})=0\) et l'application est bien définie.
+
+    Sur \( \eL[X]/(\mu)\) nous considérons la base \( \{ 1,\bar X,\ldots, \bar X^{\deg(\mu)-1} \}\), et \( \End(E_{\eL})\) nous considérons une base qui commence\footnote{Théorème de la base incomplète \ref{ThonmnWKs}\ref{ITEMooFVJXooGzzpOu}.} par \( \{ f_{\eL}^k \}_{k=0,\ldots, \deg(\mu)-1}\). Montrons tout de même que cette partie est libre (sinon le théorème de la base incomplète ne s'applique pas) : si \( \sum_k\lambda_kf_{\eL}^k=0\) alors
+    \begin{equation}        \label{EQooSFHVooLxqUEl}
+        \sum_k\pr\big( \lambda_k f_{\eL}^k\big)=0.
+    \end{equation}
+    Pour détailler ce que cela implique, nous calculons ceci :
+    \begin{equation}
+        (\lambda f_{\eL})(\tau e_i)=\lambda f_{\eL}(\tau e_i)=\sum_a \lambda f_{ia}e_a,
+    \end{equation}
+    par conséquent \( \pr(\lambda f_{\eL})e_i=\sum_a\pr(\lambda f_{ia})e_a\), et comme \( \pr\) est \( \eK\)-linéaire et que \( f_{ai}\in \eK\),
+    \begin{equation}
+        \pr(\lambda f_{\eL})e_i=\pr(\lambda)\sum_a f_{ai}e_a=\pr(\lambda)\pr(f_\eL)e_i=\pr(\lambda)f(e_i).
+    \end{equation}
+    Appliquer la projection \( \pr\) à l'équation \eqref{EQooSFHVooLxqUEl} donne alors \( \sum_k\pr(\lambda)_kf^k=0\). Mais comme les \( f^k\) sont linéairement indépendantes sur \( \eK\) nous avons pour tout \( k\) : \( \pr(\lambda_k)=0\) (égalité dans \( \eK\)). En nous souvenant de la remarque \ref{REMooBEXGooLgpHzg} nous en déduisons \( \lambda_k=0\) dans \( \eL\).
+
+    Dans les choix de bases faits, l'application \( \tilde g\) a la forme
+    \begin{equation}
+        \tilde g=\begin{pmatrix}
+            \begin{matrix}
+                1    &       &       \\
+                    &   1    &       \\
+                    &       &   1
+            \end{matrix}\\ 
+            \begin{matrix}
+                *    &   *    &   *    \\
+                *    &   *    &   *    \\
+                *    &   *    &   *    
+            \end{matrix}
+        \end{pmatrix},
+    \end{equation}
+    qui est injective.
+
+    Vu que \( \tilde g\) est injective, \( \mu\) est le polynôme minimal de \( f_{\eL}\) et donc \( \mu=\mu_{\eL}\).
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Frobenius et Jordan}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Matrice compagnon}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooOSVAooGevsda}
+    Soit le polynôme \( P=X^n-a_{n-1}X^{n-1}-\ldots-a_1X-a_0\) dans \( \eK[X]\). La \defe{matrice compagnon}{matrice!compagnon} de \( P\) est la matrice\nomenclature[A]{\( C(P)\)}{matrice compagnon} donnée par
+    \begin{equation}
+        C(P)=\begin{pmatrix}
+            0    &   \cdots    &   \cdots    &   0    &   a_0\\  
+            1    &   0    &       &   \vdots    &   a_1\\  
+            0    &   \ddots    &   \ddots    &   \vdots    &   \vdots\\  
+            \vdots    &   \ddots    &   \ddots    &   0    &   a_{n-2}\\  
+            0    &   \cdots    &   0    &   1    &   a_{n-1}    
+        \end{pmatrix}
+    \end{equation}
+    si \( n\geq 2\) et par \( (a_0)\) si \( n=1\). 
+
+    Une matrice est dite compagnon si elle a cette forme.
+\end{definition}
+
+\begin{proposition}
+    Si \( f\) est l'endomorphisme associé à la matrice \( C(P)\) nous avons
+    \begin{equation}
+        f(e_i)=\begin{cases}
+            e_{i+1}    &   \text{si } i<n\\
+            (a_0,\ldots, a_{n-1})    &    \text{si } i=n.
+        \end{cases}
+    \end{equation}
+    De plus l'endomorphisme \( f\) vérifie \( P(f)e_1=0\).
+\end{proposition}
+
+\begin{lemma}[\cite{RapportArgreg2011}] \label{LemkVNisk}
+    Un polynôme sur un corps commutatif est le polynôme caractéristique de sa matrice compagnon. En d'autres termes nous avons \( \chi_{C(P)}=P\).
+\end{lemma}
+
+\begin{proof}
+    Nous notons \( f\) l'endomorphisme associé à \( C(P)\). La propriété \( P(f)e_1=0\) nous indique que le polynôme minimal ponctuel de \( f\) en \( e_1\) divise \( P\). L'ensemble des puissances de \( f\) appliquées à \( e_1\), \( \big( f^i(e_1) \big)_{i=1,\ldots, n-1}\) est libre, donc le polynôme minimal ponctuel en \( e_1\) est de degré \( n\) au minimum. En reprenant les notations du théorème \ref{ThoCCHkoU}, nous avons \( I_{e_1}=(P)\) parce que \( P\) est de degré minimum dans \( I_{e_1}\) et \( \chi_f\in I_{e_1}\).
+
+    Donc \( P\) divise \( \chi_f\) et est de degré égal à celui de \( \chi_f\). Étant donné qu'ils sont tous deux unitaires, ils sont égaux.
+\end{proof}
+
+\begin{remark}  \label{RemmQjZOA}
+    Les matrices compagnons ne sont pas les seules dont le polynôme caractéristique est égal au polynôme minimal. En fait les matrices dont le polynôme caractéristique est égale au polynôme minimal sont denses dans les matrices. En effet une matrice dont le polynôme minimal n'est pas égal au polynôme caractéristique a un polynôme caractéristique avec une racine double. Il est possible, en modifiant arbitrairement peu la matrice de séparer la racine double en deux racines distinctes.
+\end{remark}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Réduction de Frobenius}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}       \label{LEMooKUQDooKFeIYq}
+    Soit un endomorphisme \( f\colon E\to E\) sur l'espace vectoriel de dimension finie \( n\). Nous notons \( \mu\) et \( \chi\) les polynômes minimal et caractéristique. Si \( f\) est cyclique, alors \( \mu=\chi\).
+\end{lemma}
+Le théorème \ref{THOooGLMSooYewNxW} donnera une version plus complète de ce lemme.
+
+\begin{proof}
+    Soit \( v\) un vecteur cyclique de \( f\), c'est à dire que \( \{ f^k(v) \}_{k=0,\ldots, n-1}\) est libre. Donc si \( P\) est un polynôme de degré jusqu'à \( n-1\) nous ne pouvons pas avoir \( P(f)=0\) parce que, appliqué à \( v\), ce serait une combinaisons nulle non triviale des \( f^k(v)\). Donc le polynôme minimal est au minimum de degré \( n\). Mais le polynôme caractéristique est annulateur de degré \( n\) (Cayley-Hamilton \ref{ThoCalYWLbJQ}), donc il est le polynôme minimal.
+\end{proof}
+
+\begin{theorem}[Réduction de Frobenius \cite{AutourFrobCompa,Vialivs,MoncetIVS}]        \label{THOooDOWUooOzxzxm}
+    Soit \( E\), un \( \eK\)-espace vectoriel, et \( f\in \End(E)\). Alors il existe une suite de sous-espaces \( E_1,\ldots, E_r\) stables par \( f\) tels que
+    \begin{enumerate}
+        \item   \label{ItemmpwjnSs}
+            \( E=\bigoplus_{i=1}^rE_i\);
+        \item
+            pour chaque \( E_i\), l'endomorphisme restreint \( f_i=f|_{E_i}\) est cyclique;
+        \item
+            si \( \mu_i\) est le polynôme minimal de \( f_i\) alors \( \mu_{i+1}\) divise \( \mu_i\);
+    \end{enumerate}
+    Une telle décomposition vérifie automatiquement \( \mu_1=\mu_f\) et \( \mu_1\cdots \mu_r=\chi_f\), et la suite \( (\mu_i)_{i=1,\ldots, r}\) ne dépend que de \( f\) et non du choix de la décomposition du point \ref{ItemmpwjnSs}.
+\end{theorem}
+   \index{réduction!Frobénius}
+   \index{Frobénius!réduction}
+
+Les polynômes \( \mu_i\) sont les \defe{invariants de similitude}{invariant!de similitude} de l'endomorphisme \( f\).
+
+\begin{proof}
+    Nous commençons par montrer que si une telle décomposition existe, alors
+    \begin{subequations}    \label{subEqzcGouz}
+        \begin{align}
+            \chi_f=\prod_{i=1}^r\mu_i  \label{EqTaxsvb}\\
+            \mu_f=\mu_1
+        \end{align}
+    \end{subequations}
+    où \( \chi_f\) est le polynôme caractéristique de \( f\) et \( \mu_f\) est le polynôme minimal. D'abord le polynôme caractéristique de \( f\) devra être égal au produit des polynômes caractéristique des \( f|_{E_i}\), mais ces derniers endomorphismes étant cycliques\footnote{Définition \ref{DEFooFEIFooNSGhQE}.}, leurs polynôme caractéristiques sont égaux à leurs polynômes minimaux (lemme \ref{LEMooKUQDooKFeIYq}). Cela prouve l'égalité \eqref{EqTaxsvb}. Ensuite tous les \( \mu_i\) doivent diviser le polynôme minimal, donc \( \ppcm(\mu_1,\ldots, \mu_r)\) divise \(\mu_f\). Cependant le polynôme minimal doit contenir une et une seule fois chacun des facteurs irréductibles du polynôme caractéristique, et chacun de ces facteurs sont dans les polynômes \( \mu_i\). Par conséquent \( \ppcm(\mu_1,\ldots, \mu_r)=\mu_f\). Mais par ailleurs \( \mu_1=\ppcm(\mu_1,\ldots, \mu_r)\) parce qu'on a supposé \( \mu_{i+1}\divides \mu_i\), donc \( \mu_1=\mu_f\).
+    
+    Soit \( d\), le degré du polynôme minimal de \( f\) et \( y\in E\) tel que \( \mu_f=\mu_{f,y}\) (voir lemme \ref{LemSYsJJj}). Le plus petit espace stable sous \( f\) contenant \( y\) est
+    \begin{equation}
+        E_y=\Span\{ y,f(y),\ldots, f^{d-1}(y) \}.
+    \end{equation}
+    Nous notons \( e_i=f^{i-1}(y)\). Notons que les vecteurs donnés forment bien une base de \( E_y\) parce que si les \( e_i\) n'était pas linéairement indépendants, alors nous aurions des \( a_k\) tels que \( \sum_ka_ke_k=0\) et avec lesquels
+    \begin{equation}
+        \big( \sum_ka_kX^k \big)(f)y=0,
+    \end{equation}
+    ce qui contredirait la minimalité de \( \mu_{f,y}\).
+
+    La difficulté du théorème est de trouver un complément de \( E_y\) qui soit également stable sous \( f\). Nous commençons par étendre\quext{Pour autant que j'aie compris, cette extension manque dans \cite{AutourFrobCompa}. Corrigez moi si je me trompe.} \( \{ e_1,\ldots, e_d \}\) en une base \( \{ e_1,\ldots, e_n \}\) de \( E\). Ensuite nous allons montrer que
+    \begin{equation}
+        E=E_y\oplus F
+    \end{equation}
+    avec
+    \begin{equation}
+        F=\{ x\in E\tq  e^*_d\big( f^k(x) \big)=0\forall k\in \eN \}.
+    \end{equation}
+    Par construction, \( F\) est invariant sous \( f\). Montrons pour commencer que \( E_y\cap F=\{ 0 \}\). Un élément de \( E_y\) s'écrit
+    \begin{equation}
+        z=a_1e_1+\cdots +a_ke_k
+    \end{equation}
+    avec \( k\leq d\). Étant donné que \( f\) décale les vecteurs de base, nous avons \( e^*_d\big( f^{d-k}(z) \big)=a_k\). Du coup \( z\in F\) si et seulement si \( a_1=\ldots=a_d=0\), c'est à dire que \( E_y\cap F=\{ 0 \}\).
+
+    Nous montrons maintenant que \( \dim F=n-d\). Pour cela nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            T\colon \eK[F]&\to E^* \\
+            g&\mapsto e^*_d\circ g. 
+        \end{aligned}
+    \end{equation}
+    Cette application est injective. En effet un élément général de \( \eK[f]\) est
+    \begin{equation}
+        g=a_1\id+a_2f+\cdots +a_pf^{p-1}
+    \end{equation}
+    avec \( p\leq d\). Si \( T(g)=0\), alors nous avons en particulier
+    \begin{equation}
+        0=T(g)e_{_d-p+1}=e^*_d(a_1e_{d-p+1}+a_2e_{d-p+2}+\cdots +a_pe_d)=a_p.
+    \end{equation}
+    Donc \( a_p=0\) et en appliquant maintenant \( T(g)\) à \( e_{d-p}\) nous obtenons \( a_{p-1}=0\). Au final nous trouvons que \( g=0\) et donc que \( T\) est injective.
+
+    Étant donné que \( \dim\eK[f]=d\) et que \( T\) est injective, \( \dim\Image(T)=d\). Nous regardons l'orthogonal de l'image :
+    \begin{subequations}
+        \begin{align}
+            (\Image(T))^{\perp}&=\{ x\in E\tq T(g)x=0\forall g\in\eK[f] \}\\
+            &=\{ x\in E\tq e^*_d\big( g(x) \big)=0\forall g\in \eK[f] \}\\
+            &=F.
+        \end{align}
+    \end{subequations}
+    Par conséquent \( F^{\perp}=\Image(T)\). Vu que \( \dim\Image(T)=d\), nous avons donc \( \dim F=n-d\) et il est établi que \( E=E_y\oplus F\). 
+
+    Nous avons donc trouvé \( F\), stable par \( f\) et tel que \( E=E_y\oplus F\). Nous devons maintenant nous assurer que cette décomposition tombe bien pour les polynômes minimaux. Si \( P_1\) est le polynôme minimal de \( f|_{E_yj}\), alors par le lemme \ref{LemAGZNNa} nous avons \( P_1=\mu_{f,y}=\mu_f\) parce que \( f|_{E_y}\) est cyclique sur \( E_y\). Mettons \( P_2\), le polynôme minimal de \( f|_F\). Étant attendu que \( F\) est stable par \( f\), le polynôme \( P_2\) divise \( P_1\). En recommençant la construction sur \( F\), nous construisons un nouvel espace \( F'\) stable sous \( F\) et vérifiant \( \mu_{f|_{F'}}=P_2\), etc.
+
+    Nous passons maintenant à la partie unicité du théorème. Soient deux suites \( F_1,\ldots, F_r\) et \( G_1,\ldots, G_s\) de sous-espaces stables par \( f\) et vérifiant
+    \begin{enumerate}
+        \item
+            \( E=\bigoplus_{i=1}^rF_i\),
+        \item
+            \( f|_{F_i}\) est cyclique,
+        \item
+            \( \mu_{f|_{F_{i+1}}}\) divise \( \mu_{f|_{F_i}}\),
+    \end{enumerate}
+    et, \emph{mutatis mutandis}, les mêmes conditions pour la famille \( \{ G_i \}\). Nous posons \( P_i=\mu_{f_{F_i}}\) et \( Q_i=\mu_{f|_{G_i}}\). Nous allons montrer par récurrence que \( P_i=Q_i\) et \( \dim F_i=\dim G_i\). Il ne sera cependant pas garanti que \( F_i=G_i\). D'abord, \( P_1=Q_1\) parce qu'ils sont tous deux égaux à \( \mu_f\) par les relations \eqref{subEqzcGouz}. Nous supposons que \( P_i=Q_i\) pour \( i\leq 1\leq j-1\) et nous tentons de montrer que \( P_j=Q_j\).
+
+    Nous avons 
+    \begin{equation}    \label{EqMrCtZO}
+        P_j(f)=P_j(f)|_{F_1}\oplus\ldots\oplus P_j(f)|_{F_{j-1}}.
+    \end{equation}
+    En effet étant donné que \( P_{j+k}\) divise \( P_j\), nous avons\footnote{En vertu du lemme \ref{LemQWvhYb}.} \( P_{j}(f)=A(f)\circ P_{j+k}(f)\), mais \( P_{j+k}(f)F_{j+k}=0\), donc \( P_j(f)F_{j+k}=0\). Les espaces \( G_i\) n'ayant a priori aucun rapport avec les polynômes \( P_i\), nous écrivons
+    \begin{equation}    \label{EqJreLiO}
+        P_j(f)=P_j(f)|_{G_1}\oplus\ldots\oplus P_j(f)|_{G_{j-1}}\oplus P_j(f)|_{G_j}\oplus\ldots\oplus P_j(f)|_{G_s}.
+    \end{equation}
+    Pour \( 1\leq i\leq j-1\), nous avons supposé \( P_i=Q_i\). Étant donné que \( f|_{F_i}\) est semblable à \( C_{_i}\) et \( f|_{G_i}\) est semblable à \( C_{Q_i}\), la matrice de \( f|_{E_i}\) est semblable à la matrice de \( f|_{G_i}\). En particulier,
+    \begin{equation}
+        \dim P_j(f)F_i=\dim P_j(f)G_i.
+    \end{equation}
+    En prenant les dimensions des images dans les égalités \eqref{EqMrCtZO} et \eqref{EqJreLiO}, nous trouvons que
+    \begin{equation}
+        P_j(f)|_{G_j}=\ldots=P_j(f)|_{G_s}=0.
+    \end{equation}
+    Par conséquent \( P_j\in I_{f|G_j}\) et donc \( P_j\) divise \( Q_j\), qui est générateur de \( I_{f|_{G_j}}\). La situation étant symétrique entre \( P\) et \( Q\), nous montrons de même que \( Q_j\) divise \( P_j\) et donc que \( P_j=Q_j\).
+
+    Ceci achève la démonstration du théorème de réduction de Frobenius.
+
+\end{proof}
+
+\begin{remark}      \label{REMooPVLEooYDRXQI}
+    Sous forme matricielle, ce théorème dit que toute matrice est semblable à une matrice de la forme bloc-diagonale
+    \begin{equation}
+        f=\begin{pmatrix}
+            C_{\mu_1}    &       &       \\
+                &   \ddots    &       \\
+                &       &   C_{\mu_r}
+        \end{pmatrix}
+    \end{equation}
+    où les \( C_{\mu_i}\) sont les matrices compagnon (définition \ref{DEFooOSVAooGevsda}). 
+
+    En particulier, et ceci est très important, deux applications sont semblables si et seulement si elles ont même suite d'invariants de similitude.
+\end{remark}
+
+
+\begin{remark}
+    Si nous travaillons sur \( \eR\), la réduite de Frobenius restera une matrice réelle, même si les valeurs propres sont complexes. En effet le procédé de Frobenius ne regarde absolument pas les valeurs propres, mais seulement les facteurs irréductibles du polynôme caractéristique. La réduite de Frobenius ne tente pas de résoudre ces polynômes, mais se contente d'en utiliser les matrices compagnon.
+
+    La situation sera différente dans le cas de la forme normale de Jordan.
+\end{remark}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Forme normale de Jordan}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Il existe une preuve directe de la réduction de Jordan ne nécessitant pas la réduction de Frobenius\cite{LecLinAlgAllen}. Cette dernière passe par les espaces caractéristiques\footnote{Aussi appelés «espaces propres généralisés».} et est à mon avis plus compliquée que la démonstration de Frobenius elle-même. Nous allons donc nous contenter de donner la réduction de Jordan comme un cas particulier de Frobenius.
+
+\begin{theorem}[Réduction de Jordan]        \label{ThoGGMYooPzMVpe}
+    Soit \( E\) un espace vectoriel sur \( \eK\), et \( f\in\End(E)\) un endomorphisme dont le polynôme caractéristique \( \chi_f\) est scindé\footnote{C'est pour cette hypothèse que \( \eK=\eR\) n'est pas le bon cadre.}. Il existe une base de \( E\) dans laquelle la matrice de \( f\) s'écrit sous la forme
+    \begin{equation}
+        M=\begin{pmatrix}
+            J_{n_1}(\lambda_1)    &       &       \\
+                &   \ddots    &       \\
+                &       &   J_{n_k}(\lambda_k)
+        \end{pmatrix}
+    \end{equation}
+    où les \( \lambda_i\) sont les valeurs propres de \( f\) (avec éventuelle répétitions) et \( J_n(\lambda)\) représente le bloc \( n\times n\)
+    \begin{equation}
+        J_n(\lambda)=\begin{pmatrix}
+            \lambda    &   1    &       &       &   \\  
+                &   \lambda    &   1    &       &   \\  
+                &       &   \lambda    &       &   \\  
+                &       &       &   \ddots    &   1\\  
+                &       &       &       &   \lambda    
+        \end{pmatrix}.
+    \end{equation}
+    En d'autres termes, \( J_n(\lambda)_{ii}=\lambda\) et \( J_n(\lambda)_{i-1,i}=1\).    
+\end{theorem}
+\index{réduction!Jordan}
+\index{Jordan!réduction}
+
+\begin{proof}
+    Nous commençons par le cas où \( f\) est nilpotente; nous notons \( M\) sa matrice. Dans ce cas la seule valeur propre est zéro et le polynôme caractéristique est \( X^m\) pour un certain \( m\). Nous savons par le lemme \ref{LemkVNisk} que (la matrice de) \( f\) est semblable à sa matrice compagnon. En l'occurrence pour \( f\) nous avons
+    \begin{equation}
+        C_{X^m}=\begin{pmatrix}
+             0   &       &       &  0     \\
+             1   &   \ddots    &       &   \vdots    \\
+                &   \ddots    &   \ddots    &    \vdots   \\ 
+                &       &   1    &   0     
+         \end{pmatrix}.
+    \end{equation}
+    Ensuite le changement de base (qui est une similitude) \( (e_1,\ldots, e_n)\mapsto(e_n,\ldots, e_1)\) montre que \( C_{X^m}\) est semblable à un bloc de Jordan \( J_m(0)\).
+
+    Supposons à présent que \( f\) ne soit pas nilpotente. Par l'hypothèse de polynôme caractéristique scindé, nous supposons que \( f\) a \( m\) valeurs propres distinctes et que son polynôme caractéristique est
+    \begin{equation}
+        \chi_f=(X-\lambda_1)^{l_1}\ldots (X-\lambda_m)^{l_m}.
+    \end{equation}
+    Le lemme des noyaux (théorème \ref{ThoDecompNoyayzzMWod}) nous enseigne que
+    \begin{equation}
+        E=\bigoplus_{i=1}^m\underbrace{\ker(f-\mu_i\mtu)^{l_i}}_{F_i}.
+    \end{equation}
+    La restriction de \( f-\lambda_i\mtu\) à \( F_i\) est par construction un endomorphisme nilpotent, et donc peut s'écrire comme un bloc de Jordan avec des zéros sur la diagonale. En utilisant la décomposition
+    \begin{equation}
+        f|_{F_i}=(f-\lambda_i\mtu)|_{F_i}+\lambda_i\mtu_{F_i},
+    \end{equation}
+    nous voyons que \( f|_{F_i}\) s'écrit comme un bloc de Jordan avec \( \lambda_i\) sur la diagonale.
+\end{proof}
+
+\begin{remark}
+    Nous pouvons calculer la forme normale de Jordan pour une matrice complexe ou réelle, mais dans les deux cas nous devons nous attendre à obtenir une matrice complexe parce que les valeurs propres d'une matrice réelle peuvent être complexes. Cependant nous demandons que le polynôme caractéristique de \( f\) soit scindé sur \( \eK\). En pratique, la décomposition de Jordan n'est garantie que sur les corps algébriquement clos, c'est à dire sur \( \eC\).
+
+    La suite des invariants de similitude sur laquelle repose Frobenius, elle, est disponible sur tout corps, y compris \( \eR\).
+\end{remark}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Commutant et endomorphismes cycliques}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Endomorphisme cyclique}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}\label{LemSGmdnE}
+    Si \( A\) est la matrice de l'endomorphisme \( f\) alors nous avons équivalence des propriétés suivantes :
+    \begin{enumerate}
+        \item
+            La matrice \( A\) est cyclique.
+        \item
+            L'endomorphisme \( f\) est cyclique.
+    \end{enumerate}
+\end{lemma}
+
+Si \( f\) est un endomorphisme de l'espace vectoriel \( E\) et si \( x\in E\), nous notons 
+\begin{equation}
+    E_{f,x}=\Span\{ f^k(x)\tq k\in \eN \}.
+\end{equation}
+
+\begin{definition}
+    Soit \( E\) un espace vectoriel de dimension finie sur un corps \( \eK\) et un endomorphisme \( f\colon E\to E\). Le \defe{commutant}{commutant} de \( f\) est l'ensemble des endomorphismes de \( E\) qui commutent avec \( f\) :
+    \begin{equation}
+        \comC(f)=\{ g\in\aL(E,E)\tq g\circ f=f\circ g \}.
+    \end{equation}
+\end{definition}
+Il n'est pas très compliqué de vérifier que \( \comC(f)\) est un sous-espace vectoriel de \( \aL(E,E)\).
+
+Notons l'inclusion évidente \( \eK[f]\subset \comC(f)\). L'inclusion inverse va un peu nous occuper durant les prochaines pages.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Commutant : cas diagonalisable}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[\cite{ooKPTNooMmncYA}]      \label{PROPooRHHEooIRGmtl}
+    Si \( f\) est diagonalisable, alors
+    \begin{equation}        \label{EQooOTFLooPUKAos}
+        \dim\big( \comC(f) \big)=\sum_{\lambda\in\Spec(f)}\dim(E_{\lambda})^2.
+    \end{equation}
+    où les \( E_{\lambda} \) sont les espaces propres de \( f\).
+\end{proposition}
+
+\begin{proof}
+    D'abord su \( g\in\comC(f)\) alors \( E_{\lambda}\) est stable par \( g\). En effet si \( v\in E_{\lambda}\) alors \( f\big( g(v) \big)=g\big( f(v) \big)=g(\lambda v)=\lambda g(v)\), ce qui montre que \( g(v)\) est un vecteur propre de \( f\) pour la valeur propre \( \lambda\), et donc que \( g(v)\in E_{\lambda}\).
+
+    Nous considérons ensuite l'application 
+    \begin{equation}
+        \begin{aligned}
+            \psi\colon \comC(f)&\to \End(E_1)\times \ldots\times \End(E_r) \\
+            g&\mapsto  g|_{E_1}\times \ldots\times g|_{E_r}
+        \end{aligned}
+    \end{equation}
+    qui est bien définie parce que \( g\) se restreint aux espaces propres de \( f\). Nous allons noter \( \psi(g)_{\lambda}\) la restriction de \( g\) à \( E_{\lambda}\).
+    \begin{subproof}
+    \item[\( \psi\) est injective]
+
+        Supposons que \( g,h\in\comC(f)\) tels que \( \psi(g)=\psi(h)\). Vu que \( f\) est diagonalisable nous pouvons décomposer \( x\in  E\) en ses composantes sur les espaces propres\footnote{Théorème \ref{ThoDigLEQEXR}\ref{ITEMooZNJFooEiqDYp}.} :
+        \begin{equation}
+            x=\sum_{\lambda\in\Spec(f)}x_{\lambda}
+        \end{equation}
+        avec \( x_{\lambda}\in E_{\lambda}\).  Nous avons alors
+        \begin{equation}
+            g(x)=\sum_{\lambda}g(x_{\lambda})=\sum_{\lambda}\psi(g)_{\lambda}(x_{\lambda}).
+        \end{equation}
+        Vu que nous avons \( \psi(g)_{\lambda}=\psi(h)_{\lambda}\), nous avons aussi
+        \begin{equation}
+            g(x)=\sum_{\lambda}\psi(g)_{\lambda}(x_{\lambda})=\sum_{\lambda}\psi(h)_{\lambda}(x_{\lambda})=\sum_{\lambda}h(x_{\lambda})=h(x).
+        \end{equation}
+        Cela prouve \( g=h\) et donc que \( \psi\) est injective.
+    \item[\( \psi\) est surjective]
+        Si nous avons pour chaque \( \lambda\in\Spec(f)\) un endomorphisme \( g_{\lambda}\) de \( E_{\lambda}\) alors en posant
+        \begin{equation}
+            g(x)=\sum_{\lambda\in\Spec(f)}g_{\lambda}(x_{\lambda})
+        \end{equation}
+        alors nous avons bien
+        \begin{equation}
+            \psi(g)=\big( g_{\lambda_1},\ldots, g_{\lambda_r} \big).
+        \end{equation}
+    \end{subproof}
+    Nous pouvons donc conclure en écrivant
+    \begin{equation}
+        \dim\big( \comC(f) \big)=\sum_{\lambda\in\Spec(f)}\dim\big( \End(E_{\lambda}) \big)= \sum_{\lambda\in\Spec(f)}\dim(E_{\lambda})^2.
+    \end{equation}
+\end{proof}
+
+\begin{remark}      \label{REMooUGFQooVzCOvV}
+    Nous avons alors immédiatement
+    \begin{equation}
+        \dim\big( \comC(f) \big)\geq\dim(E)
+    \end{equation}
+    lorsque \( f\) est diagonalisable.
+\end{remark}
+
+En suivant la notation \eqref{EqooOAYDooEpZELo}, un endomorphisme est cyclique lorsqu'il existe \( x\in E\) tel que \( E_x=E\). 
+
+\begin{proposition}[\cite{ooKPTNooMmncYA}]      \label{PropooQALUooTluDif}
+    Si \( f\) est un endomorphisme diagonalisable d'un espace vectoriel \( E\) de dimension \( n\). Nous avons équivalence entre les faits suivants.
+    \begin{enumerate}
+        \item\label{ITEMooSOYYooZVibjrii}
+            Le polynôme minimal est égal au polynôme caractéristique : \( \mu_f=\chi_f\)
+        \item\label{ITEMooSOYYooZVibjrvi}
+            L'endomorphisme \( f\) est cyclique.
+        \item\label{ITEMooSOYYooZVibjrv}
+            \( \comC(f)=\eK[f]\).
+        \item\label{ITEMooSOYYooZVibjriv}
+            \( \dim\big( \comC(f) \big)=n\)
+        \item\label{ITEMooSOYYooZVibjriii}
+            L'endomorphisme \( f\) possède \( n\) valeurs propres distinctes.
+        \item   \label{ITEMooSOYYooZVibjri}
+            \( \dim\big( \eK[f] \big)=n\)
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Le point important de cette proposition sont les équivalences \ref{ITEMooSOYYooZVibjrii}-\ref{ITEMooSOYYooZVibjrv}. Les autres sont des intermédiaires. En particulier, dans le cas diagonalisable, nous allons voir que le point \ref{ITEMooSOYYooZVibjriii} est essentiellement une reformulation de \ref{ITEMooSOYYooZVibjrii}.
+    \begin{subproof}
+        \item[\ref{ITEMooSOYYooZVibjriv} implique \ref{ITEMooSOYYooZVibjriii}]
+            Par la formule \eqref{EQooOTFLooPUKAos}, les espaces propres de \( f\) ont dimension \( 1\). Par conséquent \( f\) possède \( n\) valeurs propres distinctes.
+        \item[\ref{ITEMooSOYYooZVibjriii} implique \ref{ITEMooSOYYooZVibjri}]
+            Le théorème \ref{ThoDigLEQEXR} nous dit que le polynôme minimal est scindé à racines simples. Vu que \( f\) possède \( n\) valeurs propres distinctes, \( \mu\) est de degré \( n\).  Par l'isomorphisme \( \eK[f]=\eK[X]/(\mu)\) de la proposition \ref{PropooCFZDooROVlaA} nous avons \(\dim\big( \eK[f] \big)= \deg(\mu)=n\) par la proposition \ref{CorsLGiEN}.
+        \item[\ref{ITEMooSOYYooZVibjri} implique \ref{ITEMooSOYYooZVibjrii}]
+            Par l'isomorphisme \( \eK[f]=\eK[X]/(\mu)\) de la proposition \ref{PropooCFZDooROVlaA} et la proposition \ref{CorsLGiEN} nous avons \(n=\dim\big( \eK[f] \big)= \deg(\mu)\). Vu que \( \chi\) est un polynôme annulateur (Caley-Hamilton \ref{ThoCalYWLbJQ}), il est divisé par \( \mu\). Maintenant \( \mu\) et \( \chi\) sont des polynômes unitaires de degré \( n\) et \( \mu\) divise \( \chi\). Ils sont donc égaux.
+        \item[\ref{ITEMooSOYYooZVibjrii} implique \ref{ITEMooSOYYooZVibjrvi}]
+            Le fait que $f$ soit diagonalisable permet d'utiliser le théorème \ref{ThoDigLEQEXR} pour dire que \( \mu\) est scindé à racines simples. L'égalisation avec \( \chi \) nous permet de dire que \( f\) possède \( n\) valeurs propres distinctes. Soient \( \{ e_1,\ldots, e_n \}\) une base de diagonalisation, et prouvons que le vecteur \( v=e_1+\cdots +e_n\) est cyclique. Nous avons
+            \begin{equation}
+                f^k(v)=\sum_{i=1}^n\lambda_i^ke_i.
+            \end{equation}
+            Pour prouver que cette famille (avec \( k=0,\ldots, n-1\)) est libre\footnote{Ce sera alors une base parce que \( n\) vecteurs libres dans un espace de dimension \( n\) est toujours une base, théorème \ref{ThoMGQZooIgrXjy}\ref{ItemHIVAooPnTlsBi}.} nous en prenons une combinaison linéaire nulle et nous prouvons que les coefficients sont tous nuls. Soit donc
+            \begin{equation}
+                    0=\sum_{l=0}^{n-1}a_lf^l(v)=\sum_{l=0}^{n-1}a_l\sum_{i=1}^n\lambda_i^le_i=\sum_{i=1}^n\Big( \sum_{l=0}^{n-1}a_l\lambda_i^l \Big)e_i.
+            \end{equation}
+            Vu que cela est nul, nous avons pour tout \( i\) :
+            \begin{equation}
+                \sum_{l=0}^{n-1}a_l\lambda_i^l=0.
+            \end{equation}
+            En posant la matrice \( A_{ij}=\lambda_i^j\), cela revient à étudier le système \( \sum_j A_{ij}a_j=0\). Ce système n'a des solutions non nulles que si \( \det(A)= 0\); sinon il possède une unique solution et elle est \( a_j=0\) pour tout \( j\). Nous devons donc calculer le déterminant
+            \begin{equation}
+                \det\begin{pmatrix}
+                    1&\lambda_1&\lambda_1^2&\cdots&\lambda_1^{n-1}\\  
+                    \vdots&\vdots&\vdots&&\vdots\\  
+                    1&\lambda_n&\lambda_n^2&\cdots&\lambda_n^{n-1}  
+                \end{pmatrix}.
+            \end{equation}
+            Il s'agit du déterminant de Vandermonde déjà étudié par la proposition \ref{PropnuUvtj}. Nous avons \( \det(A)=\prod_{1\leq i<j\leq n}(\lambda_j-\lambda_i)\). Cela est bien non nul du fait que toutes les valeurs propres soient distinctes.
+        \item[\ref{ITEMooSOYYooZVibjrvi} implique \ref{ITEMooSOYYooZVibjrv}]
+            Soit \( v\) un vecteur cyclique de \( f\). Un endomorphisme \( g\) donne lieu à un polynôme par le fait suivant : il existe des uniques \( a_k\) (\( k=0,\ldots, n-1\)) tels que
+            \begin{equation}
+                g(v)=\sum_{k=0}^{n-1}a_kf^k(v).
+            \end{equation}
+            Cela donne une application linéaire
+            \begin{equation}
+                \begin{aligned}
+                    \psi\colon \comC(f)&\to \eK[f] \\
+                    g&\mapsto P\tq P(f)v=g(v). 
+                \end{aligned}
+            \end{equation}
+            C'est une application injective parce que si \( \psi(g)=0\) alors \( g(v)=0\) et pour tout \( k\) nous avons \( g\big( f^k(v) \big)=f^k\big( g(v) \big)=0\). L'endomorphisme \( g\) s'annulant sur une base, est nul.
+        \item[\ref{ITEMooSOYYooZVibjrv} implique \ref{ITEMooSOYYooZVibjriv}]
+            Si \( n_1,\ldots, n_r\) sont les dimensions des différents espaces propres de \( f\), nous avons les inégalités
+            \begin{equation}
+                \dim\big( \eK[f] \big)=\deg(\mu)\leq n=n_1+\cdots +n_r\leq n_1^2+\cdots +n_r^2=\dim\big( \comC(f) \big).
+            \end{equation}
+            Par hypothèse d'égalité entre le premier et le dernier terme de cette suite d'inégalités, toutes les inégalités sont des égalités et en particulier \( \dim\big( \comC(f) \big)=n\).
+        \end{subproof}
+            Nous avons fini de prouver toutes les équivalences demandées.
+\end{proof}
+
+\begin{example}
+    Pour mieux comprendre pourquoi le fait d'avoir \( n\) valeurs propres distinctes est équivalent à être cyclique, notons que si deux valeurs propres sont identiques, alors un morceau de la matrice de \( f\) serait par exemple \( \begin{pmatrix}
+          2  &   0    \\ 
+        0    &   2    
+    \end{pmatrix}\), et dans ce cas n'importe quelle combinaison \( ae_i+be_j\) reste proportionnelle à elle-même après application de \( f\). Si nous avons des valeurs propres différentes par contre, nous avons par exemples dans \( \eR^2\) la matrice \( \begin{pmatrix}
+        1    &   0    \\ 
+        0    &   2    
+    \end{pmatrix}\) qui donne \( f(e_1+e_2)=e_1+2e_2\). La partie \( \{ e_1+e_2,e_1+2e_2 \}\) est une base.
+\end{example}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Commutant : cas général}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous considérons encore un espace vectoriel \( E\) de dimension finie \( n\) et un endomorphisme \( f\colon E\to E\). Nous notons \( \mu\) sont polynôme minimal et \( \mu_x\) le polynôme minimal ponctuel en \( x\).
+
+\begin{lemma}[\cite{ooEFHLooXpAOFz,AutourFrobCompa,ooEPEFooQiPESf}]       \label{LEMooDFFDooJTQkRu}
+    Nous avons
+    \begin{equation}
+        \dim\big( \comC(f) \big)\geq \dim(E)
+    \end{equation}
+\end{lemma}
+
+\begin{proof}
+    Si \( f\) est donnée, l'espace \( \comC(f)\) est l'espace des solutions de \( fg=gf\). Supposons avoir choisit une base de \( E\) et notons \( A\) la matrice de \( f\) et \( X\) celle de \( g\). L'équation est \( AX-XA=0\).
+    \begin{subproof}
+        \item[Si \( A\) est trigonalisable]
+            Nous supposons avoir choisi la base de telle sorte que \( A\) soit triangulaire supérieure, et nous allons nous contenter de chercher les solutions \( X\) qui sont également triangulaires supérieure. S'il y en a déjà plus que \( n\), a fortiori le résultat sera vrai.
+
+            Le produit de deux matrices triangulaires supérieures étant une matrice triangulaire supérieure, l'équation \( AX-XA\) contient, pour les coefficients de \( X\), \( n(n+1)/2\) équations. Mais il se fait que les termes diagonaux ne sont pas de vraies équations parce que
+            \begin{equation}
+                (AX-XA)_{kk}=\sum_i\big( A_{ki}X_{ik}-X_{ki}A_{ik} \big)=\sum_{k\leq i\leq k}(A_{ki}X_{ik}-X_{ki}A_{ik})=0.
+            \end{equation}
+            Nous avons donc au maximum 
+            \begin{equation}
+                \frac{ n(x+1) }{2}-n
+            \end{equation}
+            équations linéairement indépendantes pour un minimum de \( n(n+1)/2\) inconnues. L'espace des solutions est donc de dimension au minimum \( n\).
+
+            Cela a l'air d'être une majoration assez large, mais il existe des cas d'égalité.
+
+        \item[Si \( A\) n'est pas trigonalisable]
+
+            La preuve que nous donnons ici est valable même pour les endomorphismes trigonalisables.
+
+            Nous considérons le résultat de Frobenius \ref{THOooDOWUooOzxzxm}. Nous avons donc la structure suivante:
+            \begin{itemize}
+                \item 
+            une décomposition en somme directe \( E=E_1\oplus\ldots\oplus E_r\),
+        \item
+            les espaces \( E_i\) sont fixés par \( f\),
+        \item
+            les endomorphismes \( f_i=f|_{E_i}\) sont cycliques
+        \item
+            le polynôme minimal de \( f_i\) est \( \mu_i\) et \( \prod_{i=1}^r\mu_i=\chi_f\).
+            \end{itemize}
+            Les endomorphismes \( f_i^k\) commutent évidemment avec \( f_j\), et la partie \( \{ f_i^k \}_{k=0,\ldots, \deg(\mu_i)-1}\) est libre. Libre en tout cas en tant que partie de \( \End(E_i)\). Mais en prolongeant par \( 0\) sur \( E\), ça reste libre en tant que partie de \( \End(E)\).
+
+            Bien entendu les \( f_j^k\) et les \( f_i^k\) (\( i\neq j\)) sont linéairement indépendants dans \( \End(E)\) parce qu'ils n'agissent pas sur les mêmes vecteurs. Donc les endomorphismes \( f_i^{k_i}\) avec \( k_i=0,\ldots, \deg(\mu_i)-1\) forment une partie libre de \( \End(E)\) composée d'endomorphismes qui commutent avec \( f\). Il y en a en tout
+            \begin{equation}
+                \sum_{i=1}^r\deg(\mu_i)=\deg(\chi_f)=n.
+            \end{equation}
+            Par conséquent \( \dim\big( \comC(f) \big)\geq \dim(E)\).
+    \end{subproof}
+\end{proof}
+
+\begin{theorem}[\cite{ooRJDSooXpVtMD}]      \label{THOooGLMSooYewNxW}
+    Soit un endomorphisme \( f\colon E\to E\) sur l'espace vectoriel de dimension finie \( n\). Nous notons \( \mu\) et \( \chi\) les polynômes minimal et caractéristique. Nous avons équivalence entre les faits suivants :
+    \begin{enumerate}
+        \item   \label{ITEMooLRXIooLWaYqJii}
+            \( \mu=\chi\),
+        \item   \label{ITEMooLRXIooLWaYqJi}
+            \( f\) est cyclique,
+        \item   \label{ITEMooLRXIooLWaYqJiii}
+            \( \comC(f)=\eK[f]\).
+    \end{enumerate}
+\end{theorem}
+
+\begin{proof}
+    Plusieurs implications. Notons que \ref{ITEMooLRXIooLWaYqJii} implique \ref{ITEMooLRXIooLWaYqJii} a déjà été démontré par le lemme \ref{LEMooKUQDooKFeIYq}.
+    \begin{subproof}
+        \item[\ref{ITEMooLRXIooLWaYqJii} implique \ref{ITEMooLRXIooLWaYqJi}]
+            Conformément à ce que nous permet le lemme \ref{LemSYsJJj} nous choisissons\footnote{Dans toute la suite,nous devrions écrire \( \mu_f\) et \( \mu_{f,a}\) mais nous omettons d'indiquer explicitement la dépendance en \( f\).} \( a\in E\) de telle sorte à avoir \( \mu_a=\mu\). De plus pour \( x\in E\) nous considérons l'application
+            \begin{equation}
+                \begin{aligned}
+                    \varphi_x\colon \eK[X]&\to E \\
+                    P&\mapsto P(f)x. 
+                \end{aligned}
+            \end{equation}
+            Nous avons \( \varphi_a(P)=P(f)a\) et vu que \( E_{a}\) est engendré par les \( f^k(a)\) nous avons \( \varphi_a\big( \eK[X] \big)=E_a\). De plus l'application \( \varphi_a\) passe aux classes pour \( (\mu_a)\). Pour rappel, un élément de \( \eK[X]/(\mu_a)\) est de la forme
+            \begin{equation}
+                \bar P=\{ P+Q\mu_a \}_{Q\in \eK[X]}.
+            \end{equation}
+            Nous considérons donc l'application
+            \begin{equation}
+                \varphi_a\colon \frac{ \eK[X] }{ (\mu_a) }\to E_a
+            \end{equation}
+            et nous prouvons que c'est un isomorphisme d'espace vectoriel.
+            \begin{subproof}
+                \item[Linéaire]
+                    Parce que \( (\lambda P+Q)(f)=(\lambda P)(f)+Q(f)\).
+                \item[Injectif]
+                    Si \( \varphi_a(\bar P)=0\) alors \( \varphi_a(P)=0\) (dans la deuxième, \( \varphi_a\) est l'application définie sur les polynômes et non sur les classes), ce qui montrer que \( P\) est annulateur de \( a\). Mais par définition \ref{DEFooUICRooBGYhqQ} du polynôme minimal ponctuel, \( \mu_a\) est générateur de \( \ker(\varphi_a)\); donc il existe \( Q\in \eK[X]\) tel que \( P=Q\mu_a\). En d'autres termes, du point de vue du quotient, \( \bar P=0\).
+                \item[Surjectif]
+                    Si \( x\in E_a\) alors il existe des coefficients \( x_k\in \eK\) tels que \( x=\sum_{k=0}^{\deg(\mu_a)-1}x_kf^k(a)\), c'est à dire \( x=P(f)a=\varphi_a(P)\).
+            \end{subproof}
+            Mais par hypothèse et par choix de \( a\) nous avons \( \mu_a=\mu=\chi\), donc en fait \( E_a=\eK[X]/(\chi)\). Mais nous savons que \( \deg(\chi)=\dim(E)\) et que \( \dim\big( \eK[X]/P \big)=\deg(P)\) par la proposition \ref{PropooCFZDooROVlaA}. Au final nous avons \( \dim(E_a)=\deg(\chi)=\dim(E)\). Et par conséquent \( E_a=E\). Cela prouver que \( a\) est un vecteur cyclique pour \( f\).
+
+        \item[\ref{ITEMooLRXIooLWaYqJi} implique \ref{ITEMooLRXIooLWaYqJiii}]
+            Soit \( g\in \comC(f)\); nous devons prouver que \( g\) est un polynôme de \( f\). Par hypothèse nous avons un vecteur cyclique que nous notons \( v\). Nous avons un polynôme \( P\) (dépendant de \( g\)) tel que \( g(v)=P(f)v\). Nous allons voir que \( g=P(f)\). Soit \( y\in E\) et un polynôme \( Q\) tel que \( y=Q(f)v\); en notant que \( g\) commute avec \( P(f)\) nous avons 
+            \begin{equation}
+                g(y)=g\big( Q(f)v \big)=Q(f)\big( g(v) \big)=Q(f)\big( P(f)v \big)=P(f)Q(f)v=P(f)y.
+            \end{equation}
+            Donc \( g=P(f)\).
+
+        \item[\ref{ITEMooLRXIooLWaYqJiii} implique \ref{ITEMooLRXIooLWaYqJii}]
+
+            Nous avons les inégalités :
+            \begin{equation}
+                n\leq \dim\big( \comC(f) \big)=\dim\big( \eK[f] \big)=\deg(\mu)\leq \deg(\chi)=n.
+            \end{equation}
+            La première est le lemme \ref{LEMooDFFDooJTQkRu}. Toutes les inégalités sont des égalités. En particulier \( \deg(\mu)=n\), ce qui signifie que \( \mu=\chi\) parce que \( \mu\) est un polynôme diviseur de \( \chi\), de même degré que \( \chi\) et unitaire tout comme \( \chi\).
+
+    \end{subproof}
+\end{proof}
+
+\begin{corollary}[\cite{ooEPEFooQiPESf}]        \label{CORooAKQEooSliXPp}
+    En suivant les notations sur les extensions de corps de base de la section \ref{SECooAUOWooNdYTZf}, l'endomorphisme \( f\colon E\to F\) est cyclique si et seulement si l'endomorphisme \( f_{\eL}\colon E_{\eL}\to F_{\eL}\) est cyclique.
+\end{corollary}
+
+\begin{proof}
+    Nous savons par le théorème \ref{THOooGLMSooYewNxW} qu'un endomorphisme est cyclique si et seulement si son polynôme minimal est égal à son polynôme caractéristique. Or par les propositions \ref{PROPooZAZFooUFdCUv} et \ref{PROPooXVZMooXcJrsJ}, nous savons que ces polynômes sont identiques pour \( f\) et pour \( f_{\eL}\).
+\end{proof}
+
+\begin{theorem}[Similitude et extension de corps\cite{ooEPEFooQiPESf}]      \label{THOooHUFBooReKZWG}
+    Les applications linéaires \( f,g\colon E\to E\) sont semblables si et seulement si \( f_{\eL}\) et \( g_{\eL}\) le sont.
+\end{theorem}
+
+\begin{proof}
+    En ce qui concerne le sens direct, s'il existe \( m\in\GL(E)\) tel que \( f=mgm^{-1}\) alors il suffit d'appliquer le lemme \ref{LEMooWZGSooONEnjZ} pour avoir \( f_{\eL}=m_{\eL}g_{\eL}m_{\eL}^{-1}\).
+
+    Nous considérons les invariants de similitude de \( f\) du théorème \ref{THOooDOWUooOzxzxm}. Il existe une unique suite de polynômes unitaires \( \mu_i\) ($i=1,\ldots, s$) tels que \( \mu_i\divides \mu_{i+1}\) et pour laquelle nous avons une décomposition \( E=E_1\oplus \ldots\oplus E_s\) pour laquelle \( f|_{E_i}\colon E_1\to E_i\) est cyclique et de polynôme minimal \( \mu\).
+
+    Nous avons aussi \( E_{\eL}=(E_1)_{\eL}\oplus\ldots \oplus (E_s)_{\eL}\) et les \( (E_i)_{\eL}\) sont stables sous \( f_{\eL}\) qui y sera également cyclique (corollaire  \ref{CORooAKQEooSliXPp}). De plus le polynôme minimal de \( f_{\eL}|_{(E_i)_{\eL}}\) est également \( \mu_i\).
+
+    Autrement dit, la suite \( \mu_i\) est également la suite des invariants de similitude de \( f_{\eL}\). La remarque \ref{REMooPVLEooYDRXQI} nous permet de conclure que \( f\) et \( g\) sont semblables si et seulement si \( f_{\eL}\) et \( g_{\eL}\) le sont.
+\end{proof}
diff --git a/tex/frido/141_EspacesVectos.tex b/tex/frido/141_EspacesVectos.tex
new file mode 100644
index 000000000..8cf8c1df1
--- /dev/null
+++ b/tex/frido/141_EspacesVectos.tex
@@ -0,0 +1,843 @@
+% This is part of Mes notes de mathématique
+% Copyright (c) 2011-2017
+%   Laurent Claessens, Carlotta Donadello
+% See the file fdl-1.3.txt for copying conditions.
+
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorème de Burnside}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{lemma}       \label{LemwXXzIt}
+    Soit \( P\), un polynôme sur \( \eK\). Une racine de \( P\) est une racine simple si et seulement si elle n'est pas racine de \( P'\).
+\end{lemma}
+
+\begin{theorem}     \label{ThoBurnsideoPuCtS}
+    Toute représentation d'un groupe abélien d'exposant fini sur \( \eC^n\) a une image finie.
+\end{theorem}
+
+\begin{proof}
+    Dans le cas d'un groupe abélien, la démonstration est facile. Étant donné que \( G\) est d'exposant fini, il existe \( \alpha\in \eN^*\) tel que \( g^{\alpha}=e\) pour tout \( g\in G\). Le polynôme \( P(X)=X^{\alpha}-1\) est scindé à racines simples. En effet tout polynôme sur \( \eC\) est scindé. Le fait qu'il soit à racines simples provient du lemme \ref{LemwXXzIt} parce que si \( a^{\alpha}=1\), alors il n'est pas possible d'avoir \( \alpha a^{\alpha-1}=0\).
+
+    Par ailleurs \( P(g)=0\). Le fait que nous ayons un polynôme annulateur de \( g\) scindé à racines simples implique que \( g\) est diagonalisable (théorème \ref{ThoDigLEQEXR}). Le fait que \( G\) soit abélien montre qu'il existe une base de \( \eC^n\) dans laquelle tous les éléments de \( G\) sont diagonaux. Nous devons par conséquent montrer qu'il existe un nombre fini de matrices de la forme
+    \begin{equation}
+        \begin{pmatrix}
+            \lambda_1    &       &       \\
+                &   \ddots    &       \\
+                &       &   \lambda_n
+        \end{pmatrix}.
+    \end{equation}
+    Nous savons que \( \lambda_i^{\alpha}=1\) parce que \( g^{\alpha}=\mtu\), par conséquent chacun des \( \lambda_i\) est une racine de l'unité dont il n'existe qu'un nombre fini.
+\end{proof}
+
+\begin{theorem}[Burnside\cite{fJhCTE,ooFBZQooXyHIWK}]\label{ThooJLTit}
+    Un sous-groupe de \( \GL(n,\eC)\) est fini si et seulement s'il est d'exposant\footnote{Définition \ref{DefvtSAyb}.} fini.
+\end{theorem}
+\index{exposant}
+\index{racine!de l'unité}
+\index{endomorphisme!diagonalisable}
+
+\begin{proof}
+    Soit \( G\) un sous-groupe de \( \GL(n,\eC)\). Si \( G\) est fini, l'ordre de ses éléments divise \( | G |\) (corollaire \ref{CorpZItFX}) au théorème de Lagrange et l'exposant est le PPCM qui est donc fini également.
+
+    Nous supposons maintenant que l'ordre de \( G\) est fini. Nous notons \( e\) l'exposant de \( G\). En particulier, tous les éléments de \( G\) sont des racines du polynôme \( X^e-1\).
+    
+    \begin{subproof}
+        \item[Générateurs]
+
+            Le groupe \( G\) est une partie de \( \eM(n,\eC)\) dont nous considérons l'algèbre engendrée (définition \ref{DefkAXaWY}) \( \mG\). Soit \( C_1,\ldots, C_r\) une famille génératrice de \( \mG\) constituée d'éléments de \( G\) et la fonction
+            \begin{equation}
+                \begin{aligned}
+                    \tau\colon G&\to \eC^r \\
+                    A&\mapsto \big( \tr(AC_1),\ldots, \tr(AC_r) \big). 
+                \end{aligned}
+            \end{equation}
+
+        \item[\( \tau\) est injective] Supposons que \( \tau(A)=\tau(B)\). Alors pour tout générateur \( C_i\) nous avons \( \tr(AC_i)=\tr(BC_i)\) et par linéarité de la trace, nous avons
+            \begin{equation}    \label{EqnCYmKW}
+                \tr(AM)=\tr(BM)
+            \end{equation}
+            pour tout \( M\in G\). Notons par ailleurs
+            \begin{equation}
+                N=AB^{-1}-\mtu,
+            \end{equation}
+            qui est diagonalisable parce que \( AB^{-1}\in G\) et donc est annulé par le polynôme \( X^e-1\) qui est scindé à racines simples. Du coup \( AB^{-1}\) est diagonalisable; posons \( PAB^{-1}P^{-1}=D\), alors \( P\big( AB^{-1}-\mtu \big)P^{-1}=D-\mtu\) qui est encore diagonale. Donc \( N\) est diagonalisable.
+
+            Par ailleurs nous avons
+            \begin{subequations}
+                \begin{align}
+                    \tr\big( (AB^{-1})^p \big)&=\tr\big( AB^{-1}(AB^{-1})^{p-1} \big)\\
+                    &=\tr\big( BB^{-1}(AB^{-1})^{p-1} \big) &\text{\eqref{EqnCYmKW}}\\
+                    &=\tr\big( (AB^{-1})^{p-1} \big).
+                \end{align}
+            \end{subequations}
+            En continuant nous obtenons
+            \begin{equation}
+                \tr\big(  (AB^{-1})^p \big)=\tr(\mtu)=n.
+            \end{equation}
+            
+            D'autre part, 
+            \begin{equation}
+                N^k=(AB^{-1}-\mtu)^k=\sum_{p=0}^k{p\choose k}(-1)^{k-p}(AB^{-1})^p
+            \end{equation}
+            En prenant la trace, et en tenant compte du fait que \( \tr\big( (AB^{-1})^p \big)=n\),
+            \begin{equation}
+                \tr(N^k)=\sum_{p=0}^k{p\choose k}(-1)^{k-p}n=n(1-1)^k=0.
+            \end{equation}
+            Donc la trace de \( N^k\) est nulle et le lemme \ref{LemzgNOjY} nous enseigne que \( N\) est alors nilpotente. Étant donné qu'elle est aussi diagonalisable, elle est nulle. Nous en concluons que \( AB^{-1}=\mtu\) et donc que \( A=B\). La fonction \( \tau\) est donc injective.
+
+        \item[Nombre fini de valeurs]
+
+            Les éléments de \( G\) sont annulés par \( X^e-1\) qui est un polynôme scindé à racines simples. Dons le polynôme minimal d'un élément de \( G\) est (a fortiori) scindé à racines simples et le théorème \ref{ThoDigLEQEXR} nous assure alors que ces éléments sont diagonalisables. Du coup les valeur propres des matrices de \( G\) sont des racines \( e\)ièmes de l'unité. Par conséquent les traces des éléments de \( G\) ne peuvent prendre qu'un nombre fini de valeurs : toutes les sommes de \( n\) racines \( e\)ièmes de l'unité. Mais vu que les \( C_i\) sont dans \( G\), nous avons
+            \begin{equation}
+                \Image(\tau)=\{ \tr(A)\tq A\in G \}^r,
+            \end{equation}
+            qui est un ensemble fini. Par conséquent \( G\) est fini parce que \( \tau\) est injective.
+    \end{subproof}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de Lie-Kolchin}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Contrairement à ce que l'on peut parfois croire, il n'est pas vrai que toute matrice à coefficient réel est diagonalisable, même pas sur \( \eC\). La raison est qu'une telle matrice peut très bien avoir des valeurs propres multiples.
+
+\begin{example} \label{ExBRXUooIlUnSx}
+    Le théorème \ref{ThoDigLEQEXR} nous donne une façon simple de trouver des matrices non diagonalisables sur \( \eC\) : il suffit que le polynôme minimal ne soit pas scindé à racines simples. Par exemple
+    \begin{equation}
+        A=\begin{pmatrix}
+            1    &   1    \\ 
+            0    &   1    
+        \end{pmatrix},
+    \end{equation}
+    dont le polynôme caractéristique est \( \chi_A=(1-X)^2\). Ce polynôme n'a manifestement pas des racines simples. Nous pouvons faire le calcul explicite pour montrer que \( A\) n'est pas diagonalisable. D'abord l'unique valeur propre de \( A\) est \( 1\) et nous pouvons sans peine résoudre
+    \begin{equation}
+        \begin{pmatrix}
+            1    &   1    \\ 
+            0    &   1    
+        \end{pmatrix}\begin{pmatrix}
+            x    \\ 
+            y    
+        \end{pmatrix}=\begin{pmatrix}
+            x    \\ 
+            y    
+        \end{pmatrix}
+    \end{equation}
+    qui revient au système
+    \begin{subequations}
+        \begin{numcases}{}
+            x+y=x\\
+            y=y.
+        \end{numcases}
+    \end{subequations}
+    La première équation donne directement \( y=0\). Le seul espace propre est de dimension \( 1\) et est engendré par \( \begin{pmatrix}
+        1    \\ 
+        0    
+    \end{pmatrix}\).
+\end{example}
+
+La remarque \ref{RemBOGooCLMwyb} donne un exemple un peu plus avancé, qui montre la multiplicité algébrique et géométrique d'une racine d'un polynôme caractéristique.
+
+\begin{lemma}[Trigonalisation simultanée]   \label{LemSLGPooIghEPI}
+    Une famille de matrices de \( \GL(n,\eC)\) commutant deux à deux est simultanément trigonalisable.
+\end{lemma}
+\index{trigonalisation!simultanée}
+
+\begin{proof}
+    Commençons par enfoncer une porte ouverte par la proposition \ref{PropKNVFooQflQsJ} : sur \( \GL(n,\eC)\) toutes les matrices sont trigonalisables parce que tous les polynômes sont scindés.
+
+    Nous effectuons la démonstration par récurrence sur la dimension. Si \( n=1\) alors toues les matrice sont triangulaires et nous ne nous posons pas de questions. Nous supposons donc \( n>1\).
+
+    Soit la famille \( (A_i)_{i\in I}\) dans \( \GL(n,\eC)\) et \( A_0\) un de ses éléments. Nous nommons \( \lambda_1,\ldots, \lambda_r\) les valeurs propres distinctes de \( A_0\). Le théorème de décomposition primaire \ref{ThoSpectraluRMLok} nous donne la somme directe d'espaces caractéristiques\footnote{Définition \ref{DefFBNIooCGbIix}.}
+    \begin{equation}
+        E=F_{\lambda_1}(A_0)\oplus\ldots\oplus F_{\lambda_r}(A_0).
+    \end{equation}
+    Nous pouvons supposer que cette somme n'est pas réduite à un seul terme. En effet si tel était le cas, \( A_0\) serait un multiple de l'identité parce que \( A_0\) n'aurait qu'une seule valeur propre et les sommes dans la décomposition de Dunford \ref{ThoRURcpW}\ref{ItemThoRURcpWiii} se réduisent à un seul terme (et \( p_i=\id\)). En particulier les dimensions des espaces \( F_{\lambda}(A_0)\) sont strictement plus petites que \( n\).
+
+    Vu que tous les \( A_i\) commutent avec \( A_0\), les espaces \( F_{\lambda}(A_0)\) sont stables par les \( A_i\) et nous pouvons trigonaliser les \( A_i\) simultanément sur chacun des \( F_{\lambda}(A_0)\) en utilisant l'hypothèse de récurrence.
+\end{proof}
+
+\begin{theorem}[Lie-Kolchin\cite{PAXrsMn}]  \label{ThoUWQBooCvutTO}
+    Tout sous-groupe connexe et résoluble de \( \GL(n,\eC)\) est conjugué à un groupe de matrices triangulaires.
+\end{theorem}
+\index{trigonalisation!simultanée}
+\index{théorème!Lie-Kolchin}
+
+\begin{proof}
+    Soit \( G\) un sous-groupe connexe et résoluble de \( \GL(n,\eC)\).
+    
+    \begin{subproof}
+        \item[Si sous-espace non trivial stable par \( G\)]
+
+    Nous commençons par voir ce qu'il se passe s'il existe un sous-espace vectoriel non trivial \( V\) de \( \eC^n\) stabilisé par \( G\). Pour cela nous considérons une base de \( \eC^n\) dont les premiers éléments forment une base de \( V\) (base incomplète, théorème \ref{ThonmnWKs}). Les éléments de \( G\) s'écrivent, dans cette base,
+    \begin{equation}    \label{EqGOKTooEaGACG}
+        \begin{pmatrix}
+            g_1    &   *    \\ 
+            0    &   g_2    
+        \end{pmatrix}.
+    \end{equation}
+    Les matrices \( g_1\) et \( g_2\) sont carrés. Nous considérons alors l'application \( \psi\) définie par
+    \begin{equation}
+        \begin{aligned}
+            \psi\colon G&\to \GL(V) \\
+            g&\mapsto g_1.
+        \end{aligned}
+    \end{equation}
+    Cela est un morphisme de groupes parce que
+    \begin{equation}
+        \begin{pmatrix}
+            g_1    &   *    \\ 
+            0    &   g_2    
+        \end{pmatrix}\begin{pmatrix}
+            h_1    &   *    \\ 
+            0    &   h_2    
+        \end{pmatrix}=
+        \begin{pmatrix}
+            g_1h_1    &   *    \\ 
+            0    &   g_2h_2    
+        \end{pmatrix},
+    \end{equation}
+    de telle sorte que \( \psi(gh)=\psi(g)\psi(h)\).
+
+    Le groupe \( \psi(G)\) est connexe et résoluble. En effet \( \psi(G)\) est connexe en tant qu'image d'un connexe par une application continue (proposition \ref{PropGWMVzqb}). Et il est résoluble en tant qu'image d'un groupe résoluble par un homomorphisme par la proposition \ref{PropBNEZooJMDFIB}. Vu que \( \psi(G)\) est un sous-groupe résoluble et connexe de \( \GL(V)\) et que la dimension de \( V\) est strictement plis petite que celle de \( \eC^n\), une récurrence sur la dimension indique que \( \psi(G)\) est conjugué à un groupe de matrices triangulaires. C'est à dire qu'il existe une base de \( V\) dans laquelle toutes les matrices \( g_1\) (avec \( g\in G\)) sont triangulaires supérieures.
+
+    On fait de même avec l'application \( g\mapsto g_2\), ce qui donne une base du supplémentaire de \( V\) dans laquelle les matrices \( g_2\) sont triangulaires. 
+
+    En couplant ces deux bases, nous obtenons une base de \( \eC^n\) dans laquelle toutes les matrices \eqref{EqGOKTooEaGACG} (c'est à dire toutes les matrices de \( G\)) sont triangulaires supérieures.
+
+    \item[Sinon]
+
+    Nous supposons à présent que \( \eC^n\) n'a pas de sous-espaces non triviaux stables sous \( G\). Nous posons \( m=\min\{ k\tq D^k(G)=\{ e \} \}\), qui existe parce que \( G\) et résoluble et que sa suite dérivée termine sur \( {e}\) (proposition \ref{PropRWYZooTarnmm}).
+
+\item[Si \( m=1\)]
+
+    Si \( m=1\) alors \( G\) est abélien et il existe une base de \( G\) dans laquelle toutes les matrices de \( G\) sont triangulaires (lemme \ref{LemSLGPooIghEPI}). Le premier vecteur d'une telle base serait stable par \( G\), mais comme nous avons supposé qu'il n'y avait pas de sous-espaces non triviaux stabilisés par \( G\), il faut déduire que ce vecteur stable est à lui tout seul non trivial, c'est à dire que \( n=1\). Dans ce cas, le théorème est démontré.
+
+\item[Si \( m>1\)]
+
+    Nous devons maintenant traiter le cas où \( m>1\). Nous posons \( H=D^{m-1}(G)\); cela est un sous-groupe normal et abélien de \( G\). Encore une fois le résultat de trigonalisation simultanée \ref{LemSLGPooIghEPI} donne une base dans laquelle tous les éléments de \( H\) sont triangulaires. En particulier le premier élément de cette base est un vecteur propre commun à toutes les matrices de \( H\).
+
+    Soit \( V\) le sous-espace engendré par tous les vecteurs propres communs de \( H\). Nous venons de voir que \( V\) n'est pas vide. Nous allons montrer que \( V\) est stable par \( G\). Soient \( h\in H\), \( v\in V\) et \( g\in G\) :
+    \begin{equation}    \label{EqPMOBooVLIhrJ}
+        h\big( g(v) \big)=g\underbrace{g^{-1}hg}_{\in H}(v)=g(\lambda v)=\lambda g(v)
+    \end{equation}
+    parce que \( v\) est vecteur propre de \( g^{-1} hg\). Ce que le calcul \eqref{EqPMOBooVLIhrJ} montre est que \( g(v)\) est vecteur propre de \( h\) pour la valeur propre \( \lambda\). Donc \( g(v)\in V\) et \( V\) est stabilisé par \( G\). Mais comme il n'existe pas d'espaces non triviaux stabilisés par \( G\), nous en déduisons que \( V=\eC^n\). Donc tous les vecteurs de \( \eC^n\) sont vecteurs propres communs de \( H\). Autrement dit on a une base de diagonalisation simultanée de \( H\).
+
+\item[\( H\) est dans le centre de \( G\)]
+
+    Montrons à présent que \( H\) est dans le centre de \( G\), c'est à dire que pour tout \( g\in G\) et \( h\in H\) il faut \( ghg^{-1}=h\). D'abord \( ghg^{-1}\) est une matrice diagonale (parce que elle est dans \( H\)) ayant les mêmes valeurs propres que \( h\). En effet si \( \lambda\) est valeur propre de \( ghg^{-1}\) pour le vecteur propre \( v\), alors
+    \begin{subequations}
+        \begin{align}
+            (ghg^{-1})(v)&=\lambda v\\
+            h\big( g^{-1} v \big)&=\lambda \big( g^{-1}v \big),
+        \end{align}
+    \end{subequations}
+    c'est à dire que \( \lambda\) est également valeur propre de \( h\), pour le vecteur propre \( g^{-1} v\). Mais comme \( h\) a un nombre fini de valeurs propres, il n'y a qu'un nombre fini de matrices diagonales ayant les mêmes valeurs propres que \( h\). L'ensemble \( \AD(G)h\) est donc un ensemble fini. D'autre part, l'application \( g\mapsto g^{-1}hg\) est continue, et \( G\) est connexe, donc l'ensemble \( \AD(G)h\) est connexe. Un ensemble fini et connexe dans \( \GL(n,\eC)\) est nécessairement réduit à un seul point. Cela prouve que \( ghg^{-1}=h\) pour tout \( g\in G\) et \( h\in H\).
+
+\item[Espaces propres stables pour tout \( G\)]
+
+        Soit \( h\in H\) et \( W\) un espace propre de \( h\) (ça existe non vide parce que \( H\) est triangularisé, voir plus haut). Alors nous allons prouver que \( W\) est stable pour tous les éléments de \( G\). En effet si \( w\in W\) avec \( h(w)=\lambda w\) alors en permutant \( g\) et \( h\),
+        \begin{equation}
+            hg(w)=g(hw)=\lambda g(w),
+        \end{equation}
+        donc \( g(w)\) est aussi vecteur propre de \( h\) pour la valeurs propre \( \lambda\), c'est à dire que \( g(w)\in W\). Vu que nous supposons que \( \eC^n\) n'a pas d'espaces invariants non triviaux, nous devons conclure que \( W=\eC^n\), c'est à dire que \( H\) est composé d'homothéties. C'est à dire que pour tout \( h\in H\) nous avons \( h=\lambda_h\mtu\).
+
+    \item[Contradiction sur la minimalité de \( m\)]
+
+        Les éléments d'un groupe dérivé sont de déterminant \( 1\) parce que \( \det(g_1g_2g_1^{-1}g_2^{-1})=1\). Par conséquent pour tout \( h\), le nombre \( \lambda_h\) est une racine \( n\)\ieme de l'unité. Vu qu'il n'y a qu'une quantité finie de racines \( n\)\ieme de l'unité, le groupe \( H\) est fini et connexe et donc une fois de plus réduit à un élément, c'est à dire \( H=\{ e \}\). Cela contredit la minimalité de \( m\) et donc produit une contradiction. Nous devons donc avoir \( m=1\).
+
+    \item[Conclusion]
+
+        Nous avons vu que si \( \eC^n\) avait un sous-espace non trivial fixé par \( G\) alors le théorème était démontré. Par ailleurs si \( \eC^n\) n'a pas un tel sous-espace, soit \( m=1\) (et alors le théorème est également prouvé), soit \( m>1\) et alors on a une contradiction.
+
+        Bref, le théorème est prouvé sous peine de contradiction.
+    \end{subproof}
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Formes bilinéaires et quadratiques}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\label{SecTQkRXIu}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Généralités}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}[\cite{RUAoonJAym}]   \label{DefBSIoouvuKR}
+    Soit un espace vectoriel \( E\) et \( \eF\) un corps de caractéristique différente de \( 2\). Une \defe{forme quadratique}{forme!quadratique} sur \( E\) est une application \( q\colon V\to \eF\) pour laquelle il existe une forme bilinéaire symétrique \( b\colon V\times V\to \eF\) satisfaisant \( q(x)=b(x,x)\) pour tout \( x\in V\).
+
+    L'ensemble des formes quadratiques réelles sur \( E\) est noté \( Q(E)\)\nomenclature[B]{\( Q(E)\)}{formes quadratiques réelles sur \( E\)}.
+\end{definition}
+
+\begin{lemma}       \label{LEMooLKNTooSfLSHt}
+    Si \( q\) est une forme quadratique, il existe une unique forme bilinéaire \( b\) telle que \( q(x)=b(x,x)\).
+\end{lemma}
+
+\begin{proof}
+    L'existence n'est pas en cause : c'est la définition d'une forme quadratique. Pour l'unicité, étant donné une forme quadratique, la forme bilinéaire \( b\) doit forcément vérifier l'\defe{identités de polarisation}{identité!polarisation}\index{polarisation (identité)} :
+\begin{equation}    \label{EqMrbsop}
+    b(x,y)=\frac{ 1 }{2}\big( q(x)+q(y)-q(x-y) \big).
+\end{equation}
+Elle est donc déterminée par \( q\).
+\end{proof}
+Notons la division par \( 2\) qui est le pourquoi de la demande de la caractéristique différente de \( 2\) pour \( \eF\) dans la définition de forme quadratique.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Topologie}
+%---------------------------------------------------------------------------------------------------------------------------
+
+La topologie considérée sur \( Q(E)\) est celle de la norme
+\begin{equation}    \label{EqZYBooZysmVh}
+    N(q)=\sup_{\| x \|_E=1}| q(x) |,
+\end{equation}
+qui du point de vue de \( S_n(\eR)\) est
+\begin{equation}    
+    N(A)=\sup_{\| x \|_E=1}| x^tAx |.
+\end{equation}
+Notons que à droite, c'est la valeur absolue usuelle sur \( \eR\).
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Matrice associée}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}     \label{PropcnJyXZ}
+    Soit $M$, une matrice symétrique. Nous avons
+    \begin{enumerate}
+        \item
+        $\det M>0$ et $\tr(M)>0$ implique $M$ définie positive,
+        \item
+        $\det M>0$ et $\tr(M)<0$ implique $M$ définie négative,
+    \item   \label{ItemluuFPN}
+        $\det M<0$ implique ni semi définie positive, ni définie négative 
+        \item
+        $\det M=0$ implique $M$ semi-définie positive ou semi-définie négative.
+    \end{enumerate}
+\end{proposition}
+
+Si une base \( \{ e_i \}_{i=1,\ldots, n}\) de l'espace vectoriel \( E\) est donnée, la \defe{matrice associée}{matrice!associée à une forme quadratique}\index{forme!quadratique!matrice associée} à la forme bilinéaire \( b\) sur \( E\) est la matrice d'éléments
+\begin{equation}
+    B_{ij}=b(e_i,e_j).
+\end{equation}
+Notons que la matrice associée à une forme bilinéaire (ou quadratique associée) est uniquement valable pour une base donnée. Si nous changeons de base, la matrice change. Cependant lorsque nous travaillons sur \( \eR^n\), la base canonique est tellement canonique que nous allons nous permettre de parler de «la» matrice associée à une forme bilinéaire. 
+
+Si \( B_{ij}\) est la matrice associée à la forme bilinéaire \( b\) alors la valeur de \( b(u,v)\) se calcule avec la formule
+\begin{equation}
+    b(x,y)=\sum_{i,j}B_{ij}x_iy_j
+\end{equation}
+lorsque \( x_i\) et \( y_j\) sont les coordonnées de \( x\) et \( y\) dans la base choisie.
+
+\begin{proposition} \label{PropFSXooRUMzdb}
+    Soit \( \{ e_i \}\) une base de \( E\). L'application
+    \begin{equation}
+        \begin{aligned}
+            \phi\colon Q(E)&\to S(n,\eR) \\
+            q&\mapsto \big(   b(e_i,e_j)   \big)_{i,j}
+        \end{aligned}
+    \end{equation}
+    où \( b\) est forme bilinéaire associée à \( q\) est une bijection linéaire et continue.
+\end{proposition}
+
+\begin{proof}
+    Si \( \phi(q)=\phi(q')\); alors
+    \begin{equation}
+        q(x)=\sum_{i,j}\phi(q)_{ij}x_ix_j=\sum_{i,j}\phi(q')_{ij}x_ix_j=q'(x).
+    \end{equation}
+    Donc \( q=q'\). L'application \( \phi\) est donc injective
+
+    De plus elle est surjective parce que si \( B\in S(n,\eR)\) alors la forme quadratique
+    \begin{equation}
+        q(x)=\sum_{i,j}B_{ij}x_ix_j
+    \end{equation}
+    a évidemment \( B\) comme matrice associée. L'application \( \phi\) est donc surjective.
+
+    Notre application \( \phi\) est de plus linéaire parce que l'association d'une forme quadratique à la forme bilinéaire associée est linéaire.
+
+    En ce qui concerne la continuité, nous la prouvons en zéro en considérant une suite convergente \( q_n\stackrel{Q(E)}{\longrightarrow}0\). C'est à dire que
+    \begin{equation}
+        \sup_{\| x \|=1}| q_n(x) |\to 0.
+    \end{equation}
+    Nous rappelons l'identité de polarisation : 
+    \begin{equation}
+        b_n(x,y)=\frac{ 1 }{2}\big( q_n(x-y)-q(x)-q(y) \big).
+    \end{equation}
+    En ce qui concerne deux des trois termes, il n'y a pas de problèmes :
+    \begin{equation}
+        \big| \phi(q_n)_{ij} \big|=\big| b_n(e_i,e_j) \big|\leq\frac{ 1 }{2}\big| b_n(e_i-e_j) \big|+\frac{ 1 }{2}\big| q_n(e_i) \big|+\frac{ 1 }{2}\big| q_n(e_j) \big|.
+    \end{equation}
+    Si \( n\) est assez grand, nous avons tout de suite
+    \begin{equation}
+        \big| \phi(q_n)_{ij} \big|\leq \frac{ 1 }{2}\big| q_n(e_i-e_j) \big|+\epsilon.
+    \end{equation}
+    Nous définissons \( e_{ij}\) et \( \alpha_{ij}\) de telle sorte que \( e_i-e_j=\alpha_{ij}e_{ij}\) avec \( \| e_{ij} \|=1\). Si \( \alpha=\max\{ \alpha_{ij},1 \}\) alors nous avons
+    \begin{equation}
+        q_n(e_i-e_j)=\alpha_{ij}^2q_n(e_{ij})\leq \alpha^2q_n(e_{ij}).
+    \end{equation}
+    Il suffit maintenant de prendre \( n\) assez grand pour avoir \( \sup_{\| x \|=1}| q_n(x) |\leq \frac{ \epsilon }{ \alpha^2 }\) pour avoir
+    \begin{equation}
+        \big| \phi(q_n)_{ij} \big|\leq \frac{ \epsilon }{2}+\frac{ \epsilon }{ \alpha^2 }.
+    \end{equation}
+\end{proof}
+
+\begin{proposition}\label{PropFWYooQXfcVY}
+    Dans la base de diagonalisation de sa matrice associée, une forme quadratique a la forme
+    \begin{equation}
+        q(x)=\sum_i\lambda_ix_i^2
+    \end{equation}
+    où les \( \lambda_i\) sont les valeurs propres de la matrice associée à \( q\).
+\end{proposition}
+
+\begin{proof}
+Soit \( q\) une forme quadratique et \( b\) la forme bilinéaire associée. Si \( \{ f_i \}\) est une base de diagonalisation de la matrice de \( b\) alors dans cette base nous avons
+\begin{equation}
+    q(x)=b(x,x)=\sum_{ij}x_ix_jb(f_i,f_j)=\sum_i\lambda_ix_i^2
+\end{equation}
+où les \( \lambda_i\) sont les valeurs propres de la matrice de \( b\).
+\end{proof}
+Notons que si nous choisissons une autre base de diagonalisation, les \( \lambda_i\) ne changement pas (à part l'ordre éventuellement). Cela pour dire que nous nous permettrons de parler des \defe{valeurs propres}{valeur propre!d'une forme quadratique} d'une forme quadratique comme étant les valeurs propres de la matrice associée.
+
+\begin{proposition}     \label{PROPooUAAFooEGVDRC}
+    Une application linéaire est définie positive si et seulement si sa matrice associée l'est.
+\end{proposition}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Dégénérescence}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit \( b\), une forme bilinéaire symétrique non dégénérée  sur l'espace vectoriel \( E\) de dimension \( n\) sur \( \eK\) où \( \eK\) est un corps de caractéristique différente de \( 2\). Nous notons \( q\) la forme quadratique associée.
+
+\begin{definition}
+    Une forme bilinéaire est \defe{non dégénérée}{forme!bilinéaire!non dégénérée} \( b(x,z)=0\) pour tout \( z\) implique \( x=0\).
+\end{definition}
+
+\begin{lemma}   \label{LemyKJpVP}
+    Soit \( b\) une forme bilinéaire non dégénérée. Si \( x\) et \( y\) sont tels que \( b(x,z)=b(y,z)\) pour tout \( z\), alors \( x=y\).
+\end{lemma}
+
+\begin{proof}
+    C'est immédiat du fait de la linéarité en le premier argument et de la non-dégénérescence : si \( b(x,z)-b(y,z)=0\) alors
+    \begin{equation}
+        b(x-y,z)=0
+    \end{equation}
+    pour tout \( z\), ce qui implique \( x-y=0\).
+\end{proof}
+
+\begin{proposition}
+    La forme bilinéaire \( b\) est non-dénénérée si et seulement si sa matrice associée est inversible.
+\end{proposition}
+
+\begin{proof}
+    Nous savons que la matrice associée est symétrique et qu'elle peut donc être diagonalisée (théorème \ref{ThoeTMXla}). En nous plaçant dans une base de diagonalisation, nous devons prouver que la forme est non-dégénérée si et seulement si les éléments diagonaux de la matrice sont tous non nuls.
+
+    Écrivons \( b(x,z)\) en choisissant pour \( z\) le vecteur de base \( e_k\) de composantes \( (e_k)_j=\delta_{kj}\) :
+    \begin{equation}
+            b(x,e_k)=\sum_{ij}x_i(e_k)_j
+            =\sum_i b_{ik}x_i
+            =b_{kk}x_k.
+    \end{equation}
+    Si \( b\) est dégénérée et si \( x\) est un vecteur non nul (disons que la composante \( x_i\) est non nulle) de \( E\) tel que \( b(x,z)=0\) pour tout \( z\in E\), alors \( b_{ii}=0\), ce qui montre que la matrice de \( b\) n'est pas inversible.
+
+    Réciproquement si la matrice de \( b\) est inversible, alors tous les \( b_{kk}\) sont différents de zéro, et le seul vecteur \( x\) tel que \( b_{kk}x_k=0\) pour tout \( k\) est le vecteur nul.
+\end{proof}
+
+
+\begin{definition}[Isotropie]   \label{DefVKMnUEM}
+    Un vecteur est \defe{isotrope}{isotrope (vecteur)} pour \( b\) s'il est perpendiculaire à lui-même; en d'autres termes, \( x\) est isotrope si et seulement si \( b(x,x)=0\). Un sous-espace \( W\subset E\) est \defe{totalement isotrope}{isotrope!totalement} si pour tout \( x,y\in W\), nous avons \( b(x,y)=0\).
+
+    Le \defe{cône isotrope}{isotrope!cône} de \( b\) est l'ensemble de ses vecteurs isotropes :
+    \begin{equation}
+        C(b)=\{ x\in E\tq b(x,x)=0 \}.
+    \end{equation}
+\end{definition}
+Nous introduisons quelque notations. D'abord pour \( y\in E\) nous notons
+\begin{equation}
+    \begin{aligned}
+        \Phi_y\colon E&\to \eR \\
+        x&\mapsto b(x,y) 
+    \end{aligned}
+\end{equation}
+et ensuite
+\begin{equation}
+    \begin{aligned}
+        \Phi\colon E&\to E^* \\
+        y&\mapsto \Phi_y. 
+    \end{aligned}
+\end{equation}
+\begin{definition}
+    Le fait pour une forme bilinéaire \( b\) d'être dégénérée signifie que l'application \( \Phi\) n'est pas injective. Le \defe{noyau}{noyau!d'une forme bilinéaire} de la forme bilinéaire est celui de \( \Phi\), c'est à dire
+    \begin{equation}
+        \ker(b)=\{ z\in E\tq b(z,y)=0\,\forall y\in E \}.
+    \end{equation}
+    Autrement dit, \( \ker(b)=E^{\perp}\) où le perpendiculaire est pris par rapport à \( b\).
+\end{definition}
+Notons tout de même que nous utilisons la notation \( \perp\) même si \( b\) est dégénérée et éventuellement pas positive; c'est à dire même si la formule \( (x,y)\mapsto b(x,y)\) ne fournit pas un produit scalaire.
+
+\begin{proposition}[\cite{RTzQrdx}]     \label{PropHIWjdMX}
+    Soit \( b\) une forme bilinéaire et symétrique. Alors
+    \begin{enumerate}
+        \item
+            \( \ker(b)\subset C(b)\) (cône d'isotropie, définition \ref{DefVKMnUEM})
+        \item
+            si \( b\) est positive alors \( \ker(b)=C(b)\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    \begin{enumerate}
+        \item
+            Si \( z\in\ker(b)\) alors pour tout \( y\in E\) nous avons \( b(z,y)=0\). En particulier pour \( y=z\) nous avons \( b(z,z,)=0\) et donc \( z\in C(b)\).
+        \item
+            Soit \( b\) positive et \( x\in C(b)\). Par l'inégalité de Cauchy-Schwarz (proposition \ref{ThoAYfEHG}) nous avons
+            \begin{equation}
+                | b(x,y) |\leq \sqrt{   b(x,x)b(y,y) }=0.
+            \end{equation}
+            Donc pour tout \( y\) nous avons \( b(x,y)=0\).
+    \end{enumerate}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Inégalité de Minkowski}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Ce qui est couramment nommé «inégalité de Minkowski» est la proposition \ref{PropInegMinkKUpRHg} dans les espaces \( L^p\). Nous allons en donner ici un cas très particulier.
+
+\begin{proposition} \label{PropACHooLtsMUL}
+    Si \( q\) est une forme quadratique sur \( \eR^n\) et si \( x,y\in \eR^n\) alors
+    \begin{equation}
+        \sqrt{q(x+y)}\leq\sqrt{q(x)}+\sqrt{q(y)}.
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    La proposition \ref{PropFWYooQXfcVY} nous permet de «diagonaliser» la forme quadratique \( q\). Quitte à ne plus avoir une base orthonormale, nous pouvons renormaliser les vecteurs de base pour avoir
+    \begin{equation}
+        q(x)=\sum_ix_i^2.
+    \end{equation}
+    Le résultat n'est donc rien d'autre que l'inégalité triangulaire pour la norme euclidienne usuelle, laquelle est démontrée dans la proposition \ref{PropEQRooQXazLz}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Ellipsoïde}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}   \label{LemYVWoohcjIX}
+    Toute matrice peut être décomposée de façon unique en une partie symétrique et une partie antisymétrique. Cette décomposition est donnée par
+\begin{equation}\label{subEqHIQooyhiWM}
+    \begin{aligned}[]
+            S&=\frac{ M+M^t }{ 2 },&A&=\frac{ M-M^t }{ 2 }
+    \end{aligned}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+    L'existence est une vérification immédiate de \( S+A=M\) en utilisant \eqref{subEqHIQooyhiWM}. Pour l'unicité, si \( S+A=S'+A'\) alors \( S-S'=A-A'\). Mais \( S-S'\) est symétrique et \( A-A'\) est antisymétrique; l'égalité implique l'annulation des deux membres, c'est à dire \( S=S'\) et \( A=A'\).
+\end{proof}
+
+\begin{definition}  \label{DefOEPooqfXsE}
+    Un \defe{ellipsoïde}{ellipsoïde} dans \( \eR^n\) centré en \( v\) est le lieu des points \( x\) vérifiant l'équation
+    \begin{equation}\label{EqSNWooXfbTH}
+        (x-v)^t M(x-v)=1
+    \end{equation}
+    où \( M\) est une matrice symétrique strictement définie positive\footnote{Définition \ref{DefAWAooCMPuVM}.}.
+
+    Lorsque nous parlons d'ellipsoïde \emph{plein}, il suffit de changer l'égalité en une inégalité.
+\end{definition}
+Une autre façon d'écrire la relation \eqref{EqSNWooXfbTH} est d'écrire \( \langle (x-v),M(x,v)\rangle\) en utilisant le produit scalaire.
+
+\begin{remark}
+    Le fait que \( M\) soit symétrique n'est pas tout à fait obligatoire; la chose important est que toutes les valeurs propres soient strictement positives. En effet si \( M\) a toutes ses valeurs propres strictement positives, nous nommons \( S\) la partie symétrique de \( M\) et \( A\) la partie antisymétrique (lemme \ref{LemYVWoohcjIX}). Alors pour tout \( x\in \eR^n\) nous avons
+    \begin{equation}
+        x^tAx=\langle x, Ax\rangle =\langle A^tx,x \rangle =-\langle Ax, x\rangle =-\langle x,Ax\rangle ,
+    \end{equation}
+    donc \( x^tAx=0\). L'équation \( x^tMx=1\) est donc équivalente à \( x^tSx=1\) (elles ont les mêmes solutions).
+    
+    De plus \( S\) reste strictement définie positive parce que pour tout \( x\in \eR^n\) nous avons 
+    \begin{equation}
+        0<x^tMx=x^tSx.
+    \end{equation}
+\end{remark}
+
+\begin{proposition}\label{PropWDRooQdJiIr}
+    Si \( \ellE\) est un ellipsoïde centrée à l'origine, il existe une base de \( \eR^n\) dans laquelle son équation est :
+    \begin{equation}
+        \sum_{i=1}^n\frac{ x_i^2 }{ a_i^2 }=1.
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Nous avons une matrice symétrique strictement définie positive \( S\) telle que l'équation soit \( \langle x, Sx\rangle =1\). Le théorème spectral \ref{ThoeTMXla} nous fournit une base orthonormale \( \{ e_i \}\) dans laquelle \( Se_i=\lambda_ie_i\) avec \( \lambda_i>0\). En substituant dans l'équation \( \langle x, Sx\rangle =1\) nous trouvons l'équation
+    \begin{equation}
+        \sum_i\lambda_ix_i^2=1.
+    \end{equation}
+    En posant \( a_i=\frac{1}{ \sqrt{\lambda_i} }\), nous trouvons le résultat.  Cette définition des \( a_i\) est toujours possible parce que \( \lambda_i>0\).
+\end{proof}
+
+\begin{corollary}   \label{CorKGJooOmcBzh}
+    Un ellipsoïde plein centré en l'origine admet une équation de la forme \( q(x)\leq 1\) où \( q\) est une forme quadratique strictement définie positive.
+\end{corollary}
+Pour rappel de notation, l'ensemble des formes quadratiques strictement définies positives sur l'espace vectoriel \( E\) est noté \( Q^{++}(E)\).
+
+\begin{proof}
+    Soit \( \{ e_i \}\) une base de \( \eR^n\) telle que l'ellipsoïde \( \ellE\) ait pour équation
+    \begin{equation}
+        \sum_{i=1}^n\frac{ x_i^2 }{ a_i^2 }\leq 1.
+    \end{equation}
+    Nous considérons la forme quadratique
+    \begin{equation}
+        \begin{aligned}
+            q\colon \eR^n&\to \eR \\
+            x&\mapsto \sum_{i=1}^n\frac{ \langle x, e_i\rangle^2 }{ a_i^2 }. 
+        \end{aligned}
+    \end{equation}
+    Nous avons évidemment \( \ellE=\{ x\in \eR^n\tq q(x)\leq 1 \}\). De plus la forme \( q\) est strictement définie positive parce que dès que \( x\neq 0\), au moins un des produits scalaires \( \langle x, e_i\rangle \) est non nul et \( q(x)> 0\).
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorème spectral auto-adjoint}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}      \label{DEFooYNEQooGQgbCf}
+    Si \( E\) est un espace euclidien, un endomorphisme \( f\colon E\to E\) est \defe{auto-adjoint}{endomorphisme!auto-adjoint} si pour tout \( x,y\in E\) nous avons \( \langle x, f(y)\rangle=\langle f(x), y\rangle  \). 
+\end{definition}
+L'ensemble des opérateurs auto-adjoints de \( E\) est noté \( \gS(E)\)\nomenclature[A]{\( \gS(E)\)}{Les opérateurs auto-adjoints de $E$}. Cette notation provient du fait que dans \( \eR^n\) muni du produit scalaire usuel, les opérateurs auto-adjoints sont les matrices symétriques.
+
+\begin{theorem}[Théorème spectral auto-adjoint] \label{ThoRSBahHH}
+    Un endomorphisme auto-adjoint d'un espace euclidien
+    \begin{enumerate}
+        \item
+            est diagonalisable dans une base orthonormée,
+        \item
+            a son spectre réel.
+    \end{enumerate}
+\end{theorem}
+\index{théorème!spectral!autoadjoint}
+\index{diagonalisation!endomorphisme auto-adjoint}
+
+\begin{proof}
+    Nous procédons par récurrence sur la dimension de \( E\), et nous commençons par \( n=1\)\footnote{Dans \cite{KXjFWKA}, l'auteur commence avec \( n=0\) mais moi je n'en ai \wikipedia{en}{Vacuous_truth}{pas le courage.}.}. Soit donc \( f\colon E\to E\) avec \( \langle f(x), y\rangle =\langle x, f(y)\rangle \). Étant donné que \( f\) est également linéaire, il existe \( \lambda\in \eR\) tel que \( f(x)=\lambda x\) pour tout \( x\in E\). Tous les vecteurs de \( E\) sont donc vecteurs propres de \( f\).
+
+    Passons à la récurrence. Nous considérons \( \dim(E)=n+1\) et \( f\in\gS(E)\). Nous considérons la forme bilinéaire symétrique \( \Phi_f\) et la forme quadratique associée \( \phi_f\). Pour rappel,
+    \begin{subequations}
+        \begin{align}
+        \Phi_f(x,y)=\langle x, f(y)\rangle \\
+        \phi_f(x)=\Phi_f(x,x).
+        \end{align}
+    \end{subequations}
+    Et nous allons laisser tomber les indices \( f\) pour noter simplement \( \Phi\) et \( \phi\). Étant donné que \( \overline{ B(0,1) }\) est compacte et que \( \phi\) est continue, il existe \( x_0\in\overline{ B(0,1) }\) tel que 
+    \begin{equation}
+        \lambda=\phi(x_0)=\sup_{x\in\overline{ B(0,1) }}\phi(x).
+    \end{equation}
+    Notons aussi que \( \| x_0 \|=1\) : le maximum est pris sur le bord. Nous posons
+    \begin{equation}
+        g=\lambda\id-f
+    \end{equation}
+    ainsi que 
+    \begin{equation}
+        \Phi_1(x,y)=\langle x, g(y)\rangle .
+    \end{equation}
+    Cela est une forme bilinéaire et symétrique parce que
+    \begin{equation}
+        \Phi_1(y,x)=\langle y, g(x)\rangle =\langle g(y), x\rangle =\langle x, g(y)\rangle =\Phi_1(x,y)
+    \end{equation}
+    où nous avons utilisé le fait que \( g\) était auto-adjoint et la symétrie du produit scalaire. De plus \( \Phi_1\) est semi-définie positive parce que
+    \begin{equation}
+        \Phi_1(x,x)=\langle x, \lambda x-f(x)\rangle =\lambda\| x \|^2-\phi(x).
+    \end{equation}
+    Vu que \( \lambda\) est le maximum, nous avons tout de suite \( \Phi_1(x)\geq 0\) tant que \( \| x \|=1\). Et si \( x\) n'est pas de norme \( 1\), c'est le même prix parce qu'on se ramène à \( \| x \|=1\) en multipliant par un nombre positif. Attention cependant : 
+    \begin{equation}
+        \Phi_1(x_0,x_0)=\lambda\| x_0 \|^2-\phi(x_0)=0.
+    \end{equation}
+    Donc \( \Phi_1\) a un noyau contenant \( x_0\) par la proposition \ref{PropHIWjdMX}. Nous en déduisons que \( \Image(g)\neq E\) en effet, \( x_0\in\Image(g)^{\perp}\), mais nous avons la proposition \ref{PropXrTDIi} sur les dimensions : 
+    \begin{equation}
+        \dim E=\dim(\Image(g))+\dim( \Image(g)^{\perp}).
+    \end{equation}
+    Vu que \( \Image(g)^{\perp}\) est un espace vectoriel non réduit à \( \{ 0 \}\), la dimension de \( \Image(g)\) ne peut pas être celle de \( E\). L'endomorphisme \( g\) n'étant pas surjectif, il ne peut pas être injectif non plus parce que nous sommes en dimension finie; il existe donc \( e_1\in E\) tel que \( g(e_1)=0\) et tant qu'à faire nous choisissons \( \| e_1 \|=1\) (ici la norme est bien celle de l'espace euclidien considéré). Par définition,
+    \begin{equation}
+        f(e_1)=\lambda e_1,
+    \end{equation}
+    c'est à dire que \( \lambda\in\Spec(f)\). Et \( \phi\) étant une forme quadratique réelle nous avons \( \lambda\in \eR\).
+
+    Nous posons à présent \( H=\Span\{ e_1 \}^{\perp}\). C'est un sous-espace stable par \( f\) parce que si \( x\in H\) alors
+    \begin{equation}
+        \langle e_1, f(x)\rangle =\langle f(e_1j),x\rangle =\lambda\langle e_1, x\rangle =0.
+    \end{equation}
+    Nous pouvons donc considérer la restriction de \( f\) à \( H\) : \( f_H\colon H\to H\). Cet endomorphisme est bilinéaire et symétrique sur l'espace \( H\) de dimension inférieure à celle de \( E\), donc la récurrence nous donne une base orthonormée
+    \begin{equation}
+        \{ e_2,\ldots, e_n \}
+    \end{equation}
+    de vecteurs propres de \( f_H\). De plus les valeurs propres sont réelles, toujours par récurrence. Donc
+    \begin{equation}
+        \Spec(f)=\{ \lambda \}\cup\Spec(f_H)\subset \eR.
+    \end{equation}
+    Notons pour être complet que si \( i\geq 2\) alors
+    \begin{equation}
+        \langle e_1, e_i\rangle =0
+    \end{equation}
+    parce que le vecteur \( e_i\) est par construction choisit dans l'espace \( H=e_1^{\perp}\). Nous avons donc bien une base orthonormée de \( E\) construite sur des vecteurs propres de \( f\).
+\end{proof}
+
+\begin{corollary}   \label{CorSMHpoVK}
+    Soit \( E\) un espace vectoriel ainsi que \( \phi\) et \( \psi\) des formes quadratiques sur \( E\) avec \( \psi\) définie positive. Alors il existe une base \( \psi\)-orthonormale dans laquelle \( \phi\) est diagonale.
+\end{corollary}
+
+\begin{proof}
+    Il suffit de considérer l'espace euclidien \( E\) muni du produit scalaire \( \langle x, y\rangle =\psi(x,y)\). Ensuite nous diagonalisons la matrice (symétrique) de \( \phi\) pour ce produit scalaire à l'aide du théorème \ref{ThoRSBahHH}.
+\end{proof}
+
+\begin{definition}      \label{DefYNWUFc}
+    Dans le cas de \( V=\eR^m\) nous avons un produit scalaire canonique. Soient $u$ et $v$, deux vecteurs de $\eR^m$. Le \defe{produit scalaire}{produit!scalaire!sur \( \eR^n\)} de $u$ et $v$, noté $\langle u, v\rangle $ ou $u\cdot v$ est le réel
+	\begin{equation}		\label{EqDefProdScalsumii}
+		\langle u, v\rangle =\sum_{k=1}^m u_kv_k=u_1v_1+u_2v_2+\cdots+u_mv_n.
+	\end{equation}
+\end{definition}
+
+Calculons par exemple le produit scalaire de deux vecteurs de la base canonique : $\langle e_i, e_j\rangle $. En utilisant la formule de définition et le fait que $(e_i)_k=\delta_{ik}$, nous avons
+\begin{equation}
+	\langle e_i, e_j\rangle =\sum_{k=1}^m\delta_{ik}\delta_{jk}.
+\end{equation}
+Nous pouvons effectuer la somme sur $k$ en remarquant qu'à cause du $\delta_{ik}$, seul le terme avec $k=i$ n'est pas nul. Effectuer la somme revient donc à remplacer tous les $k$ par des $i$ :
+\begin{equation}
+	\langle e_i, e_j\rangle =\delta_{ii}\delta_{ji}=\delta_{ji}.
+\end{equation}
+
+Une des propriétés intéressantes du produit scalaire est qu'il permet de décomposer un vecteur dans une base, comme nous le montre la proposition suivante.
+
+\begin{proposition}		\label{PropScalCompDec}
+	Si nous notons $v_i$ les composantes du vecteur $v$, c'est à dire si $v=\sum_{i=1}^m v_ie_i$, alors nous avons $v_j=\langle v, e_j\rangle $.
+\end{proposition}
+
+\begin{proof}
+	\begin{equation}		\label{Eqvejscalcomp}
+		v\cdot e_j=\sum_{i=1}^m\langle v_ie_i, e_j\rangle =\sum_{i=1}^mv_i\langle e_i, e_j\rangle =\sum_{i=1}^mv_i\delta_{ij}
+	\end{equation}
+	En effectuant la somme sur $i$ dans le membre de droite de l'équation \eqref{Eqvejscalcomp}, tous les termes sont nuls sauf celui où $i=j$; il reste donc
+	\begin{equation}
+		v\cdot e_j=v_j.
+	\end{equation}
+\end{proof}
+
+Le produit scalaire ne dépend en réalité pas de la base orthogonale choisie. 
+
+\begin{lemma}
+	Si $\{ e_i \}$ est la base canonique, et si $\{ f_i \}$ est une autre base orthonormale, alors si $u$ et $v$ sont deux vecteurs de $\eR^m$, nous avons
+	\begin{equation}
+		\sum_i u_iv_j=\sum_iu'_iv'_j
+	\end{equation}
+	où $u_i$ sont les composantes de $u$ dans la base $\{ e_i \}$ et $u'_i$ sont celles dans la base $\{ f_i \}$.
+\end{lemma}
+
+\begin{proof}
+	La preuve demande un peu d'algèbre linéaire. Étant donné que $\{ f_i \}$ est une base orthonormale, il existe une matrice $A$ orthogonale ($AA^t=\mtu$) telle que $u'_i=\sum_jA_{ij}u_j$ et idem pour $v$. Nous avons alors
+	\begin{equation}
+		\begin{aligned}[]
+			\sum_iu'_iv'_j&=\sum_i\left( \sum_jA_{ij} u_j\right)\left( \sum_k A_{ik}v_k \right)\\
+			&=\sum_{ijk}A_{ij}A_{ik}u_jv_k\\
+			&=\sum_{jk}\underbrace{\sum_i(A^t)_{ji}A_{ik}}_{=\delta_{jk}}u_jv_k\\
+			&=\sum_{jk}\delta_{jk}u_jv_k\\
+			&=\sum_ku_jv_k.
+		\end{aligned}
+	\end{equation}	
+\end{proof}
+
+Cette proposition nous permet de réellement parler du produit scalaire entre deux vecteurs de façon intrinsèque sans nous soucier de la base dans laquelle nous regardons les vecteurs.
+
+Nous dirons que deux vecteurs sont \defe{orthogonaux}{orthogonal} lorsque leur produit scalaire est nul. Nous écrivons que $u\perp v$ lorsque $\langle u, v\rangle =0$.
+\begin{definition}	\label{DefNormeEucleApp}
+	La \defe{norme euclidienne}{norme!euclidienne!dans $\eR^m$} d'un élément de $\eR^m$ est définie par $\| u \|=\sqrt{u\cdot u}$.
+\end{definition}
+
+Cette définition est motivée par le fait que le produit scalaire $u\cdot u$ donne exactement la norme usuelle donnée par le théorème de Pythagore :
+\begin{equation}
+	u\cdot u=\sum_{i=1}^mu_iu_i=\sum_{i=1}^m u_i^2=u_1^2+u_2^2+\cdots+u_m^2.
+\end{equation}
+
+Le fait que $e_i\cdot e_j=\delta_{ij}$ signifie que la base canonique est \defe{orthonormée}{orthonormé}, c'est à dire que les vecteurs de la base canonique sont orthogonaux deux à deux et qu'ils ont tout $1$ comme norme.
+
+\begin{lemma}\label{LemSclNormeXi}
+	Pour tout $u\in\eR^m$, il existe un $\xi\in\eR^m$ tel que $\| u \|=\xi\cdot u$ et $\| \xi \|=1$.
+\end{lemma}
+
+\begin{proof}
+	Vérifions que le vecteur $\xi=u/\| u \|$ ait les propriétés requises. D'abord $\| \xi \|=1$ parce que $u\cdot u=\| u \|^2$. Ensuite
+	\begin{equation}
+		\xi\cdot u=\frac{ u\cdot u }{ \| u \| }=\frac{ \| u \|^2 }{ \| u \| }=\| u \|.
+	\end{equation}
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Mini introduction au produit tensoriel}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\label{SeOOpHsn}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Définitions}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit \( E\), un espace vectoriel de dimension finie. Si \( \alpha\) et \( \beta\) sont deux formes linéaires sur un espace vectoriel \( E\), nous définissons \( \alpha\otimes \beta\) comme étant la \( 2\)-forme donnée par
+\begin{equation}
+    (\alpha\otimes \beta)(u,v)=\alpha(u)\beta(v).
+\end{equation}
+Si \( a\) et \( b\) sont des vecteurs de \( E\), ils sont vus comme des formes sur \( E\) via le produit scalaire et nous avons
+\begin{equation}
+    (a\otimes b)(u,v)=(a\cdot u)(b\cdot v).
+\end{equation}
+Cette dernière équation nous incite à pousser un peu plus loin la définition de \( a\otimes b\) et de simplement voir cela comme la matrice de composantes
+\begin{equation}
+    (a\otimes b)_{ij}=a_ib_j.
+\end{equation}
+Cette façon d'écrire a l'avantage de ne pas demander de se souvenir qui est une vecteur ligne, qui est un vecteur colonne et où il faut mettre la transposée. Évidemment \( (a\otimes b)\) est soit \( ab^t\) soit \( a^tb\) suivant que \( a\) et \( b\) soient ligne ou colonne.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Application d'opérateurs}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}   \label{LemMyKPzY}
+    Soient \( x,y\in E\) et \( A,B\) deux opérateurs linéaires sur \( E\) vus comme matrices. Alors
+    \begin{equation}        \label{EqXdxvSu}
+        (Ax\otimes By)=A(x\otimes y)B^t.
+    \end{equation}
+\end{lemma}
+
+\begin{proof}
+    Calculons la composante \( ij\) de la matrice \( (Ax\otimes By)\). Nous avons
+    \begin{subequations}
+        \begin{align}
+            (Ax\otimes By)_{ij}&=(Ax)_i(By)_j\\
+            &=\sum_{kl}A_{ik}x_kB_{jl}y_l\\
+            &=A_{ik}(x\otimes y)_{kl}B_{jl}\\
+            &=\big( A(x\otimes y)B^t \big)_{ij}.
+        \end{align}
+    \end{subequations}
+\end{proof}
+
+% TODO: Ajouter un texte sur les équations de plan, et pourquoi ax+by+cz+d=0 est perpendiculaire au vecteur (a,b,c).
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Méthode de Gauss pour résoudre des systèmes d'équations linéaires}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Pour résoudre un système d'équations linéaires, on procède comme suit:
+\begin{enumerate}
+\item Écrire le système sous forme matricielle. \[\text{p.ex. } \begin{cases} 2x+3y &= 5 \\ x+2y &= 4 \end{cases} \Leftrightarrow \left(\begin{array}{cc|c} 2 & 3 & 5 \\ 1 & 2 & 4 \end{array}\right) \]
+\item Se ramener à une matrice avec un maximum de $0$ dans la partie de gauche en utilisant les transformations admissibles:
+\begin{enumerate}
+\item Remplacer une ligne par elle-même + un multiple d'une autre;
+\[\text{p.ex. } \left(\begin{array}{cc|c} 2 & 3 & 5 \\ 1 & 2 & 4 \end{array}\right)  \stackrel{L_1  - 2. L_2 \mapsto L_1'}{\Longrightarrow} \left(\begin{array}{cc|c} 0 & -1 & -3 \\ 1 & 2 & 4 \end{array}\right) \]
+\item Remplacer une ligne par un multiple d'elle-même;
+\[\text{p.ex. } \left(\begin{array}{cc|c} 0 & -1 & -3 \\ 1 & 2 & 4 \end{array}\right)  \stackrel{-L_1  \mapsto L_1'}{\Longrightarrow} \left(\begin{array}{cc|c} 0 & 1 & 3 \\ 1 & 2 & 4 \end{array}\right) \]
+\item Permuter des lignes.
+\[\text{p.ex. } \left(\begin{array}{cc|c} 0 & 1 & 3 \\ 1 & 0 & -2 \end{array}\right)  \stackrel{L_1  \mapsto L_2' \text{ et } L_2  \mapsto L_1'}{\Longrightarrow} \left(\begin{array}{cc|c} 1 & 0 & -2 \\ 0 & 1 & 3  \end{array}\right) \]
+\end{enumerate}
+\item Retransformer la matrice obtenue en système d'équations.
+\[\text{p.ex. }  \left(\begin{array}{cc|c} 1 & 0 & -2 \\ 0 & 1 & 3  \end{array}\right) \Leftrightarrow \begin{cases} x &= -2 \\ y &= 3 \end{cases}  \]
+\end{enumerate}
+
+\begin{remark}
+\begin{itemize}
+\item Si on obtient une ligne de zéros, on peut l'enlever:
+\[\text{p.ex. }  \left(\begin{array}{ccc|c} 3 & 4 & -2 & 2 \\ 4 & -1 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \Leftrightarrow  \left(\begin{array}{ccc|c} 3 & 4 & -2 & 2 \\ 4 & -1 & 3 & 0 \end{array}\right) \]
+\item Si on obtient une ligne de zéros suivie d'un nombre non-nul, le système d'équations n'a pas de solution:
+\[\text{p.ex. }  \left(\begin{array}{ccc|c} 3 & 4 & -2 & 2 \\ 4 & -1 & 3 & 0 \\ 0 & 0 & 0 & 7 \end{array}\right) \Leftrightarrow  \begin{cases} \cdots \\ \cdots \\ 0x + 0y + 0z = 7 \end{cases} \Rightarrow \textbf{Impossible} \]
+\item Si on moins d'équations que d'inconnues, alors il y a une infinité de solutions qui dépendent d'un ou plusieurs paramètres:
+\[\text{p.ex. }  \left(\begin{array}{ccc|c} 1 & 0 & -2 & 2 \\ 0 & 1 & 3 & 0 \end{array}\right) \Leftrightarrow  \begin{cases} x - 2z = 2 \\ y + 3z = 0 \end{cases} \Leftrightarrow  \begin{cases} x = 2 + 2\lambda \\ y = -3\lambda \\ z = \lambda \end{cases} \]
+\end{itemize}
+\end{remark}
diff --git a/tex/frido/144_EspacesVectos.tex b/tex/frido/144_EspacesVectos.tex
new file mode 100644
index 000000000..2a1159895
--- /dev/null
+++ b/tex/frido/144_EspacesVectos.tex
@@ -0,0 +1,1648 @@
+% This is part of Mes notes de mathématique
+% Copyright (c) 2008-2017
+%   Laurent Claessens
+% See the file fdl-1.3.txt for copying conditions.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Déterminants}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\label{SecGYzHWs}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Formes multilinéaires alternées}
+%---------------------------------------------------------------------------------------------------------------------------
+
+%  Lire http://www.les-mathematiques.net/phorum/read.php?2,302266
+
+\begin{definition}\index{déterminant!forme linéaire alternée}
+    Soit \( E\), un \( \eK\)-espace vectoriel. Une forme linéaire \defe{alternée}{forme linéaire!alternée}\index{alternée!forme linéaire} sur \( E\) est une application linéaire \( f\colon E\to \eK\) telle que \( f(v_1,\ldots, v_k)=0\) dès que \( v_i=v_j\) pour certains \( i\neq j\).
+\end{definition}
+
+\begin{lemma}   \label{LemHiHNey}
+    Une forme linéaire alternée est antisymétrique. Si \( \eK\) est de caractéristique différente de \( 2\), alors une forme antisymétrique est alternée.
+\end{lemma}
+
+\begin{proof}
+    Soit \( f\) une forme alternée; quitte à fixer toutes les autres variables, nous pouvons travailler avec une \( 2\)-forme et simplement montrer que \( f(x,y)=-f(y,x)\). Pour ce faire nous écrivons
+    \begin{equation}
+        0=f(x+y,x+y)=f(x,x)+f(x,y)+f(y,x)+f(y,y)=f(x,y)+f(y,x).
+    \end{equation}
+    
+    Pour la réciproque, si \( f\) est antisymétrique, alors \( f(x,x)=-f(x,x)\). Cela montre que \( f(x,x)=0\) lorsque \( \eK\) est de caractéristique différente de deux.
+\end{proof}
+
+\begin{proposition}[\cite{GQolaof}] \label{ProprbjihK}
+    Soit \( E\), un \( \eK\)-espace vectoriel de dimension \( n\), où la caractéristique de \( \eK\) n'est pas deux. L'espace des \( n\)-formes multilinéaires alternées sur \( E\) est de \( \eK\)-dimension \( 1\).
+\end{proposition}
+\index{groupe!permutation}
+\index{groupe!et géométrie}
+\index{espace!vectoriel!dimension}
+\index{rang}
+\index{déterminant}
+\index{dimension!\( n\)-formes multilinéaires alternées}
+
+\begin{proof}
+    Soit \( \{ e_i \}\), une base de \( E\) et \( f\colon E\to \eK\) une \( n\)-forme linéaire alternée, puis \( (v_1,\ldots, v_n)\) des vecteurs de \( E\). Nous pouvons les écrire dans la base
+    \begin{equation}
+        v_j=\sum_{i=1}^n\alpha_{ij}e_i
+    \end{equation}
+    et alors exprimer \( f\) par
+    \begin{subequations}
+        \begin{align}
+            f(v_1,\ldots, v_n)&=f\big( \sum_{i_1=1}^n\alpha_{1i_1}e_{i_1},\ldots, \sum_{i_n=1}^n\alpha_{ni_n}e_{i_n} \big)\\
+            &=\sum_{i,j}\alpha_{1i_1}\ldots \alpha_{ni_n}f(e_{i_1},\ldots, e_{i_n}).
+        \end{align}
+    \end{subequations}
+    Étant donné que \( f\) est alternée, les seuls termes de la somme sont ceux dont les \( i_k\) sont tous différents, c'est à dire ceux où \( \{ i_1,\ldots, i_n \}=\{ 1,\ldots, n \}\). Il y a donc un terme par élément du groupe des permutations \( S_n\) et
+    \begin{equation}
+        f(v_1,\ldots, v_n)=\sum_{\sigma\in S_n}\alpha_{\sigma(1)1}\ldots \alpha_{\sigma(n)n}f(e_{\sigma(1)},\ldots, e_{\sigma(n)}).
+    \end{equation}
+    En utilisant encore une fois le fait que la forme \( f\) soit alternée, \( f=f(e_1,\ldots, e_n)\Pi\) où
+    \begin{equation}
+        \Pi(v_1,\ldots, v_n)=\sum_{\sigma\in S_n}\epsilon(\sigma)\alpha_{\sigma(1)1}\ldots \alpha_{\sigma(n)n}.
+    \end{equation}
+    Pour rappel, la donnée des \( v_i\) est dans les nombres \( \alpha_{ij}\).
+    
+    L'espace des \( n\)-formes alternées est donc \emph{au plus} de dimension \( 1\). Pour montrer qu'il est exactement de dimension \( 1\), il faut et suffit de prouver que \( \Pi\) est alternée. Par le lemme \ref{LemHiHNey}, il suffit de prouver que cette forme est antisymétrique\footnote{C'est ici que joue l'hypothèse sur la caractéristique de \( \eK\).}. 
+
+    Soient donc \( v_1,\ldots, v_n\) tels que \( v_i=v_j\). En posant \( \tau=(1i)\) et \( \tau'=(2j)\) et en sommant sur \( \sigma\tau\tau'\) au lieu de \( \sigma\), nous pouvons supposer que \( i=1\) et \( j=2\). Montrons que \( \Pi(v,v,v_3,\ldots, v_n)=0\) en tenant compte que \( \alpha_{i1}=\alpha_{i2}\) :
+    \begin{subequations}
+        \begin{align}
+            \Pi(v,v,v_3,\ldots, v_n)&=\sum_{\sigma\in S_n}\epsilon(\sigma)\alpha_{\sigma(1)1}\alpha_{\sigma(2)2}\alpha_{\sigma(3)3}\ldots \alpha_{\sigma(n)n}\\
+            &=\sum_{\sigma\in S_n}\epsilon(\sigma\tau)\alpha_{\sigma\tau(1)1}\alpha_{\sigma\tau(2)2}\alpha_{\sigma\tau(3)3}\ldots \alpha_{\sigma\tau(n)n}&\text{où } \tau=(12)\\
+            &=-\sum_{\sigma\in S_n}\epsilon(\sigma)\alpha_{\sigma(1)1}\alpha_{\sigma(2)2}\alpha_{\sigma(3)3}\ldots \alpha_{\sigma(n)n} \\
+            &=-\Pi(v,v,v_3,\ldots, v_n).
+        \end{align}
+    \end{subequations}
+\end{proof}
+
+\begin{lemma}   \label{LemcDOTzM}
+    Soit \( \eK\) un corps fini autre que \( \eF_2\)\quext{Je ne comprends pas très bien à quel moment joue cette hypothèse.}, soit un groupe abélien \( M\) et un morphisme \( \varphi\colon \GL(n,\eK)\to M\). Alors il existe un unique morphisme \( \delta\colon \eK^*\to M\) tel que \( \varphi=\delta\circ\det\).
+\end{lemma}
+
+\begin{proof}
+    D'abord le groupe dérivé de \( \GL(n,\eK)\) est \( \SL(n,\eK)\) parce que les éléments de \( D\big( \GL(n,\eK) \big)\) sont de la forme \( ghg^{-1}h^{-1}\) dont le déterminant est \( 1\).
+    
+    De plus le groupe \( \SL(n,\eK)\) est normal dans \( \GL(n,\eK)\). Par conséquent \( \GL(n,\eK)/\SL(n,\eK)\) est un groupe et nous pouvons définir l'application relevée
+    \begin{equation}
+        \tilde \varphi\colon \frac{ \GL(n,\eK) }{ \SL(n,\eK) }\to M
+    \end{equation}
+    vérifiant \( \varphi=\tilde \varphi\circ\pi\) où \( \pi\) est la projection. 
+
+    Nous pouvons faire la même chose avec l'application
+    \begin{equation}
+        \det\colon \GL(n,\eK)\to \eK^*
+    \end{equation}
+    qui est un morphisme de groupes dont le noyau est \( \SL(n,\eK)\). Cela nous donne une application
+    \begin{equation}
+        \tilde \det\colon \frac{ \GL(n,\eK) }{ \SL(n,\eK) }\to \eK^*
+    \end{equation}
+    telle que \( \det=\tilde \det\circ\pi\). Cette application \( \tilde \det\) est un isomorphisme. En effet elle est surjective parce que le déterminant l'est et elle est injective parce que son noyau est précisément ce par quoi on prend le quotient. Par conséquent \( \tilde \det \) possède un inverse et nous pouvons écrire
+    \begin{equation}
+        \varphi=\tilde \varphi\circ\tilde \det^{-1}\circ\tilde \det\circ\pi.
+    \end{equation}
+    État donné que \( \tilde \det\circ\pi=\det\), nous avons alors \( \varphi=\delta\circ\det\) avec \( \delta=\tilde \varphi\circ\tilde \det^{-1}\). Ceci conclut la partie existence de la preuve.
+
+    En ce qui concerne l'unicité, nous considérons \( \delta'\colon \eK^*\to M\) telle que \( \varphi=\delta'\circ\det\). Pour tout \( u\in \GL(n,\eK)\) nous avons \( \delta'(\det(u))=\varphi(u)=\delta(\det(u))\). L'application \( \det\) étant surjective depuis \( \GL(n,\eK)\) vers \( \eK^*\), nous avons \( \delta'=\delta\).
+\end{proof}
+
+\begin{theorem}
+    Soit \( p\geq 3\) un nombre premier et \( V\), un \( \eF_p\)-espace vectoriel de dimension finie \( n\). Pour tout \( u\in\GL(V)\) nous avons
+    \begin{equation}
+        \epsilon(u)=\left(\frac{\det(u)}{p}\right).
+    \end{equation}
+\end{theorem}
+Ici \( \epsilon\) est la signature de \( u \) vue comme une permutation des éléments de \( \eF_p\).
+
+\begin{proof}
+    Commençons par prouver que
+    \begin{equation}
+        \epsilon\colon \GL(V)\to \{ -1,1 \}.
+    \end{equation}
+    est un morphisme. Si nous notons \( \bar u\in S(V)\) l'élément du groupe symétrique correspondant à la matrice \( u\in \GL(V)\), alors nous avons \( \overline{ uv }=\bar u\circ\bar v\), et la signature étant un homomorphisme (proposition \ref{ProphIuJrC}), 
+    \begin{equation}
+        \epsilon(uv)=\epsilon(\bar u\circ\bar v)=\epsilon(\bar u)\epsilon(\bar v).
+    \end{equation}
+    Par ailleurs \( \{ -1,1 \}\) est abélien, donc le lemme \ref{LemcDOTzM} s'applique et nous pouvons considérer un morphisme \( \delta\colon \eF_p^*\to \{ -1,1 \}\) tel que \( \epsilon=\delta\circ\det\).
+
+    Nous allons utiliser le lemme \ref{Lemoabzrn} pour montrer que \( \delta\) est le symbole de Legendre. Pour cela il nous faudrait trouver un \( x\in \eF_p^*\) tel que \( \delta(x)=-1\). Étant donné que \( \det\) est surjective, nous cherchons ce \( x\) sous la forme \( x=\det(u)\). Par conséquent nous aurions
+    \begin{equation}
+        \delta(x)=(\delta\circ\det)(u)=\epsilon(u),
+    \end{equation}
+    et notre problème revient à trouver une matrice \( u\in\GL(V)\) dont la permutation associée soit de signature \( -1\).
+
+    Soit \( n=\dim V\); en conséquence de la proposition \ref{PropHfrNCB}\ref{ItemiEFRTg}, l'espace \( \eE_q=\eF_{p^n}\) est un \( \eF_p\)-espace vectoriel de dimension \( n\) et est donc isomorphe en tant qu'espace vectoriel à \( V\). Étant donné que \( \eF_q\) est un corps fini, nous savons que \( \eF_q^*\) est un groupe cyclique à \( q-1\) éléments. Soit \( y\), un générateur de \( \eF_q^*\) et l'application
+    \begin{equation}
+        \begin{aligned}
+            \beta\colon \eF_q&\to \eF_q \\
+            x&\mapsto yx. 
+        \end{aligned}
+    \end{equation}
+    Cela est manifestement \( \eF_p\)-linéaire (ici \( y\) et \( x\) sont des classes de polynômes et \( \eF_p\) est le corps des coefficients). L'application \( \beta\) fixe zéro et à part zéro, agit comme le cycle
+    \begin{equation}
+        (1,y,y^2,\ldots, y^{q-2}).
+    \end{equation}
+    Nous savons qu'un cycle de longueur \( n\) est de signature \( (-1)^{n+1}\). Ici le cycle est de longueur \( q-1\) qui est pair (parce que \( p\geq 3\)) et par conséquent, l'application \( \beta\) est de signature \( -1\).
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Déterminant d'une famille de vecteurs}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous considérons un corps \( \eK\) et l'espace vectoriel \( E\) de dimension \( n\) sur \( \eK\).
+
+\begin{definition}[Déterminant d'une famille de vecteurs\cite{MathAgreg}]\label{DEFooODDFooSNahPb}
+    Le \defe{déterminant}{déterminant!d'une famille de vecteurs} \( (v_1,\ldots, v_n)\) dans la base \( B\) est l'élément de \( \eK\)
+    \begin{equation}        \label{EQooOJEXooXUpwfZ}
+        \det_{(e_1,\ldots, e_n)}(v_1,\ldots, v_n)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^ne^*_{\sigma(i)}(v_i)
+    \end{equation}
+    où la somme porte sur le groupe symétrique, \( \epsilon(\sigma)\) est la signature de la permutation \( \sigma\) et \( e_k^*\) est le dual de \( e_k\).
+
+    Nous le notons \( \det_{(e_1,\ldots, e_n)}(v_1,\ldots, v_n)\).
+\end{definition}
+
+\begin{lemma}[\cite{MathAgreg}]     \label{LemJMWCooELZuho}
+    Les propriétés du déterminant. Soit \( B\) une base de \( E\).
+    \begin{enumerate}
+        \item
+            L'application \( \det_B\colon E^n\to \eK\) est \( n\)-linéaire et alternée.
+        \item
+            Pour toute base, \( \det_B(B)=1\).
+        \item
+            Le déterminant ne change pas si on remplace un vecteur par une combinaison linéaire des autres :
+            \begin{equation}
+                \det_B(v_1,\ldots, v_n)=\det_B\big( v_1+\sum_{s=2}^na_sv_s,v_2,\ldots, v_n \big).
+            \end{equation}
+        \item
+            Si on permute les vecteurs,
+            \begin{equation}
+                \det_B(v_1,\ldots, v_n)=\epsilon(\sigma)\det_B(v_{\sigma(1)},\ldots, v_{\sigma(n)}).
+            \end{equation}
+        \item
+            Si \( B'\) est une autre base :
+            \begin{equation}        \label{EqAWICooBLTTOY}
+                \det_B=\det_B(B')\det_{B'}
+            \end{equation}
+        \item
+            Nous avons aussi la formule \( \det_{B}(B')\det_{B'}(B)=1\).
+        \item\label{ItemDWFLooDUePAf}
+            Les vecteurs \( \{ v_1,\ldots, v_n \}\) forment une base si et seulement si \( \det_B(v_1,\ldots, v_n)\neq 0\).
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Point par point.
+    \begin{enumerate}
+        \item
+            En posant \( v_1=x_1+\lambda x_2\) nous avons
+            \begin{subequations}
+                \begin{align}
+                    \det_B(x_1+\lambda x_2,v_2,\ldots, v_n)&=\sum_{\sigma}\epsilon(\sigma)\prod_{i=1}^ne^*_{\sigma(i)}(v_i)\\
+                    &=\sum_{\sigma}\epsilon(\sigma)\Big( e^*_{\sigma(1)}(x_1+\lambda x_2) \Big)\prod_{i=2}^ne^*_{\sigma(i)}(v_i).
+                \end{align}
+            \end{subequations}
+            À partir de là, la linéarité de \( e^*_{\sigma(1)}\) montre que \( \det_B\) est linéaire en son premier argument. Pour les autres argument, le même calcul tient.
+
+            En ce qui concerne le fait d'être alternée, permuter \( v_k\) et \( v_l\) revient à calculer \( \det_B( v_{\sigma_{kl}(1)},\ldots, v_{\sigma_{kl}(n)} )\), c'est à dire changer la somme \( \sum_{\sigma}\) en \( \sum_{\sigma\circ\sigma_{kl}}\). Cela ajoute \( 1\) à \( \epsilon(\sigma)\) vu que l'on ajoute une permutation.
+        \item
+            Nous avons
+            \begin{equation}
+                \det_B(B)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^n\underbrace{e_{\sigma(i)}^*(e_i)}_{=\delta_{\sigma(i),i}}.
+            \end{equation}
+            Si \( \sigma\) n'est pas l'identité, le produit contient forcément un facteur nul. Il ne reste de la somme que \( \sigma=\id\) et le résultat est \( 1\).
+        \item
+            Vu que \( \det_B\) est linéaire en tous ses arguments, 
+            \begin{equation}
+                \det_B\big( v_1+\sum_{s=2}^na_sv_s,v_2,\ldots, v_n \big)=\det_B(v_1,\ldots, v_n)+\sum_{s=2}^na_s\det_B(v_s,v_2,\ldots, v_n).
+            \end{equation}
+            Chacun des termes de la somme est nul parce qu'il y a répétition de \( v_s\) parmi les arguments alors que la forme est alternée.
+        \item
+            Nous devons calculer \( \det_B(v_{\sigma(1)},\ldots, v_{\sigma(n)})\), et pour y voir plus clair nous posons \( w_i=v_{\sigma(i)}\). Alors :
+            \begin{subequations}
+                \begin{align}
+                    \det_B(v_{\sigma(1)},\ldots, v_{\sigma(n)})&=\sum_{\sigma'}\epsilon(\sigma')\prod_{i=1}^ne^*_{\sigma'(i)}(w_i)\\
+                    &=\sum_{\sigma'}\epsilon(\sigma')\prod_{i=1}^ne^*_{\sigma'(i)}(v_{\sigma(i)})\\
+                    &=\sum_{\sigma'}\epsilon(\sigma')\prod_{i=1}^ne^*_{\sigma^{-1}\sigma'(i)}(v_i)\\
+                    &=\sum_{\sigma'}\epsilon(\sigma\sigma')\prod_{i=1}^ne^*_{\sigma'(i)}(v_i)\\
+                    &=\epsilon(\sigma)\det_B(v_1,\ldots, v_n).
+                \end{align}
+            \end{subequations}
+            Justifications : nous avons d'abord modifié l'ordre des éléments du produit et ensuite l'ordre des éléments de la somme. Nous avons ensuite utilisé le fait que \( \epsilon\colon S_n\to \{ 0,1 \}\) était un morphisme de groupe (proposition \ref{ProphIuJrC}).
+        \item
+            Étant donné que l'espace des formes multilinéaires alternées est de dimension \( 1\), il existe un \( \lambda\in \eK\) tel que \( \det_B=\lambda\det_{B'}\). Appliquons cela à \( B'\) :
+            \begin{equation}
+                \det_B(B')=\lambda\det_{B'}(B'),
+            \end{equation}
+            donc \( \lambda=\det_B(B')\).
+        \item
+            Il suffit d'appliquer l'égalité précédente à \( B\) en nous souvenant que \( \det_B(B)=1\).
+        \item
+            Si \( B'=\{ v_1,\ldots, v_n \}\) est une base alors \( \det_B(B')\neq 0\), sinon il n'est pas possible d'avoir \( \det_B(B')\det_{B'}(B)=1\).
+
+            À l'inverse, si \( B'\) n'est pas une base, c'est que \( \{ v_1,\ldots, v_n \}\) est liée par le théorème \ref{ThoMGQZooIgrXjy}\ref{ItemHIVAooPnTlsBi}. Il y a donc moyen de remplacer un des vecteurs par une combinaison linéaire des autres. Le déterminant s'annule alors.
+    \end{enumerate}
+\end{proof}
+
+D'après la proposition \ref{ProprbjihK}, il existe une unique forme \( n\)-linéaire alternée égale à \( 1\) sur \( B\), et c'est \( \det_B\colon E^n\to \eK\).
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Déterminant d'un endomorphisme}
+%---------------------------------------------------------------------------------------------------------------------------
+
+L'interprétation géométrique du déterminant en termes d'aires et de volumes est donnée après la théorème \ref{ThoBVIJooMkifod}.
+
+\begin{lemma}       \label{LEMooQTRVooAKzucd}
+    Si \( f\colon E\to E\) est un endomorphisme, si \( B\) et \( B'\) sont deux bases, alors \( \det_B\big( f(B) \big)=\det_{B'}\big( f(B') \big)  \).
+\end{lemma}
+
+\begin{proof}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            \varphi\colon E^n&\to \eK \\
+            v_1,\ldots, v_n&\mapsto \det_B\big( f(v_1),\ldots, f(v_n) \big) 
+        \end{aligned}
+    \end{equation}
+    est \( n\)-linéaire et alternée; il existe donc \( \lambda\in \eK\) tel que \( \varphi=\lambda\det_B\). En appliquant cela à \( B\) :
+    \begin{equation}
+        \det_B\big( f(B) \big)=\lambda \det_B(B)=\lambda.
+    \end{equation}
+    Nous avons donc déjà prouvé que \( \lambda=\det_B\big( f(B) \big)\), c'est à dire
+    \begin{equation}
+        \det_B\big( f(v) \big)=\det_B\big( f(B) \big)\det_B(v).
+    \end{equation}
+    
+    Nous allons maintenant introduire \( B'\) là où il y a du \( v\) en utilisant les formules \eqref{EqAWICooBLTTOY} :
+    \begin{subequations}
+        \begin{align}
+            \det_B\big( f(v) \big)&=\det_B(B')\det_{B'}\big( f(v) \big)\\
+            \det_B(v)=\det_B(B')\det_{B'}(v).
+        \end{align}
+    \end{subequations}
+    Nous obtenons
+    \begin{equation}
+        \det_{B'}\big( f(v) \big)=\det_B\big( f(B) \big)\det_{B'}(v).
+    \end{equation}
+    Et on applique cela à \( v=B'\) :
+    \begin{equation}
+        \det_{B'}\big( f(B') \big)=\det_B\big( f(B) \big)\underbrace{\det_{B'}(B')}_{=1}.
+    \end{equation}
+\end{proof}
+
+Cette proposition nous permet de définir le déterminant d'un endomorphisme de la façon suivante sans préciser la base.
+\begin{definition}[\cite{MathAgreg}]        \label{DefCOZEooGhRfxA}
+    Si \( f\colon E\to E\) est un endomorphisme, le \defe{déterminant}{déterminant!d'un endomorphisme} de \( f\) est
+    \begin{equation}
+        \det(f)=\det_B\big( f(B) \big)
+    \end{equation}
+\end{definition}
+
+Couplé à la formule \ref{EQooOJEXooXUpwfZ}, nous pouvons écrire la formule pratique à utiliser le plus souvent. Si \( \{ e_i \}_{i=1,\ldots, n}\) est une base orthonormée de \( E\) et si \( f\colon E\to E\) est un endomorphisme,
+\begin{equation}
+    \det(f)=\sum_{\sigma\in S_n}\epsilon(\epsilon)\prod_{i=1}^n\langle e_{\sigma(i)}, f(e_i)\rangle.
+\end{equation}
+Et si vous avez tout suivi, vous aurez remarqué que les produits scalaires impliqués dans cette formule sont les éléments de la matrice de \( f\) dans la base \( \{ e_i \}\) parce que \( \langle e_i, f(e_j)\rangle \) est la composante \( i\) de l'image de \( e_j\) par \( f\). Si la matrice est composée en mettant en colonne les images des vecteurs de base, le compte est bon.
+
+\begin{proposition}     \label{PropYQNMooZjlYlA}
+    Principales propriétés géométriques du déterminant d'un endomorphisme.
+    \begin{enumerate}
+        \item   \label{ItemUPLNooYZMRJy}
+            Si \( f\) et \( g\)  sont des endomorphismes, alors \( \det(f\circ g)=\det(f)\det(g)\).
+        \item       \label{ITEMooNZNLooODdXeH}
+            L'endomorphisme \( f\) est un automorphisme\footnote{Endomorphisme inversible, définition \ref{DEFooOAOGooKuJSup}.} si et seulement si \( \det(f)\neq 0\).\index{déterminant!et inversibilité}
+        \item   \label{ITEMooZMVXooLGjvCy}
+            Si \( \det(f)\neq 0\) alors \( \det(f^{-1})=\det(f)^{-1}\).
+        \item       \label{ItemooPJVYooYSwqaE}
+            L'application \( \det\colon \GL(E)\to \eK\setminus\{ 0 \}\) est un morphisme de groupe.
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Point par point.
+    \begin{enumerate}
+        \item
+            Nous considérons l'application
+            \begin{equation}
+                \begin{aligned}
+                    \varphi\colon E^n&\to \eK \\
+                    v&\mapsto \det_B\big( f(v) \big). 
+                \end{aligned}
+            \end{equation}
+            Comme d'habitude nous avons \( \varphi(v)=\lambda\det_B(v)\). En appliquant à \( B\) et en nous souvenant que \( \det_B(B)=1\) nous avons
+                $\det_B\big( f(B) \big)=\lambda$. Autrement dit :
+                \begin{equation}
+                    \lambda=\det(f).
+                \end{equation}
+            Calculons à présent \( \varphi\big( g(B) \big)\) : d'une part,
+            \begin{equation}
+                \varphi\big( g(B) \big)=\det_B\big( (f\circ g)(B) \big)
+            \end{equation}
+            et d'autre part,
+            \begin{equation}
+                \varphi\big( g(B) \big)=\lambda\det_B\big( g(B) \big)=\lambda\det(g)
+            \end{equation}
+            En égalisant et en reprenant la la valeur déjà trouvée de \( \lambda\),
+            \begin{equation}
+                \det\big(f\circ g)(B) \big)=\det(f)\det(g),
+            \end{equation}
+            ce qu'il fallait.
+        \item
+            Supposons que \( f\) soit un automorphisme. Alors si \( B\) est une base, \( f(B) \) est une base. Par conséquent \( \det(f)=\det_B\big( f(B) \big)\neq 0\) parce que \( f(B)\) est une base (lemme \ref{LemJMWCooELZuho}\ref{ItemDWFLooDUePAf}).
+
+            Réciproquement, supposons que \( \det(f)\neq 0\). Alors si \( B\) est une base quelconque nous avons \( \det_B\big( f(B) \big)\neq 0\), ce qui est uniquement possible lorsque \( f(B)\) est une base. L'application \( f\) transforme donc toute base en une base et est alors un automorphisme d'espace vectoriel.
+        \item
+            Vu que le déterminant de l'identité est \( 1\) et que \( f\) est inversible, \( 1=\det(f\circ f^{-1})=\det(f)\det(f^{-1})\).
+    \end{enumerate}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Déterminant d'une matrice}
+%---------------------------------------------------------------------------------------------------------------------------
+
+La proposition \ref{PROPooCESFooGOZBNI} nous assure que toute application linéaire possède une matrice associée (dans une base). Vu de la matrice, voici quelque conséquences de la proposition \ref{PropYQNMooZjlYlA}.
+\begin{proposition}
+    Quelques propriétés du déterminant vu de la matrice.
+    \begin{enumerate}
+        \item
+            Si on permute deux lignes ou deux colonnes d'une matrice, alors le déterminant change de signe.
+        \item
+            Si on multiplie une ligne ou une colonne d'une matrice par un nombre $\lambda$, alors le déterminant est multiplié par $\lambda$.
+        \item
+            Si deux lignes ou deux colonnes sont proportionnelles, alors le déterminant est nul.
+        \item
+            Si on ajoute à une ligne une combinaison linéaire des autres lignes, alors le déterminant ne change pas (idem pour les colonnes).
+    \end{enumerate}
+\end{proposition}
+
+Le déterminant d'une matrice et d'une application linéaire est la définition \ref{DefCOZEooGhRfxA}, et les principales propriétés algébriques sont données dans la proposition \ref{PropYQNMooZjlYlA}.
+
+En dimension deux, le déterminant de la matrice
+    $\begin{pmatrix}
+        a    &   b    \\ 
+        c    &   d    
+    \end{pmatrix}$
+est le nombre
+\begin{equation}        \label{EQooQRGVooChwRMd}
+     \det\begin{pmatrix}
+         a   &   b    \\ 
+         c   &   d    
+     \end{pmatrix}=\begin{vmatrix}
+          a  &   b    \\ 
+        c    &   d    
+    \end{vmatrix}=ad-cb.
+\end{equation}
+Ce nombre détermine entre autres le nombre de solutions que va avoir le système d'équations linéaires associé à la matrice.
+
+Pour une matrice $3\times 3$, nous avons le même concept, mais un peu plus compliqué; nous avons la formule
+\begin{equation}
+    \det
+    \begin{pmatrix}
+        a_{11}    &   a_{12}    &   a_{13}    \\
+        a_{21}    &   a_{22}    &   a_{23}    \\
+        a_{31}    &   a_{32}    &   a_{33}    
+    \end{pmatrix}
+    =
+    \begin{vmatrix}
+        a_{11}    &   a_{12}    &   a_{13}    \\
+        a_{21}    &   a_{22}    &   a_{23}    \\
+        a_{31}    &   a_{32}    &   a_{33}    
+    \end{vmatrix}=
+    a_{11}\begin{vmatrix}
+        a_{22}  &   a_{23}    \\ 
+        a_{32}    &   a_{33}    
+    \end{vmatrix}-
+    a_{12}\begin{vmatrix}
+        a_{21}  &   a_{23}    \\ 
+        a_{31}    &   a_{33}
+    \end{vmatrix}+
+    a_{13}\begin{vmatrix}
+        a_{21}  &   a_{22}    \\ 
+        a_{31}    &   a_{32}
+    \end{vmatrix}.
+\end{equation}
+
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Déterminant de Vandermonde}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[\cite{fJhCTE}]  \label{PropnuUvtj}
+    Le \defe{déterminant de Vandermonde}{déterminant!Vandermonde}\index{Vandermonde (déterminant)} est le polynôme en \( n\) variables donné par
+    \begin{equation}
+        V(T_1,\ldots, T_n)=\det\begin{pmatrix}
+             1   &   1    &   \ldots    &   1    \\
+             T_1   &   T_2    &   \ldots    &   T_n    \\
+             \vdots   &   \ddots    &   \ddots    &   \vdots    \\ 
+             T_1^{n-1}   &   T_2^{n-1}    &   \ldots    &   T_n^{n-1}     
+         \end{pmatrix}=\prod_{1\leq i<j\leq n}(T_j-T_i).
+    \end{equation}
+    Notez que l'inégalité du milieu est stricte (sinon d'ailleurs l'expression serait nulle).
+\end{proposition}
+
+\begin{proof}
+    Nous considérons le polynôme
+    \begin{equation}
+        f(X)=V(T_1,\ldots, T_{n-1},X)\in \big( \eK[T_1,\ldots, T_{n-1}] \big)[X].
+    \end{equation}
+    C'est un polynôme de degré au plus \( n-1\) en \( X\) et il s'annule aux points \( T_1,\ldots, T_{n-1}\). Par conséquent il existe \( \alpha\in \eK[T_1,\ldots, T_{n-1}]\) tel que
+    \begin{equation}    \label{EqeVxRwO}
+        f=\alpha(X-T_{n-1})\ldots(X-T_1).
+    \end{equation}
+    Nous trouvons \( \alpha\) en écrivant \( f(0)\). D'une part la formule \eqref{EqeVxRwO} nous donne
+    \begin{equation}    \label{EqblwWMj}
+        f(0)=\alpha(-1)^{n-1}T_1\ldots T_{n-1}.
+    \end{equation}
+    D'autre par la définition donne
+    \begin{subequations}
+        \begin{align}
+            f(0)&=\det\begin{pmatrix}
+                 1   &   \cdots    &   1    &   1    \\
+                 T_1      &       &   T_{n-1}    &   0    \\
+                 \vdots   &       &   \vdots    &   \vdots    \\ 
+                 T_1^{n-1}   &   \cdots    &   T_{n-1}^{n-1}    &   0     
+             \end{pmatrix}\\
+             &=(-1)^{n-1}\det\begin{pmatrix}
+                 T_1   &   \ldots    &   T_{n-1}    \\
+                 \vdots   &   \ddots    &   \vdots    \\
+                 T_1^{n-1}   &   \ldots    &   T_{n-1}^{n-1}
+             \end{pmatrix}\\
+             &=(-1)^{n-1}T_1\ldots T_{n-1}\det\begin{pmatrix}
+                 1   &   \cdots    &   1    \\
+                 \vdots   &   \ddots    &   \vdots    \\
+                 T_1^{n-1}   &   \cdots    &   T_{n-1}^{n-1}
+             \end{pmatrix}\\
+             &=(-1)^{n-1}T_1\ldots T_{n-1}V(T_1,\ldots, T_{n-1})
+        \end{align}
+    \end{subequations}
+    En égalisant avec \eqref{EqblwWMj}, nous trouvons \( \alpha=V(T_1,\ldots, T_{n-1})\), et donc
+    \begin{equation}
+        f=V(T_1,\ldots, T_{n-1})\prod_{j\leq n-1}(X-T_j)
+    \end{equation}
+    Enfin, une récurrence montre que
+    \begin{subequations}
+        \begin{align}
+            V(T_1,\ldots, T_n)&=f(T_n)\\
+            &=V(T_1,\ldots, T_{n-1})\prod_{j\leq n-1}(T_n-T_j)\\
+            &=\prod_{k\leq n}\prod_{j\leq k-1}(T_k-T_j)\\
+            &=\prod_{1\leq j<k\leq n}(T_i-T_j).
+        \end{align}
+    \end{subequations}
+\end{proof}
+
+\begin{example}
+    Le déterminant de Vandermonde (proposition \ref{PropnuUvtj}) est alterné, semi-symétrique et non symétrique. Le fait qu'il soit alterné est le fait qu'il soit un déterminant. Étant donné qu'il est alterné, il est semi-symétrique parce que sur \( A_n\), nous avons \( \epsilon=1\). Étant donné qu'il est alterné, il change de signe sous l'action des éléments impairs de \( S_n\) et n'est donc pas symétrique.
+\end{example}
+
+\begin{proposition}\index{action de groupe} \label{PropUDqXax}
+    Un polynôme semi-symétrique \( f\in \eK[T_1,\ldots, T_n]\) se décompose de façon unique en
+    \begin{equation}
+        f=P+VQ
+    \end{equation}
+    où \( P\) et \( Q\) sont deux polynômes symétriques.
+\end{proposition}
+\index{groupe!permutation}
+\index{polynôme!symétrique}
+
+\begin{proof}
+
+    Nous commençons par prouver l'unicité en montrant que si \( f=PVQ\) avec \( P\) et \( Q\) symétrique, alors \( P\) et \( Q\) sont donnés par des formules explicites en termes de \( f\).
+
+
+    Si \( \sigma_1\) et \( \sigma_2\) sont deux permutations impaires de \( \{ 1,\ldots, n \}\), alors \( \sigma_1\cdot f=\sigma_2\cdot f\) parce que l'élément \( \sigma_2^{-1}\sigma_1\) est pair (proposition \ref{ProphIuJrC}), de telle sorte que \( \sigma_2^{-1}\sigma_1\cdot f=f\). Nous posons donc \( g=\tau\cdot f\) où \( \tau\) est une permutation impaire quelconque -- par exemple une transposition.
+
+    Vu que \( V\) est alternée et que \( \tau\) est une transposition nous avons
+    \begin{equation}
+        g=\tau\cdot f=P-VQ.
+    \end{equation}
+    Donc \( f+g=2P\) et \( f-g=2VQ\). Cela donne \( P\) et \( Q\) en terme de \( f\) et \( g\), et donc l'unicité.
+
+    Attention : cela ne donne pas un moyen de prouver l'existence parce que rien ne prouve pour l'instant que \( f-g\) peut effectivement être écrit sous la forme \( VQ\), c'est à dire que \( f-g\) soit divisible par \( V\). C'est cela que nous allons nous atteler à démontrer maintenant.
+
+    Nous commençons par prouver que \( f+g\) est symétrique et \( f-g\) alterné. Si \( \sigma\) est une transposition,
+    \begin{equation}
+        \sigma\cdot(f+g)=\sigma\cdot f+\sigma\tau\cdot f=g+f
+    \end{equation}
+    parce que \( \sigma\tau\) est pair. De la même façon,
+    \begin{equation}
+        \sigma\cdot(f-g)=g-f=\epsilon(\sigma)(f-g).
+    \end{equation}
+    Dans les deux cas nous concluons en utilisant le fait que toute permutation est un produit de transpositions (proposition \ref{PropPWIJbu}) et que \( \epsilon\) est un homomorphisme.
+
+    Soient maintenant deux entiers \( h<k\) dans \( \{ 1,\ldots, n \}\) et l'anneau
+    \begin{equation}
+        \big( \eK[T_1,\ldots, \hat T_k,\ldots, T_n] \big)[T_k].
+    \end{equation}
+    Cet anneau contient le polynôme \( T_k-T_h\) où \( T_k\) est la variable et \( T_h\) est un coefficient. Nous faisons la division euclidienne de \( f-g\) par  \( T_k-T_h\) parce que nous avons dans l'idée de faire arriver le déterminant de Vandermonde et donc le produit de toutes les différences \( T_k-T_h\) :
+    \begin{equation}    \label{EqSHdgrG}
+        f-g=(T_k-T_h)q+r
+    \end{equation}
+    où \( \deg_{T_k}r<1\), c'est à dire que \( r\) ne dépends pas de \( T_k\). Nous revoyons maintenant l'égalité \eqref{EqSHdgrG} dans \( \eK[T_1,\ldots, T_n]\) et nous y appliquons la transposition \( \tau_{kh}\). Nous savons que \( \tau_{kh}(f-g)=-(f-g)\) et \( \tau_{kh}(T_k-T_h)=-(T_k-T_h)\), et donc
+    \begin{equation}    \label{EqVOhjKB}
+        -(f-g)=-(T_k-T_h)\tau_{kh}\cdot   q+\tau_{kh}\cdot r
+    \end{equation}
+    où \(\tau_{kh}\cdot r\) ne dépend pas de \( T_h\). Nous appliquons à \eqref{EqVOhjKB} l'application
+    \begin{equation}
+        \begin{aligned}
+            t\alpha\colon \eK[T_1,\ldots, T_n]&\to \eK[T_1,\ldots, \hat T_k,\ldots, T_n] \\
+            \alpha(PT_1,\ldots, \hat T_k,\ldots, T_n)&=P(T_1,\ldots, T_h,\ldots, T_n). 
+        \end{aligned}
+    \end{equation}
+    Cette application vérifie \( \alpha\big( \tau_{kh}\cdot r \big)=\alpha(r)\) et nous avons
+    \begin{equation}
+        -\alpha(f-g)=\alpha(r).
+    \end{equation}
+    Puis en appliquant \( \alpha\) à la relation \( f-g=(T_k-T_h)q+r\), nous trouvons
+    \begin{equation}
+        \alpha(f-g)=\alpha(r),
+    \end{equation}
+    et par conséquent \( \alpha(r)=0\). Ici nous utilisons l'hypothèse de caractéristique différente de deux. Dire que \( \alpha(r)=0\), c'est dire que \( r\) est divisible par \( T_k-T_h\), mais \( r\) étant de degré zéro en \( T_k\), nous avons \( r=0\). Par conséquent \( T_k-T_h\) divise \( f-g\) pour tout \( h<k\), et nous pouvons définir un polynôme \( Q\) par
+    \begin{equation}    \label{EqrnbgdA}
+        f-g=2Q\prod_{h<k}\prod_{k\leq n}(T_k-T_h)=2Q(T_1,\ldots, T_n)V(T_1,\ldots, T_n),
+    \end{equation}
+    où nous avons utilisé la formule du déterminant de Vandermonde de la proposition \ref{PropnuUvtj}.
+
+    Étant donné que \( f+g\) est un polynôme symétrique, nous allons aussi poser \( f+g=2P\) avec \( P\) symétrique.
+
+    Montrons à présent que \( Q\) est un polynôme symétrique. Soit \( \sigma\in S_n\); vu que nous savons déjà que \( f-g\) est alternée, nous avons
+    \begin{equation}    \label{EqpSPEyq}
+        \sigma\cdot (f-g)=\epsilon(\sigma)(f-g)=\epsilon(\sigma)2QV,
+    \end{equation}
+    Mais en appliquant \( \sigma\) à l'équation \eqref{EqrnbgdA},
+    \begin{subequations}
+        \begin{align}
+            \sigma\cdot (f-g)&=2(\sigma\cdot V)(T_1,\ldots, ,T_n)(\sigma\cdot Q)(T_1,\ldots,T_n)\\
+            &=2\epsilon(\sigma)V(T_1,\ldots, T_n)(\sigma\cdot Q)(T_1,\ldots, T_n).
+        \end{align}
+    \end{subequations}
+    En égalisant avec \eqref{EqpSPEyq} et en se souvenant que l'anneau \( \eK[T_1,\ldots, T_n]\) était intègre (théorème \ref{ThoBUEDrJ}), nous simplifions par \( 2\epsilon(\sigma)V\) pour obtenir
+    \begin{equation}
+        Q=\sigma\cdot Q,
+    \end{equation}
+    c'est à dire que \( Q\) est symétrique.
+
+    Au final nous avons \( f+q=2P\) et \( f-g=2VQ\) avec \( P\) et \( Q\) symétriques. En faisant la somme,
+    \begin{equation}
+        f=P+VQ.
+    \end{equation}
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Déterminant de Gram}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Si \( x_1,\ldots, x_r\) sont des vecteurs d'un espace vectoriel, alors le \defe{déterminant de Gram}{déterminant!Gram}\index{Gram (déterminant)} est le déterminant
+\begin{equation}
+    G(x_1,\ldots, x_r)=\det\big( \langle x_i, x_j\rangle  \big).
+\end{equation}
+Notons que la matrice est une matrice symétrique.
+
+\begin{proposition}\label{PropMsZhIK}
+    Si \( F\) est un sous-espace vectoriel de base \( \{ x_1,\ldots, x_n \}\) et si \( x\) est un vecteur, alors le déterminant de Gram est un moyen de calculer la distance entre \( x\) et \( F\) par 
+    \begin{equation}
+        d(x,F)^2=\frac{ G(x,x_1,\ldots, x_n)}{ G(x_1,\ldots, x_n) }.
+    \end{equation}
+\end{proposition}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Déterminant de Cauchy}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soient des nombres \( a_i\) et \( b_i\) (\( i=1,\ldots, n\)) tels que \( a_i+b_j\neq 0\) pour tout couple \( (i,j)\). Le \defe{déterminant de Cauchy}{déterminant!de Cauchy}\index{Cauchy!déterminant} est 
+\begin{equation}
+    D_n=\det\left( \frac{1}{ a_i+b_j } \right).
+\end{equation}
+
+\begin{proposition}[\cite{RollandRobertjyYDzY}] \label{ProptoDYKA}
+    Le déterminant de Cauchy est donné par la formule
+    \begin{equation}
+        D_n=\frac{ \prod_{i<j}(a_j-a_i)\prod_{i<j}(b_j-b_i) }{ \prod_{ij}(a_i+b_j) }.
+    \end{equation}
+\end{proposition}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Matrice de Sylvester}
+%---------------------------------------------------------------------------------------------------------------------------
+\label{subsecSQBJfr}
+
+La définition est pompée de \wikipedia{fr}{Matrice_de_Sylvester}{wikipédia}. Soient \( P\) et \( Q\) deux polynômes non nuls, de degrés respectifs \( m\) et \( n\) :
+\begin{subequations}
+    \begin{align}
+        P(x)=p_0+p_1x+\cdots +p_nx^n\\
+        Q(x)=q_0+q_1x+\cdots +q_mx^m.
+    \end{align}
+\end{subequations}
+La \defe{matrice de Sylvester}{matrice!de Sylvester}\index{Sylvester (matrice)} associée à \( P\) et \( Q\) est la matrice carrée \( m+n\times m+n\) définie ainsi :
+\begin{enumerate}
+    \item
+la première ligne est formée des coefficients de \( P\), suivis de 0 :
+\begin{equation}
+\begin{pmatrix} p_n & p_{n-1} & \cdots & p_1 & p_0 & 0 & \cdots & 0 \end{pmatrix} ;
+\end{equation}
+\item la seconde ligne s'obtient à partir de la première par permutation circulaire vers la droite ;
+\item les $(m-2)$ lignes suivantes s'obtiennent en répétant la même opération ;
+\item la ligne $(m+1)$ est formée des coefficients de \( Q\), suivis de 0 :
+    \begin{equation}
+    \begin{pmatrix} q_m & q_{m-1} & \cdots & q_1 & q_0 & 0 & \cdots & 0 \end{pmatrix} ;
+    \end{equation}
+    \item les $(m-1)$ lignes suivantes sont formées par des permutations circulaires.
+\end{enumerate}
+
+Ainsi dans le cas $n=4$ et $m=3$, la matrice obtenue est
+\begin{equation}    \label{EqPEgtle}
+S_{p,q}=\begin{pmatrix} 
+p_4 & p_3 & p_2 & p_1 & p_0 & 0 & 0 \\
+0 & p_4 & p_3 & p_2 & p_1 & p_0 & 0 \\
+0 & 0 & p_4 & p_3 & p_2 & p_1 & p_0 \\
+q_3 & q_2 & q_1 & q_0 & 0 & 0 & 0 \\
+0 & q_3 & q_2 & q_1 & q_0 & 0 & 0 \\
+0 & 0 & q_3 & q_2 & q_1 & q_0 & 0 \\
+0 & 0 & 0 & q_3 & q_2 & q_1 & q_0 \\
+\end{pmatrix}.
+\end{equation}
+Le déterminant de la matrice de Sylvester associée à \( P\) et \( Q\) est appelé le \defe{résultant}{résultant} de \( P\) et \( Q\) et noté \( \res(P,Q)\)\nomenclature[A]{\( \res(P,Q)\)}{résultat des polynômes \( P\) et \( Q\)}.
+
+Attention : si \( P\) est de degré \( n\) et \( Q\) de degré \( m\), il y a \( m\) lignes pour \( P\) et \( n\) pour \( Q\) dans le déterminant du résultant (et non le contraire).
+
+\begin{lemma}[\cite{QQuRUzA}]       \label{LemBFrhgnA}
+    Si \( P\) et \( Q\) sont deux polynômes de degrés \( n\) et \( m\) à coefficients dans l'anneau \( \eA\), alors pour tout \( \lambda\in \eA\),
+    \begin{subequations}
+        \begin{align}
+            \res(\lambda P,Q)&=\lambda^m\res(P,Q)\\
+            \res(P,\lambda Q)&=\lambda^n\res(P,Q).
+        \end{align}
+    \end{subequations}
+\end{lemma}
+
+\begin{proof}
+    Cela est simplement un comptage de nombre de lignes. Il y a \( m\) lignes contenant les coefficients de \( P\); donc prendre \( \lambda P\) revient à multiplier \( m\) lignes dans un déterminant et donc le multiplier par \( \lambda^m\).
+\end{proof}
+
+L'équation de Bézout \eqref{EqkbbzAi}\index{théorème!Bézout!utilisation} peut être traitée avec une matrice de Sylvester. Soient \( P\) et \( Q\), deux polynômes donnés et à résoudre l'équation 
+\begin{equation}    \label{EqSsyXOo}
+    xP+yQ=0
+\end{equation}
+par rapport aux polynômes inconnus \( x\) et \( y\) dont les degrés sont \( \deg(x)<\deg(Q)\) et \( \deg(y)<\deg(P)\). Si nous notons \( \tilde x\) et \( \tilde y\) la liste des coefficients de \( x\) et \( y\) (dans l'ordre décroissant de degré), nous pouvons récrire l'équation \eqref{EqSsyXOo} sous la forme
+\begin{equation}
+    S_{PQ}^t\begin{pmatrix}
+        \tilde x    \\ 
+        \tilde y    
+    \end{pmatrix}=0.
+\end{equation}
+Pour s'en convaincre, écrivons pour les polynômes de l'exemple \eqref{EqPEgtle} :
+\begin{equation}
+    \begin{pmatrix}
+        p_4    &   0    &   0    &   q_3    &   0    &   0    &   0\\ 
+        p_3    &   p_4    &   0    &   q_2    &   q_3    &   0    &   0\\ 
+        p_2    &   p_3    &   p_4    &   q_1    &   q_2    &   q_3    &   0\\ 
+        p_1    &   p_2    &   p_3    &   q_0    &   q_1    &   q_2    &   q_3\\ 
+        p_0    &   p_1    &   p_2    &   0    &   q_0    &   q_1    &   q_2\\ 
+        0    &   p_0    &   p_1    &   0    &   0    &   q_0    &   q_1\\ 
+        0    &   0    &   p_0    &   0    &   0    &   0    &   q_0\\    
+    \end{pmatrix}\begin{pmatrix}
+        x_2    \\ 
+        x_1  \\ 
+        x_0  \\    
+        y_3   \\ 
+        y_2    \\ 
+        y_1    \\ 
+        y_0    
+    \end{pmatrix}=
+    \begin{pmatrix}
+        x_2p_4+y_2q_3    \\ 
+        p_3x_2+p_4x_1+q_2y_3+q_3y_2  \\ 
+          \\    
+           \\ 
+        \vdots    \\ 
+            \\ 
+            
+    \end{pmatrix}
+\end{equation}
+Nous voyons que sur la ligne numéro \( k\) (en partant du bas et en numérotant de à partir de zéro) nous avons les produits \( p_ix_j\) et \( q_iy_j\) avec \( i+j=k\). La colonne de droite représente donc bien les coefficients du polynôme \( xP+yQ\).
+
+
+\begin{proposition} \label{PropAPxzcUl}
+    Le résultant de deux polynômes est non nul si et seulement si les deux polynômes sont premiers entre eux.
+\end{proposition}
+\index{déterminant!résultant}
+
+Un polynôme \( P\) a une racine double en \( a\) si et seulement si \( P\) et \( P'\) ont \( a\) comme racine commune, ce qui revient à dire que \( P\) et \( P'\) ne sont pas premiers entre eux. 
+
+Une application importante de ces résultats sera le théorème de Rothstein-Trager \ref{ThoXJFatfu} sur l'intégration de fractions rationnelles.
+
+\begin{example}
+    Si nous prenons \( P=aX^2+bX+c\) et \( P'=2aX+b\) alors la taille de la matrice de Sylvester sera \( 2+1=3\) et
+    \begin{equation}
+        S_{P,P'}=\begin{pmatrix}
+              a  &   b    &   c    \\
+            2a    &   b    &   0    \\
+            0    &   2a    &   b
+        \end{pmatrix}.
+    \end{equation}
+    Le résultant est alors
+    \begin{equation}
+        \res(P,P')=-a(b^2-4ac).
+    \end{equation}
+    Donc un polynôme du second degré a une racine double si et seulement si \( b^2-4ac=0\). Cela est un résultat connu depuis longtemps mais qui fait toujours plaisir à revoir.
+\end{example}
+
+La matrice de Sylvester permet aussi de récrire l'équation de Bézout pour les polynômes; voir le théorème \ref{ThoBezoutOuGmLB} et la discussion qui s'ensuit.
+
+Une proposition importante du résultant est qu'il peut s'exprimer à l'aide des racines des polynômes.
+\begin{proposition} \label{PropNDBOGNx}
+    Si
+    \begin{subequations}
+        \begin{align}
+        P(X)&=a_p\prod_{i=1}^p(X-\alpha_i)\\
+        Q(X)&=b_q\prod_{j=1}^q(X-\beta_i)
+        \end{align}
+    \end{subequations}
+    alors nous avons les expressions suivantes pour le résultant :
+    \begin{equation}        \label{EqCFUumjx}
+        \res(P,Q)=a_p^qb_q^p\prod_{i=1}^p\prod_{j=1}^q(\beta_j-\alpha_i)=b_q^p\prod_{j=1}^qP(\beta_j)=(-1)^{pq}a_p^q\prod_{i=1}^pQ(\alpha_i).
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Si \( P\) et \( Q\) ne sont pas premiers entre eux, d'une part la proposition \ref{PropAPxzcUl} nous dit que \( \res(P,Q)=0\) et d'autre part, \( P\) et \( Q\) ont un facteur irréductible en commun, ce qui  signifie que nous devons avoir un des \( X-\alpha_i\) égal à un des \( X-\beta_j\). Autrement dit, nous avons \( \alpha_i=\beta_j\) pour un couple \( (i,j)\). Par conséquent tous les membres de l'équation \eqref{EqCFUumjx} sont nuls.
+
+    Nous supposons donc que \( P\) et \( Q\) sont premiers entre eux. Nous commençons par supposer que les polynômes \( P\) et \( Q\) sont unitaires, c'est à dire que \( a_p=b_q=1\). Nous considérons alors l'anneau
+    \begin{equation}
+        \eA=\eZ[\alpha_1,\ldots, \alpha_p,\beta_1,\ldots, \beta_q].
+    \end{equation}
+    Dans cet anneau, l'élément \( \beta_j-\alpha_i\) est irréductible (tout comme \( X-Y\) est irréductible dans \( \eZ[X,Y]\)). Le résultant \( R=\res(P,Q)\) est un élément de \( \eA\) parce que tous leurs coefficients peuvent être exprimés à l'aide des \( \alpha_i\) et des \( \beta_j\). Dans \( \eA\), l'élément \( \beta_j-\alpha_i\) divise \( R\). En effet lorsque \( \beta_j=\alpha_i\), le déterminant définissant le résultant est nul, ce qui signifie que \( \beta_j-\alpha_i\) est un facteur irréductible de \( R\).
+
+    Par conséquent il existe un polynôme \( T\in \eA\) tel que
+    \begin{equation}
+        R=\lambda(\alpha_1,\ldots, \beta_q)\prod_{i=1}^p\prod_{j=1}^r(\beta_j-\alpha_i).
+    \end{equation}
+    Comptons les degrés. Pour donner une idée de ce calcul de degré, voici comment se présente, au niveau des dimensions, le déterminant :
+    \begin{equation}  \label{EqJCaATOH}
+    \xymatrix{%
+        \ar@{<->}[rrr]^{p+1}&&&& \ar@{<->}[r]^{q-1}  &\\
+        a_p\ar@{.}[rrd] &a_{p-1}\ar@{.}[rr]  &  & a_0\ar@{.}[rrd] & 0\ar@{.}[r]&0&\ar@{<->}[d]^q \\
+        0\ar@{.}[r]&0&a_p\ar@{.}[rr]&&a_1&a_0&\\
+        \ar@{<->}[rrrrr]_{p+q}&&&&&&
+       }
+    \end{equation}
+    si les \( a_i\) sont les coefficients de \( P\). Mais chacun des \( a_i\) est de degré \( 1\) en les \( \alpha_i\), donc le déterminant dans son ensemble est de degré \( q\) en les \( \alpha_i\), parce que \( R\) contient \( q\) lignes telles que \eqref{EqJCaATOH}. Le même raisonnement montre que \( R\) est de degré \( p\) en les \( \beta_j\). Par ailleurs le polynôme \( \prod_{i=1}^p\prod_{j=1}^r(\beta_j-\alpha_i)\) est de degré \( p\) en les \( \beta_j\) et \( q\) en les \( \alpha_i\). Nous en déduisons que \( T\) doit être un polynôme ne dépendant pas de \( \alpha_i\) ou de \( \beta_j\).
+
+    Nous pouvons donc calculer la valeur de \( T\) en choisissant un cas particulier. Avec \( P(X)=X^p\) et \( Q(X)=X^q+1\), il est vite vu que \( R(P,Q)=1\) et donc que \( T=1\).
+
+    Si les polynômes \( P\) et \( Q\) ne sont pas unitaires, le lemme \ref{LemBFrhgnA} nous permet de conclure. 
+
+\end{proof}
+
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de Kronecker}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous considérons \( K_n\) l'ensemble des polynômes de \( \eZ[X]\)
+\begin{enumerate}
+    \item
+        unitaires de degré \( n\),
+    \item
+        dont les racines dans \( \eC\) sont de modules plus petits ou égaux à \( 1\),
+    \item
+        et qui ne sont pas divisés par \( X\).
+\end{enumerate}
+Un tel polynôme s'écrit sous la forme
+\begin{equation}
+    P=X^n+\sum_{k=0}^{n-1}a_kX^k.
+\end{equation}
+
+\begin{theorem}[Kronecker\cite{KXjFWKA}]    \label{ThoOWMNAVp}
+    Les racines des éléments de \( K_n\) sont des racines de l'unité.
+\end{theorem}
+\index{théorème!Kronecker}
+\index{polynôme!à plusieurs indéterminées}
+\index{résultant!utilisation}
+\index{polynôme!symétrique}
+
+\begin{proof}
+    Vu que \( \eC\) est algébriquement clos 
+    nous pouvons considérer les racines \( \alpha_1,\ldots, \alpha_n\) de \( P\) dans \( \eC\). Nous les considérons avec leurs multiplicités.
+%TODO : lorsqu'on aura démontré que \eC est algébriquement clos, il faudra le référentier ici.
+
+    Soit \( R=X^n+\sum_{k=0}^{n-1}b_kX^k\) un élément de \( K_n\) dont nous notons \( \beta_1,\ldots, \beta_n\) les racines dans \( \eC\). Les relations coefficients-racines stipulent que
+    \begin{equation}
+        b_k=\sum_{1\leq i_1<\ldots <i_{n-k}\leq n}\prod_{j=1}^{n-k}\beta_{i_j}.
+    \end{equation}
+    En prenant le module et en se souvenant que \( | \beta_{l} |\leq 1\) pour tout \( l\), nous trouvons que
+    \begin{equation}
+        | b_k |\leq\binom{ n }{ n-k }.
+    \end{equation}
+    Mais comme \( b_k\in \eZ\), nous avons
+    \begin{equation}
+        b_k\in\big\{    -\binom{ n }{ n-k },-\binom{ n }{ n-k }+1,\ldots, 0,\cdots,\binom{ n }{ n-k }   \big\}
+    \end{equation}
+    qui est de cardinal \( \binom{ n }{ n-k }+1\). Nous avons donc
+    \begin{equation}
+        \Card(K_n)\leq\prod_{k=0}^{n-1}\big( 1+\binom{ n }{ n-k } \big)<\infty.
+    \end{equation}
+    La conclusion jusqu'ici est que \( K_n\) est un ensemble fini.
+
+    Pour chaque \( k\in \eN^*\) nous considérons les polynômes
+    \begin{subequations}
+        \begin{align}
+            P_k&=\prod_{i=1}^n(X-\alpha_i^k)\\
+            Q_k&=X^k-Y\in \eZ[X,Y],
+        \end{align}
+    \end{subequations}
+    et puis nous considérons le résultant \( R_k=\res_X(P,Q_k)\in \eZ[Y]\) :
+    \begin{equation}
+        R_k=\res_X(P,Q_k)=
+        \begin{pmatrix}
+            1&a_{n-1}&\cdots&a_0&0&\cdots&0&0&0\\  
+            0&1&a_{n-1}&\cdots&a_0&0&\cdots&0&0\\  
+            \vdots&\ddots&\ddots&\ddots&&\ddots&\\  
+            0&\cdots&0&1&a_{n-1}&\cdots&a_0&0&0\\
+            0&\cdots&0&0&1&a_{n-1}&\cdots&a_0&0\\
+            0&\cdots&0&0&0&1&a_{n-1}&\cdots&a_0\\
+        \\
+                    1&0&\ldots&0&-Y&0&\ldots&0&0\\
+                    0&1&0&\ldots&0&-Y&0&\ldots&0\\
+                &&\ddots&&&\ddots&\ddots\\
+                    0&\cdots&0&1&0&\cdots&0&-Y&0\\
+                    0&0&\cdots&0&1&0&\cdots&0&-Y
+        \end{pmatrix}
+    \end{equation}
+    Cela est un polynôme en \( Y\) dont le terme de plus haut degré est \( (-1)^nY^n\). Les petites formules de la proposition \ref{PropNDBOGNx} nous permettent d'exprimer \( R_k(Y)\) en termes des racines de \( P\) :
+    \begin{equation}
+        R_k(Y)=\prod_{i=1}^nQ_k(\alpha_i)=\prod_{i=1}^n(\alpha_i^k-Y)=(-1)^n\prod_{i=1}^n(Y-\alpha_i^k)=(-1)^nP_k(Y).
+    \end{equation}
+    Vu que \( P\in K_n\) nous savons que les \( \alpha_i\) ne sont pas tous nuls; donc \( P_k\in K_n\). Cependant nous avons vu que \( K_n\) est un ensemble fini; donc parmi les \( P_k\), il y a des doublons (et pas un peu)\quext{Ici dans \cite{KXjFWKA}, il déduit qu'on a un \( k\) tel que \( P_k=P_1=P\). Mois je vois pourquoi on a un \( k\) et un \( l\) tels que \( P_k=P_l\), mais pourquoi on peut en trouver un spécialement égal au premier ? Une réponse à cette question permettrait de solidement réduire la lourdeur de la suite de la preuve.}. Nous regardons même l'ensemble des \( P_{2^n}\) dans lequel nous pouvons en trouver deux les mêmes. Soit \( l>k\) tels que \( P_{2^k}=P_{2^l}\). Si \( \alpha\) est racine de \( P_{2^k}\), alors il est de la forme \( \alpha=\beta^{2^k}\) pour une certaine racines \( \beta\) de \( P\). Par conséquent
+    \begin{equation}    \label{EqBEgJtzm}
+        \alpha^{2^l/2^k}=\alpha^{2^{l-k}}
+    \end{equation}
+    est racine de \( P_{2^l}\). Notons que dans cette expression il n'y a pas de problèmes de définition d'exposant fractionnaire dans \( \eC\) parce que \( l>k\). Vu que \eqref{EqBEgJtzm} est racine de \( P_{2^l}\), il est aussi racine de \( P_{2^k}\). Donc
+    \begin{equation}
+        \big( \alpha^{2^{l-k}} \big)^{2^{l-k}}=\alpha^{2^{2(l-k)}}
+    \end{equation}
+    est racine de \( P_{2^l}\) et donc de \( P_{2^k}\). Au final nous savons que tous les nombres de la forme \( \alpha^{2^{n(l-k)}}\) sont racines de \( P_{2^k}\). Mais comme \( P_{2^k}\) a un nombre fini de racines, nous pouvons en trouver deux égales. Si nous avons
+    \begin{equation}
+        \alpha^{2^{n(l-k)}}=\alpha^{2^{m(l-k)}}
+    \end{equation}
+    pour certains entiers \( m>n\), alors
+    \begin{equation}
+        \alpha^{2^{n(l-k)}-2^{m(l-k)}}=1,
+    \end{equation}
+    ce qui prouver que \( \alpha\) est une racine de l'unité. Nous avons donc prouvé que toutes les racines de \( P_{2^k}\) sont des racines de l'unité et donc que les racines de \( P\) sont racines de l'unité.
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Produit scalaire}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}      \label{DEFooEEQGooNiPjHz}
+    Une \defe{forme bilinéaire}{forme!bilinéaire} sur un espace vectoriel \( E\) est une application \( b\colon E\times E\to \eK\) telle que
+    \begin{enumerate}
+        \item
+            \( b(u,v)=b(v,u)\),
+        \item
+            \( b(u+v,w)=b(u,w)+b(v,w)\),
+        \item
+            \( b(\lambda u,v)=\lambda b(u,v)\)
+    \end{enumerate}
+    pour tout \( u,v,w\in E\) et \( \lambda\in \eK\) où \( \eK\) est un corps commutatif.
+\end{definition}
+
+\begin{definition}      \label{DEFooJIAQooZkBtTy}
+    Si $g$ est une application bilinéaire sur un espace vectoriel \( E\) nous disons qu'elle est
+    \begin{enumerate}
+        \item
+            \defe{définie positive}{application!définie positive} si $g(u,u)\geq 0$ pour tout $u\in E$ et $g(u,u)=0$ si et seulement si $u=0$.
+        \item
+            \defe{semi-définie positive}{application!semi-définie positive} si $g(u,u)\geq 0$ pour tout $u\in E$. Nous dirons aussi parfois qu'elle est simplement «positive».
+        \end{enumerate}
+\end{definition}
+Cela est évidemment à lier à la définition \ref{DefAWAooCMPuVM} et la proposition \ref{PROPooUAAFooEGVDRC} : une application bilinéaires est définie positive si et seulement si sa matrice symétrique associée l'est.
+
+\begin{definition}\label{DefVJIeTFj}
+    Un \defe{produit scalaire}{produit!scalaire!en général} sur un espace vectoriel \( E\) est une forme bilinéaire\footnote{Définition \ref{DEFooEEQGooNiPjHz}.} symétrique strictement définie positive\footnote{Définition \ref{DEFooJIAQooZkBtTy}.}.
+\end{definition}
+
+Étant donné que l'inégalité de Cauchy-Schwarz sera surtout utilisée dans le cas où un produit scalaire est bel et bien donné, nous l'énonçons et le démontrons avec des notations adaptée à l'usage. Le produit scalaire sera noté \( X\cdot Y\) pour \( b(X,Y)\) si \( b\) est la forme bilinéaire.
+\begin{theorem}[Inégalité de Cauchy-Schwarz]      \label{ThoAYfEHG}
+	Si $X$ et $Y$ sont des vecteurs, alors
+    \begin{equation}        \label{EQooZDSHooWPcryG}
+		| X\cdot Y |\leq\| X \|\| Y \|.
+	\end{equation}
+    Nous avons une égalité si et seulement si \( X\) et \( Y\) sont multiples l'un de l'autre.
+\end{theorem}
+\index{Cauchy-Schwarz}
+\index{inégalité!Cauchy-Schwarz}
+
+%TODO : mettre au point les notations.
+\begin{proof}
+	Étant donné que les deux membres de l'inéquation sont positifs, nous allons travailler en passant au carré afin d'éviter les racines carrés dans le second membre.
+
+	Nous considérons le polynôme
+	\begin{equation}
+		P(t)=\| X+tY \|^2=(X+tY)\cdot(X+tY)=X\cdot X+tX\cdot Y+tY\cdot X+t^2Y\cdot Y.
+	\end{equation}
+	En ordonnant les termes selon les puissance de $t$,
+	\begin{equation}
+		P(t)=\| Y \|^2t^2+2(X\cdot Y)t+\| X \|^2.
+	\end{equation}
+	Cela est un polynôme du second degré en $t$. Par conséquent le discriminant\footnote{Le fameux $b^2-4ac$.} doit être négatif. Nous avons donc
+	\begin{equation}
+		\Delta=4(X\cdot Y)^2-4\| X \|^2\| Y \|^2\leq 0,
+	\end{equation}
+	ce qui donne immédiatement
+	\begin{equation}
+		(X\cdot Y)^2\leq\| X \|^2\| Y^2 \|.
+	\end{equation}
+
+    En ce qui concerne le cas d'égalité, si nous avons \( X\cdot Y=\| X \|\| Y \|\), alors le discriminant \( \Delta\) ci-dessus est nul et le polynôme \( P\) admet une racine double \( t_0\). Pour cette valeur nous avons
+    \begin{equation}
+        P(t_0)=| X+t_0Y |=0,
+    \end{equation}
+    ce qui implique \( X+t_0Y=0\) et donc que \( X\) et \( Y\) sont liés.
+\end{proof}
+
+Vu que nous allons voir un pâté d'espaces avec des produits scalaires, nous leur donnons un nom.
+\begin{definition}\label{DefLZMcvfj}
+    Un espace vectoriel \defe{euclidien}{euclidien!espace} est un espace vectoriel de dimension finie muni d'un produit scalaire (définition \ref{DefVJIeTFj}).
+\end{definition}
+
+\begin{definition}[Isométrie, thème \ref{THMooVUCLooCrdbxm}]      \label{DEFooGGTYooXsHIZj}
+    Une \defe{isométrie}{isométrie!forme bilinéaire} d'une forme bilinéaire \( b\) sur l'espace vectoriel \( E\) est une application bijective \( f\colon E\to E\) telle que \( b\big( f(u),f(v) \big)=b(u,v)\) pour tout \( u,v\in E\).
+\end{definition}
+En particulier une isométrie d'un espace euclidien est une application bijective qui préserve le produit scalaire.
+
+\begin{proposition} \label{PropEQRooQXazLz}
+    Si \( x,y\mapsto x\cdot y\) est un produit scalaire sur un espace vectoriel \( E\), alors \( N(x)=\sqrt{x\cdot x}\) est une norme vérifiant l'identité du parallélogramme :
+    \begin{equation}        \label{EqYCLtWfJ}
+        \| x-y \|^2+\| x+y \|^2=2\| x \|^2+2\| y \|^2.
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+
+    Prouvons l'inégalité triangulaire\index{inégalité!triangulaire!produit scalaire}. Si \( x,y\in E\) nous avons
+    \begin{equation}
+        \| x+y \|=\sqrt{\| x \|^2+\| y \|^2+2x\cdot y}.
+    \end{equation}
+    Par l'inégalité de Cauchy-Schwarz, théorème \ref{ThoAYfEHG} nous avons aussi
+    \begin{equation}
+        \| x \|^2+\| y \|^2+2x\cdot y\leq \| x \|^2+\| y \|^2+2\| x \|\| y \|=\big( \| x \|+\| y \| \big)^2,
+    \end{equation}
+    donc
+    \begin{equation}
+        \| x+y \|\leq \sqrt{\big( \| x \|+\| y \| \big)^2}=\| x \|+\| y \|.
+    \end{equation}
+
+    La seconde assertion est seulement un calcul :
+			\begin{equation}
+				\begin{aligned}[]
+					\| x-y \|^2+\| x+y \|^2&=(x-y)\cdot (x-y)+(x+y)\cdot(x+y)\\
+					&=x\cdot x-x\cdot y-y\cdot x+y\cdot y\\
+					&\quad +x\cdot x+x\cdot y+y\cdot x+y\cdot y\\
+					&=2x\cdot x+2y\cdot y\\
+					&=2\| x \|^2+2\| y \|^2.
+				\end{aligned}
+			\end{equation}
+\end{proof}
+
+Le produit scalaire permet de donner une norme via la formule suivante :
+\begin{equation}
+    \| x \|^2=x\cdot x.
+\end{equation}
+
+\begin{lemma}[\cite{KXjFWKA}]   \label{LemLPOHUme}
+    Soit \( V\) un espace vectoriel muni d'un produit scalaire et de la norme associée. Si \( x,y\in V\) satisfont à \( \| x+y \|=\| x \|+\| y \|\), alors il existe \( \lambda\geq 0\) tel que \( x=\lambda y\).
+\end{lemma}
+
+\begin{proof}
+    Quitte à raisonner avec \( x/\| x \|\) et \( y/\| y \|\), nous supposons que \( \| x \|=\| y \|=1\). Dans ce cas l'hypothèse signifie que \( \| x+y \|^2=4\). D'autre part en écrivant la norme en terme de produit scalaire,
+    \begin{equation}
+        \| x+y \|^2=\| x \|^2+\| y \|^2+2\langle x, y\rangle ,
+    \end{equation}
+    ce qui nous mène à affirmer que \( \langle x, y\rangle =1=\| x \|\| y \|\). Nous sommes donc dans le cas d'égalité de l'inégalité de Cauchy-Schwarz\footnote{Théorème \ref{ThoAYfEHG}.}, ce qui nous donne un \( \lambda\) tel que \( x=\lambda y\). Étant donné que \( \| x \|=\| y \|=1\) nous avons obligatoirement \( \lambda=\pm 1\), mais si \( \lambda=-1\) alors \( \langle x, y\rangle =-1\), ce qui est le contraire de ce qu'on a prétendu plus haut. Par soucis de cohérence, nous allons donc croire que \( \lambda=1\).
+\end{proof}
+
+
+\begin{proposition}			\label{PropVectsOrthLibres}
+	si $v_1,\cdots,v_k$ sont des vecteurs non nuls, orthogonaux deux à deux, alors ces vecteurs forment une famille libre.
+\end{proposition}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Projection et angles}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[Propriétés du produit scalaire]
+	Si $X$ et $Y$ sont des vecteurs de $\eR^3$, alors
+	\begin{description}
+		\item[Symétrie] $X\cdot Y=Y\cdot X$;
+		\item[Linéarité] $(\lambda X+\mu X')\cdot Y=\lambda(X\cdot Y)+\mu(X'\cdot Y)$ pour tout $\lambda$ et $\mu$ dans $\eR$;
+		\item[Défini positif] $X\cdot X\geq 0$ et $X\cdot X=0$ si et seulement si $X=0$.
+	\end{description}
+\end{proposition}
+Note : lorsque nous écrivons $X=0$, nous voulons voulons dire $X=\begin{pmatrix}
+	0	\\ 
+	0	\\ 
+	0	
+\end{pmatrix}$.
+
+
+\begin{definition}
+	La \defe{norme}{norme!vecteur} du vecteur $X$, notée $\| X \|$, est définie par 
+	\begin{equation}
+		\| X \|=\sqrt{X\cdot X}=\sqrt{x^2+y^2+z^2}
+	\end{equation}
+	si $X=(x,y,z)$. Cette norme sera parfois nommée «norme euclidienne».
+\end{definition}
+Cette définition est motivée par le théorème de Pythagore. Le nombre $X\cdot X$ est bien la longueur de la «flèche» $X$. Plus intrigante est la définition suivante :
+\begin{definition}
+	Deux vecteurs $X$ et $Y$ sont \defe{orthogonaux}{orthogonal!vecteur} si $X\cdot Y=0$. 
+\end{definition}
+Cette définition de l'orthogonalité est motivée par la proposition suivante.
+
+\begin{proposition}		\label{PropProjScal}
+	Si nous écrivons $\pr_Y$  l'opération de projection sur la droite qui sous-tend $Y$, alors nous avons
+	\begin{equation}
+		\| \pr_YX \|=\frac{ X\cdot Y }{ \| Y \| }.
+	\end{equation}
+\end{proposition}
+
+\begin{proof}
+	Les vecteurs $X$ et $Y$ sont des flèches dans l'espace. Nous pouvons choisir un système d'axe orthogonal tel que les coordonnées de $X$ et $Y$ soient
+	\begin{equation}
+		\begin{aligned}[]
+			X&=\begin{pmatrix}
+				x	\\ 
+				y	\\ 
+				0	
+			\end{pmatrix},
+			&Y&=\begin{pmatrix}
+				l	\\ 
+				0	\\ 
+				0	
+			\end{pmatrix}
+		\end{aligned}
+	\end{equation}
+	où $l$ est la longueur du vecteur $Y$. Pour ce faire, il suffit de mettre le premier axe le long de $Y$, le second dans le plan qui contient $X$ et $Y$, et enfin le troisième axe dans le plan perpendiculaire aux deux premiers.
+
+	Un simple calcul montre que $X\cdot Y=xl+y\cdot 0+0\cdot 0=xl$. Par ailleurs, nous avons $\| \pr_YX \|=x$. Par conséquent,
+	\begin{equation}
+		\| \pr_YX \|=\frac{ X\cdot Y }{ l }=\frac{ X\cdot Y }{ \| Y \| }.
+	\end{equation}
+\end{proof}
+
+\begin{corollary}
+	Si la norme de $Y$ est $1$, alors le nombre $X\cdot Y$ est la longueur de la projection de $X$ sur $Y$.
+\end{corollary}
+
+\begin{proof}
+	Poser $\| Y \|=1$ dans la proposition \ref{PropProjScal}.
+\end{proof}
+
+\begin{remark}
+    Outre l'orthogonalité, le produit scalaire permet de savoir l'angle entre deux vecteurs à travers la définition \ref{DEFooSVDZooPWHwFQ}. D'autres interprétations géométriques du déterminant sont listées dans le thème \ref{THMooUXJMooOroxbI}.
+\end{remark}
+
+Nous sommes maintenant en mesure de déterminer, pour deux vecteurs quelconques $u$ et $v$, la projection orthogonale de $u$ sur $v$. Ce sera le vecteur $\bar u$ parallèle à $v$ tel que $u-\bar u$ est orthogonal à $v$. Nous avons donc
+\begin{equation}
+    \bar u=\lambda v
+\end{equation}
+et 
+\begin{equation}
+    (u-\lambda v)\cdot v=0.
+\end{equation}
+La seconde équation donne $u\cdot v-\lambda v\cdot v=0$, ce qui fournit $\lambda$ en fonction de $u$ et $v$ :
+\begin{equation}
+    \lambda=\frac{ u\cdot v }{ \| v \|^2 }.
+\end{equation}
+Nous avons par conséquent
+\begin{equation}
+    \bar u=\frac{ u\cdot v }{ \| v \|^2 }v.
+\end{equation}
+Armés de cette interprétation graphique du produit scalaire, nous comprenons pourquoi nous disons que deux vecteurs sont orthogonaux lorsque leur produit scalaire est nul.
+
+Nous pouvons maintenant savoir quel est le coefficient directeur d'une droite orthogonale à une droite donnée. En effet, supposons que la première droite soit parallèle au vecteur $X$ et la seconde au vecteur $Y$. Les droites seront perpendiculaires si $X\cdot Y=0$, c'est à dire si
+\begin{equation}
+	\begin{pmatrix}
+		x_1	\\ 
+		y_1	
+	\end{pmatrix}\cdot\begin{pmatrix}
+		y_1	\\ 
+		y_2	
+	\end{pmatrix}=0.
+\end{equation}
+Cette équation se développe en 
+\begin{equation}		\label{Eqxuyukljsca}
+	x_1y_1=-x_2y_2.
+\end{equation}
+Le coefficient directeur de la première droite est $\frac{ x_2 }{ x_1 }$. Isolons cette quantité dans l'équation \eqref{Eqxuyukljsca} :
+\begin{equation}
+	\frac{ x_2 }{ x_1 }=-\frac{ y_1 }{ y_2 }.
+\end{equation}
+Donc le coefficient directeur de la première est l'inverse et l'opposé du coefficient directeur de la seconde.
+
+\begin{example}
+	Soit la droite $d\equiv y=2x+3$. Le coefficient directeur de cette droite est $2$. Donc le coefficient directeur d'une droite perpendiculaires doit être $-\frac{ 1 }{ 2 }$.
+\end{example}
+
+\begin{proof}[Preuve alternative]
+	La preuve peut également être donnée en ne faisant pas référence au produit scalaire. Il suffit d'écrire toutes les quantités en termes des coordonnées de $X$ et $Y$. Si nous posons
+	\begin{equation}
+		\begin{aligned}[]
+			X&=\begin{pmatrix}
+				x_1	\\ 
+				x_2	\\ 
+				x_2	
+			\end{pmatrix},
+			&Y&=\begin{pmatrix}
+				y_1	\\ 
+				y_2	\\ 
+				y_3	
+			\end{pmatrix},
+		\end{aligned}
+	\end{equation}
+	l'inégalité à prouver devient
+	\begin{equation}
+		(x_1y_1+x_2y_2+x_3y_3)^2\leq (x_1^2+x_2^2+x_3^2)(y_1^2+y_2^2+y_3^2).
+	\end{equation}
+	Nous considérons la fonction
+	\begin{equation}
+		\varphi(t)=(x_1+ty_1)^2+(x_2+ty_2)^2+(x_3+ty_3)^2
+	\end{equation}
+	En tant que norme, cette fonction est évidement positive pour tout $t$. En regroupant les termes de chaque puissance de $t$, nous avons
+	\begin{equation}
+		\varphi(t)=(y_1^2+y_2^2+y_3^2)t^2+2(x_1y_1+x_2y_2+x_3y_3)t+(x_1^2+x_2^2+x_3^2).
+	\end{equation}
+	Cela est un polynôme du second degré en $t$. Par conséquent le discriminant doit être négatif. Nous avons donc
+	\begin{equation}
+		4(x_1y_1+x_2y_2+x_3y_3)^2-(x_1^2+x_2^2+x_3^2)(y_1^2+y_2^2+y_3^2)\leq 0.
+	\end{equation}
+	La thèse en découle aussitôt.
+\end{proof}
+
+\begin{proposition}
+	La norme euclidienne a les propriétés suivantes :
+	\begin{enumerate}
+		\item
+			Pour tout vecteur $X$ et réel $\lambda$,  $\| \lambda X \|=| \lambda |\| X \|$. Attention à ne pas oublier la valeur absolue !
+		\item
+			Pour tout vecteurs $X$ et $Y$, $\| X+Y \|\leq \| X \|+\| Y \|$.
+	\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+	Nous ne prouvons pas le premier point.
+    % TODO : faire la preuve
+    Pour le second, nous avons les inégalités suivantes :
+	\begin{subequations}
+		\begin{align}
+			\| X+Y \|^2&=\| X \|^2+\| Y \|^2+2X\cdot Y\\
+			&\leq\| X \|^2+\| Y \|^2+2|X\cdot Y|\\
+			&\leq\| X \|^2+\| Y \|^2+2\| X \|\| Y \|\\
+			&=\big( \| X \|+\| Y \| \big)^2
+		\end{align}
+	\end{subequations}
+    Nous avons utilisé d'abord la majoration $| x |\geq x$ qui est évident pour tout nombre $x$; et ensuite l'inégalité de Cauchy-Schwarz \ref{ThoAYfEHG}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Procédé de Gram-Schmidt}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[Procédé de Gram-Schmidt]    \label{PropUMtEqkb}
+    Un espace euclidien possède une base orthonormée.
+\end{proposition}
+\index{espace!euclidien}
+\index{Gram-Schmidt}
+
+\begin{proof}
+    Soit \( E\) un espace euclidien et \( \{ v_1,\ldots, v_n \}\), une base quelconque de \( E\). Nous posons d'abord
+    \begin{equation}
+        \begin{aligned}[]
+            f_1&=v_1,&e_1&=\frac{ f_1 }{ \| f_1 \| }.
+        \end{aligned}
+    \end{equation}
+    Ensuite
+    \begin{equation}
+        \begin{aligned}[]
+            f_2&=v_2-\langle v_2, e_1\rangle e_1,&e_2&=\frac{ f_2 }{ \| f_2 \| }.
+        \end{aligned}
+    \end{equation}
+    Notons que \( \{ e_1,e_2 \}\) est une base de \( \Span\{ v_1,v_2 \}\). De plus elle est orthogonale :
+    \begin{equation}
+        \langle e_1, f_2\rangle =\langle e_1, v_2\rangle -\langle v_2, e_1\rangle \underbrace{\langle e_1, e_1\rangle}_{=1} =0.
+    \end{equation}
+    Le fait que \( \| e_1 \|=\| e_2 \|=1\) est par construction. Nous avons donc donné une base orthonormée de \( \Span\{ v_1,v_2 \}\).
+
+    Nous continuons par récurrence en posant
+    \begin{equation}
+        \begin{aligned}[]
+            f_k&=v_k-\sum_{i=1}^{k-1}\langle v_k, e_i\rangle e_i,&e_k&=\frac{ f_k }{ \| f_k \| }.
+        \end{aligned}
+    \end{equation}
+    Pour tout \( j<k\) nous avons
+    \begin{equation}
+        \langle e_j, f_k\rangle =\langle e_j, v_k\rangle -\sum_{i=1}^{k-1}\langle v_k, e_i\rangle \underbrace{\langle e_i, e_j\rangle}_{=\delta_{ij}} =0
+    \end{equation}
+\end{proof}
+Cet algorithme de Gram-Schmidt nous donne non seulement l'existence de bases orthonormée pour tout espace euclidien, mais aussi le moyen d'en construire à partir de n'importe quelle base.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Hermitien, orthogonal, adjoint}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}  \label{DefMZQxmQ}
+Si \( E\) est un espace vectoriel sur \( \eC\), nous disons qu'une application \( \langle ., .\rangle \colon E\times E\to \eC\) est un \defe{produit scalaire hermitien}{produit!scalaire!hermitien}\index{hermitien!produit scalaire} si pour tout \( u,v\in E\) nous avons
+\begin{enumerate}
+    \item
+        \( \langle u, v\rangle =\overline{ \langle v, u\rangle  }\)
+    \item
+        \( \lambda\langle u, v\rangle =\langle \lambda u, v\rangle =\langle u, \bar \lambda v\rangle \)
+    \item
+        \( \langle u, u\rangle \in \eR^+\) et \( \langle u, u\rangle =0\) si et seulement si \( u=0\).
+\end{enumerate}
+Un espace vectoriel sur \( \eC\) muni d'un produit scalaire hermitien est un espace vectoriel hermitien.
+\end{definition}
+
+\begin{example}
+    L'ensemble \( E=\eC^n\) vu comme espace vectoriel de dimension \( n\) sur \( \eC\)  est muni d'une forme sesquilinéaire
+    \begin{equation}    \label{EqFormSesqQrjyPH}
+        \langle x, y\rangle =\sum_{k=1}^nx_k\bar y_k
+    \end{equation}
+    pour tout \( x,y\in\eC^n\). Cela est un espace vectoriel hermitien.
+\end{example}
+
+\begin{definition}      \label{DEFooROVNooFlTbSK}
+    Soit \( E\) un espace vectoriel hermitien et \( A\in\End(E)\). L'opérateur \defe{adjoint}{opérateur!adjoint} de \( A\) est l'opérateur noté \( A^*\) définit par
+    \begin{equation}    \label{EQooHWYKooFzAGgB}
+        \langle Av, w\rangle =\langle v, A^*w\rangle 
+    \end{equation}
+    pour tout \( v,w\in \eC^n\).
+\end{definition}
+
+La prescription \eqref{EQooHWYKooFzAGgB} définit bien l'opérateur \( A^*\) de façon unique. En effet si \( \{ e_i \}\) est une base orthonormée alors
+\begin{equation}
+    \langle e_i, A^*e_j\rangle =(A^*e_j)_i=\langle Ae_i, e_j\rangle =(Ae_i)_j.
+\end{equation}
+Autrement dit, \( (A^*e_j)_i=(Ae_i)_j\).
+
+\begin{proposition}     \label{PROPooSHZMooGwdfBd}
+    En ce qui concerne le déterminant,
+    \begin{equation}
+        \det(A^*)=\det(A)^*
+    \end{equation}
+    où l'étoile à droite dénote la conjugaison complexe dans \( \eC\).
+\end{proposition}
+
+\begin{proof}
+    Écrivons la définition \eqref{EQooOJEXooXUpwfZ} avec l'expression explicite \eqref{EQooOJEXooXUpwfZ} du déterminant. Le tout avec la base canonique :
+    \begin{equation}
+            \det(A)=\det_{(e_1,\ldots, e_n)}(Ae_1,\ldots, Ae_n)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^ne_{\sigma(i)}^*(Ae_i).
+    \end{equation}
+    Mais nous pouvons développer :
+    \begin{equation}
+        e^*_{\sigma(i)}(Ae_i)=\langle e_{\sigma(i)}, Ae_i\rangle =\langle A^*e_{\sigma(i)}, e_i\rangle =\langle e_i, A^*e_{\sigma(i)}\rangle^*=e_i^*(A^*e_{\sigma(i)})^*.
+    \end{equation}
+    Notez que dans la dernière expression, les trois \( {}^*\) ont trois significations différentes. Par conséquent,
+    \begin{equation}
+        \det(A)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^{n}e_i^*(A^*e_{\sigma(i)})^*.
+    \end{equation}
+    Mais \( e_i^*(A^*e_{\sigma(i)})=e_{\sigma(j)}^*(A^*e_{j})\) pour \( j=\sigma(i)\), donc le produit ne change pas si on déplace le \( \sigma\) :
+    \begin{equation}
+        \det(A)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^{n}e_{\sigma(i)}^*(A^*e_{i})^*=\det(A^*)^*.
+    \end{equation}
+\end{proof}
+
+\begin{definition}      \label{DEFooKEBHooWwCKRK}
+    Un opérateur \( A\) est \defe{hermitien}{opérateur!hermitien} si \( A^*=A\). On dit aussi \defe{autoadjoint}{opérateur!autoadjoint}.
+\end{definition}
+
+Le mot «hermitien» est réservé aux opérateurs sur des espaces hermitiens, c'est à dire des espaces vectoriels sur \( \eC\). Le mot «autoadjoint» par contre est plutôt utilisé dans le cadre d'opérateurs sur les espaces réels\footnote{Voir la définition \ref{DEFooYNEQooGQgbCf}.}. En conséquence de quoi, ces deux mots sont synonymes, mais il est préférable d'utiliser «hermitien» lorsque l'espace vectoriel est sur \( \eC\) et «autoadjoint» lorsqu'il est sur \( \eR\).
+
+\begin{remark}
+    Le fait d'être hermitien n'implique en rien le fait d'être inversible.
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Opérateur orthogonal, matrice orthogonale}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooYKCSooURQDoS}
+    Un opérateur est \defe{orthogonal}{orthogonal!opérateur} lorsque \( A^*=A^{-1}\) où \( A^*\) est l'adjoint de \( A\) définit en \ref{DEFooROVNooFlTbSK}.
+\end{definition}
+
+\begin{definition}      \label{DEFooUHANooLVBVID}
+    Une matrice \( U\) est \defe{orthogonale}{matrice!orthogonale}\index{orthogonal!matrice} si \( U^t=U^{-1}\). Le \defe{groupe orthogonal}{groupe!orthogonal} noté \( \gO(n)\) est l'ensemble des matrices orthogonales \( n\times n\).
+\end{definition}
+
+\begin{lemma}       \label{LEMooSSALooSBFzJb}
+    Soit un opérateur \( A\colon \eR^n\to \eR^n\) muni du produit scalaire usuel. Il est orthogonal si et seulement si sa matrice dans la base canonique est orthogonale\footnote{Définition \ref{DEFooUHANooLVBVID}.}.
+\end{lemma}
+
+\begin{proof}
+    Soit la base canonique \( \{ e_i \}_{i=1,\ldots, n}\) de \( \eR^n\). Nous avons
+    \begin{equation}
+        \langle AA^*e_i, e_j\rangle =\langle e_i, e_j\rangle =\delta_{ij},
+    \end{equation}
+    donc \( \big( (AA^*)e_j \big)_j=\delta_{ij}\), ou encore \( (AA^*)_{ij}=\delta_{ij}\), ce qui signifie que la matrice $AA^*$ est l'identité.
+\end{proof}
+
+
+Pour rappel, une isométrique d'un espace euclidien est une bijection qui préserve le produit scalaire, définition \ref{DEFooGGTYooXsHIZj}.
+
+\begin{proposition}[Thème \ref{THMooVUCLooCrdbxm}]     \label{PropKBCXooOuEZcS}
+    À propos de matrices orthogonales.
+    \begin{enumerate}
+        \item
+            L'ensemble des matrice réelles orthogonales forme un groupe noté \( \gO(n,\eR)\)\nomenclature[B]{\( \gO(n,\eR)\)}{le groupe des matrices orthogonales}.
+        \item
+            Si \( A\) est une matrice orthogonale, alors \( \det(A)=\pm 1\).
+        \item       \label{ITEMooOWMBooHUatNb}
+            Le groupe \( \gO(n)\) est le groupe des isométries de \( \eR^n\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Si \( A\) et \( B\) sont orthogonales, alors
+    \begin{equation}
+        (AB)(AB)^t=ABB^tA^t=A\mtu A^t=\mtu.
+    \end{equation}
+    Vu que \( \mtu\) est orthogonale, nous avons bien un groupe.
+
+    En ce qui concerne le déterminant, \( AA^t=\mtu\) donne \( \det(A)\det(A^t)=1\) ou encore \( \det(A)^2=1\). D'où le fait que \( \det(A)=\pm 1\).
+
+    En ce qui concerne le déterminant, nous savons par la proposition \ref{PROPooSHZMooGwdfBd} que \( \det(A)=\det(A^*)\) (nous sommes dans le cas réel). Étant donné que le déterminant est un morphisme et que \( AA^*=\id\) avec \( \det(\id)=1\) nous avons aussi \( \det(A)^2=1\), c'est à dire \( \det(A)=\pm1\).
+
+    D'autre part si \( A\) est une isométrie de \( \eR^n\) alors pour tout \( x,y\in \eR^n\) nous avons \( \langle Ax, Ay\rangle =\langle x, y\rangle \). En particulier,
+    \begin{equation}
+        \langle A^tAx, y\rangle =\langle x, y\rangle 
+    \end{equation}
+    pour tout \( x,y\in \eR^n\). En prenant \( y=e_i\) nous trouvons
+    \begin{equation}
+        (A^tAx)_i=x_i,
+    \end{equation}
+    ce qui signifie que pour tout \( x\), \( A^tAx=x\), ou encore que \( A^tA\) est l'identité.
+
+    Réciproquement si \( A^tA\) est l'identité nous avons
+    \begin{equation}
+        \langle a, y\rangle =\langle A^tAx, y\rangle =\langle Ax, Ay\rangle ,
+    \end{equation}
+    ce qui prouve que \( A\) est une isométrie.
+\end{proof}
+
+\begin{definition}      \label{DEFooJLNQooBKTYUy}
+    Le sous-groupe des matrices orthogonales de déterminant \( 1\) est le groupe \defe{spécial orthogonal}{groupe!spécial orthogonal} noté \( \SO(n)\).
+\end{definition}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Directions conservées}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Généralités}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous savons qu'une application \emph{linéaire} $A\colon \eR^3\to \eR^3$ est complètement définie par la donnée de son action sur les trois vecteurs de base, c'est à dire par la donnée de
+\begin{equation}
+	\begin{aligned}[]
+		Ae_1,&&Ae_2&&\text{et}&&Ae_3.
+	\end{aligned}
+\end{equation}
+Nous allons former la matrice de $A$ en mettant simplement les vecteurs $Ae_1$, $Ae_2$ et $Ae_3$ en colonne. Donc la matrice
+\begin{equation}		\label{EqExempleALin}
+	A=\begin{pmatrix}
+		3	&	0	&	0	\\
+		0	&	1	&	0	\\
+		0	&	1	&	0
+	\end{pmatrix}
+\end{equation}
+signifie que l'application linéaire $A$ envoie le vecteur $e_1$ sur $\begin{pmatrix}
+	3	\\ 
+	0	\\ 
+	0	
+\end{pmatrix}$, le vecteur $e_2$ sur $\begin{pmatrix}
+	0	\\ 
+	0	\\ 
+	1	
+\end{pmatrix}$ et le vecteur $e_3$ sur $\begin{pmatrix}
+	0	\\ 
+	1	\\ 
+	0	
+\end{pmatrix}$.
+Pour savoir comment $A$ agit sur n'importe quel vecteur, on applique la règle de produit vecteur$\times$matrice :
+\begin{equation}
+	\begin{pmatrix}
+		1	&	2	&	3	\\
+		4	&	5	&	6	\\
+		7	&	8	&	9
+	\end{pmatrix}\begin{pmatrix}
+		x	\\ 
+		y	\\ 
+		z	
+	\end{pmatrix}=
+	\begin{pmatrix}
+		x+2y+3z	\\ 
+		4x+5y+6z	\\ 
+		7x+8y+9z	
+	\end{pmatrix}.
+\end{equation}
+
+Une chose intéressante est de savoir quelles sont les directions invariantes de la transformation linéaire. Par exemple, on peut lire sur la matrice \eqref{EqExempleALin} que la direction $\begin{pmatrix}
+	1	\\ 
+	0	\\ 
+	0	
+\end{pmatrix}$ est invariante : elle est simplement multipliée par $3$. Dans cette direction, la transformation est juste une dilatation. Afin de savoir si $v$ est un vecteur d'une direction conservée, il faut voir s'il existe un nombre $\lambda$ tel que $Av=\lambda v$, c'est à dire voir si $v$ est simplement dilaté.
+
+L'équation $Av=\lambda v$ se récrit $(A-\lambda\mtu)v=0$, c'est à dire qu'il faut résoudre l'équation
+\begin{equation}
+	(A-\lambda\mtu)\begin{pmatrix}
+		x	\\ 
+		y	\\ 
+		z	
+	\end{pmatrix}=
+	\begin{pmatrix}
+		0	\\ 
+		0	\\ 
+		0	
+	\end{pmatrix}.
+\end{equation}
+Nous savons qu'une telle équation ne peut avoir de solutions que si $\det(A-\lambda\mtu)=0$. La première étape est donc de trouver les $\lambda$ qui vérifient cette condition.
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Comment trouver la matrice d'une symétrie donnée ?}
+%---------------------------------------------------------------------------------------------------------------------------
+\label{SubSecMtrSym}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Symétrie par rapport à un plan}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Comment trouver par exemple la matrice $A$ qui donne la symétrie autour du plan $z=0$ ? La définition d'une telle symétrie est que les vecteurs du plan $z=0$ ne bougent pas, tandis que les vecteurs perpendiculaires changent de signe. Ces informations vont permettre de trouver comment $A$ agit sur une base de $\eR^3$. En effet :
+\begin{enumerate}
+
+	\item
+		Le vecteur $\begin{pmatrix}
+			1	\\ 
+			0	\\ 
+			0	
+		\end{pmatrix}$ est dans le plan $z=0$, donc il ne bouge pas,
+
+	\item
+		le vecteur $\begin{pmatrix}
+			0	\\ 
+			1	\\ 
+			0	
+		\end{pmatrix}$ est également dans le plan, donc il ne bouge pas non plus,
+
+	\item
+		et le vecteur $\begin{pmatrix}
+			0	\\ 
+			0	\\ 
+			1	
+		\end{pmatrix}$ est perpendiculaire au plan $z=0$, donc il va changer de signe.
+
+\end{enumerate}
+Cela nous donne directement les valeurs de $A$ sur la base canonique et nous permet d'écrire 
+\begin{equation}
+	A=\begin{pmatrix}
+		1	&	0	&	0	\\
+		0	&	1	&	0	\\
+		0	&	0	&	-1
+	\end{pmatrix}.
+\end{equation}
+Pour écrire cela, nous avons juste mit en colonne les images des vecteurs de base. Les deux premiers n'ont pas changé et le troisième a changé.
+
+Et si maintenant on donne un plan moins facile que $z=0$ ? Le principe reste le même : il faudra trouver deux vecteurs qui sont dans le plan (et dire qu'ils ne bougent pas), et puis un vecteur qui est perpendiculaire au plan\footnote{Pour le trouver, penser au produit vectoriel.}, et dire qu'il change de signe.
+
+Voyons ce qu'il en est pour le plan $x=-z$. Il faut trouver deux vecteurs linéairement indépendants dans ce plan. Prenons par exemple
+\begin{equation}		\label{EqffudE}
+	\begin{aligned}[]
+		f_1&=\begin{pmatrix}
+			0	\\ 
+			1	\\ 
+			0	
+		\end{pmatrix},&f_2&=\begin{pmatrix}
+			1	\\ 
+			0	\\ 
+			-1	
+		\end{pmatrix}.
+	\end{aligned}
+\end{equation}
+Nous avons 
+\begin{equation}
+	\begin{aligned}[]
+		Af_1&=f_1\\
+		Af_2&=f_2.
+	\end{aligned}
+\end{equation}
+Afin de trouver un vecteur perpendiculaire au plan, calculons le produit vectoriel :
+\begin{equation}
+	f_3=f_1\times f_2=\begin{vmatrix}
+		e_1	&	e_2	&	e_3	\\
+		0	&	1	&	0	\\
+		1	&	0	&	-1
+	\end{vmatrix}=-e_1-e_3=\begin{pmatrix}
+		-1	\\ 
+		0	\\ 
+		-1	
+	\end{pmatrix}.
+\end{equation}
+Nous avons 
+\begin{equation}
+	Af_3=-f_3.
+\end{equation}
+Afin de trouver la matrice $A$, il faut trouver $Ae_1$, $Ae_2$ et $Ae_3$. Pour ce faire, il faut d'abord écrire $\{ e_1,e_2,e_3 \}$ en fonction de $\{ f_1,f_2,f_3 \}$. La première des équation \eqref{EqffudE} dit que 
+\begin{equation}
+	f_1=e_2.
+\end{equation}
+Ensuite, nous avons
+\begin{equation}
+	\begin{aligned}[]
+		f_2&=e_1-e_3\\
+		f_3&=-e_1-e_3.
+	\end{aligned}
+\end{equation}
+La somme de ces deux équations donne $-2e_3=f_2+f_3$, c'est à dire
+\begin{equation}
+	e_3=-\frac{ f_2+f_3 }{ 2 }
+\end{equation}
+Et enfin, nous avons
+\begin{equation}
+	e_1=\frac{ f_2-f_3 }{ 2 }.
+\end{equation}
+
+Maintenant nous pouvons calculer les images de $e_1$, $e_2$ et $e_3$ en faisant
+\begin{equation}
+	\begin{aligned}[]
+		Ae_1&=\frac{ Af_2-Af_3 }{ 2 }=\frac{1 }{2}\begin{pmatrix}
+			0	\\ 
+			0	\\ 
+			-2	
+		\end{pmatrix}=\begin{pmatrix}
+			0	\\ 
+			0	\\ 
+			-1	
+		\end{pmatrix},\\
+		Ae_2&=Af_1=f_1=\begin{pmatrix}
+			0	\\ 
+			1	\\ 
+			0	
+		\end{pmatrix},\\
+		Ae_3&=-\frac{ f_2-f_3 }{ 2 }=-\frac{ 1 }{2}\begin{pmatrix}
+			2	\\ 
+			0	\\ 
+			0	
+		\end{pmatrix}=\begin{pmatrix}
+			-1	\\ 
+			0	\\ 
+			0	
+		\end{pmatrix}.
+	\end{aligned}
+\end{equation}
+La matrice $A$ s'écrit maintenant en mettant les trois images trouvées en colonnes :
+\begin{equation}
+	A=\begin{pmatrix}
+		0	&	0	&	-1	\\
+		0	&	1	&	0	\\
+		-1	&	0	&	0
+	\end{pmatrix}.
+\end{equation}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Symétrie par rapport à une droite}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Le principe est exactement le même : il faut trouver trois vecteurs $f_1$, $f_2$ et $f_3$ sur lesquels on connaît l'action de la symétrie. Ensuite il faudra exprimer $e_1$, $e_2$ et $e_3$ en termes de $f_1$, $f_2$ et $f_3$.
+
+Le seul problème est de trouver les trois vecteurs $f_i$. Le premier est tout trouvé : c'est n'importe quel vecteur sur la droite. Pour les deux autres, il faut un peu ruser parce qu'il faut impérativement qu'ils soient perpendiculaire à la droite. Pour trouver $f_2$, on peut écrire
+\begin{equation}
+	f_2=\begin{pmatrix}
+		1	\\ 
+		0	\\ 
+		x	
+	\end{pmatrix},
+\end{equation}
+et puis fixer le $x$ pour que le produit scalaire de $f_2$ avec $f_1$ soit nul. S'il n'y a pas moyen (genre si $f_1$ a sa troisième composante nulle), essayer avec $\begin{pmatrix}
+	x	\\ 
+	1	\\ 
+	0	
+\end{pmatrix}$. Une fois que $f_2$ est trouvé (il y a des milliards de choix possibles), trouver $f_3$ est super facile : prendre le produit vectoriel entre $f_1$ et $f_2$.
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{En résumé}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+La marche à suivre est
+
+\begin{enumerate}
+
+	\item
+		Trouver trois vecteurs $f_1$, $f_2$ et $f_3$ sur lesquels on connaît l'action de la symétrie. Typiquement : des vecteurs qui sont sur l'axe ou le plan de symétrie, et puis des perpendiculaires. Pour la perpendiculaire, penser au produit scalaire et au produit vectoriel.
+
+	\item
+		Exprimer la base canonique $e_1$, $e_2$ et $e_3$ en termes de $f_1$, $f_2$, $f_3$.
+
+	\item
+		Trouver $Ae_1$, $Ae_2$ et $Ae_3$ en utilisant leur expression en termes des $f_i$, et le fait que l'on connaisse l'action de $A$ sur les $f_i$.
+
+	\item
+		La matrice s'obtient en mettant les images des $e_i$ en colonnes.
+\end{enumerate}
diff --git a/tex/frido/55_EspacesVectos.tex b/tex/frido/55_EspacesVectos.tex
new file mode 100644
index 000000000..f0de72244
--- /dev/null
+++ b/tex/frido/55_EspacesVectos.tex
@@ -0,0 +1,1035 @@
+% This is part of Mes notes de mathématique
+% Copyright (c) 2008-2017
+%   Laurent Claessens
+% See the file fdl-1.3.txt for copying conditions.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Parties libres, génératrices, bases et dimension}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}[Partie libre]
+    Si \( E\) est un espace vectoriel, une partie \( A\) de \( E\) est \defe{libre}{libre!partie} si pour tout choix d'un nombre fini d'éléments \( \{ u_i \}_{i=1,\ldots, n}\), l'égalité
+    \begin{equation}
+        a_1 u_1+\cdots +a_nu_n=0
+    \end{equation}
+    implique \( a_i=0\) pour tout \( i\) (ici les \( a_i\) sont dans le corps de base).
+
+    Une partie infinie est libre si toute ses parties finies le sont.
+\end{definition}
+
+\begin{remark}
+    Notons que le vecteur nul n'est dans aucune partie libre, ne fut-ce que parce que \( a0=0\) n'implique pas \( a=0\).
+\end{remark}
+
+Si \( A\) est une partie de l'espace vectoriel \( E\) nous notons \( \Span(A)\)\nomenclature[A]{$\Span(A)$}{l'ensemble des combinaisons linéaires finies d'éléments de \( A\)} l'ensemble des combinaisons linéaires finies d'éléments de \( A\). Les coefficients de ces combinaisons linéaires sont dans le corps de base \( \eK\).
+
+\begin{definition}[Partie génératrice]
+    Une partie $B$ d'un espace vectoriel \( E\) est \defe{génératrice}{partie!génératrice} si \( \Span(B)=E\).
+\end{definition}
+
+\begin{remark}
+    Ces définitions demandent des commentaires en dimension infinie\footnote{Nous n'avons pas encore définit le concept de dimension, mais nous nous adressons au lecteur trop pressé.}.
+
+    \begin{enumerate}
+        \item
+    Tout élément peut être écrit comme combinaison linéaire finie d'une partie génératrice. Cela ne signifie pas que nous pouvons extraire une partie finie qui convient pour tous les éléments à la fois. Lorsque l'espace est de dimension infinie, ceci est particulièrement important.
+\item
+    La définition séparée de liberté dans le cas des parties infinies a son importance lorsqu'on parle d'espaces vectoriels de dimension infinies (en dimension finie, aucune partie infinie n'est libre) parce que cela fera une différence entre une base algébrique et une base hilbertienne par exemple.
+    \end{enumerate}
+\end{remark}
+
+\begin{definition}[Base]
+    Une \defe{base}{base} de l'espace vectoriel \( E\) est une partie à la fois génératrice et libre.
+\end{definition}
+
+\begin{proposition}[\cite{MonCerveau}]      \label{PROPooEIQIooXfWDDV}
+    Tout élément non nul d'un espace vectoriel se décompose de façon unique en combinaison linéaire finie d'éléments d'une base.
+\end{proposition}
+
+\begin{proof}
+    Soit un espace vectoriel \( E\) et une base \( \{ e_i \}_{i\in I}\) où \( I\) est un ensemble a priori quelconque. Soit \( v\in E\). Vu que \( E=\Span\{ e_i \}_{i\in I}\), il existe une partie finie \( J\) de \( I\) et des coefficients \( \{ v_j \}_{j\in J}\) dans \( \eK\) tels que
+    \begin{equation}
+        v=\sum_{j\in J}v_je_j.
+    \end{equation}
+    Cela donne l'existence.
+
+    En ce qui concerne l'unicité, soient \( J \) et \( K\) des parties finies de \( I\) et des coefficients \( \{ v_j \}_{j\in J}\) et \( \{ w_{k} \}_{k\in K}\) tels que
+    \begin{equation}
+        v=\sum_{j\in J}v_je_j=\sum_{k\in K}v_{k}e_{k}.
+    \end{equation}
+    Nous posons \( L=J\cup K\) et, pour \( l\in L\),
+    \begin{equation}
+        \alpha_l=\begin{cases}
+            v_l    &   \text{si } l\in J\setminus K\\
+            w_l    &    \text{si } l\in K\setminus J\\
+            v_l-w_l    &    \text{si } l\in K\cap J.
+        \end{cases}
+    \end{equation}
+    Nous avons alors
+    \begin{equation}
+        \sum_{l\in L}\alpha_le_l=0,
+    \end{equation}
+    ce qui implique que \( \alpha_l=0\) pour tout \( l\in L\) parce que la partie \( \{ e_i \}_{i\in I}\) est libre et que \( L\) est finie.
+
+    L'unicité de la décomposition de \( v\) signifie que
+    \begin{equation}
+        \{ j\in J \tq v_j\neq 0 \}=\{ k\in K\tq w_k\neq 0 \}
+    \end{equation}
+    et que pour \( l\) dans cet ensemble, \( v_l=w_l\).
+
+    Soit \( j\in J\); il y a deux possibilités : \( j\in J\setminus K\) et \( j\in J\cap K\). Dans le premier cas nous avons déjà vu que \( \alpha_j=v_j=0\). Dans le second cas, \( \alpha_j=v_j-w_j=0\), c'est à dire \( v_j=w_j\).
+
+    Donc \( j\in J\) vérifiant \( v_j\neq 0\) implique \( j\in J\cap K\) et l'égalité des coefficients. Idem avec \( k\in K\) tel que \( w_k\neq 0\) implique \( k\in J\cap K\).
+\end{proof}
+
+\begin{definition}
+    Un espace vectoriel est \defe{de type fini}{type!fini!espace vectoriel} s'il contient une partie génératrice finie. 
+\end{definition}
+Nous verrons dans les résultats qui suivent que cette définition est en réalité inutile parce qu'une espace vectoriel sera de type fini si et seulement s'il est de dimension finie.
+
+\begin{lemma}       \label{LemytHnlD}
+    Si \( E\) a une famille génératrice de cardinal \( n\), alors toute famille de \( n+1\) éléments est liée.
+\end{lemma}
+
+\begin{proof}
+    Nous procédons par récurrence sur \( n\) -- qui n'est pas exactement la dimension d'un espace vectoriel fixé. Pour \( n=1\), nous avons \( E=\langle e\rangle\) et donc si \( v_1,v_2\in E\) nous avons \( v_1=\lambda_1 e\), \( v_2=\lambda_2e\) et donc \( \lambda_2v_1-\lambda_1v1=0\). Cela prouve le résultat dans le cas de la dimension \( 1\).
+
+    Supposons maintenant que le résultat soit vrai pour \( k<n\), c'est à dire que pour tout espace vectoriel contenant une partie génératrice de cardinal \( k<n\), les parties de \( k+1\) éléments sont liées. Soit maintenant un espace vectoriel muni d'une partie génératrice \( G=\{ e_1,\ldots, e_n \}\) de \( n\) éléments, et montrons que toute partie \( V=\{ v_1,\ldots, v_{n+1} \}\) contenant \( n+1\) éléments est liée. Dans nos notations nous supposons que les \( e_i\) sont des vecteurs distincts et les \( v_i\) également. Nous les supposons également tous non nuls. Étant donné que \( \{ e_i \}\) est génératrice nous pouvons définir les nombres \( \lambda_{ij}\) par
+    \begin{equation}
+        v_i=\sum_{k=1}^n\lambda_{ij}e_j
+    \end{equation}
+    Vu que
+    \begin{equation}
+        v_{n+1}=\sum_{k=1}^n\lambda_{n+1,k}e_k\neq 0,
+    \end{equation}
+    quitte à changer la numérotation des \( e_i\) nous pouvons supposer que \( \lambda_{n+1,n}\neq 0\). Nous considérons les vecteurs
+    \begin{equation}
+        w_i=\lambda_{n+1,n}v_i-\lambda_{i,n}v_{n+1}.
+    \end{equation}
+    En calculant un peu,
+    \begin{subequations}
+        \begin{align}
+            w_i&=\lambda_{n+1,n}\sum_k\lambda_{i,k}e_k-\lambda_{i,n}\sum_k\lambda_{n+1,k}e_k\\
+            &=\sum_{k=1}^{n-1}\big( \lambda_{n+1,n}\lambda_{i,k}-\lambda_{i,n}\lambda_{n+1,} \big)e_k
+        \end{align}
+    \end{subequations}
+    parce que les termes en \( e_n\) se sont simplifiés. Donc la famille \( \{ w_1,\ldots, w_n \}\) est une famille de \( n\) vecteurs dans l'espace vectoriel \( \Span\{ e_1,\ldots, e_{n-1} \}\); elle est donc liée par l'hypothèse de récurrence. Il existe donc des nombres \( \alpha_1,\ldots, \alpha_n\in \eK\) non tous nuls tels que
+    \begin{equation}        \label{EqOQGGoU}
+        0=\sum_{i=1}^n\alpha_iw_i=\sum_{i=1}^n\alpha_i\lambda_{n+1,n}v_i-\left( \sum_{i=1}^n\alpha_i\lambda_{i,n} \right)v_{n+1}.
+    \end{equation}
+    Vu que \( \lambda_{n+1,n}\neq 0\) et que parmi les \( \alpha_i\) au moins un est non nul, nous avons au moins un des produits \( \alpha_i\lambda_{n+1,n}\) qui est non nul. Par conséquent \eqref{EqOQGGoU} est une combinaison linéaire nulle non triviale des vecteurs de \( \{ v_1,\ldots, v_{n+1} \}\). Cette partie est donc liée.
+\end{proof}
+
+\begin{lemma}   \label{LemkUfzHl}
+    Soit \( L\) une partie libre et \( G\) une partie génératrice. Soit \( B\) une partie maximale parmi les parties libres \( L'\) telles que \( L\subset L'\subset G\). Alors \( B\) est une base.
+\end{lemma}
+Qu'entend-on par «maximale» ? La partie \( B\) doit être libre, contenir \( L\), être contenue dans \( G\) et de plus avoir la propriété que \( \forall x\in G\setminus B\), la partie \( B\cup\{ x \}\) est liée.
+
+\begin{proof}
+    D'abord si \( G\) est une base, alors toutes les parties de \( G\) sont libres et le maximum est \( B=G\). Dans ce cas le résultat est évident. Nous supposons donc que \( G\) est liée.
+
+    La partie \( B=\{ b_1,\ldots, b_l \}\) est libre parce qu'on l'a prise parmi les libres. Montrons que \( B\) est génératrice. Soit \( x\in G\setminus B\); par hypothèse de maximalité, \( B\cup\{ x \}\) est liée, c'est à dire qu'il existe des nombres \( \lambda_i\), \( \lambda_x\) non tous nuls tels que
+    \begin{equation}    \label{EqxfkevM}
+        \sum_{i=1}^l\lambda_ib_i+\lambda_xx=0.
+    \end{equation}
+    Si \( \lambda_x=0\) alors un de \( \lambda_i\) doit être non nul et l'équation \eqref{EqxfkevM} devient une combinaison linéaire nulle non triviale des \( b_i\), ce qui est impossible parce que \( B\) est libre. Donc \( \lambda_x\neq 0\) et
+    \begin{equation}
+        x=\frac{1}{ \lambda_x }\sum_{i=1}^l\lambda_ib_i.
+    \end{equation}
+    Donc tous les éléments de \( G\setminus B\) sont des combinaisons linéaires des éléments de \( B\), et par conséquent, \( G\) étant génératrice, tous les éléments de \( E\) sont combinaisons linéaires d'éléments de \( B\). 
+\end{proof}
+
+\begin{theorem}[Théorème de la base incomplète] \label{ThonmnWKs}
+    Soit \( E\) un espace vectoriel de type fini sur le corps \( \eK\).
+    \begin{enumerate}
+        \item   \label{ItemBazxTZ}
+            Si \( L\) est une partie libre et si \( G\) est une partie génératrice contenant \( L\), alors il existe une base \( B\) telle que \( L\subset B\subset G\).
+        \item       \label{ITEMooFVJXooGzzpOu}
+            Toute partie libre peut être étendue en une base.
+        \item
+            Toutes les bases sont finies et ont même cardinal.
+    \end{enumerate}
+\end{theorem}
+\index{théorème!base incomplète}
+
+\begin{proof}
+    Point par point.
+    \begin{enumerate}
+        \item
+    Vu que \( E\) est de type fini, il admet une partie génératrice \( G\) de cardinal fini \( n\). Donc une partie libre est de cardinal au plus \( n\) par le lemme \ref{LemytHnlD}. Soit \( L\), une partie libre contenue dans \( G\) (ça existe : par exemple \( L=\emptyset\)). La partie \( B\) maximalement libre contenue dans \( G\) et contenant \( L\) est une base par le lemme \ref{LemkUfzHl}.
+\item
+Notons que puisque \( E\) lui-même est générateur, le point \ref{ItemBazxTZ} implique que toute partie libre peut être étendue en une base.
+\item
+    Soient \( B\) et \( B'\), deux bases. En particulier \( B\) est génératrice et \( B'\) est libre, donc le lemme \ref{LemytHnlD} indique que \( \Card(B')\leq \Card(B)\). Par symétrie on a l'inégalité inverse. Donc \( \Card(B)=\Card(B')\).
+    \end{enumerate}
+\end{proof}
+
+\begin{remark}      \label{REMooYGJEooEcZQKa}
+    Le théorème de la base incomplete \ref{ThonmnWKs}\ref{ITEMooFVJXooGzzpOu} est ce qui permet de construire une base d'une espace vectoriel en « commençant par» une base d'un sous-espace. En effet si \( H\) est un sous-espace de \( E\) alors une base de \( H\) est une partie libre de \( E\) et donc peut être étendue en une base de \( E\).
+\end{remark}
+
+\begin{definition}
+    La \defe{dimension}{dimension} d'un espace vectoriel de type fini est la cardinal d'une de ses bases.
+\end{definition}
+\index{dimension!définition}
+
+\begin{example}
+    Il existe une infinité de bases de $\eR^m$. On peut démontrer que le cardinal de toute base de $\eR^m$ est $m$, c'est à dire que toute base de $\eR^m$ possède exactement $m$ éléments.
+
+    La base de $\eR^m$ qu'on dit \defe{canonique}{canonique!base}\index{base!canonique de $\eR^m$} (c.à.d. celle qu'on utilise tout le temps) est $\mathcal{B}=\{e_1,\ldots, e_m\}$, où le vecteur $e_j$ est 
+    \begin{equation}\nonumber
+      e_j=
+    \begin{array}{cc}
+      \begin{pmatrix}
+        0\\\vdots\\0\\1\\ 0\\\vdots\\0
+      \end{pmatrix} & 
+      \begin{matrix}
+        \quad\\\quad\\\leftarrow\textrm{j-ème} \quad\\\quad\\\quad\\
+      \end{matrix}
+    \end{array}.
+    \end{equation}
+    La composante numéro $j$ de $e_i$ est $1$ si $i=j$ et $0$ si $i\neq j$. Cela s'écrit $(e_i)_j=\delta_{ij}$ où $\delta$ est le \defe{symbole de Kronecker}{Kronecker} défini par
+    \begin{equation}
+        \delta_{ij}=\begin{cases}
+            1	&	\text{si }i=j\\
+            0	&	 \text{si }i\neq j
+        \end{cases}
+    \end{equation}
+    Les éléments de la base canonique de $\eR^m$ peuvent donc être écrits $e_i=\sum_{k=1}^m\delta_{ik}e_k$.
+\end{example}
+
+Le théorème suivant est essentiellement une reformulation du théorème \ref{ThonmnWKs}.
+\begin{theorem} \label{ThoBaseIncompjblieG}     \label{ThoMGQZooIgrXjy}
+    Soit \( E\) un espace vectoriel de dimension finie et \( \{ e_i \}_{i\in I}\) une partie génératrice de \( E\).
+
+    \begin{enumerate}
+        \item
+            Il existe \( J\subset I\) tel que \( \{ e_i \}_{i\in J}\) est une base. Autrement dit : de toute partie génératrice nous pouvons extraire une base.
+        \item
+            Soit \( \{ f_1,\ldots, f_l \}\) une partie libre. Alors nous pouvons la compléter en utilisant des éléments \( e_i\). C'est à dire qu'il existe \( J\subset I\) tel que \( \{ f_k \}\cup\{ e_i \}_{i\in J}\) soit une base.
+        \item       \label{ItemHIVAooPnTlsBi}
+            Si \( E\) est un espace vectoriel de dimension \( n\), alors toute partie libre contenant \( n\) éléments est une base.
+    \end{enumerate}
+\end{theorem}
+
+\begin{proof}
+    Nous ne prouvons que le troisième point. Une partie libre contenant \( n\) éléments peut être étendue en une base; si ladite extension est non triviale (c'est à dire qu'on ajoute vraiment au moins un élément) une telle base contiendra une partie de \( n+1\) éléments qui serait liée par le lemme \ref{LemytHnlD}.
+\end{proof}
+
+Soit \( F\) un sous-espace vectoriel de l'espace vectoriel \( E\). La \defe{codimension}{codimension} de \( F\) dans \( E\) est
+\begin{equation}
+    \codim_E(F)=\dim(E/F).
+\end{equation}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Applications linéaires}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Définition}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooULVAooXJuRmr}
+    Soient des espaces vectoriels \( E \) et \( F\) sur le corps \( \eK\). Une application \( T\colon E\to F\) est dite \defe{linéaire}{linéaire!application} si
+    \begin{itemize}
+        \item $T(x+y)=T(x)+T(y)$ pour tout $x$ et $y$ dans \( E\),
+        \item $T(\lambda x)=\lambda T(x)$ pour tout $\lambda$ dans $\eK$ et \( x\) dans \( E\).
+    \end{itemize}
+\end{definition}
+Si vous avez bien suivi, les égalités dans la définition \ref{DEFooULVAooXJuRmr} sont des égalités dans \( F\).
+
+
+\begin{example}
+Pour tout $b$ dans $\eR$ la fonction $T_b(x)= bx$ est une application linéaire de $\eR$ dans $\eR$. En effet,
+\begin{itemize}
+\item  $T_b(x+y)= b(x+y)= bx + by = T_b(x)+T_b(y)$,
+\item $T_b(ax)=b(ax)= abx = a T_b(x)$.
+\end{itemize}
+De la même façon on peut montrer que la fonction $T_{\lambda}$ définie par $T_{\lambda}(x)=bx$ est un application linéaire de $\eR^m$ dans $\eR^m$ pour tout $\lambda$ dans $\eR$ et $m$ dans $\eN$.
+\end{example}
+
+\begin{definition}[Quelque ensembles d'applications linéaires]      \label{DEFooOAOGooKuJSup}
+    Soient \( E\) et \( F\) des espaces vectoriels. 
+    \begin{itemize}
+        \item
+        L'ensemble des applications linéaires de \( E\) vers \( F\) est noté $\aL(E,F)$\nomenclature{$\aL(E,F)$}{Ensemble des applications linéaires de $E$ dans $F$}.
+        \item Une application linéaire \( E\to E\) est un \defe{endomorphisme}{endomorphisme} de \( E\). L'ensemble des endomorphisme de \( E\) est noté \( \End(E)\)\nomenclature[B]{$\End(E)$}{les endomorphismes de \( E\)}.
+        \item Un endomorphisme bijectif est un \defe{automorphisme}{automorphisme!d'espace vectoriel}. L'ensemble des automorphismes de \( E\) est noté \( \Aut(E)\)\nomenclature[B]{$\Aut(E)$}{automorphisme de l'espace vectoriel \( E\)}.
+        \item
+            Une application linéaire bijective \( E\to F\) est un \defe{isomorphisme}{isomorphisme!espaces vectoriels} d'espace vectoriel. L'ensemble des isomorphismes \( E\to F\) est noté \( \GL(E,F)\).
+    \end{itemize}
+\end{definition}
+
+\begin{remark}
+    Les ensembles définis en \ref{DEFooOAOGooKuJSup} concernent la structure d'espace vectoriel seulement. Lorsque nous verrons la notion d'espace vectoriel normé,nous demanderons de plus la continuité, laquelle n'est pas automatique en dimension infinie. Voir aussi les définitions \ref{DEFooTLQUooJvknvi}.
+\end{remark}
+
+\begin{definition}  \label{DefDQRooVGbzSm}
+    Si \( V\) et \( W\) sont des espaces vectoriels nous munissons \( \aL(V,W)\) d'une structure d'espace vectoriel en définissant la somme et le produit par un scalaire de la façon suivante. Si $T$ et $U$ sont des élément de $\aL(V,W)$ et si $\lambda$ est un réel, nous définissons les éléments $T+U$ et $\lambda T$ par
+    \begin{enumerate}
+        \item
+            $(T+U)(x)=T(x)+U(x)$;
+        \item
+            $(\lambda T)(x)=\lambda T(x)$
+    \end{enumerate}
+    pour tout \( x\in V\).
+\end{definition}
+
+\begin{normaltext}
+    Pour bien faire, il faudrait vérifier que la définition \ref{DefDQRooVGbzSm} produit effectivement une structure d'espace vectoriel sur \( \aL(V,W)\).
+\end{normaltext}
+
+\begin{proposition}
+    Si \( E\) et \( F\) sont des espaces vectoriels de dimension \( n\) et si \( \{ e_i \}_{i=1,\ldots, n}\) et \( \{ f_i \}_{i=1,\ldots, n}\) sont des bases respectivement de \( F\) et \( F\), alors il existe une unique application linéaire \( A\colon E\to F\) telle que \( E(e_i)=f_i\) pour tout \( i\).
+\end{proposition}
+
+\begin{proof}
+    En deux parties.\begin{subproof}
+        \item[Existence]
+            Soit \( v\in E\). Vu que \( \{ e_i \}\) est une base, il se décompose de façon unique en \( v=\sum_iv_ie_i\). Alors définir
+            \begin{equation}
+                A(v)=\sum_iv_if_i
+            \end{equation}
+            est une bonne définition et satisfait aux exigences.
+        \item[Unicité]
+            Soient \( A\) et \( B\) satisfaisant aux exigences. Alors pour tout \( i\) nous avons \( A(e_i)=B(e_i)\). Si \( v\in E\) s'écrit de la forme \( v=\sum_iv_ie_i\) alors la linéarité impose \( A(v)=\sum_iv_iA(e_i)=\sum_iv_iB(e_i)=B(v)\). Donc \( A=B\).
+    \end{subproof}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Rang}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}\label{DefALUAooSPcmyK}
+    Si  $T\colon V\to W$ est une application linéaire, son \defe{rang}{rang} est la dimension de son image. 
+\end{definition}
+
+Le théorème suivant est valable également en dimension infinie; ce sera une des rares incursions en dimension infinie de ce chapitre.
+\begin{theorem}[Théorème du rang]       \label{ThoGkkffA}
+    Soient \( E\) et \( F\) deux espaces vectoriels (de dimensions finies ou non) et soit \( f\colon E\to F\) une application linéaire. Alors le rang\footnote{Définition \ref{DefALUAooSPcmyK}.} de \( f\) est égal à la codimension du noyau, c'est à dire
+       \begin{equation}
+           \rang(f)+\dim\ker f=\dim E.
+       \end{equation}
+
+       Dans le cas de dimension infinie afin d'éviter les problèmes d'arithmétique avec l'infini nous énonçons le théorème en disant que si \( (u_s)_{s\in S}\) est une base de \( \ker(f)\) et si \( \big( f(v_t) \big)_{t\in T}\) est une base de \( \Image(f)\) alors  \( (u_s)_{s\in s}\cup (v_t)_{t\in T}\) est une base de \( E\).
+\end{theorem}
+\index{théorème!du rang}
+
+\begin{proof}
+    Nous devons montrer que 
+    \begin{equation}
+          (u_s)_{s\in S}\cup (v_t)_{t\in T}
+    \end{equation}
+    est libre et générateur.
+
+    Soit \( x\in E\). Nous définissons les nombres \( x_t\) par la décomposition de \( f(x)\) dans la base \( \big( f(v_t) \big)\) :
+    \begin{equation}
+        f(x)=\sum_{t\in T}x_tf(v_t).
+    \end{equation}
+    Ensuite le vecteur \( x=\sum_tx_tv_t\) est dans le noyau de \( f\), par conséquent nous le décomposons dans la base \( (u_s)\) :
+    \begin{equation}
+        x-\sum_tx_tv_t=\sum_s\in S x_su_s.
+    \end{equation}
+    Par conséquent
+    \begin{equation}
+        x=\sum_sx_su_s+\sum_tx_tv_t.
+    \end{equation}
+    
+    En ce qui concerne la liberté nous écrivons
+    \begin{equation}
+        \sum_tx_tv_t+\sum_sx_su_s=0.
+    \end{equation}
+    En appliquant \( f\) nous trouvons que 
+    \begin{equation}
+        \sum_tx_tf(v_t)=0
+    \end{equation}
+    et donc que les \( x_t\) doivent être nuls. Nous restons avec \( \sum_sx_su_s=0\) qui à son tour implique que \( x_s=0\).
+\end{proof}
+Un exemple d'utilisation de ce théorème en dimension infinie sera donné dans le cadre du théorème de Fréchet-Riesz, théorème \ref{ThoQgTovL}.
+
+\begin{corollary}       \label{CORooCCXHooALmxKk}
+    Si \( E\) est un espace vectoriel de dimension finie, alors un endomorphisme de \( E\) est bijectif si et seulement si il est injectif si et seulement si il est surjectif.
+\end{corollary}
+
+\begin{proof}
+    Si un endomorphisme \( f\colon E\to E\) est surjectif, alors \( \rang(f)=\dim(E)\), ce qui donne, par le théorème du rang \ref{ThoGkkffA}, \( \dim\big( \ker(f) \big)=0\), c'est à dire que \( f\) est injectif.
+
+    De la même façon, si \( f\) est injective, alors \( \dim\big( \ker(f) \big)=0\), ce qui donne \( \rang(f)=\dim(E)\) ou encore que \( f\) est surjective.
+\end{proof}
+
+Une conséquence du théorème du rang est que les endomorphismes ont un inverse à gauche et à droite égaux (lorsqu'ils existent).
+\begin{corollary}
+    Soit un endomorphisme \( f\) d'un espace vectoriel de dimension finie. Si \( f\) admet un inverse à gauche, alors
+    \begin{enumerate}
+        \item
+            \( f\) est bijective,
+        \item
+            \( f\) admet également un inverse à droite,
+        \item
+            ils sont égaux.
+    \end{enumerate}
+    Tout cela tient également en remplaçant «gauche» par «droite».
+\end{corollary}
+
+\begin{proof}
+    Soit \( g\), un inverse à gauche de \( f\) : \( gf=\id\). Cela implique que \( f\) est injective et que \( g\) est surjective, et donc qu'elles sont toutes deux bijectives par le corollaire \ref{CORooCCXHooALmxKk}. Vu que \( f\) est bijective, elle admet également un inverse à droite, soit \( h\). Nous avons : \( gf=\id\) et \( fh=\id\).
+
+    Alors \( gfh=h\) parce que \( gf=\id\), mais également \( gfh=g\) parce que \( fh=\id\). Donc \( g=h\).
+\end{proof}
+C'est ce corollaire qui nous permet d'écrire \( f^{-1}\) sans plus de précisions dès que \( f\) est une bijection.
+
+\begin{example}[Ne fonctionne pas en dimension infinie]
+    Tout cela ne fonctionne pas en dimension infinie. Par exemple avec une base \( \{ e_k \}_{k\in \eN}\) nous pouvons considérer l'opérateur
+    \begin{equation}
+        f(e_k)=e_{k+1}.
+    \end{equation}
+    Il est injectif, mais pas surjectif. Si on pose
+    \begin{equation}
+        g(e_k)=\begin{cases}
+            e_{k-1}    &   \text{si } k\geq 1\\
+            0    &    \text{si } k=0
+        \end{cases}
+    \end{equation}
+    alors nous avons \( gf=\id\), mais pas \( fg=\id\) parce que ce \( (fg)(e_0)=0\).
+\end{example}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Matrices équivalentes et similaires}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}  \label{DefBLELooTvlHoB}
+    Deux relations d'équivalence entre les matrices.
+    \begin{enumerate}
+        \item   \label{ItemPFXCooOUbSCt}
+    Deux matrices \( A\) et \( B\) sont \defe{équivalentes}{matrice!équivalence} dans \( \eM(n,\eK)\) s'il existe \( P,Q\in\GL(n,\eK)\) telles que \( A=PBQ^{-1}\). 
+\item
+    Deux matrices sont \defe{semblables}{matrices!similitude} s'il existe une matrice \( P\in \GL(n,\eK)\) telle que \( A=PBP^{-1}\).
+    \end{enumerate}
+\end{definition}
+
+\begin{lemma}   \label{LemZMxxnfM}
+    Une matrice de rang \( r\) dans \( \eM(n,\eK)\) est équivalente à la matrice par blocs
+    \begin{equation}
+        J_r=\begin{pmatrix}
+            \mtu_r    &   0    \\ 
+            0    &   0    
+        \end{pmatrix}.
+    \end{equation}
+\end{lemma}
+\index{rang!classe d'équivalence}
+
+\begin{proof}
+    Nous devons prouver que pour toute matrice \( A\in\eM(n,\eK)\) de rang \( r\), il existe \( P,Q\in\GL(n,\eK)\) telles que \(QAP=J_r\). Soit \( \{ e_i \}\) la base canonique de \( \eK^n\), puis \( \{ f_i \}\) une base telle que \( Af_i=0\) dès que \( i>r\).
+
+    Nous considérons la matrice inversible \( P\) telle que \( Pe_i=f_i\). Elle vérifie
+    \begin{equation}
+        APe_i=Af_i=\begin{cases}
+            0    &   \text{si } i>r\\
+            \neq 0    &    \text{sinon}.
+        \end{cases}
+    \end{equation}
+    La matrice \( AP\) se présente donc sous la forme
+    \begin{equation}
+        AP=\begin{pmatrix}
+            M    &   0    \\ 
+            *    &   0    
+        \end{pmatrix}
+    \end{equation}
+    où \( M\) est une matrice \( r\times r\). Nous considérons maintenant une base \( \{ g_i \}_{i=1,\ldots, n}\) dont les \( r\) premiers éléments sont les \( r\) premières colonnes de \( AP\) et une matrice inversible \( Q\) telle que \( Qg_i=e_i\). Alors
+    \begin{equation}
+        QAPe_i=\begin{cases}
+            e_i    &   \text{si } i<r\\
+            0    &    \text{sinon}.
+        \end{cases}.
+    \end{equation}
+    Cela signifie que \( QAP\) est la matrice \( J_r\).
+\end{proof}
+
+\begin{corollary}[Équivalence et rang]      \label{CorGOUYooErfOIe}
+    Deux matrices sont équivalentes\footnote{Définition \ref{DefBLELooTvlHoB}\ref{ItemPFXCooOUbSCt}.} si et seulement si elles ont même rang.
+\end{corollary}
+
+\begin{proof}
+    D'abord il y a des implicites dans l'énoncé. Vu que nous voulons soit pas hypothèse soit par conclusion que les matrices $A$ et \( B\) soient équivalentes, nous supposons qu'elles ont même dimension. Soient donc \( A\) et \( B\) deux matrices carré représentant des applications linéaires de l'espace vectoriel \( E=\eK^n\) vers lui-même.
+
+    Par le lemme \ref{LemZMxxnfM}, deux matrices de même rang \( r\) sont équivalentes à \( J_r\). Elles sont donc équivalentes entre elles.
+
+    Inversement, supposons que \( A\) et \( B\) soient deux matrices équivalentes : \( A=PBQ^{-1}\) avec \( P\) et \( B\) inversibles. Alors
+    \begin{subequations}
+        \begin{align}
+            \Image(PBQ^{-1})&=\{ PBQ^{-1}v\tq v\in E \}\\
+            &=PB\underbrace{\{ Q^{-1}v\tq v\in E \}}_{=E}\\
+            &=P\big( B(E) \big).
+        \end{align}
+    \end{subequations}
+    L'ensemble \( B(E)\) est un sous-espace vectoriel de \( E\). Vu que le rang de \( P\) est maximum, la dimension de \( P\big( B(E) \big)\) est la même que celle de \( B(E)\). Par conséquent
+    \begin{equation}
+        \dim\Big( \Image(PBQ^{-1}) \Big)=\dim\big( B(E) \big)=\rang(B).
+    \end{equation}
+    Le membre de gauche de cela n'est autre que \( \rang(A)=\dim\big( \Image(PBQ^{-1}) \big)\).
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Matrice d'une application linéaire}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Écriture dans une base}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{example}\label{ex_affine}
+	Soit $m=n$. On fixe $\lambda$ dans $\eR$ et $v$ dans $\eR^m$. L'application $U_{\lambda}$ de $\eR^m$ dans $\eR^m$ définie par $U_{\lambda}(x)=\lambda x+v$ n'est pas une application linéaire, parce que 
+\[
+U_{\lambda}(ax)=\lambda(ax)+v\neq \lambda(bx+v)=a U_{\lambda}(x).
+\]
+\end{example}
+
+\begin{example}\label{exampleT_A}
+	Soit $A$ une matrice fixée de $\mathcal{M}_{n\times m}$\nomenclature{$\mathcal{M}_{n\times m}$}{l'ensemble des matrices $n\times m$}. La fonction $T_A\colon \eR^m\to \eR^n$ définie par $T_A(x)=Ax$ est une application linéaire. En effet, 
+\begin{itemize}
+\item  $T_A(x+y)= A(x+y)= Ax + Ay = T_A(x)+T_A(y)$,
+\item $T_A(ax)=A(ax)= a(Ax) = a T_A(x)$.
+\end{itemize}
+\end{example}
+
+On peut observer que, si on identifie $\mathcal{M}_{1\times 1}$ et $\eR$, on obtient le premier exemple comme cas particulier.
+
+\begin{proposition}     \label{PROPooCESFooGOZBNI}
+ Toute application linéaire $T$ de $\eR^m$ dans $\eR^n$ s'écrit de manière unique par rapport aux bases canoniques de $\eR^m$ et $\eR^n$ sous la forme
+\[
+T(x)=Ax,
+\]
+avec $A$ dans $\mathcal{M}_{n\times m}$.
+\end{proposition}
+
+\begin{proof}
+  Soit $x$ un vecteur dans $\eR^m$. On peut écrire $x$ sous la forme $ x=\sum_{i=1}^{m}x_i e_i$. Comme $T$ est une application linéaire on a
+\[
+T(x)=\sum_{i=1}^{m}x_iT(e_i).
+\]
+Les images de la base de $\eR^m$, $T(e_j), \, j=1,\ldots,m$, sont des éléments de $\eR^n$, donc on peut les écrire sous la forme de vecteurs
+\[
+T(e_i)=
+\begin{pmatrix}
+  a_{1i}\\
+\vdots\\
+a_{ni}
+\end{pmatrix}.
+\] 
+On obtient alors
+\[
+T(x)=\sum_{i=1}^{m}x_iT(e_i)=\sum_{i=1}^{m}x_i\begin{pmatrix}
+  a_{1i}\\
+\vdots\\
+a_{ni}
+\end{pmatrix}=
+\begin{pmatrix}
+  a_{11} \ldots a_{1m}\\
+\vdots \ddots \vdots\\
+ a_{n1} \ldots a_{nm}\\
+\end{pmatrix}
+\begin{pmatrix}
+  x_1\\
+\vdots\\
+x_m
+\end{pmatrix}=Ax.
+\]
+\end{proof}
+
+\begin{definition}
+  Une application $S: \eR^m\to\eR^n$ est dite \defe{affine}{affine (application)} si elle est la somme d'une application linéaire et d'une application constante. Autrement dit, $S$ est affine s'il existe $T: \eR^m\to\eR^n$, linéaire, telle que $S(x)-T(x)$ soit un vecteur constant dans $\eR^n$. 
+\end{definition}
+
+\begin{example}
+	Les exemples les plus courants d'applications affines sont les droites et les plans ne passant pas par l'origine.
+	\begin{description}
+		\item[Les droites] Une droite dans $\eR^2$ (ou $\eR^3$) qui ne passe pas par l'origine est le graphe d'une fonction de la forme $s(x)=ax+b$ (ou $s(t)=u x +v$, avec $u$ et $v$  dans $\eR^2$). On reconnait ici la fonction de l'exemple \ref{ex_affine}.
+			
+		\item[Les plans]
+			De la même façon nous savons que tout plan qui ne passe pas par l'origine dans $\eR^3$ est le graphe d'une application affine, $P(x,y)= (a,b)^T\cdot(x,y)^T+(c,d)^T$.
+	\end{description}
+\end{example}
+
+\begin{lemma}[\cite{ooEPEFooQiPESf}]        \label{LEMooDAACooElDsYb}
+    Soit une application linéaire \( f\colon E\to F\).
+    \begin{enumerate}
+        \item       \label{ITEMooEZEWooZGoqsZ}
+            L'application \( f\) est injective si et seulement s'il existe \( g\colon F\to E\) telle que \( g\circ f=\id|_E\).
+        \item
+            L'application \( f\) est surjective si et seulement s'il existe \( g\colon F\to E\) telle que \( f\circ g=\id|_F\).
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Nous démontrons séparément les deux affirmations.
+    \begin{enumerate}
+        \item
+            Si \( f\) est injective, alors \( f\colon E\to \Image(f)\) est un isomorphisme. Si $V$ est un supplémentaire de \( \Image(f)\) dans \( F\) (c'est à dire \( F=\Image(f)\oplus V\)) alors nous pouvons poser \( g(x+v)=f^{-1}(x)\) où \( x+v\) est la décomposition (unique) d'un élément de \( F\) en \( x\in\Image(f)\) et \( v\in V\). Avec cela nous avons bien \( g\circ f=\id\).
+
+            Inversement, s'il existe \( g\colon F\to E\) telle que \( g\circ f=\id\) alors \( f\colon E\to E\) doit être injective. Parce que si \( f(x)=0\) avec \( x\neq 0\) alors \( (g\circ f)(x)=0\neq x\).
+        \item
+            Si \( f\) est surjective nous pouvons choisir des éléments \( x_1,\ldots, x_p\) dans \( E\) tels que \( \{ f(x_i) \}\) soit une base de \( F\). Ensuite nous définissons
+            \begin{equation}
+                \begin{aligned}
+                    g\colon F&\to E \\
+                    \sum_ka_kf(x_k)&\mapsto \sum_kx_k. 
+                \end{aligned}
+            \end{equation}
+            Cela donne \(  f\circ g=\id|_F\) parce que si \( v\in F\) alors \( v=\sum_kv_kf(x_k)\) avec \( v_k\in \eK\), et nous avons
+            \begin{equation}
+                (f\circ g)(v)=\sum_kv_k(f\circ g)f(x_k)=f\left( \sum_kv_kx_k \right)=\sum_kf(x_k)=v.
+            \end{equation}
+            
+            Inversement, s'il existe \( g\colon F\to E\) tel que \( f\circ g=\id\) alors \( f\) soit être surjective parce que 
+            \begin{equation}
+                F=\Image(f\circ g)=f\big( \Image(g) \big)\subset \Image(f).
+            \end{equation}
+    \end{enumerate}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Changement de base : vecteurs de base}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit un espace vectoriel \( V\) muni d'une base \( \{ e_i \}\) et une application linéaire inversible \( Q\colon V\to V\) dont nous notons \( (Q_{ij})\) les coordonnées dans la base \( \{ e_i \}\). Nous considérons aussi la base \( e'_i=Qe_i\).
+
+Nous exprimons les vecteurs \( e'_i\) en termes des vecteurs \( e_i\) et inversement de la façon suivante. D'abord
+\begin{equation}
+        e'_i=Qe_i=\sum_{kl}Q_{kl}(e_i)_ke_l=\sum_lQ_{il}e_l
+\end{equation}
+donc
+\begin{equation}        \label{EQooQECYooVWEoLx}
+    e'_i=\sum_kQ_{ik}e_k.
+\end{equation}
+
+\begin{normaltext}      \label{TEXTooOBOMooOVqUVG}
+    Afin de ne pas se tromper dans les indices, il est bon d'utiliser systématiquement les noms ``$ijk\ldots$'' pour les indices de la base \( \{ e_i \}\) et les noms ``\( \alpha\beta\gamma\cdots\)'' pour les indices de la base \( \{ e'_{\alpha} \}\). Nous écrivons donc
+    \begin{equation}        \label{EQooZQBZooTzLkLF}
+        e'_{\alpha}=\sum_iQ_{i\alpha}e_i.
+    \end{equation}
+\end{normaltext}
+
+Pour trouver la transformation inverse, nous multiplions par \( (Q^{-1})_{\alpha j}\) et nous sommons sur \( \alpha\) :
+\begin{equation}
+    \sum_{\alpha}(Q^{-1})_{\alpha j}e'_{\alpha}=\sum_{i \alpha}Q_{i\alpha}Q^{-1}_{\alpha j}e_i=e_j
+\end{equation}
+donc
+\begin{equation}        \label{EQooEZBXooGxpJiO}
+    e_j=\sum_{\alpha}(Q^{-1})_{\alpha j}e'_{\alpha}.
+\end{equation}
+
+Quelque remarques :
+\begin{enumerate}
+    \item
+        Notons que nous ne pouvons pas écrire formellement quelque chose comme \( f=Qe\). 
+    \item
+        La matrice \( Q\) vient toujours avec les indices de type \( Q_{i\alpha}\) (latin en premier et grec en second) tandis que la matrice de l'application inverse \( Q^{-1}\) vient toujours avec les indices de type \( (Q^{-1})_{\alpha i}\).
+
+        Ceci est un puissant outil pour ne pas se tromper lors des manipulations d'indices et en particulier lorsque nous renommons des indices.
+\end{enumerate}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Changement de base : coordonnées}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Un vecteur \( x\) peut s'écrire des façons suivantes dans les deux bases considérées :
+\begin{equation}
+    x=\sum_ix_ie_i=\sum_{\alpha}x'_{\alpha}e'_{\alpha}.
+\end{equation}
+En remplaçant \( e_i\) par sa valeur \eqref{EQooEZBXooGxpJiO} nous obtenons l'égalité
+\begin{equation}
+    \sum_{k\alpha}x_k(Q^{-1})_{\alpha k}e'_{\alpha}=\sum_{\alpha}x'_{\alpha}e'_{\alpha}
+\end{equation}
+et par unicité de la décomposition d'un vecteur dans une base nous déduisons
+\begin{equation}
+    x'_{\alpha}=\sum_{k}(Q^{-1})_{\alpha k}x_k.
+\end{equation}
+De même en replaçant \( e'_{\alpha}\) par sa valeur \eqref{EQooZQBZooTzLkLF} nous trouvons
+\begin{equation}        \label{EQooOEHJooCDKUcy}
+    x_i=\sum_{\alpha}Q_{i\alpha}x'_{\alpha}.
+\end{equation}
+
+Attention : il n'est pas question d'écrire quelque chose comme \( x=Qx'\) parce qu'il n'y a pas de vecteur \( x'\).
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Changement de base : matrice d'une application linéaire}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit une application linéaire \( f\colon V\to V\) de matrice \( A\) dans la base \( \{ e_i \}\) et de matrice \( A'\) dans la base \( \{ e'_{\alpha} \}\). Notre but est maintenant d'exprimer \( A'\) en termes de \( A\) et de \( Q\).
+
+Nous avons l'égalité
+\begin{equation}
+    f(x)=\sum_{kl}A_{kl}x_le_k=\sum_{\alpha\beta}A'_{\alpha\beta}x'_{\beta}e'_{\alpha}.
+\end{equation}
+Il suffit d'y substituer les valeurs \eqref{EQooOEHJooCDKUcy} et \eqref{EQooEZBXooGxpJiO} :
+\begin{equation}
+    f(x)=\sum_{kl}A_{kl}\sum_{\beta}Q_{l\beta}x'_{\beta}\sum_{\alpha}(Q^{-1})_{\alpha k}e'_{\alpha}=\sum_{\alpha\beta}(Q^{-1}AQ)_{\alpha\beta}x'_{\beta}e'_{\alpha}.
+\end{equation}
+Cela est égal à \( \sum_{\alpha\beta}A'_{\alpha\beta}x'_{\beta}e'_{\alpha}\) pour tout \( x\). En tenant encore compte de l'unicité de la décomposition du vecteur \( f(x)\) dans la base \( \{ e'_{\alpha} \}\) nous déduisons
+\begin{equation}        \label{EQooWWOUooZBeTBe}
+    A'=Q^{-1} AQ.
+\end{equation}
+
+\begin{remark}
+    L'égalité \eqref{EQooWWOUooZBeTBe} est une égalité de matrices, et non une égalité d'applications linéaire. Il n'est pas question d'écrire quelque chose comme \( f'=Q^{-1}\circ f\circ Q\).
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Changement de base : matrice d'une forme bilinéaire}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit une forme bilinéaire \( q\colon V\times V\to \eR\) de matrice \( (q_{ij})\) dans la base \( \{ e_i \}\) et de matrice \( q'_{\alpha\beta}\) dans la base \( \{ e'_{\alpha} \}\). Nous notons le changement de variables \(e'_{\alpha}=\sum_{i}Q_{i\alpha}e_i\). Nous avons l'égalité
+\begin{equation}
+    q(x,y)=\sum_{ij}q_{ij}x_iy_j=\sum_{\alpha\beta}q'_{\alpha\beta}x'_{\alpha}y'_{\beta}.
+\end{equation}
+En remplaçant \( x_i\) et \( y_j\) par leurs valeurs en termes des \( x'_{\alpha}\) et \( y'_{\beta}\) nous trouvons
+\begin{equation}
+    q(x,y)=\sum_{ij\alpha\beta}q_{ij}Q_{i\alpha}x'_{\alpha}Q_{j\beta}y'_{\beta},
+\end{equation}
+c'est à dire
+\begin{equation}        \label{EQooQLLFooToyvoH}
+    q'=Q^tqQ.
+\end{equation}
+Encore une fois, cette égalité est une égalité de matrices, et non de formes bilinéaires, et encore moins d'applications linéaires. 
+\begin{remark}
+    La «loi de transformation» \eqref{EQooQLLFooToyvoH} n'est pas la même que la loi de transformation \eqref{EQooWWOUooZBeTBe}.
+\end{remark}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Dualité}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{proposition} \label{PropEJBZooTNFPRj}
+    Si $A$ est la matrice d'une application linéaire, alors le rang de cette application linéaire est égal à la taille de la plus grande matrice carré de déterminant non nul contenue dans $A$.
+\end{proposition}
+
+\begin{definition}  \label{DefJPGSHpn}
+    Si \( E\) est un espace vectoriel sur \( \eK\), le \defe{dual algébrique}{dual!algébrique} de \( E\) est l'ensemble des formes linéaires sur \( E\). Le \defe{dual topologique}{dual!topologique} est l'ensemble des formes linéaires continues de \( E\) vers \( \eK\).
+\end{definition}
+
+En dimension infinie, ces deux notions ne coïncident pas, voir la proposition \ref{PROPooQZYVooYJVlBd} et la remarque \ref{RemOAXNooSMTDuN}.
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Orthogonal}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit \( E\), un espace vectoriel, et \( F\) une sous-espace de \( E\). L'\defe{orthogonal}{orthogonal!sous-espace} de \( F\) est la partie \( F^{\perp}\subset E^*\) donnée par
+\begin{equation}    \label{Eqiiyple}
+    F^{\perp}=\{ \alpha\in E^*\tq \forall x\in F,\alpha(x)=0 \}.
+\end{equation}
+Cette définition d'orthogonal via le dual n'est pas du pur snobisme. En effet, la définition «usuelle» qui ne parle pas de dual,
+\begin{equation}
+    F^{\perp}=\{ y\in E\tq \forall x\in F,y\cdot x=0 \},
+\end{equation}
+demande la donnée d'un produit scalaire. Évidemment dans le cas de \( \eR^n\) munie du produit scalaire usuel et de l'identification usuelle entre \( \eR^n\) et \( (\eR^n)^*\) via une base, les deux notions d'orthogonal coïncident.
+
+Si \( B\subset E^*\), on note \( B^o\)\nomenclature[G]{\( B^o\)}{orthogonal dans le dual} son orthogonal :
+\begin{equation}
+    B^o=\{ x\in E\tq \omega(x)=0\,\forall \omega\in B \}.
+\end{equation}
+Notons qu'on le note \( B^o\) et non \( B^{\perp}\) parce qu'on veut un peu s'abstraire du fait que \( (E^*)^*=E\). Du coup on impose que \( B\) soit dans un dual et on prend une notation précise pour dire qu'on remonte au pré-dual et non qu'on va au dual du dual.
+
+La définition \eqref{Eqiiyple} est intrinsèque : elle ne dépend que de la structure d'espace vectoriel.
+
+\begin{proposition} \label{PropXrTDIi}
+    Soit \( E\) un espace vectoriel, et \( F\) un sous-espace vectoriel. Alors nous avons
+    \begin{equation}
+        \dim F+\dim F^{\perp}=\dim E.
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Soit \( \{ e_1,\ldots, e_p \}\) une base de \( F\) que nous complétons en une base \( \{ e_1,\ldots, e_n \}\) de \( E\) par le théorème \ref{ThonmnWKs}. Soit \( \{ e_1^*,\ldots, e^*_n \}\) la base duale. Alors nous prouvons que \( \{ e^*_{p+1},\ldots, e_n^* \}\) est une base de \( F^{\perp}\). 
+    
+    Déjà c'est une partie libre en tant que partie d'une base.
+
+    Ensuite ce sont des éléments de \( F^{perp}\) parce que si \( i\leq p\) et si \( k\geq 1\) nous avons \( e^*_{p+k}(e_i^*)=0\); donc oui, \( e^*_{p+k}\in F^{\perp}\).
+
+    Enfin \( F^{\perp}\subset\Span\{ e_{p+1}^*,e_n^* \}\) parce que si \( \omega=\sum_{k=1}^n\omega_ke_k^*\), alors \( \omega(e_i)=\omega_i\), mais nous savons que si \( \omega\in F^{\perp}\), alors \( \omega(e_i)=0\) pour \( i\leq p\). Donc \( \omega=\sum_{i=p+1}^n\omega_ke^*_k\).
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Transposée}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DefooZLPAooKTITdd}
+    Si \( f\colon E\to F\) est une application linéaire entre deux espaces vectoriels, la \defe{transposée}{transposée} est l'application \( f^t\colon F^*\to E^*\) donnée par
+    \begin{equation}
+        f^t(\omega)(x)=\omega\big( f(x) \big).
+    \end{equation}
+    pour tout \( \omega\in F^*\) et \( x\in E\).
+\end{definition}
+
+\begin{lemma}
+    Soit \( E\) muni de la base \( \{ e_i \}\) et \( F\) muni de la base \( \{ g_i \}\) et une application \( f\colon E\to F\). Si \( A\) est la matrice de \( f\) dans ces bases, alors \( A^t\) est la matrice de \( f^t\) dans les bases \( \{ e^*_i \}\) et \( \{ g^*_i \}\) de \( E^*\) et \( F^*\).
+\end{lemma}
+
+\begin{proof}
+    Nous allons montrer que les formes \( f^t(g^*_i)\) et \( \sum_k(A^t)_{ik}g^*_k\) sont égales en les appliquant à un vecteur.
+
+    Par définition de la matrice d'une application linéaire dans une base,
+    \begin{equation}
+        f^t(g_i^*)=\sum_j(f^t)_{ij}e^*j,
+    \end{equation}
+    et
+    \begin{equation}
+        f(e_k)=\sum_lA_{klg_l}.
+    \end{equation}
+    Du coup, si \( x=\sum_kx_ke_k\), nous avons
+    \begin{equation}    \label{EqCzwftH}
+        f^t(g_i^*)x=\sum_{kl}x_kg_i^*A_{kl}g_l=\sum_{kl}x_kA_{kl}\delta_{il}=\sum_k x_kA_{ki}=\sum_k(A^t)_{ik}x_k.
+    \end{equation}
+    D'autre part, 
+    \begin{equation}    \label{EqWlQlrR}
+        \sum_k(A^t)_{ik}g_k^*x=\sum_{kl}(A^t)_{ik}g^*_kx_le_l=\sum_k(A^t)_{ik}x_k.
+    \end{equation}
+    Le fait que \eqref{EqCzwftH} et \eqref{EqWlQlrR} donnent le même résultat prouve le lemme.
+\end{proof}
+En corollaire, les rangs de \( f\) et de \( f^t\) sont égaux parce que le rang est donné par la plus grande matrice carré de déterminant non nul. Nous prouvons cependant ce résultat de façon plus intrinsèque.
+
+\begin{lemma}[\href{http://gilles.dubois10.free.fr/algebre_lineaire/dualite.html}{Gilles Dubois}]   \label{LemSEpTcW}
+    Si \( f\colon E\to F\) est une application linéaire, alors
+    \begin{equation}
+        \rang(f)=\rang(f^t).
+    \end{equation}
+\end{lemma}
+
+\begin{proof}
+    Nous posons \( \dim\ker(f)=p\) et donc \( \rang(f)=n-p\). Soit \( \{ e_1,\ldots, e_p \}\) une base de \( \ker(f)\) que l'on complète en une base \( \{ e_1,\ldots, e_n \}\) de \( E\). Nous considérons maintenant les vecteurs
+    \begin{equation}
+        g_i=f(e_{p+i})
+    \end{equation}
+    pour \( i=1,\ldots, n-p\). C'est à dire que les \( g_i\) sont les images des vecteurs qui ne sont pas dans le noyau de \( f\). Prouvons qu'ils forment une famille libre. Si
+    \begin{equation}
+        \sum_{k=1}^{n-p}a_kf(e_{p+k})=0,
+    \end{equation}
+    alors \( f\big( \sum_ka_ke_{p+k} \big)=0\), ce qui signifierait que \( \sum_ka_ke_{p+k}\) se trouve dans le noyau de \( f\), ce qui est impossible par construction de la base \( \{ e_i \}_{i=1,\ldots, n}\). Étant donné que les vecteurs \( g_1,\ldots, g_{n-p}\) sont libres, nous les complétons en une base
+    \begin{equation}
+        \{ \underbrace{g_1,\ldots, g_{n-p}}_{\text{images}},\underbrace{g_{n-p+1},\ldots, g_r}_{\text{complétion}} \}
+    \end{equation}
+    de \( F\).
+
+    Nous prouvons maintenant que \( \rang(f^t)\geq n-p\) en montrant que les formes \( \{ g_i^* \}_{i=1,\ldots, n-p}\) est libre (et donc l'espace image de \( f^f\) est au moins de dimension \( n-p\)). Pour cela nous prouvons que \( f^t(g_i^*)=e^*_{i+p}\). En effet
+    \begin{equation}
+        f^t(g^*_i)e_k=g_i^*(fe_k),
+    \end{equation}
+    Si \( k=1,\ldots, p\), alors \( fe_k=0\) et donc \( g_i^*(fe_k)=0\); si \( k=p+l\) alors
+    \begin{equation}
+        f^t(g_i^*)e_k=g_i^*(fe_{k+l})=g^*_i(g_l)=\delta_{i,l}=\delta_{i,k-p}=\delta_{k,i+p}.
+    \end{equation}
+    Donc \( f^t(g_i^*)=e^*_{i+p}\). Cela prouve que les formes \( f^t(g_i^*)\) sont libres et donc que
+    \begin{equation}
+        \rang(f^t)\geq n-p=\rang(f).
+    \end{equation}
+    En appliquant le même raisonnement à \( f^t\) au lieu de \( f\), nous trouvons
+    \begin{equation}
+        \rang\big( (f^t)^t \big)\geq \rang(f^t)
+    \end{equation}
+    et donc, vu que \( (f^t)^t=f\), nous obtenons \( \rang(f)=\rang(f^t)\).
+    
+\end{proof}
+
+\begin{proposition}[\cite{DualMarcSAge}]        \label{PropWOPIooBHFDdP}
+    Si \( f\) est une application linéaire entre les espaces vectoriels \( E\) et \( F\), alors nous avons
+    \begin{equation}
+        \Image(f^t)=\ker(f)^{\perp}.
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Soient donc l'application \( f\colon E\to F\) et sa transposée \( f^t\colon F^*\to E^*\). Nous commençons par prouver que \( \Image(f^{t})\subset(\ker f)^{\perp}\). Pour cela nous prenons \( \omega\in \Image(f^t)\), c'est à dire \( \omega=\alpha\circ f\) pour un certain élément \( \alpha\in F^*\). Si \( z\in\ker(f)\), alors \( \omega(z)=(\alpha\circ f)(z)=0\), c'est à dire que \( \omega\in (\ker f)^{\perp}\).
+
+    Pour prouver qu'il y a égalité, nous n'allons pas démontrer l'inclusion inverse, mais plutôt prouver que les dimensions sont égales. Après, on sait que si \( A\subset B\) et si \( \dim A=\dim B\), alors \( A=B\). Nous avons
+    \begin{subequations}
+        \begin{align}
+            \dim\big( \Image(f^t) \big)&=\rang(f^t)\\
+            &=\rang(f)  &\text{lemme \ref{LemSEpTcW}}\\
+            &=\dim(E)-\dim\ker(f)   &\text{théorème \ref{ThoGkkffA}}\\
+            &=\dim\big( (\ker f)^{\perp} \big)  &\text{proposition \ref{PropXrTDIi}}.
+        \end{align}
+    \end{subequations}
+\end{proof}
+
+\begin{lemma}[\cite{ooEPEFooQiPESf}]
+    Soit \( \eK\) un corps, \( E\) et \( F\) deux \( \eK\)-espaces vectoriels de dimension finie et une application linéaire \( f\colon E\to F\). L'application \( f\) est injective si et seulement si sa transposée\footnote{Définition \ref{DefooZLPAooKTITdd}.} \( f^t\) est surjective.
+\end{lemma}
+
+\begin{proof}
+    Supposons que \( f\) soit injective. Alors par le lemme \ref{LEMooDAACooElDsYb}, il existe \( g\colon F\to E\) tel que \( g\circ f=\id|_E\). Nous avons alors aussi \( (g\circ f)^t=\id|_{E^*}\), mais \( (g\circ f)^t=f^t\circ g^t\), donc \( f^t\) est surjective.
+
+    Inversement, nous supposons que \( f^t\colon F^*\to E^*\) est surjective. Alors en nous souvenant que \( E\) et \( F\) sont de dimension finie et en faisons jouer les identifications \( (f^t)^t=f\) et \( (E^*)^*=E\) nous pouvons savons qu'il existe \( s\colon E^*\to F^*\) tel que \( f^t\circ s=\id|_{E^*}\). En passant à la transposée,
+    \begin{equation}
+        s^t\circ f=\id|_{E},
+    \end{equation}
+    qui implique que \( f\) est injective.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Transposée : sans le dual}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{probleme}        \label{PROBooWNTKooNErOvt}
+    J'aimerais une confirmation de ce qui est écrit dans ici à propos du fait que si \( f\) est une application linéaire, \( f^t\) n'est pas bien définie.
+\end{probleme}
+
+Il est légitime, si \( f\colon V\to V\) est une application linéaire, de dire que sa transposée soit l'application linéaire \( f^t\colon V\to V\) dont la matrice est la matrice transposée de celle de \( f\). Lorsque nous travaillons sur \( \eR^n\) muni de la base canonique, cela ne pose pas de problèmes et nous pouvons écrire des égalités du type \( \langle x, Ay\rangle =\langle A^tx, y\rangle \).
+
+Hélas nous allons voir que cette façon de définir une transposée est mauvaise. Nous reprenons les notations de \ref{TEXTooOBOMooOVqUVG}. Nous avons
+\begin{equation}
+    f(x)=\sum_{kl}A_{kl}x_le_k=\sum_{\alpha\beta}A'_{\alpha\beta}x'_{\beta}e'_{\alpha}.
+\end{equation}
+Nous écrivons l'application transposée par rapport à la première base :
+\begin{equation}
+    f^{t_1}(x)=\sum_{kl}A_{lk}x_le_k
+\end{equation}      
+et par rapport à la seconde base :
+\begin{equation}        \label{EQooFXDQooEjYCrP}
+    f^{t_2}(x)=\sum_{\alpha\beta}A'_{\beta\alpha}x'_{\beta}e'_{\alpha}.
+\end{equation}
+En replaçant dans \( f^{(t_1)}\) les nombres \( x_l\) et les vecteurs \( e_k\) par les valeurs en termes des \( x'_{\beta}\) et \( e'_{\alpha}\) nous obtenons
+\begin{equation}
+    f^{t_1}(x)=\sum_{\alpha\beta}(Q^{-1}A^tQ)_{\alpha\beta}x'_{\beta}e'_{\alpha}.
+\end{equation}
+Nous avons égalité avec \eqref{EQooFXDQooEjYCrP} seulement lorsque \( A'_{\beta\alpha}=(Q^{-1}A^tQ)_{\alpha\beta}\), mais sachant que \( A'=Q^{-1}AQ\) nous avons besoin que
+\begin{equation}
+    Q^{-1}=Q^t
+\end{equation}
+(cela étant une égalité de matrice).
+
+\begin{normaltext}      \label{NooMZVRooExWVKJ}
+    Autrement dit, la façon «usuelle» de voir la transposée d'une application linéaire ne fonctionne dans les livres pour enfant uniquement parce qu'on n'y considère toujours \( \eR^n\) muni de la base canonique ou de bases orthonormées.
+     
+    Notons que nous avons tout de mêmes le notions d'opérateur adjoint et autoadjoint pour parler d'application orthogonale sans passer par la transposée, voir \ref{DEFooYKCSooURQDoS}.
+\end{normaltext}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Polynômes de Lagrange}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit \( E=\eR_n[X]\) l'ensemble des polynômes à coefficients réels de degré au plus \( n\). Soient les \( n+1\) réels distincts \( a_0,\ldots, a_n\). Nous considérons les formes linéaires associées \( f_i\in E^*\),
+\begin{equation}
+    f_i(P)=P(a_i).
+\end{equation}
+\begin{lemma}
+    Ces formes forment une base de \( E^*\).
+\end{lemma}
+
+\begin{proof}
+    Nous prouvons que l'orthogonal est réduit au nul :
+    \begin{equation}
+        \Span\{ f_0,\ldots, f_n \}^{\perp}=\{ 0 \}
+    \end{equation}
+    pour que la proposition \ref{PropXrTDIi} conclue. Si \( P\in\Span\{ f_i \}^{\perp}\), alors \( f_i(P)=0\) pour tout \( i\), ce qui fait que \( P(a_i)=0\) pour tout \( i=0,\ldots, n\). Un polynôme de degré au plus \( n\) qui s'annule en \( n+1\) points est automatiquement le polynôme nul.
+\end{proof}
+
+Les \defe{polynômes de Lagrange}{Lagrange!polynôme}\index{polynôme!Lagrange} sont les polynômes de la base (pré)duale de la base \( \{ f_i \}\).
+
+\begin{proposition}
+    Les polynômes de Lagrange sont donnés par
+    \begin{equation}
+        P_i=\prod_{k\neq i}\frac{ X-a_k }{ a_i-a_k }.
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Il suffit de vérifier que \( f_j(P_i)=\delta_{ij}\). Nous avons
+    \begin{equation}
+        f_j(P_i)=P_i(a_j)=\prod_{k\neq i}\frac{ a_j-a_k }{ a_i-a_k }.
+    \end{equation}
+    Si \( j\neq i\) alors un des termes est nul. Si au contraire \( i=j\), tous les termes valent \( 1\).
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Dual de \texorpdfstring{$ \eM(n,\eK)$}{M(n,K)}}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[\cite{KXjFWKA}]     \label{PropHOjJpCa}
+    Soit \( \eK\), un corps. Les formes linéaires sur \( \eM(n,\eK)\) sont les applications de la forme
+    \begin{equation}
+        \begin{aligned}
+            f_A\colon \eM_n(\eK)&\to \eK \\
+            M&\mapsto \tr(AM). 
+        \end{aligned}
+    \end{equation}
+\end{proposition}
+\index{trace!dual de \( \eM(n,\eK)\)}
+\index{dual!de \( \eM(n,\eK)\)}
+
+
+\begin{proof}
+    Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eM(n,\eK)&\to \eM(n,\eK)' \\
+            A&\mapsto f_A 
+        \end{aligned}
+    \end{equation}
+    et nous voulons prouver que c'est une bijection. Étant donné que nous sommes en dimension finie, nous avons égalité des dimensions de \( \eM_n(\eK)\) et \( \eM_n(\eK)'\), et il suffit de prouver que \( f\) est injective. Soit donc \( A\) telle que \( f_A=0\). Nous l'appliquons à la matrice \( (E_{ij})_{kl}=\delta_{ik}\delta_{jl}\) :
+    \begin{equation}
+            0=f_A(E_{ij})
+            =\sum_{k}(AE_{ij})_{kk}
+            =\sum_{kl}A_{kl}\delta_{il}\delta_{jk}
+            =A_{ij}.
+    \end{equation}
+    Donc \( A=0\).
+\end{proof}
+
+\begin{corollary}[\cite{KXjFWKA}]
+    Soit \( \eK\) un corps et \( \phi\in\eM(n,\eK)^*\) telle que pour tout \( X,Y\in \eM(n,\eK)\) on ait
+    \begin{equation}
+        \phi(XY)=\phi(YX).
+    \end{equation}
+    Alors il existe \( \lambda\in \eK\) tel que \( \phi=\lambda\Tr\).
+\end{corollary}
+\index{trace!unicité pour la propriété de trace}
+
+\begin{proof}
+    La proposition \ref{PropHOjJpCa} nous donne une matrice \( A\in \eM(n,\eK)\) telle que \( \phi=f_A\). L'hypothèse nous dit que \( f_A(XY)=f_A(YX)\), c'est à dire
+    \begin{equation}
+        \Tr(AXY)=\Tr(AYX)
+    \end{equation}
+    pour toute matrices \( X,Y\in \eM(n,\eK)\). L'invariance cyclique de la trace nous dit que \( \Tr(AXY)=\Tr(XAY)\), ce qui signifie que
+    \begin{equation}
+        \Tr\big( (AX-XA)Y \big)=0
+    \end{equation}
+    ou encore que \( f_{AX-XA}=0\) pour tout \( X\). La fonction \( f\) étant injective nous en déduisons que la matrice \( A\) doit satisfaire
+    \begin{equation}
+        AX=XA
+    \end{equation}
+    pour tout \( X\in\eM\). En particulier en prenant la fameuse matrice \( E_{ij}\) et en calculant un peu,
+    \begin{equation}
+        A_{li}\delta_{jm}=\delta_{il}A_{jm}
+    \end{equation}
+    pour tout \( i,j,l,m\). Cela implique que \( A_{ll}=A_{mm}\) pour tout \( l\) et \( m\) et que \( A_{jm}=0\) dès que \( j\neq m\). Il existe donc \( \lambda\in \eK\) tel que \( A=\lambda\mtu\). En fin de compte,
+    \begin{equation}
+        \phi(X)=f_{\lambda\mtu}(X)=\lambda\Tr(X).
+    \end{equation}
+\end{proof}
+
+\begin{corollary}[\cite{KXjFWKA}]       \label{CorICUOooPsZQrg}
+    Soit \( \eK\) un corps. Tout hyperplan de \( \eM(n,\eK)\) coupe \( \GL(n,\eK)\).
+\end{corollary}
+\index{groupe!linéaire!hyperplan}
+
+\begin{proof}
+    Soit \( \mH\) un hyperplan de \( \eM\). Il existe une forme linéaire \( \phi\) sur \( \eM(n,\eK)\) telle que \( \mH=\ker(\phi)\). Encore une fois la proposition \ref{PropHOjJpCa} nous donne \( A\in \eM\) telle que \( \phi=f_A\); nous notons \( r\) le rang de \( A\). Par le lemme \ref{LemZMxxnfM} nous avons \( A=PJ_rQ\) avec \( P,Q\in \GL(n,\eK)\) et
+    \begin{equation}
+        J_r=\begin{pmatrix}
+            \mtu_r    &   0    \\ 
+            0    &   0    
+        \end{pmatrix}.
+    \end{equation}
+    Pour tout \( X\in \eM\) nous avons
+    \begin{equation}
+        \phi(X)=\Tr(AX)=\Tr(PJ_rQX)=\Tr(J_rQXP).
+    \end{equation}
+    Ce que nous cherchons est \( X\in \GL(n,\eK)\) telle que \( \phi(X)=0\). Nous commençons par trouver \( Y\in\GL(n,\eK)\) telle que \( \Tr(J_rY)=0\). Celle-là est facile : c'est
+    \begin{equation}
+        Y=\begin{pmatrix}
+            0    &   1    \\ 
+            \mtu_{n-1}    &   0    
+        \end{pmatrix}.
+    \end{equation}
+    Les éléments diagonaux de \( J_rY\) sont tous nuls. Par conséquent en posant \( X=Q^{-1}YP^{-1}\) nous avons notre matrice inversible dans le noyau de \( \phi\).
+\end{proof}
+\index{hyperplan!de \( \eM(n,\eK)\)}
diff --git a/tex/frido/57_EspacesVectos.tex b/tex/frido/57_EspacesVectos.tex
new file mode 100644
index 000000000..605f9ee9e
--- /dev/null
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@@ -0,0 +1,1791 @@
+% This is part of Mes notes de mathématique
+% Copyright (c) 2008-2017
+%   Laurent Claessens
+% See the file fdl-1.3.txt for copying conditions.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Valeur propre et vecteur propre}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}      \label{DefooMMKZooVcskCc}
+    Soit un \( \eK\)-espace vectoriel \( E\) et un endomorphisme \( A\colon V\to V\). Un \defe{vecteur propre}{vecteur!propre} de \( A\) est un vecteur \( v \neq 0\) tel que \( Av=\lambda v\) pour un certain \( \lambda\in \eK\). Dans ce cas, \( \lambda\) est la \defe{valeur propre}{valeur!propre} de \( v\).
+
+    L'\defe{espace propre}{espace!propre} de \( A\) pour la valeur \( \lambda\)\footnote{Nous laissons au lecteur le soin de vérifier que c'est bien un sous-espace vectoriel de \( E\).} est l'ensemble des vecteurs propres de \( A\) pour la valeur propre \( \lambda\) et zéro.
+\end{definition}
+L'ensemble de valeurs propres de l'endomorphisme \( u\) est son \defe{spectre}{spectre!d'un endomorphisme} et est noté \( \Spec(u)\).
+
+\begin{remark}
+    Le nombre zéro peut être une valeur propre; c'est le vecteur zéro qui ne peut pas être vecteur propre. La matrice nulle est une matrice diagonalisable.
+\end{remark}
+
+\begin{lemma}       \label{LemjcztYH}
+    Soit \( u\) un endomorphisme et \( E_{\lambda}(u)\)\nomenclature[A]{\( E_{\lambda}(u)\)}{Espace propre de \( u\)} ses espaces propres. La somme des \( V_{\lambda}\) est directe.
+\end{lemma}
+
+\begin{proof}
+    Soit \( v_i\in V_{\lambda_i}\) un choix de vecteurs propres de \( u\). Si la somme n'est pas directe, nous pouvons considérer une combinaison linéaire des \( v_i\) qui soit nulle :
+    \begin{equation}
+        v_1+\cdots+v_p=0.
+    \end{equation}
+    Appliquons \( (A-\lambda_1\mtu)\) à cette égalité :
+    \begin{equation}
+        (\lambda_2-\lambda_1)v_1+\cdots+(\lambda_p-\lambda_1)v_p=0.
+    \end{equation}
+    En appliquant encore successivement les opérateurs \( (A-\lambda_i\mtu)\) nous réduisons le nombre de termes jusqu'à obtenir \( v_p=0\).
+\end{proof}
+
+\begin{example} \label{ExICOJcFp}
+    Sur \( \eR^2\), nous considérons la matrice \( A=\begin{pmatrix}
+        1    &   0    \\ 
+        1    &   1    
+    \end{pmatrix}\) qui a pour polynôme caractéristique le polynôme \( \chi_A=(X-1)^2\). Le nombre \( \lambda=1\) est une racine double de ce polynôme, et pourtant il n'y a qu'une seule dimension d'espace propre :
+    \begin{equation}
+        \begin{pmatrix}
+            1    &   0    \\ 
+            1    &   1    
+        \end{pmatrix}\begin{pmatrix}
+            x    \\ 
+            y    
+        \end{pmatrix}=\begin{pmatrix}
+            x    \\ 
+            y    
+        \end{pmatrix}
+    \end{equation}
+    entraine \( x=0\).
+
+    Ici la multiplicité algébrique est différente de la multiplicité géométrique.
+\end{example}
+
+\begin{proposition}[\cite{RombaldiO}]   \label{PropTVKbxU}
+    Soit \( E\), un espace vectoriel sur un corps infini et \( (F_k)_{k=1,\ldots, r}\), des sous-espaces vectoriels propres\footnote{Définition \ref{DefooMMKZooVcskCc}.} de \( E\) tels que \( \bigcup_{i=1}^rF_i=E\). Alors \( E=F_k\) pour un certain \( k\).
+
+    Autrement dit, l'union finie de sous-espaces propres ne peut être égal à l'espace complet.
+\end{proposition}
+
+La proposition suivante donne une utilisation amusante de la notion de polynôme caractéristique
+\begin{proposition}[\cite{ooNGUJooPphdsT}]
+    Soit un espace vectoriel \( V\) de dimension finie pour lequel il existe un endomorphisme \( f\colon V\to V\) tel que \( (f\circ f)(v)=-v\) pour tout \( v\in V\). Alors la dimension de \( V\) est paire.
+\end{proposition}
+
+\begin{proof}
+    Cherchons les valeurs propres de \( f\) en résolvant l'équation \( f(v)=\lambda v\). Nous appliquons \( f\) à cette égalité :
+    \begin{equation}
+        -v=\lambda f(v)=\lambda^2v.
+    \end{equation}
+    Donc \( \lambda\) ne peut pas être réel. Nous avons montré que \( f\) n'a pas de valeurs propres réelles. Or le polynôme caractéristique de \( f\) est de degré égal à la dimension. Si la dimension est impaire, le polynôme caractéristique est de degré impair, et possède donc une racine réelle. Autrement dit, l'absence de racines réelles au polynôme caractéristique indique une dimension paire.
+\end{proof}
+
+Une autre preuve possible est d'utiliser le déterminant : si la dimension de \( V\) est \( n\) nous avons :
+\begin{equation}
+    \det(f^2)=\det(-\id)=(-1)^n.
+\end{equation}
+Donc \( (-1)^n\) est positif, ce qui montre que \( n\) est pair.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Polynôme d'endomorphismes}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Soit \( A\) un anneau commutatif et \( \eK\), un corps commutatif. L'injection canonique \( A\to A[X]\) se prolonge en une injection
+\begin{equation}
+   \eM(A)\to\eM\big( A[X] \big).
+\end{equation}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Polynômes d'endomorphismes}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit \( u\in\End(E)\) où \( E\) est un \( \eK\)-espace vectoriel. Nous considérons l'application
+\begin{equation}    \label{EqOVKooeMJuv}
+    \begin{aligned}
+        \varphi_u\colon \eK[X]&\to \End(E) \\
+        P&\mapsto P(u). 
+    \end{aligned}
+\end{equation}
+L'image de \( \varphi_u\) est un sous-espace vectoriel. En effet si \( A=\varphi_u(P)\) et \( B=\varphi_u(Q)\), alors \( A+B=\varphi_u(P+Q)\) et \( \lambda A=(\lambda P)(u)\). En particulier c'est un espace fermé.
+
+Soit \( u\) un endomorphisme d'un \( \eK\)-espace vectoriel \( E\) et \( P\), un polynôme. Nous disons que \( P\) est un polynôme \defe{annulateur}{polynôme!annulateur} de \( u\) si \( P(u)=0\) en tant que endomorphisme de \( E\).
+
+\begin{lemma}       \label{LemQWvhYb}
+    Si \( P\) et \( Q\) sont des polynômes dans \( \eK[X]\) et si \( u\) est un endomorphisme d'un \( \eK\)-espace vectoriel \( E\), nous avons
+    \begin{equation}
+        (PQ)(u)=P(u)\circ Q(u).
+    \end{equation}
+\end{lemma}
+
+\begin{proof}
+    Si \( P=\sum_i a_iX^i\) et \( Q=\sum_j b_jX^j\), alors le coefficient de \( X^k\) dans \( PQ\) est
+    \begin{equation}        \label{EqCoefGPyVcv}
+        \sum_la_lb_{k-l}.
+    \end{equation}
+    Par conséquent \( (PQ)(u)\) contient \( \sum_la_lb_{k-l}u^k\). Par ailleurs \( P(u)\circ Q(u)\) est donné par
+    \begin{equation}
+        \sum_ia_iu^i\left( \sum_jb_ju^j \right)(x)=\sum_{ij}a_ib_ju^{i+j}(x).
+    \end{equation}
+    Le coefficient du terme en \( u^k\) est bien le même que celui donné par \eqref{EqCoefGPyVcv}.
+\end{proof}
+
+\begin{theorem}[Décomposition des noyaux ou lemme des noyaux]       \label{ThoDecompNoyayzzMWod}
+    Soit \( u\) un endomorphisme du \( \eK\)-espace vectoriel \( E\). Soit \( P\in\eK[X]\) un polynôme tel que \( P(u)=0\). Nous supposons que \( P\) s'écrive comme le produit \( P=P_1\ldots P_n\) de polynômes deux à deux étrangers\footnote{Définition \ref{DefDSFooZVbNAX}.}. Alors
+    \begin{equation}
+        E=\ker P_1(u)\oplus\ldots\oplus\ker P_n(u).
+    \end{equation}
+    De plus les projecteurs associés à cette décomposition sont des polynômes en \( u\).
+\end{theorem}
+\index{lemme!des noyaux}
+Ce résultat est utilisé pour prouver que toute représentation est décomposable en représentations irréductibles, proposition \ref{PropHeyoAN} ainsi que pour le théorème \ref{ThoDigLEQEXR} qui dit que si le polynôme minimal d'un endomorphisme est scindé à racine simple alors il est diagonalisable.
+
+\begin{proof}
+    Nous posons 
+    \begin{equation}
+        Q_i=\prod_{j\neq i}P_i.
+    \end{equation}
+    Par le lemme \ref{LemuALZHn} ces polynômes sont étrangers entre eux et le théorème de Bézout (théorème \ref{ThoBezoutOuGmLB}) donne l'existence de polynômes \( R_i\) tels que
+    \begin{equation}
+        R_1Q_1+\cdots+R_nQ_n=1.
+    \end{equation}
+    Si nous appliquons cette égalité à \( u\) et ensuite à \( x\in E\) nous trouvons
+    \begin{equation}        \label{EqqVcpUy}
+        \sum_{i=1}^n(R_iQ_i)(u)(x)=x,
+    \end{equation}
+    et en particulier si nous posons \( E_i=\Image\big(P_iQ_i(u)\big)\) nous avons
+    \begin{equation}
+        E=\sum_{i=1}^nE_i.
+    \end{equation}
+    Cette dernière somme n'est éventuellement pas une somme directe. Si \( i\neq j\), alors \( Q_iQ_j\) est multiple de \( P\) et nous avons, en utilisant le lemme \ref{LemQWvhYb}, 
+    \begin{equation}
+        (R_iQ_i)(u)\circ (R_jQ_j)(u)=\big( R_iQ_iR_jQ_j \big)(u)=S_{ij}(u)\circ P(u)=0
+    \end{equation}
+    où \( S_{ij}\) est un polynôme. 
+
+    Nous pouvons voir \( E\) comme un \( \eK\)-module et appliquer le théorème \ref{ThoProjModpAlsUR}. Les opérateurs \( R_iQ_i(u)\) ont l'identité comme somme et sont orthogonaux, et nous avons donc la décomposition en somme directe :
+    \begin{equation}
+        E=\bigoplus_{i=1}^nR_iQ_i(u)E.
+    \end{equation}
+
+    Afin de terminer la preuve, nous devons montrer que \( R_iQ_i(u)E=\ker P_i(u)\). D'abord nous avons
+    \begin{equation}
+        P_iR_iQ_i(u)=(R_iP)(u)=R_i(u)\circ P(u)=0,
+    \end{equation}
+    par conséquent \( \Image(R_iQ_i(u))\subset \ker P_i(u)\). Pour obtenir l'inclusion inverse, nous reprenons l'équation \eqref{EqqVcpUy} avec \( x\in\ker P_i(u)\). Elle se réduit à
+    \begin{equation}
+        (R_iQ_i)(u)x=x.
+    \end{equation}
+    Par conséquent \( x\in\Image\big( R_iQ_i(u) \big)\).
+\end{proof}
+
+\begin{corollary}   \label{CorKiSCkC}
+    Soit \( E\), un \( \eK\)-espace vectoriel de dimension finie et \( f\), un endomorphisme semi-simple dont la décomposition du polynôme minimal \( \mu_f\) en facteurs irréductibles sur \( \eK[X]\) est \( \mu_f=M_1^{\alpha_1}\cdots M_r^{\alpha_r}\). Si \( F\) est un sous-espace stable par \( f\), alors
+    \begin{equation}
+        F=\bigoplus_{i=1}^r\ker M_i^{\alpha_i}(f)\cap F
+    \end{equation}
+\end{corollary}
+
+\begin{proof}
+    Nous posons \( E_i=\ker M_i^{\alpha_i}(f)\) et \( F_i=E_i\cap F\). Les polynômes \( M_i^{\alpha_i}\) sont deux à deux étrangers et \( \mu_f(f)=0\), donc le lemme des noyaux (\ref{ThoDecompNoyayzzMWod}) s'applique et
+    \begin{equation}
+        E=E_1\oplus\ldots\oplus E_r.
+    \end{equation}
+    Nous pouvons décomposer \( x\in F\) en termes de cette somme :
+    \begin{equation}     \label{EqbBbrdi}
+        x=x_1+\cdots +x_r
+    \end{equation}
+    avec \( x_i\in E_i\). Toujours selon le lemme des noyaux, les projections sur les espaces \( E_i\) sont des polynômes en \( f\). Par conséquent \( F\) est stable sous toutes ces projections \( \pr_i\colon E\to E_i\), et en appliquant \( \pr_i\) à \eqref{EqbBbrdi}, \( \pr_i(x)=x_i\). Vu que \( x\in F\), le membre de gauche est encore dans \( F\) et \( x_i\in E_i\cap F\). Nous avons donc
+    \begin{equation}
+        F\subset\bigoplus_{i=1}^rF_i.
+    \end{equation}
+    L'inclusion inverse est immédiate parce que \( F_i\subset F\) pour chaque \( i\).
+\end{proof}
+
+\begin{lemma}   \label{LemVISooHxMdbr}
+    Si \( x\) est un vecteur propre de valeur propre \( \lambda\) pour l'endomorphisme \( u\) et si \( P\) est un polynôme, alors \( x\) est vecteur propre de \( u\) pour la valeur propre \( P(\lambda)\).
+\end{lemma}
+
+\begin{proof}
+    C'est un simple calcul de \( P(u)x\) en ayant noté \( P(X)=\sum_{k=0}^nc_kX^n\) :
+    \begin{equation}
+        P(u)x=\sum_{k=0}^nc_ku^k(x)=\sum_{k=0}^nc_k\lambda^ku=P(\lambda)x.
+    \end{equation}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Calcul effectif de l'exponentielle d'une matrice}
+%---------------------------------------------------------------------------------------------------------------------------
+\label{SUBSECooGAHVooBRUFub}
+
+Nous reprenons l'exemple de \cite{MneimneReduct}. Soit \( A\) une matrice dont le polynôme minimum s'écrit
+\begin{equation}
+    P(X)=(X-1)^2(X-2).
+\end{equation}
+Par le théorème \ref{ThoDecompNoyayzzMWod} de décomposition des noyaux nous avons
+\index{théorème!décomposition des noyaux!et exponentielle de matrice}
+\begin{equation}
+    E=\ker(A-1)^2\oplus\ker(A-2).
+\end{equation}
+En suivant les notations de ce théorème nous avons \( P_1(X)=(X-1)^2\), \( P_2(X)=X-2\) et
+\begin{subequations}
+    \begin{align}
+        Q_1(X)&=X-2\\
+        Q_2(X)&=(X-1)^2.
+    \end{align}
+\end{subequations}
+Les polynômes \( R_i\) dont l'existence est assurée par le théorème de Bézout sont
+\begin{equation}
+    \begin{aligned}[]
+        R_1(X)&=-X\\
+        R_2(X)&=1.
+    \end{aligned}
+\end{equation}
+Nous avons
+\begin{equation}
+    R_1Q_1+R_2Q_2=1.
+\end{equation}
+Le projecteur \( p_i\) sur \( \ker P_i\) est \( R_iQ_i\) :
+\begin{equation}
+    \begin{aligned}[]
+        p_1&=-A(A-2)=\pr_{\ker(u-1)^2}\\
+        p_2&=(A-1)^2=\pr_{\ker(u-2)}.
+    \end{aligned}
+\end{equation}
+Passons maintenant au calcul de l'exponentielle. Nous avons évidemment
+\begin{equation}
+    e^A=e^Ap_1+e^Ap_2.
+\end{equation}
+Étant donné que \( p_1\) est le projecteur sur le noyau de \( (A-1)^2\), nous avons
+\begin{equation}
+    e^Ap_1=ee^{A-1}p_1=ep_1+e(u-1)1=ep_1=-Ae(A-2).
+\end{equation}
+En effet \( e^{A-1}p_1=\sum_{k=0}^{\infty}(A-1)^k\circ p_1\). De la même façon nous avons
+\begin{equation}
+    e^Ap_2=e^2e^{A-2}p_2=e^2p_2=e^2(A-1)^2.
+\end{equation}
+Au final,
+\begin{equation}
+    e^A=-Ae(A-2)+e^2(A-1)^2.
+\end{equation}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Polynôme minimal et minimal ponctuel}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemmaDef}        \label{DefooOHUXooNkPWaB}
+    Soit un endomorphisme \( f\colon E\to E\) d'un \( \eK\)-espace vectoriel de dimension finie. Il existe un unique polynôme annulateur normalisé de degré minimum.
+
+    Il est nommé le \defe{polynôme minimal}{polynôme!minimal} de \( f\) et il est noté \( \mu_f\) ou simplement \( \mu\) lorsque la dépendance en \( f\) est claire.
+\end{lemmaDef}
+
+\begin{proof}
+    Pour l'unicité, soient \( P\) et \( Q\) deux polynômes annulateur de \( f\) de même degré \( N\) et ayant tous deux \( 1\) comme coefficient de \( x^N\). Alors \( P-Q\) est de degré \( N-1\) tout en étant encore annulateur.
+
+    Pour l'existence, les endomorphismes \( \id\), \( f\), \( f^2\), \ldots ne peuvent pas être tous linéairement indépendants parce que la dimension de \( \End(E)\) est finie. Il existe donc un nombre \( N\) et des coefficients \( a_k\) tels que \( \sum_{k=0}^Na_kf^k=0\). Le polynôme \( P(X)=\sum_{k=0}^Na_kX^k\) est donc annulateur de \( f\).
+
+    Une autre façon de le dire est que l'application linéaire \( \varphi\colon \eK[X]\to \End(E)\) donnée par \( \varphi(P)=P(f)\) est un endomorphisme d'un espace vectoriel de dimension infinie vers un espace vectoriel de dimension finie. Il ne peut donc pas être injectif et possède donc un noyau non réduit à zéro.
+\end{proof}
+
+\begin{remark}
+    La preuve donnée ci-dessus montre que \( \deg(\mu)\leq \dim(E)^2\). Comme conséquence du théorème de Caley-Hamilton \ref{ThoCalYWLbJQ} nous verrons qu'en réalité le degré du polynôme minimal est majoré par la dimension de l'espace.
+\end{remark}
+
+\begin{example}[Pas en dimension infinie]
+    L'endomorphisme de dérivation
+\end{example}
+
+
+Dans la suite, l'endomorphisme \( f\) du \( \eK\)-espace vectoriel \( E\) de dimension \( n\) est fixé. Pour \( x\in E\) nous notons
+\begin{equation}            \label{EqooOAYDooEpZELo}
+    E_x=\{ P(f)x\tq P\in \eK[X] \}.
+\end{equation}
+Nous considérons le morphisme d'algèbres
+\begin{equation}
+    \begin{aligned}
+        \varphi\colon \eK[X]&\to \End(E) \\
+        P&\mapsto P(f) 
+    \end{aligned}
+\end{equation}
+et si \( x\in E\) est donné nous considérons le morphisme de \( \eK\)-espaces vectoriels
+\begin{equation}
+    \begin{aligned}
+        \varphi_x\colon \eK[X]&\to E \\
+        P&\mapsto P(f)x. 
+    \end{aligned}
+\end{equation}
+Les noyaux de ces applications sont des idéaux, entre autres par le lemme \ref{LemQWvhYb}. Ils ont donc un unique générateur unitaire (chacun) par le théorème \ref{ThoCCHkoU}. En termes de vocabulaire, l'ensemble
+\begin{equation}
+    \ker(\phi)=\{  Q\in\eK[X]\tq Q(f)=0  \}
+\end{equation}
+est l'\defe{idéal annulateur}{polynôme!annulateur} de \( f\) et un polynôme \( Q\) tel que \( Q(f)=0\) est une polynôme annulateur de \( f\).
+
+\begin{definition}      \label{DEFooUICRooBGYhqQ}
+    Le générateur unitaire de \( \ker(\varphi_x)\) est le \defe{polynôme minimal ponctuel}{polynôme!minimal!ponctuel} de \( f\) en \( x\). Il sera noté \( \mu_{f,x}\) ou \( \mu_x\) lorsque la dépendance en \( f\) est claire dans le contexte.
+\end{definition}
+Nous notons \( \mu\) le générateur unitaire du noyau de \( \varphi\) et \( \mu_x\) celui de \( \varphi_x\). Vu que \( \mu\in\ker(\varphi_x)\) pour tout \( x\) nous avons \( \mu_x\divides \mu\) pour tout \( x\).
+
+\begin{example}[Pas en dimension infinie]       \label{ExooDTUJooIMqSKn}
+    En dimension infinie, il n'y a pas toujours de polynôme annulateur. Si \( E\) est un espace vectoriel de dimension infine ayant une base dénombrable \( \{ e_i \}_{i\in \eN}\) alors l'opérateur donné par \( f(e_i)=e_{i+1}\) n'a pas de polynôme annulateur. Même pas ponctuel en quel que point que ce soir.
+
+    De même l'opérateur donné par \( g(e_1)=0\) et \( g(e_i)=e_{i-1}\) si \( i\neq 1\) n'a pas de polynôme annulateur, mais il a un polynôme annulateur ponctuel évident en \( x=e_1\). L'exemple \ref{ExooLRHCooMYLQTU} donnera un habillage à peine subtil à cet exemple.
+\end{example}
+
+\begin{proposition}     \label{PropAnnncEcCxj}
+    Si \( P\) est un polynôme tel que \( P(f)=0\), alors le polynôme minimal \( \mu_f\) divise \( P\). Autrement dit, le polynôme minimal engendre l'idéal des polynômes annulateurs.
+\end{proposition}
+
+\begin{proof}
+    L'ensemble \( \ker(\varphi)=\{ Q\in \eK[X]\tq Q(u)=0 \} \) est un idéal par le lemme \ref{LemQWvhYb}. Le polynôme minimal de \( u\) est un élément de degré plus bas dans \( I\) et par conséquent \( I=(\mu_u)\) par le théorème \ref{ThoCCHkoU}. Nous concluons que \( \mu_u\) divise tous les éléments de \( I\).
+\end{proof}
+
+La proposition suivante permet de caractériser le polynôme minimal.
+\begin{proposition}[\cite{ooEPEFooQiPESf}]      \label{PROPooVUJPooMzxzjE}
+    Soit une application linéaire \( f\) sur un \( \eK\)-espace vectoriel. Il existe un unique polynôme unitaire\quext{À mon avis, «unitaire» manque dans \cite{ooEPEFooQiPESf}.} \( P\in \eK[X]\) tel que
+    \begin{enumerate}
+        \item
+            \( P(f)=0\);
+        \item
+            l'application
+            \begin{equation}        \label{EQooIBMDooVTaEhf}
+                \begin{aligned}
+                    \varphi\colon \frac{ \eK[X] }{ (P) }&\to \End(E) \\
+                    \bar Q&\mapsto Q(f) 
+                \end{aligned}
+            \end{equation}
+            est injective.
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    En ce qui concerne l'existence, il existe le polynôme minimal de \( f\) qui satisfait les conditions. Pour l'unicité nous travaillons maintenant.
+
+    Supposons que l'application \eqref{EQooIBMDooVTaEhf} soit injective. Alors pour tout \( Q\in \eK[X]\) tel que \( Q(f)=0\) nous avons \( \bar Q=0\), c'est à dire \( Q=PR\) pour un certain \( R\in \eK[X]\). Autrement dit : \( P\) est un générateur unitaire de l'idéal annulateur de \( f\). Le théorème \ref{ThoCCHkoU}\ref{ITEMooASHKooZqkiCH} nous dit alors que \( P=\mu\) parce que \( \mu\) est également générateur unitaire.
+\end{proof}
+
+\begin{lemma}[\cite{ooRJDSooXpVtMD}]\label{LemSYsJJj}
+    Soit \( f\colon E\to E\) un endomorphisme de l'espace vectoriel \( E\). Il existe un élément \( x\in E\) tel que \( \mu_{f,x}=\mu_f\).
+\end{lemma}
+
+\begin{proof}
+    Soit une décomposition en irréductibles du polynôme minimal \( \mu=P_1^{\alpha_1}\ldots P_r^{\alpha_r}\). Nous notons \( E_i=\ker\big( P_i^{\alpha_i}(f) \big)\). Les polynômes \( P_i\) sont étrangers deux à deux (un diviseur commun aurait a fortiori été un diviseur et aurait contredit l'irréductibilité). Le lemme des noyaux \ref{ThoDecompNoyayzzMWod} nous donne la somme directe
+    \begin{equation}
+        E=\bigoplus_{i=1}^r\ker\big( P_i^{\alpha_i}(f) \big).
+    \end{equation}
+    Si \( x_i\in E_i\) alors \( \mu_{x_i}\) est une puissance de \( P_i\). En effet \( \mu_{x_i}\divides \mu\) et est donc un produit des puissances des \( P_j\). Or si \( (QP_j)(f)x_i=0\) alors \( (P_jQ)(f)x_i=0\), ce qui donne \( Q(f)x_i\in E_j\cap E_i=\{ 0 \}\). Donc \( \mu_{x_i}\) n'est pas de la forme \( QP_j\) pour \( j\neq i\). Nous en déduisons que \( \mu_{x_i}\) est une puissance de \( P_i\) dès que \( x_i\in E_i\). Nous choisissons \( x_i\in E_i\) tel que \( \mu_{x_i}=P_i^{\alpha_i}\).
+
+    Nous posons enfin \( a=x_1+\cdots +x_r\); par définition du polynôme annulateur \( \mu_a\), nous avons
+    \begin{equation}        \label{EqooVIGGooSfuvwB}
+        0=\mu_a(f)a=\mu_a(f)x_1+\cdots +\mu_a(f)x_r.
+    \end{equation}
+    Mais \( m_a(f)x_j\in E_i\), et la somme des \( E_j\) est directe, donc l'annulation de la somme \eqref{EqooVIGGooSfuvwB} implique l'annulation de chacun des termes : \( \mu_a(f)x_i=0\) pour tout \( i\). Cela prouve que \( \mu_{x_i}\divides \mu_a\). Mais comme les \( \mu_{x_i}\) sont premiers deux à deux (parce que ce sont les \( P_i^{\alpha_i}\)), nous avons que le produit divise encore \( \mu_a\) :
+    \begin{equation}
+        \prod_{i=1}^r\mu_{x_i}\divides \mu_a,
+    \end{equation}
+    c'est à dire \( \mu\divides \mu_a\). Comme nous avons aussi \( \mu_a\divides \mu\), nous déduisons \( \mu_a=\mu\).
+\end{proof}
+
+\begin{definition}[Matrices, endomorphismes et vecteurs cycliques]      \label{DEFooFEIFooNSGhQE}
+    Une matrice est \defe{cyclique}{cyclique!matrice}\index{matrice!cyclique} si elle est semblable à une matrice compagnon. Un endomorphisme \( f\colon E\to E\) est \defe{cyclique}{cyclique!endomorphisme}\index{endomorphisme!cyclique} s'il existe un vecteur \( x\in E\) tel que \( \{ f^k(x) \}_{k=0,\ldots, n-1} \) est une base de \( E\). Un vecteur ayant cette propriété est un \defe{vecteur cyclique}{vecteur!cyclique} pour \( f\).
+\end{definition}
+
+\begin{lemma}   \label{LemAGZNNa}
+    Soit \( E\) un espace vectoriel de dimension finie et un endomorphisme cyclique\footnote{Voir la définition \ref{DEFooFEIFooNSGhQE}.} \( f\) de \( E\). Soit un vecteur cyclique \( v\) de \( f\), alors le polynôme minimal de \( f\) est égal au polynôme minimal de \( f\) au point \( v\) : \( \mu_{f}=\mu_{f,v}\).
+\end{lemma}
+
+\begin{proof}
+    Montrons que \( \mu_{f,v}\) est un polynôme annulateur de \( f\), ce qui prouvera que \( \mu_f\) divise \( \mu_{f,v}\) par la proposition \ref{PropAnnncEcCxj}. Étant donné que \( v\) est cyclique, tout élément de \( E\) s'écrit sous la forme \( x=Q(f)v\). Prenons un polynôme \( P\) annulateur de \( f\) en \( v\) : \( P(f)v=0\). Nous montrons que \( P\) est alors un polynôme annulateur de \( f\). En effet, nous avons
+    \begin{equation}
+        P(f)x=\big( P(f)\circ Q(f) \big)v=\big( Q(f)\circ P(f) \big)v=0
+    \end{equation}
+    où nous avons utilisé le lemme \ref{LemQWvhYb}.
+\end{proof}
+
+\begin{lemma}[\cite{ooRJDSooXpVtMD}]
+    Soit \( a\in E\) tel que \( \mu_a=\mu\). Alors \( E_a\) est un sous-espace stable pour \( f\) pour lequel il existe un supplémentaire stable.
+\end{lemma}
+
+\begin{proof}
+    Soit \( l=\deg(\mu)=\deg(\mu_a)\). L'espace \( E_a\) étant engendré par les \( f^k(a)\) nous savons que \( e_1=a\), \( e_2=f(a)\),\ldots, \( e_l=k^{l-1}(a)\) forment une base de \( E_a\). Nous pouvons la compléter en une base \( \{ e_1,\ldots, e_n \}\) de \( E\). Et nous posons\footnote{ici, comme presque partout, \( e^*_{l}\) est le dual de \( e_l\), c'est à dire l'application linéaire sur \( E\) donnée par \( e^*_l(e_i)=\delta_{li}\). }
+    \begin{subequations}
+        \begin{align}
+            G&=\{ x\in E\tq e^*_l\big( f^k(x) \big)=0\,\forall k\geq 0 \}\\
+            &=\bigcap_{k\geq 0}\ker\{ e^*_l\circ f^k \}\\
+            &=\bigcap_{k=0}^{l-1}\ker(  e^*_l\circ f^k ).
+        \end{align}
+    \end{subequations}
+    La dernière égalité est due au fait que \( l\) soit le degré de \( \mu\). Du coup \( f^l\) est une combinaison linéaire des \( f^i\) avec \( i\leq l-1\).
+
+    Nous avons \( f(G)\subset G\) et de plus \( E_a\cap G=\{ 0 \}\) parce qu'un élément de \( E_a\) est une combinaison linéaire d'éléments de la forme \( f^j(a)\) (\( j\leq l\)). Après application de \( f^{l-j}\), ces éléments obtiennent une composante \( f^l(a)=e_l\). De plus \( G\) est un sous-espace vectoriel du fait que \( e^*_l\circ f^i\) est une application linéaire. 
+    
+    Montrons enfin que \( \dim(G)=n-l\). Pour cela nous remarquons que \( G\) est une intersection d'hyperplans, et nous montrons que les équations définissant ces hyperplans sont linéairement indépendantes. Soit donc 
+    \begin{equation}        \label{EqooOHESooRtBUfc}
+        \sum_{j=0}^{l-1}\lambda_j\big( e^*_l\circ f^j \big)=0
+    \end{equation}
+    et montrons que \( \lambda_j=0\) pour tout $j$ est l'unique solution. Soit \( x\in E\) et appliquons l'opération \eqref{EqooOHESooRtBUfc} au vecteur \( f^i(x)\); le résultat est zéro :
+    \begin{equation}
+        0=\sum_{j=0}^{l-1}\lambda_j(e^*_l\circ f^i\circ f^j)=(e^*_l\circ f^i)P(u)
+    \end{equation}
+    où nous avons posé \( P(X)=\sum_{j=0}^{l-1}\lambda_jX^j\). Appliquons cela à \( a\) : pour tout \( i\) nous avons
+    \begin{equation}
+        (e^*_l\circ f^i)\big( P(f)a \big)=0.
+    \end{equation}
+    Mais par définition de \( E_a\), l'élément \(P(f)a \) est dans \( E_a\). Nous en déduisons que 
+    \begin{equation}
+        P(f)a\in G\cap E_a=\{ 0 \},
+    \end{equation}
+    c'est à dire que \( P\) est un polynôme annulateur de \( a\). Mais \( P\) est de degré \( l-1\) alors que le polynôme minimal de \( a\) est de degré \( l\). Par conséquent \( P=0\) et \( \lambda_j=0\) pour tout \( j\).
+\end{proof}
+
+\begin{definition}  \label{DEFooBOHVooSOopJN}
+    Un endomorphisme d'un espace vectoriel est \defe{semi-simple}{semi-simple!endomorphisme} si tout sous-espace stable par \( u\) possède un supplémentaire stable.
+\end{definition}
+
+\begin{lemma}   \label{LemrFINYT}
+    Si le polynôme minimal d'un endomorphisme est irréductible, alors il est semi-simple\footnote{Définition \ref{DEFooBOHVooSOopJN}.}.
+\end{lemma}
+
+\begin{proof}
+    Soit \( f\), un endomorphisme dont le polynôme minimal est irréductible et \( F\), un sous-espace stable par \( f\). Nous devons en trouver un supplémentaire stable. Si \( F=E\), il n'y a pas de problèmes. Sinon nous considérons \( u_1\in E\setminus F\) et
+    \begin{equation}
+        E_{u_1}=\{ P(f)u_1\tq P\in \eK[X] \},
+    \end{equation}
+    qui est un espace stable par \( f\). 
+
+    Montrons que \( E_{u_1}\cap F=\{ 0 \}\). Pour cela nous regardons l'idéal
+    \begin{equation}
+        I_{u_1}=\{ P\in \eK[X]\tq P(f)u_1=0 \}.
+    \end{equation}
+    Cela est un idéal non réduit à \( \{ 0 \}\) parce que le polynôme minimal de \( f\) par exemple est dans \( I_{u_1}\). Soit \( P_{u_1}\) un générateur unitaire de \( I_{u_1}\). Étant donné que \( \mu_f\in I_{u_1}\), nous avons que \( P_{u_1}\) divise \( \mu_f\) et donc \( P_{u_1}=\mu_f\) parce que \( \mu_f\) est irréductible par hypothèse.
+
+    Soit \( y\in E_{u_1}\cap F\). Par définition il existe \( P\in\eK[X]\) tel que \( y=P(f)u_1\) et si \( y\neq 0\), ce la signifie que \( P\notin I_{u_1}\), c'est à dire que \( P_{u_1} \) ne divise pas \( P\). Étant donné que \( P_{u_1}\) est irréductible cela implique que \( P_{u_1}\) et \( P\) sont premiers entre eux (ils n'ont pas d'autre \( \pgcd\) que \( 1\)).
+
+    Nous utilisons maintenant Bézout (théorème \ref{ThoBezoutOuGmLB}) qui nous donne \( A,B\in \eK[X]\) tels que 
+    \begin{equation}
+        AP+BP_{u_1}=1.
+    \end{equation}
+    Nous appliquons cette égalité à \( f\) et puis à \( u_1\):
+    \begin{equation}
+        u_1=A(f)\circ \underbrace{P(f)u_1}_{=y}+B(f)\circ \underbrace{P_{u_1}(u_1)}_{=0}=A(f)y.
+    \end{equation}
+    Mais \( y\in F\), donc \( A(f)y\in F\). Nous aurions donc \( u_1\in F\), ce qui est impossible par choix. Nous avons maintenant que l'espace \( E_{u_1}\oplus F\) est stable sous \( f\). Si cet espace est \( E\) alors nous arrêtons. Sinon nous reprenons le raisonnement avec \( E_{u_1}\oplus F\) en guise de \( F\) et en prenant \( u_2\in E\setminus(E_{u_1}\oplus F)\). Étant donné que \( E\) est de dimension finie, ce procédé s'arrête à un certain moment et nous aurons
+    \begin{equation}
+        E=F\oplus E_{u_1}\oplus\ldots\oplus E_{u_k}
+    \end{equation}
+    où chacun des \( E_{u_i}\) sont stables.
+\end{proof}
+
+\begin{theorem} \label{ThoFgsxCE}
+    Un endomorphisme est semi-simple si et seulement si son polynôme minimal est produit de polynômes irréductibles distincts deux à deux.
+\end{theorem}
+\index{anneau!principal}
+
+\begin{proof}
+
+    Supposons que \( f\) soit semi-simple et que son polynôme minimal soit donné par \( \mu_f=M_1^{\alpha_1}\ldots M_r^{\alpha_r}\) où les \( M_i\) sont des polynômes irréductibles deux à deux distincts. Nous devons montrer que \( \alpha_i=1\) pour tout \( i\). Soit \( i\) tel que \( \alpha_i\geq 1\) et \( N\in \eK[X]\) tel que \( \mu_f=M^2N\) où l'on a noté \( M=M_i\). Nous étudions l'espace
+    \begin{equation}
+        F=\ker M(f)
+    \end{equation}
+    qui est stable par \( f\), et qui possède donc un supplémentaire \( S\) également stable par \( f\). Nous allons montrer que \( MN\) est un polynôme annulateur de \( f\).
+
+    D'abord nous prenons \( x\in S\). Étant donné que \( F\) est le noyau de \( M(f)\),
+    \begin{equation}
+        M(f)\big( MN(f)x \big)=\mu_f(f)x=0,
+    \end{equation}
+    ce qui signifie que \( MN(f)x\in F\). Mais vu que \( S\) est stable par \( f\) nous avons aussi que \( MN(f)x\in S\). Finalement \( MN(f)x\in F\cap S=\{ 0 \}\). Autrement dit, \( MN(f)\) s'annule sur \( S\).
+
+    Prenons maintenant \( y\in F\). Nous avons
+    \begin{equation}
+        MN(f)=N(f)\big( M(f)y \big)=0
+    \end{equation}
+    parce que \( y\in F=\ker M(f)\).
+
+    Nous avons prouvé que \( MN(f)\) s'annule partout et donc que \( MN(f)\) est un polynôme annulateur de \( f\), ce qui contredit la minimalité de \( \mu_f=M^2N\).
+
+    Nous passons au sens inverse. Soit \( m_f=M_1\ldots M_r\) une décomposition du polynôme minimal de l'endomorphisme \( f\) en irréductibles distincts deux à deux. Soit \( F\) un sous-espace vectoriel stable par \( f\). Nous notons
+    \begin{equation}
+        E_i=\ker(M_i(f))
+    \end{equation}
+    et \( f_i=f|_{E_i}\). Par le lemme \ref{CorKiSCkC} nous avons
+    \begin{equation}
+        F=\bigoplus_{i=1}^r(F\cap E_i).
+    \end{equation}
+    Les espaces \( E_i\) sont stables par \( f\) et étant donné que \( M_i\) est irréductible, il est le polynôme minimal de \( f_i\). En effet, \( M_i\) est annulateur de \( f_i\), ce qui montre que le minimal de \( f_i\) divise \( M_i\). Mais \( M_i\) étant irréductible, \( M_i\) est le polynôme minimal. Étant donné que \( \mu_{f_i}=M_i\), l'endomorphisme \( f_i\) est semi-simple par le lemme \ref{LemrFINYT}.
+
+    L'espace \( F\cap E_i\) étant stable par l'endomorphisme semi-simple \( f_i\), il possède un supplémentaire stable que nous notons \( S_i\)~:
+    \begin{equation}
+        E_i=S_i\oplus(F\cap E_i).
+    \end{equation}
+    Étant donné que sur chaque \( S_i\) nous avons \( f|_{S_i}=f_i\), l'espace \( S=S_1\oplus\ldots\oplus S_r\) est stable par \( f\). Du coup nous avons
+    \begin{subequations}
+        \begin{align}
+            E&=E_1\oplus\ldots\oplus E_r\\
+            &=\big( S_1\oplus(F\cap E_1) \big)\oplus\ldots\oplus\big( S_r\oplus(F\cap E_r) \big)\\
+            &=\big( \bigoplus_{i=1}^rS_i \big)\oplus\big( \bigoplus_{i=1}^rF\cap E_i \big)\\
+            &=S\oplus F,
+        \end{align}
+    \end{subequations}
+    ce qui montre que \( F\) a bien un supplémentaire stable par \( f\) et donc que \( f\) est semi-simple.
+\end{proof}
+
+\begin{example}[L'espace engendré par \( \mtu\), \( A\), \( A^2\),\ldots]
+    Soit \( A\) une matrice, et 
+    \begin{equation}
+        V=\Span\{A^k\tq k\in \eN \}.
+    \end{equation}
+    Nous montrons que \( \dim(V)\) est le degré du polynôme minimal de \( A\).
+
+    D'abord l'idéal annulateur de \( A\) est engendré par le polynôme minimal\footnote{Proposition \ref{PropAnnncEcCxj}.} que nous notons
+        $\mu=\sum_{k=0}^pa_kX^k$.
+    La partie \( \{ \mtu,\ldots, A^{p-1} \}\) est libre parce qu'une combinaison linéaire nulle de cela serait un polynôme annulateur en \( A\) de degré plus petit que \( p\). Donc \( \dim(V)\geq p\).
+
+    La partie \( \{ \mtu,A,\ldots, A^p \}\) est liée à cause du polynôme minimal. Isoler \( A^p\) dans \( \mu(A)=0\) donne un polynôme \( f\) de degré \( p-1\) tel que \( A^p=f(A)\).
+
+    Nous allons montrer à présent que la famille \( \{ \mtu,A,\ldots, A^{p-1} \}\) est génératrice (alors \( \dim(V)\leq p\)). Soit un entier \( q\geq p\)et de division euclidienne\footnote{Théorème \ref{ThoDivisEuclide}.} \( np+r=q\) avec \( r<p\). Nous avons \( A^q=A^{np}A^r\). D'une part
+    \begin{equation}
+        A^{np}=(A^p)^n=f(A)^n
+    \end{equation}
+    est de degré \( n(p-1)\). Par conséquent
+    \begin{equation}
+        A^q=f(A)^nA^r
+    \end{equation}
+    qui est de degré \( n(p-1)+r=q-n\). Autrement dit il existe un polynôme \( g_1\) de degré \( q-n\) tel que \( A^q=g_1(A)\). Si \( q-n>p-1\) alors nous pouvons recommencer et obtenir un polynôme \( g_2\) de degré strictement inférieur à celui de \( g_1\) tel que \( A^q=g_2(A)\). Au bout du compte, il existe un polynôme \( g\) de degré au maximum \( p-1\) tel que \( A^q=g(A)\). Cela prouve que la partie \( \{ \mtu,A,\ldots, A^{p-1} \}\) est génératrice de \( V\).
+
+    La dimension de \( V\) est donc \( p\), le degré du polynôme minimal.
+\end{example}
+
+\begin{proposition}     \label{PropooCFZDooROVlaA}
+    Soit \( f\) un endomorphisme d'un espace vectoriel de dimension finie. Nous avons l'isomorphisme d'espace vectoriel
+    \begin{equation}
+        \eK[f]\simeq\frac{ \eK[X] }{ (\mu_f) }
+    \end{equation}
+    La dimension en est \( \deg(\mu_f)\).
+\end{proposition}
+
+\begin{proof}
+    Notons avant de commencer que \( (\mu)\) est l'idéal engendré par \( \mu\). Les classes dont il est question dans le quotient \( \eK[X]/(\mu)\) sont 
+    \begin{equation}
+        \bar P=\{ P+S\mu \}_{S\in \eK[X]}.
+    \end{equation}
+    Nous allons montrer que l'application suivante fournit l'isomorphisme : 
+    \begin{equation}
+        \begin{aligned}
+            \psi\colon \frac{ \eK[X] }{ (\mu) }&\to \eK[f] \\
+            \bar P&\mapsto P(f). 
+        \end{aligned}
+    \end{equation}
+    \begin{subproof}
+        \item[\( \psi\) est bien définie]
+            Si \( Q\in \bar P\) alors \( Q=P+S\mu\) pour un certain \( S\in \eK[X]\). Du coup nous avons
+            \begin{equation}
+                \psi(\bar Q)=P(f)+(S\mu)(f).
+            \end{equation}
+            Mais \( \mu(f)=0\) donc le deuxième terme est nul. Donc \( \psi(\bar P)\) est bien définit.
+        \item[Injectif]
+            Si \( \psi(\bar P)=0\) nous avons \( P(f)=0\), ce qui signifie que \( P=S\mu\) pour un polynôme \( S\). Par conséquent \( P\in (\mu)\) et donc \( \bar P=0\).
+        \item[Surjectif]
+            Soit \( P\in \eK[X]\). L'élément \( P(f) \) de \( \eK[f]\) est dans l'image de \( \psi\) parce que c'est \( \psi(\bar P)\).
+    \end{subproof}
+    En ce qui concerne la dimension, le corollaire \ref{CorsLGiEN} en parle déjà : une base est donné par les projections de \( 1,X,\ldots, X^{\deg(\mu_a)-1}\).
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Polynôme caractéristique}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}  \label{DefOWQooXbybYD}
+    Soit un anneau commutatif \( A\). Si \( u\in\eM_n(A)\), nous définissons le \defe{polynôme caractéristique de \( u\)}{polynôme!caractéristique}\index{caractéristique!polynôme} :
+    \begin{equation}    \label{Eqkxbdfu}
+        \chi_u(X)=\det(X\mtu_n-u).
+    \end{equation} 
+    Nous définissons de même le polynôme caractéristique d'un endomorphisme \( u\colon E\to E\).
+\end{definition}
+
+\begin{lemma}       \label{LemooWCZMooZqyaHd}
+    Le polynôme caractéristique \( \chi_u\) est unitaire et a pour degré la dimension de l'espace vectoriel \( E\)..
+\end{lemma}
+
+\begin{theorem}     \label{ThoNhbrUL}
+    Soit \( E\) un \(\eK\)-espace vectoriel de dimension finie \( n\) et un endomorphisme \( u\in\End(E)\). Alors
+    \begin{enumerate}
+        \item
+            Le polynôme caractéristique divise \( (\mu_u)^n\) dans \(\eK[X]\).
+        \item
+            Les polynômes caractéristiques et minimaux ont mêmes facteurs irréductibles dans \(\eK[X]\).
+        \item
+            Les polynômes caractéristiques et minimaux ont mêmes racines dans \(\eK[X]\).
+        \item
+            Le polynôme caractéristique est scindé si et seulement si le polynôme minimal est scindé.
+    \end{enumerate}
+\end{theorem}
+
+\begin{theorem} \label{ThoWDGooQUGSTL}
+    Soit \( u\in\End(E)\) et \( \lambda\in\eK\). Les conditions suivantes sont équivalentes
+    \begin{enumerate}
+        \item\label{ItemeXHXhHi}
+            \( \lambda\in\Spec(u)\)
+        \item\label{ItemeXHXhHii}
+            \( \chi_u(\lambda)=0\)
+        \item\label{ItemeXHXhHiii}
+            \( \mu_u(\lambda)=0\).
+    \end{enumerate}
+\end{theorem}
+
+\begin{proof}
+    \ref{ItemeXHXhHi} \( \Leftrightarrow\) \ref{ItemeXHXhHii}. Dire que \( \lambda\) est dans le spectre de \( u\) signifie que l'opérateur \( u-\lambda\mtu\) n'est pas inversible, ce qui est équivalent à dire que \( \det(u-\lambda\mtu)\) est nul par la proposition \ref{PropYQNMooZjlYlA}\ref{ItemUPLNooYZMRJy} ou encore que \( \lambda\) est une racine du polynôme caractéristique de \( u\). 
+
+    \ref{ItemeXHXhHii} \( \Leftrightarrow\) \ref{ItemeXHXhHiii}. Cela est une application directe du théorème \ref{ThoNhbrUL} qui précise que le polynôme caractéristique a les mêmes racines dans \(\eK\) que le polynôme minimal.
+\end{proof}
+
+\begin{definition}
+    Si \( \lambda\in\eK\) est une racine de \( \chi_u\), l'ordre de l'annulation est la \defe{multiplicité algébrique}{multiplicité!valeur propre!algébrique} de la valeur propre \( \lambda\) de \( u\). À ne pas confondre avec la \defe{multiplicité géométrique}{multiplicité!valeur propre!géométrique} qui sera la dimension de l'espace propre.
+\end{definition}
+
+\begin{proposition}[\cite{RombaldiO}]\label{PropNrZGhT}
+    Soit \( f\), un endomorphisme de \( E\) et \( x\in E\). Alors
+    \begin{enumerate}
+        \item
+            L'espace \( E_{f,x}\) est stable par \( f\).
+        \item\label{ItemfzKOCo}
+            L'espace \( E_{f,x}\) est de dimension
+            \begin{equation}
+                p_{f,x}=\dim E_{f,x}=\deg(\mu_{f,x})
+            \end{equation}
+            où \( \mu_{f,x}\) est le générateur unitaire de \( I_{f,x}\).
+        \item   \label{ItemKHNExH}
+            Le polynôme caractéristique de \( f|_{E_{f,x}}\) est \( \mu_{f,x}\).
+        \item   \label{ItemHMviZw}
+            Nous avons
+            \begin{equation}
+                \chi_{f|_{E_{f,x}}}(f)x=\mu_{f,x}(f)x=0.
+            \end{equation}
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Le fait que \( E_{f,x}\) soit stable par \( f\) est classique. Le point \ref{ItemHMviZw} est un une application du point \ref{ItemKHNExH}. Les deux gros morceaux sont donc les points \ref{ItemfzKOCo} et \ref{ItemKHNExH}.
+
+    Étant donné que \( \mu_{f,x}\) est de degré minimal dans \( I_{f,x}\), l'ensemble
+    \begin{equation}
+        B=\{ f^k(x)\tq 0\leq k\leq p_{f,x}-1 \}
+    \end{equation}
+    est libre. En effet une combinaison nulle des vecteurs de \( B\) donnerait un polynôme en \( f\) de degré inférieur à \( p_{f,x}\) annulant \( x\). Nous écrivons
+    \begin{equation}
+        \mu_{f,x}(X)=X^{p_{f,x}}-\sum_{i=0}^{p_{f,x}-1}a_iX^k. 
+    \end{equation}
+    Étant donné que \( \mu_{f,x}(f)x=0\) et que la somme du membre de droite est dans \( \Span(B)\), nous avons \( f^{p_{f,x}}(x)\in\Span(B)\). Nous prouvons par récurrence que \( f^{p_{f,x}+k}(x)\in\Span(B)\). En effet en appliquant \( f^k\) à l'égalité
+    \begin{equation}
+        0=f^{p_{f,x}}(x)-\sum_{i=0}^{p_{f,x}-1}a_if^i(x)
+    \end{equation}
+    nous trouvons
+    \begin{equation}
+        f^{p_{f,x}+k}(x)=\sum_{i=0}^{p_{f,x}-1}a_if^{i+k}(x),
+    \end{equation}
+    alors que par hypothèse de récurrence le membre de droite est dans \( \Span(B)\). L'ensemble \( B\) est alors générateur de \( E_{f,x}\) et donc une base d'icelui. Nous avons donc bien \( \dim(E_{f,x})=p_{f,x}\).
+
+    Nous montrons maintenant que \( \mu_{f,x}\) est annulateur de \( f\) au point \( x\). Nous savons que
+    \begin{equation}
+        \mu_{f,x}(f)x=0.
+    \end{equation}
+    En y appliquant \( f^k\) et en profitant de la commutativité des polynômes sur les endomorphismes (proposition \ref{LemQWvhYb}), nous avons
+    \begin{equation}
+        0=f^k\big( \mu_{f,x}(f)x \big)=\mu_{f,x}(f)f^k(x),
+    \end{equation}
+    de telle sorte que \( \mu_{f,x}(f)\) est nul sur \( B\) et donc est nul sur \( E_{f,x}\). Autrement dit,
+    \begin{equation}
+        \mu_{f,x}\big( f|_{E_{f,x}} \big)=0.
+    \end{equation}
+    Montrons que \( \mu_{f,x}\) est même minimal pour \( f|_{E_{f,x}}\). Sot \( Q\), un polynôme non nul de degré \( p_{f,x}-1\) annulant \( f|_{E_{f,x}}\). En particulier \( Q(f)x=0\), alors qu'une telle relation signifierait que \( B\) est un système lié, alors que nous avons montré que c'était un système libre. Nous concluons que \( \mu_{f,x}\) est le polynôme minimal de \( f|_{E_{f,x}}\).
+\end{proof}
+
+Cette histoire de densité permet de donner une démonstration alternative du théorème de Cayley-Hamilton.
+\begin{theorem}[Cayley-Hamlilton]   \label{ThoCalYWLbJQ}
+    Le polynôme caractéristique est un polynôme annulateur.
+\end{theorem}
+\index{théorème!Cayley-Hamilton}
+
+Une démonstration plus simple via la densité des diagonalisables est donnée en théorème \ref{ThoHZTooWDjTYI}.
+\begin{proof}
+    Nous devons prouver que \( \chi_f(f)x=0\) pour tout \( x\in E\). Pour cela nous nous fixons un \( x\in E\), nous considérons l'espace \( E_{f,x}\) et \( \chi_{f,x}\), le polynôme caractéristique de \( f|_{E_{f,x}}\). Étant donné que \( E_{f,x}\) est stable par \( f\), le polynôme caractéristique de \( f|_{E_{j,x}}\) divise \( \chi_f\), c'est à dire qu'il existe un polynôme \( Q_x\) tel que
+    \begin{equation}
+        \chi_f=Q_x\chi_{f,x},
+    \end{equation}
+    et donc aussi
+    \begin{equation}
+        \chi_f(f)x=Q_x(f)\big( \chi_{f,x}(f)x \big)=0
+    \end{equation}
+    parce que la proposition \ref{PropNrZGhT} nous indique que \( \chi_{f,x}\) est un polynôme annulateur de \( f|_{E_{f,x}}\).
+\end{proof}
+
+\begin{corollary}
+    Le degré du polynôme minimal est majoré par la dimension de l'espace.
+\end{corollary}
+
+\begin{proof}
+    Le polynôme minimal engendre l'idéal des polynôme annulateurs (proposition \ref{PropAnnncEcCxj}), et divise donc le polynôme caractéristique. Or le degré du polynôme caractéristique est la dimension de l'espace par le lemme \ref{LemooWCZMooZqyaHd}.
+\end{proof}
+
+\begin{example}[Calcul de l'inverse d'un endomorphisme]
+    Le polynôme de Cayley-Hamilton donne un moyen de calculer l'inverse d'un endomorphisme inversible pourvu que l'on sache son polynôme caractéristique. En effet, supposons que
+    \begin{equation}
+        \chi_f(X)=\sum_{k=0}^na_kX^k.
+    \end{equation}
+    Nous aurons alors
+    \begin{equation}
+        0=\chi_f(f)=\sum_{k=0}^na_kf^k.
+    \end{equation}
+    Nous appliquons \( f^{-1}\) à cette dernière égalité en sachant que \( f^{-1}(0)=0\) :
+    \begin{equation}
+        0=a_0f^{-1}+\sum_{k=1}^na_kf^{k-1},
+    \end{equation}
+    et donc
+    \begin{equation}
+        u^{-1}=-\frac{1}{ \det(f) }\sum_{k=1}^na_kf^{k-1}
+    \end{equation}
+    où nous avons utilisé le fait que \( a_0=\chi_f(0)=\det(f)\).
+\end{example}
+
+\begin{proposition}\label{PropooBYZCooBmYLSc}
+    Si \( (X-z)^l\) (\( l\geq 1\)) est la plus grande puissance de \( (X-z)\) dans le polynôme caractéristique d'un endomorphisme \( u\) alors 
+    \begin{equation}
+        1\leq \dim(E_e)\leq l.
+    \end{equation}
+    C'est à dire que nous avons au moins un vecteur propre pour chaque racine du polynôme caractéristique.
+\end{proposition}
+
+\begin{proof}
+    Si $(X-z)$ divise \( \chi_u\) alors en posant \( \chi_u=(X-z)P(X)\) nous avons
+    \begin{equation}
+        \det(u-X\mtu)=(X-z)P(X),
+    \end{equation}
+    ce qui, évalué en \( X=z\), donne \( \det(u-z\mtu)=0\). L'annulation du déterminant étant équivalente à l'existence d'un noyau non trivial, nous avons \( v\neq 0\) dans \( E\) tel que \( (u-z\mtu)v=0\). Cela donne \( u(v)=zv\) et donc que \( v\) est vecteur propre de \( u\) pour la valeur propre \( z\). Donc aussi \( \dim(E_z)\geq 1\).
+
+    Si \( \dim(E_z)=k\) alors le théorème de la base incomplète \ref{ThonmnWKs} nous permet d'écrire une base de \( E\) dont les \( k\) premiers vecteurs forment une base de \( E_z\). Dans cette base, la matrice de \( u\) est de la forme
+    \begin{equation}
+        \begin{pmatrix}
+             z   &       &       &   *    \\
+                &   \ddots    &       &   \vdots    \\
+                &       &   z    &   *    \\ 
+                &       &       &   *     
+         \end{pmatrix}
+    \end{equation}
+    où les étoiles représentent des blocs a priori non nuls. En tout cas il est vu sous cette forme que \( (X-z\mtu)^k\) divise \( \chi_u\).
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Diagonalisation et trigonalisation}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Ici encore \( \eK\) est un corps commutatif.
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Matrices semblables}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}[matrices semblables] \label{DefCQNFooSDhDpB}
+    Sur l'ensemble \( \eM_n(\eK)\) des matrices \( n\times n\) à coefficients dans \(\eK\) nous introduisons la relation d'équivalence \( A\sim B\) si et seulement s'il existe une matrice \( P\in\GL(n,\eK)\) telle que \( B=P^{-1}AP\). Deux matrices équivalentes en ce sens sont dites \defe{semblables}{semblables!matrices}.
+\end{definition}
+
+Le polynôme caractéristique est un invariant sous les similitudes. En effet si \( P\) est une matrice inversible,
+\begin{subequations}
+    \begin{align}
+        \chi_{PAP^{-1}}&=\det(PAP^{-1}-\lambda X)\\
+        &=\det\big( P^{-1}(PAP^{-1}-\lambda X)P^{-1} \big)\\
+        &=\det(A-\lambda X).
+    \end{align}
+\end{subequations}
+
+La permutation de lignes ou de colonnes ne sont pas de similitudes, comme le montrent les exemples suivants :
+\begin{equation}
+    \begin{aligned}[]
+        A&=\begin{pmatrix}
+            1    &   2    \\ 
+            3    &   4    
+        \end{pmatrix}&
+        B&=\begin{pmatrix}
+            2    &   1    \\ 
+            4    &   3    
+        \end{pmatrix}.
+    \end{aligned}
+\end{equation}
+Nous avons \( \chi_A=x^2-5x-2\) tandis que \( \chi_B=x^2-5x+2\) alors que le polynôme caractéristique est un invariant de similitude.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Endomorphismes nilpotents}
+%---------------------------------------------------------------------------------------------------------------------------
+
+La \defe{trace}{trace!matrice} d'une matrice \( A\in \eM(n,\eK)\) est la somme de ses éléments diagonaux :
+\begin{equation}
+    \tr(A)=\sum_{i=1}^nA_{ii}.
+\end{equation}
+Une propriété importante est son invariance cyclique.
+\begin{lemma}   \label{LemhbZTay}
+    Si \( A\) et \( B\) sont des matrices carré, alors \( \tr(AB)=\tr(BA)\).
+
+    La trace est un invariant de similitude.
+\end{lemma}
+
+\begin{proof}
+    C'est un simple calcul :
+    \begin{equation}
+            \tr(AB)=\sum_{ik}A_{ik}B_{ki}
+            =\sum_{ik}A_{ki}B_{ik} 
+            =\sum_{ik}B_{ik}A_{ki}
+            =\sum_i(BA)_{ii}
+            =\tr(BA)
+    \end{equation}
+    où nous avons simplement renommé les indices \( i\leftrightarrow k\).
+
+    En particulier, la trace est un invariant de similitude parce que \( \tr(ABA^{-1})=\tr(A^{-1} AB)=\tr(B)\). 
+\end{proof}
+La trace étant un invariant de similitude, nous pouvons donc définir la \defe{trace}{trace!endomorphisme} comme étant la trace de sa matrice dans une base quelconque. Si la matrice est diagonalisable, alors la trace est la somme des valeurs propres.
+
+\begin{lemma}[\cite{fJhCTE}]   \label{LemzgNOjY}
+    L'endomorphisme \( u\in\End(\eC^n)\) est nilpotent si et seulement si \( \tr(u^p)=0\) pour tout \( p\).
+\end{lemma}
+
+\begin{proof}
+    Supposons que \( u\) est nilpotent. Alors ses valeurs propres sont toutes nulles et celles de \( u^p\) le sont également. La trace étant la somme des valeurs propres, nous avons alors tout de suite \( \tr(u^p)=0\).
+
+    Supposons maintenant que \( \tr(u^p)=0\) pour tout \( p\). Le polynôme caractéristique \eqref{Eqkxbdfu} est
+    \begin{equation}    \label{EqfnCqWq}
+        \chi_u=(-1)^nX^{\alpha}(X-\lambda_1)^{\alpha_1}\ldots (X-\alpha_r)^{\alpha_r}.
+    \end{equation}
+    où les \( \lambda_i\) (\( i=1,\ldots, r\)) sont les valeurs propres non nulles distinctes de \( u\).
+
+    Il est vite vu que le coefficient de \( X^{n-1}\) dans \( \chi_u\) est \( -\tr(u)\) parce que le coefficient de \( X^{n-1}\) se calcule en prenant tous les $X$ sauf une fois \( -\lambda_i\). D'autre part le polynôme caractéristique de \( u^p \) est le même que celui de \( u\), en remplaçant \( \lambda_i\) par \( \lambda_i^p\); cela est dû au fait que si \( v\) est vecteur propre de valeur propre \( \lambda\), alors \( u^pv=\lambda^pv\).
+
+    Par l'équation \eqref{EqfnCqWq}, nous voyons que le coefficient du terme \( X^{n-1}\) dans les polynôme caractéristique est 
+    \begin{equation}        \label{eqSoDSKH}
+        0=\tr(u^p)=\alpha_1\lambda_1^p+\cdots +\alpha_r\lambda_r^p.
+    \end{equation}
+    Donc les nombres \( (\alpha_1,\ldots, \alpha_r)\) est une solution non triviale\footnote{Si \( \alpha_1=\ldots=\alpha_r=0\), alors les valeurs propres sont toutes nulles et la matrice est en réalité nulle dès le départ.} du système
+    \begin{subequations}    \label{EqDpvTnu}
+        \begin{numcases}{}
+            \alpha_1X_1+\cdots +\lambda_rX_r=0\\
+            \qquad\vdots\\
+            \lambda^r_1X_1+\cdots +\lambda_r^rX_r=0.
+        \end{numcases}
+    \end{subequations}
+    Cela sont les équations \eqref{eqSoDSKH} écrites avec \( p=1,\ldots, r\). Le déterminant de ce système est
+    \begin{equation}
+        \lambda_1\ldots\lambda_r\det\begin{pmatrix}
+             1   &   \ldots    &   1    \\
+             \lambda_1   &   \ldots    &   \lambda_1    \\
+             \vdots   &       &   \vdots    \\ 
+             \lambda_1^{r-1}   &   \ldots    &   \lambda_r^{r-1}
+         \end{pmatrix}\neq 0,
+    \end{equation}
+    qui est un déterminant de Vandermonde (proposition \ref{PropnuUvtj}) valant
+    \begin{equation}
+        0=\lambda_1\ldots\lambda_r\prod_{1\leq i\leq j\leq r}(\lambda_i-\lambda_j).
+    \end{equation}
+    Étant donné que les \( \lambda_i\) sont distincts et non nuls, nous avons une contradiction et nous devons conclure que \( (\alpha_1,\ldots, \alpha_r)\) était une solution triviale du système \eqref{EqDpvTnu}.
+\end{proof}
+
+\begin{proposition}[\cite{SVSFooIOYShq}]    \label{PropMWWJooVIXdJp}
+    Soit un \( \eK\)-espace vectoriel \( E\). Un endomorphisme \( u\in\End(E)\) est nilpotent si et seulement s'il existe une base de \( E\) dans laquelle la matrice de \( u\) est strictement triangulaire supérieure.
+\end{proposition}
+
+\begin{proof}
+    \begin{subproof}
+       \item[\( \Rightarrow\)]
+           Nous faisons la démonstration par récurrence sur la dimension de \( E\). Lorsque \( n=1\) nous avons \( u=(a)\) avec \( a\in \eK\). Vu que \( a^k=0\) pour un certain \( k\) nous avons \( a=0\) parce qu'un corps est toujours un anneau intègre\footnote{Lemme \ref{LemAnnCorpsnonInterdivzer}.}. 
+
+           Lorsque \( \dim(E)=n\) nous savons que \( u\) a un noyau non réduit au vecteur nul (parce qu'il est nilpotent). Soit donc un vecteur non nul \( x\in\ker(u)\) et une base
+           \begin{equation}
+               \{ x,e_2,\ldots, e_n \}
+           \end{equation}
+           donnée par le théorème de la base incomplète \ref{ThonmnWKs}. La matrice de \( u\) dans cette base s'écrit
+           \begin{equation}
+               \begin{pmatrix}
+                       \begin{array}[]{c|c}
+                           0&\begin{matrix} 
+                               * &   *    &   *    
+                           \end{matrix}\\
+                           \hline
+                           \begin{matrix}
+                               0 \\ 
+                               0 \\ 
+                               0 
+                           \end{matrix}&
+                           \begin{pmatrix}
+                                &       &       \\
+                                &   A    &       \\
+                                &       &   
+                           \end{pmatrix}
+                       \end{array}
+               \end{pmatrix}.
+           \end{equation}
+           Un tout petit peu de calcul de produit de matrice montre que la matrice de \( u^k\) est de la forme
+           \begin{equation}
+               \begin{pmatrix}
+                       \begin{array}[]{c|c}
+                           0&\begin{matrix} 
+                               * &   *    &   *    
+                           \end{matrix}\\
+                           \hline
+                           \begin{matrix}
+                               0 \\ 
+                               0 \\ 
+                               0 
+                           \end{matrix}&
+                           \begin{pmatrix}
+                                &       &       \\
+                                &   A^k    &       \\
+                                &       &   
+                           \end{pmatrix}
+                       \end{array}
+               \end{pmatrix}.
+           \end{equation}
+           Étant donné que \( u\) est nilpotente, la matrice \( A\) l'est aussi. L'hypothèse de récurrence dit alors que \( A\) est strictement triangulaire supérieure (ou en tout cas peut le devenir par un changement de base adéquat).
+
+       \item[\( \Leftarrow\)]
+
+            Lorsqu'une matrice est triangulaire supérieure stricte, elle applique
+            \begin{equation}
+                \Span\{ e_1,\ldots, e_k \}\to\Span\{ e_1,\ldots, e_{k-1} \}.
+            \end{equation}
+            Donc tout vecteur finit sur zéro si on lui applique \( u\) assez souvent.
+    \end{subproof}
+\end{proof}
+
+\begin{proposition}[Thème \ref{THEMEooPQKDooTAVKFH}]     \label{PROPooWTFWooXHlmhp}
+    Soit \( E\) un espace de Banach (espace vectoriel normé complet). Si \( A\in\aL(E,E)\) est nilpotente, alors \( (\mtu-A)\) est inversible et son inverse est donné par
+    \begin{equation}
+        (\mtu-A)^{-1}=\sum_{k=0}^{\infty}A^k,
+    \end{equation}
+    où l'infini peut évidemment être remplacé par l'ordre de nilpotence de \( A\).
+\end{proposition}
+
+\begin{proof}
+    En ce qui concerne la convergence de la somme, elle ne fait pas de doutes parce que \( A\) étant nilpotente, la somme contient seulement une quantité finie de termes non nuls.
+
+    Montrons à présent que la somme est l'inverse de \( \mtu-A\) en multipliant terme à terme :
+    \begin{equation}
+        \sum_{k=0}^nA^k(\mtu-A)=\sum_{k=0}^n(A^k-A^{k+1})=\mtu-A^{n+1}.
+    \end{equation}
+    Par conséquent 
+    \begin{equation}
+        \| \mtu-\sum_{k=0}^nA^k(\mtu-A) \|=\| A^{n+1} \|\to 0.
+    \end{equation}
+    La dernière limite est en réalité une égalité pour \( n\) assez grand.
+\end{proof}
+
+\begin{proposition}
+    Soit \( A\in\GL(n,\eC)\). La suite \( (A^k)_{k\in \eZ}\) est bornée si et seulement si \( A\) est diagonalisable et \( \Spec(A)\subset \gS^1\).
+\end{proposition}
+
+\begin{proof}
+    Si \( A\) est diagonalisable avec les valeurs propres \( \lambda_i\) de norme \( 1\) dans \( \eC\), alors \( A^k\) est la matrice diagonale avec les \( \lambda_i^k\) sur la diagonale. Cela reste borné pour toute valeur entière de \( k\).
+
+    En ce qui concerne l'autre sens, nous supposons encore que
+    \begin{equation}
+        A=\begin{pmatrix}
+            \lambda_1\mtu+N_1    &       &       \\
+                &   \ddots    &       \\
+                &       &   \lambda_s\mtu+N_s
+        \end{pmatrix},
+    \end{equation}
+    et nous regardons un des blocs. Nous voulons prouver que \( N=0\) et que \( | \lambda |=1\).
+    
+    Nous commençons par regarder ce qu'implique le fait que \( (\lambda \mtu+N)^n\) reste borné pour \( n>0\). En notant \( r\) l'ordre de nilpotence de \( N\), nous avons le développement
+    \begin{equation}
+        (\lambda\mtu+N)^n=\sum_{k=0}^{r-1}\binom{ n }{ k }N^k\lambda^{n-k}.
+    \end{equation}
+    Par la proposition \ref{PropMWWJooVIXdJp}, une matrice nilpotente s'écrit dans une base sous la forme
+    \begin{equation}
+        N=\begin{pmatrix}
+             0   &   1    &       &       \\
+                &   0    &   1    &       \\
+                & &   \ddots   &   \ddots    &      \\ 
+                &&       &   0    &   1     \\
+                &&       &      &   0     
+         \end{pmatrix}
+    \end{equation}
+    et effectuer \( A^k\) revient à décaler la diagonale de \( 1\). Donc la famille
+    \begin{equation}
+        \{ \mtu,N,\ldots, N^{r-1} \}
+    \end{equation}
+    est libre. Par conséquent la suite \( (\lambda\mtu+N)^n\) restera bornée si et seulement si chacun des termes 
+    \begin{equation}    \label{EqXRDVDCM}
+        \binom{ n }{ k }N^k\lambda^{n-k}
+    \end{equation}
+    reste borné. Le premier terme étant \( \lambda^n\mtu\), nous avons obligatoirement \( | \lambda |\leq 1\). Si \( | \lambda |<1\), alors le coefficient \( \binom{ n }{ k }\lambda^{n-k}\) tend vers zéro. Si \( | \lambda |=1\) par contre ce coefficient tend vers l'infini et la seule façon pour que \eqref{EqXRDVDCM} reste borné est que \( N=0\). Nous avons donc deux possibilités :
+    \begin{itemize}
+        \item \( | \lambda |<1\)
+        \item \( | \lambda |=1\) et \( N=0\).
+    \end{itemize}
+
+    Nous nous tournons maintenant sur la contrainte que \( (\lambda\mtu+N)^n\) doive rester borné pour \( n<0\). Nous avons
+    \begin{equation}
+        \lambda\mtu+N=\lambda(\mtu+\lambda^{-1}N),
+    \end{equation}
+    et nous pouvons appliquer la proposition \ref{PROPooWTFWooXHlmhp} à l'opérateur nilpotent \( -\lambda^{-1} N\) pour avoir
+    \begin{equation}
+        (\mtu+\lambda^{-1}N)^{-1}=\mtu+\sum_{k=1}^{\infty}(-\lambda)^{-1}N^k.
+    \end{equation}
+    Ceci pour dire que \( (\lambda\mtu+N)^{-1}=\lambda^{-1}(\mtu+\lambda^{-1}N')\) pour une autre matrice nilpotente \( N'\). Le travail déjà fait, appliqué à \( \lambda^{-1}\) et \( N'\), nous donne deux possibilités :
+    \begin{itemize}
+        \item \( | \lambda^{-1} |<1\)
+        \item \( | \lambda^{-1} |=1\) et \( N'=0\).
+    \end{itemize}
+    La possibilité \( | \lambda^{-1} |<1\) est exclue parce qu'elle impliquerait \( | \lambda |>1\) qui avait déjà été exclu. Il ne reste donc que la possibilité \( | \lambda |=1\) et \( N=N'=0\).
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Endomorphismes diagonalisables}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}  \label{DefCNJqsmo}
+    Une matrice est \defe{diagonalisable}{diagonalisable} si elle est semblable à une matrice diagonale.
+\end{definition}
+
+\begin{lemma}
+    Une matrice triangulaire supérieure avec des \( 1\) sur la diagonale n'est diagonalisable que si elle est diagonale (c'est à dire si elle est la matrice unité).
+\end{lemma}
+
+\begin{proof}
+    Si \( A\) est une matrice triangulaire supérieure de taille \( n\) telle que \( A_{ii}=1\), alors \( \det(A-\lambda\mtu)=(1-\lambda)^n\), ce qui signifie que \( \Spec(A)=\{ 1 \}\). Pour la diagonaliser, il faudrait une matrice \( P\in\GL(n,\eK)\) telle que \( \mtu=P^{-1}AP\), ce qui est uniquement possible si \( A=\mtu\).
+\end{proof}
+
+\begin{lemma}       \label{LemgnaEOk}
+    Soit \( F\) un sous-espace stable par \( u\). Soit une décomposition du polynôme minimal
+    \begin{equation}
+        \mu_u=P_1^{n_1}\ldots P_r^{n_r}
+    \end{equation}
+    où les \( P_i\) sont des polynômes irréductibles unitaires distincts. Si nous posons \( E_i=\ker P_i^{n_i}\), alors
+    \begin{equation}
+        F=(F\cap E_1)\oplus\ldots \oplus(F\cap E_r).
+    \end{equation}
+\end{lemma}
+
+\begin{theorem}     \label{ThoDigLEQEXR}
+    Soit \( E\), un espace vectoriel de dimension \( n\) sur le corps commutatif \( \eK\) et \( u\in\End(E)\). Les propriétés suivantes sont équivalentes.
+    \begin{enumerate}
+        \item\label{ItemThoDigLEQEXRiv}
+            L'endomorphisme \( u\) est diagonalisable.
+        \item       \label{ItemThoDigLEQEXRi}
+            Il existe un polynôme \( P\in\eK[X]\) non constant, scindé sur \(\eK\) dont toutes les racines sont simples tel que \( P(u)=0\).
+        \item\label{ItemThoDigLEQEXRii}
+            Le polynôme minimal \( \mu_u\) est scindé sur \(\eK\) et toutes ses racines sont simples\footnote{Le polynôme \emph{caractéristique}, lui, n'a pas spécialement ses racines simples; il peut encore être de la forme
+            \begin{equation}
+                \chi_u(X)=\prod_{i=1}^r(X-\lambda_i)^{\alpha_i},
+        \end{equation}
+        mais alors \( \dim(E_{\lambda_i})=\alpha_i\). }.
+        \item\label{ItemThoDigLEQEXRiii}
+            Tout sous-espace de \( E\) possède un supplémentaire stable par \( u\).
+        \item       \label{ITEMooZNJFooEiqDYp}
+            Dans une base adaptée, la matrice de \( u\) est diagonale et les éléments diagonaux sont ses valeurs propres.
+    \end{enumerate}
+\end{theorem}
+\index{diagonalisable!et polynôme minimum scindé}
+
+\begin{proof}
+    Plein d'implications à prouver.
+    \begin{subproof}
+    \item[\ref{ItemThoDigLEQEXRi} implique \ref{ItemThoDigLEQEXRii}] Étant donné que \( P(u)=0\), il est dans l'idéal des polynôme annulateurs de \( u\), et le polynôme minimal \( \mu_u\) le divise parce que l'idéal des polynôme annulateurs est généré par \( \mu_u\) par le théorème \ref{ThoCCHkoU}.
+
+    \item[\ref{ItemThoDigLEQEXRii} implique \ref{ItemThoDigLEQEXRiv}] Étant donné que le polynôme minimal est scindé à racines simples, il s'écrit sous forme de produits de monômes tous distincts, c'est à dire
+    \begin{equation}
+        \mu_u(X)=(X-\lambda_1)\ldots(X-\lambda_r)
+    \end{equation}
+    où les \( \lambda_i\) sont des éléments distincts de \( \eK\). Étant donné que \( \mu_u(u)=0\), le théorème de décomposition des noyaux (théorème \ref{ThoDecompNoyayzzMWod}) nous enseigne que
+    \begin{equation}
+        E=\ker(u-\lambda_1)\oplus\ldots\oplus\ker(u-\lambda_r).
+    \end{equation}
+    Mais \( \ker(u-\lambda_i)\) est l'espace propre \( E_{\lambda_i}(u)\). Donc \( u\) est diagonalisable.
+
+\item[\ref{ItemThoDigLEQEXRiv} implique \ref{ItemThoDigLEQEXRiii}] Soit \( \{ e_1,\ldots, e_n \}\) une base qui diagonalise \( u\), soit \( F\) un sous-espace de \( E\) un \( \{ f_1,\ldots, f_r \}\) une base de \( F\). Par le théorème \ref{ThoBaseIncompjblieG} (qui généralise le théorème de la base incomplète), nous pouvons compléter la base de \( F\) par des éléments de la base \( \{ e_i \}\). Le complément ainsi construit est invariant par \( u\).
+
+\item[\ref{ItemThoDigLEQEXRiii} implique \ref{ItemThoDigLEQEXRiv}] En dimension un, tout endomorphisme est diagonalisable, nous supposons donc que \( \dim E=n\geq 2\). Nous procédons par récurrence sur le nombre de vecteurs propres connus de \( u\). Supposons avoir déjà trouvé \( p\) vecteurs propres \( e_1,\ldots, e_p\) de \( u\). Considérons \( H\), un hyperplan qui contient les vecteurs \( e_1,\ldots, e_p\). Soit \( F\) un supplémentaire de \( H\) stable par \( u\); par construction \( \dim F=1\) et si \( e_{p+1}\in F\), il doit être vecteur propre de \( u\).
+
+\item[\ref{ItemThoDigLEQEXRiv} implique \ref{ItemThoDigLEQEXRi}] Nous supposons maintenant que \( u\) est diagonalisable. Soient \( \lambda_1,\ldots, \lambda_r\) les valeurs propres deux à deux distinctes, et considérons le polynôme
+    \begin{equation}
+        P(x)=(X-\lambda_1)\ldots (X-\lambda_r).
+    \end{equation}
+    Alors \( P(u)=0\). En effet si \( e_i\) est un vecteur propre pour la valeur propre \( \lambda_i\), 
+    \begin{equation}
+        P(u)e_i=\prod_{j\neq i}(u-\lambda_j)\circ(u-\lambda_i)e_i=0
+    \end{equation}
+    par le lemme \ref{LemQWvhYb}. Par conséquent \( P(u)\) s'annule sur une base.
+
+\item[\ref{ITEMooZNJFooEiqDYp} implique \ref{ItemThoDigLEQEXRi}]
+    Si la matrice \( A\) est diagonale alors le polynôme \( P=\prod_{i=1}^n(A-A_{ii}\mtu)\) est annulateur de \( A\).
+        \item[\ref{ItemThoDigLEQEXRii} implique \ref{ITEMooZNJFooEiqDYp}]
+            le polynôme minimal de \( u\) s'écrit 
+            \begin{equation}
+                \mu=(X-\lambda_1)\ldots(X-\lambda_r),
+            \end{equation}
+            et les espaces $E_i$ du lemme \ref{LemgnaEOk} sont les espaces propres \( E_i=\ker(u-\lambda_i)\). Nous avons donc une somme directe
+            \begin{equation}
+                E=E_1\oplus\ldots\oplus E_r.
+            \end{equation}
+            Dans chacun des espaces propres, $u$ a une matrice diagonale avec la valeur propre correspondante sur la diagonale. Une base de \( E\) constituée d'une base de chacun des espaces propres est donc une base comme nous en cherchons.
+    \end{subproof}
+\end{proof}
+
+\begin{corollary}       \label{CorQeVqsS}
+    Si \( u\) est diagonalisable et si \( F\) est une sous-espace stable par \( u\), alors
+    \begin{equation}
+        F=\bigoplus_{\lambda}E_{\lambda}(u)\cap F
+    \end{equation}
+    où \( E_{\lambda}(u)\) est l'espace propre de \( u\) pour la valeur propre \( \lambda\). En particulier la restriction de \( u\) à \( F\), \( u|_F\) est diagonalisable.
+\end{corollary}
+
+\begin{proof}
+    Par le théorème \ref{ThoDigLEQEXR}, le polynôme \( \mu_u\) est scindé et ne possède que des racines simples. Notons le
+    \begin{equation}
+        \mu_u(X)=(X-\lambda_1)\ldots (X-\lambda_r).
+    \end{equation}
+    Les espaces \( E_i\) du lemme \ref{LemgnaEOk} sont maintenant les espaces propres.
+
+    En ce qui concerne la diagonalisabilité de \( u|_F\), notons que nous avons une base de \( F\) composée de vecteurs dans les espaces \( E_{\lambda}(u)\). Cette base de \( F\) est une base de vecteurs propres de \( u\).
+\end{proof}
+
+\begin{lemma}
+    Soit \( E\) un \( \eK\)-espace vectoriel et \( u\in\End(E)\). Si \( \Card\big( \Spec(u) \big)=\dim(E)\) alors \( u\) est diagonalisable.
+\end{lemma}
+
+\begin{proof}
+    Soient \( \lambda_1,\ldots, \lambda_n\) les valeurs propres distinctes de \( u\). Nous savons que les espaces propres correspondants sont en somme directe (lemme \ref{LemjcztYH}). Par conséquent \( \Span\{ E_{\lambda_i}(u) \}\) est de dimension \( n\) est \( u\) est diagonalisable.
+\end{proof}
+
+Voici un résultat de diagonalisation simultanée. Nous donnerons un résultat de trigonalisation simultanée dans le lemme \ref{LemSLGPooIghEPI}.
+\begin{proposition}[Diagonalisation simultanée]     \label{PropGqhAMei}
+    Soit \( (u_i)_{i\in I}\) une famille d'endomorphismes qui commutent deux à deux.
+    \begin{enumerate}
+        \item       \label{ItemGqhAMei}
+            Si \( i,j\in I\) alors tout sous-espace propre de \( u_i\) est stable par \( u_j\). Autrement dit \( u_j\big(E_{\lambda}(u)\big)\subset E_{\lambda}(u)\).
+        \item
+            Si les \( u_i\) sont diagonalisables, alors ils le sont simultanément.
+    \end{enumerate}
+\end{proposition}
+\index{diagonalisation!simultanée}
+
+\begin{proof}
+    Supposons que \( u_i\) et \( u_j\) commutent et soit \( x\) un vecteur propre de \( u_i\) : \( u_ix=\lambda x\). Nous montrons que \( u_jx\in E_{\lambda}(u)\). Nous avons
+    \begin{equation}
+        u_i\big( u_j(x) \big)=u_j\big( u_i(x) \big)=\lambda u_j(x).
+    \end{equation}
+    Par conséquent \( u_j(x)\) est vecteur propre de \( u_i\) de valeur propre \( \lambda\).
+
+    Montrons maintenant l'affirmation à propos des endomorphismes simultanément diagonalisables. Si \( \dim E=1\), le résultat est évident. Nous supposons également qu'aucun des \( u_i\) n'est multiple de l'identité. Nous effectuons une récurrence sur la dimension.
+
+    Soit \( u_0\) un des \( u_i\) et considérons ses valeurs propres deux à deux distinctes \( \lambda_1,\ldots, \lambda_r\). Pour chaque \( k\) nous avons
+    \begin{equation}
+        E_{\lambda_k}(u_0)\neq E,
+    \end{equation}
+    sinon \( u_0\) serait un multiple de l'identité. Par contre le fait que \( u_0\) soit diagonalisable permet de décomposer \( E\) en espaces propres de \( u_0\) :
+    \begin{equation}
+        E=\bigoplus_{k}E_{\lambda_k}(u_0).
+    \end{equation}
+    Ce que nous allons faire est de simultanément diagonaliser les \( (u_i)_{i\in I}\) sur chacun des \( E_{\lambda_k}\) séparément. Par le point \ref{ItemGqhAMei}, nous avons \( u_i\colon E_{\lambda_k}(u_0)\to E_{\lambda_k}(u_0)\), et nous pouvons considérer la famille d'opérateurs
+    \begin{equation}
+        \left( u_i|_{E_{\lambda_k}(u_0)} \right)_{i\in I}.
+    \end{equation}
+    Ce sont tous des opérateurs qui commutent et qui agissent sur un espace de dimension plus petite. Par hypothèse de récurrence nous avons une base de \( E_{\lambda_k}(u_0)\) qui diagonalise tous les \( u_i\).
+\end{proof}
+
+\begin{example}     \label{ExewINgYo}
+    Soit un espace vectoriel sur un corps \( \eK\). Un opérateur \defe{involutif}{involution} est un opérateur différent de l'identité dont le carré est l'identité. Typiquement une symétrie orthogonale dans \( \eR^3\). Le polynôme caractéristique d'une involution est \( X^2-1=(X+1)(X-1)\).
+    
+    Tant que \( 1\neq -1\), \( X^1-1\) est donc scindé à racines simples et les involutions sont diagonalisables (\ref{ThoDigLEQEXR}). Cependant si le corps est de caractéristique \( 2\), alors \( X^2-1=(X+1)^2\) et l'involution n'est plus diagonalisable.
+
+    Par exemple si le corps est de caractéristique \( 2\), nous avons
+    \begin{subequations}
+        \begin{align}
+            A&=\begin{pmatrix}
+                1    &   1    \\ 
+                0    &   1    
+            \end{pmatrix}\\
+            A^1&=\begin{pmatrix}
+                1    &   2    \\ 
+                0    &   1    
+            \end{pmatrix}=\begin{pmatrix}
+                1    &   0    \\ 
+                0    &   1    
+            \end{pmatrix}.
+        \end{align}
+    \end{subequations}
+    Ce \( A\) est donc une involution mais n'est pas diagonalisable.
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Diagonalisation : cas complexe, pas toujours}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Il n'est pas vrai qu'une matrice de \( \eM(n,\eC)\) soit toujours diagonalisable. En effet le théorème \ref{ThoDigLEQEXR}\ref{ItemThoDigLEQEXRii} dit qu'une matrice est diagonalisable si et seulement si son polynôme minimal est scindé à racines simples. Certes sur \( \eC\) le polynôme minimal sera scindé, mais il ne sera pas spécialement à racines simples.
+
+\begin{example}
+    La matrice
+    \begin{equation}
+        A=\begin{pmatrix}
+            0    &   1    \\ 
+            0    &   0    
+        \end{pmatrix}
+    \end{equation}
+    a pour polynôme caractéristique \( \chi_A(X)=X^2\). Cela est également son polynôme minimal, et ce n'est pas à racine simple. 
+
+    Il est par ailleurs facile de voir que le seul espace propre de \( A\) est \( \Span\{ (1,0) \}\) (ici le span est sur \( \eC\)). Donc l'espace \( \eC^2\) ne possède pas de base de vecteurs propres de \( A\).
+\end{example}
+
+Ce qui est vrai, c'est que le polynôme caractéristique a des racines, et que ces racines correspondent à des vecteurs propres. Mais il n'y a pas toujours autant de vecteurs propres que la multiplicité des racines.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Trigonalisation : généralités}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}[\cite{MQMKooPBfnZN}]
+    Une matrice dans \( \eM(n,\eK)\) est \defe{trigonalisable}{matrice!trigonalisable} lorsqu'elle est semblable\footnote{Définition \ref{DefCQNFooSDhDpB}.} à une matrice triangulaire supérieure.
+\end{definition}
+
+\begin{proposition}[Trigonalisation et polynôme caractéristique scindé] \label{PropKNVFooQflQsJ}
+    Soit \( u\) un endomorphisme d'un espace vectoriel \( E\) sur le corps \( \eK\). Les faits suivants sont équivalents.
+    \begin{enumerate}
+        \item   \label{ItemZKDMooOrTHkwi}
+            L'endomorphisme \( u\) est trigonalisable (auquel cas les valeurs propres sont sur la diagonale).
+        \item   \label{ItemZKDMooOrTHkwii}
+            Le polynôme caractéristique de \( u\) est scindé\footnote{Définition \ref{DefCPLSooQaHJKQ}.}.
+    \end{enumerate}
+\end{proposition}
+\index{trigonalisation!et polynôme caractéristique}
+
+\begin{proof}
+    \begin{subproof}
+        \item[\ref{ItemZKDMooOrTHkwii}\( \Rightarrow\)\ref{ItemZKDMooOrTHkwi}]
+            Nous avons par hypothèse que
+            \begin{equation}
+                \chi_u(X)=\prod_{i=1}^r(X-\lambda_i)^{\alpha_i}
+            \end{equation}
+            où les \( \lambda_i\) sont les valeurs propres de \( u\). Le théorème de Cayley-Hamilton \ref{ThoCalYWLbJQ} dit que \( \chi_u(u)=0\), ce qui permet d'utiliser le théorème de décomposition des noyaux \ref{ThoDecompNoyayzzMWod} :
+            \begin{equation}
+                E=\ker(X-\lambda_1)^{\alpha_1}\oplus\ldots\oplus\ker(X-\lambda_r)^{\alpha_r}.
+            \end{equation}
+            Les espaces \( F_{\lambda_i}(u)=\ker(X-\lambda_i)^{\alpha_i}\) sont les espaces caractéristiques de \( u\), ce qui fait que \( u-\lambda_i\mtu\) est nilpotent sur \( F_{\lambda_i}(u)\). L'endomorphisme \( u-\lambda_i\mtu\) est donc strictement trigonalisable supérieur sur son bloc\footnote{Proposition \ref{PropMWWJooVIXdJp}.}. Cela signifie que \( u\) est triangulaire supérieure avec les valeurs propres sur la diagonale.
+
+        \item[\ref{ItemZKDMooOrTHkwi}\( \Rightarrow\)\ref{ItemZKDMooOrTHkwii}]
+
+            C'est immédiat parce que le déterminant d'une matrice triangulaire est le produit des éléments de sa diagonale.
+    \end{subproof}
+\end{proof}
+
+\begin{remark}
+    La méthode des pivots de Gauss\footnote{Le lemme \ref{LemZMxxnfM}.} certes permet de trigonaliser n'importe quoi, mais elle ne correspond pas à un changement de base. Autrement dit, les pivots de Gauss ne sont pas de similitudes.
+
+    C'est là qu'il faut bien avoir en tête la différence entre \emph{équivalence} et \emph{similarité} (définition \ref{DefBLELooTvlHoB}). Lorsqu'on parle de changement de base, de matrice trigonalisable ou diagonalisable, nous parlons de similarité et non d'équivalence.
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Trigonalisation : cas complexe}
+%---------------------------------------------------------------------------------------------------------------------------
+
+La proposition \ref{PropKNVFooQflQsJ} dit déjà que tous les endomorphismes sont trigonalisables sur \( \eC\). Nous allons aller plus loin et montrer que la trigonalisation peut être effectuée à l'aide d'une matrice unitaire. 
+
+Une démonstration alternative passant par le polynôme caractéristique sera présentée dans la remarque \ref{RemXFZTooXkGzQg} utilisant la proposition \ref{PropKNVFooQflQsJ}.
+\begin{lemma}[Lemme de Schur complexe, trigonisation\cite{NormHKNPKRqV}]  \label{LemSchurComplHAftTq}
+    Si \( A\in\eM(n,\eC)\), il existe une matrice unitaire \( U\) telle que \( UAU^{-1}\) soit triangulaire supérieure.
+\end{lemma}
+\index{lemme!Schur complexe}
+%TODO : Le lemme de Schur est souvent énoncé en disant que si p est une représentation irréductible, alors les seuls endomorphismes de V commutant avec tous les p(g) sont les multiples de l'idenditié. Quel est le lien avec ceci ?
+
+\begin{proof}
+    Étant donné que \( \eC\) est algébriquement clos, nous pouvons toujours considérer un vecteur propre \( v_1\) de \( A\), de valeur propre \( \lambda_1\). Nous pouvons utiliser un procédé de Gram-Schmidt pour construire une base orthonormée \( \{ v,u_2,\ldots, u_n \}\) de \( \eR^n\), et la matrice (unitaire)
+    \begin{equation}
+        Q=\begin{pmatrix}
+             \uparrow   &   \uparrow    &       &   \uparrow    \\
+             v   &   u_2    &   \cdots    &   u_n    \\ 
+             \downarrow   &   \downarrow    &       &   \downarrow
+         \end{pmatrix}.
+    \end{equation}
+    Nous avons \( Q^{-1}AQe_1=Q^{-1} Av=\lambda Q^{-1} v=\lambda e_1\), par conséquent la matrice \( Q^{-1} AQ\) est de la forme
+    \begin{equation}
+        Q^{-1}AQ=\begin{pmatrix}
+            \lambda_1    &   *    \\ 
+            0    &   A_1    
+        \end{pmatrix}
+    \end{equation}
+    où \( *\) représente une ligne quelconque et \( A_1\) est une matrice de \( \eM(n-1,\eC)\). Nous pouvons donc répéter le processus sur \( A_1\) et obtenir une matrice triangulaire supérieure (nous utilisons le fait qu'un produit de matrices orthogonales est une matrice orthogonale).  
+\end{proof}
+En particulier les matrices hermitiennes, anti-hermitiennes et unitaires sont trigonalisables par une matrice unitaire, qui peut être choisie de déterminant \( 1\).
+
+\begin{lemma}       \label{LEMooRCFGooPPXiKi}
+    Soit \( A\in \eM(n,\eC)\) et une matrice unitaire \( U\) telle que \( A=UTU^{-1}\) où \( T\) est triangulaire. 
+    \begin{enumerate}
+        \item
+            En ce qui concerne les polynômes caractéristiques, \( \chi_A=\chi_T\).
+        \item
+            Pour les spectres, \( \Spec(A)=\Spec(T)\).
+        \item
+            Les valeurs propres de \( A\) sont les éléments diagonaux de \( T\).
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Vu que \( U\) commute évidemment avec \( \mtu\) nous avons
+    \begin{equation}
+        \chi_A(\lambda)=\det(A-\lambda \mtu)=\det(UTU^{-1}-\lambda\mtu)=\det\big( U(T-\lambda\mtu)U^{-1} \big).
+    \end{equation}
+    À ce niveau nous utilisons le fait que le déterminant soit multiplicatif \ref{PropYQNMooZjlYlA} pour conclure :
+    \begin{equation}
+        \chi_A(\lambda)=\det\big( U(T-\lambda\mtu)U^{-1} \big)=\det(U)\det(T-\lambda\mtu)\det(U^{-1})=\det(T-\lambda\mtu)=\chi_T(\lambda).
+    \end{equation}
+
+    Pour les spectres, l'égalité des polynômes caractéristique implique l'égalité des spectres parce que les valeurs propres sont les racines du polynôme caractéristique par le théorème \ref{ThoWDGooQUGSTL}.
+    
+    Les valeurs propres d'une matrice triangulaire sont les valeurs sur la diagonale.
+\end{proof}
+
+\begin{remark}
+    Le lemme mentionne le fait que les valeurs propres de \( A\) sont les éléments diagonaux de \( T\). Mais attention : ceci ne dit rien au niveau des multiplicités géométriques. Un nombre peut être cinq fois sur la diagonale de \( T\) alors que l'espace propre correspondant pour \( A\) n'est que de dimension \( 1\). Exemple : la matrice
+    \begin{equation}
+        A=\begin{pmatrix}
+            1    &   1    \\ 
+            0    &   1    
+        \end{pmatrix}
+    \end{equation}
+    a deux \( 1\) sur la diagonale. Le nombre \( 1\) est bien une valeur propre de \( A\), mais le système
+    \begin{equation}
+        A\begin{pmatrix}
+            x    \\ 
+            y    
+        \end{pmatrix}=\begin{pmatrix}
+            x    \\ 
+            y    
+        \end{pmatrix}
+    \end{equation}
+    donne \( y=0\) et donc un espace propre de dimension seulement \( 1\).
+\end{remark}
+
+\begin{remark}  \label{RemXFZTooXkGzQg}
+    Si \( \eK\) est algébriquement clos (comme \( \eC\) par exemple), alors tous les polynômes sont scindés et toutes les matrices sont trigonalisables\footnote{La proposition \ref{PropKNVFooQflQsJ} montre cela, et le lemme de Schur complexe \ref{LemSchurComplHAftTq} va un peu plus loin et précise que la trigonalisation peut être faite par une matrice unitaire.}. Un exemple un peu simple de cela est la matrice
+    \begin{equation}
+        u=\begin{pmatrix}
+            0    &   -1    \\ 
+            1    &   0    
+        \end{pmatrix}.
+    \end{equation}
+    Le polynôme caractéristique est \( \chi_u(X)=X^2+1\) et les valeurs propres sont \( \pm i\). Il est vite vu que dans la base
+    \begin{equation}
+        \{ \begin{pmatrix}
+        i    \\ 
+    1    
+\end{pmatrix}, \begin{pmatrix}
+1    \\ 
+i    
+\end{pmatrix}\}
+    \end{equation}
+    de \( \eC^2\), la matrice \( u\) se note \( \begin{pmatrix}
+        i    &   0    \\ 
+        0    &   -i    
+    \end{pmatrix}\).
+\end{remark}
+
+\begin{remark}  \label{RemREOSooGEDJWX}
+    Cela nous donne une autre façon de prouver qu'une matrice nilpotente de \( \eM(n,\eC)\) ou \( \eM(n,\eR)\) est trigonalisable\cite{KDUFooVxwqlC}. D'abord dans \( \eM(n,\eC)\), toutes les matrices sont trigonalisables\footnote{Parce que le polynôme caractéristique est scindé, voir la proposition \ref{PropKNVFooQflQsJ}..}, et les valeurs propres arrivent sur la diagonale. Mais comme les valeurs propres d'une matrice nilpotente sont zéro, elle est triangulaire stricte. Par ailleurs son polynôme caractéristique est alors \( X^n\).
+
+    Ensuite si \( u\in \eM(n,\eR)\) nous pouvons voir \( u\) comme une matrice dans \( \eM(n,\eC)\) et y calculer son polynôme caractéristique qui sera tout de même \( X^n\). Ce polynôme étant scindé, la proposition \ref{PropKNVFooQflQsJ} nous assure que \( u\) est trigonalisable. Une fois de plus, les valeurs propres étant sur la diagonale, elle est triangulaire supérieure stricte.
+\end{remark}
+
+\begin{corollary}   \label{CorUNZooAZULXT}
+    Le polynôme caractéristique\footnote{Définition \ref{DefOWQooXbybYD}.} sur \( \eC\) d'une matrice s'écrit sous la forme
+    \begin{equation}
+        \chi_A(X)=\prod_{i=1}^r(X-\lambda_i)^{m_i}
+    \end{equation}
+    où les \( \lambda_i\) sont les valeurs propres distinctes de \( A\) et \( m_i\) sont les multiplicités correspondantes.
+\end{corollary}
+\index{polynôme!caractéristique}
+
+\begin{proof}
+    Le lemme \ref{LemSchurComplHAftTq} nous donne l'existence d'une base de trigonalisation; dans cette base les valeurs propres de \( A\) sont sur la diagonale et nous avons 
+    \begin{equation}
+        \chi_A(X)=\det(A-X\mtu)=\det\begin{pmatrix}
+            X-\lambda_1    &   *    &   *    \\
+            0    &   \ddots    &   *    \\
+            0    &   0    &   X-\lambda_r
+        \end{pmatrix},
+    \end{equation}
+    qui vaut bien le produit annoncé.
+\end{proof}
+
+\begin{corollary}       \label{CORooTPDHooXazTuZ}
+    Si \( A\in \eM(n,\eC)\) et \( k\in \eN\) alors
+    \begin{equation}
+        \Spec(A^k)=\{ \lambda^k\tq \lambda\in \Spec(A) \}.
+    \end{equation}
+\end{corollary}
+
+\begin{proof}
+    Par le lemme \ref{LemSchurComplHAftTq} nous avons une matrice unitaire \( U\) et une triangulaire \( T\) telles que \( A=UTU^{-1}\). En passant à a puissance \( k\) nous avons aussi
+    \begin{equation}
+        A^k=UT^kU^{-1}.
+    \end{equation}
+    Donc le spectre de \( A^k\) est celui de \( T^k\) (lemme \ref{LEMooRCFGooPPXiKi} et le fait qu'une puissance d'une matrice triangulaire est encore triangulaire). Or les éléments diagonaux de \( T^k\) sont les puissance \( k\)\ieme des éléments diagonaux de \( T\), qui sont les valeurs propres de \( A\).
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Diagonalisation : cas complexe, ce qu'on a}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}[Théorème spectral hermitien]      \label{LEMooVCEOooIXnTpp}
+    Pour un opérateur hermitien\footnote{Définition \ref{DEFooKEBHooWwCKRK}.},
+    \begin{enumerate}
+        \item
+            le spectre est réel,
+        \item
+            deux vecteurs propres à des valeurs propres distinctes sont orthogonales\footnote{Pour la forme \eqref{EqFormSesqQrjyPH}.}.
+    \end{enumerate}
+\end{lemma}
+\index{spectre!matrice hermitienne}
+
+\begin{proof}
+    Soit \( v\) un vecteur de valeur propre \( \lambda\). Nous avons d'une part 
+    \begin{equation}
+        \langle Av, v\rangle =\lambda\langle v, v\rangle =\lambda\| v \|^2,
+    \end{equation}
+    et d'autre part, en utilisant le fait que \( A\) est hermitien,
+    \begin{equation}
+        \langle Av, v\rangle =\langle v, A^*v\rangle =\langle v, Av\rangle =\bar\lambda\| v \|^2,
+    \end{equation}
+    par conséquent \( \lambda=\bar\lambda\) parce que \( v\neq 0\).
+
+    Soient \( \lambda_i\) et \( v_i\) (\( i=1,2\)) deux valeurs propres de \( A\) avec leurs vecteurs propres correspondants. Alors d'une part
+    \begin{equation}
+        \langle Av_1, v_2\rangle =\lambda_1\langle v_1, v_2\rangle ,
+    \end{equation}
+    et d'autre part
+    \begin{equation}
+        \langle Av_1, v_2\rangle =\langle v_1, Av_2\rangle =\lambda_2\langle v_1, v_2\rangle .
+    \end{equation}
+    Nous avons utilisé le fait que \( \lambda_2\) était réel. Par conséquent, soit \( \lambda_1=\lambda_2\), soit \( \langle v_1, v_2\rangle =0\).
+\end{proof}
+
+\begin{remark}      \label{REMooMLBCooTuKFmz}
+    Un opérateur de la forme \( A^*A\) est évidemment hermitien. De plus ses valeurs propres sont toutes positives parce que si \( A^*Ax=\lambda v\) alors
+    \begin{equation}
+        0\leq \langle Av, Av\rangle =\langle A^*Av, v\rangle =\lambda\langle v, v\rangle .
+    \end{equation}
+    Donc \( \lambda\geq 0\).
+\end{remark}
+
+\begin{definition}  \label{DefWQNooKEeJzv}
+    Un endomorphisme est \defe{normal}{normal!endomorphisme}\index{matrice!normale} s'il commute avec son adjoint.
+\end{definition}
+
+\begin{theorem}[Théorème spectral pour les matrices normales\footnote{Définition \ref{DefWQNooKEeJzv}}\cite{LecLinAlgAllen,OMzxpxE}]\index{théorème!spectral!matrices normales}  \index{diagonalisation!cas complexe}  \label{ThogammwA}
+    Soit \( A\in\eM(n,\eC)\) une matrice de valeurs propres \( \lambda_1,\ldots, \lambda_n\) (non spécialement distinctes). Alors les conditions suivantes sont équivalentes :
+    \begin{enumerate}
+        \item   \label{ItemJZhFPSi}
+            \( A\) est normale,
+        \item   \label{ItemJZhFPSii}
+            \( A\) se diagonalise par une matrice unitaire,
+        \item
+            \( \sum_{i,j=1}^n| A_{ij} |^2=\sum_{j=1}^n| \lambda_j |^2\),
+        \item
+            il existe une base orthonormale de vecteurs propres de \( A\).
+    \end{enumerate}
+\end{theorem}
+
+\begin{proof}
+    Nous allons nous contenter de prouver \ref{ItemJZhFPSi}\( \Leftrightarrow\)\ref{ItemJZhFPSii}.
+    %TODO : le reste.
+
+    Soit \( Q\) la matrice unitaire donnée par la décomposition de Schur (lemme \ref{LemSchurComplHAftTq}) : \( A=QTQ^{-1}\). Étant donné que \( A\) est normale nous avons
+    \begin{equation}
+        QTT^*Q^{-1}=QT^*TQ^{-1},
+    \end{equation}
+    ce qui montre que \( T\) est également normale. Or une matrice triangulaire supérieure normale est diagonale. En effet nous avons \( T_{ij}=0\) lorsque \( i>j\) et
+    \begin{equation}
+        (TT^*)_{ii}=(T^*T)_{ii}=\sum_{k=1}^n| T_{ki} |^2=\sum_{k=1}^n| T_{ik} |^2.
+    \end{equation}
+    Écrivons cela pour \( i=1\) en tenant compte de \( | T_{k1} |^2=0\) pour \( k=2,\ldots, n\),
+    \begin{equation}
+        | T_{11} |^2=| T_{11} |^2+| T_{12} |^2+\cdots+| T_{1n} |^2,
+    \end{equation}
+    ce qui implique que \( T_{11}\) est le seul non nul parmi les \( T_{1k}\). En continuant de la sorte avec \( i=2,\ldots, n\) nous trouvons que \( T\) est diagonale.
+
+    Dans l'autre sens, si \( A\) se diagonalise par une matrice unitaire, \( UAU^*=D\), nous avons
+    \begin{equation}
+        DD^*=UAA^*U^*
+    \end{equation}
+    et 
+    \begin{equation}
+        D^*D=UA^*AU^*,
+    \end{equation}
+    qui ce prouve que \( A\) est normale.
+\end{proof}
+
+Tant que nous en sommes à parler de spectre de matrices hermitiennes\ldots Soit une matrice inversible \( A\in \GL(n,\eC)\). La matrice \( A^*A\) est hermitienne\footnote{Définition \ref{DEFooKEBHooWwCKRK}.} et le théorème \ref{LEMooVCEOooIXnTpp} nous assure que ses valeurs propres sont réelles. Par la remarque \ref{REMooMLBCooTuKFmz}, ses valeurs propres sont même positives.
+
+\begin{lemma}[\cite{ooLMMRooUXhOdx}]   \label{LEMooHUGEooVYhZdZ}
+    Si \( A\) est une matrice carré et inversible,
+    \begin{equation}
+        \Spec(A^*A)=\Spec(AA^*)
+    \end{equation}
+\end{lemma}
+
+\begin{proof}
+    Nous allons montrer l'égalité des polynômes caractéristiques. D'abord une simple multiplication montre que
+    \begin{equation}
+        (A^*A-\lambda\mtu)A^{-1}=A^{-1}(AA^*-\lambda\mtu).
+    \end{equation}
+    Nous prenons le déterminant de cette égalité en utilisant les propriétés \ref{PropYQNMooZjlYlA}\ref{ItemUPLNooYZMRJy} et \ref{ITEMooZMVXooLGjvCy} :
+    \begin{equation}
+        \det(A^*A-\lambda\mtu)\det(A^{-1})=\det(A^{-1})\det(AA^*-\lambda\mtu).
+    \end{equation}
+    En simplifiant par \( \det(A^{-1})\) (qui est non nul parce que \( A\) est inversible) nous obtenons l'égalité des polynômes caractéristiques et donc l'égalité des spectres.
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Diagonalisation : cas réel}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}[Lemme de Schur réel]  \label{LemSchureRelnrqfiy}
+    Soit \( A\in\eM(n,\eR)\). Il existe une matrice orthogonale \( Q\) telle que \( Q^{-1}AQ\) soit de la forme
+    \begin{equation}        \label{EqMtrTSqRTA}
+        QAQ^{-1}=\begin{pmatrix}
+            \lambda_1    &   *    &   *    &   *    &   *\\  
+            0    &   \ddots    &   \ddots    &   \ddots    &   \vdots\\  
+            0    &   0    &   \lambda_r    &   *    &   *\\  
+            0    &   0    &   0    &   \begin{pmatrix}
+                a_1    &   b_1    \\ 
+                c_1    &   d_1    
+            \end{pmatrix}&   *\\  
+            0    &   0    &  0     &   0    &   \begin{pmatrix}
+                a_s    &   b_s    \\ 
+                c_s    &   d_s    
+            \end{pmatrix}
+        \end{pmatrix}.
+    \end{equation}
+    Le déterminant de \( A\) est le produit des déterminants des blocs diagonaux et les valeurs propres de \( A\) sont les \( \lambda_1,\ldots, \lambda_r\) et celles de ces blocs.
+\end{lemma}
+\index{lemme!Schur réel}
+
+\begin{proof}
+    Si la matrice \( A\) a des valeurs propres réelles, nous procédons comme dans le cas complexe. Cela nous fournit le partie véritablement triangulaire avec les valeurs propres \( \lambda_1,\ldots, \lambda_r\) sur la diagonale. Supposons donc que \( A\) n'a pas de valeurs propres réelles. Soit donc \( \alpha+i\beta \) une valeur propre (\( \beta\neq 0\)) et \( u+iv\) un vecteur propre correspondant où \( u\) et \( v\) sont des vecteurs réels. Nous avons
+    \begin{equation}
+        Au+iAv=A(u+iv)=(\alpha+i\beta)(u+iv)=\alpha u-\beta v+i(\alpha v+\beta v),
+    \end{equation}
+    et en égalisant les parties réelles et imaginaires,
+    \begin{subequations}
+        \begin{align}
+            Au&=\alpha u-\beta v\\
+            Av&=\alpha v+\beta u.
+        \end{align}
+    \end{subequations}
+    Sur ces relations nous voyons que ni \( u\) ni \( v\) ne sont nuls. De plus \( u\) et \( v\) sont linéairement indépendants (sur \( \eR\)), en effet si \( v=\lambda u\) nous aurions \( Au=\alpha u-\beta\lambda u=(\alpha-\beta\lambda)u\), ce qui serait une valeur propre réelle alors que nous avions supposé avoir déjà épuisé toutes les valeurs propres réelles.
+
+    Étant donné que \( u\) et \( v\) sont deux vecteurs réels non nuls et linéairement indépendants, nous pouvons trouver une base orthonormée \( \{ q_1,q_2 \}\) de \( \Span\{ u,v \}\). Nous pouvons étendre ces deux vecteurs en une base orthonormée \( \{ q_1,q_2,q_3,\ldots, q_n \}\) de \( \eR^n\). Nous considérons à présent la matrice orthogonale dont les colonnes sont formées de ces vecteurs : \( Q=[q_1\,q_2\,\ldots q_n]\).
+
+    L'espace \( \Span\{ e_1,e_2 \}\) est stable par \( Q^{-1} AQ\), en effet nous avons
+    \begin{equation}
+        Q^{-1} AQe_1=Q^{-1} Aq_1=Q^{-1}(aq_1+bq_2)=ae_1+be_2.
+    \end{equation}
+    La matrice \( Q^{-1}AQ\) est donc de la forme
+    \begin{equation}
+        Q^{-1} AQ=\begin{pmatrix}
+            \begin{pmatrix}
+                \cdot    &   \cdot    \\ 
+                \cdot    &   \cdot    
+            \end{pmatrix}&   C_1    \\ 
+            0    &   A_1    
+        \end{pmatrix}
+    \end{equation}
+    où \( C_1\) est une matrice réelle \( 2\times (n-1)\) quelconque et \( A_1\) est une matrice réelle \( (n-2)\times (n-2)\). Nous pouvons appliquer une récurrence sur la dimension pour poursuivre.
+
+    Notons que si \( A\) n'a pas de valeurs propres réelles, elle est automatiquement d'ordre pair parce que les valeurs propres complexes viennent par couple complexes conjuguées.
+
+    En ce qui concerne les valeurs propres, il est facile de voir en regardant \eqref{EqMtrTSqRTA} que les valeurs propres sont celles des blocs diagonaux. Étant donné que \( QAQ^{-1}\) et \( A\) ont même polynôme caractéristique, ce sont les valeurs propres de \( A\).
+\end{proof}
+
+\begin{theorem}[Théorème spectral, matrice symétrique\cite{KXjFWKA}] \label{ThoeTMXla}
+    Une matrice symétrique réelle,
+    \begin{enumerate}
+        \item       \label{ITEMooJWHLooSfhNSW}
+            a un spectre contenu dans \( \eR\)
+        \item
+            est diagonalisable par une matrice orthogonale.
+    \end{enumerate}
+    Si \( M\) est une matrice symétrique réelle alors \( \eR^n\) possède une base orthonormée de vecteurs propres de \( M\).
+\end{theorem}
+\index{diagonalisation!cas réel}
+\index{rang!diagonalisation}
+\index{endomorphisme!diagonalisation}
+\index{spectre!matrice symétrique réelle}
+\index{théorème!spectral!matrice symétrique}
+
+\begin{proof}
+    Soit \( A\) une matrice réelle symétrique. Si \( \lambda\) est une valeur propre complexe pour le vecteur propre complexe \( v\), alors d'une part \( \langle Av, v\rangle =\lambda\langle v, v\rangle \) et d'autre part \( \langle Av, v\rangle =\langle v, Av\rangle =\bar\lambda\langle v, v\rangle \). Par conséquent \( \lambda=\bar\lambda\).
+    
+    Le lemme de Schur réel \ref{LemSchureRelnrqfiy} donne une matrice orthogonale qui trigonalise \( A\). Les valeurs propres étant toutes réelles, la matrice \( QAQ^{-1}\) est même triangulaire (il n'y a pas de blocs dans la forme \eqref{EqMtrTSqRTA}). Prouvons que \( QAQ^{-1}\) est symétrique :
+    \begin{equation}
+        (QAQ^{-1})^t=(Q^{-1})^tA^tQ^t=QA^tQ^{-1}=QAQ^{-1}
+    \end{equation}
+    où nous avons utilisé le fait que \( Q\) était orthogonale (\( Q^{-1}=Q^t\)) et que \( A\) était symétrique (\( A^t=A\)). Une matrice triangulaire supérieure symétrique est obligatoirement une matrice diagonale.
+
+    En ce qui concerne la base de vecteurs propres, soit \( \{ e_i \}_{i=1,\ldots, n}\) la base canonique de \( \eR^n\) et \( Q\) une matrice orthogonale e telle que \( A=Q^tDQ\) avec \( D\) diagonale. Nous posons \( f_i=Q^te_i\) et en tenant compte du fait que \( Q^t=Q^{-1}\) nous avons \( Af_i=Q^tDQQ^te_i=Q^t\lambda_i e_i=\lambda_if_i\). Donc les \( f_i\) sont des vecteurs propres de \( A\). De plus ils sont orthonormés parce que
+    \begin{equation}
+        \langle f_i, f_j\rangle =\langle Q^te_i, Q^te_j\rangle =\langle e_i, Q^tQe_j\rangle =\langle e_i, e_j\rangle =\delta_{ij}.
+    \end{equation}
+\end{proof}
+Le théorème spectral pour les opérateurs auto-adjoints sera traité plus bas parce qu'il a besoin de choses sur les formes bilinéaires, théorème \ref{ThoRSBahHH}.
+% et les choses sur la dégénérescences utilisent le théorème spectral, cas réel. Donc l'enchaînement est très loumapotiste.
+
+\begin{remark}  \label{RemGKDZfxu}
+    Une matrice symétrique est diagonalisable par une matrice orthogonale. Nous pouvons en réalité nous arranger pour diagonaliser par une matrice de \( \SO(n)\). Plus généralement si \( A\) est une matrice diagonalisable par une matrice \( P\in\GL^+(n,\eR)\) alors elle est diagonalisable par une matrice de \( \GL^-(n,\eR)\) en changeant le signe de la première ligne de \( P\). Et inversement.
+
+    En effet, si nous avons \( P^tDP=A\), alors en notant \( *\) les quantités qui ne dépendent pas de \( a\), \( b\) ou~\( c\),
+    \begin{equation}
+        \begin{aligned}[]
+        \begin{pmatrix}
+            a    &   *    &   *    \\
+            b    &   *    &   *    \\
+            c    &   *    &   *
+        \end{pmatrix}
+        \begin{pmatrix}
+            \lambda_1    &       &       \\
+                &   \lambda_2    &       \\
+                &       &   \lambda_3
+            \end{pmatrix}
+            \begin{pmatrix}
+                a    &   b    &   c    \\
+                *    &   *    &   *    \\
+                *    &   *    &   *
+            \end{pmatrix}&=
+        \begin{pmatrix}
+            a    &   *    &   *    \\
+            b    &   *    &   *    \\
+            c    &   *    &   *
+        \end{pmatrix}
+        \begin{pmatrix}
+            \lambda_1a    &   \lambda_1b    &   \lambda_1c    \\
+            *    &   *    &   *    \\
+            *    &   *    &   *
+        \end{pmatrix}\\
+        &=\begin{pmatrix}
+            \lambda_1 a^2+*   &   \lambda_1ab+*    &   \lambda_1ac  +*  \\
+            \ldots    &   \ldots    &   \ldots    \\
+            \ldots    &   \ldots    &   \ldots
+        \end{pmatrix}.
+        \end{aligned}
+    \end{equation}
+    Nous voyons donc que si nous changeons les signes de \( a\), \( b\) et \( c\) en même temps, le résultat ne change pas.
+\end{remark}
+
+\begin{definition}[Matrice définie positive, opérateur définit positif]    \label{DefAWAooCMPuVM}
+    Un opérateur sur un espace vectoriel sur \( \eC\) ou \( \eR\) est \defe{définit positif}{opérateur!définit positif} si toutes ses valeurs propres sont réelles et strictement positives.  Il est \defe{semi-définie positive}{semi-définie positive} si ses valeurs propres sont réelles positives ou nulles.
+\end{definition}
+Afin d'éviter l'une ou l'autre confusion, nous disons souvent \emph{strictement} définie positive pour positive.
+
+Nous notons \( S^+(n,\eR)\)\nomenclature[A]{\( S^+(n,\eR)\)}{matrices symétriques semi-définies positives} l'ensemble des matrices réelles \( n\times n\) semi-définies positives. L'ensemble \( S^{++}(n,\eR)\)\nomenclature[A]{\( S^{++}(n,\eR)\)}{matrices symétriques strictement définies positives} est l'ensemble des matrices symétriques strictement définies positives.
+
+\begin{remark}
+    Nous ne définissons pas la notion de matrice définie positive pour une matrice non symétrique.
+\end{remark}
+
+Lorsqu'un énoncé parle d'une matrice symétrique, le premier réflexe est de la diagonaliser : considérer une matrice orthogonale \( T\) telle que \( T^tMT=D\) avec \( D\) diagonale. Et les valeurs propres sur la diagonale : \( D_{kl}=\delta_{kl}\lambda_k\). Les matrices symétriques définies positives ont cependant des propriétés même en dehors de leur base de diagonalisation.
+
+\begin{lemma}   \label{LemWZFSooYvksjw}
+    Soit une matrice symétrique \( M\).
+    \begin{enumerate}
+        \item       \label{ITEMooSKRAooOgHbGA}
+           Elle est strictement définie positive si et seulement si \( \langle x, Mx\rangle >0\) pour tout \( x\) non nul dans \( \eR^n\).
+        \item       \label{ITEMooMOZYooWcrewZ}
+           Elle est semi définie positive si et seulement si \( \langle x, Mx\rangle \geq 0\) pour tout \( x\) non nul dans \( \eR^n\).
+       \item        \label{ITEMooRRMFooHSOHxZ}
+           Si elle est seulement définie positive, alors \( \langle x, Mx\rangle \geq \lambda\| x \|^2\) dès que \( \lambda\geq 0\) minore toutes les valeurs propres.
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Démonstration en trois parties.
+    \begin{subproof}
+    \item[\ref{ITEMooSKRAooOgHbGA}]
+    Soit \( \{ e_i \}_{i=1,\ldots, n}\) une base orthonormée de vecteurs propres de \( M\) dont l'existence est assurée par le théorème spectral \ref{ThoeTMXla}. Nous nommons \( x_i\) les coordonnées de \( x\) dans cette base. Alors,
+    \begin{equation}
+        \langle x,Mx \rangle =\sum_{i,j}x_i\langle e_i, x_jMe_j\rangle =\sum_{i,j}x_ix_j\langle e_i, \lambda_je_j\rangle =\sum_{ij}x_ix_j\lambda_j\delta_{ij}=\sum_i\lambda_ix_i^2
+    \end{equation}
+    où les \( \lambda_i\) sont les valeurs propres de \( M\). Cela est strictement positif pour tout \( x\) si et seulement si tous les \( \lambda_i\) sont strictement positifs.
+\item[\ref{ITEMooMOZYooWcrewZ}]
+
+    Nous avons encore 
+    \begin{equation}
+        \langle x, Mx\rangle =\sum_{i}\lambda_ix_i^2.
+    \end{equation}
+    Cela est plus grand ou égal à zéro si et seulement si tous les \( \lambda_i\) sont plus grands ou égaux à zéro.
+        
+\item[\ref{ITEMooRRMFooHSOHxZ}]
+
+        Soit une matrice orthogonale \( T\) diagonalisant \( M\), c'est à dire telle que \( T^tMT=D\) avec \( D\) diagonale. Nous allons vérifier que 
+        \begin{equation}
+            \langle Tx, Mtx\rangle \geq \lambda\| Tx \|^2
+        \end{equation}
+        pour tout \( x\). Vu que \( T\) est une bijection\footnote{Une matrice orthogonale a un déterminant \( \pm 1\).}, cela impliquera le résultat pour tout \( x\). Si nous considérons la base de diagonalisation \( \{ e_k \}\) pour les valeurs propres \( \lambda_k\), nous avons le calcul
+        \begin{subequations}
+            \begin{align}
+                \langle Tx, MTx\rangle &=\langle x, T^tMTx\rangle \\
+                &=\langle x, Dx\rangle \\
+                &=\sum_k\langle x, x_kDe_k\rangle \\
+                &=\sum_k\lambda_kx_k \underbrace{\langle x, e_k\rangle }_{=x_k}\\
+                &\geq \sum_k\lambda| x_k |^2\\
+                &=\lambda\| x \|^2\\
+                &=\lambda\| Tx \|^2.
+            \end{align}
+        \end{subequations}
+        Au dernier passage nous avons utilisé le fait que \( T\) est une isométrie (proposition \ref{PropKBCXooOuEZcS}).
+
+    \end{subproof}
+\end{proof}
+
+Les personnes qui aiment les vecteurs lignes et colonnes écriront des inégalités comme
+\begin{equation}
+    x^tMx\geq x^tx.
+\end{equation}
+Tout à l'autre bout du spectre des personnes névrosées des notations, on trouvera des inégalités comme
+\begin{equation}
+    M(x\otimes x)\geq x\cdot x.
+\end{equation}
+Le penchant personnel de l'auteur de ces lignes est la notation avec le produit tensoriel. Si vous aimez ça, vous pouvez lire \ref{SeOOpHsn}.
+
+La notation adoptée ici avec le produit scalaire \( \langle x, Mx\rangle \) est entre les deux. Elle a l'avantage de n'être pas technologique comme le produit tensoriel (si vous y mettez les pieds, vous devez savoir ce que vous faites), tout en évitant de se casser la tête à savoir qui est un vecteur ligne ou un vecteur colonne.
+
+\begin{corollary}
+    Une matrice symétrique strictement définie positive est inversible.
+\end{corollary}
+
+\begin{proof}
+    Si \( Ax=0\) alors \( \langle Ax, x\rangle =0\). Mais dans le cas d'une matrice strictement définie positive, cela implique \( x=0\) par le lemme \ref{LemWZFSooYvksjw}.
+\end{proof}
+
+\begin{lemma}
+    Pour une base quelconque, les éléments diagonaux d'une matrice symétrique semi-définie positive sont positifs. Si la matrice est strictement définie positive, alors les éléments diagonaux sont strictement positifs.
+\end{lemma}
+
+\begin{proof}
+    Il s'agit d'une application du lemme \ref{LemWZFSooYvksjw}. Si \( A\) est définie positive et que \( \{ e_i \}\) est une base, alors
+    \begin{equation}
+        A_{ii}=\langle Ae_i, e_i\rangle \geq \lambda\| e_i \|^2=\lambda\geq 0.
+    \end{equation}
+    Si \( A\) est strictement définie positive, alors \( \lambda\) peut être choisit strictement positif.
+\end{proof}
+ 
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Pseudo-réduction simultanée}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{corollary}[Pseudo-réduction simultanée\cite{JMYQgLO}]  \label{CorNHKnLVA}
+    Soient \( A,B\in \gS(n,\eR)\) avec \( A\) définie positive\footnote{Définition \ref{DefAWAooCMPuVM}.}. Alors il existe \( Q\in \GL(n,\eR)\) tell que \( Q^tBQ\) soit diagonale et \( Q^tAQ=\mtu\).
+\end{corollary}
+
+\begin{proof}
+    Nous allons noter \( x\cdot y\) le produit scalaire usuel de \( \eR^n\) et \( \{ e_i \}_{i=1,\ldots, n}\) sa base canonique.
+
+    Vu que \( A\) est définie positive, nous avons que l'expression\footnote{On peut aussi l'écrire de façon plus matricielle sous la forme \( \langle x, y\rangle =x^tAy\).} \( \langle x, y\rangle =x\cdot Ay\) est un produit scalaire sur \( \eR^n\). Autrement dit, \( E\) muni de cette forme bilinéaire symétrique est un espace euclidien, ce qui fait dire à la proposition \ref{PropUMtEqkb} qu'il existe une base de \( \eR^n\) orthonormée \( \{ f_i \}_{i=1,\ldots, n}\) pour ce produit scalaire, c'est à dire qu'il existe une matrice \( P\in \GL(n,\eR)\) telle que \( P^tAP=\mtu\). Ici, \( P\) est la matrice de changement de base de la base canonique à notre base orthonormée, c'est à dire la matrice qui fait \( Pe_i=f_i\) pour tout \( i\). Voyons cela avec un peu de détails.
+
+    Pour savoir ce que valent les éléments de la matrice \( P^tAP\), nous nous souvenons que \( P^tAPe_j\) est un vecteur dont les coordonnées sont les éléments de la \( j\)\ieme colonne de \( P^tAP\). Nous avons donc \( (P^tAP)_{ij}=e_i\cdot P^tAPe_i\). Calculons :
+    \begin{equation}
+            (P^tAP)_{ij}=e_i\cdot P^tAPe_i
+            =Pe_i\cdot APe_j
+        =f_i\cdot Af_j
+        =\langle f_i, f_j\rangle 
+        =\delta_{ij}
+    \end{equation}
+    où nous avons utilisé le fait que \( A\) était auto-adjointe pour la passer de l'autre côté du produit scalaire (usuel). Au final nous avons effectivement \( P^tAP=\mtu\).
+
+    La matrice \( P^tBP\) est une matrice symétrique, donc le théorème spectral \ref{ThoeTMXla} nous donne une matrice \( R\in \gO(n,\eR)\) telle que \( R^tP^tBPR\) soit diagonale. En posant maintenant \( Q=PR\) nous avons la matrice cherchée.
+\end{proof}
+Note : nous avons prouvé la pseudo-réduction simultanée comme corollaire du théorème de diagonalisation des matrices symétriques \ref{ThoeTMXla}. Il aurait aussi pu être vu comme un corollaire du théorème spectral \ref{ThoRSBahHH} sur les opérateurs auto-adjoints via son corollaire \ref{CorSMHpoVK}.
diff --git a/tex/frido/59_EspacesVectos.tex b/tex/frido/59_EspacesVectos.tex
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+% This is part of Mes notes de mathématique
+% Copyright (c) 2008-2017
+%   Laurent Claessens, Carlotta Donadello
+% See the file fdl-1.3.txt for copying conditions.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Sommes de familles infinies}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\label{SECooHHDXooUgLhHR}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Convergence commutative}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit \( x_k\) une suite dans un espace vectoriel normé \( E\). Nous disons que la suite \defe{converge commutativement}{convergence!commutative} vers \( x\in E\) si \( \lim_{n\to \infty}\| x_n-x \| =0\) et si pour toute bijection \( \tau\colon \eN\to \eN\) nous avons aussi
+    \begin{equation}
+        \lim_{n\to \infty} \| x_{\tau(k)}-x \|=0.
+    \end{equation}
+    La notion de convergence commutative est surtout intéressante pour les séries. La somme
+    \begin{equation}
+        \sum_{k=0}^{\infty}x_k
+    \end{equation}
+    converge commutativement vers \( x\) si \( \lim_{N\to \infty} \| x-\sum_{k=0}^Nx_k \|=0\) et si pour toute bijection \( \tau\colon \eN\to \eN\) nous avons
+    \begin{equation}
+        \lim_{N\to \infty} \| x-\sum_{k=0}^Nx_{\tau(k)} \|=0.
+    \end{equation}
+\end{definition}
+
+Nous démontrons maintenant qu'une série converge commutativement si et seulement si elle converge absolument.
+
+Pour le sens inverse, nous avons la proposition suivante.
+\begin{proposition}
+    Soit \( \sum_{k=0}^{\infty}a_k\) une série réelle qui converge mais qui ne converge pas absolument. Alors pour tout \( b\in \eR\), il existe une bijection \( \tau\colon \eN\to \eN\) telle que \( \sum_{i=0}^{\infty}a_{\tau(i)}=b\).
+\end{proposition}
+Pour une preuve, voir \href{http://gilles.dubois10.free.fr/analyse_reelle/seriescomconv.html}{chez Gilles Dubois}.
+
+\begin{proposition} \label{PopriXWvIY}
+    Soit \( (a_i)_{i\in \eN}\) une suite dans \( \eC\) convergent absolument. Alors elle converge commutativement.
+\end{proposition}
+
+\begin{proof}
+    Soit \( \epsilon>0\). Nous posons \( \sum_{i=0}^\infty a_i=a\) et nous considérons \( N\) tel que
+    \begin{equation}
+        | \sum_{i=0}^Na_i-a |<\epsilon.
+    \end{equation}
+    Étant donné que la série des \( | a_i |\) converge, il existe \( N_1\) tel que pour tout \( p,q>N_1\) nous ayons \( \sum_{i=p}^q| a_i |<\epsilon\). Nous considérons maintenant une bijection \( \tau\colon \eN\to \eN \). Prouvons que la série \( \sum_{i=0}^{\infty}| a_{\tau(i)} |\) converge. Nous choisissons \( M\) de telle sorte que pour tout \( n>M\), \( \tau(n)>N_1\); alors si \( p,q>M\) nous avons
+    \begin{equation}
+        \sum_{i=p}^q| a_{\tau(i)} |<\epsilon.
+    \end{equation}
+    Par conséquent la somme de la suite \( (a_{\tau(i)})\) converge. Nous devons montrer à présent qu'elle converge vers la même limite que la somme «usuelle» \( \lim_{N\to \infty} \sum_{i=0}^Na_i\).
+
+    Soit \( n>\max\{ M,N \}\). Alors
+    \begin{equation}
+        \sum_{k=0}^na_{\tau(k)}-\sum_{k=0}^na_k=\sum_{k=0}^Ma_{\tau(k)}-\sum_{k=0}^Na_k+\underbrace{\sum_{M+1}^na_{\tau(k)}}_{<\epsilon}-\underbrace{\sum_{k=N+1}^na_k}_{<\epsilon}.
+    \end{equation}
+    Par construction les deux derniers termes sont plus petits que \( \epsilon\) parce que \( M\) et \( N\) sont les constantes de Cauchy pour les séries \( \sum a_{\tau(i)}\) et \( \sum a_i\). Afin de traiter les deux premiers termes, quitte à redéfinir \( M\), nous supposons que \( \{ 1,\ldots, N \}\subset \tau\{ 1,\ldots, M \}\); par conséquent tous les \( a_i\) avec \( i<N\) sont atteints par les \( a_{\tau(i)}\) avec \( i<M\). Dans ce cas, les termes qui restent dans la différence
+    \begin{equation}
+        \sum_{k=0}a_{\tau(k)}-\sum_{k=0}^Na_k
+    \end{equation}
+    sont des \( a_k\) avec \( k>N\). Cette différence est donc en valeur absolue plus petite que \( \epsilon\), et nous avons en fin de compte que
+    \begin{equation}
+        \left| \sum_{k=0}^na_{\tau(k)}-\sum_{k=0}^na_k \right| <\epsilon.
+    \end{equation}
+\end{proof}
+
+\begin{proposition}     \label{PropyFJXpr}
+    Soit \( \sum_{i=0}^{\infty}a_i\) une série qui converge mais qui ne converge pas absolument. Pour tout \( b\in \eR\), il existe une bijection \( \tau\colon \eN\to \eN\) telle que \( \sum_{i=}^{\infty}a_{\tau(i)}=b\).
+\end{proposition}
+
+Les propositions \ref{PopriXWvIY} et \ref{PropyFJXpr} disent entre autres qu'une série dans \( \eC\) est commutativement sommable si et seulement si elle est absolument sommable.
+
+Soit \( (a_i)_{i\in I}\) une famille de nombres complexes indexée par un ensemble \( I\) quelconque. Nous allons nous intéresser à la somme \( \sum_{i\in I}a_i\).
+
+
+Soit \( \{ a_i \}_{i\in I}\) des nombres positifs. Nous définissons la somme
+\begin{equation}
+    \sum_{i\in I}a_i=\sup_{ J\text{ fini}}\sum_{j\in J}a_j.
+\end{equation}
+Notons que cela est une définition qui ne fonctionne bien que pour les sommes de nombres positifs. Si \( a_i=(-1)^i\), alors selon la définition nous aurions \( \sum_i(-1)^i=\infty\). Nous ne voulons évidemment pas un tel résultat.
+
+Dans le cas de familles de nombres réels positifs, nous avons une première définition de la somme. 
+\begin{definition}  \label{DefHYgkkA}
+Soit \( (a_i)_{i\in I}\) une famille de nombres réels positifs indexés par un ensemble quelconque \( I\). Nous définissons
+\begin{equation}
+    \sum_{i\in I}a_i=\sup_{ J\text{ fini dans } I}\sum_{j\in J}a_j.
+\end{equation}
+\end{definition}
+
+\begin{definition}  \label{DefIkoheE}
+    Si \( \{ v_i \}_{i\in I}\) est une famille de vecteurs dans un espace vectoriel normé indexée par un ensemble quelconque \( I\). Nous disons que cette famille est \defe{sommable}{famille!sommable} de somme \( v\) si pour tout \( \epsilon>0\), il existe un \( J_0\) fini dans \( I\) tel que pour tout ensemble fini \( K\) tel que \( J_0\subset K\) nous avons
+    \begin{equation}
+        \| \sum_{j\in K}v_j-v \|<\epsilon.
+    \end{equation}
+\end{definition}
+Notons que cette définition implique la convergence commutative.
+
+\begin{example}
+    La suite \( a_i=(-1)^i\) n'est pas sommable parce que quel que soit \( J_0\) fini dans \( \eN\), nous pouvons trouver \( J\) fini contenant \( J_0\) tel que \( \sum_{j\in J}(-1)^j>10\). Pour cela il suffit d'ajouter à \( J_0\) suffisamment de termes pairs. De la même façon en ajoutant des termes impairs, on peut obtenir \( \sum_{j\in J'}(-1)^i<-10\).
+\end{example}
+
+\begin{example}
+    De temps en temps, la somme peut sortir d'un espace. Si nous considérons l'espace des polynômes \( \mathopen[ 0 , 1 \mathclose]\to \eR\) muni de la norme uniforme, la somme de l'ensemble
+    \begin{equation}
+        \{ 1,-1,\pm\frac{ x^n }{ n! } \}_{n\in \eN}
+    \end{equation}
+    est zéro.
+
+    Par contre la somme de l'ensemble \( \{ 1,\frac{ x^n }{ n! } \}_{n\in \eN}\) est l'exponentielle qui n'est pas un polynôme.
+\end{example}
+
+\begin{example}
+    Au sens de la définition \ref{DefIkoheE} la famille
+    \begin{equation}
+        \frac{ (-1)^n }{ n }
+    \end{equation}
+    n'est pas sommable. En effet la somme des termes pairs est \( \infty\) alors que la somme des termes impairs est \( -\infty\). Quel que soit \( J_0\in \eN\), nous pouvons concocter, en ajoutant des termes pairs, un \( J\) avec \( J_0\subset J\) tel que \( \sum_{j\in J}(-1)^j/j\) soit arbitrairement grand. En ajoutant des termes négatifs, nous pouvons également rendre \( \sum_{j\in J}(-1)^j/j\) arbitrairement petit.
+\end{example}
+
+\begin{proposition} \label{PropVQCooYiWTs}
+    Si \( (a_{ij})\) est une famille de nombres positifs indexés par \( \eN\times \eN\) alors
+    \begin{equation}
+        \sum_{(i,j)\in \eN^2}a_{ij}=\sum_{i=1}^{\infty}\Big( \sum_{j=1}^{\infty}a_{ij} \Big)
+    \end{equation}
+    où la somme de gauche est celle de la définition \ref{DefHYgkkA}.
+\end{proposition}
+%TODO : cette proposition peut être vue comme une application de Fubini pour la mesure de comptage. Le faire et référentier ici.
+
+\begin{proof}
+    Nous considérons \( J_{m,n}=\{ 0,\ldots, m \}\times \{ 0,\ldots, n \}\) et nous avons pour tout \( m\) et \( n\) :
+    \begin{equation}
+        \sum_{(i,j)\in \eN^2}a_{ij}\geq \sum_{(i,j)\in J_{m,n}}a_{ij}=\sum_{i=1}^m\Big( \sum_{j=1}^na_{ij} \Big).
+    \end{equation}
+    Si nous fixons \( m\) et que nous prenons la limite \( n\to \infty\) (qui commute avec la somme finie sur \( i\)) nous trouvons
+    \begin{equation}
+        \sum_{(i,j)\in \eN^2}a_{ij}\geq =\sum_{i=1}^m\Big( \sum_{j=1}^{\infty}a_{ij} \Big).
+    \end{equation}
+    Cela étant valable pour tout \( m\), c'est encore valable à la limite \( m\to \infty\) et donc
+    \begin{equation}
+        \sum_{(i,j)\in \eN^2}a_{ij}\geq \sum_{i=1}^{\infty}\Big( \sum_{j=1}^{\infty}a_{ij} \Big).
+    \end{equation}
+    
+    Pour l'inégalité inverse, il faut remarquer que si \( J\) est fini dans \( \eN^2\), il est forcément contenu dans \( J_{m,n}\) pour \( m\) et \( n\) assez grand. Alors
+    \begin{equation}
+        \sum_{(i,j)\in J}a_{ij}\leq \sum_{(i,j)\in J_{m,n}}a_{ij}=\sum_{i=1}^m\sum_{j=1}^na_{ij}\leq \sum_{i=1}^{\infty}\Big( \sum_{j=1}^{\infty}a_{ij} \Big).
+    \end{equation}
+    Cette inégalité étant valable pour tout ensemble fini \( J\subset \eN^2\), elle reste valable pour le supremum.    
+\end{proof}
+
+La définition générale de la somme \ref{DefIkoheE} est compatible avec la définition usuelle dans les cas où cette dernière s'applique.
+\begin{proposition}[commutative sommabilité]\label{PropoWHdjw}
+    Soit \( I\) un ensemble dénombrable et une bijection \( \tau\colon \eN\to I\). Soit \( (a_i)_{i\in I}\) une famille dans un espace vectoriel normé. Alors
+    \begin{equation}
+        \sum_{k=0}^{\infty}a_{\tau(k)}=\sum_{i\in I}a_i
+    \end{equation}
+    dès que le membre de droite existe. Le membre de gauche est définit par la limite usuelle.
+\end{proposition}
+
+\begin{proof}
+    Nous posons \( a=\sum_{i\in I}a_i\). Soit \( \epsilon>0\) et \( J_0\) comme dans la définition. Nous choisissons
+    \begin{equation}
+        N>\max_{j\in J_0}\{ \tau^{-1}(j) \}.
+    \end{equation}
+    En tant que sommes sur des ensembles finis, nous avons l'égalité
+    \begin{equation}
+        \sum_{k=0}^Na_{\tau(k)}=\sum_{j\in J_0}a_j
+    \end{equation}
+    où \( J\) est un sous-ensemble de \( I\) contenant \( J_0\). Soit \( J\) fini dans \( I\) tel que \( J_0\subset J\). Nous avons alors
+    \begin{equation}
+        \| \sum_{k=0}^Na_{\tau(k)}-a \|=\| \sum_{j\in J}a_j-a \|<\epsilon.
+    \end{equation}
+    Nous avons prouvé que pour tout \( \epsilon\), il existe \( N\) tel que \( n>N\) implique \( \| \sum_{k=0}^na_{\tau(k)}-a\| <\epsilon\).
+\end{proof}
+
+\begin{corollary}
+    Nous pouvons permuter une somme dénombrable et une fonction linéaire continue. C'est à dire que si \( f\) est une fonction linéaire continue sur l'espace vectoriel normé \( E\) et \( (a_i)_{i\in I}\) une famille sommable dans \( E\) alors
+    \begin{equation}
+        f\left( \sum_{i\in I}a_i \right)=\sum_{i\in I}f(a_i).
+    \end{equation}
+\end{corollary}
+
+\begin{proof}
+    En utilisant une bijection \( \tau\) entre \( I\) et \( \eN\) avec la proposition \ref{PropoWHdjw} ainsi que le résultat connu à propos des sommes sur \( \eN\), nous avons
+    \begin{subequations}
+        \begin{align}
+            f\left( \sum_{i\in I}a_i \right)&=f\left( \sum_{k=0}^{\infty}a_{\tau(k)} \right)\\
+            &=\sum_{k=0}^{\infty}f(a_{\tau(k)}) \label{SUBEQooCVUTooPmnHER}\\
+            &=\sum_{i\in I}f(a_i).
+        \end{align}
+    \end{subequations}
+    Notons que le passage à \eqref{SUBEQooCVUTooPmnHER} n'est pas du tout une trivialité à deux francs cinquante. Il s'agit d'écrire la somme comme la limite des sommes partielles, et de permuter \( f\) avec la limite en invoquant la continuité, puis de permuter \( f\) avec la somme partielle en invoquant sa linéarité.
+
+    Ah, tiens et tant qu'on y est à dire qu'il y a des chose évidentes qui ne le sont pas, oui, il existe des applications linéaires non continues, voir le thème \ref{THEMEooYCBUooEnFdUg}.
+\end{proof}
+
+La proposition suivante nous enseigne que les sommes infinies peuvent être manipulée de façon usuelle.
+\begin{proposition} \label{PropMpBStL}
+    Soit \( I\) un ensemble dénombrable. Soient \( (a_i)_{i\in I}\) et \( (b_i)_{i\in I}\), deux familles de réels positifs telles que \( a_i<b_i\) et telles que \( (b_i)\) est sommable. Alors \( (a_i)\) est sommable.
+
+    Si \( (a_i)_{i\in I}\) est une famille de complexes telle que \( (| a_i |)\) est sommable, alors \( (a_i)\) est sommable.
+\end{proposition}
+
+\begin{proposition}[\cite{MonCerveau}]     \label{PROPooWLEDooJogXpQ}
+    Soit un espace vectoriel normé \( E\) et une famille sommable\footnote{Définition \ref{DefIkoheE}.} \( \{ v_i \}_{i\in I}\) d'éléments de \( E\). Soit \( f\colon E\to \eC\) une application sur laquelle nous supposons
+    \begin{enumerate}
+        \item
+            \( f\) est linéaire et continue;
+        \item
+            la partie \( \{ f(v_i)_{i\in I} \} \) est sommable.
+    \end{enumerate}
+    Alors nous pouvons permuter la somme et \( f\) :
+    \begin{equation}        \label{EQooONHXooKqIEbY}
+        f\big( \sum_{i\in I}v_i \big)=\sum_{i\in I}f(v_i).
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Soit \( \epsilon>0\); vu que les familles \( \{ v_i \}_{i\in I}\) et \( \{ f(v_i) \}_{i\in I}\) sont sommables, nous pouvons considérer les parties finies \( J_1\) et \( J_2\) de \( I\) telles que
+    \begin{equation}
+        \big\| \sum_{j\in J_1}v_j-\sum_{i\in I}v_i \big\|\leq \epsilon
+    \end{equation}
+    et
+    \begin{equation}
+        \big\| \sum_{j\in J_2}f(v_j)-\sum_{i\in I}f(v_i) \big\|\leq \epsilon
+    \end{equation}
+    Ensuite nous posons \( J=J_1\cup J_2\). Avec cela nous calculons un peu avec les majorations usuelles :
+    \begin{equation}
+        \| f(\sum_{i\in I}v_i) -\sum_{i\in I}f(v_i) \|\leq \| f(\sum_{i\in I}v_i)- f(\sum_{j\in J}v_j) \|+  \| f(\sum_{j\in J}v_j)-\sum_i\in If(v_i) \|.
+    \end{equation}
+    Le second terme est majoré par \( \epsilon\), tandis que le premier, en utilisant la linéarité de \( f\) possède la majoration
+    \begin{equation}
+        \| f(\sum_{i\in I}v_i)- f(\sum_{j\in J}v_j) \|=\| f(\sum_{i\in I}v_i-\sum_{j\in J}v_j) \|\leq \| f \| \| \sum_{i\in I}v_i- \sum_{j\in J}v_j\|\leq \epsilon\| f \|.
+    \end{equation}
+    Donc pour tout \( \epsilon>0\) nous avons
+    \begin{equation}
+        \| f(\sum_{i\in I}v_i) -\sum_{i\in I}f(v_i) \|\leq \epsilon(1+\| f \|).
+    \end{equation}
+    D'où l'égalité \eqref{EQooONHXooKqIEbY}.
+\end{proof}
+
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Fonctions}		\label{Sect_fonctions}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Soient $(V,\| . \|_V)$ et $(W,\| . \|_W)$ deux espaces vectoriels normés, et une fonction $f$ de $V$ dans $W$. Il est maintenant facile de définir les notions de limites et de continuité pour de telles fonctions en copiant les définitions données pour les fonctions de $\eR$ dans $\eR$ en changeant simplement les valeurs absolues par les normes sur $V$ et $W$.
+
+La caractérisation suivante est un recopiage de la définition \ref{DefOLNtrxB} lorsque la topologie est donnée par des boules.
+\begin{proposition}\label{PropHOCWooSzrMjl}
+	Soit $f\colon V\to W$ une fonction de domaine \( \Domaine(f)\subset V\) et soit $a$ un point d'accumulation de $\Domaine(f)$. 
+    La fonction \( f\) admet une limite en $a\in V$ si et seulement s'il existe un élément $\ell\in W$ tel que pour tout $\varepsilon>0$, il existe un $\delta>0$ tel que pour tout $x\in \Domaine(f)$,
+    \begin{equation}        \label{EqDefLimzxmasubV}
+		0<\| x-a \|_V<\delta\,\Rightarrow\,\| f(x)-\ell \|_W<\varepsilon.
+	\end{equation}
+	Dans ce cas, nous écrivons $\lim_{x\to a} f(x)=\ell$ et nous disons que $\ell$ est la \defe{limite}{limite} de $f$ lorsque $x$ tend vers $a$.
+\end{proposition}
+
+\begin{remark}
+    Le fait que nous limitions la formule \eqref{EqDefLimzxmasubV} aux \( x\) dans le domaine de \( f\) n'est pas anodin. Considérons la fonction \( f(x)=\sqrt{x^2-4}\), de domaine \( | x |\geq 2\). Nous avons
+    \begin{equation}
+        \lim_{x\to 2} \sqrt{x^2-4}=0.
+    \end{equation}
+    Nous ne pouvons pas dire que cette limite n'existe pas en justifiant que la limite à gauche n'existe pas. Les points \( x<2\) sont hors du domaine de \( f\) et ne comptent dons pas dans l'appréciation de l'existence de la limite.
+
+    Vous verrez plus tard que ceci provient de la \wikipedia{fr}{Topologie_induite}{topologie induite} de \( \eR\) sur l'ensemble \( \mathopen[ 2 , \infty [\).
+\end{remark}
diff --git a/tex/frido/60_EspacesVectos.tex b/tex/frido/60_EspacesVectos.tex
new file mode 100644
index 000000000..dbeac1e0f
--- /dev/null
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@@ -0,0 +1,431 @@
+% This is part of Mes notes de mathématique
+% Copyright (c) 2011-2017
+%   Laurent Claessens, Carlotta Donadello
+% See the file fdl-1.3.txt for copying conditions.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Sous espaces caractéristiques}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+% TODO : lire le blog de Pierre Bernard; en particulier celle-ci : http://allken-bernard.org/pierre/weblog/?p=2299
+
+Lorsqu'un opérateur n'est pas diagonalisable, les valeurs propres jouent quand même un rôle important.
+
+\begin{definition}  \label{DefFBNIooCGbIix}
+    Soit \( E\) un \( \eK\)-espace vectoriel  \( f\in\End(E)\). Pour \( \lambda\in \eK\) nous définissons
+    \begin{equation}
+        F_{\lambda}(f)=\{ v\in E\tq (f-\lambda\mtu)^nv=0, n\in\eN \}
+    \end{equation}
+    et nous appelons ça un \defe{sous-espace caractéristique}{sous-espace!caractéristique} de \( f\).
+\end{definition}
+L'espace \( F_{\lambda}(f)\) est l'ensemble de nilpotence de l'opérateur \( f-\lambda\mtu\) et
+
+\begin{lemma}   \label{LemBLPooHMAoyJ}
+    L'ensemble \( F_{\lambda}(f)\) est non vide si et seulement si \( \lambda\) est une valeur propre de \( f\). L'espace \( F_{\lambda}(f)\) est invariant sous \( f\).
+\end{lemma}
+
+\begin{proof}
+    Si \( F_{\lambda}(f)\) est non vide, nous considérons \( v\in F_{\lambda}(f)\) et \( n\) le plus petit entier non nul tel que \( (f-\lambda)^nv=0\). Alors \( (f-\lambda)^{n-1}v\) est un vecteur propre de \( f\) pour la valeur propre \( \lambda\). Inversement si \( v\) est une valeur propre de \( f\) pour la valeur propre \( \lambda\), alors \( v\in F_{\lambda}(f)\).
+
+    En ce qui concerne l'invariance, remarquons que \( f\) commute avec \( f-\lambda\mtu\). Si \( x\in F_{\lambda}(f)\) il existe \( n\) tel que \( (f-\lambda\mtu)^nx=0\). Nous avons aussi
+    \begin{equation}
+        (f-\lambda\mtu)^nf(x)=f\big( (f-\lambda\mtu)^nx \big)=0,
+    \end{equation}
+    par conséquent \( f(x)\in F_{\lambda}(f)\).
+\end{proof}
+
+\begin{remark}  \label{RemBOGooCLMwyb}
+    Toute matrice sur \( \eC\) n'est pas diagonalisable : nous en avons déjà donné une exemple simple en \ref{ExBRXUooIlUnSx}. Nous en voyons maintenant un moins simple. Considérons en effet l'endomorphisme \( f\) donné par la matrice
+    \begin{equation}
+        \begin{pmatrix}
+            a&    \alpha    &   \beta    \\
+            0    &   a    &   \gamma    \\
+            0    &   0    &   b
+        \end{pmatrix}
+    \end{equation}
+    où \( a\neq b\), \( \alpha\neq 0\), \( \beta\) et \( \gamma\) sont des nombres complexes quelconques.
+    Son polynôme caractéristique est 
+    \begin{equation}
+        \chi_f(\lambda)=(a-\lambda)^2(b-\lambda),
+    \end{equation}
+    et les valeurs propres sont donc \( a\) et \( b\). Nous trouvons les vecteurs propres pour la valeur \( a\) en résolvant
+    \begin{equation}
+        \begin{pmatrix}
+            a    &   \alpha    &   \beta    \\
+            0    &   a    &   \gamma    \\
+            0    &   0    &   b
+        \end{pmatrix}\begin{pmatrix}
+            x    \\ 
+            y    \\ 
+            z    
+        \end{pmatrix}=\begin{pmatrix}
+            ax    \\ 
+            ay    \\ 
+            az    
+        \end{pmatrix}.
+    \end{equation}
+    L'espace propre \( E_a(f)\) est réduit à une seule dimension générée par \( (1,0,0)\). De la même façon l'espace propre correspondant à la valeur propre \( b\) est donné par 
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{1}{ b-a }\left( \beta+\frac{ \alpha\gamma }{ b-a } \right)    \\ 
+            \frac{ \gamma }{ b-a }    \\ 
+            1    
+        \end{pmatrix}.
+    \end{equation}
+    Il n'y a donc pas trois vecteurs propres linéairement indépendants, et l'opérateur \( f\) n'est pas diagonalisable.
+
+    Par contre nous pouvons voir que
+    \begin{equation}
+        (f-\alpha\mtu)^2\begin{pmatrix}
+             0   \\ 
+            1    \\ 
+            0    
+        \end{pmatrix}=
+        \begin{pmatrix}
+            a    &   \alpha    &   \beta    \\
+            0    &   a    &   \gamma    \\
+            0    &   0    &   b
+        \end{pmatrix}
+        \begin{pmatrix}
+            \alpha    \\ 
+            0    \\ 
+            0    
+        \end{pmatrix}-\begin{pmatrix}
+            a\alpha    \\ 
+            0    \\ 
+            0    
+        \end{pmatrix}=\begin{pmatrix}
+            0    \\ 
+            0    \\ 
+            0    
+        \end{pmatrix},
+    \end{equation}
+    de telle sorte que le vecteur \( (0,1,0)\) soit également dans l'espace caractéristique \( F_a(f)\).
+
+    Dans cet exemple, la multiplicité algébrique de la racine \( a\) du polynôme caractéristique vaut \( 2\) tandis que sa multiplicité géométrique vaut seulement \( 1\).
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorèmes de décomposition}
+%---------------------------------------------------------------------------------------------------------------------------
+
+%TODO : Je crois qu'on peut remplacer l'hypothèse de corps algébriquement clos par le polynôme caractéristique scindé.
+\begin{theorem}[Théorème spectral, décomposition primaire]\index{théorème!spectral}     \label{ThoSpectraluRMLok}
+    Soit \( E\) espace vectoriel de dimension finie sur le corps algébriquement clos \( \eK\) et \( f\in\End(E)\). Alors
+    \begin{equation}    \label{EqCTFHooBSGhYK}
+        E=F_{\lambda_1}(f)\oplus\ldots\oplus F_{\lambda_k}(f)
+    \end{equation}
+    où la somme est sur les valeurs propres distinctes de \( f\).
+
+    Les projecteurs sur les espaces caractéristique forment un système complet et orthogonal.
+\end{theorem}
+\index{décomposition!primaire}
+\index{décomposition!spectrale}
+\index{décomposition!sous-espaces caractéristiques}
+
+\begin{proof}
+    Soit \( P\) le polynôme caractéristique de \( f\) et une décomposition
+    \begin{equation}
+        P=(f-\lambda_1)^{\alpha_1}\ldots(f-\lambda_r)^{\alpha_r}
+    \end{equation}
+    en facteurs irréductibles. La le théorème de noyaux (\ref{ThoDecompNoyayzzMWod}) nous avons
+    \begin{equation}        \label{EqDeFVSaYv}
+        E=\ker(f-\lambda_1)^{\alpha_1}\oplus\ldots\oplus\ker(f-\lambda_r)^{\alpha_r}.
+    \end{equation}
+    Les projecteurs sont des polynômes en \( f\) et forment un système orthogonal. Il nous reste à prouver que \( \ker(f-\lambda_i)^{\alpha_i}=F_{\lambda_i}(f)\). L'inclusion
+    \begin{equation}    \label{EqzmNxPi}
+        \ker(f-\lambda_i)^{\alpha_i}\subset F_{\lambda_i}(f)
+    \end{equation}
+    est évidente. Nous devons montrer l'inclusion inverse. Prouvons que la somme des \( F_{\lambda_i}(f)\) est directe. Si \( v\in F_{\lambda_i}(f)\cap F_{\lambda_j}(f)\), alors il existe \( v_1=(f-\lambda_i)^nv\neq 0\) avec \( (f-\lambda_i)v_1=0\). Étant donné que \( (f-\lambda_i)\) commute avec \( (f-\lambda_j)\), ce \( v_1\) est encore dans \( F_{\lambda_j}(f)\) et par conséquent il existe \( w=(f-\lambda_j)^mv_1\) non nul tel que 
+    \begin{subequations}
+        \begin{numcases}{}
+            (f-\lambda_i)w=0\\
+            (f-\lambda_j)w=0.
+        \end{numcases}
+    \end{subequations}
+    Ce \( w\) serait donc un vecteur propre simultané pour les valeurs propres \( \lambda_i\) et \( \lambda_j\), ce qui est impossible parce que les espaces propres sont linéairement indépendants. Les espaces \( F_{\lambda_i}\) sont donc en somme directe et
+    \begin{equation}
+        \sum_i\dim F_{\lambda_i}(f)\leq \dim E.
+    \end{equation}
+    En tenant compte de l'inclusion \eqref{EqzmNxPi} nous avons même
+    \begin{equation}
+        \dim E=\sum_i\dim\ker(f-\lambda_i)^{\alpha_i}\leq\sum_i F_{\lambda_i}(f)\leq \dim E.
+    \end{equation}
+    Par conséquent nous avons \( \dim\ker(f-\lambda_i)^{\alpha_i}=\dim F_{\lambda_i}(f)\) et l'égalité des deux espaces.
+\end{proof}
+
+
+\begin{probleme}
+    Dans le cas où le corps n'est pas algébriquement clos, il paraît qu'il faut remplacer «diagonalisable» par «semi-simple».
+\end{probleme}
+%TODO : peut-être qu'il y a la réponse dans http://www.math.jussieu.fr/~romagny/agreg/dvt/endom_semi_simples.pdf
+
+Si l'espace vectoriel est sur un corps algébriquement clos, alors les endomorphismes semi-simples\footnote{Définition \ref{DEFooBOHVooSOopJN}.} sont les endomorphismes diagonaux.
+
+
+%TODO : Je crois qu'on peut remplacer l'hypothèse de corps algébriquement clos par le polynôme caractéristique scindé.
+\begin{theorem}[Décomposition de Dunford] \label{ThoRURcpW}
+    Soit \( E\) un espace vectoriel sur le corps algébriquement clos \( \eK\) et \( u\in\End(E)\) un endomorphisme de \( E\). 
+    
+    \begin{enumerate}
+        \item
+            
+            L'endomorphisme \( u\) se décompose de façon unique sous la forme
+            \begin{equation}
+                u=s+n
+            \end{equation}
+            où \( s\) est diagonalisable, \( n\) est nilpotent et \( [s,n]=0\).
+        \item
+            Les endomorphismes \( s\) et \( n\) sont des polynômes en \( u\) et commutent avec \( u\).
+        \item   \label{ItemThoRURcpWiii}
+            Les parties \( s\) et \( n\) sont données par
+            \begin{subequations}
+                \begin{align}
+                    s&=\sum_i\lambda_ip_i\\
+                    n&=\sum_i(s-\lambda_i\mtu)p_i
+                \end{align}
+            \end{subequations}
+            où les sommes sont sur les valeurs propres distinctes\footnote{C'est à dire sur les sous-espaces caractéristiques.} de \( f\) et où \( p_i\colon E\to F_{\lambda_i}(u)\) est la projection de \( E\) sur \( F_{\lambda_i}(u)\).
+    \end{enumerate}
+\end{theorem}
+\index{décomposition!Dunford}
+\index{Dunford!décomposition}
+\index{réduction!d'endomorphisme}
+\index{endomorphisme!sous-espace stable}
+\index{polynôme!d'endomorphisme!décomposition de Dunford}
+\index{endomorphisme!diagonalisable!Dunford}
+\index{endomorphisme!nilpotent!Dunford}
+%TODO : comprendre comment on calcule des exponentielles de matrices avec Dunford.
+
+\begin{proof}
+    Le théorème spectral \ref{ThoSpectraluRMLok} nous indique que
+    \begin{equation}
+        E=\bigoplus_iF_{\lambda_i}(f).
+    \end{equation}
+    Nous considérons l'endomorphisme \( s\) de \( E\) qui consiste à dilater d'un facteur \( \lambda\) l'espace caractéristique \( F_{\lambda}(f)\) :
+    \begin{equation}
+        s=\sum_i\lambda_ip_i
+    \end{equation}
+    où \( p_i\colon E\to F_{\lambda_i}(u)\) est la projection de \( E\) sur \( F_{\lambda_i}(u)\).
+
+    Nous allons prouver que \( [s,f]=0\) et \( n=f-s\) est nilpotent. Cela impliquera que \( [s,n]=0\).
+
+    Si \( x\in F_{\lambda}(f)\), alors nous avons \( sf(x)=\lambda f(x)\) parce que \( f(x)\in F_{\lambda}(f)\) tandis que \( fs(x)=f(\lambda x)=\lambda f(x)\). Par conséquent \( f\) commute avec \( s\).
+
+    Pour montrer que \( f-s\) est nilpotent, nous en considérons la restriction
+    \begin{equation}
+        f-s\colon F_{\lambda}(f)\to F_{\lambda}(f).
+    \end{equation}
+    Cet opérateur est égal à \( f-\lambda\mtu\) et est par conséquent nilpotent.
+
+    Prouvons à présent l'unicité. Soit \( u=s'+n'\) une autre décomposition qui satisfait aux conditions : \( s'\) est diagonalisable, \( n'\) est nilpotent et \( [n',s']=0\). Commençons par prouver que \( s'\) et \( n'\) commutent avec \( u\). En multipliant \( u=s'+n'\) par \( s'\) nous avons
+    \begin{equation}
+        s'u=s'^2+s'n'=s'^2+n's'=(s'+n')s'=us',
+    \end{equation}
+    par conséquent \( [u,s']=0\). Nous faisons la même chose avec \( n'\) pour trouver \( [u,n']=0\). Notons que pour obtenir ce résultat nous avons utilisé le fait que \( n'\) et \( s'\) commutent, mais pas leur propriétés de nilpotence et de diagonalisibilité.
+    
+    
+    Si \( s'+n'=s+n\) est une autre décomposition, \( s'\) et \( n'\) commutent avec \( u\), et par conséquent avec tous les polynômes en \( u\). Ils commutent en particulier avec \( n\) et \( s\). Les endomorphismes \( s\) et \( s'\) sont alors deux endomorphismes diagonalisables qui commutent. Par la proposition \ref{PropGqhAMei}, ils sont simultanément diagonalisables. Dans la base de simultanée diagonalisation, la matrice de l'opérateur \( s'-s=n-n'\) est donc diagonale. Mais \( n-n'\) est également nilpotent, en effet si \( A\) et \( B\) sont deux opérateurs nilpotents,
+    \begin{equation}
+        (A+B)^n=\sum_{k=0}^n\binom{k}{n}A^kB^{n-k}.
+    \end{equation}
+    Si \( n\) est assez grand, au moins un parmi \( A^k\) ou \( B^{n-k}\) est nul.
+
+    Maintenant que \( n-n'\) est diagonal et nilpotent, il est nul et \( n=n'\). Nous avons alors immédiatement aussi \( s=s'\).
+
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Diverses conséquences}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{theorem}
+    Soit une matrice \( A\in \eM(n,\eC)\). On a que la suite \( (A^kx)\) tends vers zéro pour tout \( x\) si et seulement si \( \rho(A)<1\) où \( \rho(A)\)\index{rayon!spectral} est le rayon spectral de $A$
+\end{theorem}
+\index{décomposition!Dunford!exponentielle de matrice}
+
+\begin{proof}
+    Dans le sens direct, il suffit de prendre comme \( x\), un vecteur propre de \( A\). Dans ce cas nous avons \( A^kx=\lambda^kx\). Mais \( \lambda^kx\) ne tend vers zéro que si \( \lambda<1\). Donc toute les valeurs propres de \( A\) doivent être plus petite que \( 1\) et \( \rho(A)<1\).
+
+    Pour l'autre sens nous utilisons la décomposition de Dunford (théorème \ref{ThoRURcpW}) : il existe une matrice inversible \( P\) telle que
+    \begin{equation}
+        A=P^{-1}(D+N)P
+    \end{equation}
+    où \( D\) est diagonale, \( N\) est nilpotente et \( [D,N]=0\). Étant donné que \( D+N\) est triangulaire, son polynôme caractéristique que
+    \begin{equation}
+        \chi_{D+N}(\lambda)=\prod_i D_{ii}-\lambda.
+    \end{equation}
+    Par similitude, c'est le même polynôme caractéristique que celui de \( A\) et nous savons alors que la diagonale de \( D\) contient les valeurs propres de \( A\).
+
+    Par ailleurs nous avons
+    \begin{subequations}
+        \begin{align}
+            A^k&=P^{-1}(D+N)^kP\\
+            &=P^{-1}\sum_{j=0}^k{j\choose k}D^{j-k}N^jP\\
+            &=P^{-1}\sum_{j=0}^{n-1}{j\choose k}D^{j-k}N^jP
+        \end{align}
+    \end{subequations}
+    où nous avons utilité le fait que \( D\) et \( N\) commutent ainsi que \( N^{n-1}=0\) parce que \( N\) est nilpotente. Nous utilisons la norme matricielle usuelle, pour laquelle \( \| D \|=\rho(D)=\rho(A)\). Nous avons alors
+    \begin{equation}
+        \| (D+N)^k \|\leq \sum_{j=0}^k{j\choose k}\rho(D)^{k-j}\| N \|^j.
+    \end{equation}
+    Du coup si \( \rho(D)<1\) alors \( \| (D+N)^k \|\to 0\) (et c'est même un si et seulement si).
+\end{proof}
+
+Une application de la décomposition de Jordan est l'existence d'un logarithme pour les matrices. La proposition suivant va d'une certaine manière donner un logarithme pour les matrices inversibles complexes. Dans le cas des matrices réelles \( m\) telles que \( \| m-\mtu \|<1\), nous donnerons au lemme \ref{LemQZIQxaB} une formule pour le logarithme sous forme d'une série; ce logarithme sera réel.
+\begin{proposition} \label{PropKKdmnkD}
+    Toute matrice inversible complexe est une exponentielle.
+\end{proposition}
+\index{exponentielle!de matrice}
+\index{décomposition!Jordan!et exponentielle de matrice}
+
+\begin{proof}
+    Soit \( A\in \GL(n,\eC)\); nous allons donner une matrice \( B\in \eM(n,\eC)\) telle que \( A=\exp(B)\). D'abord remarquons qu'il suffit de prouver le résultat pour une matrice par classe de similitude. En effet si \( A=\exp(B)\) et si \( M\) est inversible alors 
+    \begin{subequations}    \label{EqqACuGK}
+        \begin{align}
+            \exp(MBM^{-1})&=\sum_k\frac{1}{ k! }(MBM^{-1})^k\\
+            &=\sum_k\frac{1}{ k! }MB^kM^{-1}\\
+            &=M\exp(B)M^{-1}.
+        \end{align}
+    \end{subequations}
+    Donc \( MAM^{-1}=\exp(MBM^{-1})\). Nous pouvons donc nous contenter de trouver un logarithme pour les blocs de Jordan. Nous supposons donc que \( A=(\mtu+N)\) avec \( N^m=0\). 
+    En nous inspirant de \eqref{EqweEZnV}, nous posons\footnote{Le logarithme d'un nombre n'est pas encore définit à ce moment, mais cela ne nous empêche pas de poser une définition ici pour une application des réels vers les matrices.}
+    \begin{equation}
+        D(t)=tN-\frac{ t^2 }{ 2 }N^2+\cdots +(-1)^m\frac{ t^{m-1} }{ m-1 }N^{m-1}
+    \end{equation}
+    et nous allons prouver que \(  e^{D(1)}=\mtu+N\). Notons que \( N\) étant nilpotente, cette somme ainsi que toutes celles qui viennent sont finies. Il n'y a donc pas de problèmes de convergences dans cette preuve (si ce n'est les passages des équations \eqref{EqqACuGK}).
+
+    Nous posons \( S(t)= e^{D(t)}\) (la somme est finie), et nous avons
+    \begin{equation}
+        S'(t)=D'(t) e^{D(t)}
+    \end{equation}
+    Afin d'obtenir une expression qui donne \( S'\) en termes de \( S\), nous multiplions par \( (\mtu+tN)\) en remarquant que \( (\mtu+tN)D'(t)=N\) nous avons
+    \begin{equation}
+        (\mtu+tN)S'(t)=NS(t).
+    \end{equation}
+    En dérivant à nouveau,
+    \begin{equation}    \label{EqKjccqP}
+        (\mtu+tN)S''(t)=0.
+    \end{equation}
+    La matrice \( (\mtu+tN)\) est inversible parce que son noyau est réduit à \( \{ 0 \}\). En effet si \( (\mtu+tN)x=0\), alors \( Nx=-\frac{1}{ t }x\), ce qui est impossible parce que \( N\) est nilpotente. Ce que dit l'équation \eqref{EqKjccqP} est alors que \( S''(t)=0\). Si nous développons \( S(t)\) en puissances de \( t\) nous nous arrêtons au terme d'ordre \( 1\) et nous avons
+    \begin{equation}
+        S(t)=S(0)+tS'(0)=\mtu+tD'(0)=1+tN.
+    \end{equation}
+    En \( t=1\) nous trouvons \( S(1)=\mtu+N\). La matrice \( D(1)\) donnée est donc bien un logarithme de $\mtu+N$.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Diagonalisabilité d'exponentielle}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[\cite{fJhCTE}]      \label{PropCOMNooIErskN}
+    Si \( A\in \eM(n,\eR)\) a un polynôme caractéristique scindé, alors \( A\) est diagonalisable si et seulement si \( e^A\) est diagonalisable.
+\end{proposition}
+\index{décomposition!Dunford!application}
+\index{exponentielle!de matrice}
+\index{diagonalisable!exponentielle}
+
+\begin{proof}
+    Si \( A\) est diagonalisable, alors il existe une matrice inversible \( M\) telle que \( D=M^{-1}AM\) soit diagonale (c'est la définition \ref{DefCNJqsmo}). Dans ce cas nous avons aussi \( (M^{-1}AM)^k=M^{-1}A^kM\) et donc \( M^{-1}e^AM=e^{M^{-1}AM}=e^D\) qui est diagonale.
+
+    La partie difficile est donc le contraire. 
+    
+    \begin{subproof}
+        \item[Qui est diagonalisable et comment ?]
+            Nous supposons que \( e^A\) est diagonalisable et nous écrivons la décomposition de Dunford (théorème \ref{ThoRURcpW}) :
+            \begin{equation}
+                A=S+N
+            \end{equation}
+            où \( S\) est diagonalisable, \( N\) est nilpotente, \( [S,N]=0\). Nous avons besoin de prouver que \( N=0\).
+    
+            Les matrices \( A\) est \( S\) commutent; en passant au développement nous en déduisons que \( A\) et \( e^S\) commutent, puis encore en passant au développement que \( e^A\) et \( e^S\) commutent. Vu que \( S\) est diagonalisable, \( e^S\) l'est et par hypothèse \( e^A\) est également diagonalisable. Donc \( e^A\) et \( e^{-S}\) sont simultanément diagonalisables par la proposition \ref{PropGqhAMei}.
+
+            Étant donné que \( A\) et \( S\) commutent, nous avons \( e^N=e^{A-S}=e^Ae^{-S}\), et nous en déduisons que \( e^N\) est diagonalisable vu que les deux facteurs \( e^A\) et \( e^{-S}\) sont simultanément diagonalisables.
+
+        \item[Unipotence]
+
+            Si \( r\) est le degré de nilpotence de \( N\), nous avons
+            \begin{equation}    \label{EqQHjvLZQ}
+                e^N-\mtu=N+\frac{ N^2 }{2}+\cdots +\frac{ N^{r-1} }{ (r-1)! }.
+            \end{equation}
+            Donc
+            \begin{equation}
+                (e^N-\mtu)^k=\left( N+\frac{ N^2 }{2}+\cdots +\frac{ N^{r-1} }{ (r-1)! } \right)^k
+            \end{equation}
+            où le membre de droite est un polynôme en \( N\) dont le terme de plus bas degré est de degré \( k\). Donc \( (e^N-\mtu)\) est nilpotente et \( e^N\) est unipotente.
+
+            Si \( M\) est la matrice qui diagonalise \( e^N\), alors la matrice diagonale \( M^{-1}e^NM\) est tout autant unipotente que \( e^N\) elle-même. En effet,
+            \begin{subequations}
+                \begin{align}
+                    (M^{-1}e^NM-\mtu)^r&=\sum_{k=0}^r\binom{ r }{ k }(-1)^{r-k}M^{-1}(e^N)^kM\\
+                    &=M^{-1}\left( \sum_{k=0}^r\binom{ r }{ k }(-1)^{r-k}(e^N)^k \right)M\\
+                    &=M^{-1}(e^N-\mtu)^rM\\
+                    &=0.
+                \end{align}
+            \end{subequations}
+
+            La matrice \( M^{-1}e^NM\) est donc une matrice diagonale et unipotente; donc \( M^{-1}e^NM=\mtu\), ce qui donne immédiatement que \( e^N=\mtu\).
+
+        \item[Polynômes annulateurs]
+
+            En reprenant le développement \eqref{EqQHjvLZQ} sachant que \( e^N=\mtu\), nous savons que
+            \begin{equation}
+                N+\frac{ N^2 }{2}+\cdots +\frac{ N^{r-1} }{ (r-1)! }=0.
+            \end{equation}
+            Dit en termes pompeux (mais non moins porteurs de sens), le polynôme
+            \begin{equation}
+                Q(X)=X+\frac{ X^2 }{2}+\cdots +\frac{ X^{r-1} }{ (r-1)! }
+            \end{equation}
+            est un polynôme annulateur de \( N\).
+            
+            La proposition \ref{PropAnnncEcCxj} stipule que le polynôme minimal d'un endomorphisme divise tous les polynômes annulateurs. Dans notre cas, \( X^r\) est un polynôme annulateur et donc le polynôme minimal de \( N\) est de la forme \( X^k\). Donc il est \( X^r\) lui-même.
+            
+            Nous avons donc \( X^r\divides Q\). Mais \( Q\) est un polynôme contenant le monôme \( X\) donc \( X^r\) ne peut diviser \( Q\) que si \( r=1\). Nous en concluons que \( X\) est un polynôme annulateur de \( N\). C'est à dire que \( N=0\).
+
+        \item[Conclusion]
+
+            Vu que Dunford\footnote{Théorème \ref{ThoRURcpW}.} dit que \( A=S+N\) et que nous venons de prouver que \( N=0\), nous concluons que \( A=S\) avec \( S\) diagonalisable.
+
+    \end{subproof}
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Valeurs singulières}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit \( M\) une matrice \( m\times n\) sur \( \eK\) (\( \eK\) est \( \eR\) ou \( \eC\)). Un nombre réel \( \sigma\) est une \defe{valeur singulière}{valeur!singulière} de \( M\) s'il existent des vecteurs unitaires \( u\in \eK^m\), \( v\in \eK^n\) tels que
+    \begin{subequations}
+        \begin{align}
+            Mv&=\sigma u\\
+            M^*u&=\sigma v.
+        \end{align}
+    \end{subequations}
+\end{definition}
+
+\begin{theorem}[Décomposition en valeurs singulières]
+    Soit \( M\in \eM(m\times n,\eK)\) où \( \eK=\eR,\eC\). Alors \( M\) se décompose en
+    \begin{equation}
+        M=ADB
+    \end{equation}
+    où
+    il existe deux matrices unitaires \( A\in \gU(m\times m)\), \( B\in \gU(n\times n)\) et une matrice (pseudo)diagonale \( D\in \eM(m\times n)\) tels que
+    \begin{enumerate}
+        \item 
+            \( A\in\gU(m\times m)\), \( B\in\gU(n\times n)\) sont deux matrices unitaires;,
+        \item
+            \( D\) est (pseudo)diagonale,
+        \item
+            les éléments diagonaux de \( D\) sont les valeurs singulières de \( M\),
+        \item
+            le nombre d'éléments non nuls sur la diagonale de \( D\) est le rang de \( M\).
+    \end{enumerate}
+\end{theorem}
+
+\begin{corollary}
+    Soit \( M\in \eM(n,\eC)\). Il existe un isomorphisme \( f\colon \eC^n\to \eC^n\) tel que \( fM\) soit autoadjoint.
+\end{corollary}
+
+\begin{proof}
+    Si \( M=ADB\) est la décomposition de \( M\) en valeurs singulières, alors nous pouvons prendre \( f=\overline{ B }^tA^{-1}\) qui est une matrice inversible. Pour la vérification que ce \( f\) répond bien à la question, ne pas oublier que \( D\) est réelle, même si \( M\) ne l'est pas.
+\end{proof}

From 2fac2474457d65ffe391d950cbd52bf71636dc4c Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 24 Jun 2017 23:45:19 +0200
Subject: [PATCH 51/64] =?UTF-8?q?(organisation)=20Renomme=20encore=20un=20?=
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diff --git a/mazhe.tex b/mazhe.tex
index 3257ae4cc..c7f17df78 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -163,14 +163,14 @@ \chapter{Topologie générale}
 \input{54_topologieR}
 
 \chapter{Espaces vectoriels}
-\input{55_espace_vecto_norme}
-\input{144_espace_vecto_norme}
-\input{56_espace_vecto_norme}
-\input{57_espace_vecto_norme}
-\input{59_EspacesVecto}
-\input{60_EspacesVecto}
-\input{138_EspacesVecto}
-\input{141_EspacesVecto}
+\input{55_EspacesVectos}
+\input{144_EspacesVectos}
+\input{56_EspacesVectos}
+\input{57_EspacesVectos}
+\input{59_EspacesVectos}
+\input{60_EspacesVectos}
+\input{138_EspacesVectos}
+\input{141_EspacesVectos}
 
 \chapter{Espaces vectoriels normés}
 \input{181_espace_vecto_norme}
diff --git a/tex/frido/138_EspacesVecto.tex b/tex/frido/138_EspacesVecto.tex
deleted file mode 100644
index 3faadbb73..000000000
--- a/tex/frido/138_EspacesVecto.tex
+++ /dev/null
@@ -1,980 +0,0 @@
-% This is part of Mes notes de mathématique
-% Copyright (c) 2011-2016
-%   Laurent Claessens, Carlotta Donadello
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Extension du corps de base}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\label{SECooAUOWooNdYTZf}
-
-Nous avons discuté dans la section \ref{SECooLQVJooTGeqiR} de ce qui arrive au corps lorsqu'on l'étend. Dans cette sections nous allons étudier ce qui arrive aux applications linéaires entre deux \( \eK\)-espaces vectoriels lorsque nous étendons le corps \( \eK\) en un corps \( \eL\).
-
-Soit donc un corps \( \eK\) et deux \( \eK\)-espaces vectoriels \( E\) et \( F\), et entrons dans le vif du sujet\footnote{Le sujet étant le corps étendu.}. Soit \( \eK\) un corps (commutatif) et une extension \( \eL\) de \( \eK\). Soient \( E\) et \( F\), des \( \eK\)-espaces vectoriels de dimension finie. 
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Extension des applications linéaires}
-%---------------------------------------------------------------------------------------------------------------------------
-
-
-\begin{definition}[\cite{ooAFBYooYvTCCN}]
-    L'espace vectoriel obtenu par \defe{extension du corps de base}{extension!corps de base} de \( E\) est l'espace vectoriel
-    \begin{equation}
-        E_{\eL}=\eL\otimes_{\eK}E.
-    \end{equation}
-    Ce dernier est le quotient \( \eL\otimes_{\eK}E=(\eL\times E)/\sim\) par la relation d'équivalence
-    \begin{equation}
-        (\lambda,v)\sim\big( a\lambda,\frac{1}{ a }v \big)
-    \end{equation}
-    pour tout \( a\in \eK\). Nous noterons \( [\lambda,v]\) ou \( \lambda\otimes v\) ou encore \( \lambda\otimes_{\eK}v\) la classe de \( (\lambda,v)\).
-\end{definition}
-Un élément de \( E_{\eL}\) est de la forme \( \sum_k[\lambda_k,v_k]\) avec \( \lambda_k\in \eL\) et \( v_k\in E\). Si \( f\colon E\to F\) est une applications linéaire nous définissons
-\begin{equation}
-    \begin{aligned}
-        f_{\eL}\colon E_{\eL}&\to F_{\eL} \\
-        [\lambda,v]&\mapsto [\lambda,f(v)]. 
-    \end{aligned}
-\end{equation}
-
-\begin{remark}
-    Si deux vecteurs de \( E_{\eL}\) sont linéairement indépendants pour \( \eK\), ils ne le sont pas spécialement pour \( \eL\). Par exemple si \( \eC\) est vu comme \( \eR\)-espace vectoriel, alors \( \{ 1,i \}\) est une partie libre. Mais dans \( \eC\) vu comme \( \eC\)-espace vectoriel, la partie \( \{ 1,i \}\) n'est pas libre.
-\end{remark}
-
-Nous définissons aussi l'injection canonique
-\begin{equation}
-    \begin{aligned}
-        \iota\colon E&\to E_{\eL} \\
-        v&\mapsto [1,v]. 
-    \end{aligned}
-\end{equation}
-
-\begin{proposition}[\cite{ooEPEFooQiPESf}]      \label{PropooWECLooHPzIHw}
-    Injectivité et surjectivité respectées.
-    \begin{enumerate}
-        \item
-            L'application \( f_{\eL}\) est injective si et seulement si \( f\) est injective.    
-        \item
-            L'application \( f_{\eL}\) est surjective si et seulement si \( f\) est surjective.    
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Supposons pour commencer que \( f_{\eL}\) est injective.
-    Le diagramme
-    \begin{equation}
-        \xymatrix{%
-            E \ar[r]^-{f}\ar[d]_-{\tau}      &   F\ar[d]^{\tau}\\
-            E_{\eL} \ar[r]_{f_{\eL}}  &   F_{\eL}
-           }
-    \end{equation}
-    est un diagramme commutatif. En effet 
-    \begin{equation}
-        (\tau\circ f)(v)=[1,f(v)]
-    \end{equation}
-    tandis que 
-    \begin{equation}
-        (f_{\eL\circ\tau})(v)=f_{\eL}[1,v]=[1,f(v)].
-    \end{equation}
-    Donc si \( f(v)=0\) avec \( v\neq 0\) nous aurions \( (\tau\circ f)(v)=0\) et donc aussi \( (f_{\eL}\circ \tau)(v)=0\), alors que \( \tau(v)\neq 0\) dans \( E_{\eL}\).
-
-    Réciproquement, nous supposons que \( f\) est injective et nous prouvons que \( f_{\eL}\) est injective. Par le lemme \ref{LEMooDAACooElDsYb}\ref{ITEMooEZEWooZGoqsZ}, nous savons qu'il existe \( g\colon F\to E\) telle que \( f\circ g=\id|_F\). Nous en déduisons que \( f_{\eL}\circ g_{\eL}=\id|_{F_{\eL}}\) parce que si \( [\lambda,v]\in F_{\eL}\) alors
-    \begin{equation}
-        (f_{\eL}\circ g_{\eL})[\lambda,v]=f_{\eL}[\lambda,g(v)]=[\lambda,(f\circ g)(v)]=[\lambda,v].
-    \end{equation}
-    Notons que \( g\) est injective, donc \( g_{\eL}\) est injective et l'égalité \( f_{\eL}\circ g_{\eL}=\id|_{F_{\eL}} \) implique que \( f_{\eL}\) est également injective.
-\end{proof}
-
-\begin{proposition}[\cite{MonCerveau}] \label{PROPooMHARooUycAts}
-    Soit \( \{ e_i \}_{i=1,\ldots, p}\) une base de \( E\). Alors \( \{ 1\otimes e_i \}_i\) est une base de \( E_{\eL}=\eL\otimes_{\eK}E\).
-\end{proposition}
-
-\begin{proof}
-    L'espace vectoriel \( E\) peut être écrit comme somme directe \( E=\bigoplus_i\eK e_i\). Si \( \lambda\in \eL\) et \( k\in \eK\) nous avons
-    \begin{equation}
-        \lambda\otimes ke_i=\frac{ \lambda }{ k }\otimes e_i=\frac{ \lambda }{ k }(1\otimes e_i).
-    \end{equation}
-    Cela pour introduire que l'application
-    \begin{equation}
-        \begin{aligned}
-            \psi\colon \eL\otimes_{\eK}E&\to \bigoplus_i\eL(1\otimes e_i) \\
-            \sum_k \lambda_k\otimes v_k&\mapsto \oplus_i \sum_k(\lambda_k v_{ik})(1\otimes e_i)
-        \end{aligned}
-    \end{equation}
-    où \( v_k=\sum_i v_{ik}e_i\) avec \( v_{ik}\in \eK\) est un isomorphisme de \( \eL\)-espaces vectoriels. La surjectivité est facile. En ce qui concerne l'injectivité, si
-    \begin{equation}
-        \sum_i\sum_k(\lambda_kv_{ik})(1\otimes e_i)=0
-    \end{equation}
-    alors les choses suivantes sont nulles également :
-    \begin{equation}
-        \sum_i\sum_k(\lambda_kv_{ik})(1\otimes e_i)=\sum_{ik}(\lambda_k\otimes v_{ik}e_i)=\sum_k(\lambda_k\otimes \sum_iv_{ik}e_i)=\sum_k(\lambda_k\otimes v_k).
-    \end{equation}
-    Le dernier est l'argument de \( \psi\). Le fait que ce soit nul implique que \( \psi\) est injective.
-\end{proof}
-
-\begin{remark}
-    Nous n'avons pas dû prouver que chacun des \( \lambda_k\otimes v_k\) était nul. Et encore heureux, parce que cela pouvait très bien être faux, vu qu'il y a plusieurs façons de noter un élément de \( E_{\eL}\) sous la forme de tels termes.
-\end{remark}
-
-\begin{corollary}       \label{CORooTQGHooIKhNtr}
-    La \( \eL\)-dimension de \( E_{\eL}\) est égale à la \( \eK\)-dimension de \( E\).
-\end{corollary}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Projections}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{probleme}
-    Nous allons définir \( \pr\colon \aL(E_{\eL},F_{\eL})\to \aL(E,F)\) en faisant appel à des bases et en prouvant que les choses définies ne dépendent pas des bases choisies. Il y a sûrement une façon plus «intrinsèque» de faire.
-\end{probleme}
-
-
-Nous savons que \( \eL\) est un \( \eK\)-espace vectoriel dans lequel nous pouvons voir \( \eK\) comme un sous-espace (lemme \ref{LemooOLIIooXzdppM}). Dans cette optique nous choisissons dans \( \eL\) un supplémentaire de \( \eK\), c'est à dire un sous-espace vectoriel de \( \eL\) tel que
-\begin{equation}
-    \eL=\eK\oplus V.
-\end{equation}
-Nous avons alors naturellement une projection \( \pr\colon \eL\to \eK\).
-
-Soit \( \{ e_i \}\) une base de \( E \) et \(\{ e_a \}\) une  de\( F\). Nous noterons également \( e_i\) et \( e_a\) les éléments \( \tau e_i\) et \( \tau e_a\) correspondants. Grâce à la proposition \ref{PROPooMHARooUycAts}, ce sont des bases de \( E_{\eL}\) et \( F_{\eL}\). Si la fonction \( f\colon E_{\eL}\to F_{\eL}\) s'écrit dans ce ces bases comme
-\begin{equation}
-    f(e_i)=\sum_af_{ai}e_a
-\end{equation}
-alors nous définissons \( \pr(f)\) par 
-\begin{equation}        \label{EQooSAFRooJnfkLO}
-    (\pr f)e_i=\sum_a\pr(f_{ai})e_a.
-\end{equation}
-
-\begin{proposition}[\cite{MonCerveau}]      \label{PROPooOEHTooHyjuZQ}
-    L'application \( \pr\) définie en \eqref{EQooSAFRooJnfkLO} est indépendante du choix des bases.
-\end{proposition}
-
-\begin{proof}
-    Notons que dans ce qui suit, les sommes sur \( a\) ou \( b\) et celles sur \( i\) ou \( j\) ne vont pas jusqu'au même indice (dimensions de \( E\) et \( F\)). De plus nous manipulons deux choses qui se notent \( \pr\). La première est la projection \( \pr\colon \eL\to \eK\) qui ne dépend que d'un choix de supplémentaire et que nous supposons fixée ici. D'autre part il y a \( \pr\colon E_{\eL}\to E\) qui dépend a priori des bases choisies.
-
-    Nous choisissons de nouvelles bases qui sont liées aux anciennes bases par
-    \begin{subequations}
-        \begin{numcases}{}
-            e'_b=\sum_aB_{ab}e_a\\
-            e'_i=\sum_jA_{ji}e_j.
-        \end{numcases}
-    \end{subequations}
-    Les matrices \( A\) et \( B\) sont dans \( \GL(\eK)\). Nous allons écrire l'opérateur \( \pr'\) qui correspond à ces bases et montrer que pour toute application linéaire \( f\colon E_{\eL}\to F_{\eL} \) nous avons \( \pr(f)=\pr'(f)\). Nous avons :
-    \begin{subequations}
-        \begin{align}
-            f(e'_j)&=\sum_iA_{ji}f(e_i)\\
-            &=\sum_a\sum_b\sum_iA_{ji}f_{ai}(B^{-1})_{ba}e'b\\
-            &=\sum_b\Big( \sum_{ai}A_{ji}f_{ai}(B^{-1})_{ba} \Big)e'b,
-        \end{align}
-    \end{subequations}
-    ce qui fait que
-    \begin{equation}        \label{EQooUQNBooMWHRbD}
-        (\pr'f)e'_j=\sum_b\Big( \pr\big( A_{ji}f_{ai}(B^{-1})_{ba} \big) \Big)e'_b.
-    \end{equation}
-    Nous calculons maintenant \( (\pr'f)e_j\) en substituant \( e_j=\sum_l(A^{-1})_{lj}e'_l\) et en utilisant \eqref{EQooUQNBooMWHRbD} et la linéarité de \( \pr'\) et la \( \eK\)-linéarité de \( \pr\colon \eL\to \eK\) :
-    \begin{subequations}
-        \begin{align}
-            (\pr'f)\Big( \sum_l(A^{-1})_{lj}e'_l \Big)
-            &=\sum_l(A^{-1})_{lj}\sum_b\sum_{ai}\pr\big(A_{li}f_{ai}(B^{-1})_{ba}\big)e_b\\
-            &=\sum_a\pr(f_{aj})e_a\\
-            &=(\pr f)e_j.
-        \end{align}
-    \end{subequations}
-    Donc \( \pr=\pr'\).
-\end{proof}
-
-Note au passage comme toujours : il y a un abus systématique de notation entre \( e_i\in E\) et \( \tau(e_i)=1\otimes e_i\in E_{\eL}\).
-
-\begin{remark}[\cite{MonCerveau}]       \label{REMooBEXGooLgpHzg}
-    L'opération \( \pr\colon \aL(E_{\eL},F_{\eL})\to \aL(E,F)\) ne dépend pas des bases choisies un peu partout. Mais elle dépend de l'application \( pr\colon \eL\to \eK\) déjà construite. Et celle-là dépend du choix d'un supplémentaire $V$ qui fournit \( \eL=\eK\oplus V\).
-
-    Si \( \pr(\lambda)=0\) pour un de ces choix, cela n'implique nullement que \( \lambda=0\). Penser à \( i\in \eC\) si la projection \( \pr\colon \eC\to \eR\) est l'application \( (x+iy)\mapsto x\) parallèle à l'axe des imaginaires.
-
-    Par contre si \( \pr(\lambda)=0\) pour tout choix de \( V\), alors nous avons bien \( \lambda=0\). Dans la suit nous «fixons» un choix de \( V\) générique, et lorsque nous rencontrerons l'égalité \( \pr(\lambda)=0\) nous en déduirons \( \lambda=0\).
-\end{remark}
-
-\begin{proposition} \label{PROPooPWDKooFNFWRI}
-    Si \( f\colon E\to F\) et si \( f_{\eL}e_j=\sum_a(f_{\eL})_{aj}e_a\) et si \( f(e_j)=\sum_af_{aj}e_a\) alors
-    \begin{enumerate}
-        \item
-            \( \pr f_{\eL}=f\),
-        \item       \label{ITEMooNMPYooXosGhI}
-            \( (f_{\eL})_{ja}=f_{ja} \in \eK\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Nous avons
-    \begin{equation}
-        f_{\eL}(e_i)=\sum_a f_{ai}(1\otimes e_a)=\sum_a f_{ai}\tau(e_a),
-    \end{equation}
-    donc
-    \begin{equation}
-        (\pr f_{\eL})e_i=\sum_a\pr(f_{ai})e_a=\sum_af_{ai}e_a=f(e_i).
-    \end{equation}
-    Cela prouve que \( \pr f_{\eL}=f\).
-
-    Par ailleurs, 
-    \begin{equation}        \label{EQooIOTFooNAdkit}
-        f_{\eL}(\tau e_i)=f_{\eL}(1\otimes e_i)=1\otimes f(e_i)=\tau\big( f(e_i) \big)=\sum_af_{ai}\tau(e_a)
-    \end{equation}
-    alors que par définition,
-    \begin{equation}        \label{EQooMYSCooPFWATG}
-        f_{\eL}(\tau e_i)=\sum_a(f_{\eL})_{ai}\tau(e_a).
-    \end{equation}
-    Les éléments \( \tau(e_a)\) formant une base\footnote{Encore la proposition \ref{PROPooMHARooUycAts}.}, la comparaison de \eqref{EQooIOTFooNAdkit} avec \eqref{EQooMYSCooPFWATG} donne \( (f_{\eL})_{ai}=f_{ai}\in \eK\).
-\end{proof}
-
-\begin{lemma}       \label{LEMooWZGSooONEnjZ}
-    Soient
-    \begin{enumerate}
-        \item
-            Une base \( \{ e_i \}\) de \( E\) et une application linéaire \( f\colon E\to F\);
-        \item
-            une base \( \{ e_a \}\) de \( F\) et une application linéaire \( g\colon G\to F\);
-        \item
-            une base \( \{ e_{\alpha} \} \) de \( G\) et une application linéaire \( \tilde h\colon G_{\eL}\to E_{\eL}\).
-    \end{enumerate}
-    Alors nous avons
-    \begin{equation}
-        \pr(f_{\eL}\circ \tilde h)=\pr(f_{\eL})\circ\pr(\tilde h).
-    \end{equation}
-\end{lemma}
-
-\begin{proof}
-    Pour écrire \( \pr(f_{\eL}\circ \tilde h)\) à partir de la définition \eqref{EQooSAFRooJnfkLO} nous commençons par écrire
-    \begin{equation}
-        (f_{\eL}\circ \tilde h)e_{\alpha}=\sum_a(f_{\eL}\circ \tilde h)_{a\alpha}e_a=\sum_{ai}(f_{\eL})_{ai}(\tilde h)_{i\alpha}e_a=\sum_a\Big( \sum_{i}f_{ai}(\tilde h)_{i\alpha} \Big)e_a
-    \end{equation}
-    où nous avons utilisé le fait que \( (f_{\eL})_{ai}=f_{ai}\). Donc, en utilisant la \( \eK\)-linéarité de \( \pr\),
-    \begin{equation}        \label{EQooZGCGooQsCBQH}
-        \pr(f_{\eL}\circ \tilde h)e_{\alpha}=\sum_a\sum_i\pr\Big( f_{ai}(\tilde h)_{i\alpha} \Big)e_a=\sum_a\sum_if_{ai}\pr\Big( (\tilde h)_{i\alpha} \Big)e_a.
-    \end{equation}
-    D'autre part,
-    \begin{equation}
-        \begin{aligned}[]
-            \pr(f_{\eL})\circ \pr(\tilde h)e_{\alpha}&=\pr(f_{\eL})\sum_i\pr\Big( (\tilde h)_{i\alpha} \Big)e_i\\
-            &=\sum_i\pr\Big( (\tilde h)_{i\alpha} \Big)\sum_af_{ai}e_a\\
-            &=\sum_{ai}\pr\Big( (\tilde h)_{i\alpha} \Big)f_{ai}e_a,
-        \end{aligned}
-    \end{equation}
-    et c'est égal à \eqref{EQooZGCGooQsCBQH}.
-\end{proof}
-
-\begin{remark}
-    Nous n'avons en général pas \( \pr(xy)=\pr(x)\pr(y)\) pour tout \( x,y\in \eL\). Par exemple si \( \eK=\eR\) et \( \eL=\eC\) avec la projection canonique,
-    \begin{equation}
-        \pr(i\cdot i)=\pr(-1)=-1
-    \end{equation}
-    alors que \( \pr(i)=0\).
-\end{remark}
-
-\begin{proposition}
-    Soit \( f\in\aL(E,F)\) et \( g\in\aL(F,E)\). Alors il existe \( h\colon G\to E\) tel que \( f\circ h=g\) si et seulement s'il existe \( \tilde g\colon G_{\eL}\to E_{\eL}\) tel que \( f_{\eL}\circ \tilde g=g_{\eL}\).
-\end{proposition}
-
-\begin{proof}
-    Dans le sens direct, il suffit de poser \( \tilde h=h_{\eL}\).
-
-    Dans le sens inverse, si nous avons \( \tilde h\colon G_{\eL}\to E_{\eL}\) tel que \( f_{\eL}\circ\tilde h=g_{\eL}\) alors en appliquant \( \pr\) des deux côtés et en utilisant le lemme \ref{LEMooWZGSooONEnjZ},
-    \begin{equation}
-        \pr(f_{\eL})\circ\pr(\tilde h)=\pr(g_{\eL})
-    \end{equation}
-    c'est à dire
-    \begin{equation}
-        f\circ\pr(\tilde h)=g,
-    \end{equation}
-    c'est à dire que l'application \( \pr\tilde h\colon G\to E\) est la réponse à la proposition.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Rang, polynôme minimal, polynôme caractéristique}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[Stabilité du rang par extension des scalaires\cite{ooEPEFooQiPESf}]     \label{PROPooJFQDooZSsxMf}
-    Si \( f\colon E\to F\) est linéaire alors nous avons
-    \begin{equation}
-        \rang(f)=\rang(f_{\eL}).
-    \end{equation}
-    où à droite nous considérons le rang de l'application \( \eL\)-linéaire \( f_{\eL}\colon E_{\eL}\to F_{\eL}\).
-\end{proposition}
-
-\begin{proof}
-    Il existe un supplémentaire \( V\) tel que \( E=\ker(f)\oplus V\) avec \( \dim(V)=\rang(f)\). Nous pouvons factoriser \( f\) en 
-    \begin{equation}
-        f=f_2\circ f_1
-    \end{equation}
-    avec \( f_1\colon E\to V\) est la projection parallèle à \( \ker(f)\) et est surjective (vers \( V\)) parce que \( \dim(V)=\rang(f)=\dim\big( \Image(f) \big)\). De plus \( f_2\colon V\to F\) est injective parce que si \( v\in V\) est tel que \( f_2(v)=0\) alors on aurait
-    \begin{equation}
-        f(v)=(f_2\circ f_1)(v)=f_2(v)=0.
-    \end{equation}
-    Cela donne \( v\in\ker(f)\cap V=\{ 0 \}\). Par la proposition \ref{PropooWECLooHPzIHw}, les applications \( (f_1)_{\eL}\) et \( (f_2)_{\eL}\) sont respectivement surjective et injective.
-
-    L'application \( (f_2)_{\eL}\colon V_{\eL}\to F_{\eL}\) est forcément surjective sur son image, donc
-    \begin{equation}
-        (f_2)_{\eL}\colon V_{\eL}\to \Image(f_{\eL}) 
-    \end{equation}
-    est un isomorphisme de \( \eL\)-espaces vectoriels. Nous avons alors les égalités
-    \begin{equation}        \label{EQooWLOIooKlYWTL}
-        \dim_{\eL}(V_{\eL})=\dim_{\eL}\big( \Image(f_{\eL}) \big)=\rang(f_{\eL}).
-    \end{equation}
-    Mais aussi, par les définitions posées plus haut,
-    \begin{equation}        \label{EQooEVCGooAGjmoU}
-        \dim(V)=\rang(f)=\dim\big( \Image(f) \big).
-    \end{equation}
-    Mais le corollaire \ref{CORooTQGHooIKhNtr} nous dit que \( \dim_{\eL}(V_{\eL})=\dim_{\eK}(V)\). Donc il y a égalité des deux lignes \eqref{EQooWLOIooKlYWTL} et \eqref{EQooEVCGooAGjmoU} donne \( \rang(f)=\rang(f_{\eL})\).
-\end{proof}
-
-\begin{proposition}     \label{PROPooZAZFooUFdCUv}
-    Nous avons
-    \begin{enumerate}
-        \item
-            \( \det(f)=\det(f_{\eL})\)
-        \item
-            \( \chi_f=\chi_{f_{\eL}}\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Dès que l'on a des bases nous avons \( (f_{\eL})_{ai}=f_{ai}\) par la proposition \ref{PROPooPWDKooFNFWRI}\ref{ITEMooNMPYooXosGhI}. Le nombre \( \det(f)\in \eK\) est un polynôme en les \( f_{ai}\). Entendons nous : il existe un polynôme indépendant de \( f\) et de \( \eK\) et de \( \eL\) donnant le déterminant de n'importe quelle matrice. Donc \( \det(f)=\det(f_{\eL})\).
-
-    Même chose pour le polynôme caractéristique (définition \ref{DefOWQooXbybYD}) : les coefficient de ce polynôme sont des polynôme en les \( f_{ai}\) qui sont indépendants de \( \eL\), de \( \eK\) et de \( f\).
-
-    Notons que \( \chi_{f_{\eL}}\) est un polynôme à coefficients dans \( \eK\).
-\end{proof}
-
-La situation est très différente avec le polynôme minimal\footnote{Définition \ref{DefCVMooFGSAgL}.}. Autant il existe une «recette» pour créer le polynôme caractéristique, il n'en n'existe pas pour le polynôme minimal (ou en tout cas, il ne suffit pas d'appliquer des polynôme en les coefficients de la matrice). La proposition suivante montre que le polynôme minimal est conservé par extension de corps, mais que pour le voir, il faut travailler plus.
-
-\begin{proposition}[\cite{ooEPEFooQiPESf,MonCerveau}]      \label{PROPooXVZMooXcJrsJ}
-    Soit \( \eL\) une extension du corps \( \eK\) et une application linéaire \( f\colon E\to F\) entre deux \( \eK\)-espaces vectoriels. Alors \( \mu_f=\mu_{f_{\eL}}\).
-\end{proposition}
-
-\begin{proof}
-    Nous allons montrer que l'application
-    \begin{equation}
-        \begin{aligned}
-            \tilde g\colon \frac{ \eL[X] }{ (\mu) }&\to \End(E_{\eL}) \\
-            \bar P&\mapsto P(f_{\eL}) 
-        \end{aligned}
-    \end{equation}
-    est bien définie et injective. La proposition \ref{PROPooVUJPooMzxzjE} nous dira alors que \( \mu\) est le polynôme minimal de \( f_{\eL}\).
-
-    Pour prouver que l'application \( \tilde g\) est bien définie, nous commençons par prouver que  \( P(f_{\eL})=P(f)_{\eL}\) :
-    \begin{subequations}
-        \begin{align}
-            P(f_{\eL})\lambda\otimes v&=\sum_ka_kf_{\eL}^k\lambda\otimes v\\
-            &=\lambda\otimes \sum_ka_kf^k(v)\\
-            &=\lambda\otimes P(f)v\\
-            &=P(f)_{\eL}\lambda\otimes v.
-        \end{align}
-    \end{subequations}
-    Par conséquent \( \mu(f_{\eL})=0\) et l'application est bien définie.
-
-    Sur \( \eL[X]/(\mu)\) nous considérons la base \( \{ 1,\bar X,\ldots, \bar X^{\deg(\mu)-1} \}\), et \( \End(E_{\eL})\) nous considérons une base qui commence\footnote{Théorème de la base incomplète \ref{ThonmnWKs}\ref{ITEMooFVJXooGzzpOu}.} par \( \{ f_{\eL}^k \}_{k=0,\ldots, \deg(\mu)-1}\). Montrons tout de même que cette partie est libre (sinon le théorème de la base incomplète ne s'applique pas) : si \( \sum_k\lambda_kf_{\eL}^k=0\) alors
-    \begin{equation}        \label{EQooSFHVooLxqUEl}
-        \sum_k\pr\big( \lambda_k f_{\eL}^k\big)=0.
-    \end{equation}
-    Pour détailler ce que cela implique, nous calculons ceci :
-    \begin{equation}
-        (\lambda f_{\eL})(\tau e_i)=\lambda f_{\eL}(\tau e_i)=\sum_a \lambda f_{ia}e_a,
-    \end{equation}
-    par conséquent \( \pr(\lambda f_{\eL})e_i=\sum_a\pr(\lambda f_{ia})e_a\), et comme \( \pr\) est \( \eK\)-linéaire et que \( f_{ai}\in \eK\),
-    \begin{equation}
-        \pr(\lambda f_{\eL})e_i=\pr(\lambda)\sum_a f_{ai}e_a=\pr(\lambda)\pr(f_\eL)e_i=\pr(\lambda)f(e_i).
-    \end{equation}
-    Appliquer la projection \( \pr\) à l'équation \eqref{EQooSFHVooLxqUEl} donne alors \( \sum_k\pr(\lambda)_kf^k=0\). Mais comme les \( f^k\) sont linéairement indépendantes sur \( \eK\) nous avons pour tout \( k\) : \( \pr(\lambda_k)=0\) (égalité dans \( \eK\)). En nous souvenant de la remarque \ref{REMooBEXGooLgpHzg} nous en déduisons \( \lambda_k=0\) dans \( \eL\).
-
-    Dans les choix de bases faits, l'application \( \tilde g\) a la forme
-    \begin{equation}
-        \tilde g=\begin{pmatrix}
-            \begin{matrix}
-                1    &       &       \\
-                    &   1    &       \\
-                    &       &   1
-            \end{matrix}\\ 
-            \begin{matrix}
-                *    &   *    &   *    \\
-                *    &   *    &   *    \\
-                *    &   *    &   *    
-            \end{matrix}
-        \end{pmatrix},
-    \end{equation}
-    qui est injective.
-
-    Vu que \( \tilde g\) est injective, \( \mu\) est le polynôme minimal de \( f_{\eL}\) et donc \( \mu=\mu_{\eL}\).
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Frobenius et Jordan}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Matrice compagnon}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooOSVAooGevsda}
-    Soit le polynôme \( P=X^n-a_{n-1}X^{n-1}-\ldots-a_1X-a_0\) dans \( \eK[X]\). La \defe{matrice compagnon}{matrice!compagnon} de \( P\) est la matrice\nomenclature[A]{\( C(P)\)}{matrice compagnon} donnée par
-    \begin{equation}
-        C(P)=\begin{pmatrix}
-            0    &   \cdots    &   \cdots    &   0    &   a_0\\  
-            1    &   0    &       &   \vdots    &   a_1\\  
-            0    &   \ddots    &   \ddots    &   \vdots    &   \vdots\\  
-            \vdots    &   \ddots    &   \ddots    &   0    &   a_{n-2}\\  
-            0    &   \cdots    &   0    &   1    &   a_{n-1}    
-        \end{pmatrix}
-    \end{equation}
-    si \( n\geq 2\) et par \( (a_0)\) si \( n=1\). 
-
-    Une matrice est dite compagnon si elle a cette forme.
-\end{definition}
-
-\begin{proposition}
-    Si \( f\) est l'endomorphisme associé à la matrice \( C(P)\) nous avons
-    \begin{equation}
-        f(e_i)=\begin{cases}
-            e_{i+1}    &   \text{si } i<n\\
-            (a_0,\ldots, a_{n-1})    &    \text{si } i=n.
-        \end{cases}
-    \end{equation}
-    De plus l'endomorphisme \( f\) vérifie \( P(f)e_1=0\).
-\end{proposition}
-
-\begin{lemma}[\cite{RapportArgreg2011}] \label{LemkVNisk}
-    Un polynôme sur un corps commutatif est le polynôme caractéristique de sa matrice compagnon. En d'autres termes nous avons \( \chi_{C(P)}=P\).
-\end{lemma}
-
-\begin{proof}
-    Nous notons \( f\) l'endomorphisme associé à \( C(P)\). La propriété \( P(f)e_1=0\) nous indique que le polynôme minimal ponctuel de \( f\) en \( e_1\) divise \( P\). L'ensemble des puissances de \( f\) appliquées à \( e_1\), \( \big( f^i(e_1) \big)_{i=1,\ldots, n-1}\) est libre, donc le polynôme minimal ponctuel en \( e_1\) est de degré \( n\) au minimum. En reprenant les notations du théorème \ref{ThoCCHkoU}, nous avons \( I_{e_1}=(P)\) parce que \( P\) est de degré minimum dans \( I_{e_1}\) et \( \chi_f\in I_{e_1}\).
-
-    Donc \( P\) divise \( \chi_f\) et est de degré égal à celui de \( \chi_f\). Étant donné qu'ils sont tous deux unitaires, ils sont égaux.
-\end{proof}
-
-\begin{remark}  \label{RemmQjZOA}
-    Les matrices compagnons ne sont pas les seules dont le polynôme caractéristique est égal au polynôme minimal. En fait les matrices dont le polynôme caractéristique est égale au polynôme minimal sont denses dans les matrices. En effet une matrice dont le polynôme minimal n'est pas égal au polynôme caractéristique a un polynôme caractéristique avec une racine double. Il est possible, en modifiant arbitrairement peu la matrice de séparer la racine double en deux racines distinctes.
-\end{remark}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Réduction de Frobenius}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}       \label{LEMooKUQDooKFeIYq}
-    Soit un endomorphisme \( f\colon E\to E\) sur l'espace vectoriel de dimension finie \( n\). Nous notons \( \mu\) et \( \chi\) les polynômes minimal et caractéristique. Si \( f\) est cyclique, alors \( \mu=\chi\).
-\end{lemma}
-Le théorème \ref{THOooGLMSooYewNxW} donnera une version plus complète de ce lemme.
-
-\begin{proof}
-    Soit \( v\) un vecteur cyclique de \( f\), c'est à dire que \( \{ f^k(v) \}_{k=0,\ldots, n-1}\) est libre. Donc si \( P\) est un polynôme de degré jusqu'à \( n-1\) nous ne pouvons pas avoir \( P(f)=0\) parce que, appliqué à \( v\), ce serait une combinaisons nulle non triviale des \( f^k(v)\). Donc le polynôme minimal est au minimum de degré \( n\). Mais le polynôme caractéristique est annulateur de degré \( n\) (Cayley-Hamilton \ref{ThoCalYWLbJQ}), donc il est le polynôme minimal.
-\end{proof}
-
-\begin{theorem}[Réduction de Frobenius \cite{AutourFrobCompa,Vialivs,MoncetIVS}]        \label{THOooDOWUooOzxzxm}
-    Soit \( E\), un \( \eK\)-espace vectoriel, et \( f\in \End(E)\). Alors il existe une suite de sous-espaces \( E_1,\ldots, E_r\) stables par \( f\) tels que
-    \begin{enumerate}
-        \item   \label{ItemmpwjnSs}
-            \( E=\bigoplus_{i=1}^rE_i\);
-        \item
-            pour chaque \( E_i\), l'endomorphisme restreint \( f_i=f|_{E_i}\) est cyclique;
-        \item
-            si \( \mu_i\) est le polynôme minimal de \( f_i\) alors \( \mu_{i+1}\) divise \( \mu_i\);
-    \end{enumerate}
-    Une telle décomposition vérifie automatiquement \( \mu_1=\mu_f\) et \( \mu_1\cdots \mu_r=\chi_f\), et la suite \( (\mu_i)_{i=1,\ldots, r}\) ne dépend que de \( f\) et non du choix de la décomposition du point \ref{ItemmpwjnSs}.
-\end{theorem}
-   \index{réduction!Frobénius}
-   \index{Frobénius!réduction}
-
-Les polynômes \( \mu_i\) sont les \defe{invariants de similitude}{invariant!de similitude} de l'endomorphisme \( f\).
-
-\begin{proof}
-    Nous commençons par montrer que si une telle décomposition existe, alors
-    \begin{subequations}    \label{subEqzcGouz}
-        \begin{align}
-            \chi_f=\prod_{i=1}^r\mu_i  \label{EqTaxsvb}\\
-            \mu_f=\mu_1
-        \end{align}
-    \end{subequations}
-    où \( \chi_f\) est le polynôme caractéristique de \( f\) et \( \mu_f\) est le polynôme minimal. D'abord le polynôme caractéristique de \( f\) devra être égal au produit des polynômes caractéristique des \( f|_{E_i}\), mais ces derniers endomorphismes étant cycliques\footnote{Définition \ref{DEFooFEIFooNSGhQE}.}, leurs polynôme caractéristiques sont égaux à leurs polynômes minimaux (lemme \ref{LEMooKUQDooKFeIYq}). Cela prouve l'égalité \eqref{EqTaxsvb}. Ensuite tous les \( \mu_i\) doivent diviser le polynôme minimal, donc \( \ppcm(\mu_1,\ldots, \mu_r)\) divise \(\mu_f\). Cependant le polynôme minimal doit contenir une et une seule fois chacun des facteurs irréductibles du polynôme caractéristique, et chacun de ces facteurs sont dans les polynômes \( \mu_i\). Par conséquent \( \ppcm(\mu_1,\ldots, \mu_r)=\mu_f\). Mais par ailleurs \( \mu_1=\ppcm(\mu_1,\ldots, \mu_r)\) parce qu'on a supposé \( \mu_{i+1}\divides \mu_i\), donc \( \mu_1=\mu_f\).
-    
-    Soit \( d\), le degré du polynôme minimal de \( f\) et \( y\in E\) tel que \( \mu_f=\mu_{f,y}\) (voir lemme \ref{LemSYsJJj}). Le plus petit espace stable sous \( f\) contenant \( y\) est
-    \begin{equation}
-        E_y=\Span\{ y,f(y),\ldots, f^{d-1}(y) \}.
-    \end{equation}
-    Nous notons \( e_i=f^{i-1}(y)\). Notons que les vecteurs donnés forment bien une base de \( E_y\) parce que si les \( e_i\) n'était pas linéairement indépendants, alors nous aurions des \( a_k\) tels que \( \sum_ka_ke_k=0\) et avec lesquels
-    \begin{equation}
-        \big( \sum_ka_kX^k \big)(f)y=0,
-    \end{equation}
-    ce qui contredirait la minimalité de \( \mu_{f,y}\).
-
-    La difficulté du théorème est de trouver un complément de \( E_y\) qui soit également stable sous \( f\). Nous commençons par étendre\quext{Pour autant que j'aie compris, cette extension manque dans \cite{AutourFrobCompa}. Corrigez moi si je me trompe.} \( \{ e_1,\ldots, e_d \}\) en une base \( \{ e_1,\ldots, e_n \}\) de \( E\). Ensuite nous allons montrer que
-    \begin{equation}
-        E=E_y\oplus F
-    \end{equation}
-    avec
-    \begin{equation}
-        F=\{ x\in E\tq  e^*_d\big( f^k(x) \big)=0\forall k\in \eN \}.
-    \end{equation}
-    Par construction, \( F\) est invariant sous \( f\). Montrons pour commencer que \( E_y\cap F=\{ 0 \}\). Un élément de \( E_y\) s'écrit
-    \begin{equation}
-        z=a_1e_1+\cdots +a_ke_k
-    \end{equation}
-    avec \( k\leq d\). Étant donné que \( f\) décale les vecteurs de base, nous avons \( e^*_d\big( f^{d-k}(z) \big)=a_k\). Du coup \( z\in F\) si et seulement si \( a_1=\ldots=a_d=0\), c'est à dire que \( E_y\cap F=\{ 0 \}\).
-
-    Nous montrons maintenant que \( \dim F=n-d\). Pour cela nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            T\colon \eK[F]&\to E^* \\
-            g&\mapsto e^*_d\circ g. 
-        \end{aligned}
-    \end{equation}
-    Cette application est injective. En effet un élément général de \( \eK[f]\) est
-    \begin{equation}
-        g=a_1\id+a_2f+\cdots +a_pf^{p-1}
-    \end{equation}
-    avec \( p\leq d\). Si \( T(g)=0\), alors nous avons en particulier
-    \begin{equation}
-        0=T(g)e_{_d-p+1}=e^*_d(a_1e_{d-p+1}+a_2e_{d-p+2}+\cdots +a_pe_d)=a_p.
-    \end{equation}
-    Donc \( a_p=0\) et en appliquant maintenant \( T(g)\) à \( e_{d-p}\) nous obtenons \( a_{p-1}=0\). Au final nous trouvons que \( g=0\) et donc que \( T\) est injective.
-
-    Étant donné que \( \dim\eK[f]=d\) et que \( T\) est injective, \( \dim\Image(T)=d\). Nous regardons l'orthogonal de l'image :
-    \begin{subequations}
-        \begin{align}
-            (\Image(T))^{\perp}&=\{ x\in E\tq T(g)x=0\forall g\in\eK[f] \}\\
-            &=\{ x\in E\tq e^*_d\big( g(x) \big)=0\forall g\in \eK[f] \}\\
-            &=F.
-        \end{align}
-    \end{subequations}
-    Par conséquent \( F^{\perp}=\Image(T)\). Vu que \( \dim\Image(T)=d\), nous avons donc \( \dim F=n-d\) et il est établi que \( E=E_y\oplus F\). 
-
-    Nous avons donc trouvé \( F\), stable par \( f\) et tel que \( E=E_y\oplus F\). Nous devons maintenant nous assurer que cette décomposition tombe bien pour les polynômes minimaux. Si \( P_1\) est le polynôme minimal de \( f|_{E_yj}\), alors par le lemme \ref{LemAGZNNa} nous avons \( P_1=\mu_{f,y}=\mu_f\) parce que \( f|_{E_y}\) est cyclique sur \( E_y\). Mettons \( P_2\), le polynôme minimal de \( f|_F\). Étant attendu que \( F\) est stable par \( f\), le polynôme \( P_2\) divise \( P_1\). En recommençant la construction sur \( F\), nous construisons un nouvel espace \( F'\) stable sous \( F\) et vérifiant \( \mu_{f|_{F'}}=P_2\), etc.
-
-    Nous passons maintenant à la partie unicité du théorème. Soient deux suites \( F_1,\ldots, F_r\) et \( G_1,\ldots, G_s\) de sous-espaces stables par \( f\) et vérifiant
-    \begin{enumerate}
-        \item
-            \( E=\bigoplus_{i=1}^rF_i\),
-        \item
-            \( f|_{F_i}\) est cyclique,
-        \item
-            \( \mu_{f|_{F_{i+1}}}\) divise \( \mu_{f|_{F_i}}\),
-    \end{enumerate}
-    et, \emph{mutatis mutandis}, les mêmes conditions pour la famille \( \{ G_i \}\). Nous posons \( P_i=\mu_{f_{F_i}}\) et \( Q_i=\mu_{f|_{G_i}}\). Nous allons montrer par récurrence que \( P_i=Q_i\) et \( \dim F_i=\dim G_i\). Il ne sera cependant pas garanti que \( F_i=G_i\). D'abord, \( P_1=Q_1\) parce qu'ils sont tous deux égaux à \( \mu_f\) par les relations \eqref{subEqzcGouz}. Nous supposons que \( P_i=Q_i\) pour \( i\leq 1\leq j-1\) et nous tentons de montrer que \( P_j=Q_j\).
-
-    Nous avons 
-    \begin{equation}    \label{EqMrCtZO}
-        P_j(f)=P_j(f)|_{F_1}\oplus\ldots\oplus P_j(f)|_{F_{j-1}}.
-    \end{equation}
-    En effet étant donné que \( P_{j+k}\) divise \( P_j\), nous avons\footnote{En vertu du lemme \ref{LemQWvhYb}.} \( P_{j}(f)=A(f)\circ P_{j+k}(f)\), mais \( P_{j+k}(f)F_{j+k}=0\), donc \( P_j(f)F_{j+k}=0\). Les espaces \( G_i\) n'ayant a priori aucun rapport avec les polynômes \( P_i\), nous écrivons
-    \begin{equation}    \label{EqJreLiO}
-        P_j(f)=P_j(f)|_{G_1}\oplus\ldots\oplus P_j(f)|_{G_{j-1}}\oplus P_j(f)|_{G_j}\oplus\ldots\oplus P_j(f)|_{G_s}.
-    \end{equation}
-    Pour \( 1\leq i\leq j-1\), nous avons supposé \( P_i=Q_i\). Étant donné que \( f|_{F_i}\) est semblable à \( C_{_i}\) et \( f|_{G_i}\) est semblable à \( C_{Q_i}\), la matrice de \( f|_{E_i}\) est semblable à la matrice de \( f|_{G_i}\). En particulier,
-    \begin{equation}
-        \dim P_j(f)F_i=\dim P_j(f)G_i.
-    \end{equation}
-    En prenant les dimensions des images dans les égalités \eqref{EqMrCtZO} et \eqref{EqJreLiO}, nous trouvons que
-    \begin{equation}
-        P_j(f)|_{G_j}=\ldots=P_j(f)|_{G_s}=0.
-    \end{equation}
-    Par conséquent \( P_j\in I_{f|G_j}\) et donc \( P_j\) divise \( Q_j\), qui est générateur de \( I_{f|_{G_j}}\). La situation étant symétrique entre \( P\) et \( Q\), nous montrons de même que \( Q_j\) divise \( P_j\) et donc que \( P_j=Q_j\).
-
-    Ceci achève la démonstration du théorème de réduction de Frobenius.
-
-\end{proof}
-
-\begin{remark}      \label{REMooPVLEooYDRXQI}
-    Sous forme matricielle, ce théorème dit que toute matrice est semblable à une matrice de la forme bloc-diagonale
-    \begin{equation}
-        f=\begin{pmatrix}
-            C_{\mu_1}    &       &       \\
-                &   \ddots    &       \\
-                &       &   C_{\mu_r}
-        \end{pmatrix}
-    \end{equation}
-    où les \( C_{\mu_i}\) sont les matrices compagnon (définition \ref{DEFooOSVAooGevsda}). 
-
-    En particulier, et ceci est très important, deux applications sont semblables si et seulement si elles ont même suite d'invariants de similitude.
-\end{remark}
-
-
-\begin{remark}
-    Si nous travaillons sur \( \eR\), la réduite de Frobenius restera une matrice réelle, même si les valeurs propres sont complexes. En effet le procédé de Frobenius ne regarde absolument pas les valeurs propres, mais seulement les facteurs irréductibles du polynôme caractéristique. La réduite de Frobenius ne tente pas de résoudre ces polynômes, mais se contente d'en utiliser les matrices compagnon.
-
-    La situation sera différente dans le cas de la forme normale de Jordan.
-\end{remark}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Forme normale de Jordan}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Il existe une preuve directe de la réduction de Jordan ne nécessitant pas la réduction de Frobenius\cite{LecLinAlgAllen}. Cette dernière passe par les espaces caractéristiques\footnote{Aussi appelés «espaces propres généralisés».} et est à mon avis plus compliquée que la démonstration de Frobenius elle-même. Nous allons donc nous contenter de donner la réduction de Jordan comme un cas particulier de Frobenius.
-
-\begin{theorem}[Réduction de Jordan]        \label{ThoGGMYooPzMVpe}
-    Soit \( E\) un espace vectoriel sur \( \eK\), et \( f\in\End(E)\) un endomorphisme dont le polynôme caractéristique \( \chi_f\) est scindé\footnote{C'est pour cette hypothèse que \( \eK=\eR\) n'est pas le bon cadre.}. Il existe une base de \( E\) dans laquelle la matrice de \( f\) s'écrit sous la forme
-    \begin{equation}
-        M=\begin{pmatrix}
-            J_{n_1}(\lambda_1)    &       &       \\
-                &   \ddots    &       \\
-                &       &   J_{n_k}(\lambda_k)
-        \end{pmatrix}
-    \end{equation}
-    où les \( \lambda_i\) sont les valeurs propres de \( f\) (avec éventuelle répétitions) et \( J_n(\lambda)\) représente le bloc \( n\times n\)
-    \begin{equation}
-        J_n(\lambda)=\begin{pmatrix}
-            \lambda    &   1    &       &       &   \\  
-                &   \lambda    &   1    &       &   \\  
-                &       &   \lambda    &       &   \\  
-                &       &       &   \ddots    &   1\\  
-                &       &       &       &   \lambda    
-        \end{pmatrix}.
-    \end{equation}
-    En d'autres termes, \( J_n(\lambda)_{ii}=\lambda\) et \( J_n(\lambda)_{i-1,i}=1\).    
-\end{theorem}
-\index{réduction!Jordan}
-\index{Jordan!réduction}
-
-\begin{proof}
-    Nous commençons par le cas où \( f\) est nilpotente; nous notons \( M\) sa matrice. Dans ce cas la seule valeur propre est zéro et le polynôme caractéristique est \( X^m\) pour un certain \( m\). Nous savons par le lemme \ref{LemkVNisk} que (la matrice de) \( f\) est semblable à sa matrice compagnon. En l'occurrence pour \( f\) nous avons
-    \begin{equation}
-        C_{X^m}=\begin{pmatrix}
-             0   &       &       &  0     \\
-             1   &   \ddots    &       &   \vdots    \\
-                &   \ddots    &   \ddots    &    \vdots   \\ 
-                &       &   1    &   0     
-         \end{pmatrix}.
-    \end{equation}
-    Ensuite le changement de base (qui est une similitude) \( (e_1,\ldots, e_n)\mapsto(e_n,\ldots, e_1)\) montre que \( C_{X^m}\) est semblable à un bloc de Jordan \( J_m(0)\).
-
-    Supposons à présent que \( f\) ne soit pas nilpotente. Par l'hypothèse de polynôme caractéristique scindé, nous supposons que \( f\) a \( m\) valeurs propres distinctes et que son polynôme caractéristique est
-    \begin{equation}
-        \chi_f=(X-\lambda_1)^{l_1}\ldots (X-\lambda_m)^{l_m}.
-    \end{equation}
-    Le lemme des noyaux (théorème \ref{ThoDecompNoyayzzMWod}) nous enseigne que
-    \begin{equation}
-        E=\bigoplus_{i=1}^m\underbrace{\ker(f-\mu_i\mtu)^{l_i}}_{F_i}.
-    \end{equation}
-    La restriction de \( f-\lambda_i\mtu\) à \( F_i\) est par construction un endomorphisme nilpotent, et donc peut s'écrire comme un bloc de Jordan avec des zéros sur la diagonale. En utilisant la décomposition
-    \begin{equation}
-        f|_{F_i}=(f-\lambda_i\mtu)|_{F_i}+\lambda_i\mtu_{F_i},
-    \end{equation}
-    nous voyons que \( f|_{F_i}\) s'écrit comme un bloc de Jordan avec \( \lambda_i\) sur la diagonale.
-\end{proof}
-
-\begin{remark}
-    Nous pouvons calculer la forme normale de Jordan pour une matrice complexe ou réelle, mais dans les deux cas nous devons nous attendre à obtenir une matrice complexe parce que les valeurs propres d'une matrice réelle peuvent être complexes. Cependant nous demandons que le polynôme caractéristique de \( f\) soit scindé sur \( \eK\). En pratique, la décomposition de Jordan n'est garantie que sur les corps algébriquement clos, c'est à dire sur \( \eC\).
-
-    La suite des invariants de similitude sur laquelle repose Frobenius, elle, est disponible sur tout corps, y compris \( \eR\).
-\end{remark}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Commutant et endomorphismes cycliques}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Endomorphisme cyclique}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}\label{LemSGmdnE}
-    Si \( A\) est la matrice de l'endomorphisme \( f\) alors nous avons équivalence des propriétés suivantes :
-    \begin{enumerate}
-        \item
-            La matrice \( A\) est cyclique.
-        \item
-            L'endomorphisme \( f\) est cyclique.
-    \end{enumerate}
-\end{lemma}
-
-Si \( f\) est un endomorphisme de l'espace vectoriel \( E\) et si \( x\in E\), nous notons 
-\begin{equation}
-    E_{f,x}=\Span\{ f^k(x)\tq k\in \eN \}.
-\end{equation}
-
-\begin{definition}
-    Soit \( E\) un espace vectoriel de dimension finie sur un corps \( \eK\) et un endomorphisme \( f\colon E\to E\). Le \defe{commutant}{commutant} de \( f\) est l'ensemble des endomorphismes de \( E\) qui commutent avec \( f\) :
-    \begin{equation}
-        \comC(f)=\{ g\in\aL(E,E)\tq g\circ f=f\circ g \}.
-    \end{equation}
-\end{definition}
-Il n'est pas très compliqué de vérifier que \( \comC(f)\) est un sous-espace vectoriel de \( \aL(E,E)\).
-
-Notons l'inclusion évidente \( \eK[f]\subset \comC(f)\). L'inclusion inverse va un peu nous occuper durant les prochaines pages.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Commutant : cas diagonalisable}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[\cite{ooKPTNooMmncYA}]      \label{PROPooRHHEooIRGmtl}
-    Si \( f\) est diagonalisable, alors
-    \begin{equation}        \label{EQooOTFLooPUKAos}
-        \dim\big( \comC(f) \big)=\sum_{\lambda\in\Spec(f)}\dim(E_{\lambda})^2.
-    \end{equation}
-    où les \( E_{\lambda} \) sont les espaces propres de \( f\).
-\end{proposition}
-
-\begin{proof}
-    D'abord su \( g\in\comC(f)\) alors \( E_{\lambda}\) est stable par \( g\). En effet si \( v\in E_{\lambda}\) alors \( f\big( g(v) \big)=g\big( f(v) \big)=g(\lambda v)=\lambda g(v)\), ce qui montre que \( g(v)\) est un vecteur propre de \( f\) pour la valeur propre \( \lambda\), et donc que \( g(v)\in E_{\lambda}\).
-
-    Nous considérons ensuite l'application 
-    \begin{equation}
-        \begin{aligned}
-            \psi\colon \comC(f)&\to \End(E_1)\times \ldots\times \End(E_r) \\
-            g&\mapsto  g|_{E_1}\times \ldots\times g|_{E_r}
-        \end{aligned}
-    \end{equation}
-    qui est bien définie parce que \( g\) se restreint aux espaces propres de \( f\). Nous allons noter \( \psi(g)_{\lambda}\) la restriction de \( g\) à \( E_{\lambda}\).
-    \begin{subproof}
-    \item[\( \psi\) est injective]
-
-        Supposons que \( g,h\in\comC(f)\) tels que \( \psi(g)=\psi(h)\). Vu que \( f\) est diagonalisable nous pouvons décomposer \( x\in  E\) en ses composantes sur les espaces propres\footnote{Théorème \ref{ThoDigLEQEXR}\ref{ITEMooZNJFooEiqDYp}.} :
-        \begin{equation}
-            x=\sum_{\lambda\in\Spec(f)}x_{\lambda}
-        \end{equation}
-        avec \( x_{\lambda}\in E_{\lambda}\).  Nous avons alors
-        \begin{equation}
-            g(x)=\sum_{\lambda}g(x_{\lambda})=\sum_{\lambda}\psi(g)_{\lambda}(x_{\lambda}).
-        \end{equation}
-        Vu que nous avons \( \psi(g)_{\lambda}=\psi(h)_{\lambda}\), nous avons aussi
-        \begin{equation}
-            g(x)=\sum_{\lambda}\psi(g)_{\lambda}(x_{\lambda})=\sum_{\lambda}\psi(h)_{\lambda}(x_{\lambda})=\sum_{\lambda}h(x_{\lambda})=h(x).
-        \end{equation}
-        Cela prouve \( g=h\) et donc que \( \psi\) est injective.
-    \item[\( \psi\) est surjective]
-        Si nous avons pour chaque \( \lambda\in\Spec(f)\) un endomorphisme \( g_{\lambda}\) de \( E_{\lambda}\) alors en posant
-        \begin{equation}
-            g(x)=\sum_{\lambda\in\Spec(f)}g_{\lambda}(x_{\lambda})
-        \end{equation}
-        alors nous avons bien
-        \begin{equation}
-            \psi(g)=\big( g_{\lambda_1},\ldots, g_{\lambda_r} \big).
-        \end{equation}
-    \end{subproof}
-    Nous pouvons donc conclure en écrivant
-    \begin{equation}
-        \dim\big( \comC(f) \big)=\sum_{\lambda\in\Spec(f)}\dim\big( \End(E_{\lambda}) \big)= \sum_{\lambda\in\Spec(f)}\dim(E_{\lambda})^2.
-    \end{equation}
-\end{proof}
-
-\begin{remark}      \label{REMooUGFQooVzCOvV}
-    Nous avons alors immédiatement
-    \begin{equation}
-        \dim\big( \comC(f) \big)\geq\dim(E)
-    \end{equation}
-    lorsque \( f\) est diagonalisable.
-\end{remark}
-
-En suivant la notation \eqref{EqooOAYDooEpZELo}, un endomorphisme est cyclique lorsqu'il existe \( x\in E\) tel que \( E_x=E\). 
-
-\begin{proposition}[\cite{ooKPTNooMmncYA}]      \label{PropooQALUooTluDif}
-    Si \( f\) est un endomorphisme diagonalisable d'un espace vectoriel \( E\) de dimension \( n\). Nous avons équivalence entre les faits suivants.
-    \begin{enumerate}
-        \item\label{ITEMooSOYYooZVibjrii}
-            Le polynôme minimal est égal au polynôme caractéristique : \( \mu_f=\chi_f\)
-        \item\label{ITEMooSOYYooZVibjrvi}
-            L'endomorphisme \( f\) est cyclique.
-        \item\label{ITEMooSOYYooZVibjrv}
-            \( \comC(f)=\eK[f]\).
-        \item\label{ITEMooSOYYooZVibjriv}
-            \( \dim\big( \comC(f) \big)=n\)
-        \item\label{ITEMooSOYYooZVibjriii}
-            L'endomorphisme \( f\) possède \( n\) valeurs propres distinctes.
-        \item   \label{ITEMooSOYYooZVibjri}
-            \( \dim\big( \eK[f] \big)=n\)
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Le point important de cette proposition sont les équivalences \ref{ITEMooSOYYooZVibjrii}-\ref{ITEMooSOYYooZVibjrv}. Les autres sont des intermédiaires. En particulier, dans le cas diagonalisable, nous allons voir que le point \ref{ITEMooSOYYooZVibjriii} est essentiellement une reformulation de \ref{ITEMooSOYYooZVibjrii}.
-    \begin{subproof}
-        \item[\ref{ITEMooSOYYooZVibjriv} implique \ref{ITEMooSOYYooZVibjriii}]
-            Par la formule \eqref{EQooOTFLooPUKAos}, les espaces propres de \( f\) ont dimension \( 1\). Par conséquent \( f\) possède \( n\) valeurs propres distinctes.
-        \item[\ref{ITEMooSOYYooZVibjriii} implique \ref{ITEMooSOYYooZVibjri}]
-            Le théorème \ref{ThoDigLEQEXR} nous dit que le polynôme minimal est scindé à racines simples. Vu que \( f\) possède \( n\) valeurs propres distinctes, \( \mu\) est de degré \( n\).  Par l'isomorphisme \( \eK[f]=\eK[X]/(\mu)\) de la proposition \ref{PropooCFZDooROVlaA} nous avons \(\dim\big( \eK[f] \big)= \deg(\mu)=n\) par la proposition \ref{CorsLGiEN}.
-        \item[\ref{ITEMooSOYYooZVibjri} implique \ref{ITEMooSOYYooZVibjrii}]
-            Par l'isomorphisme \( \eK[f]=\eK[X]/(\mu)\) de la proposition \ref{PropooCFZDooROVlaA} et la proposition \ref{CorsLGiEN} nous avons \(n=\dim\big( \eK[f] \big)= \deg(\mu)\). Vu que \( \chi\) est un polynôme annulateur (Caley-Hamilton \ref{ThoCalYWLbJQ}), il est divisé par \( \mu\). Maintenant \( \mu\) et \( \chi\) sont des polynômes unitaires de degré \( n\) et \( \mu\) divise \( \chi\). Ils sont donc égaux.
-        \item[\ref{ITEMooSOYYooZVibjrii} implique \ref{ITEMooSOYYooZVibjrvi}]
-            Le fait que $f$ soit diagonalisable permet d'utiliser le théorème \ref{ThoDigLEQEXR} pour dire que \( \mu\) est scindé à racines simples. L'égalisation avec \( \chi \) nous permet de dire que \( f\) possède \( n\) valeurs propres distinctes. Soient \( \{ e_1,\ldots, e_n \}\) une base de diagonalisation, et prouvons que le vecteur \( v=e_1+\cdots +e_n\) est cyclique. Nous avons
-            \begin{equation}
-                f^k(v)=\sum_{i=1}^n\lambda_i^ke_i.
-            \end{equation}
-            Pour prouver que cette famille (avec \( k=0,\ldots, n-1\)) est libre\footnote{Ce sera alors une base parce que \( n\) vecteurs libres dans un espace de dimension \( n\) est toujours une base, théorème \ref{ThoMGQZooIgrXjy}\ref{ItemHIVAooPnTlsBi}.} nous en prenons une combinaison linéaire nulle et nous prouvons que les coefficients sont tous nuls. Soit donc
-            \begin{equation}
-                    0=\sum_{l=0}^{n-1}a_lf^l(v)=\sum_{l=0}^{n-1}a_l\sum_{i=1}^n\lambda_i^le_i=\sum_{i=1}^n\Big( \sum_{l=0}^{n-1}a_l\lambda_i^l \Big)e_i.
-            \end{equation}
-            Vu que cela est nul, nous avons pour tout \( i\) :
-            \begin{equation}
-                \sum_{l=0}^{n-1}a_l\lambda_i^l=0.
-            \end{equation}
-            En posant la matrice \( A_{ij}=\lambda_i^j\), cela revient à étudier le système \( \sum_j A_{ij}a_j=0\). Ce système n'a des solutions non nulles que si \( \det(A)= 0\); sinon il possède une unique solution et elle est \( a_j=0\) pour tout \( j\). Nous devons donc calculer le déterminant
-            \begin{equation}
-                \det\begin{pmatrix}
-                    1&\lambda_1&\lambda_1^2&\cdots&\lambda_1^{n-1}\\  
-                    \vdots&\vdots&\vdots&&\vdots\\  
-                    1&\lambda_n&\lambda_n^2&\cdots&\lambda_n^{n-1}  
-                \end{pmatrix}.
-            \end{equation}
-            Il s'agit du déterminant de Vandermonde déjà étudié par la proposition \ref{PropnuUvtj}. Nous avons \( \det(A)=\prod_{1\leq i<j\leq n}(\lambda_j-\lambda_i)\). Cela est bien non nul du fait que toutes les valeurs propres soient distinctes.
-        \item[\ref{ITEMooSOYYooZVibjrvi} implique \ref{ITEMooSOYYooZVibjrv}]
-            Soit \( v\) un vecteur cyclique de \( f\). Un endomorphisme \( g\) donne lieu à un polynôme par le fait suivant : il existe des uniques \( a_k\) (\( k=0,\ldots, n-1\)) tels que
-            \begin{equation}
-                g(v)=\sum_{k=0}^{n-1}a_kf^k(v).
-            \end{equation}
-            Cela donne une application linéaire
-            \begin{equation}
-                \begin{aligned}
-                    \psi\colon \comC(f)&\to \eK[f] \\
-                    g&\mapsto P\tq P(f)v=g(v). 
-                \end{aligned}
-            \end{equation}
-            C'est une application injective parce que si \( \psi(g)=0\) alors \( g(v)=0\) et pour tout \( k\) nous avons \( g\big( f^k(v) \big)=f^k\big( g(v) \big)=0\). L'endomorphisme \( g\) s'annulant sur une base, est nul.
-        \item[\ref{ITEMooSOYYooZVibjrv} implique \ref{ITEMooSOYYooZVibjriv}]
-            Si \( n_1,\ldots, n_r\) sont les dimensions des différents espaces propres de \( f\), nous avons les inégalités
-            \begin{equation}
-                \dim\big( \eK[f] \big)=\deg(\mu)\leq n=n_1+\cdots +n_r\leq n_1^2+\cdots +n_r^2=\dim\big( \comC(f) \big).
-            \end{equation}
-            Par hypothèse d'égalité entre le premier et le dernier terme de cette suite d'inégalités, toutes les inégalités sont des égalités et en particulier \( \dim\big( \comC(f) \big)=n\).
-        \end{subproof}
-            Nous avons fini de prouver toutes les équivalences demandées.
-\end{proof}
-
-\begin{example}
-    Pour mieux comprendre pourquoi le fait d'avoir \( n\) valeurs propres distinctes est équivalent à être cyclique, notons que si deux valeurs propres sont identiques, alors un morceau de la matrice de \( f\) serait par exemple \( \begin{pmatrix}
-          2  &   0    \\ 
-        0    &   2    
-    \end{pmatrix}\), et dans ce cas n'importe quelle combinaison \( ae_i+be_j\) reste proportionnelle à elle-même après application de \( f\). Si nous avons des valeurs propres différentes par contre, nous avons par exemples dans \( \eR^2\) la matrice \( \begin{pmatrix}
-        1    &   0    \\ 
-        0    &   2    
-    \end{pmatrix}\) qui donne \( f(e_1+e_2)=e_1+2e_2\). La partie \( \{ e_1+e_2,e_1+2e_2 \}\) est une base.
-\end{example}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Commutant : cas général}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous considérons encore un espace vectoriel \( E\) de dimension finie \( n\) et un endomorphisme \( f\colon E\to E\). Nous notons \( \mu\) sont polynôme minimal et \( \mu_x\) le polynôme minimal ponctuel en \( x\).
-
-\begin{lemma}[\cite{ooEFHLooXpAOFz,AutourFrobCompa,ooEPEFooQiPESf}]       \label{LEMooDFFDooJTQkRu}
-    Nous avons
-    \begin{equation}
-        \dim\big( \comC(f) \big)\geq \dim(E)
-    \end{equation}
-\end{lemma}
-
-\begin{proof}
-    Si \( f\) est donnée, l'espace \( \comC(f)\) est l'espace des solutions de \( fg=gf\). Supposons avoir choisit une base de \( E\) et notons \( A\) la matrice de \( f\) et \( X\) celle de \( g\). L'équation est \( AX-XA=0\).
-    \begin{subproof}
-        \item[Si \( A\) est trigonalisable]
-            Nous supposons avoir choisi la base de telle sorte que \( A\) soit triangulaire supérieure, et nous allons nous contenter de chercher les solutions \( X\) qui sont également triangulaires supérieure. S'il y en a déjà plus que \( n\), a fortiori le résultat sera vrai.
-
-            Le produit de deux matrices triangulaires supérieures étant une matrice triangulaire supérieure, l'équation \( AX-XA\) contient, pour les coefficients de \( X\), \( n(n+1)/2\) équations. Mais il se fait que les termes diagonaux ne sont pas de vraies équations parce que
-            \begin{equation}
-                (AX-XA)_{kk}=\sum_i\big( A_{ki}X_{ik}-X_{ki}A_{ik} \big)=\sum_{k\leq i\leq k}(A_{ki}X_{ik}-X_{ki}A_{ik})=0.
-            \end{equation}
-            Nous avons donc au maximum 
-            \begin{equation}
-                \frac{ n(x+1) }{2}-n
-            \end{equation}
-            équations linéairement indépendantes pour un minimum de \( n(n+1)/2\) inconnues. L'espace des solutions est donc de dimension au minimum \( n\).
-
-            Cela a l'air d'être une majoration assez large, mais il existe des cas d'égalité.
-
-        \item[Si \( A\) n'est pas trigonalisable]
-
-            La preuve que nous donnons ici est valable même pour les endomorphismes trigonalisables.
-
-            Nous considérons le résultat de Frobenius \ref{THOooDOWUooOzxzxm}. Nous avons donc la structure suivante:
-            \begin{itemize}
-                \item 
-            une décomposition en somme directe \( E=E_1\oplus\ldots\oplus E_r\),
-        \item
-            les espaces \( E_i\) sont fixés par \( f\),
-        \item
-            les endomorphismes \( f_i=f|_{E_i}\) sont cycliques
-        \item
-            le polynôme minimal de \( f_i\) est \( \mu_i\) et \( \prod_{i=1}^r\mu_i=\chi_f\).
-            \end{itemize}
-            Les endomorphismes \( f_i^k\) commutent évidemment avec \( f_j\), et la partie \( \{ f_i^k \}_{k=0,\ldots, \deg(\mu_i)-1}\) est libre. Libre en tout cas en tant que partie de \( \End(E_i)\). Mais en prolongeant par \( 0\) sur \( E\), ça reste libre en tant que partie de \( \End(E)\).
-
-            Bien entendu les \( f_j^k\) et les \( f_i^k\) (\( i\neq j\)) sont linéairement indépendants dans \( \End(E)\) parce qu'ils n'agissent pas sur les mêmes vecteurs. Donc les endomorphismes \( f_i^{k_i}\) avec \( k_i=0,\ldots, \deg(\mu_i)-1\) forment une partie libre de \( \End(E)\) composée d'endomorphismes qui commutent avec \( f\). Il y en a en tout
-            \begin{equation}
-                \sum_{i=1}^r\deg(\mu_i)=\deg(\chi_f)=n.
-            \end{equation}
-            Par conséquent \( \dim\big( \comC(f) \big)\geq \dim(E)\).
-    \end{subproof}
-\end{proof}
-
-\begin{theorem}[\cite{ooRJDSooXpVtMD}]      \label{THOooGLMSooYewNxW}
-    Soit un endomorphisme \( f\colon E\to E\) sur l'espace vectoriel de dimension finie \( n\). Nous notons \( \mu\) et \( \chi\) les polynômes minimal et caractéristique. Nous avons équivalence entre les faits suivants :
-    \begin{enumerate}
-        \item   \label{ITEMooLRXIooLWaYqJii}
-            \( \mu=\chi\),
-        \item   \label{ITEMooLRXIooLWaYqJi}
-            \( f\) est cyclique,
-        \item   \label{ITEMooLRXIooLWaYqJiii}
-            \( \comC(f)=\eK[f]\).
-    \end{enumerate}
-\end{theorem}
-
-\begin{proof}
-    Plusieurs implications. Notons que \ref{ITEMooLRXIooLWaYqJii} implique \ref{ITEMooLRXIooLWaYqJii} a déjà été démontré par le lemme \ref{LEMooKUQDooKFeIYq}.
-    \begin{subproof}
-        \item[\ref{ITEMooLRXIooLWaYqJii} implique \ref{ITEMooLRXIooLWaYqJi}]
-            Conformément à ce que nous permet le lemme \ref{LemSYsJJj} nous choisissons\footnote{Dans toute la suite,nous devrions écrire \( \mu_f\) et \( \mu_{f,a}\) mais nous omettons d'indiquer explicitement la dépendance en \( f\).} \( a\in E\) de telle sorte à avoir \( \mu_a=\mu\). De plus pour \( x\in E\) nous considérons l'application
-            \begin{equation}
-                \begin{aligned}
-                    \varphi_x\colon \eK[X]&\to E \\
-                    P&\mapsto P(f)x. 
-                \end{aligned}
-            \end{equation}
-            Nous avons \( \varphi_a(P)=P(f)a\) et vu que \( E_{a}\) est engendré par les \( f^k(a)\) nous avons \( \varphi_a\big( \eK[X] \big)=E_a\). De plus l'application \( \varphi_a\) passe aux classes pour \( (\mu_a)\). Pour rappel, un élément de \( \eK[X]/(\mu_a)\) est de la forme
-            \begin{equation}
-                \bar P=\{ P+Q\mu_a \}_{Q\in \eK[X]}.
-            \end{equation}
-            Nous considérons donc l'application
-            \begin{equation}
-                \varphi_a\colon \frac{ \eK[X] }{ (\mu_a) }\to E_a
-            \end{equation}
-            et nous prouvons que c'est un isomorphisme d'espace vectoriel.
-            \begin{subproof}
-                \item[Linéaire]
-                    Parce que \( (\lambda P+Q)(f)=(\lambda P)(f)+Q(f)\).
-                \item[Injectif]
-                    Si \( \varphi_a(\bar P)=0\) alors \( \varphi_a(P)=0\) (dans la deuxième, \( \varphi_a\) est l'application définie sur les polynômes et non sur les classes), ce qui montrer que \( P\) est annulateur de \( a\). Mais par définition \ref{DEFooUICRooBGYhqQ} du polynôme minimal ponctuel, \( \mu_a\) est générateur de \( \ker(\varphi_a)\); donc il existe \( Q\in \eK[X]\) tel que \( P=Q\mu_a\). En d'autres termes, du point de vue du quotient, \( \bar P=0\).
-                \item[Surjectif]
-                    Si \( x\in E_a\) alors il existe des coefficients \( x_k\in \eK\) tels que \( x=\sum_{k=0}^{\deg(\mu_a)-1}x_kf^k(a)\), c'est à dire \( x=P(f)a=\varphi_a(P)\).
-            \end{subproof}
-            Mais par hypothèse et par choix de \( a\) nous avons \( \mu_a=\mu=\chi\), donc en fait \( E_a=\eK[X]/(\chi)\). Mais nous savons que \( \deg(\chi)=\dim(E)\) et que \( \dim\big( \eK[X]/P \big)=\deg(P)\) par la proposition \ref{PropooCFZDooROVlaA}. Au final nous avons \( \dim(E_a)=\deg(\chi)=\dim(E)\). Et par conséquent \( E_a=E\). Cela prouver que \( a\) est un vecteur cyclique pour \( f\).
-
-        \item[\ref{ITEMooLRXIooLWaYqJi} implique \ref{ITEMooLRXIooLWaYqJiii}]
-            Soit \( g\in \comC(f)\); nous devons prouver que \( g\) est un polynôme de \( f\). Par hypothèse nous avons un vecteur cyclique que nous notons \( v\). Nous avons un polynôme \( P\) (dépendant de \( g\)) tel que \( g(v)=P(f)v\). Nous allons voir que \( g=P(f)\). Soit \( y\in E\) et un polynôme \( Q\) tel que \( y=Q(f)v\); en notant que \( g\) commute avec \( P(f)\) nous avons 
-            \begin{equation}
-                g(y)=g\big( Q(f)v \big)=Q(f)\big( g(v) \big)=Q(f)\big( P(f)v \big)=P(f)Q(f)v=P(f)y.
-            \end{equation}
-            Donc \( g=P(f)\).
-
-        \item[\ref{ITEMooLRXIooLWaYqJiii} implique \ref{ITEMooLRXIooLWaYqJii}]
-
-            Nous avons les inégalités :
-            \begin{equation}
-                n\leq \dim\big( \comC(f) \big)=\dim\big( \eK[f] \big)=\deg(\mu)\leq \deg(\chi)=n.
-            \end{equation}
-            La première est le lemme \ref{LEMooDFFDooJTQkRu}. Toutes les inégalités sont des égalités. En particulier \( \deg(\mu)=n\), ce qui signifie que \( \mu=\chi\) parce que \( \mu\) est un polynôme diviseur de \( \chi\), de même degré que \( \chi\) et unitaire tout comme \( \chi\).
-
-    \end{subproof}
-\end{proof}
-
-\begin{corollary}[\cite{ooEPEFooQiPESf}]        \label{CORooAKQEooSliXPp}
-    En suivant les notations sur les extensions de corps de base de la section \ref{SECooAUOWooNdYTZf}, l'endomorphisme \( f\colon E\to F\) est cyclique si et seulement si l'endomorphisme \( f_{\eL}\colon E_{\eL}\to F_{\eL}\) est cyclique.
-\end{corollary}
-
-\begin{proof}
-    Nous savons par le théorème \ref{THOooGLMSooYewNxW} qu'un endomorphisme est cyclique si et seulement si son polynôme minimal est égal à son polynôme caractéristique. Or par les propositions \ref{PROPooZAZFooUFdCUv} et \ref{PROPooXVZMooXcJrsJ}, nous savons que ces polynômes sont identiques pour \( f\) et pour \( f_{\eL}\).
-\end{proof}
-
-\begin{theorem}[Similitude et extension de corps\cite{ooEPEFooQiPESf}]      \label{THOooHUFBooReKZWG}
-    Les applications linéaires \( f,g\colon E\to E\) sont semblables si et seulement si \( f_{\eL}\) et \( g_{\eL}\) le sont.
-\end{theorem}
-
-\begin{proof}
-    En ce qui concerne le sens direct, s'il existe \( m\in\GL(E)\) tel que \( f=mgm^{-1}\) alors il suffit d'appliquer le lemme \ref{LEMooWZGSooONEnjZ} pour avoir \( f_{\eL}=m_{\eL}g_{\eL}m_{\eL}^{-1}\).
-
-    Nous considérons les invariants de similitude de \( f\) du théorème \ref{THOooDOWUooOzxzxm}. Il existe une unique suite de polynômes unitaires \( \mu_i\) ($i=1,\ldots, s$) tels que \( \mu_i\divides \mu_{i+1}\) et pour laquelle nous avons une décomposition \( E=E_1\oplus \ldots\oplus E_s\) pour laquelle \( f|_{E_i}\colon E_1\to E_i\) est cyclique et de polynôme minimal \( \mu\).
-
-    Nous avons aussi \( E_{\eL}=(E_1)_{\eL}\oplus\ldots \oplus (E_s)_{\eL}\) et les \( (E_i)_{\eL}\) sont stables sous \( f_{\eL}\) qui y sera également cyclique (corollaire  \ref{CORooAKQEooSliXPp}). De plus le polynôme minimal de \( f_{\eL}|_{(E_i)_{\eL}}\) est également \( \mu_i\).
-
-    Autrement dit, la suite \( \mu_i\) est également la suite des invariants de similitude de \( f_{\eL}\). La remarque \ref{REMooPVLEooYDRXQI} nous permet de conclure que \( f\) et \( g\) sont semblables si et seulement si \( f_{\eL}\) et \( g_{\eL}\) le sont.
-\end{proof}
diff --git a/tex/frido/141_EspacesVecto.tex b/tex/frido/141_EspacesVecto.tex
deleted file mode 100644
index 8cf8c1df1..000000000
--- a/tex/frido/141_EspacesVecto.tex
+++ /dev/null
@@ -1,843 +0,0 @@
-% This is part of Mes notes de mathématique
-% Copyright (c) 2011-2017
-%   Laurent Claessens, Carlotta Donadello
-% See the file fdl-1.3.txt for copying conditions.
-
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorème de Burnside}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{lemma}       \label{LemwXXzIt}
-    Soit \( P\), un polynôme sur \( \eK\). Une racine de \( P\) est une racine simple si et seulement si elle n'est pas racine de \( P'\).
-\end{lemma}
-
-\begin{theorem}     \label{ThoBurnsideoPuCtS}
-    Toute représentation d'un groupe abélien d'exposant fini sur \( \eC^n\) a une image finie.
-\end{theorem}
-
-\begin{proof}
-    Dans le cas d'un groupe abélien, la démonstration est facile. Étant donné que \( G\) est d'exposant fini, il existe \( \alpha\in \eN^*\) tel que \( g^{\alpha}=e\) pour tout \( g\in G\). Le polynôme \( P(X)=X^{\alpha}-1\) est scindé à racines simples. En effet tout polynôme sur \( \eC\) est scindé. Le fait qu'il soit à racines simples provient du lemme \ref{LemwXXzIt} parce que si \( a^{\alpha}=1\), alors il n'est pas possible d'avoir \( \alpha a^{\alpha-1}=0\).
-
-    Par ailleurs \( P(g)=0\). Le fait que nous ayons un polynôme annulateur de \( g\) scindé à racines simples implique que \( g\) est diagonalisable (théorème \ref{ThoDigLEQEXR}). Le fait que \( G\) soit abélien montre qu'il existe une base de \( \eC^n\) dans laquelle tous les éléments de \( G\) sont diagonaux. Nous devons par conséquent montrer qu'il existe un nombre fini de matrices de la forme
-    \begin{equation}
-        \begin{pmatrix}
-            \lambda_1    &       &       \\
-                &   \ddots    &       \\
-                &       &   \lambda_n
-        \end{pmatrix}.
-    \end{equation}
-    Nous savons que \( \lambda_i^{\alpha}=1\) parce que \( g^{\alpha}=\mtu\), par conséquent chacun des \( \lambda_i\) est une racine de l'unité dont il n'existe qu'un nombre fini.
-\end{proof}
-
-\begin{theorem}[Burnside\cite{fJhCTE,ooFBZQooXyHIWK}]\label{ThooJLTit}
-    Un sous-groupe de \( \GL(n,\eC)\) est fini si et seulement s'il est d'exposant\footnote{Définition \ref{DefvtSAyb}.} fini.
-\end{theorem}
-\index{exposant}
-\index{racine!de l'unité}
-\index{endomorphisme!diagonalisable}
-
-\begin{proof}
-    Soit \( G\) un sous-groupe de \( \GL(n,\eC)\). Si \( G\) est fini, l'ordre de ses éléments divise \( | G |\) (corollaire \ref{CorpZItFX}) au théorème de Lagrange et l'exposant est le PPCM qui est donc fini également.
-
-    Nous supposons maintenant que l'ordre de \( G\) est fini. Nous notons \( e\) l'exposant de \( G\). En particulier, tous les éléments de \( G\) sont des racines du polynôme \( X^e-1\).
-    
-    \begin{subproof}
-        \item[Générateurs]
-
-            Le groupe \( G\) est une partie de \( \eM(n,\eC)\) dont nous considérons l'algèbre engendrée (définition \ref{DefkAXaWY}) \( \mG\). Soit \( C_1,\ldots, C_r\) une famille génératrice de \( \mG\) constituée d'éléments de \( G\) et la fonction
-            \begin{equation}
-                \begin{aligned}
-                    \tau\colon G&\to \eC^r \\
-                    A&\mapsto \big( \tr(AC_1),\ldots, \tr(AC_r) \big). 
-                \end{aligned}
-            \end{equation}
-
-        \item[\( \tau\) est injective] Supposons que \( \tau(A)=\tau(B)\). Alors pour tout générateur \( C_i\) nous avons \( \tr(AC_i)=\tr(BC_i)\) et par linéarité de la trace, nous avons
-            \begin{equation}    \label{EqnCYmKW}
-                \tr(AM)=\tr(BM)
-            \end{equation}
-            pour tout \( M\in G\). Notons par ailleurs
-            \begin{equation}
-                N=AB^{-1}-\mtu,
-            \end{equation}
-            qui est diagonalisable parce que \( AB^{-1}\in G\) et donc est annulé par le polynôme \( X^e-1\) qui est scindé à racines simples. Du coup \( AB^{-1}\) est diagonalisable; posons \( PAB^{-1}P^{-1}=D\), alors \( P\big( AB^{-1}-\mtu \big)P^{-1}=D-\mtu\) qui est encore diagonale. Donc \( N\) est diagonalisable.
-
-            Par ailleurs nous avons
-            \begin{subequations}
-                \begin{align}
-                    \tr\big( (AB^{-1})^p \big)&=\tr\big( AB^{-1}(AB^{-1})^{p-1} \big)\\
-                    &=\tr\big( BB^{-1}(AB^{-1})^{p-1} \big) &\text{\eqref{EqnCYmKW}}\\
-                    &=\tr\big( (AB^{-1})^{p-1} \big).
-                \end{align}
-            \end{subequations}
-            En continuant nous obtenons
-            \begin{equation}
-                \tr\big(  (AB^{-1})^p \big)=\tr(\mtu)=n.
-            \end{equation}
-            
-            D'autre part, 
-            \begin{equation}
-                N^k=(AB^{-1}-\mtu)^k=\sum_{p=0}^k{p\choose k}(-1)^{k-p}(AB^{-1})^p
-            \end{equation}
-            En prenant la trace, et en tenant compte du fait que \( \tr\big( (AB^{-1})^p \big)=n\),
-            \begin{equation}
-                \tr(N^k)=\sum_{p=0}^k{p\choose k}(-1)^{k-p}n=n(1-1)^k=0.
-            \end{equation}
-            Donc la trace de \( N^k\) est nulle et le lemme \ref{LemzgNOjY} nous enseigne que \( N\) est alors nilpotente. Étant donné qu'elle est aussi diagonalisable, elle est nulle. Nous en concluons que \( AB^{-1}=\mtu\) et donc que \( A=B\). La fonction \( \tau\) est donc injective.
-
-        \item[Nombre fini de valeurs]
-
-            Les éléments de \( G\) sont annulés par \( X^e-1\) qui est un polynôme scindé à racines simples. Dons le polynôme minimal d'un élément de \( G\) est (a fortiori) scindé à racines simples et le théorème \ref{ThoDigLEQEXR} nous assure alors que ces éléments sont diagonalisables. Du coup les valeur propres des matrices de \( G\) sont des racines \( e\)ièmes de l'unité. Par conséquent les traces des éléments de \( G\) ne peuvent prendre qu'un nombre fini de valeurs : toutes les sommes de \( n\) racines \( e\)ièmes de l'unité. Mais vu que les \( C_i\) sont dans \( G\), nous avons
-            \begin{equation}
-                \Image(\tau)=\{ \tr(A)\tq A\in G \}^r,
-            \end{equation}
-            qui est un ensemble fini. Par conséquent \( G\) est fini parce que \( \tau\) est injective.
-    \end{subproof}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Lie-Kolchin}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Contrairement à ce que l'on peut parfois croire, il n'est pas vrai que toute matrice à coefficient réel est diagonalisable, même pas sur \( \eC\). La raison est qu'une telle matrice peut très bien avoir des valeurs propres multiples.
-
-\begin{example} \label{ExBRXUooIlUnSx}
-    Le théorème \ref{ThoDigLEQEXR} nous donne une façon simple de trouver des matrices non diagonalisables sur \( \eC\) : il suffit que le polynôme minimal ne soit pas scindé à racines simples. Par exemple
-    \begin{equation}
-        A=\begin{pmatrix}
-            1    &   1    \\ 
-            0    &   1    
-        \end{pmatrix},
-    \end{equation}
-    dont le polynôme caractéristique est \( \chi_A=(1-X)^2\). Ce polynôme n'a manifestement pas des racines simples. Nous pouvons faire le calcul explicite pour montrer que \( A\) n'est pas diagonalisable. D'abord l'unique valeur propre de \( A\) est \( 1\) et nous pouvons sans peine résoudre
-    \begin{equation}
-        \begin{pmatrix}
-            1    &   1    \\ 
-            0    &   1    
-        \end{pmatrix}\begin{pmatrix}
-            x    \\ 
-            y    
-        \end{pmatrix}=\begin{pmatrix}
-            x    \\ 
-            y    
-        \end{pmatrix}
-    \end{equation}
-    qui revient au système
-    \begin{subequations}
-        \begin{numcases}{}
-            x+y=x\\
-            y=y.
-        \end{numcases}
-    \end{subequations}
-    La première équation donne directement \( y=0\). Le seul espace propre est de dimension \( 1\) et est engendré par \( \begin{pmatrix}
-        1    \\ 
-        0    
-    \end{pmatrix}\).
-\end{example}
-
-La remarque \ref{RemBOGooCLMwyb} donne un exemple un peu plus avancé, qui montre la multiplicité algébrique et géométrique d'une racine d'un polynôme caractéristique.
-
-\begin{lemma}[Trigonalisation simultanée]   \label{LemSLGPooIghEPI}
-    Une famille de matrices de \( \GL(n,\eC)\) commutant deux à deux est simultanément trigonalisable.
-\end{lemma}
-\index{trigonalisation!simultanée}
-
-\begin{proof}
-    Commençons par enfoncer une porte ouverte par la proposition \ref{PropKNVFooQflQsJ} : sur \( \GL(n,\eC)\) toutes les matrices sont trigonalisables parce que tous les polynômes sont scindés.
-
-    Nous effectuons la démonstration par récurrence sur la dimension. Si \( n=1\) alors toues les matrice sont triangulaires et nous ne nous posons pas de questions. Nous supposons donc \( n>1\).
-
-    Soit la famille \( (A_i)_{i\in I}\) dans \( \GL(n,\eC)\) et \( A_0\) un de ses éléments. Nous nommons \( \lambda_1,\ldots, \lambda_r\) les valeurs propres distinctes de \( A_0\). Le théorème de décomposition primaire \ref{ThoSpectraluRMLok} nous donne la somme directe d'espaces caractéristiques\footnote{Définition \ref{DefFBNIooCGbIix}.}
-    \begin{equation}
-        E=F_{\lambda_1}(A_0)\oplus\ldots\oplus F_{\lambda_r}(A_0).
-    \end{equation}
-    Nous pouvons supposer que cette somme n'est pas réduite à un seul terme. En effet si tel était le cas, \( A_0\) serait un multiple de l'identité parce que \( A_0\) n'aurait qu'une seule valeur propre et les sommes dans la décomposition de Dunford \ref{ThoRURcpW}\ref{ItemThoRURcpWiii} se réduisent à un seul terme (et \( p_i=\id\)). En particulier les dimensions des espaces \( F_{\lambda}(A_0)\) sont strictement plus petites que \( n\).
-
-    Vu que tous les \( A_i\) commutent avec \( A_0\), les espaces \( F_{\lambda}(A_0)\) sont stables par les \( A_i\) et nous pouvons trigonaliser les \( A_i\) simultanément sur chacun des \( F_{\lambda}(A_0)\) en utilisant l'hypothèse de récurrence.
-\end{proof}
-
-\begin{theorem}[Lie-Kolchin\cite{PAXrsMn}]  \label{ThoUWQBooCvutTO}
-    Tout sous-groupe connexe et résoluble de \( \GL(n,\eC)\) est conjugué à un groupe de matrices triangulaires.
-\end{theorem}
-\index{trigonalisation!simultanée}
-\index{théorème!Lie-Kolchin}
-
-\begin{proof}
-    Soit \( G\) un sous-groupe connexe et résoluble de \( \GL(n,\eC)\).
-    
-    \begin{subproof}
-        \item[Si sous-espace non trivial stable par \( G\)]
-
-    Nous commençons par voir ce qu'il se passe s'il existe un sous-espace vectoriel non trivial \( V\) de \( \eC^n\) stabilisé par \( G\). Pour cela nous considérons une base de \( \eC^n\) dont les premiers éléments forment une base de \( V\) (base incomplète, théorème \ref{ThonmnWKs}). Les éléments de \( G\) s'écrivent, dans cette base,
-    \begin{equation}    \label{EqGOKTooEaGACG}
-        \begin{pmatrix}
-            g_1    &   *    \\ 
-            0    &   g_2    
-        \end{pmatrix}.
-    \end{equation}
-    Les matrices \( g_1\) et \( g_2\) sont carrés. Nous considérons alors l'application \( \psi\) définie par
-    \begin{equation}
-        \begin{aligned}
-            \psi\colon G&\to \GL(V) \\
-            g&\mapsto g_1.
-        \end{aligned}
-    \end{equation}
-    Cela est un morphisme de groupes parce que
-    \begin{equation}
-        \begin{pmatrix}
-            g_1    &   *    \\ 
-            0    &   g_2    
-        \end{pmatrix}\begin{pmatrix}
-            h_1    &   *    \\ 
-            0    &   h_2    
-        \end{pmatrix}=
-        \begin{pmatrix}
-            g_1h_1    &   *    \\ 
-            0    &   g_2h_2    
-        \end{pmatrix},
-    \end{equation}
-    de telle sorte que \( \psi(gh)=\psi(g)\psi(h)\).
-
-    Le groupe \( \psi(G)\) est connexe et résoluble. En effet \( \psi(G)\) est connexe en tant qu'image d'un connexe par une application continue (proposition \ref{PropGWMVzqb}). Et il est résoluble en tant qu'image d'un groupe résoluble par un homomorphisme par la proposition \ref{PropBNEZooJMDFIB}. Vu que \( \psi(G)\) est un sous-groupe résoluble et connexe de \( \GL(V)\) et que la dimension de \( V\) est strictement plis petite que celle de \( \eC^n\), une récurrence sur la dimension indique que \( \psi(G)\) est conjugué à un groupe de matrices triangulaires. C'est à dire qu'il existe une base de \( V\) dans laquelle toutes les matrices \( g_1\) (avec \( g\in G\)) sont triangulaires supérieures.
-
-    On fait de même avec l'application \( g\mapsto g_2\), ce qui donne une base du supplémentaire de \( V\) dans laquelle les matrices \( g_2\) sont triangulaires. 
-
-    En couplant ces deux bases, nous obtenons une base de \( \eC^n\) dans laquelle toutes les matrices \eqref{EqGOKTooEaGACG} (c'est à dire toutes les matrices de \( G\)) sont triangulaires supérieures.
-
-    \item[Sinon]
-
-    Nous supposons à présent que \( \eC^n\) n'a pas de sous-espaces non triviaux stables sous \( G\). Nous posons \( m=\min\{ k\tq D^k(G)=\{ e \} \}\), qui existe parce que \( G\) et résoluble et que sa suite dérivée termine sur \( {e}\) (proposition \ref{PropRWYZooTarnmm}).
-
-\item[Si \( m=1\)]
-
-    Si \( m=1\) alors \( G\) est abélien et il existe une base de \( G\) dans laquelle toutes les matrices de \( G\) sont triangulaires (lemme \ref{LemSLGPooIghEPI}). Le premier vecteur d'une telle base serait stable par \( G\), mais comme nous avons supposé qu'il n'y avait pas de sous-espaces non triviaux stabilisés par \( G\), il faut déduire que ce vecteur stable est à lui tout seul non trivial, c'est à dire que \( n=1\). Dans ce cas, le théorème est démontré.
-
-\item[Si \( m>1\)]
-
-    Nous devons maintenant traiter le cas où \( m>1\). Nous posons \( H=D^{m-1}(G)\); cela est un sous-groupe normal et abélien de \( G\). Encore une fois le résultat de trigonalisation simultanée \ref{LemSLGPooIghEPI} donne une base dans laquelle tous les éléments de \( H\) sont triangulaires. En particulier le premier élément de cette base est un vecteur propre commun à toutes les matrices de \( H\).
-
-    Soit \( V\) le sous-espace engendré par tous les vecteurs propres communs de \( H\). Nous venons de voir que \( V\) n'est pas vide. Nous allons montrer que \( V\) est stable par \( G\). Soient \( h\in H\), \( v\in V\) et \( g\in G\) :
-    \begin{equation}    \label{EqPMOBooVLIhrJ}
-        h\big( g(v) \big)=g\underbrace{g^{-1}hg}_{\in H}(v)=g(\lambda v)=\lambda g(v)
-    \end{equation}
-    parce que \( v\) est vecteur propre de \( g^{-1} hg\). Ce que le calcul \eqref{EqPMOBooVLIhrJ} montre est que \( g(v)\) est vecteur propre de \( h\) pour la valeur propre \( \lambda\). Donc \( g(v)\in V\) et \( V\) est stabilisé par \( G\). Mais comme il n'existe pas d'espaces non triviaux stabilisés par \( G\), nous en déduisons que \( V=\eC^n\). Donc tous les vecteurs de \( \eC^n\) sont vecteurs propres communs de \( H\). Autrement dit on a une base de diagonalisation simultanée de \( H\).
-
-\item[\( H\) est dans le centre de \( G\)]
-
-    Montrons à présent que \( H\) est dans le centre de \( G\), c'est à dire que pour tout \( g\in G\) et \( h\in H\) il faut \( ghg^{-1}=h\). D'abord \( ghg^{-1}\) est une matrice diagonale (parce que elle est dans \( H\)) ayant les mêmes valeurs propres que \( h\). En effet si \( \lambda\) est valeur propre de \( ghg^{-1}\) pour le vecteur propre \( v\), alors
-    \begin{subequations}
-        \begin{align}
-            (ghg^{-1})(v)&=\lambda v\\
-            h\big( g^{-1} v \big)&=\lambda \big( g^{-1}v \big),
-        \end{align}
-    \end{subequations}
-    c'est à dire que \( \lambda\) est également valeur propre de \( h\), pour le vecteur propre \( g^{-1} v\). Mais comme \( h\) a un nombre fini de valeurs propres, il n'y a qu'un nombre fini de matrices diagonales ayant les mêmes valeurs propres que \( h\). L'ensemble \( \AD(G)h\) est donc un ensemble fini. D'autre part, l'application \( g\mapsto g^{-1}hg\) est continue, et \( G\) est connexe, donc l'ensemble \( \AD(G)h\) est connexe. Un ensemble fini et connexe dans \( \GL(n,\eC)\) est nécessairement réduit à un seul point. Cela prouve que \( ghg^{-1}=h\) pour tout \( g\in G\) et \( h\in H\).
-
-\item[Espaces propres stables pour tout \( G\)]
-
-        Soit \( h\in H\) et \( W\) un espace propre de \( h\) (ça existe non vide parce que \( H\) est triangularisé, voir plus haut). Alors nous allons prouver que \( W\) est stable pour tous les éléments de \( G\). En effet si \( w\in W\) avec \( h(w)=\lambda w\) alors en permutant \( g\) et \( h\),
-        \begin{equation}
-            hg(w)=g(hw)=\lambda g(w),
-        \end{equation}
-        donc \( g(w)\) est aussi vecteur propre de \( h\) pour la valeurs propre \( \lambda\), c'est à dire que \( g(w)\in W\). Vu que nous supposons que \( \eC^n\) n'a pas d'espaces invariants non triviaux, nous devons conclure que \( W=\eC^n\), c'est à dire que \( H\) est composé d'homothéties. C'est à dire que pour tout \( h\in H\) nous avons \( h=\lambda_h\mtu\).
-
-    \item[Contradiction sur la minimalité de \( m\)]
-
-        Les éléments d'un groupe dérivé sont de déterminant \( 1\) parce que \( \det(g_1g_2g_1^{-1}g_2^{-1})=1\). Par conséquent pour tout \( h\), le nombre \( \lambda_h\) est une racine \( n\)\ieme de l'unité. Vu qu'il n'y a qu'une quantité finie de racines \( n\)\ieme de l'unité, le groupe \( H\) est fini et connexe et donc une fois de plus réduit à un élément, c'est à dire \( H=\{ e \}\). Cela contredit la minimalité de \( m\) et donc produit une contradiction. Nous devons donc avoir \( m=1\).
-
-    \item[Conclusion]
-
-        Nous avons vu que si \( \eC^n\) avait un sous-espace non trivial fixé par \( G\) alors le théorème était démontré. Par ailleurs si \( \eC^n\) n'a pas un tel sous-espace, soit \( m=1\) (et alors le théorème est également prouvé), soit \( m>1\) et alors on a une contradiction.
-
-        Bref, le théorème est prouvé sous peine de contradiction.
-    \end{subproof}
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Formes bilinéaires et quadratiques}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\label{SecTQkRXIu}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Généralités}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}[\cite{RUAoonJAym}]   \label{DefBSIoouvuKR}
-    Soit un espace vectoriel \( E\) et \( \eF\) un corps de caractéristique différente de \( 2\). Une \defe{forme quadratique}{forme!quadratique} sur \( E\) est une application \( q\colon V\to \eF\) pour laquelle il existe une forme bilinéaire symétrique \( b\colon V\times V\to \eF\) satisfaisant \( q(x)=b(x,x)\) pour tout \( x\in V\).
-
-    L'ensemble des formes quadratiques réelles sur \( E\) est noté \( Q(E)\)\nomenclature[B]{\( Q(E)\)}{formes quadratiques réelles sur \( E\)}.
-\end{definition}
-
-\begin{lemma}       \label{LEMooLKNTooSfLSHt}
-    Si \( q\) est une forme quadratique, il existe une unique forme bilinéaire \( b\) telle que \( q(x)=b(x,x)\).
-\end{lemma}
-
-\begin{proof}
-    L'existence n'est pas en cause : c'est la définition d'une forme quadratique. Pour l'unicité, étant donné une forme quadratique, la forme bilinéaire \( b\) doit forcément vérifier l'\defe{identités de polarisation}{identité!polarisation}\index{polarisation (identité)} :
-\begin{equation}    \label{EqMrbsop}
-    b(x,y)=\frac{ 1 }{2}\big( q(x)+q(y)-q(x-y) \big).
-\end{equation}
-Elle est donc déterminée par \( q\).
-\end{proof}
-Notons la division par \( 2\) qui est le pourquoi de la demande de la caractéristique différente de \( 2\) pour \( \eF\) dans la définition de forme quadratique.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Topologie}
-%---------------------------------------------------------------------------------------------------------------------------
-
-La topologie considérée sur \( Q(E)\) est celle de la norme
-\begin{equation}    \label{EqZYBooZysmVh}
-    N(q)=\sup_{\| x \|_E=1}| q(x) |,
-\end{equation}
-qui du point de vue de \( S_n(\eR)\) est
-\begin{equation}    
-    N(A)=\sup_{\| x \|_E=1}| x^tAx |.
-\end{equation}
-Notons que à droite, c'est la valeur absolue usuelle sur \( \eR\).
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Matrice associée}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}     \label{PropcnJyXZ}
-    Soit $M$, une matrice symétrique. Nous avons
-    \begin{enumerate}
-        \item
-        $\det M>0$ et $\tr(M)>0$ implique $M$ définie positive,
-        \item
-        $\det M>0$ et $\tr(M)<0$ implique $M$ définie négative,
-    \item   \label{ItemluuFPN}
-        $\det M<0$ implique ni semi définie positive, ni définie négative 
-        \item
-        $\det M=0$ implique $M$ semi-définie positive ou semi-définie négative.
-    \end{enumerate}
-\end{proposition}
-
-Si une base \( \{ e_i \}_{i=1,\ldots, n}\) de l'espace vectoriel \( E\) est donnée, la \defe{matrice associée}{matrice!associée à une forme quadratique}\index{forme!quadratique!matrice associée} à la forme bilinéaire \( b\) sur \( E\) est la matrice d'éléments
-\begin{equation}
-    B_{ij}=b(e_i,e_j).
-\end{equation}
-Notons que la matrice associée à une forme bilinéaire (ou quadratique associée) est uniquement valable pour une base donnée. Si nous changeons de base, la matrice change. Cependant lorsque nous travaillons sur \( \eR^n\), la base canonique est tellement canonique que nous allons nous permettre de parler de «la» matrice associée à une forme bilinéaire. 
-
-Si \( B_{ij}\) est la matrice associée à la forme bilinéaire \( b\) alors la valeur de \( b(u,v)\) se calcule avec la formule
-\begin{equation}
-    b(x,y)=\sum_{i,j}B_{ij}x_iy_j
-\end{equation}
-lorsque \( x_i\) et \( y_j\) sont les coordonnées de \( x\) et \( y\) dans la base choisie.
-
-\begin{proposition} \label{PropFSXooRUMzdb}
-    Soit \( \{ e_i \}\) une base de \( E\). L'application
-    \begin{equation}
-        \begin{aligned}
-            \phi\colon Q(E)&\to S(n,\eR) \\
-            q&\mapsto \big(   b(e_i,e_j)   \big)_{i,j}
-        \end{aligned}
-    \end{equation}
-    où \( b\) est forme bilinéaire associée à \( q\) est une bijection linéaire et continue.
-\end{proposition}
-
-\begin{proof}
-    Si \( \phi(q)=\phi(q')\); alors
-    \begin{equation}
-        q(x)=\sum_{i,j}\phi(q)_{ij}x_ix_j=\sum_{i,j}\phi(q')_{ij}x_ix_j=q'(x).
-    \end{equation}
-    Donc \( q=q'\). L'application \( \phi\) est donc injective
-
-    De plus elle est surjective parce que si \( B\in S(n,\eR)\) alors la forme quadratique
-    \begin{equation}
-        q(x)=\sum_{i,j}B_{ij}x_ix_j
-    \end{equation}
-    a évidemment \( B\) comme matrice associée. L'application \( \phi\) est donc surjective.
-
-    Notre application \( \phi\) est de plus linéaire parce que l'association d'une forme quadratique à la forme bilinéaire associée est linéaire.
-
-    En ce qui concerne la continuité, nous la prouvons en zéro en considérant une suite convergente \( q_n\stackrel{Q(E)}{\longrightarrow}0\). C'est à dire que
-    \begin{equation}
-        \sup_{\| x \|=1}| q_n(x) |\to 0.
-    \end{equation}
-    Nous rappelons l'identité de polarisation : 
-    \begin{equation}
-        b_n(x,y)=\frac{ 1 }{2}\big( q_n(x-y)-q(x)-q(y) \big).
-    \end{equation}
-    En ce qui concerne deux des trois termes, il n'y a pas de problèmes :
-    \begin{equation}
-        \big| \phi(q_n)_{ij} \big|=\big| b_n(e_i,e_j) \big|\leq\frac{ 1 }{2}\big| b_n(e_i-e_j) \big|+\frac{ 1 }{2}\big| q_n(e_i) \big|+\frac{ 1 }{2}\big| q_n(e_j) \big|.
-    \end{equation}
-    Si \( n\) est assez grand, nous avons tout de suite
-    \begin{equation}
-        \big| \phi(q_n)_{ij} \big|\leq \frac{ 1 }{2}\big| q_n(e_i-e_j) \big|+\epsilon.
-    \end{equation}
-    Nous définissons \( e_{ij}\) et \( \alpha_{ij}\) de telle sorte que \( e_i-e_j=\alpha_{ij}e_{ij}\) avec \( \| e_{ij} \|=1\). Si \( \alpha=\max\{ \alpha_{ij},1 \}\) alors nous avons
-    \begin{equation}
-        q_n(e_i-e_j)=\alpha_{ij}^2q_n(e_{ij})\leq \alpha^2q_n(e_{ij}).
-    \end{equation}
-    Il suffit maintenant de prendre \( n\) assez grand pour avoir \( \sup_{\| x \|=1}| q_n(x) |\leq \frac{ \epsilon }{ \alpha^2 }\) pour avoir
-    \begin{equation}
-        \big| \phi(q_n)_{ij} \big|\leq \frac{ \epsilon }{2}+\frac{ \epsilon }{ \alpha^2 }.
-    \end{equation}
-\end{proof}
-
-\begin{proposition}\label{PropFWYooQXfcVY}
-    Dans la base de diagonalisation de sa matrice associée, une forme quadratique a la forme
-    \begin{equation}
-        q(x)=\sum_i\lambda_ix_i^2
-    \end{equation}
-    où les \( \lambda_i\) sont les valeurs propres de la matrice associée à \( q\).
-\end{proposition}
-
-\begin{proof}
-Soit \( q\) une forme quadratique et \( b\) la forme bilinéaire associée. Si \( \{ f_i \}\) est une base de diagonalisation de la matrice de \( b\) alors dans cette base nous avons
-\begin{equation}
-    q(x)=b(x,x)=\sum_{ij}x_ix_jb(f_i,f_j)=\sum_i\lambda_ix_i^2
-\end{equation}
-où les \( \lambda_i\) sont les valeurs propres de la matrice de \( b\).
-\end{proof}
-Notons que si nous choisissons une autre base de diagonalisation, les \( \lambda_i\) ne changement pas (à part l'ordre éventuellement). Cela pour dire que nous nous permettrons de parler des \defe{valeurs propres}{valeur propre!d'une forme quadratique} d'une forme quadratique comme étant les valeurs propres de la matrice associée.
-
-\begin{proposition}     \label{PROPooUAAFooEGVDRC}
-    Une application linéaire est définie positive si et seulement si sa matrice associée l'est.
-\end{proposition}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Dégénérescence}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit \( b\), une forme bilinéaire symétrique non dégénérée  sur l'espace vectoriel \( E\) de dimension \( n\) sur \( \eK\) où \( \eK\) est un corps de caractéristique différente de \( 2\). Nous notons \( q\) la forme quadratique associée.
-
-\begin{definition}
-    Une forme bilinéaire est \defe{non dégénérée}{forme!bilinéaire!non dégénérée} \( b(x,z)=0\) pour tout \( z\) implique \( x=0\).
-\end{definition}
-
-\begin{lemma}   \label{LemyKJpVP}
-    Soit \( b\) une forme bilinéaire non dégénérée. Si \( x\) et \( y\) sont tels que \( b(x,z)=b(y,z)\) pour tout \( z\), alors \( x=y\).
-\end{lemma}
-
-\begin{proof}
-    C'est immédiat du fait de la linéarité en le premier argument et de la non-dégénérescence : si \( b(x,z)-b(y,z)=0\) alors
-    \begin{equation}
-        b(x-y,z)=0
-    \end{equation}
-    pour tout \( z\), ce qui implique \( x-y=0\).
-\end{proof}
-
-\begin{proposition}
-    La forme bilinéaire \( b\) est non-dénénérée si et seulement si sa matrice associée est inversible.
-\end{proposition}
-
-\begin{proof}
-    Nous savons que la matrice associée est symétrique et qu'elle peut donc être diagonalisée (théorème \ref{ThoeTMXla}). En nous plaçant dans une base de diagonalisation, nous devons prouver que la forme est non-dégénérée si et seulement si les éléments diagonaux de la matrice sont tous non nuls.
-
-    Écrivons \( b(x,z)\) en choisissant pour \( z\) le vecteur de base \( e_k\) de composantes \( (e_k)_j=\delta_{kj}\) :
-    \begin{equation}
-            b(x,e_k)=\sum_{ij}x_i(e_k)_j
-            =\sum_i b_{ik}x_i
-            =b_{kk}x_k.
-    \end{equation}
-    Si \( b\) est dégénérée et si \( x\) est un vecteur non nul (disons que la composante \( x_i\) est non nulle) de \( E\) tel que \( b(x,z)=0\) pour tout \( z\in E\), alors \( b_{ii}=0\), ce qui montre que la matrice de \( b\) n'est pas inversible.
-
-    Réciproquement si la matrice de \( b\) est inversible, alors tous les \( b_{kk}\) sont différents de zéro, et le seul vecteur \( x\) tel que \( b_{kk}x_k=0\) pour tout \( k\) est le vecteur nul.
-\end{proof}
-
-
-\begin{definition}[Isotropie]   \label{DefVKMnUEM}
-    Un vecteur est \defe{isotrope}{isotrope (vecteur)} pour \( b\) s'il est perpendiculaire à lui-même; en d'autres termes, \( x\) est isotrope si et seulement si \( b(x,x)=0\). Un sous-espace \( W\subset E\) est \defe{totalement isotrope}{isotrope!totalement} si pour tout \( x,y\in W\), nous avons \( b(x,y)=0\).
-
-    Le \defe{cône isotrope}{isotrope!cône} de \( b\) est l'ensemble de ses vecteurs isotropes :
-    \begin{equation}
-        C(b)=\{ x\in E\tq b(x,x)=0 \}.
-    \end{equation}
-\end{definition}
-Nous introduisons quelque notations. D'abord pour \( y\in E\) nous notons
-\begin{equation}
-    \begin{aligned}
-        \Phi_y\colon E&\to \eR \\
-        x&\mapsto b(x,y) 
-    \end{aligned}
-\end{equation}
-et ensuite
-\begin{equation}
-    \begin{aligned}
-        \Phi\colon E&\to E^* \\
-        y&\mapsto \Phi_y. 
-    \end{aligned}
-\end{equation}
-\begin{definition}
-    Le fait pour une forme bilinéaire \( b\) d'être dégénérée signifie que l'application \( \Phi\) n'est pas injective. Le \defe{noyau}{noyau!d'une forme bilinéaire} de la forme bilinéaire est celui de \( \Phi\), c'est à dire
-    \begin{equation}
-        \ker(b)=\{ z\in E\tq b(z,y)=0\,\forall y\in E \}.
-    \end{equation}
-    Autrement dit, \( \ker(b)=E^{\perp}\) où le perpendiculaire est pris par rapport à \( b\).
-\end{definition}
-Notons tout de même que nous utilisons la notation \( \perp\) même si \( b\) est dégénérée et éventuellement pas positive; c'est à dire même si la formule \( (x,y)\mapsto b(x,y)\) ne fournit pas un produit scalaire.
-
-\begin{proposition}[\cite{RTzQrdx}]     \label{PropHIWjdMX}
-    Soit \( b\) une forme bilinéaire et symétrique. Alors
-    \begin{enumerate}
-        \item
-            \( \ker(b)\subset C(b)\) (cône d'isotropie, définition \ref{DefVKMnUEM})
-        \item
-            si \( b\) est positive alors \( \ker(b)=C(b)\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    \begin{enumerate}
-        \item
-            Si \( z\in\ker(b)\) alors pour tout \( y\in E\) nous avons \( b(z,y)=0\). En particulier pour \( y=z\) nous avons \( b(z,z,)=0\) et donc \( z\in C(b)\).
-        \item
-            Soit \( b\) positive et \( x\in C(b)\). Par l'inégalité de Cauchy-Schwarz (proposition \ref{ThoAYfEHG}) nous avons
-            \begin{equation}
-                | b(x,y) |\leq \sqrt{   b(x,x)b(y,y) }=0.
-            \end{equation}
-            Donc pour tout \( y\) nous avons \( b(x,y)=0\).
-    \end{enumerate}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Inégalité de Minkowski}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Ce qui est couramment nommé «inégalité de Minkowski» est la proposition \ref{PropInegMinkKUpRHg} dans les espaces \( L^p\). Nous allons en donner ici un cas très particulier.
-
-\begin{proposition} \label{PropACHooLtsMUL}
-    Si \( q\) est une forme quadratique sur \( \eR^n\) et si \( x,y\in \eR^n\) alors
-    \begin{equation}
-        \sqrt{q(x+y)}\leq\sqrt{q(x)}+\sqrt{q(y)}.
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    La proposition \ref{PropFWYooQXfcVY} nous permet de «diagonaliser» la forme quadratique \( q\). Quitte à ne plus avoir une base orthonormale, nous pouvons renormaliser les vecteurs de base pour avoir
-    \begin{equation}
-        q(x)=\sum_ix_i^2.
-    \end{equation}
-    Le résultat n'est donc rien d'autre que l'inégalité triangulaire pour la norme euclidienne usuelle, laquelle est démontrée dans la proposition \ref{PropEQRooQXazLz}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Ellipsoïde}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}   \label{LemYVWoohcjIX}
-    Toute matrice peut être décomposée de façon unique en une partie symétrique et une partie antisymétrique. Cette décomposition est donnée par
-\begin{equation}\label{subEqHIQooyhiWM}
-    \begin{aligned}[]
-            S&=\frac{ M+M^t }{ 2 },&A&=\frac{ M-M^t }{ 2 }
-    \end{aligned}
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-    L'existence est une vérification immédiate de \( S+A=M\) en utilisant \eqref{subEqHIQooyhiWM}. Pour l'unicité, si \( S+A=S'+A'\) alors \( S-S'=A-A'\). Mais \( S-S'\) est symétrique et \( A-A'\) est antisymétrique; l'égalité implique l'annulation des deux membres, c'est à dire \( S=S'\) et \( A=A'\).
-\end{proof}
-
-\begin{definition}  \label{DefOEPooqfXsE}
-    Un \defe{ellipsoïde}{ellipsoïde} dans \( \eR^n\) centré en \( v\) est le lieu des points \( x\) vérifiant l'équation
-    \begin{equation}\label{EqSNWooXfbTH}
-        (x-v)^t M(x-v)=1
-    \end{equation}
-    où \( M\) est une matrice symétrique strictement définie positive\footnote{Définition \ref{DefAWAooCMPuVM}.}.
-
-    Lorsque nous parlons d'ellipsoïde \emph{plein}, il suffit de changer l'égalité en une inégalité.
-\end{definition}
-Une autre façon d'écrire la relation \eqref{EqSNWooXfbTH} est d'écrire \( \langle (x-v),M(x,v)\rangle\) en utilisant le produit scalaire.
-
-\begin{remark}
-    Le fait que \( M\) soit symétrique n'est pas tout à fait obligatoire; la chose important est que toutes les valeurs propres soient strictement positives. En effet si \( M\) a toutes ses valeurs propres strictement positives, nous nommons \( S\) la partie symétrique de \( M\) et \( A\) la partie antisymétrique (lemme \ref{LemYVWoohcjIX}). Alors pour tout \( x\in \eR^n\) nous avons
-    \begin{equation}
-        x^tAx=\langle x, Ax\rangle =\langle A^tx,x \rangle =-\langle Ax, x\rangle =-\langle x,Ax\rangle ,
-    \end{equation}
-    donc \( x^tAx=0\). L'équation \( x^tMx=1\) est donc équivalente à \( x^tSx=1\) (elles ont les mêmes solutions).
-    
-    De plus \( S\) reste strictement définie positive parce que pour tout \( x\in \eR^n\) nous avons 
-    \begin{equation}
-        0<x^tMx=x^tSx.
-    \end{equation}
-\end{remark}
-
-\begin{proposition}\label{PropWDRooQdJiIr}
-    Si \( \ellE\) est un ellipsoïde centrée à l'origine, il existe une base de \( \eR^n\) dans laquelle son équation est :
-    \begin{equation}
-        \sum_{i=1}^n\frac{ x_i^2 }{ a_i^2 }=1.
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Nous avons une matrice symétrique strictement définie positive \( S\) telle que l'équation soit \( \langle x, Sx\rangle =1\). Le théorème spectral \ref{ThoeTMXla} nous fournit une base orthonormale \( \{ e_i \}\) dans laquelle \( Se_i=\lambda_ie_i\) avec \( \lambda_i>0\). En substituant dans l'équation \( \langle x, Sx\rangle =1\) nous trouvons l'équation
-    \begin{equation}
-        \sum_i\lambda_ix_i^2=1.
-    \end{equation}
-    En posant \( a_i=\frac{1}{ \sqrt{\lambda_i} }\), nous trouvons le résultat.  Cette définition des \( a_i\) est toujours possible parce que \( \lambda_i>0\).
-\end{proof}
-
-\begin{corollary}   \label{CorKGJooOmcBzh}
-    Un ellipsoïde plein centré en l'origine admet une équation de la forme \( q(x)\leq 1\) où \( q\) est une forme quadratique strictement définie positive.
-\end{corollary}
-Pour rappel de notation, l'ensemble des formes quadratiques strictement définies positives sur l'espace vectoriel \( E\) est noté \( Q^{++}(E)\).
-
-\begin{proof}
-    Soit \( \{ e_i \}\) une base de \( \eR^n\) telle que l'ellipsoïde \( \ellE\) ait pour équation
-    \begin{equation}
-        \sum_{i=1}^n\frac{ x_i^2 }{ a_i^2 }\leq 1.
-    \end{equation}
-    Nous considérons la forme quadratique
-    \begin{equation}
-        \begin{aligned}
-            q\colon \eR^n&\to \eR \\
-            x&\mapsto \sum_{i=1}^n\frac{ \langle x, e_i\rangle^2 }{ a_i^2 }. 
-        \end{aligned}
-    \end{equation}
-    Nous avons évidemment \( \ellE=\{ x\in \eR^n\tq q(x)\leq 1 \}\). De plus la forme \( q\) est strictement définie positive parce que dès que \( x\neq 0\), au moins un des produits scalaires \( \langle x, e_i\rangle \) est non nul et \( q(x)> 0\).
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorème spectral auto-adjoint}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}      \label{DEFooYNEQooGQgbCf}
-    Si \( E\) est un espace euclidien, un endomorphisme \( f\colon E\to E\) est \defe{auto-adjoint}{endomorphisme!auto-adjoint} si pour tout \( x,y\in E\) nous avons \( \langle x, f(y)\rangle=\langle f(x), y\rangle  \). 
-\end{definition}
-L'ensemble des opérateurs auto-adjoints de \( E\) est noté \( \gS(E)\)\nomenclature[A]{\( \gS(E)\)}{Les opérateurs auto-adjoints de $E$}. Cette notation provient du fait que dans \( \eR^n\) muni du produit scalaire usuel, les opérateurs auto-adjoints sont les matrices symétriques.
-
-\begin{theorem}[Théorème spectral auto-adjoint] \label{ThoRSBahHH}
-    Un endomorphisme auto-adjoint d'un espace euclidien
-    \begin{enumerate}
-        \item
-            est diagonalisable dans une base orthonormée,
-        \item
-            a son spectre réel.
-    \end{enumerate}
-\end{theorem}
-\index{théorème!spectral!autoadjoint}
-\index{diagonalisation!endomorphisme auto-adjoint}
-
-\begin{proof}
-    Nous procédons par récurrence sur la dimension de \( E\), et nous commençons par \( n=1\)\footnote{Dans \cite{KXjFWKA}, l'auteur commence avec \( n=0\) mais moi je n'en ai \wikipedia{en}{Vacuous_truth}{pas le courage.}.}. Soit donc \( f\colon E\to E\) avec \( \langle f(x), y\rangle =\langle x, f(y)\rangle \). Étant donné que \( f\) est également linéaire, il existe \( \lambda\in \eR\) tel que \( f(x)=\lambda x\) pour tout \( x\in E\). Tous les vecteurs de \( E\) sont donc vecteurs propres de \( f\).
-
-    Passons à la récurrence. Nous considérons \( \dim(E)=n+1\) et \( f\in\gS(E)\). Nous considérons la forme bilinéaire symétrique \( \Phi_f\) et la forme quadratique associée \( \phi_f\). Pour rappel,
-    \begin{subequations}
-        \begin{align}
-        \Phi_f(x,y)=\langle x, f(y)\rangle \\
-        \phi_f(x)=\Phi_f(x,x).
-        \end{align}
-    \end{subequations}
-    Et nous allons laisser tomber les indices \( f\) pour noter simplement \( \Phi\) et \( \phi\). Étant donné que \( \overline{ B(0,1) }\) est compacte et que \( \phi\) est continue, il existe \( x_0\in\overline{ B(0,1) }\) tel que 
-    \begin{equation}
-        \lambda=\phi(x_0)=\sup_{x\in\overline{ B(0,1) }}\phi(x).
-    \end{equation}
-    Notons aussi que \( \| x_0 \|=1\) : le maximum est pris sur le bord. Nous posons
-    \begin{equation}
-        g=\lambda\id-f
-    \end{equation}
-    ainsi que 
-    \begin{equation}
-        \Phi_1(x,y)=\langle x, g(y)\rangle .
-    \end{equation}
-    Cela est une forme bilinéaire et symétrique parce que
-    \begin{equation}
-        \Phi_1(y,x)=\langle y, g(x)\rangle =\langle g(y), x\rangle =\langle x, g(y)\rangle =\Phi_1(x,y)
-    \end{equation}
-    où nous avons utilisé le fait que \( g\) était auto-adjoint et la symétrie du produit scalaire. De plus \( \Phi_1\) est semi-définie positive parce que
-    \begin{equation}
-        \Phi_1(x,x)=\langle x, \lambda x-f(x)\rangle =\lambda\| x \|^2-\phi(x).
-    \end{equation}
-    Vu que \( \lambda\) est le maximum, nous avons tout de suite \( \Phi_1(x)\geq 0\) tant que \( \| x \|=1\). Et si \( x\) n'est pas de norme \( 1\), c'est le même prix parce qu'on se ramène à \( \| x \|=1\) en multipliant par un nombre positif. Attention cependant : 
-    \begin{equation}
-        \Phi_1(x_0,x_0)=\lambda\| x_0 \|^2-\phi(x_0)=0.
-    \end{equation}
-    Donc \( \Phi_1\) a un noyau contenant \( x_0\) par la proposition \ref{PropHIWjdMX}. Nous en déduisons que \( \Image(g)\neq E\) en effet, \( x_0\in\Image(g)^{\perp}\), mais nous avons la proposition \ref{PropXrTDIi} sur les dimensions : 
-    \begin{equation}
-        \dim E=\dim(\Image(g))+\dim( \Image(g)^{\perp}).
-    \end{equation}
-    Vu que \( \Image(g)^{\perp}\) est un espace vectoriel non réduit à \( \{ 0 \}\), la dimension de \( \Image(g)\) ne peut pas être celle de \( E\). L'endomorphisme \( g\) n'étant pas surjectif, il ne peut pas être injectif non plus parce que nous sommes en dimension finie; il existe donc \( e_1\in E\) tel que \( g(e_1)=0\) et tant qu'à faire nous choisissons \( \| e_1 \|=1\) (ici la norme est bien celle de l'espace euclidien considéré). Par définition,
-    \begin{equation}
-        f(e_1)=\lambda e_1,
-    \end{equation}
-    c'est à dire que \( \lambda\in\Spec(f)\). Et \( \phi\) étant une forme quadratique réelle nous avons \( \lambda\in \eR\).
-
-    Nous posons à présent \( H=\Span\{ e_1 \}^{\perp}\). C'est un sous-espace stable par \( f\) parce que si \( x\in H\) alors
-    \begin{equation}
-        \langle e_1, f(x)\rangle =\langle f(e_1j),x\rangle =\lambda\langle e_1, x\rangle =0.
-    \end{equation}
-    Nous pouvons donc considérer la restriction de \( f\) à \( H\) : \( f_H\colon H\to H\). Cet endomorphisme est bilinéaire et symétrique sur l'espace \( H\) de dimension inférieure à celle de \( E\), donc la récurrence nous donne une base orthonormée
-    \begin{equation}
-        \{ e_2,\ldots, e_n \}
-    \end{equation}
-    de vecteurs propres de \( f_H\). De plus les valeurs propres sont réelles, toujours par récurrence. Donc
-    \begin{equation}
-        \Spec(f)=\{ \lambda \}\cup\Spec(f_H)\subset \eR.
-    \end{equation}
-    Notons pour être complet que si \( i\geq 2\) alors
-    \begin{equation}
-        \langle e_1, e_i\rangle =0
-    \end{equation}
-    parce que le vecteur \( e_i\) est par construction choisit dans l'espace \( H=e_1^{\perp}\). Nous avons donc bien une base orthonormée de \( E\) construite sur des vecteurs propres de \( f\).
-\end{proof}
-
-\begin{corollary}   \label{CorSMHpoVK}
-    Soit \( E\) un espace vectoriel ainsi que \( \phi\) et \( \psi\) des formes quadratiques sur \( E\) avec \( \psi\) définie positive. Alors il existe une base \( \psi\)-orthonormale dans laquelle \( \phi\) est diagonale.
-\end{corollary}
-
-\begin{proof}
-    Il suffit de considérer l'espace euclidien \( E\) muni du produit scalaire \( \langle x, y\rangle =\psi(x,y)\). Ensuite nous diagonalisons la matrice (symétrique) de \( \phi\) pour ce produit scalaire à l'aide du théorème \ref{ThoRSBahHH}.
-\end{proof}
-
-\begin{definition}      \label{DefYNWUFc}
-    Dans le cas de \( V=\eR^m\) nous avons un produit scalaire canonique. Soient $u$ et $v$, deux vecteurs de $\eR^m$. Le \defe{produit scalaire}{produit!scalaire!sur \( \eR^n\)} de $u$ et $v$, noté $\langle u, v\rangle $ ou $u\cdot v$ est le réel
-	\begin{equation}		\label{EqDefProdScalsumii}
-		\langle u, v\rangle =\sum_{k=1}^m u_kv_k=u_1v_1+u_2v_2+\cdots+u_mv_n.
-	\end{equation}
-\end{definition}
-
-Calculons par exemple le produit scalaire de deux vecteurs de la base canonique : $\langle e_i, e_j\rangle $. En utilisant la formule de définition et le fait que $(e_i)_k=\delta_{ik}$, nous avons
-\begin{equation}
-	\langle e_i, e_j\rangle =\sum_{k=1}^m\delta_{ik}\delta_{jk}.
-\end{equation}
-Nous pouvons effectuer la somme sur $k$ en remarquant qu'à cause du $\delta_{ik}$, seul le terme avec $k=i$ n'est pas nul. Effectuer la somme revient donc à remplacer tous les $k$ par des $i$ :
-\begin{equation}
-	\langle e_i, e_j\rangle =\delta_{ii}\delta_{ji}=\delta_{ji}.
-\end{equation}
-
-Une des propriétés intéressantes du produit scalaire est qu'il permet de décomposer un vecteur dans une base, comme nous le montre la proposition suivante.
-
-\begin{proposition}		\label{PropScalCompDec}
-	Si nous notons $v_i$ les composantes du vecteur $v$, c'est à dire si $v=\sum_{i=1}^m v_ie_i$, alors nous avons $v_j=\langle v, e_j\rangle $.
-\end{proposition}
-
-\begin{proof}
-	\begin{equation}		\label{Eqvejscalcomp}
-		v\cdot e_j=\sum_{i=1}^m\langle v_ie_i, e_j\rangle =\sum_{i=1}^mv_i\langle e_i, e_j\rangle =\sum_{i=1}^mv_i\delta_{ij}
-	\end{equation}
-	En effectuant la somme sur $i$ dans le membre de droite de l'équation \eqref{Eqvejscalcomp}, tous les termes sont nuls sauf celui où $i=j$; il reste donc
-	\begin{equation}
-		v\cdot e_j=v_j.
-	\end{equation}
-\end{proof}
-
-Le produit scalaire ne dépend en réalité pas de la base orthogonale choisie. 
-
-\begin{lemma}
-	Si $\{ e_i \}$ est la base canonique, et si $\{ f_i \}$ est une autre base orthonormale, alors si $u$ et $v$ sont deux vecteurs de $\eR^m$, nous avons
-	\begin{equation}
-		\sum_i u_iv_j=\sum_iu'_iv'_j
-	\end{equation}
-	où $u_i$ sont les composantes de $u$ dans la base $\{ e_i \}$ et $u'_i$ sont celles dans la base $\{ f_i \}$.
-\end{lemma}
-
-\begin{proof}
-	La preuve demande un peu d'algèbre linéaire. Étant donné que $\{ f_i \}$ est une base orthonormale, il existe une matrice $A$ orthogonale ($AA^t=\mtu$) telle que $u'_i=\sum_jA_{ij}u_j$ et idem pour $v$. Nous avons alors
-	\begin{equation}
-		\begin{aligned}[]
-			\sum_iu'_iv'_j&=\sum_i\left( \sum_jA_{ij} u_j\right)\left( \sum_k A_{ik}v_k \right)\\
-			&=\sum_{ijk}A_{ij}A_{ik}u_jv_k\\
-			&=\sum_{jk}\underbrace{\sum_i(A^t)_{ji}A_{ik}}_{=\delta_{jk}}u_jv_k\\
-			&=\sum_{jk}\delta_{jk}u_jv_k\\
-			&=\sum_ku_jv_k.
-		\end{aligned}
-	\end{equation}	
-\end{proof}
-
-Cette proposition nous permet de réellement parler du produit scalaire entre deux vecteurs de façon intrinsèque sans nous soucier de la base dans laquelle nous regardons les vecteurs.
-
-Nous dirons que deux vecteurs sont \defe{orthogonaux}{orthogonal} lorsque leur produit scalaire est nul. Nous écrivons que $u\perp v$ lorsque $\langle u, v\rangle =0$.
-\begin{definition}	\label{DefNormeEucleApp}
-	La \defe{norme euclidienne}{norme!euclidienne!dans $\eR^m$} d'un élément de $\eR^m$ est définie par $\| u \|=\sqrt{u\cdot u}$.
-\end{definition}
-
-Cette définition est motivée par le fait que le produit scalaire $u\cdot u$ donne exactement la norme usuelle donnée par le théorème de Pythagore :
-\begin{equation}
-	u\cdot u=\sum_{i=1}^mu_iu_i=\sum_{i=1}^m u_i^2=u_1^2+u_2^2+\cdots+u_m^2.
-\end{equation}
-
-Le fait que $e_i\cdot e_j=\delta_{ij}$ signifie que la base canonique est \defe{orthonormée}{orthonormé}, c'est à dire que les vecteurs de la base canonique sont orthogonaux deux à deux et qu'ils ont tout $1$ comme norme.
-
-\begin{lemma}\label{LemSclNormeXi}
-	Pour tout $u\in\eR^m$, il existe un $\xi\in\eR^m$ tel que $\| u \|=\xi\cdot u$ et $\| \xi \|=1$.
-\end{lemma}
-
-\begin{proof}
-	Vérifions que le vecteur $\xi=u/\| u \|$ ait les propriétés requises. D'abord $\| \xi \|=1$ parce que $u\cdot u=\| u \|^2$. Ensuite
-	\begin{equation}
-		\xi\cdot u=\frac{ u\cdot u }{ \| u \| }=\frac{ \| u \|^2 }{ \| u \| }=\| u \|.
-	\end{equation}
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Mini introduction au produit tensoriel}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\label{SeOOpHsn}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Définitions}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit \( E\), un espace vectoriel de dimension finie. Si \( \alpha\) et \( \beta\) sont deux formes linéaires sur un espace vectoriel \( E\), nous définissons \( \alpha\otimes \beta\) comme étant la \( 2\)-forme donnée par
-\begin{equation}
-    (\alpha\otimes \beta)(u,v)=\alpha(u)\beta(v).
-\end{equation}
-Si \( a\) et \( b\) sont des vecteurs de \( E\), ils sont vus comme des formes sur \( E\) via le produit scalaire et nous avons
-\begin{equation}
-    (a\otimes b)(u,v)=(a\cdot u)(b\cdot v).
-\end{equation}
-Cette dernière équation nous incite à pousser un peu plus loin la définition de \( a\otimes b\) et de simplement voir cela comme la matrice de composantes
-\begin{equation}
-    (a\otimes b)_{ij}=a_ib_j.
-\end{equation}
-Cette façon d'écrire a l'avantage de ne pas demander de se souvenir qui est une vecteur ligne, qui est un vecteur colonne et où il faut mettre la transposée. Évidemment \( (a\otimes b)\) est soit \( ab^t\) soit \( a^tb\) suivant que \( a\) et \( b\) soient ligne ou colonne.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Application d'opérateurs}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}   \label{LemMyKPzY}
-    Soient \( x,y\in E\) et \( A,B\) deux opérateurs linéaires sur \( E\) vus comme matrices. Alors
-    \begin{equation}        \label{EqXdxvSu}
-        (Ax\otimes By)=A(x\otimes y)B^t.
-    \end{equation}
-\end{lemma}
-
-\begin{proof}
-    Calculons la composante \( ij\) de la matrice \( (Ax\otimes By)\). Nous avons
-    \begin{subequations}
-        \begin{align}
-            (Ax\otimes By)_{ij}&=(Ax)_i(By)_j\\
-            &=\sum_{kl}A_{ik}x_kB_{jl}y_l\\
-            &=A_{ik}(x\otimes y)_{kl}B_{jl}\\
-            &=\big( A(x\otimes y)B^t \big)_{ij}.
-        \end{align}
-    \end{subequations}
-\end{proof}
-
-% TODO: Ajouter un texte sur les équations de plan, et pourquoi ax+by+cz+d=0 est perpendiculaire au vecteur (a,b,c).
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Méthode de Gauss pour résoudre des systèmes d'équations linéaires}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Pour résoudre un système d'équations linéaires, on procède comme suit:
-\begin{enumerate}
-\item Écrire le système sous forme matricielle. \[\text{p.ex. } \begin{cases} 2x+3y &= 5 \\ x+2y &= 4 \end{cases} \Leftrightarrow \left(\begin{array}{cc|c} 2 & 3 & 5 \\ 1 & 2 & 4 \end{array}\right) \]
-\item Se ramener à une matrice avec un maximum de $0$ dans la partie de gauche en utilisant les transformations admissibles:
-\begin{enumerate}
-\item Remplacer une ligne par elle-même + un multiple d'une autre;
-\[\text{p.ex. } \left(\begin{array}{cc|c} 2 & 3 & 5 \\ 1 & 2 & 4 \end{array}\right)  \stackrel{L_1  - 2. L_2 \mapsto L_1'}{\Longrightarrow} \left(\begin{array}{cc|c} 0 & -1 & -3 \\ 1 & 2 & 4 \end{array}\right) \]
-\item Remplacer une ligne par un multiple d'elle-même;
-\[\text{p.ex. } \left(\begin{array}{cc|c} 0 & -1 & -3 \\ 1 & 2 & 4 \end{array}\right)  \stackrel{-L_1  \mapsto L_1'}{\Longrightarrow} \left(\begin{array}{cc|c} 0 & 1 & 3 \\ 1 & 2 & 4 \end{array}\right) \]
-\item Permuter des lignes.
-\[\text{p.ex. } \left(\begin{array}{cc|c} 0 & 1 & 3 \\ 1 & 0 & -2 \end{array}\right)  \stackrel{L_1  \mapsto L_2' \text{ et } L_2  \mapsto L_1'}{\Longrightarrow} \left(\begin{array}{cc|c} 1 & 0 & -2 \\ 0 & 1 & 3  \end{array}\right) \]
-\end{enumerate}
-\item Retransformer la matrice obtenue en système d'équations.
-\[\text{p.ex. }  \left(\begin{array}{cc|c} 1 & 0 & -2 \\ 0 & 1 & 3  \end{array}\right) \Leftrightarrow \begin{cases} x &= -2 \\ y &= 3 \end{cases}  \]
-\end{enumerate}
-
-\begin{remark}
-\begin{itemize}
-\item Si on obtient une ligne de zéros, on peut l'enlever:
-\[\text{p.ex. }  \left(\begin{array}{ccc|c} 3 & 4 & -2 & 2 \\ 4 & -1 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right) \Leftrightarrow  \left(\begin{array}{ccc|c} 3 & 4 & -2 & 2 \\ 4 & -1 & 3 & 0 \end{array}\right) \]
-\item Si on obtient une ligne de zéros suivie d'un nombre non-nul, le système d'équations n'a pas de solution:
-\[\text{p.ex. }  \left(\begin{array}{ccc|c} 3 & 4 & -2 & 2 \\ 4 & -1 & 3 & 0 \\ 0 & 0 & 0 & 7 \end{array}\right) \Leftrightarrow  \begin{cases} \cdots \\ \cdots \\ 0x + 0y + 0z = 7 \end{cases} \Rightarrow \textbf{Impossible} \]
-\item Si on moins d'équations que d'inconnues, alors il y a une infinité de solutions qui dépendent d'un ou plusieurs paramètres:
-\[\text{p.ex. }  \left(\begin{array}{ccc|c} 1 & 0 & -2 & 2 \\ 0 & 1 & 3 & 0 \end{array}\right) \Leftrightarrow  \begin{cases} x - 2z = 2 \\ y + 3z = 0 \end{cases} \Leftrightarrow  \begin{cases} x = 2 + 2\lambda \\ y = -3\lambda \\ z = \lambda \end{cases} \]
-\end{itemize}
-\end{remark}
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-% This is part of Mes notes de mathématique
-% Copyright (c) 2008-2017
-%   Laurent Claessens
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Déterminants}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\label{SecGYzHWs}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Formes multilinéaires alternées}
-%---------------------------------------------------------------------------------------------------------------------------
-
-%  Lire http://www.les-mathematiques.net/phorum/read.php?2,302266
-
-\begin{definition}\index{déterminant!forme linéaire alternée}
-    Soit \( E\), un \( \eK\)-espace vectoriel. Une forme linéaire \defe{alternée}{forme linéaire!alternée}\index{alternée!forme linéaire} sur \( E\) est une application linéaire \( f\colon E\to \eK\) telle que \( f(v_1,\ldots, v_k)=0\) dès que \( v_i=v_j\) pour certains \( i\neq j\).
-\end{definition}
-
-\begin{lemma}   \label{LemHiHNey}
-    Une forme linéaire alternée est antisymétrique. Si \( \eK\) est de caractéristique différente de \( 2\), alors une forme antisymétrique est alternée.
-\end{lemma}
-
-\begin{proof}
-    Soit \( f\) une forme alternée; quitte à fixer toutes les autres variables, nous pouvons travailler avec une \( 2\)-forme et simplement montrer que \( f(x,y)=-f(y,x)\). Pour ce faire nous écrivons
-    \begin{equation}
-        0=f(x+y,x+y)=f(x,x)+f(x,y)+f(y,x)+f(y,y)=f(x,y)+f(y,x).
-    \end{equation}
-    
-    Pour la réciproque, si \( f\) est antisymétrique, alors \( f(x,x)=-f(x,x)\). Cela montre que \( f(x,x)=0\) lorsque \( \eK\) est de caractéristique différente de deux.
-\end{proof}
-
-\begin{proposition}[\cite{GQolaof}] \label{ProprbjihK}
-    Soit \( E\), un \( \eK\)-espace vectoriel de dimension \( n\), où la caractéristique de \( \eK\) n'est pas deux. L'espace des \( n\)-formes multilinéaires alternées sur \( E\) est de \( \eK\)-dimension \( 1\).
-\end{proposition}
-\index{groupe!permutation}
-\index{groupe!et géométrie}
-\index{espace!vectoriel!dimension}
-\index{rang}
-\index{déterminant}
-\index{dimension!\( n\)-formes multilinéaires alternées}
-
-\begin{proof}
-    Soit \( \{ e_i \}\), une base de \( E\) et \( f\colon E\to \eK\) une \( n\)-forme linéaire alternée, puis \( (v_1,\ldots, v_n)\) des vecteurs de \( E\). Nous pouvons les écrire dans la base
-    \begin{equation}
-        v_j=\sum_{i=1}^n\alpha_{ij}e_i
-    \end{equation}
-    et alors exprimer \( f\) par
-    \begin{subequations}
-        \begin{align}
-            f(v_1,\ldots, v_n)&=f\big( \sum_{i_1=1}^n\alpha_{1i_1}e_{i_1},\ldots, \sum_{i_n=1}^n\alpha_{ni_n}e_{i_n} \big)\\
-            &=\sum_{i,j}\alpha_{1i_1}\ldots \alpha_{ni_n}f(e_{i_1},\ldots, e_{i_n}).
-        \end{align}
-    \end{subequations}
-    Étant donné que \( f\) est alternée, les seuls termes de la somme sont ceux dont les \( i_k\) sont tous différents, c'est à dire ceux où \( \{ i_1,\ldots, i_n \}=\{ 1,\ldots, n \}\). Il y a donc un terme par élément du groupe des permutations \( S_n\) et
-    \begin{equation}
-        f(v_1,\ldots, v_n)=\sum_{\sigma\in S_n}\alpha_{\sigma(1)1}\ldots \alpha_{\sigma(n)n}f(e_{\sigma(1)},\ldots, e_{\sigma(n)}).
-    \end{equation}
-    En utilisant encore une fois le fait que la forme \( f\) soit alternée, \( f=f(e_1,\ldots, e_n)\Pi\) où
-    \begin{equation}
-        \Pi(v_1,\ldots, v_n)=\sum_{\sigma\in S_n}\epsilon(\sigma)\alpha_{\sigma(1)1}\ldots \alpha_{\sigma(n)n}.
-    \end{equation}
-    Pour rappel, la donnée des \( v_i\) est dans les nombres \( \alpha_{ij}\).
-    
-    L'espace des \( n\)-formes alternées est donc \emph{au plus} de dimension \( 1\). Pour montrer qu'il est exactement de dimension \( 1\), il faut et suffit de prouver que \( \Pi\) est alternée. Par le lemme \ref{LemHiHNey}, il suffit de prouver que cette forme est antisymétrique\footnote{C'est ici que joue l'hypothèse sur la caractéristique de \( \eK\).}. 
-
-    Soient donc \( v_1,\ldots, v_n\) tels que \( v_i=v_j\). En posant \( \tau=(1i)\) et \( \tau'=(2j)\) et en sommant sur \( \sigma\tau\tau'\) au lieu de \( \sigma\), nous pouvons supposer que \( i=1\) et \( j=2\). Montrons que \( \Pi(v,v,v_3,\ldots, v_n)=0\) en tenant compte que \( \alpha_{i1}=\alpha_{i2}\) :
-    \begin{subequations}
-        \begin{align}
-            \Pi(v,v,v_3,\ldots, v_n)&=\sum_{\sigma\in S_n}\epsilon(\sigma)\alpha_{\sigma(1)1}\alpha_{\sigma(2)2}\alpha_{\sigma(3)3}\ldots \alpha_{\sigma(n)n}\\
-            &=\sum_{\sigma\in S_n}\epsilon(\sigma\tau)\alpha_{\sigma\tau(1)1}\alpha_{\sigma\tau(2)2}\alpha_{\sigma\tau(3)3}\ldots \alpha_{\sigma\tau(n)n}&\text{où } \tau=(12)\\
-            &=-\sum_{\sigma\in S_n}\epsilon(\sigma)\alpha_{\sigma(1)1}\alpha_{\sigma(2)2}\alpha_{\sigma(3)3}\ldots \alpha_{\sigma(n)n} \\
-            &=-\Pi(v,v,v_3,\ldots, v_n).
-        \end{align}
-    \end{subequations}
-\end{proof}
-
-\begin{lemma}   \label{LemcDOTzM}
-    Soit \( \eK\) un corps fini autre que \( \eF_2\)\quext{Je ne comprends pas très bien à quel moment joue cette hypothèse.}, soit un groupe abélien \( M\) et un morphisme \( \varphi\colon \GL(n,\eK)\to M\). Alors il existe un unique morphisme \( \delta\colon \eK^*\to M\) tel que \( \varphi=\delta\circ\det\).
-\end{lemma}
-
-\begin{proof}
-    D'abord le groupe dérivé de \( \GL(n,\eK)\) est \( \SL(n,\eK)\) parce que les éléments de \( D\big( \GL(n,\eK) \big)\) sont de la forme \( ghg^{-1}h^{-1}\) dont le déterminant est \( 1\).
-    
-    De plus le groupe \( \SL(n,\eK)\) est normal dans \( \GL(n,\eK)\). Par conséquent \( \GL(n,\eK)/\SL(n,\eK)\) est un groupe et nous pouvons définir l'application relevée
-    \begin{equation}
-        \tilde \varphi\colon \frac{ \GL(n,\eK) }{ \SL(n,\eK) }\to M
-    \end{equation}
-    vérifiant \( \varphi=\tilde \varphi\circ\pi\) où \( \pi\) est la projection. 
-
-    Nous pouvons faire la même chose avec l'application
-    \begin{equation}
-        \det\colon \GL(n,\eK)\to \eK^*
-    \end{equation}
-    qui est un morphisme de groupes dont le noyau est \( \SL(n,\eK)\). Cela nous donne une application
-    \begin{equation}
-        \tilde \det\colon \frac{ \GL(n,\eK) }{ \SL(n,\eK) }\to \eK^*
-    \end{equation}
-    telle que \( \det=\tilde \det\circ\pi\). Cette application \( \tilde \det\) est un isomorphisme. En effet elle est surjective parce que le déterminant l'est et elle est injective parce que son noyau est précisément ce par quoi on prend le quotient. Par conséquent \( \tilde \det \) possède un inverse et nous pouvons écrire
-    \begin{equation}
-        \varphi=\tilde \varphi\circ\tilde \det^{-1}\circ\tilde \det\circ\pi.
-    \end{equation}
-    État donné que \( \tilde \det\circ\pi=\det\), nous avons alors \( \varphi=\delta\circ\det\) avec \( \delta=\tilde \varphi\circ\tilde \det^{-1}\). Ceci conclut la partie existence de la preuve.
-
-    En ce qui concerne l'unicité, nous considérons \( \delta'\colon \eK^*\to M\) telle que \( \varphi=\delta'\circ\det\). Pour tout \( u\in \GL(n,\eK)\) nous avons \( \delta'(\det(u))=\varphi(u)=\delta(\det(u))\). L'application \( \det\) étant surjective depuis \( \GL(n,\eK)\) vers \( \eK^*\), nous avons \( \delta'=\delta\).
-\end{proof}
-
-\begin{theorem}
-    Soit \( p\geq 3\) un nombre premier et \( V\), un \( \eF_p\)-espace vectoriel de dimension finie \( n\). Pour tout \( u\in\GL(V)\) nous avons
-    \begin{equation}
-        \epsilon(u)=\left(\frac{\det(u)}{p}\right).
-    \end{equation}
-\end{theorem}
-Ici \( \epsilon\) est la signature de \( u \) vue comme une permutation des éléments de \( \eF_p\).
-
-\begin{proof}
-    Commençons par prouver que
-    \begin{equation}
-        \epsilon\colon \GL(V)\to \{ -1,1 \}.
-    \end{equation}
-    est un morphisme. Si nous notons \( \bar u\in S(V)\) l'élément du groupe symétrique correspondant à la matrice \( u\in \GL(V)\), alors nous avons \( \overline{ uv }=\bar u\circ\bar v\), et la signature étant un homomorphisme (proposition \ref{ProphIuJrC}), 
-    \begin{equation}
-        \epsilon(uv)=\epsilon(\bar u\circ\bar v)=\epsilon(\bar u)\epsilon(\bar v).
-    \end{equation}
-    Par ailleurs \( \{ -1,1 \}\) est abélien, donc le lemme \ref{LemcDOTzM} s'applique et nous pouvons considérer un morphisme \( \delta\colon \eF_p^*\to \{ -1,1 \}\) tel que \( \epsilon=\delta\circ\det\).
-
-    Nous allons utiliser le lemme \ref{Lemoabzrn} pour montrer que \( \delta\) est le symbole de Legendre. Pour cela il nous faudrait trouver un \( x\in \eF_p^*\) tel que \( \delta(x)=-1\). Étant donné que \( \det\) est surjective, nous cherchons ce \( x\) sous la forme \( x=\det(u)\). Par conséquent nous aurions
-    \begin{equation}
-        \delta(x)=(\delta\circ\det)(u)=\epsilon(u),
-    \end{equation}
-    et notre problème revient à trouver une matrice \( u\in\GL(V)\) dont la permutation associée soit de signature \( -1\).
-
-    Soit \( n=\dim V\); en conséquence de la proposition \ref{PropHfrNCB}\ref{ItemiEFRTg}, l'espace \( \eE_q=\eF_{p^n}\) est un \( \eF_p\)-espace vectoriel de dimension \( n\) et est donc isomorphe en tant qu'espace vectoriel à \( V\). Étant donné que \( \eF_q\) est un corps fini, nous savons que \( \eF_q^*\) est un groupe cyclique à \( q-1\) éléments. Soit \( y\), un générateur de \( \eF_q^*\) et l'application
-    \begin{equation}
-        \begin{aligned}
-            \beta\colon \eF_q&\to \eF_q \\
-            x&\mapsto yx. 
-        \end{aligned}
-    \end{equation}
-    Cela est manifestement \( \eF_p\)-linéaire (ici \( y\) et \( x\) sont des classes de polynômes et \( \eF_p\) est le corps des coefficients). L'application \( \beta\) fixe zéro et à part zéro, agit comme le cycle
-    \begin{equation}
-        (1,y,y^2,\ldots, y^{q-2}).
-    \end{equation}
-    Nous savons qu'un cycle de longueur \( n\) est de signature \( (-1)^{n+1}\). Ici le cycle est de longueur \( q-1\) qui est pair (parce que \( p\geq 3\)) et par conséquent, l'application \( \beta\) est de signature \( -1\).
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Déterminant d'une famille de vecteurs}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous considérons un corps \( \eK\) et l'espace vectoriel \( E\) de dimension \( n\) sur \( \eK\).
-
-\begin{definition}[Déterminant d'une famille de vecteurs\cite{MathAgreg}]\label{DEFooODDFooSNahPb}
-    Le \defe{déterminant}{déterminant!d'une famille de vecteurs} \( (v_1,\ldots, v_n)\) dans la base \( B\) est l'élément de \( \eK\)
-    \begin{equation}        \label{EQooOJEXooXUpwfZ}
-        \det_{(e_1,\ldots, e_n)}(v_1,\ldots, v_n)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^ne^*_{\sigma(i)}(v_i)
-    \end{equation}
-    où la somme porte sur le groupe symétrique, \( \epsilon(\sigma)\) est la signature de la permutation \( \sigma\) et \( e_k^*\) est le dual de \( e_k\).
-
-    Nous le notons \( \det_{(e_1,\ldots, e_n)}(v_1,\ldots, v_n)\).
-\end{definition}
-
-\begin{lemma}[\cite{MathAgreg}]     \label{LemJMWCooELZuho}
-    Les propriétés du déterminant. Soit \( B\) une base de \( E\).
-    \begin{enumerate}
-        \item
-            L'application \( \det_B\colon E^n\to \eK\) est \( n\)-linéaire et alternée.
-        \item
-            Pour toute base, \( \det_B(B)=1\).
-        \item
-            Le déterminant ne change pas si on remplace un vecteur par une combinaison linéaire des autres :
-            \begin{equation}
-                \det_B(v_1,\ldots, v_n)=\det_B\big( v_1+\sum_{s=2}^na_sv_s,v_2,\ldots, v_n \big).
-            \end{equation}
-        \item
-            Si on permute les vecteurs,
-            \begin{equation}
-                \det_B(v_1,\ldots, v_n)=\epsilon(\sigma)\det_B(v_{\sigma(1)},\ldots, v_{\sigma(n)}).
-            \end{equation}
-        \item
-            Si \( B'\) est une autre base :
-            \begin{equation}        \label{EqAWICooBLTTOY}
-                \det_B=\det_B(B')\det_{B'}
-            \end{equation}
-        \item
-            Nous avons aussi la formule \( \det_{B}(B')\det_{B'}(B)=1\).
-        \item\label{ItemDWFLooDUePAf}
-            Les vecteurs \( \{ v_1,\ldots, v_n \}\) forment une base si et seulement si \( \det_B(v_1,\ldots, v_n)\neq 0\).
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Point par point.
-    \begin{enumerate}
-        \item
-            En posant \( v_1=x_1+\lambda x_2\) nous avons
-            \begin{subequations}
-                \begin{align}
-                    \det_B(x_1+\lambda x_2,v_2,\ldots, v_n)&=\sum_{\sigma}\epsilon(\sigma)\prod_{i=1}^ne^*_{\sigma(i)}(v_i)\\
-                    &=\sum_{\sigma}\epsilon(\sigma)\Big( e^*_{\sigma(1)}(x_1+\lambda x_2) \Big)\prod_{i=2}^ne^*_{\sigma(i)}(v_i).
-                \end{align}
-            \end{subequations}
-            À partir de là, la linéarité de \( e^*_{\sigma(1)}\) montre que \( \det_B\) est linéaire en son premier argument. Pour les autres argument, le même calcul tient.
-
-            En ce qui concerne le fait d'être alternée, permuter \( v_k\) et \( v_l\) revient à calculer \( \det_B( v_{\sigma_{kl}(1)},\ldots, v_{\sigma_{kl}(n)} )\), c'est à dire changer la somme \( \sum_{\sigma}\) en \( \sum_{\sigma\circ\sigma_{kl}}\). Cela ajoute \( 1\) à \( \epsilon(\sigma)\) vu que l'on ajoute une permutation.
-        \item
-            Nous avons
-            \begin{equation}
-                \det_B(B)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^n\underbrace{e_{\sigma(i)}^*(e_i)}_{=\delta_{\sigma(i),i}}.
-            \end{equation}
-            Si \( \sigma\) n'est pas l'identité, le produit contient forcément un facteur nul. Il ne reste de la somme que \( \sigma=\id\) et le résultat est \( 1\).
-        \item
-            Vu que \( \det_B\) est linéaire en tous ses arguments, 
-            \begin{equation}
-                \det_B\big( v_1+\sum_{s=2}^na_sv_s,v_2,\ldots, v_n \big)=\det_B(v_1,\ldots, v_n)+\sum_{s=2}^na_s\det_B(v_s,v_2,\ldots, v_n).
-            \end{equation}
-            Chacun des termes de la somme est nul parce qu'il y a répétition de \( v_s\) parmi les arguments alors que la forme est alternée.
-        \item
-            Nous devons calculer \( \det_B(v_{\sigma(1)},\ldots, v_{\sigma(n)})\), et pour y voir plus clair nous posons \( w_i=v_{\sigma(i)}\). Alors :
-            \begin{subequations}
-                \begin{align}
-                    \det_B(v_{\sigma(1)},\ldots, v_{\sigma(n)})&=\sum_{\sigma'}\epsilon(\sigma')\prod_{i=1}^ne^*_{\sigma'(i)}(w_i)\\
-                    &=\sum_{\sigma'}\epsilon(\sigma')\prod_{i=1}^ne^*_{\sigma'(i)}(v_{\sigma(i)})\\
-                    &=\sum_{\sigma'}\epsilon(\sigma')\prod_{i=1}^ne^*_{\sigma^{-1}\sigma'(i)}(v_i)\\
-                    &=\sum_{\sigma'}\epsilon(\sigma\sigma')\prod_{i=1}^ne^*_{\sigma'(i)}(v_i)\\
-                    &=\epsilon(\sigma)\det_B(v_1,\ldots, v_n).
-                \end{align}
-            \end{subequations}
-            Justifications : nous avons d'abord modifié l'ordre des éléments du produit et ensuite l'ordre des éléments de la somme. Nous avons ensuite utilisé le fait que \( \epsilon\colon S_n\to \{ 0,1 \}\) était un morphisme de groupe (proposition \ref{ProphIuJrC}).
-        \item
-            Étant donné que l'espace des formes multilinéaires alternées est de dimension \( 1\), il existe un \( \lambda\in \eK\) tel que \( \det_B=\lambda\det_{B'}\). Appliquons cela à \( B'\) :
-            \begin{equation}
-                \det_B(B')=\lambda\det_{B'}(B'),
-            \end{equation}
-            donc \( \lambda=\det_B(B')\).
-        \item
-            Il suffit d'appliquer l'égalité précédente à \( B\) en nous souvenant que \( \det_B(B)=1\).
-        \item
-            Si \( B'=\{ v_1,\ldots, v_n \}\) est une base alors \( \det_B(B')\neq 0\), sinon il n'est pas possible d'avoir \( \det_B(B')\det_{B'}(B)=1\).
-
-            À l'inverse, si \( B'\) n'est pas une base, c'est que \( \{ v_1,\ldots, v_n \}\) est liée par le théorème \ref{ThoMGQZooIgrXjy}\ref{ItemHIVAooPnTlsBi}. Il y a donc moyen de remplacer un des vecteurs par une combinaison linéaire des autres. Le déterminant s'annule alors.
-    \end{enumerate}
-\end{proof}
-
-D'après la proposition \ref{ProprbjihK}, il existe une unique forme \( n\)-linéaire alternée égale à \( 1\) sur \( B\), et c'est \( \det_B\colon E^n\to \eK\).
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Déterminant d'un endomorphisme}
-%---------------------------------------------------------------------------------------------------------------------------
-
-L'interprétation géométrique du déterminant en termes d'aires et de volumes est donnée après la théorème \ref{ThoBVIJooMkifod}.
-
-\begin{lemma}       \label{LEMooQTRVooAKzucd}
-    Si \( f\colon E\to E\) est un endomorphisme, si \( B\) et \( B'\) sont deux bases, alors \( \det_B\big( f(B) \big)=\det_{B'}\big( f(B') \big)  \).
-\end{lemma}
-
-\begin{proof}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            \varphi\colon E^n&\to \eK \\
-            v_1,\ldots, v_n&\mapsto \det_B\big( f(v_1),\ldots, f(v_n) \big) 
-        \end{aligned}
-    \end{equation}
-    est \( n\)-linéaire et alternée; il existe donc \( \lambda\in \eK\) tel que \( \varphi=\lambda\det_B\). En appliquant cela à \( B\) :
-    \begin{equation}
-        \det_B\big( f(B) \big)=\lambda \det_B(B)=\lambda.
-    \end{equation}
-    Nous avons donc déjà prouvé que \( \lambda=\det_B\big( f(B) \big)\), c'est à dire
-    \begin{equation}
-        \det_B\big( f(v) \big)=\det_B\big( f(B) \big)\det_B(v).
-    \end{equation}
-    
-    Nous allons maintenant introduire \( B'\) là où il y a du \( v\) en utilisant les formules \eqref{EqAWICooBLTTOY} :
-    \begin{subequations}
-        \begin{align}
-            \det_B\big( f(v) \big)&=\det_B(B')\det_{B'}\big( f(v) \big)\\
-            \det_B(v)=\det_B(B')\det_{B'}(v).
-        \end{align}
-    \end{subequations}
-    Nous obtenons
-    \begin{equation}
-        \det_{B'}\big( f(v) \big)=\det_B\big( f(B) \big)\det_{B'}(v).
-    \end{equation}
-    Et on applique cela à \( v=B'\) :
-    \begin{equation}
-        \det_{B'}\big( f(B') \big)=\det_B\big( f(B) \big)\underbrace{\det_{B'}(B')}_{=1}.
-    \end{equation}
-\end{proof}
-
-Cette proposition nous permet de définir le déterminant d'un endomorphisme de la façon suivante sans préciser la base.
-\begin{definition}[\cite{MathAgreg}]        \label{DefCOZEooGhRfxA}
-    Si \( f\colon E\to E\) est un endomorphisme, le \defe{déterminant}{déterminant!d'un endomorphisme} de \( f\) est
-    \begin{equation}
-        \det(f)=\det_B\big( f(B) \big)
-    \end{equation}
-\end{definition}
-
-Couplé à la formule \ref{EQooOJEXooXUpwfZ}, nous pouvons écrire la formule pratique à utiliser le plus souvent. Si \( \{ e_i \}_{i=1,\ldots, n}\) est une base orthonormée de \( E\) et si \( f\colon E\to E\) est un endomorphisme,
-\begin{equation}
-    \det(f)=\sum_{\sigma\in S_n}\epsilon(\epsilon)\prod_{i=1}^n\langle e_{\sigma(i)}, f(e_i)\rangle.
-\end{equation}
-Et si vous avez tout suivi, vous aurez remarqué que les produits scalaires impliqués dans cette formule sont les éléments de la matrice de \( f\) dans la base \( \{ e_i \}\) parce que \( \langle e_i, f(e_j)\rangle \) est la composante \( i\) de l'image de \( e_j\) par \( f\). Si la matrice est composée en mettant en colonne les images des vecteurs de base, le compte est bon.
-
-\begin{proposition}     \label{PropYQNMooZjlYlA}
-    Principales propriétés géométriques du déterminant d'un endomorphisme.
-    \begin{enumerate}
-        \item   \label{ItemUPLNooYZMRJy}
-            Si \( f\) et \( g\)  sont des endomorphismes, alors \( \det(f\circ g)=\det(f)\det(g)\).
-        \item       \label{ITEMooNZNLooODdXeH}
-            L'endomorphisme \( f\) est un automorphisme\footnote{Endomorphisme inversible, définition \ref{DEFooOAOGooKuJSup}.} si et seulement si \( \det(f)\neq 0\).\index{déterminant!et inversibilité}
-        \item   \label{ITEMooZMVXooLGjvCy}
-            Si \( \det(f)\neq 0\) alors \( \det(f^{-1})=\det(f)^{-1}\).
-        \item       \label{ItemooPJVYooYSwqaE}
-            L'application \( \det\colon \GL(E)\to \eK\setminus\{ 0 \}\) est un morphisme de groupe.
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Point par point.
-    \begin{enumerate}
-        \item
-            Nous considérons l'application
-            \begin{equation}
-                \begin{aligned}
-                    \varphi\colon E^n&\to \eK \\
-                    v&\mapsto \det_B\big( f(v) \big). 
-                \end{aligned}
-            \end{equation}
-            Comme d'habitude nous avons \( \varphi(v)=\lambda\det_B(v)\). En appliquant à \( B\) et en nous souvenant que \( \det_B(B)=1\) nous avons
-                $\det_B\big( f(B) \big)=\lambda$. Autrement dit :
-                \begin{equation}
-                    \lambda=\det(f).
-                \end{equation}
-            Calculons à présent \( \varphi\big( g(B) \big)\) : d'une part,
-            \begin{equation}
-                \varphi\big( g(B) \big)=\det_B\big( (f\circ g)(B) \big)
-            \end{equation}
-            et d'autre part,
-            \begin{equation}
-                \varphi\big( g(B) \big)=\lambda\det_B\big( g(B) \big)=\lambda\det(g)
-            \end{equation}
-            En égalisant et en reprenant la la valeur déjà trouvée de \( \lambda\),
-            \begin{equation}
-                \det\big(f\circ g)(B) \big)=\det(f)\det(g),
-            \end{equation}
-            ce qu'il fallait.
-        \item
-            Supposons que \( f\) soit un automorphisme. Alors si \( B\) est une base, \( f(B) \) est une base. Par conséquent \( \det(f)=\det_B\big( f(B) \big)\neq 0\) parce que \( f(B)\) est une base (lemme \ref{LemJMWCooELZuho}\ref{ItemDWFLooDUePAf}).
-
-            Réciproquement, supposons que \( \det(f)\neq 0\). Alors si \( B\) est une base quelconque nous avons \( \det_B\big( f(B) \big)\neq 0\), ce qui est uniquement possible lorsque \( f(B)\) est une base. L'application \( f\) transforme donc toute base en une base et est alors un automorphisme d'espace vectoriel.
-        \item
-            Vu que le déterminant de l'identité est \( 1\) et que \( f\) est inversible, \( 1=\det(f\circ f^{-1})=\det(f)\det(f^{-1})\).
-    \end{enumerate}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Déterminant d'une matrice}
-%---------------------------------------------------------------------------------------------------------------------------
-
-La proposition \ref{PROPooCESFooGOZBNI} nous assure que toute application linéaire possède une matrice associée (dans une base). Vu de la matrice, voici quelque conséquences de la proposition \ref{PropYQNMooZjlYlA}.
-\begin{proposition}
-    Quelques propriétés du déterminant vu de la matrice.
-    \begin{enumerate}
-        \item
-            Si on permute deux lignes ou deux colonnes d'une matrice, alors le déterminant change de signe.
-        \item
-            Si on multiplie une ligne ou une colonne d'une matrice par un nombre $\lambda$, alors le déterminant est multiplié par $\lambda$.
-        \item
-            Si deux lignes ou deux colonnes sont proportionnelles, alors le déterminant est nul.
-        \item
-            Si on ajoute à une ligne une combinaison linéaire des autres lignes, alors le déterminant ne change pas (idem pour les colonnes).
-    \end{enumerate}
-\end{proposition}
-
-Le déterminant d'une matrice et d'une application linéaire est la définition \ref{DefCOZEooGhRfxA}, et les principales propriétés algébriques sont données dans la proposition \ref{PropYQNMooZjlYlA}.
-
-En dimension deux, le déterminant de la matrice
-    $\begin{pmatrix}
-        a    &   b    \\ 
-        c    &   d    
-    \end{pmatrix}$
-est le nombre
-\begin{equation}        \label{EQooQRGVooChwRMd}
-     \det\begin{pmatrix}
-         a   &   b    \\ 
-         c   &   d    
-     \end{pmatrix}=\begin{vmatrix}
-          a  &   b    \\ 
-        c    &   d    
-    \end{vmatrix}=ad-cb.
-\end{equation}
-Ce nombre détermine entre autres le nombre de solutions que va avoir le système d'équations linéaires associé à la matrice.
-
-Pour une matrice $3\times 3$, nous avons le même concept, mais un peu plus compliqué; nous avons la formule
-\begin{equation}
-    \det
-    \begin{pmatrix}
-        a_{11}    &   a_{12}    &   a_{13}    \\
-        a_{21}    &   a_{22}    &   a_{23}    \\
-        a_{31}    &   a_{32}    &   a_{33}    
-    \end{pmatrix}
-    =
-    \begin{vmatrix}
-        a_{11}    &   a_{12}    &   a_{13}    \\
-        a_{21}    &   a_{22}    &   a_{23}    \\
-        a_{31}    &   a_{32}    &   a_{33}    
-    \end{vmatrix}=
-    a_{11}\begin{vmatrix}
-        a_{22}  &   a_{23}    \\ 
-        a_{32}    &   a_{33}    
-    \end{vmatrix}-
-    a_{12}\begin{vmatrix}
-        a_{21}  &   a_{23}    \\ 
-        a_{31}    &   a_{33}
-    \end{vmatrix}+
-    a_{13}\begin{vmatrix}
-        a_{21}  &   a_{22}    \\ 
-        a_{31}    &   a_{32}
-    \end{vmatrix}.
-\end{equation}
-
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Déterminant de Vandermonde}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[\cite{fJhCTE}]  \label{PropnuUvtj}
-    Le \defe{déterminant de Vandermonde}{déterminant!Vandermonde}\index{Vandermonde (déterminant)} est le polynôme en \( n\) variables donné par
-    \begin{equation}
-        V(T_1,\ldots, T_n)=\det\begin{pmatrix}
-             1   &   1    &   \ldots    &   1    \\
-             T_1   &   T_2    &   \ldots    &   T_n    \\
-             \vdots   &   \ddots    &   \ddots    &   \vdots    \\ 
-             T_1^{n-1}   &   T_2^{n-1}    &   \ldots    &   T_n^{n-1}     
-         \end{pmatrix}=\prod_{1\leq i<j\leq n}(T_j-T_i).
-    \end{equation}
-    Notez que l'inégalité du milieu est stricte (sinon d'ailleurs l'expression serait nulle).
-\end{proposition}
-
-\begin{proof}
-    Nous considérons le polynôme
-    \begin{equation}
-        f(X)=V(T_1,\ldots, T_{n-1},X)\in \big( \eK[T_1,\ldots, T_{n-1}] \big)[X].
-    \end{equation}
-    C'est un polynôme de degré au plus \( n-1\) en \( X\) et il s'annule aux points \( T_1,\ldots, T_{n-1}\). Par conséquent il existe \( \alpha\in \eK[T_1,\ldots, T_{n-1}]\) tel que
-    \begin{equation}    \label{EqeVxRwO}
-        f=\alpha(X-T_{n-1})\ldots(X-T_1).
-    \end{equation}
-    Nous trouvons \( \alpha\) en écrivant \( f(0)\). D'une part la formule \eqref{EqeVxRwO} nous donne
-    \begin{equation}    \label{EqblwWMj}
-        f(0)=\alpha(-1)^{n-1}T_1\ldots T_{n-1}.
-    \end{equation}
-    D'autre par la définition donne
-    \begin{subequations}
-        \begin{align}
-            f(0)&=\det\begin{pmatrix}
-                 1   &   \cdots    &   1    &   1    \\
-                 T_1      &       &   T_{n-1}    &   0    \\
-                 \vdots   &       &   \vdots    &   \vdots    \\ 
-                 T_1^{n-1}   &   \cdots    &   T_{n-1}^{n-1}    &   0     
-             \end{pmatrix}\\
-             &=(-1)^{n-1}\det\begin{pmatrix}
-                 T_1   &   \ldots    &   T_{n-1}    \\
-                 \vdots   &   \ddots    &   \vdots    \\
-                 T_1^{n-1}   &   \ldots    &   T_{n-1}^{n-1}
-             \end{pmatrix}\\
-             &=(-1)^{n-1}T_1\ldots T_{n-1}\det\begin{pmatrix}
-                 1   &   \cdots    &   1    \\
-                 \vdots   &   \ddots    &   \vdots    \\
-                 T_1^{n-1}   &   \cdots    &   T_{n-1}^{n-1}
-             \end{pmatrix}\\
-             &=(-1)^{n-1}T_1\ldots T_{n-1}V(T_1,\ldots, T_{n-1})
-        \end{align}
-    \end{subequations}
-    En égalisant avec \eqref{EqblwWMj}, nous trouvons \( \alpha=V(T_1,\ldots, T_{n-1})\), et donc
-    \begin{equation}
-        f=V(T_1,\ldots, T_{n-1})\prod_{j\leq n-1}(X-T_j)
-    \end{equation}
-    Enfin, une récurrence montre que
-    \begin{subequations}
-        \begin{align}
-            V(T_1,\ldots, T_n)&=f(T_n)\\
-            &=V(T_1,\ldots, T_{n-1})\prod_{j\leq n-1}(T_n-T_j)\\
-            &=\prod_{k\leq n}\prod_{j\leq k-1}(T_k-T_j)\\
-            &=\prod_{1\leq j<k\leq n}(T_i-T_j).
-        \end{align}
-    \end{subequations}
-\end{proof}
-
-\begin{example}
-    Le déterminant de Vandermonde (proposition \ref{PropnuUvtj}) est alterné, semi-symétrique et non symétrique. Le fait qu'il soit alterné est le fait qu'il soit un déterminant. Étant donné qu'il est alterné, il est semi-symétrique parce que sur \( A_n\), nous avons \( \epsilon=1\). Étant donné qu'il est alterné, il change de signe sous l'action des éléments impairs de \( S_n\) et n'est donc pas symétrique.
-\end{example}
-
-\begin{proposition}\index{action de groupe} \label{PropUDqXax}
-    Un polynôme semi-symétrique \( f\in \eK[T_1,\ldots, T_n]\) se décompose de façon unique en
-    \begin{equation}
-        f=P+VQ
-    \end{equation}
-    où \( P\) et \( Q\) sont deux polynômes symétriques.
-\end{proposition}
-\index{groupe!permutation}
-\index{polynôme!symétrique}
-
-\begin{proof}
-
-    Nous commençons par prouver l'unicité en montrant que si \( f=PVQ\) avec \( P\) et \( Q\) symétrique, alors \( P\) et \( Q\) sont donnés par des formules explicites en termes de \( f\).
-
-
-    Si \( \sigma_1\) et \( \sigma_2\) sont deux permutations impaires de \( \{ 1,\ldots, n \}\), alors \( \sigma_1\cdot f=\sigma_2\cdot f\) parce que l'élément \( \sigma_2^{-1}\sigma_1\) est pair (proposition \ref{ProphIuJrC}), de telle sorte que \( \sigma_2^{-1}\sigma_1\cdot f=f\). Nous posons donc \( g=\tau\cdot f\) où \( \tau\) est une permutation impaire quelconque -- par exemple une transposition.
-
-    Vu que \( V\) est alternée et que \( \tau\) est une transposition nous avons
-    \begin{equation}
-        g=\tau\cdot f=P-VQ.
-    \end{equation}
-    Donc \( f+g=2P\) et \( f-g=2VQ\). Cela donne \( P\) et \( Q\) en terme de \( f\) et \( g\), et donc l'unicité.
-
-    Attention : cela ne donne pas un moyen de prouver l'existence parce que rien ne prouve pour l'instant que \( f-g\) peut effectivement être écrit sous la forme \( VQ\), c'est à dire que \( f-g\) soit divisible par \( V\). C'est cela que nous allons nous atteler à démontrer maintenant.
-
-    Nous commençons par prouver que \( f+g\) est symétrique et \( f-g\) alterné. Si \( \sigma\) est une transposition,
-    \begin{equation}
-        \sigma\cdot(f+g)=\sigma\cdot f+\sigma\tau\cdot f=g+f
-    \end{equation}
-    parce que \( \sigma\tau\) est pair. De la même façon,
-    \begin{equation}
-        \sigma\cdot(f-g)=g-f=\epsilon(\sigma)(f-g).
-    \end{equation}
-    Dans les deux cas nous concluons en utilisant le fait que toute permutation est un produit de transpositions (proposition \ref{PropPWIJbu}) et que \( \epsilon\) est un homomorphisme.
-
-    Soient maintenant deux entiers \( h<k\) dans \( \{ 1,\ldots, n \}\) et l'anneau
-    \begin{equation}
-        \big( \eK[T_1,\ldots, \hat T_k,\ldots, T_n] \big)[T_k].
-    \end{equation}
-    Cet anneau contient le polynôme \( T_k-T_h\) où \( T_k\) est la variable et \( T_h\) est un coefficient. Nous faisons la division euclidienne de \( f-g\) par  \( T_k-T_h\) parce que nous avons dans l'idée de faire arriver le déterminant de Vandermonde et donc le produit de toutes les différences \( T_k-T_h\) :
-    \begin{equation}    \label{EqSHdgrG}
-        f-g=(T_k-T_h)q+r
-    \end{equation}
-    où \( \deg_{T_k}r<1\), c'est à dire que \( r\) ne dépends pas de \( T_k\). Nous revoyons maintenant l'égalité \eqref{EqSHdgrG} dans \( \eK[T_1,\ldots, T_n]\) et nous y appliquons la transposition \( \tau_{kh}\). Nous savons que \( \tau_{kh}(f-g)=-(f-g)\) et \( \tau_{kh}(T_k-T_h)=-(T_k-T_h)\), et donc
-    \begin{equation}    \label{EqVOhjKB}
-        -(f-g)=-(T_k-T_h)\tau_{kh}\cdot   q+\tau_{kh}\cdot r
-    \end{equation}
-    où \(\tau_{kh}\cdot r\) ne dépend pas de \( T_h\). Nous appliquons à \eqref{EqVOhjKB} l'application
-    \begin{equation}
-        \begin{aligned}
-            t\alpha\colon \eK[T_1,\ldots, T_n]&\to \eK[T_1,\ldots, \hat T_k,\ldots, T_n] \\
-            \alpha(PT_1,\ldots, \hat T_k,\ldots, T_n)&=P(T_1,\ldots, T_h,\ldots, T_n). 
-        \end{aligned}
-    \end{equation}
-    Cette application vérifie \( \alpha\big( \tau_{kh}\cdot r \big)=\alpha(r)\) et nous avons
-    \begin{equation}
-        -\alpha(f-g)=\alpha(r).
-    \end{equation}
-    Puis en appliquant \( \alpha\) à la relation \( f-g=(T_k-T_h)q+r\), nous trouvons
-    \begin{equation}
-        \alpha(f-g)=\alpha(r),
-    \end{equation}
-    et par conséquent \( \alpha(r)=0\). Ici nous utilisons l'hypothèse de caractéristique différente de deux. Dire que \( \alpha(r)=0\), c'est dire que \( r\) est divisible par \( T_k-T_h\), mais \( r\) étant de degré zéro en \( T_k\), nous avons \( r=0\). Par conséquent \( T_k-T_h\) divise \( f-g\) pour tout \( h<k\), et nous pouvons définir un polynôme \( Q\) par
-    \begin{equation}    \label{EqrnbgdA}
-        f-g=2Q\prod_{h<k}\prod_{k\leq n}(T_k-T_h)=2Q(T_1,\ldots, T_n)V(T_1,\ldots, T_n),
-    \end{equation}
-    où nous avons utilisé la formule du déterminant de Vandermonde de la proposition \ref{PropnuUvtj}.
-
-    Étant donné que \( f+g\) est un polynôme symétrique, nous allons aussi poser \( f+g=2P\) avec \( P\) symétrique.
-
-    Montrons à présent que \( Q\) est un polynôme symétrique. Soit \( \sigma\in S_n\); vu que nous savons déjà que \( f-g\) est alternée, nous avons
-    \begin{equation}    \label{EqpSPEyq}
-        \sigma\cdot (f-g)=\epsilon(\sigma)(f-g)=\epsilon(\sigma)2QV,
-    \end{equation}
-    Mais en appliquant \( \sigma\) à l'équation \eqref{EqrnbgdA},
-    \begin{subequations}
-        \begin{align}
-            \sigma\cdot (f-g)&=2(\sigma\cdot V)(T_1,\ldots, ,T_n)(\sigma\cdot Q)(T_1,\ldots,T_n)\\
-            &=2\epsilon(\sigma)V(T_1,\ldots, T_n)(\sigma\cdot Q)(T_1,\ldots, T_n).
-        \end{align}
-    \end{subequations}
-    En égalisant avec \eqref{EqpSPEyq} et en se souvenant que l'anneau \( \eK[T_1,\ldots, T_n]\) était intègre (théorème \ref{ThoBUEDrJ}), nous simplifions par \( 2\epsilon(\sigma)V\) pour obtenir
-    \begin{equation}
-        Q=\sigma\cdot Q,
-    \end{equation}
-    c'est à dire que \( Q\) est symétrique.
-
-    Au final nous avons \( f+q=2P\) et \( f-g=2VQ\) avec \( P\) et \( Q\) symétriques. En faisant la somme,
-    \begin{equation}
-        f=P+VQ.
-    \end{equation}
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Déterminant de Gram}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Si \( x_1,\ldots, x_r\) sont des vecteurs d'un espace vectoriel, alors le \defe{déterminant de Gram}{déterminant!Gram}\index{Gram (déterminant)} est le déterminant
-\begin{equation}
-    G(x_1,\ldots, x_r)=\det\big( \langle x_i, x_j\rangle  \big).
-\end{equation}
-Notons que la matrice est une matrice symétrique.
-
-\begin{proposition}\label{PropMsZhIK}
-    Si \( F\) est un sous-espace vectoriel de base \( \{ x_1,\ldots, x_n \}\) et si \( x\) est un vecteur, alors le déterminant de Gram est un moyen de calculer la distance entre \( x\) et \( F\) par 
-    \begin{equation}
-        d(x,F)^2=\frac{ G(x,x_1,\ldots, x_n)}{ G(x_1,\ldots, x_n) }.
-    \end{equation}
-\end{proposition}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Déterminant de Cauchy}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soient des nombres \( a_i\) et \( b_i\) (\( i=1,\ldots, n\)) tels que \( a_i+b_j\neq 0\) pour tout couple \( (i,j)\). Le \defe{déterminant de Cauchy}{déterminant!de Cauchy}\index{Cauchy!déterminant} est 
-\begin{equation}
-    D_n=\det\left( \frac{1}{ a_i+b_j } \right).
-\end{equation}
-
-\begin{proposition}[\cite{RollandRobertjyYDzY}] \label{ProptoDYKA}
-    Le déterminant de Cauchy est donné par la formule
-    \begin{equation}
-        D_n=\frac{ \prod_{i<j}(a_j-a_i)\prod_{i<j}(b_j-b_i) }{ \prod_{ij}(a_i+b_j) }.
-    \end{equation}
-\end{proposition}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Matrice de Sylvester}
-%---------------------------------------------------------------------------------------------------------------------------
-\label{subsecSQBJfr}
-
-La définition est pompée de \wikipedia{fr}{Matrice_de_Sylvester}{wikipédia}. Soient \( P\) et \( Q\) deux polynômes non nuls, de degrés respectifs \( m\) et \( n\) :
-\begin{subequations}
-    \begin{align}
-        P(x)=p_0+p_1x+\cdots +p_nx^n\\
-        Q(x)=q_0+q_1x+\cdots +q_mx^m.
-    \end{align}
-\end{subequations}
-La \defe{matrice de Sylvester}{matrice!de Sylvester}\index{Sylvester (matrice)} associée à \( P\) et \( Q\) est la matrice carrée \( m+n\times m+n\) définie ainsi :
-\begin{enumerate}
-    \item
-la première ligne est formée des coefficients de \( P\), suivis de 0 :
-\begin{equation}
-\begin{pmatrix} p_n & p_{n-1} & \cdots & p_1 & p_0 & 0 & \cdots & 0 \end{pmatrix} ;
-\end{equation}
-\item la seconde ligne s'obtient à partir de la première par permutation circulaire vers la droite ;
-\item les $(m-2)$ lignes suivantes s'obtiennent en répétant la même opération ;
-\item la ligne $(m+1)$ est formée des coefficients de \( Q\), suivis de 0 :
-    \begin{equation}
-    \begin{pmatrix} q_m & q_{m-1} & \cdots & q_1 & q_0 & 0 & \cdots & 0 \end{pmatrix} ;
-    \end{equation}
-    \item les $(m-1)$ lignes suivantes sont formées par des permutations circulaires.
-\end{enumerate}
-
-Ainsi dans le cas $n=4$ et $m=3$, la matrice obtenue est
-\begin{equation}    \label{EqPEgtle}
-S_{p,q}=\begin{pmatrix} 
-p_4 & p_3 & p_2 & p_1 & p_0 & 0 & 0 \\
-0 & p_4 & p_3 & p_2 & p_1 & p_0 & 0 \\
-0 & 0 & p_4 & p_3 & p_2 & p_1 & p_0 \\
-q_3 & q_2 & q_1 & q_0 & 0 & 0 & 0 \\
-0 & q_3 & q_2 & q_1 & q_0 & 0 & 0 \\
-0 & 0 & q_3 & q_2 & q_1 & q_0 & 0 \\
-0 & 0 & 0 & q_3 & q_2 & q_1 & q_0 \\
-\end{pmatrix}.
-\end{equation}
-Le déterminant de la matrice de Sylvester associée à \( P\) et \( Q\) est appelé le \defe{résultant}{résultant} de \( P\) et \( Q\) et noté \( \res(P,Q)\)\nomenclature[A]{\( \res(P,Q)\)}{résultat des polynômes \( P\) et \( Q\)}.
-
-Attention : si \( P\) est de degré \( n\) et \( Q\) de degré \( m\), il y a \( m\) lignes pour \( P\) et \( n\) pour \( Q\) dans le déterminant du résultant (et non le contraire).
-
-\begin{lemma}[\cite{QQuRUzA}]       \label{LemBFrhgnA}
-    Si \( P\) et \( Q\) sont deux polynômes de degrés \( n\) et \( m\) à coefficients dans l'anneau \( \eA\), alors pour tout \( \lambda\in \eA\),
-    \begin{subequations}
-        \begin{align}
-            \res(\lambda P,Q)&=\lambda^m\res(P,Q)\\
-            \res(P,\lambda Q)&=\lambda^n\res(P,Q).
-        \end{align}
-    \end{subequations}
-\end{lemma}
-
-\begin{proof}
-    Cela est simplement un comptage de nombre de lignes. Il y a \( m\) lignes contenant les coefficients de \( P\); donc prendre \( \lambda P\) revient à multiplier \( m\) lignes dans un déterminant et donc le multiplier par \( \lambda^m\).
-\end{proof}
-
-L'équation de Bézout \eqref{EqkbbzAi}\index{théorème!Bézout!utilisation} peut être traitée avec une matrice de Sylvester. Soient \( P\) et \( Q\), deux polynômes donnés et à résoudre l'équation 
-\begin{equation}    \label{EqSsyXOo}
-    xP+yQ=0
-\end{equation}
-par rapport aux polynômes inconnus \( x\) et \( y\) dont les degrés sont \( \deg(x)<\deg(Q)\) et \( \deg(y)<\deg(P)\). Si nous notons \( \tilde x\) et \( \tilde y\) la liste des coefficients de \( x\) et \( y\) (dans l'ordre décroissant de degré), nous pouvons récrire l'équation \eqref{EqSsyXOo} sous la forme
-\begin{equation}
-    S_{PQ}^t\begin{pmatrix}
-        \tilde x    \\ 
-        \tilde y    
-    \end{pmatrix}=0.
-\end{equation}
-Pour s'en convaincre, écrivons pour les polynômes de l'exemple \eqref{EqPEgtle} :
-\begin{equation}
-    \begin{pmatrix}
-        p_4    &   0    &   0    &   q_3    &   0    &   0    &   0\\ 
-        p_3    &   p_4    &   0    &   q_2    &   q_3    &   0    &   0\\ 
-        p_2    &   p_3    &   p_4    &   q_1    &   q_2    &   q_3    &   0\\ 
-        p_1    &   p_2    &   p_3    &   q_0    &   q_1    &   q_2    &   q_3\\ 
-        p_0    &   p_1    &   p_2    &   0    &   q_0    &   q_1    &   q_2\\ 
-        0    &   p_0    &   p_1    &   0    &   0    &   q_0    &   q_1\\ 
-        0    &   0    &   p_0    &   0    &   0    &   0    &   q_0\\    
-    \end{pmatrix}\begin{pmatrix}
-        x_2    \\ 
-        x_1  \\ 
-        x_0  \\    
-        y_3   \\ 
-        y_2    \\ 
-        y_1    \\ 
-        y_0    
-    \end{pmatrix}=
-    \begin{pmatrix}
-        x_2p_4+y_2q_3    \\ 
-        p_3x_2+p_4x_1+q_2y_3+q_3y_2  \\ 
-          \\    
-           \\ 
-        \vdots    \\ 
-            \\ 
-            
-    \end{pmatrix}
-\end{equation}
-Nous voyons que sur la ligne numéro \( k\) (en partant du bas et en numérotant de à partir de zéro) nous avons les produits \( p_ix_j\) et \( q_iy_j\) avec \( i+j=k\). La colonne de droite représente donc bien les coefficients du polynôme \( xP+yQ\).
-
-
-\begin{proposition} \label{PropAPxzcUl}
-    Le résultant de deux polynômes est non nul si et seulement si les deux polynômes sont premiers entre eux.
-\end{proposition}
-\index{déterminant!résultant}
-
-Un polynôme \( P\) a une racine double en \( a\) si et seulement si \( P\) et \( P'\) ont \( a\) comme racine commune, ce qui revient à dire que \( P\) et \( P'\) ne sont pas premiers entre eux. 
-
-Une application importante de ces résultats sera le théorème de Rothstein-Trager \ref{ThoXJFatfu} sur l'intégration de fractions rationnelles.
-
-\begin{example}
-    Si nous prenons \( P=aX^2+bX+c\) et \( P'=2aX+b\) alors la taille de la matrice de Sylvester sera \( 2+1=3\) et
-    \begin{equation}
-        S_{P,P'}=\begin{pmatrix}
-              a  &   b    &   c    \\
-            2a    &   b    &   0    \\
-            0    &   2a    &   b
-        \end{pmatrix}.
-    \end{equation}
-    Le résultant est alors
-    \begin{equation}
-        \res(P,P')=-a(b^2-4ac).
-    \end{equation}
-    Donc un polynôme du second degré a une racine double si et seulement si \( b^2-4ac=0\). Cela est un résultat connu depuis longtemps mais qui fait toujours plaisir à revoir.
-\end{example}
-
-La matrice de Sylvester permet aussi de récrire l'équation de Bézout pour les polynômes; voir le théorème \ref{ThoBezoutOuGmLB} et la discussion qui s'ensuit.
-
-Une proposition importante du résultant est qu'il peut s'exprimer à l'aide des racines des polynômes.
-\begin{proposition} \label{PropNDBOGNx}
-    Si
-    \begin{subequations}
-        \begin{align}
-        P(X)&=a_p\prod_{i=1}^p(X-\alpha_i)\\
-        Q(X)&=b_q\prod_{j=1}^q(X-\beta_i)
-        \end{align}
-    \end{subequations}
-    alors nous avons les expressions suivantes pour le résultant :
-    \begin{equation}        \label{EqCFUumjx}
-        \res(P,Q)=a_p^qb_q^p\prod_{i=1}^p\prod_{j=1}^q(\beta_j-\alpha_i)=b_q^p\prod_{j=1}^qP(\beta_j)=(-1)^{pq}a_p^q\prod_{i=1}^pQ(\alpha_i).
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Si \( P\) et \( Q\) ne sont pas premiers entre eux, d'une part la proposition \ref{PropAPxzcUl} nous dit que \( \res(P,Q)=0\) et d'autre part, \( P\) et \( Q\) ont un facteur irréductible en commun, ce qui  signifie que nous devons avoir un des \( X-\alpha_i\) égal à un des \( X-\beta_j\). Autrement dit, nous avons \( \alpha_i=\beta_j\) pour un couple \( (i,j)\). Par conséquent tous les membres de l'équation \eqref{EqCFUumjx} sont nuls.
-
-    Nous supposons donc que \( P\) et \( Q\) sont premiers entre eux. Nous commençons par supposer que les polynômes \( P\) et \( Q\) sont unitaires, c'est à dire que \( a_p=b_q=1\). Nous considérons alors l'anneau
-    \begin{equation}
-        \eA=\eZ[\alpha_1,\ldots, \alpha_p,\beta_1,\ldots, \beta_q].
-    \end{equation}
-    Dans cet anneau, l'élément \( \beta_j-\alpha_i\) est irréductible (tout comme \( X-Y\) est irréductible dans \( \eZ[X,Y]\)). Le résultant \( R=\res(P,Q)\) est un élément de \( \eA\) parce que tous leurs coefficients peuvent être exprimés à l'aide des \( \alpha_i\) et des \( \beta_j\). Dans \( \eA\), l'élément \( \beta_j-\alpha_i\) divise \( R\). En effet lorsque \( \beta_j=\alpha_i\), le déterminant définissant le résultant est nul, ce qui signifie que \( \beta_j-\alpha_i\) est un facteur irréductible de \( R\).
-
-    Par conséquent il existe un polynôme \( T\in \eA\) tel que
-    \begin{equation}
-        R=\lambda(\alpha_1,\ldots, \beta_q)\prod_{i=1}^p\prod_{j=1}^r(\beta_j-\alpha_i).
-    \end{equation}
-    Comptons les degrés. Pour donner une idée de ce calcul de degré, voici comment se présente, au niveau des dimensions, le déterminant :
-    \begin{equation}  \label{EqJCaATOH}
-    \xymatrix{%
-        \ar@{<->}[rrr]^{p+1}&&&& \ar@{<->}[r]^{q-1}  &\\
-        a_p\ar@{.}[rrd] &a_{p-1}\ar@{.}[rr]  &  & a_0\ar@{.}[rrd] & 0\ar@{.}[r]&0&\ar@{<->}[d]^q \\
-        0\ar@{.}[r]&0&a_p\ar@{.}[rr]&&a_1&a_0&\\
-        \ar@{<->}[rrrrr]_{p+q}&&&&&&
-       }
-    \end{equation}
-    si les \( a_i\) sont les coefficients de \( P\). Mais chacun des \( a_i\) est de degré \( 1\) en les \( \alpha_i\), donc le déterminant dans son ensemble est de degré \( q\) en les \( \alpha_i\), parce que \( R\) contient \( q\) lignes telles que \eqref{EqJCaATOH}. Le même raisonnement montre que \( R\) est de degré \( p\) en les \( \beta_j\). Par ailleurs le polynôme \( \prod_{i=1}^p\prod_{j=1}^r(\beta_j-\alpha_i)\) est de degré \( p\) en les \( \beta_j\) et \( q\) en les \( \alpha_i\). Nous en déduisons que \( T\) doit être un polynôme ne dépendant pas de \( \alpha_i\) ou de \( \beta_j\).
-
-    Nous pouvons donc calculer la valeur de \( T\) en choisissant un cas particulier. Avec \( P(X)=X^p\) et \( Q(X)=X^q+1\), il est vite vu que \( R(P,Q)=1\) et donc que \( T=1\).
-
-    Si les polynômes \( P\) et \( Q\) ne sont pas unitaires, le lemme \ref{LemBFrhgnA} nous permet de conclure. 
-
-\end{proof}
-
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Kronecker}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous considérons \( K_n\) l'ensemble des polynômes de \( \eZ[X]\)
-\begin{enumerate}
-    \item
-        unitaires de degré \( n\),
-    \item
-        dont les racines dans \( \eC\) sont de modules plus petits ou égaux à \( 1\),
-    \item
-        et qui ne sont pas divisés par \( X\).
-\end{enumerate}
-Un tel polynôme s'écrit sous la forme
-\begin{equation}
-    P=X^n+\sum_{k=0}^{n-1}a_kX^k.
-\end{equation}
-
-\begin{theorem}[Kronecker\cite{KXjFWKA}]    \label{ThoOWMNAVp}
-    Les racines des éléments de \( K_n\) sont des racines de l'unité.
-\end{theorem}
-\index{théorème!Kronecker}
-\index{polynôme!à plusieurs indéterminées}
-\index{résultant!utilisation}
-\index{polynôme!symétrique}
-
-\begin{proof}
-    Vu que \( \eC\) est algébriquement clos 
-    nous pouvons considérer les racines \( \alpha_1,\ldots, \alpha_n\) de \( P\) dans \( \eC\). Nous les considérons avec leurs multiplicités.
-%TODO : lorsqu'on aura démontré que \eC est algébriquement clos, il faudra le référentier ici.
-
-    Soit \( R=X^n+\sum_{k=0}^{n-1}b_kX^k\) un élément de \( K_n\) dont nous notons \( \beta_1,\ldots, \beta_n\) les racines dans \( \eC\). Les relations coefficients-racines stipulent que
-    \begin{equation}
-        b_k=\sum_{1\leq i_1<\ldots <i_{n-k}\leq n}\prod_{j=1}^{n-k}\beta_{i_j}.
-    \end{equation}
-    En prenant le module et en se souvenant que \( | \beta_{l} |\leq 1\) pour tout \( l\), nous trouvons que
-    \begin{equation}
-        | b_k |\leq\binom{ n }{ n-k }.
-    \end{equation}
-    Mais comme \( b_k\in \eZ\), nous avons
-    \begin{equation}
-        b_k\in\big\{    -\binom{ n }{ n-k },-\binom{ n }{ n-k }+1,\ldots, 0,\cdots,\binom{ n }{ n-k }   \big\}
-    \end{equation}
-    qui est de cardinal \( \binom{ n }{ n-k }+1\). Nous avons donc
-    \begin{equation}
-        \Card(K_n)\leq\prod_{k=0}^{n-1}\big( 1+\binom{ n }{ n-k } \big)<\infty.
-    \end{equation}
-    La conclusion jusqu'ici est que \( K_n\) est un ensemble fini.
-
-    Pour chaque \( k\in \eN^*\) nous considérons les polynômes
-    \begin{subequations}
-        \begin{align}
-            P_k&=\prod_{i=1}^n(X-\alpha_i^k)\\
-            Q_k&=X^k-Y\in \eZ[X,Y],
-        \end{align}
-    \end{subequations}
-    et puis nous considérons le résultant \( R_k=\res_X(P,Q_k)\in \eZ[Y]\) :
-    \begin{equation}
-        R_k=\res_X(P,Q_k)=
-        \begin{pmatrix}
-            1&a_{n-1}&\cdots&a_0&0&\cdots&0&0&0\\  
-            0&1&a_{n-1}&\cdots&a_0&0&\cdots&0&0\\  
-            \vdots&\ddots&\ddots&\ddots&&\ddots&\\  
-            0&\cdots&0&1&a_{n-1}&\cdots&a_0&0&0\\
-            0&\cdots&0&0&1&a_{n-1}&\cdots&a_0&0\\
-            0&\cdots&0&0&0&1&a_{n-1}&\cdots&a_0\\
-        \\
-                    1&0&\ldots&0&-Y&0&\ldots&0&0\\
-                    0&1&0&\ldots&0&-Y&0&\ldots&0\\
-                &&\ddots&&&\ddots&\ddots\\
-                    0&\cdots&0&1&0&\cdots&0&-Y&0\\
-                    0&0&\cdots&0&1&0&\cdots&0&-Y
-        \end{pmatrix}
-    \end{equation}
-    Cela est un polynôme en \( Y\) dont le terme de plus haut degré est \( (-1)^nY^n\). Les petites formules de la proposition \ref{PropNDBOGNx} nous permettent d'exprimer \( R_k(Y)\) en termes des racines de \( P\) :
-    \begin{equation}
-        R_k(Y)=\prod_{i=1}^nQ_k(\alpha_i)=\prod_{i=1}^n(\alpha_i^k-Y)=(-1)^n\prod_{i=1}^n(Y-\alpha_i^k)=(-1)^nP_k(Y).
-    \end{equation}
-    Vu que \( P\in K_n\) nous savons que les \( \alpha_i\) ne sont pas tous nuls; donc \( P_k\in K_n\). Cependant nous avons vu que \( K_n\) est un ensemble fini; donc parmi les \( P_k\), il y a des doublons (et pas un peu)\quext{Ici dans \cite{KXjFWKA}, il déduit qu'on a un \( k\) tel que \( P_k=P_1=P\). Mois je vois pourquoi on a un \( k\) et un \( l\) tels que \( P_k=P_l\), mais pourquoi on peut en trouver un spécialement égal au premier ? Une réponse à cette question permettrait de solidement réduire la lourdeur de la suite de la preuve.}. Nous regardons même l'ensemble des \( P_{2^n}\) dans lequel nous pouvons en trouver deux les mêmes. Soit \( l>k\) tels que \( P_{2^k}=P_{2^l}\). Si \( \alpha\) est racine de \( P_{2^k}\), alors il est de la forme \( \alpha=\beta^{2^k}\) pour une certaine racines \( \beta\) de \( P\). Par conséquent
-    \begin{equation}    \label{EqBEgJtzm}
-        \alpha^{2^l/2^k}=\alpha^{2^{l-k}}
-    \end{equation}
-    est racine de \( P_{2^l}\). Notons que dans cette expression il n'y a pas de problèmes de définition d'exposant fractionnaire dans \( \eC\) parce que \( l>k\). Vu que \eqref{EqBEgJtzm} est racine de \( P_{2^l}\), il est aussi racine de \( P_{2^k}\). Donc
-    \begin{equation}
-        \big( \alpha^{2^{l-k}} \big)^{2^{l-k}}=\alpha^{2^{2(l-k)}}
-    \end{equation}
-    est racine de \( P_{2^l}\) et donc de \( P_{2^k}\). Au final nous savons que tous les nombres de la forme \( \alpha^{2^{n(l-k)}}\) sont racines de \( P_{2^k}\). Mais comme \( P_{2^k}\) a un nombre fini de racines, nous pouvons en trouver deux égales. Si nous avons
-    \begin{equation}
-        \alpha^{2^{n(l-k)}}=\alpha^{2^{m(l-k)}}
-    \end{equation}
-    pour certains entiers \( m>n\), alors
-    \begin{equation}
-        \alpha^{2^{n(l-k)}-2^{m(l-k)}}=1,
-    \end{equation}
-    ce qui prouver que \( \alpha\) est une racine de l'unité. Nous avons donc prouvé que toutes les racines de \( P_{2^k}\) sont des racines de l'unité et donc que les racines de \( P\) sont racines de l'unité.
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Produit scalaire}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}      \label{DEFooEEQGooNiPjHz}
-    Une \defe{forme bilinéaire}{forme!bilinéaire} sur un espace vectoriel \( E\) est une application \( b\colon E\times E\to \eK\) telle que
-    \begin{enumerate}
-        \item
-            \( b(u,v)=b(v,u)\),
-        \item
-            \( b(u+v,w)=b(u,w)+b(v,w)\),
-        \item
-            \( b(\lambda u,v)=\lambda b(u,v)\)
-    \end{enumerate}
-    pour tout \( u,v,w\in E\) et \( \lambda\in \eK\) où \( \eK\) est un corps commutatif.
-\end{definition}
-
-\begin{definition}      \label{DEFooJIAQooZkBtTy}
-    Si $g$ est une application bilinéaire sur un espace vectoriel \( E\) nous disons qu'elle est
-    \begin{enumerate}
-        \item
-            \defe{définie positive}{application!définie positive} si $g(u,u)\geq 0$ pour tout $u\in E$ et $g(u,u)=0$ si et seulement si $u=0$.
-        \item
-            \defe{semi-définie positive}{application!semi-définie positive} si $g(u,u)\geq 0$ pour tout $u\in E$. Nous dirons aussi parfois qu'elle est simplement «positive».
-        \end{enumerate}
-\end{definition}
-Cela est évidemment à lier à la définition \ref{DefAWAooCMPuVM} et la proposition \ref{PROPooUAAFooEGVDRC} : une application bilinéaires est définie positive si et seulement si sa matrice symétrique associée l'est.
-
-\begin{definition}\label{DefVJIeTFj}
-    Un \defe{produit scalaire}{produit!scalaire!en général} sur un espace vectoriel \( E\) est une forme bilinéaire\footnote{Définition \ref{DEFooEEQGooNiPjHz}.} symétrique strictement définie positive\footnote{Définition \ref{DEFooJIAQooZkBtTy}.}.
-\end{definition}
-
-Étant donné que l'inégalité de Cauchy-Schwarz sera surtout utilisée dans le cas où un produit scalaire est bel et bien donné, nous l'énonçons et le démontrons avec des notations adaptée à l'usage. Le produit scalaire sera noté \( X\cdot Y\) pour \( b(X,Y)\) si \( b\) est la forme bilinéaire.
-\begin{theorem}[Inégalité de Cauchy-Schwarz]      \label{ThoAYfEHG}
-	Si $X$ et $Y$ sont des vecteurs, alors
-    \begin{equation}        \label{EQooZDSHooWPcryG}
-		| X\cdot Y |\leq\| X \|\| Y \|.
-	\end{equation}
-    Nous avons une égalité si et seulement si \( X\) et \( Y\) sont multiples l'un de l'autre.
-\end{theorem}
-\index{Cauchy-Schwarz}
-\index{inégalité!Cauchy-Schwarz}
-
-%TODO : mettre au point les notations.
-\begin{proof}
-	Étant donné que les deux membres de l'inéquation sont positifs, nous allons travailler en passant au carré afin d'éviter les racines carrés dans le second membre.
-
-	Nous considérons le polynôme
-	\begin{equation}
-		P(t)=\| X+tY \|^2=(X+tY)\cdot(X+tY)=X\cdot X+tX\cdot Y+tY\cdot X+t^2Y\cdot Y.
-	\end{equation}
-	En ordonnant les termes selon les puissance de $t$,
-	\begin{equation}
-		P(t)=\| Y \|^2t^2+2(X\cdot Y)t+\| X \|^2.
-	\end{equation}
-	Cela est un polynôme du second degré en $t$. Par conséquent le discriminant\footnote{Le fameux $b^2-4ac$.} doit être négatif. Nous avons donc
-	\begin{equation}
-		\Delta=4(X\cdot Y)^2-4\| X \|^2\| Y \|^2\leq 0,
-	\end{equation}
-	ce qui donne immédiatement
-	\begin{equation}
-		(X\cdot Y)^2\leq\| X \|^2\| Y^2 \|.
-	\end{equation}
-
-    En ce qui concerne le cas d'égalité, si nous avons \( X\cdot Y=\| X \|\| Y \|\), alors le discriminant \( \Delta\) ci-dessus est nul et le polynôme \( P\) admet une racine double \( t_0\). Pour cette valeur nous avons
-    \begin{equation}
-        P(t_0)=| X+t_0Y |=0,
-    \end{equation}
-    ce qui implique \( X+t_0Y=0\) et donc que \( X\) et \( Y\) sont liés.
-\end{proof}
-
-Vu que nous allons voir un pâté d'espaces avec des produits scalaires, nous leur donnons un nom.
-\begin{definition}\label{DefLZMcvfj}
-    Un espace vectoriel \defe{euclidien}{euclidien!espace} est un espace vectoriel de dimension finie muni d'un produit scalaire (définition \ref{DefVJIeTFj}).
-\end{definition}
-
-\begin{definition}[Isométrie, thème \ref{THMooVUCLooCrdbxm}]      \label{DEFooGGTYooXsHIZj}
-    Une \defe{isométrie}{isométrie!forme bilinéaire} d'une forme bilinéaire \( b\) sur l'espace vectoriel \( E\) est une application bijective \( f\colon E\to E\) telle que \( b\big( f(u),f(v) \big)=b(u,v)\) pour tout \( u,v\in E\).
-\end{definition}
-En particulier une isométrie d'un espace euclidien est une application bijective qui préserve le produit scalaire.
-
-\begin{proposition} \label{PropEQRooQXazLz}
-    Si \( x,y\mapsto x\cdot y\) est un produit scalaire sur un espace vectoriel \( E\), alors \( N(x)=\sqrt{x\cdot x}\) est une norme vérifiant l'identité du parallélogramme :
-    \begin{equation}        \label{EqYCLtWfJ}
-        \| x-y \|^2+\| x+y \|^2=2\| x \|^2+2\| y \|^2.
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-
-    Prouvons l'inégalité triangulaire\index{inégalité!triangulaire!produit scalaire}. Si \( x,y\in E\) nous avons
-    \begin{equation}
-        \| x+y \|=\sqrt{\| x \|^2+\| y \|^2+2x\cdot y}.
-    \end{equation}
-    Par l'inégalité de Cauchy-Schwarz, théorème \ref{ThoAYfEHG} nous avons aussi
-    \begin{equation}
-        \| x \|^2+\| y \|^2+2x\cdot y\leq \| x \|^2+\| y \|^2+2\| x \|\| y \|=\big( \| x \|+\| y \| \big)^2,
-    \end{equation}
-    donc
-    \begin{equation}
-        \| x+y \|\leq \sqrt{\big( \| x \|+\| y \| \big)^2}=\| x \|+\| y \|.
-    \end{equation}
-
-    La seconde assertion est seulement un calcul :
-			\begin{equation}
-				\begin{aligned}[]
-					\| x-y \|^2+\| x+y \|^2&=(x-y)\cdot (x-y)+(x+y)\cdot(x+y)\\
-					&=x\cdot x-x\cdot y-y\cdot x+y\cdot y\\
-					&\quad +x\cdot x+x\cdot y+y\cdot x+y\cdot y\\
-					&=2x\cdot x+2y\cdot y\\
-					&=2\| x \|^2+2\| y \|^2.
-				\end{aligned}
-			\end{equation}
-\end{proof}
-
-Le produit scalaire permet de donner une norme via la formule suivante :
-\begin{equation}
-    \| x \|^2=x\cdot x.
-\end{equation}
-
-\begin{lemma}[\cite{KXjFWKA}]   \label{LemLPOHUme}
-    Soit \( V\) un espace vectoriel muni d'un produit scalaire et de la norme associée. Si \( x,y\in V\) satisfont à \( \| x+y \|=\| x \|+\| y \|\), alors il existe \( \lambda\geq 0\) tel que \( x=\lambda y\).
-\end{lemma}
-
-\begin{proof}
-    Quitte à raisonner avec \( x/\| x \|\) et \( y/\| y \|\), nous supposons que \( \| x \|=\| y \|=1\). Dans ce cas l'hypothèse signifie que \( \| x+y \|^2=4\). D'autre part en écrivant la norme en terme de produit scalaire,
-    \begin{equation}
-        \| x+y \|^2=\| x \|^2+\| y \|^2+2\langle x, y\rangle ,
-    \end{equation}
-    ce qui nous mène à affirmer que \( \langle x, y\rangle =1=\| x \|\| y \|\). Nous sommes donc dans le cas d'égalité de l'inégalité de Cauchy-Schwarz\footnote{Théorème \ref{ThoAYfEHG}.}, ce qui nous donne un \( \lambda\) tel que \( x=\lambda y\). Étant donné que \( \| x \|=\| y \|=1\) nous avons obligatoirement \( \lambda=\pm 1\), mais si \( \lambda=-1\) alors \( \langle x, y\rangle =-1\), ce qui est le contraire de ce qu'on a prétendu plus haut. Par soucis de cohérence, nous allons donc croire que \( \lambda=1\).
-\end{proof}
-
-
-\begin{proposition}			\label{PropVectsOrthLibres}
-	si $v_1,\cdots,v_k$ sont des vecteurs non nuls, orthogonaux deux à deux, alors ces vecteurs forment une famille libre.
-\end{proposition}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Projection et angles}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[Propriétés du produit scalaire]
-	Si $X$ et $Y$ sont des vecteurs de $\eR^3$, alors
-	\begin{description}
-		\item[Symétrie] $X\cdot Y=Y\cdot X$;
-		\item[Linéarité] $(\lambda X+\mu X')\cdot Y=\lambda(X\cdot Y)+\mu(X'\cdot Y)$ pour tout $\lambda$ et $\mu$ dans $\eR$;
-		\item[Défini positif] $X\cdot X\geq 0$ et $X\cdot X=0$ si et seulement si $X=0$.
-	\end{description}
-\end{proposition}
-Note : lorsque nous écrivons $X=0$, nous voulons voulons dire $X=\begin{pmatrix}
-	0	\\ 
-	0	\\ 
-	0	
-\end{pmatrix}$.
-
-
-\begin{definition}
-	La \defe{norme}{norme!vecteur} du vecteur $X$, notée $\| X \|$, est définie par 
-	\begin{equation}
-		\| X \|=\sqrt{X\cdot X}=\sqrt{x^2+y^2+z^2}
-	\end{equation}
-	si $X=(x,y,z)$. Cette norme sera parfois nommée «norme euclidienne».
-\end{definition}
-Cette définition est motivée par le théorème de Pythagore. Le nombre $X\cdot X$ est bien la longueur de la «flèche» $X$. Plus intrigante est la définition suivante :
-\begin{definition}
-	Deux vecteurs $X$ et $Y$ sont \defe{orthogonaux}{orthogonal!vecteur} si $X\cdot Y=0$. 
-\end{definition}
-Cette définition de l'orthogonalité est motivée par la proposition suivante.
-
-\begin{proposition}		\label{PropProjScal}
-	Si nous écrivons $\pr_Y$  l'opération de projection sur la droite qui sous-tend $Y$, alors nous avons
-	\begin{equation}
-		\| \pr_YX \|=\frac{ X\cdot Y }{ \| Y \| }.
-	\end{equation}
-\end{proposition}
-
-\begin{proof}
-	Les vecteurs $X$ et $Y$ sont des flèches dans l'espace. Nous pouvons choisir un système d'axe orthogonal tel que les coordonnées de $X$ et $Y$ soient
-	\begin{equation}
-		\begin{aligned}[]
-			X&=\begin{pmatrix}
-				x	\\ 
-				y	\\ 
-				0	
-			\end{pmatrix},
-			&Y&=\begin{pmatrix}
-				l	\\ 
-				0	\\ 
-				0	
-			\end{pmatrix}
-		\end{aligned}
-	\end{equation}
-	où $l$ est la longueur du vecteur $Y$. Pour ce faire, il suffit de mettre le premier axe le long de $Y$, le second dans le plan qui contient $X$ et $Y$, et enfin le troisième axe dans le plan perpendiculaire aux deux premiers.
-
-	Un simple calcul montre que $X\cdot Y=xl+y\cdot 0+0\cdot 0=xl$. Par ailleurs, nous avons $\| \pr_YX \|=x$. Par conséquent,
-	\begin{equation}
-		\| \pr_YX \|=\frac{ X\cdot Y }{ l }=\frac{ X\cdot Y }{ \| Y \| }.
-	\end{equation}
-\end{proof}
-
-\begin{corollary}
-	Si la norme de $Y$ est $1$, alors le nombre $X\cdot Y$ est la longueur de la projection de $X$ sur $Y$.
-\end{corollary}
-
-\begin{proof}
-	Poser $\| Y \|=1$ dans la proposition \ref{PropProjScal}.
-\end{proof}
-
-\begin{remark}
-    Outre l'orthogonalité, le produit scalaire permet de savoir l'angle entre deux vecteurs à travers la définition \ref{DEFooSVDZooPWHwFQ}. D'autres interprétations géométriques du déterminant sont listées dans le thème \ref{THMooUXJMooOroxbI}.
-\end{remark}
-
-Nous sommes maintenant en mesure de déterminer, pour deux vecteurs quelconques $u$ et $v$, la projection orthogonale de $u$ sur $v$. Ce sera le vecteur $\bar u$ parallèle à $v$ tel que $u-\bar u$ est orthogonal à $v$. Nous avons donc
-\begin{equation}
-    \bar u=\lambda v
-\end{equation}
-et 
-\begin{equation}
-    (u-\lambda v)\cdot v=0.
-\end{equation}
-La seconde équation donne $u\cdot v-\lambda v\cdot v=0$, ce qui fournit $\lambda$ en fonction de $u$ et $v$ :
-\begin{equation}
-    \lambda=\frac{ u\cdot v }{ \| v \|^2 }.
-\end{equation}
-Nous avons par conséquent
-\begin{equation}
-    \bar u=\frac{ u\cdot v }{ \| v \|^2 }v.
-\end{equation}
-Armés de cette interprétation graphique du produit scalaire, nous comprenons pourquoi nous disons que deux vecteurs sont orthogonaux lorsque leur produit scalaire est nul.
-
-Nous pouvons maintenant savoir quel est le coefficient directeur d'une droite orthogonale à une droite donnée. En effet, supposons que la première droite soit parallèle au vecteur $X$ et la seconde au vecteur $Y$. Les droites seront perpendiculaires si $X\cdot Y=0$, c'est à dire si
-\begin{equation}
-	\begin{pmatrix}
-		x_1	\\ 
-		y_1	
-	\end{pmatrix}\cdot\begin{pmatrix}
-		y_1	\\ 
-		y_2	
-	\end{pmatrix}=0.
-\end{equation}
-Cette équation se développe en 
-\begin{equation}		\label{Eqxuyukljsca}
-	x_1y_1=-x_2y_2.
-\end{equation}
-Le coefficient directeur de la première droite est $\frac{ x_2 }{ x_1 }$. Isolons cette quantité dans l'équation \eqref{Eqxuyukljsca} :
-\begin{equation}
-	\frac{ x_2 }{ x_1 }=-\frac{ y_1 }{ y_2 }.
-\end{equation}
-Donc le coefficient directeur de la première est l'inverse et l'opposé du coefficient directeur de la seconde.
-
-\begin{example}
-	Soit la droite $d\equiv y=2x+3$. Le coefficient directeur de cette droite est $2$. Donc le coefficient directeur d'une droite perpendiculaires doit être $-\frac{ 1 }{ 2 }$.
-\end{example}
-
-\begin{proof}[Preuve alternative]
-	La preuve peut également être donnée en ne faisant pas référence au produit scalaire. Il suffit d'écrire toutes les quantités en termes des coordonnées de $X$ et $Y$. Si nous posons
-	\begin{equation}
-		\begin{aligned}[]
-			X&=\begin{pmatrix}
-				x_1	\\ 
-				x_2	\\ 
-				x_2	
-			\end{pmatrix},
-			&Y&=\begin{pmatrix}
-				y_1	\\ 
-				y_2	\\ 
-				y_3	
-			\end{pmatrix},
-		\end{aligned}
-	\end{equation}
-	l'inégalité à prouver devient
-	\begin{equation}
-		(x_1y_1+x_2y_2+x_3y_3)^2\leq (x_1^2+x_2^2+x_3^2)(y_1^2+y_2^2+y_3^2).
-	\end{equation}
-	Nous considérons la fonction
-	\begin{equation}
-		\varphi(t)=(x_1+ty_1)^2+(x_2+ty_2)^2+(x_3+ty_3)^2
-	\end{equation}
-	En tant que norme, cette fonction est évidement positive pour tout $t$. En regroupant les termes de chaque puissance de $t$, nous avons
-	\begin{equation}
-		\varphi(t)=(y_1^2+y_2^2+y_3^2)t^2+2(x_1y_1+x_2y_2+x_3y_3)t+(x_1^2+x_2^2+x_3^2).
-	\end{equation}
-	Cela est un polynôme du second degré en $t$. Par conséquent le discriminant doit être négatif. Nous avons donc
-	\begin{equation}
-		4(x_1y_1+x_2y_2+x_3y_3)^2-(x_1^2+x_2^2+x_3^2)(y_1^2+y_2^2+y_3^2)\leq 0.
-	\end{equation}
-	La thèse en découle aussitôt.
-\end{proof}
-
-\begin{proposition}
-	La norme euclidienne a les propriétés suivantes :
-	\begin{enumerate}
-		\item
-			Pour tout vecteur $X$ et réel $\lambda$,  $\| \lambda X \|=| \lambda |\| X \|$. Attention à ne pas oublier la valeur absolue !
-		\item
-			Pour tout vecteurs $X$ et $Y$, $\| X+Y \|\leq \| X \|+\| Y \|$.
-	\end{enumerate}
-\end{proposition}
-
-\begin{proof}
-	Nous ne prouvons pas le premier point.
-    % TODO : faire la preuve
-    Pour le second, nous avons les inégalités suivantes :
-	\begin{subequations}
-		\begin{align}
-			\| X+Y \|^2&=\| X \|^2+\| Y \|^2+2X\cdot Y\\
-			&\leq\| X \|^2+\| Y \|^2+2|X\cdot Y|\\
-			&\leq\| X \|^2+\| Y \|^2+2\| X \|\| Y \|\\
-			&=\big( \| X \|+\| Y \| \big)^2
-		\end{align}
-	\end{subequations}
-    Nous avons utilisé d'abord la majoration $| x |\geq x$ qui est évident pour tout nombre $x$; et ensuite l'inégalité de Cauchy-Schwarz \ref{ThoAYfEHG}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Procédé de Gram-Schmidt}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[Procédé de Gram-Schmidt]    \label{PropUMtEqkb}
-    Un espace euclidien possède une base orthonormée.
-\end{proposition}
-\index{espace!euclidien}
-\index{Gram-Schmidt}
-
-\begin{proof}
-    Soit \( E\) un espace euclidien et \( \{ v_1,\ldots, v_n \}\), une base quelconque de \( E\). Nous posons d'abord
-    \begin{equation}
-        \begin{aligned}[]
-            f_1&=v_1,&e_1&=\frac{ f_1 }{ \| f_1 \| }.
-        \end{aligned}
-    \end{equation}
-    Ensuite
-    \begin{equation}
-        \begin{aligned}[]
-            f_2&=v_2-\langle v_2, e_1\rangle e_1,&e_2&=\frac{ f_2 }{ \| f_2 \| }.
-        \end{aligned}
-    \end{equation}
-    Notons que \( \{ e_1,e_2 \}\) est une base de \( \Span\{ v_1,v_2 \}\). De plus elle est orthogonale :
-    \begin{equation}
-        \langle e_1, f_2\rangle =\langle e_1, v_2\rangle -\langle v_2, e_1\rangle \underbrace{\langle e_1, e_1\rangle}_{=1} =0.
-    \end{equation}
-    Le fait que \( \| e_1 \|=\| e_2 \|=1\) est par construction. Nous avons donc donné une base orthonormée de \( \Span\{ v_1,v_2 \}\).
-
-    Nous continuons par récurrence en posant
-    \begin{equation}
-        \begin{aligned}[]
-            f_k&=v_k-\sum_{i=1}^{k-1}\langle v_k, e_i\rangle e_i,&e_k&=\frac{ f_k }{ \| f_k \| }.
-        \end{aligned}
-    \end{equation}
-    Pour tout \( j<k\) nous avons
-    \begin{equation}
-        \langle e_j, f_k\rangle =\langle e_j, v_k\rangle -\sum_{i=1}^{k-1}\langle v_k, e_i\rangle \underbrace{\langle e_i, e_j\rangle}_{=\delta_{ij}} =0
-    \end{equation}
-\end{proof}
-Cet algorithme de Gram-Schmidt nous donne non seulement l'existence de bases orthonormée pour tout espace euclidien, mais aussi le moyen d'en construire à partir de n'importe quelle base.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Hermitien, orthogonal, adjoint}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}  \label{DefMZQxmQ}
-Si \( E\) est un espace vectoriel sur \( \eC\), nous disons qu'une application \( \langle ., .\rangle \colon E\times E\to \eC\) est un \defe{produit scalaire hermitien}{produit!scalaire!hermitien}\index{hermitien!produit scalaire} si pour tout \( u,v\in E\) nous avons
-\begin{enumerate}
-    \item
-        \( \langle u, v\rangle =\overline{ \langle v, u\rangle  }\)
-    \item
-        \( \lambda\langle u, v\rangle =\langle \lambda u, v\rangle =\langle u, \bar \lambda v\rangle \)
-    \item
-        \( \langle u, u\rangle \in \eR^+\) et \( \langle u, u\rangle =0\) si et seulement si \( u=0\).
-\end{enumerate}
-Un espace vectoriel sur \( \eC\) muni d'un produit scalaire hermitien est un espace vectoriel hermitien.
-\end{definition}
-
-\begin{example}
-    L'ensemble \( E=\eC^n\) vu comme espace vectoriel de dimension \( n\) sur \( \eC\)  est muni d'une forme sesquilinéaire
-    \begin{equation}    \label{EqFormSesqQrjyPH}
-        \langle x, y\rangle =\sum_{k=1}^nx_k\bar y_k
-    \end{equation}
-    pour tout \( x,y\in\eC^n\). Cela est un espace vectoriel hermitien.
-\end{example}
-
-\begin{definition}      \label{DEFooROVNooFlTbSK}
-    Soit \( E\) un espace vectoriel hermitien et \( A\in\End(E)\). L'opérateur \defe{adjoint}{opérateur!adjoint} de \( A\) est l'opérateur noté \( A^*\) définit par
-    \begin{equation}    \label{EQooHWYKooFzAGgB}
-        \langle Av, w\rangle =\langle v, A^*w\rangle 
-    \end{equation}
-    pour tout \( v,w\in \eC^n\).
-\end{definition}
-
-La prescription \eqref{EQooHWYKooFzAGgB} définit bien l'opérateur \( A^*\) de façon unique. En effet si \( \{ e_i \}\) est une base orthonormée alors
-\begin{equation}
-    \langle e_i, A^*e_j\rangle =(A^*e_j)_i=\langle Ae_i, e_j\rangle =(Ae_i)_j.
-\end{equation}
-Autrement dit, \( (A^*e_j)_i=(Ae_i)_j\).
-
-\begin{proposition}     \label{PROPooSHZMooGwdfBd}
-    En ce qui concerne le déterminant,
-    \begin{equation}
-        \det(A^*)=\det(A)^*
-    \end{equation}
-    où l'étoile à droite dénote la conjugaison complexe dans \( \eC\).
-\end{proposition}
-
-\begin{proof}
-    Écrivons la définition \eqref{EQooOJEXooXUpwfZ} avec l'expression explicite \eqref{EQooOJEXooXUpwfZ} du déterminant. Le tout avec la base canonique :
-    \begin{equation}
-            \det(A)=\det_{(e_1,\ldots, e_n)}(Ae_1,\ldots, Ae_n)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^ne_{\sigma(i)}^*(Ae_i).
-    \end{equation}
-    Mais nous pouvons développer :
-    \begin{equation}
-        e^*_{\sigma(i)}(Ae_i)=\langle e_{\sigma(i)}, Ae_i\rangle =\langle A^*e_{\sigma(i)}, e_i\rangle =\langle e_i, A^*e_{\sigma(i)}\rangle^*=e_i^*(A^*e_{\sigma(i)})^*.
-    \end{equation}
-    Notez que dans la dernière expression, les trois \( {}^*\) ont trois significations différentes. Par conséquent,
-    \begin{equation}
-        \det(A)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^{n}e_i^*(A^*e_{\sigma(i)})^*.
-    \end{equation}
-    Mais \( e_i^*(A^*e_{\sigma(i)})=e_{\sigma(j)}^*(A^*e_{j})\) pour \( j=\sigma(i)\), donc le produit ne change pas si on déplace le \( \sigma\) :
-    \begin{equation}
-        \det(A)=\sum_{\sigma\in S_n}\epsilon(\sigma)\prod_{i=1}^{n}e_{\sigma(i)}^*(A^*e_{i})^*=\det(A^*)^*.
-    \end{equation}
-\end{proof}
-
-\begin{definition}      \label{DEFooKEBHooWwCKRK}
-    Un opérateur \( A\) est \defe{hermitien}{opérateur!hermitien} si \( A^*=A\). On dit aussi \defe{autoadjoint}{opérateur!autoadjoint}.
-\end{definition}
-
-Le mot «hermitien» est réservé aux opérateurs sur des espaces hermitiens, c'est à dire des espaces vectoriels sur \( \eC\). Le mot «autoadjoint» par contre est plutôt utilisé dans le cadre d'opérateurs sur les espaces réels\footnote{Voir la définition \ref{DEFooYNEQooGQgbCf}.}. En conséquence de quoi, ces deux mots sont synonymes, mais il est préférable d'utiliser «hermitien» lorsque l'espace vectoriel est sur \( \eC\) et «autoadjoint» lorsqu'il est sur \( \eR\).
-
-\begin{remark}
-    Le fait d'être hermitien n'implique en rien le fait d'être inversible.
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Opérateur orthogonal, matrice orthogonale}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooYKCSooURQDoS}
-    Un opérateur est \defe{orthogonal}{orthogonal!opérateur} lorsque \( A^*=A^{-1}\) où \( A^*\) est l'adjoint de \( A\) définit en \ref{DEFooROVNooFlTbSK}.
-\end{definition}
-
-\begin{definition}      \label{DEFooUHANooLVBVID}
-    Une matrice \( U\) est \defe{orthogonale}{matrice!orthogonale}\index{orthogonal!matrice} si \( U^t=U^{-1}\). Le \defe{groupe orthogonal}{groupe!orthogonal} noté \( \gO(n)\) est l'ensemble des matrices orthogonales \( n\times n\).
-\end{definition}
-
-\begin{lemma}       \label{LEMooSSALooSBFzJb}
-    Soit un opérateur \( A\colon \eR^n\to \eR^n\) muni du produit scalaire usuel. Il est orthogonal si et seulement si sa matrice dans la base canonique est orthogonale\footnote{Définition \ref{DEFooUHANooLVBVID}.}.
-\end{lemma}
-
-\begin{proof}
-    Soit la base canonique \( \{ e_i \}_{i=1,\ldots, n}\) de \( \eR^n\). Nous avons
-    \begin{equation}
-        \langle AA^*e_i, e_j\rangle =\langle e_i, e_j\rangle =\delta_{ij},
-    \end{equation}
-    donc \( \big( (AA^*)e_j \big)_j=\delta_{ij}\), ou encore \( (AA^*)_{ij}=\delta_{ij}\), ce qui signifie que la matrice $AA^*$ est l'identité.
-\end{proof}
-
-
-Pour rappel, une isométrique d'un espace euclidien est une bijection qui préserve le produit scalaire, définition \ref{DEFooGGTYooXsHIZj}.
-
-\begin{proposition}[Thème \ref{THMooVUCLooCrdbxm}]     \label{PropKBCXooOuEZcS}
-    À propos de matrices orthogonales.
-    \begin{enumerate}
-        \item
-            L'ensemble des matrice réelles orthogonales forme un groupe noté \( \gO(n,\eR)\)\nomenclature[B]{\( \gO(n,\eR)\)}{le groupe des matrices orthogonales}.
-        \item
-            Si \( A\) est une matrice orthogonale, alors \( \det(A)=\pm 1\).
-        \item       \label{ITEMooOWMBooHUatNb}
-            Le groupe \( \gO(n)\) est le groupe des isométries de \( \eR^n\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Si \( A\) et \( B\) sont orthogonales, alors
-    \begin{equation}
-        (AB)(AB)^t=ABB^tA^t=A\mtu A^t=\mtu.
-    \end{equation}
-    Vu que \( \mtu\) est orthogonale, nous avons bien un groupe.
-
-    En ce qui concerne le déterminant, \( AA^t=\mtu\) donne \( \det(A)\det(A^t)=1\) ou encore \( \det(A)^2=1\). D'où le fait que \( \det(A)=\pm 1\).
-
-    En ce qui concerne le déterminant, nous savons par la proposition \ref{PROPooSHZMooGwdfBd} que \( \det(A)=\det(A^*)\) (nous sommes dans le cas réel). Étant donné que le déterminant est un morphisme et que \( AA^*=\id\) avec \( \det(\id)=1\) nous avons aussi \( \det(A)^2=1\), c'est à dire \( \det(A)=\pm1\).
-
-    D'autre part si \( A\) est une isométrie de \( \eR^n\) alors pour tout \( x,y\in \eR^n\) nous avons \( \langle Ax, Ay\rangle =\langle x, y\rangle \). En particulier,
-    \begin{equation}
-        \langle A^tAx, y\rangle =\langle x, y\rangle 
-    \end{equation}
-    pour tout \( x,y\in \eR^n\). En prenant \( y=e_i\) nous trouvons
-    \begin{equation}
-        (A^tAx)_i=x_i,
-    \end{equation}
-    ce qui signifie que pour tout \( x\), \( A^tAx=x\), ou encore que \( A^tA\) est l'identité.
-
-    Réciproquement si \( A^tA\) est l'identité nous avons
-    \begin{equation}
-        \langle a, y\rangle =\langle A^tAx, y\rangle =\langle Ax, Ay\rangle ,
-    \end{equation}
-    ce qui prouve que \( A\) est une isométrie.
-\end{proof}
-
-\begin{definition}      \label{DEFooJLNQooBKTYUy}
-    Le sous-groupe des matrices orthogonales de déterminant \( 1\) est le groupe \defe{spécial orthogonal}{groupe!spécial orthogonal} noté \( \SO(n)\).
-\end{definition}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Directions conservées}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Généralités}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous savons qu'une application \emph{linéaire} $A\colon \eR^3\to \eR^3$ est complètement définie par la donnée de son action sur les trois vecteurs de base, c'est à dire par la donnée de
-\begin{equation}
-	\begin{aligned}[]
-		Ae_1,&&Ae_2&&\text{et}&&Ae_3.
-	\end{aligned}
-\end{equation}
-Nous allons former la matrice de $A$ en mettant simplement les vecteurs $Ae_1$, $Ae_2$ et $Ae_3$ en colonne. Donc la matrice
-\begin{equation}		\label{EqExempleALin}
-	A=\begin{pmatrix}
-		3	&	0	&	0	\\
-		0	&	1	&	0	\\
-		0	&	1	&	0
-	\end{pmatrix}
-\end{equation}
-signifie que l'application linéaire $A$ envoie le vecteur $e_1$ sur $\begin{pmatrix}
-	3	\\ 
-	0	\\ 
-	0	
-\end{pmatrix}$, le vecteur $e_2$ sur $\begin{pmatrix}
-	0	\\ 
-	0	\\ 
-	1	
-\end{pmatrix}$ et le vecteur $e_3$ sur $\begin{pmatrix}
-	0	\\ 
-	1	\\ 
-	0	
-\end{pmatrix}$.
-Pour savoir comment $A$ agit sur n'importe quel vecteur, on applique la règle de produit vecteur$\times$matrice :
-\begin{equation}
-	\begin{pmatrix}
-		1	&	2	&	3	\\
-		4	&	5	&	6	\\
-		7	&	8	&	9
-	\end{pmatrix}\begin{pmatrix}
-		x	\\ 
-		y	\\ 
-		z	
-	\end{pmatrix}=
-	\begin{pmatrix}
-		x+2y+3z	\\ 
-		4x+5y+6z	\\ 
-		7x+8y+9z	
-	\end{pmatrix}.
-\end{equation}
-
-Une chose intéressante est de savoir quelles sont les directions invariantes de la transformation linéaire. Par exemple, on peut lire sur la matrice \eqref{EqExempleALin} que la direction $\begin{pmatrix}
-	1	\\ 
-	0	\\ 
-	0	
-\end{pmatrix}$ est invariante : elle est simplement multipliée par $3$. Dans cette direction, la transformation est juste une dilatation. Afin de savoir si $v$ est un vecteur d'une direction conservée, il faut voir s'il existe un nombre $\lambda$ tel que $Av=\lambda v$, c'est à dire voir si $v$ est simplement dilaté.
-
-L'équation $Av=\lambda v$ se récrit $(A-\lambda\mtu)v=0$, c'est à dire qu'il faut résoudre l'équation
-\begin{equation}
-	(A-\lambda\mtu)\begin{pmatrix}
-		x	\\ 
-		y	\\ 
-		z	
-	\end{pmatrix}=
-	\begin{pmatrix}
-		0	\\ 
-		0	\\ 
-		0	
-	\end{pmatrix}.
-\end{equation}
-Nous savons qu'une telle équation ne peut avoir de solutions que si $\det(A-\lambda\mtu)=0$. La première étape est donc de trouver les $\lambda$ qui vérifient cette condition.
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Comment trouver la matrice d'une symétrie donnée ?}
-%---------------------------------------------------------------------------------------------------------------------------
-\label{SubSecMtrSym}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Symétrie par rapport à un plan}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-Comment trouver par exemple la matrice $A$ qui donne la symétrie autour du plan $z=0$ ? La définition d'une telle symétrie est que les vecteurs du plan $z=0$ ne bougent pas, tandis que les vecteurs perpendiculaires changent de signe. Ces informations vont permettre de trouver comment $A$ agit sur une base de $\eR^3$. En effet :
-\begin{enumerate}
-
-	\item
-		Le vecteur $\begin{pmatrix}
-			1	\\ 
-			0	\\ 
-			0	
-		\end{pmatrix}$ est dans le plan $z=0$, donc il ne bouge pas,
-
-	\item
-		le vecteur $\begin{pmatrix}
-			0	\\ 
-			1	\\ 
-			0	
-		\end{pmatrix}$ est également dans le plan, donc il ne bouge pas non plus,
-
-	\item
-		et le vecteur $\begin{pmatrix}
-			0	\\ 
-			0	\\ 
-			1	
-		\end{pmatrix}$ est perpendiculaire au plan $z=0$, donc il va changer de signe.
-
-\end{enumerate}
-Cela nous donne directement les valeurs de $A$ sur la base canonique et nous permet d'écrire 
-\begin{equation}
-	A=\begin{pmatrix}
-		1	&	0	&	0	\\
-		0	&	1	&	0	\\
-		0	&	0	&	-1
-	\end{pmatrix}.
-\end{equation}
-Pour écrire cela, nous avons juste mit en colonne les images des vecteurs de base. Les deux premiers n'ont pas changé et le troisième a changé.
-
-Et si maintenant on donne un plan moins facile que $z=0$ ? Le principe reste le même : il faudra trouver deux vecteurs qui sont dans le plan (et dire qu'ils ne bougent pas), et puis un vecteur qui est perpendiculaire au plan\footnote{Pour le trouver, penser au produit vectoriel.}, et dire qu'il change de signe.
-
-Voyons ce qu'il en est pour le plan $x=-z$. Il faut trouver deux vecteurs linéairement indépendants dans ce plan. Prenons par exemple
-\begin{equation}		\label{EqffudE}
-	\begin{aligned}[]
-		f_1&=\begin{pmatrix}
-			0	\\ 
-			1	\\ 
-			0	
-		\end{pmatrix},&f_2&=\begin{pmatrix}
-			1	\\ 
-			0	\\ 
-			-1	
-		\end{pmatrix}.
-	\end{aligned}
-\end{equation}
-Nous avons 
-\begin{equation}
-	\begin{aligned}[]
-		Af_1&=f_1\\
-		Af_2&=f_2.
-	\end{aligned}
-\end{equation}
-Afin de trouver un vecteur perpendiculaire au plan, calculons le produit vectoriel :
-\begin{equation}
-	f_3=f_1\times f_2=\begin{vmatrix}
-		e_1	&	e_2	&	e_3	\\
-		0	&	1	&	0	\\
-		1	&	0	&	-1
-	\end{vmatrix}=-e_1-e_3=\begin{pmatrix}
-		-1	\\ 
-		0	\\ 
-		-1	
-	\end{pmatrix}.
-\end{equation}
-Nous avons 
-\begin{equation}
-	Af_3=-f_3.
-\end{equation}
-Afin de trouver la matrice $A$, il faut trouver $Ae_1$, $Ae_2$ et $Ae_3$. Pour ce faire, il faut d'abord écrire $\{ e_1,e_2,e_3 \}$ en fonction de $\{ f_1,f_2,f_3 \}$. La première des équation \eqref{EqffudE} dit que 
-\begin{equation}
-	f_1=e_2.
-\end{equation}
-Ensuite, nous avons
-\begin{equation}
-	\begin{aligned}[]
-		f_2&=e_1-e_3\\
-		f_3&=-e_1-e_3.
-	\end{aligned}
-\end{equation}
-La somme de ces deux équations donne $-2e_3=f_2+f_3$, c'est à dire
-\begin{equation}
-	e_3=-\frac{ f_2+f_3 }{ 2 }
-\end{equation}
-Et enfin, nous avons
-\begin{equation}
-	e_1=\frac{ f_2-f_3 }{ 2 }.
-\end{equation}
-
-Maintenant nous pouvons calculer les images de $e_1$, $e_2$ et $e_3$ en faisant
-\begin{equation}
-	\begin{aligned}[]
-		Ae_1&=\frac{ Af_2-Af_3 }{ 2 }=\frac{1 }{2}\begin{pmatrix}
-			0	\\ 
-			0	\\ 
-			-2	
-		\end{pmatrix}=\begin{pmatrix}
-			0	\\ 
-			0	\\ 
-			-1	
-		\end{pmatrix},\\
-		Ae_2&=Af_1=f_1=\begin{pmatrix}
-			0	\\ 
-			1	\\ 
-			0	
-		\end{pmatrix},\\
-		Ae_3&=-\frac{ f_2-f_3 }{ 2 }=-\frac{ 1 }{2}\begin{pmatrix}
-			2	\\ 
-			0	\\ 
-			0	
-		\end{pmatrix}=\begin{pmatrix}
-			-1	\\ 
-			0	\\ 
-			0	
-		\end{pmatrix}.
-	\end{aligned}
-\end{equation}
-La matrice $A$ s'écrit maintenant en mettant les trois images trouvées en colonnes :
-\begin{equation}
-	A=\begin{pmatrix}
-		0	&	0	&	-1	\\
-		0	&	1	&	0	\\
-		-1	&	0	&	0
-	\end{pmatrix}.
-\end{equation}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Symétrie par rapport à une droite}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-Le principe est exactement le même : il faut trouver trois vecteurs $f_1$, $f_2$ et $f_3$ sur lesquels on connaît l'action de la symétrie. Ensuite il faudra exprimer $e_1$, $e_2$ et $e_3$ en termes de $f_1$, $f_2$ et $f_3$.
-
-Le seul problème est de trouver les trois vecteurs $f_i$. Le premier est tout trouvé : c'est n'importe quel vecteur sur la droite. Pour les deux autres, il faut un peu ruser parce qu'il faut impérativement qu'ils soient perpendiculaire à la droite. Pour trouver $f_2$, on peut écrire
-\begin{equation}
-	f_2=\begin{pmatrix}
-		1	\\ 
-		0	\\ 
-		x	
-	\end{pmatrix},
-\end{equation}
-et puis fixer le $x$ pour que le produit scalaire de $f_2$ avec $f_1$ soit nul. S'il n'y a pas moyen (genre si $f_1$ a sa troisième composante nulle), essayer avec $\begin{pmatrix}
-	x	\\ 
-	1	\\ 
-	0	
-\end{pmatrix}$. Une fois que $f_2$ est trouvé (il y a des milliards de choix possibles), trouver $f_3$ est super facile : prendre le produit vectoriel entre $f_1$ et $f_2$.
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{En résumé}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-La marche à suivre est
-
-\begin{enumerate}
-
-	\item
-		Trouver trois vecteurs $f_1$, $f_2$ et $f_3$ sur lesquels on connaît l'action de la symétrie. Typiquement : des vecteurs qui sont sur l'axe ou le plan de symétrie, et puis des perpendiculaires. Pour la perpendiculaire, penser au produit scalaire et au produit vectoriel.
-
-	\item
-		Exprimer la base canonique $e_1$, $e_2$ et $e_3$ en termes de $f_1$, $f_2$, $f_3$.
-
-	\item
-		Trouver $Ae_1$, $Ae_2$ et $Ae_3$ en utilisant leur expression en termes des $f_i$, et le fait que l'on connaisse l'action de $A$ sur les $f_i$.
-
-	\item
-		La matrice s'obtient en mettant les images des $e_i$ en colonnes.
-\end{enumerate}
diff --git a/tex/frido/55_espace_vecto_norme.tex b/tex/frido/55_espace_vecto_norme.tex
deleted file mode 100644
index f0de72244..000000000
--- a/tex/frido/55_espace_vecto_norme.tex
+++ /dev/null
@@ -1,1035 +0,0 @@
-% This is part of Mes notes de mathématique
-% Copyright (c) 2008-2017
-%   Laurent Claessens
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Parties libres, génératrices, bases et dimension}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}[Partie libre]
-    Si \( E\) est un espace vectoriel, une partie \( A\) de \( E\) est \defe{libre}{libre!partie} si pour tout choix d'un nombre fini d'éléments \( \{ u_i \}_{i=1,\ldots, n}\), l'égalité
-    \begin{equation}
-        a_1 u_1+\cdots +a_nu_n=0
-    \end{equation}
-    implique \( a_i=0\) pour tout \( i\) (ici les \( a_i\) sont dans le corps de base).
-
-    Une partie infinie est libre si toute ses parties finies le sont.
-\end{definition}
-
-\begin{remark}
-    Notons que le vecteur nul n'est dans aucune partie libre, ne fut-ce que parce que \( a0=0\) n'implique pas \( a=0\).
-\end{remark}
-
-Si \( A\) est une partie de l'espace vectoriel \( E\) nous notons \( \Span(A)\)\nomenclature[A]{$\Span(A)$}{l'ensemble des combinaisons linéaires finies d'éléments de \( A\)} l'ensemble des combinaisons linéaires finies d'éléments de \( A\). Les coefficients de ces combinaisons linéaires sont dans le corps de base \( \eK\).
-
-\begin{definition}[Partie génératrice]
-    Une partie $B$ d'un espace vectoriel \( E\) est \defe{génératrice}{partie!génératrice} si \( \Span(B)=E\).
-\end{definition}
-
-\begin{remark}
-    Ces définitions demandent des commentaires en dimension infinie\footnote{Nous n'avons pas encore définit le concept de dimension, mais nous nous adressons au lecteur trop pressé.}.
-
-    \begin{enumerate}
-        \item
-    Tout élément peut être écrit comme combinaison linéaire finie d'une partie génératrice. Cela ne signifie pas que nous pouvons extraire une partie finie qui convient pour tous les éléments à la fois. Lorsque l'espace est de dimension infinie, ceci est particulièrement important.
-\item
-    La définition séparée de liberté dans le cas des parties infinies a son importance lorsqu'on parle d'espaces vectoriels de dimension infinies (en dimension finie, aucune partie infinie n'est libre) parce que cela fera une différence entre une base algébrique et une base hilbertienne par exemple.
-    \end{enumerate}
-\end{remark}
-
-\begin{definition}[Base]
-    Une \defe{base}{base} de l'espace vectoriel \( E\) est une partie à la fois génératrice et libre.
-\end{definition}
-
-\begin{proposition}[\cite{MonCerveau}]      \label{PROPooEIQIooXfWDDV}
-    Tout élément non nul d'un espace vectoriel se décompose de façon unique en combinaison linéaire finie d'éléments d'une base.
-\end{proposition}
-
-\begin{proof}
-    Soit un espace vectoriel \( E\) et une base \( \{ e_i \}_{i\in I}\) où \( I\) est un ensemble a priori quelconque. Soit \( v\in E\). Vu que \( E=\Span\{ e_i \}_{i\in I}\), il existe une partie finie \( J\) de \( I\) et des coefficients \( \{ v_j \}_{j\in J}\) dans \( \eK\) tels que
-    \begin{equation}
-        v=\sum_{j\in J}v_je_j.
-    \end{equation}
-    Cela donne l'existence.
-
-    En ce qui concerne l'unicité, soient \( J \) et \( K\) des parties finies de \( I\) et des coefficients \( \{ v_j \}_{j\in J}\) et \( \{ w_{k} \}_{k\in K}\) tels que
-    \begin{equation}
-        v=\sum_{j\in J}v_je_j=\sum_{k\in K}v_{k}e_{k}.
-    \end{equation}
-    Nous posons \( L=J\cup K\) et, pour \( l\in L\),
-    \begin{equation}
-        \alpha_l=\begin{cases}
-            v_l    &   \text{si } l\in J\setminus K\\
-            w_l    &    \text{si } l\in K\setminus J\\
-            v_l-w_l    &    \text{si } l\in K\cap J.
-        \end{cases}
-    \end{equation}
-    Nous avons alors
-    \begin{equation}
-        \sum_{l\in L}\alpha_le_l=0,
-    \end{equation}
-    ce qui implique que \( \alpha_l=0\) pour tout \( l\in L\) parce que la partie \( \{ e_i \}_{i\in I}\) est libre et que \( L\) est finie.
-
-    L'unicité de la décomposition de \( v\) signifie que
-    \begin{equation}
-        \{ j\in J \tq v_j\neq 0 \}=\{ k\in K\tq w_k\neq 0 \}
-    \end{equation}
-    et que pour \( l\) dans cet ensemble, \( v_l=w_l\).
-
-    Soit \( j\in J\); il y a deux possibilités : \( j\in J\setminus K\) et \( j\in J\cap K\). Dans le premier cas nous avons déjà vu que \( \alpha_j=v_j=0\). Dans le second cas, \( \alpha_j=v_j-w_j=0\), c'est à dire \( v_j=w_j\).
-
-    Donc \( j\in J\) vérifiant \( v_j\neq 0\) implique \( j\in J\cap K\) et l'égalité des coefficients. Idem avec \( k\in K\) tel que \( w_k\neq 0\) implique \( k\in J\cap K\).
-\end{proof}
-
-\begin{definition}
-    Un espace vectoriel est \defe{de type fini}{type!fini!espace vectoriel} s'il contient une partie génératrice finie. 
-\end{definition}
-Nous verrons dans les résultats qui suivent que cette définition est en réalité inutile parce qu'une espace vectoriel sera de type fini si et seulement s'il est de dimension finie.
-
-\begin{lemma}       \label{LemytHnlD}
-    Si \( E\) a une famille génératrice de cardinal \( n\), alors toute famille de \( n+1\) éléments est liée.
-\end{lemma}
-
-\begin{proof}
-    Nous procédons par récurrence sur \( n\) -- qui n'est pas exactement la dimension d'un espace vectoriel fixé. Pour \( n=1\), nous avons \( E=\langle e\rangle\) et donc si \( v_1,v_2\in E\) nous avons \( v_1=\lambda_1 e\), \( v_2=\lambda_2e\) et donc \( \lambda_2v_1-\lambda_1v1=0\). Cela prouve le résultat dans le cas de la dimension \( 1\).
-
-    Supposons maintenant que le résultat soit vrai pour \( k<n\), c'est à dire que pour tout espace vectoriel contenant une partie génératrice de cardinal \( k<n\), les parties de \( k+1\) éléments sont liées. Soit maintenant un espace vectoriel muni d'une partie génératrice \( G=\{ e_1,\ldots, e_n \}\) de \( n\) éléments, et montrons que toute partie \( V=\{ v_1,\ldots, v_{n+1} \}\) contenant \( n+1\) éléments est liée. Dans nos notations nous supposons que les \( e_i\) sont des vecteurs distincts et les \( v_i\) également. Nous les supposons également tous non nuls. Étant donné que \( \{ e_i \}\) est génératrice nous pouvons définir les nombres \( \lambda_{ij}\) par
-    \begin{equation}
-        v_i=\sum_{k=1}^n\lambda_{ij}e_j
-    \end{equation}
-    Vu que
-    \begin{equation}
-        v_{n+1}=\sum_{k=1}^n\lambda_{n+1,k}e_k\neq 0,
-    \end{equation}
-    quitte à changer la numérotation des \( e_i\) nous pouvons supposer que \( \lambda_{n+1,n}\neq 0\). Nous considérons les vecteurs
-    \begin{equation}
-        w_i=\lambda_{n+1,n}v_i-\lambda_{i,n}v_{n+1}.
-    \end{equation}
-    En calculant un peu,
-    \begin{subequations}
-        \begin{align}
-            w_i&=\lambda_{n+1,n}\sum_k\lambda_{i,k}e_k-\lambda_{i,n}\sum_k\lambda_{n+1,k}e_k\\
-            &=\sum_{k=1}^{n-1}\big( \lambda_{n+1,n}\lambda_{i,k}-\lambda_{i,n}\lambda_{n+1,} \big)e_k
-        \end{align}
-    \end{subequations}
-    parce que les termes en \( e_n\) se sont simplifiés. Donc la famille \( \{ w_1,\ldots, w_n \}\) est une famille de \( n\) vecteurs dans l'espace vectoriel \( \Span\{ e_1,\ldots, e_{n-1} \}\); elle est donc liée par l'hypothèse de récurrence. Il existe donc des nombres \( \alpha_1,\ldots, \alpha_n\in \eK\) non tous nuls tels que
-    \begin{equation}        \label{EqOQGGoU}
-        0=\sum_{i=1}^n\alpha_iw_i=\sum_{i=1}^n\alpha_i\lambda_{n+1,n}v_i-\left( \sum_{i=1}^n\alpha_i\lambda_{i,n} \right)v_{n+1}.
-    \end{equation}
-    Vu que \( \lambda_{n+1,n}\neq 0\) et que parmi les \( \alpha_i\) au moins un est non nul, nous avons au moins un des produits \( \alpha_i\lambda_{n+1,n}\) qui est non nul. Par conséquent \eqref{EqOQGGoU} est une combinaison linéaire nulle non triviale des vecteurs de \( \{ v_1,\ldots, v_{n+1} \}\). Cette partie est donc liée.
-\end{proof}
-
-\begin{lemma}   \label{LemkUfzHl}
-    Soit \( L\) une partie libre et \( G\) une partie génératrice. Soit \( B\) une partie maximale parmi les parties libres \( L'\) telles que \( L\subset L'\subset G\). Alors \( B\) est une base.
-\end{lemma}
-Qu'entend-on par «maximale» ? La partie \( B\) doit être libre, contenir \( L\), être contenue dans \( G\) et de plus avoir la propriété que \( \forall x\in G\setminus B\), la partie \( B\cup\{ x \}\) est liée.
-
-\begin{proof}
-    D'abord si \( G\) est une base, alors toutes les parties de \( G\) sont libres et le maximum est \( B=G\). Dans ce cas le résultat est évident. Nous supposons donc que \( G\) est liée.
-
-    La partie \( B=\{ b_1,\ldots, b_l \}\) est libre parce qu'on l'a prise parmi les libres. Montrons que \( B\) est génératrice. Soit \( x\in G\setminus B\); par hypothèse de maximalité, \( B\cup\{ x \}\) est liée, c'est à dire qu'il existe des nombres \( \lambda_i\), \( \lambda_x\) non tous nuls tels que
-    \begin{equation}    \label{EqxfkevM}
-        \sum_{i=1}^l\lambda_ib_i+\lambda_xx=0.
-    \end{equation}
-    Si \( \lambda_x=0\) alors un de \( \lambda_i\) doit être non nul et l'équation \eqref{EqxfkevM} devient une combinaison linéaire nulle non triviale des \( b_i\), ce qui est impossible parce que \( B\) est libre. Donc \( \lambda_x\neq 0\) et
-    \begin{equation}
-        x=\frac{1}{ \lambda_x }\sum_{i=1}^l\lambda_ib_i.
-    \end{equation}
-    Donc tous les éléments de \( G\setminus B\) sont des combinaisons linéaires des éléments de \( B\), et par conséquent, \( G\) étant génératrice, tous les éléments de \( E\) sont combinaisons linéaires d'éléments de \( B\). 
-\end{proof}
-
-\begin{theorem}[Théorème de la base incomplète] \label{ThonmnWKs}
-    Soit \( E\) un espace vectoriel de type fini sur le corps \( \eK\).
-    \begin{enumerate}
-        \item   \label{ItemBazxTZ}
-            Si \( L\) est une partie libre et si \( G\) est une partie génératrice contenant \( L\), alors il existe une base \( B\) telle que \( L\subset B\subset G\).
-        \item       \label{ITEMooFVJXooGzzpOu}
-            Toute partie libre peut être étendue en une base.
-        \item
-            Toutes les bases sont finies et ont même cardinal.
-    \end{enumerate}
-\end{theorem}
-\index{théorème!base incomplète}
-
-\begin{proof}
-    Point par point.
-    \begin{enumerate}
-        \item
-    Vu que \( E\) est de type fini, il admet une partie génératrice \( G\) de cardinal fini \( n\). Donc une partie libre est de cardinal au plus \( n\) par le lemme \ref{LemytHnlD}. Soit \( L\), une partie libre contenue dans \( G\) (ça existe : par exemple \( L=\emptyset\)). La partie \( B\) maximalement libre contenue dans \( G\) et contenant \( L\) est une base par le lemme \ref{LemkUfzHl}.
-\item
-Notons que puisque \( E\) lui-même est générateur, le point \ref{ItemBazxTZ} implique que toute partie libre peut être étendue en une base.
-\item
-    Soient \( B\) et \( B'\), deux bases. En particulier \( B\) est génératrice et \( B'\) est libre, donc le lemme \ref{LemytHnlD} indique que \( \Card(B')\leq \Card(B)\). Par symétrie on a l'inégalité inverse. Donc \( \Card(B)=\Card(B')\).
-    \end{enumerate}
-\end{proof}
-
-\begin{remark}      \label{REMooYGJEooEcZQKa}
-    Le théorème de la base incomplete \ref{ThonmnWKs}\ref{ITEMooFVJXooGzzpOu} est ce qui permet de construire une base d'une espace vectoriel en « commençant par» une base d'un sous-espace. En effet si \( H\) est un sous-espace de \( E\) alors une base de \( H\) est une partie libre de \( E\) et donc peut être étendue en une base de \( E\).
-\end{remark}
-
-\begin{definition}
-    La \defe{dimension}{dimension} d'un espace vectoriel de type fini est la cardinal d'une de ses bases.
-\end{definition}
-\index{dimension!définition}
-
-\begin{example}
-    Il existe une infinité de bases de $\eR^m$. On peut démontrer que le cardinal de toute base de $\eR^m$ est $m$, c'est à dire que toute base de $\eR^m$ possède exactement $m$ éléments.
-
-    La base de $\eR^m$ qu'on dit \defe{canonique}{canonique!base}\index{base!canonique de $\eR^m$} (c.à.d. celle qu'on utilise tout le temps) est $\mathcal{B}=\{e_1,\ldots, e_m\}$, où le vecteur $e_j$ est 
-    \begin{equation}\nonumber
-      e_j=
-    \begin{array}{cc}
-      \begin{pmatrix}
-        0\\\vdots\\0\\1\\ 0\\\vdots\\0
-      \end{pmatrix} & 
-      \begin{matrix}
-        \quad\\\quad\\\leftarrow\textrm{j-ème} \quad\\\quad\\\quad\\
-      \end{matrix}
-    \end{array}.
-    \end{equation}
-    La composante numéro $j$ de $e_i$ est $1$ si $i=j$ et $0$ si $i\neq j$. Cela s'écrit $(e_i)_j=\delta_{ij}$ où $\delta$ est le \defe{symbole de Kronecker}{Kronecker} défini par
-    \begin{equation}
-        \delta_{ij}=\begin{cases}
-            1	&	\text{si }i=j\\
-            0	&	 \text{si }i\neq j
-        \end{cases}
-    \end{equation}
-    Les éléments de la base canonique de $\eR^m$ peuvent donc être écrits $e_i=\sum_{k=1}^m\delta_{ik}e_k$.
-\end{example}
-
-Le théorème suivant est essentiellement une reformulation du théorème \ref{ThonmnWKs}.
-\begin{theorem} \label{ThoBaseIncompjblieG}     \label{ThoMGQZooIgrXjy}
-    Soit \( E\) un espace vectoriel de dimension finie et \( \{ e_i \}_{i\in I}\) une partie génératrice de \( E\).
-
-    \begin{enumerate}
-        \item
-            Il existe \( J\subset I\) tel que \( \{ e_i \}_{i\in J}\) est une base. Autrement dit : de toute partie génératrice nous pouvons extraire une base.
-        \item
-            Soit \( \{ f_1,\ldots, f_l \}\) une partie libre. Alors nous pouvons la compléter en utilisant des éléments \( e_i\). C'est à dire qu'il existe \( J\subset I\) tel que \( \{ f_k \}\cup\{ e_i \}_{i\in J}\) soit une base.
-        \item       \label{ItemHIVAooPnTlsBi}
-            Si \( E\) est un espace vectoriel de dimension \( n\), alors toute partie libre contenant \( n\) éléments est une base.
-    \end{enumerate}
-\end{theorem}
-
-\begin{proof}
-    Nous ne prouvons que le troisième point. Une partie libre contenant \( n\) éléments peut être étendue en une base; si ladite extension est non triviale (c'est à dire qu'on ajoute vraiment au moins un élément) une telle base contiendra une partie de \( n+1\) éléments qui serait liée par le lemme \ref{LemytHnlD}.
-\end{proof}
-
-Soit \( F\) un sous-espace vectoriel de l'espace vectoriel \( E\). La \defe{codimension}{codimension} de \( F\) dans \( E\) est
-\begin{equation}
-    \codim_E(F)=\dim(E/F).
-\end{equation}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Applications linéaires}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Définition}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooULVAooXJuRmr}
-    Soient des espaces vectoriels \( E \) et \( F\) sur le corps \( \eK\). Une application \( T\colon E\to F\) est dite \defe{linéaire}{linéaire!application} si
-    \begin{itemize}
-        \item $T(x+y)=T(x)+T(y)$ pour tout $x$ et $y$ dans \( E\),
-        \item $T(\lambda x)=\lambda T(x)$ pour tout $\lambda$ dans $\eK$ et \( x\) dans \( E\).
-    \end{itemize}
-\end{definition}
-Si vous avez bien suivi, les égalités dans la définition \ref{DEFooULVAooXJuRmr} sont des égalités dans \( F\).
-
-
-\begin{example}
-Pour tout $b$ dans $\eR$ la fonction $T_b(x)= bx$ est une application linéaire de $\eR$ dans $\eR$. En effet,
-\begin{itemize}
-\item  $T_b(x+y)= b(x+y)= bx + by = T_b(x)+T_b(y)$,
-\item $T_b(ax)=b(ax)= abx = a T_b(x)$.
-\end{itemize}
-De la même façon on peut montrer que la fonction $T_{\lambda}$ définie par $T_{\lambda}(x)=bx$ est un application linéaire de $\eR^m$ dans $\eR^m$ pour tout $\lambda$ dans $\eR$ et $m$ dans $\eN$.
-\end{example}
-
-\begin{definition}[Quelque ensembles d'applications linéaires]      \label{DEFooOAOGooKuJSup}
-    Soient \( E\) et \( F\) des espaces vectoriels. 
-    \begin{itemize}
-        \item
-        L'ensemble des applications linéaires de \( E\) vers \( F\) est noté $\aL(E,F)$\nomenclature{$\aL(E,F)$}{Ensemble des applications linéaires de $E$ dans $F$}.
-        \item Une application linéaire \( E\to E\) est un \defe{endomorphisme}{endomorphisme} de \( E\). L'ensemble des endomorphisme de \( E\) est noté \( \End(E)\)\nomenclature[B]{$\End(E)$}{les endomorphismes de \( E\)}.
-        \item Un endomorphisme bijectif est un \defe{automorphisme}{automorphisme!d'espace vectoriel}. L'ensemble des automorphismes de \( E\) est noté \( \Aut(E)\)\nomenclature[B]{$\Aut(E)$}{automorphisme de l'espace vectoriel \( E\)}.
-        \item
-            Une application linéaire bijective \( E\to F\) est un \defe{isomorphisme}{isomorphisme!espaces vectoriels} d'espace vectoriel. L'ensemble des isomorphismes \( E\to F\) est noté \( \GL(E,F)\).
-    \end{itemize}
-\end{definition}
-
-\begin{remark}
-    Les ensembles définis en \ref{DEFooOAOGooKuJSup} concernent la structure d'espace vectoriel seulement. Lorsque nous verrons la notion d'espace vectoriel normé,nous demanderons de plus la continuité, laquelle n'est pas automatique en dimension infinie. Voir aussi les définitions \ref{DEFooTLQUooJvknvi}.
-\end{remark}
-
-\begin{definition}  \label{DefDQRooVGbzSm}
-    Si \( V\) et \( W\) sont des espaces vectoriels nous munissons \( \aL(V,W)\) d'une structure d'espace vectoriel en définissant la somme et le produit par un scalaire de la façon suivante. Si $T$ et $U$ sont des élément de $\aL(V,W)$ et si $\lambda$ est un réel, nous définissons les éléments $T+U$ et $\lambda T$ par
-    \begin{enumerate}
-        \item
-            $(T+U)(x)=T(x)+U(x)$;
-        \item
-            $(\lambda T)(x)=\lambda T(x)$
-    \end{enumerate}
-    pour tout \( x\in V\).
-\end{definition}
-
-\begin{normaltext}
-    Pour bien faire, il faudrait vérifier que la définition \ref{DefDQRooVGbzSm} produit effectivement une structure d'espace vectoriel sur \( \aL(V,W)\).
-\end{normaltext}
-
-\begin{proposition}
-    Si \( E\) et \( F\) sont des espaces vectoriels de dimension \( n\) et si \( \{ e_i \}_{i=1,\ldots, n}\) et \( \{ f_i \}_{i=1,\ldots, n}\) sont des bases respectivement de \( F\) et \( F\), alors il existe une unique application linéaire \( A\colon E\to F\) telle que \( E(e_i)=f_i\) pour tout \( i\).
-\end{proposition}
-
-\begin{proof}
-    En deux parties.\begin{subproof}
-        \item[Existence]
-            Soit \( v\in E\). Vu que \( \{ e_i \}\) est une base, il se décompose de façon unique en \( v=\sum_iv_ie_i\). Alors définir
-            \begin{equation}
-                A(v)=\sum_iv_if_i
-            \end{equation}
-            est une bonne définition et satisfait aux exigences.
-        \item[Unicité]
-            Soient \( A\) et \( B\) satisfaisant aux exigences. Alors pour tout \( i\) nous avons \( A(e_i)=B(e_i)\). Si \( v\in E\) s'écrit de la forme \( v=\sum_iv_ie_i\) alors la linéarité impose \( A(v)=\sum_iv_iA(e_i)=\sum_iv_iB(e_i)=B(v)\). Donc \( A=B\).
-    \end{subproof}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Rang}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}\label{DefALUAooSPcmyK}
-    Si  $T\colon V\to W$ est une application linéaire, son \defe{rang}{rang} est la dimension de son image. 
-\end{definition}
-
-Le théorème suivant est valable également en dimension infinie; ce sera une des rares incursions en dimension infinie de ce chapitre.
-\begin{theorem}[Théorème du rang]       \label{ThoGkkffA}
-    Soient \( E\) et \( F\) deux espaces vectoriels (de dimensions finies ou non) et soit \( f\colon E\to F\) une application linéaire. Alors le rang\footnote{Définition \ref{DefALUAooSPcmyK}.} de \( f\) est égal à la codimension du noyau, c'est à dire
-       \begin{equation}
-           \rang(f)+\dim\ker f=\dim E.
-       \end{equation}
-
-       Dans le cas de dimension infinie afin d'éviter les problèmes d'arithmétique avec l'infini nous énonçons le théorème en disant que si \( (u_s)_{s\in S}\) est une base de \( \ker(f)\) et si \( \big( f(v_t) \big)_{t\in T}\) est une base de \( \Image(f)\) alors  \( (u_s)_{s\in s}\cup (v_t)_{t\in T}\) est une base de \( E\).
-\end{theorem}
-\index{théorème!du rang}
-
-\begin{proof}
-    Nous devons montrer que 
-    \begin{equation}
-          (u_s)_{s\in S}\cup (v_t)_{t\in T}
-    \end{equation}
-    est libre et générateur.
-
-    Soit \( x\in E\). Nous définissons les nombres \( x_t\) par la décomposition de \( f(x)\) dans la base \( \big( f(v_t) \big)\) :
-    \begin{equation}
-        f(x)=\sum_{t\in T}x_tf(v_t).
-    \end{equation}
-    Ensuite le vecteur \( x=\sum_tx_tv_t\) est dans le noyau de \( f\), par conséquent nous le décomposons dans la base \( (u_s)\) :
-    \begin{equation}
-        x-\sum_tx_tv_t=\sum_s\in S x_su_s.
-    \end{equation}
-    Par conséquent
-    \begin{equation}
-        x=\sum_sx_su_s+\sum_tx_tv_t.
-    \end{equation}
-    
-    En ce qui concerne la liberté nous écrivons
-    \begin{equation}
-        \sum_tx_tv_t+\sum_sx_su_s=0.
-    \end{equation}
-    En appliquant \( f\) nous trouvons que 
-    \begin{equation}
-        \sum_tx_tf(v_t)=0
-    \end{equation}
-    et donc que les \( x_t\) doivent être nuls. Nous restons avec \( \sum_sx_su_s=0\) qui à son tour implique que \( x_s=0\).
-\end{proof}
-Un exemple d'utilisation de ce théorème en dimension infinie sera donné dans le cadre du théorème de Fréchet-Riesz, théorème \ref{ThoQgTovL}.
-
-\begin{corollary}       \label{CORooCCXHooALmxKk}
-    Si \( E\) est un espace vectoriel de dimension finie, alors un endomorphisme de \( E\) est bijectif si et seulement si il est injectif si et seulement si il est surjectif.
-\end{corollary}
-
-\begin{proof}
-    Si un endomorphisme \( f\colon E\to E\) est surjectif, alors \( \rang(f)=\dim(E)\), ce qui donne, par le théorème du rang \ref{ThoGkkffA}, \( \dim\big( \ker(f) \big)=0\), c'est à dire que \( f\) est injectif.
-
-    De la même façon, si \( f\) est injective, alors \( \dim\big( \ker(f) \big)=0\), ce qui donne \( \rang(f)=\dim(E)\) ou encore que \( f\) est surjective.
-\end{proof}
-
-Une conséquence du théorème du rang est que les endomorphismes ont un inverse à gauche et à droite égaux (lorsqu'ils existent).
-\begin{corollary}
-    Soit un endomorphisme \( f\) d'un espace vectoriel de dimension finie. Si \( f\) admet un inverse à gauche, alors
-    \begin{enumerate}
-        \item
-            \( f\) est bijective,
-        \item
-            \( f\) admet également un inverse à droite,
-        \item
-            ils sont égaux.
-    \end{enumerate}
-    Tout cela tient également en remplaçant «gauche» par «droite».
-\end{corollary}
-
-\begin{proof}
-    Soit \( g\), un inverse à gauche de \( f\) : \( gf=\id\). Cela implique que \( f\) est injective et que \( g\) est surjective, et donc qu'elles sont toutes deux bijectives par le corollaire \ref{CORooCCXHooALmxKk}. Vu que \( f\) est bijective, elle admet également un inverse à droite, soit \( h\). Nous avons : \( gf=\id\) et \( fh=\id\).
-
-    Alors \( gfh=h\) parce que \( gf=\id\), mais également \( gfh=g\) parce que \( fh=\id\). Donc \( g=h\).
-\end{proof}
-C'est ce corollaire qui nous permet d'écrire \( f^{-1}\) sans plus de précisions dès que \( f\) est une bijection.
-
-\begin{example}[Ne fonctionne pas en dimension infinie]
-    Tout cela ne fonctionne pas en dimension infinie. Par exemple avec une base \( \{ e_k \}_{k\in \eN}\) nous pouvons considérer l'opérateur
-    \begin{equation}
-        f(e_k)=e_{k+1}.
-    \end{equation}
-    Il est injectif, mais pas surjectif. Si on pose
-    \begin{equation}
-        g(e_k)=\begin{cases}
-            e_{k-1}    &   \text{si } k\geq 1\\
-            0    &    \text{si } k=0
-        \end{cases}
-    \end{equation}
-    alors nous avons \( gf=\id\), mais pas \( fg=\id\) parce que ce \( (fg)(e_0)=0\).
-\end{example}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Matrices équivalentes et similaires}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}  \label{DefBLELooTvlHoB}
-    Deux relations d'équivalence entre les matrices.
-    \begin{enumerate}
-        \item   \label{ItemPFXCooOUbSCt}
-    Deux matrices \( A\) et \( B\) sont \defe{équivalentes}{matrice!équivalence} dans \( \eM(n,\eK)\) s'il existe \( P,Q\in\GL(n,\eK)\) telles que \( A=PBQ^{-1}\). 
-\item
-    Deux matrices sont \defe{semblables}{matrices!similitude} s'il existe une matrice \( P\in \GL(n,\eK)\) telle que \( A=PBP^{-1}\).
-    \end{enumerate}
-\end{definition}
-
-\begin{lemma}   \label{LemZMxxnfM}
-    Une matrice de rang \( r\) dans \( \eM(n,\eK)\) est équivalente à la matrice par blocs
-    \begin{equation}
-        J_r=\begin{pmatrix}
-            \mtu_r    &   0    \\ 
-            0    &   0    
-        \end{pmatrix}.
-    \end{equation}
-\end{lemma}
-\index{rang!classe d'équivalence}
-
-\begin{proof}
-    Nous devons prouver que pour toute matrice \( A\in\eM(n,\eK)\) de rang \( r\), il existe \( P,Q\in\GL(n,\eK)\) telles que \(QAP=J_r\). Soit \( \{ e_i \}\) la base canonique de \( \eK^n\), puis \( \{ f_i \}\) une base telle que \( Af_i=0\) dès que \( i>r\).
-
-    Nous considérons la matrice inversible \( P\) telle que \( Pe_i=f_i\). Elle vérifie
-    \begin{equation}
-        APe_i=Af_i=\begin{cases}
-            0    &   \text{si } i>r\\
-            \neq 0    &    \text{sinon}.
-        \end{cases}
-    \end{equation}
-    La matrice \( AP\) se présente donc sous la forme
-    \begin{equation}
-        AP=\begin{pmatrix}
-            M    &   0    \\ 
-            *    &   0    
-        \end{pmatrix}
-    \end{equation}
-    où \( M\) est une matrice \( r\times r\). Nous considérons maintenant une base \( \{ g_i \}_{i=1,\ldots, n}\) dont les \( r\) premiers éléments sont les \( r\) premières colonnes de \( AP\) et une matrice inversible \( Q\) telle que \( Qg_i=e_i\). Alors
-    \begin{equation}
-        QAPe_i=\begin{cases}
-            e_i    &   \text{si } i<r\\
-            0    &    \text{sinon}.
-        \end{cases}.
-    \end{equation}
-    Cela signifie que \( QAP\) est la matrice \( J_r\).
-\end{proof}
-
-\begin{corollary}[Équivalence et rang]      \label{CorGOUYooErfOIe}
-    Deux matrices sont équivalentes\footnote{Définition \ref{DefBLELooTvlHoB}\ref{ItemPFXCooOUbSCt}.} si et seulement si elles ont même rang.
-\end{corollary}
-
-\begin{proof}
-    D'abord il y a des implicites dans l'énoncé. Vu que nous voulons soit pas hypothèse soit par conclusion que les matrices $A$ et \( B\) soient équivalentes, nous supposons qu'elles ont même dimension. Soient donc \( A\) et \( B\) deux matrices carré représentant des applications linéaires de l'espace vectoriel \( E=\eK^n\) vers lui-même.
-
-    Par le lemme \ref{LemZMxxnfM}, deux matrices de même rang \( r\) sont équivalentes à \( J_r\). Elles sont donc équivalentes entre elles.
-
-    Inversement, supposons que \( A\) et \( B\) soient deux matrices équivalentes : \( A=PBQ^{-1}\) avec \( P\) et \( B\) inversibles. Alors
-    \begin{subequations}
-        \begin{align}
-            \Image(PBQ^{-1})&=\{ PBQ^{-1}v\tq v\in E \}\\
-            &=PB\underbrace{\{ Q^{-1}v\tq v\in E \}}_{=E}\\
-            &=P\big( B(E) \big).
-        \end{align}
-    \end{subequations}
-    L'ensemble \( B(E)\) est un sous-espace vectoriel de \( E\). Vu que le rang de \( P\) est maximum, la dimension de \( P\big( B(E) \big)\) est la même que celle de \( B(E)\). Par conséquent
-    \begin{equation}
-        \dim\Big( \Image(PBQ^{-1}) \Big)=\dim\big( B(E) \big)=\rang(B).
-    \end{equation}
-    Le membre de gauche de cela n'est autre que \( \rang(A)=\dim\big( \Image(PBQ^{-1}) \big)\).
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Matrice d'une application linéaire}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Écriture dans une base}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{example}\label{ex_affine}
-	Soit $m=n$. On fixe $\lambda$ dans $\eR$ et $v$ dans $\eR^m$. L'application $U_{\lambda}$ de $\eR^m$ dans $\eR^m$ définie par $U_{\lambda}(x)=\lambda x+v$ n'est pas une application linéaire, parce que 
-\[
-U_{\lambda}(ax)=\lambda(ax)+v\neq \lambda(bx+v)=a U_{\lambda}(x).
-\]
-\end{example}
-
-\begin{example}\label{exampleT_A}
-	Soit $A$ une matrice fixée de $\mathcal{M}_{n\times m}$\nomenclature{$\mathcal{M}_{n\times m}$}{l'ensemble des matrices $n\times m$}. La fonction $T_A\colon \eR^m\to \eR^n$ définie par $T_A(x)=Ax$ est une application linéaire. En effet, 
-\begin{itemize}
-\item  $T_A(x+y)= A(x+y)= Ax + Ay = T_A(x)+T_A(y)$,
-\item $T_A(ax)=A(ax)= a(Ax) = a T_A(x)$.
-\end{itemize}
-\end{example}
-
-On peut observer que, si on identifie $\mathcal{M}_{1\times 1}$ et $\eR$, on obtient le premier exemple comme cas particulier.
-
-\begin{proposition}     \label{PROPooCESFooGOZBNI}
- Toute application linéaire $T$ de $\eR^m$ dans $\eR^n$ s'écrit de manière unique par rapport aux bases canoniques de $\eR^m$ et $\eR^n$ sous la forme
-\[
-T(x)=Ax,
-\]
-avec $A$ dans $\mathcal{M}_{n\times m}$.
-\end{proposition}
-
-\begin{proof}
-  Soit $x$ un vecteur dans $\eR^m$. On peut écrire $x$ sous la forme $ x=\sum_{i=1}^{m}x_i e_i$. Comme $T$ est une application linéaire on a
-\[
-T(x)=\sum_{i=1}^{m}x_iT(e_i).
-\]
-Les images de la base de $\eR^m$, $T(e_j), \, j=1,\ldots,m$, sont des éléments de $\eR^n$, donc on peut les écrire sous la forme de vecteurs
-\[
-T(e_i)=
-\begin{pmatrix}
-  a_{1i}\\
-\vdots\\
-a_{ni}
-\end{pmatrix}.
-\] 
-On obtient alors
-\[
-T(x)=\sum_{i=1}^{m}x_iT(e_i)=\sum_{i=1}^{m}x_i\begin{pmatrix}
-  a_{1i}\\
-\vdots\\
-a_{ni}
-\end{pmatrix}=
-\begin{pmatrix}
-  a_{11} \ldots a_{1m}\\
-\vdots \ddots \vdots\\
- a_{n1} \ldots a_{nm}\\
-\end{pmatrix}
-\begin{pmatrix}
-  x_1\\
-\vdots\\
-x_m
-\end{pmatrix}=Ax.
-\]
-\end{proof}
-
-\begin{definition}
-  Une application $S: \eR^m\to\eR^n$ est dite \defe{affine}{affine (application)} si elle est la somme d'une application linéaire et d'une application constante. Autrement dit, $S$ est affine s'il existe $T: \eR^m\to\eR^n$, linéaire, telle que $S(x)-T(x)$ soit un vecteur constant dans $\eR^n$. 
-\end{definition}
-
-\begin{example}
-	Les exemples les plus courants d'applications affines sont les droites et les plans ne passant pas par l'origine.
-	\begin{description}
-		\item[Les droites] Une droite dans $\eR^2$ (ou $\eR^3$) qui ne passe pas par l'origine est le graphe d'une fonction de la forme $s(x)=ax+b$ (ou $s(t)=u x +v$, avec $u$ et $v$  dans $\eR^2$). On reconnait ici la fonction de l'exemple \ref{ex_affine}.
-			
-		\item[Les plans]
-			De la même façon nous savons que tout plan qui ne passe pas par l'origine dans $\eR^3$ est le graphe d'une application affine, $P(x,y)= (a,b)^T\cdot(x,y)^T+(c,d)^T$.
-	\end{description}
-\end{example}
-
-\begin{lemma}[\cite{ooEPEFooQiPESf}]        \label{LEMooDAACooElDsYb}
-    Soit une application linéaire \( f\colon E\to F\).
-    \begin{enumerate}
-        \item       \label{ITEMooEZEWooZGoqsZ}
-            L'application \( f\) est injective si et seulement s'il existe \( g\colon F\to E\) telle que \( g\circ f=\id|_E\).
-        \item
-            L'application \( f\) est surjective si et seulement s'il existe \( g\colon F\to E\) telle que \( f\circ g=\id|_F\).
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Nous démontrons séparément les deux affirmations.
-    \begin{enumerate}
-        \item
-            Si \( f\) est injective, alors \( f\colon E\to \Image(f)\) est un isomorphisme. Si $V$ est un supplémentaire de \( \Image(f)\) dans \( F\) (c'est à dire \( F=\Image(f)\oplus V\)) alors nous pouvons poser \( g(x+v)=f^{-1}(x)\) où \( x+v\) est la décomposition (unique) d'un élément de \( F\) en \( x\in\Image(f)\) et \( v\in V\). Avec cela nous avons bien \( g\circ f=\id\).
-
-            Inversement, s'il existe \( g\colon F\to E\) telle que \( g\circ f=\id\) alors \( f\colon E\to E\) doit être injective. Parce que si \( f(x)=0\) avec \( x\neq 0\) alors \( (g\circ f)(x)=0\neq x\).
-        \item
-            Si \( f\) est surjective nous pouvons choisir des éléments \( x_1,\ldots, x_p\) dans \( E\) tels que \( \{ f(x_i) \}\) soit une base de \( F\). Ensuite nous définissons
-            \begin{equation}
-                \begin{aligned}
-                    g\colon F&\to E \\
-                    \sum_ka_kf(x_k)&\mapsto \sum_kx_k. 
-                \end{aligned}
-            \end{equation}
-            Cela donne \(  f\circ g=\id|_F\) parce que si \( v\in F\) alors \( v=\sum_kv_kf(x_k)\) avec \( v_k\in \eK\), et nous avons
-            \begin{equation}
-                (f\circ g)(v)=\sum_kv_k(f\circ g)f(x_k)=f\left( \sum_kv_kx_k \right)=\sum_kf(x_k)=v.
-            \end{equation}
-            
-            Inversement, s'il existe \( g\colon F\to E\) tel que \( f\circ g=\id\) alors \( f\) soit être surjective parce que 
-            \begin{equation}
-                F=\Image(f\circ g)=f\big( \Image(g) \big)\subset \Image(f).
-            \end{equation}
-    \end{enumerate}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Changement de base : vecteurs de base}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit un espace vectoriel \( V\) muni d'une base \( \{ e_i \}\) et une application linéaire inversible \( Q\colon V\to V\) dont nous notons \( (Q_{ij})\) les coordonnées dans la base \( \{ e_i \}\). Nous considérons aussi la base \( e'_i=Qe_i\).
-
-Nous exprimons les vecteurs \( e'_i\) en termes des vecteurs \( e_i\) et inversement de la façon suivante. D'abord
-\begin{equation}
-        e'_i=Qe_i=\sum_{kl}Q_{kl}(e_i)_ke_l=\sum_lQ_{il}e_l
-\end{equation}
-donc
-\begin{equation}        \label{EQooQECYooVWEoLx}
-    e'_i=\sum_kQ_{ik}e_k.
-\end{equation}
-
-\begin{normaltext}      \label{TEXTooOBOMooOVqUVG}
-    Afin de ne pas se tromper dans les indices, il est bon d'utiliser systématiquement les noms ``$ijk\ldots$'' pour les indices de la base \( \{ e_i \}\) et les noms ``\( \alpha\beta\gamma\cdots\)'' pour les indices de la base \( \{ e'_{\alpha} \}\). Nous écrivons donc
-    \begin{equation}        \label{EQooZQBZooTzLkLF}
-        e'_{\alpha}=\sum_iQ_{i\alpha}e_i.
-    \end{equation}
-\end{normaltext}
-
-Pour trouver la transformation inverse, nous multiplions par \( (Q^{-1})_{\alpha j}\) et nous sommons sur \( \alpha\) :
-\begin{equation}
-    \sum_{\alpha}(Q^{-1})_{\alpha j}e'_{\alpha}=\sum_{i \alpha}Q_{i\alpha}Q^{-1}_{\alpha j}e_i=e_j
-\end{equation}
-donc
-\begin{equation}        \label{EQooEZBXooGxpJiO}
-    e_j=\sum_{\alpha}(Q^{-1})_{\alpha j}e'_{\alpha}.
-\end{equation}
-
-Quelque remarques :
-\begin{enumerate}
-    \item
-        Notons que nous ne pouvons pas écrire formellement quelque chose comme \( f=Qe\). 
-    \item
-        La matrice \( Q\) vient toujours avec les indices de type \( Q_{i\alpha}\) (latin en premier et grec en second) tandis que la matrice de l'application inverse \( Q^{-1}\) vient toujours avec les indices de type \( (Q^{-1})_{\alpha i}\).
-
-        Ceci est un puissant outil pour ne pas se tromper lors des manipulations d'indices et en particulier lorsque nous renommons des indices.
-\end{enumerate}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Changement de base : coordonnées}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Un vecteur \( x\) peut s'écrire des façons suivantes dans les deux bases considérées :
-\begin{equation}
-    x=\sum_ix_ie_i=\sum_{\alpha}x'_{\alpha}e'_{\alpha}.
-\end{equation}
-En remplaçant \( e_i\) par sa valeur \eqref{EQooEZBXooGxpJiO} nous obtenons l'égalité
-\begin{equation}
-    \sum_{k\alpha}x_k(Q^{-1})_{\alpha k}e'_{\alpha}=\sum_{\alpha}x'_{\alpha}e'_{\alpha}
-\end{equation}
-et par unicité de la décomposition d'un vecteur dans une base nous déduisons
-\begin{equation}
-    x'_{\alpha}=\sum_{k}(Q^{-1})_{\alpha k}x_k.
-\end{equation}
-De même en replaçant \( e'_{\alpha}\) par sa valeur \eqref{EQooZQBZooTzLkLF} nous trouvons
-\begin{equation}        \label{EQooOEHJooCDKUcy}
-    x_i=\sum_{\alpha}Q_{i\alpha}x'_{\alpha}.
-\end{equation}
-
-Attention : il n'est pas question d'écrire quelque chose comme \( x=Qx'\) parce qu'il n'y a pas de vecteur \( x'\).
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Changement de base : matrice d'une application linéaire}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit une application linéaire \( f\colon V\to V\) de matrice \( A\) dans la base \( \{ e_i \}\) et de matrice \( A'\) dans la base \( \{ e'_{\alpha} \}\). Notre but est maintenant d'exprimer \( A'\) en termes de \( A\) et de \( Q\).
-
-Nous avons l'égalité
-\begin{equation}
-    f(x)=\sum_{kl}A_{kl}x_le_k=\sum_{\alpha\beta}A'_{\alpha\beta}x'_{\beta}e'_{\alpha}.
-\end{equation}
-Il suffit d'y substituer les valeurs \eqref{EQooOEHJooCDKUcy} et \eqref{EQooEZBXooGxpJiO} :
-\begin{equation}
-    f(x)=\sum_{kl}A_{kl}\sum_{\beta}Q_{l\beta}x'_{\beta}\sum_{\alpha}(Q^{-1})_{\alpha k}e'_{\alpha}=\sum_{\alpha\beta}(Q^{-1}AQ)_{\alpha\beta}x'_{\beta}e'_{\alpha}.
-\end{equation}
-Cela est égal à \( \sum_{\alpha\beta}A'_{\alpha\beta}x'_{\beta}e'_{\alpha}\) pour tout \( x\). En tenant encore compte de l'unicité de la décomposition du vecteur \( f(x)\) dans la base \( \{ e'_{\alpha} \}\) nous déduisons
-\begin{equation}        \label{EQooWWOUooZBeTBe}
-    A'=Q^{-1} AQ.
-\end{equation}
-
-\begin{remark}
-    L'égalité \eqref{EQooWWOUooZBeTBe} est une égalité de matrices, et non une égalité d'applications linéaire. Il n'est pas question d'écrire quelque chose comme \( f'=Q^{-1}\circ f\circ Q\).
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Changement de base : matrice d'une forme bilinéaire}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit une forme bilinéaire \( q\colon V\times V\to \eR\) de matrice \( (q_{ij})\) dans la base \( \{ e_i \}\) et de matrice \( q'_{\alpha\beta}\) dans la base \( \{ e'_{\alpha} \}\). Nous notons le changement de variables \(e'_{\alpha}=\sum_{i}Q_{i\alpha}e_i\). Nous avons l'égalité
-\begin{equation}
-    q(x,y)=\sum_{ij}q_{ij}x_iy_j=\sum_{\alpha\beta}q'_{\alpha\beta}x'_{\alpha}y'_{\beta}.
-\end{equation}
-En remplaçant \( x_i\) et \( y_j\) par leurs valeurs en termes des \( x'_{\alpha}\) et \( y'_{\beta}\) nous trouvons
-\begin{equation}
-    q(x,y)=\sum_{ij\alpha\beta}q_{ij}Q_{i\alpha}x'_{\alpha}Q_{j\beta}y'_{\beta},
-\end{equation}
-c'est à dire
-\begin{equation}        \label{EQooQLLFooToyvoH}
-    q'=Q^tqQ.
-\end{equation}
-Encore une fois, cette égalité est une égalité de matrices, et non de formes bilinéaires, et encore moins d'applications linéaires. 
-\begin{remark}
-    La «loi de transformation» \eqref{EQooQLLFooToyvoH} n'est pas la même que la loi de transformation \eqref{EQooWWOUooZBeTBe}.
-\end{remark}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Dualité}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{proposition} \label{PropEJBZooTNFPRj}
-    Si $A$ est la matrice d'une application linéaire, alors le rang de cette application linéaire est égal à la taille de la plus grande matrice carré de déterminant non nul contenue dans $A$.
-\end{proposition}
-
-\begin{definition}  \label{DefJPGSHpn}
-    Si \( E\) est un espace vectoriel sur \( \eK\), le \defe{dual algébrique}{dual!algébrique} de \( E\) est l'ensemble des formes linéaires sur \( E\). Le \defe{dual topologique}{dual!topologique} est l'ensemble des formes linéaires continues de \( E\) vers \( \eK\).
-\end{definition}
-
-En dimension infinie, ces deux notions ne coïncident pas, voir la proposition \ref{PROPooQZYVooYJVlBd} et la remarque \ref{RemOAXNooSMTDuN}.
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Orthogonal}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit \( E\), un espace vectoriel, et \( F\) une sous-espace de \( E\). L'\defe{orthogonal}{orthogonal!sous-espace} de \( F\) est la partie \( F^{\perp}\subset E^*\) donnée par
-\begin{equation}    \label{Eqiiyple}
-    F^{\perp}=\{ \alpha\in E^*\tq \forall x\in F,\alpha(x)=0 \}.
-\end{equation}
-Cette définition d'orthogonal via le dual n'est pas du pur snobisme. En effet, la définition «usuelle» qui ne parle pas de dual,
-\begin{equation}
-    F^{\perp}=\{ y\in E\tq \forall x\in F,y\cdot x=0 \},
-\end{equation}
-demande la donnée d'un produit scalaire. Évidemment dans le cas de \( \eR^n\) munie du produit scalaire usuel et de l'identification usuelle entre \( \eR^n\) et \( (\eR^n)^*\) via une base, les deux notions d'orthogonal coïncident.
-
-Si \( B\subset E^*\), on note \( B^o\)\nomenclature[G]{\( B^o\)}{orthogonal dans le dual} son orthogonal :
-\begin{equation}
-    B^o=\{ x\in E\tq \omega(x)=0\,\forall \omega\in B \}.
-\end{equation}
-Notons qu'on le note \( B^o\) et non \( B^{\perp}\) parce qu'on veut un peu s'abstraire du fait que \( (E^*)^*=E\). Du coup on impose que \( B\) soit dans un dual et on prend une notation précise pour dire qu'on remonte au pré-dual et non qu'on va au dual du dual.
-
-La définition \eqref{Eqiiyple} est intrinsèque : elle ne dépend que de la structure d'espace vectoriel.
-
-\begin{proposition} \label{PropXrTDIi}
-    Soit \( E\) un espace vectoriel, et \( F\) un sous-espace vectoriel. Alors nous avons
-    \begin{equation}
-        \dim F+\dim F^{\perp}=\dim E.
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Soit \( \{ e_1,\ldots, e_p \}\) une base de \( F\) que nous complétons en une base \( \{ e_1,\ldots, e_n \}\) de \( E\) par le théorème \ref{ThonmnWKs}. Soit \( \{ e_1^*,\ldots, e^*_n \}\) la base duale. Alors nous prouvons que \( \{ e^*_{p+1},\ldots, e_n^* \}\) est une base de \( F^{\perp}\). 
-    
-    Déjà c'est une partie libre en tant que partie d'une base.
-
-    Ensuite ce sont des éléments de \( F^{perp}\) parce que si \( i\leq p\) et si \( k\geq 1\) nous avons \( e^*_{p+k}(e_i^*)=0\); donc oui, \( e^*_{p+k}\in F^{\perp}\).
-
-    Enfin \( F^{\perp}\subset\Span\{ e_{p+1}^*,e_n^* \}\) parce que si \( \omega=\sum_{k=1}^n\omega_ke_k^*\), alors \( \omega(e_i)=\omega_i\), mais nous savons que si \( \omega\in F^{\perp}\), alors \( \omega(e_i)=0\) pour \( i\leq p\). Donc \( \omega=\sum_{i=p+1}^n\omega_ke^*_k\).
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Transposée}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DefooZLPAooKTITdd}
-    Si \( f\colon E\to F\) est une application linéaire entre deux espaces vectoriels, la \defe{transposée}{transposée} est l'application \( f^t\colon F^*\to E^*\) donnée par
-    \begin{equation}
-        f^t(\omega)(x)=\omega\big( f(x) \big).
-    \end{equation}
-    pour tout \( \omega\in F^*\) et \( x\in E\).
-\end{definition}
-
-\begin{lemma}
-    Soit \( E\) muni de la base \( \{ e_i \}\) et \( F\) muni de la base \( \{ g_i \}\) et une application \( f\colon E\to F\). Si \( A\) est la matrice de \( f\) dans ces bases, alors \( A^t\) est la matrice de \( f^t\) dans les bases \( \{ e^*_i \}\) et \( \{ g^*_i \}\) de \( E^*\) et \( F^*\).
-\end{lemma}
-
-\begin{proof}
-    Nous allons montrer que les formes \( f^t(g^*_i)\) et \( \sum_k(A^t)_{ik}g^*_k\) sont égales en les appliquant à un vecteur.
-
-    Par définition de la matrice d'une application linéaire dans une base,
-    \begin{equation}
-        f^t(g_i^*)=\sum_j(f^t)_{ij}e^*j,
-    \end{equation}
-    et
-    \begin{equation}
-        f(e_k)=\sum_lA_{klg_l}.
-    \end{equation}
-    Du coup, si \( x=\sum_kx_ke_k\), nous avons
-    \begin{equation}    \label{EqCzwftH}
-        f^t(g_i^*)x=\sum_{kl}x_kg_i^*A_{kl}g_l=\sum_{kl}x_kA_{kl}\delta_{il}=\sum_k x_kA_{ki}=\sum_k(A^t)_{ik}x_k.
-    \end{equation}
-    D'autre part, 
-    \begin{equation}    \label{EqWlQlrR}
-        \sum_k(A^t)_{ik}g_k^*x=\sum_{kl}(A^t)_{ik}g^*_kx_le_l=\sum_k(A^t)_{ik}x_k.
-    \end{equation}
-    Le fait que \eqref{EqCzwftH} et \eqref{EqWlQlrR} donnent le même résultat prouve le lemme.
-\end{proof}
-En corollaire, les rangs de \( f\) et de \( f^t\) sont égaux parce que le rang est donné par la plus grande matrice carré de déterminant non nul. Nous prouvons cependant ce résultat de façon plus intrinsèque.
-
-\begin{lemma}[\href{http://gilles.dubois10.free.fr/algebre_lineaire/dualite.html}{Gilles Dubois}]   \label{LemSEpTcW}
-    Si \( f\colon E\to F\) est une application linéaire, alors
-    \begin{equation}
-        \rang(f)=\rang(f^t).
-    \end{equation}
-\end{lemma}
-
-\begin{proof}
-    Nous posons \( \dim\ker(f)=p\) et donc \( \rang(f)=n-p\). Soit \( \{ e_1,\ldots, e_p \}\) une base de \( \ker(f)\) que l'on complète en une base \( \{ e_1,\ldots, e_n \}\) de \( E\). Nous considérons maintenant les vecteurs
-    \begin{equation}
-        g_i=f(e_{p+i})
-    \end{equation}
-    pour \( i=1,\ldots, n-p\). C'est à dire que les \( g_i\) sont les images des vecteurs qui ne sont pas dans le noyau de \( f\). Prouvons qu'ils forment une famille libre. Si
-    \begin{equation}
-        \sum_{k=1}^{n-p}a_kf(e_{p+k})=0,
-    \end{equation}
-    alors \( f\big( \sum_ka_ke_{p+k} \big)=0\), ce qui signifierait que \( \sum_ka_ke_{p+k}\) se trouve dans le noyau de \( f\), ce qui est impossible par construction de la base \( \{ e_i \}_{i=1,\ldots, n}\). Étant donné que les vecteurs \( g_1,\ldots, g_{n-p}\) sont libres, nous les complétons en une base
-    \begin{equation}
-        \{ \underbrace{g_1,\ldots, g_{n-p}}_{\text{images}},\underbrace{g_{n-p+1},\ldots, g_r}_{\text{complétion}} \}
-    \end{equation}
-    de \( F\).
-
-    Nous prouvons maintenant que \( \rang(f^t)\geq n-p\) en montrant que les formes \( \{ g_i^* \}_{i=1,\ldots, n-p}\) est libre (et donc l'espace image de \( f^f\) est au moins de dimension \( n-p\)). Pour cela nous prouvons que \( f^t(g_i^*)=e^*_{i+p}\). En effet
-    \begin{equation}
-        f^t(g^*_i)e_k=g_i^*(fe_k),
-    \end{equation}
-    Si \( k=1,\ldots, p\), alors \( fe_k=0\) et donc \( g_i^*(fe_k)=0\); si \( k=p+l\) alors
-    \begin{equation}
-        f^t(g_i^*)e_k=g_i^*(fe_{k+l})=g^*_i(g_l)=\delta_{i,l}=\delta_{i,k-p}=\delta_{k,i+p}.
-    \end{equation}
-    Donc \( f^t(g_i^*)=e^*_{i+p}\). Cela prouve que les formes \( f^t(g_i^*)\) sont libres et donc que
-    \begin{equation}
-        \rang(f^t)\geq n-p=\rang(f).
-    \end{equation}
-    En appliquant le même raisonnement à \( f^t\) au lieu de \( f\), nous trouvons
-    \begin{equation}
-        \rang\big( (f^t)^t \big)\geq \rang(f^t)
-    \end{equation}
-    et donc, vu que \( (f^t)^t=f\), nous obtenons \( \rang(f)=\rang(f^t)\).
-    
-\end{proof}
-
-\begin{proposition}[\cite{DualMarcSAge}]        \label{PropWOPIooBHFDdP}
-    Si \( f\) est une application linéaire entre les espaces vectoriels \( E\) et \( F\), alors nous avons
-    \begin{equation}
-        \Image(f^t)=\ker(f)^{\perp}.
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Soient donc l'application \( f\colon E\to F\) et sa transposée \( f^t\colon F^*\to E^*\). Nous commençons par prouver que \( \Image(f^{t})\subset(\ker f)^{\perp}\). Pour cela nous prenons \( \omega\in \Image(f^t)\), c'est à dire \( \omega=\alpha\circ f\) pour un certain élément \( \alpha\in F^*\). Si \( z\in\ker(f)\), alors \( \omega(z)=(\alpha\circ f)(z)=0\), c'est à dire que \( \omega\in (\ker f)^{\perp}\).
-
-    Pour prouver qu'il y a égalité, nous n'allons pas démontrer l'inclusion inverse, mais plutôt prouver que les dimensions sont égales. Après, on sait que si \( A\subset B\) et si \( \dim A=\dim B\), alors \( A=B\). Nous avons
-    \begin{subequations}
-        \begin{align}
-            \dim\big( \Image(f^t) \big)&=\rang(f^t)\\
-            &=\rang(f)  &\text{lemme \ref{LemSEpTcW}}\\
-            &=\dim(E)-\dim\ker(f)   &\text{théorème \ref{ThoGkkffA}}\\
-            &=\dim\big( (\ker f)^{\perp} \big)  &\text{proposition \ref{PropXrTDIi}}.
-        \end{align}
-    \end{subequations}
-\end{proof}
-
-\begin{lemma}[\cite{ooEPEFooQiPESf}]
-    Soit \( \eK\) un corps, \( E\) et \( F\) deux \( \eK\)-espaces vectoriels de dimension finie et une application linéaire \( f\colon E\to F\). L'application \( f\) est injective si et seulement si sa transposée\footnote{Définition \ref{DefooZLPAooKTITdd}.} \( f^t\) est surjective.
-\end{lemma}
-
-\begin{proof}
-    Supposons que \( f\) soit injective. Alors par le lemme \ref{LEMooDAACooElDsYb}, il existe \( g\colon F\to E\) tel que \( g\circ f=\id|_E\). Nous avons alors aussi \( (g\circ f)^t=\id|_{E^*}\), mais \( (g\circ f)^t=f^t\circ g^t\), donc \( f^t\) est surjective.
-
-    Inversement, nous supposons que \( f^t\colon F^*\to E^*\) est surjective. Alors en nous souvenant que \( E\) et \( F\) sont de dimension finie et en faisons jouer les identifications \( (f^t)^t=f\) et \( (E^*)^*=E\) nous pouvons savons qu'il existe \( s\colon E^*\to F^*\) tel que \( f^t\circ s=\id|_{E^*}\). En passant à la transposée,
-    \begin{equation}
-        s^t\circ f=\id|_{E},
-    \end{equation}
-    qui implique que \( f\) est injective.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Transposée : sans le dual}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{probleme}        \label{PROBooWNTKooNErOvt}
-    J'aimerais une confirmation de ce qui est écrit dans ici à propos du fait que si \( f\) est une application linéaire, \( f^t\) n'est pas bien définie.
-\end{probleme}
-
-Il est légitime, si \( f\colon V\to V\) est une application linéaire, de dire que sa transposée soit l'application linéaire \( f^t\colon V\to V\) dont la matrice est la matrice transposée de celle de \( f\). Lorsque nous travaillons sur \( \eR^n\) muni de la base canonique, cela ne pose pas de problèmes et nous pouvons écrire des égalités du type \( \langle x, Ay\rangle =\langle A^tx, y\rangle \).
-
-Hélas nous allons voir que cette façon de définir une transposée est mauvaise. Nous reprenons les notations de \ref{TEXTooOBOMooOVqUVG}. Nous avons
-\begin{equation}
-    f(x)=\sum_{kl}A_{kl}x_le_k=\sum_{\alpha\beta}A'_{\alpha\beta}x'_{\beta}e'_{\alpha}.
-\end{equation}
-Nous écrivons l'application transposée par rapport à la première base :
-\begin{equation}
-    f^{t_1}(x)=\sum_{kl}A_{lk}x_le_k
-\end{equation}      
-et par rapport à la seconde base :
-\begin{equation}        \label{EQooFXDQooEjYCrP}
-    f^{t_2}(x)=\sum_{\alpha\beta}A'_{\beta\alpha}x'_{\beta}e'_{\alpha}.
-\end{equation}
-En replaçant dans \( f^{(t_1)}\) les nombres \( x_l\) et les vecteurs \( e_k\) par les valeurs en termes des \( x'_{\beta}\) et \( e'_{\alpha}\) nous obtenons
-\begin{equation}
-    f^{t_1}(x)=\sum_{\alpha\beta}(Q^{-1}A^tQ)_{\alpha\beta}x'_{\beta}e'_{\alpha}.
-\end{equation}
-Nous avons égalité avec \eqref{EQooFXDQooEjYCrP} seulement lorsque \( A'_{\beta\alpha}=(Q^{-1}A^tQ)_{\alpha\beta}\), mais sachant que \( A'=Q^{-1}AQ\) nous avons besoin que
-\begin{equation}
-    Q^{-1}=Q^t
-\end{equation}
-(cela étant une égalité de matrice).
-
-\begin{normaltext}      \label{NooMZVRooExWVKJ}
-    Autrement dit, la façon «usuelle» de voir la transposée d'une application linéaire ne fonctionne dans les livres pour enfant uniquement parce qu'on n'y considère toujours \( \eR^n\) muni de la base canonique ou de bases orthonormées.
-     
-    Notons que nous avons tout de mêmes le notions d'opérateur adjoint et autoadjoint pour parler d'application orthogonale sans passer par la transposée, voir \ref{DEFooYKCSooURQDoS}.
-\end{normaltext}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Polynômes de Lagrange}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit \( E=\eR_n[X]\) l'ensemble des polynômes à coefficients réels de degré au plus \( n\). Soient les \( n+1\) réels distincts \( a_0,\ldots, a_n\). Nous considérons les formes linéaires associées \( f_i\in E^*\),
-\begin{equation}
-    f_i(P)=P(a_i).
-\end{equation}
-\begin{lemma}
-    Ces formes forment une base de \( E^*\).
-\end{lemma}
-
-\begin{proof}
-    Nous prouvons que l'orthogonal est réduit au nul :
-    \begin{equation}
-        \Span\{ f_0,\ldots, f_n \}^{\perp}=\{ 0 \}
-    \end{equation}
-    pour que la proposition \ref{PropXrTDIi} conclue. Si \( P\in\Span\{ f_i \}^{\perp}\), alors \( f_i(P)=0\) pour tout \( i\), ce qui fait que \( P(a_i)=0\) pour tout \( i=0,\ldots, n\). Un polynôme de degré au plus \( n\) qui s'annule en \( n+1\) points est automatiquement le polynôme nul.
-\end{proof}
-
-Les \defe{polynômes de Lagrange}{Lagrange!polynôme}\index{polynôme!Lagrange} sont les polynômes de la base (pré)duale de la base \( \{ f_i \}\).
-
-\begin{proposition}
-    Les polynômes de Lagrange sont donnés par
-    \begin{equation}
-        P_i=\prod_{k\neq i}\frac{ X-a_k }{ a_i-a_k }.
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Il suffit de vérifier que \( f_j(P_i)=\delta_{ij}\). Nous avons
-    \begin{equation}
-        f_j(P_i)=P_i(a_j)=\prod_{k\neq i}\frac{ a_j-a_k }{ a_i-a_k }.
-    \end{equation}
-    Si \( j\neq i\) alors un des termes est nul. Si au contraire \( i=j\), tous les termes valent \( 1\).
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Dual de \texorpdfstring{$ \eM(n,\eK)$}{M(n,K)}}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[\cite{KXjFWKA}]     \label{PropHOjJpCa}
-    Soit \( \eK\), un corps. Les formes linéaires sur \( \eM(n,\eK)\) sont les applications de la forme
-    \begin{equation}
-        \begin{aligned}
-            f_A\colon \eM_n(\eK)&\to \eK \\
-            M&\mapsto \tr(AM). 
-        \end{aligned}
-    \end{equation}
-\end{proposition}
-\index{trace!dual de \( \eM(n,\eK)\)}
-\index{dual!de \( \eM(n,\eK)\)}
-
-
-\begin{proof}
-    Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eM(n,\eK)&\to \eM(n,\eK)' \\
-            A&\mapsto f_A 
-        \end{aligned}
-    \end{equation}
-    et nous voulons prouver que c'est une bijection. Étant donné que nous sommes en dimension finie, nous avons égalité des dimensions de \( \eM_n(\eK)\) et \( \eM_n(\eK)'\), et il suffit de prouver que \( f\) est injective. Soit donc \( A\) telle que \( f_A=0\). Nous l'appliquons à la matrice \( (E_{ij})_{kl}=\delta_{ik}\delta_{jl}\) :
-    \begin{equation}
-            0=f_A(E_{ij})
-            =\sum_{k}(AE_{ij})_{kk}
-            =\sum_{kl}A_{kl}\delta_{il}\delta_{jk}
-            =A_{ij}.
-    \end{equation}
-    Donc \( A=0\).
-\end{proof}
-
-\begin{corollary}[\cite{KXjFWKA}]
-    Soit \( \eK\) un corps et \( \phi\in\eM(n,\eK)^*\) telle que pour tout \( X,Y\in \eM(n,\eK)\) on ait
-    \begin{equation}
-        \phi(XY)=\phi(YX).
-    \end{equation}
-    Alors il existe \( \lambda\in \eK\) tel que \( \phi=\lambda\Tr\).
-\end{corollary}
-\index{trace!unicité pour la propriété de trace}
-
-\begin{proof}
-    La proposition \ref{PropHOjJpCa} nous donne une matrice \( A\in \eM(n,\eK)\) telle que \( \phi=f_A\). L'hypothèse nous dit que \( f_A(XY)=f_A(YX)\), c'est à dire
-    \begin{equation}
-        \Tr(AXY)=\Tr(AYX)
-    \end{equation}
-    pour toute matrices \( X,Y\in \eM(n,\eK)\). L'invariance cyclique de la trace nous dit que \( \Tr(AXY)=\Tr(XAY)\), ce qui signifie que
-    \begin{equation}
-        \Tr\big( (AX-XA)Y \big)=0
-    \end{equation}
-    ou encore que \( f_{AX-XA}=0\) pour tout \( X\). La fonction \( f\) étant injective nous en déduisons que la matrice \( A\) doit satisfaire
-    \begin{equation}
-        AX=XA
-    \end{equation}
-    pour tout \( X\in\eM\). En particulier en prenant la fameuse matrice \( E_{ij}\) et en calculant un peu,
-    \begin{equation}
-        A_{li}\delta_{jm}=\delta_{il}A_{jm}
-    \end{equation}
-    pour tout \( i,j,l,m\). Cela implique que \( A_{ll}=A_{mm}\) pour tout \( l\) et \( m\) et que \( A_{jm}=0\) dès que \( j\neq m\). Il existe donc \( \lambda\in \eK\) tel que \( A=\lambda\mtu\). En fin de compte,
-    \begin{equation}
-        \phi(X)=f_{\lambda\mtu}(X)=\lambda\Tr(X).
-    \end{equation}
-\end{proof}
-
-\begin{corollary}[\cite{KXjFWKA}]       \label{CorICUOooPsZQrg}
-    Soit \( \eK\) un corps. Tout hyperplan de \( \eM(n,\eK)\) coupe \( \GL(n,\eK)\).
-\end{corollary}
-\index{groupe!linéaire!hyperplan}
-
-\begin{proof}
-    Soit \( \mH\) un hyperplan de \( \eM\). Il existe une forme linéaire \( \phi\) sur \( \eM(n,\eK)\) telle que \( \mH=\ker(\phi)\). Encore une fois la proposition \ref{PropHOjJpCa} nous donne \( A\in \eM\) telle que \( \phi=f_A\); nous notons \( r\) le rang de \( A\). Par le lemme \ref{LemZMxxnfM} nous avons \( A=PJ_rQ\) avec \( P,Q\in \GL(n,\eK)\) et
-    \begin{equation}
-        J_r=\begin{pmatrix}
-            \mtu_r    &   0    \\ 
-            0    &   0    
-        \end{pmatrix}.
-    \end{equation}
-    Pour tout \( X\in \eM\) nous avons
-    \begin{equation}
-        \phi(X)=\Tr(AX)=\Tr(PJ_rQX)=\Tr(J_rQXP).
-    \end{equation}
-    Ce que nous cherchons est \( X\in \GL(n,\eK)\) telle que \( \phi(X)=0\). Nous commençons par trouver \( Y\in\GL(n,\eK)\) telle que \( \Tr(J_rY)=0\). Celle-là est facile : c'est
-    \begin{equation}
-        Y=\begin{pmatrix}
-            0    &   1    \\ 
-            \mtu_{n-1}    &   0    
-        \end{pmatrix}.
-    \end{equation}
-    Les éléments diagonaux de \( J_rY\) sont tous nuls. Par conséquent en posant \( X=Q^{-1}YP^{-1}\) nous avons notre matrice inversible dans le noyau de \( \phi\).
-\end{proof}
-\index{hyperplan!de \( \eM(n,\eK)\)}
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-% This is part of Mes notes de mathématique
-% Copyright (c) 2008-2017
-%   Laurent Claessens
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Valeur propre et vecteur propre}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}      \label{DefooMMKZooVcskCc}
-    Soit un \( \eK\)-espace vectoriel \( E\) et un endomorphisme \( A\colon V\to V\). Un \defe{vecteur propre}{vecteur!propre} de \( A\) est un vecteur \( v \neq 0\) tel que \( Av=\lambda v\) pour un certain \( \lambda\in \eK\). Dans ce cas, \( \lambda\) est la \defe{valeur propre}{valeur!propre} de \( v\).
-
-    L'\defe{espace propre}{espace!propre} de \( A\) pour la valeur \( \lambda\)\footnote{Nous laissons au lecteur le soin de vérifier que c'est bien un sous-espace vectoriel de \( E\).} est l'ensemble des vecteurs propres de \( A\) pour la valeur propre \( \lambda\) et zéro.
-\end{definition}
-L'ensemble de valeurs propres de l'endomorphisme \( u\) est son \defe{spectre}{spectre!d'un endomorphisme} et est noté \( \Spec(u)\).
-
-\begin{remark}
-    Le nombre zéro peut être une valeur propre; c'est le vecteur zéro qui ne peut pas être vecteur propre. La matrice nulle est une matrice diagonalisable.
-\end{remark}
-
-\begin{lemma}       \label{LemjcztYH}
-    Soit \( u\) un endomorphisme et \( E_{\lambda}(u)\)\nomenclature[A]{\( E_{\lambda}(u)\)}{Espace propre de \( u\)} ses espaces propres. La somme des \( V_{\lambda}\) est directe.
-\end{lemma}
-
-\begin{proof}
-    Soit \( v_i\in V_{\lambda_i}\) un choix de vecteurs propres de \( u\). Si la somme n'est pas directe, nous pouvons considérer une combinaison linéaire des \( v_i\) qui soit nulle :
-    \begin{equation}
-        v_1+\cdots+v_p=0.
-    \end{equation}
-    Appliquons \( (A-\lambda_1\mtu)\) à cette égalité :
-    \begin{equation}
-        (\lambda_2-\lambda_1)v_1+\cdots+(\lambda_p-\lambda_1)v_p=0.
-    \end{equation}
-    En appliquant encore successivement les opérateurs \( (A-\lambda_i\mtu)\) nous réduisons le nombre de termes jusqu'à obtenir \( v_p=0\).
-\end{proof}
-
-\begin{example} \label{ExICOJcFp}
-    Sur \( \eR^2\), nous considérons la matrice \( A=\begin{pmatrix}
-        1    &   0    \\ 
-        1    &   1    
-    \end{pmatrix}\) qui a pour polynôme caractéristique le polynôme \( \chi_A=(X-1)^2\). Le nombre \( \lambda=1\) est une racine double de ce polynôme, et pourtant il n'y a qu'une seule dimension d'espace propre :
-    \begin{equation}
-        \begin{pmatrix}
-            1    &   0    \\ 
-            1    &   1    
-        \end{pmatrix}\begin{pmatrix}
-            x    \\ 
-            y    
-        \end{pmatrix}=\begin{pmatrix}
-            x    \\ 
-            y    
-        \end{pmatrix}
-    \end{equation}
-    entraine \( x=0\).
-
-    Ici la multiplicité algébrique est différente de la multiplicité géométrique.
-\end{example}
-
-\begin{proposition}[\cite{RombaldiO}]   \label{PropTVKbxU}
-    Soit \( E\), un espace vectoriel sur un corps infini et \( (F_k)_{k=1,\ldots, r}\), des sous-espaces vectoriels propres\footnote{Définition \ref{DefooMMKZooVcskCc}.} de \( E\) tels que \( \bigcup_{i=1}^rF_i=E\). Alors \( E=F_k\) pour un certain \( k\).
-
-    Autrement dit, l'union finie de sous-espaces propres ne peut être égal à l'espace complet.
-\end{proposition}
-
-La proposition suivante donne une utilisation amusante de la notion de polynôme caractéristique
-\begin{proposition}[\cite{ooNGUJooPphdsT}]
-    Soit un espace vectoriel \( V\) de dimension finie pour lequel il existe un endomorphisme \( f\colon V\to V\) tel que \( (f\circ f)(v)=-v\) pour tout \( v\in V\). Alors la dimension de \( V\) est paire.
-\end{proposition}
-
-\begin{proof}
-    Cherchons les valeurs propres de \( f\) en résolvant l'équation \( f(v)=\lambda v\). Nous appliquons \( f\) à cette égalité :
-    \begin{equation}
-        -v=\lambda f(v)=\lambda^2v.
-    \end{equation}
-    Donc \( \lambda\) ne peut pas être réel. Nous avons montré que \( f\) n'a pas de valeurs propres réelles. Or le polynôme caractéristique de \( f\) est de degré égal à la dimension. Si la dimension est impaire, le polynôme caractéristique est de degré impair, et possède donc une racine réelle. Autrement dit, l'absence de racines réelles au polynôme caractéristique indique une dimension paire.
-\end{proof}
-
-Une autre preuve possible est d'utiliser le déterminant : si la dimension de \( V\) est \( n\) nous avons :
-\begin{equation}
-    \det(f^2)=\det(-\id)=(-1)^n.
-\end{equation}
-Donc \( (-1)^n\) est positif, ce qui montre que \( n\) est pair.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Polynôme d'endomorphismes}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Soit \( A\) un anneau commutatif et \( \eK\), un corps commutatif. L'injection canonique \( A\to A[X]\) se prolonge en une injection
-\begin{equation}
-   \eM(A)\to\eM\big( A[X] \big).
-\end{equation}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Polynômes d'endomorphismes}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit \( u\in\End(E)\) où \( E\) est un \( \eK\)-espace vectoriel. Nous considérons l'application
-\begin{equation}    \label{EqOVKooeMJuv}
-    \begin{aligned}
-        \varphi_u\colon \eK[X]&\to \End(E) \\
-        P&\mapsto P(u). 
-    \end{aligned}
-\end{equation}
-L'image de \( \varphi_u\) est un sous-espace vectoriel. En effet si \( A=\varphi_u(P)\) et \( B=\varphi_u(Q)\), alors \( A+B=\varphi_u(P+Q)\) et \( \lambda A=(\lambda P)(u)\). En particulier c'est un espace fermé.
-
-Soit \( u\) un endomorphisme d'un \( \eK\)-espace vectoriel \( E\) et \( P\), un polynôme. Nous disons que \( P\) est un polynôme \defe{annulateur}{polynôme!annulateur} de \( u\) si \( P(u)=0\) en tant que endomorphisme de \( E\).
-
-\begin{lemma}       \label{LemQWvhYb}
-    Si \( P\) et \( Q\) sont des polynômes dans \( \eK[X]\) et si \( u\) est un endomorphisme d'un \( \eK\)-espace vectoriel \( E\), nous avons
-    \begin{equation}
-        (PQ)(u)=P(u)\circ Q(u).
-    \end{equation}
-\end{lemma}
-
-\begin{proof}
-    Si \( P=\sum_i a_iX^i\) et \( Q=\sum_j b_jX^j\), alors le coefficient de \( X^k\) dans \( PQ\) est
-    \begin{equation}        \label{EqCoefGPyVcv}
-        \sum_la_lb_{k-l}.
-    \end{equation}
-    Par conséquent \( (PQ)(u)\) contient \( \sum_la_lb_{k-l}u^k\). Par ailleurs \( P(u)\circ Q(u)\) est donné par
-    \begin{equation}
-        \sum_ia_iu^i\left( \sum_jb_ju^j \right)(x)=\sum_{ij}a_ib_ju^{i+j}(x).
-    \end{equation}
-    Le coefficient du terme en \( u^k\) est bien le même que celui donné par \eqref{EqCoefGPyVcv}.
-\end{proof}
-
-\begin{theorem}[Décomposition des noyaux ou lemme des noyaux]       \label{ThoDecompNoyayzzMWod}
-    Soit \( u\) un endomorphisme du \( \eK\)-espace vectoriel \( E\). Soit \( P\in\eK[X]\) un polynôme tel que \( P(u)=0\). Nous supposons que \( P\) s'écrive comme le produit \( P=P_1\ldots P_n\) de polynômes deux à deux étrangers\footnote{Définition \ref{DefDSFooZVbNAX}.}. Alors
-    \begin{equation}
-        E=\ker P_1(u)\oplus\ldots\oplus\ker P_n(u).
-    \end{equation}
-    De plus les projecteurs associés à cette décomposition sont des polynômes en \( u\).
-\end{theorem}
-\index{lemme!des noyaux}
-Ce résultat est utilisé pour prouver que toute représentation est décomposable en représentations irréductibles, proposition \ref{PropHeyoAN} ainsi que pour le théorème \ref{ThoDigLEQEXR} qui dit que si le polynôme minimal d'un endomorphisme est scindé à racine simple alors il est diagonalisable.
-
-\begin{proof}
-    Nous posons 
-    \begin{equation}
-        Q_i=\prod_{j\neq i}P_i.
-    \end{equation}
-    Par le lemme \ref{LemuALZHn} ces polynômes sont étrangers entre eux et le théorème de Bézout (théorème \ref{ThoBezoutOuGmLB}) donne l'existence de polynômes \( R_i\) tels que
-    \begin{equation}
-        R_1Q_1+\cdots+R_nQ_n=1.
-    \end{equation}
-    Si nous appliquons cette égalité à \( u\) et ensuite à \( x\in E\) nous trouvons
-    \begin{equation}        \label{EqqVcpUy}
-        \sum_{i=1}^n(R_iQ_i)(u)(x)=x,
-    \end{equation}
-    et en particulier si nous posons \( E_i=\Image\big(P_iQ_i(u)\big)\) nous avons
-    \begin{equation}
-        E=\sum_{i=1}^nE_i.
-    \end{equation}
-    Cette dernière somme n'est éventuellement pas une somme directe. Si \( i\neq j\), alors \( Q_iQ_j\) est multiple de \( P\) et nous avons, en utilisant le lemme \ref{LemQWvhYb}, 
-    \begin{equation}
-        (R_iQ_i)(u)\circ (R_jQ_j)(u)=\big( R_iQ_iR_jQ_j \big)(u)=S_{ij}(u)\circ P(u)=0
-    \end{equation}
-    où \( S_{ij}\) est un polynôme. 
-
-    Nous pouvons voir \( E\) comme un \( \eK\)-module et appliquer le théorème \ref{ThoProjModpAlsUR}. Les opérateurs \( R_iQ_i(u)\) ont l'identité comme somme et sont orthogonaux, et nous avons donc la décomposition en somme directe :
-    \begin{equation}
-        E=\bigoplus_{i=1}^nR_iQ_i(u)E.
-    \end{equation}
-
-    Afin de terminer la preuve, nous devons montrer que \( R_iQ_i(u)E=\ker P_i(u)\). D'abord nous avons
-    \begin{equation}
-        P_iR_iQ_i(u)=(R_iP)(u)=R_i(u)\circ P(u)=0,
-    \end{equation}
-    par conséquent \( \Image(R_iQ_i(u))\subset \ker P_i(u)\). Pour obtenir l'inclusion inverse, nous reprenons l'équation \eqref{EqqVcpUy} avec \( x\in\ker P_i(u)\). Elle se réduit à
-    \begin{equation}
-        (R_iQ_i)(u)x=x.
-    \end{equation}
-    Par conséquent \( x\in\Image\big( R_iQ_i(u) \big)\).
-\end{proof}
-
-\begin{corollary}   \label{CorKiSCkC}
-    Soit \( E\), un \( \eK\)-espace vectoriel de dimension finie et \( f\), un endomorphisme semi-simple dont la décomposition du polynôme minimal \( \mu_f\) en facteurs irréductibles sur \( \eK[X]\) est \( \mu_f=M_1^{\alpha_1}\cdots M_r^{\alpha_r}\). Si \( F\) est un sous-espace stable par \( f\), alors
-    \begin{equation}
-        F=\bigoplus_{i=1}^r\ker M_i^{\alpha_i}(f)\cap F
-    \end{equation}
-\end{corollary}
-
-\begin{proof}
-    Nous posons \( E_i=\ker M_i^{\alpha_i}(f)\) et \( F_i=E_i\cap F\). Les polynômes \( M_i^{\alpha_i}\) sont deux à deux étrangers et \( \mu_f(f)=0\), donc le lemme des noyaux (\ref{ThoDecompNoyayzzMWod}) s'applique et
-    \begin{equation}
-        E=E_1\oplus\ldots\oplus E_r.
-    \end{equation}
-    Nous pouvons décomposer \( x\in F\) en termes de cette somme :
-    \begin{equation}     \label{EqbBbrdi}
-        x=x_1+\cdots +x_r
-    \end{equation}
-    avec \( x_i\in E_i\). Toujours selon le lemme des noyaux, les projections sur les espaces \( E_i\) sont des polynômes en \( f\). Par conséquent \( F\) est stable sous toutes ces projections \( \pr_i\colon E\to E_i\), et en appliquant \( \pr_i\) à \eqref{EqbBbrdi}, \( \pr_i(x)=x_i\). Vu que \( x\in F\), le membre de gauche est encore dans \( F\) et \( x_i\in E_i\cap F\). Nous avons donc
-    \begin{equation}
-        F\subset\bigoplus_{i=1}^rF_i.
-    \end{equation}
-    L'inclusion inverse est immédiate parce que \( F_i\subset F\) pour chaque \( i\).
-\end{proof}
-
-\begin{lemma}   \label{LemVISooHxMdbr}
-    Si \( x\) est un vecteur propre de valeur propre \( \lambda\) pour l'endomorphisme \( u\) et si \( P\) est un polynôme, alors \( x\) est vecteur propre de \( u\) pour la valeur propre \( P(\lambda)\).
-\end{lemma}
-
-\begin{proof}
-    C'est un simple calcul de \( P(u)x\) en ayant noté \( P(X)=\sum_{k=0}^nc_kX^n\) :
-    \begin{equation}
-        P(u)x=\sum_{k=0}^nc_ku^k(x)=\sum_{k=0}^nc_k\lambda^ku=P(\lambda)x.
-    \end{equation}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Calcul effectif de l'exponentielle d'une matrice}
-%---------------------------------------------------------------------------------------------------------------------------
-\label{SUBSECooGAHVooBRUFub}
-
-Nous reprenons l'exemple de \cite{MneimneReduct}. Soit \( A\) une matrice dont le polynôme minimum s'écrit
-\begin{equation}
-    P(X)=(X-1)^2(X-2).
-\end{equation}
-Par le théorème \ref{ThoDecompNoyayzzMWod} de décomposition des noyaux nous avons
-\index{théorème!décomposition des noyaux!et exponentielle de matrice}
-\begin{equation}
-    E=\ker(A-1)^2\oplus\ker(A-2).
-\end{equation}
-En suivant les notations de ce théorème nous avons \( P_1(X)=(X-1)^2\), \( P_2(X)=X-2\) et
-\begin{subequations}
-    \begin{align}
-        Q_1(X)&=X-2\\
-        Q_2(X)&=(X-1)^2.
-    \end{align}
-\end{subequations}
-Les polynômes \( R_i\) dont l'existence est assurée par le théorème de Bézout sont
-\begin{equation}
-    \begin{aligned}[]
-        R_1(X)&=-X\\
-        R_2(X)&=1.
-    \end{aligned}
-\end{equation}
-Nous avons
-\begin{equation}
-    R_1Q_1+R_2Q_2=1.
-\end{equation}
-Le projecteur \( p_i\) sur \( \ker P_i\) est \( R_iQ_i\) :
-\begin{equation}
-    \begin{aligned}[]
-        p_1&=-A(A-2)=\pr_{\ker(u-1)^2}\\
-        p_2&=(A-1)^2=\pr_{\ker(u-2)}.
-    \end{aligned}
-\end{equation}
-Passons maintenant au calcul de l'exponentielle. Nous avons évidemment
-\begin{equation}
-    e^A=e^Ap_1+e^Ap_2.
-\end{equation}
-Étant donné que \( p_1\) est le projecteur sur le noyau de \( (A-1)^2\), nous avons
-\begin{equation}
-    e^Ap_1=ee^{A-1}p_1=ep_1+e(u-1)1=ep_1=-Ae(A-2).
-\end{equation}
-En effet \( e^{A-1}p_1=\sum_{k=0}^{\infty}(A-1)^k\circ p_1\). De la même façon nous avons
-\begin{equation}
-    e^Ap_2=e^2e^{A-2}p_2=e^2p_2=e^2(A-1)^2.
-\end{equation}
-Au final,
-\begin{equation}
-    e^A=-Ae(A-2)+e^2(A-1)^2.
-\end{equation}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Polynôme minimal et minimal ponctuel}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemmaDef}        \label{DefooOHUXooNkPWaB}
-    Soit un endomorphisme \( f\colon E\to E\) d'un \( \eK\)-espace vectoriel de dimension finie. Il existe un unique polynôme annulateur normalisé de degré minimum.
-
-    Il est nommé le \defe{polynôme minimal}{polynôme!minimal} de \( f\) et il est noté \( \mu_f\) ou simplement \( \mu\) lorsque la dépendance en \( f\) est claire.
-\end{lemmaDef}
-
-\begin{proof}
-    Pour l'unicité, soient \( P\) et \( Q\) deux polynômes annulateur de \( f\) de même degré \( N\) et ayant tous deux \( 1\) comme coefficient de \( x^N\). Alors \( P-Q\) est de degré \( N-1\) tout en étant encore annulateur.
-
-    Pour l'existence, les endomorphismes \( \id\), \( f\), \( f^2\), \ldots ne peuvent pas être tous linéairement indépendants parce que la dimension de \( \End(E)\) est finie. Il existe donc un nombre \( N\) et des coefficients \( a_k\) tels que \( \sum_{k=0}^Na_kf^k=0\). Le polynôme \( P(X)=\sum_{k=0}^Na_kX^k\) est donc annulateur de \( f\).
-
-    Une autre façon de le dire est que l'application linéaire \( \varphi\colon \eK[X]\to \End(E)\) donnée par \( \varphi(P)=P(f)\) est un endomorphisme d'un espace vectoriel de dimension infinie vers un espace vectoriel de dimension finie. Il ne peut donc pas être injectif et possède donc un noyau non réduit à zéro.
-\end{proof}
-
-\begin{remark}
-    La preuve donnée ci-dessus montre que \( \deg(\mu)\leq \dim(E)^2\). Comme conséquence du théorème de Caley-Hamilton \ref{ThoCalYWLbJQ} nous verrons qu'en réalité le degré du polynôme minimal est majoré par la dimension de l'espace.
-\end{remark}
-
-\begin{example}[Pas en dimension infinie]
-    L'endomorphisme de dérivation
-\end{example}
-
-
-Dans la suite, l'endomorphisme \( f\) du \( \eK\)-espace vectoriel \( E\) de dimension \( n\) est fixé. Pour \( x\in E\) nous notons
-\begin{equation}            \label{EqooOAYDooEpZELo}
-    E_x=\{ P(f)x\tq P\in \eK[X] \}.
-\end{equation}
-Nous considérons le morphisme d'algèbres
-\begin{equation}
-    \begin{aligned}
-        \varphi\colon \eK[X]&\to \End(E) \\
-        P&\mapsto P(f) 
-    \end{aligned}
-\end{equation}
-et si \( x\in E\) est donné nous considérons le morphisme de \( \eK\)-espaces vectoriels
-\begin{equation}
-    \begin{aligned}
-        \varphi_x\colon \eK[X]&\to E \\
-        P&\mapsto P(f)x. 
-    \end{aligned}
-\end{equation}
-Les noyaux de ces applications sont des idéaux, entre autres par le lemme \ref{LemQWvhYb}. Ils ont donc un unique générateur unitaire (chacun) par le théorème \ref{ThoCCHkoU}. En termes de vocabulaire, l'ensemble
-\begin{equation}
-    \ker(\phi)=\{  Q\in\eK[X]\tq Q(f)=0  \}
-\end{equation}
-est l'\defe{idéal annulateur}{polynôme!annulateur} de \( f\) et un polynôme \( Q\) tel que \( Q(f)=0\) est une polynôme annulateur de \( f\).
-
-\begin{definition}      \label{DEFooUICRooBGYhqQ}
-    Le générateur unitaire de \( \ker(\varphi_x)\) est le \defe{polynôme minimal ponctuel}{polynôme!minimal!ponctuel} de \( f\) en \( x\). Il sera noté \( \mu_{f,x}\) ou \( \mu_x\) lorsque la dépendance en \( f\) est claire dans le contexte.
-\end{definition}
-Nous notons \( \mu\) le générateur unitaire du noyau de \( \varphi\) et \( \mu_x\) celui de \( \varphi_x\). Vu que \( \mu\in\ker(\varphi_x)\) pour tout \( x\) nous avons \( \mu_x\divides \mu\) pour tout \( x\).
-
-\begin{example}[Pas en dimension infinie]       \label{ExooDTUJooIMqSKn}
-    En dimension infinie, il n'y a pas toujours de polynôme annulateur. Si \( E\) est un espace vectoriel de dimension infine ayant une base dénombrable \( \{ e_i \}_{i\in \eN}\) alors l'opérateur donné par \( f(e_i)=e_{i+1}\) n'a pas de polynôme annulateur. Même pas ponctuel en quel que point que ce soir.
-
-    De même l'opérateur donné par \( g(e_1)=0\) et \( g(e_i)=e_{i-1}\) si \( i\neq 1\) n'a pas de polynôme annulateur, mais il a un polynôme annulateur ponctuel évident en \( x=e_1\). L'exemple \ref{ExooLRHCooMYLQTU} donnera un habillage à peine subtil à cet exemple.
-\end{example}
-
-\begin{proposition}     \label{PropAnnncEcCxj}
-    Si \( P\) est un polynôme tel que \( P(f)=0\), alors le polynôme minimal \( \mu_f\) divise \( P\). Autrement dit, le polynôme minimal engendre l'idéal des polynômes annulateurs.
-\end{proposition}
-
-\begin{proof}
-    L'ensemble \( \ker(\varphi)=\{ Q\in \eK[X]\tq Q(u)=0 \} \) est un idéal par le lemme \ref{LemQWvhYb}. Le polynôme minimal de \( u\) est un élément de degré plus bas dans \( I\) et par conséquent \( I=(\mu_u)\) par le théorème \ref{ThoCCHkoU}. Nous concluons que \( \mu_u\) divise tous les éléments de \( I\).
-\end{proof}
-
-La proposition suivante permet de caractériser le polynôme minimal.
-\begin{proposition}[\cite{ooEPEFooQiPESf}]      \label{PROPooVUJPooMzxzjE}
-    Soit une application linéaire \( f\) sur un \( \eK\)-espace vectoriel. Il existe un unique polynôme unitaire\quext{À mon avis, «unitaire» manque dans \cite{ooEPEFooQiPESf}.} \( P\in \eK[X]\) tel que
-    \begin{enumerate}
-        \item
-            \( P(f)=0\);
-        \item
-            l'application
-            \begin{equation}        \label{EQooIBMDooVTaEhf}
-                \begin{aligned}
-                    \varphi\colon \frac{ \eK[X] }{ (P) }&\to \End(E) \\
-                    \bar Q&\mapsto Q(f) 
-                \end{aligned}
-            \end{equation}
-            est injective.
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    En ce qui concerne l'existence, il existe le polynôme minimal de \( f\) qui satisfait les conditions. Pour l'unicité nous travaillons maintenant.
-
-    Supposons que l'application \eqref{EQooIBMDooVTaEhf} soit injective. Alors pour tout \( Q\in \eK[X]\) tel que \( Q(f)=0\) nous avons \( \bar Q=0\), c'est à dire \( Q=PR\) pour un certain \( R\in \eK[X]\). Autrement dit : \( P\) est un générateur unitaire de l'idéal annulateur de \( f\). Le théorème \ref{ThoCCHkoU}\ref{ITEMooASHKooZqkiCH} nous dit alors que \( P=\mu\) parce que \( \mu\) est également générateur unitaire.
-\end{proof}
-
-\begin{lemma}[\cite{ooRJDSooXpVtMD}]\label{LemSYsJJj}
-    Soit \( f\colon E\to E\) un endomorphisme de l'espace vectoriel \( E\). Il existe un élément \( x\in E\) tel que \( \mu_{f,x}=\mu_f\).
-\end{lemma}
-
-\begin{proof}
-    Soit une décomposition en irréductibles du polynôme minimal \( \mu=P_1^{\alpha_1}\ldots P_r^{\alpha_r}\). Nous notons \( E_i=\ker\big( P_i^{\alpha_i}(f) \big)\). Les polynômes \( P_i\) sont étrangers deux à deux (un diviseur commun aurait a fortiori été un diviseur et aurait contredit l'irréductibilité). Le lemme des noyaux \ref{ThoDecompNoyayzzMWod} nous donne la somme directe
-    \begin{equation}
-        E=\bigoplus_{i=1}^r\ker\big( P_i^{\alpha_i}(f) \big).
-    \end{equation}
-    Si \( x_i\in E_i\) alors \( \mu_{x_i}\) est une puissance de \( P_i\). En effet \( \mu_{x_i}\divides \mu\) et est donc un produit des puissances des \( P_j\). Or si \( (QP_j)(f)x_i=0\) alors \( (P_jQ)(f)x_i=0\), ce qui donne \( Q(f)x_i\in E_j\cap E_i=\{ 0 \}\). Donc \( \mu_{x_i}\) n'est pas de la forme \( QP_j\) pour \( j\neq i\). Nous en déduisons que \( \mu_{x_i}\) est une puissance de \( P_i\) dès que \( x_i\in E_i\). Nous choisissons \( x_i\in E_i\) tel que \( \mu_{x_i}=P_i^{\alpha_i}\).
-
-    Nous posons enfin \( a=x_1+\cdots +x_r\); par définition du polynôme annulateur \( \mu_a\), nous avons
-    \begin{equation}        \label{EqooVIGGooSfuvwB}
-        0=\mu_a(f)a=\mu_a(f)x_1+\cdots +\mu_a(f)x_r.
-    \end{equation}
-    Mais \( m_a(f)x_j\in E_i\), et la somme des \( E_j\) est directe, donc l'annulation de la somme \eqref{EqooVIGGooSfuvwB} implique l'annulation de chacun des termes : \( \mu_a(f)x_i=0\) pour tout \( i\). Cela prouve que \( \mu_{x_i}\divides \mu_a\). Mais comme les \( \mu_{x_i}\) sont premiers deux à deux (parce que ce sont les \( P_i^{\alpha_i}\)), nous avons que le produit divise encore \( \mu_a\) :
-    \begin{equation}
-        \prod_{i=1}^r\mu_{x_i}\divides \mu_a,
-    \end{equation}
-    c'est à dire \( \mu\divides \mu_a\). Comme nous avons aussi \( \mu_a\divides \mu\), nous déduisons \( \mu_a=\mu\).
-\end{proof}
-
-\begin{definition}[Matrices, endomorphismes et vecteurs cycliques]      \label{DEFooFEIFooNSGhQE}
-    Une matrice est \defe{cyclique}{cyclique!matrice}\index{matrice!cyclique} si elle est semblable à une matrice compagnon. Un endomorphisme \( f\colon E\to E\) est \defe{cyclique}{cyclique!endomorphisme}\index{endomorphisme!cyclique} s'il existe un vecteur \( x\in E\) tel que \( \{ f^k(x) \}_{k=0,\ldots, n-1} \) est une base de \( E\). Un vecteur ayant cette propriété est un \defe{vecteur cyclique}{vecteur!cyclique} pour \( f\).
-\end{definition}
-
-\begin{lemma}   \label{LemAGZNNa}
-    Soit \( E\) un espace vectoriel de dimension finie et un endomorphisme cyclique\footnote{Voir la définition \ref{DEFooFEIFooNSGhQE}.} \( f\) de \( E\). Soit un vecteur cyclique \( v\) de \( f\), alors le polynôme minimal de \( f\) est égal au polynôme minimal de \( f\) au point \( v\) : \( \mu_{f}=\mu_{f,v}\).
-\end{lemma}
-
-\begin{proof}
-    Montrons que \( \mu_{f,v}\) est un polynôme annulateur de \( f\), ce qui prouvera que \( \mu_f\) divise \( \mu_{f,v}\) par la proposition \ref{PropAnnncEcCxj}. Étant donné que \( v\) est cyclique, tout élément de \( E\) s'écrit sous la forme \( x=Q(f)v\). Prenons un polynôme \( P\) annulateur de \( f\) en \( v\) : \( P(f)v=0\). Nous montrons que \( P\) est alors un polynôme annulateur de \( f\). En effet, nous avons
-    \begin{equation}
-        P(f)x=\big( P(f)\circ Q(f) \big)v=\big( Q(f)\circ P(f) \big)v=0
-    \end{equation}
-    où nous avons utilisé le lemme \ref{LemQWvhYb}.
-\end{proof}
-
-\begin{lemma}[\cite{ooRJDSooXpVtMD}]
-    Soit \( a\in E\) tel que \( \mu_a=\mu\). Alors \( E_a\) est un sous-espace stable pour \( f\) pour lequel il existe un supplémentaire stable.
-\end{lemma}
-
-\begin{proof}
-    Soit \( l=\deg(\mu)=\deg(\mu_a)\). L'espace \( E_a\) étant engendré par les \( f^k(a)\) nous savons que \( e_1=a\), \( e_2=f(a)\),\ldots, \( e_l=k^{l-1}(a)\) forment une base de \( E_a\). Nous pouvons la compléter en une base \( \{ e_1,\ldots, e_n \}\) de \( E\). Et nous posons\footnote{ici, comme presque partout, \( e^*_{l}\) est le dual de \( e_l\), c'est à dire l'application linéaire sur \( E\) donnée par \( e^*_l(e_i)=\delta_{li}\). }
-    \begin{subequations}
-        \begin{align}
-            G&=\{ x\in E\tq e^*_l\big( f^k(x) \big)=0\,\forall k\geq 0 \}\\
-            &=\bigcap_{k\geq 0}\ker\{ e^*_l\circ f^k \}\\
-            &=\bigcap_{k=0}^{l-1}\ker(  e^*_l\circ f^k ).
-        \end{align}
-    \end{subequations}
-    La dernière égalité est due au fait que \( l\) soit le degré de \( \mu\). Du coup \( f^l\) est une combinaison linéaire des \( f^i\) avec \( i\leq l-1\).
-
-    Nous avons \( f(G)\subset G\) et de plus \( E_a\cap G=\{ 0 \}\) parce qu'un élément de \( E_a\) est une combinaison linéaire d'éléments de la forme \( f^j(a)\) (\( j\leq l\)). Après application de \( f^{l-j}\), ces éléments obtiennent une composante \( f^l(a)=e_l\). De plus \( G\) est un sous-espace vectoriel du fait que \( e^*_l\circ f^i\) est une application linéaire. 
-    
-    Montrons enfin que \( \dim(G)=n-l\). Pour cela nous remarquons que \( G\) est une intersection d'hyperplans, et nous montrons que les équations définissant ces hyperplans sont linéairement indépendantes. Soit donc 
-    \begin{equation}        \label{EqooOHESooRtBUfc}
-        \sum_{j=0}^{l-1}\lambda_j\big( e^*_l\circ f^j \big)=0
-    \end{equation}
-    et montrons que \( \lambda_j=0\) pour tout $j$ est l'unique solution. Soit \( x\in E\) et appliquons l'opération \eqref{EqooOHESooRtBUfc} au vecteur \( f^i(x)\); le résultat est zéro :
-    \begin{equation}
-        0=\sum_{j=0}^{l-1}\lambda_j(e^*_l\circ f^i\circ f^j)=(e^*_l\circ f^i)P(u)
-    \end{equation}
-    où nous avons posé \( P(X)=\sum_{j=0}^{l-1}\lambda_jX^j\). Appliquons cela à \( a\) : pour tout \( i\) nous avons
-    \begin{equation}
-        (e^*_l\circ f^i)\big( P(f)a \big)=0.
-    \end{equation}
-    Mais par définition de \( E_a\), l'élément \(P(f)a \) est dans \( E_a\). Nous en déduisons que 
-    \begin{equation}
-        P(f)a\in G\cap E_a=\{ 0 \},
-    \end{equation}
-    c'est à dire que \( P\) est un polynôme annulateur de \( a\). Mais \( P\) est de degré \( l-1\) alors que le polynôme minimal de \( a\) est de degré \( l\). Par conséquent \( P=0\) et \( \lambda_j=0\) pour tout \( j\).
-\end{proof}
-
-\begin{definition}  \label{DEFooBOHVooSOopJN}
-    Un endomorphisme d'un espace vectoriel est \defe{semi-simple}{semi-simple!endomorphisme} si tout sous-espace stable par \( u\) possède un supplémentaire stable.
-\end{definition}
-
-\begin{lemma}   \label{LemrFINYT}
-    Si le polynôme minimal d'un endomorphisme est irréductible, alors il est semi-simple\footnote{Définition \ref{DEFooBOHVooSOopJN}.}.
-\end{lemma}
-
-\begin{proof}
-    Soit \( f\), un endomorphisme dont le polynôme minimal est irréductible et \( F\), un sous-espace stable par \( f\). Nous devons en trouver un supplémentaire stable. Si \( F=E\), il n'y a pas de problèmes. Sinon nous considérons \( u_1\in E\setminus F\) et
-    \begin{equation}
-        E_{u_1}=\{ P(f)u_1\tq P\in \eK[X] \},
-    \end{equation}
-    qui est un espace stable par \( f\). 
-
-    Montrons que \( E_{u_1}\cap F=\{ 0 \}\). Pour cela nous regardons l'idéal
-    \begin{equation}
-        I_{u_1}=\{ P\in \eK[X]\tq P(f)u_1=0 \}.
-    \end{equation}
-    Cela est un idéal non réduit à \( \{ 0 \}\) parce que le polynôme minimal de \( f\) par exemple est dans \( I_{u_1}\). Soit \( P_{u_1}\) un générateur unitaire de \( I_{u_1}\). Étant donné que \( \mu_f\in I_{u_1}\), nous avons que \( P_{u_1}\) divise \( \mu_f\) et donc \( P_{u_1}=\mu_f\) parce que \( \mu_f\) est irréductible par hypothèse.
-
-    Soit \( y\in E_{u_1}\cap F\). Par définition il existe \( P\in\eK[X]\) tel que \( y=P(f)u_1\) et si \( y\neq 0\), ce la signifie que \( P\notin I_{u_1}\), c'est à dire que \( P_{u_1} \) ne divise pas \( P\). Étant donné que \( P_{u_1}\) est irréductible cela implique que \( P_{u_1}\) et \( P\) sont premiers entre eux (ils n'ont pas d'autre \( \pgcd\) que \( 1\)).
-
-    Nous utilisons maintenant Bézout (théorème \ref{ThoBezoutOuGmLB}) qui nous donne \( A,B\in \eK[X]\) tels que 
-    \begin{equation}
-        AP+BP_{u_1}=1.
-    \end{equation}
-    Nous appliquons cette égalité à \( f\) et puis à \( u_1\):
-    \begin{equation}
-        u_1=A(f)\circ \underbrace{P(f)u_1}_{=y}+B(f)\circ \underbrace{P_{u_1}(u_1)}_{=0}=A(f)y.
-    \end{equation}
-    Mais \( y\in F\), donc \( A(f)y\in F\). Nous aurions donc \( u_1\in F\), ce qui est impossible par choix. Nous avons maintenant que l'espace \( E_{u_1}\oplus F\) est stable sous \( f\). Si cet espace est \( E\) alors nous arrêtons. Sinon nous reprenons le raisonnement avec \( E_{u_1}\oplus F\) en guise de \( F\) et en prenant \( u_2\in E\setminus(E_{u_1}\oplus F)\). Étant donné que \( E\) est de dimension finie, ce procédé s'arrête à un certain moment et nous aurons
-    \begin{equation}
-        E=F\oplus E_{u_1}\oplus\ldots\oplus E_{u_k}
-    \end{equation}
-    où chacun des \( E_{u_i}\) sont stables.
-\end{proof}
-
-\begin{theorem} \label{ThoFgsxCE}
-    Un endomorphisme est semi-simple si et seulement si son polynôme minimal est produit de polynômes irréductibles distincts deux à deux.
-\end{theorem}
-\index{anneau!principal}
-
-\begin{proof}
-
-    Supposons que \( f\) soit semi-simple et que son polynôme minimal soit donné par \( \mu_f=M_1^{\alpha_1}\ldots M_r^{\alpha_r}\) où les \( M_i\) sont des polynômes irréductibles deux à deux distincts. Nous devons montrer que \( \alpha_i=1\) pour tout \( i\). Soit \( i\) tel que \( \alpha_i\geq 1\) et \( N\in \eK[X]\) tel que \( \mu_f=M^2N\) où l'on a noté \( M=M_i\). Nous étudions l'espace
-    \begin{equation}
-        F=\ker M(f)
-    \end{equation}
-    qui est stable par \( f\), et qui possède donc un supplémentaire \( S\) également stable par \( f\). Nous allons montrer que \( MN\) est un polynôme annulateur de \( f\).
-
-    D'abord nous prenons \( x\in S\). Étant donné que \( F\) est le noyau de \( M(f)\),
-    \begin{equation}
-        M(f)\big( MN(f)x \big)=\mu_f(f)x=0,
-    \end{equation}
-    ce qui signifie que \( MN(f)x\in F\). Mais vu que \( S\) est stable par \( f\) nous avons aussi que \( MN(f)x\in S\). Finalement \( MN(f)x\in F\cap S=\{ 0 \}\). Autrement dit, \( MN(f)\) s'annule sur \( S\).
-
-    Prenons maintenant \( y\in F\). Nous avons
-    \begin{equation}
-        MN(f)=N(f)\big( M(f)y \big)=0
-    \end{equation}
-    parce que \( y\in F=\ker M(f)\).
-
-    Nous avons prouvé que \( MN(f)\) s'annule partout et donc que \( MN(f)\) est un polynôme annulateur de \( f\), ce qui contredit la minimalité de \( \mu_f=M^2N\).
-
-    Nous passons au sens inverse. Soit \( m_f=M_1\ldots M_r\) une décomposition du polynôme minimal de l'endomorphisme \( f\) en irréductibles distincts deux à deux. Soit \( F\) un sous-espace vectoriel stable par \( f\). Nous notons
-    \begin{equation}
-        E_i=\ker(M_i(f))
-    \end{equation}
-    et \( f_i=f|_{E_i}\). Par le lemme \ref{CorKiSCkC} nous avons
-    \begin{equation}
-        F=\bigoplus_{i=1}^r(F\cap E_i).
-    \end{equation}
-    Les espaces \( E_i\) sont stables par \( f\) et étant donné que \( M_i\) est irréductible, il est le polynôme minimal de \( f_i\). En effet, \( M_i\) est annulateur de \( f_i\), ce qui montre que le minimal de \( f_i\) divise \( M_i\). Mais \( M_i\) étant irréductible, \( M_i\) est le polynôme minimal. Étant donné que \( \mu_{f_i}=M_i\), l'endomorphisme \( f_i\) est semi-simple par le lemme \ref{LemrFINYT}.
-
-    L'espace \( F\cap E_i\) étant stable par l'endomorphisme semi-simple \( f_i\), il possède un supplémentaire stable que nous notons \( S_i\)~:
-    \begin{equation}
-        E_i=S_i\oplus(F\cap E_i).
-    \end{equation}
-    Étant donné que sur chaque \( S_i\) nous avons \( f|_{S_i}=f_i\), l'espace \( S=S_1\oplus\ldots\oplus S_r\) est stable par \( f\). Du coup nous avons
-    \begin{subequations}
-        \begin{align}
-            E&=E_1\oplus\ldots\oplus E_r\\
-            &=\big( S_1\oplus(F\cap E_1) \big)\oplus\ldots\oplus\big( S_r\oplus(F\cap E_r) \big)\\
-            &=\big( \bigoplus_{i=1}^rS_i \big)\oplus\big( \bigoplus_{i=1}^rF\cap E_i \big)\\
-            &=S\oplus F,
-        \end{align}
-    \end{subequations}
-    ce qui montre que \( F\) a bien un supplémentaire stable par \( f\) et donc que \( f\) est semi-simple.
-\end{proof}
-
-\begin{example}[L'espace engendré par \( \mtu\), \( A\), \( A^2\),\ldots]
-    Soit \( A\) une matrice, et 
-    \begin{equation}
-        V=\Span\{A^k\tq k\in \eN \}.
-    \end{equation}
-    Nous montrons que \( \dim(V)\) est le degré du polynôme minimal de \( A\).
-
-    D'abord l'idéal annulateur de \( A\) est engendré par le polynôme minimal\footnote{Proposition \ref{PropAnnncEcCxj}.} que nous notons
-        $\mu=\sum_{k=0}^pa_kX^k$.
-    La partie \( \{ \mtu,\ldots, A^{p-1} \}\) est libre parce qu'une combinaison linéaire nulle de cela serait un polynôme annulateur en \( A\) de degré plus petit que \( p\). Donc \( \dim(V)\geq p\).
-
-    La partie \( \{ \mtu,A,\ldots, A^p \}\) est liée à cause du polynôme minimal. Isoler \( A^p\) dans \( \mu(A)=0\) donne un polynôme \( f\) de degré \( p-1\) tel que \( A^p=f(A)\).
-
-    Nous allons montrer à présent que la famille \( \{ \mtu,A,\ldots, A^{p-1} \}\) est génératrice (alors \( \dim(V)\leq p\)). Soit un entier \( q\geq p\)et de division euclidienne\footnote{Théorème \ref{ThoDivisEuclide}.} \( np+r=q\) avec \( r<p\). Nous avons \( A^q=A^{np}A^r\). D'une part
-    \begin{equation}
-        A^{np}=(A^p)^n=f(A)^n
-    \end{equation}
-    est de degré \( n(p-1)\). Par conséquent
-    \begin{equation}
-        A^q=f(A)^nA^r
-    \end{equation}
-    qui est de degré \( n(p-1)+r=q-n\). Autrement dit il existe un polynôme \( g_1\) de degré \( q-n\) tel que \( A^q=g_1(A)\). Si \( q-n>p-1\) alors nous pouvons recommencer et obtenir un polynôme \( g_2\) de degré strictement inférieur à celui de \( g_1\) tel que \( A^q=g_2(A)\). Au bout du compte, il existe un polynôme \( g\) de degré au maximum \( p-1\) tel que \( A^q=g(A)\). Cela prouve que la partie \( \{ \mtu,A,\ldots, A^{p-1} \}\) est génératrice de \( V\).
-
-    La dimension de \( V\) est donc \( p\), le degré du polynôme minimal.
-\end{example}
-
-\begin{proposition}     \label{PropooCFZDooROVlaA}
-    Soit \( f\) un endomorphisme d'un espace vectoriel de dimension finie. Nous avons l'isomorphisme d'espace vectoriel
-    \begin{equation}
-        \eK[f]\simeq\frac{ \eK[X] }{ (\mu_f) }
-    \end{equation}
-    La dimension en est \( \deg(\mu_f)\).
-\end{proposition}
-
-\begin{proof}
-    Notons avant de commencer que \( (\mu)\) est l'idéal engendré par \( \mu\). Les classes dont il est question dans le quotient \( \eK[X]/(\mu)\) sont 
-    \begin{equation}
-        \bar P=\{ P+S\mu \}_{S\in \eK[X]}.
-    \end{equation}
-    Nous allons montrer que l'application suivante fournit l'isomorphisme : 
-    \begin{equation}
-        \begin{aligned}
-            \psi\colon \frac{ \eK[X] }{ (\mu) }&\to \eK[f] \\
-            \bar P&\mapsto P(f). 
-        \end{aligned}
-    \end{equation}
-    \begin{subproof}
-        \item[\( \psi\) est bien définie]
-            Si \( Q\in \bar P\) alors \( Q=P+S\mu\) pour un certain \( S\in \eK[X]\). Du coup nous avons
-            \begin{equation}
-                \psi(\bar Q)=P(f)+(S\mu)(f).
-            \end{equation}
-            Mais \( \mu(f)=0\) donc le deuxième terme est nul. Donc \( \psi(\bar P)\) est bien définit.
-        \item[Injectif]
-            Si \( \psi(\bar P)=0\) nous avons \( P(f)=0\), ce qui signifie que \( P=S\mu\) pour un polynôme \( S\). Par conséquent \( P\in (\mu)\) et donc \( \bar P=0\).
-        \item[Surjectif]
-            Soit \( P\in \eK[X]\). L'élément \( P(f) \) de \( \eK[f]\) est dans l'image de \( \psi\) parce que c'est \( \psi(\bar P)\).
-    \end{subproof}
-    En ce qui concerne la dimension, le corollaire \ref{CorsLGiEN} en parle déjà : une base est donné par les projections de \( 1,X,\ldots, X^{\deg(\mu_a)-1}\).
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Polynôme caractéristique}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}  \label{DefOWQooXbybYD}
-    Soit un anneau commutatif \( A\). Si \( u\in\eM_n(A)\), nous définissons le \defe{polynôme caractéristique de \( u\)}{polynôme!caractéristique}\index{caractéristique!polynôme} :
-    \begin{equation}    \label{Eqkxbdfu}
-        \chi_u(X)=\det(X\mtu_n-u).
-    \end{equation} 
-    Nous définissons de même le polynôme caractéristique d'un endomorphisme \( u\colon E\to E\).
-\end{definition}
-
-\begin{lemma}       \label{LemooWCZMooZqyaHd}
-    Le polynôme caractéristique \( \chi_u\) est unitaire et a pour degré la dimension de l'espace vectoriel \( E\)..
-\end{lemma}
-
-\begin{theorem}     \label{ThoNhbrUL}
-    Soit \( E\) un \(\eK\)-espace vectoriel de dimension finie \( n\) et un endomorphisme \( u\in\End(E)\). Alors
-    \begin{enumerate}
-        \item
-            Le polynôme caractéristique divise \( (\mu_u)^n\) dans \(\eK[X]\).
-        \item
-            Les polynômes caractéristiques et minimaux ont mêmes facteurs irréductibles dans \(\eK[X]\).
-        \item
-            Les polynômes caractéristiques et minimaux ont mêmes racines dans \(\eK[X]\).
-        \item
-            Le polynôme caractéristique est scindé si et seulement si le polynôme minimal est scindé.
-    \end{enumerate}
-\end{theorem}
-
-\begin{theorem} \label{ThoWDGooQUGSTL}
-    Soit \( u\in\End(E)\) et \( \lambda\in\eK\). Les conditions suivantes sont équivalentes
-    \begin{enumerate}
-        \item\label{ItemeXHXhHi}
-            \( \lambda\in\Spec(u)\)
-        \item\label{ItemeXHXhHii}
-            \( \chi_u(\lambda)=0\)
-        \item\label{ItemeXHXhHiii}
-            \( \mu_u(\lambda)=0\).
-    \end{enumerate}
-\end{theorem}
-
-\begin{proof}
-    \ref{ItemeXHXhHi} \( \Leftrightarrow\) \ref{ItemeXHXhHii}. Dire que \( \lambda\) est dans le spectre de \( u\) signifie que l'opérateur \( u-\lambda\mtu\) n'est pas inversible, ce qui est équivalent à dire que \( \det(u-\lambda\mtu)\) est nul par la proposition \ref{PropYQNMooZjlYlA}\ref{ItemUPLNooYZMRJy} ou encore que \( \lambda\) est une racine du polynôme caractéristique de \( u\). 
-
-    \ref{ItemeXHXhHii} \( \Leftrightarrow\) \ref{ItemeXHXhHiii}. Cela est une application directe du théorème \ref{ThoNhbrUL} qui précise que le polynôme caractéristique a les mêmes racines dans \(\eK\) que le polynôme minimal.
-\end{proof}
-
-\begin{definition}
-    Si \( \lambda\in\eK\) est une racine de \( \chi_u\), l'ordre de l'annulation est la \defe{multiplicité algébrique}{multiplicité!valeur propre!algébrique} de la valeur propre \( \lambda\) de \( u\). À ne pas confondre avec la \defe{multiplicité géométrique}{multiplicité!valeur propre!géométrique} qui sera la dimension de l'espace propre.
-\end{definition}
-
-\begin{proposition}[\cite{RombaldiO}]\label{PropNrZGhT}
-    Soit \( f\), un endomorphisme de \( E\) et \( x\in E\). Alors
-    \begin{enumerate}
-        \item
-            L'espace \( E_{f,x}\) est stable par \( f\).
-        \item\label{ItemfzKOCo}
-            L'espace \( E_{f,x}\) est de dimension
-            \begin{equation}
-                p_{f,x}=\dim E_{f,x}=\deg(\mu_{f,x})
-            \end{equation}
-            où \( \mu_{f,x}\) est le générateur unitaire de \( I_{f,x}\).
-        \item   \label{ItemKHNExH}
-            Le polynôme caractéristique de \( f|_{E_{f,x}}\) est \( \mu_{f,x}\).
-        \item   \label{ItemHMviZw}
-            Nous avons
-            \begin{equation}
-                \chi_{f|_{E_{f,x}}}(f)x=\mu_{f,x}(f)x=0.
-            \end{equation}
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Le fait que \( E_{f,x}\) soit stable par \( f\) est classique. Le point \ref{ItemHMviZw} est un une application du point \ref{ItemKHNExH}. Les deux gros morceaux sont donc les points \ref{ItemfzKOCo} et \ref{ItemKHNExH}.
-
-    Étant donné que \( \mu_{f,x}\) est de degré minimal dans \( I_{f,x}\), l'ensemble
-    \begin{equation}
-        B=\{ f^k(x)\tq 0\leq k\leq p_{f,x}-1 \}
-    \end{equation}
-    est libre. En effet une combinaison nulle des vecteurs de \( B\) donnerait un polynôme en \( f\) de degré inférieur à \( p_{f,x}\) annulant \( x\). Nous écrivons
-    \begin{equation}
-        \mu_{f,x}(X)=X^{p_{f,x}}-\sum_{i=0}^{p_{f,x}-1}a_iX^k. 
-    \end{equation}
-    Étant donné que \( \mu_{f,x}(f)x=0\) et que la somme du membre de droite est dans \( \Span(B)\), nous avons \( f^{p_{f,x}}(x)\in\Span(B)\). Nous prouvons par récurrence que \( f^{p_{f,x}+k}(x)\in\Span(B)\). En effet en appliquant \( f^k\) à l'égalité
-    \begin{equation}
-        0=f^{p_{f,x}}(x)-\sum_{i=0}^{p_{f,x}-1}a_if^i(x)
-    \end{equation}
-    nous trouvons
-    \begin{equation}
-        f^{p_{f,x}+k}(x)=\sum_{i=0}^{p_{f,x}-1}a_if^{i+k}(x),
-    \end{equation}
-    alors que par hypothèse de récurrence le membre de droite est dans \( \Span(B)\). L'ensemble \( B\) est alors générateur de \( E_{f,x}\) et donc une base d'icelui. Nous avons donc bien \( \dim(E_{f,x})=p_{f,x}\).
-
-    Nous montrons maintenant que \( \mu_{f,x}\) est annulateur de \( f\) au point \( x\). Nous savons que
-    \begin{equation}
-        \mu_{f,x}(f)x=0.
-    \end{equation}
-    En y appliquant \( f^k\) et en profitant de la commutativité des polynômes sur les endomorphismes (proposition \ref{LemQWvhYb}), nous avons
-    \begin{equation}
-        0=f^k\big( \mu_{f,x}(f)x \big)=\mu_{f,x}(f)f^k(x),
-    \end{equation}
-    de telle sorte que \( \mu_{f,x}(f)\) est nul sur \( B\) et donc est nul sur \( E_{f,x}\). Autrement dit,
-    \begin{equation}
-        \mu_{f,x}\big( f|_{E_{f,x}} \big)=0.
-    \end{equation}
-    Montrons que \( \mu_{f,x}\) est même minimal pour \( f|_{E_{f,x}}\). Sot \( Q\), un polynôme non nul de degré \( p_{f,x}-1\) annulant \( f|_{E_{f,x}}\). En particulier \( Q(f)x=0\), alors qu'une telle relation signifierait que \( B\) est un système lié, alors que nous avons montré que c'était un système libre. Nous concluons que \( \mu_{f,x}\) est le polynôme minimal de \( f|_{E_{f,x}}\).
-\end{proof}
-
-Cette histoire de densité permet de donner une démonstration alternative du théorème de Cayley-Hamilton.
-\begin{theorem}[Cayley-Hamlilton]   \label{ThoCalYWLbJQ}
-    Le polynôme caractéristique est un polynôme annulateur.
-\end{theorem}
-\index{théorème!Cayley-Hamilton}
-
-Une démonstration plus simple via la densité des diagonalisables est donnée en théorème \ref{ThoHZTooWDjTYI}.
-\begin{proof}
-    Nous devons prouver que \( \chi_f(f)x=0\) pour tout \( x\in E\). Pour cela nous nous fixons un \( x\in E\), nous considérons l'espace \( E_{f,x}\) et \( \chi_{f,x}\), le polynôme caractéristique de \( f|_{E_{f,x}}\). Étant donné que \( E_{f,x}\) est stable par \( f\), le polynôme caractéristique de \( f|_{E_{j,x}}\) divise \( \chi_f\), c'est à dire qu'il existe un polynôme \( Q_x\) tel que
-    \begin{equation}
-        \chi_f=Q_x\chi_{f,x},
-    \end{equation}
-    et donc aussi
-    \begin{equation}
-        \chi_f(f)x=Q_x(f)\big( \chi_{f,x}(f)x \big)=0
-    \end{equation}
-    parce que la proposition \ref{PropNrZGhT} nous indique que \( \chi_{f,x}\) est un polynôme annulateur de \( f|_{E_{f,x}}\).
-\end{proof}
-
-\begin{corollary}
-    Le degré du polynôme minimal est majoré par la dimension de l'espace.
-\end{corollary}
-
-\begin{proof}
-    Le polynôme minimal engendre l'idéal des polynôme annulateurs (proposition \ref{PropAnnncEcCxj}), et divise donc le polynôme caractéristique. Or le degré du polynôme caractéristique est la dimension de l'espace par le lemme \ref{LemooWCZMooZqyaHd}.
-\end{proof}
-
-\begin{example}[Calcul de l'inverse d'un endomorphisme]
-    Le polynôme de Cayley-Hamilton donne un moyen de calculer l'inverse d'un endomorphisme inversible pourvu que l'on sache son polynôme caractéristique. En effet, supposons que
-    \begin{equation}
-        \chi_f(X)=\sum_{k=0}^na_kX^k.
-    \end{equation}
-    Nous aurons alors
-    \begin{equation}
-        0=\chi_f(f)=\sum_{k=0}^na_kf^k.
-    \end{equation}
-    Nous appliquons \( f^{-1}\) à cette dernière égalité en sachant que \( f^{-1}(0)=0\) :
-    \begin{equation}
-        0=a_0f^{-1}+\sum_{k=1}^na_kf^{k-1},
-    \end{equation}
-    et donc
-    \begin{equation}
-        u^{-1}=-\frac{1}{ \det(f) }\sum_{k=1}^na_kf^{k-1}
-    \end{equation}
-    où nous avons utilisé le fait que \( a_0=\chi_f(0)=\det(f)\).
-\end{example}
-
-\begin{proposition}\label{PropooBYZCooBmYLSc}
-    Si \( (X-z)^l\) (\( l\geq 1\)) est la plus grande puissance de \( (X-z)\) dans le polynôme caractéristique d'un endomorphisme \( u\) alors 
-    \begin{equation}
-        1\leq \dim(E_e)\leq l.
-    \end{equation}
-    C'est à dire que nous avons au moins un vecteur propre pour chaque racine du polynôme caractéristique.
-\end{proposition}
-
-\begin{proof}
-    Si $(X-z)$ divise \( \chi_u\) alors en posant \( \chi_u=(X-z)P(X)\) nous avons
-    \begin{equation}
-        \det(u-X\mtu)=(X-z)P(X),
-    \end{equation}
-    ce qui, évalué en \( X=z\), donne \( \det(u-z\mtu)=0\). L'annulation du déterminant étant équivalente à l'existence d'un noyau non trivial, nous avons \( v\neq 0\) dans \( E\) tel que \( (u-z\mtu)v=0\). Cela donne \( u(v)=zv\) et donc que \( v\) est vecteur propre de \( u\) pour la valeur propre \( z\). Donc aussi \( \dim(E_z)\geq 1\).
-
-    Si \( \dim(E_z)=k\) alors le théorème de la base incomplète \ref{ThonmnWKs} nous permet d'écrire une base de \( E\) dont les \( k\) premiers vecteurs forment une base de \( E_z\). Dans cette base, la matrice de \( u\) est de la forme
-    \begin{equation}
-        \begin{pmatrix}
-             z   &       &       &   *    \\
-                &   \ddots    &       &   \vdots    \\
-                &       &   z    &   *    \\ 
-                &       &       &   *     
-         \end{pmatrix}
-    \end{equation}
-    où les étoiles représentent des blocs a priori non nuls. En tout cas il est vu sous cette forme que \( (X-z\mtu)^k\) divise \( \chi_u\).
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Diagonalisation et trigonalisation}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Ici encore \( \eK\) est un corps commutatif.
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Matrices semblables}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}[matrices semblables] \label{DefCQNFooSDhDpB}
-    Sur l'ensemble \( \eM_n(\eK)\) des matrices \( n\times n\) à coefficients dans \(\eK\) nous introduisons la relation d'équivalence \( A\sim B\) si et seulement s'il existe une matrice \( P\in\GL(n,\eK)\) telle que \( B=P^{-1}AP\). Deux matrices équivalentes en ce sens sont dites \defe{semblables}{semblables!matrices}.
-\end{definition}
-
-Le polynôme caractéristique est un invariant sous les similitudes. En effet si \( P\) est une matrice inversible,
-\begin{subequations}
-    \begin{align}
-        \chi_{PAP^{-1}}&=\det(PAP^{-1}-\lambda X)\\
-        &=\det\big( P^{-1}(PAP^{-1}-\lambda X)P^{-1} \big)\\
-        &=\det(A-\lambda X).
-    \end{align}
-\end{subequations}
-
-La permutation de lignes ou de colonnes ne sont pas de similitudes, comme le montrent les exemples suivants :
-\begin{equation}
-    \begin{aligned}[]
-        A&=\begin{pmatrix}
-            1    &   2    \\ 
-            3    &   4    
-        \end{pmatrix}&
-        B&=\begin{pmatrix}
-            2    &   1    \\ 
-            4    &   3    
-        \end{pmatrix}.
-    \end{aligned}
-\end{equation}
-Nous avons \( \chi_A=x^2-5x-2\) tandis que \( \chi_B=x^2-5x+2\) alors que le polynôme caractéristique est un invariant de similitude.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Endomorphismes nilpotents}
-%---------------------------------------------------------------------------------------------------------------------------
-
-La \defe{trace}{trace!matrice} d'une matrice \( A\in \eM(n,\eK)\) est la somme de ses éléments diagonaux :
-\begin{equation}
-    \tr(A)=\sum_{i=1}^nA_{ii}.
-\end{equation}
-Une propriété importante est son invariance cyclique.
-\begin{lemma}   \label{LemhbZTay}
-    Si \( A\) et \( B\) sont des matrices carré, alors \( \tr(AB)=\tr(BA)\).
-
-    La trace est un invariant de similitude.
-\end{lemma}
-
-\begin{proof}
-    C'est un simple calcul :
-    \begin{equation}
-            \tr(AB)=\sum_{ik}A_{ik}B_{ki}
-            =\sum_{ik}A_{ki}B_{ik} 
-            =\sum_{ik}B_{ik}A_{ki}
-            =\sum_i(BA)_{ii}
-            =\tr(BA)
-    \end{equation}
-    où nous avons simplement renommé les indices \( i\leftrightarrow k\).
-
-    En particulier, la trace est un invariant de similitude parce que \( \tr(ABA^{-1})=\tr(A^{-1} AB)=\tr(B)\). 
-\end{proof}
-La trace étant un invariant de similitude, nous pouvons donc définir la \defe{trace}{trace!endomorphisme} comme étant la trace de sa matrice dans une base quelconque. Si la matrice est diagonalisable, alors la trace est la somme des valeurs propres.
-
-\begin{lemma}[\cite{fJhCTE}]   \label{LemzgNOjY}
-    L'endomorphisme \( u\in\End(\eC^n)\) est nilpotent si et seulement si \( \tr(u^p)=0\) pour tout \( p\).
-\end{lemma}
-
-\begin{proof}
-    Supposons que \( u\) est nilpotent. Alors ses valeurs propres sont toutes nulles et celles de \( u^p\) le sont également. La trace étant la somme des valeurs propres, nous avons alors tout de suite \( \tr(u^p)=0\).
-
-    Supposons maintenant que \( \tr(u^p)=0\) pour tout \( p\). Le polynôme caractéristique \eqref{Eqkxbdfu} est
-    \begin{equation}    \label{EqfnCqWq}
-        \chi_u=(-1)^nX^{\alpha}(X-\lambda_1)^{\alpha_1}\ldots (X-\alpha_r)^{\alpha_r}.
-    \end{equation}
-    où les \( \lambda_i\) (\( i=1,\ldots, r\)) sont les valeurs propres non nulles distinctes de \( u\).
-
-    Il est vite vu que le coefficient de \( X^{n-1}\) dans \( \chi_u\) est \( -\tr(u)\) parce que le coefficient de \( X^{n-1}\) se calcule en prenant tous les $X$ sauf une fois \( -\lambda_i\). D'autre part le polynôme caractéristique de \( u^p \) est le même que celui de \( u\), en remplaçant \( \lambda_i\) par \( \lambda_i^p\); cela est dû au fait que si \( v\) est vecteur propre de valeur propre \( \lambda\), alors \( u^pv=\lambda^pv\).
-
-    Par l'équation \eqref{EqfnCqWq}, nous voyons que le coefficient du terme \( X^{n-1}\) dans les polynôme caractéristique est 
-    \begin{equation}        \label{eqSoDSKH}
-        0=\tr(u^p)=\alpha_1\lambda_1^p+\cdots +\alpha_r\lambda_r^p.
-    \end{equation}
-    Donc les nombres \( (\alpha_1,\ldots, \alpha_r)\) est une solution non triviale\footnote{Si \( \alpha_1=\ldots=\alpha_r=0\), alors les valeurs propres sont toutes nulles et la matrice est en réalité nulle dès le départ.} du système
-    \begin{subequations}    \label{EqDpvTnu}
-        \begin{numcases}{}
-            \alpha_1X_1+\cdots +\lambda_rX_r=0\\
-            \qquad\vdots\\
-            \lambda^r_1X_1+\cdots +\lambda_r^rX_r=0.
-        \end{numcases}
-    \end{subequations}
-    Cela sont les équations \eqref{eqSoDSKH} écrites avec \( p=1,\ldots, r\). Le déterminant de ce système est
-    \begin{equation}
-        \lambda_1\ldots\lambda_r\det\begin{pmatrix}
-             1   &   \ldots    &   1    \\
-             \lambda_1   &   \ldots    &   \lambda_1    \\
-             \vdots   &       &   \vdots    \\ 
-             \lambda_1^{r-1}   &   \ldots    &   \lambda_r^{r-1}
-         \end{pmatrix}\neq 0,
-    \end{equation}
-    qui est un déterminant de Vandermonde (proposition \ref{PropnuUvtj}) valant
-    \begin{equation}
-        0=\lambda_1\ldots\lambda_r\prod_{1\leq i\leq j\leq r}(\lambda_i-\lambda_j).
-    \end{equation}
-    Étant donné que les \( \lambda_i\) sont distincts et non nuls, nous avons une contradiction et nous devons conclure que \( (\alpha_1,\ldots, \alpha_r)\) était une solution triviale du système \eqref{EqDpvTnu}.
-\end{proof}
-
-\begin{proposition}[\cite{SVSFooIOYShq}]    \label{PropMWWJooVIXdJp}
-    Soit un \( \eK\)-espace vectoriel \( E\). Un endomorphisme \( u\in\End(E)\) est nilpotent si et seulement s'il existe une base de \( E\) dans laquelle la matrice de \( u\) est strictement triangulaire supérieure.
-\end{proposition}
-
-\begin{proof}
-    \begin{subproof}
-       \item[\( \Rightarrow\)]
-           Nous faisons la démonstration par récurrence sur la dimension de \( E\). Lorsque \( n=1\) nous avons \( u=(a)\) avec \( a\in \eK\). Vu que \( a^k=0\) pour un certain \( k\) nous avons \( a=0\) parce qu'un corps est toujours un anneau intègre\footnote{Lemme \ref{LemAnnCorpsnonInterdivzer}.}. 
-
-           Lorsque \( \dim(E)=n\) nous savons que \( u\) a un noyau non réduit au vecteur nul (parce qu'il est nilpotent). Soit donc un vecteur non nul \( x\in\ker(u)\) et une base
-           \begin{equation}
-               \{ x,e_2,\ldots, e_n \}
-           \end{equation}
-           donnée par le théorème de la base incomplète \ref{ThonmnWKs}. La matrice de \( u\) dans cette base s'écrit
-           \begin{equation}
-               \begin{pmatrix}
-                       \begin{array}[]{c|c}
-                           0&\begin{matrix} 
-                               * &   *    &   *    
-                           \end{matrix}\\
-                           \hline
-                           \begin{matrix}
-                               0 \\ 
-                               0 \\ 
-                               0 
-                           \end{matrix}&
-                           \begin{pmatrix}
-                                &       &       \\
-                                &   A    &       \\
-                                &       &   
-                           \end{pmatrix}
-                       \end{array}
-               \end{pmatrix}.
-           \end{equation}
-           Un tout petit peu de calcul de produit de matrice montre que la matrice de \( u^k\) est de la forme
-           \begin{equation}
-               \begin{pmatrix}
-                       \begin{array}[]{c|c}
-                           0&\begin{matrix} 
-                               * &   *    &   *    
-                           \end{matrix}\\
-                           \hline
-                           \begin{matrix}
-                               0 \\ 
-                               0 \\ 
-                               0 
-                           \end{matrix}&
-                           \begin{pmatrix}
-                                &       &       \\
-                                &   A^k    &       \\
-                                &       &   
-                           \end{pmatrix}
-                       \end{array}
-               \end{pmatrix}.
-           \end{equation}
-           Étant donné que \( u\) est nilpotente, la matrice \( A\) l'est aussi. L'hypothèse de récurrence dit alors que \( A\) est strictement triangulaire supérieure (ou en tout cas peut le devenir par un changement de base adéquat).
-
-       \item[\( \Leftarrow\)]
-
-            Lorsqu'une matrice est triangulaire supérieure stricte, elle applique
-            \begin{equation}
-                \Span\{ e_1,\ldots, e_k \}\to\Span\{ e_1,\ldots, e_{k-1} \}.
-            \end{equation}
-            Donc tout vecteur finit sur zéro si on lui applique \( u\) assez souvent.
-    \end{subproof}
-\end{proof}
-
-\begin{proposition}[Thème \ref{THEMEooPQKDooTAVKFH}]     \label{PROPooWTFWooXHlmhp}
-    Soit \( E\) un espace de Banach (espace vectoriel normé complet). Si \( A\in\aL(E,E)\) est nilpotente, alors \( (\mtu-A)\) est inversible et son inverse est donné par
-    \begin{equation}
-        (\mtu-A)^{-1}=\sum_{k=0}^{\infty}A^k,
-    \end{equation}
-    où l'infini peut évidemment être remplacé par l'ordre de nilpotence de \( A\).
-\end{proposition}
-
-\begin{proof}
-    En ce qui concerne la convergence de la somme, elle ne fait pas de doutes parce que \( A\) étant nilpotente, la somme contient seulement une quantité finie de termes non nuls.
-
-    Montrons à présent que la somme est l'inverse de \( \mtu-A\) en multipliant terme à terme :
-    \begin{equation}
-        \sum_{k=0}^nA^k(\mtu-A)=\sum_{k=0}^n(A^k-A^{k+1})=\mtu-A^{n+1}.
-    \end{equation}
-    Par conséquent 
-    \begin{equation}
-        \| \mtu-\sum_{k=0}^nA^k(\mtu-A) \|=\| A^{n+1} \|\to 0.
-    \end{equation}
-    La dernière limite est en réalité une égalité pour \( n\) assez grand.
-\end{proof}
-
-\begin{proposition}
-    Soit \( A\in\GL(n,\eC)\). La suite \( (A^k)_{k\in \eZ}\) est bornée si et seulement si \( A\) est diagonalisable et \( \Spec(A)\subset \gS^1\).
-\end{proposition}
-
-\begin{proof}
-    Si \( A\) est diagonalisable avec les valeurs propres \( \lambda_i\) de norme \( 1\) dans \( \eC\), alors \( A^k\) est la matrice diagonale avec les \( \lambda_i^k\) sur la diagonale. Cela reste borné pour toute valeur entière de \( k\).
-
-    En ce qui concerne l'autre sens, nous supposons encore que
-    \begin{equation}
-        A=\begin{pmatrix}
-            \lambda_1\mtu+N_1    &       &       \\
-                &   \ddots    &       \\
-                &       &   \lambda_s\mtu+N_s
-        \end{pmatrix},
-    \end{equation}
-    et nous regardons un des blocs. Nous voulons prouver que \( N=0\) et que \( | \lambda |=1\).
-    
-    Nous commençons par regarder ce qu'implique le fait que \( (\lambda \mtu+N)^n\) reste borné pour \( n>0\). En notant \( r\) l'ordre de nilpotence de \( N\), nous avons le développement
-    \begin{equation}
-        (\lambda\mtu+N)^n=\sum_{k=0}^{r-1}\binom{ n }{ k }N^k\lambda^{n-k}.
-    \end{equation}
-    Par la proposition \ref{PropMWWJooVIXdJp}, une matrice nilpotente s'écrit dans une base sous la forme
-    \begin{equation}
-        N=\begin{pmatrix}
-             0   &   1    &       &       \\
-                &   0    &   1    &       \\
-                & &   \ddots   &   \ddots    &      \\ 
-                &&       &   0    &   1     \\
-                &&       &      &   0     
-         \end{pmatrix}
-    \end{equation}
-    et effectuer \( A^k\) revient à décaler la diagonale de \( 1\). Donc la famille
-    \begin{equation}
-        \{ \mtu,N,\ldots, N^{r-1} \}
-    \end{equation}
-    est libre. Par conséquent la suite \( (\lambda\mtu+N)^n\) restera bornée si et seulement si chacun des termes 
-    \begin{equation}    \label{EqXRDVDCM}
-        \binom{ n }{ k }N^k\lambda^{n-k}
-    \end{equation}
-    reste borné. Le premier terme étant \( \lambda^n\mtu\), nous avons obligatoirement \( | \lambda |\leq 1\). Si \( | \lambda |<1\), alors le coefficient \( \binom{ n }{ k }\lambda^{n-k}\) tend vers zéro. Si \( | \lambda |=1\) par contre ce coefficient tend vers l'infini et la seule façon pour que \eqref{EqXRDVDCM} reste borné est que \( N=0\). Nous avons donc deux possibilités :
-    \begin{itemize}
-        \item \( | \lambda |<1\)
-        \item \( | \lambda |=1\) et \( N=0\).
-    \end{itemize}
-
-    Nous nous tournons maintenant sur la contrainte que \( (\lambda\mtu+N)^n\) doive rester borné pour \( n<0\). Nous avons
-    \begin{equation}
-        \lambda\mtu+N=\lambda(\mtu+\lambda^{-1}N),
-    \end{equation}
-    et nous pouvons appliquer la proposition \ref{PROPooWTFWooXHlmhp} à l'opérateur nilpotent \( -\lambda^{-1} N\) pour avoir
-    \begin{equation}
-        (\mtu+\lambda^{-1}N)^{-1}=\mtu+\sum_{k=1}^{\infty}(-\lambda)^{-1}N^k.
-    \end{equation}
-    Ceci pour dire que \( (\lambda\mtu+N)^{-1}=\lambda^{-1}(\mtu+\lambda^{-1}N')\) pour une autre matrice nilpotente \( N'\). Le travail déjà fait, appliqué à \( \lambda^{-1}\) et \( N'\), nous donne deux possibilités :
-    \begin{itemize}
-        \item \( | \lambda^{-1} |<1\)
-        \item \( | \lambda^{-1} |=1\) et \( N'=0\).
-    \end{itemize}
-    La possibilité \( | \lambda^{-1} |<1\) est exclue parce qu'elle impliquerait \( | \lambda |>1\) qui avait déjà été exclu. Il ne reste donc que la possibilité \( | \lambda |=1\) et \( N=N'=0\).
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Endomorphismes diagonalisables}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}  \label{DefCNJqsmo}
-    Une matrice est \defe{diagonalisable}{diagonalisable} si elle est semblable à une matrice diagonale.
-\end{definition}
-
-\begin{lemma}
-    Une matrice triangulaire supérieure avec des \( 1\) sur la diagonale n'est diagonalisable que si elle est diagonale (c'est à dire si elle est la matrice unité).
-\end{lemma}
-
-\begin{proof}
-    Si \( A\) est une matrice triangulaire supérieure de taille \( n\) telle que \( A_{ii}=1\), alors \( \det(A-\lambda\mtu)=(1-\lambda)^n\), ce qui signifie que \( \Spec(A)=\{ 1 \}\). Pour la diagonaliser, il faudrait une matrice \( P\in\GL(n,\eK)\) telle que \( \mtu=P^{-1}AP\), ce qui est uniquement possible si \( A=\mtu\).
-\end{proof}
-
-\begin{lemma}       \label{LemgnaEOk}
-    Soit \( F\) un sous-espace stable par \( u\). Soit une décomposition du polynôme minimal
-    \begin{equation}
-        \mu_u=P_1^{n_1}\ldots P_r^{n_r}
-    \end{equation}
-    où les \( P_i\) sont des polynômes irréductibles unitaires distincts. Si nous posons \( E_i=\ker P_i^{n_i}\), alors
-    \begin{equation}
-        F=(F\cap E_1)\oplus\ldots \oplus(F\cap E_r).
-    \end{equation}
-\end{lemma}
-
-\begin{theorem}     \label{ThoDigLEQEXR}
-    Soit \( E\), un espace vectoriel de dimension \( n\) sur le corps commutatif \( \eK\) et \( u\in\End(E)\). Les propriétés suivantes sont équivalentes.
-    \begin{enumerate}
-        \item\label{ItemThoDigLEQEXRiv}
-            L'endomorphisme \( u\) est diagonalisable.
-        \item       \label{ItemThoDigLEQEXRi}
-            Il existe un polynôme \( P\in\eK[X]\) non constant, scindé sur \(\eK\) dont toutes les racines sont simples tel que \( P(u)=0\).
-        \item\label{ItemThoDigLEQEXRii}
-            Le polynôme minimal \( \mu_u\) est scindé sur \(\eK\) et toutes ses racines sont simples\footnote{Le polynôme \emph{caractéristique}, lui, n'a pas spécialement ses racines simples; il peut encore être de la forme
-            \begin{equation}
-                \chi_u(X)=\prod_{i=1}^r(X-\lambda_i)^{\alpha_i},
-        \end{equation}
-        mais alors \( \dim(E_{\lambda_i})=\alpha_i\). }.
-        \item\label{ItemThoDigLEQEXRiii}
-            Tout sous-espace de \( E\) possède un supplémentaire stable par \( u\).
-        \item       \label{ITEMooZNJFooEiqDYp}
-            Dans une base adaptée, la matrice de \( u\) est diagonale et les éléments diagonaux sont ses valeurs propres.
-    \end{enumerate}
-\end{theorem}
-\index{diagonalisable!et polynôme minimum scindé}
-
-\begin{proof}
-    Plein d'implications à prouver.
-    \begin{subproof}
-    \item[\ref{ItemThoDigLEQEXRi} implique \ref{ItemThoDigLEQEXRii}] Étant donné que \( P(u)=0\), il est dans l'idéal des polynôme annulateurs de \( u\), et le polynôme minimal \( \mu_u\) le divise parce que l'idéal des polynôme annulateurs est généré par \( \mu_u\) par le théorème \ref{ThoCCHkoU}.
-
-    \item[\ref{ItemThoDigLEQEXRii} implique \ref{ItemThoDigLEQEXRiv}] Étant donné que le polynôme minimal est scindé à racines simples, il s'écrit sous forme de produits de monômes tous distincts, c'est à dire
-    \begin{equation}
-        \mu_u(X)=(X-\lambda_1)\ldots(X-\lambda_r)
-    \end{equation}
-    où les \( \lambda_i\) sont des éléments distincts de \( \eK\). Étant donné que \( \mu_u(u)=0\), le théorème de décomposition des noyaux (théorème \ref{ThoDecompNoyayzzMWod}) nous enseigne que
-    \begin{equation}
-        E=\ker(u-\lambda_1)\oplus\ldots\oplus\ker(u-\lambda_r).
-    \end{equation}
-    Mais \( \ker(u-\lambda_i)\) est l'espace propre \( E_{\lambda_i}(u)\). Donc \( u\) est diagonalisable.
-
-\item[\ref{ItemThoDigLEQEXRiv} implique \ref{ItemThoDigLEQEXRiii}] Soit \( \{ e_1,\ldots, e_n \}\) une base qui diagonalise \( u\), soit \( F\) un sous-espace de \( E\) un \( \{ f_1,\ldots, f_r \}\) une base de \( F\). Par le théorème \ref{ThoBaseIncompjblieG} (qui généralise le théorème de la base incomplète), nous pouvons compléter la base de \( F\) par des éléments de la base \( \{ e_i \}\). Le complément ainsi construit est invariant par \( u\).
-
-\item[\ref{ItemThoDigLEQEXRiii} implique \ref{ItemThoDigLEQEXRiv}] En dimension un, tout endomorphisme est diagonalisable, nous supposons donc que \( \dim E=n\geq 2\). Nous procédons par récurrence sur le nombre de vecteurs propres connus de \( u\). Supposons avoir déjà trouvé \( p\) vecteurs propres \( e_1,\ldots, e_p\) de \( u\). Considérons \( H\), un hyperplan qui contient les vecteurs \( e_1,\ldots, e_p\). Soit \( F\) un supplémentaire de \( H\) stable par \( u\); par construction \( \dim F=1\) et si \( e_{p+1}\in F\), il doit être vecteur propre de \( u\).
-
-\item[\ref{ItemThoDigLEQEXRiv} implique \ref{ItemThoDigLEQEXRi}] Nous supposons maintenant que \( u\) est diagonalisable. Soient \( \lambda_1,\ldots, \lambda_r\) les valeurs propres deux à deux distinctes, et considérons le polynôme
-    \begin{equation}
-        P(x)=(X-\lambda_1)\ldots (X-\lambda_r).
-    \end{equation}
-    Alors \( P(u)=0\). En effet si \( e_i\) est un vecteur propre pour la valeur propre \( \lambda_i\), 
-    \begin{equation}
-        P(u)e_i=\prod_{j\neq i}(u-\lambda_j)\circ(u-\lambda_i)e_i=0
-    \end{equation}
-    par le lemme \ref{LemQWvhYb}. Par conséquent \( P(u)\) s'annule sur une base.
-
-\item[\ref{ITEMooZNJFooEiqDYp} implique \ref{ItemThoDigLEQEXRi}]
-    Si la matrice \( A\) est diagonale alors le polynôme \( P=\prod_{i=1}^n(A-A_{ii}\mtu)\) est annulateur de \( A\).
-        \item[\ref{ItemThoDigLEQEXRii} implique \ref{ITEMooZNJFooEiqDYp}]
-            le polynôme minimal de \( u\) s'écrit 
-            \begin{equation}
-                \mu=(X-\lambda_1)\ldots(X-\lambda_r),
-            \end{equation}
-            et les espaces $E_i$ du lemme \ref{LemgnaEOk} sont les espaces propres \( E_i=\ker(u-\lambda_i)\). Nous avons donc une somme directe
-            \begin{equation}
-                E=E_1\oplus\ldots\oplus E_r.
-            \end{equation}
-            Dans chacun des espaces propres, $u$ a une matrice diagonale avec la valeur propre correspondante sur la diagonale. Une base de \( E\) constituée d'une base de chacun des espaces propres est donc une base comme nous en cherchons.
-    \end{subproof}
-\end{proof}
-
-\begin{corollary}       \label{CorQeVqsS}
-    Si \( u\) est diagonalisable et si \( F\) est une sous-espace stable par \( u\), alors
-    \begin{equation}
-        F=\bigoplus_{\lambda}E_{\lambda}(u)\cap F
-    \end{equation}
-    où \( E_{\lambda}(u)\) est l'espace propre de \( u\) pour la valeur propre \( \lambda\). En particulier la restriction de \( u\) à \( F\), \( u|_F\) est diagonalisable.
-\end{corollary}
-
-\begin{proof}
-    Par le théorème \ref{ThoDigLEQEXR}, le polynôme \( \mu_u\) est scindé et ne possède que des racines simples. Notons le
-    \begin{equation}
-        \mu_u(X)=(X-\lambda_1)\ldots (X-\lambda_r).
-    \end{equation}
-    Les espaces \( E_i\) du lemme \ref{LemgnaEOk} sont maintenant les espaces propres.
-
-    En ce qui concerne la diagonalisabilité de \( u|_F\), notons que nous avons une base de \( F\) composée de vecteurs dans les espaces \( E_{\lambda}(u)\). Cette base de \( F\) est une base de vecteurs propres de \( u\).
-\end{proof}
-
-\begin{lemma}
-    Soit \( E\) un \( \eK\)-espace vectoriel et \( u\in\End(E)\). Si \( \Card\big( \Spec(u) \big)=\dim(E)\) alors \( u\) est diagonalisable.
-\end{lemma}
-
-\begin{proof}
-    Soient \( \lambda_1,\ldots, \lambda_n\) les valeurs propres distinctes de \( u\). Nous savons que les espaces propres correspondants sont en somme directe (lemme \ref{LemjcztYH}). Par conséquent \( \Span\{ E_{\lambda_i}(u) \}\) est de dimension \( n\) est \( u\) est diagonalisable.
-\end{proof}
-
-Voici un résultat de diagonalisation simultanée. Nous donnerons un résultat de trigonalisation simultanée dans le lemme \ref{LemSLGPooIghEPI}.
-\begin{proposition}[Diagonalisation simultanée]     \label{PropGqhAMei}
-    Soit \( (u_i)_{i\in I}\) une famille d'endomorphismes qui commutent deux à deux.
-    \begin{enumerate}
-        \item       \label{ItemGqhAMei}
-            Si \( i,j\in I\) alors tout sous-espace propre de \( u_i\) est stable par \( u_j\). Autrement dit \( u_j\big(E_{\lambda}(u)\big)\subset E_{\lambda}(u)\).
-        \item
-            Si les \( u_i\) sont diagonalisables, alors ils le sont simultanément.
-    \end{enumerate}
-\end{proposition}
-\index{diagonalisation!simultanée}
-
-\begin{proof}
-    Supposons que \( u_i\) et \( u_j\) commutent et soit \( x\) un vecteur propre de \( u_i\) : \( u_ix=\lambda x\). Nous montrons que \( u_jx\in E_{\lambda}(u)\). Nous avons
-    \begin{equation}
-        u_i\big( u_j(x) \big)=u_j\big( u_i(x) \big)=\lambda u_j(x).
-    \end{equation}
-    Par conséquent \( u_j(x)\) est vecteur propre de \( u_i\) de valeur propre \( \lambda\).
-
-    Montrons maintenant l'affirmation à propos des endomorphismes simultanément diagonalisables. Si \( \dim E=1\), le résultat est évident. Nous supposons également qu'aucun des \( u_i\) n'est multiple de l'identité. Nous effectuons une récurrence sur la dimension.
-
-    Soit \( u_0\) un des \( u_i\) et considérons ses valeurs propres deux à deux distinctes \( \lambda_1,\ldots, \lambda_r\). Pour chaque \( k\) nous avons
-    \begin{equation}
-        E_{\lambda_k}(u_0)\neq E,
-    \end{equation}
-    sinon \( u_0\) serait un multiple de l'identité. Par contre le fait que \( u_0\) soit diagonalisable permet de décomposer \( E\) en espaces propres de \( u_0\) :
-    \begin{equation}
-        E=\bigoplus_{k}E_{\lambda_k}(u_0).
-    \end{equation}
-    Ce que nous allons faire est de simultanément diagonaliser les \( (u_i)_{i\in I}\) sur chacun des \( E_{\lambda_k}\) séparément. Par le point \ref{ItemGqhAMei}, nous avons \( u_i\colon E_{\lambda_k}(u_0)\to E_{\lambda_k}(u_0)\), et nous pouvons considérer la famille d'opérateurs
-    \begin{equation}
-        \left( u_i|_{E_{\lambda_k}(u_0)} \right)_{i\in I}.
-    \end{equation}
-    Ce sont tous des opérateurs qui commutent et qui agissent sur un espace de dimension plus petite. Par hypothèse de récurrence nous avons une base de \( E_{\lambda_k}(u_0)\) qui diagonalise tous les \( u_i\).
-\end{proof}
-
-\begin{example}     \label{ExewINgYo}
-    Soit un espace vectoriel sur un corps \( \eK\). Un opérateur \defe{involutif}{involution} est un opérateur différent de l'identité dont le carré est l'identité. Typiquement une symétrie orthogonale dans \( \eR^3\). Le polynôme caractéristique d'une involution est \( X^2-1=(X+1)(X-1)\).
-    
-    Tant que \( 1\neq -1\), \( X^1-1\) est donc scindé à racines simples et les involutions sont diagonalisables (\ref{ThoDigLEQEXR}). Cependant si le corps est de caractéristique \( 2\), alors \( X^2-1=(X+1)^2\) et l'involution n'est plus diagonalisable.
-
-    Par exemple si le corps est de caractéristique \( 2\), nous avons
-    \begin{subequations}
-        \begin{align}
-            A&=\begin{pmatrix}
-                1    &   1    \\ 
-                0    &   1    
-            \end{pmatrix}\\
-            A^1&=\begin{pmatrix}
-                1    &   2    \\ 
-                0    &   1    
-            \end{pmatrix}=\begin{pmatrix}
-                1    &   0    \\ 
-                0    &   1    
-            \end{pmatrix}.
-        \end{align}
-    \end{subequations}
-    Ce \( A\) est donc une involution mais n'est pas diagonalisable.
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Diagonalisation : cas complexe, pas toujours}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Il n'est pas vrai qu'une matrice de \( \eM(n,\eC)\) soit toujours diagonalisable. En effet le théorème \ref{ThoDigLEQEXR}\ref{ItemThoDigLEQEXRii} dit qu'une matrice est diagonalisable si et seulement si son polynôme minimal est scindé à racines simples. Certes sur \( \eC\) le polynôme minimal sera scindé, mais il ne sera pas spécialement à racines simples.
-
-\begin{example}
-    La matrice
-    \begin{equation}
-        A=\begin{pmatrix}
-            0    &   1    \\ 
-            0    &   0    
-        \end{pmatrix}
-    \end{equation}
-    a pour polynôme caractéristique \( \chi_A(X)=X^2\). Cela est également son polynôme minimal, et ce n'est pas à racine simple. 
-
-    Il est par ailleurs facile de voir que le seul espace propre de \( A\) est \( \Span\{ (1,0) \}\) (ici le span est sur \( \eC\)). Donc l'espace \( \eC^2\) ne possède pas de base de vecteurs propres de \( A\).
-\end{example}
-
-Ce qui est vrai, c'est que le polynôme caractéristique a des racines, et que ces racines correspondent à des vecteurs propres. Mais il n'y a pas toujours autant de vecteurs propres que la multiplicité des racines.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Trigonalisation : généralités}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}[\cite{MQMKooPBfnZN}]
-    Une matrice dans \( \eM(n,\eK)\) est \defe{trigonalisable}{matrice!trigonalisable} lorsqu'elle est semblable\footnote{Définition \ref{DefCQNFooSDhDpB}.} à une matrice triangulaire supérieure.
-\end{definition}
-
-\begin{proposition}[Trigonalisation et polynôme caractéristique scindé] \label{PropKNVFooQflQsJ}
-    Soit \( u\) un endomorphisme d'un espace vectoriel \( E\) sur le corps \( \eK\). Les faits suivants sont équivalents.
-    \begin{enumerate}
-        \item   \label{ItemZKDMooOrTHkwi}
-            L'endomorphisme \( u\) est trigonalisable (auquel cas les valeurs propres sont sur la diagonale).
-        \item   \label{ItemZKDMooOrTHkwii}
-            Le polynôme caractéristique de \( u\) est scindé\footnote{Définition \ref{DefCPLSooQaHJKQ}.}.
-    \end{enumerate}
-\end{proposition}
-\index{trigonalisation!et polynôme caractéristique}
-
-\begin{proof}
-    \begin{subproof}
-        \item[\ref{ItemZKDMooOrTHkwii}\( \Rightarrow\)\ref{ItemZKDMooOrTHkwi}]
-            Nous avons par hypothèse que
-            \begin{equation}
-                \chi_u(X)=\prod_{i=1}^r(X-\lambda_i)^{\alpha_i}
-            \end{equation}
-            où les \( \lambda_i\) sont les valeurs propres de \( u\). Le théorème de Cayley-Hamilton \ref{ThoCalYWLbJQ} dit que \( \chi_u(u)=0\), ce qui permet d'utiliser le théorème de décomposition des noyaux \ref{ThoDecompNoyayzzMWod} :
-            \begin{equation}
-                E=\ker(X-\lambda_1)^{\alpha_1}\oplus\ldots\oplus\ker(X-\lambda_r)^{\alpha_r}.
-            \end{equation}
-            Les espaces \( F_{\lambda_i}(u)=\ker(X-\lambda_i)^{\alpha_i}\) sont les espaces caractéristiques de \( u\), ce qui fait que \( u-\lambda_i\mtu\) est nilpotent sur \( F_{\lambda_i}(u)\). L'endomorphisme \( u-\lambda_i\mtu\) est donc strictement trigonalisable supérieur sur son bloc\footnote{Proposition \ref{PropMWWJooVIXdJp}.}. Cela signifie que \( u\) est triangulaire supérieure avec les valeurs propres sur la diagonale.
-
-        \item[\ref{ItemZKDMooOrTHkwi}\( \Rightarrow\)\ref{ItemZKDMooOrTHkwii}]
-
-            C'est immédiat parce que le déterminant d'une matrice triangulaire est le produit des éléments de sa diagonale.
-    \end{subproof}
-\end{proof}
-
-\begin{remark}
-    La méthode des pivots de Gauss\footnote{Le lemme \ref{LemZMxxnfM}.} certes permet de trigonaliser n'importe quoi, mais elle ne correspond pas à un changement de base. Autrement dit, les pivots de Gauss ne sont pas de similitudes.
-
-    C'est là qu'il faut bien avoir en tête la différence entre \emph{équivalence} et \emph{similarité} (définition \ref{DefBLELooTvlHoB}). Lorsqu'on parle de changement de base, de matrice trigonalisable ou diagonalisable, nous parlons de similarité et non d'équivalence.
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Trigonalisation : cas complexe}
-%---------------------------------------------------------------------------------------------------------------------------
-
-La proposition \ref{PropKNVFooQflQsJ} dit déjà que tous les endomorphismes sont trigonalisables sur \( \eC\). Nous allons aller plus loin et montrer que la trigonalisation peut être effectuée à l'aide d'une matrice unitaire. 
-
-Une démonstration alternative passant par le polynôme caractéristique sera présentée dans la remarque \ref{RemXFZTooXkGzQg} utilisant la proposition \ref{PropKNVFooQflQsJ}.
-\begin{lemma}[Lemme de Schur complexe, trigonisation\cite{NormHKNPKRqV}]  \label{LemSchurComplHAftTq}
-    Si \( A\in\eM(n,\eC)\), il existe une matrice unitaire \( U\) telle que \( UAU^{-1}\) soit triangulaire supérieure.
-\end{lemma}
-\index{lemme!Schur complexe}
-%TODO : Le lemme de Schur est souvent énoncé en disant que si p est une représentation irréductible, alors les seuls endomorphismes de V commutant avec tous les p(g) sont les multiples de l'idenditié. Quel est le lien avec ceci ?
-
-\begin{proof}
-    Étant donné que \( \eC\) est algébriquement clos, nous pouvons toujours considérer un vecteur propre \( v_1\) de \( A\), de valeur propre \( \lambda_1\). Nous pouvons utiliser un procédé de Gram-Schmidt pour construire une base orthonormée \( \{ v,u_2,\ldots, u_n \}\) de \( \eR^n\), et la matrice (unitaire)
-    \begin{equation}
-        Q=\begin{pmatrix}
-             \uparrow   &   \uparrow    &       &   \uparrow    \\
-             v   &   u_2    &   \cdots    &   u_n    \\ 
-             \downarrow   &   \downarrow    &       &   \downarrow
-         \end{pmatrix}.
-    \end{equation}
-    Nous avons \( Q^{-1}AQe_1=Q^{-1} Av=\lambda Q^{-1} v=\lambda e_1\), par conséquent la matrice \( Q^{-1} AQ\) est de la forme
-    \begin{equation}
-        Q^{-1}AQ=\begin{pmatrix}
-            \lambda_1    &   *    \\ 
-            0    &   A_1    
-        \end{pmatrix}
-    \end{equation}
-    où \( *\) représente une ligne quelconque et \( A_1\) est une matrice de \( \eM(n-1,\eC)\). Nous pouvons donc répéter le processus sur \( A_1\) et obtenir une matrice triangulaire supérieure (nous utilisons le fait qu'un produit de matrices orthogonales est une matrice orthogonale).  
-\end{proof}
-En particulier les matrices hermitiennes, anti-hermitiennes et unitaires sont trigonalisables par une matrice unitaire, qui peut être choisie de déterminant \( 1\).
-
-\begin{lemma}       \label{LEMooRCFGooPPXiKi}
-    Soit \( A\in \eM(n,\eC)\) et une matrice unitaire \( U\) telle que \( A=UTU^{-1}\) où \( T\) est triangulaire. 
-    \begin{enumerate}
-        \item
-            En ce qui concerne les polynômes caractéristiques, \( \chi_A=\chi_T\).
-        \item
-            Pour les spectres, \( \Spec(A)=\Spec(T)\).
-        \item
-            Les valeurs propres de \( A\) sont les éléments diagonaux de \( T\).
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Vu que \( U\) commute évidemment avec \( \mtu\) nous avons
-    \begin{equation}
-        \chi_A(\lambda)=\det(A-\lambda \mtu)=\det(UTU^{-1}-\lambda\mtu)=\det\big( U(T-\lambda\mtu)U^{-1} \big).
-    \end{equation}
-    À ce niveau nous utilisons le fait que le déterminant soit multiplicatif \ref{PropYQNMooZjlYlA} pour conclure :
-    \begin{equation}
-        \chi_A(\lambda)=\det\big( U(T-\lambda\mtu)U^{-1} \big)=\det(U)\det(T-\lambda\mtu)\det(U^{-1})=\det(T-\lambda\mtu)=\chi_T(\lambda).
-    \end{equation}
-
-    Pour les spectres, l'égalité des polynômes caractéristique implique l'égalité des spectres parce que les valeurs propres sont les racines du polynôme caractéristique par le théorème \ref{ThoWDGooQUGSTL}.
-    
-    Les valeurs propres d'une matrice triangulaire sont les valeurs sur la diagonale.
-\end{proof}
-
-\begin{remark}
-    Le lemme mentionne le fait que les valeurs propres de \( A\) sont les éléments diagonaux de \( T\). Mais attention : ceci ne dit rien au niveau des multiplicités géométriques. Un nombre peut être cinq fois sur la diagonale de \( T\) alors que l'espace propre correspondant pour \( A\) n'est que de dimension \( 1\). Exemple : la matrice
-    \begin{equation}
-        A=\begin{pmatrix}
-            1    &   1    \\ 
-            0    &   1    
-        \end{pmatrix}
-    \end{equation}
-    a deux \( 1\) sur la diagonale. Le nombre \( 1\) est bien une valeur propre de \( A\), mais le système
-    \begin{equation}
-        A\begin{pmatrix}
-            x    \\ 
-            y    
-        \end{pmatrix}=\begin{pmatrix}
-            x    \\ 
-            y    
-        \end{pmatrix}
-    \end{equation}
-    donne \( y=0\) et donc un espace propre de dimension seulement \( 1\).
-\end{remark}
-
-\begin{remark}  \label{RemXFZTooXkGzQg}
-    Si \( \eK\) est algébriquement clos (comme \( \eC\) par exemple), alors tous les polynômes sont scindés et toutes les matrices sont trigonalisables\footnote{La proposition \ref{PropKNVFooQflQsJ} montre cela, et le lemme de Schur complexe \ref{LemSchurComplHAftTq} va un peu plus loin et précise que la trigonalisation peut être faite par une matrice unitaire.}. Un exemple un peu simple de cela est la matrice
-    \begin{equation}
-        u=\begin{pmatrix}
-            0    &   -1    \\ 
-            1    &   0    
-        \end{pmatrix}.
-    \end{equation}
-    Le polynôme caractéristique est \( \chi_u(X)=X^2+1\) et les valeurs propres sont \( \pm i\). Il est vite vu que dans la base
-    \begin{equation}
-        \{ \begin{pmatrix}
-        i    \\ 
-    1    
-\end{pmatrix}, \begin{pmatrix}
-1    \\ 
-i    
-\end{pmatrix}\}
-    \end{equation}
-    de \( \eC^2\), la matrice \( u\) se note \( \begin{pmatrix}
-        i    &   0    \\ 
-        0    &   -i    
-    \end{pmatrix}\).
-\end{remark}
-
-\begin{remark}  \label{RemREOSooGEDJWX}
-    Cela nous donne une autre façon de prouver qu'une matrice nilpotente de \( \eM(n,\eC)\) ou \( \eM(n,\eR)\) est trigonalisable\cite{KDUFooVxwqlC}. D'abord dans \( \eM(n,\eC)\), toutes les matrices sont trigonalisables\footnote{Parce que le polynôme caractéristique est scindé, voir la proposition \ref{PropKNVFooQflQsJ}..}, et les valeurs propres arrivent sur la diagonale. Mais comme les valeurs propres d'une matrice nilpotente sont zéro, elle est triangulaire stricte. Par ailleurs son polynôme caractéristique est alors \( X^n\).
-
-    Ensuite si \( u\in \eM(n,\eR)\) nous pouvons voir \( u\) comme une matrice dans \( \eM(n,\eC)\) et y calculer son polynôme caractéristique qui sera tout de même \( X^n\). Ce polynôme étant scindé, la proposition \ref{PropKNVFooQflQsJ} nous assure que \( u\) est trigonalisable. Une fois de plus, les valeurs propres étant sur la diagonale, elle est triangulaire supérieure stricte.
-\end{remark}
-
-\begin{corollary}   \label{CorUNZooAZULXT}
-    Le polynôme caractéristique\footnote{Définition \ref{DefOWQooXbybYD}.} sur \( \eC\) d'une matrice s'écrit sous la forme
-    \begin{equation}
-        \chi_A(X)=\prod_{i=1}^r(X-\lambda_i)^{m_i}
-    \end{equation}
-    où les \( \lambda_i\) sont les valeurs propres distinctes de \( A\) et \( m_i\) sont les multiplicités correspondantes.
-\end{corollary}
-\index{polynôme!caractéristique}
-
-\begin{proof}
-    Le lemme \ref{LemSchurComplHAftTq} nous donne l'existence d'une base de trigonalisation; dans cette base les valeurs propres de \( A\) sont sur la diagonale et nous avons 
-    \begin{equation}
-        \chi_A(X)=\det(A-X\mtu)=\det\begin{pmatrix}
-            X-\lambda_1    &   *    &   *    \\
-            0    &   \ddots    &   *    \\
-            0    &   0    &   X-\lambda_r
-        \end{pmatrix},
-    \end{equation}
-    qui vaut bien le produit annoncé.
-\end{proof}
-
-\begin{corollary}       \label{CORooTPDHooXazTuZ}
-    Si \( A\in \eM(n,\eC)\) et \( k\in \eN\) alors
-    \begin{equation}
-        \Spec(A^k)=\{ \lambda^k\tq \lambda\in \Spec(A) \}.
-    \end{equation}
-\end{corollary}
-
-\begin{proof}
-    Par le lemme \ref{LemSchurComplHAftTq} nous avons une matrice unitaire \( U\) et une triangulaire \( T\) telles que \( A=UTU^{-1}\). En passant à a puissance \( k\) nous avons aussi
-    \begin{equation}
-        A^k=UT^kU^{-1}.
-    \end{equation}
-    Donc le spectre de \( A^k\) est celui de \( T^k\) (lemme \ref{LEMooRCFGooPPXiKi} et le fait qu'une puissance d'une matrice triangulaire est encore triangulaire). Or les éléments diagonaux de \( T^k\) sont les puissance \( k\)\ieme des éléments diagonaux de \( T\), qui sont les valeurs propres de \( A\).
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Diagonalisation : cas complexe, ce qu'on a}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}[Théorème spectral hermitien]      \label{LEMooVCEOooIXnTpp}
-    Pour un opérateur hermitien\footnote{Définition \ref{DEFooKEBHooWwCKRK}.},
-    \begin{enumerate}
-        \item
-            le spectre est réel,
-        \item
-            deux vecteurs propres à des valeurs propres distinctes sont orthogonales\footnote{Pour la forme \eqref{EqFormSesqQrjyPH}.}.
-    \end{enumerate}
-\end{lemma}
-\index{spectre!matrice hermitienne}
-
-\begin{proof}
-    Soit \( v\) un vecteur de valeur propre \( \lambda\). Nous avons d'une part 
-    \begin{equation}
-        \langle Av, v\rangle =\lambda\langle v, v\rangle =\lambda\| v \|^2,
-    \end{equation}
-    et d'autre part, en utilisant le fait que \( A\) est hermitien,
-    \begin{equation}
-        \langle Av, v\rangle =\langle v, A^*v\rangle =\langle v, Av\rangle =\bar\lambda\| v \|^2,
-    \end{equation}
-    par conséquent \( \lambda=\bar\lambda\) parce que \( v\neq 0\).
-
-    Soient \( \lambda_i\) et \( v_i\) (\( i=1,2\)) deux valeurs propres de \( A\) avec leurs vecteurs propres correspondants. Alors d'une part
-    \begin{equation}
-        \langle Av_1, v_2\rangle =\lambda_1\langle v_1, v_2\rangle ,
-    \end{equation}
-    et d'autre part
-    \begin{equation}
-        \langle Av_1, v_2\rangle =\langle v_1, Av_2\rangle =\lambda_2\langle v_1, v_2\rangle .
-    \end{equation}
-    Nous avons utilisé le fait que \( \lambda_2\) était réel. Par conséquent, soit \( \lambda_1=\lambda_2\), soit \( \langle v_1, v_2\rangle =0\).
-\end{proof}
-
-\begin{remark}      \label{REMooMLBCooTuKFmz}
-    Un opérateur de la forme \( A^*A\) est évidemment hermitien. De plus ses valeurs propres sont toutes positives parce que si \( A^*Ax=\lambda v\) alors
-    \begin{equation}
-        0\leq \langle Av, Av\rangle =\langle A^*Av, v\rangle =\lambda\langle v, v\rangle .
-    \end{equation}
-    Donc \( \lambda\geq 0\).
-\end{remark}
-
-\begin{definition}  \label{DefWQNooKEeJzv}
-    Un endomorphisme est \defe{normal}{normal!endomorphisme}\index{matrice!normale} s'il commute avec son adjoint.
-\end{definition}
-
-\begin{theorem}[Théorème spectral pour les matrices normales\footnote{Définition \ref{DefWQNooKEeJzv}}\cite{LecLinAlgAllen,OMzxpxE}]\index{théorème!spectral!matrices normales}  \index{diagonalisation!cas complexe}  \label{ThogammwA}
-    Soit \( A\in\eM(n,\eC)\) une matrice de valeurs propres \( \lambda_1,\ldots, \lambda_n\) (non spécialement distinctes). Alors les conditions suivantes sont équivalentes :
-    \begin{enumerate}
-        \item   \label{ItemJZhFPSi}
-            \( A\) est normale,
-        \item   \label{ItemJZhFPSii}
-            \( A\) se diagonalise par une matrice unitaire,
-        \item
-            \( \sum_{i,j=1}^n| A_{ij} |^2=\sum_{j=1}^n| \lambda_j |^2\),
-        \item
-            il existe une base orthonormale de vecteurs propres de \( A\).
-    \end{enumerate}
-\end{theorem}
-
-\begin{proof}
-    Nous allons nous contenter de prouver \ref{ItemJZhFPSi}\( \Leftrightarrow\)\ref{ItemJZhFPSii}.
-    %TODO : le reste.
-
-    Soit \( Q\) la matrice unitaire donnée par la décomposition de Schur (lemme \ref{LemSchurComplHAftTq}) : \( A=QTQ^{-1}\). Étant donné que \( A\) est normale nous avons
-    \begin{equation}
-        QTT^*Q^{-1}=QT^*TQ^{-1},
-    \end{equation}
-    ce qui montre que \( T\) est également normale. Or une matrice triangulaire supérieure normale est diagonale. En effet nous avons \( T_{ij}=0\) lorsque \( i>j\) et
-    \begin{equation}
-        (TT^*)_{ii}=(T^*T)_{ii}=\sum_{k=1}^n| T_{ki} |^2=\sum_{k=1}^n| T_{ik} |^2.
-    \end{equation}
-    Écrivons cela pour \( i=1\) en tenant compte de \( | T_{k1} |^2=0\) pour \( k=2,\ldots, n\),
-    \begin{equation}
-        | T_{11} |^2=| T_{11} |^2+| T_{12} |^2+\cdots+| T_{1n} |^2,
-    \end{equation}
-    ce qui implique que \( T_{11}\) est le seul non nul parmi les \( T_{1k}\). En continuant de la sorte avec \( i=2,\ldots, n\) nous trouvons que \( T\) est diagonale.
-
-    Dans l'autre sens, si \( A\) se diagonalise par une matrice unitaire, \( UAU^*=D\), nous avons
-    \begin{equation}
-        DD^*=UAA^*U^*
-    \end{equation}
-    et 
-    \begin{equation}
-        D^*D=UA^*AU^*,
-    \end{equation}
-    qui ce prouve que \( A\) est normale.
-\end{proof}
-
-Tant que nous en sommes à parler de spectre de matrices hermitiennes\ldots Soit une matrice inversible \( A\in \GL(n,\eC)\). La matrice \( A^*A\) est hermitienne\footnote{Définition \ref{DEFooKEBHooWwCKRK}.} et le théorème \ref{LEMooVCEOooIXnTpp} nous assure que ses valeurs propres sont réelles. Par la remarque \ref{REMooMLBCooTuKFmz}, ses valeurs propres sont même positives.
-
-\begin{lemma}[\cite{ooLMMRooUXhOdx}]   \label{LEMooHUGEooVYhZdZ}
-    Si \( A\) est une matrice carré et inversible,
-    \begin{equation}
-        \Spec(A^*A)=\Spec(AA^*)
-    \end{equation}
-\end{lemma}
-
-\begin{proof}
-    Nous allons montrer l'égalité des polynômes caractéristiques. D'abord une simple multiplication montre que
-    \begin{equation}
-        (A^*A-\lambda\mtu)A^{-1}=A^{-1}(AA^*-\lambda\mtu).
-    \end{equation}
-    Nous prenons le déterminant de cette égalité en utilisant les propriétés \ref{PropYQNMooZjlYlA}\ref{ItemUPLNooYZMRJy} et \ref{ITEMooZMVXooLGjvCy} :
-    \begin{equation}
-        \det(A^*A-\lambda\mtu)\det(A^{-1})=\det(A^{-1})\det(AA^*-\lambda\mtu).
-    \end{equation}
-    En simplifiant par \( \det(A^{-1})\) (qui est non nul parce que \( A\) est inversible) nous obtenons l'égalité des polynômes caractéristiques et donc l'égalité des spectres.
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Diagonalisation : cas réel}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}[Lemme de Schur réel]  \label{LemSchureRelnrqfiy}
-    Soit \( A\in\eM(n,\eR)\). Il existe une matrice orthogonale \( Q\) telle que \( Q^{-1}AQ\) soit de la forme
-    \begin{equation}        \label{EqMtrTSqRTA}
-        QAQ^{-1}=\begin{pmatrix}
-            \lambda_1    &   *    &   *    &   *    &   *\\  
-            0    &   \ddots    &   \ddots    &   \ddots    &   \vdots\\  
-            0    &   0    &   \lambda_r    &   *    &   *\\  
-            0    &   0    &   0    &   \begin{pmatrix}
-                a_1    &   b_1    \\ 
-                c_1    &   d_1    
-            \end{pmatrix}&   *\\  
-            0    &   0    &  0     &   0    &   \begin{pmatrix}
-                a_s    &   b_s    \\ 
-                c_s    &   d_s    
-            \end{pmatrix}
-        \end{pmatrix}.
-    \end{equation}
-    Le déterminant de \( A\) est le produit des déterminants des blocs diagonaux et les valeurs propres de \( A\) sont les \( \lambda_1,\ldots, \lambda_r\) et celles de ces blocs.
-\end{lemma}
-\index{lemme!Schur réel}
-
-\begin{proof}
-    Si la matrice \( A\) a des valeurs propres réelles, nous procédons comme dans le cas complexe. Cela nous fournit le partie véritablement triangulaire avec les valeurs propres \( \lambda_1,\ldots, \lambda_r\) sur la diagonale. Supposons donc que \( A\) n'a pas de valeurs propres réelles. Soit donc \( \alpha+i\beta \) une valeur propre (\( \beta\neq 0\)) et \( u+iv\) un vecteur propre correspondant où \( u\) et \( v\) sont des vecteurs réels. Nous avons
-    \begin{equation}
-        Au+iAv=A(u+iv)=(\alpha+i\beta)(u+iv)=\alpha u-\beta v+i(\alpha v+\beta v),
-    \end{equation}
-    et en égalisant les parties réelles et imaginaires,
-    \begin{subequations}
-        \begin{align}
-            Au&=\alpha u-\beta v\\
-            Av&=\alpha v+\beta u.
-        \end{align}
-    \end{subequations}
-    Sur ces relations nous voyons que ni \( u\) ni \( v\) ne sont nuls. De plus \( u\) et \( v\) sont linéairement indépendants (sur \( \eR\)), en effet si \( v=\lambda u\) nous aurions \( Au=\alpha u-\beta\lambda u=(\alpha-\beta\lambda)u\), ce qui serait une valeur propre réelle alors que nous avions supposé avoir déjà épuisé toutes les valeurs propres réelles.
-
-    Étant donné que \( u\) et \( v\) sont deux vecteurs réels non nuls et linéairement indépendants, nous pouvons trouver une base orthonormée \( \{ q_1,q_2 \}\) de \( \Span\{ u,v \}\). Nous pouvons étendre ces deux vecteurs en une base orthonormée \( \{ q_1,q_2,q_3,\ldots, q_n \}\) de \( \eR^n\). Nous considérons à présent la matrice orthogonale dont les colonnes sont formées de ces vecteurs : \( Q=[q_1\,q_2\,\ldots q_n]\).
-
-    L'espace \( \Span\{ e_1,e_2 \}\) est stable par \( Q^{-1} AQ\), en effet nous avons
-    \begin{equation}
-        Q^{-1} AQe_1=Q^{-1} Aq_1=Q^{-1}(aq_1+bq_2)=ae_1+be_2.
-    \end{equation}
-    La matrice \( Q^{-1}AQ\) est donc de la forme
-    \begin{equation}
-        Q^{-1} AQ=\begin{pmatrix}
-            \begin{pmatrix}
-                \cdot    &   \cdot    \\ 
-                \cdot    &   \cdot    
-            \end{pmatrix}&   C_1    \\ 
-            0    &   A_1    
-        \end{pmatrix}
-    \end{equation}
-    où \( C_1\) est une matrice réelle \( 2\times (n-1)\) quelconque et \( A_1\) est une matrice réelle \( (n-2)\times (n-2)\). Nous pouvons appliquer une récurrence sur la dimension pour poursuivre.
-
-    Notons que si \( A\) n'a pas de valeurs propres réelles, elle est automatiquement d'ordre pair parce que les valeurs propres complexes viennent par couple complexes conjuguées.
-
-    En ce qui concerne les valeurs propres, il est facile de voir en regardant \eqref{EqMtrTSqRTA} que les valeurs propres sont celles des blocs diagonaux. Étant donné que \( QAQ^{-1}\) et \( A\) ont même polynôme caractéristique, ce sont les valeurs propres de \( A\).
-\end{proof}
-
-\begin{theorem}[Théorème spectral, matrice symétrique\cite{KXjFWKA}] \label{ThoeTMXla}
-    Une matrice symétrique réelle,
-    \begin{enumerate}
-        \item       \label{ITEMooJWHLooSfhNSW}
-            a un spectre contenu dans \( \eR\)
-        \item
-            est diagonalisable par une matrice orthogonale.
-    \end{enumerate}
-    Si \( M\) est une matrice symétrique réelle alors \( \eR^n\) possède une base orthonormée de vecteurs propres de \( M\).
-\end{theorem}
-\index{diagonalisation!cas réel}
-\index{rang!diagonalisation}
-\index{endomorphisme!diagonalisation}
-\index{spectre!matrice symétrique réelle}
-\index{théorème!spectral!matrice symétrique}
-
-\begin{proof}
-    Soit \( A\) une matrice réelle symétrique. Si \( \lambda\) est une valeur propre complexe pour le vecteur propre complexe \( v\), alors d'une part \( \langle Av, v\rangle =\lambda\langle v, v\rangle \) et d'autre part \( \langle Av, v\rangle =\langle v, Av\rangle =\bar\lambda\langle v, v\rangle \). Par conséquent \( \lambda=\bar\lambda\).
-    
-    Le lemme de Schur réel \ref{LemSchureRelnrqfiy} donne une matrice orthogonale qui trigonalise \( A\). Les valeurs propres étant toutes réelles, la matrice \( QAQ^{-1}\) est même triangulaire (il n'y a pas de blocs dans la forme \eqref{EqMtrTSqRTA}). Prouvons que \( QAQ^{-1}\) est symétrique :
-    \begin{equation}
-        (QAQ^{-1})^t=(Q^{-1})^tA^tQ^t=QA^tQ^{-1}=QAQ^{-1}
-    \end{equation}
-    où nous avons utilisé le fait que \( Q\) était orthogonale (\( Q^{-1}=Q^t\)) et que \( A\) était symétrique (\( A^t=A\)). Une matrice triangulaire supérieure symétrique est obligatoirement une matrice diagonale.
-
-    En ce qui concerne la base de vecteurs propres, soit \( \{ e_i \}_{i=1,\ldots, n}\) la base canonique de \( \eR^n\) et \( Q\) une matrice orthogonale e telle que \( A=Q^tDQ\) avec \( D\) diagonale. Nous posons \( f_i=Q^te_i\) et en tenant compte du fait que \( Q^t=Q^{-1}\) nous avons \( Af_i=Q^tDQQ^te_i=Q^t\lambda_i e_i=\lambda_if_i\). Donc les \( f_i\) sont des vecteurs propres de \( A\). De plus ils sont orthonormés parce que
-    \begin{equation}
-        \langle f_i, f_j\rangle =\langle Q^te_i, Q^te_j\rangle =\langle e_i, Q^tQe_j\rangle =\langle e_i, e_j\rangle =\delta_{ij}.
-    \end{equation}
-\end{proof}
-Le théorème spectral pour les opérateurs auto-adjoints sera traité plus bas parce qu'il a besoin de choses sur les formes bilinéaires, théorème \ref{ThoRSBahHH}.
-% et les choses sur la dégénérescences utilisent le théorème spectral, cas réel. Donc l'enchaînement est très loumapotiste.
-
-\begin{remark}  \label{RemGKDZfxu}
-    Une matrice symétrique est diagonalisable par une matrice orthogonale. Nous pouvons en réalité nous arranger pour diagonaliser par une matrice de \( \SO(n)\). Plus généralement si \( A\) est une matrice diagonalisable par une matrice \( P\in\GL^+(n,\eR)\) alors elle est diagonalisable par une matrice de \( \GL^-(n,\eR)\) en changeant le signe de la première ligne de \( P\). Et inversement.
-
-    En effet, si nous avons \( P^tDP=A\), alors en notant \( *\) les quantités qui ne dépendent pas de \( a\), \( b\) ou~\( c\),
-    \begin{equation}
-        \begin{aligned}[]
-        \begin{pmatrix}
-            a    &   *    &   *    \\
-            b    &   *    &   *    \\
-            c    &   *    &   *
-        \end{pmatrix}
-        \begin{pmatrix}
-            \lambda_1    &       &       \\
-                &   \lambda_2    &       \\
-                &       &   \lambda_3
-            \end{pmatrix}
-            \begin{pmatrix}
-                a    &   b    &   c    \\
-                *    &   *    &   *    \\
-                *    &   *    &   *
-            \end{pmatrix}&=
-        \begin{pmatrix}
-            a    &   *    &   *    \\
-            b    &   *    &   *    \\
-            c    &   *    &   *
-        \end{pmatrix}
-        \begin{pmatrix}
-            \lambda_1a    &   \lambda_1b    &   \lambda_1c    \\
-            *    &   *    &   *    \\
-            *    &   *    &   *
-        \end{pmatrix}\\
-        &=\begin{pmatrix}
-            \lambda_1 a^2+*   &   \lambda_1ab+*    &   \lambda_1ac  +*  \\
-            \ldots    &   \ldots    &   \ldots    \\
-            \ldots    &   \ldots    &   \ldots
-        \end{pmatrix}.
-        \end{aligned}
-    \end{equation}
-    Nous voyons donc que si nous changeons les signes de \( a\), \( b\) et \( c\) en même temps, le résultat ne change pas.
-\end{remark}
-
-\begin{definition}[Matrice définie positive, opérateur définit positif]    \label{DefAWAooCMPuVM}
-    Un opérateur sur un espace vectoriel sur \( \eC\) ou \( \eR\) est \defe{définit positif}{opérateur!définit positif} si toutes ses valeurs propres sont réelles et strictement positives.  Il est \defe{semi-définie positive}{semi-définie positive} si ses valeurs propres sont réelles positives ou nulles.
-\end{definition}
-Afin d'éviter l'une ou l'autre confusion, nous disons souvent \emph{strictement} définie positive pour positive.
-
-Nous notons \( S^+(n,\eR)\)\nomenclature[A]{\( S^+(n,\eR)\)}{matrices symétriques semi-définies positives} l'ensemble des matrices réelles \( n\times n\) semi-définies positives. L'ensemble \( S^{++}(n,\eR)\)\nomenclature[A]{\( S^{++}(n,\eR)\)}{matrices symétriques strictement définies positives} est l'ensemble des matrices symétriques strictement définies positives.
-
-\begin{remark}
-    Nous ne définissons pas la notion de matrice définie positive pour une matrice non symétrique.
-\end{remark}
-
-Lorsqu'un énoncé parle d'une matrice symétrique, le premier réflexe est de la diagonaliser : considérer une matrice orthogonale \( T\) telle que \( T^tMT=D\) avec \( D\) diagonale. Et les valeurs propres sur la diagonale : \( D_{kl}=\delta_{kl}\lambda_k\). Les matrices symétriques définies positives ont cependant des propriétés même en dehors de leur base de diagonalisation.
-
-\begin{lemma}   \label{LemWZFSooYvksjw}
-    Soit une matrice symétrique \( M\).
-    \begin{enumerate}
-        \item       \label{ITEMooSKRAooOgHbGA}
-           Elle est strictement définie positive si et seulement si \( \langle x, Mx\rangle >0\) pour tout \( x\) non nul dans \( \eR^n\).
-        \item       \label{ITEMooMOZYooWcrewZ}
-           Elle est semi définie positive si et seulement si \( \langle x, Mx\rangle \geq 0\) pour tout \( x\) non nul dans \( \eR^n\).
-       \item        \label{ITEMooRRMFooHSOHxZ}
-           Si elle est seulement définie positive, alors \( \langle x, Mx\rangle \geq \lambda\| x \|^2\) dès que \( \lambda\geq 0\) minore toutes les valeurs propres.
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Démonstration en trois parties.
-    \begin{subproof}
-    \item[\ref{ITEMooSKRAooOgHbGA}]
-    Soit \( \{ e_i \}_{i=1,\ldots, n}\) une base orthonormée de vecteurs propres de \( M\) dont l'existence est assurée par le théorème spectral \ref{ThoeTMXla}. Nous nommons \( x_i\) les coordonnées de \( x\) dans cette base. Alors,
-    \begin{equation}
-        \langle x,Mx \rangle =\sum_{i,j}x_i\langle e_i, x_jMe_j\rangle =\sum_{i,j}x_ix_j\langle e_i, \lambda_je_j\rangle =\sum_{ij}x_ix_j\lambda_j\delta_{ij}=\sum_i\lambda_ix_i^2
-    \end{equation}
-    où les \( \lambda_i\) sont les valeurs propres de \( M\). Cela est strictement positif pour tout \( x\) si et seulement si tous les \( \lambda_i\) sont strictement positifs.
-\item[\ref{ITEMooMOZYooWcrewZ}]
-
-    Nous avons encore 
-    \begin{equation}
-        \langle x, Mx\rangle =\sum_{i}\lambda_ix_i^2.
-    \end{equation}
-    Cela est plus grand ou égal à zéro si et seulement si tous les \( \lambda_i\) sont plus grands ou égaux à zéro.
-        
-\item[\ref{ITEMooRRMFooHSOHxZ}]
-
-        Soit une matrice orthogonale \( T\) diagonalisant \( M\), c'est à dire telle que \( T^tMT=D\) avec \( D\) diagonale. Nous allons vérifier que 
-        \begin{equation}
-            \langle Tx, Mtx\rangle \geq \lambda\| Tx \|^2
-        \end{equation}
-        pour tout \( x\). Vu que \( T\) est une bijection\footnote{Une matrice orthogonale a un déterminant \( \pm 1\).}, cela impliquera le résultat pour tout \( x\). Si nous considérons la base de diagonalisation \( \{ e_k \}\) pour les valeurs propres \( \lambda_k\), nous avons le calcul
-        \begin{subequations}
-            \begin{align}
-                \langle Tx, MTx\rangle &=\langle x, T^tMTx\rangle \\
-                &=\langle x, Dx\rangle \\
-                &=\sum_k\langle x, x_kDe_k\rangle \\
-                &=\sum_k\lambda_kx_k \underbrace{\langle x, e_k\rangle }_{=x_k}\\
-                &\geq \sum_k\lambda| x_k |^2\\
-                &=\lambda\| x \|^2\\
-                &=\lambda\| Tx \|^2.
-            \end{align}
-        \end{subequations}
-        Au dernier passage nous avons utilisé le fait que \( T\) est une isométrie (proposition \ref{PropKBCXooOuEZcS}).
-
-    \end{subproof}
-\end{proof}
-
-Les personnes qui aiment les vecteurs lignes et colonnes écriront des inégalités comme
-\begin{equation}
-    x^tMx\geq x^tx.
-\end{equation}
-Tout à l'autre bout du spectre des personnes névrosées des notations, on trouvera des inégalités comme
-\begin{equation}
-    M(x\otimes x)\geq x\cdot x.
-\end{equation}
-Le penchant personnel de l'auteur de ces lignes est la notation avec le produit tensoriel. Si vous aimez ça, vous pouvez lire \ref{SeOOpHsn}.
-
-La notation adoptée ici avec le produit scalaire \( \langle x, Mx\rangle \) est entre les deux. Elle a l'avantage de n'être pas technologique comme le produit tensoriel (si vous y mettez les pieds, vous devez savoir ce que vous faites), tout en évitant de se casser la tête à savoir qui est un vecteur ligne ou un vecteur colonne.
-
-\begin{corollary}
-    Une matrice symétrique strictement définie positive est inversible.
-\end{corollary}
-
-\begin{proof}
-    Si \( Ax=0\) alors \( \langle Ax, x\rangle =0\). Mais dans le cas d'une matrice strictement définie positive, cela implique \( x=0\) par le lemme \ref{LemWZFSooYvksjw}.
-\end{proof}
-
-\begin{lemma}
-    Pour une base quelconque, les éléments diagonaux d'une matrice symétrique semi-définie positive sont positifs. Si la matrice est strictement définie positive, alors les éléments diagonaux sont strictement positifs.
-\end{lemma}
-
-\begin{proof}
-    Il s'agit d'une application du lemme \ref{LemWZFSooYvksjw}. Si \( A\) est définie positive et que \( \{ e_i \}\) est une base, alors
-    \begin{equation}
-        A_{ii}=\langle Ae_i, e_i\rangle \geq \lambda\| e_i \|^2=\lambda\geq 0.
-    \end{equation}
-    Si \( A\) est strictement définie positive, alors \( \lambda\) peut être choisit strictement positif.
-\end{proof}
- 
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Pseudo-réduction simultanée}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{corollary}[Pseudo-réduction simultanée\cite{JMYQgLO}]  \label{CorNHKnLVA}
-    Soient \( A,B\in \gS(n,\eR)\) avec \( A\) définie positive\footnote{Définition \ref{DefAWAooCMPuVM}.}. Alors il existe \( Q\in \GL(n,\eR)\) tell que \( Q^tBQ\) soit diagonale et \( Q^tAQ=\mtu\).
-\end{corollary}
-
-\begin{proof}
-    Nous allons noter \( x\cdot y\) le produit scalaire usuel de \( \eR^n\) et \( \{ e_i \}_{i=1,\ldots, n}\) sa base canonique.
-
-    Vu que \( A\) est définie positive, nous avons que l'expression\footnote{On peut aussi l'écrire de façon plus matricielle sous la forme \( \langle x, y\rangle =x^tAy\).} \( \langle x, y\rangle =x\cdot Ay\) est un produit scalaire sur \( \eR^n\). Autrement dit, \( E\) muni de cette forme bilinéaire symétrique est un espace euclidien, ce qui fait dire à la proposition \ref{PropUMtEqkb} qu'il existe une base de \( \eR^n\) orthonormée \( \{ f_i \}_{i=1,\ldots, n}\) pour ce produit scalaire, c'est à dire qu'il existe une matrice \( P\in \GL(n,\eR)\) telle que \( P^tAP=\mtu\). Ici, \( P\) est la matrice de changement de base de la base canonique à notre base orthonormée, c'est à dire la matrice qui fait \( Pe_i=f_i\) pour tout \( i\). Voyons cela avec un peu de détails.
-
-    Pour savoir ce que valent les éléments de la matrice \( P^tAP\), nous nous souvenons que \( P^tAPe_j\) est un vecteur dont les coordonnées sont les éléments de la \( j\)\ieme colonne de \( P^tAP\). Nous avons donc \( (P^tAP)_{ij}=e_i\cdot P^tAPe_i\). Calculons :
-    \begin{equation}
-            (P^tAP)_{ij}=e_i\cdot P^tAPe_i
-            =Pe_i\cdot APe_j
-        =f_i\cdot Af_j
-        =\langle f_i, f_j\rangle 
-        =\delta_{ij}
-    \end{equation}
-    où nous avons utilisé le fait que \( A\) était auto-adjointe pour la passer de l'autre côté du produit scalaire (usuel). Au final nous avons effectivement \( P^tAP=\mtu\).
-
-    La matrice \( P^tBP\) est une matrice symétrique, donc le théorème spectral \ref{ThoeTMXla} nous donne une matrice \( R\in \gO(n,\eR)\) telle que \( R^tP^tBPR\) soit diagonale. En posant maintenant \( Q=PR\) nous avons la matrice cherchée.
-\end{proof}
-Note : nous avons prouvé la pseudo-réduction simultanée comme corollaire du théorème de diagonalisation des matrices symétriques \ref{ThoeTMXla}. Il aurait aussi pu être vu comme un corollaire du théorème spectral \ref{ThoRSBahHH} sur les opérateurs auto-adjoints via son corollaire \ref{CorSMHpoVK}.
diff --git a/tex/frido/59_EspacesVecto.tex b/tex/frido/59_EspacesVecto.tex
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-% This is part of Mes notes de mathématique
-% Copyright (c) 2008-2017
-%   Laurent Claessens, Carlotta Donadello
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Sommes de familles infinies}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\label{SECooHHDXooUgLhHR}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Convergence commutative}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit \( x_k\) une suite dans un espace vectoriel normé \( E\). Nous disons que la suite \defe{converge commutativement}{convergence!commutative} vers \( x\in E\) si \( \lim_{n\to \infty}\| x_n-x \| =0\) et si pour toute bijection \( \tau\colon \eN\to \eN\) nous avons aussi
-    \begin{equation}
-        \lim_{n\to \infty} \| x_{\tau(k)}-x \|=0.
-    \end{equation}
-    La notion de convergence commutative est surtout intéressante pour les séries. La somme
-    \begin{equation}
-        \sum_{k=0}^{\infty}x_k
-    \end{equation}
-    converge commutativement vers \( x\) si \( \lim_{N\to \infty} \| x-\sum_{k=0}^Nx_k \|=0\) et si pour toute bijection \( \tau\colon \eN\to \eN\) nous avons
-    \begin{equation}
-        \lim_{N\to \infty} \| x-\sum_{k=0}^Nx_{\tau(k)} \|=0.
-    \end{equation}
-\end{definition}
-
-Nous démontrons maintenant qu'une série converge commutativement si et seulement si elle converge absolument.
-
-Pour le sens inverse, nous avons la proposition suivante.
-\begin{proposition}
-    Soit \( \sum_{k=0}^{\infty}a_k\) une série réelle qui converge mais qui ne converge pas absolument. Alors pour tout \( b\in \eR\), il existe une bijection \( \tau\colon \eN\to \eN\) telle que \( \sum_{i=0}^{\infty}a_{\tau(i)}=b\).
-\end{proposition}
-Pour une preuve, voir \href{http://gilles.dubois10.free.fr/analyse_reelle/seriescomconv.html}{chez Gilles Dubois}.
-
-\begin{proposition} \label{PopriXWvIY}
-    Soit \( (a_i)_{i\in \eN}\) une suite dans \( \eC\) convergent absolument. Alors elle converge commutativement.
-\end{proposition}
-
-\begin{proof}
-    Soit \( \epsilon>0\). Nous posons \( \sum_{i=0}^\infty a_i=a\) et nous considérons \( N\) tel que
-    \begin{equation}
-        | \sum_{i=0}^Na_i-a |<\epsilon.
-    \end{equation}
-    Étant donné que la série des \( | a_i |\) converge, il existe \( N_1\) tel que pour tout \( p,q>N_1\) nous ayons \( \sum_{i=p}^q| a_i |<\epsilon\). Nous considérons maintenant une bijection \( \tau\colon \eN\to \eN \). Prouvons que la série \( \sum_{i=0}^{\infty}| a_{\tau(i)} |\) converge. Nous choisissons \( M\) de telle sorte que pour tout \( n>M\), \( \tau(n)>N_1\); alors si \( p,q>M\) nous avons
-    \begin{equation}
-        \sum_{i=p}^q| a_{\tau(i)} |<\epsilon.
-    \end{equation}
-    Par conséquent la somme de la suite \( (a_{\tau(i)})\) converge. Nous devons montrer à présent qu'elle converge vers la même limite que la somme «usuelle» \( \lim_{N\to \infty} \sum_{i=0}^Na_i\).
-
-    Soit \( n>\max\{ M,N \}\). Alors
-    \begin{equation}
-        \sum_{k=0}^na_{\tau(k)}-\sum_{k=0}^na_k=\sum_{k=0}^Ma_{\tau(k)}-\sum_{k=0}^Na_k+\underbrace{\sum_{M+1}^na_{\tau(k)}}_{<\epsilon}-\underbrace{\sum_{k=N+1}^na_k}_{<\epsilon}.
-    \end{equation}
-    Par construction les deux derniers termes sont plus petits que \( \epsilon\) parce que \( M\) et \( N\) sont les constantes de Cauchy pour les séries \( \sum a_{\tau(i)}\) et \( \sum a_i\). Afin de traiter les deux premiers termes, quitte à redéfinir \( M\), nous supposons que \( \{ 1,\ldots, N \}\subset \tau\{ 1,\ldots, M \}\); par conséquent tous les \( a_i\) avec \( i<N\) sont atteints par les \( a_{\tau(i)}\) avec \( i<M\). Dans ce cas, les termes qui restent dans la différence
-    \begin{equation}
-        \sum_{k=0}a_{\tau(k)}-\sum_{k=0}^Na_k
-    \end{equation}
-    sont des \( a_k\) avec \( k>N\). Cette différence est donc en valeur absolue plus petite que \( \epsilon\), et nous avons en fin de compte que
-    \begin{equation}
-        \left| \sum_{k=0}^na_{\tau(k)}-\sum_{k=0}^na_k \right| <\epsilon.
-    \end{equation}
-\end{proof}
-
-\begin{proposition}     \label{PropyFJXpr}
-    Soit \( \sum_{i=0}^{\infty}a_i\) une série qui converge mais qui ne converge pas absolument. Pour tout \( b\in \eR\), il existe une bijection \( \tau\colon \eN\to \eN\) telle que \( \sum_{i=}^{\infty}a_{\tau(i)}=b\).
-\end{proposition}
-
-Les propositions \ref{PopriXWvIY} et \ref{PropyFJXpr} disent entre autres qu'une série dans \( \eC\) est commutativement sommable si et seulement si elle est absolument sommable.
-
-Soit \( (a_i)_{i\in I}\) une famille de nombres complexes indexée par un ensemble \( I\) quelconque. Nous allons nous intéresser à la somme \( \sum_{i\in I}a_i\).
-
-
-Soit \( \{ a_i \}_{i\in I}\) des nombres positifs. Nous définissons la somme
-\begin{equation}
-    \sum_{i\in I}a_i=\sup_{ J\text{ fini}}\sum_{j\in J}a_j.
-\end{equation}
-Notons que cela est une définition qui ne fonctionne bien que pour les sommes de nombres positifs. Si \( a_i=(-1)^i\), alors selon la définition nous aurions \( \sum_i(-1)^i=\infty\). Nous ne voulons évidemment pas un tel résultat.
-
-Dans le cas de familles de nombres réels positifs, nous avons une première définition de la somme. 
-\begin{definition}  \label{DefHYgkkA}
-Soit \( (a_i)_{i\in I}\) une famille de nombres réels positifs indexés par un ensemble quelconque \( I\). Nous définissons
-\begin{equation}
-    \sum_{i\in I}a_i=\sup_{ J\text{ fini dans } I}\sum_{j\in J}a_j.
-\end{equation}
-\end{definition}
-
-\begin{definition}  \label{DefIkoheE}
-    Si \( \{ v_i \}_{i\in I}\) est une famille de vecteurs dans un espace vectoriel normé indexée par un ensemble quelconque \( I\). Nous disons que cette famille est \defe{sommable}{famille!sommable} de somme \( v\) si pour tout \( \epsilon>0\), il existe un \( J_0\) fini dans \( I\) tel que pour tout ensemble fini \( K\) tel que \( J_0\subset K\) nous avons
-    \begin{equation}
-        \| \sum_{j\in K}v_j-v \|<\epsilon.
-    \end{equation}
-\end{definition}
-Notons que cette définition implique la convergence commutative.
-
-\begin{example}
-    La suite \( a_i=(-1)^i\) n'est pas sommable parce que quel que soit \( J_0\) fini dans \( \eN\), nous pouvons trouver \( J\) fini contenant \( J_0\) tel que \( \sum_{j\in J}(-1)^j>10\). Pour cela il suffit d'ajouter à \( J_0\) suffisamment de termes pairs. De la même façon en ajoutant des termes impairs, on peut obtenir \( \sum_{j\in J'}(-1)^i<-10\).
-\end{example}
-
-\begin{example}
-    De temps en temps, la somme peut sortir d'un espace. Si nous considérons l'espace des polynômes \( \mathopen[ 0 , 1 \mathclose]\to \eR\) muni de la norme uniforme, la somme de l'ensemble
-    \begin{equation}
-        \{ 1,-1,\pm\frac{ x^n }{ n! } \}_{n\in \eN}
-    \end{equation}
-    est zéro.
-
-    Par contre la somme de l'ensemble \( \{ 1,\frac{ x^n }{ n! } \}_{n\in \eN}\) est l'exponentielle qui n'est pas un polynôme.
-\end{example}
-
-\begin{example}
-    Au sens de la définition \ref{DefIkoheE} la famille
-    \begin{equation}
-        \frac{ (-1)^n }{ n }
-    \end{equation}
-    n'est pas sommable. En effet la somme des termes pairs est \( \infty\) alors que la somme des termes impairs est \( -\infty\). Quel que soit \( J_0\in \eN\), nous pouvons concocter, en ajoutant des termes pairs, un \( J\) avec \( J_0\subset J\) tel que \( \sum_{j\in J}(-1)^j/j\) soit arbitrairement grand. En ajoutant des termes négatifs, nous pouvons également rendre \( \sum_{j\in J}(-1)^j/j\) arbitrairement petit.
-\end{example}
-
-\begin{proposition} \label{PropVQCooYiWTs}
-    Si \( (a_{ij})\) est une famille de nombres positifs indexés par \( \eN\times \eN\) alors
-    \begin{equation}
-        \sum_{(i,j)\in \eN^2}a_{ij}=\sum_{i=1}^{\infty}\Big( \sum_{j=1}^{\infty}a_{ij} \Big)
-    \end{equation}
-    où la somme de gauche est celle de la définition \ref{DefHYgkkA}.
-\end{proposition}
-%TODO : cette proposition peut être vue comme une application de Fubini pour la mesure de comptage. Le faire et référentier ici.
-
-\begin{proof}
-    Nous considérons \( J_{m,n}=\{ 0,\ldots, m \}\times \{ 0,\ldots, n \}\) et nous avons pour tout \( m\) et \( n\) :
-    \begin{equation}
-        \sum_{(i,j)\in \eN^2}a_{ij}\geq \sum_{(i,j)\in J_{m,n}}a_{ij}=\sum_{i=1}^m\Big( \sum_{j=1}^na_{ij} \Big).
-    \end{equation}
-    Si nous fixons \( m\) et que nous prenons la limite \( n\to \infty\) (qui commute avec la somme finie sur \( i\)) nous trouvons
-    \begin{equation}
-        \sum_{(i,j)\in \eN^2}a_{ij}\geq =\sum_{i=1}^m\Big( \sum_{j=1}^{\infty}a_{ij} \Big).
-    \end{equation}
-    Cela étant valable pour tout \( m\), c'est encore valable à la limite \( m\to \infty\) et donc
-    \begin{equation}
-        \sum_{(i,j)\in \eN^2}a_{ij}\geq \sum_{i=1}^{\infty}\Big( \sum_{j=1}^{\infty}a_{ij} \Big).
-    \end{equation}
-    
-    Pour l'inégalité inverse, il faut remarquer que si \( J\) est fini dans \( \eN^2\), il est forcément contenu dans \( J_{m,n}\) pour \( m\) et \( n\) assez grand. Alors
-    \begin{equation}
-        \sum_{(i,j)\in J}a_{ij}\leq \sum_{(i,j)\in J_{m,n}}a_{ij}=\sum_{i=1}^m\sum_{j=1}^na_{ij}\leq \sum_{i=1}^{\infty}\Big( \sum_{j=1}^{\infty}a_{ij} \Big).
-    \end{equation}
-    Cette inégalité étant valable pour tout ensemble fini \( J\subset \eN^2\), elle reste valable pour le supremum.    
-\end{proof}
-
-La définition générale de la somme \ref{DefIkoheE} est compatible avec la définition usuelle dans les cas où cette dernière s'applique.
-\begin{proposition}[commutative sommabilité]\label{PropoWHdjw}
-    Soit \( I\) un ensemble dénombrable et une bijection \( \tau\colon \eN\to I\). Soit \( (a_i)_{i\in I}\) une famille dans un espace vectoriel normé. Alors
-    \begin{equation}
-        \sum_{k=0}^{\infty}a_{\tau(k)}=\sum_{i\in I}a_i
-    \end{equation}
-    dès que le membre de droite existe. Le membre de gauche est définit par la limite usuelle.
-\end{proposition}
-
-\begin{proof}
-    Nous posons \( a=\sum_{i\in I}a_i\). Soit \( \epsilon>0\) et \( J_0\) comme dans la définition. Nous choisissons
-    \begin{equation}
-        N>\max_{j\in J_0}\{ \tau^{-1}(j) \}.
-    \end{equation}
-    En tant que sommes sur des ensembles finis, nous avons l'égalité
-    \begin{equation}
-        \sum_{k=0}^Na_{\tau(k)}=\sum_{j\in J_0}a_j
-    \end{equation}
-    où \( J\) est un sous-ensemble de \( I\) contenant \( J_0\). Soit \( J\) fini dans \( I\) tel que \( J_0\subset J\). Nous avons alors
-    \begin{equation}
-        \| \sum_{k=0}^Na_{\tau(k)}-a \|=\| \sum_{j\in J}a_j-a \|<\epsilon.
-    \end{equation}
-    Nous avons prouvé que pour tout \( \epsilon\), il existe \( N\) tel que \( n>N\) implique \( \| \sum_{k=0}^na_{\tau(k)}-a\| <\epsilon\).
-\end{proof}
-
-\begin{corollary}
-    Nous pouvons permuter une somme dénombrable et une fonction linéaire continue. C'est à dire que si \( f\) est une fonction linéaire continue sur l'espace vectoriel normé \( E\) et \( (a_i)_{i\in I}\) une famille sommable dans \( E\) alors
-    \begin{equation}
-        f\left( \sum_{i\in I}a_i \right)=\sum_{i\in I}f(a_i).
-    \end{equation}
-\end{corollary}
-
-\begin{proof}
-    En utilisant une bijection \( \tau\) entre \( I\) et \( \eN\) avec la proposition \ref{PropoWHdjw} ainsi que le résultat connu à propos des sommes sur \( \eN\), nous avons
-    \begin{subequations}
-        \begin{align}
-            f\left( \sum_{i\in I}a_i \right)&=f\left( \sum_{k=0}^{\infty}a_{\tau(k)} \right)\\
-            &=\sum_{k=0}^{\infty}f(a_{\tau(k)}) \label{SUBEQooCVUTooPmnHER}\\
-            &=\sum_{i\in I}f(a_i).
-        \end{align}
-    \end{subequations}
-    Notons que le passage à \eqref{SUBEQooCVUTooPmnHER} n'est pas du tout une trivialité à deux francs cinquante. Il s'agit d'écrire la somme comme la limite des sommes partielles, et de permuter \( f\) avec la limite en invoquant la continuité, puis de permuter \( f\) avec la somme partielle en invoquant sa linéarité.
-
-    Ah, tiens et tant qu'on y est à dire qu'il y a des chose évidentes qui ne le sont pas, oui, il existe des applications linéaires non continues, voir le thème \ref{THEMEooYCBUooEnFdUg}.
-\end{proof}
-
-La proposition suivante nous enseigne que les sommes infinies peuvent être manipulée de façon usuelle.
-\begin{proposition} \label{PropMpBStL}
-    Soit \( I\) un ensemble dénombrable. Soient \( (a_i)_{i\in I}\) et \( (b_i)_{i\in I}\), deux familles de réels positifs telles que \( a_i<b_i\) et telles que \( (b_i)\) est sommable. Alors \( (a_i)\) est sommable.
-
-    Si \( (a_i)_{i\in I}\) est une famille de complexes telle que \( (| a_i |)\) est sommable, alors \( (a_i)\) est sommable.
-\end{proposition}
-
-\begin{proposition}[\cite{MonCerveau}]     \label{PROPooWLEDooJogXpQ}
-    Soit un espace vectoriel normé \( E\) et une famille sommable\footnote{Définition \ref{DefIkoheE}.} \( \{ v_i \}_{i\in I}\) d'éléments de \( E\). Soit \( f\colon E\to \eC\) une application sur laquelle nous supposons
-    \begin{enumerate}
-        \item
-            \( f\) est linéaire et continue;
-        \item
-            la partie \( \{ f(v_i)_{i\in I} \} \) est sommable.
-    \end{enumerate}
-    Alors nous pouvons permuter la somme et \( f\) :
-    \begin{equation}        \label{EQooONHXooKqIEbY}
-        f\big( \sum_{i\in I}v_i \big)=\sum_{i\in I}f(v_i).
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Soit \( \epsilon>0\); vu que les familles \( \{ v_i \}_{i\in I}\) et \( \{ f(v_i) \}_{i\in I}\) sont sommables, nous pouvons considérer les parties finies \( J_1\) et \( J_2\) de \( I\) telles que
-    \begin{equation}
-        \big\| \sum_{j\in J_1}v_j-\sum_{i\in I}v_i \big\|\leq \epsilon
-    \end{equation}
-    et
-    \begin{equation}
-        \big\| \sum_{j\in J_2}f(v_j)-\sum_{i\in I}f(v_i) \big\|\leq \epsilon
-    \end{equation}
-    Ensuite nous posons \( J=J_1\cup J_2\). Avec cela nous calculons un peu avec les majorations usuelles :
-    \begin{equation}
-        \| f(\sum_{i\in I}v_i) -\sum_{i\in I}f(v_i) \|\leq \| f(\sum_{i\in I}v_i)- f(\sum_{j\in J}v_j) \|+  \| f(\sum_{j\in J}v_j)-\sum_i\in If(v_i) \|.
-    \end{equation}
-    Le second terme est majoré par \( \epsilon\), tandis que le premier, en utilisant la linéarité de \( f\) possède la majoration
-    \begin{equation}
-        \| f(\sum_{i\in I}v_i)- f(\sum_{j\in J}v_j) \|=\| f(\sum_{i\in I}v_i-\sum_{j\in J}v_j) \|\leq \| f \| \| \sum_{i\in I}v_i- \sum_{j\in J}v_j\|\leq \epsilon\| f \|.
-    \end{equation}
-    Donc pour tout \( \epsilon>0\) nous avons
-    \begin{equation}
-        \| f(\sum_{i\in I}v_i) -\sum_{i\in I}f(v_i) \|\leq \epsilon(1+\| f \|).
-    \end{equation}
-    D'où l'égalité \eqref{EQooONHXooKqIEbY}.
-\end{proof}
-
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Fonctions}		\label{Sect_fonctions}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Soient $(V,\| . \|_V)$ et $(W,\| . \|_W)$ deux espaces vectoriels normés, et une fonction $f$ de $V$ dans $W$. Il est maintenant facile de définir les notions de limites et de continuité pour de telles fonctions en copiant les définitions données pour les fonctions de $\eR$ dans $\eR$ en changeant simplement les valeurs absolues par les normes sur $V$ et $W$.
-
-La caractérisation suivante est un recopiage de la définition \ref{DefOLNtrxB} lorsque la topologie est donnée par des boules.
-\begin{proposition}\label{PropHOCWooSzrMjl}
-	Soit $f\colon V\to W$ une fonction de domaine \( \Domaine(f)\subset V\) et soit $a$ un point d'accumulation de $\Domaine(f)$. 
-    La fonction \( f\) admet une limite en $a\in V$ si et seulement s'il existe un élément $\ell\in W$ tel que pour tout $\varepsilon>0$, il existe un $\delta>0$ tel que pour tout $x\in \Domaine(f)$,
-    \begin{equation}        \label{EqDefLimzxmasubV}
-		0<\| x-a \|_V<\delta\,\Rightarrow\,\| f(x)-\ell \|_W<\varepsilon.
-	\end{equation}
-	Dans ce cas, nous écrivons $\lim_{x\to a} f(x)=\ell$ et nous disons que $\ell$ est la \defe{limite}{limite} de $f$ lorsque $x$ tend vers $a$.
-\end{proposition}
-
-\begin{remark}
-    Le fait que nous limitions la formule \eqref{EqDefLimzxmasubV} aux \( x\) dans le domaine de \( f\) n'est pas anodin. Considérons la fonction \( f(x)=\sqrt{x^2-4}\), de domaine \( | x |\geq 2\). Nous avons
-    \begin{equation}
-        \lim_{x\to 2} \sqrt{x^2-4}=0.
-    \end{equation}
-    Nous ne pouvons pas dire que cette limite n'existe pas en justifiant que la limite à gauche n'existe pas. Les points \( x<2\) sont hors du domaine de \( f\) et ne comptent dons pas dans l'appréciation de l'existence de la limite.
-
-    Vous verrez plus tard que ceci provient de la \wikipedia{fr}{Topologie_induite}{topologie induite} de \( \eR\) sur l'ensemble \( \mathopen[ 2 , \infty [\).
-\end{remark}
diff --git a/tex/frido/60_EspacesVecto.tex b/tex/frido/60_EspacesVecto.tex
deleted file mode 100644
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-% This is part of Mes notes de mathématique
-% Copyright (c) 2011-2017
-%   Laurent Claessens, Carlotta Donadello
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Sous espaces caractéristiques}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-% TODO : lire le blog de Pierre Bernard; en particulier celle-ci : http://allken-bernard.org/pierre/weblog/?p=2299
-
-Lorsqu'un opérateur n'est pas diagonalisable, les valeurs propres jouent quand même un rôle important.
-
-\begin{definition}  \label{DefFBNIooCGbIix}
-    Soit \( E\) un \( \eK\)-espace vectoriel  \( f\in\End(E)\). Pour \( \lambda\in \eK\) nous définissons
-    \begin{equation}
-        F_{\lambda}(f)=\{ v\in E\tq (f-\lambda\mtu)^nv=0, n\in\eN \}
-    \end{equation}
-    et nous appelons ça un \defe{sous-espace caractéristique}{sous-espace!caractéristique} de \( f\).
-\end{definition}
-L'espace \( F_{\lambda}(f)\) est l'ensemble de nilpotence de l'opérateur \( f-\lambda\mtu\) et
-
-\begin{lemma}   \label{LemBLPooHMAoyJ}
-    L'ensemble \( F_{\lambda}(f)\) est non vide si et seulement si \( \lambda\) est une valeur propre de \( f\). L'espace \( F_{\lambda}(f)\) est invariant sous \( f\).
-\end{lemma}
-
-\begin{proof}
-    Si \( F_{\lambda}(f)\) est non vide, nous considérons \( v\in F_{\lambda}(f)\) et \( n\) le plus petit entier non nul tel que \( (f-\lambda)^nv=0\). Alors \( (f-\lambda)^{n-1}v\) est un vecteur propre de \( f\) pour la valeur propre \( \lambda\). Inversement si \( v\) est une valeur propre de \( f\) pour la valeur propre \( \lambda\), alors \( v\in F_{\lambda}(f)\).
-
-    En ce qui concerne l'invariance, remarquons que \( f\) commute avec \( f-\lambda\mtu\). Si \( x\in F_{\lambda}(f)\) il existe \( n\) tel que \( (f-\lambda\mtu)^nx=0\). Nous avons aussi
-    \begin{equation}
-        (f-\lambda\mtu)^nf(x)=f\big( (f-\lambda\mtu)^nx \big)=0,
-    \end{equation}
-    par conséquent \( f(x)\in F_{\lambda}(f)\).
-\end{proof}
-
-\begin{remark}  \label{RemBOGooCLMwyb}
-    Toute matrice sur \( \eC\) n'est pas diagonalisable : nous en avons déjà donné une exemple simple en \ref{ExBRXUooIlUnSx}. Nous en voyons maintenant un moins simple. Considérons en effet l'endomorphisme \( f\) donné par la matrice
-    \begin{equation}
-        \begin{pmatrix}
-            a&    \alpha    &   \beta    \\
-            0    &   a    &   \gamma    \\
-            0    &   0    &   b
-        \end{pmatrix}
-    \end{equation}
-    où \( a\neq b\), \( \alpha\neq 0\), \( \beta\) et \( \gamma\) sont des nombres complexes quelconques.
-    Son polynôme caractéristique est 
-    \begin{equation}
-        \chi_f(\lambda)=(a-\lambda)^2(b-\lambda),
-    \end{equation}
-    et les valeurs propres sont donc \( a\) et \( b\). Nous trouvons les vecteurs propres pour la valeur \( a\) en résolvant
-    \begin{equation}
-        \begin{pmatrix}
-            a    &   \alpha    &   \beta    \\
-            0    &   a    &   \gamma    \\
-            0    &   0    &   b
-        \end{pmatrix}\begin{pmatrix}
-            x    \\ 
-            y    \\ 
-            z    
-        \end{pmatrix}=\begin{pmatrix}
-            ax    \\ 
-            ay    \\ 
-            az    
-        \end{pmatrix}.
-    \end{equation}
-    L'espace propre \( E_a(f)\) est réduit à une seule dimension générée par \( (1,0,0)\). De la même façon l'espace propre correspondant à la valeur propre \( b\) est donné par 
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{1}{ b-a }\left( \beta+\frac{ \alpha\gamma }{ b-a } \right)    \\ 
-            \frac{ \gamma }{ b-a }    \\ 
-            1    
-        \end{pmatrix}.
-    \end{equation}
-    Il n'y a donc pas trois vecteurs propres linéairement indépendants, et l'opérateur \( f\) n'est pas diagonalisable.
-
-    Par contre nous pouvons voir que
-    \begin{equation}
-        (f-\alpha\mtu)^2\begin{pmatrix}
-             0   \\ 
-            1    \\ 
-            0    
-        \end{pmatrix}=
-        \begin{pmatrix}
-            a    &   \alpha    &   \beta    \\
-            0    &   a    &   \gamma    \\
-            0    &   0    &   b
-        \end{pmatrix}
-        \begin{pmatrix}
-            \alpha    \\ 
-            0    \\ 
-            0    
-        \end{pmatrix}-\begin{pmatrix}
-            a\alpha    \\ 
-            0    \\ 
-            0    
-        \end{pmatrix}=\begin{pmatrix}
-            0    \\ 
-            0    \\ 
-            0    
-        \end{pmatrix},
-    \end{equation}
-    de telle sorte que le vecteur \( (0,1,0)\) soit également dans l'espace caractéristique \( F_a(f)\).
-
-    Dans cet exemple, la multiplicité algébrique de la racine \( a\) du polynôme caractéristique vaut \( 2\) tandis que sa multiplicité géométrique vaut seulement \( 1\).
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorèmes de décomposition}
-%---------------------------------------------------------------------------------------------------------------------------
-
-%TODO : Je crois qu'on peut remplacer l'hypothèse de corps algébriquement clos par le polynôme caractéristique scindé.
-\begin{theorem}[Théorème spectral, décomposition primaire]\index{théorème!spectral}     \label{ThoSpectraluRMLok}
-    Soit \( E\) espace vectoriel de dimension finie sur le corps algébriquement clos \( \eK\) et \( f\in\End(E)\). Alors
-    \begin{equation}    \label{EqCTFHooBSGhYK}
-        E=F_{\lambda_1}(f)\oplus\ldots\oplus F_{\lambda_k}(f)
-    \end{equation}
-    où la somme est sur les valeurs propres distinctes de \( f\).
-
-    Les projecteurs sur les espaces caractéristique forment un système complet et orthogonal.
-\end{theorem}
-\index{décomposition!primaire}
-\index{décomposition!spectrale}
-\index{décomposition!sous-espaces caractéristiques}
-
-\begin{proof}
-    Soit \( P\) le polynôme caractéristique de \( f\) et une décomposition
-    \begin{equation}
-        P=(f-\lambda_1)^{\alpha_1}\ldots(f-\lambda_r)^{\alpha_r}
-    \end{equation}
-    en facteurs irréductibles. La le théorème de noyaux (\ref{ThoDecompNoyayzzMWod}) nous avons
-    \begin{equation}        \label{EqDeFVSaYv}
-        E=\ker(f-\lambda_1)^{\alpha_1}\oplus\ldots\oplus\ker(f-\lambda_r)^{\alpha_r}.
-    \end{equation}
-    Les projecteurs sont des polynômes en \( f\) et forment un système orthogonal. Il nous reste à prouver que \( \ker(f-\lambda_i)^{\alpha_i}=F_{\lambda_i}(f)\). L'inclusion
-    \begin{equation}    \label{EqzmNxPi}
-        \ker(f-\lambda_i)^{\alpha_i}\subset F_{\lambda_i}(f)
-    \end{equation}
-    est évidente. Nous devons montrer l'inclusion inverse. Prouvons que la somme des \( F_{\lambda_i}(f)\) est directe. Si \( v\in F_{\lambda_i}(f)\cap F_{\lambda_j}(f)\), alors il existe \( v_1=(f-\lambda_i)^nv\neq 0\) avec \( (f-\lambda_i)v_1=0\). Étant donné que \( (f-\lambda_i)\) commute avec \( (f-\lambda_j)\), ce \( v_1\) est encore dans \( F_{\lambda_j}(f)\) et par conséquent il existe \( w=(f-\lambda_j)^mv_1\) non nul tel que 
-    \begin{subequations}
-        \begin{numcases}{}
-            (f-\lambda_i)w=0\\
-            (f-\lambda_j)w=0.
-        \end{numcases}
-    \end{subequations}
-    Ce \( w\) serait donc un vecteur propre simultané pour les valeurs propres \( \lambda_i\) et \( \lambda_j\), ce qui est impossible parce que les espaces propres sont linéairement indépendants. Les espaces \( F_{\lambda_i}\) sont donc en somme directe et
-    \begin{equation}
-        \sum_i\dim F_{\lambda_i}(f)\leq \dim E.
-    \end{equation}
-    En tenant compte de l'inclusion \eqref{EqzmNxPi} nous avons même
-    \begin{equation}
-        \dim E=\sum_i\dim\ker(f-\lambda_i)^{\alpha_i}\leq\sum_i F_{\lambda_i}(f)\leq \dim E.
-    \end{equation}
-    Par conséquent nous avons \( \dim\ker(f-\lambda_i)^{\alpha_i}=\dim F_{\lambda_i}(f)\) et l'égalité des deux espaces.
-\end{proof}
-
-
-\begin{probleme}
-    Dans le cas où le corps n'est pas algébriquement clos, il paraît qu'il faut remplacer «diagonalisable» par «semi-simple».
-\end{probleme}
-%TODO : peut-être qu'il y a la réponse dans http://www.math.jussieu.fr/~romagny/agreg/dvt/endom_semi_simples.pdf
-
-Si l'espace vectoriel est sur un corps algébriquement clos, alors les endomorphismes semi-simples\footnote{Définition \ref{DEFooBOHVooSOopJN}.} sont les endomorphismes diagonaux.
-
-
-%TODO : Je crois qu'on peut remplacer l'hypothèse de corps algébriquement clos par le polynôme caractéristique scindé.
-\begin{theorem}[Décomposition de Dunford] \label{ThoRURcpW}
-    Soit \( E\) un espace vectoriel sur le corps algébriquement clos \( \eK\) et \( u\in\End(E)\) un endomorphisme de \( E\). 
-    
-    \begin{enumerate}
-        \item
-            
-            L'endomorphisme \( u\) se décompose de façon unique sous la forme
-            \begin{equation}
-                u=s+n
-            \end{equation}
-            où \( s\) est diagonalisable, \( n\) est nilpotent et \( [s,n]=0\).
-        \item
-            Les endomorphismes \( s\) et \( n\) sont des polynômes en \( u\) et commutent avec \( u\).
-        \item   \label{ItemThoRURcpWiii}
-            Les parties \( s\) et \( n\) sont données par
-            \begin{subequations}
-                \begin{align}
-                    s&=\sum_i\lambda_ip_i\\
-                    n&=\sum_i(s-\lambda_i\mtu)p_i
-                \end{align}
-            \end{subequations}
-            où les sommes sont sur les valeurs propres distinctes\footnote{C'est à dire sur les sous-espaces caractéristiques.} de \( f\) et où \( p_i\colon E\to F_{\lambda_i}(u)\) est la projection de \( E\) sur \( F_{\lambda_i}(u)\).
-    \end{enumerate}
-\end{theorem}
-\index{décomposition!Dunford}
-\index{Dunford!décomposition}
-\index{réduction!d'endomorphisme}
-\index{endomorphisme!sous-espace stable}
-\index{polynôme!d'endomorphisme!décomposition de Dunford}
-\index{endomorphisme!diagonalisable!Dunford}
-\index{endomorphisme!nilpotent!Dunford}
-%TODO : comprendre comment on calcule des exponentielles de matrices avec Dunford.
-
-\begin{proof}
-    Le théorème spectral \ref{ThoSpectraluRMLok} nous indique que
-    \begin{equation}
-        E=\bigoplus_iF_{\lambda_i}(f).
-    \end{equation}
-    Nous considérons l'endomorphisme \( s\) de \( E\) qui consiste à dilater d'un facteur \( \lambda\) l'espace caractéristique \( F_{\lambda}(f)\) :
-    \begin{equation}
-        s=\sum_i\lambda_ip_i
-    \end{equation}
-    où \( p_i\colon E\to F_{\lambda_i}(u)\) est la projection de \( E\) sur \( F_{\lambda_i}(u)\).
-
-    Nous allons prouver que \( [s,f]=0\) et \( n=f-s\) est nilpotent. Cela impliquera que \( [s,n]=0\).
-
-    Si \( x\in F_{\lambda}(f)\), alors nous avons \( sf(x)=\lambda f(x)\) parce que \( f(x)\in F_{\lambda}(f)\) tandis que \( fs(x)=f(\lambda x)=\lambda f(x)\). Par conséquent \( f\) commute avec \( s\).
-
-    Pour montrer que \( f-s\) est nilpotent, nous en considérons la restriction
-    \begin{equation}
-        f-s\colon F_{\lambda}(f)\to F_{\lambda}(f).
-    \end{equation}
-    Cet opérateur est égal à \( f-\lambda\mtu\) et est par conséquent nilpotent.
-
-    Prouvons à présent l'unicité. Soit \( u=s'+n'\) une autre décomposition qui satisfait aux conditions : \( s'\) est diagonalisable, \( n'\) est nilpotent et \( [n',s']=0\). Commençons par prouver que \( s'\) et \( n'\) commutent avec \( u\). En multipliant \( u=s'+n'\) par \( s'\) nous avons
-    \begin{equation}
-        s'u=s'^2+s'n'=s'^2+n's'=(s'+n')s'=us',
-    \end{equation}
-    par conséquent \( [u,s']=0\). Nous faisons la même chose avec \( n'\) pour trouver \( [u,n']=0\). Notons que pour obtenir ce résultat nous avons utilisé le fait que \( n'\) et \( s'\) commutent, mais pas leur propriétés de nilpotence et de diagonalisibilité.
-    
-    
-    Si \( s'+n'=s+n\) est une autre décomposition, \( s'\) et \( n'\) commutent avec \( u\), et par conséquent avec tous les polynômes en \( u\). Ils commutent en particulier avec \( n\) et \( s\). Les endomorphismes \( s\) et \( s'\) sont alors deux endomorphismes diagonalisables qui commutent. Par la proposition \ref{PropGqhAMei}, ils sont simultanément diagonalisables. Dans la base de simultanée diagonalisation, la matrice de l'opérateur \( s'-s=n-n'\) est donc diagonale. Mais \( n-n'\) est également nilpotent, en effet si \( A\) et \( B\) sont deux opérateurs nilpotents,
-    \begin{equation}
-        (A+B)^n=\sum_{k=0}^n\binom{k}{n}A^kB^{n-k}.
-    \end{equation}
-    Si \( n\) est assez grand, au moins un parmi \( A^k\) ou \( B^{n-k}\) est nul.
-
-    Maintenant que \( n-n'\) est diagonal et nilpotent, il est nul et \( n=n'\). Nous avons alors immédiatement aussi \( s=s'\).
-
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Diverses conséquences}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{theorem}
-    Soit une matrice \( A\in \eM(n,\eC)\). On a que la suite \( (A^kx)\) tends vers zéro pour tout \( x\) si et seulement si \( \rho(A)<1\) où \( \rho(A)\)\index{rayon!spectral} est le rayon spectral de $A$
-\end{theorem}
-\index{décomposition!Dunford!exponentielle de matrice}
-
-\begin{proof}
-    Dans le sens direct, il suffit de prendre comme \( x\), un vecteur propre de \( A\). Dans ce cas nous avons \( A^kx=\lambda^kx\). Mais \( \lambda^kx\) ne tend vers zéro que si \( \lambda<1\). Donc toute les valeurs propres de \( A\) doivent être plus petite que \( 1\) et \( \rho(A)<1\).
-
-    Pour l'autre sens nous utilisons la décomposition de Dunford (théorème \ref{ThoRURcpW}) : il existe une matrice inversible \( P\) telle que
-    \begin{equation}
-        A=P^{-1}(D+N)P
-    \end{equation}
-    où \( D\) est diagonale, \( N\) est nilpotente et \( [D,N]=0\). Étant donné que \( D+N\) est triangulaire, son polynôme caractéristique que
-    \begin{equation}
-        \chi_{D+N}(\lambda)=\prod_i D_{ii}-\lambda.
-    \end{equation}
-    Par similitude, c'est le même polynôme caractéristique que celui de \( A\) et nous savons alors que la diagonale de \( D\) contient les valeurs propres de \( A\).
-
-    Par ailleurs nous avons
-    \begin{subequations}
-        \begin{align}
-            A^k&=P^{-1}(D+N)^kP\\
-            &=P^{-1}\sum_{j=0}^k{j\choose k}D^{j-k}N^jP\\
-            &=P^{-1}\sum_{j=0}^{n-1}{j\choose k}D^{j-k}N^jP
-        \end{align}
-    \end{subequations}
-    où nous avons utilité le fait que \( D\) et \( N\) commutent ainsi que \( N^{n-1}=0\) parce que \( N\) est nilpotente. Nous utilisons la norme matricielle usuelle, pour laquelle \( \| D \|=\rho(D)=\rho(A)\). Nous avons alors
-    \begin{equation}
-        \| (D+N)^k \|\leq \sum_{j=0}^k{j\choose k}\rho(D)^{k-j}\| N \|^j.
-    \end{equation}
-    Du coup si \( \rho(D)<1\) alors \( \| (D+N)^k \|\to 0\) (et c'est même un si et seulement si).
-\end{proof}
-
-Une application de la décomposition de Jordan est l'existence d'un logarithme pour les matrices. La proposition suivant va d'une certaine manière donner un logarithme pour les matrices inversibles complexes. Dans le cas des matrices réelles \( m\) telles que \( \| m-\mtu \|<1\), nous donnerons au lemme \ref{LemQZIQxaB} une formule pour le logarithme sous forme d'une série; ce logarithme sera réel.
-\begin{proposition} \label{PropKKdmnkD}
-    Toute matrice inversible complexe est une exponentielle.
-\end{proposition}
-\index{exponentielle!de matrice}
-\index{décomposition!Jordan!et exponentielle de matrice}
-
-\begin{proof}
-    Soit \( A\in \GL(n,\eC)\); nous allons donner une matrice \( B\in \eM(n,\eC)\) telle que \( A=\exp(B)\). D'abord remarquons qu'il suffit de prouver le résultat pour une matrice par classe de similitude. En effet si \( A=\exp(B)\) et si \( M\) est inversible alors 
-    \begin{subequations}    \label{EqqACuGK}
-        \begin{align}
-            \exp(MBM^{-1})&=\sum_k\frac{1}{ k! }(MBM^{-1})^k\\
-            &=\sum_k\frac{1}{ k! }MB^kM^{-1}\\
-            &=M\exp(B)M^{-1}.
-        \end{align}
-    \end{subequations}
-    Donc \( MAM^{-1}=\exp(MBM^{-1})\). Nous pouvons donc nous contenter de trouver un logarithme pour les blocs de Jordan. Nous supposons donc que \( A=(\mtu+N)\) avec \( N^m=0\). 
-    En nous inspirant de \eqref{EqweEZnV}, nous posons\footnote{Le logarithme d'un nombre n'est pas encore définit à ce moment, mais cela ne nous empêche pas de poser une définition ici pour une application des réels vers les matrices.}
-    \begin{equation}
-        D(t)=tN-\frac{ t^2 }{ 2 }N^2+\cdots +(-1)^m\frac{ t^{m-1} }{ m-1 }N^{m-1}
-    \end{equation}
-    et nous allons prouver que \(  e^{D(1)}=\mtu+N\). Notons que \( N\) étant nilpotente, cette somme ainsi que toutes celles qui viennent sont finies. Il n'y a donc pas de problèmes de convergences dans cette preuve (si ce n'est les passages des équations \eqref{EqqACuGK}).
-
-    Nous posons \( S(t)= e^{D(t)}\) (la somme est finie), et nous avons
-    \begin{equation}
-        S'(t)=D'(t) e^{D(t)}
-    \end{equation}
-    Afin d'obtenir une expression qui donne \( S'\) en termes de \( S\), nous multiplions par \( (\mtu+tN)\) en remarquant que \( (\mtu+tN)D'(t)=N\) nous avons
-    \begin{equation}
-        (\mtu+tN)S'(t)=NS(t).
-    \end{equation}
-    En dérivant à nouveau,
-    \begin{equation}    \label{EqKjccqP}
-        (\mtu+tN)S''(t)=0.
-    \end{equation}
-    La matrice \( (\mtu+tN)\) est inversible parce que son noyau est réduit à \( \{ 0 \}\). En effet si \( (\mtu+tN)x=0\), alors \( Nx=-\frac{1}{ t }x\), ce qui est impossible parce que \( N\) est nilpotente. Ce que dit l'équation \eqref{EqKjccqP} est alors que \( S''(t)=0\). Si nous développons \( S(t)\) en puissances de \( t\) nous nous arrêtons au terme d'ordre \( 1\) et nous avons
-    \begin{equation}
-        S(t)=S(0)+tS'(0)=\mtu+tD'(0)=1+tN.
-    \end{equation}
-    En \( t=1\) nous trouvons \( S(1)=\mtu+N\). La matrice \( D(1)\) donnée est donc bien un logarithme de $\mtu+N$.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Diagonalisabilité d'exponentielle}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[\cite{fJhCTE}]      \label{PropCOMNooIErskN}
-    Si \( A\in \eM(n,\eR)\) a un polynôme caractéristique scindé, alors \( A\) est diagonalisable si et seulement si \( e^A\) est diagonalisable.
-\end{proposition}
-\index{décomposition!Dunford!application}
-\index{exponentielle!de matrice}
-\index{diagonalisable!exponentielle}
-
-\begin{proof}
-    Si \( A\) est diagonalisable, alors il existe une matrice inversible \( M\) telle que \( D=M^{-1}AM\) soit diagonale (c'est la définition \ref{DefCNJqsmo}). Dans ce cas nous avons aussi \( (M^{-1}AM)^k=M^{-1}A^kM\) et donc \( M^{-1}e^AM=e^{M^{-1}AM}=e^D\) qui est diagonale.
-
-    La partie difficile est donc le contraire. 
-    
-    \begin{subproof}
-        \item[Qui est diagonalisable et comment ?]
-            Nous supposons que \( e^A\) est diagonalisable et nous écrivons la décomposition de Dunford (théorème \ref{ThoRURcpW}) :
-            \begin{equation}
-                A=S+N
-            \end{equation}
-            où \( S\) est diagonalisable, \( N\) est nilpotente, \( [S,N]=0\). Nous avons besoin de prouver que \( N=0\).
-    
-            Les matrices \( A\) est \( S\) commutent; en passant au développement nous en déduisons que \( A\) et \( e^S\) commutent, puis encore en passant au développement que \( e^A\) et \( e^S\) commutent. Vu que \( S\) est diagonalisable, \( e^S\) l'est et par hypothèse \( e^A\) est également diagonalisable. Donc \( e^A\) et \( e^{-S}\) sont simultanément diagonalisables par la proposition \ref{PropGqhAMei}.
-
-            Étant donné que \( A\) et \( S\) commutent, nous avons \( e^N=e^{A-S}=e^Ae^{-S}\), et nous en déduisons que \( e^N\) est diagonalisable vu que les deux facteurs \( e^A\) et \( e^{-S}\) sont simultanément diagonalisables.
-
-        \item[Unipotence]
-
-            Si \( r\) est le degré de nilpotence de \( N\), nous avons
-            \begin{equation}    \label{EqQHjvLZQ}
-                e^N-\mtu=N+\frac{ N^2 }{2}+\cdots +\frac{ N^{r-1} }{ (r-1)! }.
-            \end{equation}
-            Donc
-            \begin{equation}
-                (e^N-\mtu)^k=\left( N+\frac{ N^2 }{2}+\cdots +\frac{ N^{r-1} }{ (r-1)! } \right)^k
-            \end{equation}
-            où le membre de droite est un polynôme en \( N\) dont le terme de plus bas degré est de degré \( k\). Donc \( (e^N-\mtu)\) est nilpotente et \( e^N\) est unipotente.
-
-            Si \( M\) est la matrice qui diagonalise \( e^N\), alors la matrice diagonale \( M^{-1}e^NM\) est tout autant unipotente que \( e^N\) elle-même. En effet,
-            \begin{subequations}
-                \begin{align}
-                    (M^{-1}e^NM-\mtu)^r&=\sum_{k=0}^r\binom{ r }{ k }(-1)^{r-k}M^{-1}(e^N)^kM\\
-                    &=M^{-1}\left( \sum_{k=0}^r\binom{ r }{ k }(-1)^{r-k}(e^N)^k \right)M\\
-                    &=M^{-1}(e^N-\mtu)^rM\\
-                    &=0.
-                \end{align}
-            \end{subequations}
-
-            La matrice \( M^{-1}e^NM\) est donc une matrice diagonale et unipotente; donc \( M^{-1}e^NM=\mtu\), ce qui donne immédiatement que \( e^N=\mtu\).
-
-        \item[Polynômes annulateurs]
-
-            En reprenant le développement \eqref{EqQHjvLZQ} sachant que \( e^N=\mtu\), nous savons que
-            \begin{equation}
-                N+\frac{ N^2 }{2}+\cdots +\frac{ N^{r-1} }{ (r-1)! }=0.
-            \end{equation}
-            Dit en termes pompeux (mais non moins porteurs de sens), le polynôme
-            \begin{equation}
-                Q(X)=X+\frac{ X^2 }{2}+\cdots +\frac{ X^{r-1} }{ (r-1)! }
-            \end{equation}
-            est un polynôme annulateur de \( N\).
-            
-            La proposition \ref{PropAnnncEcCxj} stipule que le polynôme minimal d'un endomorphisme divise tous les polynômes annulateurs. Dans notre cas, \( X^r\) est un polynôme annulateur et donc le polynôme minimal de \( N\) est de la forme \( X^k\). Donc il est \( X^r\) lui-même.
-            
-            Nous avons donc \( X^r\divides Q\). Mais \( Q\) est un polynôme contenant le monôme \( X\) donc \( X^r\) ne peut diviser \( Q\) que si \( r=1\). Nous en concluons que \( X\) est un polynôme annulateur de \( N\). C'est à dire que \( N=0\).
-
-        \item[Conclusion]
-
-            Vu que Dunford\footnote{Théorème \ref{ThoRURcpW}.} dit que \( A=S+N\) et que nous venons de prouver que \( N=0\), nous concluons que \( A=S\) avec \( S\) diagonalisable.
-
-    \end{subproof}
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Valeurs singulières}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit \( M\) une matrice \( m\times n\) sur \( \eK\) (\( \eK\) est \( \eR\) ou \( \eC\)). Un nombre réel \( \sigma\) est une \defe{valeur singulière}{valeur!singulière} de \( M\) s'il existent des vecteurs unitaires \( u\in \eK^m\), \( v\in \eK^n\) tels que
-    \begin{subequations}
-        \begin{align}
-            Mv&=\sigma u\\
-            M^*u&=\sigma v.
-        \end{align}
-    \end{subequations}
-\end{definition}
-
-\begin{theorem}[Décomposition en valeurs singulières]
-    Soit \( M\in \eM(m\times n,\eK)\) où \( \eK=\eR,\eC\). Alors \( M\) se décompose en
-    \begin{equation}
-        M=ADB
-    \end{equation}
-    où
-    il existe deux matrices unitaires \( A\in \gU(m\times m)\), \( B\in \gU(n\times n)\) et une matrice (pseudo)diagonale \( D\in \eM(m\times n)\) tels que
-    \begin{enumerate}
-        \item 
-            \( A\in\gU(m\times m)\), \( B\in\gU(n\times n)\) sont deux matrices unitaires;,
-        \item
-            \( D\) est (pseudo)diagonale,
-        \item
-            les éléments diagonaux de \( D\) sont les valeurs singulières de \( M\),
-        \item
-            le nombre d'éléments non nuls sur la diagonale de \( D\) est le rang de \( M\).
-    \end{enumerate}
-\end{theorem}
-
-\begin{corollary}
-    Soit \( M\in \eM(n,\eC)\). Il existe un isomorphisme \( f\colon \eC^n\to \eC^n\) tel que \( fM\) soit autoadjoint.
-\end{corollary}
-
-\begin{proof}
-    Si \( M=ADB\) est la décomposition de \( M\) en valeurs singulières, alors nous pouvons prendre \( f=\overline{ B }^tA^{-1}\) qui est une matrice inversible. Pour la vérification que ce \( f\) répond bien à la question, ne pas oublier que \( D\) est réelle, même si \( M\) ne l'est pas.
-\end{proof}

From 2ce6579505edba5045b89e6a2a2c00c16ca81433 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sat, 24 Jun 2017 23:53:52 +0200
Subject: [PATCH 52/64] =?UTF-8?q?(organisation)=20Renomme=20les=20fichiers?=
 =?UTF-8?q?=20d'analyse=20r=C3=A9elle=20pour=20le=20coh=C3=A9rence.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 commons.py                                         |  2 +-
 mazhe.tex                                          | 14 +++++++-------
 .../{173_differentielle.tex => 173_analyseR.tex}   |  0
 .../{64_Chap_analyse_R.tex => 64_analyseR.tex}     |  0
 .../{65_Chap_analyse_R.tex => 65_analyseR.tex}     |  0
 ...hap_calcul_differentiel.tex => 66_analyseR.tex} |  0
 ...hap_calcul_differentiel.tex => 67_analyseR.tex} |  0
 ...hap_calcul_differentiel.tex => 68_analyseR.tex} |  0
 ...hap_calcul_differentiel.tex => 69_analyseR.tex} |  0
 9 files changed, 8 insertions(+), 8 deletions(-)
 rename tex/frido/{173_differentielle.tex => 173_analyseR.tex} (100%)
 rename tex/frido/{64_Chap_analyse_R.tex => 64_analyseR.tex} (100%)
 rename tex/frido/{65_Chap_analyse_R.tex => 65_analyseR.tex} (100%)
 rename tex/frido/{66_Chap_calcul_differentiel.tex => 66_analyseR.tex} (100%)
 rename tex/frido/{67_Chap_calcul_differentiel.tex => 67_analyseR.tex} (100%)
 rename tex/frido/{68_Chap_calcul_differentiel.tex => 68_analyseR.tex} (100%)
 rename tex/frido/{69_Chap_calcul_differentiel.tex => 69_analyseR.tex} (100%)

diff --git a/commons.py b/commons.py
index d29d3f94a..3f6098fd1 100644
--- a/commons.py
+++ b/commons.py
@@ -20,7 +20,7 @@
 
 
 ok_hash=[]
-ok_hash.append("<++>")
+ok_hash.append("7dd1f22241df25fe18b534a41346d88aaa2f6584")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
diff --git a/mazhe.tex b/mazhe.tex
index c7f17df78..e8added00 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -179,13 +179,13 @@ \chapter{Espaces affines}
 \input{140_EspacesAffines}
 
 \chapter{Analyse réelle}
-\input{64_Chap_analyse_R}
-\input{65_Chap_analyse_R}
-\input{66_Chap_calcul_differentiel}
-\input{173_differentielle}
-\input{67_Chap_calcul_differentiel}
-\input{68_Chap_calcul_differentiel}
-\input{69_Chap_calcul_differentiel}
+\input{64_analyseR}
+\input{65_analyseR}
+\input{66_analyseR}
+\input{173_analyseR}
+\input{67_analyseR}
+\input{68_analyseR}
+\input{69_analyseR}
 
 % les tribus ont besoin de la notion de somme infinie (qui est dans le chapitre espaces vectoriels) pour séparer une somme sur NxN en deux sommes sur N
 \chapter{Tribus, théorie de la mesure}
diff --git a/tex/frido/173_differentielle.tex b/tex/frido/173_analyseR.tex
similarity index 100%
rename from tex/frido/173_differentielle.tex
rename to tex/frido/173_analyseR.tex
diff --git a/tex/frido/64_Chap_analyse_R.tex b/tex/frido/64_analyseR.tex
similarity index 100%
rename from tex/frido/64_Chap_analyse_R.tex
rename to tex/frido/64_analyseR.tex
diff --git a/tex/frido/65_Chap_analyse_R.tex b/tex/frido/65_analyseR.tex
similarity index 100%
rename from tex/frido/65_Chap_analyse_R.tex
rename to tex/frido/65_analyseR.tex
diff --git a/tex/frido/66_Chap_calcul_differentiel.tex b/tex/frido/66_analyseR.tex
similarity index 100%
rename from tex/frido/66_Chap_calcul_differentiel.tex
rename to tex/frido/66_analyseR.tex
diff --git a/tex/frido/67_Chap_calcul_differentiel.tex b/tex/frido/67_analyseR.tex
similarity index 100%
rename from tex/frido/67_Chap_calcul_differentiel.tex
rename to tex/frido/67_analyseR.tex
diff --git a/tex/frido/68_Chap_calcul_differentiel.tex b/tex/frido/68_analyseR.tex
similarity index 100%
rename from tex/frido/68_Chap_calcul_differentiel.tex
rename to tex/frido/68_analyseR.tex
diff --git a/tex/frido/69_Chap_calcul_differentiel.tex b/tex/frido/69_analyseR.tex
similarity index 100%
rename from tex/frido/69_Chap_calcul_differentiel.tex
rename to tex/frido/69_analyseR.tex

From 01bdc6d875477329f2256cb42f33c2320daac212 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sun, 25 Jun 2017 00:03:46 +0200
Subject: [PATCH 53/64] =?UTF-8?q?(Taylor)=20D=C3=A9place=20les=20parties?=
 =?UTF-8?q?=20sur=20les=20d=C3=A9veloppements=20de=20Taylor=20avant=20la?=
 =?UTF-8?q?=20partie=20sur=20les=20fonctions=20convexes.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/frido/4_dev_limite.tex        | 629 -----------------------------
 tex/frido/68_analyseR.tex         | 637 +++++++++++++++++++++++++++++-
 tex/frido/79_inversion_locale.tex |   2 +-
 3 files changed, 635 insertions(+), 633 deletions(-)

diff --git a/tex/frido/4_dev_limite.tex b/tex/frido/4_dev_limite.tex
index 42ff5d242..ab36ce77e 100644
--- a/tex/frido/4_dev_limite.tex
+++ b/tex/frido/4_dev_limite.tex
@@ -134,635 +134,6 @@ \section{Nombres de Bell}
 \end{proof}
 
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Développement asymptotique, théorème de Taylor}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\label{AppSecTaylorR}
-
-\begin{theorem}[Théorème de Taylor\cite{TrenchRealAnalisys,ooCNZAooJEcgHZ}]		\label{ThoTaylor}
-Soit $I\subset$ un intervalle non vide et non réduit à un point de $\eR$ ainsi que $a\in I$. Soit une fonction $f\colon I\to \eR$ telle que $f^{(n)}(a)$ existe. Alors il existe une fonction $\alpha$ définie sur $I$ et à valeurs dans $\eR$ vérifiant les deux conditions suivantes :
-\begin{subequations}		\label{SubEqsDevTauil}
-	\begin{align}
-		f(x)&= \sum_{k=0}^n\frac{ f^{(k)}(a) }{ k! }(x-a)^k +\alpha(x)(x-a)^{n},	\\	\label{subeqfTepseqb}
-		\lim_{t\to a}\alpha(t)&=0
-	\end{align}
-\end{subequations}
-pour tout \( x\in I\). Ici $f^{(k)}$ dénote la $k$-ième dérivée de $f$ (en particulier, $f^{(0)}=f$, $f^{(1)}=f'$).\nomenclature{$f^{(n)}$}{La $n$-ième dérivée de la fonction $f$}
-\end{theorem}
-
-Nous insistons sur le fait que la formule \eqref{subeqfTepseqb} est une égalité, et non une approximation. Ce qui serait une approximation serait de récrire la formule dans le terme contenant $\alpha$.
-
-Le polynôme $T^a_{f,n}$ est le \defe{polynôme de Taylor}{Taylor} de $f$ au point $a$ à l'ordre $n$. 
-
-Les conditions \eqref{SubEqsDevTauil} sont souvent aussi énoncées sous la forme qu'il existe une fonction \( \alpha\) telle que
-\begin{subequations}    \label{SUBEQooPYABooKpDgdu}
-    \begin{numcases}{}
-        \lim_{t\to 0} \frac{ \alpha(t) }{ t^n }=0\\
-        f(a+h)=f(a)+hf'(a)+\frac{ h^2 }{2}f''(a)+\cdots+ \frac{ h^n }{ n! }f^{(n)}(a)+\alpha(h).
-    \end{numcases}
-\end{subequations}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Fonctions «petit o» }
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous voulons formaliser l'idée d'une fonction qui tend vers zéro \og plus vite\fg{} qu'une autre. Nous disons que $f\in o\big(\varphi(x)\big)$ si
-\begin{equation}
-    \lim_{x\to 0} \frac{ f(x) }{ \varphi(x) }=0.
-\end{equation}
-En particulier, nous disons que $f\in o(x)$ lorsque $\lim_{x\to 0} f(x)/x=0$.
-
-\begin{remark}
-    À titre personnel, l'auteur de ces lignes déconseille d'utiliser cette notation qui est un peu casse-figure pour qui ne la maîtrise pas bien.
-\end{remark}
-
-En termes de notations, nous définissons l'ensemble $o(x)$\nomenclature{$o(x)$}{fonction tendant rapidement vers zéro} l'ensemble des fonctions $f$ telles que
-\begin{equation}
-	\lim_{x\to 0} \frac{ f(x) }{ x }=0.
-\end{equation}
-Plus généralement si $g$ est une fonction telle que $\lim_{x\to 0} g(x)=0$, nous disons $f\in o(g)$ si
-\begin{equation}
-	\lim_{x\to 0} \frac{ f(x) }{ g(x) }=0.
-\end{equation}
-De façon intuitive, l'ensemble $o(g)$ est l'ensemble des fonctions qui tendent vers zéro «plus vite» que $g$.
-
-Nous pouvons donner un énoncé alternatif au théorème \ref{ThoTaylor} en définissant $h(x)=\epsilon(x+a)x^n$. Cette fonction est définie exprès pour avoir
-\begin{equation}
-	h(x-a)=\epsilon(x)(x-a)^n,
-\end{equation}
-et donc
-\begin{equation}
-	\lim_{x\to 0} \frac{ h(x) }{ x^n }=\lim_{x\to 0} \epsilon(x-a)=\lim_{x\to a}\epsilon(x)=0. 
-\end{equation}
-Donc $h\in o(x^n)$.
-
-Le théorème dit donc qu'il existe une fonction $\alpha\in o(x^n)$ telle que
-\begin{equation}
-	f(x)=T^a_{f,n}(x)+\alpha(x-a).
-\end{equation}
-pour tout $x\in I$. 
-
-\begin{example}
-    L'exemple \ref{EXooXLYJooKVqhTE} du développement du cosinus sera traité quand les fonctions trigonométriques seront définies.
-\end{example}
-
-\begin{proposition}[Ordre deux sur \( \eR^n\)\cite{MonCerveau}]         \label{PROPooTOXIooMMlghF}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\) ainsi qu'une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\). Alors il existe une fonction \( \alpha\colon \eR^n\to \eR\) telle que
-    \begin{subequations}
-        \begin{numcases}{}
-            f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(h)\\
-            \lim_{h\to 0} \alpha(h)=0.
-        \end{numcases}
-    \end{subequations}
-    Ici, la notation \( (d^2f)_a(h,h)\) réfère à ce qui est expliqué en \ref{NORMooZAOEooGqjpLH}.
-\end{proposition}
-
-\begin{proof}
-    Dans la suite nous considérons \( t\) et \( h\) tels que toutes les expressions suivantes aient un sens, c'est à dire que tous les trucs comme \( a+th\) restent dans \( \Omega\). Pour \( h\in \eR^n\) nous nommons \( e_h\) le vecteur unitaire dans la direction de \( h\), c'est à dire \( e_h=h/\| h \|\) et nous posons
-    \begin{equation}
-        k_h(t)=f(a+te_h).
-    \end{equation}
-    et nous lui appliquons Taylor \ref{ThoTaylor} à l'ordre deux : il existe une fonction \( \beta_h\) telle que
-    \begin{equation}        \label{EQooETDFooAmiRcV}
-        k_h(x)=k_h(0)+xk_h'(0)+\frac{ x^2 }{2}k''_h(0)+x^2\beta_h(x).
-    \end{equation}
-    avec \( \lim_{x\to 0} \beta_h(x)=0\).
-
-    En ce qui concerne les dérivées de \( k_h\) nous avons 
-    \begin{equation}
-        k'_h(0)=df_a(e_h)
-    \end{equation}
-    et
-    \begin{equation}
-        k_h''(0)=(d^2f)_{a}(e_h,e_h).
-    \end{equation}
-    Il est maintenant temps d'écrire \( f(a+h)=k(\| h \|)\) et de substituer les dérivées de \( k\) par les différentielles de \( f\) dans \eqref{EQooETDFooAmiRcV} :
-    \begin{equation}        \label{EQooUSUGooYPscxV}
-            f(a+h)=k(\| h \|)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h^2 \|\beta_{h}(\| h \|).
-    \end{equation}
-    Il reste à voir que la fonction \( \alpha\colon h\mapsto \beta_h(\| h \|)\) tend vers zéro pour \( h\to 0\). En prenant la limite \( h\to 0\) dans \eqref{EQooUSUGooYPscxV}, il est manifeste que la limite du membre de gauche existe et vaut \( f(a)\). Donc la limite du membre de droite doit exister et valoir également \( f(a)\). Nous en déduisons que la limite de
-    \begin{equation}
-        df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\beta_h(\| h \|)
-    \end{equation}
-    existe et vaut zéro. La limite des deux premiers termes existe et vaut zéro, donc la limite du troisième existe et vaut zéro :
-    \begin{equation}
-        \lim_{h\to 0} \| h \|^2\beta_h(\| h \|)=0.
-    \end{equation}
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Autres formulations}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{example}		\label{ExempleUtlDev}
-	Une des façons les plus courantes d'utiliser les formules \eqref{SubEqsDevTauil} est de développer $f(a+t)$ pour des petits $t$ en posant $x=a+t$ dans la formule :
-	\begin{equation}	\label{EqDevfautouraeps}
-		f(a+t)=f(a)+f'(a)t+f''(a)\frac{ t^2 }{ 2 }+\epsilon(a+t)t^2
-	\end{equation}
-	avec $\lim_{t\to 0} \epsilon(a+t)=0$. Ici, la fonction $T$ dont on parle dans le théorème est $T_{f,2}^a(a+t)=f(a)+f'(a)t+f''(a)\frac{ t^2 }{2}$.
-
-	Lorsque $x$ et $y$ sont deux nombres «proches\footnote{par exemple dans une limite $(x,y)\to(h,h)$.}», nous pouvons développer $f(y)$ autour de $f(x)$ :
-	\begin{equation}		\label{Eqfydevfx}
-		f(y)=f(x)+f'(x)(y-x)+f''(x)\frac{ (y-x)^2 }{ 2 }+\epsilon(y-x)(y-x)^2,
-	\end{equation}
-	et donc écrire
-	\begin{equation}
-		f(x)-f(y)=-f'(x)(y-x)-f''(x)\frac{ (y-x)^2 }{ 2 }-\epsilon(y-x)(y-x)^2.
-	\end{equation}
-	De cette manière nous obtenons une formule qui ne contient plus que $y$ dans la différence $y-x$.
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Formule et reste}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}     \label{PropDevTaylorPol}
-    Soient $f\colon I\subset\eR\to \eR$ et $a\in\Int(I)$. Soit un entier $k\geq 1$. Si $f$ est $k$ fois dérivable en $a$, alors il existe un et un seul polynôme $P$ de degré $\leq k$ tel que
-    \begin{equation}
-        f(x)-P(x-a)\in o\big( | x-a |^k \big)
-    \end{equation}
-    lorsque $x\to a$, $x\neq a$. Ce polynôme  est donné par
-    \begin{equation}
-        P(h)=f(a)+f'(a)h+\frac{ f''(a) }{ 2! }h^2+\cdots+\frac{ f^{(k)}(a) }{ k! }h^k.
-    \end{equation}
-    Notons encore deux façons alternatives d'écrire le résultat. Si \( f\in C^k\) il existe une fonction \( \alpha\) telle que \( \lim_{t\to 0} \alpha(t)=0\) et
-    \begin{equation}
-        f(x)=\sum_{n=0}^k\frac{ f^{(n)}(a) }{ n! }(x-a)^n+(x-a)^n\alpha(x-a).
-    \end{equation}
-    Si \( f\in C^{k+1}\) alors
-    \begin{equation}        \label{EquQtpoN}
-        f(x)=\sum_{n=0}^k\frac{ f^{(n)}(a) }{ n! }(x-a)^n+(x-a)^{n+1}\xi(x-a)
-    \end{equation}
-    où \( \xi\) est une fonction telle que \( \xi(t)\) tend vers une constante lorsque \( t\to 0\).
-\end{proposition}
-
-La proposition suivant donne une intéressante façon de trouver le reste d'un développement de Taylor.
-\begin{proposition}     \label{PropResteTaylorc}
-Soient $I$, un intervalle dans $\eR$ et $f\colon I\to \eR$ une fonction de classe $C^k$ sur $I$ telle que $f^{(k+1)}$ existe sur $I$. Soient $a\in\Int(I)$ et $x\in I$. Alors il existe $c\in\mathopen] x , a \mathclose[$ tel que
-\begin{equation}
-    f(x)=\sum_{n=0}^k\frac{ f^{(n)}(a) }{ n! }(x-a)^n+\frac{ f^{(k+1)}(c) }{ (k+1)! }(x-a)^{k+1}.
-\end{equation}
-\end{proposition}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Reste intégral}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[Formule de Taylor avec reste intégral\cite{VBYOJrU}]\label{PropAXaSClx}
-    Soient \( X\) et \( Y\) des espaces normés et un ouvert \( \mO\subset X\). Si \( f\in C^m(\mO,Y)\) et si \( [p,x]\subset \mO\) alors
-    \begin{equation}
-        \begin{aligned}[]
-            f(x)=f(p)&+\sum_{k=1}^{m-1}\frac{1}{ k! }(d^kf)_p (x-p)^k \\
-            &+\frac{1}{ (m-1)! }\int_0^1(1-t)^{m-1}(d^mf)_{ p+t(x-p) }(x-p)^m \
-        \end{aligned}
-    \end{equation}
-    où \( \omega_pu^k\) signifie \( \omega_p(u,\ldots, u)\) lorsque \( \omega\in \Omega^k\).
-\end{proposition}
-\index{formule!Taylor!reste intégral}
-Comme expliqué dans l'exemple \ref{ExZHZYcNH}, toute ces applications de différentielles se réduisent à des termes de la forme
-\begin{equation}
-    f^{(k)}(p)(x-p)^k
-\end{equation}
-dans le cas d'une fonction \( \eR\to\eR\).
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Développement limité autour de zéro}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Dans cette sections nous supposons toujours que les fonctions sont définies sur un intervalle ouvert de $\eR$, $I$, contenant \( 0\).
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Généralités}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit \( f\colon I\to 0\) une fonction définie sur un ouvert \( I\) autour de zéro. Nous disons que \( f\) admet un \defe{développement limité}{développement!limité!en zéro} autour de \( 0\) à l'ordre \( n\) s'il existe une fonction \( \alpha\colon I\to \eR\) telle que
-    \begin{subequations}
-        \begin{numcases}{}
-            f(x)=P_n(x)+x^n\alpha(x)\\
-            \lim_{x\to 0} \alpha(x)=0
-        \end{numcases}
-    \end{subequations}
-    où \( P(x)=a_0+a_1x+\cdots +a_nx^n\) est une polynôme de degré \( n\). Le polynôme \( P_n\) est appelé la \defe{partie régulière}{partie!régulière} du développement.
-\end{definition}
-La fonction \( \alpha\) est appelé le \defe{reste}{reste!d'un développement limité} du développement et sera parfois noté \( \alpha_f\). Lorsque \( P\) est la partie régulière d'un développement limité de \( f\) nous notons parfois \( f\sim P\).
-
-\begin{proposition}[Troncature]
-    Si \( f\) admet un développement limité d'ordre \( n\) alors il admet également un développement limité d'ordre \( n'\) pour tout \( n'<n\). Ce dernier s'obtient en tronquant le polynôme d'ordre \( n\) à l'ordre \( n'\).
-\end{proposition}
-
-\begin{proposition}[Unicité]
-    Si \( f\) admet une développement limité alors ce dernier est unique : il existe un unique polynôme \( P_n\) d'ordre \( n\) et une unique fonction \( \alpha\) vérifiant simultanément les deux conditions
-    \begin{subequations}
-        \begin{numcases}{}
-            f(x)=P_n(x)+x^n\alpha(x),\\
-            \lim_{x\to 0} \alpha(x)=0.
-        \end{numcases}
-    \end{subequations}
-\end{proposition}
-
-\begin{example} \label{ExTHGooCBcnAy}
-    En ce qui concerne les séries géométriques de raison \( x\) nous savons les formules
-    \begin{equation}
-        1+x+x^2+\cdots +x^n=\frac{ 1-x^{n+1} }{ 1-x }
-    \end{equation}
-    et
-    \begin{equation}
-        1+x+x^2+x^3+\cdots=\frac{ 1 }{ 1-x }
-    \end{equation}
-    pour tout \( x\in\mathopen] -\infty , 1 \mathclose[\). Comparant les deux, il est naturel d'essayer de prendre \( 1+x+x^2+\cdots +x^n\) comme développement limité de la fonction \( f(x)=\frac{1}{ 1-x }\). Pour voir si cela fonctionne, il faut vérifier si «le reste» est bien de la forme \( x^n\alpha(x)\) avec \( \lim_{x\to 0} \alpha(x)=0\).
-
-    Le reste en question est donné par
-    \begin{equation}
-        \frac{1}{ 1-x }-1-x-x^2-\ldots-x^n=\frac{1}{ 1-x }-\frac{ 1-x^{n+1} }{ 1-x }=\frac{ x^{n+1} }{ 1-x }=x^n\frac{ x }{ 1-x }.
-    \end{equation}
-    En posant \( \alpha(x)=\frac{ x }{ 1-x }\) nous avons donc bien
-    \begin{equation}
-        f(x)=\frac{1}{ 1-x }=1+x+x^2+\cdots +x^n+x^n\frac{ x }{ 1-x }
-    \end{equation}
-    et \( \lim_{x\to 0} \frac{ x }{ 1-x }=0\). Cela est le développement limité de \( f\) à l'ordre \( n\) autour de \( 0\).
-\end{example}
-
-La formule des accroissements finis est un cas particulier de développement fini. Supposons que \( f\) soit dérivable en \( 0\). En effet nous pouvons facilement trouver la fonction \( \alpha\) qui convient. Sachant que \( f(0)+xf'(0)\) donne l'approximation affine de \( f\) autour de \( 0\), nous cherchons \( \alpha\) en écrivant
-\begin{equation}
-    f(x)=f(0)+xf'(0)+x\alpha(x).
-\end{equation}
-Cela nous pousse à définir
-\begin{equation}    \label{EqDCFooKozKrt}
-    \alpha(x)=\frac{ f(x)-f(0) }{ x }-f'(0).
-\end{equation}
-Notons que cette fonction n'est pas définie en \( x=0\), mais cela n'a pas d'importance : seule la limite \( \lim_{x\to 0} \alpha(x)\) nous intéresse. Par définition de la dérivée,
-\begin{equation}
-    \lim_{x\to 0} \alpha(x)=\lim_{x\to 0} \frac{ f(x)-f(0) }{ x }-f'(0)=0.
-\end{equation}
-
-En conclusion si \( f\) est dérivable, son développement limité à l'ordre \(  1\) est donné par
-\begin{equation}
-    f(x)=f(0)+xf'(0)+x\alpha(x)
-\end{equation}
-où \( \alpha(x)\) est donnée par la formule \eqref{EqDCFooKozKrt}.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Formule de Taylor-Young}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Plus généralement nous avons la proposition suivante qui donne le développement limité de toute fonction dérivable \( n\) fois.
-
-\begin{proposition}[Formule de Taylor-Young]    \label{PropVDGooCexFwy}
-    Soit \( f\) une fonction \( n\) fois dérivable sur un intervalle \( I\) contenant \( 0\). Alors il existe une fonction \( \alpha\colon I\to \eR\) telle que
-    \begin{equation}        \label{EQooBKZDooTqYyIB}
-        f(x)=f(0)+f'(0)x+\frac{ f''(0) }{ 2 }x^2+\frac{ f^{(3)}(0) }{ 3! }x^3+\cdots +\frac{ f^{(n)}(0) }{ n! }x^n+x^n\alpha(x)
-    \end{equation}
-    et
-    \begin{equation}
-        \lim_{x\to 0} \alpha(x)=0.
-    \end{equation}
-\end{proposition}
-
-Cette proposition nous permet de trouver facilement des développements limités. Dans l'exemple \ref{ExTHGooCBcnAy} nous avons dû utiliser des astuces et des formules pour déterminer le développement limité de \( \frac{1}{ 1-x }\). Au contraire la formule \eqref{EQooBKZDooTqYyIB} nous permet de trouver le polynôme en calculant seulement de dérivées.
-
-\begin{example}
-    Utilisation de la formule \eqref{EQooBKZDooTqYyIB} pour déterminer le développement limité de la fonction
-    \begin{equation}
-        f(x)=\frac{1}{ 1-x }.
-    \end{equation}
-    Il faut calculer les dérivées successives de \( f\) :
-    \begin{subequations}
-        \begin{align}
-            f(x)&=\frac{1}{ 1-x }\\
-            f'(x)&=\frac{ 1 }{ (1-x)^2 }\\
-            f''(x)&=\frac{ 2 }{ (1-x)^3 }
-        \end{align}
-    \end{subequations}
-    Avec ces résultats, nous devinons que 
-    \begin{equation}
-        f^{(n)}(x)=\frac{ n! }{ (1-x)^{n+1} }.
-    \end{equation}
-    Pour en être sûr nous le prouvons par récurrence. La dérivée de \(\frac{ n! }{ (1-x)^{n+1} } \) est donnée par
-    \begin{equation}
-        \frac{ n!(n+1)(1-x)^n }{ (1-x)^{2n+2} }=\frac{(n+1)! }{ (1-x)^{n+2} }.
-    \end{equation}
-    Évaluées en \( x=0\), les dérivées successives de \( f\) sont \( f(0)=0\), \( f'(0)=1\), \( f''(0)=2\),\ldots,\( f^{(n)}(0)=n!\). Utilisant la formule \eqref{EQooBKZDooTqYyIB} nous avons
-    \begin{equation}
-        f(x)=1+x+x^2+\cdots +x^n+x^n\alpha(x),
-    \end{equation}
-    conformément à ce que nous avions déjà trouvé.
-\end{example}
-\begin{remark}
-  Pour alléger la notation et ne pas écrire \(\ldots +x^n\alpha(x)\) nous pouvons aussi écrire
-    \begin{equation}
-         f(x)\sim 1+x+x^2+\cdots +x^n,
-    \end{equation}
-    mais il est interdit d'écrire
-    \begin{equation}
-         f(x)= 1+x+x^2+\cdots +x^n
-    \end{equation}
-    en mettant un signe d'égalité entre une fonction et son développement limité\footnote{Il faut cependant \^etre très prudents avec la notation abrégée. Elle pourrait nous faire oublier des informations importantes, voir les développements des fonction trigonométriques pour un exemple.}.
-\end{remark}
-
-Notons cependant que la proposition \ref{PropVDGooCexFwy} ne donne pas de moyen simple de trouver la fonction \( \alpha\). Si la fonction $f$ est très régulière dans l'intervalle $I$ on a le résultat suivant.
-
-\begin{proposition}[Reste dans la forme de Lagrange]
-    Si la fonction $f$ est dérivable $n+1$ fois dans $I$ alors il existe $\bar x$ dans l'intervalle \( \mathopen[ 0 , x \mathclose]\) tel que
-  \begin{equation}
-    f(x) = P_n(x) + \frac{1}{(n+1)!} f^{n+1}(\bar x) x^{n+1}.
-  \end{equation}
-\end{proposition}
-Voici quelque développements limités à savoir. Ils sont calculables en utilisant la formule de Taylor-Young (proposition \ref{PropVDGooCexFwy}).
-\begin{framed}
-    \begin{subequations}  \label{SUBEQSooIFXBooIOkveI}
-    \begin{align}
-      e^x&=\sum_{k=0}^n\frac{ x^k }{ k! }+x^n\alpha(x)&\text{ordre } n\\
-      \cos(x)&=\sum_{k=0}^p\frac{ (-1)^kx^{2k} }{ (2k)! }+x^{2p+1}\alpha(x)&\text{ordre } 2p+1    \label{subeqCBQooTZGxQpcos}\\
-      \sin(x)&=\sum_{k=0}^p\frac{ (-1)^kx^{2k+1} }{ (2k+1)! }+x^{p+2}\alpha(x)&\text{ordre } 2p+1 \label{subeqCBQooTZGxQpsin}\\
-      \ln(1+x)&=\sum_{k=0}^{n-1}(-1)^k\frac{ x^{k+1} }{ k+1 }+x^n\alpha(x)&\text{ordre } n\\
-      (1+x)^l&=\sum_{k=0}^l\binom{ l }{ k }x^k&\text{exact si } l\text{ est entier.}\\
-      (1+x)^{\alpha}&=1+\sum_{k=1}^n\frac{ \alpha(\alpha-1)\ldots(\alpha-k+1) }{ k! }x^k+x^n\alpha(x)&\text{ordre } n.\label{subEqJFNooQQodFQ}
-    \end{align}
-  \end{subequations}
-\end{framed}
-Il est fortement suggéré au lecteur de vérifier ces développements par lui-même.
-
-\begin{remark}
-  Quelque remarques.
-  \begin{enumerate}
-  \item
-    Les développements de sinus et de cosinus ont un terme sur deux qui est nul. C'est pour cela qu'en ayant une polynôme de degré \( 2p\), nous avons le développement d'ordre \( 2p+1\).
-  \item
-    En ce qui concerne le développement de \( \ln(1+x)\), il faut noter que la somme va jusqu'à \( k=n-1\) (et non \( k=n\)) parce que nous voulons aller jusqu'à \( x^n\) et que nous écrivons \( x^{k+1}\).
-  \item
-    Si \( l\) est entier, le développement de \( (1+x)^l\) est exact. Dans le développement de \( (1+x)^{\alpha}\), nous reconnaissons la formule de \( \binom{ k }{\alpha }\), sauf que nous ne pouvons pas l'écrire avec cette notation lorsque \( \alpha\) n'est pas entier.
-  \item Le développement limité en $0$ d'une fonction paire ne contient que les puissances de $x$ d'exposant paire. Voir comme exemple le développement de la fonction cosinus.
-  \end{enumerate}
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Règles de calcul}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Les règles suivantes permettent de calculer les développements limités des fonctions qu'on peut écrire comme combinaison des fonctions dans le tableau des équation s\eqref{SUBEQSooIFXBooIOkveI} de la section précédente. 
-\begin{remark}
-  Il est toujours possible de calculer le développement limité d'une fonction par la formule de Taylor-Young. Les règles suivantes peuvent nous economiser de l'effort et du temps. 
-\end{remark}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Linéarité des développements limités}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-L'opération qui consiste à prendre le développement limité d'une fonction est une opération linéaire : connaissant les développements limités de \( f\) et de \( g\), il suffit de les sommer pour obtenir celui de \( f+g\). De m\^eme, si $\lambda$ est une constante, le développement limité de $\lambda f$ est le développement limité de $f$ fois $\lambda$. 
-\begin{proposition}
-Soient $\lambda$ et $\mu$ dans $\eR$.  Si \( f\) et \( g\) sont deux fonctions acceptant des développements limités d'ordre \( n\)
-    \begin{subequations}    \label{EqJPPooCHihNn}
-        \begin{align}
-            f(x)&=P(x)+x^n\alpha_f(x)\\
-            g(x)&=Q(x)+x^n\beta(x)
-        \end{align}
-    \end{subequations}
-    avec \( \lim_{x\to 0} \alpha(x)=\lim_{x\to 0} \beta(x)=0\), alors la fonction \( \lambda f+\mu g\) admet le développement limité
-    \begin{equation}    \label{EqCJFooVpyCtz}
-        (f+g)(x)=(\lambda P+\mu Q)(x)+(\lambda \alpha+\mu\beta)(x).
-    \end{equation}
-\end{proposition}
-\begin{remark}
-  La forme explicite du reste ne nous interesse pas. Dans la pratique on écrira toujours $(f+g)(x)=(P+Q)(x)+\alpha(x)$, où on appelle $\alpha$ une fonction apportune telle que $\lim_{x\to 0} \alpha(x)=0$.
-\end{remark}
-\begin{proof}
-    Vu les définitions \eqref{EqJPPooCHihNn} des polynômes \( P\), \( Q\) et des restes \( \alpha\) et \( \beta\), l'égalité \eqref{EqCJFooVpyCtz} est une conséquence de la linéarité de la dérivation et de la proposition \ref{PropVDGooCexFwy}
-
-    De plus \( P+Q\) est un polynôme de degré \( n\) dès que \( P\) et \( Q\) sont des polynômes de degré \( n\), et
-    \begin{equation}
-        \lim_{x\to 0} (\lambda \alpha+\mu\beta)(x)=\lim_{x\to 0} \lambda\alpha(x)+\lim_{x\to 0} \mu\beta(x)=0.
-    \end{equation}
-    Par conséquent \(\lambda \alpha+\mu\beta\) est la fonction de reste de \( \lambda f+\mu g\).
-\end{proof}
-
-\begin{example} \label{ExKPBooJmdFvY}
-    Calculer le développement de la fonction
-    \begin{equation}
-        f(x)=3\sqrt[3]{1+x}+ e^{-2x}.
-    \end{equation}
-    Le développement de \( \sqrt[3]{1+x}\) est donné par la formule \eqref{subEqJFNooQQodFQ} avec \( \alpha=\frac{1}{ 3 }\). Nous avons donc dans un premier temps
-    \begin{subequations}
-        \begin{align}
-            \sqrt[3]{1+x}&=1+\frac{ 1 }{ 3 }x+\frac{ \frac{1}{ 3 }\left( \frac{1}{ 3 }-1 \right) }{ 2 }x^2+\frac{ \frac{1}{ 3 }\left( \frac{1}{ 3 }-1 \right)\left( \frac{1}{ 3 }-2 \right) }{ 6 }x^3+x^3\alpha(x)\\
-            &=1+\frac{1}{ 3 }x-\frac{1}{ 9 }x^2+\frac{ 5 }{ 81 }x^3+x^3\alpha(x).
-        \end{align}
-    \end{subequations}
-    Nous avons alors
-    \begin{subequations}
-        \begin{align}
-            3\sqrt[3]{1+x}+ e^{-2x}&=3\Big[  1+\frac{1}{ 3 }x-\frac{1}{ 9 }x^2+\frac{ 5 }{ 81 }x^3+x^3\alpha(x)\Big]+1-2x+2x^2-\frac{ 4 }{ 3 }x^3+x^3\beta(x)\\
-            &=4-x+\frac{ 5 }{ 3 }x^2-\frac{ 31 }{ 27 }x^3+x^3\big( \alpha(x)+\beta(x) \big).
-        \end{align}
-    \end{subequations}
-
-\end{example}
-
-La condition \( \lim_{x\to 0} \alpha(x)=0\) signifie que l'approximation qui consiste à remplacer \( f(x) \) par le polynôme n'est pas une trop mauvaise approximation lorsque \( x\) est petit. Cela ne signifie rien de plus. En particulier si \( x\) est grand, l'approximation polynômiale peut-être (et est souvent) très mauvaise.
-
-À ce propos, notez qu'un polynôme tend toujours vers \( \pm\infty\) lorsque \( x\) est grand. Une approximation polynômiale d'une fonction bornée est donc toujours (très) mauvaise pour les grandes valeurs de \( x\).
-
-À titre d'exemple nous avons tracé sur la figure \ref{LabelFigWUYooCISzeB} la fonction
-\begin{equation}
-    f(x)=3\sqrt[3]{x+1}+ e^{-2x}
-\end{equation}
-et ses développements limités d'ordre \( 1\) à \( 3\). Il est particulièrement visible que l'approximation est assez bonne pour la partie gauche du graphe sur laquelle la fonction est bien croissante, alors qu'elle est franchement mauvaise sur la droite où le graphe ressemble plutôt à une constante\footnote{Pouvez-vous cependant dire que vaut \( \lim_{x\to \infty} f(x)\) ?}.
-
-\newcommand{\CaptionFigWUYooCISzeB}{Les développements limités d'ordre de plus en plus grand de la fonction de l'exemple \ref{ExKPBooJmdFvY}. La fonction est en bleu et les «approximations» sont en rouge.}
-\input{auto/pictures_tex/Fig_WUYooCISzeB.pstricks}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Développement limité d'un quotient}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-\begin{proposition}
-    Si \( P_f\) est le polynôme du développement limité de \( f\) à l'ordre \( n\) et \( P_g\) celui de \( g\), alors nous obtenons le développement limité de \( f/g\) à l'ordre \( n\) en effectuant la division selon les puissances croissantes de \( P_f\) par \( P_g\).
-\end{proposition}
-Attention : il s'agit bien de faire une division selon les puissances croissantes, et non une divisions euclidienne. La division euclidienne de \( A\) par \( B\) consiste à écrire \( A=BQ+R\) avec le reste \( R\) de degré le plus \emph{petit} possible. Ici nous voulons avoir un reste de degré le plus \emph{grand} possible.
-
-\begin{example}
-    Cherchons le développement limité à l'ordre \( 5\) de \( \tan(x)=\frac{ \sin(x) }{ \cos(x) }\). Nous utilisons les formules \eqref{subeqCBQooTZGxQpcos} et \eqref{subeqCBQooTZGxQpsin} pour savoir les développements de sinus et cosinus\footnote{Encore une fois, vous êtes très fortement encouragés à calculer vous-même ces développements à partir de la formule de Taylor-Young \ref{EQooBKZDooTqYyIB}.} :
-    \begin{subequations}
-        \begin{align}
-            \sin(x)&=x-\frac{ x^3 }{ 6 }+\frac{ x^5 }{ 120 }+x^5\alpha_1(x)\\
-            \cos(x)&=1-\frac{ x^2 }{ 2 }+\frac{ x^4 }{ 24 }+x^5\alpha_2(x).
-        \end{align}
-    \end{subequations}
-    Nous calculons alors la division des deux polynômes, en classant les puissances dans l'ordre croissant (c'est le sens inverse de ce qui est fait pour la divisions euclidienne !) :
-    \begin{equation*}
-        \begin{array}[]{ccccccccccc|c}
-            &x&-&\frac{1}{ 6 }x^3&+&\frac{1}{ 120 }x^5&&&&&&1-\frac{ 1 }{2}x^2+\frac{1}{ 24 }x^4\\
-            \cline{12-12}
-            -\Big( &x&-&\frac{ 1 }{2}x^3&+&\frac{1}{ 24 }x^5&\Big)& & && &x+\frac{1}{ 3 }x^3+\frac{ 2 }{ 15 }x^5\\
-            \cline{2-6}
-            & & &\frac{1}{ 3 }x^3&-&\frac{1}{ 30 }x^5& & & & && \\
-            &&-\Big(  &\frac{1}{ 3 }x^3&-&\frac{1}{ 6 }x^5&+&\frac{1}{ 72 }x^7&\Big) & & \\
-            \cline{4-8}
-            & & & & &\frac{ 2 }{ 15 }x^5&-&\frac{1}{ 72 }x^7& & & \\
-            & & &  &-\Big(  &\frac{ 2 }{ 15 }x^5&-&\frac{1}{ 15 }x^7&+&\frac{1}{ 180 }x^9&\Big)& \\
-            \cline{6-10}
-            & & & & & & &\frac{ 29 }{ 360 }x^7&-&\frac{1}{ 180 }x^9&& \\
-        \end{array}
-    \end{equation*}
-    Nous avons continué la division jusqu'à obtenir un reste de degré plus grand que \( 5\). Le développement à l'ordre $5$ de la fonction tangente autour de zéro est alors
-    \begin{equation}
-        \tan(x)=x+\frac{1}{ 3 }x^3+\frac{ 2 }{ 15 }x^5+x^5\alpha(x).
-    \end{equation}
-    Notons que, vu que le reste ne nous intéresse pas vraiment, nous aurions pu ne pas calculer les coefficients des termes en \( x^7\) et \( x^8\). La dernière soustraction était également inutile.
-\end{example}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Développement limité d'une fonction composée}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-
-\begin{proposition}
-    Soient \( f\) et \( g\) des fonctions admettant des développements limités d'ordre $n$ au voisinage de $0$. Nous supposons que \( \lim_{x\to 0} g(x)=0\). Alors la composée \( f\big( g(x) \big)\) admet un développement limité d'ordre $n$ au voisinage de $0$ qui s'obtient en substituant le développement de \( g\) à chaque <<\(x \)>> du développement de \( f\), et en supprimant tous les termes de degré plus élevé que $n$.
-\end{proposition}
-
-\begin{example}\label{compose1}
-    Pour trouver le développement de la fonction \( f(x)= e^{-2x}\), il suffit d'écrire celui de \( e^t\) et de remplacer ensuite $t$ par \( -2x\). Le développement à l'ordre \( 3\) de la fonction exponentielle est :
-    \begin{equation}
-        e^t=1+t+\frac{ t^2 }{2}+\frac{ t^3 }{ 6 }+t^3\alpha(t).
-    \end{equation}
-    Le développement de \( f(x)= e^{-2x}\) sera donc 
-    \begin{equation}
-        f(x)=1-2x+\frac{ 4x^2 }{ 2 }-\frac{ 8x^3 }{ 6 }-8x^3\alpha(-2x).
-    \end{equation}
-    Donc le polynôme de degré \( 3\) partie régulière de \( g\) est :
-    \begin{equation}
-        1-2x+2x^2-\frac{ 4 }{ 3 }x^3,
-    \end{equation}
-    et la fonction reste correspondante est :
-    \begin{equation}
-        \alpha_g(x)=-8\alpha(-2x).
-    \end{equation}
-\end{example}
-
-\begin{example}
-    Nous savons les développements
-    \begin{equation}
-        f(x)=\ln(1+x)\sim x-\frac{ x^2 }{ 2 }+\frac{ x^3 }{ 3 }
-    \end{equation}
-    et
-    \begin{equation}
-        \sin(x)\sim x-\frac{ x^3 }{ 6 }.
-    \end{equation}
-    Nous obtenons le développement d'ordre \( 3\) de la fonction \( x\mapsto \ln\big( 1+\sin(x) \big)\) en écrivant
-    \begin{equation}    \label{EqGXMooWKQkIL}
-        \ln\big( 1+\sin(x) \big)\sim \big( x-\frac{ x^3 }{ 6 } \big)-\frac{ 1 }{2}\left( x-\frac{ x^3 }{ 6 } \right)^2+\frac{1}{ 3 }\left( x-\frac{ x^3 }{ 6 } \right)^3.
-    \end{equation}
-    Il s'agit maintenant de trouver les termes qui sont de degré inférieur ou égale à \( 3\).
-
-    D'abord
-    \begin{equation}
-        \left( x-\frac{ x^3 }{ 6 } \right)^2=x^2-\frac{ x^4 }{ 3 }+\frac{ x^6 }{ 36 }\sim x^2
-    \end{equation}
-    Nous avons alors aussi
-    \begin{equation}
-        \left( x-\frac{ x^3 }{ 6 } \right)^6\sim x^2\left( x-\frac{ x^3 }{ 6 } \right)\sim x^3.
-    \end{equation}
-    En replaçant tout ça dans \eqref{EqGXMooWKQkIL} nous trouvons
-    \begin{equation}
-        \ln\big( 1+\sin(x) \big)\sim x-\frac{ x^2 }{2}+\frac{ x^3 }{ 6 }.
-    \end{equation}
-\end{example}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Développement au voisinage de $x_0\neq 0$}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Il est intéressant de développer une fonction au voisinage de zéro lorsque nous nous intéressons à son comportement pour les \( x\) pas très grands. Il est toutefois souvent souhaitable de savoir le comportement d'une fonction au voisinage d'autres valeurs que zéro.
-
-Pour développer la fonction \( f\) autour de \( x_0\), nous considérons la fonction \( h\mapsto f(x_0+h)\) que nous développons autour de zéro (pour \( h\)). L'objectif est de trouver une polynôme \( P\) et une fonction \( \alpha\) tels que
-\begin{subequations}
-    \begin{numcases}{}
-        f(x)=P(x)+(x-x_0)^n\alpha(x)\\
-        \lim_{x\to x_0} \alpha(x)=0.
-    \end{numcases}
-\end{subequations}
-En pratique, le développement limité à l'ordre $n$ d'une fonction autour d'un point $x_0$ quelconque à l'intérieur de son domaine prend la forme suivante, qui généralise la formule de Taylor-Young vue dans la proposition \ref{PropVDGooCexFwy}
-\begin{proposition}[Formule de Taylor-Young, cas général]    
-    Soit \( f\) une fonction \( n\) fois dérivable sur un intervalle \( I\) contenant \(x_0\). Alors il existe une fonction \( \alpha\colon I\to \eR\) telle que
-    \begin{equation}    \label{EqTJRooUbsyzJ}
-      \begin{aligned}
-        f(x)=f(x_0)+&f'(x_0)(x-x_0)+\frac{ f''(x_0) }{ 2 }(x-x_0)^2+\\
-        &+\frac{ f^{(3)}(x_0) }{ 3! }(x-x_0)^3+\cdots +\frac{ f^{(n)}(x_0) }{ n! }(x-x_0)^n+(x-x_0)^n\alpha(x-x_0)
-      \end{aligned}
-    \end{equation}
-    et
-    \begin{equation}
-        \lim_{t\to 0} \alpha(t)=0.
-    \end{equation}
-  
-\end{proposition}
-\begin{example}\label{developcosenpisur3}
-    Développer la fonction \( \cos\) autour de \( x=\frac{ \pi }{ 3 }\). Nous développons autour de \( h=0\) la fonction \( \cos(\frac{ \pi }{ 3 }+h)\) :
-    \begin{equation}
-        \cos\big( \frac{ \pi }{ 3 }+h \big)\sim \cos\big( \frac{ \pi }{ 3 } \big)+h\cos'(\frac{ \pi }{ 3 })+\frac{ h^2 }{2}\cos''\big( \frac{ \pi }{ 3 } \big)=\frac{ 1 }{2}-\frac{ \sqrt{3} }{2}h-\frac{1}{ 4 }h^2.
-    \end{equation}
-    Il est aussi possible d'écrire cela en notant \( x=x_0+h\), c'est à dire en remplaçant \( h\) par \( x-\frac{ \pi }{ 3 }\) :
-    \begin{equation}
-        \cos(x)\sim\frac{ 1 }{2}-\frac{ \sqrt{3} }{ 2 }(x-\frac{ \pi }{ 3 })-\frac{1}{ 4 }(x-\frac{ \pi }{ 3 })^2.
-    \end{equation}
-\end{example}
-
-Pour donner une idée nous avons dessiné sur le graphe suivant la fonction sinus et ses développements d'ordre \( 4\) autour de zéro et autour de \( 3\pi/4\).
-\begin{center}
-   \input{auto/pictures_tex/Fig_WJBooMTAhtl.pstricks}
-\end{center}
-
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Application au calcul de limites}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Lors d'un calcul de limite, développer une partie d'une expression peut être utile.
-
-\begin{example}
-    À calculer :
-    \begin{equation}
-        \lim_{x\to 0} \frac{ \ln(1+x) }{ x }.
-    \end{equation}
-    Cela est une indétermination de type \( \frac{ 0 }{ 0 }\). Le développement limité du numérateur nous donne une fonction \( \alpha(x)\) telle que \( \lim_{x\to 0} \alpha(x)=0\) et
-    \begin{equation}
-        \frac{ \ln(1+x) }{ x }=\frac{ x-\frac{ x^2 }{2}+x^2\alpha(x) }{ x }=1-\frac{ x }{ 2 }+x\alpha(x).
-    \end{equation}
-    Sur le membre de droite la limite est facile à calculer :
-    \begin{equation}
-        \lim_{x\to 0} \frac{ \ln(1+x) }{ x }=\lim_{x\to 0} \Big( 1-\frac{ x }{ 2 }+x\alpha(x) \Big) =1.
-    \end{equation}
-\end{example}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Développement au voisinage de l'infini}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Il est souvent utile de connaître le comportement d'une fonction pour les grandes valeurs de \( x\) et de déterminer ses asymptotes éventuelles. La technique que nous allons utiliser consiste à poser \( x=\frac{1}{ h }\) et de développer la fonction ``auxiliaire'' $g(h) = f(1/h)$ autour de \( h=0\). La limite avec \( h\to 0^+\) donnera le comportement pour \( x\to \infty\) et la limite \( h\to 0^-\) donnera le comportement pour \( x\to -\infty\).
-
-Dans le cas d'une développement autour de \( \pm\infty\) nous ne parlons plus de développement \emph{limité} mais de \defe{développement asymptotique}{développement!asymptotique}.
-
-\begin{example}	\label{ExBCDookjljhjk}
-    Calculer
-    \begin{equation}\label{EqABCoolkjh}
-        \lim_{x\to \infty}  e^{1/x}\sqrt{1+4x^2}-2x.
-    \end{equation}
-    Nous allons effectuer un développement asymptotique de la partie «difficile» de l'expression posant d'abord $x=1/h$. Si $f(x)=e^{1/x}\sqrt{1-4x^2}$ alors
-    \begin{equation}
-	g(h)=\frac{1}{|h|}e^h\sqrt{h^2+4}=\frac{1}{h}\big(  1+h+h\alpha(h) \big)\big( 2+h\beta(h) \big).
-    \end{equation}
-    La première parenthèse est le développement de $e^h$ et la seconde celui de $\sqrt{h^2+4}$. Nous nous apprêtons à faire la limite $x\to\infty$ qui correspond à $h\to 0^+$, nous pouvons donc supposer que $h>0$ et omettre la valeur absolue. En effectuant le produit et en regroupant tous les termes contenant $h^2$, $\alpha(h)$ ou $\beta(h)$ dans un seul terme $h\gamma(h)$,
-    \begin{equation}
-	f(h)=\frac{1}{h}\big(  2+2h+h\gamma(h) \big)=\frac{2}{h}+2+\gamma(h)=2x+2+\gamma(1/x)
-    \end{equation}
-    où $\gamma$ est une fonction vérifiant $\lim_{t\to 0}\gamma(t)=0$.
-
-    Nous sommes maintenant en mesure de calculer la limite \eqref{EqABCoolkjh} :
-    \begin{equation}
-	\lim_{x\to\infty}e^{1/x}\sqrt{1+x^2}-2x= \lim_{x\to \infty}\big(  2x+2+\gamma(1/x)-2x \big)=2.
-    \end{equation}
-\end{example}
-
 
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Étude d'asymptote}
diff --git a/tex/frido/68_analyseR.tex b/tex/frido/68_analyseR.tex
index 8b221c051..40b6bf488 100644
--- a/tex/frido/68_analyseR.tex
+++ b/tex/frido/68_analyseR.tex
@@ -788,6 +788,636 @@ \subsection{Ordre supérieur}
 \end{proposition}
 % TODO : une preuve serait importante.
 
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Développement asymptotique, théorème de Taylor}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\label{AppSecTaylorR}
+
+\begin{theorem}[Théorème de Taylor\cite{TrenchRealAnalisys,ooCNZAooJEcgHZ}]		\label{ThoTaylor}
+Soit $I\subset$ un intervalle non vide et non réduit à un point de $\eR$ ainsi que $a\in I$. Soit une fonction $f\colon I\to \eR$ telle que $f^{(n)}(a)$ existe. Alors il existe une fonction $\alpha$ définie sur $I$ et à valeurs dans $\eR$ vérifiant les deux conditions suivantes :
+\begin{subequations}		\label{SubEqsDevTauil}
+	\begin{align}
+		f(x)&= \sum_{k=0}^n\frac{ f^{(k)}(a) }{ k! }(x-a)^k +\alpha(x)(x-a)^{n},	\\	\label{subeqfTepseqb}
+		\lim_{t\to a}\alpha(t)&=0
+	\end{align}
+\end{subequations}
+pour tout \( x\in I\). Ici $f^{(k)}$ dénote la $k$-ième dérivée de $f$ (en particulier, $f^{(0)}=f$, $f^{(1)}=f'$).\nomenclature{$f^{(n)}$}{La $n$-ième dérivée de la fonction $f$}
+\end{theorem}
+
+Nous insistons sur le fait que la formule \eqref{subeqfTepseqb} est une égalité, et non une approximation. Ce qui serait une approximation serait de récrire la formule dans le terme contenant $\alpha$.
+
+Le polynôme $T^a_{f,n}$ est le \defe{polynôme de Taylor}{Taylor} de $f$ au point $a$ à l'ordre $n$. 
+
+Les conditions \eqref{SubEqsDevTauil} sont souvent aussi énoncées sous la forme qu'il existe une fonction \( \alpha\) telle que
+\begin{subequations}    \label{SUBEQooPYABooKpDgdu}
+    \begin{numcases}{}
+        \lim_{t\to 0} \frac{ \alpha(t) }{ t^n }=0\\
+        f(a+h)=f(a)+hf'(a)+\frac{ h^2 }{2}f''(a)+\cdots+ \frac{ h^n }{ n! }f^{(n)}(a)+\alpha(h).
+    \end{numcases}
+\end{subequations}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Fonctions «petit o» }
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous voulons formaliser l'idée d'une fonction qui tend vers zéro \og plus vite\fg{} qu'une autre. Nous disons que $f\in o\big(\varphi(x)\big)$ si
+\begin{equation}
+    \lim_{x\to 0} \frac{ f(x) }{ \varphi(x) }=0.
+\end{equation}
+En particulier, nous disons que $f\in o(x)$ lorsque $\lim_{x\to 0} f(x)/x=0$.
+
+\begin{remark}
+    À titre personnel, l'auteur de ces lignes déconseille d'utiliser cette notation qui est un peu casse-figure pour qui ne la maîtrise pas bien.
+\end{remark}
+
+En termes de notations, nous définissons l'ensemble $o(x)$\nomenclature{$o(x)$}{fonction tendant rapidement vers zéro} l'ensemble des fonctions $f$ telles que
+\begin{equation}
+	\lim_{x\to 0} \frac{ f(x) }{ x }=0.
+\end{equation}
+Plus généralement si $g$ est une fonction telle que $\lim_{x\to 0} g(x)=0$, nous disons $f\in o(g)$ si
+\begin{equation}
+	\lim_{x\to 0} \frac{ f(x) }{ g(x) }=0.
+\end{equation}
+De façon intuitive, l'ensemble $o(g)$ est l'ensemble des fonctions qui tendent vers zéro «plus vite» que $g$.
+
+Nous pouvons donner un énoncé alternatif au théorème \ref{ThoTaylor} en définissant $h(x)=\epsilon(x+a)x^n$. Cette fonction est définie exprès pour avoir
+\begin{equation}
+	h(x-a)=\epsilon(x)(x-a)^n,
+\end{equation}
+et donc
+\begin{equation}
+	\lim_{x\to 0} \frac{ h(x) }{ x^n }=\lim_{x\to 0} \epsilon(x-a)=\lim_{x\to a}\epsilon(x)=0. 
+\end{equation}
+Donc $h\in o(x^n)$.
+
+Le théorème dit donc qu'il existe une fonction $\alpha\in o(x^n)$ telle que
+\begin{equation}
+	f(x)=T^a_{f,n}(x)+\alpha(x-a).
+\end{equation}
+pour tout $x\in I$. 
+
+\begin{example}
+    L'exemple \ref{EXooXLYJooKVqhTE} du développement du cosinus sera traité quand les fonctions trigonométriques seront définies.
+\end{example}
+
+\begin{proposition}[Ordre deux sur \( \eR^n\)\cite{MonCerveau}]         \label{PROPooTOXIooMMlghF}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\) ainsi qu'une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\). Alors il existe une fonction \( \alpha\colon \eR^n\to \eR\) telle que
+    \begin{subequations}
+        \begin{numcases}{}
+            f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(h)\\
+            \lim_{h\to 0} \alpha(h)=0.
+        \end{numcases}
+    \end{subequations}
+    Ici, la notation \( (d^2f)_a(h,h)\) réfère à ce qui est expliqué en \ref{NORMooZAOEooGqjpLH}.
+\end{proposition}
+
+\begin{proof}
+    Dans la suite nous considérons \( t\) et \( h\) tels que toutes les expressions suivantes aient un sens, c'est à dire que tous les trucs comme \( a+th\) restent dans \( \Omega\). Pour \( h\in \eR^n\) nous nommons \( e_h\) le vecteur unitaire dans la direction de \( h\), c'est à dire \( e_h=h/\| h \|\) et nous posons
+    \begin{equation}
+        k_h(t)=f(a+te_h).
+    \end{equation}
+    et nous lui appliquons Taylor \ref{ThoTaylor} à l'ordre deux : il existe une fonction \( \beta_h\) telle que
+    \begin{equation}        \label{EQooETDFooAmiRcV}
+        k_h(x)=k_h(0)+xk_h'(0)+\frac{ x^2 }{2}k''_h(0)+x^2\beta_h(x).
+    \end{equation}
+    avec \( \lim_{x\to 0} \beta_h(x)=0\).
+
+    En ce qui concerne les dérivées de \( k_h\) nous avons 
+    \begin{equation}
+        k'_h(0)=df_a(e_h)
+    \end{equation}
+    et
+    \begin{equation}
+        k_h''(0)=(d^2f)_{a}(e_h,e_h).
+    \end{equation}
+    Il est maintenant temps d'écrire \( f(a+h)=k(\| h \|)\) et de substituer les dérivées de \( k\) par les différentielles de \( f\) dans \eqref{EQooETDFooAmiRcV} :
+    \begin{equation}        \label{EQooUSUGooYPscxV}
+            f(a+h)=k(\| h \|)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h^2 \|\beta_{h}(\| h \|).
+    \end{equation}
+    Il reste à voir que la fonction \( \alpha\colon h\mapsto \beta_h(\| h \|)\) tend vers zéro pour \( h\to 0\). En prenant la limite \( h\to 0\) dans \eqref{EQooUSUGooYPscxV}, il est manifeste que la limite du membre de gauche existe et vaut \( f(a)\). Donc la limite du membre de droite doit exister et valoir également \( f(a)\). Nous en déduisons que la limite de
+    \begin{equation}
+        df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\beta_h(\| h \|)
+    \end{equation}
+    existe et vaut zéro. La limite des deux premiers termes existe et vaut zéro, donc la limite du troisième existe et vaut zéro :
+    \begin{equation}
+        \lim_{h\to 0} \| h \|^2\beta_h(\| h \|)=0.
+    \end{equation}
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Autres formulations}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{example}		\label{ExempleUtlDev}
+	Une des façons les plus courantes d'utiliser les formules \eqref{SubEqsDevTauil} est de développer $f(a+t)$ pour des petits $t$ en posant $x=a+t$ dans la formule :
+	\begin{equation}	\label{EqDevfautouraeps}
+		f(a+t)=f(a)+f'(a)t+f''(a)\frac{ t^2 }{ 2 }+\epsilon(a+t)t^2
+	\end{equation}
+	avec $\lim_{t\to 0} \epsilon(a+t)=0$. Ici, la fonction $T$ dont on parle dans le théorème est $T_{f,2}^a(a+t)=f(a)+f'(a)t+f''(a)\frac{ t^2 }{2}$.
+
+	Lorsque $x$ et $y$ sont deux nombres «proches\footnote{par exemple dans une limite $(x,y)\to(h,h)$.}», nous pouvons développer $f(y)$ autour de $f(x)$ :
+	\begin{equation}		\label{Eqfydevfx}
+		f(y)=f(x)+f'(x)(y-x)+f''(x)\frac{ (y-x)^2 }{ 2 }+\epsilon(y-x)(y-x)^2,
+	\end{equation}
+	et donc écrire
+	\begin{equation}
+		f(x)-f(y)=-f'(x)(y-x)-f''(x)\frac{ (y-x)^2 }{ 2 }-\epsilon(y-x)(y-x)^2.
+	\end{equation}
+	De cette manière nous obtenons une formule qui ne contient plus que $y$ dans la différence $y-x$.
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Formule et reste}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}     \label{PropDevTaylorPol}
+    Soient $f\colon I\subset\eR\to \eR$ et $a\in\Int(I)$. Soit un entier $k\geq 1$. Si $f$ est $k$ fois dérivable en $a$, alors il existe un et un seul polynôme $P$ de degré $\leq k$ tel que
+    \begin{equation}
+        f(x)-P(x-a)\in o\big( | x-a |^k \big)
+    \end{equation}
+    lorsque $x\to a$, $x\neq a$. Ce polynôme  est donné par
+    \begin{equation}
+        P(h)=f(a)+f'(a)h+\frac{ f''(a) }{ 2! }h^2+\cdots+\frac{ f^{(k)}(a) }{ k! }h^k.
+    \end{equation}
+    Notons encore deux façons alternatives d'écrire le résultat. Si \( f\in C^k\) il existe une fonction \( \alpha\) telle que \( \lim_{t\to 0} \alpha(t)=0\) et
+    \begin{equation}
+        f(x)=\sum_{n=0}^k\frac{ f^{(n)}(a) }{ n! }(x-a)^n+(x-a)^n\alpha(x-a).
+    \end{equation}
+    Si \( f\in C^{k+1}\) alors
+    \begin{equation}        \label{EquQtpoN}
+        f(x)=\sum_{n=0}^k\frac{ f^{(n)}(a) }{ n! }(x-a)^n+(x-a)^{n+1}\xi(x-a)
+    \end{equation}
+    où \( \xi\) est une fonction telle que \( \xi(t)\) tend vers une constante lorsque \( t\to 0\).
+\end{proposition}
+
+La proposition suivant donne une intéressante façon de trouver le reste d'un développement de Taylor.
+\begin{proposition}     \label{PropResteTaylorc}
+Soient $I$, un intervalle dans $\eR$ et $f\colon I\to \eR$ une fonction de classe $C^k$ sur $I$ telle que $f^{(k+1)}$ existe sur $I$. Soient $a\in\Int(I)$ et $x\in I$. Alors il existe $c\in\mathopen] x , a \mathclose[$ tel que
+\begin{equation}
+    f(x)=\sum_{n=0}^k\frac{ f^{(n)}(a) }{ n! }(x-a)^n+\frac{ f^{(k+1)}(c) }{ (k+1)! }(x-a)^{k+1}.
+\end{equation}
+\end{proposition}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Reste intégral}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[Formule de Taylor avec reste intégral\cite{VBYOJrU}]\label{PropAXaSClx}
+    Soient \( X\) et \( Y\) des espaces normés et un ouvert \( \mO\subset X\). Si \( f\in C^m(\mO,Y)\) et si \( [p,x]\subset \mO\) alors
+    \begin{equation}
+        \begin{aligned}[]
+            f(x)=f(p)&+\sum_{k=1}^{m-1}\frac{1}{ k! }(d^kf)_p (x-p)^k \\
+            &+\frac{1}{ (m-1)! }\int_0^1(1-t)^{m-1}(d^mf)_{ p+t(x-p) }(x-p)^m \
+        \end{aligned}
+    \end{equation}
+    où \( \omega_pu^k\) signifie \( \omega_p(u,\ldots, u)\) lorsque \( \omega\in \Omega^k\).
+\end{proposition}
+\index{formule!Taylor!reste intégral}
+Comme expliqué dans l'exemple \ref{ExZHZYcNH}, toute ces applications de différentielles se réduisent à des termes de la forme
+\begin{equation}
+    f^{(k)}(p)(x-p)^k
+\end{equation}
+dans le cas d'une fonction \( \eR\to\eR\).
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Développement limité autour de zéro}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Dans cette sections nous supposons toujours que les fonctions sont définies sur un intervalle ouvert de $\eR$, $I$, contenant \( 0\).
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Généralités}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit \( f\colon I\to 0\) une fonction définie sur un ouvert \( I\) autour de zéro. Nous disons que \( f\) admet un \defe{développement limité}{développement!limité!en zéro} autour de \( 0\) à l'ordre \( n\) s'il existe une fonction \( \alpha\colon I\to \eR\) telle que
+    \begin{subequations}
+        \begin{numcases}{}
+            f(x)=P_n(x)+x^n\alpha(x)\\
+            \lim_{x\to 0} \alpha(x)=0
+        \end{numcases}
+    \end{subequations}
+    où \( P(x)=a_0+a_1x+\cdots +a_nx^n\) est une polynôme de degré \( n\). Le polynôme \( P_n\) est appelé la \defe{partie régulière}{partie!régulière} du développement.
+\end{definition}
+La fonction \( \alpha\) est appelé le \defe{reste}{reste!d'un développement limité} du développement et sera parfois noté \( \alpha_f\). Lorsque \( P\) est la partie régulière d'un développement limité de \( f\) nous notons parfois \( f\sim P\).
+
+\begin{proposition}[Troncature]
+    Si \( f\) admet un développement limité d'ordre \( n\) alors il admet également un développement limité d'ordre \( n'\) pour tout \( n'<n\). Ce dernier s'obtient en tronquant le polynôme d'ordre \( n\) à l'ordre \( n'\).
+\end{proposition}
+
+\begin{proposition}[Unicité]
+    Si \( f\) admet une développement limité alors ce dernier est unique : il existe un unique polynôme \( P_n\) d'ordre \( n\) et une unique fonction \( \alpha\) vérifiant simultanément les deux conditions
+    \begin{subequations}
+        \begin{numcases}{}
+            f(x)=P_n(x)+x^n\alpha(x),\\
+            \lim_{x\to 0} \alpha(x)=0.
+        \end{numcases}
+    \end{subequations}
+\end{proposition}
+
+\begin{example} \label{ExTHGooCBcnAy}
+    En ce qui concerne les séries géométriques de raison \( x\) nous savons les formules
+    \begin{equation}
+        1+x+x^2+\cdots +x^n=\frac{ 1-x^{n+1} }{ 1-x }
+    \end{equation}
+    et
+    \begin{equation}
+        1+x+x^2+x^3+\cdots=\frac{ 1 }{ 1-x }
+    \end{equation}
+    pour tout \( x\in\mathopen] -\infty , 1 \mathclose[\). Comparant les deux, il est naturel d'essayer de prendre \( 1+x+x^2+\cdots +x^n\) comme développement limité de la fonction \( f(x)=\frac{1}{ 1-x }\). Pour voir si cela fonctionne, il faut vérifier si «le reste» est bien de la forme \( x^n\alpha(x)\) avec \( \lim_{x\to 0} \alpha(x)=0\).
+
+    Le reste en question est donné par
+    \begin{equation}
+        \frac{1}{ 1-x }-1-x-x^2-\ldots-x^n=\frac{1}{ 1-x }-\frac{ 1-x^{n+1} }{ 1-x }=\frac{ x^{n+1} }{ 1-x }=x^n\frac{ x }{ 1-x }.
+    \end{equation}
+    En posant \( \alpha(x)=\frac{ x }{ 1-x }\) nous avons donc bien
+    \begin{equation}
+        f(x)=\frac{1}{ 1-x }=1+x+x^2+\cdots +x^n+x^n\frac{ x }{ 1-x }
+    \end{equation}
+    et \( \lim_{x\to 0} \frac{ x }{ 1-x }=0\). Cela est le développement limité de \( f\) à l'ordre \( n\) autour de \( 0\).
+\end{example}
+
+La formule des accroissements finis est un cas particulier de développement fini. Supposons que \( f\) soit dérivable en \( 0\). En effet nous pouvons facilement trouver la fonction \( \alpha\) qui convient. Sachant que \( f(0)+xf'(0)\) donne l'approximation affine de \( f\) autour de \( 0\), nous cherchons \( \alpha\) en écrivant
+\begin{equation}
+    f(x)=f(0)+xf'(0)+x\alpha(x).
+\end{equation}
+Cela nous pousse à définir
+\begin{equation}    \label{EqDCFooKozKrt}
+    \alpha(x)=\frac{ f(x)-f(0) }{ x }-f'(0).
+\end{equation}
+Notons que cette fonction n'est pas définie en \( x=0\), mais cela n'a pas d'importance : seule la limite \( \lim_{x\to 0} \alpha(x)\) nous intéresse. Par définition de la dérivée,
+\begin{equation}
+    \lim_{x\to 0} \alpha(x)=\lim_{x\to 0} \frac{ f(x)-f(0) }{ x }-f'(0)=0.
+\end{equation}
+
+En conclusion si \( f\) est dérivable, son développement limité à l'ordre \(  1\) est donné par
+\begin{equation}
+    f(x)=f(0)+xf'(0)+x\alpha(x)
+\end{equation}
+où \( \alpha(x)\) est donnée par la formule \eqref{EqDCFooKozKrt}.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Formule de Taylor-Young}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Plus généralement nous avons la proposition suivante qui donne le développement limité de toute fonction dérivable \( n\) fois.
+
+\begin{proposition}[Formule de Taylor-Young]    \label{PropVDGooCexFwy}
+    Soit \( f\) une fonction \( n\) fois dérivable sur un intervalle \( I\) contenant \( 0\). Alors il existe une fonction \( \alpha\colon I\to \eR\) telle que
+    \begin{equation}        \label{EQooBKZDooTqYyIB}
+        f(x)=f(0)+f'(0)x+\frac{ f''(0) }{ 2 }x^2+\frac{ f^{(3)}(0) }{ 3! }x^3+\cdots +\frac{ f^{(n)}(0) }{ n! }x^n+x^n\alpha(x)
+    \end{equation}
+    et
+    \begin{equation}
+        \lim_{x\to 0} \alpha(x)=0.
+    \end{equation}
+\end{proposition}
+
+Cette proposition nous permet de trouver facilement des développements limités. Dans l'exemple \ref{ExTHGooCBcnAy} nous avons dû utiliser des astuces et des formules pour déterminer le développement limité de \( \frac{1}{ 1-x }\). Au contraire la formule \eqref{EQooBKZDooTqYyIB} nous permet de trouver le polynôme en calculant seulement de dérivées.
+
+\begin{example}
+    Utilisation de la formule \eqref{EQooBKZDooTqYyIB} pour déterminer le développement limité de la fonction
+    \begin{equation}
+        f(x)=\frac{1}{ 1-x }.
+    \end{equation}
+    Il faut calculer les dérivées successives de \( f\) :
+    \begin{subequations}
+        \begin{align}
+            f(x)&=\frac{1}{ 1-x }\\
+            f'(x)&=\frac{ 1 }{ (1-x)^2 }\\
+            f''(x)&=\frac{ 2 }{ (1-x)^3 }
+        \end{align}
+    \end{subequations}
+    Avec ces résultats, nous devinons que 
+    \begin{equation}
+        f^{(n)}(x)=\frac{ n! }{ (1-x)^{n+1} }.
+    \end{equation}
+    Pour en être sûr nous le prouvons par récurrence. La dérivée de \(\frac{ n! }{ (1-x)^{n+1} } \) est donnée par
+    \begin{equation}
+        \frac{ n!(n+1)(1-x)^n }{ (1-x)^{2n+2} }=\frac{(n+1)! }{ (1-x)^{n+2} }.
+    \end{equation}
+    Évaluées en \( x=0\), les dérivées successives de \( f\) sont \( f(0)=0\), \( f'(0)=1\), \( f''(0)=2\),\ldots,\( f^{(n)}(0)=n!\). Utilisant la formule \eqref{EQooBKZDooTqYyIB} nous avons
+    \begin{equation}
+        f(x)=1+x+x^2+\cdots +x^n+x^n\alpha(x),
+    \end{equation}
+    conformément à ce que nous avions déjà trouvé.
+\end{example}
+\begin{remark}
+  Pour alléger la notation et ne pas écrire \(\ldots +x^n\alpha(x)\) nous pouvons aussi écrire
+    \begin{equation}
+         f(x)\sim 1+x+x^2+\cdots +x^n,
+    \end{equation}
+    mais il est interdit d'écrire
+    \begin{equation}
+         f(x)= 1+x+x^2+\cdots +x^n
+    \end{equation}
+    en mettant un signe d'égalité entre une fonction et son développement limité\footnote{Il faut cependant \^etre très prudents avec la notation abrégée. Elle pourrait nous faire oublier des informations importantes, voir les développements des fonction trigonométriques pour un exemple.}.
+\end{remark}
+
+Notons cependant que la proposition \ref{PropVDGooCexFwy} ne donne pas de moyen simple de trouver la fonction \( \alpha\). Si la fonction $f$ est très régulière dans l'intervalle $I$ on a le résultat suivant.
+
+\begin{proposition}[Reste dans la forme de Lagrange]
+    Si la fonction $f$ est dérivable $n+1$ fois dans $I$ alors il existe $\bar x$ dans l'intervalle \( \mathopen[ 0 , x \mathclose]\) tel que
+  \begin{equation}
+    f(x) = P_n(x) + \frac{1}{(n+1)!} f^{n+1}(\bar x) x^{n+1}.
+  \end{equation}
+\end{proposition}
+Voici quelque développements limités à savoir. Ils sont calculables en utilisant la formule de Taylor-Young (proposition \ref{PropVDGooCexFwy}).
+\begin{framed}
+    \begin{subequations}  \label{SUBEQSooIFXBooIOkveI}
+    \begin{align}
+      e^x&=\sum_{k=0}^n\frac{ x^k }{ k! }+x^n\alpha(x)&\text{ordre } n\\
+      \cos(x)&=\sum_{k=0}^p\frac{ (-1)^kx^{2k} }{ (2k)! }+x^{2p+1}\alpha(x)&\text{ordre } 2p+1    \label{subeqCBQooTZGxQpcos}\\
+      \sin(x)&=\sum_{k=0}^p\frac{ (-1)^kx^{2k+1} }{ (2k+1)! }+x^{p+2}\alpha(x)&\text{ordre } 2p+1 \label{subeqCBQooTZGxQpsin}\\
+      \ln(1+x)&=\sum_{k=0}^{n-1}(-1)^k\frac{ x^{k+1} }{ k+1 }+x^n\alpha(x)&\text{ordre } n\\
+      (1+x)^l&=\sum_{k=0}^l\binom{ l }{ k }x^k&\text{exact si } l\text{ est entier.}\\
+      (1+x)^{\alpha}&=1+\sum_{k=1}^n\frac{ \alpha(\alpha-1)\ldots(\alpha-k+1) }{ k! }x^k+x^n\alpha(x)&\text{ordre } n.\label{subEqJFNooQQodFQ}
+    \end{align}
+  \end{subequations}
+\end{framed}
+Il est fortement suggéré au lecteur de vérifier ces développements par lui-même.
+
+\begin{remark}
+  Quelque remarques.
+  \begin{enumerate}
+  \item
+    Les développements de sinus et de cosinus ont un terme sur deux qui est nul. C'est pour cela qu'en ayant une polynôme de degré \( 2p\), nous avons le développement d'ordre \( 2p+1\).
+  \item
+    En ce qui concerne le développement de \( \ln(1+x)\), il faut noter que la somme va jusqu'à \( k=n-1\) (et non \( k=n\)) parce que nous voulons aller jusqu'à \( x^n\) et que nous écrivons \( x^{k+1}\).
+  \item
+    Si \( l\) est entier, le développement de \( (1+x)^l\) est exact. Dans le développement de \( (1+x)^{\alpha}\), nous reconnaissons la formule de \( \binom{ k }{\alpha }\), sauf que nous ne pouvons pas l'écrire avec cette notation lorsque \( \alpha\) n'est pas entier.
+  \item Le développement limité en $0$ d'une fonction paire ne contient que les puissances de $x$ d'exposant paire. Voir comme exemple le développement de la fonction cosinus.
+  \end{enumerate}
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Règles de calcul}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Les règles suivantes permettent de calculer les développements limités des fonctions qu'on peut écrire comme combinaison des fonctions dans le tableau des équation s\eqref{SUBEQSooIFXBooIOkveI} de la section précédente. 
+\begin{remark}
+  Il est toujours possible de calculer le développement limité d'une fonction par la formule de Taylor-Young. Les règles suivantes peuvent nous economiser de l'effort et du temps. 
+\end{remark}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Linéarité des développements limités}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+L'opération qui consiste à prendre le développement limité d'une fonction est une opération linéaire : connaissant les développements limités de \( f\) et de \( g\), il suffit de les sommer pour obtenir celui de \( f+g\). De m\^eme, si $\lambda$ est une constante, le développement limité de $\lambda f$ est le développement limité de $f$ fois $\lambda$. 
+\begin{proposition}
+Soient $\lambda$ et $\mu$ dans $\eR$.  Si \( f\) et \( g\) sont deux fonctions acceptant des développements limités d'ordre \( n\)
+    \begin{subequations}    \label{EqJPPooCHihNn}
+        \begin{align}
+            f(x)&=P(x)+x^n\alpha_f(x)\\
+            g(x)&=Q(x)+x^n\beta(x)
+        \end{align}
+    \end{subequations}
+    avec \( \lim_{x\to 0} \alpha(x)=\lim_{x\to 0} \beta(x)=0\), alors la fonction \( \lambda f+\mu g\) admet le développement limité
+    \begin{equation}    \label{EqCJFooVpyCtz}
+        (f+g)(x)=(\lambda P+\mu Q)(x)+(\lambda \alpha+\mu\beta)(x).
+    \end{equation}
+\end{proposition}
+\begin{remark}
+  La forme explicite du reste ne nous interesse pas. Dans la pratique on écrira toujours $(f+g)(x)=(P+Q)(x)+\alpha(x)$, où on appelle $\alpha$ une fonction apportune telle que $\lim_{x\to 0} \alpha(x)=0$.
+\end{remark}
+\begin{proof}
+    Vu les définitions \eqref{EqJPPooCHihNn} des polynômes \( P\), \( Q\) et des restes \( \alpha\) et \( \beta\), l'égalité \eqref{EqCJFooVpyCtz} est une conséquence de la linéarité de la dérivation et de la proposition \ref{PropVDGooCexFwy}
+
+    De plus \( P+Q\) est un polynôme de degré \( n\) dès que \( P\) et \( Q\) sont des polynômes de degré \( n\), et
+    \begin{equation}
+        \lim_{x\to 0} (\lambda \alpha+\mu\beta)(x)=\lim_{x\to 0} \lambda\alpha(x)+\lim_{x\to 0} \mu\beta(x)=0.
+    \end{equation}
+    Par conséquent \(\lambda \alpha+\mu\beta\) est la fonction de reste de \( \lambda f+\mu g\).
+\end{proof}
+
+\begin{example} \label{ExKPBooJmdFvY}
+    Calculer le développement de la fonction
+    \begin{equation}
+        f(x)=3\sqrt[3]{1+x}+ e^{-2x}.
+    \end{equation}
+    Le développement de \( \sqrt[3]{1+x}\) est donné par la formule \eqref{subEqJFNooQQodFQ} avec \( \alpha=\frac{1}{ 3 }\). Nous avons donc dans un premier temps
+    \begin{subequations}
+        \begin{align}
+            \sqrt[3]{1+x}&=1+\frac{ 1 }{ 3 }x+\frac{ \frac{1}{ 3 }\left( \frac{1}{ 3 }-1 \right) }{ 2 }x^2+\frac{ \frac{1}{ 3 }\left( \frac{1}{ 3 }-1 \right)\left( \frac{1}{ 3 }-2 \right) }{ 6 }x^3+x^3\alpha(x)\\
+            &=1+\frac{1}{ 3 }x-\frac{1}{ 9 }x^2+\frac{ 5 }{ 81 }x^3+x^3\alpha(x).
+        \end{align}
+    \end{subequations}
+    Nous avons alors
+    \begin{subequations}
+        \begin{align}
+            3\sqrt[3]{1+x}+ e^{-2x}&=3\Big[  1+\frac{1}{ 3 }x-\frac{1}{ 9 }x^2+\frac{ 5 }{ 81 }x^3+x^3\alpha(x)\Big]+1-2x+2x^2-\frac{ 4 }{ 3 }x^3+x^3\beta(x)\\
+            &=4-x+\frac{ 5 }{ 3 }x^2-\frac{ 31 }{ 27 }x^3+x^3\big( \alpha(x)+\beta(x) \big).
+        \end{align}
+    \end{subequations}
+
+\end{example}
+
+La condition \( \lim_{x\to 0} \alpha(x)=0\) signifie que l'approximation qui consiste à remplacer \( f(x) \) par le polynôme n'est pas une trop mauvaise approximation lorsque \( x\) est petit. Cela ne signifie rien de plus. En particulier si \( x\) est grand, l'approximation polynômiale peut-être (et est souvent) très mauvaise.
+
+À ce propos, notez qu'un polynôme tend toujours vers \( \pm\infty\) lorsque \( x\) est grand. Une approximation polynômiale d'une fonction bornée est donc toujours (très) mauvaise pour les grandes valeurs de \( x\).
+
+À titre d'exemple nous avons tracé sur la figure \ref{LabelFigWUYooCISzeB} la fonction
+\begin{equation}
+    f(x)=3\sqrt[3]{x+1}+ e^{-2x}
+\end{equation}
+et ses développements limités d'ordre \( 1\) à \( 3\). Il est particulièrement visible que l'approximation est assez bonne pour la partie gauche du graphe sur laquelle la fonction est bien croissante, alors qu'elle est franchement mauvaise sur la droite où le graphe ressemble plutôt à une constante\footnote{Pouvez-vous cependant dire que vaut \( \lim_{x\to \infty} f(x)\) ?}.
+
+\newcommand{\CaptionFigWUYooCISzeB}{Les développements limités d'ordre de plus en plus grand de la fonction de l'exemple \ref{ExKPBooJmdFvY}. La fonction est en bleu et les «approximations» sont en rouge.}
+\input{auto/pictures_tex/Fig_WUYooCISzeB.pstricks}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Développement limité d'un quotient}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+\begin{proposition}
+    Si \( P_f\) est le polynôme du développement limité de \( f\) à l'ordre \( n\) et \( P_g\) celui de \( g\), alors nous obtenons le développement limité de \( f/g\) à l'ordre \( n\) en effectuant la division selon les puissances croissantes de \( P_f\) par \( P_g\).
+\end{proposition}
+Attention : il s'agit bien de faire une division selon les puissances croissantes, et non une divisions euclidienne. La division euclidienne de \( A\) par \( B\) consiste à écrire \( A=BQ+R\) avec le reste \( R\) de degré le plus \emph{petit} possible. Ici nous voulons avoir un reste de degré le plus \emph{grand} possible.
+
+\begin{example}
+    Cherchons le développement limité à l'ordre \( 5\) de \( \tan(x)=\frac{ \sin(x) }{ \cos(x) }\). Nous utilisons les formules \eqref{subeqCBQooTZGxQpcos} et \eqref{subeqCBQooTZGxQpsin} pour savoir les développements de sinus et cosinus\footnote{Encore une fois, vous êtes très fortement encouragés à calculer vous-même ces développements à partir de la formule de Taylor-Young \ref{EQooBKZDooTqYyIB}.} :
+    \begin{subequations}
+        \begin{align}
+            \sin(x)&=x-\frac{ x^3 }{ 6 }+\frac{ x^5 }{ 120 }+x^5\alpha_1(x)\\
+            \cos(x)&=1-\frac{ x^2 }{ 2 }+\frac{ x^4 }{ 24 }+x^5\alpha_2(x).
+        \end{align}
+    \end{subequations}
+    Nous calculons alors la division des deux polynômes, en classant les puissances dans l'ordre croissant (c'est le sens inverse de ce qui est fait pour la divisions euclidienne !) :
+    \begin{equation*}
+        \begin{array}[]{ccccccccccc|c}
+            &x&-&\frac{1}{ 6 }x^3&+&\frac{1}{ 120 }x^5&&&&&&1-\frac{ 1 }{2}x^2+\frac{1}{ 24 }x^4\\
+            \cline{12-12}
+            -\Big( &x&-&\frac{ 1 }{2}x^3&+&\frac{1}{ 24 }x^5&\Big)& & && &x+\frac{1}{ 3 }x^3+\frac{ 2 }{ 15 }x^5\\
+            \cline{2-6}
+            & & &\frac{1}{ 3 }x^3&-&\frac{1}{ 30 }x^5& & & & && \\
+            &&-\Big(  &\frac{1}{ 3 }x^3&-&\frac{1}{ 6 }x^5&+&\frac{1}{ 72 }x^7&\Big) & & \\
+            \cline{4-8}
+            & & & & &\frac{ 2 }{ 15 }x^5&-&\frac{1}{ 72 }x^7& & & \\
+            & & &  &-\Big(  &\frac{ 2 }{ 15 }x^5&-&\frac{1}{ 15 }x^7&+&\frac{1}{ 180 }x^9&\Big)& \\
+            \cline{6-10}
+            & & & & & & &\frac{ 29 }{ 360 }x^7&-&\frac{1}{ 180 }x^9&& \\
+        \end{array}
+    \end{equation*}
+    Nous avons continué la division jusqu'à obtenir un reste de degré plus grand que \( 5\). Le développement à l'ordre $5$ de la fonction tangente autour de zéro est alors
+    \begin{equation}
+        \tan(x)=x+\frac{1}{ 3 }x^3+\frac{ 2 }{ 15 }x^5+x^5\alpha(x).
+    \end{equation}
+    Notons que, vu que le reste ne nous intéresse pas vraiment, nous aurions pu ne pas calculer les coefficients des termes en \( x^7\) et \( x^8\). La dernière soustraction était également inutile.
+\end{example}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Développement limité d'une fonction composée}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+
+\begin{proposition}
+    Soient \( f\) et \( g\) des fonctions admettant des développements limités d'ordre $n$ au voisinage de $0$. Nous supposons que \( \lim_{x\to 0} g(x)=0\). Alors la composée \( f\big( g(x) \big)\) admet un développement limité d'ordre $n$ au voisinage de $0$ qui s'obtient en substituant le développement de \( g\) à chaque <<\(x \)>> du développement de \( f\), et en supprimant tous les termes de degré plus élevé que $n$.
+\end{proposition}
+
+\begin{example}\label{compose1}
+    Pour trouver le développement de la fonction \( f(x)= e^{-2x}\), il suffit d'écrire celui de \( e^t\) et de remplacer ensuite $t$ par \( -2x\). Le développement à l'ordre \( 3\) de la fonction exponentielle est :
+    \begin{equation}
+        e^t=1+t+\frac{ t^2 }{2}+\frac{ t^3 }{ 6 }+t^3\alpha(t).
+    \end{equation}
+    Le développement de \( f(x)= e^{-2x}\) sera donc 
+    \begin{equation}
+        f(x)=1-2x+\frac{ 4x^2 }{ 2 }-\frac{ 8x^3 }{ 6 }-8x^3\alpha(-2x).
+    \end{equation}
+    Donc le polynôme de degré \( 3\) partie régulière de \( g\) est :
+    \begin{equation}
+        1-2x+2x^2-\frac{ 4 }{ 3 }x^3,
+    \end{equation}
+    et la fonction reste correspondante est :
+    \begin{equation}
+        \alpha_g(x)=-8\alpha(-2x).
+    \end{equation}
+\end{example}
+
+\begin{example}
+    Nous savons les développements
+    \begin{equation}
+        f(x)=\ln(1+x)\sim x-\frac{ x^2 }{ 2 }+\frac{ x^3 }{ 3 }
+    \end{equation}
+    et
+    \begin{equation}
+        \sin(x)\sim x-\frac{ x^3 }{ 6 }.
+    \end{equation}
+    Nous obtenons le développement d'ordre \( 3\) de la fonction \( x\mapsto \ln\big( 1+\sin(x) \big)\) en écrivant
+    \begin{equation}    \label{EqGXMooWKQkIL}
+        \ln\big( 1+\sin(x) \big)\sim \big( x-\frac{ x^3 }{ 6 } \big)-\frac{ 1 }{2}\left( x-\frac{ x^3 }{ 6 } \right)^2+\frac{1}{ 3 }\left( x-\frac{ x^3 }{ 6 } \right)^3.
+    \end{equation}
+    Il s'agit maintenant de trouver les termes qui sont de degré inférieur ou égale à \( 3\).
+
+    D'abord
+    \begin{equation}
+        \left( x-\frac{ x^3 }{ 6 } \right)^2=x^2-\frac{ x^4 }{ 3 }+\frac{ x^6 }{ 36 }\sim x^2
+    \end{equation}
+    Nous avons alors aussi
+    \begin{equation}
+        \left( x-\frac{ x^3 }{ 6 } \right)^6\sim x^2\left( x-\frac{ x^3 }{ 6 } \right)\sim x^3.
+    \end{equation}
+    En replaçant tout ça dans \eqref{EqGXMooWKQkIL} nous trouvons
+    \begin{equation}
+        \ln\big( 1+\sin(x) \big)\sim x-\frac{ x^2 }{2}+\frac{ x^3 }{ 6 }.
+    \end{equation}
+\end{example}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Développement au voisinage de $x_0\neq 0$}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Il est intéressant de développer une fonction au voisinage de zéro lorsque nous nous intéressons à son comportement pour les \( x\) pas très grands. Il est toutefois souvent souhaitable de savoir le comportement d'une fonction au voisinage d'autres valeurs que zéro.
+
+Pour développer la fonction \( f\) autour de \( x_0\), nous considérons la fonction \( h\mapsto f(x_0+h)\) que nous développons autour de zéro (pour \( h\)). L'objectif est de trouver une polynôme \( P\) et une fonction \( \alpha\) tels que
+\begin{subequations}
+    \begin{numcases}{}
+        f(x)=P(x)+(x-x_0)^n\alpha(x)\\
+        \lim_{x\to x_0} \alpha(x)=0.
+    \end{numcases}
+\end{subequations}
+En pratique, le développement limité à l'ordre $n$ d'une fonction autour d'un point $x_0$ quelconque à l'intérieur de son domaine prend la forme suivante, qui généralise la formule de Taylor-Young vue dans la proposition \ref{PropVDGooCexFwy}
+\begin{proposition}[Formule de Taylor-Young, cas général]    
+    Soit \( f\) une fonction \( n\) fois dérivable sur un intervalle \( I\) contenant \(x_0\). Alors il existe une fonction \( \alpha\colon I\to \eR\) telle que
+    \begin{equation}    \label{EqTJRooUbsyzJ}
+      \begin{aligned}
+        f(x)=f(x_0)+&f'(x_0)(x-x_0)+\frac{ f''(x_0) }{ 2 }(x-x_0)^2+\\
+        &+\frac{ f^{(3)}(x_0) }{ 3! }(x-x_0)^3+\cdots +\frac{ f^{(n)}(x_0) }{ n! }(x-x_0)^n+(x-x_0)^n\alpha(x-x_0)
+      \end{aligned}
+    \end{equation}
+    et
+    \begin{equation}
+        \lim_{t\to 0} \alpha(t)=0.
+    \end{equation}
+  
+\end{proposition}
+\begin{example}\label{developcosenpisur3}
+    Développer la fonction \( \cos\) autour de \( x=\frac{ \pi }{ 3 }\). Nous développons autour de \( h=0\) la fonction \( \cos(\frac{ \pi }{ 3 }+h)\) :
+    \begin{equation}
+        \cos\big( \frac{ \pi }{ 3 }+h \big)\sim \cos\big( \frac{ \pi }{ 3 } \big)+h\cos'(\frac{ \pi }{ 3 })+\frac{ h^2 }{2}\cos''\big( \frac{ \pi }{ 3 } \big)=\frac{ 1 }{2}-\frac{ \sqrt{3} }{2}h-\frac{1}{ 4 }h^2.
+    \end{equation}
+    Il est aussi possible d'écrire cela en notant \( x=x_0+h\), c'est à dire en remplaçant \( h\) par \( x-\frac{ \pi }{ 3 }\) :
+    \begin{equation}
+        \cos(x)\sim\frac{ 1 }{2}-\frac{ \sqrt{3} }{ 2 }(x-\frac{ \pi }{ 3 })-\frac{1}{ 4 }(x-\frac{ \pi }{ 3 })^2.
+    \end{equation}
+\end{example}
+
+Pour donner une idée nous avons dessiné sur le graphe suivant la fonction sinus et ses développements d'ordre \( 4\) autour de zéro et autour de \( 3\pi/4\).
+\begin{center}
+   \input{auto/pictures_tex/Fig_WJBooMTAhtl.pstricks}
+\end{center}
+
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Application au calcul de limites}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Lors d'un calcul de limite, développer une partie d'une expression peut être utile.
+
+\begin{example}
+    À calculer :
+    \begin{equation}
+        \lim_{x\to 0} \frac{ \ln(1+x) }{ x }.
+    \end{equation}
+    Cela est une indétermination de type \( \frac{ 0 }{ 0 }\). Le développement limité du numérateur nous donne une fonction \( \alpha(x)\) telle que \( \lim_{x\to 0} \alpha(x)=0\) et
+    \begin{equation}
+        \frac{ \ln(1+x) }{ x }=\frac{ x-\frac{ x^2 }{2}+x^2\alpha(x) }{ x }=1-\frac{ x }{ 2 }+x\alpha(x).
+    \end{equation}
+    Sur le membre de droite la limite est facile à calculer :
+    \begin{equation}
+        \lim_{x\to 0} \frac{ \ln(1+x) }{ x }=\lim_{x\to 0} \Big( 1-\frac{ x }{ 2 }+x\alpha(x) \Big) =1.
+    \end{equation}
+\end{example}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Développement au voisinage de l'infini}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Il est souvent utile de connaître le comportement d'une fonction pour les grandes valeurs de \( x\) et de déterminer ses asymptotes éventuelles. La technique que nous allons utiliser consiste à poser \( x=\frac{1}{ h }\) et de développer la fonction ``auxiliaire'' $g(h) = f(1/h)$ autour de \( h=0\). La limite avec \( h\to 0^+\) donnera le comportement pour \( x\to \infty\) et la limite \( h\to 0^-\) donnera le comportement pour \( x\to -\infty\).
+
+Dans le cas d'une développement autour de \( \pm\infty\) nous ne parlons plus de développement \emph{limité} mais de \defe{développement asymptotique}{développement!asymptotique}.
+
+\begin{example}	\label{ExBCDookjljhjk}
+    Calculer
+    \begin{equation}\label{EqABCoolkjh}
+        \lim_{x\to \infty}  e^{1/x}\sqrt{1+4x^2}-2x.
+    \end{equation}
+    Nous allons effectuer un développement asymptotique de la partie «difficile» de l'expression posant d'abord $x=1/h$. Si $f(x)=e^{1/x}\sqrt{1-4x^2}$ alors
+    \begin{equation}
+	g(h)=\frac{1}{|h|}e^h\sqrt{h^2+4}=\frac{1}{h}\big(  1+h+h\alpha(h) \big)\big( 2+h\beta(h) \big).
+    \end{equation}
+    La première parenthèse est le développement de $e^h$ et la seconde celui de $\sqrt{h^2+4}$. Nous nous apprêtons à faire la limite $x\to\infty$ qui correspond à $h\to 0^+$, nous pouvons donc supposer que $h>0$ et omettre la valeur absolue. En effectuant le produit et en regroupant tous les termes contenant $h^2$, $\alpha(h)$ ou $\beta(h)$ dans un seul terme $h\gamma(h)$,
+    \begin{equation}
+	f(h)=\frac{1}{h}\big(  2+2h+h\gamma(h) \big)=\frac{2}{h}+2+\gamma(h)=2x+2+\gamma(1/x)
+    \end{equation}
+    où $\gamma$ est une fonction vérifiant $\lim_{t\to 0}\gamma(t)=0$.
+
+    Nous sommes maintenant en mesure de calculer la limite \eqref{EqABCoolkjh} :
+    \begin{equation}
+	\lim_{x\to\infty}e^{1/x}\sqrt{1+x^2}-2x= \lim_{x\to \infty}\big(  2x+2+\gamma(1/x)-2x \big)=2.
+    \end{equation}
+\end{example}
+
+
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Fonctions convexes}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
@@ -1426,9 +2056,10 @@ \subsection{En dimension supérieure}
 \end{proof}
 <++>
 
-\begin{proposition}[\cite{CLTooTlwZoz}] %\label{PropHRLooTqIJPS}
-    Si \( f\colon \eR^n\to \eR\) est de classe \( C^2\), elle est convexe si et seulement si sa matrice hessienne est définie positive en tout point.
-\end{proposition}
+\begin{corollary}       \label{CORooMBQMooWBAIIH}
+    Avec la hessienne\ldots en cours d'écriture.
+\end{corollary}
+<++>
 
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Quelque inégalités}
diff --git a/tex/frido/79_inversion_locale.tex b/tex/frido/79_inversion_locale.tex
index b6897e162..e32c80639 100644
--- a/tex/frido/79_inversion_locale.tex
+++ b/tex/frido/79_inversion_locale.tex
@@ -383,7 +383,7 @@ \subsection{Algorithme du gradient à pas optimal}
     \begin{subproof}
     \item[\( f\) est strictement convexe]
         
-        Nous utilisons la proposition \ref{PropHRLooTqIJPS}. La fonction \( f\) s'écrit
+        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
     \begin{equation}
         f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
     \end{equation}

From 7cd43d119878840252cbc856f6a80782e628a9f6 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sun, 25 Jun 2017 00:14:18 +0200
Subject: [PATCH 54/64] (organisation) Vide le fichier 4_dev_limite.tex dans
 76_series_fonctions.tex

---
 mazhe.tex                         |   1 -
 tex/frido/4_dev_limite.tex        | 190 ------------------------------
 tex/frido/76_series_fonctions.tex | 184 +++++++++++++++++++++++++++++
 3 files changed, 184 insertions(+), 191 deletions(-)
 delete mode 100644 tex/frido/4_dev_limite.tex

diff --git a/mazhe.tex b/mazhe.tex
index e8added00..904a4a56d 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -213,7 +213,6 @@ \chapter{Intégration}
 \chapter{Suites et séries de fonctions}
 \input{75_series_fonctions}
 \input{174_series_fonctions}
-\input{4_dev_limite}
 \input{76_series_fonctions}
 
 \chapter{Trigonométrie, isométries}
diff --git a/tex/frido/4_dev_limite.tex b/tex/frido/4_dev_limite.tex
deleted file mode 100644
index ab36ce77e..000000000
--- a/tex/frido/4_dev_limite.tex
+++ /dev/null
@@ -1,190 +0,0 @@
-% This is part of Analyse Starter CTU
-% Copyright (c) 2014,2017
-%   Laurent Claessens,Carlotta Donadello
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Nombres de Bell}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{theorem}[Nombres de Bell\cite{KXjFWKA}]  \label{ThoYFAzwSg}
-    Soit \( n\geq 1\) et \( B_n\) le nombre de partitions distinctes de l'ensemble \( \{ 1,\ldots, n \}\) avec la convention que \( B_0=0\). Alors
-    \begin{enumerate}
-        \item
-            La série entière
-            \begin{equation}    \label{EqYCMGBmP}
-                \sum_{n=0}^{\infty}\frac{ B_n }{ n! }x^n
-            \end{equation}
-            a un rayon de convergence \( R>0\) et sa somme est donnée par
-            \begin{equation}
-                f(x)= e^{ e^{x}-1}
-            \end{equation}
-            pour tout \( x\in\mathopen] -R , R \mathclose[\).
-        \item
-            Pour tout \( k\in \eN\),
-            \begin{equation}
-                B_n=\frac{1}{ e }\sum_{k=0}^{\infty}\frac{ k^n }{ k! }.
-            \end{equation}
-            \item
-                Le rayon de convergence de la série \eqref{EqYCMGBmP} est en réalité infini : \( R=\infty\).
-    \end{enumerate}
-\end{theorem}
-\index{anneau!de séries formelles}
-\index{dénombrement!partitions de \( \{ 1,\ldots, n \} \)}
-\index{série!numérique}
-\index{série!entière}
-\index{limite!inversion}
-
-\begin{proof}
-    \begin{enumerate}
-        \item
-            Soit \( n\geq 1\) et \( 0\leq k\leq n\). Nous notons \( E_k\) l'ensemble des partitions de \( \{ 1,\ldots, n+1 \}\) pour lesquelles le «paquet» contenant \( n+1\) soit de cardinal \( k+1\). Calculons le cardinal de \( E_k\).
-
-            Pour construire un élément de \( E_k\), il faut d'abord prendre le nombre \( n+1\) et lui adjoindre \( k\) éléments choisis dans \( \{ 1,\ldots, n \}\), ce qui donne \( n\choose k\) possibilités. Ensuite il faut trouver une partition des \( (n+1)-(k+1)=n-k\) éléments restants, ce qui fait \( B_{n-k}\) possibilités. Donc
-            \begin{equation}
-                \Card(E_k)={n\choose k}B_{n-k}.
-            \end{equation}
-            L'intérêt des ensembles \( E_k\) est que \( \{ E_0,\ldots, E_n \}\) est une partition de l'ensemble des partitions de \( \{ 1,\ldots, n+1 \}\), c'est à dire que \( B_{n+1}=\sum_{k=0}^n\Card(E_k)\), ce qui va nous donner une relation de récurrence pour les \( B_n\) :
-\begin{equation}
-                    B_{n+1}=\sum_{k=0}^n\Card(E_k)
-                   =\sum_{k=0}^n{n\choose k}B_{n-k}
-                    =\sum_{l=0}^n{n\choose n-l}B_l 
-                    =\sum_{l=0}^n{n\choose l}B_l.
-\end{equation}
-où nous avons utilisé un petit changement de variables \( l=n-k\). Afin d'étudier la convergence de la série \eqref{EqYCMGBmP}, nous allons montrer par récurrence que pour tout \( n\), \( B_n<n!\). D'abord pour \( n=0\) c'est bon : \( B_1=1\) parce que la seule partition de \( \{ 1 \}\) est \( \{ 1 \}\). Supposons que l'inégalité soit vraie pour une certaine valeur \( k\), et montrons qu'elle est vraie pour la valeur \( k+1\) :
-\begin{equation}
-                    B_{k+1}=\sum_{l=0}^n{n\choose k}B_k
-                   \leq \sum_{l=0}^n{n\choose k}k!
-                    =k!\sum_{l=0}^k\underbrace{\frac{1}{ (n-k)! }}_{\leq 1}
-                    \leq n!(n+1)
-                    =(n+1)!
-\end{equation}
-            où nous avons utilisé la formule \( {n\choose k}=\frac{ n! }{ k!(n-k)! }\).
-
-            Donc pour tout \( x\in \eR\) nous avons
-            \begin{equation}
-                0\leq \frac{ B_n }{ n! }| x^n |\leq | x |^n,
-            \end{equation}
-            et donc la série a un rayon de convergence au moins aussi grand que celui de la série géométrique, c'est à dire que \( 1\). Donc \( R\geq 1\). Nous nommons \( R\) ce rayon de convergence.
-
-        \item
-
-            Soit \( x\in\mathopen] -R , R \mathclose[\). Pour une telle valeur de \( x\) à l'intérieur du disque de convergence, la proposition \ref{ProptzOIuG} nous permet de dériver terme à terme la série\footnote{C'est ici qu'on utilise la convention \( B_0=0\) et ça aura une influence sur le choix de la constante \( K\) plus bas.}
-                \begin{equation}
-                    f(x)=\sum_{k=0}^{\infty}\frac{ B_k }{ k! }x^k=1+\sum_{k=0}^{\infty}\frac{ B_{k+1} }{ (k+1)! }x^{k+1},
-                \end{equation}
-                pour obtenir
-                \begin{equation}
-                    f'(x)=\sum_{k=0}^{\infty}\frac{ B_{k+1} }{ (k+1) }(k+1)x^k=\sum_{k=0}^{\infty}\frac{ B_{k+1} }{ k! }x^k=\sum_{k=0}^{\infty}\left( \sum_{l=0}^k{k\choose l}B_l \right)\frac{ x^k }{ k! }=\sum_{k=0}^{\infty}\left( \sum_{l=0}^k\frac{ B_l }{ l!(l-k)! } \right)x^k.
-                \end{equation}
-                En cette expression, nous reconnaissons un produit de Cauchy (proposition \ref{ThokPTXYC}) avec \( a_l=\frac{ B_l }{ l! }\) et \( b_n=\frac{ 1 }{ n! }\). Vu que ce sont deux séries ayant un rayon de convergence plus grand que zéro, le produit a encore un rayon de convergence plus grand que zéro et nous pouvons prendre le produit des séries :
-                \begin{equation}
-                    f'(x)=\left( \sum_{l=0}^{\infty}\frac{ B_l }{ l! }x^l \right)\left( \sum_{k=0}^{\infty}\frac{1}{ k! }x^k \right)=f(x) e^{x}.
-                \end{equation}
-            Étudions l'équation différentielle \( y'=ye^x\). D'abord par un argument en lacet de chaussure\footnote{Genre ce qui est fait pour prouver \ref{ThoRWOZooYJOGgR}\ref{ItemYTLTooSnfhOu}.}, une solution est de classe \(  C^{\infty}\). Ensuite si une solution est non nulle, elle est de signe constant. En effet si \( y(x_0)<0\) et \( y(x_1)=0\) (on choisit \( x_1\) minimum pour cette propriété parmi les nombres plus grands que \( x_0\)) alors il existe\footnote{Théorème de Rolle \ref{ThoRolle}.} un \( t\in\mathopen] x_0 , x_1 \mathclose[\) tel que \( y'(t)>0\), ce qui donnerait \( y(t)>0\), ce qui contredirait la minimalité de \( x_1\).
-
-                Nous prétendons\footnote{Ou alors on utilise le théorème \ref{ThoNYEXqxO} avec \( M(x)=e^x\) dans les cas \( n=1\) et \( I=\mathopen] -R , R \mathclose[\).} que cette équation différentielle a un espace de solutions de dimension \( 1\). En effet, si \( y'=ye^x\) et \( g'=ge^x\) alors en posant \( \varphi=y/g\) nous obtenons tout de suite \( \varphi'=0\), ce qui signifie que \( \varphi\) est constante, ou encore que \( y\) et \( g\) sont multiples l'un de l'autre.
-
-                 Si nous en trouvons une non nulle par n'importe quel moyen, c'est bon. Une solution étant dérivable est continue, donc l'équation \( f'=f e^{x}\) nous indique que \( f'\) est continue. Une solution non nulle va automatiquement accepter un petit voisinage sur lequel la manipulation suivante a un sens :
-                    \begin{equation}
-                        \frac{ f'(x) }{ f(x) }= e^{x},
-                    \end{equation}
-                    donc \( \ln\big( | f(x) | \big)= e^{x}+C\) et \( f(x)=K e^{ e^{x}}\) pour une certaine constante. Il est vite vérifié que cette fonction est une solution de l'équation différentielle \( y'(x)=y(x) e^{x}\) et par unicité, toutes les solutions sont de cette forme. Autrement dit, l'espace des solution est l'espace vectoriel \( \Span\{ x\mapsto e^{e^x} \}\). Étant donné que \( f(0)=0\), nous devons choisir \( K=\frac{1}{ e }\) et donc 
-                    \begin{equation}
-                        f(x)=\frac{1}{ e } e^{e^x}= e^{e^x-1}.
-                    \end{equation}
-
-                \item
-
-                    Nous commençons par écrire la fonction \( f\) comme une série de puissance. La partie simple du calcul : pour \( x\in \mathopen] -R , R \mathclose[\), nous avons
-                        \begin{equation}    \label{EqODjgjDN}
-                        e^{e^x}=\sum_{k=0}^{\infty}\frac{ (e^x)^k }{ k! }=\sum_{k=0}^{\infty}\frac{1}{ k! }\sum_{l=0}^{\infty}\frac{ (kx)^l }{ l! }=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{k^l}{k! }\frac{ x^l }{ l! }.
-                    \end{equation}
-                    Notons que cela n'est pas une série de puissance en \( x\) parce qu'il y a la double somme. Nous allons inverser les sommes au moyen du théorème de Fubini sous la forme du corollaire \ref{CorTKZKwP}. Pour cela nous considérons la fonction
-                    \begin{equation}
-                        \begin{aligned}
-                            a\colon \eN\times \eN&\to \eR \\
-                            (k,l)&\mapsto \frac{ (kx)^l }{ k!l! } 
-                        \end{aligned}
-                    \end{equation}
-                    et nous mettons la mesure de comptage\footnote{Nous passons outre les avertissements et menaces de Arnaud Girand.} sur \( \eN\) et \( \eN^2\). Nous commençons donc à vérifier l'intégrabilité variable par variable de \( | a |\) :
-                    \begin{subequations}    \label{SubEqsFHsBfhk}
-                        \begin{align}
-                            \int_{\eN}\left( \int_{\eN}| a(k,l) |dm(l) \right)dm(k)&=\sum_{k=0}^{\infty}\frac{1}{ k! }\frac{ (k| x |)^l }{ l! }\\
-                            &=\sum_{k=0}^{\infty}\frac{1}{ k! } e^{k| x |}.
-                        \end{align}
-                    \end{subequations}
-                    Nous devons montrer que cette dernière somme va bien. Pour cela nous posons \( u_k=\frac{ e^{k| x |} }{ k! }\) et nous remarquons que \( \frac{ u_{k+1} }{ u_k }\to 0\). Donc la double intégrale \eqref{SubEqsFHsBfhk} converge, ergo \( a\in L^1(\eN\times \eN)\), ce qui nous permet d'utiliser le théorème de Fubini \ref{ThoFubinioYLtPI} pour inverser les \sout{sommes} \sout{intégrales} sommes dans l'équation \eqref{EqODjgjDN} :
-                    \begin{equation}
-                        \frac{1}{ e }e^{e^x}=\frac{1}{ e }\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{1}{ k! }\frac{1}{ l! }(kx)^l=\sum_{l=0}\frac{1}{ e }\frac{1}{ l! }\left( \sum_{k=0}^{\infty}\frac{ k^l }{ k! } \right)x^l.
-                    \end{equation}
-                    Cela est un développement en série entière pour la fonction \( \frac{1}{ e } e^{e^x}\), dont nous savions déjà le développement \eqref{EqYCMGBmP}; par unicité du développement nous pouvons identifier les coefficients :
-                    \begin{equation}
-                        B_l=\frac{1}{ e }\sum_{k=0}^{\infty}\frac{ k^l }{ k! }.
-                    \end{equation}
-                    
-                \item
-
-                    Le développement \eqref{EqODjgjDN} étant en réalité valable pour tout \( x\) et tous les calculs subséquents l'étant aussi, le développement
-                    \begin{equation}
-                        e^{e^x-1}=\sum_{n=0}^{\infty}\frac{ B_n }{ n! }x^n
-                    \end{equation}
-                    est en fait valable pour tout \( x\), ce qui donne à la série entière un rayon de convergence infini.
-    \end{enumerate}
-\end{proof}
-
-
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Étude d'asymptote}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Lorsqu'une fonction tend vers l'infini pour \( x\to \infty\), une question qui peut venir est : à quelle vitesse tend-t-elle vers l'infini ?
-
-Il est «visible» que la fonction logarithme ne tend pas très vite vers l'infini : certes
-\begin{equation}
-    \lim_{x\to \infty} \ln(x)=+\infty,
-\end{equation}
-mais par exemple \( \ln(100000)\simeq 11.5\) tandis que \(  e^{100000}\simeq 10^{43429}\). Sans contestations possibles, l'exponentielle croit plus vite que le logarithme.
-
-Soient \( f\) et \( g\) deux fonctions dont la limite \( x\to \infty\) est \( \infty\). Si
-\begin{equation}
-    \lim_{x\to \infty} \frac{ f(x) }{ g(x) }=0
-\end{equation}
-nous disons que \( g\) tend vers \( \infty\) plus vite que \( f\); si
-\begin{equation}
-    \lim_{x\to \infty} \frac{ f(x) }{ g(x) }=\infty
-\end{equation}
-nous disons que \( f\) tend vers \( \infty\) plus vite que \( g\), et si
-\begin{equation}
-    \lim_{x\to \infty} \frac{ f(x) }{ g(x) }=a\in \eR
-\end{equation}
-avec \( a\neq 0\) alors nous disons que \( f\) tend vers l'infini à la même vitesse que \( ag(x)\).
-
-\begin{example}
-    La fonction \( x\mapsto x^2\) tend vers l'infini plus vite que la fonction \( x\mapsto \sqrt{x}\).
-\end{example}
-
-Dans cette section nous allons nous contenter de déterminer les fonctions qui tendent vers l'infini aussi vite qu'une droite oblique, que nous appellons asymptote et que nous voulons déterminer.
-
-
-\begin{example}
-    Déterminer les asymptotes obliques (s'ils existent) de la fonction
-    \begin{equation}
-        f(x)= e^{1/x}\sqrt{1+4x^2}.
-    \end{equation}
-    Tout d'abord nous remarquons que \( \lim_{x\to \infty} f(x)=\infty\). Nous sommes donc en présence d'une branche du graphe qui tend vers l'infini. Ensuite,
-    \begin{equation}
-        \lim_{x\to \infty} \frac{ f(x) }{ x }=\lim_{x\to \infty}  e^{1/x}\sqrt{\frac{1}{ x^2 }+4}=2.
-    \end{equation}
-    Donc le graphe de \( f\) tend vers l'infini à la même vitesse que le graphe de la fonction \( y=2x\). Nous aurons donc une asymptote oblique de coefficient directeur \( 2\). De façon imagée, nous pouvons penser que le graphe de \( f\) et celui de \( y=2x\) sont presque parallèles si \( x\) est assez grand. Afin de déterminer l'ordonnée à l'origine de l'asymptote, il nous reste à voir quelle est la «distance» entre le graphe de \( f\) et celui de \( y=2x\) :
-    \begin{equation}
-        \lim_{x\to \infty} f(x)-2x=\lim_{x\to \infty}  e^{1/x}\sqrt{1+4x^2}-2x.
-    \end{equation}
-    Cette limite a été calculée dans l'exemple \ref{ExBCDookjljhjk} et vaut $2$.
-
-	Nous concluons que le graphe de la fonction $f$ admet l'asymptote
-    \begin{equation}
-	y=2x+2.
-    \end{equation}
-\end{example}
diff --git a/tex/frido/76_series_fonctions.tex b/tex/frido/76_series_fonctions.tex
index 8d9a7972d..e9c2d6824 100644
--- a/tex/frido/76_series_fonctions.tex
+++ b/tex/frido/76_series_fonctions.tex
@@ -3,6 +3,190 @@
 %   Laurent Claessens
 % See the file fdl-1.3.txt for copying conditions.
 
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Nombres de Bell}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{theorem}[Nombres de Bell\cite{KXjFWKA}]  \label{ThoYFAzwSg}
+    Soit \( n\geq 1\) et \( B_n\) le nombre de partitions distinctes de l'ensemble \( \{ 1,\ldots, n \}\) avec la convention que \( B_0=0\). Alors
+    \begin{enumerate}
+        \item
+            La série entière
+            \begin{equation}    \label{EqYCMGBmP}
+                \sum_{n=0}^{\infty}\frac{ B_n }{ n! }x^n
+            \end{equation}
+            a un rayon de convergence \( R>0\) et sa somme est donnée par
+            \begin{equation}
+                f(x)= e^{ e^{x}-1}
+            \end{equation}
+            pour tout \( x\in\mathopen] -R , R \mathclose[\).
+        \item
+            Pour tout \( k\in \eN\),
+            \begin{equation}
+                B_n=\frac{1}{ e }\sum_{k=0}^{\infty}\frac{ k^n }{ k! }.
+            \end{equation}
+            \item
+                Le rayon de convergence de la série \eqref{EqYCMGBmP} est en réalité infini : \( R=\infty\).
+    \end{enumerate}
+\end{theorem}
+\index{anneau!de séries formelles}
+\index{dénombrement!partitions de \( \{ 1,\ldots, n \} \)}
+\index{série!numérique}
+\index{série!entière}
+\index{limite!inversion}
+
+\begin{proof}
+    \begin{enumerate}
+        \item
+            Soit \( n\geq 1\) et \( 0\leq k\leq n\). Nous notons \( E_k\) l'ensemble des partitions de \( \{ 1,\ldots, n+1 \}\) pour lesquelles le «paquet» contenant \( n+1\) soit de cardinal \( k+1\). Calculons le cardinal de \( E_k\).
+
+            Pour construire un élément de \( E_k\), il faut d'abord prendre le nombre \( n+1\) et lui adjoindre \( k\) éléments choisis dans \( \{ 1,\ldots, n \}\), ce qui donne \( n\choose k\) possibilités. Ensuite il faut trouver une partition des \( (n+1)-(k+1)=n-k\) éléments restants, ce qui fait \( B_{n-k}\) possibilités. Donc
+            \begin{equation}
+                \Card(E_k)={n\choose k}B_{n-k}.
+            \end{equation}
+            L'intérêt des ensembles \( E_k\) est que \( \{ E_0,\ldots, E_n \}\) est une partition de l'ensemble des partitions de \( \{ 1,\ldots, n+1 \}\), c'est à dire que \( B_{n+1}=\sum_{k=0}^n\Card(E_k)\), ce qui va nous donner une relation de récurrence pour les \( B_n\) :
+\begin{equation}
+                    B_{n+1}=\sum_{k=0}^n\Card(E_k)
+                   =\sum_{k=0}^n{n\choose k}B_{n-k}
+                    =\sum_{l=0}^n{n\choose n-l}B_l 
+                    =\sum_{l=0}^n{n\choose l}B_l.
+\end{equation}
+où nous avons utilisé un petit changement de variables \( l=n-k\). Afin d'étudier la convergence de la série \eqref{EqYCMGBmP}, nous allons montrer par récurrence que pour tout \( n\), \( B_n<n!\). D'abord pour \( n=0\) c'est bon : \( B_1=1\) parce que la seule partition de \( \{ 1 \}\) est \( \{ 1 \}\). Supposons que l'inégalité soit vraie pour une certaine valeur \( k\), et montrons qu'elle est vraie pour la valeur \( k+1\) :
+\begin{equation}
+                    B_{k+1}=\sum_{l=0}^n{n\choose k}B_k
+                   \leq \sum_{l=0}^n{n\choose k}k!
+                    =k!\sum_{l=0}^k\underbrace{\frac{1}{ (n-k)! }}_{\leq 1}
+                    \leq n!(n+1)
+                    =(n+1)!
+\end{equation}
+            où nous avons utilisé la formule \( {n\choose k}=\frac{ n! }{ k!(n-k)! }\).
+
+            Donc pour tout \( x\in \eR\) nous avons
+            \begin{equation}
+                0\leq \frac{ B_n }{ n! }| x^n |\leq | x |^n,
+            \end{equation}
+            et donc la série a un rayon de convergence au moins aussi grand que celui de la série géométrique, c'est à dire que \( 1\). Donc \( R\geq 1\). Nous nommons \( R\) ce rayon de convergence.
+
+        \item
+
+            Soit \( x\in\mathopen] -R , R \mathclose[\). Pour une telle valeur de \( x\) à l'intérieur du disque de convergence, la proposition \ref{ProptzOIuG} nous permet de dériver terme à terme la série\footnote{C'est ici qu'on utilise la convention \( B_0=0\) et ça aura une influence sur le choix de la constante \( K\) plus bas.}
+                \begin{equation}
+                    f(x)=\sum_{k=0}^{\infty}\frac{ B_k }{ k! }x^k=1+\sum_{k=0}^{\infty}\frac{ B_{k+1} }{ (k+1)! }x^{k+1},
+                \end{equation}
+                pour obtenir
+                \begin{equation}
+                    f'(x)=\sum_{k=0}^{\infty}\frac{ B_{k+1} }{ (k+1) }(k+1)x^k=\sum_{k=0}^{\infty}\frac{ B_{k+1} }{ k! }x^k=\sum_{k=0}^{\infty}\left( \sum_{l=0}^k{k\choose l}B_l \right)\frac{ x^k }{ k! }=\sum_{k=0}^{\infty}\left( \sum_{l=0}^k\frac{ B_l }{ l!(l-k)! } \right)x^k.
+                \end{equation}
+                En cette expression, nous reconnaissons un produit de Cauchy (proposition \ref{ThokPTXYC}) avec \( a_l=\frac{ B_l }{ l! }\) et \( b_n=\frac{ 1 }{ n! }\). Vu que ce sont deux séries ayant un rayon de convergence plus grand que zéro, le produit a encore un rayon de convergence plus grand que zéro et nous pouvons prendre le produit des séries :
+                \begin{equation}
+                    f'(x)=\left( \sum_{l=0}^{\infty}\frac{ B_l }{ l! }x^l \right)\left( \sum_{k=0}^{\infty}\frac{1}{ k! }x^k \right)=f(x) e^{x}.
+                \end{equation}
+            Étudions l'équation différentielle \( y'=ye^x\). D'abord par un argument en lacet de chaussure\footnote{Genre ce qui est fait pour prouver \ref{ThoRWOZooYJOGgR}\ref{ItemYTLTooSnfhOu}.}, une solution est de classe \(  C^{\infty}\). Ensuite si une solution est non nulle, elle est de signe constant. En effet si \( y(x_0)<0\) et \( y(x_1)=0\) (on choisit \( x_1\) minimum pour cette propriété parmi les nombres plus grands que \( x_0\)) alors il existe\footnote{Théorème de Rolle \ref{ThoRolle}.} un \( t\in\mathopen] x_0 , x_1 \mathclose[\) tel que \( y'(t)>0\), ce qui donnerait \( y(t)>0\), ce qui contredirait la minimalité de \( x_1\).
+
+                Nous prétendons\footnote{Ou alors on utilise le théorème \ref{ThoNYEXqxO} avec \( M(x)=e^x\) dans les cas \( n=1\) et \( I=\mathopen] -R , R \mathclose[\).} que cette équation différentielle a un espace de solutions de dimension \( 1\). En effet, si \( y'=ye^x\) et \( g'=ge^x\) alors en posant \( \varphi=y/g\) nous obtenons tout de suite \( \varphi'=0\), ce qui signifie que \( \varphi\) est constante, ou encore que \( y\) et \( g\) sont multiples l'un de l'autre.
+
+                 Si nous en trouvons une non nulle par n'importe quel moyen, c'est bon. Une solution étant dérivable est continue, donc l'équation \( f'=f e^{x}\) nous indique que \( f'\) est continue. Une solution non nulle va automatiquement accepter un petit voisinage sur lequel la manipulation suivante a un sens :
+                    \begin{equation}
+                        \frac{ f'(x) }{ f(x) }= e^{x},
+                    \end{equation}
+                    donc \( \ln\big( | f(x) | \big)= e^{x}+C\) et \( f(x)=K e^{ e^{x}}\) pour une certaine constante. Il est vite vérifié que cette fonction est une solution de l'équation différentielle \( y'(x)=y(x) e^{x}\) et par unicité, toutes les solutions sont de cette forme. Autrement dit, l'espace des solution est l'espace vectoriel \( \Span\{ x\mapsto e^{e^x} \}\). Étant donné que \( f(0)=0\), nous devons choisir \( K=\frac{1}{ e }\) et donc 
+                    \begin{equation}
+                        f(x)=\frac{1}{ e } e^{e^x}= e^{e^x-1}.
+                    \end{equation}
+
+                \item
+
+                    Nous commençons par écrire la fonction \( f\) comme une série de puissance. La partie simple du calcul : pour \( x\in \mathopen] -R , R \mathclose[\), nous avons
+                        \begin{equation}    \label{EqODjgjDN}
+                        e^{e^x}=\sum_{k=0}^{\infty}\frac{ (e^x)^k }{ k! }=\sum_{k=0}^{\infty}\frac{1}{ k! }\sum_{l=0}^{\infty}\frac{ (kx)^l }{ l! }=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{k^l}{k! }\frac{ x^l }{ l! }.
+                    \end{equation}
+                    Notons que cela n'est pas une série de puissance en \( x\) parce qu'il y a la double somme. Nous allons inverser les sommes au moyen du théorème de Fubini sous la forme du corollaire \ref{CorTKZKwP}. Pour cela nous considérons la fonction
+                    \begin{equation}
+                        \begin{aligned}
+                            a\colon \eN\times \eN&\to \eR \\
+                            (k,l)&\mapsto \frac{ (kx)^l }{ k!l! } 
+                        \end{aligned}
+                    \end{equation}
+                    et nous mettons la mesure de comptage\footnote{Nous passons outre les avertissements et menaces de Arnaud Girand.} sur \( \eN\) et \( \eN^2\). Nous commençons donc à vérifier l'intégrabilité variable par variable de \( | a |\) :
+                    \begin{subequations}    \label{SubEqsFHsBfhk}
+                        \begin{align}
+                            \int_{\eN}\left( \int_{\eN}| a(k,l) |dm(l) \right)dm(k)&=\sum_{k=0}^{\infty}\frac{1}{ k! }\frac{ (k| x |)^l }{ l! }\\
+                            &=\sum_{k=0}^{\infty}\frac{1}{ k! } e^{k| x |}.
+                        \end{align}
+                    \end{subequations}
+                    Nous devons montrer que cette dernière somme va bien. Pour cela nous posons \( u_k=\frac{ e^{k| x |} }{ k! }\) et nous remarquons que \( \frac{ u_{k+1} }{ u_k }\to 0\). Donc la double intégrale \eqref{SubEqsFHsBfhk} converge, ergo \( a\in L^1(\eN\times \eN)\), ce qui nous permet d'utiliser le théorème de Fubini \ref{ThoFubinioYLtPI} pour inverser les \sout{sommes} \sout{intégrales} sommes dans l'équation \eqref{EqODjgjDN} :
+                    \begin{equation}
+                        \frac{1}{ e }e^{e^x}=\frac{1}{ e }\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{1}{ k! }\frac{1}{ l! }(kx)^l=\sum_{l=0}\frac{1}{ e }\frac{1}{ l! }\left( \sum_{k=0}^{\infty}\frac{ k^l }{ k! } \right)x^l.
+                    \end{equation}
+                    Cela est un développement en série entière pour la fonction \( \frac{1}{ e } e^{e^x}\), dont nous savions déjà le développement \eqref{EqYCMGBmP}; par unicité du développement nous pouvons identifier les coefficients :
+                    \begin{equation}
+                        B_l=\frac{1}{ e }\sum_{k=0}^{\infty}\frac{ k^l }{ k! }.
+                    \end{equation}
+                    
+                \item
+
+                    Le développement \eqref{EqODjgjDN} étant en réalité valable pour tout \( x\) et tous les calculs subséquents l'étant aussi, le développement
+                    \begin{equation}
+                        e^{e^x-1}=\sum_{n=0}^{\infty}\frac{ B_n }{ n! }x^n
+                    \end{equation}
+                    est en fait valable pour tout \( x\), ce qui donne à la série entière un rayon de convergence infini.
+    \end{enumerate}
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Étude d'asymptote}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Lorsqu'une fonction tend vers l'infini pour \( x\to \infty\), une question qui peut venir est : à quelle vitesse tend-t-elle vers l'infini ?
+
+Il est «visible» que la fonction logarithme ne tend pas très vite vers l'infini : certes
+\begin{equation}
+    \lim_{x\to \infty} \ln(x)=+\infty,
+\end{equation}
+mais par exemple \( \ln(100000)\simeq 11.5\) tandis que \(  e^{100000}\simeq 10^{43429}\). Sans contestations possibles, l'exponentielle croit plus vite que le logarithme.
+
+Soient \( f\) et \( g\) deux fonctions dont la limite \( x\to \infty\) est \( \infty\). Si
+\begin{equation}
+    \lim_{x\to \infty} \frac{ f(x) }{ g(x) }=0
+\end{equation}
+nous disons que \( g\) tend vers \( \infty\) plus vite que \( f\); si
+\begin{equation}
+    \lim_{x\to \infty} \frac{ f(x) }{ g(x) }=\infty
+\end{equation}
+nous disons que \( f\) tend vers \( \infty\) plus vite que \( g\), et si
+\begin{equation}
+    \lim_{x\to \infty} \frac{ f(x) }{ g(x) }=a\in \eR
+\end{equation}
+avec \( a\neq 0\) alors nous disons que \( f\) tend vers l'infini à la même vitesse que \( ag(x)\).
+
+\begin{example}
+    La fonction \( x\mapsto x^2\) tend vers l'infini plus vite que la fonction \( x\mapsto \sqrt{x}\).
+\end{example}
+
+Dans cette section nous allons nous contenter de déterminer les fonctions qui tendent vers l'infini aussi vite qu'une droite oblique, que nous appellons asymptote et que nous voulons déterminer.
+
+
+\begin{example}
+    Déterminer les asymptotes obliques (s'ils existent) de la fonction
+    \begin{equation}
+        f(x)= e^{1/x}\sqrt{1+4x^2}.
+    \end{equation}
+    Tout d'abord nous remarquons que \( \lim_{x\to \infty} f(x)=\infty\). Nous sommes donc en présence d'une branche du graphe qui tend vers l'infini. Ensuite,
+    \begin{equation}
+        \lim_{x\to \infty} \frac{ f(x) }{ x }=\lim_{x\to \infty}  e^{1/x}\sqrt{\frac{1}{ x^2 }+4}=2.
+    \end{equation}
+    Donc le graphe de \( f\) tend vers l'infini à la même vitesse que le graphe de la fonction \( y=2x\). Nous aurons donc une asymptote oblique de coefficient directeur \( 2\). De façon imagée, nous pouvons penser que le graphe de \( f\) et celui de \( y=2x\) sont presque parallèles si \( x\) est assez grand. Afin de déterminer l'ordonnée à l'origine de l'asymptote, il nous reste à voir quelle est la «distance» entre le graphe de \( f\) et celui de \( y=2x\) :
+    \begin{equation}
+        \lim_{x\to \infty} f(x)-2x=\lim_{x\to \infty}  e^{1/x}\sqrt{1+4x^2}-2x.
+    \end{equation}
+    Cette limite a été calculée dans l'exemple \ref{ExBCDookjljhjk} et vaut $2$.
+
+	Nous concluons que le graphe de la fonction $f$ admet l'asymptote
+    \begin{equation}
+	y=2x+2.
+    \end{equation}
+\end{example}
+
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Développement en série}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

From aa1b000e6f69e1cf6762fd1d79344c1c075a8d44 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sun, 25 Jun 2017 07:01:15 +0200
Subject: [PATCH 55/64] =?UTF-8?q?(fonctions=20convexes)=20Convexit=C3=A9?=
 =?UTF-8?q?=20et=20diff=C3=A9rentielle=20seconde.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 lst_actu.py               |  4 +--
 tex/frido/68_analyseR.tex | 52 ++++++++++++++++++++++++++++++---------
 2 files changed, 42 insertions(+), 14 deletions(-)

diff --git a/lst_actu.py b/lst_actu.py
index 6a3793fc0..e0974a276 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -16,9 +16,7 @@
 myRequest.original_filename="mazhe.tex"
 
 myRequest.ok_filenames_list=["e_mazhe"]
-myRequest.ok_filenames_list.extend(["79_inversion_locale"])
-myRequest.ok_filenames_list.extend(["68_Chap_calcul_differentiel"])
-myRequest.ok_filenames_list.extend(["78_inversion_locale"])
+myRequest.ok_filenames_list.extend(["68_analyseR"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
 
diff --git a/tex/frido/68_analyseR.tex b/tex/frido/68_analyseR.tex
index 40b6bf488..0cf93d967 100644
--- a/tex/frido/68_analyseR.tex
+++ b/tex/frido/68_analyseR.tex
@@ -904,6 +904,13 @@ \subsection{Fonctions «petit o» }
     \end{equation}
 \end{proof}
 
+\begin{proposition}     \label{PROPooWWMYooPOmSds}
+Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^1\) et deux fois différentiable sur \( \mathopen] x , x+h \mathclose[\). Alors il existe \( \theta\in \mathopen] 0 , 1 \mathclose[\) tel que
+    \begin{equation}
+        f(x+h)=f(x)+df_x(h)+\frac{ 1 }{2}(d^2f)_{x+\theta h}(h,h).
+    \end{equation}
+\end{proposition}
+
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Autres formulations}
 %---------------------------------------------------------------------------------------------------------------------------
@@ -1923,7 +1930,7 @@ \subsection{En dimension supérieure}
     \end{enumerate}
 \end{definition}
 
-\begin{proposition}[\cite{ooLJMHooMSBWki}]
+\begin{proposition}[\cite{ooLJMHooMSBWki}]      \label{PROPooYNNHooSHLvHp}
     Soit \( \Omega\) ouvert dans \( \eR^n\) et \( U\) convexe dans \( \Omega\), et une fonction différentiable \( f\colon U\to \eR\).
     \begin{enumerate}
         \item       \label{ITEMooRVIVooIayuPS}
@@ -1942,7 +1949,7 @@ \subsection{En dimension supérieure}
 \begin{proof}
     Nous avons quatre petites choses à démontrer.
     \begin{subproof}
-    \item[\ref{ITEMooRVIVooIayuPS}, sens direct]
+    \item[\ref{ITEMooRVIVooIayuPS} sens direct]
         Soit une fonction convexe \( f\). Nous avons :
         \begin{equation}
             f\big( (1-\theta)x+\theta y \big)\leq (1-\theta)f(x)+\theta f(y),
@@ -1964,7 +1971,7 @@ \subsection{En dimension supérieure}
             df_x(y-x)\leq f(y)-f(x)
         \end{equation}
         par le lemme \ref{LemdfaSurLesPartielles}.
-    \item[\ref{ITEMooRVIVooIayuPS}, sens inverse]
+    \item[\ref{ITEMooRVIVooIayuPS} sens inverse]
         Pour tout \( a\neq b\) dans \( U\) nous avons
         \begin{equation}        \label{EQooEALSooJOszWr}
             f(b)\geq f(a)+df_a(b-a).
@@ -1981,7 +1988,7 @@ \subsection{En dimension supérieure}
         \begin{equation}
             \theta f(x)+(1-\theta)f(y)\geq f\big( \theta x+(1-\theta)y \big).
         \end{equation}
-    \item[\ref{ITEMooCWEWooFtNnKl}, sens direct]
+    \item[\ref{ITEMooCWEWooFtNnKl} sens direct]
         Nous avons encore l'équation \eqref{EQooAXXFooHWtiJh}, avec une inégalité stricte. Par contre, ça ne va pas être suffisant parce que le passage à la limite ne conserve pas les inégalités strictes. Nous devons donc être plus malins. 
 
         Soient \( 0<\theta<\omega<1\). Nous avons \( (1-\theta)x+\theta y\in \mathopen[ x , (1-\omega)x+\omega y \mathclose]\), donc nous pouvons écrire \( (1-\theta)x+\theta y\) sous la forme \( (1-s)x+s\big( (1-\omega)x+\omega y \big)\). Il se fait que c'est bon pour \( s=\theta/\omega\) (et aussi que nous avons \( \theta/\omega<1\)). Donc nous avons
@@ -2006,7 +2013,7 @@ \subsection{En dimension supérieure}
 Avant de lire la proposition suivante, il faut relire la proposition \ref{PROPooFWZYooUQwzjW} et ce qui s'y rapporte. 
 Lire aussi la remarque \ref{REMooVRPQooIybxmp} qui indique
 qu'il n'y a pas de réciproque dans l'énoncé \ref{ITEMooHAGQooYZyhQk}.     
-\begin{proposition}
+\begin{proposition}[\cite{ooLJMHooMSBWki}]
     Soit une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) et un convexe \( U\subset \Omega\).
     \begin{enumerate}
         \item       \label{ITEMooZQCAooIFjHOn}
@@ -2024,17 +2031,16 @@ \subsection{En dimension supérieure}
     \end{enumerate}
 \end{proposition}
 
-\begin{remark}
+\begin{remark}      \label{REMooYCRKooEQNIkC}
     Notons que la condition \eqref{EQooIBDCooJYdiBb} n'est pas équivalente à demander \( (d^2f)_x(h,h)\geq 0\) pour tout \( h\). En effet nous ne demandons la positivité que dans les directions atteignables comme différence de deux éléments de \( U\). La partie \( U\) n'est pas spécialement ouverte; elle pourrait n'être qu'une droite dans \( \eR^3\). Dans ce cas, demander que \( f\) (qui est \( C^2\) sur l'ouvert \( \Omega\)) soit convexe sur \( U\) ne demande que la positivité de \( (d^2f)_x\) appliqué à des vecteurs situés sur la droite \( U\).
 \end{remark}
 
 \begin{proof}
-
     Il y a trois parties à démontrer.
     \begin{subproof}
     \item[\ref{ITEMooZQCAooIFjHOn} sens direct]
 
-        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci à un jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
+        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci au jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
         \begin{equation}
             f(x+th)=f(x)+tdf_x(h)+\frac{ t^2 }{2}(d^2_x)(h,h)+t^2\| h \|^2\alpha(th)
         \end{equation}
@@ -2050,11 +2056,35 @@ \subsection{En dimension supérieure}
         \begin{equation}
             0\leq (d^2f)_x(h,h)+2\| h \|^2\alpha(th).
         \end{equation}
-        En prenant \( t\to 0\) nous avons bien pour tout \( h\) : \( (d^2f)_x(h,h)\geq 0\).
+        En prenant \( t\to 0\) nous avons bien  \( (d^2f)_x(y-x,y-x)\geq 0\).
+
+    \item[\ref{ITEMooZQCAooIFjHOn} sens inverse]
+
+        Soient \( x,y\in U\). Nous écrivons Taylor en version de la proposition \ref{PROPooWWMYooPOmSds} :
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\frac{ 1 }{2}(d^2f)_z(y-x,y-x)
+        \end{equation}
+    pour un certain \( z\in\mathopen] x , y \mathclose[\). En vertu de ce qui a été dit dans la remarque \ref{REMooYCRKooEQNIkC} nous ne pouvons pas évoquer l'hypothèse \eqref{EQooIBDCooJYdiBb} pour conclure que \( (d^2f)_z(y-x,y-x)\geq 0\). Il y a deux manières de nous sortir du problème :
+        \begin{itemize}
+            \item Trouver \( s\in U\) tel que \( y-x=s-z\).
+            \item Trouver un multiple de \( y-x\) qui soit de la forme \( y-x\).
+        \end{itemize}
+        La première approche ne fonctionne pas parce que \( s=y-x+z\) n'est pas garanti d'être dans \( U\); par exemple avec \( x=1\), \( z=2\), \( y=3\) et \( U=\mathopen[ 0 , 3 \mathclose]\). Dans ce cas \( s=4\notin U\).
+
+        Heureusement nous avons \( z=\theta x+(1-\theta)y\), donc \( z-x=(1-\theta)(y-x)\). Dans ce cas la bilinéarité de \( (d^2f)_z\) donne\footnote{Si vous avez bien suivi, la bilinéarité est contenue dans la proposition \ref{PROPooFWZYooUQwzjW}.}
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{\geq 0}.
+        \end{equation}
+        Nous en déduisons que \( f\) est convexe par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooRVIVooIayuPS}.
+    \item[\ref{ITEMooHAGQooYZyhQk}]
+
+        Le raisonnement que nous venons de faire pour le sens inverse de \ref{ITEMooZQCAooIFjHOn} tient encore, et nous avons
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{> 0}
+        \end{equation}
+        d'où nous déduisons la stricte convexité de \( f\) par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooCWEWooFtNnKl}.
     \end{subproof}
-    <++>
 \end{proof}
-<++>
 
 \begin{corollary}       \label{CORooMBQMooWBAIIH}
     Avec la hessienne\ldots en cours d'écriture.

From 98862c2b956d7f0134579ae5fd02d55895db99a2 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Sun, 25 Jun 2017 19:13:43 +0200
Subject: [PATCH 56/64] =?UTF-8?q?(recherche=20extrema)=20D=C3=A9place=20la?=
 =?UTF-8?q?=20recherche=20d'extrema=20au-dessus=20des=20fonctions=20convex?=
 =?UTF-8?q?es.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Nous en aurons besoin.
---
 lst_actu.py                       |   4 +
 tex/frido/57_EspacesVectos.tex    |   6 +-
 tex/frido/61_representations.tex  |   6 +-
 tex/frido/68_analyseR.tex         | 569 ++++++++++++++++++++++++++++++
 tex/frido/79_inversion_locale.tex | 569 ------------------------------
 5 files changed, 579 insertions(+), 575 deletions(-)

diff --git a/lst_actu.py b/lst_actu.py
index e0974a276..f64125b16 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -17,6 +17,10 @@
 
 myRequest.ok_filenames_list=["e_mazhe"]
 myRequest.ok_filenames_list.extend(["68_analyseR"])
+myRequest.ok_filenames_list.extend(["61_representations"])
+myRequest.ok_filenames_list.extend(["78_inversion_locale"])
+myRequest.ok_filenames_list.extend(["57_EspacesVectos"])
+myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
 
diff --git a/tex/frido/57_EspacesVectos.tex b/tex/frido/57_EspacesVectos.tex
index 605f9ee9e..3ff6228fa 100644
--- a/tex/frido/57_EspacesVectos.tex
+++ b/tex/frido/57_EspacesVectos.tex
@@ -1671,7 +1671,11 @@ \subsection{Diagonalisation : cas réel}
 \end{definition}
 Afin d'éviter l'une ou l'autre confusion, nous disons souvent \emph{strictement} définie positive pour positive.
 
-Nous notons \( S^+(n,\eR)\)\nomenclature[A]{\( S^+(n,\eR)\)}{matrices symétriques semi-définies positives} l'ensemble des matrices réelles \( n\times n\) semi-définies positives. L'ensemble \( S^{++}(n,\eR)\)\nomenclature[A]{\( S^{++}(n,\eR)\)}{matrices symétriques strictement définies positives} est l'ensemble des matrices symétriques strictement définies positives.
+\begin{normaltext}      \label{NORMooAJLHooQhwpvr}
+    Nous nommons \( S^+(n,\eR)\) l'ensemble des matrices réelles symétriques \( n\times n\) et \( S^{++}(n,\eR)\) le sous-ensemble de \( S^+(n,\eR)\) des matrices strictement définies positives.
+    \nomenclature[B]{\( S^+(n,\eR)\)}{matrices symétriques définies positives}
+    \nomenclature[B]{\( S^{++}(n,\eR)\)}{matrices symétriques strictement définies positives}
+\end{normaltext}
 
 \begin{remark}
     Nous ne définissons pas la notion de matrice définie positive pour une matrice non symétrique.
diff --git a/tex/frido/61_representations.tex b/tex/frido/61_representations.tex
index 8f68a53ff..36deab17f 100644
--- a/tex/frido/61_representations.tex
+++ b/tex/frido/61_representations.tex
@@ -865,11 +865,7 @@ \subsection{Racine carré d'une matrice symétrique positive}
 \subsection{Décomposition polaires : cas réel}
 %---------------------------------------------------------------------------------------------------------------------------
 
-\begin{normaltext}      \label{NORMooAJLHooQhwpvr}
-    Nous nommons \( S^+(n,\eR)\) l'ensemble des matrices \( n\times n\) symétriques réelles définies positives et \( S^{++}(n,\eR)\) le sous-ensemble de \( S^+(n,\eR)\) des matrices strictement définies positives.
-    \nomenclature[B]{\( S^+(n,\eR)\)}{matrices symétriques définies positives}
-    \nomenclature[B]{\( S^{++}(n,\eR)\)}{matrices symétriques strictement définies positives}
-\end{normaltext}
+Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
 
 \begin{lemma}   \label{LemMGUSooPqjguE}
     La partie \( S^+(n,\eR)\) est fermée dans \( \eM(n,\eR)\).
diff --git a/tex/frido/68_analyseR.tex b/tex/frido/68_analyseR.tex
index 0cf93d967..1403e2409 100644
--- a/tex/frido/68_analyseR.tex
+++ b/tex/frido/68_analyseR.tex
@@ -1424,6 +1424,575 @@ \section{Développement au voisinage de l'infini}
     \end{equation}
 \end{example}
 
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Recherche d'extrema}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema à une variable}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
+\end{definition}
+
+La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
+\begin{proposition}     \label{PROPooNVKXooXtKkuz}
+Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
+\end{proposition}
+
+La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
+
+Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
+\begin{proposition}
+Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
+\begin{equation}
+    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
+\end{equation}
+et que
+\begin{equation}
+    f^{(k)}(a)\neq 0.
+\end{equation}
+Dans ce cas,
+\begin{enumerate}
+
+\item
+Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
+\item
+Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
+
+\end{enumerate}
+\end{proposition}
+
+Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
+
+Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
+
+Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
+
+\begin{example}
+	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
+\end{example}
+ 
+Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
+\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
+\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
+
+Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
+
+Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
+
+\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
+\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
+ 
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema libre}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooYJLZooLkEAYf}
+Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
+\end{definition}
+
+Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
+
+Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
+\begin{equation}
+    d^2f(a)=\begin{pmatrix}
+    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
+    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
+\end{pmatrix}.
+\end{equation}
+Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
+
+\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
+\end{proposition}
+
+\begin{proof}
+    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
+    \begin{equation}
+        \begin{aligned}
+            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
+            t&\mapsto f(a+tu) 
+        \end{aligned}
+    \end{equation}
+    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
+    \begin{equation}
+        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
+    \end{equation}
+\end{proof}
+
+\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
+    \begin{enumerate}
+        \item   \label{ITEMooCVFVooWltGqI}
+            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
+        \item\label{ItemPropoExtreRn}
+            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
+    \end{enumerate}
+\end{proposition}
+\index{extrema}
+
+\begin{proof}
+    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
+    \begin{equation}
+        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
+    \end{equation}
+    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
+    \end{equation}
+    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
+    \end{equation}
+    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
+    \begin{equation}
+        (d^2f)_a(e_h,e_h)\geq 0.
+    \end{equation}
+    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
+\end{proof}
+
+La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
+
+Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
+
+La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
+\begin{enumerate}
+    \item
+        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
+    \item
+        écrire $d^2f(a)$ pour chacun des candidats
+    \item
+        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
+    \item
+        conclure.
+\end{enumerate}
+
+Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
+
+\begin{example}
+    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
+    \begin{equation}
+        df=\big(4x^3-4y;4y^3-4x),
+    \end{equation}
+    et la matrice des dérivées secondes est
+    \begin{equation}
+        M=d^2f(x,y)=\begin{pmatrix}
+            12x^2    &   -4    \\ 
+            -4    &   12y^2    
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
+
+    Pour le point \( (0,0)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            0    &   -4    \\ 
+            -4    &   0    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
+
+    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            12    &   -4    \\ 
+            -4    &   12    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Un peu de recettes de cuisine}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{enumerate}
+\item Rechercher les points critiques, càd les $(x,y)$ tels que
+\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
+En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
+\item Déterminer la nature des points critiques: «test» des dérivées secondes:
+\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
+\begin{enumerate}
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
+\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
+\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
+\end{enumerate}
+\end{enumerate}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Extrema liés}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
+
+Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
+
+\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
+    Soit \( A\), un ouvert de \( \eR^n\) et
+    \begin{enumerate}
+        \item
+            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
+        \item 
+            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
+        \item
+            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
+        \item
+            un extrema local $a\in M$ de $f$ relativement à $M$.
+    \end{enumerate}
+    Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
+    \begin{equation}
+        L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_r)=f(x_1,\ldots,x_n)+\sum_{i=1}^r\lambda_iG_i(x_1,\ldots,x_n).
+    \end{equation}
+    Autrement dit, si \( a\) est un extrema lié, alors \( \nabla f(a)\) est une combinaisons des \( \nabla G_i(a)\), ou encore il existe des \( \lambda_i\) tels que
+    \begin{equation}    \label{EqRDsSXyZ}
+        df(a)=\sum_i\lambda_idG_i(a).
+    \end{equation}
+\end{theorem}
+\index{théorème!inversion locale!utilisation}
+\index{extrema!lié}
+\index{théorème!extrema!lié}
+\index{application!différentiable!extrema lié}
+\index{variété}
+\index{rang!différentielle}
+\index{forme!linéaire!différentielle}
+La fonction $L$ est le \defe{lagrangien}{lagrangien} du problème et les variables \( \lambda_i\) sont les \defe{multiplicateurs de Lagrange}{multiplicateur!de Lagrange}\index{Lagrange!multiplicateur}.
+
+\begin{proof}
+    Si \( r=n\) alors les vecteurs linéairement indépendantes \( \nabla G_i(a) \) forment une base de \( \eR^n\) et donc évidemment les \( \lambda_i\) existent. Nous supposons donc maintenant que \( r<n\). Nous notons \( (z_i)_{i=1\ldots n}\) les coordonnées sur \( \eR^n\).
+    
+    La matrice
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{ \partial G_1 }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_1 }{ \partial z_n }(a)    \\
+            \vdots    &   \ddots    &   \vdots    \\
+            \frac{ \partial G_r }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_r }{ \partial z_n }(a)
+        \end{pmatrix}
+    \end{equation}
+    est de rang \( r\) parce que les lignes sont par hypothèses linéairement indépendantes. Nous nommons \( (y_i)_{i=1,\ldots, r}\) un choix de \( r\) parmi les \( (z_i)\) tels que
+    \begin{equation}
+        \det\begin{pmatrix}
+            \frac{ \partial G_1 }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_1 }{ \partial y_r }    \\
+            \vdots    &   \ddots    &   \vdots    \\
+            \frac{ \partial G_r }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_r }{ \partial y_r }
+        \end{pmatrix}\neq 0.
+    \end{equation}
+    Nous identifions \( \eR^n\) à \( \eR^s\times \eR^r\) dans lequel \( \eR^r\) est la partie générée par les \( (y_i)_{i=1,\ldots, r}\). Nous nommons \( (x_j)_{j=1,\ldots, s}\) les coordonnées sur \( \eR^s\). Autrement dit, les coordonnées sur \( \eR^n\) sont \( x_1,\ldots, x_s,y_1,\ldots, y_r\). Dans ces coordonnées, nous nommons \( a=(\alpha,\beta)\) avec \( \alpha\in \eR^s\) et \( \beta\in \eR^r\).
+
+    Si nous notons \( G=(G_1,\ldots, G_r)\), le théorème de la fonction implicite (théorème \ref{ThoAcaWho})  nous dit qu'il existe un voisinage \( U'\) de \( \alpha\in \eR^n\), un voisinage \( V'\) de \( \beta\in \eR^r\) et une fonction \( \varphi\colon U'\to V'\) de classe \( C^1\) telle que si \( (x,y)\in U'\times V'\), alors
+    \begin{equation}
+        g(x,y)=0
+    \end{equation}
+    si et seulement si \( y=\varphi(x)\). Nous posons maintenant
+    \begin{subequations}
+        \begin{align}
+            \psi(x)&=(x,\varphi(x))\\
+            h(x)&=f\big( \psi(x) \big).
+        \end{align}
+    \end{subequations}
+    Nous avons \( \psi(\alpha)=a\) et \( \psi(x)\in M\) pour tout \( x\in U'\). La fonction \( h\) a donc un extrema local en \( \alpha\) et donc les dérivées partielles de \( h\) y sont nulles. Cela signifie que
+    \begin{equation}
+        0=\frac{ \partial h }{ \partial x_i }(\alpha)=\sum_{j=1}^n\frac{ \partial f }{ \partial x_j }\frac{ \partial x_j }{ \partial x_i }+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }\frac{ \partial \varphi_k }{ \partial x_i },
+    \end{equation}
+    c'est à dire
+    \begin{equation}
+        \frac{ \partial f }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
+    \end{equation}
+    pour tout \( i=1,\ldots, s\). D'autre part pour tout $k$, la fonction \( l_k(x)=G_k\big( x,\varphi(x) \big)\) est constante et vaut zéro; ses dérivées partielles sont donc nulles :
+    \begin{equation}
+        \frac{ \partial l }{ \partial x_i }(\alpha)=\frac{ \partial G_k }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial G_k }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
+    \end{equation}
+    pour tout \( i=1,\ldots, s\) et \( k=1,\ldots, r\).
+    
+    Les \( s\) premières colonnes de la matrice
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{ \partial f }{ \partial x_1 }   &   \cdots    &   \frac{ \partial f }{ \partial x_s }    &   \frac{ \partial f }{ \partial y_1 }    &   \cdots    &   \frac{ \partial f }{ \partial y_r }\\  
+            \frac{ \partial G_1 }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial x_s }    &   \frac{ \partial G_1 }{ \partial y_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial y_r }\\
+            \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots\\
+            \frac{ \partial G_r }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_r }{ \partial x_s }    &   \frac{ \partial G_r }{ \partial y_1 }    &  \cdots   & \frac{ \partial G_r }{ \partial y_r }  
+        \end{pmatrix}
+    \end{equation}
+    s'expriment en terme des \( r\) dernières. La matrice est donc au maximum de rang \( r\). Notons que la première ligne est \( \nabla f\) et les \( r\) suivantes sont les \( \nabla G_i\). Vu que ces lignes sont des vecteurs liés, il existe \( \mu_0,\ldots, \mu_r\) tels que
+    \begin{equation}
+        \mu_0\nabla f+\sum_{i=1}^r\mu_i\nabla G_i=0.
+    \end{equation}
+    Par hypothèse les \( \nabla G_i\) sont linéairement indépendants, ce qui nous dit que \( \mu_0\neq 0\). Donc nous avons ce qu'il nous faut :
+    \begin{equation}
+        \nabla f(a)=\sum_i\frac{ \mu_i }{ \mu_0 } \nabla G_i(a).
+    \end{equation}
+
+    Notons qu'au vu de l'expression \eqref{EqRDsSXyZ}, le fait que les formes \( \{ dG_i(a) \}_{1\leq i\leq r}\) forment une partie libre dans \( (\eR^n)^*\) implique que les \( \lambda_i\) sont uniques.
+\end{proof}
+
+La proposition suivante est la même que \ref{ThoRGJosS}.
+\begin{proposition} \label{PropfPPUxh}
+    Soit \( U\), un ouvert de \( \eR^n\) et des fonctions de classe \( C^1\) \( f,g_1,\ldots, g_r\colon U\to \eR\). Nous considérons
+    \begin{equation}
+        \Gamma=\{ x\in U\tq g_1(x)=\ldots=g_r(x)=0 \}.
+    \end{equation}
+    Soit \( a\) un extrémum de \( f|_{\Gamma}\). Supposons que les formes \( dg_1,\ldots, dg_r\) soient linéairement indépendantes en \( a\). Alors il existe \( \lambda_1,\ldots, \lambda_r\) dans \( \eR\) tel que
+    \begin{equation}
+        df_a=\sum_{i=1}^r\lambda_i(dg_i)_a.
+    \end{equation}
+\end{proposition}
+
+En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
+
+L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
+
+\begin{lemma}       \label{LemmeMinSCimpliqueS}
+    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
+\end{lemma}
+
+\begin{proof}
+    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Algorithme du gradient à pas optimal}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
+
+Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
+\begin{itemize}
+    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
+    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
+\end{itemize}
+
+\begin{definition}  \label{DefQXPooYSygGP}
+    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
+    \begin{equation}
+        \{ x\in P\tq f(x)\leq M \}
+    \end{equation}
+    est borné.
+\end{definition}
+En langage imagé la coercivité de \( f\) s'exprime par la limite
+\begin{equation}
+    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
+\end{equation}
+
+
+Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
+\begin{proposition}     \label{PROPooYRLDooTwzfWU}
+    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Alors :
+    \begin{enumerate}
+        \item
+            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
+        \item
+            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
+        \item
+            Ils sont égaux : \( \bar x=x^*\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+
+    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
+
+    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
+
+    \begin{subproof}
+    \item[\( f\) est strictement convexe]
+        
+        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
+    \end{equation}
+    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
+
+\item[\( f\) est coercive]
+    Montrons à présent que \( f\) est coercive. Nous avons :
+    \begin{subequations}
+        \begin{align}
+            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
+            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
+            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
+        \end{align}
+    \end{subequations}
+    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
+    \end{subproof}
+
+    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
+    \begin{equation}
+        \{ x\in \eR^n\tq f(x)\leq M \}
+    \end{equation}
+    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
+
+    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
+    \begin{equation}
+        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
+    \end{equation}
+    ce qui contredit la minimalité de \( f(\bar x_1)\).
+
+    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
+    \begin{equation}
+        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
+    \end{equation}
+    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
+\end{proof}
+
+\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
+    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
+    \begin{equation}
+        x_{k+1}=x_k+t_kd_k
+    \end{equation}
+    où
+    \begin{itemize}
+        \item 
+    \( d_k=-(\nabla f)(x_k)\) 
+\item
+    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
+    \end{itemize}
+
+    Alors pour tout \( k\geq 0\) nous avons
+    \begin{equation}
+        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
+    \end{equation}
+    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
+\end{proposition}
+
+\begin{proof}
+    Décomposition en plusieurs points.
+    \begin{subproof}
+    \item[Existence de \( \bar x\)]
+        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
+    \item[Si \( (\nabla f)(x_k)=0\)]
+    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
+    \end{equation}
+    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
+    \begin{equation}
+        (\nabla f)(a)=Aa+b.
+    \end{equation}
+    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
+
+    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
+        \item[\( t_k\) est bien défini]
+
+            Pour \( t\in \eR\) nous avons
+            \begin{equation}    \label{EqKEHooYaazQi}
+                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
+            \end{equation}
+            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
+            \begin{equation}    \label{EqVWJooWmDSER}
+                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
+            \end{equation}
+            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
+
+        \item[La valeur de \( d_{k+1}\)]
+
+            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
+            \begin{equation}
+                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
+            \end{equation}
+            parce que \( -Ax_k-b=d_k\).
+            
+            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
+            \begin{equation}
+                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
+            \end{equation}
+            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
+        
+        \item[Calcul de \( f(x_{k+1})\)] 
+
+            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
+            \begin{equation}
+                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
+            \end{equation}
+            
+        \item[Encore du calcul \ldots]
+
+            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
+            \begin{equation}
+                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
+            \end{equation}
+            Ayant cela en tête nous pouvons calculer :
+            \begin{subequations}
+                \begin{align}
+                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
+                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
+                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
+                    &=2\big( f(x_k)-\bar f \big)
+                \end{align}
+            \end{subequations}
+            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
+
+        \item[Erreur sur la valeur du minimum]
+
+            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
+            \begin{subequations}
+                \begin{align}
+                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
+                \end{align}
+            \end{subequations}
+            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
+            \begin{equation}
+                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
+            \end{equation}
+            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
+            \begin{equation}
+                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
+            \end{equation}
+            Par récurrence nous avons alors
+            \begin{equation}    \label{eqANKooNPfCFj}
+                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
+            \end{equation}
+            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
+
+        \item[Erreur sur la position du minimum]
+
+            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
+            \begin{subequations}
+                \begin{align}
+                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
+                \end{align}
+            \end{subequations}
+            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
+
+            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
+            \begin{equation}    \label{EqVMRooUMXjig}
+                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
+            \end{equation}
+            
+        \item[Conclusion]
+
+            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
+            \begin{equation}
+                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
+            \end{equation}
+            c'est à dire
+            \begin{equation}
+                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
+            \end{equation}
+    \end{subproof}
+\end{proof}
+
+Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
 
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Fonctions convexes}
diff --git a/tex/frido/79_inversion_locale.tex b/tex/frido/79_inversion_locale.tex
index e32c80639..1616656eb 100644
--- a/tex/frido/79_inversion_locale.tex
+++ b/tex/frido/79_inversion_locale.tex
@@ -3,575 +3,6 @@
 %   Laurent Claessens
 % See the file fdl-1.3.txt for copying conditions.
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Recherche d'extrema}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema à une variable}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
-\end{definition}
-
-La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
-\begin{proposition}     \label{PROPooNVKXooXtKkuz}
-Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
-\end{proposition}
-
-La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
-
-Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
-\begin{proposition}
-Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
-\begin{equation}
-    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
-\end{equation}
-et que
-\begin{equation}
-    f^{(k)}(a)\neq 0.
-\end{equation}
-Dans ce cas,
-\begin{enumerate}
-
-\item
-Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
-\item
-Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
-
-\end{enumerate}
-\end{proposition}
-
-Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
-
-Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
-
-Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
-
-\begin{example}
-	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
-\end{example}
- 
-Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
-\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
-\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
-
-Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
-
-Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
-
-\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
-\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
- 
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema libre}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooYJLZooLkEAYf}
-Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
-\end{definition}
-
-Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
-
-Pour rappel, dans le cas d'une fonction à deux variables, $d^2f(a)$ est la matrice (et donc l'application linéaire)
-\begin{equation}
-    d^2f(a)=\begin{pmatrix}
-    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
-    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
-\end{pmatrix}.
-\end{equation}
-Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
-
-\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
-\end{proposition}
-
-\begin{proof}
-    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
-    \begin{equation}
-        \begin{aligned}
-            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
-            t&\mapsto f(a+tu) 
-        \end{aligned}
-    \end{equation}
-    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
-    \begin{equation}
-        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
-    \end{equation}
-\end{proof}
-
-\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
-    \begin{enumerate}
-        \item   \label{ITEMooCVFVooWltGqI}
-            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est \href{http://fr.wikipedia.org/wiki/Matrice_définie_positive}{strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}}, alors $a$ est un minimum local strict de $f$,
-        \item\label{ItemPropoExtreRn}
-            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
-    \end{enumerate}
-\end{proposition}
-\index{extrema}
-
-\begin{proof}
-    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
-    \begin{equation}
-        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
-    \end{equation}
-    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
-    \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
-    \end{equation}
-    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
-    \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
-    \end{equation}
-    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
-    \begin{equation}
-        (d^2f)_a(e_h,e_h)\geq 0.
-    \end{equation}
-    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
-\end{proof}
-
-La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
-
-Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
-
-La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
-\begin{enumerate}
-    \item
-        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
-    \item
-        écrire $d^2f(a)$ pour chacun des candidats
-    \item
-        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
-    \item
-        conclure.
-\end{enumerate}
-
-Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
-
-\begin{example}
-    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
-    \begin{equation}
-        df=\big(4x^3-4y;4y^3-4x),
-    \end{equation}
-    et la matrice des dérivées secondes est
-    \begin{equation}
-        M=d^2f(x,y)=\begin{pmatrix}
-            12x^2    &   -4    \\ 
-            -4    &   12y^2    
-        \end{pmatrix}.
-    \end{equation}
-    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
-
-    Pour le point \( (0,0)\) nous avons
-    \begin{equation}
-        M=\begin{pmatrix}
-            0    &   -4    \\ 
-            -4    &   0    
-        \end{pmatrix},
-    \end{equation}
-    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
-
-    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
-    \begin{equation}
-        M=\begin{pmatrix}
-            12    &   -4    \\ 
-            -4    &   12    
-        \end{pmatrix},
-    \end{equation}
-    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Un peu de recettes de cuisine}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{enumerate}
-\item Rechercher les points critiques, càd les $(x,y)$ tels que
-\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
-En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
-\item Déterminer la nature des points critiques: «test» des dérivées secondes:
-\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
-\begin{enumerate}
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
-\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
-\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
-\end{enumerate}
-\end{enumerate}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Extrema liés}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
-
-Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
-
-\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
-    Soit \( A\), un ouvert de \( \eR^n\) et
-    \begin{enumerate}
-        \item
-            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
-        \item 
-            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
-        \item
-            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
-        \item
-            un extrema local $a\in M$ de $f$ relativement à $M$.
-    \end{enumerate}
-    Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
-    \begin{equation}
-        L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_r)=f(x_1,\ldots,x_n)+\sum_{i=1}^r\lambda_iG_i(x_1,\ldots,x_n).
-    \end{equation}
-    Autrement dit, si \( a\) est un extrema lié, alors \( \nabla f(a)\) est une combinaisons des \( \nabla G_i(a)\), ou encore il existe des \( \lambda_i\) tels que
-    \begin{equation}    \label{EqRDsSXyZ}
-        df(a)=\sum_i\lambda_idG_i(a).
-    \end{equation}
-\end{theorem}
-\index{théorème!inversion locale!utilisation}
-\index{extrema!lié}
-\index{théorème!extrema!lié}
-\index{application!différentiable!extrema lié}
-\index{variété}
-\index{rang!différentielle}
-\index{forme!linéaire!différentielle}
-La fonction $L$ est le \defe{lagrangien}{lagrangien} du problème et les variables \( \lambda_i\) sont les \defe{multiplicateurs de Lagrange}{multiplicateur!de Lagrange}\index{Lagrange!multiplicateur}.
-
-\begin{proof}
-    Si \( r=n\) alors les vecteurs linéairement indépendantes \( \nabla G_i(a) \) forment une base de \( \eR^n\) et donc évidemment les \( \lambda_i\) existent. Nous supposons donc maintenant que \( r<n\). Nous notons \( (z_i)_{i=1\ldots n}\) les coordonnées sur \( \eR^n\).
-    
-    La matrice
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{ \partial G_1 }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_1 }{ \partial z_n }(a)    \\
-            \vdots    &   \ddots    &   \vdots    \\
-            \frac{ \partial G_r }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_r }{ \partial z_n }(a)
-        \end{pmatrix}
-    \end{equation}
-    est de rang \( r\) parce que les lignes sont par hypothèses linéairement indépendantes. Nous nommons \( (y_i)_{i=1,\ldots, r}\) un choix de \( r\) parmi les \( (z_i)\) tels que
-    \begin{equation}
-        \det\begin{pmatrix}
-            \frac{ \partial G_1 }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_1 }{ \partial y_r }    \\
-            \vdots    &   \ddots    &   \vdots    \\
-            \frac{ \partial G_r }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_r }{ \partial y_r }
-        \end{pmatrix}\neq 0.
-    \end{equation}
-    Nous identifions \( \eR^n\) à \( \eR^s\times \eR^r\) dans lequel \( \eR^r\) est la partie générée par les \( (y_i)_{i=1,\ldots, r}\). Nous nommons \( (x_j)_{j=1,\ldots, s}\) les coordonnées sur \( \eR^s\). Autrement dit, les coordonnées sur \( \eR^n\) sont \( x_1,\ldots, x_s,y_1,\ldots, y_r\). Dans ces coordonnées, nous nommons \( a=(\alpha,\beta)\) avec \( \alpha\in \eR^s\) et \( \beta\in \eR^r\).
-
-    Si nous notons \( G=(G_1,\ldots, G_r)\), le théorème de la fonction implicite (théorème \ref{ThoAcaWho})  nous dit qu'il existe un voisinage \( U'\) de \( \alpha\in \eR^n\), un voisinage \( V'\) de \( \beta\in \eR^r\) et une fonction \( \varphi\colon U'\to V'\) de classe \( C^1\) telle que si \( (x,y)\in U'\times V'\), alors
-    \begin{equation}
-        g(x,y)=0
-    \end{equation}
-    si et seulement si \( y=\varphi(x)\). Nous posons maintenant
-    \begin{subequations}
-        \begin{align}
-            \psi(x)&=(x,\varphi(x))\\
-            h(x)&=f\big( \psi(x) \big).
-        \end{align}
-    \end{subequations}
-    Nous avons \( \psi(\alpha)=a\) et \( \psi(x)\in M\) pour tout \( x\in U'\). La fonction \( h\) a donc un extrema local en \( \alpha\) et donc les dérivées partielles de \( h\) y sont nulles. Cela signifie que
-    \begin{equation}
-        0=\frac{ \partial h }{ \partial x_i }(\alpha)=\sum_{j=1}^n\frac{ \partial f }{ \partial x_j }\frac{ \partial x_j }{ \partial x_i }+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }\frac{ \partial \varphi_k }{ \partial x_i },
-    \end{equation}
-    c'est à dire
-    \begin{equation}
-        \frac{ \partial f }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
-    \end{equation}
-    pour tout \( i=1,\ldots, s\). D'autre part pour tout $k$, la fonction \( l_k(x)=G_k\big( x,\varphi(x) \big)\) est constante et vaut zéro; ses dérivées partielles sont donc nulles :
-    \begin{equation}
-        \frac{ \partial l }{ \partial x_i }(\alpha)=\frac{ \partial G_k }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial G_k }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
-    \end{equation}
-    pour tout \( i=1,\ldots, s\) et \( k=1,\ldots, r\).
-    
-    Les \( s\) premières colonnes de la matrice
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{ \partial f }{ \partial x_1 }   &   \cdots    &   \frac{ \partial f }{ \partial x_s }    &   \frac{ \partial f }{ \partial y_1 }    &   \cdots    &   \frac{ \partial f }{ \partial y_r }\\  
-            \frac{ \partial G_1 }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial x_s }    &   \frac{ \partial G_1 }{ \partial y_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial y_r }\\
-            \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots\\
-            \frac{ \partial G_r }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_r }{ \partial x_s }    &   \frac{ \partial G_r }{ \partial y_1 }    &  \cdots   & \frac{ \partial G_r }{ \partial y_r }  
-        \end{pmatrix}
-    \end{equation}
-    s'expriment en terme des \( r\) dernières. La matrice est donc au maximum de rang \( r\). Notons que la première ligne est \( \nabla f\) et les \( r\) suivantes sont les \( \nabla G_i\). Vu que ces lignes sont des vecteurs liés, il existe \( \mu_0,\ldots, \mu_r\) tels que
-    \begin{equation}
-        \mu_0\nabla f+\sum_{i=1}^r\mu_i\nabla G_i=0.
-    \end{equation}
-    Par hypothèse les \( \nabla G_i\) sont linéairement indépendants, ce qui nous dit que \( \mu_0\neq 0\). Donc nous avons ce qu'il nous faut :
-    \begin{equation}
-        \nabla f(a)=\sum_i\frac{ \mu_i }{ \mu_0 } \nabla G_i(a).
-    \end{equation}
-
-    Notons qu'au vu de l'expression \eqref{EqRDsSXyZ}, le fait que les formes \( \{ dG_i(a) \}_{1\leq i\leq r}\) forment une partie libre dans \( (\eR^n)^*\) implique que les \( \lambda_i\) sont uniques.
-\end{proof}
-
-La proposition suivante est la même que \ref{ThoRGJosS}.
-\begin{proposition} \label{PropfPPUxh}
-    Soit \( U\), un ouvert de \( \eR^n\) et des fonctions de classe \( C^1\) \( f,g_1,\ldots, g_r\colon U\to \eR\). Nous considérons
-    \begin{equation}
-        \Gamma=\{ x\in U\tq g_1(x)=\ldots=g_r(x)=0 \}.
-    \end{equation}
-    Soit \( a\) un extrémum de \( f|_{\Gamma}\). Supposons que les formes \( dg_1,\ldots, dg_r\) soient linéairement indépendantes en \( a\). Alors il existe \( \lambda_1,\ldots, \lambda_r\) dans \( \eR\) tel que
-    \begin{equation}
-        df_a=\sum_{i=1}^r\lambda_i(dg_i)_a.
-    \end{equation}
-\end{proposition}
-
-En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
-
-L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
-
-\begin{lemma}       \label{LemmeMinSCimpliqueS}
-    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
-\end{lemma}
-
-\begin{proof}
-    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Algorithme du gradient à pas optimal}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
-
-Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
-\begin{itemize}
-    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
-    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
-\end{itemize}
-
-\begin{definition}  \label{DefQXPooYSygGP}
-    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
-    \begin{equation}
-        \{ x\in P\tq f(x)\leq M \}
-    \end{equation}
-    est borné.
-\end{definition}
-En langage imagé la coercivité de \( f\) s'exprime par la limite
-\begin{equation}
-    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
-\end{equation}
-
-Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
-\begin{proposition}     \label{PROPooYRLDooTwzfWU}
-    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Alors :
-    \begin{enumerate}
-        \item
-            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
-        \item
-            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
-        \item
-            Ils sont égaux : \( \bar x=x^*\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-
-    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
-
-    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
-
-    \begin{subproof}
-    \item[\( f\) est strictement convexe]
-        
-        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
-    \end{equation}
-    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
-
-\item[\( f\) est coercive]
-    Montrons à présent que \( f\) est coercive. Nous avons :
-    \begin{subequations}
-        \begin{align}
-            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
-            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
-            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
-        \end{align}
-    \end{subequations}
-    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
-    \end{subproof}
-
-    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
-    \begin{equation}
-        \{ x\in \eR^n\tq f(x)\leq M \}
-    \end{equation}
-    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
-
-    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
-    \begin{equation}
-        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
-    \end{equation}
-    ce qui contredit la minimalité de \( f(\bar x_1)\).
-
-    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
-    \begin{equation}
-        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
-    \end{equation}
-    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
-\end{proof}
-
-\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
-    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
-    \begin{equation}
-        x_{k+1}=x_k+t_kd_k
-    \end{equation}
-    où
-    \begin{itemize}
-        \item 
-    \( d_k=-(\nabla f)(x_k)\) 
-\item
-    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
-    \end{itemize}
-
-    Alors pour tout \( k\geq 0\) nous avons
-    \begin{equation}
-        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
-    \end{equation}
-    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
-\end{proposition}
-
-\begin{proof}
-    Décomposition en plusieurs points.
-    \begin{subproof}
-    \item[Existence de \( \bar x\)]
-        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
-    \item[Si \( (\nabla f)(x_k)=0\)]
-    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
-    \end{equation}
-    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
-    \begin{equation}
-        (\nabla f)(a)=Aa+b.
-    \end{equation}
-    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
-
-    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
-        \item[\( t_k\) est bien défini]
-
-            Pour \( t\in \eR\) nous avons
-            \begin{equation}    \label{EqKEHooYaazQi}
-                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
-            \end{equation}
-            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
-            \begin{equation}    \label{EqVWJooWmDSER}
-                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
-            \end{equation}
-            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
-
-        \item[La valeur de \( d_{k+1}\)]
-
-            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
-            \begin{equation}
-                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
-            \end{equation}
-            parce que \( -Ax_k-b=d_k\).
-            
-            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
-            \begin{equation}
-                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
-            \end{equation}
-            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
-        
-        \item[Calcul de \( f(x_{k+1})\)] 
-
-            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
-            \begin{equation}
-                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
-            \end{equation}
-            
-        \item[Encore du calcul \ldots]
-
-            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
-            \begin{equation}
-                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
-            \end{equation}
-            Ayant cela en tête nous pouvons calculer :
-            \begin{subequations}
-                \begin{align}
-                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
-                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
-                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
-                    &=2\big( f(x_k)-\bar f \big)
-                \end{align}
-            \end{subequations}
-            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
-
-        \item[Erreur sur la valeur du minimum]
-
-            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
-            \begin{subequations}
-                \begin{align}
-                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
-                \end{align}
-            \end{subequations}
-            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
-            \begin{equation}
-                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
-            \end{equation}
-            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
-            \begin{equation}
-                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
-            \end{equation}
-            Par récurrence nous avons alors
-            \begin{equation}    \label{eqANKooNPfCFj}
-                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
-            \end{equation}
-            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
-
-        \item[Erreur sur la position du minimum]
-
-            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
-            \begin{subequations}
-                \begin{align}
-                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
-                \end{align}
-            \end{subequations}
-            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
-
-            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
-            \begin{equation}    \label{EqVMRooUMXjig}
-                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
-            \end{equation}
-            
-        \item[Conclusion]
-
-            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
-            \begin{equation}
-                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
-            \end{equation}
-            c'est à dire
-            \begin{equation}
-                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
-            \end{equation}
-    \end{subproof}
-\end{proof}
-
-Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
-
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Formes quadratiques, signature, et lemme de Morse}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

From 1290bc49fd62b50927724912db53428cada008ad Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Mon, 26 Jun 2017 18:47:28 +0200
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 tex/frido/69_analyseR.tex | 32 --------------------------------
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diff --git a/tex/frido/69_analyseR.tex b/tex/frido/69_analyseR.tex
index 5b31430b6..416b9525c 100644
--- a/tex/frido/69_analyseR.tex
+++ b/tex/frido/69_analyseR.tex
@@ -7,38 +7,6 @@
 \section{Suites de fonctions}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Convergence de suites de fonctions}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous considérons un espace normé \( (\Omega,\| . \|)\). Nous disons qu'une suite de fonctions \( f_n\) \defe{converge}{convergence!en norme} vers \( f\) pour la norme \( \| . \|\) si \( \forall \epsilon>0\), \( \exists N\) tel que \( n\geq N\) implique \( \| f_n-f \|<\epsilon\).
-
-Dans le cas particulier de la norme 
-\begin{equation}
-    \| f \|_{\infty}=\sup_{x\in\Omega}| f(x) |,
-\end{equation}
-nous parlons que \defe{convergence uniforme}{convergence!uniforme!suite de fonctions}.
-
-\begin{theorem}[Critère de Cauchy]  \label{ThoCauchyZelUF}
-    Une suite de fonctions  \( (f_n)_{n\in\eN}\) sur \( \Omega\) converge en norme sur \( \Omega\) si et seulement si \( \forall\epsilon>0\), \( \exists N\) tel que
-    \begin{equation}
-        \| f_n-f_m \|<\epsilon
-    \end{equation}
-    pour \( n,m>N\).
-\end{theorem}
-
-\begin{corollary}       \label{CorCauchyCkXnvY}
-    La série \( \sum f_n\) converge en norme sur \( \Omega\) si et seulement si \( \exists N\) tel que
-    \begin{equation}
-        \| f_n+\cdots+f_m \|\leq \epsilon
-    \end{equation}
-    pour tout \( n,m>N\).
-\end{corollary}
-
-\begin{proof}
-    L'hypothèse montre que la suite des sommes partielles de la série \( \sum f_n\) vérifie le critère de Cauchy du théorème \ref{ThoCauchyZelUF}.
-\end{proof}
-
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Convergence uniforme}
 %---------------------------------------------------------------------------------------------------------------------------

From 7a171279802cebe13a3c659bf3d8e4e92a4722f9 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Mon, 26 Jun 2017 23:01:54 +0200
Subject: [PATCH 58/64] =?UTF-8?q?(organisation)=20D=C3=A9place=20de=20nomb?=
 =?UTF-8?q?reux=20r=C3=A9sultats=20pour=20les=20r=C3=A9f=C3=A9rences=20ver?=
 =?UTF-8?q?s=20le=20futur.?=
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Content-Transfer-Encoding: 8bit

Il faut mettre la recherche d'extrema avant les fonctions convexes...
---
 commons.py                         |   11 +-
 lst_actu.py                        |    2 -
 mazhe.tex                          |    1 -
 tex/frido/53_topologie.tex         |    6 -
 tex/frido/61_representations.tex   |  115 ++
 tex/frido/68_analyseR.tex          | 2183 ++++++++++++++++++++++------
 tex/frido/77_series_fonctions.tex  |  142 ++
 tex/frido/78_inversion_locale.tex  | 1610 --------------------
 tex/front_back_matter/38_theme.tex |    2 +-
 tex/front_back_matter/64_theme.tex |   13 +-
 10 files changed, 2049 insertions(+), 2036 deletions(-)
 delete mode 100644 tex/frido/78_inversion_locale.tex

diff --git a/commons.py b/commons.py
index 3f6098fd1..5bc462899 100644
--- a/commons.py
+++ b/commons.py
@@ -20,13 +20,22 @@
 
 
 ok_hash=[]
-ok_hash.append("7dd1f22241df25fe18b534a41346d88aaa2f6584")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
 ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("<++>")
+ok_hash.append("7dd1f22241df25fe18b534a41346d88aaa2f6584")
+ok_hash.append("8a8f415da5dd19372c11e752715bc537114bdd87")
+ok_hash.append("d07c64f2b7c0b9c2de776dc065f762b54fbfbfba")
+ok_hash.append("a66228915d6108a67f2a66b8212b9700b2cdacb9")
 ok_hash.append("ccced7c771d6b95a06cf3339fc4ae85a7624e8ca")
 ok_hash.append("77a4d1ada815b1388561bd7b44b18ebc93c38af6")
 ok_hash.append("246a74658264789058fbfbd84086f5ef1016df1b")
diff --git a/lst_actu.py b/lst_actu.py
index f64125b16..b7a9629b5 100644
--- a/lst_actu.py
+++ b/lst_actu.py
@@ -17,9 +17,7 @@
 
 myRequest.ok_filenames_list=["e_mazhe"]
 myRequest.ok_filenames_list.extend(["68_analyseR"])
-myRequest.ok_filenames_list.extend(["61_representations"])
 myRequest.ok_filenames_list.extend(["78_inversion_locale"])
-myRequest.ok_filenames_list.extend(["57_EspacesVectos"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
 myRequest.ok_filenames_list.extend(["<++>"])
diff --git a/mazhe.tex b/mazhe.tex
index 904a4a56d..7f86f93b2 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -229,7 +229,6 @@ \chapter{Arc paramétré}
 
 \chapter{Suite de l'analyse}
 \input{77_series_fonctions}
-\input{78_inversion_locale}
 \input{79_inversion_locale}
 \input{80_Newton}
 
diff --git a/tex/frido/53_topologie.tex b/tex/frido/53_topologie.tex
index 9d7abe9fb..10ef56880 100644
--- a/tex/frido/53_topologie.tex
+++ b/tex/frido/53_topologie.tex
@@ -410,12 +410,6 @@ \subsection{Norme}
 
 Un espace vectoriel normé est alors immédiatement un espace vectoriel topologique
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Espace vectoriel normé}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-La définition d'une norme sur un espace vectoriel est la définition \ref{DefNorme}.
-
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Quelque exemples}
 %---------------------------------------------------------------------------------------------------------------------------
diff --git a/tex/frido/61_representations.tex b/tex/frido/61_representations.tex
index 36deab17f..654835275 100644
--- a/tex/frido/61_representations.tex
+++ b/tex/frido/61_representations.tex
@@ -1005,6 +1005,121 @@ \subsection{Décomposition polaires : cas réel}
     Si \( A\in\eM(n,\eR)\) alors la décomposition polaire \ref{ThoLHebUAU} nous donne \( A=SQ\) où \( S\) est symétrique définie positive et \( Q\) est orthogonale. La matrice \( S\) peut ensuite être diagonalisée par le théorème \ref{ThoeTMXla} : \( S=RDR^{-1}\) où \( D\) est diagonale et \( R\) est orthogonale. Avec ces deux décompositions en main, \( A=SQ=RDR^{-1}Q\). La matrice \( R^{-1}Q\) est orthogonale.
 \end{proof}
 
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Décomposition polaire}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{normaltext}      \label{NomDJMUooTRUVkS}
+    Nous allons montrer que l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
+            A&\mapsto \sqrt{A} 
+        \end{aligned}
+    \end{equation}
+    est une difféomorphisme.
+
+    Cependant \( S^{++}(n,\eR)\) n'est pas un ouvert de \( \eM(n,\eR)\) et nous ne savons pas ce qu'est la différentielle d'une application non définie sur un ouvert. Nous allons donc en réalité montrer que l'application racine carré existe sur un voisinage de chacun des points de \( S^{++}(n,\eR)\). Et comme une union quelconque d'ouverts est un ouvert, la fonction \( f\) sera bien définie sur un ouvert de \( \eM(n,\eR)\).
+\end{normaltext}
+
+\begin{lemma}       \label{LemLBFOooDdNcgy}
+    L'application 
+    \begin{equation}
+        \begin{aligned}
+            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
+            A&\mapsto A^2 
+        \end{aligned}
+    \end{equation}
+    est un \(  C^{\infty}\)-difféomorphisme.
+\end{lemma}
+
+\begin{proof}
+    Prouvons d'abord que \( f\) prend ses valeurs dans \( S^{++}(n,\eR)\). Si \( A\in S^{++}(n,\eR)\) alors par la diagonalisation \ref{ThoeTMXla} elle s'écrit \( A=QDQ^{-1}\) où \( D\) est diagonale avec des nombres strictement positifs sur la diagonale. Avec cela, \( A^2=QD^2Q^{-1}\) où \( D^2\) contient encore des nombres strictement positifs sur la diagonale.
+
+    L'application \( f\) étant essentiellement des polynôme en les entrées de \( A\), elle est de classe \( C^{\infty}\).
+
+    Passons à l'étude de la différentielle. Comme mentionné en \ref{NomDJMUooTRUVkS} nous allons en réalité voir \( f\) sur un ouvert de \( \eM(n,\eR)\) autour de \( A\in S^{++}(n,\eR)\). Par conséquent si \( A\in S^{++}(n,\eR)\),
+    \begin{subequations}
+        \begin{align}
+            df\colon S^{++}(n,\eR)&\to \aL\big( \eM(n,\eR),\eM(n,\eR) \big)\\
+            df_A\colon \eM(n,\eR)&\to \eM(n,\eR).
+        \end{align}
+    \end{subequations}
+    Le calcul de \( df_A\) est facile. Soit \( u\in \eM(n,\eR)\) et faisons le calcul en utilisant la formule du lemme \eqref{LemdfaSurLesPartielles} :
+    \begin{subequations}
+        \begin{align}
+            df_A(u)&=\Dsdd{ f(A+tu) }{t}{0}\\
+            &=\Dsdd{ A^2+tAu+tuA+t^2u^2 }{t}{0}\\
+            &=Au+uA.
+        \end{align}
+    \end{subequations}
+    Nous allons utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} à la fonction \( f\). Dans la suite, \( A\) est une matrice de \( S^{++}(n,\eR)\).
+
+    \begin{subproof}
+        \item[\( df_A\) est injective]
+            Soit \( M\in \eM(n,\eR)\) dans le noyau de \( df_A\). En posant \( M'=A^{-1}MQ\) nous avons \( M=QM'Q^{-1}\) et on applique \( df_A\) à \( QM'Q^{-1}\) :
+            \begin{equation}
+                df_A(QM'Q^{-1})=Q\big( DM+MD \big)Q^{-1}.
+            \end{equation}
+            où \( D=\begin{pmatrix}
+                \lambda_1    &       &       \\
+                    &   \ddots    &       \\
+                    &       &   \lambda_n
+                \end{pmatrix}\) avec \( \lambda_i>0\). La matrice \( D\) est inversible. Nous avons \( M'=-DM'D^{-1}\), et en coordonnées,
+                \begin{subequations}
+                    \begin{align}
+                        M'_{ij}&=-\sum_{kl}D_{ikM'_{kl}}D^{-1}_{lj}\\
+                        &=-\sum_{kl}\lambda_i\delta_{ik}M'_{kl}\frac{1}{ \lambda_j }\delta_{lj}\\
+                        &=-\frac{ \lambda_i }{ \lambda_i }M'_{ij}.
+                    \end{align}
+                \end{subequations}
+                C'est à dire que \( M'_{ij}=-\frac{ \lambda_i }{ \lambda_j }M'_{ij}\) avec \( -\frac{ \lambda_i }{ \lambda_j }<0\). Cela implique \( M'=0\) et par conséquent \( M=0\).
+            \item[\( df_A\) est surjective]
+                Soit \( N\in \eM(n,\eR)\); nous cherchons \( M\in \eM(n,\eR)\) tel que \( df_A(M)=N\). Nous posons \( N'=Q^{-1} NQ\) et \( M=QM'Q^{-1}\), ce qui nous donne à résoudre \( df_D(M')=N'\). Passons en coordonnées :
+                \begin{subequations}
+                    \begin{align}
+                        (DM'+M'D)_{ij}&=\sum_k(\delta_{ik}\lambda_iM'_{kj}+M'_{ik}\delta_{kj}\lambda_j)\\&
+                        &=M'_{ij}(\lambda_i+\lambda_j)
+                    \end{align}
+                \end{subequations}
+                où \( \lambda_i+\lambda_j\neq 0\). Il suffit donc de prendre la matrice \( M'\) donnée par
+                \begin{equation}
+                    M'_{ij}=\frac{1}{ \lambda_i+\lambda_j }N'_{ij}
+                \end{equation}
+                pour que \( df_A(M')=N'\).
+    \end{subproof}
+    
+    Le théorème d'inversion locale donne un voisinage \( V\) de $A$ dans \( \eM(n,\eR)\) et un voisinage \( W\) de \( A^2\) dans \( \eM(n,\eR)\) tels que \( f\colon V\to W\) soit une bijection  et \( f^{-1}\colon W\to V\) soit de même régularité, en l'occurrence \( C^{\infty}\).
+\end{proof}
+
+\begin{remark}
+    Oui, il y a des matrices non symétriques qui ont une unique racine carré.
+\end{remark}
+
+La proposition suivante, qui dépend du le théorème d'inversion locale par le lemme \ref{LemLBFOooDdNcgy}, donne plus de régularité à la décomposition polaire donnée dans le théorème \ref{ThoLHebUAU}.
+\begin{proposition}[Décomposition polaire : cas réel (suite)]       \label{PropWCXAooDuFMjn}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \gO(n,\eR)\times S^{++}(n,\eR)&\to \GL(n,\eR) \\
+            (Q,S)&\mapsto SQ 
+        \end{aligned}
+    \end{equation}
+    est un difféomorphisme de classe \( C^{\infty}\).
+\end{proposition}
+
+\begin{proof}
+    Si \( M\) est donnée dans \( \GL(n,\eR)\) alors la décomposition polaire\footnote{Proposition \ref{ThoLHebUAU}.} \( M=QS\) est donnée par \( S=\sqrt{MM^t}\) et \( Q=MS^{-1}\). Autrement dit, si nous considérons la fonction de décomposition polaire
+    \begin{equation}
+        f\colon \gO(n,\eR)\times S^{++}(n,\eR)\to \GL(n,\eR)
+    \end{equation}
+    alors
+    \begin{equation}
+        f^{-1}(M)=\big(  M(\sqrt{MM^t})^{-1},\sqrt{MM^t}  \big).
+    \end{equation}
+    Nous avons vu dans le lemme \ref{LemLBFOooDdNcgy} que la racine carré était un \( C^{\infty}\)-difféomorphisme. Le reste n'étant que des produits de matrice, la régularité est de mise.
+\end{proof}
+
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Enveloppe convexe}
 %---------------------------------------------------------------------------------------------------------------------------
diff --git a/tex/frido/68_analyseR.tex b/tex/frido/68_analyseR.tex
index 1403e2409..7bd03006e 100644
--- a/tex/frido/68_analyseR.tex
+++ b/tex/frido/68_analyseR.tex
@@ -634,10 +634,10 @@ \subsection{Différentielle seconde, fonction de classe \( C^2\)}
 \begin{equation}
   \frac{1}{g} \Delta_g  \left(\frac{1}{h} \Delta_h f(v)\right) = \frac{1}{h} \Delta_h \left(\frac{1}{g} \Delta_g f(v)\right).
 \end{equation}
-On utilise alors le théorème des accroissements finis
-\[
+On utilise alors le théorème des accroissements finis \ref{ThoAccFinis}
+\begin{equation}
 \frac{1}{h} \Delta_h f(v)=\frac{1}{h}\big(f(x+h,y)-f(x,y)\big)=\frac{1}{h}\partial_1f(x+t_1h,y )h=\partial_1f(x+t_1h, y),
-\]
+\end{equation}
 pour un certain $t_1$ dans $]0,1[$. De même on obtient 
 \[
 \frac{1}{g} \Delta_g f(v)= \partial_2 f(x, y+t_2g),
@@ -1424,222 +1424,1575 @@ \section{Développement au voisinage de l'infini}
     \end{equation}
 \end{example}
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Recherche d'extrema}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Complétude avec la norme uniforme}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema à une variable}
-%---------------------------------------------------------------------------------------------------------------------------
+\begin{proposition}[Limite uniforme de fonctions continues]\label{PropCZslHBx}
+    Soit \( X\) un espace topologique et \( (Y,d)\) un espace métrique. Si une suite de fonctions \( f_n\colon X\to Y\) continues converge uniformément, alors la limite est séquentiellement continue\footnote{Si \( X\) est métrique, alors c'est la continuité usuelle par la proposition \ref{PropFnContParSuite}.}.
+\end{proposition}
 
-\begin{definition}
-Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
-\end{definition}
+\begin{proof}
+    Soit \( a\in X\) et prouvons que \( f\) est séquentiellement continue en \( a\). Pour cela nous considérons une suite \( x_n\to a\) dans \( X\). Nous savons que \( f(x_n)\stackrel{Y}{\longrightarrow}f(x)\). Pour tout \(k\in \eN\), tout \( n\in \eN\) et tout \( x\in X\) nous avons la majoration
+    \begin{equation}
+        \big\| f(x_n)-f(x) \big\|\leq \big\| f(x_n)-f_k(x_n) \big\|+\big\| f_k(x_n)-f_k(x) \big\|+\big\| f_k(x)-f(x) \big\|\leq 2\| f-f_k \|_{\infty}+\big\| f_k(x_n)-f_k(x) \big\|.    
+    \end{equation}
+    Soit \( \epsilon>0\). Si nous choisissons \( k\) suffisamment grand la premier terme est plus petit que \( \epsilon\). Et par continuité de \( f_k\), en prenant \( n\) assez grand, le dernier terme est également plus petit que \( \epsilon\).
+\end{proof}
 
-La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
-\begin{proposition}     \label{PROPooNVKXooXtKkuz}
-Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
+\begin{proposition} \label{PropSYMEZGU}
+    Soit \( X\) un espace topologique métrique \( (Y,d)\) un espace espace métrique complet. Alors les espaces
+    \begin{enumerate}
+        \item
+            \( \big( C^0_b(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et bornées \( X\to Y\),
+        \item
+            \( \big( C^0_0(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et s'annulant à l'infini
+        \item
+            \( \big( C^k_0(X,Y),\| . \|_{\infty} \big)\) des fonctions de classe \( C^k\) et s'annulant à l'infini
+    \end{enumerate} 
+    sont complets.
 \end{proposition}
 
-La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
+\begin{proof}
+    Soit \( (f_n)\) une suite de Cauchy dans \( C(X,Y)\), c'est à dire que pour tout \( \epsilon>0\) il existe \( N\in \eN\) tel que si \( k,l>N\) nous avons \( \| f_k-f_l \|_{\infty}\leq \epsilon\). Cette suite vérifie le critère de Cauchy uniforme \ref{PropNTEynwq} et donc converge uniformément vers une fonction \( f\colon X\to Y\). La continuité (ou l'aspect \( C^k\)) de la fonction \( f\) découle de la convergence uniforme et de la proposition \ref{PropCZslHBx} (c'est pour avoir l'équivalence entre la continuité séquentielle et la continuité normale que nous avons pris l'hypothèse d'espace métrique).
 
-Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
-\begin{proposition}
-Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
-\begin{equation}
-    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
-\end{equation}
-et que
-\begin{equation}
-    f^{(k)}(a)\neq 0.
-\end{equation}
-Dans ce cas,
-\begin{enumerate}
+    Si les fonctions \( f_k\) sont bornées ou s'annulent à l'infini, la convergence uniforme implique que la limite le sera également.
+\end{proof}
+    Notons que si \( X\) est compact, les fonctions continues sont bornées par le théorème \ref{ThoImCompCotComp} et nous pouvons simplement dire que \( C^0(X,Y)\) est complet, sans préciser que nous parlons des fonctions bornées.
 
-\item
-Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
-\item
-Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
 
-\end{enumerate}
-\end{proposition}
+\begin{lemma}       \label{LemdLKKnd}
+    Soit \( A\) compact et \( B\) complet. L'ensemble des fonctions continues de \( A\) vers \( B\) muni de la norme uniforme est complet.
 
-Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
+    Dit de façon courte : \( \big( C(A,B),\| . \|_{\infty} \big)\) est complet.
+\end{lemma}
 
-Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
+\begin{proof}
+    Soit \( (f_k)\) une suite de Cauchy de fonctions dans \( C(A,B)\). Pour chaque \( x\in A \) nous avons
+    \begin{equation}
+        \| f_k(x)-f_l(x) \|_B\leq \| f_k-f_l \|_{\infty},
+    \end{equation}
+    de telle sorte que la suite \( (f_k(x))\) est de Cauchy dans \( B\) et converge donc vers un élément de \( B\). La suite de Cauchy \( (f_k)\) converge donc ponctuellement vers une fonction \( f\colon A\to B\). Nous devons encore voir que cette fonction est continue; ce sera l'uniformité de la norme qui donnera la continuité. En effet soit \( x_n\to x\) une suite dans \( A\) convergent vers \( x\in A\). Pour chaque \( k\in \eN\) nous avons
+    \begin{equation}
+        \| f(x_n)-f(x) \|\leq \| f(x_n)-f_k(x_n) \|  +\| f_k(x_n)-f_k(x) \|+\| f_k(x)-f(x) \|.
+    \end{equation}
+    En prenant \( k\) et \( n\) assez grands, cette expression peut être rendue aussi petite que l'on veut; le premier et le troisième terme par convergence ponctuelle \( f_k\to f\), le second terme par continuité de \( f_k\). La suite \( f(x_n)\) est donc convergente vers \( f(x)\) et la fonction \( f\) est continue.
+\end{proof}
 
-Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
+\begin{probleme}
+Il serait sans doute bon de revoir cette preuve à la lumière du critère de Cauchy uniforme \ref{PropNTEynwq}.
+\end{probleme}
 
-\begin{example}
-	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
-\end{example}
- 
-Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
-\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
-\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
 
-Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
+\begin{normaltext}[\cite{ooXYZDooWKypYR}]
+    Le théorème de Stone-Weierstrass indique que les polynômes sont denses pour la topologie uniforme dans les fonctions continues. Donc il existe des limites uniformes de fonctions \( C^{\infty}\) qui ne sont même pas dérivables. Les espaces de type \( C^p\) munis de \( \| . \|_{\infty}\) ne sont donc pas complets sans quelque hypothèses. Voir la proposition \ref{PropSYMEZGU} et le thème \ref{THMooOCXTooWenIJE}.
+\end{normaltext}
 
-Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorèmes de point fixe}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
-\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
-\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
- 
 %---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema libre}
+\subsection{Points fixes attractifs et répulsifs}
 %---------------------------------------------------------------------------------------------------------------------------
 
-\begin{definition}      \label{DEFooYJLZooLkEAYf}
-Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
+\begin{definition}      \label{DEFooTMZUooMoBDGC}
+    Soit \( I\) un intervalle fermé de \( \eR\) et \( \varphi\colon I\to I\) une application \( C^1\). Soit \( a\) un point fixe de \( \varphi\). Nous disons que \( a\) est \defe{attractif}{point fixe!attractif}\index{attractif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) converge vers \( a\). Le point \( a\) sera dit \defe{répulsif}{répulsif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) diverge.
 \end{definition}
 
-Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
-
-Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
-\begin{equation}
-    d^2f(a)=\begin{pmatrix}
-    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
-    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
-\end{pmatrix}.
-\end{equation}
-Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
-
-\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
-\end{proposition}
+\begin{lemma}[\cite{DemaillyNum}]
+    Soit \( a\) un point fixe de \( \varphi\).
+    \begin{enumerate}
+        \item
+    Si \( | \varphi'(a) |<1\) alors \( a\) est attractif et la convergence est au moins exponentielle.
+\item
+    Si \( | \varphi'(a) |>1\) alors \( a\) est répulsif et la divergence est au moins exponentielle.
+    \end{enumerate}
+\end{lemma}
 
 \begin{proof}
-    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
+    Si \( | \varphi'(a)<1 |\) alors il existe \( k\) tel que \( | \varphi'(a) |<k<1\) et par continuité il existe un voisinage \( V\) de \( a\) dans lequel \( | \varphi'(x) |<k\) pour tout \( x\in V\). En utilisant le théorème des accroissements finis nous avons
     \begin{equation}
-        \begin{aligned}
-            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
-            t&\mapsto f(a+tu) 
-        \end{aligned}
+        | x_n-a |=\big| f(x_{n-1}-a) \big|\leq k| x_{n-1}-a |
     \end{equation}
-    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
+    et par récurrence
     \begin{equation}
-        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
+        | x_n-a |\leq k^n| x_0-a |.
     \end{equation}
+
+    Le cas \( | \varphi'(a)>1 |\) se traite de façon similaire.
 \end{proof}
 
-\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
-    \begin{enumerate}
-        \item   \label{ITEMooCVFVooWltGqI}
-            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
-        \item\label{ItemPropoExtreRn}
-            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
-    \end{enumerate}
-\end{proposition}
-\index{extrema}
+\begin{remark}
+    Dans le cas \(| \varphi'(a) |=1\), nous ne pouvons rien conclure. Si \( \varphi(x)=\sin(x)\) nous avons \( \sin(x)<x\) et le point \( a=0\) est attractif. A contrario, si \( \varphi(x)=\sinh(x)\) nous avons \( |\sinh(x)|>|x|\) et le point \( a=0\) est répulsif.
+\end{remark}
 
-\begin{proof}
-    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Picard}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooRSLCooAsWisu}
+    Une application \( f\colon (X,\| . \|_X)\to (Y,\| . \|_Y)\) entre deux espaces métriques est une \defe{contraction}{contraction} si elle est \( k\)-\defe{Lipschitz}{Lipschitz} pour un certain \( 0\leq k<1\), c'est à dire si pour tout \( x,y\in X\) nous avons
     \begin{equation}
-        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
+        \| f(x)-f(y) \|_Y\leq k\| x-y \|_{X}.
     \end{equation}
-    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
+\end{definition}
+
+\begin{theorem}[Picard \cite{ClemKetl,NourdinAnal}\footnote{Il me semble qu'à la page 100 de \cite{NourdinAnal}, l'hypothèse H1 qui est prouvée ne prouve pas Hn dans le cas \( n=1\). Merci de m'écrire si vous pouvez confirmer ou infirmer. La preuve donnée ici ne contient pas cette «erreur».}.]     \label{ThoEPVkCL}
+    Soit \( X\) un espace métrique complet et \( f\colon X\to X\) une application contractante, de constante de Lipschitz \( k\). Alors \( f\) admet un unique point fixe, nommé \( \xi\). Ce dernier est donné par la limite de la suite définie par récurrence 
+    \begin{subequations}
+        \begin{numcases}{}
+            x_0\in X\\
+            x_{n+1}=f(x_n).
+        \end{numcases}
+    \end{subequations}
+    De plus nous pouvons majorer l'erreur par
+    \begin{equation}    \label{EqKErdim}
+        \| x_n-x \|\leq \frac{ k^n }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k^n }{ 1-k }\| x_1-x_0 \|.
+    \end{equation}
+
+    Soit \( r>0\), \( a\in X\) tels que la fonction \( f\) laisse la boule \( K=\overline{ B(a,r) }\) invariante (c'est à dire que \( f\) se restreint à \( f\colon K\to K\)). Nous considérons les suites \( (u_n)\) et \( (v_n)\) définies par
+    \begin{subequations}
+        \begin{numcases}{}
+            u_0=v_0\in K\\
+            u_{n+1}=f(v_n), v_{n+1}\in B(u_n,\epsilon).
+        \end{numcases}
+    \end{subequations}
+    Alors le point fixe \( \xi\) de \( f\) est dans \( K\) et la suite \( (v_n)\) satisfait l'estimation
     \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
+        \| v_n-\xi \|\leq \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k }.
     \end{equation}
-    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
+\end{theorem}
+\index{théorème!Picard}
+\index{point fixe!Picard}
+
+La première inégalité \eqref{EqKErdim} donne une estimation de l'erreur calculable en cours de processus; la seconde donne une estimation de l'erreur calculable avant de commencer.
+
+\begin{proof}
+    
+    Nous commençons par l'unicité du point fixe. Si \( a\) et \( b\) sont des points fixes, alors \( f(a)=a\) et \( f(b)=b\). Par conséquent
     \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
+        \| f(a)-f(b) \|=\| a-b \|,
     \end{equation}
-    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
+    ce qui contredit le fait que \( f\) soit une contraction.
+
+    En ce qui concerne l'existence, notons que si la suite des \( x_n\) converge dans \( X\), alors la limite est un point fixe. En effet en prenant la limite des deux côtés de l'équation \( x_{n+1}=f(x_n)\), nous obtenons \( \xi=f(\xi)\), c'est à dire que \( \xi\) est un point fixe de \( f\). Notons que nous avons utilisé ici la continuité de \( f\), laquelle est une conséquence du fait qu'elle soit Lipschitz. Nous allons donc porter nos efforts à prouver que la suite est de Cauchy (et donc convergente parce que \( X\) est complet). Nous commençons par prouver que \( \| x_{n+1}-x_n \|\leq k^n\| x_0-x_1 \|\). En effet pour tout \( n\) nous avons
     \begin{equation}
-        (d^2f)_a(e_h,e_h)\geq 0.
+        \| x_{n+1}-x_n \|=\| f(x_n)-f(x_{n-1}) \|\leq k\| x_n-x_{n-1} \|.
     \end{equation}
-    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
-\end{proof}
+    La relation cherchée s'obtient alors par récurrence. Soient \( q>p\). En utilisant une somme télescopique,
+    \begin{subequations}
+        \begin{align}
+            \| x_q-x_p \|&\leq \sum_{l=p}^{q-1}\| x_{l+1}-x_l \|\\
+            &\leq\left( \sum_{l=p}^{q-1}k^l \right)\| x_1-x_0 \|\\
+            &\leq\left(\sum_{l=p}^{\infty}k^l\right)\| x_1-x_0 \|.
+        \end{align}
+    \end{subequations}
+    Étant donné que \( k<1\), la parenthèse est la queue d'une série qui converge, et donc tend vers zéro lorsque \( p\) tend vers l'infini.
 
-La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
+    En ce qui concerne les inégalités \eqref{EqKErdim}, nous refaisons une somme télescopique :
+    \begin{subequations}
+        \begin{align}
+            \| x_{n+p}-x_n \|&\leq \| x_{n+p}-x_{n+p-1} \|+\cdots +\| x_{n+1}-x_n \|\\
+            &\leq k^p\| x_n-x_{n-1} \|+k^{p-1}\| x_n-x_{n-1} \|+\cdots +k\| x_n-x_{n-1} \|\\
+            &=k(1+\cdots +k^{p-1})\| x_n-x_{n-1}\|  \\
+            &\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|.
+        \end{align}
+    \end{subequations}
+    En prenant la limite \( p\to \infty\) nous trouvons
+    \begin{equation}        \label{EqlUMVGW}
+        \| \xi-x_n \|\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k }{ 1-k }\| x_1-x_0 \|.
+    \end{equation}
 
-Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
+    Nous passons maintenant à la seconde partie du théorème en supposant que \( f\) se restreigne en une fonction \( f\colon K\to K\). D'abord \( K\) est encore un espace métrique complet, donc la première partie du théorème s'y applique et \( f\) y a un unique point fixe.
+    
+    Nous allons montrer la relation par récurrence. Tout d'abord pour \( n=1\) nous avons
+    \begin{equation}
+        \| v_1-\xi \|\leq\| v_1-u_1 \|+\| u_1-\xi \|\leq \epsilon+\frac{ k }{ 1-k }\| u_1-u_0 \|
+    \end{equation}
+    où nous avons utilisé l'estimation \eqref{EqlUMVGW}, qui reste valable en remplaçant \( x_1\) par \( u_1\)\footnote{Elle n'est cependant pas spécialement valable si on remplace \( x_n\) par \( u_n\).}. Nous pouvons maintenant faire la récurrence :
+    \begin{subequations}
+        \begin{align}
+            \| v_{n+1}-\xi \|&\leq \| v_{n+1}-u_{n+1} \|+\| u_{n+1}-\xi \|\\
+            &\leq \epsilon+k\| v_n-\xi \|\\
+            &\leq \epsilon+k\left( \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k } \right)\\
+            &=\frac{ \epsilon }{ 1-k }+\frac{ k^{n+1} }{ 1-k }\| u_1-u_0 \|.
+        \end{align}
+    \end{subequations}
+\end{proof}
 
-La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
-\begin{enumerate}
-    \item
-        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
-    \item
-        écrire $d^2f(a)$ pour chacun des candidats
-    \item
-        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
-    \item
-        conclure.
-\end{enumerate}
+\begin{remark}
+    Ce théorème comporte deux parties d'intérêts différents. La première partie est un théorème de point fixe usuel, qui sera utilisé pour prouver l'existence de certaines équations différentielles.
 
-Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
+    La seconde partie est intéressante d'un point de vie numérique. En effet, ce qu'elle nous enseigne est que si à chaque pas de calcul de la récurrence \( x_{n+1}=f(x_n)\) nous commettons une erreur d'ordre de grandeur \( \epsilon\), alors le procédé (la suite \( (v_n)\)) ne converge plus spécialement vers le point fixe, mais tend vers le point fixe avec une erreur majorée par \( \epsilon/(k-1)\).
+\end{remark}
 
-\begin{example}
-    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
+\begin{remark}
+Au final l'erreur minimale qu'on peut atteindre est de l'ordre de \( \epsilon\). Évidemment si on commet une faute de calcul de l'ordre de \( \epsilon\) à chaque pas, on ne peut pas espérer mieux.
+\end{remark}
+
+\begin{remark}  \label{remIOHUJm}
+    Si \( f\) elle-même n'est pas contractante, mais si \( f^p\) est contractante pour un certain \( p\in \eN\) alors la conclusion du théorème de Picard reste valide et \( f\) a le même unique point fixe que \( f^p\). En effet nommons \( x\) le point fixe de \( f\) : \( f^p(x)=x\). Nous avons alors
     \begin{equation}
-        df=\big(4x^3-4y;4y^3-4x),
+        f^p\big( f(x) \big)=f\big( f^p(x) \big)=f(x),
     \end{equation}
-    et la matrice des dérivées secondes est
-    \begin{equation}
-        M=d^2f(x,y)=\begin{pmatrix}
-            12x^2    &   -4    \\ 
-            -4    &   12y^2    
-        \end{pmatrix}.
+    ce qui prouve que \( f(x)\) est un point fixe de \( f^p\). Par unicité nous avons alors \( f(x)=x\), c'est à dire que \( x\) est également un point fixe de \( f\).
+\end{remark}
+
+Si la fonction n'est pas Lipschitz mais presque, nous avons une variante.
+\begin{proposition}
+    Soit \( E\) un ensemble compact\footnote{Notez cette hypothèse plus forte} et si \( f\colon E\to E\) est une fonction telle que
+    \begin{equation}        \label{EqLJRVvN}
+        \| f(x)-f(y) \|< \| x-y \|
     \end{equation}
-    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
+    pour tout \( x\neq y\) dans \( E\) alors \( f\) possède un unique point fixe.
+\end{proposition}
 
-    Pour le point \( (0,0)\) nous avons
+\begin{proof}
+    La suite \( x_{n+1}=f(x_n)\) possède une sous suite convergente. La limite de cette sous suite est un point fixe de \( f\) parce que \( f\) est continue. L'unicité est due à l'aspect strict de l'inégalité \eqref{EqLJRVvN}.
+\end{proof}
+
+\begin{theorem}[Équation de Fredholm]\index{Fredholm!équation}\index{équation!Fredholm}     \label{ThoagJPZJ}
+    Soit \( K\colon \mathopen[ a , b \mathclose]\times \mathopen[ a , b \mathclose]\to \eR\) et \( \varphi\colon \mathopen[ a , b \mathclose]\to \eR\), deux fonctions continues. Alors si \( \lambda\) est suffisamment petit, l'équation
     \begin{equation}
-        M=\begin{pmatrix}
-            0    &   -4    \\ 
-            -4    &   0    
-        \end{pmatrix},
+        f(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x)
     \end{equation}
-    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
+    admet une unique solution qui sera de plus continue sur \( \mathopen[ a , b \mathclose]\).
+\end{theorem}
 
-    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
+\begin{proof}
+    Nous considérons l'ensemble \( \mF\) des fonctions continues \( \mathopen[ a , b \mathclose]\to\mathopen[ a , b \mathclose]\) muni de la norme uniforme. Le lemme \ref{LemdLKKnd} implique que \( \mF\) est complet. Nous considérons l'application \( \Phi\colon \mF\to \mF\) donnée par
     \begin{equation}
-        M=\begin{pmatrix}
-            12    &   -4    \\ 
-            -4    &   12    
-        \end{pmatrix},
+        \Phi(f)(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x). 
     \end{equation}
-    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
-\end{example}
+    Nous montrons que \( \Phi^p\) est une application contractante pour un certain \( p\). Pour tout \( x\in \mathopen[ a , b \mathclose]\) nous avons
+    \begin{subequations}
+        \begin{align}
+            \| \Phi(f)-\Phi(g) \|_{\infty}&\leq \| \Phi(f)(x)-\Phi(g)(x) \|\\
+            &=| \lambda |\Big\| \int_a^bK(x,y)\big( f(y)-g(y) \big)dy  \Big\|\\
+            &\leq | \lambda |\| K \|_{\infty}| b-a |\| f-g \|_{\infty}
+        \end{align}
+    \end{subequations}
+    Si \( \lambda\) est assez petit, et si \( p\) est assez grand, l'application \( \Phi^p\) est donc une contraction. Elle possède donc un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+\end{proof}
 
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Brouwer}
 %---------------------------------------------------------------------------------------------------------------------------
-\subsection{Un peu de recettes de cuisine}
-%---------------------------------------------------------------------------------------------------------------------------
+\label{subSecZCCmMnQ}
 
-\begin{enumerate}
-\item Rechercher les points critiques, càd les $(x,y)$ tels que
-\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
-En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
-\item Déterminer la nature des points critiques: «test» des dérivées secondes:
-\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
-\begin{enumerate}
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
-\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
-\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
-\end{enumerate}
-\end{enumerate}
+\begin{proposition}
+    Soit \( f\colon \mathopen[ a , b \mathclose]\to \mathopen[ a , b \mathclose]\) une fonction continue. Alors \( f\) accepte un point fixe.
+\end{proposition}
 
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Extrema liés}
-%---------------------------------------------------------------------------------------------------------------------------
+\begin{proof}
+    En effet si nous considérons \( g(x)=f(x)-x\) alors nous avons \( g(a)=f(a)-a\geq 0\) et \( g(b)=f(b)-b\leq 0\). Si \( g(a)\) ou \( g(b)\) est nul, la proposition est démontrée; nous supposons donc que \( g(a)>0\) et \( g(b)<0\). La proposition découle à présent du théorème des valeurs intermédiaires \ref{ThoValInter}.
+\end{proof}
 
-Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
+\begin{example}
+    La fonction \( x\mapsto\cos(x)\) est continue entre \( \mathopen[ -1 , 1 \mathclose]\) et \( \mathopen[ -1 , 1 \mathclose]\). Elle admet donc un point fixe. Par conséquent il existe (au moins) une solution à l'équation \( \cos(x)=x\).
+\end{example}
 
-Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
+\begin{proposition}[Brouwer dans \( \eR^n\) version \(  C^{\infty}\) via Stokes]     \label{PropDRpYwv}
+    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction \(  C^{\infty}\). Alors \( f\) admet un point fixe.
+\end{proposition}
+\index{point fixe!Brouwer}
 
-\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
-    Soit \( A\), un ouvert de \( \eR^n\) et
-    \begin{enumerate}
-        \item
-            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
-        \item 
-            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
-        \item
-            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
-        \item
+\begin{proof}
+    Supposons que \( f\) ne possède pas de points fixes. Alors pour tout \( x\in B\) nous considérons la ligne droite partant de \( x\) dans la direction de \( f(x)\) (cette droite existe parce que \( x\) et \( f(x)\) sont supposés distincts). Cette ligne intersecte \( \partial B\) en un point que nous appelons \( g(x)\). Prouvons que cette fonction est \( C^k\) dès que \( f\) est \( C^k\) (y compris avec \( k=\infty\)).
+
+   Le point \( g(x) \) est la solution du système
+    \begin{subequations}
+        \begin{numcases}{}
+        g(x)-f(x)=\lambda\big( x-f(x) \big)\\
+        \| g(x) \|^2=1\\
+        \lambda\geq 0.
+        \end{numcases}
+    \end{subequations}
+    En substituant nous obtenons l'équation
+    \begin{equation}
+        P_x(\lambda)=\| \lambda\big( x-f(x) \big)+f(x) \|^2-1=0,
+    \end{equation}
+    ou encore
+    \begin{equation}
+        \lambda^2\| x-f(x) \|^2+2\lambda\big( x-f(x) \big)\cdot f(x)+\| f(x) \|^2-1=0.
+    \end{equation}
+    En tenant compte du fait que \( \| f(x)<1 \|\) (pare que les images de \( f\) sont dans \( \mB\)), nous trouvons que \( P_x(0)\leq 0\) et \( P_x(1)\leq 0\). De même \( \lim_{\lambda\to\infty} P_x(\lambda)=+\infty\). Par conséquent le polynôme de second degré \( P_x\) a exactement deux racines distinctes \( \lambda_1\leq 0\) et \( \lambda_2\geq 1\). La racine que nous cherchons est la seconde. Le discriminant est strictement positif, donc pas besoin d'avoir peur de la racine dans
+    \begin{equation}
+        \lambda(x)=\frac{ -\big( x-f(x) \big)\cdot f(x)+\sqrt{   \Delta_x  } }{ \| x-f(x) \|^2 }
+    \end{equation}
+    où 
+    \begin{equation}
+        \Delta_x=4\Big( \big( x-f(x) \big)\cdot f(x) \Big)^2-4\| x-f(x) \|^2\big( \| f(x) \|^2-1 \big).
+    \end{equation}
+    Notons que la fonction \( \lambda(x)\) est \( C^k\) dès que \( f\) est \( C^k\); et en particulier elle est \( C^{\infty}\) si \( f\) l'est.
+
+    En résumé la fonction \( g\) ainsi définie vérifie deux propriétés :
+    \begin{enumerate}
+        \item
+            elle est \(  C^{\infty}\);
+        \item
+            elle est l'identité sur \( \partial B\).
+    \end{enumerate}
+    La suite de la preuve consiste à montrer qu'une telle rétraction sur \( B\) ne peut pas exister\footnote{Notons qu'il n'existe pas non plus de rétractions continues sur \( B\), mais pour le montrer il faut utiliser d'autres méthodes que Stokes, ou alors présenter les choses dans un autre ordre.}.
+
+    Nous considérons une forme de volume \( \omega\) sur \( \partial B\) : l'intégrale de \( \omega\) sur \( \partial B\) est la surface de \( \partial B\) qui est non nulle. Nous avons alors
+    \begin{equation}
+        0<\int_{\partial B}\omega
+        =\int_{\partial B}g^*\omega
+        =\int_Bd(g^*\omega)
+        =\int_Bg^*(d\omega)
+        =0
+    \end{equation}
+    Justifications :
+    \begin{itemize}
+        \item 
+            L'intégrale \( \int_{\partial B}\omega\) est la surface de \( \partial B\) et est donc non nulle.
+        \item
+            La fonction \( g\) est l'identité sur \( \partial B\). Nous avons donc \( \omega=g^*\omega\).
+        \item
+            Le lemme \ref{LemdwLGFG}.
+        \item
+            La forme \( \omega\) est de volume, par conséquent de degré maximum et \( d\omega=0\).
+    \end{itemize}
+\end{proof}
+
+Un des points délicats est de se ramener au cas de fonctions \( C^{\infty}\). Pour la régularisation par convolution, voir \cite{AllardBrouwer}; pour celle utilisant le théorème de Weierstrass, voir \cite{KuttlerTopInAl}.
+\begin{theorem}[Brouwer dans \( \eR^n\) version continue]\label{ThoRGjGdO}
+    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction continue\footnote{Une fonction continue sur un fermé de \( \eR^n\) est à comprendre pour la topologie induite.}. Alors \( f\) admet un point fixe.
+\end{theorem}
+\index{théorème!Brouwer}
+
+\begin{proof}
+    Nous commençons par définir une suite de fonctions
+    \begin{equation}
+        f_k(x)=\frac{ f(x) }{ 1+\frac{1}{ k } }.
+    \end{equation}
+    Nous avons \( \| f_k-f \|_{\infty}\leq \frac{1}{ 1+k }\) où la norme est la norme uniforme sur \( B\). Par le théorème de Weierstrass \ref{ThoWmAzSMF} il existe une suite de fonctions \(  C^{\infty}\) \( g_k\) telles que
+    \begin{equation}
+        \|  g_k-f_k\|_{\infty}\leq\frac{1}{ 1+k }.
+    \end{equation}
+    Vérifions que cette fonction \( g_k\) soit bien une fonction qui prend ses valeurs dans \( B\) :
+    \begin{subequations}
+        \begin{align}
+            \| g_k(x) \|&\leq \| g_k(x)-f_k(x) \|+\| f_k(x) \|\\
+            &\leq \frac{1}{ 1+k }+\frac{ \| f(x) \| }{ 1+\frac{1}{ k } }\\
+            &\leq \frac{1}{ 1+k}+\frac{1}{ 1+\frac{1}{ k } }\\
+            &=1.
+        \end{align}
+    \end{subequations}
+    Par la version \(  C^{\infty}\) du théorème (proposition \ref{PropDRpYwv}), \( g_k\) admet un point fixe que l'on nomme \( x_k\).
+
+    Étant donné que \( x_k\) est dans le compact \( B\), quitte à prendre une sous suite nous supposons que la suite \( (x_k)\) converge vers un élément \( x\in B\). Nous montrons maintenant que \( x\) est un point fixe de \( f\) :
+    \begin{subequations}
+        \begin{align}
+            \| f(x)-x \|&=\| f(x)-g_k(x)+g_k(x)-x_k+x_k-x \|\\
+            &\leq \| f(x)-g_k(x) \| +\underbrace{\| g_k(x)-x_k \|}_{=0}+\| x_k-x \|\\
+            &\leq \frac{1}{ 1+k }+\| x_k-x \|.
+        \end{align}
+    \end{subequations}
+    En prenant le limite \( k\to\infty\) le membre de droite tend vers zéro et nous obtenons \( f(x)=x\).
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Théorème de Schauder}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Une conséquence du théorème de Brouwer est le théorème de Schauder qui est valide en dimension infinie.
+
+\begin{theorem}[Théorème de Schauder\cite{ooWWBQooKIciWi}]\index{théorème!Schauder}       \label{ThovHJXIU}
+    Soit \( E\), un espace vectoriel normé, \( K\) un convexe compact de \( E\) et \( f\colon K\to K\) une fonction continue. Alors \( f\) admet un point fixe.
+\end{theorem}
+\index{théorème!Schauder}
+\index{point fixe!Schauder}
+
+\begin{proof}
+    Étant donné que \( f\colon K\to K\) est continue, elle y est uniformément continue. Si nous choisissons \( \epsilon\) alors il existe \( \delta>0\) tel que 
+    \begin{equation}
+        \| f(x)-f(y) \|\leq \epsilon
+    \end{equation}
+    dès que \( \| x-y \|\leq \delta\). La compacité de \( K\) permet de choisir un recouvrement fini par des ouverts de la forme
+    \begin{equation}    \label{EqKNPUVR}
+        K\subset \bigcup_{1\leq i\leq p}B(x_j,\delta)
+    \end{equation}
+    où \( \{ x_1,\ldots, x_p \}\subset K\). Nous considérons maintenant \( L=\Span\{ f(x_j)\tq 1\leq j\leq p \}\) et
+    \begin{equation}
+        K^*=K\cap L.
+    \end{equation}
+    Le fait que \( K\) et \( L\) soient convexes implique que \( K^*\) est convexe. L'ensemble \( K^*\) est également compact parce qu'il s'agit d'une partie fermée de \( K\) qui est compact (lemme \ref{LemnAeACf}). Notons en particulier que \( K^*\) est contenu dans un espace vectoriel de dimension finie, ce qui n'est pas le cas de \( K\).
+
+    Nous allons à présent construire une sorte de partition de l'unité subordonnée au recouvrement \eqref{EqKNPUVR} sur \( K\) (voir le lemme \ref{LemGPmRGZ}). Nous commençons par définir
+    \begin{equation}
+        \psi_j(x)=\begin{cases}
+            0    &   \text{si } \| x-x_j \|\geq \delta\\
+            1-\frac{ \| x-x_j \| }{ \delta }    &    \text{sinon}.
+        \end{cases}
+    \end{equation}
+    pour chaque \( 1\leq j\leq p\). Notons que \( \psi_j\) est une fonction positive, nulle en-dehors de \( B(x_j,\delta)\). En particulier la fonction suivante est bien définie :
+    \begin{equation}
+        \varphi_j(x)=\frac{ \psi_j(x) }{ \sum_{k=1}^p\psi_k(x) }
+    \end{equation}
+    et nous avons \( \sum_{j=1}^p\varphi_j(x)=1\). Les fonctions \( \varphi_j\) sont continues sur \( K\) et nous définissons finalement
+    \begin{equation}
+        g(x)=\sum_{j=1}^p\varphi_j(x)f(x_j).
+    \end{equation}
+    Pour chaque \( x\in K\), l'élément \( g(x)\) est une combinaison des éléments \( f(x_j)\in K^*\). Étant donné que \( K^*\) est convexe et que la somme des coefficients \( \varphi_j(x)\) vaut un, nous avons que \( g\) prend ses valeurs dans \( K^*\) par la proposition \ref{PropPoNpPz}.
+
+    Nous considérons seulement la restriction \( g\colon K^*\to K^*\) qui est continue sur un compact contenu dans un espace vectoriel de dimension finie. Le théorème de Brouwer nous enseigne alors que \( g\) a un point fixe (proposition \ref{ThoRGjGdO}). Nous nommons \( y\) ce point fixe. Notons que \( y\) est fonction du \( \epsilon\) choisit au début de la construction, via le \( \delta\) qui avait conditionné la partition de l'unité.
+
+    Nous avons
+    \begin{subequations}        \label{EqoXuTzE}
+        \begin{align}
+            f(y)-y&=f(y)-g(y)\\
+            &=\sum_{j=1}^p\varphi_j(y)f(y)-\sum_{j=1}^p\varphi_j(y)f(x_j)\\
+            &=\sum_{j=1}^p\varphi(j)(y)\big( f(y)-f(x_j) \big).
+        \end{align}
+    \end{subequations}
+    Par construction, \( \varphi_j(y)\neq 0\) seulement si \( \| y-x_j \|\leq \delta\) et par conséquent seulement si \( \| f(y)-f(x_j) \|\leq \epsilon\). D'autre par nous avons \( \varphi_j(y)\geq 0\); en prenant la norme de \eqref{EqoXuTzE} nous trouvons
+    \begin{equation}
+        \| f(y)-y \|\leq \sum_{j=1}^p\| \varphi_j(y)\big( f(y)-f(x_j) \big) \|\leq \sum_{j=1}^p\varphi_j(y)\epsilon=\epsilon.
+    \end{equation}
+    Nous nous souvenons maintenant que \( y\) était fonction de \( \epsilon\). Soit \( y_m\) le \( y\) qui correspond à \( \epsilon=2^{-m}\). Nous avons alors
+    \begin{equation}
+        \| f(y_m)-y_m \|\leq 2^{-m}.
+    \end{equation}
+    L'élément \( y_m\) est dans \( K^*\) qui est compact, donc quitte à choisir une sous suite nous pouvons supposer que \( y_m\) est une suite qui converge vers \( y^*\in K\)\footnote{Notons que même dans la sous suite nous avons \( \| f(y_m)-y_m \|\leq 2^{-m}\), avec le même «\( m\)» des deux côtés de l'inégalité.}. Nous avons les majorations
+    \begin{equation}
+        \| f(y^*)-y^* \|\leq \| f(y^*)-f(y_m) \|+\| f(y_m)-y_m \|+\| y_m-y^* \|.
+    \end{equation}
+    Si \( m\) est assez grand, les trois termes du membre de droite peuvent être rendus arbitrairement petits, d'où nous concluons que
+    \begin{equation}
+        f(y^*)=y^*
+    \end{equation}
+    et donc que \( f\) possède un point fixe.
+\end{proof}
+
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de Markov-Kakutani et mesure de Haar}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit \( G\) un groupe topologique. Une \defe{mesure de Haar}{mesure!de Haar} sur \( G\) est une mesure \( \mu\) telle que 
+    \begin{enumerate}
+        \item
+            \( \mu(gA)=\mu(A)\) pour tout mesurable \( A\) et tout \( g\in G\),
+        \item
+            \( \mu(K)<\infty\) pour tout compact \( K\subset G\).
+    \end{enumerate}
+    Si de plus le groupe \( G\) lui-même est compact nous demandons que la mesure soit normalisée : \( \mu(G)=1\).
+\end{definition}
+
+Le théorème suivant nous donne l'existence d'une mesure de Haar sur un groupe compact.
+\begin{theorem}[Markov-Katutani\cite{BeaakPtFix}]\index{théorème!Markov-Takutani}   \label{ThoeJCdMP}
+    Soit \( E\) un espace vectoriel normé et \( L\), une partie non vide, convexe, fermée et bornée de \( E'\). Soit \( T\colon L\to L\) une application continue. Alors \( T\) a un point fixe.
+\end{theorem}
+
+\begin{proof}
+    Nous considérons un point \( x_0\in L\) et la suite
+    \begin{equation}
+        x_n=\frac{1}{ n+1 }\sum_{i=0}^n T^ix_0.
+    \end{equation}
+    La somme des coefficients devant les \( T^i(x_0)\) étant \( 1\), la convexité de \( L\) montre que \( x_n\in L\). Nous considérons l'ensemble
+    \begin{equation}
+        C=\bigcap_{n\in \eN}\overline{ \{ x_m\tq m\geq n \} }.
+    \end{equation}
+    Le lemme \ref{LemooynkH} indique que \( C\) n'est pas vide, et de plus il existe une sous suite de \( (x_n)\) qui converge vers un élément \( x\in C\). Nous avons
+    \begin{equation}
+        \lim_{n\to \infty} x_{\sigma(n)}(v)=x(v)
+    \end{equation}
+    pour tout \( v\in E\). Montrons que \( x\) est un point fixe de \( T\). Nous avons
+    \begin{subequations}
+        \begin{align}
+            \| (Tx_{\sigma(k)}-x_{\sigma(k)})v \|&=\Big\| T\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v)-\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v) \Big\|\\
+            &=\Big\| \frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^{i+1}x_0(v)-T^ix_0(v) \Big\|\\
+            &=\frac{1}{ 1+\sigma(k) }\big\| T^{\sigma(k)+1}x_0(v)-x_0(v) \big\|\\
+            &\leq\frac{ 2M }{ \sigma(k)+1 }
+        \end{align}
+    \end{subequations}
+    où \( M=\sum_{y\in L}\| y(v) \|<\infty\) parce que \( L\) est borné. En prenant \( k\to\infty\) nous trouvons
+    \begin{equation}
+        \lim_{k\to \infty} \big( Tx_{\sigma(k)}-x_{\sigma(k)} \big)v=0,
+    \end{equation}
+    ce qui signifie que \( Tx=x\) parce que \( T\) est continue.
+\end{proof}
+
+Le théorème suivant est une conséquence du théorème de Markov-Katutani.
+\begin{theorem} \label{ThoBZBooOTxqcI}
+    Si \( G\) est un groupe topologique compact possédant une base dénombrable de topologie alors \( G\) accepte une unique mesure de Haar normalisée. De plus elle est unimodulaire :
+    \begin{equation}
+        \mu(Ag)=\mu(gA)=\mu(A)
+    \end{equation}
+    pour tout mesurables \( A\subset G\) et tout élément \( g\in G\).
+\end{theorem}
+\index{mesure!de Haar}
+
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorèmes de point fixes et équations différentielles}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Théorème de Cauchy-Lipschitz}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous démontrons ici deux théorèmes de Cauchy-Lipschitz. De nombreuses propriétés annexes seront démontrées dans le chapitre sur les équations différentielles, section \ref{SECooNKICooDnOFTD}.
+
+\begin{theorem}[Cauchy-Lipschitz\cite{SandrineCL,ZPNooLNyWjX}] \label{ThokUUlgU}
+    Nous considérons l'équation différentielle
+    \begin{subequations}        \label{XtiXON}
+        \begin{numcases}{}
+            y'(t)=f\big( t,y(t) \big)\\
+            y(t_0)=y_0
+        \end{numcases}
+    \end{subequations}
+    avec \( f\colon U=I\times \Omega\to \eR^n\) où \( I\) est ouvert dans \( \eR\) et \( \Omega\) ouvert dans \( \eR^n\). Nous supposons que \( f\) est continue sur \( U\) et localement Lipschitz\footnote{Définition \ref{DefJSFFooEOCogV}. Notons que nous ne supposons pas que \( f\) soit une contraction.} par rapport à \( y\). 
+    
+    Alors il existe un intervalle \( J\subset I\) sur lequel la solution au problème est unique. De plus toute solution du problème est une restriction de cette solution à une partie de \( J\). La solution sur \( J\) (dite «solution maximale») est de classe \( C^1\).
+\end{theorem}
+\index{théorème!Cauchy-Lipschitz}
+
+% Il serait tentant de mettre ce théorème dans la partie sur les équations différentielles, mais ce n'est pas aussi simple :
+% Il est utilisé pour calculer la transformée de Fourier de la Gaussienne (lemme LEMooPAAJooCsoyAJ) dans le chapitre sur la transformée de Fourier.
+
+\begin{proof}
+    Nous divisions la preuve en plusieurs étapes (même pas toutes simples).
+    \begin{subproof}
+    \item[Cylindre de sécurité et espace fonctionnel]
+
+    Précisons l'espace fonctionnel \( \mF\) adéquat. Soient \( V\) et \( W\) les voisinages de \( t_0\) et \( y_0\) sur lesquels \( f\) est localement Lipschitz. Nous considérons les quantités suivantes :
+    \begin{enumerate}
+        \item
+            \( M=\sup_{V\times W}f\) ;
+        \item
+            \( r>0\) tel que \( \overline{ B(y_0,r) }\subset V\)
+        \item
+            \( T>0\) tel que \( \overline{ B(t_0,T) }\subset W\) et \( T<r/M\).
+    \end{enumerate}
+    Nous considérons alors l'ensemble
+    \begin{equation}
+        \mF=C^0\big( \overline{ B(t_0,T) },\overline{ B(y_0,r) } \big)
+    \end{equation}
+    que nous munissons de la norme uniforme. Par le lemme \ref{LemdLKKnd} l'espace \( \big( \mF,\| . \|_{\infty} \big)\) est complet.
+
+    \item[Une application \( \Phi\colon \mF\to \mF\)]
+
+
+    Si \( y\) est une solution de l'équation différentielle considérée, elle vérifie
+    \begin{equation}        \label{EqPGLwcL}
+        y(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
+    \end{equation}
+    Ceci nous incite à considérer l'opérateur \( \Phi\colon \mF\to \mF\) défini par
+    \begin{equation}
+        \Phi(y)(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
+    \end{equation}
+
+    Pour que l'application \( \Phi\) soit utile nous devons montrer que pour tout \( y\in \mF\),
+    \begin{itemize}
+        \item l'application \( \Phi(y)\) est bien définie,
+        \item pour tout \( t\in\overline{ B(y_0,r) }\) nous avons \( \Phi(y)(t)\in\overline{ B(t_0,T) }\),
+        \item l'application $\Phi(y)\colon    \overline{ B(t_0,T) }\to \overline{ B(y_0,r)} $ est continue.
+    \end{itemize}
+    Attention : nous ne prétendons pas que \( \Phi\) elle-même soit continue. C'est parti.
+    \begin{subproof}
+    \item[\( \Phi(y)\) est bien définie]
+            
+        Il faut montrer que l'intégrale converge. Le calcul de \( \Phi(y)(t)\) ne se fait qu'avec \( t\in \overline{ B(t_0,T) }\). Vu que \( u\) prend ses valeurs dans \( \mathopen[ t_0 , t \mathclose]\) et que \( y\in\mF\), le nombre \( y(u)\) est toujours dans \( \overline{ B(y_0,r) }\). Ceci pour dire que dans l'intégrale, la fonction \( f\) n'est considérée que sur \( \mathopen[ t_0 , t \mathclose]\times \overline{ B(y_0,r) }\subset V\times W\). La fonction \( f\) est donc uniformément majorable, et l'intégrale ne pose pas de problèmes.
+
+    \item[\( \Phi(y)(t)\in \overline{ B(t_0,T) }\)]
+
+    Prouvons que \( \Phi(y)(t)\in\overline{ B(y_0,r) }\). Pour cela, notons que
+    \begin{equation}
+        | \Phi(y)(t)-y_0 |\leq \int_{t_0}^t |f\big( u,y(u) \big)|du\leq | t-t_0 |\| f \|_{\infty}.
+    \end{equation}
+    Étant donné que \( t\in\overline{ B(t_0,T) }\) nous avons \( | t-t_0 |\leq r/M\) et donc \( | \Phi(y)(t)-y_0 |\leq r\).
+
+    \item[\( \Phi(y)\) est continue]
+
+        Nous pourrions invoquer le théorème \ref{ThoKnuSNd}, mais nous allons le faire à la main. Soit \( s_0\in B(t_0,T)\) et prouvons que \( \Phi(y)\) est continue en \( s_0\). Pour cela nous prenons \( s\in B(s_0,\delta)\) et nous calculons :
+        \begin{equation}
+            | \Phi(y)(s)-\Phi(y)(s_0) |\leq \int_{s_0}^s|f\big( u,y(u) \big)|du\leq | s_0-s |\| f \|_{\infty}.
+        \end{equation}
+        C'est le fait que \( f\) soit bornée dans le cylindre de sécurité qui fait en sorte que cela tende vers zéro lorsque \( s\to s_0\).
+    \end{subproof}
+
+
+    
+    L'équation \eqref{EqPGLwcL} signifie que \( y\) est un point fixe de \( \Phi\). L'espace \( \mF\) étant complet le théorème de point fixe de Picard (théorème \ref{ThoEPVkCL}) s'applique. Nous allons montrer qu'il existe un \( p\in\eN\) tel que \( \Phi^p\) soit contractante. Par conséquent \( \Phi^p\) aura un unique point fixe qui sera également unique point fixe de \( \Phi\) par la remarque \ref{remIOHUJm}.
+    
+\item[Contractante]
+
+    Prouvons donc que \( \Phi^p\) est contractante pour un certain \( p\). Pour cela nous commençons par montrer la formule suivante par récurrence :
+    \begin{equation}        \label{EqRAdKxT}
+        \big\| \Phi^p(x)(t)-\Phi^p(y)(t) \big\|\leq \frac{ k^p| t-t_0 |^p }{ p! }\| x-y \|_{\infty}
+    \end{equation}
+    pour tout \( x,y\in\mF\), et pour tout \( t\in\overline{ B(t_0,T) }\). Pour \( p=0\) la formule \eqref{EqRAdKxT} est vérifiée parce que \( \| x-y \|_{\infty}\) est le supremum de \( \| x(t)-y(t) \|\) pour \( t\in\overline{ B(t_0,T) }\). Supposons que la formule soit vraie pour \( p\) et calculons pour \( p+1\). Pour tout \( t\in\overline{ B(t_0,T) }\) nous avons
+    \begin{subequations}
+        \begin{align}
+            \big\| \Phi^{p+1}(x)(t)-\Phi^{p+1}(y)(t) \big\|&\leq \left| \int_{t_0}^t\big\| f\big( u,\Phi^p(x)(u) \big)-f\big( u,\Phi^p(y)(u) \big) \big\|du \right| \\
+            &\leq \left| \int_{t_0}^tk\| \Phi^p(x)(u)-\Phi^p(y)(u) \|du \right|    \label{subIKYixF}\\
+            &\leq \left| \int_{t_0}^tk\frac{ k^p| t-t_0 | }{ p! }\| x-y \|_{\infty} \right| \label{subxkNjiV} \\
+            &=\frac{ k^{p+1}| t-t_0 |^{p+1} }{ (p+1)! }\| x-y \|_{\infty}.
+        \end{align}
+    \end{subequations}
+    Justifications :
+    \begin{itemize}
+        \item \eqref{subIKYixF} parce que \( f\) est Lipschitz.
+        \item \eqref{subxkNjiV} par hypothèse de récurrence.
+    \end{itemize}
+    La formule \eqref{EqRAdKxT} est maintenant établie. Nous pouvons maintenant montrer que \( \Phi^p\) est une contraction pour un certain \( p\). Pour tout \( t\in \overline{ B(t_0,T) }\) nous avons
+    \begin{equation}
+         \| \Phi^p(x)(t)-\Phi^p(y)(t) \|\leq \frac{ k^p }{ t! }| t-t_0 |^p\| x-y \|_{\infty}     \leq \frac{ k^pT^p }{ p! }\| x-y \|_{\infty}
+    \end{equation}
+    où nous avons utilisé le fait que \( | t-t_0 |^p<T^p\). En prenant le supremum sur \( t\) des deux côtés il vient
+    \begin{equation}
+        \| \Phi^p(x)-\Phi^p(y) \|_{\infty}\leq\frac{ k^pT^p }{ p! }\| x-y \|_{\infty}.
+    \end{equation}
+    Le membre de droite tend vers zéro lorsque \( p\to\infty\) parce que \( k<1\) et \( T^p/p!\to 0\)\footnote{C'est le terme général du développement de \(  e^{T}\) qui est une série convergente.}. Nous concluons donc que \( \Phi^p\) est une contraction pour un certain \( p\).
+
+\item[Conclusion]
+
+    L'unique point fixe de \( \Phi\) est alors l'unique solution continue de l'équation différentielle \eqref{XtiXON}. Par ailleurs l'équation elle-même \( y'=f(t,y)\) demande implicitement que \( y\) soit dérivable et donc continue. Nous concluons que l'unique point fixe de \( \Phi\) est l'unique solution de l'équation différentielle donnée. Cette dernière est automatiquement \( C^1\) parce que si \( y\) est continue alors \( u\mapsto f(u,y(u))\) est continue, c'est à dire que \( y'\) est continue.
+
+\item[Unicité]
+
+    Nous passons maintenant à la partie «prolongement maximum» du théorème. Soient \( x_1\) et \( x_2\) deux solutions maximales du problème \eqref{XtiXON} sur des intervalles \( I_1\) et \( I_2\) respectivement. Les intervalles \( I_1\) et \( I_2\) contiennent \( \overline{ B(t_0,r) }\) sur lequel \( x_1=x_2\) par unicité.
+    
+    
+    Nous allons maintenant montrer que pour tout \( t\geq t_0\) pour lequel \( x_1\) ou \( x_2\) est défini, \( x_1(t)\) et \( x_2(t)\) sont définis et sont égaux. Le raisonnement sur \( t\leq t_0\) est similaire.
+    
+    Supposons que l'ensemble des \( t\geq t_0\) tels que \( x_1=x_2\) soit ouvert à droite, c'est à dire soit de la forme \( \mathopen[ t_0 ,b [\). Dans ce cas, soit \( x_1\) soit \( x_2\) (soit les deux) cesse d'exister en \( b\). En effet si nous avions les fonctions \( x_i\) sur \(\mathopen[ t_0 , b+\epsilon [\) alors l'équation \( x_1=x_2\) définirait un fermé dans \( \mathopen[ t_0 , b+\epsilon [\). Supposons pour fixer les idées que \( x_1\) cesse d'exister : le domaine de \( x_1\) (parmi les \( t\geq 0\)) est \( \mathopen[ t_0 , b [\) et sur ce domaine nous avons \( x_1=x_2\). Dans ce cas \( x_1\) pourrait être prolongé en \( x_2\) au-delà de \( b\). Si \( x_1\) et \( x_2\) s'arrêtent d'exister en même temps en \( b\), alors nous avons bien \( x_1=x_2\).
+
+    Nous devons donc traiter le cas où \( x_1=x_2\) sur \( \mathopen[ t_0 , b \mathclose]\) alors que \( x_1\) et \( x_2\) existent sur \( \mathopen[ t_0 , b+\epsilon [\) pour un certain \( \epsilon\).
+
+    Nous pouvons appliquer le théorème d'existence locale au problème
+    \begin{subequations}
+        \begin{numcases}{}
+            y'=f(t,y)\\
+            y(b)=x_1(b).
+        \end{numcases}
+    \end{subequations}
+    Il existe un voisinage de \( b\) sur lequel la solution est unique. Sur ce voisinage nous devons donc avoir \( x_1=x_2\), ce qui contredit le fait que \( x_1\neq x_2\) en dehors de \( \mathopen[ t_0 , b \mathclose]\).
+
+    Donc \( x_1\) et \( x_2\) existent et sont égaux sur au moins \( I_1\cup I_2\).
+    \end{subproof}
+\end{proof}
+
+Le théorème de Cauchy-Lipschitz donne existence et unicité d'une solution maximale. Cependant cette solution peut ne pas exister partout où les hypothèses sur \( f\) sont remplies. En d'autres termes, il peut arriver que \( f\) soit Lipschitz jusqu'à \( t_1\), mais que la solution maximale ne soit définie que jusqu'en \( t_2<t_1\). Ce cas fait l'objet du théorème d'explosion en temps fini \ref{CorGDJQooNEIvpp}.
+
+Sous quelque hypothèses nous pouvons nous assurer de l'existence d'une solution unique sur tout \( \eR\).
+
+\begin{theorem}[Cauchy-Lipschitz global\cite{ooJZJPooAygxpk,KXjFWKA}]       \label{THOooZIVRooPSWMxg}
+    Soit un intervalle \( I\) de \( \eR\), \( y_0\in \eR^n\), \( t_0\in I\) et une fonction continue \( f\colon I\times \eR^n\to \eR^n\) telle que pour tout compact \( K\) dans \( I\), il existe \( k>0\) tel que
+    \begin{equation}
+        \| f(t,y_1)-f(t,y_2) \|\leq k\| y_1-y_2 \|
+    \end{equation}
+    pour tout \( t\in K\) et \( y_1,y_2\in \eR^n\).
+
+    Alors le problème
+    \begin{subequations}        \label{EQSooBNREooUTfbMH}
+        \begin{numcases}{}
+            y'(t)=f\big( t,y(t) \big)\\
+            y(t_0)=y_0
+        \end{numcases}
+    \end{subequations}
+    possède une unique solution \( y\colon I\to \eR^n\) sur \( I\).
+\end{theorem}
+
+\begin{proof}
+    Soit un intervalle compact \( K\) dans \( I\) et contenant \( t_0\). Nous notons \( \ell\) le diamètre de \( K\). Sur l'espace \( E=C^0(K,\eR^n)\) nous considérons la topologie uniforme : \( (E,\| . \|_{\infty})\). C'est un espace complet par le lemme \ref{LemdLKKnd} (nous utilisons le fait que \( \eR^n\) soit complet, théorème \ref{ThoTFGioqS}). Nous allons utiliser l'application suivante :
+    \begin{equation}        \label{EQooJUTBooILBKoE}
+        \begin{aligned}
+            \Phi\colon E&\to E \\
+            \Phi(y)(t)&=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds
+        \end{aligned}
+    \end{equation}
+    Démontrons quelque faits à propos de \( \Phi\).
+    \begin{subproof}
+        \item[La définition fonctionne bien]
+            Nous devons commencer par prouver que cette application est bien définie. Si \( y\in E\) alors \( f\) et \( y\) sont continues; l'application \( s\mapsto f\big(s,y(s)\big)\) est donc également continue. L'intégrale de cette fonction sur le compact \( \mathopen[ t_0 , t \mathclose]\) ne pose alors pas de problèmes. En ce qui concerne la continuité de \( \phi(y)\) sous l'hypothèse que \( y\) soit continue,
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(y)(t') \|\leq \int_t^{t'}\| f(s,y(s)) \|ds\leq M| t-t' |
+    \end{equation}
+    où \( M\) est une majoration de \( \| s\mapsto f\big( s,y(s) \big) \|_{\infty,K}\).
+
+        \item[Si \( y\) est solution alors \( \Phi(y)=y\)]
+
+            Supposons que \( y\) soit une solution de l'équation différentielle \eqref{EQSooBNREooUTfbMH}. Alors, vu que \( y'(t)=f\big( t,y(t) \big)\) nous avons :
+            \begin{equation}
+                y(t)=y_0+\int_{t_0}^ty'(s)ds=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds=\Phi(y)(t).
+            \end{equation}
+            
+        \item[Si \( \Phi(y)=y\) alors \( y\) est solution]
+
+            Nous avons, pour tout \( t\) :
+            \begin{equation}
+                y(t)=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds.
+            \end{equation}
+            Le membre de droite est dérivable par rapport à \( t\), et la dérivée fait \(  f\big( t,y(t) \big)   \). Donc le membre de gauche est également dérivable et nous avons bien
+            \begin{equation}
+                y'(t)=f\big( t,y(t) \big).
+            \end{equation}
+            De plus \( y(t_0)=y_0+\int_{t_0}^{t_0}\ldots=y_0\).
+    \end{subproof}
+    
+    Nous sommes encore avec \( K\) compact et \( E=C^0(K,\eR^n)\) muni de la norme uniforme. Nous allons montrer que \( \Phi\) est une contraction de \( E\) pour une norme bien choisie.
+
+    \begin{subproof}
+        \item[Une norme sur \( E\)]
+            Pour \( y\in E\) nous posons
+            \begin{equation}
+                \| y \|_k=\max_{t\in K}\big(  e^{-k| t-t_0 |}\| y(t) \| \big).
+            \end{equation}
+            Ce maximum est bien définit et fini parce que la fonction de \( t\) dedans est une fonction continue sur le compact \( K\). Cela est également une norme parce que si \( \| y \|_k=0\) alors \(  e^{-k| t-t_0 |}\| y(t) \|=0\) pour tout \( t\). Étant donné que l'exponentielle ne s'annule pas, \( \| y(t) \|=0\) pour tout \( t\).
+        \item[Équivalence de norme]
+
+            Nous montrons que les normes \( \| . \|_k\) et \( \| . \|_{\infty}\) sont équivalentes\footnote{Définition \ref{DefEquivNorm}} :
+            \begin{equation}        \label{EQooSQYWooBTXvDL}
+                \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\leq \| y \|_{\infty}
+            \end{equation}
+            pour tout \( y\in E\). Pour la première inégalité, \( \ell\geq | t-t_0 |\) pour tout \( t\in K\), et \( k>0\), donc
+            \begin{equation}
+                \| y(t) \| e^{-k\ell}\leq  e^{-k| t-t_0 |}\| y(t) \|.
+            \end{equation}
+            En prenant le maximum des deux côtés, \( \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\). 
+
+            En ce qui concerne la seconde inégalité dans \eqref{EQooSQYWooBTXvDL}, \( k| t-t_0 |\geq 0\) et donc \(  e^{-k| t-t_0 |}<1\).
+
+    \end{subproof}
+    Vu que les normes \( \| . \|_{\infty}\) et \( \| . \|_k\) sont équivalentes, l'espace \( (E,\| . \|_k)\) est tout autant complet que \( (E,\| . \|_{\infty})\). Nous démontrons à présent que \( \Phi\) est une contraction dans \( (E,\|  \|_k)\). 
+
+    Soient \( y,z\in E\). Si \( t\geq t_0\) nous avons
+    \begin{subequations}        \label{SUBEQSooEXVYooDkyTuB}
+        \begin{align}
+            \| \Phi(y)(t)-\Phi(z)(t) \|&\leq \int_{t_0}^t\| f\big( s,y(s) \big)-f\big( s,z(s) \big) \|ds\\
+            &\leq k\int_{t_0}^t\| y(s)-z(s) \|ds.
+        \end{align}
+    \end{subequations}
+    Il convient maintenant de remarquer que
+    \begin{equation}
+        \| y(t) \|= e^{-k| t-t_0 |} e^{k| t-t_0 |}\| y(t) \|\leq \| y \|_k e^{k| t-t_0 |}.
+    \end{equation}
+    Nous pouvons avec ça prolonger les inégalités \eqref{SUBEQSooEXVYooDkyTuB} par
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(z)(t) \|\leq k\| y-z \|_k\int_{t_0}^t e^{k| s-t_0 |}ds=k\| y-z \|_k\int_{t_0}^t e^{k(s-t_0)}ds
+    \end{equation}
+    où nous avons utilisé notre supposition \( t\geq t_0\) pour éliminer les valeurs absolues. L'intégrale peut être faite explicitement, mais nous en sommes arrivés à un niveau de fainéantise tellement inconcevable que
+
+\lstinputlisting{tex/sage/sageSnip014.sage}
+
+Au final, si \( t\geq t_0\),
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t-t_0)}-1 \big).
+    \end{equation}
+    Si \( t\leq t_0\), il faut retourner les bornes de l'intégrale avant d'y faire rentrer la norme parce que \( \| \int_0^1f \|\leq \int_0^1\| f \|\), mais ça ne marche pas avec \( \| \int_1^0f \|\). Pour \( t\leq t_0\) tout le calcul donne
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t_0-t)}-1 \big).
+    \end{equation}
+    Les deux inéquations sont valables a fortiori en mettant des valeurs absolues dans l'exponentielle, de telle sorte que pour tout \( t\in K\) nous avons
+    \begin{equation}
+        e^{-k| t_0-t |}\| \phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big( 1- e^{-k| t_0-t |} \big).
+    \end{equation}
+    En prenant le supremum sur \( t\),
+    \begin{equation}
+        \| \Phi(y)-\Phi(z) \|_k\leq \| y-z \|_k(1- e^{-k\ell}),
+    \end{equation}
+    mais \( 0<(1- e^{e-k\ell})<1\), donc \( \Phi\) est contractante pour la norme \( \| . \|_k\). Vu que \( (E,\| . \|_k)\) est complet, l'application \( \Phi\) y a un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+
+    Ce point fixe est donc l'unique solution de l'équation différentielle de départ.
+
+    \begin{subproof}
+        \item[Existence et unicité sur \( I\)]
+            Il nous reste à prouver que la solution que nous avons trouvée existe sur \( I\) : jusqu'à présent nous avons démontré l'existence et l'unicité sur n'importe quel compact dans \( I\).
+
+            Soit une suite croissante de compacts \( K_n\) contenant \( t_0\) (par exemple une suite exhaustive comme celle du lemme \ref{LemGDeZlOo}). Nous avons en particulier
+            \begin{equation}
+                I=\bigcup_{n=0}^{\infty}K_n.
+            \end{equation}
+        \item[Existence sur \( I\)]
+        
+            Soit \( y_n\) l'unique solution sur \( K_n\). Il suffit de poser
+            \begin{equation}
+                y(t)=y_n(t)
+            \end{equation}
+            pour \( n\) tel que \( t\in K_n\). Cette définition fonctionne parce que si \( t\in K_n\cap K_m\), il y a forcément un des deux qui est inclus à l'autre et le résultat d'unicité sur le plus grand des deux donne \( y_n(t)=y_m(t)\).
+
+        \item[Unicité sur \( I\)]
+
+            Soient \( y\) et \(z \) des solutions sur \( I\); vu que \( I\) n'est pas spécialement compact, le travail fait plus haut ne permet pas de conclure que \( y=z\). 
+
+            Soit \( t\in I\). Alors \( t\in K_n\) pour un certain \( n\) et \( y\) et \( z\) sont des solutions sur \( K_n\) qui est compact. L'unicité sur \( K_n\) donne \( y(t)=z(t)\).
+    \end{subproof}
+\end{proof}
+
+\begin{normaltext}
+    Il y a d'autres moyens de prouver qu'une solution existe globalement sur \( \eR\). Si \( f\) est globalement bornée, le théorème d'explosion en temps fini donne quelque garanties, voir \ref{NORMooZROGooZfsdnZ}.
+\end{normaltext}
+
+Le théorème suivant donne une version du théorème de Cauchy-Lipschitz lorsque la fonction \( f\) dépend d'un paramètre. Ce théorème n'utilise rien de fondamentalement nouveau. Nous le donnons seulement pour montrer que l'on peut choisir l'espace \( \mF\) de façon un peu maligne pour élargir le résultat. Si vous voulez un théorème de Cauchy-Lipschitz avec paramètre vraiment intéressant, allez voir le théorème \ref{PROPooPYHWooIZhQST}.
+
+\begin{theorem}[Cauchy-Lipschitz avec paramètre\cite{MonCerveau,ooXVPAooTQUIRw}]           \label{THOooDTCWooSPKeYu}
+    Soit un intervalle ouvert \( I\) de \( \eR\), un connexe ouvert \( \Omega\) de \( \eR^n\) et un intervalle ouvert \( \Lambda\) de \( \eR^d\). Soit une fonction \( f\colon I\times \Omega\times \Lambda\to \eR^n\) continue et localement Lipschitz en \( \Omega\). Soient \( t_0\in I\), \( y_0\in \Omega\) et \( \lambda_0\in \Lambda\). Il existe un voisinage compact de \( (t_0,y_0,\lambda_0)\) sur lequel le problème
+    \begin{subequations}
+        \begin{numcases}{}
+            y'_{\lambda}(t)=f\big( t,y_{\lambda}(t),\lambda \big)\\
+            y_{\lambda}(t_0)=y_0
+        \end{numcases}
+    \end{subequations}
+    possède une unique solution. De plus \( (t,\lambda)\mapsto y_{\lambda}(t)\) est continue\footnote{Ici, la surprise est que ce soit continu par rapport à \( \lambda\). Le fait qu'elle le soit par rapport à \( t\) est clair depuis le départ parce que c'est finalement rien d'autre que le Cauchy-Lipschitz vieux et connu.}.
+\end{theorem}
+
+\begin{proof}
+
+    \begin{probleme}
+        Ceci est une idée de la preuve. Je n'ai pas vérifié toutes les étapes. Soyez prudent.
+
+    \end{probleme}
+
+    D'abord nous avons un voisinage compact \( V\times \overline{ B(y_0,r) }\times \Lambda_0\) de \( (t_0,y_0,\lambda_0)\) sur lequel $f$ est bornée. Ensuite nous récrivons l'équation différentielle sous la forme
+    \begin{subequations}
+        \begin{numcases}{}
+            \frac{ \partial y }{ \partial t }(t,\lambda)=f\big( t,y(t,\lambda),\lambda \big)\\
+            y(t_0,\lambda)=y_0.
+        \end{numcases}
+    \end{subequations}
+    pour une fonction \( y\colon V\times \Lambda_0\to \eR^n\).
+
+    Nous posons \( \mF=C^0\big( V\times\Lambda_0 ,\eR^n\big)\) et nous y définissons l'application
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon \mF&\to \mF \\
+            \Phi(y)(t,\lambda)&=y_0+\int_{t_0}^tf\big( s,y(s,\lambda),\lambda \big)ds. 
+        \end{aligned}
+    \end{equation}
+    Il y a plein de vérifications à faire\cite{ooXVPAooTQUIRw}, mais je parie que \( \Phi\) est bien définie, et que une de ses puissances est une contraction de \( (\mF,\| . \|_{\infty})\). L'unique point fixe est une solution de notre problème et est dans \( C^0\), donc \( (t,\lambda)\mapsto y(t,\lambda)=y_{\lambda}(t)\) est de classe \( C^0\), c'est à dire continue.
+\end{proof}
+
+\begin{normaltext}
+    Ce théorème marque un peu la limite de ce que l'on peut faire avec la méthode des points fixes dans le cadre de Cauchy-Lipschitz : nous sommes limités à la continuité de la solution parce que les espaces \( C^p\) ne sont pas complets\footnote{Par exemple, le théorème de Stone-Weierstrass \ref{ThoGddfas} nous dit que la limite uniforme de polynômes (de classe \(  C^{\infty}\)) peut n'être que continue. Voir aussi le thème \ref{THMooOCXTooWenIJE}.}. Il n'y a donc pas d'espoir d'adapter la méthode pour prouver que si \( f\) est de classe \( C^p\) alors \( (t,\lambda)\mapsto y_{\lambda}(t)\) est de classe \( C^p\). On peut, à \( \lambda\) fixé prouver que \( t\mapsto y_{\lambda}(t)\) est de classe \( C^p\) (utiliser une récurrence), mais pas plus.
+
+    La régularité \( C^1\) de \( y\) par rapport à la condition initiale sera l'objet du théorème \ref{THOooSTHXooXqLBoT}. Ce résultat n'est vraiment pas facile et utilise des ingrédients bien autres qu'un point fixe. Ensuite la régularité \( C^p\) par rapport à la condition initiale et par rapport à un paramètre seront presque des cadeaux (proposition \ref{PROPooINLNooDVWaMn} et \ref{PROPooPYHWooIZhQST}).
+\end{normaltext}
+
+\begin{example}[\cite{ooSBHXooOMnaTC}]          \label{EXooJXIGooQtotMc}
+    Nous savons que le théorème de Picard permet de trouver le point fixe par itération de la contraction à partir d'un point quelconque. Tentons donc de résoudre
+    \begin{subequations}
+        \begin{numcases}{}
+            y'(t)=y(t)\\
+            y(0)=1
+        \end{numcases}
+    \end{subequations}
+    dont nous savons depuis l'enfance que la solution est l'exponentielle. Partons donc de la fonction constante \( y_0=1\), et appliquons la contraction \eqref{EQooJUTBooILBKoE} :
+    \begin{equation}
+        u_1=1+\int_0^1u_0(s)ds=1+t.
+    \end{equation}
+    Ensuite
+    \begin{equation}
+        u_2=1+\int_0^t(1+s)ds=1+t+\frac{ t^2 }{2}.
+    \end{equation}
+    Et on voit que les itérations suivantes vont donner l'exponentielle.
+
+    Nous sommes évidemment en droit de se dire que nous avons choisi un bon point de départ. Tentons le coup avec une fonction qui n'a rien à voir avec l'exponentielle : \( u_0(x)=\sin(x)\).
+
+    Le programme suivant permet de faire de belles investigations numériques en partant d'à peu près n'importe quelle fonction :
+
+\lstinputlisting{tex/sage/picard_exp.py}
+
+    Ce programme fait \( 30\) itérations depuis la fonction \( \sin(x)\) pour tenter d'approximer \( \exp(x)\). Pour donner une idée, après \( 7\) itérations nous avons la fonction suivante :
+    \begin{equation}
+        \frac{1}{ 60 }x^5+\frac{1}{ 24 }x^4+\frac{ 1 }{2}x^2+2x-\sin(x)+1.
+    \end{equation}
+    Nous voyons que les coefficients sont des factorielles, mais pas toujours celles correspondantes à la puissance, et qu'il manque certains termes par rapport au développement de l'exponentielle que nous connaissons. Bref, le polynôme qui se met en face de \( \sin(x)\) s'adapte tout seul pour compenser.
+
+    Et après \( 30\) itérations, ça donne quoi ? Voici un graphe de l'erreur entre \( u_{30}(x)\) et \( \exp(30)\) :
+    
+    
+\begin{center}
+   \input{auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks}
+\end{center}
+
+    Pour donner une idée, \( \exp(10)\simeq 22000\). Donc il y a une faute de \( 0.01\) sur \( 22000\). Pas mal.
+
+\end{example}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de Cauchy-Arzella}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{theorem}[Cauchy-Arzela\cite{ClemKetl}]   \label{ThoHNBooUipgPX}
+    Nous considérons le système d'équation différentielles
+    \begin{subequations}        \label{EqTXlJdH}
+        \begin{numcases}{}
+            y'=f(t,y)\\
+            y(t_0)=y_0.
+        \end{numcases}
+    \end{subequations}
+    avec \( f\colon U\to \eR^n\), continue où \( U\) est ouvert dans \( \eR\times \eR^n\). Alors il existe un voisinage fermé \( V\) de \( t_0\) sur lequel une solution \( C^1\) du problème \eqref{EqTXlJdH} existe.
+\end{theorem}
+\index{théorème!Cauchy-Arzela}
+
+\begin{proof}[Idée de la démonstration]
+    Nous considérons \( M=\| f \|_{\infty}\) et \( K\), l'ensemble des fonctions \( M\)-Lipschitz sur \( U\). Nous prouvons que \( (K,\| . \|_{\infty})\) est compact. Ensuite nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon K&\to K \\
+            \Phi(f)(t)&=x_0+\int_{t_0}^tf\big( u,f(u) \big)du. 
+        \end{aligned}
+    \end{equation}
+    Après avoir prouvé que \( \Phi\) était continue, nous concluons qu'elle a un point fixe par le théorème de Schauder \ref{ThovHJXIU}.
+\end{proof}
+
+\begin{remark}
+    Quelque remarques.
+    \begin{enumerate}
+        \item
+    Les théorème de Cauchy-Lipschitz et Cauchy-Arzella donnent des existences pour des équations différentielles du type \( y'=f(t,y)\). Et si nous avons une équation du second ordre ? Alors il y a la méthode de la réduction de l'ordre qui permet de transformer une équation différentielle d'ordre élevé en un système d'ordre \( 1\).
+\item
+    Ces théorèmes posent des \emph{conditions initiales} : la valeur de \( y\) est donnée en un point, et la méthode de la réduction de l'ordre permet de donner l'existence de solutions d'un problème d'ordre \( k\) en donnant les valeurs de \( y(0)\), \( y'(0)\), \ldots \( y^{(k-1)}(0)\). C'est à dire de la fonction et de ses dérivées en un point. Rien n'est dit sur l'existence de \emph{conditions aux bords}.
+    \end{enumerate}
+    Ces deux points sont illustrés dans les exemples \ref{EXooSHMMooHVfsMB} et \ref{EXooJNOMooYqUwTZ}.
+\end{remark}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+                    \section{Théorèmes d'inversion locale et de la fonction implicite}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Mise en situation}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Dans un certain nombre de situation, il n'est pas possible de trouver des solutions explicites aux équations qui apparaissent. Néanmoins, l'existence «théorique» d'une telle solution est souvent déjà suffisante. C'est l'objet du théorème de la fonction implicite.
+
+Prenons par exemple la fonction sur $\eR^2$ donnée par 
+\begin{equation}
+    F(x,y)=x^2+y^2-1.
+\end{equation}
+Nous pouvons bien entendu regarder l'ensemble des points donnés par $F(x,y)=0$. C'est le cercle dessiné à la figure \ref{LabelFigCercleImplicite}.
+\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite. Si on donne \( x\), nous ne pouvons pas savoir si nous parlons de \( P\) ou de \( P'\).}
+\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
+
+%\ref{LabelFigCercleImplicite}.
+%\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite.}
+%\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
+
+Nous ne pouvons pas donner le cercle sous la forme $y=y(x)$ à cause du $\pm$ qui arrive quand on prend la racine carrée. Mais si on se donne le point $P$, nous pouvons dire que \emph{autour de $P$}, le cercle est la fonction
+\begin{equation}
+    y(x)=\sqrt{1-x^2}.
+\end{equation}
+Tandis que autour du point $P'$, le cercle est la fonction
+\begin{equation}
+    y(x)=-\sqrt{1-x^2}.
+\end{equation}
+Autour de ces deux point, donc, le cercle est donné par une fonction. Il n'est par contre pas possible de donner le cercle autour du point $Q$ sous la forme d'une fonction.
+
+Ce que nous voulons faire, en général, est de voir si l'ensemble des points tels que
+\begin{equation}
+    F(x_1,\ldots,x_n,y)=0
+\end{equation}
+peut être donné par une fonction $y=y(x_1,\ldots,x_n)$. En d'autre termes, est-ce qu'il existe une fonction $y(x_1,\ldots,x_n)$ telle que
+\begin{equation}
+    F\big( x_1,\ldots,x_n,y(x_1,\ldots,x_n)\big)=0.
+\end{equation}
+
+Plus généralement, soit une fonction
+\begin{equation}
+    \begin{aligned}
+        F\colon D\subset \eR^n\times \eR^m&\to \eR^m \\
+        (x,y)&\mapsto \big( F_1(x,y),\ldots, F_m(x,y) \big) 
+    \end{aligned}
+\end{equation}
+avec $x = (x_1,\ldots, x_n)$ et $y = (y_1,\ldots,y_m)$. Pour chaque $x$ fixé, on s'intéresse aux solutions du système de $m$ équations $F(x,y) = 0$ pour les inconnues $y$ ; en particulier, on voudrait pouvoir écrire $y = \varphi(x)$ vérifiant $F(x,\varphi(x)) = 0$.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème d'inversion locale}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma} \label{LemGZoqknC}
+    Soit \( E\) un espace de Banach (métrique complet) et \( \mO\) un ouvert de \( E\). Nous considérons une \( \lambda\)-contraction \( \varphi\colon \mO\to E\). Alors l'application
+    \begin{equation}
+    f\colon x\mapsto x+\varphi(x)
+    \end{equation}
+    est un homéomorphisme entre \( \mO\) et un ouvert de \( E\). De plus \( f^{-1}\) est Lipschitz de constante plus petite ou égale à \( (1-\lambda)^{-1}\).
+\end{lemma}
+Cette proposition utilise le théorème de point fixe de Picard \ref{ThoEPVkCL},
+et sera utilisée pour démontrer le théorème d'inversion locale \ref{ThoXWpzqCn}.
+% note que garder deux lignes ici est important pour vérifier les références vers le futur : la seconde ligne peut être ignorée, pas la seconde.
+
+\begin{proof}
+        Soient \( x_1,x_2\in\mO\). Nous posons \( y_1=f(x_1)\) et \( y_2=f(x_2)\). En vertu de l'inégalité de la proposition \ref{PropNmNNm} nous avons
+        \begin{subequations}    \label{subEqEBJsBfz}
+            \begin{align}
+            \big\| f(x_2)-f(x_1) \big\|&=\big\| x_2+\varphi(x_2)-x_1-\varphi(x_1) \big\|\\
+        &\geq \Big|        \| x_2-x_1 \|-\big\| \varphi(x_2)-\varphi(x_1) \big\|  \Big|\\
+    &\geq   (1-\lambda)\| x_2-x_1 \|.
+            \end{align}
+        \end{subequations}
+        À la dernière ligne les valeurs absolues sont enlevées parce que nous savons que ce qui est à l'intérieur est positif. Cela nous dit d'abord que \( f\) est injective parce que \( f(x_2)=f(x_1)\) implique \( x_2=x_1\). Donc \( f\) est inversible sur son image. Nous posons \( A=f(\mO)\) et nous devons prouver que que \( f^{-1}\colon A\to \mO\) est continue, Lipschitz de constante majorée par \( (1-\lambda)^{-1}\) et que \( A\) est ouvert.
+
+    Les inéquations \eqref{subEqEBJsBfz} nous disent que
+    \begin{equation}
+    \big\| f^{-1}(y_1)-f^{-1}(y_2) \big\|\leq \frac{ \| y_1-y_2 \| }{ 1-\lambda },
+    \end{equation}
+    c'est à dire que
+    \begin{equation}
+        f^{-1}\big( B(y,r) \big)\subset B\big( f^{-1}(y),\frac{ r }{ 1-\lambda } \big),
+    \end{equation}
+    ce qui signifie que \( f^{-1}\) est Lipschitz de constante souhaitée et donc continue.
+
+    Il reste à prouver que \( f(\mO)\) est ouvert. Pour cela nous prenons \( y_0=f(x_0)\) dans \( f(\mO)\) est nous prouvons qu'il existe \( \epsilon\) tel que \( B(y_0,\epsilon)\) soit dans \( f(\mO)\). Il faut donc que pour tout \( y\in B(y_0,\epsilon)\), l'équation \( f(x)=y\) ait une solution. Nous considérons l'application
+    \begin{equation}
+        L_y\colon x\mapsto y-\varphi(x).
+    \end{equation}
+    Ce que nous cherchons est un point fixe de \( L_y\) parce que si \( L_y(x)=x\) alors \( y=x+\varphi(x)=f(x)\). Vu que
+    \begin{equation}
+        \big\| L_y(x)-L_y(x') \big\|=\big\| \varphi(x)-\varphi(x') \big\|\leq\lambda\| x-x' \|,
+    \end{equation}
+    l'application \( L_y\) est une contraction de constante \( \lambda\). Par ailleurs \( x_0\) est un point fixe de \( L_{y_0}\), donc en vertu de la caractérisation \eqref{EqDZvtUbn} des fonctions Lipschitziennes, 
+    \begin{equation}
+        L_{y_0}\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B\big( L_{y_0}(x_0),\lambda\delta \big) }=\overline{ B(x_0,\lambda\delta) }.
+    \end{equation}
+    Vu que pour tout \( y\) et \( x\) nous avons \( L_y(x)=L_{y_0}(x)+y-y_0\),
+    \begin{equation}
+    L_y\big( \overline{ B(x_0,\delta) } \big)=L_{y_0}\big( \overline{ B(x_0,\delta) } \big)+(y-y_0)\subset \overline{ B(x_0,\lambda\delta) }+(y-y_0)\subset \overline{ B(x_0),\lambda\delta+\| y-y_0 \| }.
+    \end{equation}
+    Si \( \epsilon<(1-\lambda)\delta\) alors \( \lambda\delta+\| y-y_0 \|<\delta\). Un tel choix de \( \epsilon>0\) est possible parce que \( \lambda<1\). Pour une telle valeur de \( \epsilon\) nous avons
+    \begin{equation}
+        L_y\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B(x_0,\delta) }.
+    \end{equation}
+    Par conséquent \( L_y\) est une contraction sur l'espace métrique complet \( \overline{ B(x_0,\delta) }\), ce qui signifie que \( L_y\) y possède un point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+\end{proof}
+
+Le théorème d'inversion locale s'énonce de la façon suivante dans \( \eR^n\) :
+\begin{theorem}[Inversion locale dans \( \eR^n\)]    \label{THOooQGGWooPBRNEX}      % Ne pas mettre de label ici parce qu'il faut référencer l'autre, celui dans Banach.
+    Soit \( f\in C^k(\eR^n,\eR^n)\) et \( x_0\in \eR^n\). Si \( df_{x_0}\) est inversible, alors il existe un voisinage ouvert \( U\) de \( x_0\) et \( V\) de \( f(x_0)\) tels que \( f\colon U\to V\) soit un \( C^k\)-difféomorphisme. (c'est à dire que \( f^{-1}\) est également de classe \( C^k\))
+\end{theorem}
+
+Nous allons le démontrer dans le cas un peu plus général (mais pas plus cher\footnote{Sauf la justification de la régularité de l'application \( A\mapsto A^{-1}\)}) des espaces de Banach en tant que conséquence du théorème de point fixe de Picard \ref{ThoEPVkCL}.
+
+\begin{theorem}[Inversion locale dans un espace de Banach\cite{OWTzoEK}] \label{ThoXWpzqCn}
+    Soit une fonction \( f\in C^p(E,F)\) avec \( p\geq 1\) entre deux espaces de Banach. Soit \( x_0\in E\) tel que \( df_{x_0}\) soit une bijection bicontinue\footnote{En dimension finie, une application linéaire est toujours continue et d'inverse continu.}. Alors il existe un voisinage ouvert \( V\) de \( x_0\) et \( W\) de \( f(x_0)\) tels que
+    \begin{enumerate}
+        \item
+        \( f\colon V\to W\) soit une bijection,
+    \item
+        \( f^{-1}\colon W\to V\) soit de classe \( C^p\).
+    \end{enumerate}
+\end{theorem}
+\index{application!différentiable}
+\index{théorème!inversion locale}
+
+\begin{proof}
+    Nous commençons par simplifier un peu le problème. Pour cela, nous considérons la translation \( T\colon x\mapsto x+x_0 \) et l'application linéaire
+    \begin{equation}
+        \begin{aligned}
+            L\colon \eR^n&\to \eR^n \\
+            x&\mapsto (df_{x_0})^{-1}x
+        \end{aligned}
+    \end{equation}
+    qui sont tout deux des difféomorphismes (\( L\) en est un par hypothèse d'inversibilité). Quitte à travailler avec la fonction \( k=L\circ f\circ T\), nous pouvons supposer que \( x_0=0\) et que \( df_{x_0}=\mtu\). Pour comprendre cela il faut utiliser deux fois la formule de différentielle de fonction composée de la proposition \ref{EqDiffCompose} :
+    \begin{equation}
+        dk_0(u)=dL_{(f\circ T)(0)}\Big( df_{T(0)}dT_0(u) \Big).
+    \end{equation}
+    Vu que \( L\) est linéaire, sa différentielle est elle-même, c'est à dire \( dL_{(f\circ T)(0)}=(df_{x_0})^{-1}\), et par ailleurs \( dT_0=\mtu\), donc
+    \begin{equation}
+        dk_0(u)=(df_{x_0})^{-1}\Big( df_{x_0}(u) \Big)=u,
+    \end{equation}
+    ce qui signifie bien que \( dk_0=\mtu\). Pour tout cela nous avons utilisé en plein le fait que \( df_{x_0}\) était inversible.
+
+Nous posons \( g=f-\mtu\), c'est à dire \( g(x)=f(x)-x\), qui a la propriété \( dg_0=0\). Étant donné que \( g\) est de classe \( C^1\), l'application\footnote{Ici \( \GL(F)\) est l'ensemble des applications linéaires, inversibles et continues de \( F\) dans lui-même. Ce ne sont pas spécialement des matrices parce que nous n'avons pas d'hypothèses sur la dimension de \( F\), finie ou non.}
+    \begin{equation}
+        \begin{aligned}
+            dg\colon E&\to \GL(F) \\
+            x&\mapsto dg_x 
+        \end{aligned}
+    \end{equation}
+    est continue. En conséquence de quoi nous avons un voisinage \( U'\) de \( 0 \) pour lequel
+    \begin{equation}    \label{EqSGTOfvx}
+        \sup_{x\in U'}\| dg_x \|<\frac{ 1 }{2}.
+    \end{equation}
+    Maintenant le théorème des accroissements finis \ref{ThoNAKKght} (\ref{val_medio_2} pour la dimension finie) nous indique que pour tout \( x,x'\in U'\) nous avons\footnote{Ici nous supposons avoir choisi \( U'\) convexe afin que tous les \( a\in \mathopen[ x , x' \mathclose]\) soient bien dans \( U'\) et donc soumis à l'inéquation \eqref{EqSGTOfvx}, ce qui est toujours possible, il suffit de prendre une boule.}
+    \begin{equation}
+        \| g(x')-g(x) \|\leq \sup_{a\in\mathopen[ x , x' \mathclose]}\| dg_a \| \cdot \| x-x' \|\leq \frac{ 1 }{2}\| x-x' \|,
+    \end{equation}
+    ce qui prouve que \( g\) est une contraction au moins sur l'ouvert \( U'\). Nous allons aussi donner une idée de la façon dont \( f\) fonctionne : si \( x_1,x_2\in U'\) alors
+    \begin{subequations}
+        \begin{align}
+            \| x_1-x_2 \|&=\| g(x_1)-f(x_1)-g(x_2)+f(x_2) \| \\
+            &\leq \| g(x_1)-g(x_2) \|+\| f(x_1)-f(x_2) \|\\
+            &\leq \frac{ 1 }{2}\| x_1-x_2 \|+\| f(x_1)-f(x_2) \|,
+        \end{align}
+    \end{subequations}
+    ce qui montre que
+    \begin{equation}
+        \| x_1-x_2 \|\leq 2\| f(x_1)-f(x_2) \|.
+    \end{equation}
+    Maintenant que nous savons que \( g\) est contractante de constante \( \frac{ 1 }{2}\) et que \( f=g+\mtu\) nous pouvons utiliser la proposition \ref{LemGZoqknC} pour conclure que \( f\) est un homéomorphisme sur un ouvert \( U\) (partie de \( U'\)) de \( E\) et \( f^{-1}\) a une constante de Lipschitz plus petite ou égale à \( (1-\frac{ 1 }{2})^{-1}=2\).
+
+    Nous allons maintenant prouver que \( f^{-1}\) est différentiable et que sa différentielle est donnée par \( (df^{-1})_{f(x)}=(df_x)^{-1}\).
+
+    Soient \( a,b\in U\) et \( u=b-a\). Étant donné que \( f\) est différentiable en \( a\), il existe une fonction \( \alpha\in o(\| u \|)\) telle que
+    \begin{equation}
+        f(b)-f(a)-df_a(u)=\alpha(u).
+    \end{equation}
+    En notant \( y_a=f(a)\) et \( y_b=f(b)\) et en appliquant \( (df_a)^{-1}\) à cette dernière équation,
+    \begin{equation}
+        (df_a)^{-1}(y_b-y_a)-u=(df_a)^{-1} \big( \alpha(u) \big).
+    \end{equation}
+    Vu que \( df_a\) est bornée (et son inverse aussi), le membre de droite est encore une fonction \( \beta\) ayant la propriété \( \lim_{u\to 0}\beta(u)/\| u \|=0\); en réordonnant les termes,
+    \begin{equation}
+        b-a=(df_a)^{-1}(y_b-y_a)+\beta(u)
+    \end{equation}
+    et donc
+    \begin{equation}
+        f^{-1}(y_b)-f^{-1}(y_a)-(df_a)^{-1}(y_b-y_a)=\beta(u),
+    \end{equation}
+    ce qui prouve que \( f^{-1}\) est différentiable et que \( (df^{-1})_{y_a}=(df_a)^{-1}\).
+
+    La différentielle \( df^{-1}\) est donc obtenue par la chaine
+    \begin{equation}
+    \xymatrix{%
+        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}     &   U'\ar[r]^-{df}&\GL(F)\ar[r]^-{\Inv}&\GL(F)
+       }
+    \end{equation}
+    où l'application \( \Inv\colon \GL(F)\to \GL(F)\) est l'application \( X\mapsto X^{-1}\) qui est de classe \(  C^{\infty}\) par le théorème \ref{ThoCINVBTJ}. D'autre part, par hypothèse \( df\) est une application de classe \( C^{k-1}\) et donc au minimum \( C^0\) parce que \( k\geq 1\). Enfin, l'application \( f^{-1}\colon f(U)\to U\) est continue (parce que la proposition \ref{LemGZoqknC} précise que \( f\) est un homéomorphisme). Donc toute la chaine est continue et \( df^{-1}\) est continue. Cela entraine immédiatement que \( f^{-1}\) est \( C^1\) et donc que toute la chaine est \( C^1\).
+
+    Par récurrence nous obtenons la chaine
+    \begin{equation}
+    \xymatrix{%
+        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}_-{C^{k-1}}     &   U'\ar[r]^-{df}_-{C^{k-1}}&\GL(F)\ar[r]^-{\Inv}_-{ C^{\infty}}&\GL(F)
+       }
+    \end{equation}
+    qui prouve que \( df^{-1}\) est \( C^{k-1} \) et donc que \( f^{-1}\) est \( C^k\). La récurrence s'arrête ici parce que \( df\) n'est pas mieux que \( C^{k-1}\).
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de la fonction implicite}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous énonçons et le démontrons le théorème de la fonction implicite dans le cas d'espaces de Banach.
+\begin{theorem}[Théorème de la fonction implicite dans Banach\cite{SNPdukn}] \label{ThoAcaWho}
+    Soient \( E\), \( F\) et \( G\) des espaces de Banach et des ouverts \( U\subset E\), \( V\subset F\). Nous considérons une fonction \( f\colon U\times V\to G\) de classe \( C^r\) telle que\footnote{La notation \( d_y\) est la différentielle partielle de la définition \ref{VJM_CtSKT}.}
+    \begin{equation}
+        d_yf_{(x_0,y_0)}\colon F\to G
+    \end{equation}
+    soit un isomorphisme pour un certain \( (x_0,y_0)\in U\times V\).
+
+    Alors nous avons des voisinages \( U_0\) de \( x_0\) dans \( E\) et \( W_0\) de \( f(x_0,y_0)\) dans \( G\) et une fonction de classe \( C^r\) 
+    \begin{equation}
+        g\colon U_0\times W_0\to V
+    \end{equation}
+    telle que 
+    \begin{equation}
+        f\big( x,g(x,w) \big)=w
+    \end{equation}
+    pour tout \( (x,w)\in U_0\times W_0\).
+    
+    Cette fonction \( g\) est unique au sens suivant : il existe un voisinage \( V_0 \) de \( y_0\) tel que si \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) satisfont à \( f(x,y)=w\) alors \( y=g(x,w)\). Autrement dit, la fonction \( g\colon U_0\times W_0\to V_0\) est unique.
+\end{theorem}
+\index{théorème!fonction implicite dans Banach}
+
+\begin{proof}
+    Nous commençons par considérer la fonction
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon U\times V&\to E\times G \\
+            (x,y)&\mapsto \big( x,f(x,y) \big)
+        \end{aligned}
+    \end{equation}
+    et sa différentielle 
+    \begin{subequations}
+        \begin{align}
+            d\Phi_{(x_0,y_0)}(u,v)&=\Dsdd{ \big( x_0+tu,f(x_0+tu,y_0+tv) \big) }{t}{0}\\
+            &=\left( \Dsdd{ x_0+tu }{t}{0},\Dsdd{ f(x_0+tu,y_0+tv) }{t}{0} \right)\\
+            &=\left( u,df_{(x_0,y_0)}(u,v) \right).
+        \end{align}
+    \end{subequations}
+    Nous utilisons alors la proposition \ref{PropLDN_nHWDF} pour conclure que
+    \begin{equation}
+        d\Phi_{(x_0,y_0)}(u,v)=\big( u,(d_1f)_{(x_0,y_0)}(u)+(d_2f)_{(x_0,y_0)}(v) \big),
+    \end{equation}
+    mais comme par hypothèse \( (d_2f)_{(x_0,y_0)}\colon F\to G\) est un isomorphisme, l'application \( d\Phi_{(x_0,y_0)}\colon E\times F\to E\times G\) est également un isomorphisme. Par conséquent le théorème d'inversion locale \ref{ThoXWpzqCn} nous indique qu'il existe un voisinage \( \mO\) de \( (x_0,y_0)\) et \( \mP\) de \( \Phi(x_0,y_0)\) tels que \( \Phi\colon \mO\to \mP\) soit une bijection et \( \Phi^{-1}\colon \mP\to \mO\) soit de classe \( C^r\). Vu que \( \mP\) est un voisinage de
+    \begin{equation}
+        \Phi(x_0,y_0)=\big( x_0,f(x_0,y_0) \big),
+    \end{equation}
+    nous pouvons par \ref{PropDXR_KbaLC} le choisir un peu plus petit de telle sorte à avoir \( \mP=U_0\times W_0\) où \( U_0\) est un voisinage de \( x_0\) et \( W_0\) un voisinage de \( f(x_0,y_0)\). Dans ce cas nous devons obligatoirement aussi restreindre \( \mO\) à \( U_0\times V_0\) pour un certain voisinage \( V_0\) de \( y_0\). L'application \( \Phi^{-1}\) a obligatoirement la forme
+    \begin{equation}    \label{EqMHT_QrHRn}
+        \begin{aligned}
+            \Phi^{-1}\colon U_0\times W_0&\to U_0\times V_0 \\
+            (x,w)&\mapsto \big( x,g(x,w) \big) 
+        \end{aligned}
+    \end{equation}
+    pour une certaine fonction \( g\colon U_0\times W_0\to V\). Cette fonction \( g\) est la fonction cherchée parce qu'en appliquant \( \Phi\) à \eqref{EqMHT_QrHRn}, 
+    \begin{equation}
+        (x,w)=\Phi\big( x,g(x,w) \big)=\Big( x,f\big( x,g(x,w) \big) \Big),
+    \end{equation}
+    qui nous dit que pour tout \( x\in U_0\) et tout \( w\in W_0\) nous avons
+    \begin{equation}
+        f\big( x,g(x,w) \big)=w.
+    \end{equation}
+
+    Si vous avez bien suivi le sens de l'équation \eqref{EqMHT_QrHRn} alors vous avez compris l'unicité. Sinon, considérez \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) tels que \( f(x,y)=w\). Alors \( \big( x,f(x,y) \big)=(x,w)\) et 
+    \begin{equation}
+        \Phi(x,y)=(x,w).
+    \end{equation}
+    Mais vu que \( \Phi\colon U_0\times V_0\to U_0\times W_0\) est une bijection, cette relation définit de façon univoque l'élément \( (x,y)\) de \( U_0\times V_0\), qui ne sera autre que \( g(x,w)\).
+\end{proof}
+
+Le théorème de la fonction implicite s'énonce de la façon suivante pour des espaces de dimension finie.
+% Attention : avant de citer ce théorème, voir s'il est suffisant. Ici \varphi a une variable; dans l'autre énoncé il en a deux.
+\begin{theorem}[Théorème de la fonction implicite en dimension finie]   \label{ThoRYN_jvZrZ}
+    Soit une fonction \( F\colon \eR^n\times \eR^m\to \eR^m\) de classe \( C^k\) et \( (\alpha,\beta)\in \eR^n\times \eR^m\) tels que
+    \begin{enumerate}
+        \item
+            \( F(\alpha,\beta)=0\),
+        \item
+            \( \frac{ \partial (F_1,\ldots, F_m) }{ \partial (y_1,\ldots, y_m) }\neq 0\), c'est à dire que \( (d_yF)_{(\alpha,\beta)} \) est inversible.
+    \end{enumerate}
+    Alors il existe un voisinage ouvert \( V\) de \( \alpha\) dans \( \eR^n\), un voisinage ouvert \( W\) de \( \beta\) dans \( \eR^m\) et une application \( \varphi\colon V\to W\) de classe \( C^k\)  telle que pour tout \( x\in V\) on ait
+    \begin{equation}
+        F\big( x,\varphi(x) \big)=0.
+    \end{equation}
+    De plus si \( (x,y)\in V\times W\) satisfait à \( F(x,y)=0\), alors \( y=\varphi(x)\).
+\end{theorem}
+\index{théorème!fonction implicite dans \( \eR^n\)}
+
+\begin{remark}\label{RemPYA_pkTEx}
+    Notons que cet énoncé est tourné un peu différemment en ce qui concerne le nombre de variables dont dépend la fonction implicite : comparez
+    \begin{subequations}
+        \begin{align}
+            f\big( x,g(x,w) \big)=w\\
+            F\big( x,\varphi(x) \big)=0.
+        \end{align}
+    \end{subequations}
+    Le deuxième est un cas particulier du premier en posant 
+    \begin{equation}
+        F(x,y)=f(x,y)-f(x_0,y_0)
+    \end{equation}
+    et donc en considérant \( w\) comme valant la constante \( f(x_0,y_0)\); dans ce cas la fonction \( g\) ne dépend plus que de la variable \( x\).
+
+\end{remark}
+
+\begin{example}
+    La remarque \ref{RemPYA_pkTEx} signifie entre autres que le théorème \ref{ThoAcaWho} est plus fort que \ref{ThoRYN_jvZrZ} parce que le premier permet de choisir la valeur d'arrivée. Parlons de l'exemple classique du cercle et de la fonction \( f(x,y)=x^2+y^2\). Nous savons que
+    \begin{equation}
+        f(\alpha,\beta)=1.
+    \end{equation}
+    Alors le théorème \ref{ThoAcaWho} nous donne une fonction \( g\) telle que
+    \begin{equation}
+        f(x,g(x,r))=r
+    \end{equation}
+    tant que \( x\) est proche de \( \alpha\), que \( r\) est proche de \( 1\) et que \( g\) donne des valeurs proches de \( \beta\).
+
+    L'énoncé \ref{ThoRYN_jvZrZ} nous oblige à travailler avec la fonction \( F(x,y)=x^2+y^2-1\), de telle sorte que
+    \begin{equation}
+        F(\alpha,\beta)=0,
+    \end{equation}
+    et que nous ayons une fonction \( \varphi\) telle que
+    \begin{equation}
+        F(x,\varphi(x))=0.
+    \end{equation}
+    La fonction \( \varphi\) ne permet donc que de trouver des points sur le cercle de rayon \( 1\).
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Exemple}
+%---------------------------------------------------------------------------------------------------------------------------
+    
+Le théorème de la fonction implicite a pour objet de donner l'existence de la fonction $\varphi$. Maintenant nous pouvons dire beaucoup de choses sur les dérivées de $\varphi$ en considérant la fonction
+\begin{equation}
+    x\mapsto F\big( x,\varphi(x) \big).
+\end{equation}
+Par définition de $\varphi$, cette fonction est toujours nulle. En particulier, nous pouvons dériver l'équation
+\begin{equation}
+    F\big( x,\varphi(x) \big)=0,
+\end{equation}
+et nous trouvons plein de choses.
+
+
+Prenons par exemple la fonction
+\begin{equation}
+    F\big( (x,y),z \big)=ze^z-x-y,
+\end{equation}
+et demandons nous ce que nous pouvons dire sur la fonction $z(x,y)$ telle que
+\begin{equation}
+    F\big( x,y,z(x,y) \big)=0,
+\end{equation}
+c'est à dire telle que
+\begin{equation}        \label{EqDefZImplExemple}
+    z(x,y) e^{z(x,y)}-x-y=0.
+\end{equation}
+pour tout $x$ et $y\in\eR$. Nous pouvons facilement trouver $z(0,0)$ parce que
+\begin{equation}
+    z(0,0) e^{z(0,0)}=0,
+\end{equation}
+donc $z(0,0)=0$.
+
+Nous pouvons dire des choses sur les dérivées de $z(x,y)$. Voyons par exemple $(\partial_xz)(x,y)$. Pour trouver cette dérivée, nous dérivons la relation \eqref{EqDefZImplExemple} par rapport à $x$. Ce que nous trouvons est
+\begin{equation}
+    (\partial_xz)e^z+ze^z(\partial_xz)-1=0.
+\end{equation}
+Cette équation peut être résolue par rapport à $\partial_xz$~:
+\begin{equation}
+    \frac{ \partial z }{ \partial x }(x,y)=\frac{1}{ e^z(1+z) }.
+\end{equation}
+Remarquez que cette équation ne donne pas tout à fait la dérivée de $z$ en fonction de $x$ et $y$, parce que $z$ apparaît dans l'expression, alors que $z$ est justement la fonction inconnue. En général, c'est la vie, nous ne pouvons pas faire mieux.
+
+Dans certains cas, on peut aller plus loin. Par exemple, nous pouvons calculer cette dérivée au point $(x,y)=(0,0)$ parce que $z(0,0)$ est connu :
+\begin{equation}
+    \frac{ \partial z }{ \partial x }(0,0)=1.
+\end{equation}
+Cela est pratique pour calculer, par exemple, le développement en Taylor de $z$ autour de $(0,0)$.
+
+\begin{example}
+    Est-ce que l'équation \( e^{y}+xy=0\) définit au moins localement une fonction \( y(x)\) ? Nous considérons la fonction
+    \begin{equation}
+        f(x,y)=\begin{pmatrix}
+            x    \\ 
+            e^{y}+xy    
+        \end{pmatrix}
+    \end{equation}
+    La différentielle de cette application est
+    \begin{equation}
+            df_{(0,0)}(u)=\frac{ d }{ dt }\Big[ f(tu_1,tu_2) \Big]_{t=0}
+            =\frac{ d }{ dt }\begin{pmatrix}
+                tu_1    \\ 
+                e^{tu_2}+t^2u_1u_2    
+            \end{pmatrix}_{t=0}
+            =\begin{pmatrix}
+                u_1    \\ 
+                u_2    
+            \end{pmatrix}.
+    \end{equation}
+    L'application \( f\) définit donc un difféomorphisme local autour des points \( (x_0,y_0)\) et \( f(x_0,y_0)\). Soit \( (u,0)\) un point dans le voisinage de \( f(x_0,y_0)\). Alors il existe un unique \( (x,y)\) tel que
+    \begin{equation}
+        f(x,y)=\begin{pmatrix}
+               x \\ 
+            e^y+xy    
+        \end{pmatrix}=
+        \begin{pmatrix}
+            u    \\ 
+                0
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons automatiquement \( x=u\) et \( e^y+xy=0\). Notons toutefois que pour que ce procédé donne effectivement une fonction implicite \( y(x)\) nous devons avoir des points de la forme \( (u,0)\) dans le voisinage de \( f(x_0,y_0)\).
+\end{example}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Recherche d'extrema}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema à une variable}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
+\end{definition}
+
+La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
+\begin{proposition}     \label{PROPooNVKXooXtKkuz}
+Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
+\end{proposition}
+
+La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
+
+Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
+\begin{proposition}
+Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
+\begin{equation}
+    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
+\end{equation}
+et que
+\begin{equation}
+    f^{(k)}(a)\neq 0.
+\end{equation}
+Dans ce cas,
+\begin{enumerate}
+
+\item
+Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
+\item
+Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
+
+\end{enumerate}
+\end{proposition}
+
+Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
+
+Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
+
+Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
+
+\begin{example}
+	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
+\end{example}
+ 
+Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
+\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
+\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
+
+Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
+
+Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
+
+\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
+\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
+ 
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema libre}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooYJLZooLkEAYf}
+Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
+\end{definition}
+
+Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
+
+Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
+\begin{equation}
+    d^2f(a)=\begin{pmatrix}
+    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
+    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
+\end{pmatrix}.
+\end{equation}
+Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
+
+\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
+\end{proposition}
+
+\begin{proof}
+    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
+    \begin{equation}
+        \begin{aligned}
+            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
+            t&\mapsto f(a+tu) 
+        \end{aligned}
+    \end{equation}
+    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
+    \begin{equation}
+        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
+    \end{equation}
+\end{proof}
+
+\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
+    \begin{enumerate}
+        \item   \label{ITEMooCVFVooWltGqI}
+            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
+        \item\label{ItemPropoExtreRn}
+            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
+    \end{enumerate}
+\end{proposition}
+\index{extrema}
+
+\begin{proof}
+    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
+    \begin{equation}
+        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
+    \end{equation}
+    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
+    \end{equation}
+    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
+    \end{equation}
+    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
+    \begin{equation}
+        (d^2f)_a(e_h,e_h)\geq 0.
+    \end{equation}
+    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
+\end{proof}
+
+La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
+
+Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
+
+La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
+\begin{enumerate}
+    \item
+        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
+    \item
+        écrire $d^2f(a)$ pour chacun des candidats
+    \item
+        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
+    \item
+        conclure.
+\end{enumerate}
+
+Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
+
+\begin{example}
+    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
+    \begin{equation}
+        df=\big(4x^3-4y;4y^3-4x),
+    \end{equation}
+    et la matrice des dérivées secondes est
+    \begin{equation}
+        M=d^2f(x,y)=\begin{pmatrix}
+            12x^2    &   -4    \\ 
+            -4    &   12y^2    
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
+
+    Pour le point \( (0,0)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            0    &   -4    \\ 
+            -4    &   0    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
+
+    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            12    &   -4    \\ 
+            -4    &   12    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Un peu de recettes de cuisine}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{enumerate}
+\item Rechercher les points critiques, càd les $(x,y)$ tels que
+\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
+En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
+\item Déterminer la nature des points critiques: «test» des dérivées secondes:
+\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
+\begin{enumerate}
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
+\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
+\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
+\end{enumerate}
+\end{enumerate}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Extrema liés}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
+
+Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
+
+\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
+    Soit \( A\), un ouvert de \( \eR^n\) et
+    \begin{enumerate}
+        \item
+            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
+        \item 
+            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
+        \item
+            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
+        \item
             un extrema local $a\in M$ de $f$ relativement à $M$.
     \end{enumerate}
     Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
@@ -1741,258 +3094,15 @@ \subsection{Extrema liés}
 
 En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
 
-L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
-
-\begin{lemma}       \label{LemmeMinSCimpliqueS}
-    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
-\end{lemma}
-
-\begin{proof}
-    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Algorithme du gradient à pas optimal}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
-
-Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
-\begin{itemize}
-    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
-    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
-\end{itemize}
-
-\begin{definition}  \label{DefQXPooYSygGP}
-    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
-    \begin{equation}
-        \{ x\in P\tq f(x)\leq M \}
-    \end{equation}
-    est borné.
-\end{definition}
-En langage imagé la coercivité de \( f\) s'exprime par la limite
-\begin{equation}
-    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
-\end{equation}
-
-
-Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
-\begin{proposition}     \label{PROPooYRLDooTwzfWU}
-    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Alors :
-    \begin{enumerate}
-        \item
-            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
-        \item
-            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
-        \item
-            Ils sont égaux : \( \bar x=x^*\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-
-    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
-
-    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
-
-    \begin{subproof}
-    \item[\( f\) est strictement convexe]
-        
-        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
-    \end{equation}
-    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
-
-\item[\( f\) est coercive]
-    Montrons à présent que \( f\) est coercive. Nous avons :
-    \begin{subequations}
-        \begin{align}
-            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
-            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
-            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
-        \end{align}
-    \end{subequations}
-    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
-    \end{subproof}
-
-    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
-    \begin{equation}
-        \{ x\in \eR^n\tq f(x)\leq M \}
-    \end{equation}
-    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
-
-    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
-    \begin{equation}
-        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
-    \end{equation}
-    ce qui contredit la minimalité de \( f(\bar x_1)\).
-
-    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
-    \begin{equation}
-        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
-    \end{equation}
-    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
-\end{proof}
-
-\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
-    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
-    \begin{equation}
-        x_{k+1}=x_k+t_kd_k
-    \end{equation}
-    où
-    \begin{itemize}
-        \item 
-    \( d_k=-(\nabla f)(x_k)\) 
-\item
-    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
-    \end{itemize}
-
-    Alors pour tout \( k\geq 0\) nous avons
-    \begin{equation}
-        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
-    \end{equation}
-    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
-\end{proposition}
-
-\begin{proof}
-    Décomposition en plusieurs points.
-    \begin{subproof}
-    \item[Existence de \( \bar x\)]
-        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
-    \item[Si \( (\nabla f)(x_k)=0\)]
-    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
-    \end{equation}
-    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
-    \begin{equation}
-        (\nabla f)(a)=Aa+b.
-    \end{equation}
-    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
-
-    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
-        \item[\( t_k\) est bien défini]
-
-            Pour \( t\in \eR\) nous avons
-            \begin{equation}    \label{EqKEHooYaazQi}
-                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
-            \end{equation}
-            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
-            \begin{equation}    \label{EqVWJooWmDSER}
-                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
-            \end{equation}
-            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
-
-        \item[La valeur de \( d_{k+1}\)]
-
-            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
-            \begin{equation}
-                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
-            \end{equation}
-            parce que \( -Ax_k-b=d_k\).
-            
-            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
-            \begin{equation}
-                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
-            \end{equation}
-            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
-        
-        \item[Calcul de \( f(x_{k+1})\)] 
-
-            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
-            \begin{equation}
-                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
-            \end{equation}
-            
-        \item[Encore du calcul \ldots]
-
-            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
-            \begin{equation}
-                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
-            \end{equation}
-            Ayant cela en tête nous pouvons calculer :
-            \begin{subequations}
-                \begin{align}
-                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
-                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
-                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
-                    &=2\big( f(x_k)-\bar f \big)
-                \end{align}
-            \end{subequations}
-            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
-
-        \item[Erreur sur la valeur du minimum]
-
-            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
-            \begin{subequations}
-                \begin{align}
-                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
-                \end{align}
-            \end{subequations}
-            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
-            \begin{equation}
-                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
-            \end{equation}
-            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
-            \begin{equation}
-                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
-            \end{equation}
-            Par récurrence nous avons alors
-            \begin{equation}    \label{eqANKooNPfCFj}
-                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
-            \end{equation}
-            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
-
-        \item[Erreur sur la position du minimum]
-
-            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
-            \begin{subequations}
-                \begin{align}
-                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
-                \end{align}
-            \end{subequations}
-            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
-
-            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
-            \begin{equation}    \label{EqVMRooUMXjig}
-                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
-            \end{equation}
-            
-        \item[Conclusion]
-
-            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
-            \begin{equation}
-                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
-            \end{equation}
-            c'est à dire
-            \begin{equation}
-                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
-            \end{equation}
-    \end{subproof}
-\end{proof}
-
-Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
+L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
+
+\begin{lemma}       \label{LemmeMinSCimpliqueS}
+    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
+\end{lemma}
+
+\begin{proof}
+    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
+\end{proof}
 
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Fonctions convexes}
@@ -2658,7 +3768,10 @@ \subsection{En dimension supérieure}
 \begin{corollary}       \label{CORooMBQMooWBAIIH}
     Avec la hessienne\ldots en cours d'écriture.
 \end{corollary}
-<++>
+
+\begin{proof}
+    Cela va utiliser la proposition \ref{PropoExtreRn}.
+\end{proof}
 
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Quelque inégalités}
@@ -2775,3 +3888,247 @@ \subsubsection{Inégalité de Kantorovitch}
             \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \leq \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
     \end{equation}
 \end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Algorithme du gradient à pas optimal}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
+
+Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
+\begin{itemize}
+    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
+    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
+\end{itemize}
+
+\begin{definition}  \label{DefQXPooYSygGP}
+    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
+    \begin{equation}
+        \{ x\in P\tq f(x)\leq M \}
+    \end{equation}
+    est borné.
+\end{definition}
+En langage imagé la coercivité de \( f\) s'exprime par la limite
+\begin{equation}
+    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
+\end{equation}
+
+
+Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
+\begin{proposition}     \label{PROPooYRLDooTwzfWU}
+    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Alors :
+    \begin{enumerate}
+        \item
+            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
+        \item
+            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
+        \item
+            Ils sont égaux : \( \bar x=x^*\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+
+    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
+
+    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
+
+    \begin{subproof}
+    \item[\( f\) est strictement convexe]
+        
+        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
+    \end{equation}
+    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
+
+\item[\( f\) est coercive]
+    Montrons à présent que \( f\) est coercive. Nous avons :
+    \begin{subequations}
+        \begin{align}
+            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
+            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
+            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
+        \end{align}
+    \end{subequations}
+    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
+    \end{subproof}
+
+    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
+    \begin{equation}
+        \{ x\in \eR^n\tq f(x)\leq M \}
+    \end{equation}
+    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
+
+    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
+    \begin{equation}
+        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
+    \end{equation}
+    ce qui contredit la minimalité de \( f(\bar x_1)\).
+
+    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
+    \begin{equation}
+        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
+    \end{equation}
+    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
+\end{proof}
+
+\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
+    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
+    \begin{equation}
+        x_{k+1}=x_k+t_kd_k
+    \end{equation}
+    où
+    \begin{itemize}
+        \item 
+    \( d_k=-(\nabla f)(x_k)\) 
+\item
+    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
+    \end{itemize}
+
+    Alors pour tout \( k\geq 0\) nous avons
+    \begin{equation}
+        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
+    \end{equation}
+    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
+\end{proposition}
+
+\begin{proof}
+    Décomposition en plusieurs points.
+    \begin{subproof}
+    \item[Existence de \( \bar x\)]
+        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
+    \item[Si \( (\nabla f)(x_k)=0\)]
+    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
+    \end{equation}
+    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
+    \begin{equation}
+        (\nabla f)(a)=Aa+b.
+    \end{equation}
+    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
+
+    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
+        \item[\( t_k\) est bien défini]
+
+            Pour \( t\in \eR\) nous avons
+            \begin{equation}    \label{EqKEHooYaazQi}
+                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
+            \end{equation}
+            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
+            \begin{equation}    \label{EqVWJooWmDSER}
+                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
+            \end{equation}
+            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
+
+        \item[La valeur de \( d_{k+1}\)]
+
+            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
+            \begin{equation}
+                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
+            \end{equation}
+            parce que \( -Ax_k-b=d_k\).
+            
+            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
+            \begin{equation}
+                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
+            \end{equation}
+            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
+        
+        \item[Calcul de \( f(x_{k+1})\)] 
+
+            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
+            \begin{equation}
+                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
+            \end{equation}
+            
+        \item[Encore du calcul \ldots]
+
+            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
+            \begin{equation}
+                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
+            \end{equation}
+            Ayant cela en tête nous pouvons calculer :
+            \begin{subequations}
+                \begin{align}
+                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
+                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
+                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
+                    &=2\big( f(x_k)-\bar f \big)
+                \end{align}
+            \end{subequations}
+            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
+
+        \item[Erreur sur la valeur du minimum]
+
+            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
+            \begin{subequations}
+                \begin{align}
+                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
+                \end{align}
+            \end{subequations}
+            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
+            \begin{equation}
+                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
+            \end{equation}
+            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
+            \begin{equation}
+                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
+            \end{equation}
+            Par récurrence nous avons alors
+            \begin{equation}    \label{eqANKooNPfCFj}
+                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
+            \end{equation}
+            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
+
+        \item[Erreur sur la position du minimum]
+
+            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
+            \begin{subequations}
+                \begin{align}
+                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
+                \end{align}
+            \end{subequations}
+            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
+
+            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
+            \begin{equation}    \label{EqVMRooUMXjig}
+                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
+            \end{equation}
+            
+        \item[Conclusion]
+
+            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
+            \begin{equation}
+                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
+            \end{equation}
+            c'est à dire
+            \begin{equation}
+                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
+            \end{equation}
+    \end{subproof}
+\end{proof}
+
+Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
+
diff --git a/tex/frido/77_series_fonctions.tex b/tex/frido/77_series_fonctions.tex
index 07f53337c..7881e65b4 100644
--- a/tex/frido/77_series_fonctions.tex
+++ b/tex/frido/77_series_fonctions.tex
@@ -3,6 +3,148 @@
 %   Laurent Claessens
 % See the file fdl-1.3.txt for copying conditions.
 
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorème de Von Neumann}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{lemma}[\cite{KXjFWKA}]
+    Soit \( G\), un sous groupe fermé de \( \GL(n,\eR)\) et 
+    \begin{equation}
+        \mL_G=\{ m\in \eM(n,\eR)\tq  e^{tm}\in G\,\forall t\in\eR \}.
+    \end{equation}
+    Alors \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\).
+\end{lemma}
+
+\begin{proof}
+    Si \( m\in\mL_G\), alors \( \lambda m\in\mL_G\) par construction. Le point délicat à prouver est le fait que si \( a,b\in \mL_G\), alors \( a+b\in\mL_G\). Soit \( a\in \eM(n,\eR)\); nous savons qu'il existe une fonction \( \alpha_a\colon \eR\to \eM\) telle que
+    \begin{equation}
+        e^{ta}=\mtu+ta+\alpha_a(t)
+    \end{equation}
+    et 
+    \begin{equation}
+        \lim_{t\to 0} \frac{ \alpha(t) }{ t }=0.
+    \end{equation}
+    Si \( a\) et \( b\) sont dans \( \mL_G\), alors \(  e^{ta} e^{tb}\in G\), mais il n'est pas vrai en général que cela soit égal à \(  e^{t(a+b)}\). Pour tout \( k\in \eN\) nous avons
+    \begin{equation}
+        e^{a/k} e^{b/k}=\left( \mtu+\frac{ a }{ k }+\alpha_a(\frac{1}{ k }) \right)\left( \mtu+\frac{ b }{ k }+\alpha_b(\frac{1}{ k }) \right)=\mtu+\frac{ a+b }{2}+\beta\left( \frac{1}{ k } \right)
+    \end{equation}
+   où \( \beta\colon \eR\to \eM\) est encore une fonction vérifiant \( \beta(t)/t\to 0\). Si \( k\) est assez grand, nous avons
+   \begin{equation}
+       \left\| \frac{ a+b }{ k }+\beta(\frac{1}{ k })  \right\|<1,
+   \end{equation}
+   et nous pouvons profiter du lemme \ref{LemQZIQxaB} pour écrire alors
+   \begin{equation}
+       \left(  e^{a/k} e^{b/k} \right)^k= e^{k\ln\big(\mtu+\frac{ a+b }{ k }+\beta(\frac{1}{ k })\big)}.
+   \end{equation}
+   Ce qui se trouve dans l'exponentielle est
+   \begin{equation}
+       k\left[ \frac{ a+b }{ k }+\alpha( \frac{1}{ k })+\sigma\left( \frac{ a+b }{ k }+\alpha(\frac{1}{ k }) \right) \right].
+   \end{equation}
+   Les diverses propriétés vues montrent que le tout tend vers \( a+b\) lorsque \( k\to \infty\). Par conséquent
+   \begin{equation}
+       \lim_{k\to \infty} \left(  e^{a/k} e^{b/k} \right)^k= e^{a+b}.
+   \end{equation}
+   Ce que nous avons prouvé est que pour tout \( t\), \(  e^{t(a+b)}\) est une limite d'éléments dans \( G\) et est donc dans \( G\) parce que ce dernier est fermé.
+\end{proof}
+
+Vu que \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\), nous pouvons considérer un supplémentaire \( M\).
+
+\begin{lemma}   \label{LemHOsbREC}
+    Il n'existe pas se suites \( (m_k)\) dans \( M\setminus\{ 0 \}\) convergeant vers zéro et telle que \(  e^{m_k}\in G\) pour tout \( k\).
+\end{lemma}
+
+\begin{proof}
+    Supposons que nous ayons \( m_k\to 0\) dans \( M\setminus\{ 0 \}\) avec \(  e^{m_k}\in G\). Nous considérons les éléments \( \epsilon_k=\frac{ m_k }{ \| m_k \| }\) qui sont sur la sphère unité de \(\GL(n,\eR)\). Quitte à prendre une sous-suite, nous pouvons supposer que cette suite converge, et vu que \( M\) est fermé, ce sera vers \( \epsilon\in M\) avec \( \| \epsilon \|=1\). Pour tout \( t\in \eR\) nous avons
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}.
+    \end{equation}
+    En vertu de la décomposition d'un réel en partie entière et décimale, pour tout \( k\) nous avons \( \lambda_k\in \eZ\) et \( | \mu_k |\leq \frac{ 1 }{2}\) tel que \( t/\| m_k \|=\lambda_k+\mu_k\). Avec ça,
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}\exp\Big( \frac{ t }{ m_k }m_k \Big)=\lim_{k\to \infty}  e^{\lambda_km_k} e^{\mu_km_k}.
+    \end{equation}
+    Pour tout \( k\) nous avons \(  e^{\lambda_km_k}\in G\). De plus \( | \mu_k |\) étant borné et \( m_k\) tendant vers zéro nous avons \(  e^{\mu_km_k}\to 1\). Au final
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}\in G
+    \end{equation}
+    Cela signifie que \( \epsilon\in\mL_G\), ce qui est impossible parce que nous avions déjà dit que \( \epsilon\in M\setminus\{ 0 \}\).
+\end{proof}
+
+\begin{lemma}   \label{LemGGTtxdF}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \mL_G\times M&\to \GL(n,\eR) \\
+            l,m&\mapsto  e^{l} e^{m} 
+        \end{aligned}
+    \end{equation}
+    est un difféomorphisme local entre un voisinage de \( (0,0)\) dans \( \eM(n,\eR)\) et un voisinage de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
+\end{lemma}
+Notons que nous ne disons rien de \(  e^{\eM(n,\eR)}\). Nous n'allons pas nous embarquer à discuter si ce serait tout \( \GL(n,\eR)\)\footnote{Vu les dimensions y'a tout de même peu de chance.} ou bien si ça contiendrait ne fut-ce que \( G\).
+
+\begin{proof}
+    Le fait que \( f\) prenne ses valeurs dans \( \GL(n,\eR)\) est simplement dû au fait que les exponentielles sont toujours inversibles. Nous considérons ensuite la différentielle : si \( u\in \mL_G\) et \( v\in M\) nous avons
+    \begin{equation}
+        df_{(0,0)}(u,v)=\Dsdd{ f\big( t(u,v) \big) }{t}{0}=\Dsdd{  e^{tu} e^{tv} }{t}{0}=u+v.
+    \end{equation}
+    L'application \( df_0\) est donc une bijection entre \( \mL_G\times M\) et \( \eM(n,\eR)\). Le théorème d'inversion locale \ref{ThoXWpzqCn} nous assure alors que \( f\) est une bijection entre un voisinage de \( (0,0)\) dans \( \mL_G\times M\) et son image. Mais vu que \( df_0\) est une bijection avec \( \eM(n,\eR)\), l'image en question contient un ouvert autour de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
+\end{proof}
+
+\begin{theorem}[Von Neumann\cite{KXjFWKA,ISpsBzT,Lie_groups}]       \label{ThoOBriEoe}
+    Tout sous-groupe fermé de \( \GL(n,\eR)\) est une sous-variété de \( \GL(n,\eR)\).
+\end{theorem}
+\index{théorème!Von Neumann}
+\index{exponentielle!de matrice!utilisation}
+
+\begin{proof}
+    Soit \( G\) un tel groupe; nous devons prouver que c'est localement difféomorphe à un ouvert de \( \eR^n\). Et si on est pervers, on ne va pas faire localement difféomorphe à un ouvert de \( \eR^n\), mais à un ouvert d'un espace vectoriel de dimension finie. Nous allons être pervers.
+
+    Étant donné que pour tout \( g\in G\), l'application 
+    \begin{equation}
+        \begin{aligned}
+            L_g\colon G&\to G \\
+            h&\mapsto gh 
+        \end{aligned}
+    \end{equation}
+    est de classe \(  C^{\infty}\) et d'inverse \(  C^{\infty}\), il suffit de prouver le résultat pour un voisinage de \( \mtu\).
+
+    Supposons d'abord que \( \mL_G=\{ 0 \}\). Alors \( 0\) est un point isolé de \( \ln(G)\); en effet si ce n'était pas le cas nous aurions un élément \( m_k\) de \( \ln(G)\) dans chaque boule \( B(0,r_k)\). Nous aurions alors \( m_k=\ln(a_k)\) avec \( a_k\in G\) et donc
+    \begin{equation}
+        e^{m_k}=a_k\in G.
+    \end{equation}
+    De plus \( m_k\) appartient forcément à \( M\) parce que \( \mL_G\) est réduit à zéro. Cela nous donnerait une suite \( m_k\to 0\) dans \( M\) dont l'exponentielle reste dans \( G\). Or cela est interdit par le lemme \ref{LemHOsbREC}. Donc \( 0\) est un point isolé de \( \ln(G)\). L'application \(\ln\) étant continue\footnote{Par le lemme \ref{LemQZIQxaB}.}, nous en déduisons que \( \mtu\) est isolé dans \( G\). Par le difféomorphisme \( L_g\), tous les points de \( G\) sont isolés; ce groupe est donc discret et par voie de conséquence une variété.
+
+    Nous supposons maintenant que \( \mL_G\neq\{ 0 \}\). Nous savons par la proposition \ref{PropXFfOiOb} que 
+    \begin{equation}
+        \exp\colon \eM(n,\eR)\to \eM(n,\eR)
+    \end{equation}
+    est une application \(  C^{\infty}\) vérifiant \( d\exp_0=\id\). Nous pouvons donc utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} qui nous offre donc l'existence d'un voisinage \( U\) de \( 0\) dans \( \eM(n,\eR)\) tel que \( W=\exp(U)\) soit un ouvert de \( \GL(n,\eR)\) et que \( \exp\colon U\to W\) soit un difféomorphisme de classe \(  C^{\infty}\).
+
+    Montrons que quitte à restreindre \( U\) (et donc \( W\) qui reste par définition l'image de \( U\) par \( \exp\)), nous pouvons avoir \( \exp\big( U\cap\mL_G \big)=W\cap G\). D'abord \( \exp(\mL_G)\subset G\) par construction. Nous avons donc \( \exp\big( U\cap\mL_G \big)\subset W\cap G\). Pour trouver une restriction de \( U\) pour laquelle nous avons l'égalité, nous supposons que pour tout ouvert \( \mO\) dans \( U\), 
+    \begin{equation}
+        \exp\colon \mO\cap\mL_G\to \exp(\mO)\cap G
+    \end{equation}
+    ne soit pas surjective. Cela donnerait un élément de \( \mO\cap\complement\mL_G\) dont l'image par \( \exp\) n'est pas dans \( G\). Nous construisons ainsi une suite en considérant une boule \( B(0,\frac{1}{ k })\) inclue à \( U\) et \( x_k\in B(0,\frac{1}{ k })\cap\complement\mL_G\) vérifiant \(  e^{x_k}\in G\). Vu le choix des boules nous avons évidemment \( x_k\to 0\).
+
+    L'élément \(  e^{x_k}\) est dans \(  e^{\eM(n,\eR)}\) et le difféomorphisme du lemme \ref{LemGGTtxdF}\quext{Il me semble que l'utilisation de ce lemme manque à l'avant-dernière ligne de la preuve chez \cite{KXjFWKA}.} nous donne \( (l_k,m_k)\in \mL_G\times M\) tel que \(  e^{l_k} e^{m_k}= e^{x_k}\). À ce point nous considérons \( k\) suffisamment grand pour que \(  e^{x_k}\) soit dans la partie de l'image de \( f\) sur lequel nous avons le difféomorphisme. Plus prosaïquement, nous posons
+    \begin{equation}
+        (l_k,m_k)=f^{-1}( e^{x_k})
+    \end{equation}
+    et nous profitons de la continuité pour permuter la limite avec \( f^{-1}\) :
+    \begin{equation}
+        \lim_{k\to \infty} (l_k,m_k)=f^{-1}\big( \lim_{k\to \infty}  e^{x_k} \big)=f^{-1}(\mtu)=(0,0).
+    \end{equation}
+    En particulier \( m_k\to 0\) alors que \(  e^{m_k}= e^{x_k} e^{-l_k}\in G\). La suite \( m_k\) viole le lemme \ref{LemHOsbREC}. Nous pouvons donc restreindre \( U\) de telle façon à avoir
+    \begin{equation}
+        \exp\big( U\cap\mL_G \big)=W\cap G.
+    \end{equation}
+    Nous avons donc un ouvert de \( \mL_G\) (l'ouvert \( U\cap\mL_G\)) qui est difféomorphe avec l'ouvert \( W\cap G\) de \( G\). Donc \( G\) est une variété et accepte \( \mL_G\) comme carte locale.
+
+\end{proof}
+
+\begin{remark}
+    En termes savants, nous avons surtout montré que si \( G\) est un groupe de Lie d'algèbre de Lie \( \lG\), alors l'exponentielle donne un difféomorphisme local entre \( \lG\) et \( G\).
+\end{remark}
+
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Densité des polynômes}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
diff --git a/tex/frido/78_inversion_locale.tex b/tex/frido/78_inversion_locale.tex
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-% This is part of Mes notes de mathématique
-% Copyright (c) 2011-2017
-%   Laurent Claessens
-% See the file fdl-1.3.txt for copying conditions.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Complétude avec la norme uniforme}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{proposition}[Limite uniforme de fonctions continues]\label{PropCZslHBx}
-    Soit \( X\) un espace topologique et \( (Y,d)\) un espace métrique. Si une suite de fonctions \( f_n\colon X\to Y\) continues converge uniformément, alors la limite est séquentiellement continue\footnote{Si \( X\) est métrique, alors c'est la continuité usuelle par la proposition \ref{PropFnContParSuite}.}.
-\end{proposition}
-
-\begin{proof}
-    Soit \( a\in X\) et prouvons que \( f\) est séquentiellement continue en \( a\). Pour cela nous considérons une suite \( x_n\to a\) dans \( X\). Nous savons que \( f(x_n)\stackrel{Y}{\longrightarrow}f(x)\). Pour tout \(k\in \eN\), tout \( n\in \eN\) et tout \( x\in X\) nous avons la majoration
-    \begin{equation}
-        \big\| f(x_n)-f(x) \big\|\leq \big\| f(x_n)-f_k(x_n) \big\|+\big\| f_k(x_n)-f_k(x) \big\|+\big\| f_k(x)-f(x) \big\|\leq 2\| f-f_k \|_{\infty}+\big\| f_k(x_n)-f_k(x) \big\|.    
-    \end{equation}
-    Soit \( \epsilon>0\). Si nous choisissons \( k\) suffisamment grand la premier terme est plus petit que \( \epsilon\). Et par continuité de \( f_k\), en prenant \( n\) assez grand, le dernier terme est également plus petit que \( \epsilon\).
-\end{proof}
-
-\begin{proposition} \label{PropSYMEZGU}
-    Soit \( X\) un espace topologique métrique \( (Y,d)\) un espace espace métrique complet. Alors les espaces
-    \begin{enumerate}
-        \item
-            \( \big( C^0_b(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et bornées \( X\to Y\),
-        \item
-            \( \big( C^0_0(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et s'annulant à l'infini
-        \item
-            \( \big( C^k_0(X,Y),\| . \|_{\infty} \big)\) des fonctions de classe \( C^k\) et s'annulant à l'infini
-    \end{enumerate} 
-    sont complets.
-\end{proposition}
-
-\begin{proof}
-    Soit \( (f_n)\) une suite de Cauchy dans \( C(X,Y)\), c'est à dire que pour tout \( \epsilon>0\) il existe \( N\in \eN\) tel que si \( k,l>N\) nous avons \( \| f_k-f_l \|_{\infty}\leq \epsilon\). Cette suite vérifie le critère de Cauchy uniforme \ref{PropNTEynwq} et donc converge uniformément vers une fonction \( f\colon X\to Y\). La continuité (ou l'aspect \( C^k\)) de la fonction \( f\) découle de la convergence uniforme et de la proposition \ref{PropCZslHBx} (c'est pour avoir l'équivalence entre la continuité séquentielle et la continuité normale que nous avons pris l'hypothèse d'espace métrique).
-
-    Si les fonctions \( f_k\) sont bornées ou s'annulent à l'infini, la convergence uniforme implique que la limite le sera également.
-\end{proof}
-    Notons que si \( X\) est compact, les fonctions continues sont bornées par le théorème \ref{ThoImCompCotComp} et nous pouvons simplement dire que \( C^0(X,Y)\) est complet, sans préciser que nous parlons des fonctions bornées.
-
-
-\begin{lemma}       \label{LemdLKKnd}
-    Soit \( A\) compact et \( B\) complet. L'ensemble des fonctions continues de \( A\) vers \( B\) muni de la norme uniforme est complet.
-
-    Dit de façon courte : \( \big( C(A,B),\| . \|_{\infty} \big)\) est complet.
-\end{lemma}
-
-\begin{proof}
-    Soit \( (f_k)\) une suite de Cauchy de fonctions dans \( C(A,B)\). Pour chaque \( x\in A \) nous avons
-    \begin{equation}
-        \| f_k(x)-f_l(x) \|_B\leq \| f_k-f_l \|_{\infty},
-    \end{equation}
-    de telle sorte que la suite \( (f_k(x))\) est de Cauchy dans \( B\) et converge donc vers un élément de \( B\). La suite de Cauchy \( (f_k)\) converge donc ponctuellement vers une fonction \( f\colon A\to B\). Nous devons encore voir que cette fonction est continue; ce sera l'uniformité de la norme qui donnera la continuité. En effet soit \( x_n\to x\) une suite dans \( A\) convergent vers \( x\in A\). Pour chaque \( k\in \eN\) nous avons
-    \begin{equation}
-        \| f(x_n)-f(x) \|\leq \| f(x_n)-f_k(x_n) \|  +\| f_k(x_n)-f_k(x) \|+\| f_k(x)-f(x) \|.
-    \end{equation}
-    En prenant \( k\) et \( n\) assez grands, cette expression peut être rendue aussi petite que l'on veut; le premier et le troisième terme par convergence ponctuelle \( f_k\to f\), le second terme par continuité de \( f_k\). La suite \( f(x_n)\) est donc convergente vers \( f(x)\) et la fonction \( f\) est continue.
-\end{proof}
-
-\begin{probleme}
-Il serait sans doute bon de revoir cette preuve à la lumière du critère de Cauchy uniforme \ref{PropNTEynwq}.
-\end{probleme}
-
-
-\begin{normaltext}[\cite{ooXYZDooWKypYR}]
-    Le théorème de Stone-Weierstrass indique que les polynômes sont denses pour la topologie uniforme dans les fonctions continues. Donc il existe des limites uniformes de fonctions \( C^{\infty}\) qui ne sont même pas dérivables. Les espaces de type \( C^p\) munis de \( \| . \|_{\infty}\) ne sont donc pas complets sans quelque hypothèses. Voir la proposition \ref{PropSYMEZGU} et le thème \ref{THMooOCXTooWenIJE}.
-\end{normaltext}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorèmes de point fixe}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Points fixes attractifs et répulsifs}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooTMZUooMoBDGC}
-    Soit \( I\) un intervalle fermé de \( \eR\) et \( \varphi\colon I\to I\) une application \( C^1\). Soit \( a\) un point fixe de \( \varphi\). Nous disons que \( a\) est \defe{attractif}{point fixe!attractif}\index{attractif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) converge vers \( a\). Le point \( a\) sera dit \defe{répulsif}{répulsif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) diverge.
-\end{definition}
-
-\begin{lemma}[\cite{DemaillyNum}]
-    Soit \( a\) un point fixe de \( \varphi\).
-    \begin{enumerate}
-        \item
-    Si \( | \varphi'(a) |<1\) alors \( a\) est attractif et la convergence est au moins exponentielle.
-\item
-    Si \( | \varphi'(a) |>1\) alors \( a\) est répulsif et la divergence est au moins exponentielle.
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Si \( | \varphi'(a)<1 |\) alors il existe \( k\) tel que \( | \varphi'(a) |<k<1\) et par continuité il existe un voisinage \( V\) de \( a\) dans lequel \( | \varphi'(x) |<k\) pour tout \( x\in V\). En utilisant le théorème des accroissements finis nous avons
-    \begin{equation}
-        | x_n-a |=\big| f(x_{n-1}-a) \big|\leq k| x_{n-1}-a |
-    \end{equation}
-    et par récurrence
-    \begin{equation}
-        | x_n-a |\leq k^n| x_0-a |.
-    \end{equation}
-
-    Le cas \( | \varphi'(a)>1 |\) se traite de façon similaire.
-\end{proof}
-
-\begin{remark}
-    Dans le cas \(| \varphi'(a) |=1\), nous ne pouvons rien conclure. Si \( \varphi(x)=\sin(x)\) nous avons \( \sin(x)<x\) et le point \( a=0\) est attractif. A contrario, si \( \varphi(x)=\sinh(x)\) nous avons \( |\sinh(x)|>|x|\) et le point \( a=0\) est répulsif.
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Picard}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooRSLCooAsWisu}
-    Une application \( f\colon (X,\| . \|_X)\to (Y,\| . \|_Y)\) entre deux espaces métriques est une \defe{contraction}{contraction} si elle est \( k\)-\defe{Lipschitz}{Lipschitz} pour un certain \( 0\leq k<1\), c'est à dire si pour tout \( x,y\in X\) nous avons
-    \begin{equation}
-        \| f(x)-f(y) \|_Y\leq k\| x-y \|_{X}.
-    \end{equation}
-\end{definition}
-
-\begin{theorem}[Picard \cite{ClemKetl,NourdinAnal}\footnote{Il me semble qu'à la page 100 de \cite{NourdinAnal}, l'hypothèse H1 qui est prouvée ne prouve pas Hn dans le cas \( n=1\). Merci de m'écrire si vous pouvez confirmer ou infirmer. La preuve donnée ici ne contient pas cette «erreur».}.]     \label{ThoEPVkCL}
-    Soit \( X\) un espace métrique complet et \( f\colon X\to X\) une application contractante, de constante de Lipschitz \( k\). Alors \( f\) admet un unique point fixe, nommé \( \xi\). Ce dernier est donné par la limite de la suite définie par récurrence 
-    \begin{subequations}
-        \begin{numcases}{}
-            x_0\in X\\
-            x_{n+1}=f(x_n).
-        \end{numcases}
-    \end{subequations}
-    De plus nous pouvons majorer l'erreur par
-    \begin{equation}    \label{EqKErdim}
-        \| x_n-x \|\leq \frac{ k^n }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k^n }{ 1-k }\| x_1-x_0 \|.
-    \end{equation}
-
-    Soit \( r>0\), \( a\in X\) tels que la fonction \( f\) laisse la boule \( K=\overline{ B(a,r) }\) invariante (c'est à dire que \( f\) se restreint à \( f\colon K\to K\)). Nous considérons les suites \( (u_n)\) et \( (v_n)\) définies par
-    \begin{subequations}
-        \begin{numcases}{}
-            u_0=v_0\in K\\
-            u_{n+1}=f(v_n), v_{n+1}\in B(u_n,\epsilon).
-        \end{numcases}
-    \end{subequations}
-    Alors le point fixe \( \xi\) de \( f\) est dans \( K\) et la suite \( (v_n)\) satisfait l'estimation
-    \begin{equation}
-        \| v_n-\xi \|\leq \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k }.
-    \end{equation}
-\end{theorem}
-\index{théorème!Picard}
-\index{point fixe!Picard}
-
-La première inégalité \eqref{EqKErdim} donne une estimation de l'erreur calculable en cours de processus; la seconde donne une estimation de l'erreur calculable avant de commencer.
-
-\begin{proof}
-    
-    Nous commençons par l'unicité du point fixe. Si \( a\) et \( b\) sont des points fixes, alors \( f(a)=a\) et \( f(b)=b\). Par conséquent
-    \begin{equation}
-        \| f(a)-f(b) \|=\| a-b \|,
-    \end{equation}
-    ce qui contredit le fait que \( f\) soit une contraction.
-
-    En ce qui concerne l'existence, notons que si la suite des \( x_n\) converge dans \( X\), alors la limite est un point fixe. En effet en prenant la limite des deux côtés de l'équation \( x_{n+1}=f(x_n)\), nous obtenons \( \xi=f(\xi)\), c'est à dire que \( \xi\) est un point fixe de \( f\). Notons que nous avons utilisé ici la continuité de \( f\), laquelle est une conséquence du fait qu'elle soit Lipschitz. Nous allons donc porter nos efforts à prouver que la suite est de Cauchy (et donc convergente parce que \( X\) est complet). Nous commençons par prouver que \( \| x_{n+1}-x_n \|\leq k^n\| x_0-x_1 \|\). En effet pour tout \( n\) nous avons
-    \begin{equation}
-        \| x_{n+1}-x_n \|=\| f(x_n)-f(x_{n-1}) \|\leq k\| x_n-x_{n-1} \|.
-    \end{equation}
-    La relation cherchée s'obtient alors par récurrence. Soient \( q>p\). En utilisant une somme télescopique,
-    \begin{subequations}
-        \begin{align}
-            \| x_q-x_p \|&\leq \sum_{l=p}^{q-1}\| x_{l+1}-x_l \|\\
-            &\leq\left( \sum_{l=p}^{q-1}k^l \right)\| x_1-x_0 \|\\
-            &\leq\left(\sum_{l=p}^{\infty}k^l\right)\| x_1-x_0 \|.
-        \end{align}
-    \end{subequations}
-    Étant donné que \( k<1\), la parenthèse est la queue d'une série qui converge, et donc tend vers zéro lorsque \( p\) tend vers l'infini.
-
-    En ce qui concerne les inégalités \eqref{EqKErdim}, nous refaisons une somme télescopique :
-    \begin{subequations}
-        \begin{align}
-            \| x_{n+p}-x_n \|&\leq \| x_{n+p}-x_{n+p-1} \|+\cdots +\| x_{n+1}-x_n \|\\
-            &\leq k^p\| x_n-x_{n-1} \|+k^{p-1}\| x_n-x_{n-1} \|+\cdots +k\| x_n-x_{n-1} \|\\
-            &=k(1+\cdots +k^{p-1})\| x_n-x_{n-1}\|  \\
-            &\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|.
-        \end{align}
-    \end{subequations}
-    En prenant la limite \( p\to \infty\) nous trouvons
-    \begin{equation}        \label{EqlUMVGW}
-        \| \xi-x_n \|\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k }{ 1-k }\| x_1-x_0 \|.
-    \end{equation}
-
-    Nous passons maintenant à la seconde partie du théorème en supposant que \( f\) se restreigne en une fonction \( f\colon K\to K\). D'abord \( K\) est encore un espace métrique complet, donc la première partie du théorème s'y applique et \( f\) y a un unique point fixe.
-    
-    Nous allons montrer la relation par récurrence. Tout d'abord pour \( n=1\) nous avons
-    \begin{equation}
-        \| v_1-\xi \|\leq\| v_1-u_1 \|+\| u_1-\xi \|\leq \epsilon+\frac{ k }{ 1-k }\| u_1-u_0 \|
-    \end{equation}
-    où nous avons utilisé l'estimation \eqref{EqlUMVGW}, qui reste valable en remplaçant \( x_1\) par \( u_1\)\footnote{Elle n'est cependant pas spécialement valable si on remplace \( x_n\) par \( u_n\).}. Nous pouvons maintenant faire la récurrence :
-    \begin{subequations}
-        \begin{align}
-            \| v_{n+1}-\xi \|&\leq \| v_{n+1}-u_{n+1} \|+\| u_{n+1}-\xi \|\\
-            &\leq \epsilon+k\| v_n-\xi \|\\
-            &\leq \epsilon+k\left( \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k } \right)\\
-            &=\frac{ \epsilon }{ 1-k }+\frac{ k^{n+1} }{ 1-k }\| u_1-u_0 \|.
-        \end{align}
-    \end{subequations}
-\end{proof}
-
-\begin{remark}
-    Ce théorème comporte deux parties d'intérêts différents. La première partie est un théorème de point fixe usuel, qui sera utilisé pour prouver l'existence de certaines équations différentielles.
-
-    La seconde partie est intéressante d'un point de vie numérique. En effet, ce qu'elle nous enseigne est que si à chaque pas de calcul de la récurrence \( x_{n+1}=f(x_n)\) nous commettons une erreur d'ordre de grandeur \( \epsilon\), alors le procédé (la suite \( (v_n)\)) ne converge plus spécialement vers le point fixe, mais tend vers le point fixe avec une erreur majorée par \( \epsilon/(k-1)\).
-\end{remark}
-
-\begin{remark}
-Au final l'erreur minimale qu'on peut atteindre est de l'ordre de \( \epsilon\). Évidemment si on commet une faute de calcul de l'ordre de \( \epsilon\) à chaque pas, on ne peut pas espérer mieux.
-\end{remark}
-
-\begin{remark}  \label{remIOHUJm}
-    Si \( f\) elle-même n'est pas contractante, mais si \( f^p\) est contractante pour un certain \( p\in \eN\) alors la conclusion du théorème de Picard reste valide et \( f\) a le même unique point fixe que \( f^p\). En effet nommons \( x\) le point fixe de \( f\) : \( f^p(x)=x\). Nous avons alors
-    \begin{equation}
-        f^p\big( f(x) \big)=f\big( f^p(x) \big)=f(x),
-    \end{equation}
-    ce qui prouve que \( f(x)\) est un point fixe de \( f^p\). Par unicité nous avons alors \( f(x)=x\), c'est à dire que \( x\) est également un point fixe de \( f\).
-\end{remark}
-
-Si la fonction n'est pas Lipschitz mais presque, nous avons une variante.
-\begin{proposition}
-    Soit \( E\) un ensemble compact\footnote{Notez cette hypothèse plus forte} et si \( f\colon E\to E\) est une fonction telle que
-    \begin{equation}        \label{EqLJRVvN}
-        \| f(x)-f(y) \|< \| x-y \|
-    \end{equation}
-    pour tout \( x\neq y\) dans \( E\) alors \( f\) possède un unique point fixe.
-\end{proposition}
-
-\begin{proof}
-    La suite \( x_{n+1}=f(x_n)\) possède une sous suite convergente. La limite de cette sous suite est un point fixe de \( f\) parce que \( f\) est continue. L'unicité est due à l'aspect strict de l'inégalité \eqref{EqLJRVvN}.
-\end{proof}
-
-\begin{theorem}[Équation de Fredholm]\index{Fredholm!équation}\index{équation!Fredholm}     \label{ThoagJPZJ}
-    Soit \( K\colon \mathopen[ a , b \mathclose]\times \mathopen[ a , b \mathclose]\to \eR\) et \( \varphi\colon \mathopen[ a , b \mathclose]\to \eR\), deux fonctions continues. Alors si \( \lambda\) est suffisamment petit, l'équation
-    \begin{equation}
-        f(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x)
-    \end{equation}
-    admet une unique solution qui sera de plus continue sur \( \mathopen[ a , b \mathclose]\).
-\end{theorem}
-
-\begin{proof}
-    Nous considérons l'ensemble \( \mF\) des fonctions continues \( \mathopen[ a , b \mathclose]\to\mathopen[ a , b \mathclose]\) muni de la norme uniforme. Le lemme \ref{LemdLKKnd} implique que \( \mF\) est complet. Nous considérons l'application \( \Phi\colon \mF\to \mF\) donnée par
-    \begin{equation}
-        \Phi(f)(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x). 
-    \end{equation}
-    Nous montrons que \( \Phi^p\) est une application contractante pour un certain \( p\). Pour tout \( x\in \mathopen[ a , b \mathclose]\) nous avons
-    \begin{subequations}
-        \begin{align}
-            \| \Phi(f)-\Phi(g) \|_{\infty}&\leq \| \Phi(f)(x)-\Phi(g)(x) \|\\
-            &=| \lambda |\Big\| \int_a^bK(x,y)\big( f(y)-g(y) \big)dy  \Big\|\\
-            &\leq | \lambda |\| K \|_{\infty}| b-a |\| f-g \|_{\infty}
-        \end{align}
-    \end{subequations}
-    Si \( \lambda\) est assez petit, et si \( p\) est assez grand, l'application \( \Phi^p\) est donc une contraction. Elle possède donc un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Brouwer}
-%---------------------------------------------------------------------------------------------------------------------------
-\label{subSecZCCmMnQ}
-
-\begin{proposition}
-    Soit \( f\colon \mathopen[ a , b \mathclose]\to \mathopen[ a , b \mathclose]\) une fonction continue. Alors \( f\) accepte un point fixe.
-\end{proposition}
-
-\begin{proof}
-    En effet si nous considérons \( g(x)=f(x)-x\) alors nous avons \( g(a)=f(a)-a\geq 0\) et \( g(b)=f(b)-b\leq 0\). Si \( g(a)\) ou \( g(b)\) est nul, la proposition est démontrée; nous supposons donc que \( g(a)>0\) et \( g(b)<0\). La proposition découle à présent du théorème des valeurs intermédiaires \ref{ThoValInter}.
-\end{proof}
-
-\begin{example}
-    La fonction \( x\mapsto\cos(x)\) est continue entre \( \mathopen[ -1 , 1 \mathclose]\) et \( \mathopen[ -1 , 1 \mathclose]\). Elle admet donc un point fixe. Par conséquent il existe (au moins) une solution à l'équation \( \cos(x)=x\).
-\end{example}
-
-\begin{proposition}[Brouwer dans \( \eR^n\) version \(  C^{\infty}\) via Stokes]     \label{PropDRpYwv}
-    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction \(  C^{\infty}\). Alors \( f\) admet un point fixe.
-\end{proposition}
-\index{point fixe!Brouwer}
-
-\begin{proof}
-    Supposons que \( f\) ne possède pas de points fixes. Alors pour tout \( x\in B\) nous considérons la ligne droite partant de \( x\) dans la direction de \( f(x)\) (cette droite existe parce que \( x\) et \( f(x)\) sont supposés distincts). Cette ligne intersecte \( \partial B\) en un point que nous appelons \( g(x)\). Prouvons que cette fonction est \( C^k\) dès que \( f\) est \( C^k\) (y compris avec \( k=\infty\)).
-
-   Le point \( g(x) \) est la solution du système
-    \begin{subequations}
-        \begin{numcases}{}
-        g(x)-f(x)=\lambda\big( x-f(x) \big)\\
-        \| g(x) \|^2=1\\
-        \lambda\geq 0.
-        \end{numcases}
-    \end{subequations}
-    En substituant nous obtenons l'équation
-    \begin{equation}
-        P_x(\lambda)=\| \lambda\big( x-f(x) \big)+f(x) \|^2-1=0,
-    \end{equation}
-    ou encore
-    \begin{equation}
-        \lambda^2\| x-f(x) \|^2+2\lambda\big( x-f(x) \big)\cdot f(x)+\| f(x) \|^2-1=0.
-    \end{equation}
-    En tenant compte du fait que \( \| f(x)<1 \|\) (pare que les images de \( f\) sont dans \( \mB\)), nous trouvons que \( P_x(0)\leq 0\) et \( P_x(1)\leq 0\). De même \( \lim_{\lambda\to\infty} P_x(\lambda)=+\infty\). Par conséquent le polynôme de second degré \( P_x\) a exactement deux racines distinctes \( \lambda_1\leq 0\) et \( \lambda_2\geq 1\). La racine que nous cherchons est la seconde. Le discriminant est strictement positif, donc pas besoin d'avoir peur de la racine dans
-    \begin{equation}
-        \lambda(x)=\frac{ -\big( x-f(x) \big)\cdot f(x)+\sqrt{   \Delta_x  } }{ \| x-f(x) \|^2 }
-    \end{equation}
-    où 
-    \begin{equation}
-        \Delta_x=4\Big( \big( x-f(x) \big)\cdot f(x) \Big)^2-4\| x-f(x) \|^2\big( \| f(x) \|^2-1 \big).
-    \end{equation}
-    Notons que la fonction \( \lambda(x)\) est \( C^k\) dès que \( f\) est \( C^k\); et en particulier elle est \( C^{\infty}\) si \( f\) l'est.
-
-    En résumé la fonction \( g\) ainsi définie vérifie deux propriétés :
-    \begin{enumerate}
-        \item
-            elle est \(  C^{\infty}\);
-        \item
-            elle est l'identité sur \( \partial B\).
-    \end{enumerate}
-    La suite de la preuve consiste à montrer qu'une telle rétraction sur \( B\) ne peut pas exister\footnote{Notons qu'il n'existe pas non plus de rétractions continues sur \( B\), mais pour le montrer il faut utiliser d'autres méthodes que Stokes, ou alors présenter les choses dans un autre ordre.}.
-
-    Nous considérons une forme de volume \( \omega\) sur \( \partial B\) : l'intégrale de \( \omega\) sur \( \partial B\) est la surface de \( \partial B\) qui est non nulle. Nous avons alors
-    \begin{equation}
-        0<\int_{\partial B}\omega
-        =\int_{\partial B}g^*\omega
-        =\int_Bd(g^*\omega)
-        =\int_Bg^*(d\omega)
-        =0
-    \end{equation}
-    Justifications :
-    \begin{itemize}
-        \item 
-            L'intégrale \( \int_{\partial B}\omega\) est la surface de \( \partial B\) et est donc non nulle.
-        \item
-            La fonction \( g\) est l'identité sur \( \partial B\). Nous avons donc \( \omega=g^*\omega\).
-        \item
-            Le lemme \ref{LemdwLGFG}.
-        \item
-            La forme \( \omega\) est de volume, par conséquent de degré maximum et \( d\omega=0\).
-    \end{itemize}
-\end{proof}
-
-Un des points délicats est de se ramener au cas de fonctions \( C^{\infty}\). Pour la régularisation par convolution, voir \cite{AllardBrouwer}; pour celle utilisant le théorème de Weierstrass, voir \cite{KuttlerTopInAl}.
-\begin{theorem}[Brouwer dans \( \eR^n\) version continue]\label{ThoRGjGdO}
-    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction continue. Alors \( f\) admet un point fixe.
-\end{theorem}
-\index{théorème!Brouwer}
-
-\begin{proof}
-    Nous commençons par définir une suite de fonctions
-    \begin{equation}
-        f_k(x)=\frac{ f(x) }{ 1+\frac{1}{ k } }.
-    \end{equation}
-    Nous avons \( \| f_k-f \|_{\infty}\leq \frac{1}{ 1+k }\) où la norme est la norme uniforme sur \( B\). Par le théorème de Weierstrass \ref{ThoWmAzSMF} il existe une suite de fonctions \(  C^{\infty}\) \( g_k\) telles que
-    \begin{equation}
-        \|  g_k-f_k\|_{\infty}\leq\frac{1}{ 1+k }.
-    \end{equation}
-    Vérifions que cette fonction \( g_k\) soit bien une fonction qui prend ses valeurs dans \( B\) :
-    \begin{subequations}
-        \begin{align}
-            \| g_k(x) \|&\leq \| g_k(x)-f_k(x) \|+\| f_k(x) \|\\
-            &\leq \frac{1}{ 1+k }+\frac{ \| f(x) \| }{ 1+\frac{1}{ k } }\\
-            &\leq \frac{1}{ 1+k}+\frac{1}{ 1+\frac{1}{ k } }\\
-            &=1.
-        \end{align}
-    \end{subequations}
-    Par la version \(  C^{\infty}\) du théorème (proposition \ref{PropDRpYwv}), \( g_k\) admet un point fixe que l'on nomme \( x_k\).
-
-    Étant donné que \( x_k\) est dans le compact \( B\), quitte à prendre une sous suite nous supposons que la suite \( (x_k)\) converge vers un élément \( x\in B\). Nous montrons maintenant que \( x\) est un point fixe de \( f\) :
-    \begin{subequations}
-        \begin{align}
-            \| f(x)-x \|&=\| f(x)-g_k(x)+g_k(x)-x_k+x_k-x \|\\
-            &\leq \| f(x)-g_k(x) \| +\underbrace{\| g_k(x)-x_k \|}_{=0}+\| x_k-x \|\\
-            &\leq \frac{1}{ 1+k }+\| x_k-x \|.
-        \end{align}
-    \end{subequations}
-    En prenant le limite \( k\to\infty\) le membre de droite tend vers zéro et nous obtenons \( f(x)=x\).
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Théorème de Schauder}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Une conséquence du théorème de Brouwer est le théorème de Schauder qui est valide en dimension infinie.
-
-\begin{theorem}[Théorème de Schauder\cite{ooWWBQooKIciWi}]\index{théorème!Schauder}       \label{ThovHJXIU}
-    Soit \( E\), un espace vectoriel normé, \( K\) un convexe compact de \( E\) et \( f\colon K\to K\) une fonction continue. Alors \( f\) admet un point fixe.
-\end{theorem}
-\index{théorème!Schauder}
-\index{point fixe!Schauder}
-
-\begin{proof}
-    Étant donné que \( f\colon K\to K\) est continue, elle y est uniformément continue. Si nous choisissons \( \epsilon\) alors il existe \( \delta>0\) tel que 
-    \begin{equation}
-        \| f(x)-f(y) \|\leq \epsilon
-    \end{equation}
-    dès que \( \| x-y \|\leq \delta\). La compacité de \( K\) permet de choisir un recouvrement fini par des ouverts de la forme
-    \begin{equation}    \label{EqKNPUVR}
-        K\subset \bigcup_{1\leq i\leq p}B(x_j,\delta)
-    \end{equation}
-    où \( \{ x_1,\ldots, x_p \}\subset K\). Nous considérons maintenant \( L=\Span\{ f(x_j)\tq 1\leq j\leq p \}\) et
-    \begin{equation}
-        K^*=K\cap L.
-    \end{equation}
-    Le fait que \( K\) et \( L\) soient convexes implique que \( K^*\) est convexe. L'ensemble \( K^*\) est également compact parce qu'il s'agit d'une partie fermée de \( K\) qui est compact (lemme \ref{LemnAeACf}). Notons en particulier que \( K^*\) est contenu dans un espace vectoriel de dimension finie, ce qui n'est pas le cas de \( K\).
-
-    Nous allons à présent construire une sorte de partition de l'unité subordonnée au recouvrement \eqref{EqKNPUVR} sur \( K\) (voir le lemme \ref{LemGPmRGZ}). Nous commençons par définir
-    \begin{equation}
-        \psi_j(x)=\begin{cases}
-            0    &   \text{si } \| x-x_j \|\geq \delta\\
-            1-\frac{ \| x-x_j \| }{ \delta }    &    \text{sinon}.
-        \end{cases}
-    \end{equation}
-    pour chaque \( 1\leq j\leq p\). Notons que \( \psi_j\) est une fonction positive, nulle en-dehors de \( B(x_j,\delta)\). En particulier la fonction suivante est bien définie :
-    \begin{equation}
-        \varphi_j(x)=\frac{ \psi_j(x) }{ \sum_{k=1}^p\psi_k(x) }
-    \end{equation}
-    et nous avons \( \sum_{j=1}^p\varphi_j(x)=1\). Les fonctions \( \varphi_j\) sont continues sur \( K\) et nous définissons finalement
-    \begin{equation}
-        g(x)=\sum_{j=1}^p\varphi_j(x)f(x_j).
-    \end{equation}
-    Pour chaque \( x\in K\), l'élément \( g(x)\) est une combinaison des éléments \( f(x_j)\in K^*\). Étant donné que \( K^*\) est convexe et que la somme des coefficients \( \varphi_j(x)\) vaut un, nous avons que \( g\) prend ses valeurs dans \( K^*\) par la proposition \ref{PropPoNpPz}.
-
-    Nous considérons seulement la restriction \( g\colon K^*\to K^*\) qui est continue sur un compact contenu dans un espace vectoriel de dimension finie. Le théorème de Brouwer nous enseigne alors que \( g\) a un point fixe (proposition \ref{ThoRGjGdO}). Nous nommons \( y\) ce point fixe. Notons que \( y\) est fonction du \( \epsilon\) choisit au début de la construction, via le \( \delta\) qui avait conditionné la partition de l'unité.
-
-    Nous avons
-    \begin{subequations}        \label{EqoXuTzE}
-        \begin{align}
-            f(y)-y&=f(y)-g(y)\\
-            &=\sum_{j=1}^p\varphi_j(y)f(y)-\sum_{j=1}^p\varphi_j(y)f(x_j)\\
-            &=\sum_{j=1}^p\varphi(j)(y)\big( f(y)-f(x_j) \big).
-        \end{align}
-    \end{subequations}
-    Par construction, \( \varphi_j(y)\neq 0\) seulement si \( \| y-x_j \|\leq \delta\) et par conséquent seulement si \( \| f(y)-f(x_j) \|\leq \epsilon\). D'autre par nous avons \( \varphi_j(y)\geq 0\); en prenant la norme de \eqref{EqoXuTzE} nous trouvons
-    \begin{equation}
-        \| f(y)-y \|\leq \sum_{j=1}^p\| \varphi_j(y)\big( f(y)-f(x_j) \big) \|\leq \sum_{j=1}^p\varphi_j(y)\epsilon=\epsilon.
-    \end{equation}
-    Nous nous souvenons maintenant que \( y\) était fonction de \( \epsilon\). Soit \( y_m\) le \( y\) qui correspond à \( \epsilon=2^{-m}\). Nous avons alors
-    \begin{equation}
-        \| f(y_m)-y_m \|\leq 2^{-m}.
-    \end{equation}
-    L'élément \( y_m\) est dans \( K^*\) qui est compact, donc quitte à choisir une sous suite nous pouvons supposer que \( y_m\) est une suite qui converge vers \( y^*\in K\)\footnote{Notons que même dans la sous suite nous avons \( \| f(y_m)-y_m \|\leq 2^{-m}\), avec le même «\( m\)» des deux côtés de l'inégalité.}. Nous avons les majorations
-    \begin{equation}
-        \| f(y^*)-y^* \|\leq \| f(y^*)-f(y_m) \|+\| f(y_m)-y_m \|+\| y_m-y^* \|.
-    \end{equation}
-    Si \( m\) est assez grand, les trois termes du membre de droite peuvent être rendus arbitrairement petits, d'où nous concluons que
-    \begin{equation}
-        f(y^*)=y^*
-    \end{equation}
-    et donc que \( f\) possède un point fixe.
-\end{proof}
-
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Markov-Kakutani et mesure de Haar}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit \( G\) un groupe topologique. Une \defe{mesure de Haar}{mesure!de Haar} sur \( G\) est une mesure \( \mu\) telle que 
-    \begin{enumerate}
-        \item
-            \( \mu(gA)=\mu(A)\) pour tout mesurable \( A\) et tout \( g\in G\),
-        \item
-            \( \mu(K)<\infty\) pour tout compact \( K\subset G\).
-    \end{enumerate}
-    Si de plus le groupe \( G\) lui-même est compact nous demandons que la mesure soit normalisée : \( \mu(G)=1\).
-\end{definition}
-
-Le théorème suivant nous donne l'existence d'une mesure de Haar sur un groupe compact.
-\begin{theorem}[Markov-Katutani\cite{BeaakPtFix}]\index{théorème!Markov-Takutani}   \label{ThoeJCdMP}
-    Soit \( E\) un espace vectoriel normé et \( L\), une partie non vide, convexe, fermée et bornée de \( E'\). Soit \( T\colon L\to L\) une application continue. Alors \( T\) a un point fixe.
-\end{theorem}
-
-\begin{proof}
-    Nous considérons un point \( x_0\in L\) et la suite
-    \begin{equation}
-        x_n=\frac{1}{ n+1 }\sum_{i=0}^n T^ix_0.
-    \end{equation}
-    La somme des coefficients devant les \( T^i(x_0)\) étant \( 1\), la convexité de \( L\) montre que \( x_n\in L\). Nous considérons l'ensemble
-    \begin{equation}
-        C=\bigcap_{n\in \eN}\overline{ \{ x_m\tq m\geq n \} }.
-    \end{equation}
-    Le lemme \ref{LemooynkH} indique que \( C\) n'est pas vide, et de plus il existe une sous suite de \( (x_n)\) qui converge vers un élément \( x\in C\). Nous avons
-    \begin{equation}
-        \lim_{n\to \infty} x_{\sigma(n)}(v)=x(v)
-    \end{equation}
-    pour tout \( v\in E\). Montrons que \( x\) est un point fixe de \( T\). Nous avons
-    \begin{subequations}
-        \begin{align}
-            \| (Tx_{\sigma(k)}-x_{\sigma(k)})v \|&=\Big\| T\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v)-\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v) \Big\|\\
-            &=\Big\| \frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^{i+1}x_0(v)-T^ix_0(v) \Big\|\\
-            &=\frac{1}{ 1+\sigma(k) }\big\| T^{\sigma(k)+1}x_0(v)-x_0(v) \big\|\\
-            &\leq\frac{ 2M }{ \sigma(k)+1 }
-        \end{align}
-    \end{subequations}
-    où \( M=\sum_{y\in L}\| y(v) \|<\infty\) parce que \( L\) est borné. En prenant \( k\to\infty\) nous trouvons
-    \begin{equation}
-        \lim_{k\to \infty} \big( Tx_{\sigma(k)}-x_{\sigma(k)} \big)v=0,
-    \end{equation}
-    ce qui signifie que \( Tx=x\) parce que \( T\) est continue.
-\end{proof}
-
-Le théorème suivant est une conséquence du théorème de Markov-Katutani.
-\begin{theorem} \label{ThoBZBooOTxqcI}
-    Si \( G\) est un groupe topologique compact possédant une base dénombrable de topologie alors \( G\) accepte une unique mesure de Haar normalisée. De plus elle est unimodulaire :
-    \begin{equation}
-        \mu(Ag)=\mu(gA)=\mu(A)
-    \end{equation}
-    pour tout mesurables \( A\subset G\) et tout élément \( g\in G\).
-\end{theorem}
-\index{mesure!de Haar}
-
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorèmes de point fixes et équations différentielles}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Théorème de Cauchy-Lipschitz}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous démontrons ici deux théorèmes de Cauchy-Lipschitz. De nombreuses propriétés annexes seront démontrées dans le chapitre sur les équations différentielles, section \ref{SECooNKICooDnOFTD}.
-
-\begin{theorem}[Cauchy-Lipschitz\cite{SandrineCL,ZPNooLNyWjX}] \label{ThokUUlgU}
-    Nous considérons l'équation différentielle
-    \begin{subequations}        \label{XtiXON}
-        \begin{numcases}{}
-            y'(t)=f\big( t,y(t) \big)\\
-            y(t_0)=y_0
-        \end{numcases}
-    \end{subequations}
-    avec \( f\colon U=I\times \Omega\to \eR^n\) où \( I\) est ouvert dans \( \eR\) et \( \Omega\) ouvert dans \( \eR^n\). Nous supposons que \( f\) est continue sur \( U\) et localement Lipschitz\footnote{Définition \ref{DefJSFFooEOCogV}. Notons que nous ne supposons pas que \( f\) soit une contraction.} par rapport à \( y\). 
-    
-    Alors il existe un intervalle \( J\subset I\) sur lequel la solution au problème est unique. De plus toute solution du problème est une restriction de cette solution à une partie de \( J\). La solution sur \( J\) (dite «solution maximale») est de classe \( C^1\).
-\end{theorem}
-\index{théorème!Cauchy-Lipschitz}
-
-% Il serait tentant de mettre ce théorème dans la partie sur les équations différentielles, mais ce n'est pas aussi simple :
-% Il est utilisé pour calculer la transformée de Fourier de la Gaussienne (lemme LEMooPAAJooCsoyAJ) dans le chapitre sur la transformée de Fourier.
-
-\begin{proof}
-    Nous divisions la preuve en plusieurs étapes (même pas toutes simples).
-    \begin{subproof}
-    \item[Cylindre de sécurité et espace fonctionnel]
-
-    Précisons l'espace fonctionnel \( \mF\) adéquat. Soient \( V\) et \( W\) les voisinages de \( t_0\) et \( y_0\) sur lesquels \( f\) est localement Lipschitz. Nous considérons les quantités suivantes :
-    \begin{enumerate}
-        \item
-            \( M=\sup_{V\times W}f\) ;
-        \item
-            \( r>0\) tel que \( \overline{ B(y_0,r) }\subset V\)
-        \item
-            \( T>0\) tel que \( \overline{ B(t_0,T) }\subset W\) et \( T<r/M\).
-    \end{enumerate}
-    Nous considérons alors l'ensemble
-    \begin{equation}
-        \mF=C^0\big( \overline{ B(t_0,T) },\overline{ B(y_0,r) } \big)
-    \end{equation}
-    que nous munissons de la norme uniforme. Par le lemme \ref{LemdLKKnd} l'espace \( \big( \mF,\| . \|_{\infty} \big)\) est complet.
-
-    \item[Une application \( \Phi\colon \mF\to \mF\)]
-
-
-    Si \( y\) est une solution de l'équation différentielle considérée, elle vérifie
-    \begin{equation}        \label{EqPGLwcL}
-        y(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
-    \end{equation}
-    Ceci nous incite à considérer l'opérateur \( \Phi\colon \mF\to \mF\) défini par
-    \begin{equation}
-        \Phi(y)(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
-    \end{equation}
-
-    Pour que l'application \( \Phi\) soit utile nous devons montrer que pour tout \( y\in \mF\),
-    \begin{itemize}
-        \item l'application \( \Phi(y)\) est bien définie,
-        \item pour tout \( t\in\overline{ B(y_0,r) }\) nous avons \( \Phi(y)(t)\in\overline{ B(t_0,T) }\),
-        \item l'application $\Phi(y)\colon    \overline{ B(t_0,T) }\to \overline{ B(y_0,r)} $ est continue.
-    \end{itemize}
-    Attention : nous ne prétendons pas que \( \Phi\) elle-même soit continue. C'est parti.
-    \begin{subproof}
-    \item[\( \Phi(y)\) est bien définie]
-            
-        Il faut montrer que l'intégrale converge. Le calcul de \( \Phi(y)(t)\) ne se fait qu'avec \( t\in \overline{ B(t_0,T) }\). Vu que \( u\) prend ses valeurs dans \( \mathopen[ t_0 , t \mathclose]\) et que \( y\in\mF\), le nombre \( y(u)\) est toujours dans \( \overline{ B(y_0,r) }\). Ceci pour dire que dans l'intégrale, la fonction \( f\) n'est considérée que sur \( \mathopen[ t_0 , t \mathclose]\times \overline{ B(y_0,r) }\subset V\times W\). La fonction \( f\) est donc uniformément majorable, et l'intégrale ne pose pas de problèmes.
-
-    \item[\( \Phi(y)(t)\in \overline{ B(t_0,T) }\)]
-
-    Prouvons que \( \Phi(y)(t)\in\overline{ B(y_0,r) }\). Pour cela, notons que
-    \begin{equation}
-        | \Phi(y)(t)-y_0 |\leq \int_{t_0}^t |f\big( u,y(u) \big)|du\leq | t-t_0 |\| f \|_{\infty}.
-    \end{equation}
-    Étant donné que \( t\in\overline{ B(t_0,T) }\) nous avons \( | t-t_0 |\leq r/M\) et donc \( | \Phi(y)(t)-y_0 |\leq r\).
-
-    \item[\( \Phi(y)\) est continue]
-
-        Nous pourrions invoquer le théorème \ref{ThoKnuSNd}, mais nous allons le faire à la main. Soit \( s_0\in B(t_0,T)\) et prouvons que \( \Phi(y)\) est continue en \( s_0\). Pour cela nous prenons \( s\in B(s_0,\delta)\) et nous calculons :
-        \begin{equation}
-            | \Phi(y)(s)-\Phi(y)(s_0) |\leq \int_{s_0}^s|f\big( u,y(u) \big)|du\leq | s_0-s |\| f \|_{\infty}.
-        \end{equation}
-        C'est le fait que \( f\) soit bornée dans le cylindre de sécurité qui fait en sorte que cela tende vers zéro lorsque \( s\to s_0\).
-    \end{subproof}
-
-
-    
-    L'équation \eqref{EqPGLwcL} signifie que \( y\) est un point fixe de \( \Phi\). L'espace \( \mF\) étant complet le théorème de point fixe de Picard (théorème \ref{ThoEPVkCL}) s'applique. Nous allons montrer qu'il existe un \( p\in\eN\) tel que \( \Phi^p\) soit contractante. Par conséquent \( \Phi^p\) aura un unique point fixe qui sera également unique point fixe de \( \Phi\) par la remarque \ref{remIOHUJm}.
-    
-\item[Contractante]
-
-    Prouvons donc que \( \Phi^p\) est contractante pour un certain \( p\). Pour cela nous commençons par montrer la formule suivante par récurrence :
-    \begin{equation}        \label{EqRAdKxT}
-        \big\| \Phi^p(x)(t)-\Phi^p(y)(t) \big\|\leq \frac{ k^p| t-t_0 |^p }{ p! }\| x-y \|_{\infty}
-    \end{equation}
-    pour tout \( x,y\in\mF\), et pour tout \( t\in\overline{ B(t_0,T) }\). Pour \( p=0\) la formule \eqref{EqRAdKxT} est vérifiée parce que \( \| x-y \|_{\infty}\) est le supremum de \( \| x(t)-y(t) \|\) pour \( t\in\overline{ B(t_0,T) }\). Supposons que la formule soit vraie pour \( p\) et calculons pour \( p+1\). Pour tout \( t\in\overline{ B(t_0,T) }\) nous avons
-    \begin{subequations}
-        \begin{align}
-            \big\| \Phi^{p+1}(x)(t)-\Phi^{p+1}(y)(t) \big\|&\leq \left| \int_{t_0}^t\big\| f\big( u,\Phi^p(x)(u) \big)-f\big( u,\Phi^p(y)(u) \big) \big\|du \right| \\
-            &\leq \left| \int_{t_0}^tk\| \Phi^p(x)(u)-\Phi^p(y)(u) \|du \right|    \label{subIKYixF}\\
-            &\leq \left| \int_{t_0}^tk\frac{ k^p| t-t_0 | }{ p! }\| x-y \|_{\infty} \right| \label{subxkNjiV} \\
-            &=\frac{ k^{p+1}| t-t_0 |^{p+1} }{ (p+1)! }\| x-y \|_{\infty}.
-        \end{align}
-    \end{subequations}
-    Justifications :
-    \begin{itemize}
-        \item \eqref{subIKYixF} parce que \( f\) est Lipschitz.
-        \item \eqref{subxkNjiV} par hypothèse de récurrence.
-    \end{itemize}
-    La formule \eqref{EqRAdKxT} est maintenant établie. Nous pouvons maintenant montrer que \( \Phi^p\) est une contraction pour un certain \( p\). Pour tout \( t\in \overline{ B(t_0,T) }\) nous avons
-    \begin{equation}
-         \| \Phi^p(x)(t)-\Phi^p(y)(t) \|\leq \frac{ k^p }{ t! }| t-t_0 |^p\| x-y \|_{\infty}     \leq \frac{ k^pT^p }{ p! }\| x-y \|_{\infty}
-    \end{equation}
-    où nous avons utilisé le fait que \( | t-t_0 |^p<T^p\). En prenant le supremum sur \( t\) des deux côtés il vient
-    \begin{equation}
-        \| \Phi^p(x)-\Phi^p(y) \|_{\infty}\leq\frac{ k^pT^p }{ p! }\| x-y \|_{\infty}.
-    \end{equation}
-    Le membre de droite tend vers zéro lorsque \( p\to\infty\) parce que \( k<1\) et \( T^p/p!\to 0\)\footnote{C'est le terme général du développement de \(  e^{T}\) qui est une série convergente.}. Nous concluons donc que \( \Phi^p\) est une contraction pour un certain \( p\).
-
-\item[Conclusion]
-
-    L'unique point fixe de \( \Phi\) est alors l'unique solution continue de l'équation différentielle \eqref{XtiXON}. Par ailleurs l'équation elle-même \( y'=f(t,y)\) demande implicitement que \( y\) soit dérivable et donc continue. Nous concluons que l'unique point fixe de \( \Phi\) est l'unique solution de l'équation différentielle donnée. Cette dernière est automatiquement \( C^1\) parce que si \( y\) est continue alors \( u\mapsto f(u,y(u))\) est continue, c'est à dire que \( y'\) est continue.
-
-\item[Unicité]
-
-    Nous passons maintenant à la partie «prolongement maximum» du théorème. Soient \( x_1\) et \( x_2\) deux solutions maximales du problème \eqref{XtiXON} sur des intervalles \( I_1\) et \( I_2\) respectivement. Les intervalles \( I_1\) et \( I_2\) contiennent \( \overline{ B(t_0,r) }\) sur lequel \( x_1=x_2\) par unicité.
-    
-    
-    Nous allons maintenant montrer que pour tout \( t\geq t_0\) pour lequel \( x_1\) ou \( x_2\) est défini, \( x_1(t)\) et \( x_2(t)\) sont définis et sont égaux. Le raisonnement sur \( t\leq t_0\) est similaire.
-    
-    Supposons que l'ensemble des \( t\geq t_0\) tels que \( x_1=x_2\) soit ouvert à droite, c'est à dire soit de la forme \( \mathopen[ t_0 ,b [\). Dans ce cas, soit \( x_1\) soit \( x_2\) (soit les deux) cesse d'exister en \( b\). En effet si nous avions les fonctions \( x_i\) sur \(\mathopen[ t_0 , b+\epsilon [\) alors l'équation \( x_1=x_2\) définirait un fermé dans \( \mathopen[ t_0 , b+\epsilon [\). Supposons pour fixer les idées que \( x_1\) cesse d'exister : le domaine de \( x_1\) (parmi les \( t\geq 0\)) est \( \mathopen[ t_0 , b [\) et sur ce domaine nous avons \( x_1=x_2\). Dans ce cas \( x_1\) pourrait être prolongé en \( x_2\) au-delà de \( b\). Si \( x_1\) et \( x_2\) s'arrêtent d'exister en même temps en \( b\), alors nous avons bien \( x_1=x_2\).
-
-    Nous devons donc traiter le cas où \( x_1=x_2\) sur \( \mathopen[ t_0 , b \mathclose]\) alors que \( x_1\) et \( x_2\) existent sur \( \mathopen[ t_0 , b+\epsilon [\) pour un certain \( \epsilon\).
-
-    Nous pouvons appliquer le théorème d'existence locale au problème
-    \begin{subequations}
-        \begin{numcases}{}
-            y'=f(t,y)\\
-            y(b)=x_1(b).
-        \end{numcases}
-    \end{subequations}
-    Il existe un voisinage de \( b\) sur lequel la solution est unique. Sur ce voisinage nous devons donc avoir \( x_1=x_2\), ce qui contredit le fait que \( x_1\neq x_2\) en dehors de \( \mathopen[ t_0 , b \mathclose]\).
-
-    Donc \( x_1\) et \( x_2\) existent et sont égaux sur au moins \( I_1\cup I_2\).
-    \end{subproof}
-\end{proof}
-
-Le théorème de Cauchy-Lipschitz donne existence et unicité d'une solution maximale. Cependant cette solution peut ne pas exister partout où les hypothèses sur \( f\) sont remplies. En d'autres termes, il peut arriver que \( f\) soit Lipschitz jusqu'à \( t_1\), mais que la solution maximale ne soit définie que jusqu'en \( t_2<t_1\). Ce cas fait l'objet du théorème d'explosion en temps fini \ref{CorGDJQooNEIvpp}.
-
-Sous quelque hypothèses nous pouvons nous assurer de l'existence d'une solution unique sur tout \( \eR\).
-
-\begin{theorem}[Cauchy-Lipschitz global\cite{ooJZJPooAygxpk,KXjFWKA}]       \label{THOooZIVRooPSWMxg}
-    Soit un intervalle \( I\) de \( \eR\), \( y_0\in \eR^n\), \( t_0\in I\) et une fonction continue \( f\colon I\times \eR^n\to \eR^n\) telle que pour tout compact \( K\) dans \( I\), il existe \( k>0\) tel que
-    \begin{equation}
-        \| f(t,y_1)-f(t,y_2) \|\leq k\| y_1-y_2 \|
-    \end{equation}
-    pour tout \( t\in K\) et \( y_1,y_2\in \eR^n\).
-
-    Alors le problème
-    \begin{subequations}        \label{EQSooBNREooUTfbMH}
-        \begin{numcases}{}
-            y'(t)=f\big( t,y(t) \big)\\
-            y(t_0)=y_0
-        \end{numcases}
-    \end{subequations}
-    possède une unique solution \( y\colon I\to \eR^n\) sur \( I\).
-\end{theorem}
-
-\begin{proof}
-    Soit un intervalle compact \( K\) dans \( I\) et contenant \( t_0\). Nous notons \( \ell\) le diamètre de \( K\). Sur l'espace \( E=C^0(K,\eR^n)\) nous considérons la topologie uniforme : \( (E,\| . \|_{\infty})\). C'est un espace complet par le lemme \ref{LemdLKKnd} (nous utilisons le fait que \( \eR^n\) soit complet, théorème \ref{ThoTFGioqS}). Nous allons utiliser l'application suivante :
-    \begin{equation}        \label{EQooJUTBooILBKoE}
-        \begin{aligned}
-            \Phi\colon E&\to E \\
-            \Phi(y)(t)&=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds
-        \end{aligned}
-    \end{equation}
-    Démontrons quelque faits à propos de \( \Phi\).
-    \begin{subproof}
-        \item[La définition fonctionne bien]
-            Nous devons commencer par prouver que cette application est bien définie. Si \( y\in E\) alors \( f\) et \( y\) sont continues; l'application \( s\mapsto f\big(s,y(s)\big)\) est donc également continue. L'intégrale de cette fonction sur le compact \( \mathopen[ t_0 , t \mathclose]\) ne pose alors pas de problèmes. En ce qui concerne la continuité de \( \phi(y)\) sous l'hypothèse que \( y\) soit continue,
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(y)(t') \|\leq \int_t^{t'}\| f(s,y(s)) \|ds\leq M| t-t' |
-    \end{equation}
-    où \( M\) est une majoration de \( \| s\mapsto f\big( s,y(s) \big) \|_{\infty,K}\).
-
-        \item[Si \( y\) est solution alors \( \Phi(y)=y\)]
-
-            Supposons que \( y\) soit une solution de l'équation différentielle \eqref{EQSooBNREooUTfbMH}. Alors, vu que \( y'(t)=f\big( t,y(t) \big)\) nous avons :
-            \begin{equation}
-                y(t)=y_0+\int_{t_0}^ty'(s)ds=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds=\Phi(y)(t).
-            \end{equation}
-            
-        \item[Si \( \Phi(y)=y\) alors \( y\) est solution]
-
-            Nous avons, pour tout \( t\) :
-            \begin{equation}
-                y(t)=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds.
-            \end{equation}
-            Le membre de droite est dérivable par rapport à \( t\), et la dérivée fait \(  f\big( t,y(t) \big)   \). Donc le membre de gauche est également dérivable et nous avons bien
-            \begin{equation}
-                y'(t)=f\big( t,y(t) \big).
-            \end{equation}
-            De plus \( y(t_0)=y_0+\int_{t_0}^{t_0}\ldots=y_0\).
-    \end{subproof}
-    
-    Nous sommes encore avec \( K\) compact et \( E=C^0(K,\eR^n)\) muni de la norme uniforme. Nous allons montrer que \( \Phi\) est une contraction de \( E\) pour une norme bien choisie.
-
-    \begin{subproof}
-        \item[Une norme sur \( E\)]
-            Pour \( y\in E\) nous posons
-            \begin{equation}
-                \| y \|_k=\max_{t\in K}\big(  e^{-k| t-t_0 |}\| y(t) \| \big).
-            \end{equation}
-            Ce maximum est bien définit et fini parce que la fonction de \( t\) dedans est une fonction continue sur le compact \( K\). Cela est également une norme parce que si \( \| y \|_k=0\) alors \(  e^{-k| t-t_0 |}\| y(t) \|=0\) pour tout \( t\). Étant donné que l'exponentielle ne s'annule pas, \( \| y(t) \|=0\) pour tout \( t\).
-        \item[Équivalence de norme]
-
-            Nous montrons que les normes \( \| . \|_k\) et \( \| . \|_{\infty}\) sont équivalentes\footnote{Définition \ref{DefEquivNorm}} :
-            \begin{equation}        \label{EQooSQYWooBTXvDL}
-                \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\leq \| y \|_{\infty}
-            \end{equation}
-            pour tout \( y\in E\). Pour la première inégalité, \( \ell\geq | t-t_0 |\) pour tout \( t\in K\), et \( k>0\), donc
-            \begin{equation}
-                \| y(t) \| e^{-k\ell}\leq  e^{-k| t-t_0 |}\| y(t) \|.
-            \end{equation}
-            En prenant le maximum des deux côtés, \( \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\). 
-
-            En ce qui concerne la seconde inégalité dans \eqref{EQooSQYWooBTXvDL}, \( k| t-t_0 |\geq 0\) et donc \(  e^{-k| t-t_0 |}<1\).
-
-    \end{subproof}
-    Vu que les normes \( \| . \|_{\infty}\) et \( \| . \|_k\) sont équivalentes, l'espace \( (E,\| . \|_k)\) est tout autant complet que \( (E,\| . \|_{\infty})\). Nous démontrons à présent que \( \Phi\) est une contraction dans \( (E,\|  \|_k)\). 
-
-    Soient \( y,z\in E\). Si \( t\geq t_0\) nous avons
-    \begin{subequations}        \label{SUBEQSooEXVYooDkyTuB}
-        \begin{align}
-            \| \Phi(y)(t)-\Phi(z)(t) \|&\leq \int_{t_0}^t\| f\big( s,y(s) \big)-f\big( s,z(s) \big) \|ds\\
-            &\leq k\int_{t_0}^t\| y(s)-z(s) \|ds.
-        \end{align}
-    \end{subequations}
-    Il convient maintenant de remarquer que
-    \begin{equation}
-        \| y(t) \|= e^{-k| t-t_0 |} e^{k| t-t_0 |}\| y(t) \|\leq \| y \|_k e^{k| t-t_0 |}.
-    \end{equation}
-    Nous pouvons avec ça prolonger les inégalités \eqref{SUBEQSooEXVYooDkyTuB} par
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(z)(t) \|\leq k\| y-z \|_k\int_{t_0}^t e^{k| s-t_0 |}ds=k\| y-z \|_k\int_{t_0}^t e^{k(s-t_0)}ds
-    \end{equation}
-    où nous avons utilisé notre supposition \( t\geq t_0\) pour éliminer les valeurs absolues. L'intégrale peut être faite explicitement, mais nous en sommes arrivés à un niveau de fainéantise tellement inconcevable que
-
-\lstinputlisting{tex/sage/sageSnip014.sage}
-
-Au final, si \( t\geq t_0\),
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t-t_0)}-1 \big).
-    \end{equation}
-    Si \( t\leq t_0\), il faut retourner les bornes de l'intégrale avant d'y faire rentrer la norme parce que \( \| \int_0^1f \|\leq \int_0^1\| f \|\), mais ça ne marche pas avec \( \| \int_1^0f \|\). Pour \( t\leq t_0\) tout le calcul donne
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t_0-t)}-1 \big).
-    \end{equation}
-    Les deux inéquations sont valables a fortiori en mettant des valeurs absolues dans l'exponentielle, de telle sorte que pour tout \( t\in K\) nous avons
-    \begin{equation}
-        e^{-k| t_0-t |}\| \phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big( 1- e^{-k| t_0-t |} \big).
-    \end{equation}
-    En prenant le supremum sur \( t\),
-    \begin{equation}
-        \| \Phi(y)-\Phi(z) \|_k\leq \| y-z \|_k(1- e^{-k\ell}),
-    \end{equation}
-    mais \( 0<(1- e^{e-k\ell})<1\), donc \( \Phi\) est contractante pour la norme \( \| . \|_k\). Vu que \( (E,\| . \|_k)\) est complet, l'application \( \Phi\) y a un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-
-    Ce point fixe est donc l'unique solution de l'équation différentielle de départ.
-
-    \begin{subproof}
-        \item[Existence et unicité sur \( I\)]
-            Il nous reste à prouver que la solution que nous avons trouvée existe sur \( I\) : jusqu'à présent nous avons démontré l'existence et l'unicité sur n'importe quel compact dans \( I\).
-
-            Soit une suite croissante de compacts \( K_n\) contenant \( t_0\) (par exemple une suite exhaustive comme celle du lemme \ref{LemGDeZlOo}). Nous avons en particulier
-            \begin{equation}
-                I=\bigcup_{n=0}^{\infty}K_n.
-            \end{equation}
-        \item[Existence sur \( I\)]
-        
-            Soit \( y_n\) l'unique solution sur \( K_n\). Il suffit de poser
-            \begin{equation}
-                y(t)=y_n(t)
-            \end{equation}
-            pour \( n\) tel que \( t\in K_n\). Cette définition fonctionne parce que si \( t\in K_n\cap K_m\), il y a forcément un des deux qui est inclus à l'autre et le résultat d'unicité sur le plus grand des deux donne \( y_n(t)=y_m(t)\).
-
-        \item[Unicité sur \( I\)]
-
-            Soient \( y\) et \(z \) des solutions sur \( I\); vu que \( I\) n'est pas spécialement compact, le travail fait plus haut ne permet pas de conclure que \( y=z\). 
-
-            Soit \( t\in I\). Alors \( t\in K_n\) pour un certain \( n\) et \( y\) et \( z\) sont des solutions sur \( K_n\) qui est compact. L'unicité sur \( K_n\) donne \( y(t)=z(t)\).
-    \end{subproof}
-\end{proof}
-
-\begin{normaltext}
-    Il y a d'autres moyens de prouver qu'une solution existe globalement sur \( \eR\). Si \( f\) est globalement bornée, le théorème d'explosion en temps fini donne quelque garanties, voir \ref{NORMooZROGooZfsdnZ}.
-\end{normaltext}
-
-Le théorème suivant donne une version du théorème de Cauchy-Lipschitz lorsque la fonction \( f\) dépend d'un paramètre. Ce théorème n'utilise rien de fondamentalement nouveau. Nous le donnons seulement pour montrer que l'on peut choisir l'espace \( \mF\) de façon un peu maligne pour élargir le résultat. Si vous voulez un théorème de Cauchy-Lipschitz avec paramètre vraiment intéressant, allez voir le théorème \ref{PROPooPYHWooIZhQST}.
-
-\begin{theorem}[Cauchy-Lipschitz avec paramètre\cite{MonCerveau,ooXVPAooTQUIRw}]           \label{THOooDTCWooSPKeYu}
-    Soit un intervalle ouvert \( I\) de \( \eR\), un connexe ouvert \( \Omega\) de \( \eR^n\) et un intervalle ouvert \( \Lambda\) de \( \eR^d\). Soit une fonction \( f\colon I\times \Omega\times \Lambda\to \eR^n\) continue et localement Lipschitz en \( \Omega\). Soient \( t_0\in I\), \( y_0\in \Omega\) et \( \lambda_0\in \Lambda\). Il existe un voisinage compact de \( (t_0,y_0,\lambda_0)\) sur lequel le problème
-    \begin{subequations}
-        \begin{numcases}{}
-            y'_{\lambda}(t)=f\big( t,y_{\lambda}(t),\lambda \big)\\
-            y_{\lambda}(t_0)=y_0
-        \end{numcases}
-    \end{subequations}
-    possède une unique solution. De plus \( (t,\lambda)\mapsto y_{\lambda}(t)\) est continue\footnote{Ici, la surprise est que ce soit continu par rapport à \( \lambda\). Le fait qu'elle le soit par rapport à \( t\) est clair depuis le départ parce que c'est finalement rien d'autre que le Cauchy-Lipschitz vieux et connu.}.
-\end{theorem}
-
-\begin{proof}
-
-    \begin{probleme}
-        Ceci est une idée de la preuve. Je n'ai pas vérifié toutes les étapes. Soyez prudent.
-
-    \end{probleme}
-
-    D'abord nous avons un voisinage compact \( V\times \overline{ B(y_0,r) }\times \Lambda_0\) de \( (t_0,y_0,\lambda_0)\) sur lequel $f$ est bornée. Ensuite nous récrivons l'équation différentielle sous la forme
-    \begin{subequations}
-        \begin{numcases}{}
-            \frac{ \partial y }{ \partial t }(t,\lambda)=f\big( t,y(t,\lambda),\lambda \big)\\
-            y(t_0,\lambda)=y_0.
-        \end{numcases}
-    \end{subequations}
-    pour une fonction \( y\colon V\times \Lambda_0\to \eR^n\).
-
-    Nous posons \( \mF=C^0\big( V\times\Lambda_0 ,\eR^n\big)\) et nous y définissons l'application
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon \mF&\to \mF \\
-            \Phi(y)(t,\lambda)&=y_0+\int_{t_0}^tf\big( s,y(s,\lambda),\lambda \big)ds. 
-        \end{aligned}
-    \end{equation}
-    Il y a plein de vérifications à faire\cite{ooXVPAooTQUIRw}, mais je parie que \( \Phi\) est bien définie, et que une de ses puissances est une contraction de \( (\mF,\| . \|_{\infty})\). L'unique point fixe est une solution de notre problème et est dans \( C^0\), donc \( (t,\lambda)\mapsto y(t,\lambda)=y_{\lambda}(t)\) est de classe \( C^0\), c'est à dire continue.
-\end{proof}
-
-\begin{normaltext}
-    Ce théorème marque un peu la limite de ce que l'on peut faire avec la méthode des points fixes dans le cadre de Cauchy-Lipschitz : nous sommes limités à la continuité de la solution parce que les espaces \( C^p\) ne sont pas complets\footnote{Par exemple, le théorème de Stone-Weierstrass \ref{ThoGddfas} nous dit que la limite uniforme de polynômes (de classe \(  C^{\infty}\)) peut n'être que continue. Voir aussi le thème \ref{THMooOCXTooWenIJE}.}. Il n'y a donc pas d'espoir d'adapter la méthode pour prouver que si \( f\) est de classe \( C^p\) alors \( (t,\lambda)\mapsto y_{\lambda}(t)\) est de classe \( C^p\). On peut, à \( \lambda\) fixé prouver que \( t\mapsto y_{\lambda}(t)\) est de classe \( C^p\) (utiliser une récurrence), mais pas plus.
-
-    La régularité \( C^1\) de \( y\) par rapport à la condition initiale sera l'objet du théorème \ref{THOooSTHXooXqLBoT}. Ce résultat n'est vraiment pas facile et utilise des ingrédients bien autres qu'un point fixe. Ensuite la régularité \( C^p\) par rapport à la condition initiale et par rapport à un paramètre seront presque des cadeaux (proposition \ref{PROPooINLNooDVWaMn} et \ref{PROPooPYHWooIZhQST}).
-\end{normaltext}
-
-\begin{example}[\cite{ooSBHXooOMnaTC}]          \label{EXooJXIGooQtotMc}
-    Nous savons que le théorème de Picard permet de trouver le point fixe par itération de la contraction à partir d'un point quelconque. Tentons donc de résoudre
-    \begin{subequations}
-        \begin{numcases}{}
-            y'(t)=y(t)\\
-            y(0)=1
-        \end{numcases}
-    \end{subequations}
-    dont nous savons depuis l'enfance que la solution est l'exponentielle. Partons donc de la fonction constante \( y_0=1\), et appliquons la contraction \eqref{EQooJUTBooILBKoE} :
-    \begin{equation}
-        u_1=1+\int_0^1u_0(s)ds=1+t.
-    \end{equation}
-    Ensuite
-    \begin{equation}
-        u_2=1+\int_0^t(1+s)ds=1+t+\frac{ t^2 }{2}.
-    \end{equation}
-    Et on voit que les itérations suivantes vont donner l'exponentielle.
-
-    Nous sommes évidemment en droit de se dire que nous avons choisi un bon point de départ. Tentons le coup avec une fonction qui n'a rien à voir avec l'exponentielle : \( u_0(x)=\sin(x)\).
-
-    Le programme suivant permet de faire de belles investigations numériques en partant d'à peu près n'importe quelle fonction :
-
-\lstinputlisting{tex/sage/picard_exp.py}
-
-    Ce programme fait \( 30\) itérations depuis la fonction \( \sin(x)\) pour tenter d'approximer \( \exp(x)\). Pour donner une idée, après \( 7\) itérations nous avons la fonction suivante :
-    \begin{equation}
-        \frac{1}{ 60 }x^5+\frac{1}{ 24 }x^4+\frac{ 1 }{2}x^2+2x-\sin(x)+1.
-    \end{equation}
-    Nous voyons que les coefficients sont des factorielles, mais pas toujours celles correspondantes à la puissance, et qu'il manque certains termes par rapport au développement de l'exponentielle que nous connaissons. Bref, le polynôme qui se met en face de \( \sin(x)\) s'adapte tout seul pour compenser.
-
-    Et après \( 30\) itérations, ça donne quoi ? Voici un graphe de l'erreur entre \( u_{30}(x)\) et \( \exp(30)\) :
-    
-    
-\begin{center}
-   \input{auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks}
-\end{center}
-
-    Pour donner une idée, \( \exp(10)\simeq 22000\). Donc il y a une faute de \( 0.01\) sur \( 22000\). Pas mal.
-
-\end{example}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Cauchy-Arzella}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{theorem}[Cauchy-Arzela\cite{ClemKetl}]   \label{ThoHNBooUipgPX}
-    Nous considérons le système d'équation différentielles
-    \begin{subequations}        \label{EqTXlJdH}
-        \begin{numcases}{}
-            y'=f(t,y)\\
-            y(t_0)=y_0.
-        \end{numcases}
-    \end{subequations}
-    avec \( f\colon U\to \eR^n\), continue où \( U\) est ouvert dans \( \eR\times \eR^n\). Alors il existe un voisinage fermé \( V\) de \( t_0\) sur lequel une solution \( C^1\) du problème \eqref{EqTXlJdH} existe.
-\end{theorem}
-\index{théorème!Cauchy-Arzela}
-
-\begin{proof}[Idée de la démonstration]
-    Nous considérons \( M=\| f \|_{\infty}\) et \( K\), l'ensemble des fonctions \( M\)-Lipschitz sur \( U\). Nous prouvons que \( (K,\| . \|_{\infty})\) est compact. Ensuite nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon K&\to K \\
-            \Phi(f)(t)&=x_0+\int_{t_0}^tf\big( u,f(u) \big)du. 
-        \end{aligned}
-    \end{equation}
-    Après avoir prouvé que \( \Phi\) était continue, nous concluons qu'elle a un point fixe par le théorème de Schauder \ref{ThovHJXIU}.
-\end{proof}
-
-\begin{remark}
-    Quelque remarques.
-    \begin{enumerate}
-        \item
-    Les théorème de Cauchy-Lipschitz et Cauchy-Arzella donnent des existences pour des équations différentielles du type \( y'=f(t,y)\). Et si nous avons une équation du second ordre ? Alors il y a la méthode de la réduction de l'ordre qui permet de transformer une équation différentielle d'ordre élevé en un système d'ordre \( 1\).
-\item
-    Ces théorèmes posent des \emph{conditions initiales} : la valeur de \( y\) est donnée en un point, et la méthode de la réduction de l'ordre permet de donner l'existence de solutions d'un problème d'ordre \( k\) en donnant les valeurs de \( y(0)\), \( y'(0)\), \ldots \( y^{(k-1)}(0)\). C'est à dire de la fonction et de ses dérivées en un point. Rien n'est dit sur l'existence de \emph{conditions aux bords}.
-    \end{enumerate}
-    Ces deux points sont illustrés dans les exemples \ref{EXooSHMMooHVfsMB} et \ref{EXooJNOMooYqUwTZ}.
-\end{remark}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-                    \section{Théorèmes d'inversion locale et de la fonction implicite}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Mise en situation}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Dans un certain nombre de situation, il n'est pas possible de trouver des solutions explicites aux équations qui apparaissent. Néanmoins, l'existence «théorique» d'une telle solution est souvent déjà suffisante. C'est l'objet du théorème de la fonction implicite.
-
-Prenons par exemple la fonction sur $\eR^2$ donnée par 
-\begin{equation}
-    F(x,y)=x^2+y^2-1.
-\end{equation}
-Nous pouvons bien entendu regarder l'ensemble des points donnés par $F(x,y)=0$. C'est le cercle dessiné à la figure \ref{LabelFigCercleImplicite}.
-\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite. Si on donne \( x\), nous ne pouvons pas savoir si nous parlons de \( P\) ou de \( P'\).}
-\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
-
-%\ref{LabelFigCercleImplicite}.
-%\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite.}
-%\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
-
-Nous ne pouvons pas donner le cercle sous la forme $y=y(x)$ à cause du $\pm$ qui arrive quand on prend la racine carrée. Mais si on se donne le point $P$, nous pouvons dire que \emph{autour de $P$}, le cercle est la fonction
-\begin{equation}
-    y(x)=\sqrt{1-x^2}.
-\end{equation}
-Tandis que autour du point $P'$, le cercle est la fonction
-\begin{equation}
-    y(x)=-\sqrt{1-x^2}.
-\end{equation}
-Autour de ces deux point, donc, le cercle est donné par une fonction. Il n'est par contre pas possible de donner le cercle autour du point $Q$ sous la forme d'une fonction.
-
-Ce que nous voulons faire, en général, est de voir si l'ensemble des points tels que
-\begin{equation}
-    F(x_1,\ldots,x_n,y)=0
-\end{equation}
-peut être donné par une fonction $y=y(x_1,\ldots,x_n)$. En d'autre termes, est-ce qu'il existe une fonction $y(x_1,\ldots,x_n)$ telle que
-\begin{equation}
-    F\big( x_1,\ldots,x_n,y(x_1,\ldots,x_n)\big)=0.
-\end{equation}
-
-Plus généralement, soit une fonction
-\begin{equation}
-    \begin{aligned}
-        F\colon D\subset \eR^n\times \eR^m&\to \eR^m \\
-        (x,y)&\mapsto \big( F_1(x,y),\ldots, F_m(x,y) \big) 
-    \end{aligned}
-\end{equation}
-avec $x = (x_1,\ldots, x_n)$ et $y = (y_1,\ldots,y_m)$. Pour chaque $x$ fixé, on s'intéresse aux solutions du système de $m$ équations $F(x,y) = 0$ pour les inconnues $y$ ; en particulier, on voudrait pouvoir écrire $y = \varphi(x)$ vérifiant $F(x,\varphi(x)) = 0$.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème d'inversion locale}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma} \label{LemGZoqknC}
-    Soit \( E\) un espace de Banach (métrique complet) et \( \mO\) un ouvert de \( E\). Nous considérons une \( \lambda\)-contraction \( \varphi\colon \mO\to E\). Alors l'application
-    \begin{equation}
-    f\colon x\mapsto x+\varphi(x)
-    \end{equation}
-    est un homéomorphisme entre \( \mO\) et un ouvert de \( E\). De plus \( f^{-1}\) est Lipschitz de constante plus petite ou égale à \( (1-\lambda)^{-1}\).
-\end{lemma}
-Cette proposition utilise le théorème de point fixe de Picard \ref{ThoEPVkCL},
-et sera utilisée pour démontrer le théorème d'inversion locale \ref{ThoXWpzqCn}.
-% note que garder deux lignes ici est important pour vérifier les références vers le futur : la seconde ligne peut être ignorée, pas la seconde.
-
-\begin{proof}
-        Soient \( x_1,x_2\in\mO\). Nous posons \( y_1=f(x_1)\) et \( y_2=f(x_2)\). En vertu de l'inégalité de la proposition \ref{PropNmNNm} nous avons
-        \begin{subequations}    \label{subEqEBJsBfz}
-            \begin{align}
-            \big\| f(x_2)-f(x_1) \big\|&=\big\| x_2+\varphi(x_2)-x_1-\varphi(x_1) \big\|\\
-        &\geq \Big|        \| x_2-x_1 \|-\big\| \varphi(x_2)-\varphi(x_1) \big\|  \Big|\\
-    &\geq   (1-\lambda)\| x_2-x_1 \|.
-            \end{align}
-        \end{subequations}
-        À la dernière ligne les valeurs absolues sont enlevées parce que nous savons que ce qui est à l'intérieur est positif. Cela nous dit d'abord que \( f\) est injective parce que \( f(x_2)=f(x_1)\) implique \( x_2=x_1\). Donc \( f\) est inversible sur son image. Nous posons \( A=f(\mO)\) et nous devons prouver que que \( f^{-1}\colon A\to \mO\) est continue, Lipschitz de constante majorée par \( (1-\lambda)^{-1}\) et que \( A\) est ouvert.
-
-    Les inéquations \eqref{subEqEBJsBfz} nous disent que
-    \begin{equation}
-    \big\| f^{-1}(y_1)-f^{-1}(y_2) \big\|\leq \frac{ \| y_1-y_2 \| }{ 1-\lambda },
-    \end{equation}
-    c'est à dire que
-    \begin{equation}
-        f^{-1}\big( B(y,r) \big)\subset B\big( f^{-1}(y),\frac{ r }{ 1-\lambda } \big),
-    \end{equation}
-    ce qui signifie que \( f^{-1}\) est Lipschitz de constante souhaitée et donc continue.
-
-    Il reste à prouver que \( f(\mO)\) est ouvert. Pour cela nous prenons \( y_0=f(x_0)\) dans \( f(\mO)\) est nous prouvons qu'il existe \( \epsilon\) tel que \( B(y_0,\epsilon)\) soit dans \( f(\mO)\). Il faut donc que pour tout \( y\in B(y_0,\epsilon)\), l'équation \( f(x)=y\) ait une solution. Nous considérons l'application
-    \begin{equation}
-        L_y\colon x\mapsto y-\varphi(x).
-    \end{equation}
-    Ce que nous cherchons est un point fixe de \( L_y\) parce que si \( L_y(x)=x\) alors \( y=x+\varphi(x)=f(x)\). Vu que
-    \begin{equation}
-        \big\| L_y(x)-L_y(x') \big\|=\big\| \varphi(x)-\varphi(x') \big\|\leq\lambda\| x-x' \|,
-    \end{equation}
-    l'application \( L_y\) est une contraction de constante \( \lambda\). Par ailleurs \( x_0\) est un point fixe de \( L_{y_0}\), donc en vertu de la caractérisation \eqref{EqDZvtUbn} des fonctions Lipschitziennes, 
-    \begin{equation}
-        L_{y_0}\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B\big( L_{y_0}(x_0),\lambda\delta \big) }=\overline{ B(x_0,\lambda\delta) }.
-    \end{equation}
-    Vu que pour tout \( y\) et \( x\) nous avons \( L_y(x)=L_{y_0}(x)+y-y_0\),
-    \begin{equation}
-    L_y\big( \overline{ B(x_0,\delta) } \big)=L_{y_0}\big( \overline{ B(x_0,\delta) } \big)+(y-y_0)\subset \overline{ B(x_0,\lambda\delta) }+(y-y_0)\subset \overline{ B(x_0),\lambda\delta+\| y-y_0 \| }.
-    \end{equation}
-    Si \( \epsilon<(1-\lambda)\delta\) alors \( \lambda\delta+\| y-y_0 \|<\delta\). Un tel choix de \( \epsilon>0\) est possible parce que \( \lambda<1\). Pour une telle valeur de \( \epsilon\) nous avons
-    \begin{equation}
-        L_y\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B(x_0,\delta) }.
-    \end{equation}
-    Par conséquent \( L_y\) est une contraction sur l'espace métrique complet \( \overline{ B(x_0,\delta) }\), ce qui signifie que \( L_y\) y possède un point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-\end{proof}
-
-Le théorème d'inversion locale s'énonce de la façon suivante dans \( \eR^n\) :
-\begin{theorem}[Inversion locale dans \( \eR^n\)]    \label{THOooQGGWooPBRNEX}      % Ne pas mettre de label ici parce qu'il faut référencer l'autre, celui dans Banach.
-    Soit \( f\in C^k(\eR^n,\eR^n)\) et \( x_0\in \eR^n\). Si \( df_{x_0}\) est inversible, alors il existe un voisinage ouvert \( U\) de \( x_0\) et \( V\) de \( f(x_0)\) tels que \( f\colon U\to V\) soit un \( C^k\)-difféomorphisme. (c'est à dire que \( f^{-1}\) est également de classe \( C^k\))
-\end{theorem}
-
-Nous allons le démontrer dans le cas un peu plus général (mais pas plus cher\footnote{Sauf la justification de la régularité de l'application \( A\mapsto A^{-1}\)}) des espaces de Banach en tant que conséquence du théorème de point fixe de Picard \ref{ThoEPVkCL}.
-
-\begin{theorem}[Inversion locale dans un espace de Banach\cite{OWTzoEK}] \label{ThoXWpzqCn}
-    Soit une fonction \( f\in C^p(E,F)\) avec \( p\geq 1\) entre deux espaces de Banach. Soit \( x_0\in E\) tel que \( df_{x_0}\) soit une bijection bicontinue\footnote{En dimension finie, une application linéaire est toujours continue et d'inverse continu.}. Alors il existe un voisinage ouvert \( V\) de \( x_0\) et \( W\) de \( f(x_0)\) tels que
-    \begin{enumerate}
-        \item
-        \( f\colon V\to W\) soit une bijection,
-    \item
-        \( f^{-1}\colon W\to V\) soit de classe \( C^p\).
-    \end{enumerate}
-\end{theorem}
-\index{application!différentiable}
-\index{théorème!inversion locale}
-
-\begin{proof}
-    Nous commençons par simplifier un peu le problème. Pour cela, nous considérons la translation \( T\colon x\mapsto x+x_0 \) et l'application linéaire
-    \begin{equation}
-        \begin{aligned}
-            L\colon \eR^n&\to \eR^n \\
-            x&\mapsto (df_{x_0})^{-1}x
-        \end{aligned}
-    \end{equation}
-    qui sont tout deux des difféomorphismes (\( L\) en est un par hypothèse d'inversibilité). Quitte à travailler avec la fonction \( k=L\circ f\circ T\), nous pouvons supposer que \( x_0=0\) et que \( df_{x_0}=\mtu\). Pour comprendre cela il faut utiliser deux fois la formule de différentielle de fonction composée de la proposition \ref{EqDiffCompose} :
-    \begin{equation}
-        dk_0(u)=dL_{(f\circ T)(0)}\Big( df_{T(0)}dT_0(u) \Big).
-    \end{equation}
-    Vu que \( L\) est linéaire, sa différentielle est elle-même, c'est à dire \( dL_{(f\circ T)(0)}=(df_{x_0})^{-1}\), et par ailleurs \( dT_0=\mtu\), donc
-    \begin{equation}
-        dk_0(u)=(df_{x_0})^{-1}\Big( df_{x_0}(u) \Big)=u,
-    \end{equation}
-    ce qui signifie bien que \( dk_0=\mtu\). Pour tout cela nous avons utilisé en plein le fait que \( df_{x_0}\) était inversible.
-
-Nous posons \( g=f-\mtu\), c'est à dire \( g(x)=f(x)-x\), qui a la propriété \( dg_0=0\). Étant donné que \( g\) est de classe \( C^1\), l'application\footnote{Ici \( \GL(F)\) est l'ensemble des applications linéaires, inversibles et continues de \( F\) dans lui-même. Ce ne sont pas spécialement des matrices parce que nous n'avons pas d'hypothèses sur la dimension de \( F\), finie ou non.}
-    \begin{equation}
-        \begin{aligned}
-            dg\colon E&\to \GL(F) \\
-            x&\mapsto dg_x 
-        \end{aligned}
-    \end{equation}
-    est continue. En conséquence de quoi nous avons un voisinage \( U'\) de \( 0 \) pour lequel
-    \begin{equation}    \label{EqSGTOfvx}
-        \sup_{x\in U'}\| dg_x \|<\frac{ 1 }{2}.
-    \end{equation}
-    Maintenant le théorème des accroissements finis \ref{ThoNAKKght} (\ref{val_medio_2} pour la dimension finie) nous indique que pour tout \( x,x'\in U'\) nous avons\footnote{Ici nous supposons avoir choisi \( U'\) convexe afin que tous les \( a\in \mathopen[ x , x' \mathclose]\) soient bien dans \( U'\) et donc soumis à l'inéquation \eqref{EqSGTOfvx}, ce qui est toujours possible, il suffit de prendre une boule.}
-    \begin{equation}
-        \| g(x')-g(x) \|\leq \sup_{a\in\mathopen[ x , x' \mathclose]}\| dg_a \| \cdot \| x-x' \|\leq \frac{ 1 }{2}\| x-x' \|,
-    \end{equation}
-    ce qui prouve que \( g\) est une contraction au moins sur l'ouvert \( U'\). Nous allons aussi donner une idée de la façon dont \( f\) fonctionne : si \( x_1,x_2\in U'\) alors
-    \begin{subequations}
-        \begin{align}
-            \| x_1-x_2 \|&=\| g(x_1)-f(x_1)-g(x_2)+f(x_2) \| \\
-            &\leq \| g(x_1)-g(x_2) \|+\| f(x_1)-f(x_2) \|\\
-            &\leq \frac{ 1 }{2}\| x_1-x_2 \|+\| f(x_1)-f(x_2) \|,
-        \end{align}
-    \end{subequations}
-    ce qui montre que
-    \begin{equation}
-        \| x_1-x_2 \|\leq 2\| f(x_1)-f(x_2) \|.
-    \end{equation}
-    Maintenant que nous savons que \( g\) est contractante de constante \( \frac{ 1 }{2}\) et que \( f=g+\mtu\) nous pouvons utiliser la proposition \ref{LemGZoqknC} pour conclure que \( f\) est un homéomorphisme sur un ouvert \( U\) (partie de \( U'\)) de \( E\) et \( f^{-1}\) a une constante de Lipschitz plus petite ou égale à \( (1-\frac{ 1 }{2})^{-1}=2\).
-
-    Nous allons maintenant prouver que \( f^{-1}\) est différentiable et que sa différentielle est donnée par \( (df^{-1})_{f(x)}=(df_x)^{-1}\).
-
-    Soient \( a,b\in U\) et \( u=b-a\). Étant donné que \( f\) est différentiable en \( a\), il existe une fonction \( \alpha\in o(\| u \|)\) telle que
-    \begin{equation}
-        f(b)-f(a)-df_a(u)=\alpha(u).
-    \end{equation}
-    En notant \( y_a=f(a)\) et \( y_b=f(b)\) et en appliquant \( (df_a)^{-1}\) à cette dernière équation,
-    \begin{equation}
-        (df_a)^{-1}(y_b-y_a)-u=(df_a)^{-1} \big( \alpha(u) \big).
-    \end{equation}
-    Vu que \( df_a\) est bornée (et son inverse aussi), le membre de droite est encore une fonction \( \beta\) ayant la propriété \( \lim_{u\to 0}\beta(u)/\| u \|=0\); en réordonnant les termes,
-    \begin{equation}
-        b-a=(df_a)^{-1}(y_b-y_a)+\beta(u)
-    \end{equation}
-    et donc
-    \begin{equation}
-        f^{-1}(y_b)-f^{-1}(y_a)-(df_a)^{-1}(y_b-y_a)=\beta(u),
-    \end{equation}
-    ce qui prouve que \( f^{-1}\) est différentiable et que \( (df^{-1})_{y_a}=(df_a)^{-1}\).
-
-    La différentielle \( df^{-1}\) est donc obtenue par la chaine
-    \begin{equation}
-    \xymatrix{%
-        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}     &   U'\ar[r]^-{df}&\GL(F)\ar[r]^-{\Inv}&\GL(F)
-       }
-    \end{equation}
-    où l'application \( \Inv\colon \GL(F)\to \GL(F)\) est l'application \( X\mapsto X^{-1}\) qui est de classe \(  C^{\infty}\) par le théorème \ref{ThoCINVBTJ}. D'autre part, par hypothèse \( df\) est une application de classe \( C^{k-1}\) et donc au minimum \( C^0\) parce que \( k\geq 1\). Enfin, l'application \( f^{-1}\colon f(U)\to U\) est continue (parce que la proposition \ref{LemGZoqknC} précise que \( f\) est un homéomorphisme). Donc toute la chaine est continue et \( df^{-1}\) est continue. Cela entraine immédiatement que \( f^{-1}\) est \( C^1\) et donc que toute la chaine est \( C^1\).
-
-    Par récurrence nous obtenons la chaine
-    \begin{equation}
-    \xymatrix{%
-        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}_-{C^{k-1}}     &   U'\ar[r]^-{df}_-{C^{k-1}}&\GL(F)\ar[r]^-{\Inv}_-{ C^{\infty}}&\GL(F)
-       }
-    \end{equation}
-    qui prouve que \( df^{-1}\) est \( C^{k-1} \) et donc que \( f^{-1}\) est \( C^k\). La récurrence s'arrête ici parce que \( df\) n'est pas mieux que \( C^{k-1}\).
-\end{proof}
-
-\begin{normaltext}      \label{NomDJMUooTRUVkS}
-    Nous allons montrer que l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
-            A&\mapsto \sqrt{A} 
-        \end{aligned}
-    \end{equation}
-    est une difféomorphisme.
-
-    Cependant \( S^{++}(n,\eR)\) n'est pas un ouvert de \( \eM(n,\eR)\) et nous ne savons pas ce qu'est la différentielle d'une application non définie sur un ouvert. Nous allons donc en réalité montrer que l'application racine carré existe sur un voisinage de chacun des points de \( S^{++}(n,\eR)\). Et comme une union quelconque d'ouverts est un ouvert, la fonction \( f\) sera bien définie sur un ouvert de \( \eM(n,\eR)\).
-\end{normaltext}
-
-\begin{lemma}       \label{LemLBFOooDdNcgy}
-    L'application 
-    \begin{equation}
-        \begin{aligned}
-            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
-            A&\mapsto A^2 
-        \end{aligned}
-    \end{equation}
-    est un \(  C^{\infty}\)-difféomorphisme.
-\end{lemma}
-
-\begin{proof}
-    Prouvons d'abord que \( f\) prend ses valeurs dans \( S^{++}(n,\eR)\). Si \( A\in S^{++}(n,\eR)\) alors par la diagonalisation \ref{ThoeTMXla} elle s'écrit \( A=QDQ^{-1}\) où \( D\) est diagonale avec des nombres strictement positifs sur la diagonale. Avec cela, \( A^2=QD^2Q^{-1}\) où \( D^2\) contient encore des nombres strictement positifs sur la diagonale.
-
-    L'application \( f\) étant essentiellement des polynôme en les entrées de \( A\), elle est de classe \( C^{\infty}\).
-
-    Passons à l'étude de la différentielle. Comme mentionné en \ref{NomDJMUooTRUVkS} nous allons en réalité voir \( f\) sur un ouvert de \( \eM(n,\eR)\) autour de \( A\in S^{++}(n,\eR)\). Par conséquent si \( A\in S^{++}(n,\eR)\),
-    \begin{subequations}
-        \begin{align}
-            df\colon S^{++}(n,\eR)&\to \aL\big( \eM(n,\eR),\eM(n,\eR) \big)\\
-            df_A\colon \eM(n,\eR)&\to \eM(n,\eR).
-        \end{align}
-    \end{subequations}
-    Le calcul de \( df_A\) est facile. Soit \( u\in \eM(n,\eR)\) et faisons le calcul en utilisant la formule du lemme \eqref{LemdfaSurLesPartielles} :
-    \begin{subequations}
-        \begin{align}
-            df_A(u)&=\Dsdd{ f(A+tu) }{t}{0}\\
-            &=\Dsdd{ A^2+tAu+tuA+t^2u^2 }{t}{0}\\
-            &=Au+uA.
-        \end{align}
-    \end{subequations}
-    Nous allons utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} à la fonction \( f\). Dans la suite, \( A\) est une matrice de \( S^{++}(n,\eR)\).
-
-    \begin{subproof}
-        \item[\( df_A\) est injective]
-            Soit \( M\in \eM(n,\eR)\) dans le noyau de \( df_A\). En posant \( M'=A^{-1}MQ\) nous avons \( M=QM'Q^{-1}\) et on applique \( df_A\) à \( QM'Q^{-1}\) :
-            \begin{equation}
-                df_A(QM'Q^{-1})=Q\big( DM+MD \big)Q^{-1}.
-            \end{equation}
-            où \( D=\begin{pmatrix}
-                \lambda_1    &       &       \\
-                    &   \ddots    &       \\
-                    &       &   \lambda_n
-                \end{pmatrix}\) avec \( \lambda_i>0\). La matrice \( D\) est inversible. Nous avons \( M'=-DM'D^{-1}\), et en coordonnées,
-                \begin{subequations}
-                    \begin{align}
-                        M'_{ij}&=-\sum_{kl}D_{ikM'_{kl}}D^{-1}_{lj}\\
-                        &=-\sum_{kl}\lambda_i\delta_{ik}M'_{kl}\frac{1}{ \lambda_j }\delta_{lj}\\
-                        &=-\frac{ \lambda_i }{ \lambda_i }M'_{ij}.
-                    \end{align}
-                \end{subequations}
-                C'est à dire que \( M'_{ij}=-\frac{ \lambda_i }{ \lambda_j }M'_{ij}\) avec \( -\frac{ \lambda_i }{ \lambda_j }<0\). Cela implique \( M'=0\) et par conséquent \( M=0\).
-            \item[\( df_A\) est surjective]
-                Soit \( N\in \eM(n,\eR)\); nous cherchons \( M\in \eM(n,\eR)\) tel que \( df_A(M)=N\). Nous posons \( N'=Q^{-1} NQ\) et \( M=QM'Q^{-1}\), ce qui nous donne à résoudre \( df_D(M')=N'\). Passons en coordonnées :
-                \begin{subequations}
-                    \begin{align}
-                        (DM'+M'D)_{ij}&=\sum_k(\delta_{ik}\lambda_iM'_{kj}+M'_{ik}\delta_{kj}\lambda_j)\\&
-                        &=M'_{ij}(\lambda_i+\lambda_j)
-                    \end{align}
-                \end{subequations}
-                où \( \lambda_i+\lambda_j\neq 0\). Il suffit donc de prendre la matrice \( M'\) donnée par
-                \begin{equation}
-                    M'_{ij}=\frac{1}{ \lambda_i+\lambda_j }N'_{ij}
-                \end{equation}
-                pour que \( df_A(M')=N'\).
-    \end{subproof}
-    
-    Le théorème d'inversion locale donne un voisinage \( V\) de $A$ dans \( \eM(n,\eR)\) et un voisinage \( W\) de \( A^2\) dans \( \eM(n,\eR)\) tels que \( f\colon V\to W\) soit une bijection  et \( f^{-1}\colon W\to V\) soit de même régularité, en l'occurrence \( C^{\infty}\).
-\end{proof}
-
-\begin{remark}
-    Oui, il y a des matrices non symétriques qui ont une unique racine carré.
-\end{remark}
-
-La proposition suivante, qui dépend du le théorème d'inversion locale par le lemme \ref{LemLBFOooDdNcgy}, donne plus de régularité à la décomposition polaire donnée dans le théorème \ref{ThoLHebUAU}.
-\begin{proposition}[Décomposition polaire : cas réel (suite)]       \label{PropWCXAooDuFMjn}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \gO(n,\eR)\times S^{++}(n,\eR)&\to \GL(n,\eR) \\
-            (Q,S)&\mapsto SQ 
-        \end{aligned}
-    \end{equation}
-    est un difféomorphisme de classe \( C^{\infty}\).
-\end{proposition}
-
-\begin{proof}
-    Si \( M\) est donnée dans \( \GL(n,\eR)\) alors la décomposition polaire \( M=QS\) est donnée par \( S=\sqrt{MM^t}\) et \( Q=MS^{-1}\). Autrement dit, si nous considérons la fonction de décomposition polaire
-    \begin{equation}
-        f\colon \gO(n,\eR)\times S^{++}(n,\eR)\to \GL(n,\eR)
-    \end{equation}
-    alors
-    \begin{equation}
-        f^{-1}(M)=\big(  M(\sqrt{MM^t})^{-1},\sqrt{MM^t}  \big).
-    \end{equation}
-    Nous avons vu dans le lemme \ref{LemLBFOooDdNcgy} que la racine carré était un \( C^{\infty}\)-difféomorphisme. Le reste n'étant que des produits de matrice, la régularité est de mise.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de la fonction implicite}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous énonçons et le démontrons le théorème de la fonction implicite dans le cas d'espaces de Banach.
-\begin{theorem}[Théorème de la fonction implicite dans Banach\cite{SNPdukn}] \label{ThoAcaWho}
-    Soient \( E\), \( F\) et \( G\) des espaces de Banach et des ouverts \( U\subset E\), \( V\subset F\). Nous considérons une fonction \( f\colon U\times V\to G\) de classe \( C^r\) telle que\footnote{La notation \( d_y\) est la différentielle partielle de la définition \ref{VJM_CtSKT}.}
-    \begin{equation}
-        d_yf_{(x_0,y_0)}\colon F\to G
-    \end{equation}
-    soit un isomorphisme pour un certain \( (x_0,y_0)\in U\times V\).
-
-    Alors nous avons des voisinages \( U_0\) de \( x_0\) dans \( E\) et \( W_0\) de \( f(x_0,y_0)\) dans \( G\) et une fonction de classe \( C^r\) 
-    \begin{equation}
-        g\colon U_0\times W_0\to V
-    \end{equation}
-    telle que 
-    \begin{equation}
-        f\big( x,g(x,w) \big)=w
-    \end{equation}
-    pour tout \( (x,w)\in U_0\times W_0\).
-    
-    Cette fonction \( g\) est unique au sens suivant : il existe un voisinage \( V_0 \) de \( y_0\) tel que si \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) satisfont à \( f(x,y)=w\) alors \( y=g(x,w)\). Autrement dit, la fonction \( g\colon U_0\times W_0\to V_0\) est unique.
-\end{theorem}
-\index{théorème!fonction implicite dans Banach}
-
-\begin{proof}
-    Nous commençons par considérer la fonction
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon U\times V&\to E\times G \\
-            (x,y)&\mapsto \big( x,f(x,y) \big)
-        \end{aligned}
-    \end{equation}
-    et sa différentielle 
-    \begin{subequations}
-        \begin{align}
-            d\Phi_{(x_0,y_0)}(u,v)&=\Dsdd{ \big( x_0+tu,f(x_0+tu,y_0+tv) \big) }{t}{0}\\
-            &=\left( \Dsdd{ x_0+tu }{t}{0},\Dsdd{ f(x_0+tu,y_0+tv) }{t}{0} \right)\\
-            &=\left( u,df_{(x_0,y_0)}(u,v) \right).
-        \end{align}
-    \end{subequations}
-    Nous utilisons alors la proposition \ref{PropLDN_nHWDF} pour conclure que
-    \begin{equation}
-        d\Phi_{(x_0,y_0)}(u,v)=\big( u,(d_1f)_{(x_0,y_0)}(u)+(d_2f)_{(x_0,y_0)}(v) \big),
-    \end{equation}
-    mais comme par hypothèse \( (d_2f)_{(x_0,y_0)}\colon F\to G\) est un isomorphisme, l'application \( d\Phi_{(x_0,y_0)}\colon E\times F\to E\times G\) est également un isomorphisme. Par conséquent le théorème d'inversion locale \ref{ThoXWpzqCn} nous indique qu'il existe un voisinage \( \mO\) de \( (x_0,y_0)\) et \( \mP\) de \( \Phi(x_0,y_0)\) tels que \( \Phi\colon \mO\to \mP\) soit une bijection et \( \Phi^{-1}\colon \mP\to \mO\) soit de classe \( C^r\). Vu que \( \mP\) est un voisinage de
-    \begin{equation}
-        \Phi(x_0,y_0)=\big( x_0,f(x_0,y_0) \big),
-    \end{equation}
-    nous pouvons par \ref{PropDXR_KbaLC} le choisir un peu plus petit de telle sorte à avoir \( \mP=U_0\times W_0\) où \( U_0\) est un voisinage de \( x_0\) et \( W_0\) un voisinage de \( f(x_0,y_0)\). Dans ce cas nous devons obligatoirement aussi restreindre \( \mO\) à \( U_0\times V_0\) pour un certain voisinage \( V_0\) de \( y_0\). L'application \( \Phi^{-1}\) a obligatoirement la forme
-    \begin{equation}    \label{EqMHT_QrHRn}
-        \begin{aligned}
-            \Phi^{-1}\colon U_0\times W_0&\to U_0\times V_0 \\
-            (x,w)&\mapsto \big( x,g(x,w) \big) 
-        \end{aligned}
-    \end{equation}
-    pour une certaine fonction \( g\colon U_0\times W_0\to V\). Cette fonction \( g\) est la fonction cherchée parce qu'en appliquant \( \Phi\) à \eqref{EqMHT_QrHRn}, 
-    \begin{equation}
-        (x,w)=\Phi\big( x,g(x,w) \big)=\Big( x,f\big( x,g(x,w) \big) \Big),
-    \end{equation}
-    qui nous dit que pour tout \( x\in U_0\) et tout \( w\in W_0\) nous avons
-    \begin{equation}
-        f\big( x,g(x,w) \big)=w.
-    \end{equation}
-
-    Si vous avez bien suivi le sens de l'équation \eqref{EqMHT_QrHRn} alors vous avez compris l'unicité. Sinon, considérez \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) tels que \( f(x,y)=w\). Alors \( \big( x,f(x,y) \big)=(x,w)\) et 
-    \begin{equation}
-        \Phi(x,y)=(x,w).
-    \end{equation}
-    Mais vu que \( \Phi\colon U_0\times V_0\to U_0\times W_0\) est une bijection, cette relation définit de façon univoque l'élément \( (x,y)\) de \( U_0\times V_0\), qui ne sera autre que \( g(x,w)\).
-\end{proof}
-
-Le théorème de la fonction implicite s'énonce de la façon suivante pour des espaces de dimension finie.
-% Attention : avant de citer ce théorème, voir s'il est suffisant. Ici \varphi a une variable; dans l'autre énoncé il en a deux.
-\begin{theorem}[Théorème de la fonction implicite en dimension finie]   \label{ThoRYN_jvZrZ}
-    Soit une fonction \( F\colon \eR^n\times \eR^m\to \eR^m\) de classe \( C^k\) et \( (\alpha,\beta)\in \eR^n\times \eR^m\) tels que
-    \begin{enumerate}
-        \item
-            \( F(\alpha,\beta)=0\),
-        \item
-            \( \frac{ \partial (F_1,\ldots, F_m) }{ \partial (y_1,\ldots, y_m) }\neq 0\), c'est à dire que \( (d_yF)_{(\alpha,\beta)} \) est inversible.
-    \end{enumerate}
-    Alors il existe un voisinage ouvert \( V\) de \( \alpha\) dans \( \eR^n\), un voisinage ouvert \( W\) de \( \beta\) dans \( \eR^m\) et une application \( \varphi\colon V\to W\) de classe \( C^k\)  telle que pour tout \( x\in V\) on ait
-    \begin{equation}
-        F\big( x,\varphi(x) \big)=0.
-    \end{equation}
-    De plus si \( (x,y)\in V\times W\) satisfait à \( F(x,y)=0\), alors \( y=\varphi(x)\).
-\end{theorem}
-\index{théorème!fonction implicite dans \( \eR^n\)}
-
-\begin{remark}\label{RemPYA_pkTEx}
-    Notons que cet énoncé est tourné un peu différemment en ce qui concerne le nombre de variables dont dépend la fonction implicite : comparez
-    \begin{subequations}
-        \begin{align}
-            f\big( x,g(x,w) \big)=w\\
-            F\big( x,\varphi(x) \big)=0.
-        \end{align}
-    \end{subequations}
-    Le deuxième est un cas particulier du premier en posant 
-    \begin{equation}
-        F(x,y)=f(x,y)-f(x_0,y_0)
-    \end{equation}
-    et donc en considérant \( w\) comme valant la constante \( f(x_0,y_0)\); dans ce cas la fonction \( g\) ne dépend plus que de la variable \( x\).
-
-\end{remark}
-
-\begin{example}
-    La remarque \ref{RemPYA_pkTEx} signifie entre autres que le théorème \ref{ThoAcaWho} est plus fort que \ref{ThoRYN_jvZrZ} parce que le premier permet de choisir la valeur d'arrivée. Parlons de l'exemple classique du cercle et de la fonction \( f(x,y)=x^2+y^2\). Nous savons que
-    \begin{equation}
-        f(\alpha,\beta)=1.
-    \end{equation}
-    Alors le théorème \ref{ThoAcaWho} nous donne une fonction \( g\) telle que
-    \begin{equation}
-        f(x,g(x,r))=r
-    \end{equation}
-    tant que \( x\) est proche de \( \alpha\), que \( r\) est proche de \( 1\) et que \( g\) donne des valeurs proches de \( \beta\).
-
-    L'énoncé \ref{ThoRYN_jvZrZ} nous oblige à travailler avec la fonction \( F(x,y)=x^2+y^2-1\), de telle sorte que
-    \begin{equation}
-        F(\alpha,\beta)=0,
-    \end{equation}
-    et que nous ayons une fonction \( \varphi\) telle que
-    \begin{equation}
-        F(x,\varphi(x))=0.
-    \end{equation}
-    La fonction \( \varphi\) ne permet donc que de trouver des points sur le cercle de rayon \( 1\).
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Exemple}
-%---------------------------------------------------------------------------------------------------------------------------
-    
-Le théorème de la fonction implicite a pour objet de donner l'existence de la fonction $\varphi$. Maintenant nous pouvons dire beaucoup de choses sur les dérivées de $\varphi$ en considérant la fonction
-\begin{equation}
-    x\mapsto F\big( x,\varphi(x) \big).
-\end{equation}
-Par définition de $\varphi$, cette fonction est toujours nulle. En particulier, nous pouvons dériver l'équation
-\begin{equation}
-    F\big( x,\varphi(x) \big)=0,
-\end{equation}
-et nous trouvons plein de choses.
-
-
-Prenons par exemple la fonction
-\begin{equation}
-    F\big( (x,y),z \big)=ze^z-x-y,
-\end{equation}
-et demandons nous ce que nous pouvons dire sur la fonction $z(x,y)$ telle que
-\begin{equation}
-    F\big( x,y,z(x,y) \big)=0,
-\end{equation}
-c'est à dire telle que
-\begin{equation}        \label{EqDefZImplExemple}
-    z(x,y) e^{z(x,y)}-x-y=0.
-\end{equation}
-pour tout $x$ et $y\in\eR$. Nous pouvons facilement trouver $z(0,0)$ parce que
-\begin{equation}
-    z(0,0) e^{z(0,0)}=0,
-\end{equation}
-donc $z(0,0)=0$.
-
-Nous pouvons dire des choses sur les dérivées de $z(x,y)$. Voyons par exemple $(\partial_xz)(x,y)$. Pour trouver cette dérivée, nous dérivons la relation \eqref{EqDefZImplExemple} par rapport à $x$. Ce que nous trouvons est
-\begin{equation}
-    (\partial_xz)e^z+ze^z(\partial_xz)-1=0.
-\end{equation}
-Cette équation peut être résolue par rapport à $\partial_xz$~:
-\begin{equation}
-    \frac{ \partial z }{ \partial x }(x,y)=\frac{1}{ e^z(1+z) }.
-\end{equation}
-Remarquez que cette équation ne donne pas tout à fait la dérivée de $z$ en fonction de $x$ et $y$, parce que $z$ apparaît dans l'expression, alors que $z$ est justement la fonction inconnue. En général, c'est la vie, nous ne pouvons pas faire mieux.
-
-Dans certains cas, on peut aller plus loin. Par exemple, nous pouvons calculer cette dérivée au point $(x,y)=(0,0)$ parce que $z(0,0)$ est connu :
-\begin{equation}
-    \frac{ \partial z }{ \partial x }(0,0)=1.
-\end{equation}
-Cela est pratique pour calculer, par exemple, le développement en Taylor de $z$ autour de $(0,0)$.
-
-\begin{example}
-    Est-ce que l'équation \( e^{y}+xy=0\) définit au moins localement une fonction \( y(x)\) ? Nous considérons la fonction
-    \begin{equation}
-        f(x,y)=\begin{pmatrix}
-            x    \\ 
-            e^{y}+xy    
-        \end{pmatrix}
-    \end{equation}
-    La différentielle de cette application est
-    \begin{equation}
-            df_{(0,0)}(u)=\frac{ d }{ dt }\Big[ f(tu_1,tu_2) \Big]_{t=0}
-            =\frac{ d }{ dt }\begin{pmatrix}
-                tu_1    \\ 
-                e^{tu_2}+t^2u_1u_2    
-            \end{pmatrix}_{t=0}
-            =\begin{pmatrix}
-                u_1    \\ 
-                u_2    
-            \end{pmatrix}.
-    \end{equation}
-    L'application \( f\) définit donc un difféomorphisme local autour des points \( (x_0,y_0)\) et \( f(x_0,y_0)\). Soit \( (u,0)\) un point dans le voisinage de \( f(x_0,y_0)\). Alors il existe un unique \( (x,y)\) tel que
-    \begin{equation}
-        f(x,y)=\begin{pmatrix}
-               x \\ 
-            e^y+xy    
-        \end{pmatrix}=
-        \begin{pmatrix}
-            u    \\ 
-                0
-        \end{pmatrix}.
-    \end{equation}
-    Nous avons automatiquement \( x=u\) et \( e^y+xy=0\). Notons toutefois que pour que ce procédé donne effectivement une fonction implicite \( y(x)\) nous devons avoir des points de la forme \( (u,0)\) dans le voisinage de \( f(x_0,y_0)\).
-\end{example}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Von Neumann}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}[\cite{KXjFWKA}]
-    Soit \( G\), un sous groupe fermé de \( \GL(n,\eR)\) et 
-    \begin{equation}
-        \mL_G=\{ m\in \eM(n,\eR)\tq  e^{tm}\in G\,\forall t\in\eR \}.
-    \end{equation}
-    Alors \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\).
-\end{lemma}
-
-\begin{proof}
-    Si \( m\in\mL_G\), alors \( \lambda m\in\mL_G\) par construction. Le point délicat à prouver est le fait que si \( a,b\in \mL_G\), alors \( a+b\in\mL_G\). Soit \( a\in \eM(n,\eR)\); nous savons qu'il existe une fonction \( \alpha_a\colon \eR\to \eM\) telle que
-    \begin{equation}
-        e^{ta}=\mtu+ta+\alpha_a(t)
-    \end{equation}
-    et 
-    \begin{equation}
-        \lim_{t\to 0} \frac{ \alpha(t) }{ t }=0.
-    \end{equation}
-    Si \( a\) et \( b\) sont dans \( \mL_G\), alors \(  e^{ta} e^{tb}\in G\), mais il n'est pas vrai en général que cela soit égal à \(  e^{t(a+b)}\). Pour tout \( k\in \eN\) nous avons
-    \begin{equation}
-        e^{a/k} e^{b/k}=\left( \mtu+\frac{ a }{ k }+\alpha_a(\frac{1}{ k }) \right)\left( \mtu+\frac{ b }{ k }+\alpha_b(\frac{1}{ k }) \right)=\mtu+\frac{ a+b }{2}+\beta\left( \frac{1}{ k } \right)
-    \end{equation}
-   où \( \beta\colon \eR\to \eM\) est encore une fonction vérifiant \( \beta(t)/t\to 0\). Si \( k\) est assez grand, nous avons
-   \begin{equation}
-       \left\| \frac{ a+b }{ k }+\beta(\frac{1}{ k })  \right\|<1,
-   \end{equation}
-   et nous pouvons profiter du lemme \ref{LemQZIQxaB} pour écrire alors
-   \begin{equation}
-       \left(  e^{a/k} e^{b/k} \right)^k= e^{k\ln\big(\mtu+\frac{ a+b }{ k }+\beta(\frac{1}{ k })\big)}.
-   \end{equation}
-   Ce qui se trouve dans l'exponentielle est
-   \begin{equation}
-       k\left[ \frac{ a+b }{ k }+\alpha( \frac{1}{ k })+\sigma\left( \frac{ a+b }{ k }+\alpha(\frac{1}{ k }) \right) \right].
-   \end{equation}
-   Les diverses propriétés vues montrent que le tout tend vers \( a+b\) lorsque \( k\to \infty\). Par conséquent
-   \begin{equation}
-       \lim_{k\to \infty} \left(  e^{a/k} e^{b/k} \right)^k= e^{a+b}.
-   \end{equation}
-   Ce que nous avons prouvé est que pour tout \( t\), \(  e^{t(a+b)}\) est une limite d'éléments dans \( G\) et est donc dans \( G\) parce que ce dernier est fermé.
-\end{proof}
-
-Vu que \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\), nous pouvons considérer un supplémentaire \( M\).
-
-\begin{lemma}   \label{LemHOsbREC}
-    Il n'existe pas se suites \( (m_k)\) dans \( M\setminus\{ 0 \}\) convergeant vers zéro et telle que \(  e^{m_k}\in G\) pour tout \( k\).
-\end{lemma}
-
-\begin{proof}
-    Supposons que nous ayons \( m_k\to 0\) dans \( M\setminus\{ 0 \}\) avec \(  e^{m_k}\in G\). Nous considérons les éléments \( \epsilon_k=\frac{ m_k }{ \| m_k \| }\) qui sont sur la sphère unité de \(\GL(n,\eR)\). Quitte à prendre une sous-suite, nous pouvons supposer que cette suite converge, et vu que \( M\) est fermé, ce sera vers \( \epsilon\in M\) avec \( \| \epsilon \|=1\). Pour tout \( t\in \eR\) nous avons
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}.
-    \end{equation}
-    En vertu de la décomposition d'un réel en partie entière et décimale, pour tout \( k\) nous avons \( \lambda_k\in \eZ\) et \( | \mu_k |\leq \frac{ 1 }{2}\) tel que \( t/\| m_k \|=\lambda_k+\mu_k\). Avec ça,
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}\exp\Big( \frac{ t }{ m_k }m_k \Big)=\lim_{k\to \infty}  e^{\lambda_km_k} e^{\mu_km_k}.
-    \end{equation}
-    Pour tout \( k\) nous avons \(  e^{\lambda_km_k}\in G\). De plus \( | \mu_k |\) étant borné et \( m_k\) tendant vers zéro nous avons \(  e^{\mu_km_k}\to 1\). Au final
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}\in G
-    \end{equation}
-    Cela signifie que \( \epsilon\in\mL_G\), ce qui est impossible parce que nous avions déjà dit que \( \epsilon\in M\setminus\{ 0 \}\).
-\end{proof}
-
-\begin{lemma}   \label{LemGGTtxdF}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \mL_G\times M&\to \GL(n,\eR) \\
-            l,m&\mapsto  e^{l} e^{m} 
-        \end{aligned}
-    \end{equation}
-    est un difféomorphisme local entre un voisinage de \( (0,0)\) dans \( \eM(n,\eR)\) et un voisinage de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
-\end{lemma}
-Notons que nous ne disons rien de \(  e^{\eM(n,\eR)}\). Nous n'allons pas nous embarquer à discuter si ce serait tout \( \GL(n,\eR)\)\footnote{Vu les dimensions y'a tout de même peu de chance.} ou bien si ça contiendrait ne fut-ce que \( G\).
-
-\begin{proof}
-    Le fait que \( f\) prenne ses valeurs dans \( \GL(n,\eR)\) est simplement dû au fait que les exponentielles sont toujours inversibles. Nous considérons ensuite la différentielle : si \( u\in \mL_G\) et \( v\in M\) nous avons
-    \begin{equation}
-        df_{(0,0)}(u,v)=\Dsdd{ f\big( t(u,v) \big) }{t}{0}=\Dsdd{  e^{tu} e^{tv} }{t}{0}=u+v.
-    \end{equation}
-    L'application \( df_0\) est donc une bijection entre \( \mL_G\times M\) et \( \eM(n,\eR)\). Le théorème d'inversion locale \ref{ThoXWpzqCn} nous assure alors que \( f\) est une bijection entre un voisinage de \( (0,0)\) dans \( \mL_G\times M\) et son image. Mais vu que \( df_0\) est une bijection avec \( \eM(n,\eR)\), l'image en question contient un ouvert autour de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
-\end{proof}
-
-\begin{theorem}[Von Neumann\cite{KXjFWKA,ISpsBzT,Lie_groups}]       \label{ThoOBriEoe}
-    Tout sous-groupe fermé de \( \GL(n,\eR)\) est une sous-variété de \( \GL(n,\eR)\).
-\end{theorem}
-\index{théorème!Von Neumann}
-\index{exponentielle!de matrice!utilisation}
-
-\begin{proof}
-    Soit \( G\) un tel groupe; nous devons prouver que c'est localement difféomorphe à un ouvert de \( \eR^n\). Et si on est pervers, on ne va pas faire localement difféomorphe à un ouvert de \( \eR^n\), mais à un ouvert d'un espace vectoriel de dimension finie. Nous allons être pervers.
-
-    Étant donné que pour tout \( g\in G\), l'application 
-    \begin{equation}
-        \begin{aligned}
-            L_g\colon G&\to G \\
-            h&\mapsto gh 
-        \end{aligned}
-    \end{equation}
-    est de classe \(  C^{\infty}\) et d'inverse \(  C^{\infty}\), il suffit de prouver le résultat pour un voisinage de \( \mtu\).
-
-    Supposons d'abord que \( \mL_G=\{ 0 \}\). Alors \( 0\) est un point isolé de \( \ln(G)\); en effet si ce n'était pas le cas nous aurions un élément \( m_k\) de \( \ln(G)\) dans chaque boule \( B(0,r_k)\). Nous aurions alors \( m_k=\ln(a_k)\) avec \( a_k\in G\) et donc
-    \begin{equation}
-        e^{m_k}=a_k\in G.
-    \end{equation}
-    De plus \( m_k\) appartient forcément à \( M\) parce que \( \mL_G\) est réduit à zéro. Cela nous donnerait une suite \( m_k\to 0\) dans \( M\) dont l'exponentielle reste dans \( G\). Or cela est interdit par le lemme \ref{LemHOsbREC}. Donc \( 0\) est un point isolé de \( \ln(G)\). L'application \(\ln\) étant continue\footnote{Par le lemme \ref{LemQZIQxaB}.}, nous en déduisons que \( \mtu\) est isolé dans \( G\). Par le difféomorphisme \( L_g\), tous les points de \( G\) sont isolés; ce groupe est donc discret et par voie de conséquence une variété.
-
-    Nous supposons maintenant que \( \mL_G\neq\{ 0 \}\). Nous savons par la proposition \ref{PropXFfOiOb} que 
-    \begin{equation}
-        \exp\colon \eM(n,\eR)\to \eM(n,\eR)
-    \end{equation}
-    est une application \(  C^{\infty}\) vérifiant \( d\exp_0=\id\). Nous pouvons donc utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} qui nous offre donc l'existence d'un voisinage \( U\) de \( 0\) dans \( \eM(n,\eR)\) tel que \( W=\exp(U)\) soit un ouvert de \( \GL(n,\eR)\) et que \( \exp\colon U\to W\) soit un difféomorphisme de classe \(  C^{\infty}\).
-
-    Montrons que quitte à restreindre \( U\) (et donc \( W\) qui reste par définition l'image de \( U\) par \( \exp\)), nous pouvons avoir \( \exp\big( U\cap\mL_G \big)=W\cap G\). D'abord \( \exp(\mL_G)\subset G\) par construction. Nous avons donc \( \exp\big( U\cap\mL_G \big)\subset W\cap G\). Pour trouver une restriction de \( U\) pour laquelle nous avons l'égalité, nous supposons que pour tout ouvert \( \mO\) dans \( U\), 
-    \begin{equation}
-        \exp\colon \mO\cap\mL_G\to \exp(\mO)\cap G
-    \end{equation}
-    ne soit pas surjective. Cela donnerait un élément de \( \mO\cap\complement\mL_G\) dont l'image par \( \exp\) n'est pas dans \( G\). Nous construisons ainsi une suite en considérant une boule \( B(0,\frac{1}{ k })\) inclue à \( U\) et \( x_k\in B(0,\frac{1}{ k })\cap\complement\mL_G\) vérifiant \(  e^{x_k}\in G\). Vu le choix des boules nous avons évidemment \( x_k\to 0\).
-
-    L'élément \(  e^{x_k}\) est dans \(  e^{\eM(n,\eR)}\) et le difféomorphisme du lemme \ref{LemGGTtxdF}\quext{Il me semble que l'utilisation de ce lemme manque à l'avant-dernière ligne de la preuve chez \cite{KXjFWKA}.} nous donne \( (l_k,m_k)\in \mL_G\times M\) tel que \(  e^{l_k} e^{m_k}= e^{x_k}\). À ce point nous considérons \( k\) suffisamment grand pour que \(  e^{x_k}\) soit dans la partie de l'image de \( f\) sur lequel nous avons le difféomorphisme. Plus prosaïquement, nous posons
-    \begin{equation}
-        (l_k,m_k)=f^{-1}( e^{x_k})
-    \end{equation}
-    et nous profitons de la continuité pour permuter la limite avec \( f^{-1}\) :
-    \begin{equation}
-        \lim_{k\to \infty} (l_k,m_k)=f^{-1}\big( \lim_{k\to \infty}  e^{x_k} \big)=f^{-1}(\mtu)=(0,0).
-    \end{equation}
-    En particulier \( m_k\to 0\) alors que \(  e^{m_k}= e^{x_k} e^{-l_k}\in G\). La suite \( m_k\) viole le lemme \ref{LemHOsbREC}. Nous pouvons donc restreindre \( U\) de telle façon à avoir
-    \begin{equation}
-        \exp\big( U\cap\mL_G \big)=W\cap G.
-    \end{equation}
-    Nous avons donc un ouvert de \( \mL_G\) (l'ouvert \( U\cap\mL_G\)) qui est difféomorphe avec l'ouvert \( W\cap G\) de \( G\). Donc \( G\) est une variété et accepte \( \mL_G\) comme carte locale.
-
-\end{proof}
-
-\begin{remark}
-    En termes savants, nous avons surtout montré que si \( G\) est un groupe de Lie d'algèbre de Lie \( \lG\), alors l'exponentielle donne un difféomorphisme local entre \( \lG\) et \( G\).
-\end{remark}
diff --git a/tex/front_back_matter/38_theme.tex b/tex/front_back_matter/38_theme.tex
index bc4e9ea96..87d242bcb 100644
--- a/tex/front_back_matter/38_theme.tex
+++ b/tex/front_back_matter/38_theme.tex
@@ -6,6 +6,6 @@
     \item 
         Décomposition de Dunford, théorème \ref{ThoRURcpW}. 
     \item 
-        Décomposition polaire \ref{ThoLHebUAU}.
+        Décomposition polaire \ref{ThoLHebUAU} et la proposition \ref{PropWCXAooDuFMjn} pour la régularité.
 \end{enumerate}
 
diff --git a/tex/front_back_matter/64_theme.tex b/tex/front_back_matter/64_theme.tex
index 7abc4445e..60b46dbcb 100644
--- a/tex/front_back_matter/64_theme.tex
+++ b/tex/front_back_matter/64_theme.tex
@@ -1,8 +1,17 @@
 \InternalLinks{inversion locale, fonction implicite}
-\begin{enumerate}
+       \begin{description}
+           \item[Des énoncés] 
+               \begin{enumerate}
     \item Inversion locale dans \( \eR^n\) : théorème \ref{THOooQGGWooPBRNEX}. Pour un Banach c'est le théorème \ref{ThoXWpzqCn}.
     \item
         Fonction implicite dans un Banach : théorème \ref{ThoAcaWho}.
+               \end{enumerate}
+           \item[Des utilisations]   
+               \begin{enumerate}
+                       
     \item
         Utilisé pour montrer que le flot d'une équation différentielle est un \( C^p\)-difféomorphisme local, voir \ref{NORMooWEWVooXbGmfE}. % position 1051229132
-\end{enumerate}
+    \item 
+        Pour le théorème de Von Neumann \ref{ThoOBriEoe}.
+               \end{enumerate}
+       \end{description}

From 5086bcd6e708e05112eeb5cce25547fe57b26a18 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 27 Jun 2017 05:46:14 +0200
Subject: [PATCH 59/64] =?UTF-8?q?(Stone-Weierstrass)=20Ajoute=20un=20corro?=
 =?UTF-8?q?laire=20=C3=A0=20propos=20de=20la=20densit=C3=A9=20des=20foncti?=
 =?UTF-8?q?ons=20Cinfini=20dans=20les=20fonctions=20continues=20sur=20une?=
 =?UTF-8?q?=20boule=20de=20R^n.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 tex/frido/77_series_fonctions.tex | 12 ++++++++++++
 1 file changed, 12 insertions(+)

diff --git a/tex/frido/77_series_fonctions.tex b/tex/frido/77_series_fonctions.tex
index 7881e65b4..ebb9d6879 100644
--- a/tex/frido/77_series_fonctions.tex
+++ b/tex/frido/77_series_fonctions.tex
@@ -296,6 +296,18 @@ \subsection{Théorème de Stone-Weierstrass}
 
 \end{proof}
 
+\begin{corollary}[\cite{MonCerveau}]        \label{CORooNIUJooLDrPSv}
+    Soit \( B\), la boule fermée de centre \( 0\) et de rayon \( 1\) dans \( \eR^n\). La partie \( C^{\infty}(B,\eR^n)\) est dense dans \( \big( C(B,B),\| . \|_{\infty} \big)\).
+\end{corollary}
+
+\begin{proof}
+    Soit \( f \in C(B,B)\) et \( \epsilon>0\). La fonction donnant la composante \( i\) est une fonction \( f_i\in C(B,\eR)\) et il existe donc, par le théorème de Stone-Weierstrass \ref{ThoWmAzSMF}, une fonction \( g_i\in  C^{\infty}(B,\eR)\) telle que \( \| g_i-f_i \|_{\infty}\leq \epsilon\).
+
+    La fonction \( g\) dont les composantes sont les \( g_i\) ainsi construits vérifie \( \| g-f \|_{\infty}\leq n\epsilon\).
+\end{proof}
+
+Attention toutefois que rien n'assure que les fonctions construites par le corollaire \ref{CORooNIUJooLDrPSv} prennent leurs valeurs dans \( B\).
+
 Le théorème suivant est un des énoncés les plus classiques de Stone-Weierstrass. Il découle évidement du théorème général \ref{ThoWmAzSMF} (encore qu'il faut alors bien comprendre qu'il faut traiter la fonction \( x\mapsto \sqrt{x}\) séparément). Il en existe cependant une preuve indépendante.
 %TODO : trouver cette preuve indépendante.
 \begin{theorem}     \label{ThoGddfas}   \index{théorème!Stone-Weierstrass}

From 3a290976294283018dabe4fa9450b2960d201f27 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 27 Jun 2017 07:50:47 +0200
Subject: [PATCH 60/64] =?UTF-8?q?(organisation)=20D=C3=A9place=20encore=20?=
 =?UTF-8?q?pas=20mal=20de=20r=C3=A9sultats=20pour=20avoir=20la=20coh=C3=A9?=
 =?UTF-8?q?rence=20des=20r=C3=A9f=C3=A9rences.?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 mazhe.tex                                  |    7 +-
 tex/frido/38_Chap_integrales_multiples.tex |  179 --
 tex/frido/61_representations.tex           |  115 -
 tex/frido/68_analyseR.tex                  | 2709 ----------------
 tex/frido/69_analyseR.tex                  |   70 +-
 tex/frido/77_series_fonctions.tex          | 3225 +++++++++++++++++++-
 6 files changed, 3151 insertions(+), 3154 deletions(-)

diff --git a/mazhe.tex b/mazhe.tex
index 7f86f93b2..1e6d19bad 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -223,15 +223,16 @@ \chapter{Trigonométrie, isométries}
 \chapter{Représentations et caractères}
 \input{63_representations}
 
-\chapter{Arc paramétré}
-\input{74_Chap_courbes_parametre}
-\input{152_Chap_courbes_parametre}
 
 \chapter{Suite de l'analyse}
 \input{77_series_fonctions}
 \input{79_inversion_locale}
 \input{80_Newton}
 
+\chapter{Arc paramétré}
+\input{74_Chap_courbes_parametre}
+\input{152_Chap_courbes_parametre}
+
 \chapter{Géométrie hyperbolique}
 \input{160_hyperbolique}
 
diff --git a/tex/frido/38_Chap_integrales_multiples.tex b/tex/frido/38_Chap_integrales_multiples.tex
index a13b65f6f..97c1da1d6 100644
--- a/tex/frido/38_Chap_integrales_multiples.tex
+++ b/tex/frido/38_Chap_integrales_multiples.tex
@@ -305,185 +305,6 @@ \subsection{La méthode de Rothstein-Trager}
 
 %TODO : lorsque j'aurai fait la construction du logarithme et ses propriétés, il faudra en faire référence ici. 
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Ellipsoïde de John-Loewer}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Soit \( q\) une forme quadratique sur \( \eR^n\) ainsi que \( \mB\) une base orthonormée de \( \eR^n\) dans laquelle la matrice de  \( q\) est diagonale. Dans cette base, la forme \( q\) est donnée par la proposition \ref{PropFWYooQXfcVY} :
-\begin{equation}
-    q(x)=\sum_i\lambda_ix_i
-\end{equation}
-où les \( \lambda_i\) sont les valeurs propres de \( q\).
-
-Plus généralement nous notons \( mat_{\mB}(q)\)\nomenclature[A]{\( mat_{\mB}(q)\)}{matrice de \( q\) dans la base \( \mB\)} la matrice de \( q\) dans la base \( \mB\) de \( \eR^n\).
-
-\begin{proposition} \label{PropOXWooYrDKpw}
-    Soit \( \mB\) une base orthonormée de \( \eR^n\) et l'application\footnote{L'ensemble \( Q(E)\) est l'ensemble des formes quadratiques sur \( E\).}
-    \begin{equation}
-        \begin{aligned}
-            D\colon Q(\eR^n)&\to \eR \\
-            q&\mapsto \det\big( mat_{\mB}(q) \big) .
-        \end{aligned}
-    \end{equation}
-    Alors :
-    \begin{enumerate}
-        \item
-            La valeur et \( D\) ne dépend pas du choix de la base orthonormée \( \mB\).
-        \item
-            La fonction \( D\) est donnée par la formule \( D(q)=\prod_i\lambda_i\) où les \( \lambda_i\) sont les valeurs propres de \( q\).
-        \item
-            La fonction \( D\) est continue.
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Soit \( q\) une forme quadratique sur \( \eR^n\). Nous considérons \( \mB\) une base de diagonalisation de \( q\) :
-    \begin{equation}
-        q(x)=\sum_i\lambda_ix_i
-    \end{equation}
-    où les \( x_i\) sont les composantes de \( x\) dans la base \( \mB\). Par définition, la matrice \( mat_{\mB}(q)\) est la matrice diagonale contenant les valeurs propres de \( q\).
-
-    Nous considérons aussi \( \mB_1\), une autre base orthonormées de \( \eR^n\). Nous notons \( S=mat_{\mB_1}(q)\); étant symétrique, cette matrice se diagonalise par une matrice orthogonale : il existe \( P\in\gO(n,\eR)\) telle que
-    \begin{equation}
-        S=P mat_{\mB}(q)P^t;
-    \end{equation}
-    donc \( \det(S)=\det(PP^t)\det\big( \diag(\lambda_1,\ldots, \lambda_n) \big)=\lambda_1\ldots\lambda_n\). Ceci prouve en même temps que \( D\) ne dépend pas du choix de la base et que sa valeur est le produit des valeurs propres.
-
-    Passons à la continuité. L'application déterminant \( \det\colon S_n(\eR^n)\to \eR\) est continue car polynôme en les composantes. D'autre par l'application \( mat_{\mB}\colon Q(\eR^n)\to S_n(\eR)\) est continue par la proposition \ref{PropFSXooRUMzdb}. L'application  \( D\) étant la composée de deux applications continues, elle est continue.
-\end{proof}
-
-\begin{proposition}[Ellipsoïde de John-Loewner\cite{KXjFWKA}]   \label{PropJYVooRMaPok}
-    Soit \( K\) compact dans \( \eR^n\) et d'intérieur non vide. Il existe une unique ellipsoïde\footnote{Définition \ref{DefOEPooqfXsE}.} (pleine) de volume minimal contenant \( K\).
-\end{proposition}
-\index{déterminant!utilisation}
-\index{extrema!volume d'un ellipsoïde}
-\index{convexité!utilisation}
-\index{compacité!utilisation}
-
-\begin{proof}
-    Nous subdivisons la preuve en plusieurs parties.
-    \begin{subproof}
-        \item[À propos de volume d'un ellipsoïde]
-
-            Soit \( \ellE\) un ellipsoïde. La proposition \ref{PropWDRooQdJiIr} et son corollaire \ref{CorKGJooOmcBzh} nous indiquent que 
-            \begin{equation}
-                \ellE=\{ x\in \eR^n\tq q(x)\leq 1 \}
-            \end{equation}
-            pour une certaine forme quadratique strictement définie positive \( q\). De plus il existe une base orthonormée \( \mB=\{ e_1,\ldots, e_n \}\) de \( \eR^n\) telle que 
-            \begin{equation}    \label{EqELBooQLPQUj}
-                q(x)=\sum_{i=1}^na_ix_i^2
-            \end{equation}
-            où \( x_i=\langle e_i, x\rangle \) et les \( a_i\) sont tous strictement positifs. Nous nommons \( \ellE_q\) l'éllipsoïde associée à la forme quadratique \( q\) et \( V_q\) son volume que nous allons maintenant calculer\footnote{Le volume ne change pas si nous écrivons l'inégalité stricte au lieu de large dans le domaine d'intégration; nous le faisons pour avoir un domaine ouvert.} :
-            \begin{equation}
-                V_q=\int_{\sum_ia_ix_i^2<1}dx
-            \end{equation}
-            Cette intégrale est écrite de façon plus simple en utilisant le \( C^1\)-difféomorphisme
-            \begin{equation}
-                \begin{aligned}
-                    \varphi\colon \ellE_q&\to B(0,1) \\
-                    x&\mapsto \Big( x_1\sqrt{a_1},\ldots, x_n\sqrt{a_n} \Big). 
-                \end{aligned}
-            \end{equation}
-            Le fait que \( \varphi\) prenne bien ses valeurs dans \( B(0,1)\) est un simple calcul : si \( x\in\ellE_q\), alors
-            \begin{equation}
-                \sum_i\varphi(x)_i^2=\sum_ia_ix_i^2<1.
-            \end{equation}
-            Cela nous permet d'utiliser le théorème de changement de variables \ref{THOooUMIWooZUtUSg} :
-            \begin{equation}
-                V_q=\int_{\sum_ia_ix_i^2<1}dx=\frac{1}{ \sqrt{a_1\ldots a_n} }\int_{B(0,1)}dx.
-            \end{equation}
-            %TODO : le volume de la sphère dans \eR^n. Mettre alors une référence ici.
-            La dernière intégrale est le volume de la sphère unité dans \( \eR^n\); elle n'a pas d'importance ici et nous la notons \( V_0\). La proposition \ref{PropOXWooYrDKpw} nous permet d'écrire \(V_q\) sous la forme
-            \begin{equation}
-                V_q=\frac{ V_0 }{ \sqrt{D(q)} }.
-            \end{equation}
-            
-        \item[Existence de l'ellipsoïde]
-
-            Nous voulons trouver un ellipsoïde contenant \( K\) de volume minimal, c'est à dire une forme quadratique \( q\in Q^{++}(\eR^n)\) telle que
-            \begin{itemize}
-                \item \( D(q)\) soit maximal
-                \item \( q(x)\leq 1\) pour tout \( x\in K\).
-            \end{itemize}
-            Nous considérons l'ensemble des candidats semi-définis positifs.
-            \begin{equation}
-                A=\{ q\in Q^+\tq q(x)\leq 1\forall x\in K \}.
-            \end{equation}
-            Nous allons montrer que \( A\) est convexe, compact et non vide dans \( Q(\eR^n)\); il aura ainsi un maximum de la fonction continue \( D\) définie sur \( Q(\eR^n)\). Nous montrerons ensuite que le maximum est dans \( Q^{++}\). L'unicité sera prouvée à part.
-
-            \begin{subproof}
-            \item[Non vide]
-                L'ensemble \( K\) est compact et donc borné par \( M>0\). La forme quadratique \( q\colon x\mapsto \| x \|^2/M^2\) est dans \( A\) parce que si \( x\in K\) alors 
-                \begin{equation}
-                    q(x)=\frac{ \| x \|^2 }{ M^2 }\leq 1.
-                \end{equation}
-            \item[Convexe]
-                Soient \( q,q'\in A\) et \( \lambda\in\mathopen[ 0 , 1 \mathclose]\). Nous avons encore \( \lambda q+(1-\lambda)q'\in Q^+\) parce que 
-                \begin{equation}
-                    \lambda q(x)+(1-\lambda)q'(x)\geq 0
-                \end{equation}
-                dès que \( q(x)\geq 0\) et \( q'(x)\geq 0\).
-            D'autre part si \( x\in K\) nous avons
-            \begin{equation}
-                \lambda q(x)+(1-\lambda)q'(x)\leq \lambda+(1-\lambda)=1.
-            \end{equation}
-            Donc \( \lambda q+(1-\lambda)q'\in A\).
-
-        \item[Fermé]
-
-            Pour rappel, la topologie de \( Q(\eR^n)\) est celle de la norme \eqref{EqZYBooZysmVh}. Nous considérons une suite \( (q_n)\) dans \( A\) convergeant vers \( q\in Q(\eR^n)\) et nous allons prouver que \( q\in A\), de sorte que la caractérisation séquentielle de la fermeture (proposition \ref{PropLFBXIjt}) conclue que \( A\) est fermé. En nommant \( e_x\) le vecteur unitaire dans la direction \( x\) nous avons
-            \begin{equation}
-                \big| q(x) \big|=\big| \| x \|^2q(e_x) \big|\leq \| x \|^2N(q),
-            \end{equation}
-            de sorte que notre histoire de suite convergente  donne pour tout \( x\) :
-            \begin{equation}
-                \big| q_n(x)-q(x) \big|\leq \| x \|^2N(q_n-q)\to 0.
-            \end{equation}
-            Vu que \( q_n(x)\geq 0\) pour tout \( n\), nous devons aussi avoir \( q(x)\geq 0\) et donc \( q\in Q^+\) (semi-définie positive). De la même manière si \( x\in K\) alors \( q_n(x)\leq 1\) pour tout \( n\) et donc \( q(x)\leq 1\). Par conséquent \( q\in A\) et \( A\) est fermé.
-
-        \item[Borné]
-
-            La partie \( K\) de \( \eR^n\) est borné et d'intérieur non vide, donc il existe \( a\in K\) et \( r>0\) tel que \( \overline{ B(a,r) }\subset K\). Si par ailleurs \( q\in A\) et \( x\in\overline{ B(0,r) }\) nous avons \( a+x\in K\) et donc \( q(a+x)\leq 1\). De plus \( q(-a)=q(a)\leq 1\), donc
-            \begin{equation}
-                \sqrt{q(x)}=\sqrt{q\big( x+a-a \big)}\leq \sqrt{q(x+a)}+\sqrt{q(-a)}\leq 2
-            \end{equation}
-            par l'inégalité de Minkowski \ref{PropACHooLtsMUL}. Cela prouve que si \( x\in\overline{ B(0,r) }\) alors \( q(x)\leq 4\). Si par contre \( x\in\overline{ B(0,1) }\) alors \( rx\in\overline{ B(0,r) } \) et 
-            \begin{equation}
-                0\leq q(x)=\frac{1}{ r^2 }q(rx)\leq \frac{ 4 }{ r^2 },
-            \end{equation}
-            ce qui prouve que \( N(q)\leq \frac{ 4 }{ r^2 }\) et que \( A\) est borné.
-
-
-            \end{subproof}
-
-            L'ensemble \( A\) est compact parce que fermé et borné, théorème de Borel-Lebesgue \ref{ThoXTEooxFmdI}. L'application continue \( D\colon Q(\eR^n)\to \eR\) de la proposition \ref{PropOXWooYrDKpw} admet donc un maximum sur le compact \( A\). Soit \( q_0\) ce maximum.
-
-            Nous montrons que \( q_0\in Q^{++}(\eR^d)\). Nous savons que l'application \( f\colon x\mapsto \frac{ \| x \|^2 }{ M^2 }\) est dans \( A\) et que \( D(f)>0\). Vu que \( q_0\) est maximale pour \( D\), nous avons
-            \begin{equation}
-                D(q_0)\geq D(f)>0.
-            \end{equation}
-            Donc \( q_0\in Q^{++}\).
-
-        \item[Unicité]
-
-            S'il existe une autre ellipsoïde de même volume que celle associée à la forme quadratique \( q_0\), nous avons une forme quadratique \( q\in Q^{++}\) telle que \( q(x)\leq 1\) pour tout \( x\in K\). C'est à dire que nous avons \( q_0,q\in A\) tels que \( D(q_0)=D(q)\).
-
-            Nous considérons la base canonique \( \mB_c\) de \( \eR^n\) et nous posons \( S=mat_{\mB_c}(q)\), \( S_0=mat_{\mB_c}(q_0)\). Étant donné que \( A\) est convexe, \( (q_0+q)/2\in A\) et nous allons prouver que cet élément de \( A\) contredit la maximalité de \( q_0\). En effet
-            \begin{equation}
-                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)
-            \end{equation}
-            Nous allons utiliser le lemme \ref{LemXOUooQsigHs} qui dit que le logarithme est log-concave sous la forme de l'équation \eqref{EqSPKooHFZvmB} avec \( \alpha=\beta=\frac{ 1 }{2}\) :
-            \begin{equation}    \label{eqBHJooYEUDPC}
-                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)>\sqrt{\det(S)}\sqrt{\det(S_0)}=\det(S_0)=D(q_0).
-            \end{equation}
-            Nous avons utilisé le fait que \( D(q_0)=D(q)\) qui signifie que \( \det(S_0)=\det(S)\). L'inéquation \eqref{eqBHJooYEUDPC} contredit la maximalité de \( D(q_0)\) et donne donc l'unicité.
-    \end{subproof}
-\end{proof}
-% This is part of Mes notes de mathématique
-% Copyright (c) 2011-2014
-%   Laurent Claessens
-% See the file fdl-1.3.txt for copying conditions.
-
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 \section{Rappel sur les intégrales usuelles}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
diff --git a/tex/frido/61_representations.tex b/tex/frido/61_representations.tex
index 654835275..36deab17f 100644
--- a/tex/frido/61_representations.tex
+++ b/tex/frido/61_representations.tex
@@ -1005,121 +1005,6 @@ \subsection{Décomposition polaires : cas réel}
     Si \( A\in\eM(n,\eR)\) alors la décomposition polaire \ref{ThoLHebUAU} nous donne \( A=SQ\) où \( S\) est symétrique définie positive et \( Q\) est orthogonale. La matrice \( S\) peut ensuite être diagonalisée par le théorème \ref{ThoeTMXla} : \( S=RDR^{-1}\) où \( D\) est diagonale et \( R\) est orthogonale. Avec ces deux décompositions en main, \( A=SQ=RDR^{-1}Q\). La matrice \( R^{-1}Q\) est orthogonale.
 \end{proof}
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Décomposition polaire}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{normaltext}      \label{NomDJMUooTRUVkS}
-    Nous allons montrer que l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
-            A&\mapsto \sqrt{A} 
-        \end{aligned}
-    \end{equation}
-    est une difféomorphisme.
-
-    Cependant \( S^{++}(n,\eR)\) n'est pas un ouvert de \( \eM(n,\eR)\) et nous ne savons pas ce qu'est la différentielle d'une application non définie sur un ouvert. Nous allons donc en réalité montrer que l'application racine carré existe sur un voisinage de chacun des points de \( S^{++}(n,\eR)\). Et comme une union quelconque d'ouverts est un ouvert, la fonction \( f\) sera bien définie sur un ouvert de \( \eM(n,\eR)\).
-\end{normaltext}
-
-\begin{lemma}       \label{LemLBFOooDdNcgy}
-    L'application 
-    \begin{equation}
-        \begin{aligned}
-            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
-            A&\mapsto A^2 
-        \end{aligned}
-    \end{equation}
-    est un \(  C^{\infty}\)-difféomorphisme.
-\end{lemma}
-
-\begin{proof}
-    Prouvons d'abord que \( f\) prend ses valeurs dans \( S^{++}(n,\eR)\). Si \( A\in S^{++}(n,\eR)\) alors par la diagonalisation \ref{ThoeTMXla} elle s'écrit \( A=QDQ^{-1}\) où \( D\) est diagonale avec des nombres strictement positifs sur la diagonale. Avec cela, \( A^2=QD^2Q^{-1}\) où \( D^2\) contient encore des nombres strictement positifs sur la diagonale.
-
-    L'application \( f\) étant essentiellement des polynôme en les entrées de \( A\), elle est de classe \( C^{\infty}\).
-
-    Passons à l'étude de la différentielle. Comme mentionné en \ref{NomDJMUooTRUVkS} nous allons en réalité voir \( f\) sur un ouvert de \( \eM(n,\eR)\) autour de \( A\in S^{++}(n,\eR)\). Par conséquent si \( A\in S^{++}(n,\eR)\),
-    \begin{subequations}
-        \begin{align}
-            df\colon S^{++}(n,\eR)&\to \aL\big( \eM(n,\eR),\eM(n,\eR) \big)\\
-            df_A\colon \eM(n,\eR)&\to \eM(n,\eR).
-        \end{align}
-    \end{subequations}
-    Le calcul de \( df_A\) est facile. Soit \( u\in \eM(n,\eR)\) et faisons le calcul en utilisant la formule du lemme \eqref{LemdfaSurLesPartielles} :
-    \begin{subequations}
-        \begin{align}
-            df_A(u)&=\Dsdd{ f(A+tu) }{t}{0}\\
-            &=\Dsdd{ A^2+tAu+tuA+t^2u^2 }{t}{0}\\
-            &=Au+uA.
-        \end{align}
-    \end{subequations}
-    Nous allons utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} à la fonction \( f\). Dans la suite, \( A\) est une matrice de \( S^{++}(n,\eR)\).
-
-    \begin{subproof}
-        \item[\( df_A\) est injective]
-            Soit \( M\in \eM(n,\eR)\) dans le noyau de \( df_A\). En posant \( M'=A^{-1}MQ\) nous avons \( M=QM'Q^{-1}\) et on applique \( df_A\) à \( QM'Q^{-1}\) :
-            \begin{equation}
-                df_A(QM'Q^{-1})=Q\big( DM+MD \big)Q^{-1}.
-            \end{equation}
-            où \( D=\begin{pmatrix}
-                \lambda_1    &       &       \\
-                    &   \ddots    &       \\
-                    &       &   \lambda_n
-                \end{pmatrix}\) avec \( \lambda_i>0\). La matrice \( D\) est inversible. Nous avons \( M'=-DM'D^{-1}\), et en coordonnées,
-                \begin{subequations}
-                    \begin{align}
-                        M'_{ij}&=-\sum_{kl}D_{ikM'_{kl}}D^{-1}_{lj}\\
-                        &=-\sum_{kl}\lambda_i\delta_{ik}M'_{kl}\frac{1}{ \lambda_j }\delta_{lj}\\
-                        &=-\frac{ \lambda_i }{ \lambda_i }M'_{ij}.
-                    \end{align}
-                \end{subequations}
-                C'est à dire que \( M'_{ij}=-\frac{ \lambda_i }{ \lambda_j }M'_{ij}\) avec \( -\frac{ \lambda_i }{ \lambda_j }<0\). Cela implique \( M'=0\) et par conséquent \( M=0\).
-            \item[\( df_A\) est surjective]
-                Soit \( N\in \eM(n,\eR)\); nous cherchons \( M\in \eM(n,\eR)\) tel que \( df_A(M)=N\). Nous posons \( N'=Q^{-1} NQ\) et \( M=QM'Q^{-1}\), ce qui nous donne à résoudre \( df_D(M')=N'\). Passons en coordonnées :
-                \begin{subequations}
-                    \begin{align}
-                        (DM'+M'D)_{ij}&=\sum_k(\delta_{ik}\lambda_iM'_{kj}+M'_{ik}\delta_{kj}\lambda_j)\\&
-                        &=M'_{ij}(\lambda_i+\lambda_j)
-                    \end{align}
-                \end{subequations}
-                où \( \lambda_i+\lambda_j\neq 0\). Il suffit donc de prendre la matrice \( M'\) donnée par
-                \begin{equation}
-                    M'_{ij}=\frac{1}{ \lambda_i+\lambda_j }N'_{ij}
-                \end{equation}
-                pour que \( df_A(M')=N'\).
-    \end{subproof}
-    
-    Le théorème d'inversion locale donne un voisinage \( V\) de $A$ dans \( \eM(n,\eR)\) et un voisinage \( W\) de \( A^2\) dans \( \eM(n,\eR)\) tels que \( f\colon V\to W\) soit une bijection  et \( f^{-1}\colon W\to V\) soit de même régularité, en l'occurrence \( C^{\infty}\).
-\end{proof}
-
-\begin{remark}
-    Oui, il y a des matrices non symétriques qui ont une unique racine carré.
-\end{remark}
-
-La proposition suivante, qui dépend du le théorème d'inversion locale par le lemme \ref{LemLBFOooDdNcgy}, donne plus de régularité à la décomposition polaire donnée dans le théorème \ref{ThoLHebUAU}.
-\begin{proposition}[Décomposition polaire : cas réel (suite)]       \label{PropWCXAooDuFMjn}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \gO(n,\eR)\times S^{++}(n,\eR)&\to \GL(n,\eR) \\
-            (Q,S)&\mapsto SQ 
-        \end{aligned}
-    \end{equation}
-    est un difféomorphisme de classe \( C^{\infty}\).
-\end{proposition}
-
-\begin{proof}
-    Si \( M\) est donnée dans \( \GL(n,\eR)\) alors la décomposition polaire\footnote{Proposition \ref{ThoLHebUAU}.} \( M=QS\) est donnée par \( S=\sqrt{MM^t}\) et \( Q=MS^{-1}\). Autrement dit, si nous considérons la fonction de décomposition polaire
-    \begin{equation}
-        f\colon \gO(n,\eR)\times S^{++}(n,\eR)\to \GL(n,\eR)
-    \end{equation}
-    alors
-    \begin{equation}
-        f^{-1}(M)=\big(  M(\sqrt{MM^t})^{-1},\sqrt{MM^t}  \big).
-    \end{equation}
-    Nous avons vu dans le lemme \ref{LemLBFOooDdNcgy} que la racine carré était un \( C^{\infty}\)-difféomorphisme. Le reste n'étant que des produits de matrice, la régularité est de mise.
-\end{proof}
-
 %--------------------------------------------------------------------------------------------------------------------------- 
 \subsection{Enveloppe convexe}
 %---------------------------------------------------------------------------------------------------------------------------
diff --git a/tex/frido/68_analyseR.tex b/tex/frido/68_analyseR.tex
index 7bd03006e..e2ec8aa36 100644
--- a/tex/frido/68_analyseR.tex
+++ b/tex/frido/68_analyseR.tex
@@ -1423,2712 +1423,3 @@ \section{Développement au voisinage de l'infini}
 	\lim_{x\to\infty}e^{1/x}\sqrt{1+x^2}-2x= \lim_{x\to \infty}\big(  2x+2+\gamma(1/x)-2x \big)=2.
     \end{equation}
 \end{example}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Complétude avec la norme uniforme}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{proposition}[Limite uniforme de fonctions continues]\label{PropCZslHBx}
-    Soit \( X\) un espace topologique et \( (Y,d)\) un espace métrique. Si une suite de fonctions \( f_n\colon X\to Y\) continues converge uniformément, alors la limite est séquentiellement continue\footnote{Si \( X\) est métrique, alors c'est la continuité usuelle par la proposition \ref{PropFnContParSuite}.}.
-\end{proposition}
-
-\begin{proof}
-    Soit \( a\in X\) et prouvons que \( f\) est séquentiellement continue en \( a\). Pour cela nous considérons une suite \( x_n\to a\) dans \( X\). Nous savons que \( f(x_n)\stackrel{Y}{\longrightarrow}f(x)\). Pour tout \(k\in \eN\), tout \( n\in \eN\) et tout \( x\in X\) nous avons la majoration
-    \begin{equation}
-        \big\| f(x_n)-f(x) \big\|\leq \big\| f(x_n)-f_k(x_n) \big\|+\big\| f_k(x_n)-f_k(x) \big\|+\big\| f_k(x)-f(x) \big\|\leq 2\| f-f_k \|_{\infty}+\big\| f_k(x_n)-f_k(x) \big\|.    
-    \end{equation}
-    Soit \( \epsilon>0\). Si nous choisissons \( k\) suffisamment grand la premier terme est plus petit que \( \epsilon\). Et par continuité de \( f_k\), en prenant \( n\) assez grand, le dernier terme est également plus petit que \( \epsilon\).
-\end{proof}
-
-\begin{proposition} \label{PropSYMEZGU}
-    Soit \( X\) un espace topologique métrique \( (Y,d)\) un espace espace métrique complet. Alors les espaces
-    \begin{enumerate}
-        \item
-            \( \big( C^0_b(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et bornées \( X\to Y\),
-        \item
-            \( \big( C^0_0(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et s'annulant à l'infini
-        \item
-            \( \big( C^k_0(X,Y),\| . \|_{\infty} \big)\) des fonctions de classe \( C^k\) et s'annulant à l'infini
-    \end{enumerate} 
-    sont complets.
-\end{proposition}
-
-\begin{proof}
-    Soit \( (f_n)\) une suite de Cauchy dans \( C(X,Y)\), c'est à dire que pour tout \( \epsilon>0\) il existe \( N\in \eN\) tel que si \( k,l>N\) nous avons \( \| f_k-f_l \|_{\infty}\leq \epsilon\). Cette suite vérifie le critère de Cauchy uniforme \ref{PropNTEynwq} et donc converge uniformément vers une fonction \( f\colon X\to Y\). La continuité (ou l'aspect \( C^k\)) de la fonction \( f\) découle de la convergence uniforme et de la proposition \ref{PropCZslHBx} (c'est pour avoir l'équivalence entre la continuité séquentielle et la continuité normale que nous avons pris l'hypothèse d'espace métrique).
-
-    Si les fonctions \( f_k\) sont bornées ou s'annulent à l'infini, la convergence uniforme implique que la limite le sera également.
-\end{proof}
-    Notons que si \( X\) est compact, les fonctions continues sont bornées par le théorème \ref{ThoImCompCotComp} et nous pouvons simplement dire que \( C^0(X,Y)\) est complet, sans préciser que nous parlons des fonctions bornées.
-
-
-\begin{lemma}       \label{LemdLKKnd}
-    Soit \( A\) compact et \( B\) complet. L'ensemble des fonctions continues de \( A\) vers \( B\) muni de la norme uniforme est complet.
-
-    Dit de façon courte : \( \big( C(A,B),\| . \|_{\infty} \big)\) est complet.
-\end{lemma}
-
-\begin{proof}
-    Soit \( (f_k)\) une suite de Cauchy de fonctions dans \( C(A,B)\). Pour chaque \( x\in A \) nous avons
-    \begin{equation}
-        \| f_k(x)-f_l(x) \|_B\leq \| f_k-f_l \|_{\infty},
-    \end{equation}
-    de telle sorte que la suite \( (f_k(x))\) est de Cauchy dans \( B\) et converge donc vers un élément de \( B\). La suite de Cauchy \( (f_k)\) converge donc ponctuellement vers une fonction \( f\colon A\to B\). Nous devons encore voir que cette fonction est continue; ce sera l'uniformité de la norme qui donnera la continuité. En effet soit \( x_n\to x\) une suite dans \( A\) convergent vers \( x\in A\). Pour chaque \( k\in \eN\) nous avons
-    \begin{equation}
-        \| f(x_n)-f(x) \|\leq \| f(x_n)-f_k(x_n) \|  +\| f_k(x_n)-f_k(x) \|+\| f_k(x)-f(x) \|.
-    \end{equation}
-    En prenant \( k\) et \( n\) assez grands, cette expression peut être rendue aussi petite que l'on veut; le premier et le troisième terme par convergence ponctuelle \( f_k\to f\), le second terme par continuité de \( f_k\). La suite \( f(x_n)\) est donc convergente vers \( f(x)\) et la fonction \( f\) est continue.
-\end{proof}
-
-\begin{probleme}
-Il serait sans doute bon de revoir cette preuve à la lumière du critère de Cauchy uniforme \ref{PropNTEynwq}.
-\end{probleme}
-
-
-\begin{normaltext}[\cite{ooXYZDooWKypYR}]
-    Le théorème de Stone-Weierstrass indique que les polynômes sont denses pour la topologie uniforme dans les fonctions continues. Donc il existe des limites uniformes de fonctions \( C^{\infty}\) qui ne sont même pas dérivables. Les espaces de type \( C^p\) munis de \( \| . \|_{\infty}\) ne sont donc pas complets sans quelque hypothèses. Voir la proposition \ref{PropSYMEZGU} et le thème \ref{THMooOCXTooWenIJE}.
-\end{normaltext}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorèmes de point fixe}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Points fixes attractifs et répulsifs}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooTMZUooMoBDGC}
-    Soit \( I\) un intervalle fermé de \( \eR\) et \( \varphi\colon I\to I\) une application \( C^1\). Soit \( a\) un point fixe de \( \varphi\). Nous disons que \( a\) est \defe{attractif}{point fixe!attractif}\index{attractif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) converge vers \( a\). Le point \( a\) sera dit \defe{répulsif}{répulsif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) diverge.
-\end{definition}
-
-\begin{lemma}[\cite{DemaillyNum}]
-    Soit \( a\) un point fixe de \( \varphi\).
-    \begin{enumerate}
-        \item
-    Si \( | \varphi'(a) |<1\) alors \( a\) est attractif et la convergence est au moins exponentielle.
-\item
-    Si \( | \varphi'(a) |>1\) alors \( a\) est répulsif et la divergence est au moins exponentielle.
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Si \( | \varphi'(a)<1 |\) alors il existe \( k\) tel que \( | \varphi'(a) |<k<1\) et par continuité il existe un voisinage \( V\) de \( a\) dans lequel \( | \varphi'(x) |<k\) pour tout \( x\in V\). En utilisant le théorème des accroissements finis nous avons
-    \begin{equation}
-        | x_n-a |=\big| f(x_{n-1}-a) \big|\leq k| x_{n-1}-a |
-    \end{equation}
-    et par récurrence
-    \begin{equation}
-        | x_n-a |\leq k^n| x_0-a |.
-    \end{equation}
-
-    Le cas \( | \varphi'(a)>1 |\) se traite de façon similaire.
-\end{proof}
-
-\begin{remark}
-    Dans le cas \(| \varphi'(a) |=1\), nous ne pouvons rien conclure. Si \( \varphi(x)=\sin(x)\) nous avons \( \sin(x)<x\) et le point \( a=0\) est attractif. A contrario, si \( \varphi(x)=\sinh(x)\) nous avons \( |\sinh(x)|>|x|\) et le point \( a=0\) est répulsif.
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Picard}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooRSLCooAsWisu}
-    Une application \( f\colon (X,\| . \|_X)\to (Y,\| . \|_Y)\) entre deux espaces métriques est une \defe{contraction}{contraction} si elle est \( k\)-\defe{Lipschitz}{Lipschitz} pour un certain \( 0\leq k<1\), c'est à dire si pour tout \( x,y\in X\) nous avons
-    \begin{equation}
-        \| f(x)-f(y) \|_Y\leq k\| x-y \|_{X}.
-    \end{equation}
-\end{definition}
-
-\begin{theorem}[Picard \cite{ClemKetl,NourdinAnal}\footnote{Il me semble qu'à la page 100 de \cite{NourdinAnal}, l'hypothèse H1 qui est prouvée ne prouve pas Hn dans le cas \( n=1\). Merci de m'écrire si vous pouvez confirmer ou infirmer. La preuve donnée ici ne contient pas cette «erreur».}.]     \label{ThoEPVkCL}
-    Soit \( X\) un espace métrique complet et \( f\colon X\to X\) une application contractante, de constante de Lipschitz \( k\). Alors \( f\) admet un unique point fixe, nommé \( \xi\). Ce dernier est donné par la limite de la suite définie par récurrence 
-    \begin{subequations}
-        \begin{numcases}{}
-            x_0\in X\\
-            x_{n+1}=f(x_n).
-        \end{numcases}
-    \end{subequations}
-    De plus nous pouvons majorer l'erreur par
-    \begin{equation}    \label{EqKErdim}
-        \| x_n-x \|\leq \frac{ k^n }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k^n }{ 1-k }\| x_1-x_0 \|.
-    \end{equation}
-
-    Soit \( r>0\), \( a\in X\) tels que la fonction \( f\) laisse la boule \( K=\overline{ B(a,r) }\) invariante (c'est à dire que \( f\) se restreint à \( f\colon K\to K\)). Nous considérons les suites \( (u_n)\) et \( (v_n)\) définies par
-    \begin{subequations}
-        \begin{numcases}{}
-            u_0=v_0\in K\\
-            u_{n+1}=f(v_n), v_{n+1}\in B(u_n,\epsilon).
-        \end{numcases}
-    \end{subequations}
-    Alors le point fixe \( \xi\) de \( f\) est dans \( K\) et la suite \( (v_n)\) satisfait l'estimation
-    \begin{equation}
-        \| v_n-\xi \|\leq \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k }.
-    \end{equation}
-\end{theorem}
-\index{théorème!Picard}
-\index{point fixe!Picard}
-
-La première inégalité \eqref{EqKErdim} donne une estimation de l'erreur calculable en cours de processus; la seconde donne une estimation de l'erreur calculable avant de commencer.
-
-\begin{proof}
-    
-    Nous commençons par l'unicité du point fixe. Si \( a\) et \( b\) sont des points fixes, alors \( f(a)=a\) et \( f(b)=b\). Par conséquent
-    \begin{equation}
-        \| f(a)-f(b) \|=\| a-b \|,
-    \end{equation}
-    ce qui contredit le fait que \( f\) soit une contraction.
-
-    En ce qui concerne l'existence, notons que si la suite des \( x_n\) converge dans \( X\), alors la limite est un point fixe. En effet en prenant la limite des deux côtés de l'équation \( x_{n+1}=f(x_n)\), nous obtenons \( \xi=f(\xi)\), c'est à dire que \( \xi\) est un point fixe de \( f\). Notons que nous avons utilisé ici la continuité de \( f\), laquelle est une conséquence du fait qu'elle soit Lipschitz. Nous allons donc porter nos efforts à prouver que la suite est de Cauchy (et donc convergente parce que \( X\) est complet). Nous commençons par prouver que \( \| x_{n+1}-x_n \|\leq k^n\| x_0-x_1 \|\). En effet pour tout \( n\) nous avons
-    \begin{equation}
-        \| x_{n+1}-x_n \|=\| f(x_n)-f(x_{n-1}) \|\leq k\| x_n-x_{n-1} \|.
-    \end{equation}
-    La relation cherchée s'obtient alors par récurrence. Soient \( q>p\). En utilisant une somme télescopique,
-    \begin{subequations}
-        \begin{align}
-            \| x_q-x_p \|&\leq \sum_{l=p}^{q-1}\| x_{l+1}-x_l \|\\
-            &\leq\left( \sum_{l=p}^{q-1}k^l \right)\| x_1-x_0 \|\\
-            &\leq\left(\sum_{l=p}^{\infty}k^l\right)\| x_1-x_0 \|.
-        \end{align}
-    \end{subequations}
-    Étant donné que \( k<1\), la parenthèse est la queue d'une série qui converge, et donc tend vers zéro lorsque \( p\) tend vers l'infini.
-
-    En ce qui concerne les inégalités \eqref{EqKErdim}, nous refaisons une somme télescopique :
-    \begin{subequations}
-        \begin{align}
-            \| x_{n+p}-x_n \|&\leq \| x_{n+p}-x_{n+p-1} \|+\cdots +\| x_{n+1}-x_n \|\\
-            &\leq k^p\| x_n-x_{n-1} \|+k^{p-1}\| x_n-x_{n-1} \|+\cdots +k\| x_n-x_{n-1} \|\\
-            &=k(1+\cdots +k^{p-1})\| x_n-x_{n-1}\|  \\
-            &\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|.
-        \end{align}
-    \end{subequations}
-    En prenant la limite \( p\to \infty\) nous trouvons
-    \begin{equation}        \label{EqlUMVGW}
-        \| \xi-x_n \|\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k }{ 1-k }\| x_1-x_0 \|.
-    \end{equation}
-
-    Nous passons maintenant à la seconde partie du théorème en supposant que \( f\) se restreigne en une fonction \( f\colon K\to K\). D'abord \( K\) est encore un espace métrique complet, donc la première partie du théorème s'y applique et \( f\) y a un unique point fixe.
-    
-    Nous allons montrer la relation par récurrence. Tout d'abord pour \( n=1\) nous avons
-    \begin{equation}
-        \| v_1-\xi \|\leq\| v_1-u_1 \|+\| u_1-\xi \|\leq \epsilon+\frac{ k }{ 1-k }\| u_1-u_0 \|
-    \end{equation}
-    où nous avons utilisé l'estimation \eqref{EqlUMVGW}, qui reste valable en remplaçant \( x_1\) par \( u_1\)\footnote{Elle n'est cependant pas spécialement valable si on remplace \( x_n\) par \( u_n\).}. Nous pouvons maintenant faire la récurrence :
-    \begin{subequations}
-        \begin{align}
-            \| v_{n+1}-\xi \|&\leq \| v_{n+1}-u_{n+1} \|+\| u_{n+1}-\xi \|\\
-            &\leq \epsilon+k\| v_n-\xi \|\\
-            &\leq \epsilon+k\left( \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k } \right)\\
-            &=\frac{ \epsilon }{ 1-k }+\frac{ k^{n+1} }{ 1-k }\| u_1-u_0 \|.
-        \end{align}
-    \end{subequations}
-\end{proof}
-
-\begin{remark}
-    Ce théorème comporte deux parties d'intérêts différents. La première partie est un théorème de point fixe usuel, qui sera utilisé pour prouver l'existence de certaines équations différentielles.
-
-    La seconde partie est intéressante d'un point de vie numérique. En effet, ce qu'elle nous enseigne est que si à chaque pas de calcul de la récurrence \( x_{n+1}=f(x_n)\) nous commettons une erreur d'ordre de grandeur \( \epsilon\), alors le procédé (la suite \( (v_n)\)) ne converge plus spécialement vers le point fixe, mais tend vers le point fixe avec une erreur majorée par \( \epsilon/(k-1)\).
-\end{remark}
-
-\begin{remark}
-Au final l'erreur minimale qu'on peut atteindre est de l'ordre de \( \epsilon\). Évidemment si on commet une faute de calcul de l'ordre de \( \epsilon\) à chaque pas, on ne peut pas espérer mieux.
-\end{remark}
-
-\begin{remark}  \label{remIOHUJm}
-    Si \( f\) elle-même n'est pas contractante, mais si \( f^p\) est contractante pour un certain \( p\in \eN\) alors la conclusion du théorème de Picard reste valide et \( f\) a le même unique point fixe que \( f^p\). En effet nommons \( x\) le point fixe de \( f\) : \( f^p(x)=x\). Nous avons alors
-    \begin{equation}
-        f^p\big( f(x) \big)=f\big( f^p(x) \big)=f(x),
-    \end{equation}
-    ce qui prouve que \( f(x)\) est un point fixe de \( f^p\). Par unicité nous avons alors \( f(x)=x\), c'est à dire que \( x\) est également un point fixe de \( f\).
-\end{remark}
-
-Si la fonction n'est pas Lipschitz mais presque, nous avons une variante.
-\begin{proposition}
-    Soit \( E\) un ensemble compact\footnote{Notez cette hypothèse plus forte} et si \( f\colon E\to E\) est une fonction telle que
-    \begin{equation}        \label{EqLJRVvN}
-        \| f(x)-f(y) \|< \| x-y \|
-    \end{equation}
-    pour tout \( x\neq y\) dans \( E\) alors \( f\) possède un unique point fixe.
-\end{proposition}
-
-\begin{proof}
-    La suite \( x_{n+1}=f(x_n)\) possède une sous suite convergente. La limite de cette sous suite est un point fixe de \( f\) parce que \( f\) est continue. L'unicité est due à l'aspect strict de l'inégalité \eqref{EqLJRVvN}.
-\end{proof}
-
-\begin{theorem}[Équation de Fredholm]\index{Fredholm!équation}\index{équation!Fredholm}     \label{ThoagJPZJ}
-    Soit \( K\colon \mathopen[ a , b \mathclose]\times \mathopen[ a , b \mathclose]\to \eR\) et \( \varphi\colon \mathopen[ a , b \mathclose]\to \eR\), deux fonctions continues. Alors si \( \lambda\) est suffisamment petit, l'équation
-    \begin{equation}
-        f(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x)
-    \end{equation}
-    admet une unique solution qui sera de plus continue sur \( \mathopen[ a , b \mathclose]\).
-\end{theorem}
-
-\begin{proof}
-    Nous considérons l'ensemble \( \mF\) des fonctions continues \( \mathopen[ a , b \mathclose]\to\mathopen[ a , b \mathclose]\) muni de la norme uniforme. Le lemme \ref{LemdLKKnd} implique que \( \mF\) est complet. Nous considérons l'application \( \Phi\colon \mF\to \mF\) donnée par
-    \begin{equation}
-        \Phi(f)(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x). 
-    \end{equation}
-    Nous montrons que \( \Phi^p\) est une application contractante pour un certain \( p\). Pour tout \( x\in \mathopen[ a , b \mathclose]\) nous avons
-    \begin{subequations}
-        \begin{align}
-            \| \Phi(f)-\Phi(g) \|_{\infty}&\leq \| \Phi(f)(x)-\Phi(g)(x) \|\\
-            &=| \lambda |\Big\| \int_a^bK(x,y)\big( f(y)-g(y) \big)dy  \Big\|\\
-            &\leq | \lambda |\| K \|_{\infty}| b-a |\| f-g \|_{\infty}
-        \end{align}
-    \end{subequations}
-    Si \( \lambda\) est assez petit, et si \( p\) est assez grand, l'application \( \Phi^p\) est donc une contraction. Elle possède donc un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Brouwer}
-%---------------------------------------------------------------------------------------------------------------------------
-\label{subSecZCCmMnQ}
-
-\begin{proposition}
-    Soit \( f\colon \mathopen[ a , b \mathclose]\to \mathopen[ a , b \mathclose]\) une fonction continue. Alors \( f\) accepte un point fixe.
-\end{proposition}
-
-\begin{proof}
-    En effet si nous considérons \( g(x)=f(x)-x\) alors nous avons \( g(a)=f(a)-a\geq 0\) et \( g(b)=f(b)-b\leq 0\). Si \( g(a)\) ou \( g(b)\) est nul, la proposition est démontrée; nous supposons donc que \( g(a)>0\) et \( g(b)<0\). La proposition découle à présent du théorème des valeurs intermédiaires \ref{ThoValInter}.
-\end{proof}
-
-\begin{example}
-    La fonction \( x\mapsto\cos(x)\) est continue entre \( \mathopen[ -1 , 1 \mathclose]\) et \( \mathopen[ -1 , 1 \mathclose]\). Elle admet donc un point fixe. Par conséquent il existe (au moins) une solution à l'équation \( \cos(x)=x\).
-\end{example}
-
-\begin{proposition}[Brouwer dans \( \eR^n\) version \(  C^{\infty}\) via Stokes]     \label{PropDRpYwv}
-    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction \(  C^{\infty}\). Alors \( f\) admet un point fixe.
-\end{proposition}
-\index{point fixe!Brouwer}
-
-\begin{proof}
-    Supposons que \( f\) ne possède pas de points fixes. Alors pour tout \( x\in B\) nous considérons la ligne droite partant de \( x\) dans la direction de \( f(x)\) (cette droite existe parce que \( x\) et \( f(x)\) sont supposés distincts). Cette ligne intersecte \( \partial B\) en un point que nous appelons \( g(x)\). Prouvons que cette fonction est \( C^k\) dès que \( f\) est \( C^k\) (y compris avec \( k=\infty\)).
-
-   Le point \( g(x) \) est la solution du système
-    \begin{subequations}
-        \begin{numcases}{}
-        g(x)-f(x)=\lambda\big( x-f(x) \big)\\
-        \| g(x) \|^2=1\\
-        \lambda\geq 0.
-        \end{numcases}
-    \end{subequations}
-    En substituant nous obtenons l'équation
-    \begin{equation}
-        P_x(\lambda)=\| \lambda\big( x-f(x) \big)+f(x) \|^2-1=0,
-    \end{equation}
-    ou encore
-    \begin{equation}
-        \lambda^2\| x-f(x) \|^2+2\lambda\big( x-f(x) \big)\cdot f(x)+\| f(x) \|^2-1=0.
-    \end{equation}
-    En tenant compte du fait que \( \| f(x)<1 \|\) (pare que les images de \( f\) sont dans \( \mB\)), nous trouvons que \( P_x(0)\leq 0\) et \( P_x(1)\leq 0\). De même \( \lim_{\lambda\to\infty} P_x(\lambda)=+\infty\). Par conséquent le polynôme de second degré \( P_x\) a exactement deux racines distinctes \( \lambda_1\leq 0\) et \( \lambda_2\geq 1\). La racine que nous cherchons est la seconde. Le discriminant est strictement positif, donc pas besoin d'avoir peur de la racine dans
-    \begin{equation}
-        \lambda(x)=\frac{ -\big( x-f(x) \big)\cdot f(x)+\sqrt{   \Delta_x  } }{ \| x-f(x) \|^2 }
-    \end{equation}
-    où 
-    \begin{equation}
-        \Delta_x=4\Big( \big( x-f(x) \big)\cdot f(x) \Big)^2-4\| x-f(x) \|^2\big( \| f(x) \|^2-1 \big).
-    \end{equation}
-    Notons que la fonction \( \lambda(x)\) est \( C^k\) dès que \( f\) est \( C^k\); et en particulier elle est \( C^{\infty}\) si \( f\) l'est.
-
-    En résumé la fonction \( g\) ainsi définie vérifie deux propriétés :
-    \begin{enumerate}
-        \item
-            elle est \(  C^{\infty}\);
-        \item
-            elle est l'identité sur \( \partial B\).
-    \end{enumerate}
-    La suite de la preuve consiste à montrer qu'une telle rétraction sur \( B\) ne peut pas exister\footnote{Notons qu'il n'existe pas non plus de rétractions continues sur \( B\), mais pour le montrer il faut utiliser d'autres méthodes que Stokes, ou alors présenter les choses dans un autre ordre.}.
-
-    Nous considérons une forme de volume \( \omega\) sur \( \partial B\) : l'intégrale de \( \omega\) sur \( \partial B\) est la surface de \( \partial B\) qui est non nulle. Nous avons alors
-    \begin{equation}
-        0<\int_{\partial B}\omega
-        =\int_{\partial B}g^*\omega
-        =\int_Bd(g^*\omega)
-        =\int_Bg^*(d\omega)
-        =0
-    \end{equation}
-    Justifications :
-    \begin{itemize}
-        \item 
-            L'intégrale \( \int_{\partial B}\omega\) est la surface de \( \partial B\) et est donc non nulle.
-        \item
-            La fonction \( g\) est l'identité sur \( \partial B\). Nous avons donc \( \omega=g^*\omega\).
-        \item
-            Le lemme \ref{LemdwLGFG}.
-        \item
-            La forme \( \omega\) est de volume, par conséquent de degré maximum et \( d\omega=0\).
-    \end{itemize}
-\end{proof}
-
-Un des points délicats est de se ramener au cas de fonctions \( C^{\infty}\). Pour la régularisation par convolution, voir \cite{AllardBrouwer}; pour celle utilisant le théorème de Weierstrass, voir \cite{KuttlerTopInAl}.
-\begin{theorem}[Brouwer dans \( \eR^n\) version continue]\label{ThoRGjGdO}
-    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction continue\footnote{Une fonction continue sur un fermé de \( \eR^n\) est à comprendre pour la topologie induite.}. Alors \( f\) admet un point fixe.
-\end{theorem}
-\index{théorème!Brouwer}
-
-\begin{proof}
-    Nous commençons par définir une suite de fonctions
-    \begin{equation}
-        f_k(x)=\frac{ f(x) }{ 1+\frac{1}{ k } }.
-    \end{equation}
-    Nous avons \( \| f_k-f \|_{\infty}\leq \frac{1}{ 1+k }\) où la norme est la norme uniforme sur \( B\). Par le théorème de Weierstrass \ref{ThoWmAzSMF} il existe une suite de fonctions \(  C^{\infty}\) \( g_k\) telles que
-    \begin{equation}
-        \|  g_k-f_k\|_{\infty}\leq\frac{1}{ 1+k }.
-    \end{equation}
-    Vérifions que cette fonction \( g_k\) soit bien une fonction qui prend ses valeurs dans \( B\) :
-    \begin{subequations}
-        \begin{align}
-            \| g_k(x) \|&\leq \| g_k(x)-f_k(x) \|+\| f_k(x) \|\\
-            &\leq \frac{1}{ 1+k }+\frac{ \| f(x) \| }{ 1+\frac{1}{ k } }\\
-            &\leq \frac{1}{ 1+k}+\frac{1}{ 1+\frac{1}{ k } }\\
-            &=1.
-        \end{align}
-    \end{subequations}
-    Par la version \(  C^{\infty}\) du théorème (proposition \ref{PropDRpYwv}), \( g_k\) admet un point fixe que l'on nomme \( x_k\).
-
-    Étant donné que \( x_k\) est dans le compact \( B\), quitte à prendre une sous suite nous supposons que la suite \( (x_k)\) converge vers un élément \( x\in B\). Nous montrons maintenant que \( x\) est un point fixe de \( f\) :
-    \begin{subequations}
-        \begin{align}
-            \| f(x)-x \|&=\| f(x)-g_k(x)+g_k(x)-x_k+x_k-x \|\\
-            &\leq \| f(x)-g_k(x) \| +\underbrace{\| g_k(x)-x_k \|}_{=0}+\| x_k-x \|\\
-            &\leq \frac{1}{ 1+k }+\| x_k-x \|.
-        \end{align}
-    \end{subequations}
-    En prenant le limite \( k\to\infty\) le membre de droite tend vers zéro et nous obtenons \( f(x)=x\).
-\end{proof}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Théorème de Schauder}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Une conséquence du théorème de Brouwer est le théorème de Schauder qui est valide en dimension infinie.
-
-\begin{theorem}[Théorème de Schauder\cite{ooWWBQooKIciWi}]\index{théorème!Schauder}       \label{ThovHJXIU}
-    Soit \( E\), un espace vectoriel normé, \( K\) un convexe compact de \( E\) et \( f\colon K\to K\) une fonction continue. Alors \( f\) admet un point fixe.
-\end{theorem}
-\index{théorème!Schauder}
-\index{point fixe!Schauder}
-
-\begin{proof}
-    Étant donné que \( f\colon K\to K\) est continue, elle y est uniformément continue. Si nous choisissons \( \epsilon\) alors il existe \( \delta>0\) tel que 
-    \begin{equation}
-        \| f(x)-f(y) \|\leq \epsilon
-    \end{equation}
-    dès que \( \| x-y \|\leq \delta\). La compacité de \( K\) permet de choisir un recouvrement fini par des ouverts de la forme
-    \begin{equation}    \label{EqKNPUVR}
-        K\subset \bigcup_{1\leq i\leq p}B(x_j,\delta)
-    \end{equation}
-    où \( \{ x_1,\ldots, x_p \}\subset K\). Nous considérons maintenant \( L=\Span\{ f(x_j)\tq 1\leq j\leq p \}\) et
-    \begin{equation}
-        K^*=K\cap L.
-    \end{equation}
-    Le fait que \( K\) et \( L\) soient convexes implique que \( K^*\) est convexe. L'ensemble \( K^*\) est également compact parce qu'il s'agit d'une partie fermée de \( K\) qui est compact (lemme \ref{LemnAeACf}). Notons en particulier que \( K^*\) est contenu dans un espace vectoriel de dimension finie, ce qui n'est pas le cas de \( K\).
-
-    Nous allons à présent construire une sorte de partition de l'unité subordonnée au recouvrement \eqref{EqKNPUVR} sur \( K\) (voir le lemme \ref{LemGPmRGZ}). Nous commençons par définir
-    \begin{equation}
-        \psi_j(x)=\begin{cases}
-            0    &   \text{si } \| x-x_j \|\geq \delta\\
-            1-\frac{ \| x-x_j \| }{ \delta }    &    \text{sinon}.
-        \end{cases}
-    \end{equation}
-    pour chaque \( 1\leq j\leq p\). Notons que \( \psi_j\) est une fonction positive, nulle en-dehors de \( B(x_j,\delta)\). En particulier la fonction suivante est bien définie :
-    \begin{equation}
-        \varphi_j(x)=\frac{ \psi_j(x) }{ \sum_{k=1}^p\psi_k(x) }
-    \end{equation}
-    et nous avons \( \sum_{j=1}^p\varphi_j(x)=1\). Les fonctions \( \varphi_j\) sont continues sur \( K\) et nous définissons finalement
-    \begin{equation}
-        g(x)=\sum_{j=1}^p\varphi_j(x)f(x_j).
-    \end{equation}
-    Pour chaque \( x\in K\), l'élément \( g(x)\) est une combinaison des éléments \( f(x_j)\in K^*\). Étant donné que \( K^*\) est convexe et que la somme des coefficients \( \varphi_j(x)\) vaut un, nous avons que \( g\) prend ses valeurs dans \( K^*\) par la proposition \ref{PropPoNpPz}.
-
-    Nous considérons seulement la restriction \( g\colon K^*\to K^*\) qui est continue sur un compact contenu dans un espace vectoriel de dimension finie. Le théorème de Brouwer nous enseigne alors que \( g\) a un point fixe (proposition \ref{ThoRGjGdO}). Nous nommons \( y\) ce point fixe. Notons que \( y\) est fonction du \( \epsilon\) choisit au début de la construction, via le \( \delta\) qui avait conditionné la partition de l'unité.
-
-    Nous avons
-    \begin{subequations}        \label{EqoXuTzE}
-        \begin{align}
-            f(y)-y&=f(y)-g(y)\\
-            &=\sum_{j=1}^p\varphi_j(y)f(y)-\sum_{j=1}^p\varphi_j(y)f(x_j)\\
-            &=\sum_{j=1}^p\varphi(j)(y)\big( f(y)-f(x_j) \big).
-        \end{align}
-    \end{subequations}
-    Par construction, \( \varphi_j(y)\neq 0\) seulement si \( \| y-x_j \|\leq \delta\) et par conséquent seulement si \( \| f(y)-f(x_j) \|\leq \epsilon\). D'autre par nous avons \( \varphi_j(y)\geq 0\); en prenant la norme de \eqref{EqoXuTzE} nous trouvons
-    \begin{equation}
-        \| f(y)-y \|\leq \sum_{j=1}^p\| \varphi_j(y)\big( f(y)-f(x_j) \big) \|\leq \sum_{j=1}^p\varphi_j(y)\epsilon=\epsilon.
-    \end{equation}
-    Nous nous souvenons maintenant que \( y\) était fonction de \( \epsilon\). Soit \( y_m\) le \( y\) qui correspond à \( \epsilon=2^{-m}\). Nous avons alors
-    \begin{equation}
-        \| f(y_m)-y_m \|\leq 2^{-m}.
-    \end{equation}
-    L'élément \( y_m\) est dans \( K^*\) qui est compact, donc quitte à choisir une sous suite nous pouvons supposer que \( y_m\) est une suite qui converge vers \( y^*\in K\)\footnote{Notons que même dans la sous suite nous avons \( \| f(y_m)-y_m \|\leq 2^{-m}\), avec le même «\( m\)» des deux côtés de l'inégalité.}. Nous avons les majorations
-    \begin{equation}
-        \| f(y^*)-y^* \|\leq \| f(y^*)-f(y_m) \|+\| f(y_m)-y_m \|+\| y_m-y^* \|.
-    \end{equation}
-    Si \( m\) est assez grand, les trois termes du membre de droite peuvent être rendus arbitrairement petits, d'où nous concluons que
-    \begin{equation}
-        f(y^*)=y^*
-    \end{equation}
-    et donc que \( f\) possède un point fixe.
-\end{proof}
-
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Markov-Kakutani et mesure de Haar}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit \( G\) un groupe topologique. Une \defe{mesure de Haar}{mesure!de Haar} sur \( G\) est une mesure \( \mu\) telle que 
-    \begin{enumerate}
-        \item
-            \( \mu(gA)=\mu(A)\) pour tout mesurable \( A\) et tout \( g\in G\),
-        \item
-            \( \mu(K)<\infty\) pour tout compact \( K\subset G\).
-    \end{enumerate}
-    Si de plus le groupe \( G\) lui-même est compact nous demandons que la mesure soit normalisée : \( \mu(G)=1\).
-\end{definition}
-
-Le théorème suivant nous donne l'existence d'une mesure de Haar sur un groupe compact.
-\begin{theorem}[Markov-Katutani\cite{BeaakPtFix}]\index{théorème!Markov-Takutani}   \label{ThoeJCdMP}
-    Soit \( E\) un espace vectoriel normé et \( L\), une partie non vide, convexe, fermée et bornée de \( E'\). Soit \( T\colon L\to L\) une application continue. Alors \( T\) a un point fixe.
-\end{theorem}
-
-\begin{proof}
-    Nous considérons un point \( x_0\in L\) et la suite
-    \begin{equation}
-        x_n=\frac{1}{ n+1 }\sum_{i=0}^n T^ix_0.
-    \end{equation}
-    La somme des coefficients devant les \( T^i(x_0)\) étant \( 1\), la convexité de \( L\) montre que \( x_n\in L\). Nous considérons l'ensemble
-    \begin{equation}
-        C=\bigcap_{n\in \eN}\overline{ \{ x_m\tq m\geq n \} }.
-    \end{equation}
-    Le lemme \ref{LemooynkH} indique que \( C\) n'est pas vide, et de plus il existe une sous suite de \( (x_n)\) qui converge vers un élément \( x\in C\). Nous avons
-    \begin{equation}
-        \lim_{n\to \infty} x_{\sigma(n)}(v)=x(v)
-    \end{equation}
-    pour tout \( v\in E\). Montrons que \( x\) est un point fixe de \( T\). Nous avons
-    \begin{subequations}
-        \begin{align}
-            \| (Tx_{\sigma(k)}-x_{\sigma(k)})v \|&=\Big\| T\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v)-\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v) \Big\|\\
-            &=\Big\| \frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^{i+1}x_0(v)-T^ix_0(v) \Big\|\\
-            &=\frac{1}{ 1+\sigma(k) }\big\| T^{\sigma(k)+1}x_0(v)-x_0(v) \big\|\\
-            &\leq\frac{ 2M }{ \sigma(k)+1 }
-        \end{align}
-    \end{subequations}
-    où \( M=\sum_{y\in L}\| y(v) \|<\infty\) parce que \( L\) est borné. En prenant \( k\to\infty\) nous trouvons
-    \begin{equation}
-        \lim_{k\to \infty} \big( Tx_{\sigma(k)}-x_{\sigma(k)} \big)v=0,
-    \end{equation}
-    ce qui signifie que \( Tx=x\) parce que \( T\) est continue.
-\end{proof}
-
-Le théorème suivant est une conséquence du théorème de Markov-Katutani.
-\begin{theorem} \label{ThoBZBooOTxqcI}
-    Si \( G\) est un groupe topologique compact possédant une base dénombrable de topologie alors \( G\) accepte une unique mesure de Haar normalisée. De plus elle est unimodulaire :
-    \begin{equation}
-        \mu(Ag)=\mu(gA)=\mu(A)
-    \end{equation}
-    pour tout mesurables \( A\subset G\) et tout élément \( g\in G\).
-\end{theorem}
-\index{mesure!de Haar}
-
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorèmes de point fixes et équations différentielles}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Théorème de Cauchy-Lipschitz}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous démontrons ici deux théorèmes de Cauchy-Lipschitz. De nombreuses propriétés annexes seront démontrées dans le chapitre sur les équations différentielles, section \ref{SECooNKICooDnOFTD}.
-
-\begin{theorem}[Cauchy-Lipschitz\cite{SandrineCL,ZPNooLNyWjX}] \label{ThokUUlgU}
-    Nous considérons l'équation différentielle
-    \begin{subequations}        \label{XtiXON}
-        \begin{numcases}{}
-            y'(t)=f\big( t,y(t) \big)\\
-            y(t_0)=y_0
-        \end{numcases}
-    \end{subequations}
-    avec \( f\colon U=I\times \Omega\to \eR^n\) où \( I\) est ouvert dans \( \eR\) et \( \Omega\) ouvert dans \( \eR^n\). Nous supposons que \( f\) est continue sur \( U\) et localement Lipschitz\footnote{Définition \ref{DefJSFFooEOCogV}. Notons que nous ne supposons pas que \( f\) soit une contraction.} par rapport à \( y\). 
-    
-    Alors il existe un intervalle \( J\subset I\) sur lequel la solution au problème est unique. De plus toute solution du problème est une restriction de cette solution à une partie de \( J\). La solution sur \( J\) (dite «solution maximale») est de classe \( C^1\).
-\end{theorem}
-\index{théorème!Cauchy-Lipschitz}
-
-% Il serait tentant de mettre ce théorème dans la partie sur les équations différentielles, mais ce n'est pas aussi simple :
-% Il est utilisé pour calculer la transformée de Fourier de la Gaussienne (lemme LEMooPAAJooCsoyAJ) dans le chapitre sur la transformée de Fourier.
-
-\begin{proof}
-    Nous divisions la preuve en plusieurs étapes (même pas toutes simples).
-    \begin{subproof}
-    \item[Cylindre de sécurité et espace fonctionnel]
-
-    Précisons l'espace fonctionnel \( \mF\) adéquat. Soient \( V\) et \( W\) les voisinages de \( t_0\) et \( y_0\) sur lesquels \( f\) est localement Lipschitz. Nous considérons les quantités suivantes :
-    \begin{enumerate}
-        \item
-            \( M=\sup_{V\times W}f\) ;
-        \item
-            \( r>0\) tel que \( \overline{ B(y_0,r) }\subset V\)
-        \item
-            \( T>0\) tel que \( \overline{ B(t_0,T) }\subset W\) et \( T<r/M\).
-    \end{enumerate}
-    Nous considérons alors l'ensemble
-    \begin{equation}
-        \mF=C^0\big( \overline{ B(t_0,T) },\overline{ B(y_0,r) } \big)
-    \end{equation}
-    que nous munissons de la norme uniforme. Par le lemme \ref{LemdLKKnd} l'espace \( \big( \mF,\| . \|_{\infty} \big)\) est complet.
-
-    \item[Une application \( \Phi\colon \mF\to \mF\)]
-
-
-    Si \( y\) est une solution de l'équation différentielle considérée, elle vérifie
-    \begin{equation}        \label{EqPGLwcL}
-        y(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
-    \end{equation}
-    Ceci nous incite à considérer l'opérateur \( \Phi\colon \mF\to \mF\) défini par
-    \begin{equation}
-        \Phi(y)(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
-    \end{equation}
-
-    Pour que l'application \( \Phi\) soit utile nous devons montrer que pour tout \( y\in \mF\),
-    \begin{itemize}
-        \item l'application \( \Phi(y)\) est bien définie,
-        \item pour tout \( t\in\overline{ B(y_0,r) }\) nous avons \( \Phi(y)(t)\in\overline{ B(t_0,T) }\),
-        \item l'application $\Phi(y)\colon    \overline{ B(t_0,T) }\to \overline{ B(y_0,r)} $ est continue.
-    \end{itemize}
-    Attention : nous ne prétendons pas que \( \Phi\) elle-même soit continue. C'est parti.
-    \begin{subproof}
-    \item[\( \Phi(y)\) est bien définie]
-            
-        Il faut montrer que l'intégrale converge. Le calcul de \( \Phi(y)(t)\) ne se fait qu'avec \( t\in \overline{ B(t_0,T) }\). Vu que \( u\) prend ses valeurs dans \( \mathopen[ t_0 , t \mathclose]\) et que \( y\in\mF\), le nombre \( y(u)\) est toujours dans \( \overline{ B(y_0,r) }\). Ceci pour dire que dans l'intégrale, la fonction \( f\) n'est considérée que sur \( \mathopen[ t_0 , t \mathclose]\times \overline{ B(y_0,r) }\subset V\times W\). La fonction \( f\) est donc uniformément majorable, et l'intégrale ne pose pas de problèmes.
-
-    \item[\( \Phi(y)(t)\in \overline{ B(t_0,T) }\)]
-
-    Prouvons que \( \Phi(y)(t)\in\overline{ B(y_0,r) }\). Pour cela, notons que
-    \begin{equation}
-        | \Phi(y)(t)-y_0 |\leq \int_{t_0}^t |f\big( u,y(u) \big)|du\leq | t-t_0 |\| f \|_{\infty}.
-    \end{equation}
-    Étant donné que \( t\in\overline{ B(t_0,T) }\) nous avons \( | t-t_0 |\leq r/M\) et donc \( | \Phi(y)(t)-y_0 |\leq r\).
-
-    \item[\( \Phi(y)\) est continue]
-
-        Nous pourrions invoquer le théorème \ref{ThoKnuSNd}, mais nous allons le faire à la main. Soit \( s_0\in B(t_0,T)\) et prouvons que \( \Phi(y)\) est continue en \( s_0\). Pour cela nous prenons \( s\in B(s_0,\delta)\) et nous calculons :
-        \begin{equation}
-            | \Phi(y)(s)-\Phi(y)(s_0) |\leq \int_{s_0}^s|f\big( u,y(u) \big)|du\leq | s_0-s |\| f \|_{\infty}.
-        \end{equation}
-        C'est le fait que \( f\) soit bornée dans le cylindre de sécurité qui fait en sorte que cela tende vers zéro lorsque \( s\to s_0\).
-    \end{subproof}
-
-
-    
-    L'équation \eqref{EqPGLwcL} signifie que \( y\) est un point fixe de \( \Phi\). L'espace \( \mF\) étant complet le théorème de point fixe de Picard (théorème \ref{ThoEPVkCL}) s'applique. Nous allons montrer qu'il existe un \( p\in\eN\) tel que \( \Phi^p\) soit contractante. Par conséquent \( \Phi^p\) aura un unique point fixe qui sera également unique point fixe de \( \Phi\) par la remarque \ref{remIOHUJm}.
-    
-\item[Contractante]
-
-    Prouvons donc que \( \Phi^p\) est contractante pour un certain \( p\). Pour cela nous commençons par montrer la formule suivante par récurrence :
-    \begin{equation}        \label{EqRAdKxT}
-        \big\| \Phi^p(x)(t)-\Phi^p(y)(t) \big\|\leq \frac{ k^p| t-t_0 |^p }{ p! }\| x-y \|_{\infty}
-    \end{equation}
-    pour tout \( x,y\in\mF\), et pour tout \( t\in\overline{ B(t_0,T) }\). Pour \( p=0\) la formule \eqref{EqRAdKxT} est vérifiée parce que \( \| x-y \|_{\infty}\) est le supremum de \( \| x(t)-y(t) \|\) pour \( t\in\overline{ B(t_0,T) }\). Supposons que la formule soit vraie pour \( p\) et calculons pour \( p+1\). Pour tout \( t\in\overline{ B(t_0,T) }\) nous avons
-    \begin{subequations}
-        \begin{align}
-            \big\| \Phi^{p+1}(x)(t)-\Phi^{p+1}(y)(t) \big\|&\leq \left| \int_{t_0}^t\big\| f\big( u,\Phi^p(x)(u) \big)-f\big( u,\Phi^p(y)(u) \big) \big\|du \right| \\
-            &\leq \left| \int_{t_0}^tk\| \Phi^p(x)(u)-\Phi^p(y)(u) \|du \right|    \label{subIKYixF}\\
-            &\leq \left| \int_{t_0}^tk\frac{ k^p| t-t_0 | }{ p! }\| x-y \|_{\infty} \right| \label{subxkNjiV} \\
-            &=\frac{ k^{p+1}| t-t_0 |^{p+1} }{ (p+1)! }\| x-y \|_{\infty}.
-        \end{align}
-    \end{subequations}
-    Justifications :
-    \begin{itemize}
-        \item \eqref{subIKYixF} parce que \( f\) est Lipschitz.
-        \item \eqref{subxkNjiV} par hypothèse de récurrence.
-    \end{itemize}
-    La formule \eqref{EqRAdKxT} est maintenant établie. Nous pouvons maintenant montrer que \( \Phi^p\) est une contraction pour un certain \( p\). Pour tout \( t\in \overline{ B(t_0,T) }\) nous avons
-    \begin{equation}
-         \| \Phi^p(x)(t)-\Phi^p(y)(t) \|\leq \frac{ k^p }{ t! }| t-t_0 |^p\| x-y \|_{\infty}     \leq \frac{ k^pT^p }{ p! }\| x-y \|_{\infty}
-    \end{equation}
-    où nous avons utilisé le fait que \( | t-t_0 |^p<T^p\). En prenant le supremum sur \( t\) des deux côtés il vient
-    \begin{equation}
-        \| \Phi^p(x)-\Phi^p(y) \|_{\infty}\leq\frac{ k^pT^p }{ p! }\| x-y \|_{\infty}.
-    \end{equation}
-    Le membre de droite tend vers zéro lorsque \( p\to\infty\) parce que \( k<1\) et \( T^p/p!\to 0\)\footnote{C'est le terme général du développement de \(  e^{T}\) qui est une série convergente.}. Nous concluons donc que \( \Phi^p\) est une contraction pour un certain \( p\).
-
-\item[Conclusion]
-
-    L'unique point fixe de \( \Phi\) est alors l'unique solution continue de l'équation différentielle \eqref{XtiXON}. Par ailleurs l'équation elle-même \( y'=f(t,y)\) demande implicitement que \( y\) soit dérivable et donc continue. Nous concluons que l'unique point fixe de \( \Phi\) est l'unique solution de l'équation différentielle donnée. Cette dernière est automatiquement \( C^1\) parce que si \( y\) est continue alors \( u\mapsto f(u,y(u))\) est continue, c'est à dire que \( y'\) est continue.
-
-\item[Unicité]
-
-    Nous passons maintenant à la partie «prolongement maximum» du théorème. Soient \( x_1\) et \( x_2\) deux solutions maximales du problème \eqref{XtiXON} sur des intervalles \( I_1\) et \( I_2\) respectivement. Les intervalles \( I_1\) et \( I_2\) contiennent \( \overline{ B(t_0,r) }\) sur lequel \( x_1=x_2\) par unicité.
-    
-    
-    Nous allons maintenant montrer que pour tout \( t\geq t_0\) pour lequel \( x_1\) ou \( x_2\) est défini, \( x_1(t)\) et \( x_2(t)\) sont définis et sont égaux. Le raisonnement sur \( t\leq t_0\) est similaire.
-    
-    Supposons que l'ensemble des \( t\geq t_0\) tels que \( x_1=x_2\) soit ouvert à droite, c'est à dire soit de la forme \( \mathopen[ t_0 ,b [\). Dans ce cas, soit \( x_1\) soit \( x_2\) (soit les deux) cesse d'exister en \( b\). En effet si nous avions les fonctions \( x_i\) sur \(\mathopen[ t_0 , b+\epsilon [\) alors l'équation \( x_1=x_2\) définirait un fermé dans \( \mathopen[ t_0 , b+\epsilon [\). Supposons pour fixer les idées que \( x_1\) cesse d'exister : le domaine de \( x_1\) (parmi les \( t\geq 0\)) est \( \mathopen[ t_0 , b [\) et sur ce domaine nous avons \( x_1=x_2\). Dans ce cas \( x_1\) pourrait être prolongé en \( x_2\) au-delà de \( b\). Si \( x_1\) et \( x_2\) s'arrêtent d'exister en même temps en \( b\), alors nous avons bien \( x_1=x_2\).
-
-    Nous devons donc traiter le cas où \( x_1=x_2\) sur \( \mathopen[ t_0 , b \mathclose]\) alors que \( x_1\) et \( x_2\) existent sur \( \mathopen[ t_0 , b+\epsilon [\) pour un certain \( \epsilon\).
-
-    Nous pouvons appliquer le théorème d'existence locale au problème
-    \begin{subequations}
-        \begin{numcases}{}
-            y'=f(t,y)\\
-            y(b)=x_1(b).
-        \end{numcases}
-    \end{subequations}
-    Il existe un voisinage de \( b\) sur lequel la solution est unique. Sur ce voisinage nous devons donc avoir \( x_1=x_2\), ce qui contredit le fait que \( x_1\neq x_2\) en dehors de \( \mathopen[ t_0 , b \mathclose]\).
-
-    Donc \( x_1\) et \( x_2\) existent et sont égaux sur au moins \( I_1\cup I_2\).
-    \end{subproof}
-\end{proof}
-
-Le théorème de Cauchy-Lipschitz donne existence et unicité d'une solution maximale. Cependant cette solution peut ne pas exister partout où les hypothèses sur \( f\) sont remplies. En d'autres termes, il peut arriver que \( f\) soit Lipschitz jusqu'à \( t_1\), mais que la solution maximale ne soit définie que jusqu'en \( t_2<t_1\). Ce cas fait l'objet du théorème d'explosion en temps fini \ref{CorGDJQooNEIvpp}.
-
-Sous quelque hypothèses nous pouvons nous assurer de l'existence d'une solution unique sur tout \( \eR\).
-
-\begin{theorem}[Cauchy-Lipschitz global\cite{ooJZJPooAygxpk,KXjFWKA}]       \label{THOooZIVRooPSWMxg}
-    Soit un intervalle \( I\) de \( \eR\), \( y_0\in \eR^n\), \( t_0\in I\) et une fonction continue \( f\colon I\times \eR^n\to \eR^n\) telle que pour tout compact \( K\) dans \( I\), il existe \( k>0\) tel que
-    \begin{equation}
-        \| f(t,y_1)-f(t,y_2) \|\leq k\| y_1-y_2 \|
-    \end{equation}
-    pour tout \( t\in K\) et \( y_1,y_2\in \eR^n\).
-
-    Alors le problème
-    \begin{subequations}        \label{EQSooBNREooUTfbMH}
-        \begin{numcases}{}
-            y'(t)=f\big( t,y(t) \big)\\
-            y(t_0)=y_0
-        \end{numcases}
-    \end{subequations}
-    possède une unique solution \( y\colon I\to \eR^n\) sur \( I\).
-\end{theorem}
-
-\begin{proof}
-    Soit un intervalle compact \( K\) dans \( I\) et contenant \( t_0\). Nous notons \( \ell\) le diamètre de \( K\). Sur l'espace \( E=C^0(K,\eR^n)\) nous considérons la topologie uniforme : \( (E,\| . \|_{\infty})\). C'est un espace complet par le lemme \ref{LemdLKKnd} (nous utilisons le fait que \( \eR^n\) soit complet, théorème \ref{ThoTFGioqS}). Nous allons utiliser l'application suivante :
-    \begin{equation}        \label{EQooJUTBooILBKoE}
-        \begin{aligned}
-            \Phi\colon E&\to E \\
-            \Phi(y)(t)&=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds
-        \end{aligned}
-    \end{equation}
-    Démontrons quelque faits à propos de \( \Phi\).
-    \begin{subproof}
-        \item[La définition fonctionne bien]
-            Nous devons commencer par prouver que cette application est bien définie. Si \( y\in E\) alors \( f\) et \( y\) sont continues; l'application \( s\mapsto f\big(s,y(s)\big)\) est donc également continue. L'intégrale de cette fonction sur le compact \( \mathopen[ t_0 , t \mathclose]\) ne pose alors pas de problèmes. En ce qui concerne la continuité de \( \phi(y)\) sous l'hypothèse que \( y\) soit continue,
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(y)(t') \|\leq \int_t^{t'}\| f(s,y(s)) \|ds\leq M| t-t' |
-    \end{equation}
-    où \( M\) est une majoration de \( \| s\mapsto f\big( s,y(s) \big) \|_{\infty,K}\).
-
-        \item[Si \( y\) est solution alors \( \Phi(y)=y\)]
-
-            Supposons que \( y\) soit une solution de l'équation différentielle \eqref{EQSooBNREooUTfbMH}. Alors, vu que \( y'(t)=f\big( t,y(t) \big)\) nous avons :
-            \begin{equation}
-                y(t)=y_0+\int_{t_0}^ty'(s)ds=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds=\Phi(y)(t).
-            \end{equation}
-            
-        \item[Si \( \Phi(y)=y\) alors \( y\) est solution]
-
-            Nous avons, pour tout \( t\) :
-            \begin{equation}
-                y(t)=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds.
-            \end{equation}
-            Le membre de droite est dérivable par rapport à \( t\), et la dérivée fait \(  f\big( t,y(t) \big)   \). Donc le membre de gauche est également dérivable et nous avons bien
-            \begin{equation}
-                y'(t)=f\big( t,y(t) \big).
-            \end{equation}
-            De plus \( y(t_0)=y_0+\int_{t_0}^{t_0}\ldots=y_0\).
-    \end{subproof}
-    
-    Nous sommes encore avec \( K\) compact et \( E=C^0(K,\eR^n)\) muni de la norme uniforme. Nous allons montrer que \( \Phi\) est une contraction de \( E\) pour une norme bien choisie.
-
-    \begin{subproof}
-        \item[Une norme sur \( E\)]
-            Pour \( y\in E\) nous posons
-            \begin{equation}
-                \| y \|_k=\max_{t\in K}\big(  e^{-k| t-t_0 |}\| y(t) \| \big).
-            \end{equation}
-            Ce maximum est bien définit et fini parce que la fonction de \( t\) dedans est une fonction continue sur le compact \( K\). Cela est également une norme parce que si \( \| y \|_k=0\) alors \(  e^{-k| t-t_0 |}\| y(t) \|=0\) pour tout \( t\). Étant donné que l'exponentielle ne s'annule pas, \( \| y(t) \|=0\) pour tout \( t\).
-        \item[Équivalence de norme]
-
-            Nous montrons que les normes \( \| . \|_k\) et \( \| . \|_{\infty}\) sont équivalentes\footnote{Définition \ref{DefEquivNorm}} :
-            \begin{equation}        \label{EQooSQYWooBTXvDL}
-                \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\leq \| y \|_{\infty}
-            \end{equation}
-            pour tout \( y\in E\). Pour la première inégalité, \( \ell\geq | t-t_0 |\) pour tout \( t\in K\), et \( k>0\), donc
-            \begin{equation}
-                \| y(t) \| e^{-k\ell}\leq  e^{-k| t-t_0 |}\| y(t) \|.
-            \end{equation}
-            En prenant le maximum des deux côtés, \( \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\). 
-
-            En ce qui concerne la seconde inégalité dans \eqref{EQooSQYWooBTXvDL}, \( k| t-t_0 |\geq 0\) et donc \(  e^{-k| t-t_0 |}<1\).
-
-    \end{subproof}
-    Vu que les normes \( \| . \|_{\infty}\) et \( \| . \|_k\) sont équivalentes, l'espace \( (E,\| . \|_k)\) est tout autant complet que \( (E,\| . \|_{\infty})\). Nous démontrons à présent que \( \Phi\) est une contraction dans \( (E,\|  \|_k)\). 
-
-    Soient \( y,z\in E\). Si \( t\geq t_0\) nous avons
-    \begin{subequations}        \label{SUBEQSooEXVYooDkyTuB}
-        \begin{align}
-            \| \Phi(y)(t)-\Phi(z)(t) \|&\leq \int_{t_0}^t\| f\big( s,y(s) \big)-f\big( s,z(s) \big) \|ds\\
-            &\leq k\int_{t_0}^t\| y(s)-z(s) \|ds.
-        \end{align}
-    \end{subequations}
-    Il convient maintenant de remarquer que
-    \begin{equation}
-        \| y(t) \|= e^{-k| t-t_0 |} e^{k| t-t_0 |}\| y(t) \|\leq \| y \|_k e^{k| t-t_0 |}.
-    \end{equation}
-    Nous pouvons avec ça prolonger les inégalités \eqref{SUBEQSooEXVYooDkyTuB} par
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(z)(t) \|\leq k\| y-z \|_k\int_{t_0}^t e^{k| s-t_0 |}ds=k\| y-z \|_k\int_{t_0}^t e^{k(s-t_0)}ds
-    \end{equation}
-    où nous avons utilisé notre supposition \( t\geq t_0\) pour éliminer les valeurs absolues. L'intégrale peut être faite explicitement, mais nous en sommes arrivés à un niveau de fainéantise tellement inconcevable que
-
-\lstinputlisting{tex/sage/sageSnip014.sage}
-
-Au final, si \( t\geq t_0\),
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t-t_0)}-1 \big).
-    \end{equation}
-    Si \( t\leq t_0\), il faut retourner les bornes de l'intégrale avant d'y faire rentrer la norme parce que \( \| \int_0^1f \|\leq \int_0^1\| f \|\), mais ça ne marche pas avec \( \| \int_1^0f \|\). Pour \( t\leq t_0\) tout le calcul donne
-    \begin{equation}
-        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t_0-t)}-1 \big).
-    \end{equation}
-    Les deux inéquations sont valables a fortiori en mettant des valeurs absolues dans l'exponentielle, de telle sorte que pour tout \( t\in K\) nous avons
-    \begin{equation}
-        e^{-k| t_0-t |}\| \phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big( 1- e^{-k| t_0-t |} \big).
-    \end{equation}
-    En prenant le supremum sur \( t\),
-    \begin{equation}
-        \| \Phi(y)-\Phi(z) \|_k\leq \| y-z \|_k(1- e^{-k\ell}),
-    \end{equation}
-    mais \( 0<(1- e^{e-k\ell})<1\), donc \( \Phi\) est contractante pour la norme \( \| . \|_k\). Vu que \( (E,\| . \|_k)\) est complet, l'application \( \Phi\) y a un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-
-    Ce point fixe est donc l'unique solution de l'équation différentielle de départ.
-
-    \begin{subproof}
-        \item[Existence et unicité sur \( I\)]
-            Il nous reste à prouver que la solution que nous avons trouvée existe sur \( I\) : jusqu'à présent nous avons démontré l'existence et l'unicité sur n'importe quel compact dans \( I\).
-
-            Soit une suite croissante de compacts \( K_n\) contenant \( t_0\) (par exemple une suite exhaustive comme celle du lemme \ref{LemGDeZlOo}). Nous avons en particulier
-            \begin{equation}
-                I=\bigcup_{n=0}^{\infty}K_n.
-            \end{equation}
-        \item[Existence sur \( I\)]
-        
-            Soit \( y_n\) l'unique solution sur \( K_n\). Il suffit de poser
-            \begin{equation}
-                y(t)=y_n(t)
-            \end{equation}
-            pour \( n\) tel que \( t\in K_n\). Cette définition fonctionne parce que si \( t\in K_n\cap K_m\), il y a forcément un des deux qui est inclus à l'autre et le résultat d'unicité sur le plus grand des deux donne \( y_n(t)=y_m(t)\).
-
-        \item[Unicité sur \( I\)]
-
-            Soient \( y\) et \(z \) des solutions sur \( I\); vu que \( I\) n'est pas spécialement compact, le travail fait plus haut ne permet pas de conclure que \( y=z\). 
-
-            Soit \( t\in I\). Alors \( t\in K_n\) pour un certain \( n\) et \( y\) et \( z\) sont des solutions sur \( K_n\) qui est compact. L'unicité sur \( K_n\) donne \( y(t)=z(t)\).
-    \end{subproof}
-\end{proof}
-
-\begin{normaltext}
-    Il y a d'autres moyens de prouver qu'une solution existe globalement sur \( \eR\). Si \( f\) est globalement bornée, le théorème d'explosion en temps fini donne quelque garanties, voir \ref{NORMooZROGooZfsdnZ}.
-\end{normaltext}
-
-Le théorème suivant donne une version du théorème de Cauchy-Lipschitz lorsque la fonction \( f\) dépend d'un paramètre. Ce théorème n'utilise rien de fondamentalement nouveau. Nous le donnons seulement pour montrer que l'on peut choisir l'espace \( \mF\) de façon un peu maligne pour élargir le résultat. Si vous voulez un théorème de Cauchy-Lipschitz avec paramètre vraiment intéressant, allez voir le théorème \ref{PROPooPYHWooIZhQST}.
-
-\begin{theorem}[Cauchy-Lipschitz avec paramètre\cite{MonCerveau,ooXVPAooTQUIRw}]           \label{THOooDTCWooSPKeYu}
-    Soit un intervalle ouvert \( I\) de \( \eR\), un connexe ouvert \( \Omega\) de \( \eR^n\) et un intervalle ouvert \( \Lambda\) de \( \eR^d\). Soit une fonction \( f\colon I\times \Omega\times \Lambda\to \eR^n\) continue et localement Lipschitz en \( \Omega\). Soient \( t_0\in I\), \( y_0\in \Omega\) et \( \lambda_0\in \Lambda\). Il existe un voisinage compact de \( (t_0,y_0,\lambda_0)\) sur lequel le problème
-    \begin{subequations}
-        \begin{numcases}{}
-            y'_{\lambda}(t)=f\big( t,y_{\lambda}(t),\lambda \big)\\
-            y_{\lambda}(t_0)=y_0
-        \end{numcases}
-    \end{subequations}
-    possède une unique solution. De plus \( (t,\lambda)\mapsto y_{\lambda}(t)\) est continue\footnote{Ici, la surprise est que ce soit continu par rapport à \( \lambda\). Le fait qu'elle le soit par rapport à \( t\) est clair depuis le départ parce que c'est finalement rien d'autre que le Cauchy-Lipschitz vieux et connu.}.
-\end{theorem}
-
-\begin{proof}
-
-    \begin{probleme}
-        Ceci est une idée de la preuve. Je n'ai pas vérifié toutes les étapes. Soyez prudent.
-
-    \end{probleme}
-
-    D'abord nous avons un voisinage compact \( V\times \overline{ B(y_0,r) }\times \Lambda_0\) de \( (t_0,y_0,\lambda_0)\) sur lequel $f$ est bornée. Ensuite nous récrivons l'équation différentielle sous la forme
-    \begin{subequations}
-        \begin{numcases}{}
-            \frac{ \partial y }{ \partial t }(t,\lambda)=f\big( t,y(t,\lambda),\lambda \big)\\
-            y(t_0,\lambda)=y_0.
-        \end{numcases}
-    \end{subequations}
-    pour une fonction \( y\colon V\times \Lambda_0\to \eR^n\).
-
-    Nous posons \( \mF=C^0\big( V\times\Lambda_0 ,\eR^n\big)\) et nous y définissons l'application
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon \mF&\to \mF \\
-            \Phi(y)(t,\lambda)&=y_0+\int_{t_0}^tf\big( s,y(s,\lambda),\lambda \big)ds. 
-        \end{aligned}
-    \end{equation}
-    Il y a plein de vérifications à faire\cite{ooXVPAooTQUIRw}, mais je parie que \( \Phi\) est bien définie, et que une de ses puissances est une contraction de \( (\mF,\| . \|_{\infty})\). L'unique point fixe est une solution de notre problème et est dans \( C^0\), donc \( (t,\lambda)\mapsto y(t,\lambda)=y_{\lambda}(t)\) est de classe \( C^0\), c'est à dire continue.
-\end{proof}
-
-\begin{normaltext}
-    Ce théorème marque un peu la limite de ce que l'on peut faire avec la méthode des points fixes dans le cadre de Cauchy-Lipschitz : nous sommes limités à la continuité de la solution parce que les espaces \( C^p\) ne sont pas complets\footnote{Par exemple, le théorème de Stone-Weierstrass \ref{ThoGddfas} nous dit que la limite uniforme de polynômes (de classe \(  C^{\infty}\)) peut n'être que continue. Voir aussi le thème \ref{THMooOCXTooWenIJE}.}. Il n'y a donc pas d'espoir d'adapter la méthode pour prouver que si \( f\) est de classe \( C^p\) alors \( (t,\lambda)\mapsto y_{\lambda}(t)\) est de classe \( C^p\). On peut, à \( \lambda\) fixé prouver que \( t\mapsto y_{\lambda}(t)\) est de classe \( C^p\) (utiliser une récurrence), mais pas plus.
-
-    La régularité \( C^1\) de \( y\) par rapport à la condition initiale sera l'objet du théorème \ref{THOooSTHXooXqLBoT}. Ce résultat n'est vraiment pas facile et utilise des ingrédients bien autres qu'un point fixe. Ensuite la régularité \( C^p\) par rapport à la condition initiale et par rapport à un paramètre seront presque des cadeaux (proposition \ref{PROPooINLNooDVWaMn} et \ref{PROPooPYHWooIZhQST}).
-\end{normaltext}
-
-\begin{example}[\cite{ooSBHXooOMnaTC}]          \label{EXooJXIGooQtotMc}
-    Nous savons que le théorème de Picard permet de trouver le point fixe par itération de la contraction à partir d'un point quelconque. Tentons donc de résoudre
-    \begin{subequations}
-        \begin{numcases}{}
-            y'(t)=y(t)\\
-            y(0)=1
-        \end{numcases}
-    \end{subequations}
-    dont nous savons depuis l'enfance que la solution est l'exponentielle. Partons donc de la fonction constante \( y_0=1\), et appliquons la contraction \eqref{EQooJUTBooILBKoE} :
-    \begin{equation}
-        u_1=1+\int_0^1u_0(s)ds=1+t.
-    \end{equation}
-    Ensuite
-    \begin{equation}
-        u_2=1+\int_0^t(1+s)ds=1+t+\frac{ t^2 }{2}.
-    \end{equation}
-    Et on voit que les itérations suivantes vont donner l'exponentielle.
-
-    Nous sommes évidemment en droit de se dire que nous avons choisi un bon point de départ. Tentons le coup avec une fonction qui n'a rien à voir avec l'exponentielle : \( u_0(x)=\sin(x)\).
-
-    Le programme suivant permet de faire de belles investigations numériques en partant d'à peu près n'importe quelle fonction :
-
-\lstinputlisting{tex/sage/picard_exp.py}
-
-    Ce programme fait \( 30\) itérations depuis la fonction \( \sin(x)\) pour tenter d'approximer \( \exp(x)\). Pour donner une idée, après \( 7\) itérations nous avons la fonction suivante :
-    \begin{equation}
-        \frac{1}{ 60 }x^5+\frac{1}{ 24 }x^4+\frac{ 1 }{2}x^2+2x-\sin(x)+1.
-    \end{equation}
-    Nous voyons que les coefficients sont des factorielles, mais pas toujours celles correspondantes à la puissance, et qu'il manque certains termes par rapport au développement de l'exponentielle que nous connaissons. Bref, le polynôme qui se met en face de \( \sin(x)\) s'adapte tout seul pour compenser.
-
-    Et après \( 30\) itérations, ça donne quoi ? Voici un graphe de l'erreur entre \( u_{30}(x)\) et \( \exp(30)\) :
-    
-    
-\begin{center}
-   \input{auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks}
-\end{center}
-
-    Pour donner une idée, \( \exp(10)\simeq 22000\). Donc il y a une faute de \( 0.01\) sur \( 22000\). Pas mal.
-
-\end{example}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de Cauchy-Arzella}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{theorem}[Cauchy-Arzela\cite{ClemKetl}]   \label{ThoHNBooUipgPX}
-    Nous considérons le système d'équation différentielles
-    \begin{subequations}        \label{EqTXlJdH}
-        \begin{numcases}{}
-            y'=f(t,y)\\
-            y(t_0)=y_0.
-        \end{numcases}
-    \end{subequations}
-    avec \( f\colon U\to \eR^n\), continue où \( U\) est ouvert dans \( \eR\times \eR^n\). Alors il existe un voisinage fermé \( V\) de \( t_0\) sur lequel une solution \( C^1\) du problème \eqref{EqTXlJdH} existe.
-\end{theorem}
-\index{théorème!Cauchy-Arzela}
-
-\begin{proof}[Idée de la démonstration]
-    Nous considérons \( M=\| f \|_{\infty}\) et \( K\), l'ensemble des fonctions \( M\)-Lipschitz sur \( U\). Nous prouvons que \( (K,\| . \|_{\infty})\) est compact. Ensuite nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon K&\to K \\
-            \Phi(f)(t)&=x_0+\int_{t_0}^tf\big( u,f(u) \big)du. 
-        \end{aligned}
-    \end{equation}
-    Après avoir prouvé que \( \Phi\) était continue, nous concluons qu'elle a un point fixe par le théorème de Schauder \ref{ThovHJXIU}.
-\end{proof}
-
-\begin{remark}
-    Quelque remarques.
-    \begin{enumerate}
-        \item
-    Les théorème de Cauchy-Lipschitz et Cauchy-Arzella donnent des existences pour des équations différentielles du type \( y'=f(t,y)\). Et si nous avons une équation du second ordre ? Alors il y a la méthode de la réduction de l'ordre qui permet de transformer une équation différentielle d'ordre élevé en un système d'ordre \( 1\).
-\item
-    Ces théorèmes posent des \emph{conditions initiales} : la valeur de \( y\) est donnée en un point, et la méthode de la réduction de l'ordre permet de donner l'existence de solutions d'un problème d'ordre \( k\) en donnant les valeurs de \( y(0)\), \( y'(0)\), \ldots \( y^{(k-1)}(0)\). C'est à dire de la fonction et de ses dérivées en un point. Rien n'est dit sur l'existence de \emph{conditions aux bords}.
-    \end{enumerate}
-    Ces deux points sont illustrés dans les exemples \ref{EXooSHMMooHVfsMB} et \ref{EXooJNOMooYqUwTZ}.
-\end{remark}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-                    \section{Théorèmes d'inversion locale et de la fonction implicite}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Mise en situation}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Dans un certain nombre de situation, il n'est pas possible de trouver des solutions explicites aux équations qui apparaissent. Néanmoins, l'existence «théorique» d'une telle solution est souvent déjà suffisante. C'est l'objet du théorème de la fonction implicite.
-
-Prenons par exemple la fonction sur $\eR^2$ donnée par 
-\begin{equation}
-    F(x,y)=x^2+y^2-1.
-\end{equation}
-Nous pouvons bien entendu regarder l'ensemble des points donnés par $F(x,y)=0$. C'est le cercle dessiné à la figure \ref{LabelFigCercleImplicite}.
-\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite. Si on donne \( x\), nous ne pouvons pas savoir si nous parlons de \( P\) ou de \( P'\).}
-\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
-
-%\ref{LabelFigCercleImplicite}.
-%\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite.}
-%\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
-
-Nous ne pouvons pas donner le cercle sous la forme $y=y(x)$ à cause du $\pm$ qui arrive quand on prend la racine carrée. Mais si on se donne le point $P$, nous pouvons dire que \emph{autour de $P$}, le cercle est la fonction
-\begin{equation}
-    y(x)=\sqrt{1-x^2}.
-\end{equation}
-Tandis que autour du point $P'$, le cercle est la fonction
-\begin{equation}
-    y(x)=-\sqrt{1-x^2}.
-\end{equation}
-Autour de ces deux point, donc, le cercle est donné par une fonction. Il n'est par contre pas possible de donner le cercle autour du point $Q$ sous la forme d'une fonction.
-
-Ce que nous voulons faire, en général, est de voir si l'ensemble des points tels que
-\begin{equation}
-    F(x_1,\ldots,x_n,y)=0
-\end{equation}
-peut être donné par une fonction $y=y(x_1,\ldots,x_n)$. En d'autre termes, est-ce qu'il existe une fonction $y(x_1,\ldots,x_n)$ telle que
-\begin{equation}
-    F\big( x_1,\ldots,x_n,y(x_1,\ldots,x_n)\big)=0.
-\end{equation}
-
-Plus généralement, soit une fonction
-\begin{equation}
-    \begin{aligned}
-        F\colon D\subset \eR^n\times \eR^m&\to \eR^m \\
-        (x,y)&\mapsto \big( F_1(x,y),\ldots, F_m(x,y) \big) 
-    \end{aligned}
-\end{equation}
-avec $x = (x_1,\ldots, x_n)$ et $y = (y_1,\ldots,y_m)$. Pour chaque $x$ fixé, on s'intéresse aux solutions du système de $m$ équations $F(x,y) = 0$ pour les inconnues $y$ ; en particulier, on voudrait pouvoir écrire $y = \varphi(x)$ vérifiant $F(x,\varphi(x)) = 0$.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème d'inversion locale}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma} \label{LemGZoqknC}
-    Soit \( E\) un espace de Banach (métrique complet) et \( \mO\) un ouvert de \( E\). Nous considérons une \( \lambda\)-contraction \( \varphi\colon \mO\to E\). Alors l'application
-    \begin{equation}
-    f\colon x\mapsto x+\varphi(x)
-    \end{equation}
-    est un homéomorphisme entre \( \mO\) et un ouvert de \( E\). De plus \( f^{-1}\) est Lipschitz de constante plus petite ou égale à \( (1-\lambda)^{-1}\).
-\end{lemma}
-Cette proposition utilise le théorème de point fixe de Picard \ref{ThoEPVkCL},
-et sera utilisée pour démontrer le théorème d'inversion locale \ref{ThoXWpzqCn}.
-% note que garder deux lignes ici est important pour vérifier les références vers le futur : la seconde ligne peut être ignorée, pas la seconde.
-
-\begin{proof}
-        Soient \( x_1,x_2\in\mO\). Nous posons \( y_1=f(x_1)\) et \( y_2=f(x_2)\). En vertu de l'inégalité de la proposition \ref{PropNmNNm} nous avons
-        \begin{subequations}    \label{subEqEBJsBfz}
-            \begin{align}
-            \big\| f(x_2)-f(x_1) \big\|&=\big\| x_2+\varphi(x_2)-x_1-\varphi(x_1) \big\|\\
-        &\geq \Big|        \| x_2-x_1 \|-\big\| \varphi(x_2)-\varphi(x_1) \big\|  \Big|\\
-    &\geq   (1-\lambda)\| x_2-x_1 \|.
-            \end{align}
-        \end{subequations}
-        À la dernière ligne les valeurs absolues sont enlevées parce que nous savons que ce qui est à l'intérieur est positif. Cela nous dit d'abord que \( f\) est injective parce que \( f(x_2)=f(x_1)\) implique \( x_2=x_1\). Donc \( f\) est inversible sur son image. Nous posons \( A=f(\mO)\) et nous devons prouver que que \( f^{-1}\colon A\to \mO\) est continue, Lipschitz de constante majorée par \( (1-\lambda)^{-1}\) et que \( A\) est ouvert.
-
-    Les inéquations \eqref{subEqEBJsBfz} nous disent que
-    \begin{equation}
-    \big\| f^{-1}(y_1)-f^{-1}(y_2) \big\|\leq \frac{ \| y_1-y_2 \| }{ 1-\lambda },
-    \end{equation}
-    c'est à dire que
-    \begin{equation}
-        f^{-1}\big( B(y,r) \big)\subset B\big( f^{-1}(y),\frac{ r }{ 1-\lambda } \big),
-    \end{equation}
-    ce qui signifie que \( f^{-1}\) est Lipschitz de constante souhaitée et donc continue.
-
-    Il reste à prouver que \( f(\mO)\) est ouvert. Pour cela nous prenons \( y_0=f(x_0)\) dans \( f(\mO)\) est nous prouvons qu'il existe \( \epsilon\) tel que \( B(y_0,\epsilon)\) soit dans \( f(\mO)\). Il faut donc que pour tout \( y\in B(y_0,\epsilon)\), l'équation \( f(x)=y\) ait une solution. Nous considérons l'application
-    \begin{equation}
-        L_y\colon x\mapsto y-\varphi(x).
-    \end{equation}
-    Ce que nous cherchons est un point fixe de \( L_y\) parce que si \( L_y(x)=x\) alors \( y=x+\varphi(x)=f(x)\). Vu que
-    \begin{equation}
-        \big\| L_y(x)-L_y(x') \big\|=\big\| \varphi(x)-\varphi(x') \big\|\leq\lambda\| x-x' \|,
-    \end{equation}
-    l'application \( L_y\) est une contraction de constante \( \lambda\). Par ailleurs \( x_0\) est un point fixe de \( L_{y_0}\), donc en vertu de la caractérisation \eqref{EqDZvtUbn} des fonctions Lipschitziennes, 
-    \begin{equation}
-        L_{y_0}\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B\big( L_{y_0}(x_0),\lambda\delta \big) }=\overline{ B(x_0,\lambda\delta) }.
-    \end{equation}
-    Vu que pour tout \( y\) et \( x\) nous avons \( L_y(x)=L_{y_0}(x)+y-y_0\),
-    \begin{equation}
-    L_y\big( \overline{ B(x_0,\delta) } \big)=L_{y_0}\big( \overline{ B(x_0,\delta) } \big)+(y-y_0)\subset \overline{ B(x_0,\lambda\delta) }+(y-y_0)\subset \overline{ B(x_0),\lambda\delta+\| y-y_0 \| }.
-    \end{equation}
-    Si \( \epsilon<(1-\lambda)\delta\) alors \( \lambda\delta+\| y-y_0 \|<\delta\). Un tel choix de \( \epsilon>0\) est possible parce que \( \lambda<1\). Pour une telle valeur de \( \epsilon\) nous avons
-    \begin{equation}
-        L_y\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B(x_0,\delta) }.
-    \end{equation}
-    Par conséquent \( L_y\) est une contraction sur l'espace métrique complet \( \overline{ B(x_0,\delta) }\), ce qui signifie que \( L_y\) y possède un point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-\end{proof}
-
-Le théorème d'inversion locale s'énonce de la façon suivante dans \( \eR^n\) :
-\begin{theorem}[Inversion locale dans \( \eR^n\)]    \label{THOooQGGWooPBRNEX}      % Ne pas mettre de label ici parce qu'il faut référencer l'autre, celui dans Banach.
-    Soit \( f\in C^k(\eR^n,\eR^n)\) et \( x_0\in \eR^n\). Si \( df_{x_0}\) est inversible, alors il existe un voisinage ouvert \( U\) de \( x_0\) et \( V\) de \( f(x_0)\) tels que \( f\colon U\to V\) soit un \( C^k\)-difféomorphisme. (c'est à dire que \( f^{-1}\) est également de classe \( C^k\))
-\end{theorem}
-
-Nous allons le démontrer dans le cas un peu plus général (mais pas plus cher\footnote{Sauf la justification de la régularité de l'application \( A\mapsto A^{-1}\)}) des espaces de Banach en tant que conséquence du théorème de point fixe de Picard \ref{ThoEPVkCL}.
-
-\begin{theorem}[Inversion locale dans un espace de Banach\cite{OWTzoEK}] \label{ThoXWpzqCn}
-    Soit une fonction \( f\in C^p(E,F)\) avec \( p\geq 1\) entre deux espaces de Banach. Soit \( x_0\in E\) tel que \( df_{x_0}\) soit une bijection bicontinue\footnote{En dimension finie, une application linéaire est toujours continue et d'inverse continu.}. Alors il existe un voisinage ouvert \( V\) de \( x_0\) et \( W\) de \( f(x_0)\) tels que
-    \begin{enumerate}
-        \item
-        \( f\colon V\to W\) soit une bijection,
-    \item
-        \( f^{-1}\colon W\to V\) soit de classe \( C^p\).
-    \end{enumerate}
-\end{theorem}
-\index{application!différentiable}
-\index{théorème!inversion locale}
-
-\begin{proof}
-    Nous commençons par simplifier un peu le problème. Pour cela, nous considérons la translation \( T\colon x\mapsto x+x_0 \) et l'application linéaire
-    \begin{equation}
-        \begin{aligned}
-            L\colon \eR^n&\to \eR^n \\
-            x&\mapsto (df_{x_0})^{-1}x
-        \end{aligned}
-    \end{equation}
-    qui sont tout deux des difféomorphismes (\( L\) en est un par hypothèse d'inversibilité). Quitte à travailler avec la fonction \( k=L\circ f\circ T\), nous pouvons supposer que \( x_0=0\) et que \( df_{x_0}=\mtu\). Pour comprendre cela il faut utiliser deux fois la formule de différentielle de fonction composée de la proposition \ref{EqDiffCompose} :
-    \begin{equation}
-        dk_0(u)=dL_{(f\circ T)(0)}\Big( df_{T(0)}dT_0(u) \Big).
-    \end{equation}
-    Vu que \( L\) est linéaire, sa différentielle est elle-même, c'est à dire \( dL_{(f\circ T)(0)}=(df_{x_0})^{-1}\), et par ailleurs \( dT_0=\mtu\), donc
-    \begin{equation}
-        dk_0(u)=(df_{x_0})^{-1}\Big( df_{x_0}(u) \Big)=u,
-    \end{equation}
-    ce qui signifie bien que \( dk_0=\mtu\). Pour tout cela nous avons utilisé en plein le fait que \( df_{x_0}\) était inversible.
-
-Nous posons \( g=f-\mtu\), c'est à dire \( g(x)=f(x)-x\), qui a la propriété \( dg_0=0\). Étant donné que \( g\) est de classe \( C^1\), l'application\footnote{Ici \( \GL(F)\) est l'ensemble des applications linéaires, inversibles et continues de \( F\) dans lui-même. Ce ne sont pas spécialement des matrices parce que nous n'avons pas d'hypothèses sur la dimension de \( F\), finie ou non.}
-    \begin{equation}
-        \begin{aligned}
-            dg\colon E&\to \GL(F) \\
-            x&\mapsto dg_x 
-        \end{aligned}
-    \end{equation}
-    est continue. En conséquence de quoi nous avons un voisinage \( U'\) de \( 0 \) pour lequel
-    \begin{equation}    \label{EqSGTOfvx}
-        \sup_{x\in U'}\| dg_x \|<\frac{ 1 }{2}.
-    \end{equation}
-    Maintenant le théorème des accroissements finis \ref{ThoNAKKght} (\ref{val_medio_2} pour la dimension finie) nous indique que pour tout \( x,x'\in U'\) nous avons\footnote{Ici nous supposons avoir choisi \( U'\) convexe afin que tous les \( a\in \mathopen[ x , x' \mathclose]\) soient bien dans \( U'\) et donc soumis à l'inéquation \eqref{EqSGTOfvx}, ce qui est toujours possible, il suffit de prendre une boule.}
-    \begin{equation}
-        \| g(x')-g(x) \|\leq \sup_{a\in\mathopen[ x , x' \mathclose]}\| dg_a \| \cdot \| x-x' \|\leq \frac{ 1 }{2}\| x-x' \|,
-    \end{equation}
-    ce qui prouve que \( g\) est une contraction au moins sur l'ouvert \( U'\). Nous allons aussi donner une idée de la façon dont \( f\) fonctionne : si \( x_1,x_2\in U'\) alors
-    \begin{subequations}
-        \begin{align}
-            \| x_1-x_2 \|&=\| g(x_1)-f(x_1)-g(x_2)+f(x_2) \| \\
-            &\leq \| g(x_1)-g(x_2) \|+\| f(x_1)-f(x_2) \|\\
-            &\leq \frac{ 1 }{2}\| x_1-x_2 \|+\| f(x_1)-f(x_2) \|,
-        \end{align}
-    \end{subequations}
-    ce qui montre que
-    \begin{equation}
-        \| x_1-x_2 \|\leq 2\| f(x_1)-f(x_2) \|.
-    \end{equation}
-    Maintenant que nous savons que \( g\) est contractante de constante \( \frac{ 1 }{2}\) et que \( f=g+\mtu\) nous pouvons utiliser la proposition \ref{LemGZoqknC} pour conclure que \( f\) est un homéomorphisme sur un ouvert \( U\) (partie de \( U'\)) de \( E\) et \( f^{-1}\) a une constante de Lipschitz plus petite ou égale à \( (1-\frac{ 1 }{2})^{-1}=2\).
-
-    Nous allons maintenant prouver que \( f^{-1}\) est différentiable et que sa différentielle est donnée par \( (df^{-1})_{f(x)}=(df_x)^{-1}\).
-
-    Soient \( a,b\in U\) et \( u=b-a\). Étant donné que \( f\) est différentiable en \( a\), il existe une fonction \( \alpha\in o(\| u \|)\) telle que
-    \begin{equation}
-        f(b)-f(a)-df_a(u)=\alpha(u).
-    \end{equation}
-    En notant \( y_a=f(a)\) et \( y_b=f(b)\) et en appliquant \( (df_a)^{-1}\) à cette dernière équation,
-    \begin{equation}
-        (df_a)^{-1}(y_b-y_a)-u=(df_a)^{-1} \big( \alpha(u) \big).
-    \end{equation}
-    Vu que \( df_a\) est bornée (et son inverse aussi), le membre de droite est encore une fonction \( \beta\) ayant la propriété \( \lim_{u\to 0}\beta(u)/\| u \|=0\); en réordonnant les termes,
-    \begin{equation}
-        b-a=(df_a)^{-1}(y_b-y_a)+\beta(u)
-    \end{equation}
-    et donc
-    \begin{equation}
-        f^{-1}(y_b)-f^{-1}(y_a)-(df_a)^{-1}(y_b-y_a)=\beta(u),
-    \end{equation}
-    ce qui prouve que \( f^{-1}\) est différentiable et que \( (df^{-1})_{y_a}=(df_a)^{-1}\).
-
-    La différentielle \( df^{-1}\) est donc obtenue par la chaine
-    \begin{equation}
-    \xymatrix{%
-        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}     &   U'\ar[r]^-{df}&\GL(F)\ar[r]^-{\Inv}&\GL(F)
-       }
-    \end{equation}
-    où l'application \( \Inv\colon \GL(F)\to \GL(F)\) est l'application \( X\mapsto X^{-1}\) qui est de classe \(  C^{\infty}\) par le théorème \ref{ThoCINVBTJ}. D'autre part, par hypothèse \( df\) est une application de classe \( C^{k-1}\) et donc au minimum \( C^0\) parce que \( k\geq 1\). Enfin, l'application \( f^{-1}\colon f(U)\to U\) est continue (parce que la proposition \ref{LemGZoqknC} précise que \( f\) est un homéomorphisme). Donc toute la chaine est continue et \( df^{-1}\) est continue. Cela entraine immédiatement que \( f^{-1}\) est \( C^1\) et donc que toute la chaine est \( C^1\).
-
-    Par récurrence nous obtenons la chaine
-    \begin{equation}
-    \xymatrix{%
-        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}_-{C^{k-1}}     &   U'\ar[r]^-{df}_-{C^{k-1}}&\GL(F)\ar[r]^-{\Inv}_-{ C^{\infty}}&\GL(F)
-       }
-    \end{equation}
-    qui prouve que \( df^{-1}\) est \( C^{k-1} \) et donc que \( f^{-1}\) est \( C^k\). La récurrence s'arrête ici parce que \( df\) n'est pas mieux que \( C^{k-1}\).
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de la fonction implicite}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous énonçons et le démontrons le théorème de la fonction implicite dans le cas d'espaces de Banach.
-\begin{theorem}[Théorème de la fonction implicite dans Banach\cite{SNPdukn}] \label{ThoAcaWho}
-    Soient \( E\), \( F\) et \( G\) des espaces de Banach et des ouverts \( U\subset E\), \( V\subset F\). Nous considérons une fonction \( f\colon U\times V\to G\) de classe \( C^r\) telle que\footnote{La notation \( d_y\) est la différentielle partielle de la définition \ref{VJM_CtSKT}.}
-    \begin{equation}
-        d_yf_{(x_0,y_0)}\colon F\to G
-    \end{equation}
-    soit un isomorphisme pour un certain \( (x_0,y_0)\in U\times V\).
-
-    Alors nous avons des voisinages \( U_0\) de \( x_0\) dans \( E\) et \( W_0\) de \( f(x_0,y_0)\) dans \( G\) et une fonction de classe \( C^r\) 
-    \begin{equation}
-        g\colon U_0\times W_0\to V
-    \end{equation}
-    telle que 
-    \begin{equation}
-        f\big( x,g(x,w) \big)=w
-    \end{equation}
-    pour tout \( (x,w)\in U_0\times W_0\).
-    
-    Cette fonction \( g\) est unique au sens suivant : il existe un voisinage \( V_0 \) de \( y_0\) tel que si \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) satisfont à \( f(x,y)=w\) alors \( y=g(x,w)\). Autrement dit, la fonction \( g\colon U_0\times W_0\to V_0\) est unique.
-\end{theorem}
-\index{théorème!fonction implicite dans Banach}
-
-\begin{proof}
-    Nous commençons par considérer la fonction
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon U\times V&\to E\times G \\
-            (x,y)&\mapsto \big( x,f(x,y) \big)
-        \end{aligned}
-    \end{equation}
-    et sa différentielle 
-    \begin{subequations}
-        \begin{align}
-            d\Phi_{(x_0,y_0)}(u,v)&=\Dsdd{ \big( x_0+tu,f(x_0+tu,y_0+tv) \big) }{t}{0}\\
-            &=\left( \Dsdd{ x_0+tu }{t}{0},\Dsdd{ f(x_0+tu,y_0+tv) }{t}{0} \right)\\
-            &=\left( u,df_{(x_0,y_0)}(u,v) \right).
-        \end{align}
-    \end{subequations}
-    Nous utilisons alors la proposition \ref{PropLDN_nHWDF} pour conclure que
-    \begin{equation}
-        d\Phi_{(x_0,y_0)}(u,v)=\big( u,(d_1f)_{(x_0,y_0)}(u)+(d_2f)_{(x_0,y_0)}(v) \big),
-    \end{equation}
-    mais comme par hypothèse \( (d_2f)_{(x_0,y_0)}\colon F\to G\) est un isomorphisme, l'application \( d\Phi_{(x_0,y_0)}\colon E\times F\to E\times G\) est également un isomorphisme. Par conséquent le théorème d'inversion locale \ref{ThoXWpzqCn} nous indique qu'il existe un voisinage \( \mO\) de \( (x_0,y_0)\) et \( \mP\) de \( \Phi(x_0,y_0)\) tels que \( \Phi\colon \mO\to \mP\) soit une bijection et \( \Phi^{-1}\colon \mP\to \mO\) soit de classe \( C^r\). Vu que \( \mP\) est un voisinage de
-    \begin{equation}
-        \Phi(x_0,y_0)=\big( x_0,f(x_0,y_0) \big),
-    \end{equation}
-    nous pouvons par \ref{PropDXR_KbaLC} le choisir un peu plus petit de telle sorte à avoir \( \mP=U_0\times W_0\) où \( U_0\) est un voisinage de \( x_0\) et \( W_0\) un voisinage de \( f(x_0,y_0)\). Dans ce cas nous devons obligatoirement aussi restreindre \( \mO\) à \( U_0\times V_0\) pour un certain voisinage \( V_0\) de \( y_0\). L'application \( \Phi^{-1}\) a obligatoirement la forme
-    \begin{equation}    \label{EqMHT_QrHRn}
-        \begin{aligned}
-            \Phi^{-1}\colon U_0\times W_0&\to U_0\times V_0 \\
-            (x,w)&\mapsto \big( x,g(x,w) \big) 
-        \end{aligned}
-    \end{equation}
-    pour une certaine fonction \( g\colon U_0\times W_0\to V\). Cette fonction \( g\) est la fonction cherchée parce qu'en appliquant \( \Phi\) à \eqref{EqMHT_QrHRn}, 
-    \begin{equation}
-        (x,w)=\Phi\big( x,g(x,w) \big)=\Big( x,f\big( x,g(x,w) \big) \Big),
-    \end{equation}
-    qui nous dit que pour tout \( x\in U_0\) et tout \( w\in W_0\) nous avons
-    \begin{equation}
-        f\big( x,g(x,w) \big)=w.
-    \end{equation}
-
-    Si vous avez bien suivi le sens de l'équation \eqref{EqMHT_QrHRn} alors vous avez compris l'unicité. Sinon, considérez \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) tels que \( f(x,y)=w\). Alors \( \big( x,f(x,y) \big)=(x,w)\) et 
-    \begin{equation}
-        \Phi(x,y)=(x,w).
-    \end{equation}
-    Mais vu que \( \Phi\colon U_0\times V_0\to U_0\times W_0\) est une bijection, cette relation définit de façon univoque l'élément \( (x,y)\) de \( U_0\times V_0\), qui ne sera autre que \( g(x,w)\).
-\end{proof}
-
-Le théorème de la fonction implicite s'énonce de la façon suivante pour des espaces de dimension finie.
-% Attention : avant de citer ce théorème, voir s'il est suffisant. Ici \varphi a une variable; dans l'autre énoncé il en a deux.
-\begin{theorem}[Théorème de la fonction implicite en dimension finie]   \label{ThoRYN_jvZrZ}
-    Soit une fonction \( F\colon \eR^n\times \eR^m\to \eR^m\) de classe \( C^k\) et \( (\alpha,\beta)\in \eR^n\times \eR^m\) tels que
-    \begin{enumerate}
-        \item
-            \( F(\alpha,\beta)=0\),
-        \item
-            \( \frac{ \partial (F_1,\ldots, F_m) }{ \partial (y_1,\ldots, y_m) }\neq 0\), c'est à dire que \( (d_yF)_{(\alpha,\beta)} \) est inversible.
-    \end{enumerate}
-    Alors il existe un voisinage ouvert \( V\) de \( \alpha\) dans \( \eR^n\), un voisinage ouvert \( W\) de \( \beta\) dans \( \eR^m\) et une application \( \varphi\colon V\to W\) de classe \( C^k\)  telle que pour tout \( x\in V\) on ait
-    \begin{equation}
-        F\big( x,\varphi(x) \big)=0.
-    \end{equation}
-    De plus si \( (x,y)\in V\times W\) satisfait à \( F(x,y)=0\), alors \( y=\varphi(x)\).
-\end{theorem}
-\index{théorème!fonction implicite dans \( \eR^n\)}
-
-\begin{remark}\label{RemPYA_pkTEx}
-    Notons que cet énoncé est tourné un peu différemment en ce qui concerne le nombre de variables dont dépend la fonction implicite : comparez
-    \begin{subequations}
-        \begin{align}
-            f\big( x,g(x,w) \big)=w\\
-            F\big( x,\varphi(x) \big)=0.
-        \end{align}
-    \end{subequations}
-    Le deuxième est un cas particulier du premier en posant 
-    \begin{equation}
-        F(x,y)=f(x,y)-f(x_0,y_0)
-    \end{equation}
-    et donc en considérant \( w\) comme valant la constante \( f(x_0,y_0)\); dans ce cas la fonction \( g\) ne dépend plus que de la variable \( x\).
-
-\end{remark}
-
-\begin{example}
-    La remarque \ref{RemPYA_pkTEx} signifie entre autres que le théorème \ref{ThoAcaWho} est plus fort que \ref{ThoRYN_jvZrZ} parce que le premier permet de choisir la valeur d'arrivée. Parlons de l'exemple classique du cercle et de la fonction \( f(x,y)=x^2+y^2\). Nous savons que
-    \begin{equation}
-        f(\alpha,\beta)=1.
-    \end{equation}
-    Alors le théorème \ref{ThoAcaWho} nous donne une fonction \( g\) telle que
-    \begin{equation}
-        f(x,g(x,r))=r
-    \end{equation}
-    tant que \( x\) est proche de \( \alpha\), que \( r\) est proche de \( 1\) et que \( g\) donne des valeurs proches de \( \beta\).
-
-    L'énoncé \ref{ThoRYN_jvZrZ} nous oblige à travailler avec la fonction \( F(x,y)=x^2+y^2-1\), de telle sorte que
-    \begin{equation}
-        F(\alpha,\beta)=0,
-    \end{equation}
-    et que nous ayons une fonction \( \varphi\) telle que
-    \begin{equation}
-        F(x,\varphi(x))=0.
-    \end{equation}
-    La fonction \( \varphi\) ne permet donc que de trouver des points sur le cercle de rayon \( 1\).
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Exemple}
-%---------------------------------------------------------------------------------------------------------------------------
-    
-Le théorème de la fonction implicite a pour objet de donner l'existence de la fonction $\varphi$. Maintenant nous pouvons dire beaucoup de choses sur les dérivées de $\varphi$ en considérant la fonction
-\begin{equation}
-    x\mapsto F\big( x,\varphi(x) \big).
-\end{equation}
-Par définition de $\varphi$, cette fonction est toujours nulle. En particulier, nous pouvons dériver l'équation
-\begin{equation}
-    F\big( x,\varphi(x) \big)=0,
-\end{equation}
-et nous trouvons plein de choses.
-
-
-Prenons par exemple la fonction
-\begin{equation}
-    F\big( (x,y),z \big)=ze^z-x-y,
-\end{equation}
-et demandons nous ce que nous pouvons dire sur la fonction $z(x,y)$ telle que
-\begin{equation}
-    F\big( x,y,z(x,y) \big)=0,
-\end{equation}
-c'est à dire telle que
-\begin{equation}        \label{EqDefZImplExemple}
-    z(x,y) e^{z(x,y)}-x-y=0.
-\end{equation}
-pour tout $x$ et $y\in\eR$. Nous pouvons facilement trouver $z(0,0)$ parce que
-\begin{equation}
-    z(0,0) e^{z(0,0)}=0,
-\end{equation}
-donc $z(0,0)=0$.
-
-Nous pouvons dire des choses sur les dérivées de $z(x,y)$. Voyons par exemple $(\partial_xz)(x,y)$. Pour trouver cette dérivée, nous dérivons la relation \eqref{EqDefZImplExemple} par rapport à $x$. Ce que nous trouvons est
-\begin{equation}
-    (\partial_xz)e^z+ze^z(\partial_xz)-1=0.
-\end{equation}
-Cette équation peut être résolue par rapport à $\partial_xz$~:
-\begin{equation}
-    \frac{ \partial z }{ \partial x }(x,y)=\frac{1}{ e^z(1+z) }.
-\end{equation}
-Remarquez que cette équation ne donne pas tout à fait la dérivée de $z$ en fonction de $x$ et $y$, parce que $z$ apparaît dans l'expression, alors que $z$ est justement la fonction inconnue. En général, c'est la vie, nous ne pouvons pas faire mieux.
-
-Dans certains cas, on peut aller plus loin. Par exemple, nous pouvons calculer cette dérivée au point $(x,y)=(0,0)$ parce que $z(0,0)$ est connu :
-\begin{equation}
-    \frac{ \partial z }{ \partial x }(0,0)=1.
-\end{equation}
-Cela est pratique pour calculer, par exemple, le développement en Taylor de $z$ autour de $(0,0)$.
-
-\begin{example}
-    Est-ce que l'équation \( e^{y}+xy=0\) définit au moins localement une fonction \( y(x)\) ? Nous considérons la fonction
-    \begin{equation}
-        f(x,y)=\begin{pmatrix}
-            x    \\ 
-            e^{y}+xy    
-        \end{pmatrix}
-    \end{equation}
-    La différentielle de cette application est
-    \begin{equation}
-            df_{(0,0)}(u)=\frac{ d }{ dt }\Big[ f(tu_1,tu_2) \Big]_{t=0}
-            =\frac{ d }{ dt }\begin{pmatrix}
-                tu_1    \\ 
-                e^{tu_2}+t^2u_1u_2    
-            \end{pmatrix}_{t=0}
-            =\begin{pmatrix}
-                u_1    \\ 
-                u_2    
-            \end{pmatrix}.
-    \end{equation}
-    L'application \( f\) définit donc un difféomorphisme local autour des points \( (x_0,y_0)\) et \( f(x_0,y_0)\). Soit \( (u,0)\) un point dans le voisinage de \( f(x_0,y_0)\). Alors il existe un unique \( (x,y)\) tel que
-    \begin{equation}
-        f(x,y)=\begin{pmatrix}
-               x \\ 
-            e^y+xy    
-        \end{pmatrix}=
-        \begin{pmatrix}
-            u    \\ 
-                0
-        \end{pmatrix}.
-    \end{equation}
-    Nous avons automatiquement \( x=u\) et \( e^y+xy=0\). Notons toutefois que pour que ce procédé donne effectivement une fonction implicite \( y(x)\) nous devons avoir des points de la forme \( (u,0)\) dans le voisinage de \( f(x_0,y_0)\).
-\end{example}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Recherche d'extrema}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema à une variable}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
-\end{definition}
-
-La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
-\begin{proposition}     \label{PROPooNVKXooXtKkuz}
-Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
-\end{proposition}
-
-La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
-
-Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
-\begin{proposition}
-Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
-\begin{equation}
-    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
-\end{equation}
-et que
-\begin{equation}
-    f^{(k)}(a)\neq 0.
-\end{equation}
-Dans ce cas,
-\begin{enumerate}
-
-\item
-Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
-\item
-Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
-
-\end{enumerate}
-\end{proposition}
-
-Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
-
-Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
-
-Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
-
-\begin{example}
-	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
-\end{example}
- 
-Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
-\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
-\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
-
-Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
-
-Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
-
-\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
-\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
- 
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema libre}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooYJLZooLkEAYf}
-Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
-\end{definition}
-
-Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
-
-Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
-\begin{equation}
-    d^2f(a)=\begin{pmatrix}
-    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
-    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
-\end{pmatrix}.
-\end{equation}
-Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
-
-\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
-\end{proposition}
-
-\begin{proof}
-    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
-    \begin{equation}
-        \begin{aligned}
-            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
-            t&\mapsto f(a+tu) 
-        \end{aligned}
-    \end{equation}
-    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
-    \begin{equation}
-        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
-    \end{equation}
-\end{proof}
-
-\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
-    \begin{enumerate}
-        \item   \label{ITEMooCVFVooWltGqI}
-            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
-        \item\label{ItemPropoExtreRn}
-            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
-    \end{enumerate}
-\end{proposition}
-\index{extrema}
-
-\begin{proof}
-    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
-    \begin{equation}
-        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
-    \end{equation}
-    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
-    \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
-    \end{equation}
-    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
-    \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
-    \end{equation}
-    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
-    \begin{equation}
-        (d^2f)_a(e_h,e_h)\geq 0.
-    \end{equation}
-    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
-\end{proof}
-
-La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
-
-Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
-
-La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
-\begin{enumerate}
-    \item
-        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
-    \item
-        écrire $d^2f(a)$ pour chacun des candidats
-    \item
-        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
-    \item
-        conclure.
-\end{enumerate}
-
-Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
-
-\begin{example}
-    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
-    \begin{equation}
-        df=\big(4x^3-4y;4y^3-4x),
-    \end{equation}
-    et la matrice des dérivées secondes est
-    \begin{equation}
-        M=d^2f(x,y)=\begin{pmatrix}
-            12x^2    &   -4    \\ 
-            -4    &   12y^2    
-        \end{pmatrix}.
-    \end{equation}
-    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
-
-    Pour le point \( (0,0)\) nous avons
-    \begin{equation}
-        M=\begin{pmatrix}
-            0    &   -4    \\ 
-            -4    &   0    
-        \end{pmatrix},
-    \end{equation}
-    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
-
-    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
-    \begin{equation}
-        M=\begin{pmatrix}
-            12    &   -4    \\ 
-            -4    &   12    
-        \end{pmatrix},
-    \end{equation}
-    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Un peu de recettes de cuisine}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{enumerate}
-\item Rechercher les points critiques, càd les $(x,y)$ tels que
-\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
-En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
-\item Déterminer la nature des points critiques: «test» des dérivées secondes:
-\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
-\begin{enumerate}
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
-\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
-\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
-\end{enumerate}
-\end{enumerate}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Extrema liés}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
-
-Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
-
-\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
-    Soit \( A\), un ouvert de \( \eR^n\) et
-    \begin{enumerate}
-        \item
-            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
-        \item 
-            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
-        \item
-            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
-        \item
-            un extrema local $a\in M$ de $f$ relativement à $M$.
-    \end{enumerate}
-    Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
-    \begin{equation}
-        L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_r)=f(x_1,\ldots,x_n)+\sum_{i=1}^r\lambda_iG_i(x_1,\ldots,x_n).
-    \end{equation}
-    Autrement dit, si \( a\) est un extrema lié, alors \( \nabla f(a)\) est une combinaisons des \( \nabla G_i(a)\), ou encore il existe des \( \lambda_i\) tels que
-    \begin{equation}    \label{EqRDsSXyZ}
-        df(a)=\sum_i\lambda_idG_i(a).
-    \end{equation}
-\end{theorem}
-\index{théorème!inversion locale!utilisation}
-\index{extrema!lié}
-\index{théorème!extrema!lié}
-\index{application!différentiable!extrema lié}
-\index{variété}
-\index{rang!différentielle}
-\index{forme!linéaire!différentielle}
-La fonction $L$ est le \defe{lagrangien}{lagrangien} du problème et les variables \( \lambda_i\) sont les \defe{multiplicateurs de Lagrange}{multiplicateur!de Lagrange}\index{Lagrange!multiplicateur}.
-
-\begin{proof}
-    Si \( r=n\) alors les vecteurs linéairement indépendantes \( \nabla G_i(a) \) forment une base de \( \eR^n\) et donc évidemment les \( \lambda_i\) existent. Nous supposons donc maintenant que \( r<n\). Nous notons \( (z_i)_{i=1\ldots n}\) les coordonnées sur \( \eR^n\).
-    
-    La matrice
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{ \partial G_1 }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_1 }{ \partial z_n }(a)    \\
-            \vdots    &   \ddots    &   \vdots    \\
-            \frac{ \partial G_r }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_r }{ \partial z_n }(a)
-        \end{pmatrix}
-    \end{equation}
-    est de rang \( r\) parce que les lignes sont par hypothèses linéairement indépendantes. Nous nommons \( (y_i)_{i=1,\ldots, r}\) un choix de \( r\) parmi les \( (z_i)\) tels que
-    \begin{equation}
-        \det\begin{pmatrix}
-            \frac{ \partial G_1 }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_1 }{ \partial y_r }    \\
-            \vdots    &   \ddots    &   \vdots    \\
-            \frac{ \partial G_r }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_r }{ \partial y_r }
-        \end{pmatrix}\neq 0.
-    \end{equation}
-    Nous identifions \( \eR^n\) à \( \eR^s\times \eR^r\) dans lequel \( \eR^r\) est la partie générée par les \( (y_i)_{i=1,\ldots, r}\). Nous nommons \( (x_j)_{j=1,\ldots, s}\) les coordonnées sur \( \eR^s\). Autrement dit, les coordonnées sur \( \eR^n\) sont \( x_1,\ldots, x_s,y_1,\ldots, y_r\). Dans ces coordonnées, nous nommons \( a=(\alpha,\beta)\) avec \( \alpha\in \eR^s\) et \( \beta\in \eR^r\).
-
-    Si nous notons \( G=(G_1,\ldots, G_r)\), le théorème de la fonction implicite (théorème \ref{ThoAcaWho})  nous dit qu'il existe un voisinage \( U'\) de \( \alpha\in \eR^n\), un voisinage \( V'\) de \( \beta\in \eR^r\) et une fonction \( \varphi\colon U'\to V'\) de classe \( C^1\) telle que si \( (x,y)\in U'\times V'\), alors
-    \begin{equation}
-        g(x,y)=0
-    \end{equation}
-    si et seulement si \( y=\varphi(x)\). Nous posons maintenant
-    \begin{subequations}
-        \begin{align}
-            \psi(x)&=(x,\varphi(x))\\
-            h(x)&=f\big( \psi(x) \big).
-        \end{align}
-    \end{subequations}
-    Nous avons \( \psi(\alpha)=a\) et \( \psi(x)\in M\) pour tout \( x\in U'\). La fonction \( h\) a donc un extrema local en \( \alpha\) et donc les dérivées partielles de \( h\) y sont nulles. Cela signifie que
-    \begin{equation}
-        0=\frac{ \partial h }{ \partial x_i }(\alpha)=\sum_{j=1}^n\frac{ \partial f }{ \partial x_j }\frac{ \partial x_j }{ \partial x_i }+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }\frac{ \partial \varphi_k }{ \partial x_i },
-    \end{equation}
-    c'est à dire
-    \begin{equation}
-        \frac{ \partial f }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
-    \end{equation}
-    pour tout \( i=1,\ldots, s\). D'autre part pour tout $k$, la fonction \( l_k(x)=G_k\big( x,\varphi(x) \big)\) est constante et vaut zéro; ses dérivées partielles sont donc nulles :
-    \begin{equation}
-        \frac{ \partial l }{ \partial x_i }(\alpha)=\frac{ \partial G_k }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial G_k }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
-    \end{equation}
-    pour tout \( i=1,\ldots, s\) et \( k=1,\ldots, r\).
-    
-    Les \( s\) premières colonnes de la matrice
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{ \partial f }{ \partial x_1 }   &   \cdots    &   \frac{ \partial f }{ \partial x_s }    &   \frac{ \partial f }{ \partial y_1 }    &   \cdots    &   \frac{ \partial f }{ \partial y_r }\\  
-            \frac{ \partial G_1 }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial x_s }    &   \frac{ \partial G_1 }{ \partial y_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial y_r }\\
-            \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots\\
-            \frac{ \partial G_r }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_r }{ \partial x_s }    &   \frac{ \partial G_r }{ \partial y_1 }    &  \cdots   & \frac{ \partial G_r }{ \partial y_r }  
-        \end{pmatrix}
-    \end{equation}
-    s'expriment en terme des \( r\) dernières. La matrice est donc au maximum de rang \( r\). Notons que la première ligne est \( \nabla f\) et les \( r\) suivantes sont les \( \nabla G_i\). Vu que ces lignes sont des vecteurs liés, il existe \( \mu_0,\ldots, \mu_r\) tels que
-    \begin{equation}
-        \mu_0\nabla f+\sum_{i=1}^r\mu_i\nabla G_i=0.
-    \end{equation}
-    Par hypothèse les \( \nabla G_i\) sont linéairement indépendants, ce qui nous dit que \( \mu_0\neq 0\). Donc nous avons ce qu'il nous faut :
-    \begin{equation}
-        \nabla f(a)=\sum_i\frac{ \mu_i }{ \mu_0 } \nabla G_i(a).
-    \end{equation}
-
-    Notons qu'au vu de l'expression \eqref{EqRDsSXyZ}, le fait que les formes \( \{ dG_i(a) \}_{1\leq i\leq r}\) forment une partie libre dans \( (\eR^n)^*\) implique que les \( \lambda_i\) sont uniques.
-\end{proof}
-
-La proposition suivante est la même que \ref{ThoRGJosS}.
-\begin{proposition} \label{PropfPPUxh}
-    Soit \( U\), un ouvert de \( \eR^n\) et des fonctions de classe \( C^1\) \( f,g_1,\ldots, g_r\colon U\to \eR\). Nous considérons
-    \begin{equation}
-        \Gamma=\{ x\in U\tq g_1(x)=\ldots=g_r(x)=0 \}.
-    \end{equation}
-    Soit \( a\) un extrémum de \( f|_{\Gamma}\). Supposons que les formes \( dg_1,\ldots, dg_r\) soient linéairement indépendantes en \( a\). Alors il existe \( \lambda_1,\ldots, \lambda_r\) dans \( \eR\) tel que
-    \begin{equation}
-        df_a=\sum_{i=1}^r\lambda_i(dg_i)_a.
-    \end{equation}
-\end{proposition}
-
-En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
-
-L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
-
-\begin{lemma}       \label{LemmeMinSCimpliqueS}
-    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
-\end{lemma}
-
-\begin{proof}
-    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Fonctions convexes}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}[\cite{JFihMcQ}]  \label{DefVQXRJQz}
-    Une fonction $f$ d’un intervalle $I$ de \( \eR\) vers \( \eR\) est dite \defe{convexe}{fonction!convexe}\index{convexité!fonction} lorsque, pour tous \( x_1\) et \( x_2\) de $I$ et tout $\lambda$ dans $[0, 1]$ nous avons
-    \begin{equation}        \label{EQooYNAPooFePQZy}
-        f\big(\lambda\, x_1+(1-\lambda)\, x_2\big) \leq \lambda\, f(x_1)+(1-\lambda)\, f(x_2)
-    \end{equation}
-    Si l'inégalité est stricte, alors nous disons que la fonction \( f\) est \defe{strictement convexe}{convexité!stricte}.
-
-    Une fonction est \defe{concave}{concave} si son opposée est convexe.
-\end{definition}
-
-
-\begin{normaltext}[\cite{GYfviRu}]
-    Les différents résultats pour les fonctions convexes s'adaptent généralement sans mal aux fonctions strictement convexes. Une nuance cependant : de même que les fonctions dérivables convexes sont celles qui ont une dérivée croissante, les fonctions dérivables strictement convexes sont celles qui ont une dérivée strictement croissante (proposition \ref{PropYKwTDPX}). En revanche, il ne faudrait pas croire que la dérivée seconde d'une fonction dérivable strictement convexe est nécessairement une fonction à valeurs strictement positives (voir théorème \ref{ThoGXjKeYb}) : la dérivée d'une fonction strictement croissante peut s'annuler occasionnellement, ou plus exactement peut s'annuler sur un ensemble de points d'intérieur vide. Penser à \( x\mapsto x^4\) pour un exemple de fonction strictement convexe dont la dérivée seconde s'annule.
-\end{normaltext}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Inégalité des pentes}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Dans l'étude des fonctions convexes nous allons souvent utiliser la fonction \defe{taux d'accroissement}{taux d'accroissement} qui est, pour \( \alpha\) dans le domaine de convexité de \( f\) définie par
-\begin{equation}    \label{EqRYBazWd}
-    \begin{aligned}
-        \tau_{\alpha}\colon I\setminus\{ \alpha \}&\to \eR \\
-        x&\mapsto \frac{ f(x)-f(\alpha) }{ x-\alpha }. 
-    \end{aligned}
-\end{equation}
-
-\begin{proposition}[Inégalité des pentes\cite{OJIMBtv}] \label{PropMDMGjGO}
-    Soit \( f\) une fonction convexe sur un intervalle \( I\subset \eR\). Alors pour tout \( a<b<c\) dans \( I\) nous avons\footnote{Les inégalités sont strictes si la fonction \( f\) est strictement convexe.}
-    \begin{equation}
-        \frac{ f(b)-f(a)  }{ b-a }\leq\frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
-    \end{equation}
-    En d'autres termes,
-    \begin{equation}
-        \tau_a(b)\leq\tau_a(c)\leq \tau_b(c),
-    \end{equation}
-    c'est à dire que \( \tau\) est croissante en ses deux arguments.
-\end{proposition}
-\index{inégalité!des pentes}
-
-\begin{proof}
-    D'abord les inégalités \( a<b<c\) impliquent \( 0<b-a<c-a\) et donc
-    \begin{equation}
-        \lambda=\frac{ b-a }{ c-a }<1.
-    \end{equation}
-    L'astuce est de remarquer que \( (1-\lambda)a+\lambda c=b\). Donc \( \lambda\) a toutes les bonnes propriétés pour être utilisé dans la définition de la convexité :
-    \begin{equation}
-        f\big( (1-\lambda)a+\lambda c \big)\leq \lambda f(c)+(1-\lambda)f(a),
-    \end{equation}
-    c'est à dire
-    \begin{equation}
-        f(b)-f(a)\leq \lambda\big( f(c)-f(a) \big)
-    \end{equation}
-    ou encore, en remplaçant \( \lambda\) par sa valeur :
-    \begin{equation}
-        \frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(c)-f(a) }{ c-a }.
-    \end{equation}
-    Cela fait déjà une des inégalités à savoir.
-    
-    D'autre part en partant de \( -a<-b<-c\) nous posons
-    \begin{equation}
-        0<\lambda=\frac{ c-b }{ c-a }.
-    \end{equation}
-    Nous avons à nouveau \( b=(1-\lambda)c+\lambda a\) et nous pouvons obtenir la seconde inégalité
-    \begin{equation}
-        \frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
-    \end{equation}
-\end{proof}
-
-Géométriquement, l'inégalité des pentes se comprend facilement : le coefficient angulaire de la corde du graphe augmente. Donc si \( x<y<z\), le coefficient moyen entre \( x\) et \( y\) est plus petit que celui entre \( x\) et \( z\) qui est plus petit que celui entre \( y\) et \( z\).
-
-Donc si le coefficient angulaire moyen entre \( a\) et \( b+u\) vaut celui entre \( a\) et \( b\), ce coefficient ne peut qu'être constant entra \( a\) et \( b\) : sinon il serait plus grand entre \( b\) et \( b+u\) et la moyenne sur \( a\to b+u\) serait plus grande que sa moyenne sur \( a\to b\). Mais avoir un coefficient angulaire constant signifie être une droite.
-
-En résumé, si une fonction est convexe et non strictement convexe, alors son graphe est une droite. C'est en gros cela que la proposition \ref{PROPooOCOEooEGybmS} clarifiera.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Convexité et régularité}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}[\cite{GYfviRu}]   \label{LemKLTsHIQ}
-    Une fonction convexe sur un ouvert
-    \begin{enumerate}
-        \item
-            y admet des dérivées à gauche et à droite en chaque point,
-        \item
-            y est continue.
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Soit \( I=\mathopen] a , b \mathclose[\) un intervalle sur lequel \( f\) est convexe et \( \alpha\in I\). Nous allons prouver que \( f\) est continue en \( \alpha\). Nous considérons \( \tau_{\alpha}\) le taux d'accroissement définit par \eqref{EqRYBazWd}; c'est une fonction croissante comme précisé dans l'inégalité des trois pentes \ref{PropMDMGjGO} et de plus \( \tau_{\alpha}(x)\) est bornée supérieurement par \( \tau_{\alpha}(b)\) pour \( x<\alpha\) et inférieurement par \( \tau_{\alpha}(a)\) pour \( x>\alpha\). Les limites existent donc et sont finies par la proposition \ref{PropMTmBYeU}. Autrement dit les limites
-        \begin{subequations}
-            \begin{align}
-                \lim_{x\to \alpha+} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^+} \tau_{\alpha}(x)=\inf_{t>\alpha}\tau_{\alpha}(t)\\
-                \lim_{x\to \alpha^-} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^-} \tau_{\alpha}(x)=\sup_{t<\alpha}\tau_{\alpha}(t).
-            \end{align}
-        \end{subequations}
-        existent et sont finies, c'est à dire que la fonction \( f\) admet une dérivée à gauche et à droite.
-
-        Pour tout \( x\) nous avons les inégalités
-        \begin{equation}
-            \tau_{\alpha}(a)\leq \frac{ f(x)-f(\alpha) }{ x-\alpha }\leq \tau_{\alpha}(b).
-        \end{equation}
-        En posant \( k=\max\{ \tau_{\alpha}(a),\tau_{\alpha}(b) \}\) nous avons
-        \begin{equation}
-            \big| f(x)-f(\alpha) \big|\leq k| x-\alpha |.
-        \end{equation}
-        La fonction est donc Lipschitzienne et par conséquent continue par la proposition \ref{PropFZgFTEW}.
-\end{proof}
-
-\begin{remark}
-    Les dérivées à gauche et à droite ne sont a priori pas égales. Penser par exemple à une fonction affine par morceaux dont les pentes augmentent à chaque morceau.
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Dérivées d'une fonction convexe}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[\cite{RIKpeIH,ooGCESooQzZtVC,MonCerveau}] \label{PropYKwTDPX}
-    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) 
-    \begin{enumerate}
-        \item       \label{ITEMooUTSAooJvhZNm}
-            est convexe si et seulement si sa dérivée est croissante sur \( I\).
-        \item       \label{ITEMooLLSIooFwkxtV}
-            est strictement convexe si et seulement si sa dérivée est strictement croissante sur \( I\)
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-
-
-    Pour la preuve de \ref{ITEMooUTSAooJvhZNm} et \ref{ITEMooLLSIooFwkxtV}, nous allons démontrer les énoncés «non stricts»  et indiquer ce qu'il faut changer pour obtenir les énoncés «stricts».
-    \begin{subproof}
-    \item[Sens direct]
-    Nous supposons que \( f\) est convexe. Soient \( a<b\) dans \( I\) et \( x\in\mathopen] a , b \mathclose[\). D'après l'inégalité des pentes \ref{PropMDMGjGO},
-        \begin{equation}        \label{EqATDLooIcqdDI}
-            \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(b)-f(x) }{ b-x }.
-        \end{equation}
-        En faisant la limite \( x\to a\) nous avons
-        \begin{equation}
-            f'(a)\leq \frac{ f(b)-f(a) }{ b-a }
-        \end{equation}
-        et la limite \( x\to b\) donne
-        \begin{equation}
-            \frac{ f(b)-f(a) }{ b-a }\leq f'(b).
-        \end{equation}
-        Ici les inégalités sont non a priori strictes, même si \( f\) est strictement convexe : même avec des inégalités strictes dans \eqref{EqATDLooIcqdDI}, le passage à la limite rend l'inégalité non stricte. Quoi qu'il en soit nous avons 
-        \begin{equation}        \label{EqQGVMooBpuvNr}
-            f'(a)\leq f'(b).
-        \end{equation}
-    \item[Sens direct : strict]
-         Nous savons déjà que \( f'\) est croissante. Si \eqref{EqQGVMooBpuvNr} était une égalité, alors \( f'\) serait constante sur \( \mathopen] a , b \mathclose[\) parce qu'en prenant \( c\) entre \( a\) et \( b\) nous aurions \( f'(a)\leq f'(c)\leq f'(b)\) avec \( f'(a)=f'(b)\). Donc \( f'(a)=f'(c)\). Avoir \( f'\) constante sur un intervalle est contraire à la stricte convexité.
-
-         \item[Sens réciproque]
-
-             Nous supposons que \( f'\) est croissante et nous considérons \( a<b\) dans \( I\) ainsi que \( \lambda\in \mathopen[ 0 , 1 \mathclose]\). Nous posons \( x=\lambda a+(1-\lambda)b\), et nous savons que \( a\leq x\leq b\). Le théorème des accroissements finis \ref{ThoAccFinis} donne \( c_1\in\mathopen] a , x \mathclose[\) et \( c_2\in \mathopen] x , b \mathclose[\) tels que
-                 \begin{equation}
-                     f'(c_1)=\frac{ f(x)-f(a) }{ x-a }
-                 \end{equation}
-                 et 
-                 \begin{equation}
-                     f'(c_2)=\frac{ f(b)-f(x) }{ b-x }.
-                 \end{equation}
-                 Et en plus \( c_1<c_2\). Vu que \( f'\) est croissante nous avons \( f'(c_1)\leq f'(c_2)\) et donc
-                 \begin{equation}       \label{EqSAOCooWAwClQ}
-                     \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(x) }{ b-x }.
-                 \end{equation}
-                 En remplaçant \( x\) par sa valeur en termes de \( \lambda\), \( a\) et \( b\) nous avons \( x-a=(1-\lambda)(b-a)\) et \( b-x=\lambda(b-a)\), et l'inégalité \eqref{EqSAOCooWAwClQ} nous donne
-                 \begin{equation}
-                     f(x)\leq \lambda f(a)+(1-\lambda)f(b).
-                 \end{equation}
-             \item[Sens réciproque : strict]
-                 Si \( f'\) est strictement croissante, nous avons \( f'(c_2)<f'(c_2)\) et les inégalité suivantes sont strictes, ce qui donne
-                 \begin{equation}
-                     f(x)< \lambda f(a)+(1-\lambda)f(b).
-                 \end{equation}
-    \end{subproof}
-\end{proof}
-
-\begin{theorem}[\cite{RIKpeIH}] \label{ThoGXjKeYb}
-    Une fonction \( f\) de classe \( C^2\) est convexe si et seulement si \( f''\) est positive.
-\end{theorem}
-
-\begin{proof}
-    La fonction est \( C^2\), donc \( f''\) est positive si et seulement si \( f'\) est croissante (proposition \ref{PropGFkZMwD}) alors que la proposition \ref{PropYKwTDPX} nous jure que \( f\) sera convexe si et seulement si \( f'\) est croissante.
-\end{proof}
-
-\begin{remark}      \label{REMooVRPQooIybxmp}
-    Une fonction peut être strictement convexe sans que sa dérivée seconde ne soit toujours strictement positive. En exemple : \( x\mapsto x^4\) est strictement convexe alors que sa dérivée seconde s'annule en zéro.
-\end{remark}
-
-\begin{example} \label{ExPDRooZCtkOz}
-    Quelques exemples utilisant le théorème \ref{ThoGXjKeYb}
-    \begin{enumerate}
-        \item
-    La fonction \( x\mapsto x^2\) est convexe parce que sa dérivée seconde est la constante (positive) \( 2\).
-\item La fonction \( x\mapsto\frac{1}{ x }\) est convexe sur \( \eR^+\setminus\{ 0 \}\) (sa dérivée seconde est \( 2x^{-3}\)).
-\item
-    La fonction exponentielle est également convexe.
-\item
-    La fonction \( \ln\) est concave parce que la dérivée seconde de \( -\ln\) est \( \frac{1}{ x^2 }\) qui est strictement positif.
-    \end{enumerate}
-\end{example}
-
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Graphe d'une fonction convexe}
-%---------------------------------------------------------------------------------------------------------------------------
-
-L'idée principale du graphe d'une fonction convexe est qu'il est toujours au dessus du graphe de ses tangentes (lorsqu'elles existent). Lorsqu'elles n'existent pas, le lemme \ref{LemKLTsHIQ} donne des coefficients directeurs de droites qui vont rester en dessous du graphe de la fonction.
-
-\begin{proposition}[\cite{ooKCFNooVrqYhc}]      \label{PROPooOCOEooEGybmS}
-    Une fonction convexe est strictement convexe si et seulement s'il n'existe aucun intervalle de longueur non nulle sur lequel elle coïncide avec une fonction affine.
-\end{proposition}
-
-\begin{proof}
-    Si sur l'intervalle (non réduit à un point) \( \mathopen[ x , y \mathclose]\), la fonction convexe \( f\) coïncide avec une fonction affine, alors \( f(t)=at+b\) et pour \( \lambda\in\mathopen] 0 , 1 \mathclose[\) nous avons
-        \begin{equation}
-                f\big( \lambda x+(1-\lambda)y \big)=a\lambda x+a(1-\lambda)y+b=\lambda f(x)+(1-\lambda)f(y)
-        \end{equation}
-        où nous avons remplacé \( b\) par \( \lambda b+(1-\lambda)b\). Par conséquent la fonction n'est pas strictement convexe.
-
-    Nous supposons maintenant que la fonction convexe \( f\) n'est pas strictement convexe sur l'intervalle \( I\). Il existe \( x\neq y\in I\) et \( \lambda\in \mathopen] 0 , 1 \mathclose[\) tels que
-        \begin{equation}
-            f\big( \lambda x+(1-\lambda)y \big)=\lambda f(x)+(1-\lambda)f(y).
-        \end{equation}
-    Nous posons \( z=\lambda x+(1-\lambda)y\) et \( u\in\mathopen] x , z \mathclose[\) pour écrire des inégalités des pentes entre \( x<u<z<y\). Plus précisément si nous notons \( a\to b\) la pente de \( a\) à \( b\), c'est à dire \( a\to b=\frac{ f(b)-f(a) }{ b-a }\), alors les inégalités des pentes pour \( x<u<z\) puis \( u<z<y\) donnent
-        \begin{equation}        \label{EqooBMEFooMpoEzd}
-            x\to z\leq u\to z\leq z\to y.
-        \end{equation}
-        Voyons maintenant qu'en réalité \( z\to y=x\to z\). En effet en replaçant
-        \begin{equation}
-            f(y)=\frac{ f(z)-\lambda f(x) }{ 1-\lambda }
-        \end{equation}
-        et
-        \begin{equation}
-            y=\frac{ \lambda x }{ 1-\lambda }
-        \end{equation}
-        dans l'expression \( z\to y=\frac{ f(y)-f(z) }{ y-z }\) nous obtenons
-        \begin{equation}
-            z\to y=\frac{ f(y)-f(z) }{ y-z }=\frac{ f(z)-f(x) }{ z-x }=x\to z.
-        \end{equation}
-        Les inégalités \eqref{EqooBMEFooMpoEzd} sont donc des égalités :
-        \begin{equation}
-            \frac{ f(z)-f(x) }{ z-x }=\frac{ f(z)-f(u) }{ z-u }=\frac{ f(y)-f(z) }{ y-z }.
-        \end{equation}
-        Nous avons donc montré que le nombre \( a=\frac{ f(z)-f(u) }{ z-u }\) ne dépend pas de \( u\). Nous avons alors
-        \begin{equation}
-            f(z)-f(u)=a(z-u) 
-        \end{equation}
-        ou encore :
-        \begin{equation}
-            f(u)=f(z)-a(z-u),
-        \end{equation}
-    ce qui signifie que sur \( \mathopen] x , z \mathclose[\), la fonction \( f\) est affine.
-\end{proof}
-
-\begin{proposition} \label{PROPooQPOSooDZlUAJ}
-    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) est convexe si et seulement si son graphe est au dessus de chacune de ses tangentes.
-\end{proposition}
-
-\begin{proof}
-    \begin{subproof}
-        \item[Sens direct]
-            Soient \( x,y\in I\). Nous voulons :
-            \begin{equation}
-                f(y)\geq f(x)+f'(x)(y-x).
-            \end{equation}
-            Étant donné que nous aurons besoin, dans le quotient différentiel de quelque chose comme \( f(x+t)-f(x)\) nous écrivons la définition \eqref{EQooYNAPooFePQZy} de la convexité en inversant les rôles de \( x\) et \( y\) et en manipulant un peu :
-            \begin{subequations}
-                \begin{align}
-                    f\big( ty+(1-t)x \big)\leq tf(y)+(1-t)f(x)\\
-                    f\big( x+t(y-x) \big)\leq tf(y)+(1-t)f(x)\\
-                    f\big(  x+t(y-x)  \big)=f(x)\leq tf(y)-tf(x)
-                \end{align}
-            \end{subequations}
-            Nous divisons par \( t\) :
-            \begin{equation}
-                \frac{ f\big( x+t(y-x) \big)-f(x) }{ t }\leq f(y)-f(x).
-            \end{equation}
-            Le passage à la limite \( t\to 0\) donne
-            \begin{equation}
-                (y-x)f'(x)\leq f(y)-f(x),
-            \end{equation}
-            ce qu'il fallait.
-        \item[Sens inverse]
-            Pour tout \( x,y\in I\) nous supposons avoir
-            \begin{equation}        \label{EQooEXXIooHXJnER}
-                f(y)\geq f(x)+f'(x)(y-x).
-            \end{equation}
-            Si nous supposons \( x\neq y\) et si nous posons \( z=\lambda x+(1-\lambda)y\) nous voulons prouver que
-            \begin{equation}
-                f(z)\leq \lambda f(x)+(1-\lambda)f(y).
-            \end{equation}
-            Pour cela nous écrivons l'inégalité \eqref{EQooEXXIooHXJnER} avec les couples \( (x,z)\) et \( (y,z)\) :
-            \begin{subequations}
-                \begin{align}
-                    f(x)\geq f(z)+f'(z)'(x-z)\\
-                    f(y)\geq f(z)+f'(z)'(y-z)
-                \end{align}
-            \end{subequations}
-            En multipliant la première par \( \lambda\) et la seconde par \( (1-\lambda)\) et en sommant,
-            \begin{subequations}
-                \begin{align}
-                    \lambda f(x)+(1-\lambda)f(y)&\geq \lambda f(z)+\lambda f'(z)(x-z)+(1-\lambda)f(z)+(1-\lambda)f'(z)(y-z)\\
-                    &=f(z)+f'(z)\big( \lambda(x-z)+(1-\lambda)(y-z) \big)\\
-                    &=f(z).
-                \end{align}
-            \end{subequations}
-    \end{subproof}
-\end{proof}
-
-\begin{proposition}[\cite{MonCerveau}] \label{PropNIBooSbXIKO}
-    Soit \( f\colon \eR\to \eR \) une fonction convexe et \( a\in \eR\). Il existe une constante \( c_a\in \eR\) telle que pour tout \( x\) nous ayons
-    \begin{equation}    \label{EqSKIooSeAekM}
-        f(x)-f(a)\geq c_a(x-a).
-    \end{equation}
-    Autrement dit, le graphe de la fonction \( f\) est toujours au dessus de la droite d'équation
-    \begin{equation}
-        y=f(a)+c_a(x-a).
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Les dérivées à gauche et à droite de \( f\) données par le lemme \ref{LemKLTsHIQ} sont les candidats tout cuits pour être coefficient directeur de la droite que l'on cherche. Nous allons prouver qu'en posant
-    \begin{equation}
-        c_a=\inf_{t>a}\tau_a(t),
-    \end{equation}
-    la droite \( y=f(a)+c_a(x-a)\) répond à la question\footnote{En prenant l'autre, \( c_a'=\sup_{t<a}\tau_a(t)\), ça fonctionne aussi. En pensant à une fonction affine par morceaux, on remarque qu'en choisissant un nombre entre les deux, nous avons plus facilement une inégalité stricte dans \eqref{EqSKIooSeAekM}.}.
-
-    Nous devons prouver que le nombre \( \Delta_x=f(x)-\big( f(a)+c_a(x-a) \big)\) est positif pour tout \( x\).
-    \begin{subproof}
-
-    \item[Si \( x>a\)]
-        
-        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est positif :
-        \begin{subequations}
-            \begin{align}
-                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
-                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
-                &\geq 0
-            \end{align}
-        \end{subequations}
-        parce que \( t\to\tau_a(t)\) est croissante et que \( x>a\).
-
-    \item[Si \( x<a\)]
-        
-        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est négatif :
-        \begin{subequations}
-            \begin{align}
-                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
-                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
-                &\leq 0
-            \end{align}
-        \end{subequations}
-        parce que \( t\to\tau_a(t)\) est croissante et que \( x<a\).
-    \end{subproof}
-\end{proof}
-
-\begin{proposition}[\cite{MonCerveau}] \label{PropPEJCgCH}
-    Si \( g\) est une fonction convexe, il existe deux suites réelles \( (a_n)\) et \( (b_n)\) telles que
-    \begin{equation}
-        g(x)=\sup_{n\in \eN}(a_nx+b_n).
-    \end{equation}
-\end{proposition}
-\index{fonction!convexe}
-\index{densité!de \( \eQ\) dans \( \eR\)!utilisation}
-
-\begin{proof}
-    Pour \( u\in \eR\) nous considérons \( a(u)\) et \( b(u)\) tels que la droite \( y(x)=a(u)x+b(u)\) vérifie \( y(u)=g(u)\) et \( y(x)\leq g(x)\) pour tout \( x\). Cela est possible par la proposition \ref{PropNIBooSbXIKO}. Il s'agit d'une droite coupant le graphe de \( g\) en \( x=u\) et restant en dessous. Nous considérons alors \( (u_n)\) une suite quelconque dense dans \( \eR\) (disons les rationnels pour fixer les idées) et nous posons
-    \begin{subequations}
-        \begin{numcases}{}
-            a_n=a(u_n)\\
-            b_n=b(u_n).
-        \end{numcases}
-    \end{subequations}
-    Si \( q\in \eQ\) alors \( a_nx+b_n\leq g(x)\) pour tout \( n\) et \( g(q)\) est le supremum qui est atteint pour le \( n\) tel que \( u_n=q\). Si maintenant \( x\) n'est pas dans \( \eQ\) il faut travailler plus.
-
-    Nous prenons \( (\tilde q_n)\), une sous-suite de \( (q_n)\) convergeant vers \( x\) et \( N\) suffisamment grand pour que pour tout \( n\geq N\) on ait \( | \tilde q_n-x |\leq \epsilon\) et \( | g(\tilde q_n)-g(x) |\leq \epsilon\); cela est possible grâce à la continuité de \( g\) (lemme \ref{LemKLTsHIQ}). Ensuite les sous-suites \( (\tilde a_n)\) et \( (\tilde b_n)\) sont celles qui correspondent :
-    \begin{equation}
-        \tilde a_n\tilde q_n+\tilde b_n=g(\tilde q_n).
-    \end{equation}
-    Nous considérons la majoration
-    \begin{subequations}
-        \begin{align}
-            | \tilde a_nx+\tilde b_n-g(x) |&\leq| \tilde a_nx+\tilde b_n-(\tilde a_n\tilde q_n+\tilde b_n) |+\underbrace{| \tilde a_n\tilde q_n+\tilde b_n-g(\tilde q_n) |}_{=0}+\underbrace{| g(\tilde q_n)-g(x) |}_{\leq \epsilon}\\
-            &\leq | \tilde a_n | |x-\tilde q_n |+\epsilon\\
-            &=\epsilon\big( | \tilde a_n |+1 \big).
-        \end{align}
-    \end{subequations}
-    Il nous reste à montrer que \( | \tilde a_n |\) est borné par un nombre ne dépendant pas de \( n\) (pour les \( n>N\)).
-
-    Vu que la droite de coefficient directeur \( \tilde a_n\) et passant par le point \( \big( \tilde q_n,g(\tilde q_n) \big)\) reste en dessous du graphe de \( g\), nous avons pour tout \( n\) et tout \( y\in \eR\) l'inégalité
-    \begin{equation}
-        g(y)\geq \tilde a_n(y-\tilde q_n)+g(\tilde q_n)\in \tilde a_nB(y-x,\epsilon)+B\big( g(x),\epsilon \big).
-    \end{equation}
-    Si \( \tilde a_n\) n'est pas borné vers le haut, nous prenons \( y\) tel que \( B(y-x,\epsilon)\) soit minoré par un nombre \( k\) strictement positif et nous obtenons
-    \begin{equation}
-        g(y)\geq k\tilde a_n+l
-    \end{equation}
-    avec \( k\) et \( l\) indépendants de \( n\). Cela donne \( g(y)=\infty\). Si au contraire \( \tilde a_n\) n'est pas borné vers le bas, nous prenons $y$ tel que \( B(y-x,\epsilon)\) est majoré par un nombre \( k\) strictement négatif. Nous obtenons encore \( g(y)=\infty\).
-
-    Nous concluons que \( | \tilde a_n |\) est bornée.
-\end{proof}
-
-\begin{lemma}[\cite{KXjFWKA}]   \label{LemXOUooQsigHs}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            \phi\colon S^{++}(n,\eR)&\to \eR \\
-            A&\mapsto \det(A) 
-        \end{aligned}
-    \end{equation}
-    est \defe{log-convave}{concave!log-concave}\index{log-concave}, c'est à dire que l'application \( \ln\circ\phi\) est concave. De façon équivalente, si \( A,B\in S^{++}\) et si \( \alpha+b=1\), alors
-    \begin{equation}    \label{EqSPKooHFZvmB}
-        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
-    \end{equation}
-\end{lemma}
-Ici \( S^{++}\) est l'ensemble des matrices symétriques strictement définies positives, définition \ref{DefAWAooCMPuVM}.
-
-\begin{proof}
-    Nous commençons par prouver que l'équation \eqref{EqSPKooHFZvmB} est équivalente à la log-concavité du déterminant. Pour cela il suffit de remarquer que les propriétés de croissance et d'additivité du logarithme donnent l'équivalence entre
-    \begin{equation}
-        \ln\Big( \det(\alpha A+\beta B) \Big)\geq \ln\Big( \det(\alpha A) \Big)+\ln\Big( \det(\beta B) \Big),
-    \end{equation}
-    et
-    \begin{equation}    \label{EqTJYooBWiRrn}
-        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
-    \end{equation}
-
-    Le théorème de pseudo-réduction simultanée, corollaire \ref{CorNHKnLVA}, appliqué aux matrices \( A\) et \( B\) nous donne une matrice inversible \( Q\) telle que
-    \begin{subequations}
-        \begin{numcases}{}
-            B=Q^tDQ\\
-            A=Q^tQ
-        \end{numcases}
-    \end{subequations}
-    avec 
-    \begin{equation}
-        D=\begin{pmatrix}
-            \lambda_1    &       &       \\
-                &   \ddots    &       \\
-                &       &   \lambda_n
-        \end{pmatrix},
-    \end{equation}
-    \( \lambda_i>0\). Nous avons alors
-    \begin{equation}
-        \det(A)^{\alpha}\det(B)^{\beta}=\det(Q)^{2\alpha}\det(Q)^{2\beta}\det(D)^{\beta}=\det(Q)^2\det(D)^{\beta}
-    \end{equation}
-    (parce que \( \alpha+\beta=1\)) et
-    \begin{equation}
-        \det(\alpha A+\beta B)=\det(\alpha Q^tQ+\beta Q^tDQ)=\det\big( Q^t(\alpha\mtu+\beta D)Q \big)=\det(Q)^2\det(\alpha\mtu+\beta D).
-    \end{equation}
-    L'inégalité \eqref{EqTJYooBWiRrn} qu'il nous faut prouver se réduit donc  à
-    \begin{equation}
-        \det(\alpha \mtu+\beta D)\geq \det(D)^{\beta}.
-    \end{equation}
-    Vue la forme de \( D\) nous avons
-    \begin{equation}
-        \det(\alpha\mtu+\beta D)=\prod_{i=1}^n(\alpha+\beta\lambda_i)
-    \end{equation}
-    et 
-    \begin{equation}
-        \det(D)^{\beta}=\big( \prod_{i=1}^{n}\lambda_i \big)^{\beta}.
-    \end{equation}
-    Il faut donc prouver que
-    \begin{equation}\label{EqGFLooOElciS}    
-        \prod_{i=1}^n(\alpha+\beta\lambda_i)\geq \big( \prod_{i=1}^n\lambda_i \big)^{\beta}.
-    \end{equation}
-    Cette dernière égalité de produit sera prouvée en passant au logarithme. Vu que le logarithme est concave par l'exemple \ref{ExPDRooZCtkOz}, nous avons pour chaque \( i\) que
-    \begin{equation}
-        \ln(\alpha+\beta\lambda_i)\geq \alpha\ln(1)+\beta\ln(\lambda_i)=\beta\ln(\lambda_i).
-    \end{equation}
-    En sommant cela sur \( i\) et en utilisant les propriétés de croissance et de multiplicativité du logarithme nous obtenons successivement
-    \begin{subequations}
-        \begin{align}
-            \sum_{i=1}^n\ln(\alpha+\beta\lambda_i)\geq \beta\sum_i\ln(\lambda_i)\\
-            \ln\big( \prod_i(\alpha+\beta\lambda_i) \big)\geq\ln\Big( \big( \prod_i\lambda_i \big)^{\beta} \Big)\\
-            \prod_i(\alpha+\beta\lambda_i)\geq\big( \prod_i\lambda_i \big)^{\beta},
-        \end{align}
-    \end{subequations}
-    ce qui est bien \eqref{EqGFLooOElciS}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{En dimension supérieure}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit une partie convexe \( U\) de \( \eR^n\) et une fonction \( f\colon U\to \eR\). 
-    \begin{enumerate}
-        \item
-        La fonction \( f\) est \defe{convexe}{convexe!fonction sur \( \eR^n\)} si pour tout \( x,y\in U\) avec \( x\neq y\) et pour tout \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous avons
-            \begin{equation}
-                f\big( \theta x+(1-\theta)y \big)\leq \theta f(x)+(1-\theta)f(y).
-            \end{equation}
-        \item
-            Elle est \defe{strictement convexe}{strictement!convexe!sur \( \eR^n\)} si nous avons l'inégalité stricte.
-    \end{enumerate}
-\end{definition}
-
-\begin{proposition}[\cite{ooLJMHooMSBWki}]      \label{PROPooYNNHooSHLvHp}
-    Soit \( \Omega\) ouvert dans \( \eR^n\) et \( U\) convexe dans \( \Omega\), et une fonction différentiable \( f\colon U\to \eR\).
-    \begin{enumerate}
-        \item       \label{ITEMooRVIVooIayuPS}
-            La fonction \( f\) est convexe sur \( U\) si et seulement si pour tout \( x,y\in U\),
-            \begin{equation}
-                f(y)\geq f(x)+df_x(y-x).
-            \end{equation}
-        \item       \label{ITEMooCWEWooFtNnKl}
-            La fonction \( f\) est strictement convexe sur \( U\) si et seulement si pour tout \( x,y\in U\) avec \( x\neq y\),
-            \begin{equation}
-                f(y)>f(x)+df_x(y-x).
-            \end{equation}
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Nous avons quatre petites choses à démontrer.
-    \begin{subproof}
-    \item[\ref{ITEMooRVIVooIayuPS} sens direct]
-        Soit une fonction convexe \( f\). Nous avons :
-        \begin{equation}
-            f\big( (1-\theta)x+\theta y \big)\leq (1-\theta)f(x)+\theta f(y),
-        \end{equation}
-        donc
-        \begin{equation}
-            f\big( x+\theta(y-x) \big)-f(x)\leq \theta\big( f(y)-f(x) \big)
-        \end{equation}
-        Vu que \( \theta>0\) nous pouvons diviser par \( \theta\) sans changer le sens de l'inégalité :
-        \begin{equation}        \label{EQooAXXFooHWtiJh}
-            \frac{ f\big( x+\theta(y-x) \big)-f(x) }{ \theta }\leq f(y)-f(x).
-        \end{equation}
-        Nous prenons la limite \( \theta\to 0^+\). Cette limite est égale à a limite simple \( \theta\to 0\) et vaut (parce que \( f\) est différentiable) :
-        \begin{equation}
-            \frac{ \partial f }{ \partial (y-x) }(x)\leq f(y)-f(x),
-        \end{equation}
-        et aussi
-        \begin{equation}
-            df_x(y-x)\leq f(y)-f(x)
-        \end{equation}
-        par le lemme \ref{LemdfaSurLesPartielles}.
-    \item[\ref{ITEMooRVIVooIayuPS} sens inverse]
-        Pour tout \( a\neq b\) dans \( U\) nous avons
-        \begin{equation}        \label{EQooEALSooJOszWr}
-            f(b)\geq f(a)+df_a(b-a).
-        \end{equation}
-    Pour \( x\neq y\) dans \( U\) et pour \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous écrivons \eqref{EQooEALSooJOszWr} pour les couples \( \big( \theta x+(1-\theta)y,y \big)\) et \( \big( \theta x+(1-\theta)y,x \big)\). Ça donne :
-        \begin{equation}
-            f(y)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( \theta(y-x) \big),
-        \end{equation}
-        et
-        \begin{equation}
-            f(x)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( (1-\theta)(x-y) \big).
-        \end{equation}
-        La différentielle est linéaire; en multipliant la première par \( (1-\theta)\) et la seconde par \( \theta\) et en la somme, les termes en \( df\) se simplifient et nous trouvons
-        \begin{equation}
-            \theta f(x)+(1-\theta)f(y)\geq f\big( \theta x+(1-\theta)y \big).
-        \end{equation}
-    \item[\ref{ITEMooCWEWooFtNnKl} sens direct]
-        Nous avons encore l'équation \eqref{EQooAXXFooHWtiJh}, avec une inégalité stricte. Par contre, ça ne va pas être suffisant parce que le passage à la limite ne conserve pas les inégalités strictes. Nous devons donc être plus malins. 
-
-        Soient \( 0<\theta<\omega<1\). Nous avons \( (1-\theta)x+\theta y\in \mathopen[ x , (1-\omega)x+\omega y \mathclose]\), donc nous pouvons écrire \( (1-\theta)x+\theta y\) sous la forme \( (1-s)x+s\big( (1-\omega)x+\omega y \big)\). Il se fait que c'est bon pour \( s=\theta/\omega\) (et aussi que nous avons \( \theta/\omega<1\)). Donc nous avons
-        \begin{subequations}
-            \begin{align}
-            f\big( (1-\theta)x+\theta y \big)&=f\Big( (1-\frac{ \theta }{ \omega })x+\frac{ \theta }{ \omega }\big( (1-\omega)x+\omega y \big) \Big)\\
-            &<(1-\frac{ \theta }{ \omega })f(x)+\frac{ \theta }{ \omega }f\big( (1-\omega)x+\omega y \big).
-            \end{align}
-        \end{subequations}
-        Cela nous permet d'écrire
-        \begin{equation}
-            \frac{ f\big( (1-\theta)x+\theta y \big)-f(x) }{ \theta }<\frac{ f\big( (1-\omega)x+\omega y \big) }{ \omega }<f(y)-f(x).
-        \end{equation}
-        Le seconde inégalité est le pendant de \eqref{EQooAXXFooHWtiJh}. Maintenant en passant à la limite pour \( \theta\) nous conservons une inégalité stricte par rapport à \( f(y)-f(x)\) :
-        \begin{equation}
-            df_x(y-x)<f(y)-f(x).
-        \end{equation}
-    \end{subproof}
-\end{proof}
-
-% Il faut laisser les sauts de lignes suivants, pour rechercher efficacement les références vers le futur.
-Avant de lire la proposition suivante, il faut relire la proposition \ref{PROPooFWZYooUQwzjW} et ce qui s'y rapporte. 
-Lire aussi la remarque \ref{REMooVRPQooIybxmp} qui indique
-qu'il n'y a pas de réciproque dans l'énoncé \ref{ITEMooHAGQooYZyhQk}.     
-\begin{proposition}[\cite{ooLJMHooMSBWki}]
-    Soit une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) et un convexe \( U\subset \Omega\).
-    \begin{enumerate}
-        \item       \label{ITEMooZQCAooIFjHOn}
-            La fonction \( f\) est convexe sur \( U\) si et seulement si
-            \begin{equation}        \label{EQooIBDCooJYdiBb}
-                (d^2f)_x(y-x,y-x)\geq 0
-            \end{equation}
-            pour tout \( x,y\in U\).
-        \item       \label{ITEMooHAGQooYZyhQk}
-            Si pour tout \( x\neq y\) dans \( U\) nous avons
-            \begin{equation}
-                (d^2f)_x(y-x,y-x)>0
-            \end{equation}
-            alors la fonction \( f\) est strictement convexe sur \( U\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{remark}      \label{REMooYCRKooEQNIkC}
-    Notons que la condition \eqref{EQooIBDCooJYdiBb} n'est pas équivalente à demander \( (d^2f)_x(h,h)\geq 0\) pour tout \( h\). En effet nous ne demandons la positivité que dans les directions atteignables comme différence de deux éléments de \( U\). La partie \( U\) n'est pas spécialement ouverte; elle pourrait n'être qu'une droite dans \( \eR^3\). Dans ce cas, demander que \( f\) (qui est \( C^2\) sur l'ouvert \( \Omega\)) soit convexe sur \( U\) ne demande que la positivité de \( (d^2f)_x\) appliqué à des vecteurs situés sur la droite \( U\).
-\end{remark}
-
-\begin{proof}
-    Il y a trois parties à démontrer.
-    \begin{subproof}
-    \item[\ref{ITEMooZQCAooIFjHOn} sens direct]
-
-        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci au jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
-        \begin{equation}
-            f(x+th)=f(x)+tdf_x(h)+\frac{ t^2 }{2}(d^2_x)(h,h)+t^2\| h \|^2\alpha(th)
-        \end{equation}
-        avec \( \lim_{s\to 0}\alpha(s)=0\). Le fait que \( f\) soit convexe donne
-        \begin{equation}
-            0\leq f(x+th)-f(x)-tdf_x(h),
-        \end{equation}
-        et donc
-        \begin{equation}
-            0\leq \frac{ t^2 }{2}(d^2f)_x(h,h)+f^2\| h \|^2\alpha(th).
-        \end{equation}
-        En multipliant par \( 2\) et en divisant par \( t^2\),
-        \begin{equation}
-            0\leq (d^2f)_x(h,h)+2\| h \|^2\alpha(th).
-        \end{equation}
-        En prenant \( t\to 0\) nous avons bien  \( (d^2f)_x(y-x,y-x)\geq 0\).
-
-    \item[\ref{ITEMooZQCAooIFjHOn} sens inverse]
-
-        Soient \( x,y\in U\). Nous écrivons Taylor en version de la proposition \ref{PROPooWWMYooPOmSds} :
-        \begin{equation}
-            f(y)=f(x)+df_x(y-x)+\frac{ 1 }{2}(d^2f)_z(y-x,y-x)
-        \end{equation}
-    pour un certain \( z\in\mathopen] x , y \mathclose[\). En vertu de ce qui a été dit dans la remarque \ref{REMooYCRKooEQNIkC} nous ne pouvons pas évoquer l'hypothèse \eqref{EQooIBDCooJYdiBb} pour conclure que \( (d^2f)_z(y-x,y-x)\geq 0\). Il y a deux manières de nous sortir du problème :
-        \begin{itemize}
-            \item Trouver \( s\in U\) tel que \( y-x=s-z\).
-            \item Trouver un multiple de \( y-x\) qui soit de la forme \( y-x\).
-        \end{itemize}
-        La première approche ne fonctionne pas parce que \( s=y-x+z\) n'est pas garanti d'être dans \( U\); par exemple avec \( x=1\), \( z=2\), \( y=3\) et \( U=\mathopen[ 0 , 3 \mathclose]\). Dans ce cas \( s=4\notin U\).
-
-        Heureusement nous avons \( z=\theta x+(1-\theta)y\), donc \( z-x=(1-\theta)(y-x)\). Dans ce cas la bilinéarité de \( (d^2f)_z\) donne\footnote{Si vous avez bien suivi, la bilinéarité est contenue dans la proposition \ref{PROPooFWZYooUQwzjW}.}
-        \begin{equation}
-            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{\geq 0}.
-        \end{equation}
-        Nous en déduisons que \( f\) est convexe par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooRVIVooIayuPS}.
-    \item[\ref{ITEMooHAGQooYZyhQk}]
-
-        Le raisonnement que nous venons de faire pour le sens inverse de \ref{ITEMooZQCAooIFjHOn} tient encore, et nous avons
-        \begin{equation}
-            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{> 0}
-        \end{equation}
-        d'où nous déduisons la stricte convexité de \( f\) par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooCWEWooFtNnKl}.
-    \end{subproof}
-\end{proof}
-
-\begin{corollary}       \label{CORooMBQMooWBAIIH}
-    Avec la hessienne\ldots en cours d'écriture.
-\end{corollary}
-
-\begin{proof}
-    Cela va utiliser la proposition \ref{PropoExtreRn}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Quelque inégalités}
-%---------------------------------------------------------------------------------------------------------------------------
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Inégalité de Jensen}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\index{inégalité!Jensen}
-\index{convexité!inégalité de Jensen}
-
-\begin{proposition}[Inégalité de Jensen]    \label{PropXIBooLxTkhU}
-    Soit \( f\colon \eR\to \eR\) une fonction convexe et des réels \( x_1\),\ldots,  \( x_n\). Soient des nombres positifs \( \lambda_1\),\ldots,  \( \lambda_n\) formant une combinaison convexe\footnote{Définition \ref{DefIMZooLFdIUB}.}. Alors
-    \begin{equation}
-        f\big( \sum_i\lambda_ix_i \big)\leq \sum_i\lambda_if(x_i).
-    \end{equation}
-\end{proposition}
-\index{inégalité!Jensen!pour une somme}
-
-\begin{proof}
-    Nous procédons par récurrence sur \( n\), en sachant que \( n=2\) est la définition de la convexité de \( f\). Vu que
-    \begin{equation}
-        \sum_{k=1}^n\lambda_kx_k=\lambda_nx_n+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n },
-    \end{equation}
-    nous avons
-    \begin{equation}
-        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)f\big( \sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n } \big).
-    \end{equation}
-    La chose à remarquer est que les nombres \( \frac{ \lambda_k }{ 1-\lambda_n }\) avec \( k\) allant de \( 1\) à \( n-1\) forment eux-mêmes une combinaison convexe. L'hypothèse de récurrence peut donc s'appliquer au second terme du membre de droite :
-    \begin{equation}
-        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_k }{ 1-\lambda_n }f(x_k)=\lambda_nf(x_n)+\sum_{k=1}^{n-1}\lambda_kf(x_k).
-    \end{equation}
-\end{proof}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Inégalité arithmético-géométrique}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-La proposition suivante dit que la moyenne arithmétique de nombres strictement positifs est supérieure ou égale à la moyenne géométrique.
-\begin{proposition}[Inégalité arithmético-géométrique\cite{CENooZKvihz}]    \label{PropWDPooBtHIAR}
-    Soient \( x_1\),\ldots, \( x_n\) des nombres strictement positifs. Nous posons
-    \begin{equation}
-        m_a=\frac{1}{ n }(x_1+\cdots +x_n)
-    \end{equation}
-    et
-    \begin{equation}
-        m_g=\sqrt[n]{x_1\ldots x_n}
-    \end{equation}
-    Alors \( m_g\leq m_a\) et \( m_g=m_a\) si et seulement si \( x_i=x_j\) pour tout \( i,j\).
-\end{proposition}
-\index{inégalité!arithmético-géométrique}
-
-\begin{proof}
-    Par hypothèse les nombres \( m_a\) et \( m_g\) sont tout deux strictement positifs, de telle sorte qu'il est équivalent de prouver \( \ln(m_g)\leq \ln(m_a)\) ou encore
-    \begin{equation}
-        \frac{1}{ n }\big( \ln(x_1)+\cdots +\ln(x_n) \big)\leq \ln\left( \frac{ x_1+\cdots +x_n }{ n } \right).
-    \end{equation}
-    Cela n'est rien d'autre que l'inégalité de Jensen de la proposition \ref{PropXIBooLxTkhU} appliquée à la fonction \( \ln\) et aux coefficients \( \lambda_i=\frac{1}{ n }\).
-\end{proof}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Inégalité de Kantorovitch}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-\begin{proposition}[Inégalité de Kantorovitch\cite{EYGooOoQDnt}]    \label{PropMNUooFbYkug}
-    Soit \( A\) une matrice symétrique strictement définie positive dont les plus grandes et plus petites valeurs propres sont \( \lambda_{min}\) et \( \lambda_{max}\). Alors pour tout \( x\in \eR^n\) nous avons
-    \begin{equation}
-        \langle Ax, x\rangle \langle A^{-1}x, x\rangle \leq \frac{1}{ 4 }\left( \frac{ \lambda_{min} }{ \lambda_{max} }+\frac{ \lambda_{max} }{ \lambda_{min} } \right)^2\| x^4 \|.
-    \end{equation}
-\end{proposition}
-\index{inégalité!Kantorovitch}
-
-\begin{proof}
-    Sans perte de généralité nous pouvons supposer que \( \| x \|=1\). Nous diagonalisons\footnote{Théorème spectral \ref{ThoeTMXla}.} la matrice \( A\) par la matrice orthogonale  \( P\in\gO(n,\eR)\) : \( A=PDP^{-1}\) et \( A^{-1}=PD^{-1}P^{-1}\) où \( D\) est  une matrice diagonale formée des valeurs propres de \( A\).
-
-    Nous posons \( \alpha=\sqrt{\lambda_{min}\lambda_{max}}\) et nous regardons la matrice
-    \begin{equation}
-        \frac{1}{ \alpha }A+tA^{-1}
-    \end{equation}
-    dont les valeurs propres sont
-    \begin{equation}
-        \frac{ \lambda_i }{ \alpha }+\frac{ \alpha }{ \lambda_i }
-    \end{equation}
-    parce que les vecteurs propres de \( A\) et de \( A^{-1}\) sont les mêmes (ce sont les valeurs de la diagonale de \( D\)). Nous allons quelque peu étudier la fonction
-    \begin{equation}
-        \theta(x)=\frac{ x }{ \alpha }+\frac{ \alpha }{ x }.
-    \end{equation}
-    Elle est convexe en tant que somme de deux fonctions convexes. Elle a son minimum en \( x=\alpha\) et ce minimum vaut \( \theta(\alpha)=2\). De plus
-    \begin{equation}
-        \theta(\lambda_{max})=\theta(\lambda_{min})=\sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
-    \end{equation}
-    Une fonction convexe passant deux fois par la même valeur doit forcément être plus petite que cette valeur entre les deux\footnote{Je ne suis pas certain que cette phrase soit claire, non ?} : pour tout \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\),
-    \begin{equation}
-        \theta(x)\leq  \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
-    \end{equation}
-    
-    Nous sommes maintenant en mesure de nous lancer dans l'inégalité de Kantorovitch.
-    \begin{subequations}
-        \begin{align}
-            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }&\leq\frac{ 1 }{2}\left( \frac{ \langle Ax, x\rangle  }{ \alpha }+\alpha\langle A^{-1}x, x\rangle  \right)\label{subEqUKIooCWFSkwi}\\
-            &=\frac{ 1 }{2}\langle   \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x , x\rangle \\
-            &\leq\frac{ 1 }{2}\Big\| \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x \|\| x \| \label{subEqUKIooCWFSkwiii}\\
-            &\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \label{subEqUKIooCWFSkwiv}
-        \end{align}
-    \end{subequations}
-    Justifications :
-    \begin{itemize}
-        \item \ref{subEqUKIooCWFSkwi} par l'inégalité arithmético-géométrique, proposition \ref{PropWDPooBtHIAR}. Nous avons aussi inséré \( \alpha\frac{1}{ \alpha }\) dans le produit sous la racine.
-        \item \ref{subEqUKIooCWFSkwiii} par l'inégalité de Cauchy-Schwarz, théorème \ref{ThoAYfEHG}.
-        \item \ref{subEqUKIooCWFSkwiv} par la définition de la norme opérateur de la proposition \ref{DefNFYUooBZCPTr}
-    \end{itemize}
-    La norme opérateur est la plus grande des valeurs propres. Mais les valeurs propres de \( A/\alpha+\alpha A^{-1}\) sont de la forme \( \theta(\lambda_i)\), et tous les \( \lambda_i\) sont entre \( \lambda_{min} \) et \( \lambda_{max}\). Donc la plus grande valeur propre de \( A/\alpha+\alpha A^{-1}\) est \( \theta(x)\) pour un certain \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\). Par conséquent
-    \begin{equation}
-            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \leq \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
-    \end{equation}
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Algorithme du gradient à pas optimal}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
-
-Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
-\begin{itemize}
-    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
-    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
-\end{itemize}
-
-\begin{definition}  \label{DefQXPooYSygGP}
-    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
-    \begin{equation}
-        \{ x\in P\tq f(x)\leq M \}
-    \end{equation}
-    est borné.
-\end{definition}
-En langage imagé la coercivité de \( f\) s'exprime par la limite
-\begin{equation}
-    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
-\end{equation}
-
-
-Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
-\begin{proposition}     \label{PROPooYRLDooTwzfWU}
-    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Alors :
-    \begin{enumerate}
-        \item
-            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
-        \item
-            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
-        \item
-            Ils sont égaux : \( \bar x=x^*\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-
-    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
-
-    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
-
-    \begin{subproof}
-    \item[\( f\) est strictement convexe]
-        
-        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
-    \end{equation}
-    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
-
-\item[\( f\) est coercive]
-    Montrons à présent que \( f\) est coercive. Nous avons :
-    \begin{subequations}
-        \begin{align}
-            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
-            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
-            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
-        \end{align}
-    \end{subequations}
-    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
-    \end{subproof}
-
-    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
-    \begin{equation}
-        \{ x\in \eR^n\tq f(x)\leq M \}
-    \end{equation}
-    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
-
-    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
-    \begin{equation}
-        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
-    \end{equation}
-    ce qui contredit la minimalité de \( f(\bar x_1)\).
-
-    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
-    \begin{equation}
-        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
-    \end{equation}
-    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
-\end{proof}
-
-\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
-    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
-    \begin{equation}
-        x_{k+1}=x_k+t_kd_k
-    \end{equation}
-    où
-    \begin{itemize}
-        \item 
-    \( d_k=-(\nabla f)(x_k)\) 
-\item
-    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
-    \end{itemize}
-
-    Alors pour tout \( k\geq 0\) nous avons
-    \begin{equation}
-        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
-    \end{equation}
-    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
-\end{proposition}
-
-\begin{proof}
-    Décomposition en plusieurs points.
-    \begin{subproof}
-    \item[Existence de \( \bar x\)]
-        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
-    \item[Si \( (\nabla f)(x_k)=0\)]
-    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
-    \end{equation}
-    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
-    \begin{equation}
-        (\nabla f)(a)=Aa+b.
-    \end{equation}
-    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
-
-    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
-        \item[\( t_k\) est bien défini]
-
-            Pour \( t\in \eR\) nous avons
-            \begin{equation}    \label{EqKEHooYaazQi}
-                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
-            \end{equation}
-            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
-            \begin{equation}    \label{EqVWJooWmDSER}
-                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
-            \end{equation}
-            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
-
-        \item[La valeur de \( d_{k+1}\)]
-
-            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
-            \begin{equation}
-                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
-            \end{equation}
-            parce que \( -Ax_k-b=d_k\).
-            
-            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
-            \begin{equation}
-                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
-            \end{equation}
-            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
-        
-        \item[Calcul de \( f(x_{k+1})\)] 
-
-            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
-            \begin{equation}
-                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
-            \end{equation}
-            
-        \item[Encore du calcul \ldots]
-
-            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
-            \begin{equation}
-                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
-            \end{equation}
-            Ayant cela en tête nous pouvons calculer :
-            \begin{subequations}
-                \begin{align}
-                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
-                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
-                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
-                    &=2\big( f(x_k)-\bar f \big)
-                \end{align}
-            \end{subequations}
-            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
-
-        \item[Erreur sur la valeur du minimum]
-
-            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
-            \begin{subequations}
-                \begin{align}
-                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
-                \end{align}
-            \end{subequations}
-            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
-            \begin{equation}
-                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
-            \end{equation}
-            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
-            \begin{equation}
-                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
-            \end{equation}
-            Par récurrence nous avons alors
-            \begin{equation}    \label{eqANKooNPfCFj}
-                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
-            \end{equation}
-            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
-
-        \item[Erreur sur la position du minimum]
-
-            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
-            \begin{subequations}
-                \begin{align}
-                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
-                \end{align}
-            \end{subequations}
-            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
-
-            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
-            \begin{equation}    \label{EqVMRooUMXjig}
-                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
-            \end{equation}
-            
-        \item[Conclusion]
-
-            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
-            \begin{equation}
-                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
-            \end{equation}
-            c'est à dire
-            \begin{equation}
-                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
-            \end{equation}
-    \end{subproof}
-\end{proof}
-
-Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
-
diff --git a/tex/frido/69_analyseR.tex b/tex/frido/69_analyseR.tex
index 416b9525c..7bb959e34 100644
--- a/tex/frido/69_analyseR.tex
+++ b/tex/frido/69_analyseR.tex
@@ -3,12 +3,12 @@
 %   Laurent Claessens, Carlotta Donadello
 % See the file fdl-1.3.txt for copying conditions.
  
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Suites de fonctions}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Convergence uniforme}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
 %--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Convergence uniforme}
+\subsection{Critère de Cauchy uniforme}
 %---------------------------------------------------------------------------------------------------------------------------
 
 \begin{definition}[\cite{TrenchRealAnalisys}]
@@ -67,6 +67,70 @@ \subsection{Convergence uniforme}
     La continuité de \( f_n\) nous fournit un \( \delta>0\) tel que \( \| f_n(x_0)-f_n(x) \|<\epsilon\) dès que \( \| x-x_0 \|<\delta\). Pour ce \( \delta\), nous avons alors \( \| f(x)-f(x_0) \|<\epsilon\).
 \end{proof}
 
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Complétude avec la norme uniforme}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[Limite uniforme de fonctions continues]\label{PropCZslHBx}
+    Soit \( X\) un espace topologique et \( (Y,d)\) un espace métrique. Si une suite de fonctions \( f_n\colon X\to Y\) continues converge uniformément, alors la limite est séquentiellement continue\footnote{Si \( X\) est métrique, alors c'est la continuité usuelle par la proposition \ref{PropFnContParSuite}.}.
+\end{proposition}
+
+\begin{proof}
+    Soit \( a\in X\) et prouvons que \( f\) est séquentiellement continue en \( a\). Pour cela nous considérons une suite \( x_n\to a\) dans \( X\). Nous savons que \( f(x_n)\stackrel{Y}{\longrightarrow}f(x)\). Pour tout \(k\in \eN\), tout \( n\in \eN\) et tout \( x\in X\) nous avons la majoration
+    \begin{equation}
+        \big\| f(x_n)-f(x) \big\|\leq \big\| f(x_n)-f_k(x_n) \big\|+\big\| f_k(x_n)-f_k(x) \big\|+\big\| f_k(x)-f(x) \big\|\leq 2\| f-f_k \|_{\infty}+\big\| f_k(x_n)-f_k(x) \big\|.    
+    \end{equation}
+    Soit \( \epsilon>0\). Si nous choisissons \( k\) suffisamment grand la premier terme est plus petit que \( \epsilon\). Et par continuité de \( f_k\), en prenant \( n\) assez grand, le dernier terme est également plus petit que \( \epsilon\).
+\end{proof}
+
+\begin{proposition} \label{PropSYMEZGU}
+    Soit \( X\) un espace topologique métrique \( (Y,d)\) un espace espace métrique complet. Alors les espaces
+    \begin{enumerate}
+        \item
+            \( \big( C^0_b(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et bornées \( X\to Y\),
+        \item
+            \( \big( C^0_0(X,Y),\| . \|_{\infty} \big)\) des fonctions continues et s'annulant à l'infini
+        \item
+            \( \big( C^k_0(X,Y),\| . \|_{\infty} \big)\) des fonctions de classe \( C^k\) et s'annulant à l'infini
+    \end{enumerate} 
+    sont complets.
+\end{proposition}
+
+\begin{proof}
+    Soit \( (f_n)\) une suite de Cauchy dans \( C(X,Y)\), c'est à dire que pour tout \( \epsilon>0\) il existe \( N\in \eN\) tel que si \( k,l>N\) nous avons \( \| f_k-f_l \|_{\infty}\leq \epsilon\). Cette suite vérifie le critère de Cauchy uniforme \ref{PropNTEynwq} et donc converge uniformément vers une fonction \( f\colon X\to Y\). La continuité (ou l'aspect \( C^k\)) de la fonction \( f\) découle de la convergence uniforme et de la proposition \ref{PropCZslHBx} (c'est pour avoir l'équivalence entre la continuité séquentielle et la continuité normale que nous avons pris l'hypothèse d'espace métrique).
+
+    Si les fonctions \( f_k\) sont bornées ou s'annulent à l'infini, la convergence uniforme implique que la limite le sera également.
+\end{proof}
+    Notons que si \( X\) est compact, les fonctions continues sont bornées par le théorème \ref{ThoImCompCotComp} et nous pouvons simplement dire que \( C^0(X,Y)\) est complet, sans préciser que nous parlons des fonctions bornées.
+
+
+\begin{lemma}       \label{LemdLKKnd}
+    Soit \( A\) compact et \( B\) complet. L'ensemble des fonctions continues de \( A\) vers \( B\) muni de la norme uniforme est complet.
+
+    Dit de façon courte : \( \big( C(A,B),\| . \|_{\infty} \big)\) est complet.
+\end{lemma}
+
+\begin{proof}
+    Soit \( (f_k)\) une suite de Cauchy de fonctions dans \( C(A,B)\). Pour chaque \( x\in A \) nous avons
+    \begin{equation}
+        \| f_k(x)-f_l(x) \|_B\leq \| f_k-f_l \|_{\infty},
+    \end{equation}
+    de telle sorte que la suite \( (f_k(x))\) est de Cauchy dans \( B\) et converge donc vers un élément de \( B\). La suite de Cauchy \( (f_k)\) converge donc ponctuellement vers une fonction \( f\colon A\to B\). Nous devons encore voir que cette fonction est continue; ce sera l'uniformité de la norme qui donnera la continuité. En effet soit \( x_n\to x\) une suite dans \( A\) convergent vers \( x\in A\). Pour chaque \( k\in \eN\) nous avons
+    \begin{equation}
+        \| f(x_n)-f(x) \|\leq \| f(x_n)-f_k(x_n) \|  +\| f_k(x_n)-f_k(x) \|+\| f_k(x)-f(x) \|.
+    \end{equation}
+    En prenant \( k\) et \( n\) assez grands, cette expression peut être rendue aussi petite que l'on veut; le premier et le troisième terme par convergence ponctuelle \( f_k\to f\), le second terme par continuité de \( f_k\). La suite \( f(x_n)\) est donc convergente vers \( f(x)\) et la fonction \( f\) est continue.
+\end{proof}
+
+\begin{probleme}
+Il serait sans doute bon de revoir cette preuve à la lumière du critère de Cauchy uniforme \ref{PropNTEynwq}.
+\end{probleme}
+
+
+\begin{normaltext}[\cite{ooXYZDooWKypYR}]
+    Le théorème de Stone-Weierstrass indique que les polynômes sont denses pour la topologie uniforme dans les fonctions continues. Donc il existe des limites uniformes de fonctions \( C^{\infty}\) qui ne sont même pas dérivables. Les espaces de type \( C^p\) munis de \( \| . \|_{\infty}\) ne sont donc pas complets sans quelque hypothèses. Voir la proposition \ref{PropSYMEZGU} et le thème \ref{THMooOCXTooWenIJE}.
+\end{normaltext}
+
 \begin{theorem}[Théorème de Dini\cite{JIFGuct}] \label{ThoUFPLEZh}
     Soit \( D\) un espace métrique compact et une suite de fonctions \( f_n\in C(D,\eR)\) telle que
     \begin{enumerate}
diff --git a/tex/frido/77_series_fonctions.tex b/tex/frido/77_series_fonctions.tex
index ebb9d6879..3295be401 100644
--- a/tex/frido/77_series_fonctions.tex
+++ b/tex/frido/77_series_fonctions.tex
@@ -3,148 +3,6 @@
 %   Laurent Claessens
 % See the file fdl-1.3.txt for copying conditions.
 
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorème de Von Neumann}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{lemma}[\cite{KXjFWKA}]
-    Soit \( G\), un sous groupe fermé de \( \GL(n,\eR)\) et 
-    \begin{equation}
-        \mL_G=\{ m\in \eM(n,\eR)\tq  e^{tm}\in G\,\forall t\in\eR \}.
-    \end{equation}
-    Alors \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\).
-\end{lemma}
-
-\begin{proof}
-    Si \( m\in\mL_G\), alors \( \lambda m\in\mL_G\) par construction. Le point délicat à prouver est le fait que si \( a,b\in \mL_G\), alors \( a+b\in\mL_G\). Soit \( a\in \eM(n,\eR)\); nous savons qu'il existe une fonction \( \alpha_a\colon \eR\to \eM\) telle que
-    \begin{equation}
-        e^{ta}=\mtu+ta+\alpha_a(t)
-    \end{equation}
-    et 
-    \begin{equation}
-        \lim_{t\to 0} \frac{ \alpha(t) }{ t }=0.
-    \end{equation}
-    Si \( a\) et \( b\) sont dans \( \mL_G\), alors \(  e^{ta} e^{tb}\in G\), mais il n'est pas vrai en général que cela soit égal à \(  e^{t(a+b)}\). Pour tout \( k\in \eN\) nous avons
-    \begin{equation}
-        e^{a/k} e^{b/k}=\left( \mtu+\frac{ a }{ k }+\alpha_a(\frac{1}{ k }) \right)\left( \mtu+\frac{ b }{ k }+\alpha_b(\frac{1}{ k }) \right)=\mtu+\frac{ a+b }{2}+\beta\left( \frac{1}{ k } \right)
-    \end{equation}
-   où \( \beta\colon \eR\to \eM\) est encore une fonction vérifiant \( \beta(t)/t\to 0\). Si \( k\) est assez grand, nous avons
-   \begin{equation}
-       \left\| \frac{ a+b }{ k }+\beta(\frac{1}{ k })  \right\|<1,
-   \end{equation}
-   et nous pouvons profiter du lemme \ref{LemQZIQxaB} pour écrire alors
-   \begin{equation}
-       \left(  e^{a/k} e^{b/k} \right)^k= e^{k\ln\big(\mtu+\frac{ a+b }{ k }+\beta(\frac{1}{ k })\big)}.
-   \end{equation}
-   Ce qui se trouve dans l'exponentielle est
-   \begin{equation}
-       k\left[ \frac{ a+b }{ k }+\alpha( \frac{1}{ k })+\sigma\left( \frac{ a+b }{ k }+\alpha(\frac{1}{ k }) \right) \right].
-   \end{equation}
-   Les diverses propriétés vues montrent que le tout tend vers \( a+b\) lorsque \( k\to \infty\). Par conséquent
-   \begin{equation}
-       \lim_{k\to \infty} \left(  e^{a/k} e^{b/k} \right)^k= e^{a+b}.
-   \end{equation}
-   Ce que nous avons prouvé est que pour tout \( t\), \(  e^{t(a+b)}\) est une limite d'éléments dans \( G\) et est donc dans \( G\) parce que ce dernier est fermé.
-\end{proof}
-
-Vu que \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\), nous pouvons considérer un supplémentaire \( M\).
-
-\begin{lemma}   \label{LemHOsbREC}
-    Il n'existe pas se suites \( (m_k)\) dans \( M\setminus\{ 0 \}\) convergeant vers zéro et telle que \(  e^{m_k}\in G\) pour tout \( k\).
-\end{lemma}
-
-\begin{proof}
-    Supposons que nous ayons \( m_k\to 0\) dans \( M\setminus\{ 0 \}\) avec \(  e^{m_k}\in G\). Nous considérons les éléments \( \epsilon_k=\frac{ m_k }{ \| m_k \| }\) qui sont sur la sphère unité de \(\GL(n,\eR)\). Quitte à prendre une sous-suite, nous pouvons supposer que cette suite converge, et vu que \( M\) est fermé, ce sera vers \( \epsilon\in M\) avec \( \| \epsilon \|=1\). Pour tout \( t\in \eR\) nous avons
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}.
-    \end{equation}
-    En vertu de la décomposition d'un réel en partie entière et décimale, pour tout \( k\) nous avons \( \lambda_k\in \eZ\) et \( | \mu_k |\leq \frac{ 1 }{2}\) tel que \( t/\| m_k \|=\lambda_k+\mu_k\). Avec ça,
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}\exp\Big( \frac{ t }{ m_k }m_k \Big)=\lim_{k\to \infty}  e^{\lambda_km_k} e^{\mu_km_k}.
-    \end{equation}
-    Pour tout \( k\) nous avons \(  e^{\lambda_km_k}\in G\). De plus \( | \mu_k |\) étant borné et \( m_k\) tendant vers zéro nous avons \(  e^{\mu_km_k}\to 1\). Au final
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}\in G
-    \end{equation}
-    Cela signifie que \( \epsilon\in\mL_G\), ce qui est impossible parce que nous avions déjà dit que \( \epsilon\in M\setminus\{ 0 \}\).
-\end{proof}
-
-\begin{lemma}   \label{LemGGTtxdF}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \mL_G\times M&\to \GL(n,\eR) \\
-            l,m&\mapsto  e^{l} e^{m} 
-        \end{aligned}
-    \end{equation}
-    est un difféomorphisme local entre un voisinage de \( (0,0)\) dans \( \eM(n,\eR)\) et un voisinage de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
-\end{lemma}
-Notons que nous ne disons rien de \(  e^{\eM(n,\eR)}\). Nous n'allons pas nous embarquer à discuter si ce serait tout \( \GL(n,\eR)\)\footnote{Vu les dimensions y'a tout de même peu de chance.} ou bien si ça contiendrait ne fut-ce que \( G\).
-
-\begin{proof}
-    Le fait que \( f\) prenne ses valeurs dans \( \GL(n,\eR)\) est simplement dû au fait que les exponentielles sont toujours inversibles. Nous considérons ensuite la différentielle : si \( u\in \mL_G\) et \( v\in M\) nous avons
-    \begin{equation}
-        df_{(0,0)}(u,v)=\Dsdd{ f\big( t(u,v) \big) }{t}{0}=\Dsdd{  e^{tu} e^{tv} }{t}{0}=u+v.
-    \end{equation}
-    L'application \( df_0\) est donc une bijection entre \( \mL_G\times M\) et \( \eM(n,\eR)\). Le théorème d'inversion locale \ref{ThoXWpzqCn} nous assure alors que \( f\) est une bijection entre un voisinage de \( (0,0)\) dans \( \mL_G\times M\) et son image. Mais vu que \( df_0\) est une bijection avec \( \eM(n,\eR)\), l'image en question contient un ouvert autour de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
-\end{proof}
-
-\begin{theorem}[Von Neumann\cite{KXjFWKA,ISpsBzT,Lie_groups}]       \label{ThoOBriEoe}
-    Tout sous-groupe fermé de \( \GL(n,\eR)\) est une sous-variété de \( \GL(n,\eR)\).
-\end{theorem}
-\index{théorème!Von Neumann}
-\index{exponentielle!de matrice!utilisation}
-
-\begin{proof}
-    Soit \( G\) un tel groupe; nous devons prouver que c'est localement difféomorphe à un ouvert de \( \eR^n\). Et si on est pervers, on ne va pas faire localement difféomorphe à un ouvert de \( \eR^n\), mais à un ouvert d'un espace vectoriel de dimension finie. Nous allons être pervers.
-
-    Étant donné que pour tout \( g\in G\), l'application 
-    \begin{equation}
-        \begin{aligned}
-            L_g\colon G&\to G \\
-            h&\mapsto gh 
-        \end{aligned}
-    \end{equation}
-    est de classe \(  C^{\infty}\) et d'inverse \(  C^{\infty}\), il suffit de prouver le résultat pour un voisinage de \( \mtu\).
-
-    Supposons d'abord que \( \mL_G=\{ 0 \}\). Alors \( 0\) est un point isolé de \( \ln(G)\); en effet si ce n'était pas le cas nous aurions un élément \( m_k\) de \( \ln(G)\) dans chaque boule \( B(0,r_k)\). Nous aurions alors \( m_k=\ln(a_k)\) avec \( a_k\in G\) et donc
-    \begin{equation}
-        e^{m_k}=a_k\in G.
-    \end{equation}
-    De plus \( m_k\) appartient forcément à \( M\) parce que \( \mL_G\) est réduit à zéro. Cela nous donnerait une suite \( m_k\to 0\) dans \( M\) dont l'exponentielle reste dans \( G\). Or cela est interdit par le lemme \ref{LemHOsbREC}. Donc \( 0\) est un point isolé de \( \ln(G)\). L'application \(\ln\) étant continue\footnote{Par le lemme \ref{LemQZIQxaB}.}, nous en déduisons que \( \mtu\) est isolé dans \( G\). Par le difféomorphisme \( L_g\), tous les points de \( G\) sont isolés; ce groupe est donc discret et par voie de conséquence une variété.
-
-    Nous supposons maintenant que \( \mL_G\neq\{ 0 \}\). Nous savons par la proposition \ref{PropXFfOiOb} que 
-    \begin{equation}
-        \exp\colon \eM(n,\eR)\to \eM(n,\eR)
-    \end{equation}
-    est une application \(  C^{\infty}\) vérifiant \( d\exp_0=\id\). Nous pouvons donc utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} qui nous offre donc l'existence d'un voisinage \( U\) de \( 0\) dans \( \eM(n,\eR)\) tel que \( W=\exp(U)\) soit un ouvert de \( \GL(n,\eR)\) et que \( \exp\colon U\to W\) soit un difféomorphisme de classe \(  C^{\infty}\).
-
-    Montrons que quitte à restreindre \( U\) (et donc \( W\) qui reste par définition l'image de \( U\) par \( \exp\)), nous pouvons avoir \( \exp\big( U\cap\mL_G \big)=W\cap G\). D'abord \( \exp(\mL_G)\subset G\) par construction. Nous avons donc \( \exp\big( U\cap\mL_G \big)\subset W\cap G\). Pour trouver une restriction de \( U\) pour laquelle nous avons l'égalité, nous supposons que pour tout ouvert \( \mO\) dans \( U\), 
-    \begin{equation}
-        \exp\colon \mO\cap\mL_G\to \exp(\mO)\cap G
-    \end{equation}
-    ne soit pas surjective. Cela donnerait un élément de \( \mO\cap\complement\mL_G\) dont l'image par \( \exp\) n'est pas dans \( G\). Nous construisons ainsi une suite en considérant une boule \( B(0,\frac{1}{ k })\) inclue à \( U\) et \( x_k\in B(0,\frac{1}{ k })\cap\complement\mL_G\) vérifiant \(  e^{x_k}\in G\). Vu le choix des boules nous avons évidemment \( x_k\to 0\).
-
-    L'élément \(  e^{x_k}\) est dans \(  e^{\eM(n,\eR)}\) et le difféomorphisme du lemme \ref{LemGGTtxdF}\quext{Il me semble que l'utilisation de ce lemme manque à l'avant-dernière ligne de la preuve chez \cite{KXjFWKA}.} nous donne \( (l_k,m_k)\in \mL_G\times M\) tel que \(  e^{l_k} e^{m_k}= e^{x_k}\). À ce point nous considérons \( k\) suffisamment grand pour que \(  e^{x_k}\) soit dans la partie de l'image de \( f\) sur lequel nous avons le difféomorphisme. Plus prosaïquement, nous posons
-    \begin{equation}
-        (l_k,m_k)=f^{-1}( e^{x_k})
-    \end{equation}
-    et nous profitons de la continuité pour permuter la limite avec \( f^{-1}\) :
-    \begin{equation}
-        \lim_{k\to \infty} (l_k,m_k)=f^{-1}\big( \lim_{k\to \infty}  e^{x_k} \big)=f^{-1}(\mtu)=(0,0).
-    \end{equation}
-    En particulier \( m_k\to 0\) alors que \(  e^{m_k}= e^{x_k} e^{-l_k}\in G\). La suite \( m_k\) viole le lemme \ref{LemHOsbREC}. Nous pouvons donc restreindre \( U\) de telle façon à avoir
-    \begin{equation}
-        \exp\big( U\cap\mL_G \big)=W\cap G.
-    \end{equation}
-    Nous avons donc un ouvert de \( \mL_G\) (l'ouvert \( U\cap\mL_G\)) qui est difféomorphe avec l'ouvert \( W\cap G\) de \( G\). Donc \( G\) est une variété et accepte \( \mL_G\) comme carte locale.
-
-\end{proof}
-
-\begin{remark}
-    En termes savants, nous avons surtout montré que si \( G\) est un groupe de Lie d'algèbre de Lie \( \lG\), alors l'exponentielle donne un difféomorphisme local entre \( \lG\) et \( G\).
-\end{remark}
-
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Densité des polynômes}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
@@ -328,9 +186,9 @@ \subsection{Théorème de Stone-Weierstrass}
     où \( \mu(X)\) est la mesure de \( X\) (finie par hypothèse).
 \end{proof}
 
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Primitive de fonction continue}
-%---------------------------------------------------------------------------------------------------------------------------
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Primitive de fonction continue}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
 \begin{proposition}[\cite{MQKDooSuEGxk}]    \label{PropQACVooBnHtRJ}
     Soit un intervalle compact \( K\) de \( \eR\) et une suite \( (f_n)\) de fonctions continues sur \( K\) telles que \( f_n\stackrel{unif}{\longrightarrow}f\). Si chacune des fonctions \( f_n\) a une primitive sur \( K\) alors \( f\) également.
@@ -758,3 +616,3080 @@ \subsection{Théorème de Müntz}
 \begin{example}
     Nous savons depuis le théorème \ref{ThonfVruT} que la somme des inverses des nombres premiers diverge.
 \end{example}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorèmes de point fixe}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Points fixes attractifs et répulsifs}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooTMZUooMoBDGC}
+    Soit \( I\) un intervalle fermé de \( \eR\) et \( \varphi\colon I\to I\) une application \( C^1\). Soit \( a\) un point fixe de \( \varphi\). Nous disons que \( a\) est \defe{attractif}{point fixe!attractif}\index{attractif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) converge vers \( a\). Le point \( a\) sera dit \defe{répulsif}{répulsif!point fixe} s'il existe un voisinage \( V\) de \( a\) tel que pour tout \( x_0\in V\) la suite \( x_{n+1}=\varphi(x_n)\) diverge.
+\end{definition}
+
+\begin{lemma}[\cite{DemaillyNum}]
+    Soit \( a\) un point fixe de \( \varphi\).
+    \begin{enumerate}
+        \item
+    Si \( | \varphi'(a) |<1\) alors \( a\) est attractif et la convergence est au moins exponentielle.
+\item
+    Si \( | \varphi'(a) |>1\) alors \( a\) est répulsif et la divergence est au moins exponentielle.
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Si \( | \varphi'(a)<1 |\) alors il existe \( k\) tel que \( | \varphi'(a) |<k<1\) et par continuité il existe un voisinage \( V\) de \( a\) dans lequel \( | \varphi'(x) |<k\) pour tout \( x\in V\). En utilisant le théorème des accroissements finis nous avons
+    \begin{equation}
+        | x_n-a |=\big| f(x_{n-1}-a) \big|\leq k| x_{n-1}-a |
+    \end{equation}
+    et par récurrence
+    \begin{equation}
+        | x_n-a |\leq k^n| x_0-a |.
+    \end{equation}
+
+    Le cas \( | \varphi'(a)>1 |\) se traite de façon similaire.
+\end{proof}
+
+\begin{remark}
+    Dans le cas \(| \varphi'(a) |=1\), nous ne pouvons rien conclure. Si \( \varphi(x)=\sin(x)\) nous avons \( \sin(x)<x\) et le point \( a=0\) est attractif. A contrario, si \( \varphi(x)=\sinh(x)\) nous avons \( |\sinh(x)|>|x|\) et le point \( a=0\) est répulsif.
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Picard}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooRSLCooAsWisu}
+    Une application \( f\colon (X,\| . \|_X)\to (Y,\| . \|_Y)\) entre deux espaces métriques est une \defe{contraction}{contraction} si elle est \( k\)-\defe{Lipschitz}{Lipschitz} pour un certain \( 0\leq k<1\), c'est à dire si pour tout \( x,y\in X\) nous avons
+    \begin{equation}
+        \| f(x)-f(y) \|_Y\leq k\| x-y \|_{X}.
+    \end{equation}
+\end{definition}
+
+\begin{theorem}[Picard \cite{ClemKetl,NourdinAnal}\footnote{Il me semble qu'à la page 100 de \cite{NourdinAnal}, l'hypothèse H1 qui est prouvée ne prouve pas Hn dans le cas \( n=1\). Merci de m'écrire si vous pouvez confirmer ou infirmer. La preuve donnée ici ne contient pas cette «erreur».}.]     \label{ThoEPVkCL}
+    Soit \( X\) un espace métrique complet et \( f\colon X\to X\) une application contractante, de constante de Lipschitz \( k\). Alors \( f\) admet un unique point fixe, nommé \( \xi\). Ce dernier est donné par la limite de la suite définie par récurrence 
+    \begin{subequations}
+        \begin{numcases}{}
+            x_0\in X\\
+            x_{n+1}=f(x_n).
+        \end{numcases}
+    \end{subequations}
+    De plus nous pouvons majorer l'erreur par
+    \begin{equation}    \label{EqKErdim}
+        \| x_n-x \|\leq \frac{ k^n }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k^n }{ 1-k }\| x_1-x_0 \|.
+    \end{equation}
+
+    Soit \( r>0\), \( a\in X\) tels que la fonction \( f\) laisse la boule \( K=\overline{ B(a,r) }\) invariante (c'est à dire que \( f\) se restreint à \( f\colon K\to K\)). Nous considérons les suites \( (u_n)\) et \( (v_n)\) définies par
+    \begin{subequations}
+        \begin{numcases}{}
+            u_0=v_0\in K\\
+            u_{n+1}=f(v_n), v_{n+1}\in B(u_n,\epsilon).
+        \end{numcases}
+    \end{subequations}
+    Alors le point fixe \( \xi\) de \( f\) est dans \( K\) et la suite \( (v_n)\) satisfait l'estimation
+    \begin{equation}
+        \| v_n-\xi \|\leq \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k }.
+    \end{equation}
+\end{theorem}
+\index{théorème!Picard}
+\index{point fixe!Picard}
+
+La première inégalité \eqref{EqKErdim} donne une estimation de l'erreur calculable en cours de processus; la seconde donne une estimation de l'erreur calculable avant de commencer.
+
+\begin{proof}
+    
+    Nous commençons par l'unicité du point fixe. Si \( a\) et \( b\) sont des points fixes, alors \( f(a)=a\) et \( f(b)=b\). Par conséquent
+    \begin{equation}
+        \| f(a)-f(b) \|=\| a-b \|,
+    \end{equation}
+    ce qui contredit le fait que \( f\) soit une contraction.
+
+    En ce qui concerne l'existence, notons que si la suite des \( x_n\) converge dans \( X\), alors la limite est un point fixe. En effet en prenant la limite des deux côtés de l'équation \( x_{n+1}=f(x_n)\), nous obtenons \( \xi=f(\xi)\), c'est à dire que \( \xi\) est un point fixe de \( f\). Notons que nous avons utilisé ici la continuité de \( f\), laquelle est une conséquence du fait qu'elle soit Lipschitz. Nous allons donc porter nos efforts à prouver que la suite est de Cauchy (et donc convergente parce que \( X\) est complet). Nous commençons par prouver que \( \| x_{n+1}-x_n \|\leq k^n\| x_0-x_1 \|\). En effet pour tout \( n\) nous avons
+    \begin{equation}
+        \| x_{n+1}-x_n \|=\| f(x_n)-f(x_{n-1}) \|\leq k\| x_n-x_{n-1} \|.
+    \end{equation}
+    La relation cherchée s'obtient alors par récurrence. Soient \( q>p\). En utilisant une somme télescopique,
+    \begin{subequations}
+        \begin{align}
+            \| x_q-x_p \|&\leq \sum_{l=p}^{q-1}\| x_{l+1}-x_l \|\\
+            &\leq\left( \sum_{l=p}^{q-1}k^l \right)\| x_1-x_0 \|\\
+            &\leq\left(\sum_{l=p}^{\infty}k^l\right)\| x_1-x_0 \|.
+        \end{align}
+    \end{subequations}
+    Étant donné que \( k<1\), la parenthèse est la queue d'une série qui converge, et donc tend vers zéro lorsque \( p\) tend vers l'infini.
+
+    En ce qui concerne les inégalités \eqref{EqKErdim}, nous refaisons une somme télescopique :
+    \begin{subequations}
+        \begin{align}
+            \| x_{n+p}-x_n \|&\leq \| x_{n+p}-x_{n+p-1} \|+\cdots +\| x_{n+1}-x_n \|\\
+            &\leq k^p\| x_n-x_{n-1} \|+k^{p-1}\| x_n-x_{n-1} \|+\cdots +k\| x_n-x_{n-1} \|\\
+            &=k(1+\cdots +k^{p-1})\| x_n-x_{n-1}\|  \\
+            &\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|.
+        \end{align}
+    \end{subequations}
+    En prenant la limite \( p\to \infty\) nous trouvons
+    \begin{equation}        \label{EqlUMVGW}
+        \| \xi-x_n \|\leq \frac{ k }{ 1-k }\| x_n-x_{n-1} \|\leq \frac{ k }{ 1-k }\| x_1-x_0 \|.
+    \end{equation}
+
+    Nous passons maintenant à la seconde partie du théorème en supposant que \( f\) se restreigne en une fonction \( f\colon K\to K\). D'abord \( K\) est encore un espace métrique complet, donc la première partie du théorème s'y applique et \( f\) y a un unique point fixe.
+    
+    Nous allons montrer la relation par récurrence. Tout d'abord pour \( n=1\) nous avons
+    \begin{equation}
+        \| v_1-\xi \|\leq\| v_1-u_1 \|+\| u_1-\xi \|\leq \epsilon+\frac{ k }{ 1-k }\| u_1-u_0 \|
+    \end{equation}
+    où nous avons utilisé l'estimation \eqref{EqlUMVGW}, qui reste valable en remplaçant \( x_1\) par \( u_1\)\footnote{Elle n'est cependant pas spécialement valable si on remplace \( x_n\) par \( u_n\).}. Nous pouvons maintenant faire la récurrence :
+    \begin{subequations}
+        \begin{align}
+            \| v_{n+1}-\xi \|&\leq \| v_{n+1}-u_{n+1} \|+\| u_{n+1}-\xi \|\\
+            &\leq \epsilon+k\| v_n-\xi \|\\
+            &\leq \epsilon+k\left( \frac{ k^n }{ 1-k }\| u_1-u_0 \|+\frac{ \epsilon }{ 1-k } \right)\\
+            &=\frac{ \epsilon }{ 1-k }+\frac{ k^{n+1} }{ 1-k }\| u_1-u_0 \|.
+        \end{align}
+    \end{subequations}
+\end{proof}
+
+\begin{remark}
+    Ce théorème comporte deux parties d'intérêts différents. La première partie est un théorème de point fixe usuel, qui sera utilisé pour prouver l'existence de certaines équations différentielles.
+
+    La seconde partie est intéressante d'un point de vie numérique. En effet, ce qu'elle nous enseigne est que si à chaque pas de calcul de la récurrence \( x_{n+1}=f(x_n)\) nous commettons une erreur d'ordre de grandeur \( \epsilon\), alors le procédé (la suite \( (v_n)\)) ne converge plus spécialement vers le point fixe, mais tend vers le point fixe avec une erreur majorée par \( \epsilon/(k-1)\).
+\end{remark}
+
+\begin{remark}
+Au final l'erreur minimale qu'on peut atteindre est de l'ordre de \( \epsilon\). Évidemment si on commet une faute de calcul de l'ordre de \( \epsilon\) à chaque pas, on ne peut pas espérer mieux.
+\end{remark}
+
+\begin{remark}  \label{remIOHUJm}
+    Si \( f\) elle-même n'est pas contractante, mais si \( f^p\) est contractante pour un certain \( p\in \eN\) alors la conclusion du théorème de Picard reste valide et \( f\) a le même unique point fixe que \( f^p\). En effet nommons \( x\) le point fixe de \( f\) : \( f^p(x)=x\). Nous avons alors
+    \begin{equation}
+        f^p\big( f(x) \big)=f\big( f^p(x) \big)=f(x),
+    \end{equation}
+    ce qui prouve que \( f(x)\) est un point fixe de \( f^p\). Par unicité nous avons alors \( f(x)=x\), c'est à dire que \( x\) est également un point fixe de \( f\).
+\end{remark}
+
+Si la fonction n'est pas Lipschitz mais presque, nous avons une variante.
+\begin{proposition}
+    Soit \( E\) un ensemble compact\footnote{Notez cette hypothèse plus forte} et si \( f\colon E\to E\) est une fonction telle que
+    \begin{equation}        \label{EqLJRVvN}
+        \| f(x)-f(y) \|< \| x-y \|
+    \end{equation}
+    pour tout \( x\neq y\) dans \( E\) alors \( f\) possède un unique point fixe.
+\end{proposition}
+
+\begin{proof}
+    La suite \( x_{n+1}=f(x_n)\) possède une sous suite convergente. La limite de cette sous suite est un point fixe de \( f\) parce que \( f\) est continue. L'unicité est due à l'aspect strict de l'inégalité \eqref{EqLJRVvN}.
+\end{proof}
+
+\begin{theorem}[Équation de Fredholm]\index{Fredholm!équation}\index{équation!Fredholm}     \label{ThoagJPZJ}
+    Soit \( K\colon \mathopen[ a , b \mathclose]\times \mathopen[ a , b \mathclose]\to \eR\) et \( \varphi\colon \mathopen[ a , b \mathclose]\to \eR\), deux fonctions continues. Alors si \( \lambda\) est suffisamment petit, l'équation
+    \begin{equation}
+        f(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x)
+    \end{equation}
+    admet une unique solution qui sera de plus continue sur \( \mathopen[ a , b \mathclose]\).
+\end{theorem}
+
+\begin{proof}
+    Nous considérons l'ensemble \( \mF\) des fonctions continues \( \mathopen[ a , b \mathclose]\to\mathopen[ a , b \mathclose]\) muni de la norme uniforme. Le lemme \ref{LemdLKKnd} implique que \( \mF\) est complet. Nous considérons l'application \( \Phi\colon \mF\to \mF\) donnée par
+    \begin{equation}
+        \Phi(f)(x)=\lambda\int_a^bK(x,y)f(y)dy+\varphi(x). 
+    \end{equation}
+    Nous montrons que \( \Phi^p\) est une application contractante pour un certain \( p\). Pour tout \( x\in \mathopen[ a , b \mathclose]\) nous avons
+    \begin{subequations}
+        \begin{align}
+            \| \Phi(f)-\Phi(g) \|_{\infty}&\leq \| \Phi(f)(x)-\Phi(g)(x) \|\\
+            &=| \lambda |\Big\| \int_a^bK(x,y)\big( f(y)-g(y) \big)dy  \Big\|\\
+            &\leq | \lambda |\| K \|_{\infty}| b-a |\| f-g \|_{\infty}
+        \end{align}
+    \end{subequations}
+    Si \( \lambda\) est assez petit, et si \( p\) est assez grand, l'application \( \Phi^p\) est donc une contraction. Elle possède donc un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Brouwer}
+%---------------------------------------------------------------------------------------------------------------------------
+\label{subSecZCCmMnQ}
+
+\begin{proposition}
+    Soit \( f\colon \mathopen[ a , b \mathclose]\to \mathopen[ a , b \mathclose]\) une fonction continue. Alors \( f\) accepte un point fixe.
+\end{proposition}
+
+\begin{proof}
+    En effet si nous considérons \( g(x)=f(x)-x\) alors nous avons \( g(a)=f(a)-a\geq 0\) et \( g(b)=f(b)-b\leq 0\). Si \( g(a)\) ou \( g(b)\) est nul, la proposition est démontrée; nous supposons donc que \( g(a)>0\) et \( g(b)<0\). La proposition découle à présent du théorème des valeurs intermédiaires \ref{ThoValInter}.
+\end{proof}
+
+\begin{example}
+    La fonction \( x\mapsto\cos(x)\) est continue entre \( \mathopen[ -1 , 1 \mathclose]\) et \( \mathopen[ -1 , 1 \mathclose]\). Elle admet donc un point fixe. Par conséquent il existe (au moins) une solution à l'équation \( \cos(x)=x\).
+\end{example}
+
+\begin{proposition}[Brouwer dans \( \eR^n\) version \(  C^{\infty}\) via Stokes]     \label{PropDRpYwv}
+    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction \(  C^{\infty}\). Alors \( f\) admet un point fixe.
+\end{proposition}
+\index{point fixe!Brouwer}
+
+\begin{proof}
+    Supposons que \( f\) ne possède pas de points fixes. Alors pour tout \( x\in B\) nous considérons la ligne droite partant de \( x\) dans la direction de \( f(x)\) (cette droite existe parce que \( x\) et \( f(x)\) sont supposés distincts). Cette ligne intersecte \( \partial B\) en un point que nous appelons \( g(x)\). Prouvons que cette fonction est \( C^k\) dès que \( f\) est \( C^k\) (y compris avec \( k=\infty\)).
+
+   Le point \( g(x) \) est la solution du système
+    \begin{subequations}
+        \begin{numcases}{}
+        g(x)-f(x)=\lambda\big( x-f(x) \big)\\
+        \| g(x) \|^2=1\\
+        \lambda\geq 0.
+        \end{numcases}
+    \end{subequations}
+    En substituant nous obtenons l'équation
+    \begin{equation}
+        P_x(\lambda)=\| \lambda\big( x-f(x) \big)+f(x) \|^2-1=0,
+    \end{equation}
+    ou encore
+    \begin{equation}
+        \lambda^2\| x-f(x) \|^2+2\lambda\big( x-f(x) \big)\cdot f(x)+\| f(x) \|^2-1=0.
+    \end{equation}
+    En tenant compte du fait que \( \| f(x)<1 \|\) (pare que les images de \( f\) sont dans \( \mB\)), nous trouvons que \( P_x(0)\leq 0\) et \( P_x(1)\leq 0\). De même \( \lim_{\lambda\to\infty} P_x(\lambda)=+\infty\). Par conséquent le polynôme de second degré \( P_x\) a exactement deux racines distinctes \( \lambda_1\leq 0\) et \( \lambda_2\geq 1\). La racine que nous cherchons est la seconde. Le discriminant est strictement positif, donc pas besoin d'avoir peur de la racine dans
+    \begin{equation}
+        \lambda(x)=\frac{ -\big( x-f(x) \big)\cdot f(x)+\sqrt{   \Delta_x  } }{ \| x-f(x) \|^2 }
+    \end{equation}
+    où 
+    \begin{equation}
+        \Delta_x=4\Big( \big( x-f(x) \big)\cdot f(x) \Big)^2-4\| x-f(x) \|^2\big( \| f(x) \|^2-1 \big).
+    \end{equation}
+    Notons que la fonction \( \lambda(x)\) est \( C^k\) dès que \( f\) est \( C^k\); et en particulier elle est \( C^{\infty}\) si \( f\) l'est.
+
+    En résumé la fonction \( g\) ainsi définie vérifie deux propriétés :
+    \begin{enumerate}
+        \item
+            elle est \(  C^{\infty}\);
+        \item
+            elle est l'identité sur \( \partial B\).
+    \end{enumerate}
+    La suite de la preuve consiste à montrer qu'une telle rétraction sur \( B\) ne peut pas exister\footnote{Notons qu'il n'existe pas non plus de rétractions continues sur \( B\), mais pour le montrer il faut utiliser d'autres méthodes que Stokes, ou alors présenter les choses dans un autre ordre.}.
+
+    Nous considérons une forme de volume \( \omega\) sur \( \partial B\) : l'intégrale de \( \omega\) sur \( \partial B\) est la surface de \( \partial B\) qui est non nulle. Nous avons alors
+    \begin{equation}
+        0<\int_{\partial B}\omega
+        =\int_{\partial B}g^*\omega
+        =\int_Bd(g^*\omega)
+        =\int_Bg^*(d\omega)
+        =0
+    \end{equation}
+    Justifications :
+    \begin{itemize}
+        \item 
+            L'intégrale \( \int_{\partial B}\omega\) est la surface de \( \partial B\) et est donc non nulle.
+        \item
+            La fonction \( g\) est l'identité sur \( \partial B\). Nous avons donc \( \omega=g^*\omega\).
+        \item
+            Le lemme \ref{LemdwLGFG}.
+        \item
+            La forme \( \omega\) est de volume, par conséquent de degré maximum et \( d\omega=0\).
+    \end{itemize}
+\end{proof}
+
+Un des points délicats est de se ramener au cas de fonctions \( C^{\infty}\). Pour la régularisation par convolution, voir \cite{AllardBrouwer}; pour celle utilisant le théorème de Weierstrass, voir \cite{KuttlerTopInAl}.
+\begin{theorem}[Brouwer dans \( \eR^n\) version continue]\label{ThoRGjGdO}
+    Soit \( B\) la boule fermée de centre \( 0\) et de rayon \( 1\) de \( \eR^n\) et \( f\colon B\to B\) une fonction continue\footnote{Une fonction continue sur un fermé de \( \eR^n\) est à comprendre pour la topologie induite.}. Alors \( f\) admet un point fixe.
+\end{theorem}
+\index{théorème!Brouwer}
+
+\begin{proof}
+    Nous commençons par définir une suite de fonctions
+    \begin{equation}
+        f_k(x)=\frac{ f(x) }{ 1+\frac{1}{ k } }.
+    \end{equation}
+    Nous avons \( \| f_k-f \|_{\infty}\leq \frac{1}{ 1+k }\) où la norme est la norme uniforme sur \( B\). Par le théorème de Weierstrass \ref{CORooNIUJooLDrPSv} il existe une suite de fonctions \(  C^{\infty}(B,\eR)\) que nous nommons \( g_k\) telles que
+    \begin{equation}
+        \|  g_k-f_k\|_{\infty}\leq\frac{1}{ 1+k }.
+    \end{equation}
+    Vérifions que cette fonction \( g_k\) soit bien une fonction qui prend ses valeurs dans \( B\) :
+    \begin{subequations}
+        \begin{align}
+            \| g_k(x) \|&\leq \| g_k(x)-f_k(x) \|+\| f_k(x) \|\\
+            &\leq \frac{1}{ 1+k }+\frac{ \| f(x) \| }{ 1+\frac{1}{ k } }\\
+            &\leq \frac{1}{ 1+k}+\frac{1}{ 1+\frac{1}{ k } }\\
+            &=1.
+        \end{align}
+    \end{subequations}
+    Par la version \(  C^{\infty}\) du théorème (proposition \ref{PropDRpYwv}), \( g_k\) admet un point fixe que l'on nomme \( x_k\).
+
+    Étant donné que \( x_k\) est dans le compact \( B\), quitte à prendre une sous suite nous supposons que la suite \( (x_k)\) converge vers un élément \( x\in B\). Nous montrons maintenant que \( x\) est un point fixe de \( f\) :
+    \begin{subequations}
+        \begin{align}
+            \| f(x)-x \|&=\| f(x)-g_k(x)+g_k(x)-x_k+x_k-x \|\\
+            &\leq \| f(x)-g_k(x) \| +\underbrace{\| g_k(x)-x_k \|}_{=0}+\| x_k-x \|\\
+            &\leq \frac{1}{ 1+k }+\| x_k-x \|.
+        \end{align}
+    \end{subequations}
+    En prenant le limite \( k\to\infty\) le membre de droite tend vers zéro et nous obtenons \( f(x)=x\).
+\end{proof}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Théorème de Schauder}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Une conséquence du théorème de Brouwer est le théorème de Schauder qui est valide en dimension infinie.
+
+\begin{theorem}[Théorème de Schauder\cite{ooWWBQooKIciWi}]\index{théorème!Schauder}       \label{ThovHJXIU}
+    Soit \( E\), un espace vectoriel normé, \( K\) un convexe compact de \( E\) et \( f\colon K\to K\) une fonction continue. Alors \( f\) admet un point fixe.
+\end{theorem}
+\index{théorème!Schauder}
+\index{point fixe!Schauder}
+
+\begin{proof}
+    Étant donné que \( f\colon K\to K\) est continue, elle y est uniformément continue. Si nous choisissons \( \epsilon\) alors il existe \( \delta>0\) tel que 
+    \begin{equation}
+        \| f(x)-f(y) \|\leq \epsilon
+    \end{equation}
+    dès que \( \| x-y \|\leq \delta\). La compacité de \( K\) permet de choisir un recouvrement fini par des ouverts de la forme
+    \begin{equation}    \label{EqKNPUVR}
+        K\subset \bigcup_{1\leq i\leq p}B(x_j,\delta)
+    \end{equation}
+    où \( \{ x_1,\ldots, x_p \}\subset K\). Nous considérons maintenant \( L=\Span\{ f(x_j)\tq 1\leq j\leq p \}\) et
+    \begin{equation}
+        K^*=K\cap L.
+    \end{equation}
+    Le fait que \( K\) et \( L\) soient convexes implique que \( K^*\) est convexe. L'ensemble \( K^*\) est également compact parce qu'il s'agit d'une partie fermée de \( K\) qui est compact (lemme \ref{LemnAeACf}). Notons en particulier que \( K^*\) est contenu dans un espace vectoriel de dimension finie, ce qui n'est pas le cas de \( K\).
+
+    Nous allons à présent construire une sorte de partition de l'unité subordonnée au recouvrement \eqref{EqKNPUVR} sur \( K\) (voir le lemme \ref{LemGPmRGZ}). Nous commençons par définir
+    \begin{equation}
+        \psi_j(x)=\begin{cases}
+            0    &   \text{si } \| x-x_j \|\geq \delta\\
+            1-\frac{ \| x-x_j \| }{ \delta }    &    \text{sinon}.
+        \end{cases}
+    \end{equation}
+    pour chaque \( 1\leq j\leq p\). Notons que \( \psi_j\) est une fonction positive, nulle en-dehors de \( B(x_j,\delta)\). En particulier la fonction suivante est bien définie :
+    \begin{equation}
+        \varphi_j(x)=\frac{ \psi_j(x) }{ \sum_{k=1}^p\psi_k(x) }
+    \end{equation}
+    et nous avons \( \sum_{j=1}^p\varphi_j(x)=1\). Les fonctions \( \varphi_j\) sont continues sur \( K\) et nous définissons finalement
+    \begin{equation}
+        g(x)=\sum_{j=1}^p\varphi_j(x)f(x_j).
+    \end{equation}
+    Pour chaque \( x\in K\), l'élément \( g(x)\) est une combinaison des éléments \( f(x_j)\in K^*\). Étant donné que \( K^*\) est convexe et que la somme des coefficients \( \varphi_j(x)\) vaut un, nous avons que \( g\) prend ses valeurs dans \( K^*\) par la proposition \ref{PropPoNpPz}.
+
+    Nous considérons seulement la restriction \( g\colon K^*\to K^*\) qui est continue sur un compact contenu dans un espace vectoriel de dimension finie. Le théorème de Brouwer nous enseigne alors que \( g\) a un point fixe (proposition \ref{ThoRGjGdO}). Nous nommons \( y\) ce point fixe. Notons que \( y\) est fonction du \( \epsilon\) choisit au début de la construction, via le \( \delta\) qui avait conditionné la partition de l'unité.
+
+    Nous avons
+    \begin{subequations}        \label{EqoXuTzE}
+        \begin{align}
+            f(y)-y&=f(y)-g(y)\\
+            &=\sum_{j=1}^p\varphi_j(y)f(y)-\sum_{j=1}^p\varphi_j(y)f(x_j)\\
+            &=\sum_{j=1}^p\varphi(j)(y)\big( f(y)-f(x_j) \big).
+        \end{align}
+    \end{subequations}
+    Par construction, \( \varphi_j(y)\neq 0\) seulement si \( \| y-x_j \|\leq \delta\) et par conséquent seulement si \( \| f(y)-f(x_j) \|\leq \epsilon\). D'autre par nous avons \( \varphi_j(y)\geq 0\); en prenant la norme de \eqref{EqoXuTzE} nous trouvons
+    \begin{equation}
+        \| f(y)-y \|\leq \sum_{j=1}^p\| \varphi_j(y)\big( f(y)-f(x_j) \big) \|\leq \sum_{j=1}^p\varphi_j(y)\epsilon=\epsilon.
+    \end{equation}
+    Nous nous souvenons maintenant que \( y\) était fonction de \( \epsilon\). Soit \( y_m\) le \( y\) qui correspond à \( \epsilon=2^{-m}\). Nous avons alors
+    \begin{equation}
+        \| f(y_m)-y_m \|\leq 2^{-m}.
+    \end{equation}
+    L'élément \( y_m\) est dans \( K^*\) qui est compact, donc quitte à choisir une sous suite nous pouvons supposer que \( y_m\) est une suite qui converge vers \( y^*\in K\)\footnote{Notons que même dans la sous suite nous avons \( \| f(y_m)-y_m \|\leq 2^{-m}\), avec le même «\( m\)» des deux côtés de l'inégalité.}. Nous avons les majorations
+    \begin{equation}
+        \| f(y^*)-y^* \|\leq \| f(y^*)-f(y_m) \|+\| f(y_m)-y_m \|+\| y_m-y^* \|.
+    \end{equation}
+    Si \( m\) est assez grand, les trois termes du membre de droite peuvent être rendus arbitrairement petits, d'où nous concluons que
+    \begin{equation}
+        f(y^*)=y^*
+    \end{equation}
+    et donc que \( f\) possède un point fixe.
+\end{proof}
+
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de Markov-Kakutani et mesure de Haar}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit \( G\) un groupe topologique. Une \defe{mesure de Haar}{mesure!de Haar} sur \( G\) est une mesure \( \mu\) telle que 
+    \begin{enumerate}
+        \item
+            \( \mu(gA)=\mu(A)\) pour tout mesurable \( A\) et tout \( g\in G\),
+        \item
+            \( \mu(K)<\infty\) pour tout compact \( K\subset G\).
+    \end{enumerate}
+    Si de plus le groupe \( G\) lui-même est compact nous demandons que la mesure soit normalisée : \( \mu(G)=1\).
+\end{definition}
+
+Le théorème suivant nous donne l'existence d'une mesure de Haar sur un groupe compact.
+\begin{theorem}[Markov-Katutani\cite{BeaakPtFix}]\index{théorème!Markov-Takutani}   \label{ThoeJCdMP}
+    Soit \( E\) un espace vectoriel normé et \( L\), une partie non vide, convexe, fermée et bornée de \( E'\). Soit \( T\colon L\to L\) une application continue. Alors \( T\) a un point fixe.
+\end{theorem}
+
+\begin{proof}
+    Nous considérons un point \( x_0\in L\) et la suite
+    \begin{equation}
+        x_n=\frac{1}{ n+1 }\sum_{i=0}^n T^ix_0.
+    \end{equation}
+    La somme des coefficients devant les \( T^i(x_0)\) étant \( 1\), la convexité de \( L\) montre que \( x_n\in L\). Nous considérons l'ensemble
+    \begin{equation}
+        C=\bigcap_{n\in \eN}\overline{ \{ x_m\tq m\geq n \} }.
+    \end{equation}
+    Le lemme \ref{LemooynkH} indique que \( C\) n'est pas vide, et de plus il existe une sous suite de \( (x_n)\) qui converge vers un élément \( x\in C\). Nous avons
+    \begin{equation}
+        \lim_{n\to \infty} x_{\sigma(n)}(v)=x(v)
+    \end{equation}
+    pour tout \( v\in E\). Montrons que \( x\) est un point fixe de \( T\). Nous avons
+    \begin{subequations}
+        \begin{align}
+            \| (Tx_{\sigma(k)}-x_{\sigma(k)})v \|&=\Big\| T\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v)-\frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^ix_0(v) \Big\|\\
+            &=\Big\| \frac{1}{ 1+\sigma(k) }\sum_{i=0}^{\sigma(k)}T^{i+1}x_0(v)-T^ix_0(v) \Big\|\\
+            &=\frac{1}{ 1+\sigma(k) }\big\| T^{\sigma(k)+1}x_0(v)-x_0(v) \big\|\\
+            &\leq\frac{ 2M }{ \sigma(k)+1 }
+        \end{align}
+    \end{subequations}
+    où \( M=\sum_{y\in L}\| y(v) \|<\infty\) parce que \( L\) est borné. En prenant \( k\to\infty\) nous trouvons
+    \begin{equation}
+        \lim_{k\to \infty} \big( Tx_{\sigma(k)}-x_{\sigma(k)} \big)v=0,
+    \end{equation}
+    ce qui signifie que \( Tx=x\) parce que \( T\) est continue.
+\end{proof}
+
+Le théorème suivant est une conséquence du théorème de Markov-Katutani.
+\begin{theorem} \label{ThoBZBooOTxqcI}
+    Si \( G\) est un groupe topologique compact possédant une base dénombrable de topologie alors \( G\) accepte une unique mesure de Haar normalisée. De plus elle est unimodulaire :
+    \begin{equation}
+        \mu(Ag)=\mu(gA)=\mu(A)
+    \end{equation}
+    pour tout mesurables \( A\subset G\) et tout élément \( g\in G\).
+\end{theorem}
+\index{mesure!de Haar}
+
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorèmes de point fixes et équations différentielles}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Théorème de Cauchy-Lipschitz}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous démontrons ici deux théorèmes de Cauchy-Lipschitz. De nombreuses propriétés annexes seront démontrées dans le chapitre sur les équations différentielles, section \ref{SECooNKICooDnOFTD}.
+
+\begin{theorem}[Cauchy-Lipschitz\cite{SandrineCL,ZPNooLNyWjX}] \label{ThokUUlgU}
+    Nous considérons l'équation différentielle
+    \begin{subequations}        \label{XtiXON}
+        \begin{numcases}{}
+            y'(t)=f\big( t,y(t) \big)\\
+            y(t_0)=y_0
+        \end{numcases}
+    \end{subequations}
+    avec \( f\colon U=I\times \Omega\to \eR^n\) où \( I\) est ouvert dans \( \eR\) et \( \Omega\) ouvert dans \( \eR^n\). Nous supposons que \( f\) est continue sur \( U\) et localement Lipschitz\footnote{Définition \ref{DefJSFFooEOCogV}. Notons que nous ne supposons pas que \( f\) soit une contraction.} par rapport à \( y\). 
+    
+    Alors il existe un intervalle \( J\subset I\) sur lequel la solution au problème est unique. De plus toute solution du problème est une restriction de cette solution à une partie de \( J\). La solution sur \( J\) (dite «solution maximale») est de classe \( C^1\).
+\end{theorem}
+\index{théorème!Cauchy-Lipschitz}
+
+% Il serait tentant de mettre ce théorème dans la partie sur les équations différentielles, mais ce n'est pas aussi simple :
+% Il est utilisé pour calculer la transformée de Fourier de la Gaussienne (lemme LEMooPAAJooCsoyAJ) dans le chapitre sur la transformée de Fourier.
+
+\begin{proof}
+    Nous divisions la preuve en plusieurs étapes (même pas toutes simples).
+    \begin{subproof}
+    \item[Cylindre de sécurité et espace fonctionnel]
+
+    Précisons l'espace fonctionnel \( \mF\) adéquat. Soient \( V\) et \( W\) les voisinages de \( t_0\) et \( y_0\) sur lesquels \( f\) est localement Lipschitz. Nous considérons les quantités suivantes :
+    \begin{enumerate}
+        \item
+            \( M=\sup_{V\times W}f\) ;
+        \item
+            \( r>0\) tel que \( \overline{ B(y_0,r) }\subset V\)
+        \item
+            \( T>0\) tel que \( \overline{ B(t_0,T) }\subset W\) et \( T<r/M\).
+    \end{enumerate}
+    Nous considérons alors l'ensemble
+    \begin{equation}
+        \mF=C^0\big( \overline{ B(t_0,T) },\overline{ B(y_0,r) } \big)
+    \end{equation}
+    que nous munissons de la norme uniforme. Par le lemme \ref{LemdLKKnd} l'espace \( \big( \mF,\| . \|_{\infty} \big)\) est complet.
+
+    \item[Une application \( \Phi\colon \mF\to \mF\)]
+
+
+    Si \( y\) est une solution de l'équation différentielle considérée, elle vérifie
+    \begin{equation}        \label{EqPGLwcL}
+        y(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
+    \end{equation}
+    Ceci nous incite à considérer l'opérateur \( \Phi\colon \mF\to \mF\) défini par
+    \begin{equation}
+        \Phi(y)(t)=y_0+\int_{t_0}^tf\big( u,y(u) \big)du.
+    \end{equation}
+
+    Pour que l'application \( \Phi\) soit utile nous devons montrer que pour tout \( y\in \mF\),
+    \begin{itemize}
+        \item l'application \( \Phi(y)\) est bien définie,
+        \item pour tout \( t\in\overline{ B(y_0,r) }\) nous avons \( \Phi(y)(t)\in\overline{ B(t_0,T) }\),
+        \item l'application $\Phi(y)\colon    \overline{ B(t_0,T) }\to \overline{ B(y_0,r)} $ est continue.
+    \end{itemize}
+    Attention : nous ne prétendons pas que \( \Phi\) elle-même soit continue. C'est parti.
+    \begin{subproof}
+    \item[\( \Phi(y)\) est bien définie]
+            
+        Il faut montrer que l'intégrale converge. Le calcul de \( \Phi(y)(t)\) ne se fait qu'avec \( t\in \overline{ B(t_0,T) }\). Vu que \( u\) prend ses valeurs dans \( \mathopen[ t_0 , t \mathclose]\) et que \( y\in\mF\), le nombre \( y(u)\) est toujours dans \( \overline{ B(y_0,r) }\). Ceci pour dire que dans l'intégrale, la fonction \( f\) n'est considérée que sur \( \mathopen[ t_0 , t \mathclose]\times \overline{ B(y_0,r) }\subset V\times W\). La fonction \( f\) est donc uniformément majorable, et l'intégrale ne pose pas de problèmes.
+
+    \item[\( \Phi(y)(t)\in \overline{ B(t_0,T) }\)]
+
+    Prouvons que \( \Phi(y)(t)\in\overline{ B(y_0,r) }\). Pour cela, notons que
+    \begin{equation}
+        | \Phi(y)(t)-y_0 |\leq \int_{t_0}^t |f\big( u,y(u) \big)|du\leq | t-t_0 |\| f \|_{\infty}.
+    \end{equation}
+    Étant donné que \( t\in\overline{ B(t_0,T) }\) nous avons \( | t-t_0 |\leq r/M\) et donc \( | \Phi(y)(t)-y_0 |\leq r\).
+
+    \item[\( \Phi(y)\) est continue]
+
+        Nous pourrions invoquer le théorème \ref{ThoKnuSNd}, mais nous allons le faire à la main. Soit \( s_0\in B(t_0,T)\) et prouvons que \( \Phi(y)\) est continue en \( s_0\). Pour cela nous prenons \( s\in B(s_0,\delta)\) et nous calculons :
+        \begin{equation}
+            | \Phi(y)(s)-\Phi(y)(s_0) |\leq \int_{s_0}^s|f\big( u,y(u) \big)|du\leq | s_0-s |\| f \|_{\infty}.
+        \end{equation}
+        C'est le fait que \( f\) soit bornée dans le cylindre de sécurité qui fait en sorte que cela tende vers zéro lorsque \( s\to s_0\).
+    \end{subproof}
+
+
+    
+    L'équation \eqref{EqPGLwcL} signifie que \( y\) est un point fixe de \( \Phi\). L'espace \( \mF\) étant complet le théorème de point fixe de Picard (théorème \ref{ThoEPVkCL}) s'applique. Nous allons montrer qu'il existe un \( p\in\eN\) tel que \( \Phi^p\) soit contractante. Par conséquent \( \Phi^p\) aura un unique point fixe qui sera également unique point fixe de \( \Phi\) par la remarque \ref{remIOHUJm}.
+    
+\item[Contractante]
+
+    Prouvons donc que \( \Phi^p\) est contractante pour un certain \( p\). Pour cela nous commençons par montrer la formule suivante par récurrence :
+    \begin{equation}        \label{EqRAdKxT}
+        \big\| \Phi^p(x)(t)-\Phi^p(y)(t) \big\|\leq \frac{ k^p| t-t_0 |^p }{ p! }\| x-y \|_{\infty}
+    \end{equation}
+    pour tout \( x,y\in\mF\), et pour tout \( t\in\overline{ B(t_0,T) }\). Pour \( p=0\) la formule \eqref{EqRAdKxT} est vérifiée parce que \( \| x-y \|_{\infty}\) est le supremum de \( \| x(t)-y(t) \|\) pour \( t\in\overline{ B(t_0,T) }\). Supposons que la formule soit vraie pour \( p\) et calculons pour \( p+1\). Pour tout \( t\in\overline{ B(t_0,T) }\) nous avons
+    \begin{subequations}
+        \begin{align}
+            \big\| \Phi^{p+1}(x)(t)-\Phi^{p+1}(y)(t) \big\|&\leq \left| \int_{t_0}^t\big\| f\big( u,\Phi^p(x)(u) \big)-f\big( u,\Phi^p(y)(u) \big) \big\|du \right| \\
+            &\leq \left| \int_{t_0}^tk\| \Phi^p(x)(u)-\Phi^p(y)(u) \|du \right|    \label{subIKYixF}\\
+            &\leq \left| \int_{t_0}^tk\frac{ k^p| t-t_0 | }{ p! }\| x-y \|_{\infty} \right| \label{subxkNjiV} \\
+            &=\frac{ k^{p+1}| t-t_0 |^{p+1} }{ (p+1)! }\| x-y \|_{\infty}.
+        \end{align}
+    \end{subequations}
+    Justifications :
+    \begin{itemize}
+        \item \eqref{subIKYixF} parce que \( f\) est Lipschitz.
+        \item \eqref{subxkNjiV} par hypothèse de récurrence.
+    \end{itemize}
+    La formule \eqref{EqRAdKxT} est maintenant établie. Nous pouvons maintenant montrer que \( \Phi^p\) est une contraction pour un certain \( p\). Pour tout \( t\in \overline{ B(t_0,T) }\) nous avons
+    \begin{equation}
+         \| \Phi^p(x)(t)-\Phi^p(y)(t) \|\leq \frac{ k^p }{ t! }| t-t_0 |^p\| x-y \|_{\infty}     \leq \frac{ k^pT^p }{ p! }\| x-y \|_{\infty}
+    \end{equation}
+    où nous avons utilisé le fait que \( | t-t_0 |^p<T^p\). En prenant le supremum sur \( t\) des deux côtés il vient
+    \begin{equation}
+        \| \Phi^p(x)-\Phi^p(y) \|_{\infty}\leq\frac{ k^pT^p }{ p! }\| x-y \|_{\infty}.
+    \end{equation}
+    Le membre de droite tend vers zéro lorsque \( p\to\infty\) parce que \( k<1\) et \( T^p/p!\to 0\)\footnote{C'est le terme général du développement de \(  e^{T}\) qui est une série convergente.}. Nous concluons donc que \( \Phi^p\) est une contraction pour un certain \( p\).
+
+\item[Conclusion]
+
+    L'unique point fixe de \( \Phi\) est alors l'unique solution continue de l'équation différentielle \eqref{XtiXON}. Par ailleurs l'équation elle-même \( y'=f(t,y)\) demande implicitement que \( y\) soit dérivable et donc continue. Nous concluons que l'unique point fixe de \( \Phi\) est l'unique solution de l'équation différentielle donnée. Cette dernière est automatiquement \( C^1\) parce que si \( y\) est continue alors \( u\mapsto f(u,y(u))\) est continue, c'est à dire que \( y'\) est continue.
+
+\item[Unicité]
+
+    Nous passons maintenant à la partie «prolongement maximum» du théorème. Soient \( x_1\) et \( x_2\) deux solutions maximales du problème \eqref{XtiXON} sur des intervalles \( I_1\) et \( I_2\) respectivement. Les intervalles \( I_1\) et \( I_2\) contiennent \( \overline{ B(t_0,r) }\) sur lequel \( x_1=x_2\) par unicité.
+    
+    
+    Nous allons maintenant montrer que pour tout \( t\geq t_0\) pour lequel \( x_1\) ou \( x_2\) est défini, \( x_1(t)\) et \( x_2(t)\) sont définis et sont égaux. Le raisonnement sur \( t\leq t_0\) est similaire.
+    
+    Supposons que l'ensemble des \( t\geq t_0\) tels que \( x_1=x_2\) soit ouvert à droite, c'est à dire soit de la forme \( \mathopen[ t_0 ,b [\). Dans ce cas, soit \( x_1\) soit \( x_2\) (soit les deux) cesse d'exister en \( b\). En effet si nous avions les fonctions \( x_i\) sur \(\mathopen[ t_0 , b+\epsilon [\) alors l'équation \( x_1=x_2\) définirait un fermé dans \( \mathopen[ t_0 , b+\epsilon [\). Supposons pour fixer les idées que \( x_1\) cesse d'exister : le domaine de \( x_1\) (parmi les \( t\geq 0\)) est \( \mathopen[ t_0 , b [\) et sur ce domaine nous avons \( x_1=x_2\). Dans ce cas \( x_1\) pourrait être prolongé en \( x_2\) au-delà de \( b\). Si \( x_1\) et \( x_2\) s'arrêtent d'exister en même temps en \( b\), alors nous avons bien \( x_1=x_2\).
+
+    Nous devons donc traiter le cas où \( x_1=x_2\) sur \( \mathopen[ t_0 , b \mathclose]\) alors que \( x_1\) et \( x_2\) existent sur \( \mathopen[ t_0 , b+\epsilon [\) pour un certain \( \epsilon\).
+
+    Nous pouvons appliquer le théorème d'existence locale au problème
+    \begin{subequations}
+        \begin{numcases}{}
+            y'=f(t,y)\\
+            y(b)=x_1(b).
+        \end{numcases}
+    \end{subequations}
+    Il existe un voisinage de \( b\) sur lequel la solution est unique. Sur ce voisinage nous devons donc avoir \( x_1=x_2\), ce qui contredit le fait que \( x_1\neq x_2\) en dehors de \( \mathopen[ t_0 , b \mathclose]\).
+
+    Donc \( x_1\) et \( x_2\) existent et sont égaux sur au moins \( I_1\cup I_2\).
+    \end{subproof}
+\end{proof}
+
+Le théorème de Cauchy-Lipschitz donne existence et unicité d'une solution maximale. Cependant cette solution peut ne pas exister partout où les hypothèses sur \( f\) sont remplies. En d'autres termes, il peut arriver que \( f\) soit Lipschitz jusqu'à \( t_1\), mais que la solution maximale ne soit définie que jusqu'en \( t_2<t_1\). Ce cas fait l'objet du théorème d'explosion en temps fini \ref{CorGDJQooNEIvpp}.
+
+Sous quelque hypothèses nous pouvons nous assurer de l'existence d'une solution unique sur tout \( \eR\).
+
+\begin{theorem}[Cauchy-Lipschitz global\cite{ooJZJPooAygxpk,KXjFWKA}]       \label{THOooZIVRooPSWMxg}
+    Soit un intervalle \( I\) de \( \eR\), \( y_0\in \eR^n\), \( t_0\in I\) et une fonction continue \( f\colon I\times \eR^n\to \eR^n\) telle que pour tout compact \( K\) dans \( I\), il existe \( k>0\) tel que
+    \begin{equation}
+        \| f(t,y_1)-f(t,y_2) \|\leq k\| y_1-y_2 \|
+    \end{equation}
+    pour tout \( t\in K\) et \( y_1,y_2\in \eR^n\).
+
+    Alors le problème
+    \begin{subequations}        \label{EQSooBNREooUTfbMH}
+        \begin{numcases}{}
+            y'(t)=f\big( t,y(t) \big)\\
+            y(t_0)=y_0
+        \end{numcases}
+    \end{subequations}
+    possède une unique solution \( y\colon I\to \eR^n\) sur \( I\).
+\end{theorem}
+
+\begin{proof}
+    Soit un intervalle compact \( K\) dans \( I\) et contenant \( t_0\). Nous notons \( \ell\) le diamètre de \( K\). Sur l'espace \( E=C^0(K,\eR^n)\) nous considérons la topologie uniforme : \( (E,\| . \|_{\infty})\). C'est un espace complet par le lemme \ref{LemdLKKnd} (nous utilisons le fait que \( \eR^n\) soit complet, théorème \ref{ThoTFGioqS}). Nous allons utiliser l'application suivante :
+    \begin{equation}        \label{EQooJUTBooILBKoE}
+        \begin{aligned}
+            \Phi\colon E&\to E \\
+            \Phi(y)(t)&=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds
+        \end{aligned}
+    \end{equation}
+    Démontrons quelque faits à propos de \( \Phi\).
+    \begin{subproof}
+        \item[La définition fonctionne bien]
+            Nous devons commencer par prouver que cette application est bien définie. Si \( y\in E\) alors \( f\) et \( y\) sont continues; l'application \( s\mapsto f\big(s,y(s)\big)\) est donc également continue. L'intégrale de cette fonction sur le compact \( \mathopen[ t_0 , t \mathclose]\) ne pose alors pas de problèmes. En ce qui concerne la continuité de \( \phi(y)\) sous l'hypothèse que \( y\) soit continue,
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(y)(t') \|\leq \int_t^{t'}\| f(s,y(s)) \|ds\leq M| t-t' |
+    \end{equation}
+    où \( M\) est une majoration de \( \| s\mapsto f\big( s,y(s) \big) \|_{\infty,K}\).
+
+        \item[Si \( y\) est solution alors \( \Phi(y)=y\)]
+
+            Supposons que \( y\) soit une solution de l'équation différentielle \eqref{EQSooBNREooUTfbMH}. Alors, vu que \( y'(t)=f\big( t,y(t) \big)\) nous avons :
+            \begin{equation}
+                y(t)=y_0+\int_{t_0}^ty'(s)ds=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds=\Phi(y)(t).
+            \end{equation}
+            
+        \item[Si \( \Phi(y)=y\) alors \( y\) est solution]
+
+            Nous avons, pour tout \( t\) :
+            \begin{equation}
+                y(t)=y_0+\int_{t_0}^tf\big( s,y(s) \big)ds.
+            \end{equation}
+            Le membre de droite est dérivable par rapport à \( t\), et la dérivée fait \(  f\big( t,y(t) \big)   \). Donc le membre de gauche est également dérivable et nous avons bien
+            \begin{equation}
+                y'(t)=f\big( t,y(t) \big).
+            \end{equation}
+            De plus \( y(t_0)=y_0+\int_{t_0}^{t_0}\ldots=y_0\).
+    \end{subproof}
+    
+    Nous sommes encore avec \( K\) compact et \( E=C^0(K,\eR^n)\) muni de la norme uniforme. Nous allons montrer que \( \Phi\) est une contraction de \( E\) pour une norme bien choisie.
+
+    \begin{subproof}
+        \item[Une norme sur \( E\)]
+            Pour \( y\in E\) nous posons
+            \begin{equation}
+                \| y \|_k=\max_{t\in K}\big(  e^{-k| t-t_0 |}\| y(t) \| \big).
+            \end{equation}
+            Ce maximum est bien définit et fini parce que la fonction de \( t\) dedans est une fonction continue sur le compact \( K\). Cela est également une norme parce que si \( \| y \|_k=0\) alors \(  e^{-k| t-t_0 |}\| y(t) \|=0\) pour tout \( t\). Étant donné que l'exponentielle ne s'annule pas, \( \| y(t) \|=0\) pour tout \( t\).
+        \item[Équivalence de norme]
+
+            Nous montrons que les normes \( \| . \|_k\) et \( \| . \|_{\infty}\) sont équivalentes\footnote{Définition \ref{DefEquivNorm}} :
+            \begin{equation}        \label{EQooSQYWooBTXvDL}
+                \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\leq \| y \|_{\infty}
+            \end{equation}
+            pour tout \( y\in E\). Pour la première inégalité, \( \ell\geq | t-t_0 |\) pour tout \( t\in K\), et \( k>0\), donc
+            \begin{equation}
+                \| y(t) \| e^{-k\ell}\leq  e^{-k| t-t_0 |}\| y(t) \|.
+            \end{equation}
+            En prenant le maximum des deux côtés, \( \| y \|_{\infty} e^{-k\ell}\leq \| y \|_k\). 
+
+            En ce qui concerne la seconde inégalité dans \eqref{EQooSQYWooBTXvDL}, \( k| t-t_0 |\geq 0\) et donc \(  e^{-k| t-t_0 |}<1\).
+
+    \end{subproof}
+    Vu que les normes \( \| . \|_{\infty}\) et \( \| . \|_k\) sont équivalentes, l'espace \( (E,\| . \|_k)\) est tout autant complet que \( (E,\| . \|_{\infty})\). Nous démontrons à présent que \( \Phi\) est une contraction dans \( (E,\|  \|_k)\). 
+
+    Soient \( y,z\in E\). Si \( t\geq t_0\) nous avons
+    \begin{subequations}        \label{SUBEQSooEXVYooDkyTuB}
+        \begin{align}
+            \| \Phi(y)(t)-\Phi(z)(t) \|&\leq \int_{t_0}^t\| f\big( s,y(s) \big)-f\big( s,z(s) \big) \|ds\\
+            &\leq k\int_{t_0}^t\| y(s)-z(s) \|ds.
+        \end{align}
+    \end{subequations}
+    Il convient maintenant de remarquer que
+    \begin{equation}
+        \| y(t) \|= e^{-k| t-t_0 |} e^{k| t-t_0 |}\| y(t) \|\leq \| y \|_k e^{k| t-t_0 |}.
+    \end{equation}
+    Nous pouvons avec ça prolonger les inégalités \eqref{SUBEQSooEXVYooDkyTuB} par
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(z)(t) \|\leq k\| y-z \|_k\int_{t_0}^t e^{k| s-t_0 |}ds=k\| y-z \|_k\int_{t_0}^t e^{k(s-t_0)}ds
+    \end{equation}
+    où nous avons utilisé notre supposition \( t\geq t_0\) pour éliminer les valeurs absolues. L'intégrale peut être faite explicitement, mais nous en sommes arrivés à un niveau de fainéantise tellement inconcevable que
+
+\lstinputlisting{tex/sage/sageSnip014.sage}
+
+Au final, si \( t\geq t_0\),
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t-t_0)}-1 \big).
+    \end{equation}
+    Si \( t\leq t_0\), il faut retourner les bornes de l'intégrale avant d'y faire rentrer la norme parce que \( \| \int_0^1f \|\leq \int_0^1\| f \|\), mais ça ne marche pas avec \( \| \int_1^0f \|\). Pour \( t\leq t_0\) tout le calcul donne
+    \begin{equation}
+        \| \Phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big(  e^{k(t_0-t)}-1 \big).
+    \end{equation}
+    Les deux inéquations sont valables a fortiori en mettant des valeurs absolues dans l'exponentielle, de telle sorte que pour tout \( t\in K\) nous avons
+    \begin{equation}
+        e^{-k| t_0-t |}\| \phi(y)(t)-\Phi(z)(t) \|\leq \| y-z \|_k\big( 1- e^{-k| t_0-t |} \big).
+    \end{equation}
+    En prenant le supremum sur \( t\),
+    \begin{equation}
+        \| \Phi(y)-\Phi(z) \|_k\leq \| y-z \|_k(1- e^{-k\ell}),
+    \end{equation}
+    mais \( 0<(1- e^{e-k\ell})<1\), donc \( \Phi\) est contractante pour la norme \( \| . \|_k\). Vu que \( (E,\| . \|_k)\) est complet, l'application \( \Phi\) y a un unique point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+
+    Ce point fixe est donc l'unique solution de l'équation différentielle de départ.
+
+    \begin{subproof}
+        \item[Existence et unicité sur \( I\)]
+            Il nous reste à prouver que la solution que nous avons trouvée existe sur \( I\) : jusqu'à présent nous avons démontré l'existence et l'unicité sur n'importe quel compact dans \( I\).
+
+            Soit une suite croissante de compacts \( K_n\) contenant \( t_0\) (par exemple une suite exhaustive comme celle du lemme \ref{LemGDeZlOo}). Nous avons en particulier
+            \begin{equation}
+                I=\bigcup_{n=0}^{\infty}K_n.
+            \end{equation}
+        \item[Existence sur \( I\)]
+        
+            Soit \( y_n\) l'unique solution sur \( K_n\). Il suffit de poser
+            \begin{equation}
+                y(t)=y_n(t)
+            \end{equation}
+            pour \( n\) tel que \( t\in K_n\). Cette définition fonctionne parce que si \( t\in K_n\cap K_m\), il y a forcément un des deux qui est inclus à l'autre et le résultat d'unicité sur le plus grand des deux donne \( y_n(t)=y_m(t)\).
+
+        \item[Unicité sur \( I\)]
+
+            Soient \( y\) et \(z \) des solutions sur \( I\); vu que \( I\) n'est pas spécialement compact, le travail fait plus haut ne permet pas de conclure que \( y=z\). 
+
+            Soit \( t\in I\). Alors \( t\in K_n\) pour un certain \( n\) et \( y\) et \( z\) sont des solutions sur \( K_n\) qui est compact. L'unicité sur \( K_n\) donne \( y(t)=z(t)\).
+    \end{subproof}
+\end{proof}
+
+\begin{normaltext}
+    Il y a d'autres moyens de prouver qu'une solution existe globalement sur \( \eR\). Si \( f\) est globalement bornée, le théorème d'explosion en temps fini donne quelque garanties, voir \ref{NORMooZROGooZfsdnZ}.
+\end{normaltext}
+
+Le théorème suivant donne une version du théorème de Cauchy-Lipschitz lorsque la fonction \( f\) dépend d'un paramètre. Ce théorème n'utilise rien de fondamentalement nouveau. Nous le donnons seulement pour montrer que l'on peut choisir l'espace \( \mF\) de façon un peu maligne pour élargir le résultat. Si vous voulez un théorème de Cauchy-Lipschitz avec paramètre vraiment intéressant, allez voir le théorème \ref{PROPooPYHWooIZhQST}.
+
+\begin{theorem}[Cauchy-Lipschitz avec paramètre\cite{MonCerveau,ooXVPAooTQUIRw}]           \label{THOooDTCWooSPKeYu}
+    Soit un intervalle ouvert \( I\) de \( \eR\), un connexe ouvert \( \Omega\) de \( \eR^n\) et un intervalle ouvert \( \Lambda\) de \( \eR^d\). Soit une fonction \( f\colon I\times \Omega\times \Lambda\to \eR^n\) continue et localement Lipschitz en \( \Omega\). Soient \( t_0\in I\), \( y_0\in \Omega\) et \( \lambda_0\in \Lambda\). Il existe un voisinage compact de \( (t_0,y_0,\lambda_0)\) sur lequel le problème
+    \begin{subequations}
+        \begin{numcases}{}
+            y'_{\lambda}(t)=f\big( t,y_{\lambda}(t),\lambda \big)\\
+            y_{\lambda}(t_0)=y_0
+        \end{numcases}
+    \end{subequations}
+    possède une unique solution. De plus \( (t,\lambda)\mapsto y_{\lambda}(t)\) est continue\footnote{Ici, la surprise est que ce soit continu par rapport à \( \lambda\). Le fait qu'elle le soit par rapport à \( t\) est clair depuis le départ parce que c'est finalement rien d'autre que le Cauchy-Lipschitz vieux et connu.}.
+\end{theorem}
+
+\begin{proof}
+
+    \begin{probleme}
+        Ceci est une idée de la preuve. Je n'ai pas vérifié toutes les étapes. Soyez prudent.
+
+    \end{probleme}
+
+    D'abord nous avons un voisinage compact \( V\times \overline{ B(y_0,r) }\times \Lambda_0\) de \( (t_0,y_0,\lambda_0)\) sur lequel $f$ est bornée. Ensuite nous récrivons l'équation différentielle sous la forme
+    \begin{subequations}
+        \begin{numcases}{}
+            \frac{ \partial y }{ \partial t }(t,\lambda)=f\big( t,y(t,\lambda),\lambda \big)\\
+            y(t_0,\lambda)=y_0.
+        \end{numcases}
+    \end{subequations}
+    pour une fonction \( y\colon V\times \Lambda_0\to \eR^n\).
+
+    Nous posons \( \mF=C^0\big( V\times\Lambda_0 ,\eR^n\big)\) et nous y définissons l'application
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon \mF&\to \mF \\
+            \Phi(y)(t,\lambda)&=y_0+\int_{t_0}^tf\big( s,y(s,\lambda),\lambda \big)ds. 
+        \end{aligned}
+    \end{equation}
+    Il y a plein de vérifications à faire\cite{ooXVPAooTQUIRw}, mais je parie que \( \Phi\) est bien définie, et que une de ses puissances est une contraction de \( (\mF,\| . \|_{\infty})\). L'unique point fixe est une solution de notre problème et est dans \( C^0\), donc \( (t,\lambda)\mapsto y(t,\lambda)=y_{\lambda}(t)\) est de classe \( C^0\), c'est à dire continue.
+\end{proof}
+
+\begin{normaltext}
+    Ce théorème marque un peu la limite de ce que l'on peut faire avec la méthode des points fixes dans le cadre de Cauchy-Lipschitz : nous sommes limités à la continuité de la solution parce que les espaces \( C^p\) ne sont pas complets\footnote{Par exemple, le théorème de Stone-Weierstrass \ref{ThoGddfas} nous dit que la limite uniforme de polynômes (de classe \(  C^{\infty}\)) peut n'être que continue. Voir aussi le thème \ref{THMooOCXTooWenIJE}.}. Il n'y a donc pas d'espoir d'adapter la méthode pour prouver que si \( f\) est de classe \( C^p\) alors \( (t,\lambda)\mapsto y_{\lambda}(t)\) est de classe \( C^p\). On peut, à \( \lambda\) fixé prouver que \( t\mapsto y_{\lambda}(t)\) est de classe \( C^p\) (utiliser une récurrence), mais pas plus.
+
+    La régularité \( C^1\) de \( y\) par rapport à la condition initiale sera l'objet du théorème \ref{THOooSTHXooXqLBoT}. Ce résultat n'est vraiment pas facile et utilise des ingrédients bien autres qu'un point fixe. Ensuite la régularité \( C^p\) par rapport à la condition initiale et par rapport à un paramètre seront presque des cadeaux (proposition \ref{PROPooINLNooDVWaMn} et \ref{PROPooPYHWooIZhQST}).
+\end{normaltext}
+
+\begin{example}[\cite{ooSBHXooOMnaTC}]          \label{EXooJXIGooQtotMc}
+    Nous savons que le théorème de Picard permet de trouver le point fixe par itération de la contraction à partir d'un point quelconque. Tentons donc de résoudre
+    \begin{subequations}
+        \begin{numcases}{}
+            y'(t)=y(t)\\
+            y(0)=1
+        \end{numcases}
+    \end{subequations}
+    dont nous savons depuis l'enfance que la solution est l'exponentielle. Partons donc de la fonction constante \( y_0=1\), et appliquons la contraction \eqref{EQooJUTBooILBKoE} :
+    \begin{equation}
+        u_1=1+\int_0^1u_0(s)ds=1+t.
+    \end{equation}
+    Ensuite
+    \begin{equation}
+        u_2=1+\int_0^t(1+s)ds=1+t+\frac{ t^2 }{2}.
+    \end{equation}
+    Et on voit que les itérations suivantes vont donner l'exponentielle.
+
+    Nous sommes évidemment en droit de se dire que nous avons choisi un bon point de départ. Tentons le coup avec une fonction qui n'a rien à voir avec l'exponentielle : \( u_0(x)=\sin(x)\).
+
+    Le programme suivant permet de faire de belles investigations numériques en partant d'à peu près n'importe quelle fonction :
+
+\lstinputlisting{tex/sage/picard_exp.py}
+
+    Ce programme fait \( 30\) itérations depuis la fonction \( \sin(x)\) pour tenter d'approximer \( \exp(x)\). Pour donner une idée, après \( 7\) itérations nous avons la fonction suivante :
+    \begin{equation}
+        \frac{1}{ 60 }x^5+\frac{1}{ 24 }x^4+\frac{ 1 }{2}x^2+2x-\sin(x)+1.
+    \end{equation}
+    Nous voyons que les coefficients sont des factorielles, mais pas toujours celles correspondantes à la puissance, et qu'il manque certains termes par rapport au développement de l'exponentielle que nous connaissons. Bref, le polynôme qui se met en face de \( \sin(x)\) s'adapte tout seul pour compenser.
+
+    Et après \( 30\) itérations, ça donne quoi ? Voici un graphe de l'erreur entre \( u_{30}(x)\) et \( \exp(30)\) :
+    
+    
+\begin{center}
+   \input{auto/pictures_tex/Fig_XOLBooGcrjiwoU.pstricks}
+\end{center}
+
+    Pour donner une idée, \( \exp(10)\simeq 22000\). Donc il y a une faute de \( 0.01\) sur \( 22000\). Pas mal.
+
+\end{example}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de Cauchy-Arzella}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{theorem}[Cauchy-Arzela\cite{ClemKetl}]   \label{ThoHNBooUipgPX}
+    Nous considérons le système d'équation différentielles
+    \begin{subequations}        \label{EqTXlJdH}
+        \begin{numcases}{}
+            y'=f(t,y)\\
+            y(t_0)=y_0.
+        \end{numcases}
+    \end{subequations}
+    avec \( f\colon U\to \eR^n\), continue où \( U\) est ouvert dans \( \eR\times \eR^n\). Alors il existe un voisinage fermé \( V\) de \( t_0\) sur lequel une solution \( C^1\) du problème \eqref{EqTXlJdH} existe.
+\end{theorem}
+\index{théorème!Cauchy-Arzela}
+
+\begin{proof}[Idée de la démonstration]
+    Nous considérons \( M=\| f \|_{\infty}\) et \( K\), l'ensemble des fonctions \( M\)-Lipschitz sur \( U\). Nous prouvons que \( (K,\| . \|_{\infty})\) est compact. Ensuite nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon K&\to K \\
+            \Phi(f)(t)&=x_0+\int_{t_0}^tf\big( u,f(u) \big)du. 
+        \end{aligned}
+    \end{equation}
+    Après avoir prouvé que \( \Phi\) était continue, nous concluons qu'elle a un point fixe par le théorème de Schauder \ref{ThovHJXIU}.
+\end{proof}
+
+\begin{remark}
+    Quelque remarques.
+    \begin{enumerate}
+        \item
+    Les théorème de Cauchy-Lipschitz et Cauchy-Arzella donnent des existences pour des équations différentielles du type \( y'=f(t,y)\). Et si nous avons une équation du second ordre ? Alors il y a la méthode de la réduction de l'ordre qui permet de transformer une équation différentielle d'ordre élevé en un système d'ordre \( 1\).
+\item
+    Ces théorèmes posent des \emph{conditions initiales} : la valeur de \( y\) est donnée en un point, et la méthode de la réduction de l'ordre permet de donner l'existence de solutions d'un problème d'ordre \( k\) en donnant les valeurs de \( y(0)\), \( y'(0)\), \ldots \( y^{(k-1)}(0)\). C'est à dire de la fonction et de ses dérivées en un point. Rien n'est dit sur l'existence de \emph{conditions aux bords}.
+    \end{enumerate}
+    Ces deux points sont illustrés dans les exemples \ref{EXooSHMMooHVfsMB} et \ref{EXooJNOMooYqUwTZ}.
+\end{remark}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+                    \section{Théorèmes d'inversion locale et de la fonction implicite}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Mise en situation}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Dans un certain nombre de situation, il n'est pas possible de trouver des solutions explicites aux équations qui apparaissent. Néanmoins, l'existence «théorique» d'une telle solution est souvent déjà suffisante. C'est l'objet du théorème de la fonction implicite.
+
+Prenons par exemple la fonction sur $\eR^2$ donnée par 
+\begin{equation}
+    F(x,y)=x^2+y^2-1.
+\end{equation}
+Nous pouvons bien entendu regarder l'ensemble des points donnés par $F(x,y)=0$. C'est le cercle dessiné à la figure \ref{LabelFigCercleImplicite}.
+\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite. Si on donne \( x\), nous ne pouvons pas savoir si nous parlons de \( P\) ou de \( P'\).}
+\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
+
+%\ref{LabelFigCercleImplicite}.
+%\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite.}
+%\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
+
+Nous ne pouvons pas donner le cercle sous la forme $y=y(x)$ à cause du $\pm$ qui arrive quand on prend la racine carrée. Mais si on se donne le point $P$, nous pouvons dire que \emph{autour de $P$}, le cercle est la fonction
+\begin{equation}
+    y(x)=\sqrt{1-x^2}.
+\end{equation}
+Tandis que autour du point $P'$, le cercle est la fonction
+\begin{equation}
+    y(x)=-\sqrt{1-x^2}.
+\end{equation}
+Autour de ces deux point, donc, le cercle est donné par une fonction. Il n'est par contre pas possible de donner le cercle autour du point $Q$ sous la forme d'une fonction.
+
+Ce que nous voulons faire, en général, est de voir si l'ensemble des points tels que
+\begin{equation}
+    F(x_1,\ldots,x_n,y)=0
+\end{equation}
+peut être donné par une fonction $y=y(x_1,\ldots,x_n)$. En d'autre termes, est-ce qu'il existe une fonction $y(x_1,\ldots,x_n)$ telle que
+\begin{equation}
+    F\big( x_1,\ldots,x_n,y(x_1,\ldots,x_n)\big)=0.
+\end{equation}
+
+Plus généralement, soit une fonction
+\begin{equation}
+    \begin{aligned}
+        F\colon D\subset \eR^n\times \eR^m&\to \eR^m \\
+        (x,y)&\mapsto \big( F_1(x,y),\ldots, F_m(x,y) \big) 
+    \end{aligned}
+\end{equation}
+avec $x = (x_1,\ldots, x_n)$ et $y = (y_1,\ldots,y_m)$. Pour chaque $x$ fixé, on s'intéresse aux solutions du système de $m$ équations $F(x,y) = 0$ pour les inconnues $y$ ; en particulier, on voudrait pouvoir écrire $y = \varphi(x)$ vérifiant $F(x,\varphi(x)) = 0$.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème d'inversion locale}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma} \label{LemGZoqknC}
+    Soit \( E\) un espace de Banach (métrique complet) et \( \mO\) un ouvert de \( E\). Nous considérons une \( \lambda\)-contraction \( \varphi\colon \mO\to E\). Alors l'application
+    \begin{equation}
+    f\colon x\mapsto x+\varphi(x)
+    \end{equation}
+    est un homéomorphisme entre \( \mO\) et un ouvert de \( E\). De plus \( f^{-1}\) est Lipschitz de constante plus petite ou égale à \( (1-\lambda)^{-1}\).
+\end{lemma}
+Cette proposition utilise le théorème de point fixe de Picard \ref{ThoEPVkCL},
+et sera utilisée pour démontrer le théorème d'inversion locale \ref{ThoXWpzqCn}.
+% note que garder deux lignes ici est important pour vérifier les références vers le futur : la seconde ligne peut être ignorée, pas la seconde.
+
+\begin{proof}
+        Soient \( x_1,x_2\in\mO\). Nous posons \( y_1=f(x_1)\) et \( y_2=f(x_2)\). En vertu de l'inégalité de la proposition \ref{PropNmNNm} nous avons
+        \begin{subequations}    \label{subEqEBJsBfz}
+            \begin{align}
+            \big\| f(x_2)-f(x_1) \big\|&=\big\| x_2+\varphi(x_2)-x_1-\varphi(x_1) \big\|\\
+        &\geq \Big|        \| x_2-x_1 \|-\big\| \varphi(x_2)-\varphi(x_1) \big\|  \Big|\\
+    &\geq   (1-\lambda)\| x_2-x_1 \|.
+            \end{align}
+        \end{subequations}
+        À la dernière ligne les valeurs absolues sont enlevées parce que nous savons que ce qui est à l'intérieur est positif. Cela nous dit d'abord que \( f\) est injective parce que \( f(x_2)=f(x_1)\) implique \( x_2=x_1\). Donc \( f\) est inversible sur son image. Nous posons \( A=f(\mO)\) et nous devons prouver que que \( f^{-1}\colon A\to \mO\) est continue, Lipschitz de constante majorée par \( (1-\lambda)^{-1}\) et que \( A\) est ouvert.
+
+    Les inéquations \eqref{subEqEBJsBfz} nous disent que
+    \begin{equation}
+    \big\| f^{-1}(y_1)-f^{-1}(y_2) \big\|\leq \frac{ \| y_1-y_2 \| }{ 1-\lambda },
+    \end{equation}
+    c'est à dire que
+    \begin{equation}
+        f^{-1}\big( B(y,r) \big)\subset B\big( f^{-1}(y),\frac{ r }{ 1-\lambda } \big),
+    \end{equation}
+    ce qui signifie que \( f^{-1}\) est Lipschitz de constante souhaitée et donc continue.
+
+    Il reste à prouver que \( f(\mO)\) est ouvert. Pour cela nous prenons \( y_0=f(x_0)\) dans \( f(\mO)\) est nous prouvons qu'il existe \( \epsilon\) tel que \( B(y_0,\epsilon)\) soit dans \( f(\mO)\). Il faut donc que pour tout \( y\in B(y_0,\epsilon)\), l'équation \( f(x)=y\) ait une solution. Nous considérons l'application
+    \begin{equation}
+        L_y\colon x\mapsto y-\varphi(x).
+    \end{equation}
+    Ce que nous cherchons est un point fixe de \( L_y\) parce que si \( L_y(x)=x\) alors \( y=x+\varphi(x)=f(x)\). Vu que
+    \begin{equation}
+        \big\| L_y(x)-L_y(x') \big\|=\big\| \varphi(x)-\varphi(x') \big\|\leq\lambda\| x-x' \|,
+    \end{equation}
+    l'application \( L_y\) est une contraction de constante \( \lambda\). Par ailleurs \( x_0\) est un point fixe de \( L_{y_0}\), donc en vertu de la caractérisation \eqref{EqDZvtUbn} des fonctions Lipschitziennes, 
+    \begin{equation}
+        L_{y_0}\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B\big( L_{y_0}(x_0),\lambda\delta \big) }=\overline{ B(x_0,\lambda\delta) }.
+    \end{equation}
+    Vu que pour tout \( y\) et \( x\) nous avons \( L_y(x)=L_{y_0}(x)+y-y_0\),
+    \begin{equation}
+    L_y\big( \overline{ B(x_0,\delta) } \big)=L_{y_0}\big( \overline{ B(x_0,\delta) } \big)+(y-y_0)\subset \overline{ B(x_0,\lambda\delta) }+(y-y_0)\subset \overline{ B(x_0),\lambda\delta+\| y-y_0 \| }.
+    \end{equation}
+    Si \( \epsilon<(1-\lambda)\delta\) alors \( \lambda\delta+\| y-y_0 \|<\delta\). Un tel choix de \( \epsilon>0\) est possible parce que \( \lambda<1\). Pour une telle valeur de \( \epsilon\) nous avons
+    \begin{equation}
+        L_y\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B(x_0,\delta) }.
+    \end{equation}
+    Par conséquent \( L_y\) est une contraction sur l'espace métrique complet \( \overline{ B(x_0,\delta) }\), ce qui signifie que \( L_y\) y possède un point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+\end{proof}
+
+Le théorème d'inversion locale s'énonce de la façon suivante dans \( \eR^n\) :
+\begin{theorem}[Inversion locale dans \( \eR^n\)]    \label{THOooQGGWooPBRNEX}      % Ne pas mettre de label ici parce qu'il faut référencer l'autre, celui dans Banach.
+    Soit \( f\in C^k(\eR^n,\eR^n)\) et \( x_0\in \eR^n\). Si \( df_{x_0}\) est inversible, alors il existe un voisinage ouvert \( U\) de \( x_0\) et \( V\) de \( f(x_0)\) tels que \( f\colon U\to V\) soit un \( C^k\)-difféomorphisme. (c'est à dire que \( f^{-1}\) est également de classe \( C^k\))
+\end{theorem}
+
+Nous allons le démontrer dans le cas un peu plus général (mais pas plus cher\footnote{Sauf la justification de la régularité de l'application \( A\mapsto A^{-1}\)}) des espaces de Banach en tant que conséquence du théorème de point fixe de Picard \ref{ThoEPVkCL}.
+
+\begin{theorem}[Inversion locale dans un espace de Banach\cite{OWTzoEK}] \label{ThoXWpzqCn}
+    Soit une fonction \( f\in C^p(E,F)\) avec \( p\geq 1\) entre deux espaces de Banach. Soit \( x_0\in E\) tel que \( df_{x_0}\) soit une bijection bicontinue\footnote{En dimension finie, une application linéaire est toujours continue et d'inverse continu.}. Alors il existe un voisinage ouvert \( V\) de \( x_0\) et \( W\) de \( f(x_0)\) tels que
+    \begin{enumerate}
+        \item
+        \( f\colon V\to W\) soit une bijection,
+    \item
+        \( f^{-1}\colon W\to V\) soit de classe \( C^p\).
+    \end{enumerate}
+\end{theorem}
+\index{application!différentiable}
+\index{théorème!inversion locale}
+
+\begin{proof}
+    Nous commençons par simplifier un peu le problème. Pour cela, nous considérons la translation \( T\colon x\mapsto x+x_0 \) et l'application linéaire
+    \begin{equation}
+        \begin{aligned}
+            L\colon \eR^n&\to \eR^n \\
+            x&\mapsto (df_{x_0})^{-1}x
+        \end{aligned}
+    \end{equation}
+    qui sont tout deux des difféomorphismes (\( L\) en est un par hypothèse d'inversibilité). Quitte à travailler avec la fonction \( k=L\circ f\circ T\), nous pouvons supposer que \( x_0=0\) et que \( df_{x_0}=\mtu\). Pour comprendre cela il faut utiliser deux fois la formule de différentielle de fonction composée de la proposition \ref{EqDiffCompose} :
+    \begin{equation}
+        dk_0(u)=dL_{(f\circ T)(0)}\Big( df_{T(0)}dT_0(u) \Big).
+    \end{equation}
+    Vu que \( L\) est linéaire, sa différentielle est elle-même, c'est à dire \( dL_{(f\circ T)(0)}=(df_{x_0})^{-1}\), et par ailleurs \( dT_0=\mtu\), donc
+    \begin{equation}
+        dk_0(u)=(df_{x_0})^{-1}\Big( df_{x_0}(u) \Big)=u,
+    \end{equation}
+    ce qui signifie bien que \( dk_0=\mtu\). Pour tout cela nous avons utilisé en plein le fait que \( df_{x_0}\) était inversible.
+
+Nous posons \( g=f-\mtu\), c'est à dire \( g(x)=f(x)-x\), qui a la propriété \( dg_0=0\). Étant donné que \( g\) est de classe \( C^1\), l'application\footnote{Ici \( \GL(F)\) est l'ensemble des applications linéaires, inversibles et continues de \( F\) dans lui-même. Ce ne sont pas spécialement des matrices parce que nous n'avons pas d'hypothèses sur la dimension de \( F\), finie ou non.}
+    \begin{equation}
+        \begin{aligned}
+            dg\colon E&\to \GL(F) \\
+            x&\mapsto dg_x 
+        \end{aligned}
+    \end{equation}
+    est continue. En conséquence de quoi nous avons un voisinage \( U'\) de \( 0 \) pour lequel
+    \begin{equation}    \label{EqSGTOfvx}
+        \sup_{x\in U'}\| dg_x \|<\frac{ 1 }{2}.
+    \end{equation}
+    Maintenant le théorème des accroissements finis \ref{ThoNAKKght} (\ref{val_medio_2} pour la dimension finie) nous indique que pour tout \( x,x'\in U'\) nous avons\footnote{Ici nous supposons avoir choisi \( U'\) convexe afin que tous les \( a\in \mathopen[ x , x' \mathclose]\) soient bien dans \( U'\) et donc soumis à l'inéquation \eqref{EqSGTOfvx}, ce qui est toujours possible, il suffit de prendre une boule.}
+    \begin{equation}
+        \| g(x')-g(x) \|\leq \sup_{a\in\mathopen[ x , x' \mathclose]}\| dg_a \| \cdot \| x-x' \|\leq \frac{ 1 }{2}\| x-x' \|,
+    \end{equation}
+    ce qui prouve que \( g\) est une contraction au moins sur l'ouvert \( U'\). Nous allons aussi donner une idée de la façon dont \( f\) fonctionne : si \( x_1,x_2\in U'\) alors
+    \begin{subequations}
+        \begin{align}
+            \| x_1-x_2 \|&=\| g(x_1)-f(x_1)-g(x_2)+f(x_2) \| \\
+            &\leq \| g(x_1)-g(x_2) \|+\| f(x_1)-f(x_2) \|\\
+            &\leq \frac{ 1 }{2}\| x_1-x_2 \|+\| f(x_1)-f(x_2) \|,
+        \end{align}
+    \end{subequations}
+    ce qui montre que
+    \begin{equation}
+        \| x_1-x_2 \|\leq 2\| f(x_1)-f(x_2) \|.
+    \end{equation}
+    Maintenant que nous savons que \( g\) est contractante de constante \( \frac{ 1 }{2}\) et que \( f=g+\mtu\) nous pouvons utiliser la proposition \ref{LemGZoqknC} pour conclure que \( f\) est un homéomorphisme sur un ouvert \( U\) (partie de \( U'\)) de \( E\) et \( f^{-1}\) a une constante de Lipschitz plus petite ou égale à \( (1-\frac{ 1 }{2})^{-1}=2\).
+
+    Nous allons maintenant prouver que \( f^{-1}\) est différentiable et que sa différentielle est donnée par \( (df^{-1})_{f(x)}=(df_x)^{-1}\).
+
+    Soient \( a,b\in U\) et \( u=b-a\). Étant donné que \( f\) est différentiable en \( a\), il existe une fonction \( \alpha\in o(\| u \|)\) telle que
+    \begin{equation}
+        f(b)-f(a)-df_a(u)=\alpha(u).
+    \end{equation}
+    En notant \( y_a=f(a)\) et \( y_b=f(b)\) et en appliquant \( (df_a)^{-1}\) à cette dernière équation,
+    \begin{equation}
+        (df_a)^{-1}(y_b-y_a)-u=(df_a)^{-1} \big( \alpha(u) \big).
+    \end{equation}
+    Vu que \( df_a\) est bornée (et son inverse aussi), le membre de droite est encore une fonction \( \beta\) ayant la propriété \( \lim_{u\to 0}\beta(u)/\| u \|=0\); en réordonnant les termes,
+    \begin{equation}
+        b-a=(df_a)^{-1}(y_b-y_a)+\beta(u)
+    \end{equation}
+    et donc
+    \begin{equation}
+        f^{-1}(y_b)-f^{-1}(y_a)-(df_a)^{-1}(y_b-y_a)=\beta(u),
+    \end{equation}
+    ce qui prouve que \( f^{-1}\) est différentiable et que \( (df^{-1})_{y_a}=(df_a)^{-1}\).
+
+    La différentielle \( df^{-1}\) est donc obtenue par la chaine
+    \begin{equation}
+    \xymatrix{%
+        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}     &   U'\ar[r]^-{df}&\GL(F)\ar[r]^-{\Inv}&\GL(F)
+       }
+    \end{equation}
+    où l'application \( \Inv\colon \GL(F)\to \GL(F)\) est l'application \( X\mapsto X^{-1}\) qui est de classe \(  C^{\infty}\) par le théorème \ref{ThoCINVBTJ}. D'autre part, par hypothèse \( df\) est une application de classe \( C^{k-1}\) et donc au minimum \( C^0\) parce que \( k\geq 1\). Enfin, l'application \( f^{-1}\colon f(U)\to U\) est continue (parce que la proposition \ref{LemGZoqknC} précise que \( f\) est un homéomorphisme). Donc toute la chaine est continue et \( df^{-1}\) est continue. Cela entraine immédiatement que \( f^{-1}\) est \( C^1\) et donc que toute la chaine est \( C^1\).
+
+    Par récurrence nous obtenons la chaine
+    \begin{equation}
+    \xymatrix{%
+        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}_-{C^{k-1}}     &   U'\ar[r]^-{df}_-{C^{k-1}}&\GL(F)\ar[r]^-{\Inv}_-{ C^{\infty}}&\GL(F)
+       }
+    \end{equation}
+    qui prouve que \( df^{-1}\) est \( C^{k-1} \) et donc que \( f^{-1}\) est \( C^k\). La récurrence s'arrête ici parce que \( df\) n'est pas mieux que \( C^{k-1}\).
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de la fonction implicite}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous énonçons et le démontrons le théorème de la fonction implicite dans le cas d'espaces de Banach.
+\begin{theorem}[Théorème de la fonction implicite dans Banach\cite{SNPdukn}] \label{ThoAcaWho}
+    Soient \( E\), \( F\) et \( G\) des espaces de Banach et des ouverts \( U\subset E\), \( V\subset F\). Nous considérons une fonction \( f\colon U\times V\to G\) de classe \( C^r\) telle que\footnote{La notation \( d_y\) est la différentielle partielle de la définition \ref{VJM_CtSKT}.}
+    \begin{equation}
+        d_yf_{(x_0,y_0)}\colon F\to G
+    \end{equation}
+    soit un isomorphisme pour un certain \( (x_0,y_0)\in U\times V\).
+
+    Alors nous avons des voisinages \( U_0\) de \( x_0\) dans \( E\) et \( W_0\) de \( f(x_0,y_0)\) dans \( G\) et une fonction de classe \( C^r\) 
+    \begin{equation}
+        g\colon U_0\times W_0\to V
+    \end{equation}
+    telle que 
+    \begin{equation}
+        f\big( x,g(x,w) \big)=w
+    \end{equation}
+    pour tout \( (x,w)\in U_0\times W_0\).
+    
+    Cette fonction \( g\) est unique au sens suivant : il existe un voisinage \( V_0 \) de \( y_0\) tel que si \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) satisfont à \( f(x,y)=w\) alors \( y=g(x,w)\). Autrement dit, la fonction \( g\colon U_0\times W_0\to V_0\) est unique.
+\end{theorem}
+\index{théorème!fonction implicite dans Banach}
+
+\begin{proof}
+    Nous commençons par considérer la fonction
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon U\times V&\to E\times G \\
+            (x,y)&\mapsto \big( x,f(x,y) \big)
+        \end{aligned}
+    \end{equation}
+    et sa différentielle 
+    \begin{subequations}
+        \begin{align}
+            d\Phi_{(x_0,y_0)}(u,v)&=\Dsdd{ \big( x_0+tu,f(x_0+tu,y_0+tv) \big) }{t}{0}\\
+            &=\left( \Dsdd{ x_0+tu }{t}{0},\Dsdd{ f(x_0+tu,y_0+tv) }{t}{0} \right)\\
+            &=\left( u,df_{(x_0,y_0)}(u,v) \right).
+        \end{align}
+    \end{subequations}
+    Nous utilisons alors la proposition \ref{PropLDN_nHWDF} pour conclure que
+    \begin{equation}
+        d\Phi_{(x_0,y_0)}(u,v)=\big( u,(d_1f)_{(x_0,y_0)}(u)+(d_2f)_{(x_0,y_0)}(v) \big),
+    \end{equation}
+    mais comme par hypothèse \( (d_2f)_{(x_0,y_0)}\colon F\to G\) est un isomorphisme, l'application \( d\Phi_{(x_0,y_0)}\colon E\times F\to E\times G\) est également un isomorphisme. Par conséquent le théorème d'inversion locale \ref{ThoXWpzqCn} nous indique qu'il existe un voisinage \( \mO\) de \( (x_0,y_0)\) et \( \mP\) de \( \Phi(x_0,y_0)\) tels que \( \Phi\colon \mO\to \mP\) soit une bijection et \( \Phi^{-1}\colon \mP\to \mO\) soit de classe \( C^r\). Vu que \( \mP\) est un voisinage de
+    \begin{equation}
+        \Phi(x_0,y_0)=\big( x_0,f(x_0,y_0) \big),
+    \end{equation}
+    nous pouvons par \ref{PropDXR_KbaLC} le choisir un peu plus petit de telle sorte à avoir \( \mP=U_0\times W_0\) où \( U_0\) est un voisinage de \( x_0\) et \( W_0\) un voisinage de \( f(x_0,y_0)\). Dans ce cas nous devons obligatoirement aussi restreindre \( \mO\) à \( U_0\times V_0\) pour un certain voisinage \( V_0\) de \( y_0\). L'application \( \Phi^{-1}\) a obligatoirement la forme
+    \begin{equation}    \label{EqMHT_QrHRn}
+        \begin{aligned}
+            \Phi^{-1}\colon U_0\times W_0&\to U_0\times V_0 \\
+            (x,w)&\mapsto \big( x,g(x,w) \big) 
+        \end{aligned}
+    \end{equation}
+    pour une certaine fonction \( g\colon U_0\times W_0\to V\). Cette fonction \( g\) est la fonction cherchée parce qu'en appliquant \( \Phi\) à \eqref{EqMHT_QrHRn}, 
+    \begin{equation}
+        (x,w)=\Phi\big( x,g(x,w) \big)=\Big( x,f\big( x,g(x,w) \big) \Big),
+    \end{equation}
+    qui nous dit que pour tout \( x\in U_0\) et tout \( w\in W_0\) nous avons
+    \begin{equation}
+        f\big( x,g(x,w) \big)=w.
+    \end{equation}
+
+    Si vous avez bien suivi le sens de l'équation \eqref{EqMHT_QrHRn} alors vous avez compris l'unicité. Sinon, considérez \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) tels que \( f(x,y)=w\). Alors \( \big( x,f(x,y) \big)=(x,w)\) et 
+    \begin{equation}
+        \Phi(x,y)=(x,w).
+    \end{equation}
+    Mais vu que \( \Phi\colon U_0\times V_0\to U_0\times W_0\) est une bijection, cette relation définit de façon univoque l'élément \( (x,y)\) de \( U_0\times V_0\), qui ne sera autre que \( g(x,w)\).
+\end{proof}
+
+Le théorème de la fonction implicite s'énonce de la façon suivante pour des espaces de dimension finie.
+% Attention : avant de citer ce théorème, voir s'il est suffisant. Ici \varphi a une variable; dans l'autre énoncé il en a deux.
+\begin{theorem}[Théorème de la fonction implicite en dimension finie]   \label{ThoRYN_jvZrZ}
+    Soit une fonction \( F\colon \eR^n\times \eR^m\to \eR^m\) de classe \( C^k\) et \( (\alpha,\beta)\in \eR^n\times \eR^m\) tels que
+    \begin{enumerate}
+        \item
+            \( F(\alpha,\beta)=0\),
+        \item
+            \( \frac{ \partial (F_1,\ldots, F_m) }{ \partial (y_1,\ldots, y_m) }\neq 0\), c'est à dire que \( (d_yF)_{(\alpha,\beta)} \) est inversible.
+    \end{enumerate}
+    Alors il existe un voisinage ouvert \( V\) de \( \alpha\) dans \( \eR^n\), un voisinage ouvert \( W\) de \( \beta\) dans \( \eR^m\) et une application \( \varphi\colon V\to W\) de classe \( C^k\)  telle que pour tout \( x\in V\) on ait
+    \begin{equation}
+        F\big( x,\varphi(x) \big)=0.
+    \end{equation}
+    De plus si \( (x,y)\in V\times W\) satisfait à \( F(x,y)=0\), alors \( y=\varphi(x)\).
+\end{theorem}
+\index{théorème!fonction implicite dans \( \eR^n\)}
+
+\begin{remark}\label{RemPYA_pkTEx}
+    Notons que cet énoncé est tourné un peu différemment en ce qui concerne le nombre de variables dont dépend la fonction implicite : comparez
+    \begin{subequations}
+        \begin{align}
+            f\big( x,g(x,w) \big)=w\\
+            F\big( x,\varphi(x) \big)=0.
+        \end{align}
+    \end{subequations}
+    Le deuxième est un cas particulier du premier en posant 
+    \begin{equation}
+        F(x,y)=f(x,y)-f(x_0,y_0)
+    \end{equation}
+    et donc en considérant \( w\) comme valant la constante \( f(x_0,y_0)\); dans ce cas la fonction \( g\) ne dépend plus que de la variable \( x\).
+
+\end{remark}
+
+\begin{example}
+    La remarque \ref{RemPYA_pkTEx} signifie entre autres que le théorème \ref{ThoAcaWho} est plus fort que \ref{ThoRYN_jvZrZ} parce que le premier permet de choisir la valeur d'arrivée. Parlons de l'exemple classique du cercle et de la fonction \( f(x,y)=x^2+y^2\). Nous savons que
+    \begin{equation}
+        f(\alpha,\beta)=1.
+    \end{equation}
+    Alors le théorème \ref{ThoAcaWho} nous donne une fonction \( g\) telle que
+    \begin{equation}
+        f(x,g(x,r))=r
+    \end{equation}
+    tant que \( x\) est proche de \( \alpha\), que \( r\) est proche de \( 1\) et que \( g\) donne des valeurs proches de \( \beta\).
+
+    L'énoncé \ref{ThoRYN_jvZrZ} nous oblige à travailler avec la fonction \( F(x,y)=x^2+y^2-1\), de telle sorte que
+    \begin{equation}
+        F(\alpha,\beta)=0,
+    \end{equation}
+    et que nous ayons une fonction \( \varphi\) telle que
+    \begin{equation}
+        F(x,\varphi(x))=0.
+    \end{equation}
+    La fonction \( \varphi\) ne permet donc que de trouver des points sur le cercle de rayon \( 1\).
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Exemple}
+%---------------------------------------------------------------------------------------------------------------------------
+    
+Le théorème de la fonction implicite a pour objet de donner l'existence de la fonction $\varphi$. Maintenant nous pouvons dire beaucoup de choses sur les dérivées de $\varphi$ en considérant la fonction
+\begin{equation}
+    x\mapsto F\big( x,\varphi(x) \big).
+\end{equation}
+Par définition de $\varphi$, cette fonction est toujours nulle. En particulier, nous pouvons dériver l'équation
+\begin{equation}
+    F\big( x,\varphi(x) \big)=0,
+\end{equation}
+et nous trouvons plein de choses.
+
+
+Prenons par exemple la fonction
+\begin{equation}
+    F\big( (x,y),z \big)=ze^z-x-y,
+\end{equation}
+et demandons nous ce que nous pouvons dire sur la fonction $z(x,y)$ telle que
+\begin{equation}
+    F\big( x,y,z(x,y) \big)=0,
+\end{equation}
+c'est à dire telle que
+\begin{equation}        \label{EqDefZImplExemple}
+    z(x,y) e^{z(x,y)}-x-y=0.
+\end{equation}
+pour tout $x$ et $y\in\eR$. Nous pouvons facilement trouver $z(0,0)$ parce que
+\begin{equation}
+    z(0,0) e^{z(0,0)}=0,
+\end{equation}
+donc $z(0,0)=0$.
+
+Nous pouvons dire des choses sur les dérivées de $z(x,y)$. Voyons par exemple $(\partial_xz)(x,y)$. Pour trouver cette dérivée, nous dérivons la relation \eqref{EqDefZImplExemple} par rapport à $x$. Ce que nous trouvons est
+\begin{equation}
+    (\partial_xz)e^z+ze^z(\partial_xz)-1=0.
+\end{equation}
+Cette équation peut être résolue par rapport à $\partial_xz$~:
+\begin{equation}
+    \frac{ \partial z }{ \partial x }(x,y)=\frac{1}{ e^z(1+z) }.
+\end{equation}
+Remarquez que cette équation ne donne pas tout à fait la dérivée de $z$ en fonction de $x$ et $y$, parce que $z$ apparaît dans l'expression, alors que $z$ est justement la fonction inconnue. En général, c'est la vie, nous ne pouvons pas faire mieux.
+
+Dans certains cas, on peut aller plus loin. Par exemple, nous pouvons calculer cette dérivée au point $(x,y)=(0,0)$ parce que $z(0,0)$ est connu :
+\begin{equation}
+    \frac{ \partial z }{ \partial x }(0,0)=1.
+\end{equation}
+Cela est pratique pour calculer, par exemple, le développement en Taylor de $z$ autour de $(0,0)$.
+
+\begin{example}
+    Est-ce que l'équation \( e^{y}+xy=0\) définit au moins localement une fonction \( y(x)\) ? Nous considérons la fonction
+    \begin{equation}
+        f(x,y)=\begin{pmatrix}
+            x    \\ 
+            e^{y}+xy    
+        \end{pmatrix}
+    \end{equation}
+    La différentielle de cette application est
+    \begin{equation}
+            df_{(0,0)}(u)=\frac{ d }{ dt }\Big[ f(tu_1,tu_2) \Big]_{t=0}
+            =\frac{ d }{ dt }\begin{pmatrix}
+                tu_1    \\ 
+                e^{tu_2}+t^2u_1u_2    
+            \end{pmatrix}_{t=0}
+            =\begin{pmatrix}
+                u_1    \\ 
+                u_2    
+            \end{pmatrix}.
+    \end{equation}
+    L'application \( f\) définit donc un difféomorphisme local autour des points \( (x_0,y_0)\) et \( f(x_0,y_0)\). Soit \( (u,0)\) un point dans le voisinage de \( f(x_0,y_0)\). Alors il existe un unique \( (x,y)\) tel que
+    \begin{equation}
+        f(x,y)=\begin{pmatrix}
+               x \\ 
+            e^y+xy    
+        \end{pmatrix}=
+        \begin{pmatrix}
+            u    \\ 
+                0
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons automatiquement \( x=u\) et \( e^y+xy=0\). Notons toutefois que pour que ce procédé donne effectivement une fonction implicite \( y(x)\) nous devons avoir des points de la forme \( (u,0)\) dans le voisinage de \( f(x_0,y_0)\).
+\end{example}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Décomposition polaire (régularité)}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{normaltext}      \label{NomDJMUooTRUVkS}
+    Nous allons montrer que l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
+            A&\mapsto \sqrt{A} 
+        \end{aligned}
+    \end{equation}
+    est une difféomorphisme.
+
+    Cependant \( S^{++}(n,\eR)\) n'est pas un ouvert de \( \eM(n,\eR)\) et nous ne savons pas ce qu'est la différentielle d'une application non définie sur un ouvert. Nous allons donc en réalité montrer que l'application racine carré existe sur un voisinage de chacun des points de \( S^{++}(n,\eR)\). Et comme une union quelconque d'ouverts est un ouvert, la fonction \( f\) sera bien définie sur un ouvert de \( \eM(n,\eR)\).
+\end{normaltext}
+
+\begin{lemma}       \label{LemLBFOooDdNcgy}
+    L'application 
+    \begin{equation}
+        \begin{aligned}
+            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
+            A&\mapsto A^2 
+        \end{aligned}
+    \end{equation}
+    est un \(  C^{\infty}\)-difféomorphisme.
+\end{lemma}
+
+\begin{proof}
+    Prouvons d'abord que \( f\) prend ses valeurs dans \( S^{++}(n,\eR)\). Si \( A\in S^{++}(n,\eR)\) alors par la diagonalisation \ref{ThoeTMXla} elle s'écrit \( A=QDQ^{-1}\) où \( D\) est diagonale avec des nombres strictement positifs sur la diagonale. Avec cela, \( A^2=QD^2Q^{-1}\) où \( D^2\) contient encore des nombres strictement positifs sur la diagonale.
+
+    L'application \( f\) étant essentiellement des polynôme en les entrées de \( A\), elle est de classe \( C^{\infty}\).
+
+    Passons à l'étude de la différentielle. Comme mentionné en \ref{NomDJMUooTRUVkS} nous allons en réalité voir \( f\) sur un ouvert de \( \eM(n,\eR)\) autour de \( A\in S^{++}(n,\eR)\). Par conséquent si \( A\in S^{++}(n,\eR)\),
+    \begin{subequations}
+        \begin{align}
+            df\colon S^{++}(n,\eR)&\to \aL\big( \eM(n,\eR),\eM(n,\eR) \big)\\
+            df_A\colon \eM(n,\eR)&\to \eM(n,\eR).
+        \end{align}
+    \end{subequations}
+    Le calcul de \( df_A\) est facile. Soit \( u\in \eM(n,\eR)\) et faisons le calcul en utilisant la formule du lemme \eqref{LemdfaSurLesPartielles} :
+    \begin{subequations}
+        \begin{align}
+            df_A(u)&=\Dsdd{ f(A+tu) }{t}{0}\\
+            &=\Dsdd{ A^2+tAu+tuA+t^2u^2 }{t}{0}\\
+            &=Au+uA.
+        \end{align}
+    \end{subequations}
+    Nous allons utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} à la fonction \( f\). Dans la suite, \( A\) est une matrice de \( S^{++}(n,\eR)\).
+
+    \begin{subproof}
+        \item[\( df_A\) est injective]
+            Soit \( M\in \eM(n,\eR)\) dans le noyau de \( df_A\). En posant \( M'=A^{-1}MQ\) nous avons \( M=QM'Q^{-1}\) et on applique \( df_A\) à \( QM'Q^{-1}\) :
+            \begin{equation}
+                df_A(QM'Q^{-1})=Q\big( DM+MD \big)Q^{-1}.
+            \end{equation}
+            où \( D=\begin{pmatrix}
+                \lambda_1    &       &       \\
+                    &   \ddots    &       \\
+                    &       &   \lambda_n
+                \end{pmatrix}\) avec \( \lambda_i>0\). La matrice \( D\) est inversible. Nous avons \( M'=-DM'D^{-1}\), et en coordonnées,
+                \begin{subequations}
+                    \begin{align}
+                        M'_{ij}&=-\sum_{kl}D_{ikM'_{kl}}D^{-1}_{lj}\\
+                        &=-\sum_{kl}\lambda_i\delta_{ik}M'_{kl}\frac{1}{ \lambda_j }\delta_{lj}\\
+                        &=-\frac{ \lambda_i }{ \lambda_i }M'_{ij}.
+                    \end{align}
+                \end{subequations}
+                C'est à dire que \( M'_{ij}=-\frac{ \lambda_i }{ \lambda_j }M'_{ij}\) avec \( -\frac{ \lambda_i }{ \lambda_j }<0\). Cela implique \( M'=0\) et par conséquent \( M=0\).
+            \item[\( df_A\) est surjective]
+                Soit \( N\in \eM(n,\eR)\); nous cherchons \( M\in \eM(n,\eR)\) tel que \( df_A(M)=N\). Nous posons \( N'=Q^{-1} NQ\) et \( M=QM'Q^{-1}\), ce qui nous donne à résoudre \( df_D(M')=N'\). Passons en coordonnées :
+                \begin{subequations}
+                    \begin{align}
+                        (DM'+M'D)_{ij}&=\sum_k(\delta_{ik}\lambda_iM'_{kj}+M'_{ik}\delta_{kj}\lambda_j)\\&
+                        &=M'_{ij}(\lambda_i+\lambda_j)
+                    \end{align}
+                \end{subequations}
+                où \( \lambda_i+\lambda_j\neq 0\). Il suffit donc de prendre la matrice \( M'\) donnée par
+                \begin{equation}
+                    M'_{ij}=\frac{1}{ \lambda_i+\lambda_j }N'_{ij}
+                \end{equation}
+                pour que \( df_A(M')=N'\).
+    \end{subproof}
+    
+    Le théorème d'inversion locale donne un voisinage \( V\) de $A$ dans \( \eM(n,\eR)\) et un voisinage \( W\) de \( A^2\) dans \( \eM(n,\eR)\) tels que \( f\colon V\to W\) soit une bijection  et \( f^{-1}\colon W\to V\) soit de même régularité, en l'occurrence \( C^{\infty}\).
+\end{proof}
+
+\begin{remark}
+    Oui, il y a des matrices non symétriques qui ont une unique racine carré.
+\end{remark}
+
+La proposition suivante, qui dépend du le théorème d'inversion locale par le lemme \ref{LemLBFOooDdNcgy}, donne plus de régularité à la décomposition polaire donnée dans le théorème \ref{ThoLHebUAU}.
+\begin{proposition}[Décomposition polaire : cas réel (suite)]       \label{PropWCXAooDuFMjn}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \gO(n,\eR)\times S^{++}(n,\eR)&\to \GL(n,\eR) \\
+            (Q,S)&\mapsto SQ 
+        \end{aligned}
+    \end{equation}
+    est un difféomorphisme de classe \( C^{\infty}\).
+\end{proposition}
+
+\begin{proof}
+    Si \( M\) est donnée dans \( \GL(n,\eR)\) alors la décomposition polaire\footnote{Proposition \ref{ThoLHebUAU}.} \( M=QS\) est donnée par \( S=\sqrt{MM^t}\) et \( Q=MS^{-1}\). Autrement dit, si nous considérons la fonction de décomposition polaire
+    \begin{equation}
+        f\colon \gO(n,\eR)\times S^{++}(n,\eR)\to \GL(n,\eR)
+    \end{equation}
+    alors
+    \begin{equation}
+        f^{-1}(M)=\big(  M(\sqrt{MM^t})^{-1},\sqrt{MM^t}  \big).
+    \end{equation}
+    Nous avons vu dans le lemme \ref{LemLBFOooDdNcgy} que la racine carré était un \( C^{\infty}\)-difféomorphisme. Le reste n'étant que des produits de matrice, la régularité est de mise.
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorème de Von Neumann}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{lemma}[\cite{KXjFWKA}]
+    Soit \( G\), un sous groupe fermé de \( \GL(n,\eR)\) et 
+    \begin{equation}
+        \mL_G=\{ m\in \eM(n,\eR)\tq  e^{tm}\in G\,\forall t\in\eR \}.
+    \end{equation}
+    Alors \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\).
+\end{lemma}
+
+\begin{proof}
+    Si \( m\in\mL_G\), alors \( \lambda m\in\mL_G\) par construction. Le point délicat à prouver est le fait que si \( a,b\in \mL_G\), alors \( a+b\in\mL_G\). Soit \( a\in \eM(n,\eR)\); nous savons qu'il existe une fonction \( \alpha_a\colon \eR\to \eM\) telle que
+    \begin{equation}
+        e^{ta}=\mtu+ta+\alpha_a(t)
+    \end{equation}
+    et 
+    \begin{equation}
+        \lim_{t\to 0} \frac{ \alpha(t) }{ t }=0.
+    \end{equation}
+    Si \( a\) et \( b\) sont dans \( \mL_G\), alors \(  e^{ta} e^{tb}\in G\), mais il n'est pas vrai en général que cela soit égal à \(  e^{t(a+b)}\). Pour tout \( k\in \eN\) nous avons
+    \begin{equation}
+        e^{a/k} e^{b/k}=\left( \mtu+\frac{ a }{ k }+\alpha_a(\frac{1}{ k }) \right)\left( \mtu+\frac{ b }{ k }+\alpha_b(\frac{1}{ k }) \right)=\mtu+\frac{ a+b }{2}+\beta\left( \frac{1}{ k } \right)
+    \end{equation}
+   où \( \beta\colon \eR\to \eM\) est encore une fonction vérifiant \( \beta(t)/t\to 0\). Si \( k\) est assez grand, nous avons
+   \begin{equation}
+       \left\| \frac{ a+b }{ k }+\beta(\frac{1}{ k })  \right\|<1,
+   \end{equation}
+   et nous pouvons profiter du lemme \ref{LemQZIQxaB} pour écrire alors
+   \begin{equation}
+       \left(  e^{a/k} e^{b/k} \right)^k= e^{k\ln\big(\mtu+\frac{ a+b }{ k }+\beta(\frac{1}{ k })\big)}.
+   \end{equation}
+   Ce qui se trouve dans l'exponentielle est
+   \begin{equation}
+       k\left[ \frac{ a+b }{ k }+\alpha( \frac{1}{ k })+\sigma\left( \frac{ a+b }{ k }+\alpha(\frac{1}{ k }) \right) \right].
+   \end{equation}
+   Les diverses propriétés vues montrent que le tout tend vers \( a+b\) lorsque \( k\to \infty\). Par conséquent
+   \begin{equation}
+       \lim_{k\to \infty} \left(  e^{a/k} e^{b/k} \right)^k= e^{a+b}.
+   \end{equation}
+   Ce que nous avons prouvé est que pour tout \( t\), \(  e^{t(a+b)}\) est une limite d'éléments dans \( G\) et est donc dans \( G\) parce que ce dernier est fermé.
+\end{proof}
+
+Vu que \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\), nous pouvons considérer un supplémentaire \( M\).
+
+\begin{lemma}   \label{LemHOsbREC}
+    Il n'existe pas se suites \( (m_k)\) dans \( M\setminus\{ 0 \}\) convergeant vers zéro et telle que \(  e^{m_k}\in G\) pour tout \( k\).
+\end{lemma}
+
+\begin{proof}
+    Supposons que nous ayons \( m_k\to 0\) dans \( M\setminus\{ 0 \}\) avec \(  e^{m_k}\in G\). Nous considérons les éléments \( \epsilon_k=\frac{ m_k }{ \| m_k \| }\) qui sont sur la sphère unité de \(\GL(n,\eR)\). Quitte à prendre une sous-suite, nous pouvons supposer que cette suite converge, et vu que \( M\) est fermé, ce sera vers \( \epsilon\in M\) avec \( \| \epsilon \|=1\). Pour tout \( t\in \eR\) nous avons
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}.
+    \end{equation}
+    En vertu de la décomposition d'un réel en partie entière et décimale, pour tout \( k\) nous avons \( \lambda_k\in \eZ\) et \( | \mu_k |\leq \frac{ 1 }{2}\) tel que \( t/\| m_k \|=\lambda_k+\mu_k\). Avec ça,
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}\exp\Big( \frac{ t }{ m_k }m_k \Big)=\lim_{k\to \infty}  e^{\lambda_km_k} e^{\mu_km_k}.
+    \end{equation}
+    Pour tout \( k\) nous avons \(  e^{\lambda_km_k}\in G\). De plus \( | \mu_k |\) étant borné et \( m_k\) tendant vers zéro nous avons \(  e^{\mu_km_k}\to 1\). Au final
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}\in G
+    \end{equation}
+    Cela signifie que \( \epsilon\in\mL_G\), ce qui est impossible parce que nous avions déjà dit que \( \epsilon\in M\setminus\{ 0 \}\).
+\end{proof}
+
+\begin{lemma}   \label{LemGGTtxdF}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \mL_G\times M&\to \GL(n,\eR) \\
+            l,m&\mapsto  e^{l} e^{m} 
+        \end{aligned}
+    \end{equation}
+    est un difféomorphisme local entre un voisinage de \( (0,0)\) dans \( \eM(n,\eR)\) et un voisinage de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
+\end{lemma}
+Notons que nous ne disons rien de \(  e^{\eM(n,\eR)}\). Nous n'allons pas nous embarquer à discuter si ce serait tout \( \GL(n,\eR)\)\footnote{Vu les dimensions y'a tout de même peu de chance.} ou bien si ça contiendrait ne fut-ce que \( G\).
+
+\begin{proof}
+    Le fait que \( f\) prenne ses valeurs dans \( \GL(n,\eR)\) est simplement dû au fait que les exponentielles sont toujours inversibles. Nous considérons ensuite la différentielle : si \( u\in \mL_G\) et \( v\in M\) nous avons
+    \begin{equation}
+        df_{(0,0)}(u,v)=\Dsdd{ f\big( t(u,v) \big) }{t}{0}=\Dsdd{  e^{tu} e^{tv} }{t}{0}=u+v.
+    \end{equation}
+    L'application \( df_0\) est donc une bijection entre \( \mL_G\times M\) et \( \eM(n,\eR)\). Le théorème d'inversion locale \ref{ThoXWpzqCn} nous assure alors que \( f\) est une bijection entre un voisinage de \( (0,0)\) dans \( \mL_G\times M\) et son image. Mais vu que \( df_0\) est une bijection avec \( \eM(n,\eR)\), l'image en question contient un ouvert autour de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
+\end{proof}
+
+\begin{theorem}[Von Neumann\cite{KXjFWKA,ISpsBzT,Lie_groups}]       \label{ThoOBriEoe}
+    Tout sous-groupe fermé de \( \GL(n,\eR)\) est une sous-variété de \( \GL(n,\eR)\).
+\end{theorem}
+\index{théorème!Von Neumann}
+\index{exponentielle!de matrice!utilisation}
+
+\begin{proof}
+    Soit \( G\) un tel groupe; nous devons prouver que c'est localement difféomorphe à un ouvert de \( \eR^n\). Et si on est pervers, on ne va pas faire localement difféomorphe à un ouvert de \( \eR^n\), mais à un ouvert d'un espace vectoriel de dimension finie. Nous allons être pervers.
+
+    Étant donné que pour tout \( g\in G\), l'application 
+    \begin{equation}
+        \begin{aligned}
+            L_g\colon G&\to G \\
+            h&\mapsto gh 
+        \end{aligned}
+    \end{equation}
+    est de classe \(  C^{\infty}\) et d'inverse \(  C^{\infty}\), il suffit de prouver le résultat pour un voisinage de \( \mtu\).
+
+    Supposons d'abord que \( \mL_G=\{ 0 \}\). Alors \( 0\) est un point isolé de \( \ln(G)\); en effet si ce n'était pas le cas nous aurions un élément \( m_k\) de \( \ln(G)\) dans chaque boule \( B(0,r_k)\). Nous aurions alors \( m_k=\ln(a_k)\) avec \( a_k\in G\) et donc
+    \begin{equation}
+        e^{m_k}=a_k\in G.
+    \end{equation}
+    De plus \( m_k\) appartient forcément à \( M\) parce que \( \mL_G\) est réduit à zéro. Cela nous donnerait une suite \( m_k\to 0\) dans \( M\) dont l'exponentielle reste dans \( G\). Or cela est interdit par le lemme \ref{LemHOsbREC}. Donc \( 0\) est un point isolé de \( \ln(G)\). L'application \(\ln\) étant continue\footnote{Par le lemme \ref{LemQZIQxaB}.}, nous en déduisons que \( \mtu\) est isolé dans \( G\). Par le difféomorphisme \( L_g\), tous les points de \( G\) sont isolés; ce groupe est donc discret et par voie de conséquence une variété.
+
+    Nous supposons maintenant que \( \mL_G\neq\{ 0 \}\). Nous savons par la proposition \ref{PropXFfOiOb} que 
+    \begin{equation}
+        \exp\colon \eM(n,\eR)\to \eM(n,\eR)
+    \end{equation}
+    est une application \(  C^{\infty}\) vérifiant \( d\exp_0=\id\). Nous pouvons donc utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} qui nous offre donc l'existence d'un voisinage \( U\) de \( 0\) dans \( \eM(n,\eR)\) tel que \( W=\exp(U)\) soit un ouvert de \( \GL(n,\eR)\) et que \( \exp\colon U\to W\) soit un difféomorphisme de classe \(  C^{\infty}\).
+
+    Montrons que quitte à restreindre \( U\) (et donc \( W\) qui reste par définition l'image de \( U\) par \( \exp\)), nous pouvons avoir \( \exp\big( U\cap\mL_G \big)=W\cap G\). D'abord \( \exp(\mL_G)\subset G\) par construction. Nous avons donc \( \exp\big( U\cap\mL_G \big)\subset W\cap G\). Pour trouver une restriction de \( U\) pour laquelle nous avons l'égalité, nous supposons que pour tout ouvert \( \mO\) dans \( U\), 
+    \begin{equation}
+        \exp\colon \mO\cap\mL_G\to \exp(\mO)\cap G
+    \end{equation}
+    ne soit pas surjective. Cela donnerait un élément de \( \mO\cap\complement\mL_G\) dont l'image par \( \exp\) n'est pas dans \( G\). Nous construisons ainsi une suite en considérant une boule \( B(0,\frac{1}{ k })\) inclue à \( U\) et \( x_k\in B(0,\frac{1}{ k })\cap\complement\mL_G\) vérifiant \(  e^{x_k}\in G\). Vu le choix des boules nous avons évidemment \( x_k\to 0\).
+
+    L'élément \(  e^{x_k}\) est dans \(  e^{\eM(n,\eR)}\) et le difféomorphisme du lemme \ref{LemGGTtxdF}\quext{Il me semble que l'utilisation de ce lemme manque à l'avant-dernière ligne de la preuve chez \cite{KXjFWKA}.} nous donne \( (l_k,m_k)\in \mL_G\times M\) tel que \(  e^{l_k} e^{m_k}= e^{x_k}\). À ce point nous considérons \( k\) suffisamment grand pour que \(  e^{x_k}\) soit dans la partie de l'image de \( f\) sur lequel nous avons le difféomorphisme. Plus prosaïquement, nous posons
+    \begin{equation}
+        (l_k,m_k)=f^{-1}( e^{x_k})
+    \end{equation}
+    et nous profitons de la continuité pour permuter la limite avec \( f^{-1}\) :
+    \begin{equation}
+        \lim_{k\to \infty} (l_k,m_k)=f^{-1}\big( \lim_{k\to \infty}  e^{x_k} \big)=f^{-1}(\mtu)=(0,0).
+    \end{equation}
+    En particulier \( m_k\to 0\) alors que \(  e^{m_k}= e^{x_k} e^{-l_k}\in G\). La suite \( m_k\) viole le lemme \ref{LemHOsbREC}. Nous pouvons donc restreindre \( U\) de telle façon à avoir
+    \begin{equation}
+        \exp\big( U\cap\mL_G \big)=W\cap G.
+    \end{equation}
+    Nous avons donc un ouvert de \( \mL_G\) (l'ouvert \( U\cap\mL_G\)) qui est difféomorphe avec l'ouvert \( W\cap G\) de \( G\). Donc \( G\) est une variété et accepte \( \mL_G\) comme carte locale.
+
+\end{proof}
+
+\begin{remark}
+    En termes savants, nous avons surtout montré que si \( G\) est un groupe de Lie d'algèbre de Lie \( \lG\), alors l'exponentielle donne un difféomorphisme local entre \( \lG\) et \( G\).
+\end{remark}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Recherche d'extrema}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema à une variable}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
+\end{definition}
+
+La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
+\begin{proposition}     \label{PROPooNVKXooXtKkuz}
+Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
+\end{proposition}
+
+La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
+
+Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
+\begin{proposition}
+Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
+\begin{equation}
+    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
+\end{equation}
+et que
+\begin{equation}
+    f^{(k)}(a)\neq 0.
+\end{equation}
+Dans ce cas,
+\begin{enumerate}
+
+\item
+Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
+\item
+Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
+
+\end{enumerate}
+\end{proposition}
+
+Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
+
+Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
+
+Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
+
+\begin{example}
+	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
+\end{example}
+ 
+Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
+\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
+\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
+
+Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
+
+Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
+
+\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
+\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
+ 
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema libre}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooYJLZooLkEAYf}
+Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
+\end{definition}
+
+Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
+
+Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
+\begin{equation}
+    d^2f(a)=\begin{pmatrix}
+    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
+    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
+\end{pmatrix}.
+\end{equation}
+Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
+
+\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
+\end{proposition}
+
+\begin{proof}
+    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
+    \begin{equation}
+        \begin{aligned}
+            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
+            t&\mapsto f(a+tu) 
+        \end{aligned}
+    \end{equation}
+    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
+    \begin{equation}
+        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
+    \end{equation}
+\end{proof}
+
+\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
+    \begin{enumerate}
+        \item   \label{ITEMooCVFVooWltGqI}
+            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
+        \item\label{ItemPropoExtreRn}
+            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
+    \end{enumerate}
+\end{proposition}
+\index{extrema}
+
+\begin{proof}
+    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
+    \begin{equation}
+        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
+    \end{equation}
+    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
+    \end{equation}
+    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
+    \end{equation}
+    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
+    \begin{equation}
+        (d^2f)_a(e_h,e_h)\geq 0.
+    \end{equation}
+    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
+\end{proof}
+
+La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
+
+Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
+
+La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
+\begin{enumerate}
+    \item
+        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
+    \item
+        écrire $d^2f(a)$ pour chacun des candidats
+    \item
+        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
+    \item
+        conclure.
+\end{enumerate}
+
+Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
+
+\begin{example}
+    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
+    \begin{equation}
+        df=\big(4x^3-4y;4y^3-4x),
+    \end{equation}
+    et la matrice des dérivées secondes est
+    \begin{equation}
+        M=d^2f(x,y)=\begin{pmatrix}
+            12x^2    &   -4    \\ 
+            -4    &   12y^2    
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
+
+    Pour le point \( (0,0)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            0    &   -4    \\ 
+            -4    &   0    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
+
+    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            12    &   -4    \\ 
+            -4    &   12    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Un peu de recettes de cuisine}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{enumerate}
+\item Rechercher les points critiques, càd les $(x,y)$ tels que
+\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
+En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
+\item Déterminer la nature des points critiques: «test» des dérivées secondes:
+\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
+\begin{enumerate}
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
+\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
+\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
+\end{enumerate}
+\end{enumerate}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Extrema liés}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
+
+Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
+
+\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
+    Soit \( A\), un ouvert de \( \eR^n\) et
+    \begin{enumerate}
+        \item
+            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
+        \item 
+            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
+        \item
+            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
+        \item
+            un extrema local $a\in M$ de $f$ relativement à $M$.
+    \end{enumerate}
+    Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
+    \begin{equation}
+        L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_r)=f(x_1,\ldots,x_n)+\sum_{i=1}^r\lambda_iG_i(x_1,\ldots,x_n).
+    \end{equation}
+    Autrement dit, si \( a\) est un extrema lié, alors \( \nabla f(a)\) est une combinaisons des \( \nabla G_i(a)\), ou encore il existe des \( \lambda_i\) tels que
+    \begin{equation}    \label{EqRDsSXyZ}
+        df(a)=\sum_i\lambda_idG_i(a).
+    \end{equation}
+\end{theorem}
+\index{théorème!inversion locale!utilisation}
+\index{extrema!lié}
+\index{théorème!extrema!lié}
+\index{application!différentiable!extrema lié}
+\index{variété}
+\index{rang!différentielle}
+\index{forme!linéaire!différentielle}
+La fonction $L$ est le \defe{lagrangien}{lagrangien} du problème et les variables \( \lambda_i\) sont les \defe{multiplicateurs de Lagrange}{multiplicateur!de Lagrange}\index{Lagrange!multiplicateur}.
+
+\begin{proof}
+    Si \( r=n\) alors les vecteurs linéairement indépendantes \( \nabla G_i(a) \) forment une base de \( \eR^n\) et donc évidemment les \( \lambda_i\) existent. Nous supposons donc maintenant que \( r<n\). Nous notons \( (z_i)_{i=1\ldots n}\) les coordonnées sur \( \eR^n\).
+    
+    La matrice
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{ \partial G_1 }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_1 }{ \partial z_n }(a)    \\
+            \vdots    &   \ddots    &   \vdots    \\
+            \frac{ \partial G_r }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_r }{ \partial z_n }(a)
+        \end{pmatrix}
+    \end{equation}
+    est de rang \( r\) parce que les lignes sont par hypothèses linéairement indépendantes. Nous nommons \( (y_i)_{i=1,\ldots, r}\) un choix de \( r\) parmi les \( (z_i)\) tels que
+    \begin{equation}
+        \det\begin{pmatrix}
+            \frac{ \partial G_1 }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_1 }{ \partial y_r }    \\
+            \vdots    &   \ddots    &   \vdots    \\
+            \frac{ \partial G_r }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_r }{ \partial y_r }
+        \end{pmatrix}\neq 0.
+    \end{equation}
+    Nous identifions \( \eR^n\) à \( \eR^s\times \eR^r\) dans lequel \( \eR^r\) est la partie générée par les \( (y_i)_{i=1,\ldots, r}\). Nous nommons \( (x_j)_{j=1,\ldots, s}\) les coordonnées sur \( \eR^s\). Autrement dit, les coordonnées sur \( \eR^n\) sont \( x_1,\ldots, x_s,y_1,\ldots, y_r\). Dans ces coordonnées, nous nommons \( a=(\alpha,\beta)\) avec \( \alpha\in \eR^s\) et \( \beta\in \eR^r\).
+
+    Si nous notons \( G=(G_1,\ldots, G_r)\), le théorème de la fonction implicite (théorème \ref{ThoAcaWho})  nous dit qu'il existe un voisinage \( U'\) de \( \alpha\in \eR^n\), un voisinage \( V'\) de \( \beta\in \eR^r\) et une fonction \( \varphi\colon U'\to V'\) de classe \( C^1\) telle que si \( (x,y)\in U'\times V'\), alors
+    \begin{equation}
+        g(x,y)=0
+    \end{equation}
+    si et seulement si \( y=\varphi(x)\). Nous posons maintenant
+    \begin{subequations}
+        \begin{align}
+            \psi(x)&=(x,\varphi(x))\\
+            h(x)&=f\big( \psi(x) \big).
+        \end{align}
+    \end{subequations}
+    Nous avons \( \psi(\alpha)=a\) et \( \psi(x)\in M\) pour tout \( x\in U'\). La fonction \( h\) a donc un extrema local en \( \alpha\) et donc les dérivées partielles de \( h\) y sont nulles. Cela signifie que
+    \begin{equation}
+        0=\frac{ \partial h }{ \partial x_i }(\alpha)=\sum_{j=1}^n\frac{ \partial f }{ \partial x_j }\frac{ \partial x_j }{ \partial x_i }+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }\frac{ \partial \varphi_k }{ \partial x_i },
+    \end{equation}
+    c'est à dire
+    \begin{equation}
+        \frac{ \partial f }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
+    \end{equation}
+    pour tout \( i=1,\ldots, s\). D'autre part pour tout $k$, la fonction \( l_k(x)=G_k\big( x,\varphi(x) \big)\) est constante et vaut zéro; ses dérivées partielles sont donc nulles :
+    \begin{equation}
+        \frac{ \partial l }{ \partial x_i }(\alpha)=\frac{ \partial G_k }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial G_k }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
+    \end{equation}
+    pour tout \( i=1,\ldots, s\) et \( k=1,\ldots, r\).
+    
+    Les \( s\) premières colonnes de la matrice
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{ \partial f }{ \partial x_1 }   &   \cdots    &   \frac{ \partial f }{ \partial x_s }    &   \frac{ \partial f }{ \partial y_1 }    &   \cdots    &   \frac{ \partial f }{ \partial y_r }\\  
+            \frac{ \partial G_1 }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial x_s }    &   \frac{ \partial G_1 }{ \partial y_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial y_r }\\
+            \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots\\
+            \frac{ \partial G_r }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_r }{ \partial x_s }    &   \frac{ \partial G_r }{ \partial y_1 }    &  \cdots   & \frac{ \partial G_r }{ \partial y_r }  
+        \end{pmatrix}
+    \end{equation}
+    s'expriment en terme des \( r\) dernières. La matrice est donc au maximum de rang \( r\). Notons que la première ligne est \( \nabla f\) et les \( r\) suivantes sont les \( \nabla G_i\). Vu que ces lignes sont des vecteurs liés, il existe \( \mu_0,\ldots, \mu_r\) tels que
+    \begin{equation}
+        \mu_0\nabla f+\sum_{i=1}^r\mu_i\nabla G_i=0.
+    \end{equation}
+    Par hypothèse les \( \nabla G_i\) sont linéairement indépendants, ce qui nous dit que \( \mu_0\neq 0\). Donc nous avons ce qu'il nous faut :
+    \begin{equation}
+        \nabla f(a)=\sum_i\frac{ \mu_i }{ \mu_0 } \nabla G_i(a).
+    \end{equation}
+
+    Notons qu'au vu de l'expression \eqref{EqRDsSXyZ}, le fait que les formes \( \{ dG_i(a) \}_{1\leq i\leq r}\) forment une partie libre dans \( (\eR^n)^*\) implique que les \( \lambda_i\) sont uniques.
+\end{proof}
+
+La proposition suivante est la même que \ref{ThoRGJosS}.
+\begin{proposition} \label{PropfPPUxh}
+    Soit \( U\), un ouvert de \( \eR^n\) et des fonctions de classe \( C^1\) \( f,g_1,\ldots, g_r\colon U\to \eR\). Nous considérons
+    \begin{equation}
+        \Gamma=\{ x\in U\tq g_1(x)=\ldots=g_r(x)=0 \}.
+    \end{equation}
+    Soit \( a\) un extrémum de \( f|_{\Gamma}\). Supposons que les formes \( dg_1,\ldots, dg_r\) soient linéairement indépendantes en \( a\). Alors il existe \( \lambda_1,\ldots, \lambda_r\) dans \( \eR\) tel que
+    \begin{equation}
+        df_a=\sum_{i=1}^r\lambda_i(dg_i)_a.
+    \end{equation}
+\end{proposition}
+
+En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
+
+L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
+
+\begin{lemma}       \label{LemmeMinSCimpliqueS}
+    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
+\end{lemma}
+
+\begin{proof}
+    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Fonctions convexes}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}[\cite{JFihMcQ}]  \label{DefVQXRJQz}
+    Une fonction $f$ d’un intervalle $I$ de \( \eR\) vers \( \eR\) est dite \defe{convexe}{fonction!convexe}\index{convexité!fonction} lorsque, pour tous \( x_1\) et \( x_2\) de $I$ et tout $\lambda$ dans $[0, 1]$ nous avons
+    \begin{equation}        \label{EQooYNAPooFePQZy}
+        f\big(\lambda\, x_1+(1-\lambda)\, x_2\big) \leq \lambda\, f(x_1)+(1-\lambda)\, f(x_2)
+    \end{equation}
+    Si l'inégalité est stricte, alors nous disons que la fonction \( f\) est \defe{strictement convexe}{convexité!stricte}.
+
+    Une fonction est \defe{concave}{concave} si son opposée est convexe.
+\end{definition}
+
+
+\begin{normaltext}[\cite{GYfviRu}]
+    Les différents résultats pour les fonctions convexes s'adaptent généralement sans mal aux fonctions strictement convexes. Une nuance cependant : de même que les fonctions dérivables convexes sont celles qui ont une dérivée croissante, les fonctions dérivables strictement convexes sont celles qui ont une dérivée strictement croissante (proposition \ref{PropYKwTDPX}). En revanche, il ne faudrait pas croire que la dérivée seconde d'une fonction dérivable strictement convexe est nécessairement une fonction à valeurs strictement positives (voir théorème \ref{ThoGXjKeYb}) : la dérivée d'une fonction strictement croissante peut s'annuler occasionnellement, ou plus exactement peut s'annuler sur un ensemble de points d'intérieur vide. Penser à \( x\mapsto x^4\) pour un exemple de fonction strictement convexe dont la dérivée seconde s'annule.
+\end{normaltext}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Inégalité des pentes}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Dans l'étude des fonctions convexes nous allons souvent utiliser la fonction \defe{taux d'accroissement}{taux d'accroissement} qui est, pour \( \alpha\) dans le domaine de convexité de \( f\) définie par
+\begin{equation}    \label{EqRYBazWd}
+    \begin{aligned}
+        \tau_{\alpha}\colon I\setminus\{ \alpha \}&\to \eR \\
+        x&\mapsto \frac{ f(x)-f(\alpha) }{ x-\alpha }. 
+    \end{aligned}
+\end{equation}
+
+\begin{proposition}[Inégalité des pentes\cite{OJIMBtv}] \label{PropMDMGjGO}
+    Soit \( f\) une fonction convexe sur un intervalle \( I\subset \eR\). Alors pour tout \( a<b<c\) dans \( I\) nous avons\footnote{Les inégalités sont strictes si la fonction \( f\) est strictement convexe.}
+    \begin{equation}
+        \frac{ f(b)-f(a)  }{ b-a }\leq\frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
+    \end{equation}
+    En d'autres termes,
+    \begin{equation}
+        \tau_a(b)\leq\tau_a(c)\leq \tau_b(c),
+    \end{equation}
+    c'est à dire que \( \tau\) est croissante en ses deux arguments.
+\end{proposition}
+\index{inégalité!des pentes}
+
+\begin{proof}
+    D'abord les inégalités \( a<b<c\) impliquent \( 0<b-a<c-a\) et donc
+    \begin{equation}
+        \lambda=\frac{ b-a }{ c-a }<1.
+    \end{equation}
+    L'astuce est de remarquer que \( (1-\lambda)a+\lambda c=b\). Donc \( \lambda\) a toutes les bonnes propriétés pour être utilisé dans la définition de la convexité :
+    \begin{equation}
+        f\big( (1-\lambda)a+\lambda c \big)\leq \lambda f(c)+(1-\lambda)f(a),
+    \end{equation}
+    c'est à dire
+    \begin{equation}
+        f(b)-f(a)\leq \lambda\big( f(c)-f(a) \big)
+    \end{equation}
+    ou encore, en remplaçant \( \lambda\) par sa valeur :
+    \begin{equation}
+        \frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(c)-f(a) }{ c-a }.
+    \end{equation}
+    Cela fait déjà une des inégalités à savoir.
+    
+    D'autre part en partant de \( -a<-b<-c\) nous posons
+    \begin{equation}
+        0<\lambda=\frac{ c-b }{ c-a }.
+    \end{equation}
+    Nous avons à nouveau \( b=(1-\lambda)c+\lambda a\) et nous pouvons obtenir la seconde inégalité
+    \begin{equation}
+        \frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
+    \end{equation}
+\end{proof}
+
+Géométriquement, l'inégalité des pentes se comprend facilement : le coefficient angulaire de la corde du graphe augmente. Donc si \( x<y<z\), le coefficient moyen entre \( x\) et \( y\) est plus petit que celui entre \( x\) et \( z\) qui est plus petit que celui entre \( y\) et \( z\).
+
+Donc si le coefficient angulaire moyen entre \( a\) et \( b+u\) vaut celui entre \( a\) et \( b\), ce coefficient ne peut qu'être constant entra \( a\) et \( b\) : sinon il serait plus grand entre \( b\) et \( b+u\) et la moyenne sur \( a\to b+u\) serait plus grande que sa moyenne sur \( a\to b\). Mais avoir un coefficient angulaire constant signifie être une droite.
+
+En résumé, si une fonction est convexe et non strictement convexe, alors son graphe est une droite. C'est en gros cela que la proposition \ref{PROPooOCOEooEGybmS} clarifiera.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Convexité et régularité}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}[\cite{GYfviRu}]   \label{LemKLTsHIQ}
+    Une fonction convexe sur un ouvert
+    \begin{enumerate}
+        \item
+            y admet des dérivées à gauche et à droite en chaque point,
+        \item
+            y est continue.
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Soit \( I=\mathopen] a , b \mathclose[\) un intervalle sur lequel \( f\) est convexe et \( \alpha\in I\). Nous allons prouver que \( f\) est continue en \( \alpha\). Nous considérons \( \tau_{\alpha}\) le taux d'accroissement définit par \eqref{EqRYBazWd}; c'est une fonction croissante comme précisé dans l'inégalité des trois pentes \ref{PropMDMGjGO} et de plus \( \tau_{\alpha}(x)\) est bornée supérieurement par \( \tau_{\alpha}(b)\) pour \( x<\alpha\) et inférieurement par \( \tau_{\alpha}(a)\) pour \( x>\alpha\). Les limites existent donc et sont finies par la proposition \ref{PropMTmBYeU}. Autrement dit les limites
+        \begin{subequations}
+            \begin{align}
+                \lim_{x\to \alpha+} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^+} \tau_{\alpha}(x)=\inf_{t>\alpha}\tau_{\alpha}(t)\\
+                \lim_{x\to \alpha^-} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^-} \tau_{\alpha}(x)=\sup_{t<\alpha}\tau_{\alpha}(t).
+            \end{align}
+        \end{subequations}
+        existent et sont finies, c'est à dire que la fonction \( f\) admet une dérivée à gauche et à droite.
+
+        Pour tout \( x\) nous avons les inégalités
+        \begin{equation}
+            \tau_{\alpha}(a)\leq \frac{ f(x)-f(\alpha) }{ x-\alpha }\leq \tau_{\alpha}(b).
+        \end{equation}
+        En posant \( k=\max\{ \tau_{\alpha}(a),\tau_{\alpha}(b) \}\) nous avons
+        \begin{equation}
+            \big| f(x)-f(\alpha) \big|\leq k| x-\alpha |.
+        \end{equation}
+        La fonction est donc Lipschitzienne et par conséquent continue par la proposition \ref{PropFZgFTEW}.
+\end{proof}
+
+\begin{remark}
+    Les dérivées à gauche et à droite ne sont a priori pas égales. Penser par exemple à une fonction affine par morceaux dont les pentes augmentent à chaque morceau.
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Dérivées d'une fonction convexe}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[\cite{RIKpeIH,ooGCESooQzZtVC,MonCerveau}] \label{PropYKwTDPX}
+    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) 
+    \begin{enumerate}
+        \item       \label{ITEMooUTSAooJvhZNm}
+            est convexe si et seulement si sa dérivée est croissante sur \( I\).
+        \item       \label{ITEMooLLSIooFwkxtV}
+            est strictement convexe si et seulement si sa dérivée est strictement croissante sur \( I\)
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+
+
+    Pour la preuve de \ref{ITEMooUTSAooJvhZNm} et \ref{ITEMooLLSIooFwkxtV}, nous allons démontrer les énoncés «non stricts»  et indiquer ce qu'il faut changer pour obtenir les énoncés «stricts».
+    \begin{subproof}
+    \item[Sens direct]
+    Nous supposons que \( f\) est convexe. Soient \( a<b\) dans \( I\) et \( x\in\mathopen] a , b \mathclose[\). D'après l'inégalité des pentes \ref{PropMDMGjGO},
+        \begin{equation}        \label{EqATDLooIcqdDI}
+            \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(b)-f(x) }{ b-x }.
+        \end{equation}
+        En faisant la limite \( x\to a\) nous avons
+        \begin{equation}
+            f'(a)\leq \frac{ f(b)-f(a) }{ b-a }
+        \end{equation}
+        et la limite \( x\to b\) donne
+        \begin{equation}
+            \frac{ f(b)-f(a) }{ b-a }\leq f'(b).
+        \end{equation}
+        Ici les inégalités sont non a priori strictes, même si \( f\) est strictement convexe : même avec des inégalités strictes dans \eqref{EqATDLooIcqdDI}, le passage à la limite rend l'inégalité non stricte. Quoi qu'il en soit nous avons 
+        \begin{equation}        \label{EqQGVMooBpuvNr}
+            f'(a)\leq f'(b).
+        \end{equation}
+    \item[Sens direct : strict]
+         Nous savons déjà que \( f'\) est croissante. Si \eqref{EqQGVMooBpuvNr} était une égalité, alors \( f'\) serait constante sur \( \mathopen] a , b \mathclose[\) parce qu'en prenant \( c\) entre \( a\) et \( b\) nous aurions \( f'(a)\leq f'(c)\leq f'(b)\) avec \( f'(a)=f'(b)\). Donc \( f'(a)=f'(c)\). Avoir \( f'\) constante sur un intervalle est contraire à la stricte convexité.
+
+         \item[Sens réciproque]
+
+             Nous supposons que \( f'\) est croissante et nous considérons \( a<b\) dans \( I\) ainsi que \( \lambda\in \mathopen[ 0 , 1 \mathclose]\). Nous posons \( x=\lambda a+(1-\lambda)b\), et nous savons que \( a\leq x\leq b\). Le théorème des accroissements finis \ref{ThoAccFinis} donne \( c_1\in\mathopen] a , x \mathclose[\) et \( c_2\in \mathopen] x , b \mathclose[\) tels que
+                 \begin{equation}
+                     f'(c_1)=\frac{ f(x)-f(a) }{ x-a }
+                 \end{equation}
+                 et 
+                 \begin{equation}
+                     f'(c_2)=\frac{ f(b)-f(x) }{ b-x }.
+                 \end{equation}
+                 Et en plus \( c_1<c_2\). Vu que \( f'\) est croissante nous avons \( f'(c_1)\leq f'(c_2)\) et donc
+                 \begin{equation}       \label{EqSAOCooWAwClQ}
+                     \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(x) }{ b-x }.
+                 \end{equation}
+                 En remplaçant \( x\) par sa valeur en termes de \( \lambda\), \( a\) et \( b\) nous avons \( x-a=(1-\lambda)(b-a)\) et \( b-x=\lambda(b-a)\), et l'inégalité \eqref{EqSAOCooWAwClQ} nous donne
+                 \begin{equation}
+                     f(x)\leq \lambda f(a)+(1-\lambda)f(b).
+                 \end{equation}
+             \item[Sens réciproque : strict]
+                 Si \( f'\) est strictement croissante, nous avons \( f'(c_2)<f'(c_2)\) et les inégalité suivantes sont strictes, ce qui donne
+                 \begin{equation}
+                     f(x)< \lambda f(a)+(1-\lambda)f(b).
+                 \end{equation}
+    \end{subproof}
+\end{proof}
+
+\begin{theorem}[\cite{RIKpeIH}] \label{ThoGXjKeYb}
+    Une fonction \( f\) de classe \( C^2\) est convexe si et seulement si \( f''\) est positive.
+\end{theorem}
+
+\begin{proof}
+    La fonction est \( C^2\), donc \( f''\) est positive si et seulement si \( f'\) est croissante (proposition \ref{PropGFkZMwD}) alors que la proposition \ref{PropYKwTDPX} nous jure que \( f\) sera convexe si et seulement si \( f'\) est croissante.
+\end{proof}
+
+\begin{remark}      \label{REMooVRPQooIybxmp}
+    Une fonction peut être strictement convexe sans que sa dérivée seconde ne soit toujours strictement positive. En exemple : \( x\mapsto x^4\) est strictement convexe alors que sa dérivée seconde s'annule en zéro.
+\end{remark}
+
+\begin{example} \label{ExPDRooZCtkOz}
+    Quelques exemples utilisant le théorème \ref{ThoGXjKeYb}
+    \begin{enumerate}
+        \item
+    La fonction \( x\mapsto x^2\) est convexe parce que sa dérivée seconde est la constante (positive) \( 2\).
+\item La fonction \( x\mapsto\frac{1}{ x }\) est convexe sur \( \eR^+\setminus\{ 0 \}\) (sa dérivée seconde est \( 2x^{-3}\)).
+\item
+    La fonction exponentielle est également convexe.
+\item
+    La fonction \( \ln\) est concave parce que la dérivée seconde de \( -\ln\) est \( \frac{1}{ x^2 }\) qui est strictement positif.
+    \end{enumerate}
+\end{example}
+
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Graphe d'une fonction convexe}
+%---------------------------------------------------------------------------------------------------------------------------
+
+L'idée principale du graphe d'une fonction convexe est qu'il est toujours au dessus du graphe de ses tangentes (lorsqu'elles existent). Lorsqu'elles n'existent pas, le lemme \ref{LemKLTsHIQ} donne des coefficients directeurs de droites qui vont rester en dessous du graphe de la fonction.
+
+\begin{proposition}[\cite{ooKCFNooVrqYhc}]      \label{PROPooOCOEooEGybmS}
+    Une fonction convexe est strictement convexe si et seulement s'il n'existe aucun intervalle de longueur non nulle sur lequel elle coïncide avec une fonction affine.
+\end{proposition}
+
+\begin{proof}
+    Si sur l'intervalle (non réduit à un point) \( \mathopen[ x , y \mathclose]\), la fonction convexe \( f\) coïncide avec une fonction affine, alors \( f(t)=at+b\) et pour \( \lambda\in\mathopen] 0 , 1 \mathclose[\) nous avons
+        \begin{equation}
+                f\big( \lambda x+(1-\lambda)y \big)=a\lambda x+a(1-\lambda)y+b=\lambda f(x)+(1-\lambda)f(y)
+        \end{equation}
+        où nous avons remplacé \( b\) par \( \lambda b+(1-\lambda)b\). Par conséquent la fonction n'est pas strictement convexe.
+
+    Nous supposons maintenant que la fonction convexe \( f\) n'est pas strictement convexe sur l'intervalle \( I\). Il existe \( x\neq y\in I\) et \( \lambda\in \mathopen] 0 , 1 \mathclose[\) tels que
+        \begin{equation}
+            f\big( \lambda x+(1-\lambda)y \big)=\lambda f(x)+(1-\lambda)f(y).
+        \end{equation}
+    Nous posons \( z=\lambda x+(1-\lambda)y\) et \( u\in\mathopen] x , z \mathclose[\) pour écrire des inégalités des pentes entre \( x<u<z<y\). Plus précisément si nous notons \( a\to b\) la pente de \( a\) à \( b\), c'est à dire \( a\to b=\frac{ f(b)-f(a) }{ b-a }\), alors les inégalités des pentes pour \( x<u<z\) puis \( u<z<y\) donnent
+        \begin{equation}        \label{EqooBMEFooMpoEzd}
+            x\to z\leq u\to z\leq z\to y.
+        \end{equation}
+        Voyons maintenant qu'en réalité \( z\to y=x\to z\). En effet en replaçant
+        \begin{equation}
+            f(y)=\frac{ f(z)-\lambda f(x) }{ 1-\lambda }
+        \end{equation}
+        et
+        \begin{equation}
+            y=\frac{ \lambda x }{ 1-\lambda }
+        \end{equation}
+        dans l'expression \( z\to y=\frac{ f(y)-f(z) }{ y-z }\) nous obtenons
+        \begin{equation}
+            z\to y=\frac{ f(y)-f(z) }{ y-z }=\frac{ f(z)-f(x) }{ z-x }=x\to z.
+        \end{equation}
+        Les inégalités \eqref{EqooBMEFooMpoEzd} sont donc des égalités :
+        \begin{equation}
+            \frac{ f(z)-f(x) }{ z-x }=\frac{ f(z)-f(u) }{ z-u }=\frac{ f(y)-f(z) }{ y-z }.
+        \end{equation}
+        Nous avons donc montré que le nombre \( a=\frac{ f(z)-f(u) }{ z-u }\) ne dépend pas de \( u\). Nous avons alors
+        \begin{equation}
+            f(z)-f(u)=a(z-u) 
+        \end{equation}
+        ou encore :
+        \begin{equation}
+            f(u)=f(z)-a(z-u),
+        \end{equation}
+    ce qui signifie que sur \( \mathopen] x , z \mathclose[\), la fonction \( f\) est affine.
+\end{proof}
+
+\begin{proposition} \label{PROPooQPOSooDZlUAJ}
+    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) est convexe si et seulement si son graphe est au dessus de chacune de ses tangentes.
+\end{proposition}
+
+\begin{proof}
+    \begin{subproof}
+        \item[Sens direct]
+            Soient \( x,y\in I\). Nous voulons :
+            \begin{equation}
+                f(y)\geq f(x)+f'(x)(y-x).
+            \end{equation}
+            Étant donné que nous aurons besoin, dans le quotient différentiel de quelque chose comme \( f(x+t)-f(x)\) nous écrivons la définition \eqref{EQooYNAPooFePQZy} de la convexité en inversant les rôles de \( x\) et \( y\) et en manipulant un peu :
+            \begin{subequations}
+                \begin{align}
+                    f\big( ty+(1-t)x \big)\leq tf(y)+(1-t)f(x)\\
+                    f\big( x+t(y-x) \big)\leq tf(y)+(1-t)f(x)\\
+                    f\big(  x+t(y-x)  \big)=f(x)\leq tf(y)-tf(x)
+                \end{align}
+            \end{subequations}
+            Nous divisons par \( t\) :
+            \begin{equation}
+                \frac{ f\big( x+t(y-x) \big)-f(x) }{ t }\leq f(y)-f(x).
+            \end{equation}
+            Le passage à la limite \( t\to 0\) donne
+            \begin{equation}
+                (y-x)f'(x)\leq f(y)-f(x),
+            \end{equation}
+            ce qu'il fallait.
+        \item[Sens inverse]
+            Pour tout \( x,y\in I\) nous supposons avoir
+            \begin{equation}        \label{EQooEXXIooHXJnER}
+                f(y)\geq f(x)+f'(x)(y-x).
+            \end{equation}
+            Si nous supposons \( x\neq y\) et si nous posons \( z=\lambda x+(1-\lambda)y\) nous voulons prouver que
+            \begin{equation}
+                f(z)\leq \lambda f(x)+(1-\lambda)f(y).
+            \end{equation}
+            Pour cela nous écrivons l'inégalité \eqref{EQooEXXIooHXJnER} avec les couples \( (x,z)\) et \( (y,z)\) :
+            \begin{subequations}
+                \begin{align}
+                    f(x)\geq f(z)+f'(z)'(x-z)\\
+                    f(y)\geq f(z)+f'(z)'(y-z)
+                \end{align}
+            \end{subequations}
+            En multipliant la première par \( \lambda\) et la seconde par \( (1-\lambda)\) et en sommant,
+            \begin{subequations}
+                \begin{align}
+                    \lambda f(x)+(1-\lambda)f(y)&\geq \lambda f(z)+\lambda f'(z)(x-z)+(1-\lambda)f(z)+(1-\lambda)f'(z)(y-z)\\
+                    &=f(z)+f'(z)\big( \lambda(x-z)+(1-\lambda)(y-z) \big)\\
+                    &=f(z).
+                \end{align}
+            \end{subequations}
+    \end{subproof}
+\end{proof}
+
+\begin{proposition}[\cite{MonCerveau}] \label{PropNIBooSbXIKO}
+    Soit \( f\colon \eR\to \eR \) une fonction convexe et \( a\in \eR\). Il existe une constante \( c_a\in \eR\) telle que pour tout \( x\) nous ayons
+    \begin{equation}    \label{EqSKIooSeAekM}
+        f(x)-f(a)\geq c_a(x-a).
+    \end{equation}
+    Autrement dit, le graphe de la fonction \( f\) est toujours au dessus de la droite d'équation
+    \begin{equation}
+        y=f(a)+c_a(x-a).
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Les dérivées à gauche et à droite de \( f\) données par le lemme \ref{LemKLTsHIQ} sont les candidats tout cuits pour être coefficient directeur de la droite que l'on cherche. Nous allons prouver qu'en posant
+    \begin{equation}
+        c_a=\inf_{t>a}\tau_a(t),
+    \end{equation}
+    la droite \( y=f(a)+c_a(x-a)\) répond à la question\footnote{En prenant l'autre, \( c_a'=\sup_{t<a}\tau_a(t)\), ça fonctionne aussi. En pensant à une fonction affine par morceaux, on remarque qu'en choisissant un nombre entre les deux, nous avons plus facilement une inégalité stricte dans \eqref{EqSKIooSeAekM}.}.
+
+    Nous devons prouver que le nombre \( \Delta_x=f(x)-\big( f(a)+c_a(x-a) \big)\) est positif pour tout \( x\).
+    \begin{subproof}
+
+    \item[Si \( x>a\)]
+        
+        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est positif :
+        \begin{subequations}
+            \begin{align}
+                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
+                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
+                &\geq 0
+            \end{align}
+        \end{subequations}
+        parce que \( t\to\tau_a(t)\) est croissante et que \( x>a\).
+
+    \item[Si \( x<a\)]
+        
+        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est négatif :
+        \begin{subequations}
+            \begin{align}
+                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
+                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
+                &\leq 0
+            \end{align}
+        \end{subequations}
+        parce que \( t\to\tau_a(t)\) est croissante et que \( x<a\).
+    \end{subproof}
+\end{proof}
+
+\begin{proposition}[\cite{MonCerveau}] \label{PropPEJCgCH}
+    Si \( g\) est une fonction convexe, il existe deux suites réelles \( (a_n)\) et \( (b_n)\) telles que
+    \begin{equation}
+        g(x)=\sup_{n\in \eN}(a_nx+b_n).
+    \end{equation}
+\end{proposition}
+\index{fonction!convexe}
+\index{densité!de \( \eQ\) dans \( \eR\)!utilisation}
+
+\begin{proof}
+    Pour \( u\in \eR\) nous considérons \( a(u)\) et \( b(u)\) tels que la droite \( y(x)=a(u)x+b(u)\) vérifie \( y(u)=g(u)\) et \( y(x)\leq g(x)\) pour tout \( x\). Cela est possible par la proposition \ref{PropNIBooSbXIKO}. Il s'agit d'une droite coupant le graphe de \( g\) en \( x=u\) et restant en dessous. Nous considérons alors \( (u_n)\) une suite quelconque dense dans \( \eR\) (disons les rationnels pour fixer les idées) et nous posons
+    \begin{subequations}
+        \begin{numcases}{}
+            a_n=a(u_n)\\
+            b_n=b(u_n).
+        \end{numcases}
+    \end{subequations}
+    Si \( q\in \eQ\) alors \( a_nx+b_n\leq g(x)\) pour tout \( n\) et \( g(q)\) est le supremum qui est atteint pour le \( n\) tel que \( u_n=q\). Si maintenant \( x\) n'est pas dans \( \eQ\) il faut travailler plus.
+
+    Nous prenons \( (\tilde q_n)\), une sous-suite de \( (q_n)\) convergeant vers \( x\) et \( N\) suffisamment grand pour que pour tout \( n\geq N\) on ait \( | \tilde q_n-x |\leq \epsilon\) et \( | g(\tilde q_n)-g(x) |\leq \epsilon\); cela est possible grâce à la continuité de \( g\) (lemme \ref{LemKLTsHIQ}). Ensuite les sous-suites \( (\tilde a_n)\) et \( (\tilde b_n)\) sont celles qui correspondent :
+    \begin{equation}
+        \tilde a_n\tilde q_n+\tilde b_n=g(\tilde q_n).
+    \end{equation}
+    Nous considérons la majoration
+    \begin{subequations}
+        \begin{align}
+            | \tilde a_nx+\tilde b_n-g(x) |&\leq| \tilde a_nx+\tilde b_n-(\tilde a_n\tilde q_n+\tilde b_n) |+\underbrace{| \tilde a_n\tilde q_n+\tilde b_n-g(\tilde q_n) |}_{=0}+\underbrace{| g(\tilde q_n)-g(x) |}_{\leq \epsilon}\\
+            &\leq | \tilde a_n | |x-\tilde q_n |+\epsilon\\
+            &=\epsilon\big( | \tilde a_n |+1 \big).
+        \end{align}
+    \end{subequations}
+    Il nous reste à montrer que \( | \tilde a_n |\) est borné par un nombre ne dépendant pas de \( n\) (pour les \( n>N\)).
+
+    Vu que la droite de coefficient directeur \( \tilde a_n\) et passant par le point \( \big( \tilde q_n,g(\tilde q_n) \big)\) reste en dessous du graphe de \( g\), nous avons pour tout \( n\) et tout \( y\in \eR\) l'inégalité
+    \begin{equation}
+        g(y)\geq \tilde a_n(y-\tilde q_n)+g(\tilde q_n)\in \tilde a_nB(y-x,\epsilon)+B\big( g(x),\epsilon \big).
+    \end{equation}
+    Si \( \tilde a_n\) n'est pas borné vers le haut, nous prenons \( y\) tel que \( B(y-x,\epsilon)\) soit minoré par un nombre \( k\) strictement positif et nous obtenons
+    \begin{equation}
+        g(y)\geq k\tilde a_n+l
+    \end{equation}
+    avec \( k\) et \( l\) indépendants de \( n\). Cela donne \( g(y)=\infty\). Si au contraire \( \tilde a_n\) n'est pas borné vers le bas, nous prenons $y$ tel que \( B(y-x,\epsilon)\) est majoré par un nombre \( k\) strictement négatif. Nous obtenons encore \( g(y)=\infty\).
+
+    Nous concluons que \( | \tilde a_n |\) est bornée.
+\end{proof}
+
+\begin{lemma}[\cite{KXjFWKA}]   \label{LemXOUooQsigHs}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            \phi\colon S^{++}(n,\eR)&\to \eR \\
+            A&\mapsto \det(A) 
+        \end{aligned}
+    \end{equation}
+    est \defe{log-convave}{concave!log-concave}\index{log-concave}, c'est à dire que l'application \( \ln\circ\phi\) est concave\footnote{La définition \ref{DEFooELGOooGiZQjt} du logarithme ne fonctionne que pour les réels strictement positifs. C'est le cas du déterminant d'une matrice réelle symétrique strictement définie positive.}. De façon équivalente, si \( A,B\in S^{++}\) et si \( \alpha+b=1\), alors
+    \begin{equation}    \label{EqSPKooHFZvmB}
+        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
+    \end{equation}
+\end{lemma}
+Ici \( S^{++}\) est l'ensemble des matrices symétriques strictement définies positives, définition \ref{DefAWAooCMPuVM}.
+
+\begin{proof}
+    Nous commençons par prouver que l'équation \eqref{EqSPKooHFZvmB} est équivalente à la log-concavité du déterminant. Pour cela il suffit de remarquer que les propriétés de croissance et d'additivité du logarithme donnent l'équivalence entre
+    \begin{equation}
+        \ln\Big( \det(\alpha A+\beta B) \Big)\geq \ln\Big( \det(\alpha A) \Big)+\ln\Big( \det(\beta B) \Big),
+    \end{equation}
+    et
+    \begin{equation}    \label{EqTJYooBWiRrn}
+        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
+    \end{equation}
+
+    Le théorème de pseudo-réduction simultanée, corollaire \ref{CorNHKnLVA}, appliqué aux matrices \( A\) et \( B\) nous donne une matrice inversible \( Q\) telle que
+    \begin{subequations}
+        \begin{numcases}{}
+            B=Q^tDQ\\
+            A=Q^tQ
+        \end{numcases}
+    \end{subequations}
+    avec 
+    \begin{equation}
+        D=\begin{pmatrix}
+            \lambda_1    &       &       \\
+                &   \ddots    &       \\
+                &       &   \lambda_n
+        \end{pmatrix},
+    \end{equation}
+    \( \lambda_i>0\). Nous avons alors
+    \begin{equation}
+        \det(A)^{\alpha}\det(B)^{\beta}=\det(Q)^{2\alpha}\det(Q)^{2\beta}\det(D)^{\beta}=\det(Q)^2\det(D)^{\beta}
+    \end{equation}
+    (parce que \( \alpha+\beta=1\)) et
+    \begin{equation}
+        \det(\alpha A+\beta B)=\det(\alpha Q^tQ+\beta Q^tDQ)=\det\big( Q^t(\alpha\mtu+\beta D)Q \big)=\det(Q)^2\det(\alpha\mtu+\beta D).
+    \end{equation}
+    L'inégalité \eqref{EqTJYooBWiRrn} qu'il nous faut prouver se réduit donc  à
+    \begin{equation}
+        \det(\alpha \mtu+\beta D)\geq \det(D)^{\beta}.
+    \end{equation}
+    Vue la forme de \( D\) nous avons
+    \begin{equation}
+        \det(\alpha\mtu+\beta D)=\prod_{i=1}^n(\alpha+\beta\lambda_i)
+    \end{equation}
+    et 
+    \begin{equation}
+        \det(D)^{\beta}=\big( \prod_{i=1}^{n}\lambda_i \big)^{\beta}.
+    \end{equation}
+    Il faut donc prouver que
+    \begin{equation}\label{EqGFLooOElciS}    
+        \prod_{i=1}^n(\alpha+\beta\lambda_i)\geq \big( \prod_{i=1}^n\lambda_i \big)^{\beta}.
+    \end{equation}
+    Cette dernière égalité de produit sera prouvée en passant au logarithme. Vu que le logarithme est concave par l'exemple \ref{ExPDRooZCtkOz}, nous avons pour chaque \( i\) que
+    \begin{equation}
+        \ln(\alpha+\beta\lambda_i)\geq \alpha\ln(1)+\beta\ln(\lambda_i)=\beta\ln(\lambda_i).
+    \end{equation}
+    En sommant cela sur \( i\) et en utilisant les propriétés de croissance et de multiplicativité du logarithme nous obtenons successivement
+    \begin{subequations}
+        \begin{align}
+            \sum_{i=1}^n\ln(\alpha+\beta\lambda_i)\geq \beta\sum_i\ln(\lambda_i)\\
+            \ln\big( \prod_i(\alpha+\beta\lambda_i) \big)\geq\ln\Big( \big( \prod_i\lambda_i \big)^{\beta} \Big)\\
+            \prod_i(\alpha+\beta\lambda_i)\geq\big( \prod_i\lambda_i \big)^{\beta},
+        \end{align}
+    \end{subequations}
+    ce qui est bien \eqref{EqGFLooOElciS}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{En dimension supérieure}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit une partie convexe \( U\) de \( \eR^n\) et une fonction \( f\colon U\to \eR\). 
+    \begin{enumerate}
+        \item
+        La fonction \( f\) est \defe{convexe}{convexe!fonction sur \( \eR^n\)} si pour tout \( x,y\in U\) avec \( x\neq y\) et pour tout \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous avons
+            \begin{equation}
+                f\big( \theta x+(1-\theta)y \big)\leq \theta f(x)+(1-\theta)f(y).
+            \end{equation}
+        \item
+            Elle est \defe{strictement convexe}{strictement!convexe!sur \( \eR^n\)} si nous avons l'inégalité stricte.
+    \end{enumerate}
+\end{definition}
+
+\begin{proposition}[\cite{ooLJMHooMSBWki}]      \label{PROPooYNNHooSHLvHp}
+    Soit \( \Omega\) ouvert dans \( \eR^n\) et \( U\) convexe dans \( \Omega\), et une fonction différentiable \( f\colon U\to \eR\).
+    \begin{enumerate}
+        \item       \label{ITEMooRVIVooIayuPS}
+            La fonction \( f\) est convexe sur \( U\) si et seulement si pour tout \( x,y\in U\),
+            \begin{equation}
+                f(y)\geq f(x)+df_x(y-x).
+            \end{equation}
+        \item       \label{ITEMooCWEWooFtNnKl}
+            La fonction \( f\) est strictement convexe sur \( U\) si et seulement si pour tout \( x,y\in U\) avec \( x\neq y\),
+            \begin{equation}
+                f(y)>f(x)+df_x(y-x).
+            \end{equation}
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Nous avons quatre petites choses à démontrer.
+    \begin{subproof}
+    \item[\ref{ITEMooRVIVooIayuPS} sens direct]
+        Soit une fonction convexe \( f\). Nous avons :
+        \begin{equation}
+            f\big( (1-\theta)x+\theta y \big)\leq (1-\theta)f(x)+\theta f(y),
+        \end{equation}
+        donc
+        \begin{equation}
+            f\big( x+\theta(y-x) \big)-f(x)\leq \theta\big( f(y)-f(x) \big)
+        \end{equation}
+        Vu que \( \theta>0\) nous pouvons diviser par \( \theta\) sans changer le sens de l'inégalité :
+        \begin{equation}        \label{EQooAXXFooHWtiJh}
+            \frac{ f\big( x+\theta(y-x) \big)-f(x) }{ \theta }\leq f(y)-f(x).
+        \end{equation}
+        Nous prenons la limite \( \theta\to 0^+\). Cette limite est égale à a limite simple \( \theta\to 0\) et vaut (parce que \( f\) est différentiable) :
+        \begin{equation}
+            \frac{ \partial f }{ \partial (y-x) }(x)\leq f(y)-f(x),
+        \end{equation}
+        et aussi
+        \begin{equation}
+            df_x(y-x)\leq f(y)-f(x)
+        \end{equation}
+        par le lemme \ref{LemdfaSurLesPartielles}.
+    \item[\ref{ITEMooRVIVooIayuPS} sens inverse]
+        Pour tout \( a\neq b\) dans \( U\) nous avons
+        \begin{equation}        \label{EQooEALSooJOszWr}
+            f(b)\geq f(a)+df_a(b-a).
+        \end{equation}
+    Pour \( x\neq y\) dans \( U\) et pour \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous écrivons \eqref{EQooEALSooJOszWr} pour les couples \( \big( \theta x+(1-\theta)y,y \big)\) et \( \big( \theta x+(1-\theta)y,x \big)\). Ça donne :
+        \begin{equation}
+            f(y)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( \theta(y-x) \big),
+        \end{equation}
+        et
+        \begin{equation}
+            f(x)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( (1-\theta)(x-y) \big).
+        \end{equation}
+        La différentielle est linéaire; en multipliant la première par \( (1-\theta)\) et la seconde par \( \theta\) et en la somme, les termes en \( df\) se simplifient et nous trouvons
+        \begin{equation}
+            \theta f(x)+(1-\theta)f(y)\geq f\big( \theta x+(1-\theta)y \big).
+        \end{equation}
+    \item[\ref{ITEMooCWEWooFtNnKl} sens direct]
+        Nous avons encore l'équation \eqref{EQooAXXFooHWtiJh}, avec une inégalité stricte. Par contre, ça ne va pas être suffisant parce que le passage à la limite ne conserve pas les inégalités strictes. Nous devons donc être plus malins. 
+
+        Soient \( 0<\theta<\omega<1\). Nous avons \( (1-\theta)x+\theta y\in \mathopen[ x , (1-\omega)x+\omega y \mathclose]\), donc nous pouvons écrire \( (1-\theta)x+\theta y\) sous la forme \( (1-s)x+s\big( (1-\omega)x+\omega y \big)\). Il se fait que c'est bon pour \( s=\theta/\omega\) (et aussi que nous avons \( \theta/\omega<1\)). Donc nous avons
+        \begin{subequations}
+            \begin{align}
+            f\big( (1-\theta)x+\theta y \big)&=f\Big( (1-\frac{ \theta }{ \omega })x+\frac{ \theta }{ \omega }\big( (1-\omega)x+\omega y \big) \Big)\\
+            &<(1-\frac{ \theta }{ \omega })f(x)+\frac{ \theta }{ \omega }f\big( (1-\omega)x+\omega y \big).
+            \end{align}
+        \end{subequations}
+        Cela nous permet d'écrire
+        \begin{equation}
+            \frac{ f\big( (1-\theta)x+\theta y \big)-f(x) }{ \theta }<\frac{ f\big( (1-\omega)x+\omega y \big) }{ \omega }<f(y)-f(x).
+        \end{equation}
+        Le seconde inégalité est le pendant de \eqref{EQooAXXFooHWtiJh}. Maintenant en passant à la limite pour \( \theta\) nous conservons une inégalité stricte par rapport à \( f(y)-f(x)\) :
+        \begin{equation}
+            df_x(y-x)<f(y)-f(x).
+        \end{equation}
+    \end{subproof}
+\end{proof}
+
+% Il faut laisser les sauts de lignes suivants, pour rechercher efficacement les références vers le futur.
+Avant de lire la proposition suivante, il faut relire la proposition \ref{PROPooFWZYooUQwzjW} et ce qui s'y rapporte. 
+Lire aussi la remarque \ref{REMooVRPQooIybxmp} qui indique
+qu'il n'y a pas de réciproque dans l'énoncé \ref{ITEMooHAGQooYZyhQk}.     
+\begin{proposition}[\cite{ooLJMHooMSBWki}]
+    Soit une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) et un convexe \( U\subset \Omega\).
+    \begin{enumerate}
+        \item       \label{ITEMooZQCAooIFjHOn}
+            La fonction \( f\) est convexe sur \( U\) si et seulement si
+            \begin{equation}        \label{EQooIBDCooJYdiBb}
+                (d^2f)_x(y-x,y-x)\geq 0
+            \end{equation}
+            pour tout \( x,y\in U\).
+        \item       \label{ITEMooHAGQooYZyhQk}
+            Si pour tout \( x\neq y\) dans \( U\) nous avons
+            \begin{equation}
+                (d^2f)_x(y-x,y-x)>0
+            \end{equation}
+            alors la fonction \( f\) est strictement convexe sur \( U\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{remark}      \label{REMooYCRKooEQNIkC}
+    Notons que la condition \eqref{EQooIBDCooJYdiBb} n'est pas équivalente à demander \( (d^2f)_x(h,h)\geq 0\) pour tout \( h\). En effet nous ne demandons la positivité que dans les directions atteignables comme différence de deux éléments de \( U\). La partie \( U\) n'est pas spécialement ouverte; elle pourrait n'être qu'une droite dans \( \eR^3\). Dans ce cas, demander que \( f\) (qui est \( C^2\) sur l'ouvert \( \Omega\)) soit convexe sur \( U\) ne demande que la positivité de \( (d^2f)_x\) appliqué à des vecteurs situés sur la droite \( U\).
+\end{remark}
+
+\begin{proof}
+    Il y a trois parties à démontrer.
+    \begin{subproof}
+    \item[\ref{ITEMooZQCAooIFjHOn} sens direct]
+
+        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci au jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
+        \begin{equation}
+            f(x+th)=f(x)+tdf_x(h)+\frac{ t^2 }{2}(d^2_x)(h,h)+t^2\| h \|^2\alpha(th)
+        \end{equation}
+        avec \( \lim_{s\to 0}\alpha(s)=0\). Le fait que \( f\) soit convexe donne
+        \begin{equation}
+            0\leq f(x+th)-f(x)-tdf_x(h),
+        \end{equation}
+        et donc
+        \begin{equation}
+            0\leq \frac{ t^2 }{2}(d^2f)_x(h,h)+f^2\| h \|^2\alpha(th).
+        \end{equation}
+        En multipliant par \( 2\) et en divisant par \( t^2\),
+        \begin{equation}
+            0\leq (d^2f)_x(h,h)+2\| h \|^2\alpha(th).
+        \end{equation}
+        En prenant \( t\to 0\) nous avons bien  \( (d^2f)_x(y-x,y-x)\geq 0\).
+
+    \item[\ref{ITEMooZQCAooIFjHOn} sens inverse]
+
+        Soient \( x,y\in U\). Nous écrivons Taylor en version de la proposition \ref{PROPooWWMYooPOmSds} :
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\frac{ 1 }{2}(d^2f)_z(y-x,y-x)
+        \end{equation}
+    pour un certain \( z\in\mathopen] x , y \mathclose[\). En vertu de ce qui a été dit dans la remarque \ref{REMooYCRKooEQNIkC} nous ne pouvons pas évoquer l'hypothèse \eqref{EQooIBDCooJYdiBb} pour conclure que \( (d^2f)_z(y-x,y-x)\geq 0\). Il y a deux manières de nous sortir du problème :
+        \begin{itemize}
+            \item Trouver \( s\in U\) tel que \( y-x=s-z\).
+            \item Trouver un multiple de \( y-x\) qui soit de la forme \( y-x\).
+        \end{itemize}
+        La première approche ne fonctionne pas parce que \( s=y-x+z\) n'est pas garanti d'être dans \( U\); par exemple avec \( x=1\), \( z=2\), \( y=3\) et \( U=\mathopen[ 0 , 3 \mathclose]\). Dans ce cas \( s=4\notin U\).
+
+        Heureusement nous avons \( z=\theta x+(1-\theta)y\), donc \( z-x=(1-\theta)(y-x)\). Dans ce cas la bilinéarité de \( (d^2f)_z\) donne\footnote{Si vous avez bien suivi, la bilinéarité est contenue dans la proposition \ref{PROPooFWZYooUQwzjW}.}
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{\geq 0}.
+        \end{equation}
+        Nous en déduisons que \( f\) est convexe par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooRVIVooIayuPS}.
+    \item[\ref{ITEMooHAGQooYZyhQk}]
+
+        Le raisonnement que nous venons de faire pour le sens inverse de \ref{ITEMooZQCAooIFjHOn} tient encore, et nous avons
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{> 0}
+        \end{equation}
+        d'où nous déduisons la stricte convexité de \( f\) par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooCWEWooFtNnKl}.
+    \end{subproof}
+\end{proof}
+
+\begin{corollary}       \label{CORooMBQMooWBAIIH}
+    Avec la hessienne\ldots en cours d'écriture.
+\end{corollary}
+
+\begin{proof}
+    Cela va utiliser la proposition \ref{PropoExtreRn}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Quelque inégalités}
+%---------------------------------------------------------------------------------------------------------------------------
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Inégalité de Jensen}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\index{inégalité!Jensen}
+\index{convexité!inégalité de Jensen}
+
+\begin{proposition}[Inégalité de Jensen]    \label{PropXIBooLxTkhU}
+    Soit \( f\colon \eR\to \eR\) une fonction convexe et des réels \( x_1\),\ldots,  \( x_n\). Soient des nombres positifs \( \lambda_1\),\ldots,  \( \lambda_n\) formant une combinaison convexe\footnote{Définition \ref{DefIMZooLFdIUB}.}. Alors
+    \begin{equation}
+        f\big( \sum_i\lambda_ix_i \big)\leq \sum_i\lambda_if(x_i).
+    \end{equation}
+\end{proposition}
+\index{inégalité!Jensen!pour une somme}
+
+\begin{proof}
+    Nous procédons par récurrence sur \( n\), en sachant que \( n=2\) est la définition de la convexité de \( f\). Vu que
+    \begin{equation}
+        \sum_{k=1}^n\lambda_kx_k=\lambda_nx_n+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n },
+    \end{equation}
+    nous avons
+    \begin{equation}
+        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)f\big( \sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n } \big).
+    \end{equation}
+    La chose à remarquer est que les nombres \( \frac{ \lambda_k }{ 1-\lambda_n }\) avec \( k\) allant de \( 1\) à \( n-1\) forment eux-mêmes une combinaison convexe. L'hypothèse de récurrence peut donc s'appliquer au second terme du membre de droite :
+    \begin{equation}
+        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_k }{ 1-\lambda_n }f(x_k)=\lambda_nf(x_n)+\sum_{k=1}^{n-1}\lambda_kf(x_k).
+    \end{equation}
+\end{proof}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Inégalité arithmético-géométrique}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+La proposition suivante dit que la moyenne arithmétique de nombres strictement positifs est supérieure ou égale à la moyenne géométrique.
+\begin{proposition}[Inégalité arithmético-géométrique\cite{CENooZKvihz}]    \label{PropWDPooBtHIAR}
+    Soient \( x_1\),\ldots, \( x_n\) des nombres strictement positifs. Nous posons
+    \begin{equation}
+        m_a=\frac{1}{ n }(x_1+\cdots +x_n)
+    \end{equation}
+    et
+    \begin{equation}
+        m_g=\sqrt[n]{x_1\ldots x_n}
+    \end{equation}
+    Alors \( m_g\leq m_a\) et \( m_g=m_a\) si et seulement si \( x_i=x_j\) pour tout \( i,j\).
+\end{proposition}
+\index{inégalité!arithmético-géométrique}
+
+\begin{proof}
+    Par hypothèse les nombres \( m_a\) et \( m_g\) sont tout deux strictement positifs, de telle sorte qu'il est équivalent de prouver \( \ln(m_g)\leq \ln(m_a)\) ou encore
+    \begin{equation}
+        \frac{1}{ n }\big( \ln(x_1)+\cdots +\ln(x_n) \big)\leq \ln\left( \frac{ x_1+\cdots +x_n }{ n } \right).
+    \end{equation}
+    Cela n'est rien d'autre que l'inégalité de Jensen de la proposition \ref{PropXIBooLxTkhU} appliquée à la fonction \( \ln\) et aux coefficients \( \lambda_i=\frac{1}{ n }\).
+\end{proof}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Inégalité de Kantorovitch}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+\begin{proposition}[Inégalité de Kantorovitch\cite{EYGooOoQDnt}]    \label{PropMNUooFbYkug}
+    Soit \( A\) une matrice symétrique strictement définie positive dont les plus grandes et plus petites valeurs propres sont \( \lambda_{min}\) et \( \lambda_{max}\). Alors pour tout \( x\in \eR^n\) nous avons
+    \begin{equation}
+        \langle Ax, x\rangle \langle A^{-1}x, x\rangle \leq \frac{1}{ 4 }\left( \frac{ \lambda_{min} }{ \lambda_{max} }+\frac{ \lambda_{max} }{ \lambda_{min} } \right)^2\| x^4 \|.
+    \end{equation}
+\end{proposition}
+\index{inégalité!Kantorovitch}
+
+\begin{proof}
+    Sans perte de généralité nous pouvons supposer que \( \| x \|=1\). Nous diagonalisons\footnote{Théorème spectral \ref{ThoeTMXla}.} la matrice \( A\) par la matrice orthogonale  \( P\in\gO(n,\eR)\) : \( A=PDP^{-1}\) et \( A^{-1}=PD^{-1}P^{-1}\) où \( D\) est  une matrice diagonale formée des valeurs propres de \( A\).
+
+    Nous posons \( \alpha=\sqrt{\lambda_{min}\lambda_{max}}\) et nous regardons la matrice
+    \begin{equation}
+        \frac{1}{ \alpha }A+tA^{-1}
+    \end{equation}
+    dont les valeurs propres sont
+    \begin{equation}
+        \frac{ \lambda_i }{ \alpha }+\frac{ \alpha }{ \lambda_i }
+    \end{equation}
+    parce que les vecteurs propres de \( A\) et de \( A^{-1}\) sont les mêmes (ce sont les valeurs de la diagonale de \( D\)). Nous allons quelque peu étudier la fonction
+    \begin{equation}
+        \theta(x)=\frac{ x }{ \alpha }+\frac{ \alpha }{ x }.
+    \end{equation}
+    Elle est convexe en tant que somme de deux fonctions convexes. Elle a son minimum en \( x=\alpha\) et ce minimum vaut \( \theta(\alpha)=2\). De plus
+    \begin{equation}
+        \theta(\lambda_{max})=\theta(\lambda_{min})=\sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
+    \end{equation}
+    Une fonction convexe passant deux fois par la même valeur doit forcément être plus petite que cette valeur entre les deux\footnote{Je ne suis pas certain que cette phrase soit claire, non ?} : pour tout \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\),
+    \begin{equation}
+        \theta(x)\leq  \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
+    \end{equation}
+    
+    Nous sommes maintenant en mesure de nous lancer dans l'inégalité de Kantorovitch.
+    \begin{subequations}
+        \begin{align}
+            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }&\leq\frac{ 1 }{2}\left( \frac{ \langle Ax, x\rangle  }{ \alpha }+\alpha\langle A^{-1}x, x\rangle  \right)\label{subEqUKIooCWFSkwi}\\
+            &=\frac{ 1 }{2}\langle   \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x , x\rangle \\
+            &\leq\frac{ 1 }{2}\Big\| \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x \|\| x \| \label{subEqUKIooCWFSkwiii}\\
+            &\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \label{subEqUKIooCWFSkwiv}
+        \end{align}
+    \end{subequations}
+    Justifications :
+    \begin{itemize}
+        \item \ref{subEqUKIooCWFSkwi} par l'inégalité arithmético-géométrique, proposition \ref{PropWDPooBtHIAR}. Nous avons aussi inséré \( \alpha\frac{1}{ \alpha }\) dans le produit sous la racine.
+        \item \ref{subEqUKIooCWFSkwiii} par l'inégalité de Cauchy-Schwarz, théorème \ref{ThoAYfEHG}.
+        \item \ref{subEqUKIooCWFSkwiv} par la définition de la norme opérateur de la proposition \ref{DefNFYUooBZCPTr}
+    \end{itemize}
+    La norme opérateur est la plus grande des valeurs propres. Mais les valeurs propres de \( A/\alpha+\alpha A^{-1}\) sont de la forme \( \theta(\lambda_i)\), et tous les \( \lambda_i\) sont entre \( \lambda_{min} \) et \( \lambda_{max}\). Donc la plus grande valeur propre de \( A/\alpha+\alpha A^{-1}\) est \( \theta(x)\) pour un certain \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\). Par conséquent
+    \begin{equation}
+            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \leq \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
+    \end{equation}
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Algorithme du gradient à pas optimal}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
+
+Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
+\begin{itemize}
+    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
+    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
+\end{itemize}
+
+\begin{definition}  \label{DefQXPooYSygGP}
+    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
+    \begin{equation}
+        \{ x\in P\tq f(x)\leq M \}
+    \end{equation}
+    est borné.
+\end{definition}
+En langage imagé la coercivité de \( f\) s'exprime par la limite
+\begin{equation}
+    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
+\end{equation}
+
+
+Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
+\begin{proposition}     \label{PROPooYRLDooTwzfWU}
+    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Alors :
+    \begin{enumerate}
+        \item
+            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
+        \item
+            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
+        \item
+            Ils sont égaux : \( \bar x=x^*\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+
+    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
+
+    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
+
+    \begin{subproof}
+    \item[\( f\) est strictement convexe]
+        
+        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
+    \end{equation}
+    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
+
+\item[\( f\) est coercive]
+    Montrons à présent que \( f\) est coercive. Nous avons :
+    \begin{subequations}
+        \begin{align}
+            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
+            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
+            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
+        \end{align}
+    \end{subequations}
+    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
+    \end{subproof}
+
+    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
+    \begin{equation}
+        \{ x\in \eR^n\tq f(x)\leq M \}
+    \end{equation}
+    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
+
+    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
+    \begin{equation}
+        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
+    \end{equation}
+    ce qui contredit la minimalité de \( f(\bar x_1)\).
+
+    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
+    \begin{equation}
+        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
+    \end{equation}
+    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
+\end{proof}
+
+\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
+    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
+    \begin{equation}
+        x_{k+1}=x_k+t_kd_k
+    \end{equation}
+    où
+    \begin{itemize}
+        \item 
+    \( d_k=-(\nabla f)(x_k)\) 
+\item
+    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
+    \end{itemize}
+
+    Alors pour tout \( k\geq 0\) nous avons
+    \begin{equation}
+        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
+    \end{equation}
+    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
+\end{proposition}
+
+\begin{proof}
+    Décomposition en plusieurs points.
+    \begin{subproof}
+    \item[Existence de \( \bar x\)]
+        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
+    \item[Si \( (\nabla f)(x_k)=0\)]
+    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
+    \end{equation}
+    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
+    \begin{equation}
+        (\nabla f)(a)=Aa+b.
+    \end{equation}
+    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
+
+    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
+        \item[\( t_k\) est bien défini]
+
+            Pour \( t\in \eR\) nous avons
+            \begin{equation}    \label{EqKEHooYaazQi}
+                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
+            \end{equation}
+            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
+            \begin{equation}    \label{EqVWJooWmDSER}
+                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
+            \end{equation}
+            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
+
+        \item[La valeur de \( d_{k+1}\)]
+
+            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
+            \begin{equation}
+                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
+            \end{equation}
+            parce que \( -Ax_k-b=d_k\).
+            
+            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
+            \begin{equation}
+                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
+            \end{equation}
+            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
+        
+        \item[Calcul de \( f(x_{k+1})\)] 
+
+            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
+            \begin{equation}
+                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
+            \end{equation}
+            
+        \item[Encore du calcul \ldots]
+
+            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
+            \begin{equation}
+                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
+            \end{equation}
+            Ayant cela en tête nous pouvons calculer :
+            \begin{subequations}
+                \begin{align}
+                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
+                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
+                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
+                    &=2\big( f(x_k)-\bar f \big)
+                \end{align}
+            \end{subequations}
+            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
+
+        \item[Erreur sur la valeur du minimum]
+
+            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
+            \begin{subequations}
+                \begin{align}
+                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
+                \end{align}
+            \end{subequations}
+            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
+            \begin{equation}
+                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
+            \end{equation}
+            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
+            \begin{equation}
+                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
+            \end{equation}
+            Par récurrence nous avons alors
+            \begin{equation}    \label{eqANKooNPfCFj}
+                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
+            \end{equation}
+            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
+
+        \item[Erreur sur la position du minimum]
+
+            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
+            \begin{subequations}
+                \begin{align}
+                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
+                \end{align}
+            \end{subequations}
+            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
+
+            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
+            \begin{equation}    \label{EqVMRooUMXjig}
+                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
+            \end{equation}
+            
+        \item[Conclusion]
+
+            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
+            \begin{equation}
+                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
+            \end{equation}
+            c'est à dire
+            \begin{equation}
+                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
+            \end{equation}
+    \end{subproof}
+\end{proof}
+
+Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Ellipsoïde de John-Loewer}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Soit \( q\) une forme quadratique sur \( \eR^n\) ainsi que \( \mB\) une base orthonormée de \( \eR^n\) dans laquelle la matrice de  \( q\) est diagonale. Dans cette base, la forme \( q\) est donnée par la proposition \ref{PropFWYooQXfcVY} :
+\begin{equation}
+    q(x)=\sum_i\lambda_ix_i
+\end{equation}
+où les \( \lambda_i\) sont les valeurs propres de \( q\).
+
+Plus généralement nous notons \( mat_{\mB}(q)\)\nomenclature[A]{\( mat_{\mB}(q)\)}{matrice de \( q\) dans la base \( \mB\)} la matrice de \( q\) dans la base \( \mB\) de \( \eR^n\).
+
+\begin{proposition} \label{PropOXWooYrDKpw}
+    Soit \( \mB\) une base orthonormée de \( \eR^n\) et l'application\footnote{L'ensemble \( Q(E)\) est l'ensemble des formes quadratiques sur \( E\).}
+    \begin{equation}
+        \begin{aligned}
+            D\colon Q(\eR^n)&\to \eR \\
+            q&\mapsto \det\big( mat_{\mB}(q) \big) .
+        \end{aligned}
+    \end{equation}
+    Alors :
+    \begin{enumerate}
+        \item
+            La valeur et \( D\) ne dépend pas du choix de la base orthonormée \( \mB\).
+        \item
+            La fonction \( D\) est donnée par la formule \( D(q)=\prod_i\lambda_i\) où les \( \lambda_i\) sont les valeurs propres de \( q\).
+        \item
+            La fonction \( D\) est continue.
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Soit \( q\) une forme quadratique sur \( \eR^n\). Nous considérons \( \mB\) une base de diagonalisation de \( q\) :
+    \begin{equation}
+        q(x)=\sum_i\lambda_ix_i
+    \end{equation}
+    où les \( x_i\) sont les composantes de \( x\) dans la base \( \mB\). Par définition, la matrice \( mat_{\mB}(q)\) est la matrice diagonale contenant les valeurs propres de \( q\).
+
+    Nous considérons aussi \( \mB_1\), une autre base orthonormées de \( \eR^n\). Nous notons \( S=mat_{\mB_1}(q)\); étant symétrique, cette matrice se diagonalise par une matrice orthogonale : il existe \( P\in\gO(n,\eR)\) telle que
+    \begin{equation}
+        S=P mat_{\mB}(q)P^t;
+    \end{equation}
+    donc \( \det(S)=\det(PP^t)\det\big( \diag(\lambda_1,\ldots, \lambda_n) \big)=\lambda_1\ldots\lambda_n\). Ceci prouve en même temps que \( D\) ne dépend pas du choix de la base et que sa valeur est le produit des valeurs propres.
+
+    Passons à la continuité. L'application déterminant \( \det\colon S_n(\eR^n)\to \eR\) est continue car polynôme en les composantes. D'autre par l'application \( mat_{\mB}\colon Q(\eR^n)\to S_n(\eR)\) est continue par la proposition \ref{PropFSXooRUMzdb}. L'application  \( D\) étant la composée de deux applications continues, elle est continue.
+\end{proof}
+
+\begin{proposition}[Ellipsoïde de John-Loewner\cite{KXjFWKA}]   \label{PropJYVooRMaPok}
+    Soit \( K\) compact dans \( \eR^n\) et d'intérieur non vide. Il existe une unique ellipsoïde\footnote{Définition \ref{DefOEPooqfXsE}.} (pleine) de volume minimal contenant \( K\).
+\end{proposition}
+\index{déterminant!utilisation}
+\index{extrema!volume d'un ellipsoïde}
+\index{convexité!utilisation}
+\index{compacité!utilisation}
+
+\begin{proof}
+    Nous subdivisons la preuve en plusieurs parties.
+    \begin{subproof}
+        \item[À propos de volume d'un ellipsoïde]
+
+            Soit \( \ellE\) un ellipsoïde. La proposition \ref{PropWDRooQdJiIr} et son corollaire \ref{CorKGJooOmcBzh} nous indiquent que 
+            \begin{equation}
+                \ellE=\{ x\in \eR^n\tq q(x)\leq 1 \}
+            \end{equation}
+            pour une certaine forme quadratique strictement définie positive \( q\). De plus il existe une base orthonormée \( \mB=\{ e_1,\ldots, e_n \}\) de \( \eR^n\) telle que 
+            \begin{equation}    \label{EqELBooQLPQUj}
+                q(x)=\sum_{i=1}^na_ix_i^2
+            \end{equation}
+            où \( x_i=\langle e_i, x\rangle \) et les \( a_i\) sont tous strictement positifs. Nous nommons \( \ellE_q\) l'éllipsoïde associée à la forme quadratique \( q\) et \( V_q\) son volume que nous allons maintenant calculer\footnote{Le volume ne change pas si nous écrivons l'inégalité stricte au lieu de large dans le domaine d'intégration; nous le faisons pour avoir un domaine ouvert.} :
+            \begin{equation}
+                V_q=\int_{\sum_ia_ix_i^2<1}dx
+            \end{equation}
+            Cette intégrale est écrite de façon plus simple en utilisant le \( C^1\)-difféomorphisme
+            \begin{equation}
+                \begin{aligned}
+                    \varphi\colon \ellE_q&\to B(0,1) \\
+                    x&\mapsto \Big( x_1\sqrt{a_1},\ldots, x_n\sqrt{a_n} \Big). 
+                \end{aligned}
+            \end{equation}
+            Le fait que \( \varphi\) prenne bien ses valeurs dans \( B(0,1)\) est un simple calcul : si \( x\in\ellE_q\), alors
+            \begin{equation}
+                \sum_i\varphi(x)_i^2=\sum_ia_ix_i^2<1.
+            \end{equation}
+            Cela nous permet d'utiliser le théorème de changement de variables \ref{THOooUMIWooZUtUSg} :
+            \begin{equation}
+                V_q=\int_{\sum_ia_ix_i^2<1}dx=\frac{1}{ \sqrt{a_1\ldots a_n} }\int_{B(0,1)}dx.
+            \end{equation}
+            %TODO : le volume de la sphère dans \eR^n. Mettre alors une référence ici.
+            La dernière intégrale est le volume de la sphère unité dans \( \eR^n\); elle n'a pas d'importance ici et nous la notons \( V_0\). La proposition \ref{PropOXWooYrDKpw} nous permet d'écrire \(V_q\) sous la forme
+            \begin{equation}
+                V_q=\frac{ V_0 }{ \sqrt{D(q)} }.
+            \end{equation}
+            
+        \item[Existence de l'ellipsoïde]
+
+            Nous voulons trouver un ellipsoïde contenant \( K\) de volume minimal, c'est à dire une forme quadratique \( q\in Q^{++}(\eR^n)\) telle que
+            \begin{itemize}
+                \item \( D(q)\) soit maximal
+                \item \( q(x)\leq 1\) pour tout \( x\in K\).
+            \end{itemize}
+            Nous considérons l'ensemble des candidats semi-définis positifs.
+            \begin{equation}
+                A=\{ q\in Q^+\tq q(x)\leq 1\forall x\in K \}.
+            \end{equation}
+            Nous allons montrer que \( A\) est convexe, compact et non vide dans \( Q(\eR^n)\); il aura ainsi un maximum de la fonction continue \( D\) définie sur \( Q(\eR^n)\). Nous montrerons ensuite que le maximum est dans \( Q^{++}\). L'unicité sera prouvée à part.
+
+            \begin{subproof}
+            \item[Non vide]
+                L'ensemble \( K\) est compact et donc borné par \( M>0\). La forme quadratique \( q\colon x\mapsto \| x \|^2/M^2\) est dans \( A\) parce que si \( x\in K\) alors 
+                \begin{equation}
+                    q(x)=\frac{ \| x \|^2 }{ M^2 }\leq 1.
+                \end{equation}
+            \item[Convexe]
+                Soient \( q,q'\in A\) et \( \lambda\in\mathopen[ 0 , 1 \mathclose]\). Nous avons encore \( \lambda q+(1-\lambda)q'\in Q^+\) parce que 
+                \begin{equation}
+                    \lambda q(x)+(1-\lambda)q'(x)\geq 0
+                \end{equation}
+                dès que \( q(x)\geq 0\) et \( q'(x)\geq 0\).
+            D'autre part si \( x\in K\) nous avons
+            \begin{equation}
+                \lambda q(x)+(1-\lambda)q'(x)\leq \lambda+(1-\lambda)=1.
+            \end{equation}
+            Donc \( \lambda q+(1-\lambda)q'\in A\).
+
+        \item[Fermé]
+
+            Pour rappel, la topologie de \( Q(\eR^n)\) est celle de la norme \eqref{EqZYBooZysmVh}. Nous considérons une suite \( (q_n)\) dans \( A\) convergeant vers \( q\in Q(\eR^n)\) et nous allons prouver que \( q\in A\), de sorte que la caractérisation séquentielle de la fermeture (proposition \ref{PropLFBXIjt}) conclue que \( A\) est fermé. En nommant \( e_x\) le vecteur unitaire dans la direction \( x\) nous avons
+            \begin{equation}
+                \big| q(x) \big|=\big| \| x \|^2q(e_x) \big|\leq \| x \|^2N(q),
+            \end{equation}
+            de sorte que notre histoire de suite convergente  donne pour tout \( x\) :
+            \begin{equation}
+                \big| q_n(x)-q(x) \big|\leq \| x \|^2N(q_n-q)\to 0.
+            \end{equation}
+            Vu que \( q_n(x)\geq 0\) pour tout \( n\), nous devons aussi avoir \( q(x)\geq 0\) et donc \( q\in Q^+\) (semi-définie positive). De la même manière si \( x\in K\) alors \( q_n(x)\leq 1\) pour tout \( n\) et donc \( q(x)\leq 1\). Par conséquent \( q\in A\) et \( A\) est fermé.
+
+        \item[Borné]
+
+            La partie \( K\) de \( \eR^n\) est borné et d'intérieur non vide, donc il existe \( a\in K\) et \( r>0\) tel que \( \overline{ B(a,r) }\subset K\). Si par ailleurs \( q\in A\) et \( x\in\overline{ B(0,r) }\) nous avons \( a+x\in K\) et donc \( q(a+x)\leq 1\). De plus \( q(-a)=q(a)\leq 1\), donc
+            \begin{equation}
+                \sqrt{q(x)}=\sqrt{q\big( x+a-a \big)}\leq \sqrt{q(x+a)}+\sqrt{q(-a)}\leq 2
+            \end{equation}
+            par l'inégalité de Minkowski \ref{PropACHooLtsMUL}. Cela prouve que si \( x\in\overline{ B(0,r) }\) alors \( q(x)\leq 4\). Si par contre \( x\in\overline{ B(0,1) }\) alors \( rx\in\overline{ B(0,r) } \) et 
+            \begin{equation}
+                0\leq q(x)=\frac{1}{ r^2 }q(rx)\leq \frac{ 4 }{ r^2 },
+            \end{equation}
+            ce qui prouve que \( N(q)\leq \frac{ 4 }{ r^2 }\) et que \( A\) est borné.
+
+
+            \end{subproof}
+
+            L'ensemble \( A\) est compact parce que fermé et borné, théorème de Borel-Lebesgue \ref{ThoXTEooxFmdI}. L'application continue \( D\colon Q(\eR^n)\to \eR\) de la proposition \ref{PropOXWooYrDKpw} admet donc un maximum sur le compact \( A\). Soit \( q_0\) ce maximum.
+
+            Nous montrons que \( q_0\in Q^{++}(\eR^d)\). Nous savons que l'application \( f\colon x\mapsto \frac{ \| x \|^2 }{ M^2 }\) est dans \( A\) et que \( D(f)>0\). Vu que \( q_0\) est maximale pour \( D\), nous avons
+            \begin{equation}
+                D(q_0)\geq D(f)>0.
+            \end{equation}
+            Donc \( q_0\in Q^{++}\).
+
+        \item[Unicité]
+
+            S'il existe une autre ellipsoïde de même volume que celle associée à la forme quadratique \( q_0\), nous avons une forme quadratique \( q\in Q^{++}\) telle que \( q(x)\leq 1\) pour tout \( x\in K\). C'est à dire que nous avons \( q_0,q\in A\) tels que \( D(q_0)=D(q)\).
+
+            Nous considérons la base canonique \( \mB_c\) de \( \eR^n\) et nous posons \( S=mat_{\mB_c}(q)\), \( S_0=mat_{\mB_c}(q_0)\). Étant donné que \( A\) est convexe, \( (q_0+q)/2\in A\) et nous allons prouver que cet élément de \( A\) contredit la maximalité de \( q_0\). En effet
+            \begin{equation}
+                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)
+            \end{equation}
+            Nous allons utiliser le lemme \ref{LemXOUooQsigHs} qui dit que le logarithme est log-concave sous la forme de l'équation \eqref{EqSPKooHFZvmB} avec \( \alpha=\beta=\frac{ 1 }{2}\) :
+            \begin{equation}    \label{eqBHJooYEUDPC}
+                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)>\sqrt{\det(S)}\sqrt{\det(S_0)}=\det(S_0)=D(q_0).
+            \end{equation}
+            Nous avons utilisé le fait que \( D(q_0)=D(q)\) qui signifie que \( \det(S_0)=\det(S)\). L'inéquation \eqref{eqBHJooYEUDPC} contredit la maximalité de \( D(q_0)\) et donne donc l'unicité.
+    \end{subproof}
+\end{proof}
+

From 86a379628e2c6e294670370a1cadda0cc8264a24 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 27 Jun 2017 07:55:47 +0200
Subject: [PATCH 61/64] =?UTF-8?q?(division=20cellulaire)=20Couper=20des=20?=
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 =?UTF-8?q?=202000=20lignes.?=
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---
 mazhe.tex                         |    1 +
 "r\303\251serve.tex"              |    1 -
 tex/frido/184_SuiteAnalyse.tex    | 1798 +++++++++++++++++++++++
 tex/frido/77_series_fonctions.tex | 2212 -----------------------------
 tex/frido/79_inversion_locale.tex |  419 ++++++
 5 files changed, 2218 insertions(+), 2213 deletions(-)
 create mode 100644 tex/frido/184_SuiteAnalyse.tex

diff --git a/mazhe.tex b/mazhe.tex
index 1e6d19bad..d0831f205 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -226,6 +226,7 @@ \chapter{Représentations et caractères}
 
 \chapter{Suite de l'analyse}
 \input{77_series_fonctions}
+\input{184_SuiteAnalyse}
 \input{79_inversion_locale}
 \input{80_Newton}
 
diff --git "a/r\303\251serve.tex" "b/r\303\251serve.tex"
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+% This is part of Mes notes de mathématique
+% Copyright (c) 2011-2015,2017
+%   Laurent Claessens
+% See the file fdl-1.3.txt for copying conditions.
+
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+                    \section{Théorèmes d'inversion locale et de la fonction implicite}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Mise en situation}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Dans un certain nombre de situation, il n'est pas possible de trouver des solutions explicites aux équations qui apparaissent. Néanmoins, l'existence «théorique» d'une telle solution est souvent déjà suffisante. C'est l'objet du théorème de la fonction implicite.
+
+Prenons par exemple la fonction sur $\eR^2$ donnée par 
+\begin{equation}
+    F(x,y)=x^2+y^2-1.
+\end{equation}
+Nous pouvons bien entendu regarder l'ensemble des points donnés par $F(x,y)=0$. C'est le cercle dessiné à la figure \ref{LabelFigCercleImplicite}.
+\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite. Si on donne \( x\), nous ne pouvons pas savoir si nous parlons de \( P\) ou de \( P'\).}
+\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
+
+%\ref{LabelFigCercleImplicite}.
+%\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite.}
+%\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
+
+Nous ne pouvons pas donner le cercle sous la forme $y=y(x)$ à cause du $\pm$ qui arrive quand on prend la racine carrée. Mais si on se donne le point $P$, nous pouvons dire que \emph{autour de $P$}, le cercle est la fonction
+\begin{equation}
+    y(x)=\sqrt{1-x^2}.
+\end{equation}
+Tandis que autour du point $P'$, le cercle est la fonction
+\begin{equation}
+    y(x)=-\sqrt{1-x^2}.
+\end{equation}
+Autour de ces deux point, donc, le cercle est donné par une fonction. Il n'est par contre pas possible de donner le cercle autour du point $Q$ sous la forme d'une fonction.
+
+Ce que nous voulons faire, en général, est de voir si l'ensemble des points tels que
+\begin{equation}
+    F(x_1,\ldots,x_n,y)=0
+\end{equation}
+peut être donné par une fonction $y=y(x_1,\ldots,x_n)$. En d'autre termes, est-ce qu'il existe une fonction $y(x_1,\ldots,x_n)$ telle que
+\begin{equation}
+    F\big( x_1,\ldots,x_n,y(x_1,\ldots,x_n)\big)=0.
+\end{equation}
+
+Plus généralement, soit une fonction
+\begin{equation}
+    \begin{aligned}
+        F\colon D\subset \eR^n\times \eR^m&\to \eR^m \\
+        (x,y)&\mapsto \big( F_1(x,y),\ldots, F_m(x,y) \big) 
+    \end{aligned}
+\end{equation}
+avec $x = (x_1,\ldots, x_n)$ et $y = (y_1,\ldots,y_m)$. Pour chaque $x$ fixé, on s'intéresse aux solutions du système de $m$ équations $F(x,y) = 0$ pour les inconnues $y$ ; en particulier, on voudrait pouvoir écrire $y = \varphi(x)$ vérifiant $F(x,\varphi(x)) = 0$.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème d'inversion locale}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma} \label{LemGZoqknC}
+    Soit \( E\) un espace de Banach (métrique complet) et \( \mO\) un ouvert de \( E\). Nous considérons une \( \lambda\)-contraction \( \varphi\colon \mO\to E\). Alors l'application
+    \begin{equation}
+    f\colon x\mapsto x+\varphi(x)
+    \end{equation}
+    est un homéomorphisme entre \( \mO\) et un ouvert de \( E\). De plus \( f^{-1}\) est Lipschitz de constante plus petite ou égale à \( (1-\lambda)^{-1}\).
+\end{lemma}
+Cette proposition utilise le théorème de point fixe de Picard \ref{ThoEPVkCL},
+et sera utilisée pour démontrer le théorème d'inversion locale \ref{ThoXWpzqCn}.
+% note que garder deux lignes ici est important pour vérifier les références vers le futur : la seconde ligne peut être ignorée, pas la seconde.
+
+\begin{proof}
+        Soient \( x_1,x_2\in\mO\). Nous posons \( y_1=f(x_1)\) et \( y_2=f(x_2)\). En vertu de l'inégalité de la proposition \ref{PropNmNNm} nous avons
+        \begin{subequations}    \label{subEqEBJsBfz}
+            \begin{align}
+            \big\| f(x_2)-f(x_1) \big\|&=\big\| x_2+\varphi(x_2)-x_1-\varphi(x_1) \big\|\\
+        &\geq \Big|        \| x_2-x_1 \|-\big\| \varphi(x_2)-\varphi(x_1) \big\|  \Big|\\
+    &\geq   (1-\lambda)\| x_2-x_1 \|.
+            \end{align}
+        \end{subequations}
+        À la dernière ligne les valeurs absolues sont enlevées parce que nous savons que ce qui est à l'intérieur est positif. Cela nous dit d'abord que \( f\) est injective parce que \( f(x_2)=f(x_1)\) implique \( x_2=x_1\). Donc \( f\) est inversible sur son image. Nous posons \( A=f(\mO)\) et nous devons prouver que que \( f^{-1}\colon A\to \mO\) est continue, Lipschitz de constante majorée par \( (1-\lambda)^{-1}\) et que \( A\) est ouvert.
+
+    Les inéquations \eqref{subEqEBJsBfz} nous disent que
+    \begin{equation}
+    \big\| f^{-1}(y_1)-f^{-1}(y_2) \big\|\leq \frac{ \| y_1-y_2 \| }{ 1-\lambda },
+    \end{equation}
+    c'est à dire que
+    \begin{equation}
+        f^{-1}\big( B(y,r) \big)\subset B\big( f^{-1}(y),\frac{ r }{ 1-\lambda } \big),
+    \end{equation}
+    ce qui signifie que \( f^{-1}\) est Lipschitz de constante souhaitée et donc continue.
+
+    Il reste à prouver que \( f(\mO)\) est ouvert. Pour cela nous prenons \( y_0=f(x_0)\) dans \( f(\mO)\) est nous prouvons qu'il existe \( \epsilon\) tel que \( B(y_0,\epsilon)\) soit dans \( f(\mO)\). Il faut donc que pour tout \( y\in B(y_0,\epsilon)\), l'équation \( f(x)=y\) ait une solution. Nous considérons l'application
+    \begin{equation}
+        L_y\colon x\mapsto y-\varphi(x).
+    \end{equation}
+    Ce que nous cherchons est un point fixe de \( L_y\) parce que si \( L_y(x)=x\) alors \( y=x+\varphi(x)=f(x)\). Vu que
+    \begin{equation}
+        \big\| L_y(x)-L_y(x') \big\|=\big\| \varphi(x)-\varphi(x') \big\|\leq\lambda\| x-x' \|,
+    \end{equation}
+    l'application \( L_y\) est une contraction de constante \( \lambda\). Par ailleurs \( x_0\) est un point fixe de \( L_{y_0}\), donc en vertu de la caractérisation \eqref{EqDZvtUbn} des fonctions Lipschitziennes, 
+    \begin{equation}
+        L_{y_0}\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B\big( L_{y_0}(x_0),\lambda\delta \big) }=\overline{ B(x_0,\lambda\delta) }.
+    \end{equation}
+    Vu que pour tout \( y\) et \( x\) nous avons \( L_y(x)=L_{y_0}(x)+y-y_0\),
+    \begin{equation}
+    L_y\big( \overline{ B(x_0,\delta) } \big)=L_{y_0}\big( \overline{ B(x_0,\delta) } \big)+(y-y_0)\subset \overline{ B(x_0,\lambda\delta) }+(y-y_0)\subset \overline{ B(x_0),\lambda\delta+\| y-y_0 \| }.
+    \end{equation}
+    Si \( \epsilon<(1-\lambda)\delta\) alors \( \lambda\delta+\| y-y_0 \|<\delta\). Un tel choix de \( \epsilon>0\) est possible parce que \( \lambda<1\). Pour une telle valeur de \( \epsilon\) nous avons
+    \begin{equation}
+        L_y\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B(x_0,\delta) }.
+    \end{equation}
+    Par conséquent \( L_y\) est une contraction sur l'espace métrique complet \( \overline{ B(x_0,\delta) }\), ce qui signifie que \( L_y\) y possède un point fixe par le théorème de Picard \ref{ThoEPVkCL}.
+\end{proof}
+
+Le théorème d'inversion locale s'énonce de la façon suivante dans \( \eR^n\) :
+\begin{theorem}[Inversion locale dans \( \eR^n\)]    \label{THOooQGGWooPBRNEX}      % Ne pas mettre de label ici parce qu'il faut référencer l'autre, celui dans Banach.
+    Soit \( f\in C^k(\eR^n,\eR^n)\) et \( x_0\in \eR^n\). Si \( df_{x_0}\) est inversible, alors il existe un voisinage ouvert \( U\) de \( x_0\) et \( V\) de \( f(x_0)\) tels que \( f\colon U\to V\) soit un \( C^k\)-difféomorphisme. (c'est à dire que \( f^{-1}\) est également de classe \( C^k\))
+\end{theorem}
+
+Nous allons le démontrer dans le cas un peu plus général (mais pas plus cher\footnote{Sauf la justification de la régularité de l'application \( A\mapsto A^{-1}\)}) des espaces de Banach en tant que conséquence du théorème de point fixe de Picard \ref{ThoEPVkCL}.
+
+\begin{theorem}[Inversion locale dans un espace de Banach\cite{OWTzoEK}] \label{ThoXWpzqCn}
+    Soit une fonction \( f\in C^p(E,F)\) avec \( p\geq 1\) entre deux espaces de Banach. Soit \( x_0\in E\) tel que \( df_{x_0}\) soit une bijection bicontinue\footnote{En dimension finie, une application linéaire est toujours continue et d'inverse continu.}. Alors il existe un voisinage ouvert \( V\) de \( x_0\) et \( W\) de \( f(x_0)\) tels que
+    \begin{enumerate}
+        \item
+        \( f\colon V\to W\) soit une bijection,
+    \item
+        \( f^{-1}\colon W\to V\) soit de classe \( C^p\).
+    \end{enumerate}
+\end{theorem}
+\index{application!différentiable}
+\index{théorème!inversion locale}
+
+\begin{proof}
+    Nous commençons par simplifier un peu le problème. Pour cela, nous considérons la translation \( T\colon x\mapsto x+x_0 \) et l'application linéaire
+    \begin{equation}
+        \begin{aligned}
+            L\colon \eR^n&\to \eR^n \\
+            x&\mapsto (df_{x_0})^{-1}x
+        \end{aligned}
+    \end{equation}
+    qui sont tout deux des difféomorphismes (\( L\) en est un par hypothèse d'inversibilité). Quitte à travailler avec la fonction \( k=L\circ f\circ T\), nous pouvons supposer que \( x_0=0\) et que \( df_{x_0}=\mtu\). Pour comprendre cela il faut utiliser deux fois la formule de différentielle de fonction composée de la proposition \ref{EqDiffCompose} :
+    \begin{equation}
+        dk_0(u)=dL_{(f\circ T)(0)}\Big( df_{T(0)}dT_0(u) \Big).
+    \end{equation}
+    Vu que \( L\) est linéaire, sa différentielle est elle-même, c'est à dire \( dL_{(f\circ T)(0)}=(df_{x_0})^{-1}\), et par ailleurs \( dT_0=\mtu\), donc
+    \begin{equation}
+        dk_0(u)=(df_{x_0})^{-1}\Big( df_{x_0}(u) \Big)=u,
+    \end{equation}
+    ce qui signifie bien que \( dk_0=\mtu\). Pour tout cela nous avons utilisé en plein le fait que \( df_{x_0}\) était inversible.
+
+Nous posons \( g=f-\mtu\), c'est à dire \( g(x)=f(x)-x\), qui a la propriété \( dg_0=0\). Étant donné que \( g\) est de classe \( C^1\), l'application\footnote{Ici \( \GL(F)\) est l'ensemble des applications linéaires, inversibles et continues de \( F\) dans lui-même. Ce ne sont pas spécialement des matrices parce que nous n'avons pas d'hypothèses sur la dimension de \( F\), finie ou non.}
+    \begin{equation}
+        \begin{aligned}
+            dg\colon E&\to \GL(F) \\
+            x&\mapsto dg_x 
+        \end{aligned}
+    \end{equation}
+    est continue. En conséquence de quoi nous avons un voisinage \( U'\) de \( 0 \) pour lequel
+    \begin{equation}    \label{EqSGTOfvx}
+        \sup_{x\in U'}\| dg_x \|<\frac{ 1 }{2}.
+    \end{equation}
+    Maintenant le théorème des accroissements finis \ref{ThoNAKKght} (\ref{val_medio_2} pour la dimension finie) nous indique que pour tout \( x,x'\in U'\) nous avons\footnote{Ici nous supposons avoir choisi \( U'\) convexe afin que tous les \( a\in \mathopen[ x , x' \mathclose]\) soient bien dans \( U'\) et donc soumis à l'inéquation \eqref{EqSGTOfvx}, ce qui est toujours possible, il suffit de prendre une boule.}
+    \begin{equation}
+        \| g(x')-g(x) \|\leq \sup_{a\in\mathopen[ x , x' \mathclose]}\| dg_a \| \cdot \| x-x' \|\leq \frac{ 1 }{2}\| x-x' \|,
+    \end{equation}
+    ce qui prouve que \( g\) est une contraction au moins sur l'ouvert \( U'\). Nous allons aussi donner une idée de la façon dont \( f\) fonctionne : si \( x_1,x_2\in U'\) alors
+    \begin{subequations}
+        \begin{align}
+            \| x_1-x_2 \|&=\| g(x_1)-f(x_1)-g(x_2)+f(x_2) \| \\
+            &\leq \| g(x_1)-g(x_2) \|+\| f(x_1)-f(x_2) \|\\
+            &\leq \frac{ 1 }{2}\| x_1-x_2 \|+\| f(x_1)-f(x_2) \|,
+        \end{align}
+    \end{subequations}
+    ce qui montre que
+    \begin{equation}
+        \| x_1-x_2 \|\leq 2\| f(x_1)-f(x_2) \|.
+    \end{equation}
+    Maintenant que nous savons que \( g\) est contractante de constante \( \frac{ 1 }{2}\) et que \( f=g+\mtu\) nous pouvons utiliser la proposition \ref{LemGZoqknC} pour conclure que \( f\) est un homéomorphisme sur un ouvert \( U\) (partie de \( U'\)) de \( E\) et \( f^{-1}\) a une constante de Lipschitz plus petite ou égale à \( (1-\frac{ 1 }{2})^{-1}=2\).
+
+    Nous allons maintenant prouver que \( f^{-1}\) est différentiable et que sa différentielle est donnée par \( (df^{-1})_{f(x)}=(df_x)^{-1}\).
+
+    Soient \( a,b\in U\) et \( u=b-a\). Étant donné que \( f\) est différentiable en \( a\), il existe une fonction \( \alpha\in o(\| u \|)\) telle que
+    \begin{equation}
+        f(b)-f(a)-df_a(u)=\alpha(u).
+    \end{equation}
+    En notant \( y_a=f(a)\) et \( y_b=f(b)\) et en appliquant \( (df_a)^{-1}\) à cette dernière équation,
+    \begin{equation}
+        (df_a)^{-1}(y_b-y_a)-u=(df_a)^{-1} \big( \alpha(u) \big).
+    \end{equation}
+    Vu que \( df_a\) est bornée (et son inverse aussi), le membre de droite est encore une fonction \( \beta\) ayant la propriété \( \lim_{u\to 0}\beta(u)/\| u \|=0\); en réordonnant les termes,
+    \begin{equation}
+        b-a=(df_a)^{-1}(y_b-y_a)+\beta(u)
+    \end{equation}
+    et donc
+    \begin{equation}
+        f^{-1}(y_b)-f^{-1}(y_a)-(df_a)^{-1}(y_b-y_a)=\beta(u),
+    \end{equation}
+    ce qui prouve que \( f^{-1}\) est différentiable et que \( (df^{-1})_{y_a}=(df_a)^{-1}\).
+
+    La différentielle \( df^{-1}\) est donc obtenue par la chaine
+    \begin{equation}
+    \xymatrix{%
+        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}     &   U'\ar[r]^-{df}&\GL(F)\ar[r]^-{\Inv}&\GL(F)
+       }
+    \end{equation}
+    où l'application \( \Inv\colon \GL(F)\to \GL(F)\) est l'application \( X\mapsto X^{-1}\) qui est de classe \(  C^{\infty}\) par le théorème \ref{ThoCINVBTJ}. D'autre part, par hypothèse \( df\) est une application de classe \( C^{k-1}\) et donc au minimum \( C^0\) parce que \( k\geq 1\). Enfin, l'application \( f^{-1}\colon f(U)\to U\) est continue (parce que la proposition \ref{LemGZoqknC} précise que \( f\) est un homéomorphisme). Donc toute la chaine est continue et \( df^{-1}\) est continue. Cela entraine immédiatement que \( f^{-1}\) est \( C^1\) et donc que toute la chaine est \( C^1\).
+
+    Par récurrence nous obtenons la chaine
+    \begin{equation}
+    \xymatrix{%
+        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}_-{C^{k-1}}     &   U'\ar[r]^-{df}_-{C^{k-1}}&\GL(F)\ar[r]^-{\Inv}_-{ C^{\infty}}&\GL(F)
+       }
+    \end{equation}
+    qui prouve que \( df^{-1}\) est \( C^{k-1} \) et donc que \( f^{-1}\) est \( C^k\). La récurrence s'arrête ici parce que \( df\) n'est pas mieux que \( C^{k-1}\).
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Théorème de la fonction implicite}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Nous énonçons et le démontrons le théorème de la fonction implicite dans le cas d'espaces de Banach.
+\begin{theorem}[Théorème de la fonction implicite dans Banach\cite{SNPdukn}] \label{ThoAcaWho}
+    Soient \( E\), \( F\) et \( G\) des espaces de Banach et des ouverts \( U\subset E\), \( V\subset F\). Nous considérons une fonction \( f\colon U\times V\to G\) de classe \( C^r\) telle que\footnote{La notation \( d_y\) est la différentielle partielle de la définition \ref{VJM_CtSKT}.}
+    \begin{equation}
+        d_yf_{(x_0,y_0)}\colon F\to G
+    \end{equation}
+    soit un isomorphisme pour un certain \( (x_0,y_0)\in U\times V\).
+
+    Alors nous avons des voisinages \( U_0\) de \( x_0\) dans \( E\) et \( W_0\) de \( f(x_0,y_0)\) dans \( G\) et une fonction de classe \( C^r\) 
+    \begin{equation}
+        g\colon U_0\times W_0\to V
+    \end{equation}
+    telle que 
+    \begin{equation}
+        f\big( x,g(x,w) \big)=w
+    \end{equation}
+    pour tout \( (x,w)\in U_0\times W_0\).
+    
+    Cette fonction \( g\) est unique au sens suivant : il existe un voisinage \( V_0 \) de \( y_0\) tel que si \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) satisfont à \( f(x,y)=w\) alors \( y=g(x,w)\). Autrement dit, la fonction \( g\colon U_0\times W_0\to V_0\) est unique.
+\end{theorem}
+\index{théorème!fonction implicite dans Banach}
+
+\begin{proof}
+    Nous commençons par considérer la fonction
+    \begin{equation}
+        \begin{aligned}
+            \Phi\colon U\times V&\to E\times G \\
+            (x,y)&\mapsto \big( x,f(x,y) \big)
+        \end{aligned}
+    \end{equation}
+    et sa différentielle 
+    \begin{subequations}
+        \begin{align}
+            d\Phi_{(x_0,y_0)}(u,v)&=\Dsdd{ \big( x_0+tu,f(x_0+tu,y_0+tv) \big) }{t}{0}\\
+            &=\left( \Dsdd{ x_0+tu }{t}{0},\Dsdd{ f(x_0+tu,y_0+tv) }{t}{0} \right)\\
+            &=\left( u,df_{(x_0,y_0)}(u,v) \right).
+        \end{align}
+    \end{subequations}
+    Nous utilisons alors la proposition \ref{PropLDN_nHWDF} pour conclure que
+    \begin{equation}
+        d\Phi_{(x_0,y_0)}(u,v)=\big( u,(d_1f)_{(x_0,y_0)}(u)+(d_2f)_{(x_0,y_0)}(v) \big),
+    \end{equation}
+    mais comme par hypothèse \( (d_2f)_{(x_0,y_0)}\colon F\to G\) est un isomorphisme, l'application \( d\Phi_{(x_0,y_0)}\colon E\times F\to E\times G\) est également un isomorphisme. Par conséquent le théorème d'inversion locale \ref{ThoXWpzqCn} nous indique qu'il existe un voisinage \( \mO\) de \( (x_0,y_0)\) et \( \mP\) de \( \Phi(x_0,y_0)\) tels que \( \Phi\colon \mO\to \mP\) soit une bijection et \( \Phi^{-1}\colon \mP\to \mO\) soit de classe \( C^r\). Vu que \( \mP\) est un voisinage de
+    \begin{equation}
+        \Phi(x_0,y_0)=\big( x_0,f(x_0,y_0) \big),
+    \end{equation}
+    nous pouvons par \ref{PropDXR_KbaLC} le choisir un peu plus petit de telle sorte à avoir \( \mP=U_0\times W_0\) où \( U_0\) est un voisinage de \( x_0\) et \( W_0\) un voisinage de \( f(x_0,y_0)\). Dans ce cas nous devons obligatoirement aussi restreindre \( \mO\) à \( U_0\times V_0\) pour un certain voisinage \( V_0\) de \( y_0\). L'application \( \Phi^{-1}\) a obligatoirement la forme
+    \begin{equation}    \label{EqMHT_QrHRn}
+        \begin{aligned}
+            \Phi^{-1}\colon U_0\times W_0&\to U_0\times V_0 \\
+            (x,w)&\mapsto \big( x,g(x,w) \big) 
+        \end{aligned}
+    \end{equation}
+    pour une certaine fonction \( g\colon U_0\times W_0\to V\). Cette fonction \( g\) est la fonction cherchée parce qu'en appliquant \( \Phi\) à \eqref{EqMHT_QrHRn}, 
+    \begin{equation}
+        (x,w)=\Phi\big( x,g(x,w) \big)=\Big( x,f\big( x,g(x,w) \big) \Big),
+    \end{equation}
+    qui nous dit que pour tout \( x\in U_0\) et tout \( w\in W_0\) nous avons
+    \begin{equation}
+        f\big( x,g(x,w) \big)=w.
+    \end{equation}
+
+    Si vous avez bien suivi le sens de l'équation \eqref{EqMHT_QrHRn} alors vous avez compris l'unicité. Sinon, considérez \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) tels que \( f(x,y)=w\). Alors \( \big( x,f(x,y) \big)=(x,w)\) et 
+    \begin{equation}
+        \Phi(x,y)=(x,w).
+    \end{equation}
+    Mais vu que \( \Phi\colon U_0\times V_0\to U_0\times W_0\) est une bijection, cette relation définit de façon univoque l'élément \( (x,y)\) de \( U_0\times V_0\), qui ne sera autre que \( g(x,w)\).
+\end{proof}
+
+Le théorème de la fonction implicite s'énonce de la façon suivante pour des espaces de dimension finie.
+% Attention : avant de citer ce théorème, voir s'il est suffisant. Ici \varphi a une variable; dans l'autre énoncé il en a deux.
+\begin{theorem}[Théorème de la fonction implicite en dimension finie]   \label{ThoRYN_jvZrZ}
+    Soit une fonction \( F\colon \eR^n\times \eR^m\to \eR^m\) de classe \( C^k\) et \( (\alpha,\beta)\in \eR^n\times \eR^m\) tels que
+    \begin{enumerate}
+        \item
+            \( F(\alpha,\beta)=0\),
+        \item
+            \( \frac{ \partial (F_1,\ldots, F_m) }{ \partial (y_1,\ldots, y_m) }\neq 0\), c'est à dire que \( (d_yF)_{(\alpha,\beta)} \) est inversible.
+    \end{enumerate}
+    Alors il existe un voisinage ouvert \( V\) de \( \alpha\) dans \( \eR^n\), un voisinage ouvert \( W\) de \( \beta\) dans \( \eR^m\) et une application \( \varphi\colon V\to W\) de classe \( C^k\)  telle que pour tout \( x\in V\) on ait
+    \begin{equation}
+        F\big( x,\varphi(x) \big)=0.
+    \end{equation}
+    De plus si \( (x,y)\in V\times W\) satisfait à \( F(x,y)=0\), alors \( y=\varphi(x)\).
+\end{theorem}
+\index{théorème!fonction implicite dans \( \eR^n\)}
+
+\begin{remark}\label{RemPYA_pkTEx}
+    Notons que cet énoncé est tourné un peu différemment en ce qui concerne le nombre de variables dont dépend la fonction implicite : comparez
+    \begin{subequations}
+        \begin{align}
+            f\big( x,g(x,w) \big)=w\\
+            F\big( x,\varphi(x) \big)=0.
+        \end{align}
+    \end{subequations}
+    Le deuxième est un cas particulier du premier en posant 
+    \begin{equation}
+        F(x,y)=f(x,y)-f(x_0,y_0)
+    \end{equation}
+    et donc en considérant \( w\) comme valant la constante \( f(x_0,y_0)\); dans ce cas la fonction \( g\) ne dépend plus que de la variable \( x\).
+
+\end{remark}
+
+\begin{example}
+    La remarque \ref{RemPYA_pkTEx} signifie entre autres que le théorème \ref{ThoAcaWho} est plus fort que \ref{ThoRYN_jvZrZ} parce que le premier permet de choisir la valeur d'arrivée. Parlons de l'exemple classique du cercle et de la fonction \( f(x,y)=x^2+y^2\). Nous savons que
+    \begin{equation}
+        f(\alpha,\beta)=1.
+    \end{equation}
+    Alors le théorème \ref{ThoAcaWho} nous donne une fonction \( g\) telle que
+    \begin{equation}
+        f(x,g(x,r))=r
+    \end{equation}
+    tant que \( x\) est proche de \( \alpha\), que \( r\) est proche de \( 1\) et que \( g\) donne des valeurs proches de \( \beta\).
+
+    L'énoncé \ref{ThoRYN_jvZrZ} nous oblige à travailler avec la fonction \( F(x,y)=x^2+y^2-1\), de telle sorte que
+    \begin{equation}
+        F(\alpha,\beta)=0,
+    \end{equation}
+    et que nous ayons une fonction \( \varphi\) telle que
+    \begin{equation}
+        F(x,\varphi(x))=0.
+    \end{equation}
+    La fonction \( \varphi\) ne permet donc que de trouver des points sur le cercle de rayon \( 1\).
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Exemple}
+%---------------------------------------------------------------------------------------------------------------------------
+    
+Le théorème de la fonction implicite a pour objet de donner l'existence de la fonction $\varphi$. Maintenant nous pouvons dire beaucoup de choses sur les dérivées de $\varphi$ en considérant la fonction
+\begin{equation}
+    x\mapsto F\big( x,\varphi(x) \big).
+\end{equation}
+Par définition de $\varphi$, cette fonction est toujours nulle. En particulier, nous pouvons dériver l'équation
+\begin{equation}
+    F\big( x,\varphi(x) \big)=0,
+\end{equation}
+et nous trouvons plein de choses.
+
+
+Prenons par exemple la fonction
+\begin{equation}
+    F\big( (x,y),z \big)=ze^z-x-y,
+\end{equation}
+et demandons nous ce que nous pouvons dire sur la fonction $z(x,y)$ telle que
+\begin{equation}
+    F\big( x,y,z(x,y) \big)=0,
+\end{equation}
+c'est à dire telle que
+\begin{equation}        \label{EqDefZImplExemple}
+    z(x,y) e^{z(x,y)}-x-y=0.
+\end{equation}
+pour tout $x$ et $y\in\eR$. Nous pouvons facilement trouver $z(0,0)$ parce que
+\begin{equation}
+    z(0,0) e^{z(0,0)}=0,
+\end{equation}
+donc $z(0,0)=0$.
+
+Nous pouvons dire des choses sur les dérivées de $z(x,y)$. Voyons par exemple $(\partial_xz)(x,y)$. Pour trouver cette dérivée, nous dérivons la relation \eqref{EqDefZImplExemple} par rapport à $x$. Ce que nous trouvons est
+\begin{equation}
+    (\partial_xz)e^z+ze^z(\partial_xz)-1=0.
+\end{equation}
+Cette équation peut être résolue par rapport à $\partial_xz$~:
+\begin{equation}
+    \frac{ \partial z }{ \partial x }(x,y)=\frac{1}{ e^z(1+z) }.
+\end{equation}
+Remarquez que cette équation ne donne pas tout à fait la dérivée de $z$ en fonction de $x$ et $y$, parce que $z$ apparaît dans l'expression, alors que $z$ est justement la fonction inconnue. En général, c'est la vie, nous ne pouvons pas faire mieux.
+
+Dans certains cas, on peut aller plus loin. Par exemple, nous pouvons calculer cette dérivée au point $(x,y)=(0,0)$ parce que $z(0,0)$ est connu :
+\begin{equation}
+    \frac{ \partial z }{ \partial x }(0,0)=1.
+\end{equation}
+Cela est pratique pour calculer, par exemple, le développement en Taylor de $z$ autour de $(0,0)$.
+
+\begin{example}
+    Est-ce que l'équation \( e^{y}+xy=0\) définit au moins localement une fonction \( y(x)\) ? Nous considérons la fonction
+    \begin{equation}
+        f(x,y)=\begin{pmatrix}
+            x    \\ 
+            e^{y}+xy    
+        \end{pmatrix}
+    \end{equation}
+    La différentielle de cette application est
+    \begin{equation}
+            df_{(0,0)}(u)=\frac{ d }{ dt }\Big[ f(tu_1,tu_2) \Big]_{t=0}
+            =\frac{ d }{ dt }\begin{pmatrix}
+                tu_1    \\ 
+                e^{tu_2}+t^2u_1u_2    
+            \end{pmatrix}_{t=0}
+            =\begin{pmatrix}
+                u_1    \\ 
+                u_2    
+            \end{pmatrix}.
+    \end{equation}
+    L'application \( f\) définit donc un difféomorphisme local autour des points \( (x_0,y_0)\) et \( f(x_0,y_0)\). Soit \( (u,0)\) un point dans le voisinage de \( f(x_0,y_0)\). Alors il existe un unique \( (x,y)\) tel que
+    \begin{equation}
+        f(x,y)=\begin{pmatrix}
+               x \\ 
+            e^y+xy    
+        \end{pmatrix}=
+        \begin{pmatrix}
+            u    \\ 
+                0
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons automatiquement \( x=u\) et \( e^y+xy=0\). Notons toutefois que pour que ce procédé donne effectivement une fonction implicite \( y(x)\) nous devons avoir des points de la forme \( (u,0)\) dans le voisinage de \( f(x_0,y_0)\).
+\end{example}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Décomposition polaire (régularité)}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{normaltext}      \label{NomDJMUooTRUVkS}
+    Nous allons montrer que l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
+            A&\mapsto \sqrt{A} 
+        \end{aligned}
+    \end{equation}
+    est une difféomorphisme.
+
+    Cependant \( S^{++}(n,\eR)\) n'est pas un ouvert de \( \eM(n,\eR)\) et nous ne savons pas ce qu'est la différentielle d'une application non définie sur un ouvert. Nous allons donc en réalité montrer que l'application racine carré existe sur un voisinage de chacun des points de \( S^{++}(n,\eR)\). Et comme une union quelconque d'ouverts est un ouvert, la fonction \( f\) sera bien définie sur un ouvert de \( \eM(n,\eR)\).
+\end{normaltext}
+
+\begin{lemma}       \label{LemLBFOooDdNcgy}
+    L'application 
+    \begin{equation}
+        \begin{aligned}
+            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
+            A&\mapsto A^2 
+        \end{aligned}
+    \end{equation}
+    est un \(  C^{\infty}\)-difféomorphisme.
+\end{lemma}
+
+\begin{proof}
+    Prouvons d'abord que \( f\) prend ses valeurs dans \( S^{++}(n,\eR)\). Si \( A\in S^{++}(n,\eR)\) alors par la diagonalisation \ref{ThoeTMXla} elle s'écrit \( A=QDQ^{-1}\) où \( D\) est diagonale avec des nombres strictement positifs sur la diagonale. Avec cela, \( A^2=QD^2Q^{-1}\) où \( D^2\) contient encore des nombres strictement positifs sur la diagonale.
+
+    L'application \( f\) étant essentiellement des polynôme en les entrées de \( A\), elle est de classe \( C^{\infty}\).
+
+    Passons à l'étude de la différentielle. Comme mentionné en \ref{NomDJMUooTRUVkS} nous allons en réalité voir \( f\) sur un ouvert de \( \eM(n,\eR)\) autour de \( A\in S^{++}(n,\eR)\). Par conséquent si \( A\in S^{++}(n,\eR)\),
+    \begin{subequations}
+        \begin{align}
+            df\colon S^{++}(n,\eR)&\to \aL\big( \eM(n,\eR),\eM(n,\eR) \big)\\
+            df_A\colon \eM(n,\eR)&\to \eM(n,\eR).
+        \end{align}
+    \end{subequations}
+    Le calcul de \( df_A\) est facile. Soit \( u\in \eM(n,\eR)\) et faisons le calcul en utilisant la formule du lemme \eqref{LemdfaSurLesPartielles} :
+    \begin{subequations}
+        \begin{align}
+            df_A(u)&=\Dsdd{ f(A+tu) }{t}{0}\\
+            &=\Dsdd{ A^2+tAu+tuA+t^2u^2 }{t}{0}\\
+            &=Au+uA.
+        \end{align}
+    \end{subequations}
+    Nous allons utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} à la fonction \( f\). Dans la suite, \( A\) est une matrice de \( S^{++}(n,\eR)\).
+
+    \begin{subproof}
+        \item[\( df_A\) est injective]
+            Soit \( M\in \eM(n,\eR)\) dans le noyau de \( df_A\). En posant \( M'=A^{-1}MQ\) nous avons \( M=QM'Q^{-1}\) et on applique \( df_A\) à \( QM'Q^{-1}\) :
+            \begin{equation}
+                df_A(QM'Q^{-1})=Q\big( DM+MD \big)Q^{-1}.
+            \end{equation}
+            où \( D=\begin{pmatrix}
+                \lambda_1    &       &       \\
+                    &   \ddots    &       \\
+                    &       &   \lambda_n
+                \end{pmatrix}\) avec \( \lambda_i>0\). La matrice \( D\) est inversible. Nous avons \( M'=-DM'D^{-1}\), et en coordonnées,
+                \begin{subequations}
+                    \begin{align}
+                        M'_{ij}&=-\sum_{kl}D_{ikM'_{kl}}D^{-1}_{lj}\\
+                        &=-\sum_{kl}\lambda_i\delta_{ik}M'_{kl}\frac{1}{ \lambda_j }\delta_{lj}\\
+                        &=-\frac{ \lambda_i }{ \lambda_i }M'_{ij}.
+                    \end{align}
+                \end{subequations}
+                C'est à dire que \( M'_{ij}=-\frac{ \lambda_i }{ \lambda_j }M'_{ij}\) avec \( -\frac{ \lambda_i }{ \lambda_j }<0\). Cela implique \( M'=0\) et par conséquent \( M=0\).
+            \item[\( df_A\) est surjective]
+                Soit \( N\in \eM(n,\eR)\); nous cherchons \( M\in \eM(n,\eR)\) tel que \( df_A(M)=N\). Nous posons \( N'=Q^{-1} NQ\) et \( M=QM'Q^{-1}\), ce qui nous donne à résoudre \( df_D(M')=N'\). Passons en coordonnées :
+                \begin{subequations}
+                    \begin{align}
+                        (DM'+M'D)_{ij}&=\sum_k(\delta_{ik}\lambda_iM'_{kj}+M'_{ik}\delta_{kj}\lambda_j)\\&
+                        &=M'_{ij}(\lambda_i+\lambda_j)
+                    \end{align}
+                \end{subequations}
+                où \( \lambda_i+\lambda_j\neq 0\). Il suffit donc de prendre la matrice \( M'\) donnée par
+                \begin{equation}
+                    M'_{ij}=\frac{1}{ \lambda_i+\lambda_j }N'_{ij}
+                \end{equation}
+                pour que \( df_A(M')=N'\).
+    \end{subproof}
+    
+    Le théorème d'inversion locale donne un voisinage \( V\) de $A$ dans \( \eM(n,\eR)\) et un voisinage \( W\) de \( A^2\) dans \( \eM(n,\eR)\) tels que \( f\colon V\to W\) soit une bijection  et \( f^{-1}\colon W\to V\) soit de même régularité, en l'occurrence \( C^{\infty}\).
+\end{proof}
+
+\begin{remark}
+    Oui, il y a des matrices non symétriques qui ont une unique racine carré.
+\end{remark}
+
+La proposition suivante, qui dépend du le théorème d'inversion locale par le lemme \ref{LemLBFOooDdNcgy}, donne plus de régularité à la décomposition polaire donnée dans le théorème \ref{ThoLHebUAU}.
+\begin{proposition}[Décomposition polaire : cas réel (suite)]       \label{PropWCXAooDuFMjn}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \gO(n,\eR)\times S^{++}(n,\eR)&\to \GL(n,\eR) \\
+            (Q,S)&\mapsto SQ 
+        \end{aligned}
+    \end{equation}
+    est un difféomorphisme de classe \( C^{\infty}\).
+\end{proposition}
+
+\begin{proof}
+    Si \( M\) est donnée dans \( \GL(n,\eR)\) alors la décomposition polaire\footnote{Proposition \ref{ThoLHebUAU}.} \( M=QS\) est donnée par \( S=\sqrt{MM^t}\) et \( Q=MS^{-1}\). Autrement dit, si nous considérons la fonction de décomposition polaire
+    \begin{equation}
+        f\colon \gO(n,\eR)\times S^{++}(n,\eR)\to \GL(n,\eR)
+    \end{equation}
+    alors
+    \begin{equation}
+        f^{-1}(M)=\big(  M(\sqrt{MM^t})^{-1},\sqrt{MM^t}  \big).
+    \end{equation}
+    Nous avons vu dans le lemme \ref{LemLBFOooDdNcgy} que la racine carré était un \( C^{\infty}\)-difféomorphisme. Le reste n'étant que des produits de matrice, la régularité est de mise.
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Théorème de Von Neumann}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{lemma}[\cite{KXjFWKA}]
+    Soit \( G\), un sous groupe fermé de \( \GL(n,\eR)\) et 
+    \begin{equation}
+        \mL_G=\{ m\in \eM(n,\eR)\tq  e^{tm}\in G\,\forall t\in\eR \}.
+    \end{equation}
+    Alors \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\).
+\end{lemma}
+
+\begin{proof}
+    Si \( m\in\mL_G\), alors \( \lambda m\in\mL_G\) par construction. Le point délicat à prouver est le fait que si \( a,b\in \mL_G\), alors \( a+b\in\mL_G\). Soit \( a\in \eM(n,\eR)\); nous savons qu'il existe une fonction \( \alpha_a\colon \eR\to \eM\) telle que
+    \begin{equation}
+        e^{ta}=\mtu+ta+\alpha_a(t)
+    \end{equation}
+    et 
+    \begin{equation}
+        \lim_{t\to 0} \frac{ \alpha(t) }{ t }=0.
+    \end{equation}
+    Si \( a\) et \( b\) sont dans \( \mL_G\), alors \(  e^{ta} e^{tb}\in G\), mais il n'est pas vrai en général que cela soit égal à \(  e^{t(a+b)}\). Pour tout \( k\in \eN\) nous avons
+    \begin{equation}
+        e^{a/k} e^{b/k}=\left( \mtu+\frac{ a }{ k }+\alpha_a(\frac{1}{ k }) \right)\left( \mtu+\frac{ b }{ k }+\alpha_b(\frac{1}{ k }) \right)=\mtu+\frac{ a+b }{2}+\beta\left( \frac{1}{ k } \right)
+    \end{equation}
+   où \( \beta\colon \eR\to \eM\) est encore une fonction vérifiant \( \beta(t)/t\to 0\). Si \( k\) est assez grand, nous avons
+   \begin{equation}
+       \left\| \frac{ a+b }{ k }+\beta(\frac{1}{ k })  \right\|<1,
+   \end{equation}
+   et nous pouvons profiter du lemme \ref{LemQZIQxaB} pour écrire alors
+   \begin{equation}
+       \left(  e^{a/k} e^{b/k} \right)^k= e^{k\ln\big(\mtu+\frac{ a+b }{ k }+\beta(\frac{1}{ k })\big)}.
+   \end{equation}
+   Ce qui se trouve dans l'exponentielle est
+   \begin{equation}
+       k\left[ \frac{ a+b }{ k }+\alpha( \frac{1}{ k })+\sigma\left( \frac{ a+b }{ k }+\alpha(\frac{1}{ k }) \right) \right].
+   \end{equation}
+   Les diverses propriétés vues montrent que le tout tend vers \( a+b\) lorsque \( k\to \infty\). Par conséquent
+   \begin{equation}
+       \lim_{k\to \infty} \left(  e^{a/k} e^{b/k} \right)^k= e^{a+b}.
+   \end{equation}
+   Ce que nous avons prouvé est que pour tout \( t\), \(  e^{t(a+b)}\) est une limite d'éléments dans \( G\) et est donc dans \( G\) parce que ce dernier est fermé.
+\end{proof}
+
+Vu que \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\), nous pouvons considérer un supplémentaire \( M\).
+
+\begin{lemma}   \label{LemHOsbREC}
+    Il n'existe pas se suites \( (m_k)\) dans \( M\setminus\{ 0 \}\) convergeant vers zéro et telle que \(  e^{m_k}\in G\) pour tout \( k\).
+\end{lemma}
+
+\begin{proof}
+    Supposons que nous ayons \( m_k\to 0\) dans \( M\setminus\{ 0 \}\) avec \(  e^{m_k}\in G\). Nous considérons les éléments \( \epsilon_k=\frac{ m_k }{ \| m_k \| }\) qui sont sur la sphère unité de \(\GL(n,\eR)\). Quitte à prendre une sous-suite, nous pouvons supposer que cette suite converge, et vu que \( M\) est fermé, ce sera vers \( \epsilon\in M\) avec \( \| \epsilon \|=1\). Pour tout \( t\in \eR\) nous avons
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}.
+    \end{equation}
+    En vertu de la décomposition d'un réel en partie entière et décimale, pour tout \( k\) nous avons \( \lambda_k\in \eZ\) et \( | \mu_k |\leq \frac{ 1 }{2}\) tel que \( t/\| m_k \|=\lambda_k+\mu_k\). Avec ça,
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}\exp\Big( \frac{ t }{ m_k }m_k \Big)=\lim_{k\to \infty}  e^{\lambda_km_k} e^{\mu_km_k}.
+    \end{equation}
+    Pour tout \( k\) nous avons \(  e^{\lambda_km_k}\in G\). De plus \( | \mu_k |\) étant borné et \( m_k\) tendant vers zéro nous avons \(  e^{\mu_km_k}\to 1\). Au final
+    \begin{equation}
+        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}\in G
+    \end{equation}
+    Cela signifie que \( \epsilon\in\mL_G\), ce qui est impossible parce que nous avions déjà dit que \( \epsilon\in M\setminus\{ 0 \}\).
+\end{proof}
+
+\begin{lemma}   \label{LemGGTtxdF}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \mL_G\times M&\to \GL(n,\eR) \\
+            l,m&\mapsto  e^{l} e^{m} 
+        \end{aligned}
+    \end{equation}
+    est un difféomorphisme local entre un voisinage de \( (0,0)\) dans \( \eM(n,\eR)\) et un voisinage de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
+\end{lemma}
+Notons que nous ne disons rien de \(  e^{\eM(n,\eR)}\). Nous n'allons pas nous embarquer à discuter si ce serait tout \( \GL(n,\eR)\)\footnote{Vu les dimensions y'a tout de même peu de chance.} ou bien si ça contiendrait ne fut-ce que \( G\).
+
+\begin{proof}
+    Le fait que \( f\) prenne ses valeurs dans \( \GL(n,\eR)\) est simplement dû au fait que les exponentielles sont toujours inversibles. Nous considérons ensuite la différentielle : si \( u\in \mL_G\) et \( v\in M\) nous avons
+    \begin{equation}
+        df_{(0,0)}(u,v)=\Dsdd{ f\big( t(u,v) \big) }{t}{0}=\Dsdd{  e^{tu} e^{tv} }{t}{0}=u+v.
+    \end{equation}
+    L'application \( df_0\) est donc une bijection entre \( \mL_G\times M\) et \( \eM(n,\eR)\). Le théorème d'inversion locale \ref{ThoXWpzqCn} nous assure alors que \( f\) est une bijection entre un voisinage de \( (0,0)\) dans \( \mL_G\times M\) et son image. Mais vu que \( df_0\) est une bijection avec \( \eM(n,\eR)\), l'image en question contient un ouvert autour de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
+\end{proof}
+
+\begin{theorem}[Von Neumann\cite{KXjFWKA,ISpsBzT,Lie_groups}]       \label{ThoOBriEoe}
+    Tout sous-groupe fermé de \( \GL(n,\eR)\) est une sous-variété de \( \GL(n,\eR)\).
+\end{theorem}
+\index{théorème!Von Neumann}
+\index{exponentielle!de matrice!utilisation}
+
+\begin{proof}
+    Soit \( G\) un tel groupe; nous devons prouver que c'est localement difféomorphe à un ouvert de \( \eR^n\). Et si on est pervers, on ne va pas faire localement difféomorphe à un ouvert de \( \eR^n\), mais à un ouvert d'un espace vectoriel de dimension finie. Nous allons être pervers.
+
+    Étant donné que pour tout \( g\in G\), l'application 
+    \begin{equation}
+        \begin{aligned}
+            L_g\colon G&\to G \\
+            h&\mapsto gh 
+        \end{aligned}
+    \end{equation}
+    est de classe \(  C^{\infty}\) et d'inverse \(  C^{\infty}\), il suffit de prouver le résultat pour un voisinage de \( \mtu\).
+
+    Supposons d'abord que \( \mL_G=\{ 0 \}\). Alors \( 0\) est un point isolé de \( \ln(G)\); en effet si ce n'était pas le cas nous aurions un élément \( m_k\) de \( \ln(G)\) dans chaque boule \( B(0,r_k)\). Nous aurions alors \( m_k=\ln(a_k)\) avec \( a_k\in G\) et donc
+    \begin{equation}
+        e^{m_k}=a_k\in G.
+    \end{equation}
+    De plus \( m_k\) appartient forcément à \( M\) parce que \( \mL_G\) est réduit à zéro. Cela nous donnerait une suite \( m_k\to 0\) dans \( M\) dont l'exponentielle reste dans \( G\). Or cela est interdit par le lemme \ref{LemHOsbREC}. Donc \( 0\) est un point isolé de \( \ln(G)\). L'application \(\ln\) étant continue\footnote{Par le lemme \ref{LemQZIQxaB}.}, nous en déduisons que \( \mtu\) est isolé dans \( G\). Par le difféomorphisme \( L_g\), tous les points de \( G\) sont isolés; ce groupe est donc discret et par voie de conséquence une variété.
+
+    Nous supposons maintenant que \( \mL_G\neq\{ 0 \}\). Nous savons par la proposition \ref{PropXFfOiOb} que 
+    \begin{equation}
+        \exp\colon \eM(n,\eR)\to \eM(n,\eR)
+    \end{equation}
+    est une application \(  C^{\infty}\) vérifiant \( d\exp_0=\id\). Nous pouvons donc utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} qui nous offre donc l'existence d'un voisinage \( U\) de \( 0\) dans \( \eM(n,\eR)\) tel que \( W=\exp(U)\) soit un ouvert de \( \GL(n,\eR)\) et que \( \exp\colon U\to W\) soit un difféomorphisme de classe \(  C^{\infty}\).
+
+    Montrons que quitte à restreindre \( U\) (et donc \( W\) qui reste par définition l'image de \( U\) par \( \exp\)), nous pouvons avoir \( \exp\big( U\cap\mL_G \big)=W\cap G\). D'abord \( \exp(\mL_G)\subset G\) par construction. Nous avons donc \( \exp\big( U\cap\mL_G \big)\subset W\cap G\). Pour trouver une restriction de \( U\) pour laquelle nous avons l'égalité, nous supposons que pour tout ouvert \( \mO\) dans \( U\), 
+    \begin{equation}
+        \exp\colon \mO\cap\mL_G\to \exp(\mO)\cap G
+    \end{equation}
+    ne soit pas surjective. Cela donnerait un élément de \( \mO\cap\complement\mL_G\) dont l'image par \( \exp\) n'est pas dans \( G\). Nous construisons ainsi une suite en considérant une boule \( B(0,\frac{1}{ k })\) inclue à \( U\) et \( x_k\in B(0,\frac{1}{ k })\cap\complement\mL_G\) vérifiant \(  e^{x_k}\in G\). Vu le choix des boules nous avons évidemment \( x_k\to 0\).
+
+    L'élément \(  e^{x_k}\) est dans \(  e^{\eM(n,\eR)}\) et le difféomorphisme du lemme \ref{LemGGTtxdF}\quext{Il me semble que l'utilisation de ce lemme manque à l'avant-dernière ligne de la preuve chez \cite{KXjFWKA}.} nous donne \( (l_k,m_k)\in \mL_G\times M\) tel que \(  e^{l_k} e^{m_k}= e^{x_k}\). À ce point nous considérons \( k\) suffisamment grand pour que \(  e^{x_k}\) soit dans la partie de l'image de \( f\) sur lequel nous avons le difféomorphisme. Plus prosaïquement, nous posons
+    \begin{equation}
+        (l_k,m_k)=f^{-1}( e^{x_k})
+    \end{equation}
+    et nous profitons de la continuité pour permuter la limite avec \( f^{-1}\) :
+    \begin{equation}
+        \lim_{k\to \infty} (l_k,m_k)=f^{-1}\big( \lim_{k\to \infty}  e^{x_k} \big)=f^{-1}(\mtu)=(0,0).
+    \end{equation}
+    En particulier \( m_k\to 0\) alors que \(  e^{m_k}= e^{x_k} e^{-l_k}\in G\). La suite \( m_k\) viole le lemme \ref{LemHOsbREC}. Nous pouvons donc restreindre \( U\) de telle façon à avoir
+    \begin{equation}
+        \exp\big( U\cap\mL_G \big)=W\cap G.
+    \end{equation}
+    Nous avons donc un ouvert de \( \mL_G\) (l'ouvert \( U\cap\mL_G\)) qui est difféomorphe avec l'ouvert \( W\cap G\) de \( G\). Donc \( G\) est une variété et accepte \( \mL_G\) comme carte locale.
+
+\end{proof}
+
+\begin{remark}
+    En termes savants, nous avons surtout montré que si \( G\) est un groupe de Lie d'algèbre de Lie \( \lG\), alors l'exponentielle donne un difféomorphisme local entre \( \lG\) et \( G\).
+\end{remark}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+\section{Recherche d'extrema}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema à une variable}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
+\end{definition}
+
+La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
+\begin{proposition}     \label{PROPooNVKXooXtKkuz}
+Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
+\end{proposition}
+
+La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
+
+Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
+\begin{proposition}
+Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
+\begin{equation}
+    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
+\end{equation}
+et que
+\begin{equation}
+    f^{(k)}(a)\neq 0.
+\end{equation}
+Dans ce cas,
+\begin{enumerate}
+
+\item
+Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
+\item
+Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
+
+\end{enumerate}
+\end{proposition}
+
+Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
+
+Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
+
+Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
+
+\begin{example}
+	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
+\end{example}
+ 
+Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
+\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
+\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
+
+Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
+
+Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
+
+\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
+\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
+ 
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Extrema libre}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}      \label{DEFooYJLZooLkEAYf}
+Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
+\end{definition}
+
+Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
+
+Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
+\begin{equation}
+    d^2f(a)=\begin{pmatrix}
+    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
+    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
+\end{pmatrix}.
+\end{equation}
+Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
+
+\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
+\end{proposition}
+
+\begin{proof}
+    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
+    \begin{equation}
+        \begin{aligned}
+            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
+            t&\mapsto f(a+tu) 
+        \end{aligned}
+    \end{equation}
+    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
+    \begin{equation}
+        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
+    \end{equation}
+\end{proof}
+
+\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
+    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
+    \begin{enumerate}
+        \item   \label{ITEMooCVFVooWltGqI}
+            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
+        \item\label{ItemPropoExtreRn}
+            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
+    \end{enumerate}
+\end{proposition}
+\index{extrema}
+
+\begin{proof}
+    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
+    \begin{equation}
+        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
+    \end{equation}
+    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
+    \end{equation}
+    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
+    \begin{equation}
+        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
+    \end{equation}
+    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
+    \begin{equation}
+        (d^2f)_a(e_h,e_h)\geq 0.
+    \end{equation}
+    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
+\end{proof}
+
+La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
+
+Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
+
+La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
+\begin{enumerate}
+    \item
+        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
+    \item
+        écrire $d^2f(a)$ pour chacun des candidats
+    \item
+        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
+    \item
+        conclure.
+\end{enumerate}
+
+Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
+
+\begin{example}
+    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
+    \begin{equation}
+        df=\big(4x^3-4y;4y^3-4x),
+    \end{equation}
+    et la matrice des dérivées secondes est
+    \begin{equation}
+        M=d^2f(x,y)=\begin{pmatrix}
+            12x^2    &   -4    \\ 
+            -4    &   12y^2    
+        \end{pmatrix}.
+    \end{equation}
+    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
+
+    Pour le point \( (0,0)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            0    &   -4    \\ 
+            -4    &   0    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
+
+    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
+    \begin{equation}
+        M=\begin{pmatrix}
+            12    &   -4    \\ 
+            -4    &   12    
+        \end{pmatrix},
+    \end{equation}
+    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
+\end{example}
+
+%---------------------------------------------------------------------------------------------------------------------------
+\subsection{Un peu de recettes de cuisine}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{enumerate}
+\item Rechercher les points critiques, càd les $(x,y)$ tels que
+\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
+En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
+\item Déterminer la nature des points critiques: «test» des dérivées secondes:
+\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
+\begin{enumerate}
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
+\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
+\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
+\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
+\end{enumerate}
+\end{enumerate}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Extrema liés}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
+
+Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
+
+\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
+    Soit \( A\), un ouvert de \( \eR^n\) et
+    \begin{enumerate}
+        \item
+            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
+        \item 
+            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
+        \item
+            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
+        \item
+            un extrema local $a\in M$ de $f$ relativement à $M$.
+    \end{enumerate}
+    Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
+    \begin{equation}
+        L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_r)=f(x_1,\ldots,x_n)+\sum_{i=1}^r\lambda_iG_i(x_1,\ldots,x_n).
+    \end{equation}
+    Autrement dit, si \( a\) est un extrema lié, alors \( \nabla f(a)\) est une combinaisons des \( \nabla G_i(a)\), ou encore il existe des \( \lambda_i\) tels que
+    \begin{equation}    \label{EqRDsSXyZ}
+        df(a)=\sum_i\lambda_idG_i(a).
+    \end{equation}
+\end{theorem}
+\index{théorème!inversion locale!utilisation}
+\index{extrema!lié}
+\index{théorème!extrema!lié}
+\index{application!différentiable!extrema lié}
+\index{variété}
+\index{rang!différentielle}
+\index{forme!linéaire!différentielle}
+La fonction $L$ est le \defe{lagrangien}{lagrangien} du problème et les variables \( \lambda_i\) sont les \defe{multiplicateurs de Lagrange}{multiplicateur!de Lagrange}\index{Lagrange!multiplicateur}.
+
+\begin{proof}
+    Si \( r=n\) alors les vecteurs linéairement indépendantes \( \nabla G_i(a) \) forment une base de \( \eR^n\) et donc évidemment les \( \lambda_i\) existent. Nous supposons donc maintenant que \( r<n\). Nous notons \( (z_i)_{i=1\ldots n}\) les coordonnées sur \( \eR^n\).
+    
+    La matrice
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{ \partial G_1 }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_1 }{ \partial z_n }(a)    \\
+            \vdots    &   \ddots    &   \vdots    \\
+            \frac{ \partial G_r }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_r }{ \partial z_n }(a)
+        \end{pmatrix}
+    \end{equation}
+    est de rang \( r\) parce que les lignes sont par hypothèses linéairement indépendantes. Nous nommons \( (y_i)_{i=1,\ldots, r}\) un choix de \( r\) parmi les \( (z_i)\) tels que
+    \begin{equation}
+        \det\begin{pmatrix}
+            \frac{ \partial G_1 }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_1 }{ \partial y_r }    \\
+            \vdots    &   \ddots    &   \vdots    \\
+            \frac{ \partial G_r }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_r }{ \partial y_r }
+        \end{pmatrix}\neq 0.
+    \end{equation}
+    Nous identifions \( \eR^n\) à \( \eR^s\times \eR^r\) dans lequel \( \eR^r\) est la partie générée par les \( (y_i)_{i=1,\ldots, r}\). Nous nommons \( (x_j)_{j=1,\ldots, s}\) les coordonnées sur \( \eR^s\). Autrement dit, les coordonnées sur \( \eR^n\) sont \( x_1,\ldots, x_s,y_1,\ldots, y_r\). Dans ces coordonnées, nous nommons \( a=(\alpha,\beta)\) avec \( \alpha\in \eR^s\) et \( \beta\in \eR^r\).
+
+    Si nous notons \( G=(G_1,\ldots, G_r)\), le théorème de la fonction implicite (théorème \ref{ThoAcaWho})  nous dit qu'il existe un voisinage \( U'\) de \( \alpha\in \eR^n\), un voisinage \( V'\) de \( \beta\in \eR^r\) et une fonction \( \varphi\colon U'\to V'\) de classe \( C^1\) telle que si \( (x,y)\in U'\times V'\), alors
+    \begin{equation}
+        g(x,y)=0
+    \end{equation}
+    si et seulement si \( y=\varphi(x)\). Nous posons maintenant
+    \begin{subequations}
+        \begin{align}
+            \psi(x)&=(x,\varphi(x))\\
+            h(x)&=f\big( \psi(x) \big).
+        \end{align}
+    \end{subequations}
+    Nous avons \( \psi(\alpha)=a\) et \( \psi(x)\in M\) pour tout \( x\in U'\). La fonction \( h\) a donc un extrema local en \( \alpha\) et donc les dérivées partielles de \( h\) y sont nulles. Cela signifie que
+    \begin{equation}
+        0=\frac{ \partial h }{ \partial x_i }(\alpha)=\sum_{j=1}^n\frac{ \partial f }{ \partial x_j }\frac{ \partial x_j }{ \partial x_i }+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }\frac{ \partial \varphi_k }{ \partial x_i },
+    \end{equation}
+    c'est à dire
+    \begin{equation}
+        \frac{ \partial f }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
+    \end{equation}
+    pour tout \( i=1,\ldots, s\). D'autre part pour tout $k$, la fonction \( l_k(x)=G_k\big( x,\varphi(x) \big)\) est constante et vaut zéro; ses dérivées partielles sont donc nulles :
+    \begin{equation}
+        \frac{ \partial l }{ \partial x_i }(\alpha)=\frac{ \partial G_k }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial G_k }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
+    \end{equation}
+    pour tout \( i=1,\ldots, s\) et \( k=1,\ldots, r\).
+    
+    Les \( s\) premières colonnes de la matrice
+    \begin{equation}
+        \begin{pmatrix}
+            \frac{ \partial f }{ \partial x_1 }   &   \cdots    &   \frac{ \partial f }{ \partial x_s }    &   \frac{ \partial f }{ \partial y_1 }    &   \cdots    &   \frac{ \partial f }{ \partial y_r }\\  
+            \frac{ \partial G_1 }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial x_s }    &   \frac{ \partial G_1 }{ \partial y_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial y_r }\\
+            \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots\\
+            \frac{ \partial G_r }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_r }{ \partial x_s }    &   \frac{ \partial G_r }{ \partial y_1 }    &  \cdots   & \frac{ \partial G_r }{ \partial y_r }  
+        \end{pmatrix}
+    \end{equation}
+    s'expriment en terme des \( r\) dernières. La matrice est donc au maximum de rang \( r\). Notons que la première ligne est \( \nabla f\) et les \( r\) suivantes sont les \( \nabla G_i\). Vu que ces lignes sont des vecteurs liés, il existe \( \mu_0,\ldots, \mu_r\) tels que
+    \begin{equation}
+        \mu_0\nabla f+\sum_{i=1}^r\mu_i\nabla G_i=0.
+    \end{equation}
+    Par hypothèse les \( \nabla G_i\) sont linéairement indépendants, ce qui nous dit que \( \mu_0\neq 0\). Donc nous avons ce qu'il nous faut :
+    \begin{equation}
+        \nabla f(a)=\sum_i\frac{ \mu_i }{ \mu_0 } \nabla G_i(a).
+    \end{equation}
+
+    Notons qu'au vu de l'expression \eqref{EqRDsSXyZ}, le fait que les formes \( \{ dG_i(a) \}_{1\leq i\leq r}\) forment une partie libre dans \( (\eR^n)^*\) implique que les \( \lambda_i\) sont uniques.
+\end{proof}
+
+La proposition suivante est la même que \ref{ThoRGJosS}.
+\begin{proposition} \label{PropfPPUxh}
+    Soit \( U\), un ouvert de \( \eR^n\) et des fonctions de classe \( C^1\) \( f,g_1,\ldots, g_r\colon U\to \eR\). Nous considérons
+    \begin{equation}
+        \Gamma=\{ x\in U\tq g_1(x)=\ldots=g_r(x)=0 \}.
+    \end{equation}
+    Soit \( a\) un extrémum de \( f|_{\Gamma}\). Supposons que les formes \( dg_1,\ldots, dg_r\) soient linéairement indépendantes en \( a\). Alors il existe \( \lambda_1,\ldots, \lambda_r\) dans \( \eR\) tel que
+    \begin{equation}
+        df_a=\sum_{i=1}^r\lambda_i(dg_i)_a.
+    \end{equation}
+\end{proposition}
+
+En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
+
+L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
+
+\begin{lemma}       \label{LemmeMinSCimpliqueS}
+    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
+\end{lemma}
+
+\begin{proof}
+    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
+\end{proof}
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Fonctions convexes}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+\begin{definition}[\cite{JFihMcQ}]  \label{DefVQXRJQz}
+    Une fonction $f$ d’un intervalle $I$ de \( \eR\) vers \( \eR\) est dite \defe{convexe}{fonction!convexe}\index{convexité!fonction} lorsque, pour tous \( x_1\) et \( x_2\) de $I$ et tout $\lambda$ dans $[0, 1]$ nous avons
+    \begin{equation}        \label{EQooYNAPooFePQZy}
+        f\big(\lambda\, x_1+(1-\lambda)\, x_2\big) \leq \lambda\, f(x_1)+(1-\lambda)\, f(x_2)
+    \end{equation}
+    Si l'inégalité est stricte, alors nous disons que la fonction \( f\) est \defe{strictement convexe}{convexité!stricte}.
+
+    Une fonction est \defe{concave}{concave} si son opposée est convexe.
+\end{definition}
+
+
+\begin{normaltext}[\cite{GYfviRu}]
+    Les différents résultats pour les fonctions convexes s'adaptent généralement sans mal aux fonctions strictement convexes. Une nuance cependant : de même que les fonctions dérivables convexes sont celles qui ont une dérivée croissante, les fonctions dérivables strictement convexes sont celles qui ont une dérivée strictement croissante (proposition \ref{PropYKwTDPX}). En revanche, il ne faudrait pas croire que la dérivée seconde d'une fonction dérivable strictement convexe est nécessairement une fonction à valeurs strictement positives (voir théorème \ref{ThoGXjKeYb}) : la dérivée d'une fonction strictement croissante peut s'annuler occasionnellement, ou plus exactement peut s'annuler sur un ensemble de points d'intérieur vide. Penser à \( x\mapsto x^4\) pour un exemple de fonction strictement convexe dont la dérivée seconde s'annule.
+\end{normaltext}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Inégalité des pentes}
+%---------------------------------------------------------------------------------------------------------------------------
+
+Dans l'étude des fonctions convexes nous allons souvent utiliser la fonction \defe{taux d'accroissement}{taux d'accroissement} qui est, pour \( \alpha\) dans le domaine de convexité de \( f\) définie par
+\begin{equation}    \label{EqRYBazWd}
+    \begin{aligned}
+        \tau_{\alpha}\colon I\setminus\{ \alpha \}&\to \eR \\
+        x&\mapsto \frac{ f(x)-f(\alpha) }{ x-\alpha }. 
+    \end{aligned}
+\end{equation}
+
+\begin{proposition}[Inégalité des pentes\cite{OJIMBtv}] \label{PropMDMGjGO}
+    Soit \( f\) une fonction convexe sur un intervalle \( I\subset \eR\). Alors pour tout \( a<b<c\) dans \( I\) nous avons\footnote{Les inégalités sont strictes si la fonction \( f\) est strictement convexe.}
+    \begin{equation}
+        \frac{ f(b)-f(a)  }{ b-a }\leq\frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
+    \end{equation}
+    En d'autres termes,
+    \begin{equation}
+        \tau_a(b)\leq\tau_a(c)\leq \tau_b(c),
+    \end{equation}
+    c'est à dire que \( \tau\) est croissante en ses deux arguments.
+\end{proposition}
+\index{inégalité!des pentes}
+
+\begin{proof}
+    D'abord les inégalités \( a<b<c\) impliquent \( 0<b-a<c-a\) et donc
+    \begin{equation}
+        \lambda=\frac{ b-a }{ c-a }<1.
+    \end{equation}
+    L'astuce est de remarquer que \( (1-\lambda)a+\lambda c=b\). Donc \( \lambda\) a toutes les bonnes propriétés pour être utilisé dans la définition de la convexité :
+    \begin{equation}
+        f\big( (1-\lambda)a+\lambda c \big)\leq \lambda f(c)+(1-\lambda)f(a),
+    \end{equation}
+    c'est à dire
+    \begin{equation}
+        f(b)-f(a)\leq \lambda\big( f(c)-f(a) \big)
+    \end{equation}
+    ou encore, en remplaçant \( \lambda\) par sa valeur :
+    \begin{equation}
+        \frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(c)-f(a) }{ c-a }.
+    \end{equation}
+    Cela fait déjà une des inégalités à savoir.
+    
+    D'autre part en partant de \( -a<-b<-c\) nous posons
+    \begin{equation}
+        0<\lambda=\frac{ c-b }{ c-a }.
+    \end{equation}
+    Nous avons à nouveau \( b=(1-\lambda)c+\lambda a\) et nous pouvons obtenir la seconde inégalité
+    \begin{equation}
+        \frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
+    \end{equation}
+\end{proof}
+
+Géométriquement, l'inégalité des pentes se comprend facilement : le coefficient angulaire de la corde du graphe augmente. Donc si \( x<y<z\), le coefficient moyen entre \( x\) et \( y\) est plus petit que celui entre \( x\) et \( z\) qui est plus petit que celui entre \( y\) et \( z\).
+
+Donc si le coefficient angulaire moyen entre \( a\) et \( b+u\) vaut celui entre \( a\) et \( b\), ce coefficient ne peut qu'être constant entra \( a\) et \( b\) : sinon il serait plus grand entre \( b\) et \( b+u\) et la moyenne sur \( a\to b+u\) serait plus grande que sa moyenne sur \( a\to b\). Mais avoir un coefficient angulaire constant signifie être une droite.
+
+En résumé, si une fonction est convexe et non strictement convexe, alors son graphe est une droite. C'est en gros cela que la proposition \ref{PROPooOCOEooEGybmS} clarifiera.
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Convexité et régularité}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{lemma}[\cite{GYfviRu}]   \label{LemKLTsHIQ}
+    Une fonction convexe sur un ouvert
+    \begin{enumerate}
+        \item
+            y admet des dérivées à gauche et à droite en chaque point,
+        \item
+            y est continue.
+    \end{enumerate}
+\end{lemma}
+
+\begin{proof}
+    Soit \( I=\mathopen] a , b \mathclose[\) un intervalle sur lequel \( f\) est convexe et \( \alpha\in I\). Nous allons prouver que \( f\) est continue en \( \alpha\). Nous considérons \( \tau_{\alpha}\) le taux d'accroissement définit par \eqref{EqRYBazWd}; c'est une fonction croissante comme précisé dans l'inégalité des trois pentes \ref{PropMDMGjGO} et de plus \( \tau_{\alpha}(x)\) est bornée supérieurement par \( \tau_{\alpha}(b)\) pour \( x<\alpha\) et inférieurement par \( \tau_{\alpha}(a)\) pour \( x>\alpha\). Les limites existent donc et sont finies par la proposition \ref{PropMTmBYeU}. Autrement dit les limites
+        \begin{subequations}
+            \begin{align}
+                \lim_{x\to \alpha+} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^+} \tau_{\alpha}(x)=\inf_{t>\alpha}\tau_{\alpha}(t)\\
+                \lim_{x\to \alpha^-} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^-} \tau_{\alpha}(x)=\sup_{t<\alpha}\tau_{\alpha}(t).
+            \end{align}
+        \end{subequations}
+        existent et sont finies, c'est à dire que la fonction \( f\) admet une dérivée à gauche et à droite.
+
+        Pour tout \( x\) nous avons les inégalités
+        \begin{equation}
+            \tau_{\alpha}(a)\leq \frac{ f(x)-f(\alpha) }{ x-\alpha }\leq \tau_{\alpha}(b).
+        \end{equation}
+        En posant \( k=\max\{ \tau_{\alpha}(a),\tau_{\alpha}(b) \}\) nous avons
+        \begin{equation}
+            \big| f(x)-f(\alpha) \big|\leq k| x-\alpha |.
+        \end{equation}
+        La fonction est donc Lipschitzienne et par conséquent continue par la proposition \ref{PropFZgFTEW}.
+\end{proof}
+
+\begin{remark}
+    Les dérivées à gauche et à droite ne sont a priori pas égales. Penser par exemple à une fonction affine par morceaux dont les pentes augmentent à chaque morceau.
+\end{remark}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Dérivées d'une fonction convexe}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{proposition}[\cite{RIKpeIH,ooGCESooQzZtVC,MonCerveau}] \label{PropYKwTDPX}
+    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) 
+    \begin{enumerate}
+        \item       \label{ITEMooUTSAooJvhZNm}
+            est convexe si et seulement si sa dérivée est croissante sur \( I\).
+        \item       \label{ITEMooLLSIooFwkxtV}
+            est strictement convexe si et seulement si sa dérivée est strictement croissante sur \( I\)
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+
+
+    Pour la preuve de \ref{ITEMooUTSAooJvhZNm} et \ref{ITEMooLLSIooFwkxtV}, nous allons démontrer les énoncés «non stricts»  et indiquer ce qu'il faut changer pour obtenir les énoncés «stricts».
+    \begin{subproof}
+    \item[Sens direct]
+    Nous supposons que \( f\) est convexe. Soient \( a<b\) dans \( I\) et \( x\in\mathopen] a , b \mathclose[\). D'après l'inégalité des pentes \ref{PropMDMGjGO},
+        \begin{equation}        \label{EqATDLooIcqdDI}
+            \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(b)-f(x) }{ b-x }.
+        \end{equation}
+        En faisant la limite \( x\to a\) nous avons
+        \begin{equation}
+            f'(a)\leq \frac{ f(b)-f(a) }{ b-a }
+        \end{equation}
+        et la limite \( x\to b\) donne
+        \begin{equation}
+            \frac{ f(b)-f(a) }{ b-a }\leq f'(b).
+        \end{equation}
+        Ici les inégalités sont non a priori strictes, même si \( f\) est strictement convexe : même avec des inégalités strictes dans \eqref{EqATDLooIcqdDI}, le passage à la limite rend l'inégalité non stricte. Quoi qu'il en soit nous avons 
+        \begin{equation}        \label{EqQGVMooBpuvNr}
+            f'(a)\leq f'(b).
+        \end{equation}
+    \item[Sens direct : strict]
+         Nous savons déjà que \( f'\) est croissante. Si \eqref{EqQGVMooBpuvNr} était une égalité, alors \( f'\) serait constante sur \( \mathopen] a , b \mathclose[\) parce qu'en prenant \( c\) entre \( a\) et \( b\) nous aurions \( f'(a)\leq f'(c)\leq f'(b)\) avec \( f'(a)=f'(b)\). Donc \( f'(a)=f'(c)\). Avoir \( f'\) constante sur un intervalle est contraire à la stricte convexité.
+
+         \item[Sens réciproque]
+
+             Nous supposons que \( f'\) est croissante et nous considérons \( a<b\) dans \( I\) ainsi que \( \lambda\in \mathopen[ 0 , 1 \mathclose]\). Nous posons \( x=\lambda a+(1-\lambda)b\), et nous savons que \( a\leq x\leq b\). Le théorème des accroissements finis \ref{ThoAccFinis} donne \( c_1\in\mathopen] a , x \mathclose[\) et \( c_2\in \mathopen] x , b \mathclose[\) tels que
+                 \begin{equation}
+                     f'(c_1)=\frac{ f(x)-f(a) }{ x-a }
+                 \end{equation}
+                 et 
+                 \begin{equation}
+                     f'(c_2)=\frac{ f(b)-f(x) }{ b-x }.
+                 \end{equation}
+                 Et en plus \( c_1<c_2\). Vu que \( f'\) est croissante nous avons \( f'(c_1)\leq f'(c_2)\) et donc
+                 \begin{equation}       \label{EqSAOCooWAwClQ}
+                     \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(x) }{ b-x }.
+                 \end{equation}
+                 En remplaçant \( x\) par sa valeur en termes de \( \lambda\), \( a\) et \( b\) nous avons \( x-a=(1-\lambda)(b-a)\) et \( b-x=\lambda(b-a)\), et l'inégalité \eqref{EqSAOCooWAwClQ} nous donne
+                 \begin{equation}
+                     f(x)\leq \lambda f(a)+(1-\lambda)f(b).
+                 \end{equation}
+             \item[Sens réciproque : strict]
+                 Si \( f'\) est strictement croissante, nous avons \( f'(c_2)<f'(c_2)\) et les inégalité suivantes sont strictes, ce qui donne
+                 \begin{equation}
+                     f(x)< \lambda f(a)+(1-\lambda)f(b).
+                 \end{equation}
+    \end{subproof}
+\end{proof}
+
+\begin{theorem}[\cite{RIKpeIH}] \label{ThoGXjKeYb}
+    Une fonction \( f\) de classe \( C^2\) est convexe si et seulement si \( f''\) est positive.
+\end{theorem}
+
+\begin{proof}
+    La fonction est \( C^2\), donc \( f''\) est positive si et seulement si \( f'\) est croissante (proposition \ref{PropGFkZMwD}) alors que la proposition \ref{PropYKwTDPX} nous jure que \( f\) sera convexe si et seulement si \( f'\) est croissante.
+\end{proof}
+
+\begin{remark}      \label{REMooVRPQooIybxmp}
+    Une fonction peut être strictement convexe sans que sa dérivée seconde ne soit toujours strictement positive. En exemple : \( x\mapsto x^4\) est strictement convexe alors que sa dérivée seconde s'annule en zéro.
+\end{remark}
+
+\begin{example} \label{ExPDRooZCtkOz}
+    Quelques exemples utilisant le théorème \ref{ThoGXjKeYb}
+    \begin{enumerate}
+        \item
+    La fonction \( x\mapsto x^2\) est convexe parce que sa dérivée seconde est la constante (positive) \( 2\).
+\item La fonction \( x\mapsto\frac{1}{ x }\) est convexe sur \( \eR^+\setminus\{ 0 \}\) (sa dérivée seconde est \( 2x^{-3}\)).
+\item
+    La fonction exponentielle est également convexe.
+\item
+    La fonction \( \ln\) est concave parce que la dérivée seconde de \( -\ln\) est \( \frac{1}{ x^2 }\) qui est strictement positif.
+    \end{enumerate}
+\end{example}
+
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Graphe d'une fonction convexe}
+%---------------------------------------------------------------------------------------------------------------------------
+
+L'idée principale du graphe d'une fonction convexe est qu'il est toujours au dessus du graphe de ses tangentes (lorsqu'elles existent). Lorsqu'elles n'existent pas, le lemme \ref{LemKLTsHIQ} donne des coefficients directeurs de droites qui vont rester en dessous du graphe de la fonction.
+
+\begin{proposition}[\cite{ooKCFNooVrqYhc}]      \label{PROPooOCOEooEGybmS}
+    Une fonction convexe est strictement convexe si et seulement s'il n'existe aucun intervalle de longueur non nulle sur lequel elle coïncide avec une fonction affine.
+\end{proposition}
+
+\begin{proof}
+    Si sur l'intervalle (non réduit à un point) \( \mathopen[ x , y \mathclose]\), la fonction convexe \( f\) coïncide avec une fonction affine, alors \( f(t)=at+b\) et pour \( \lambda\in\mathopen] 0 , 1 \mathclose[\) nous avons
+        \begin{equation}
+                f\big( \lambda x+(1-\lambda)y \big)=a\lambda x+a(1-\lambda)y+b=\lambda f(x)+(1-\lambda)f(y)
+        \end{equation}
+        où nous avons remplacé \( b\) par \( \lambda b+(1-\lambda)b\). Par conséquent la fonction n'est pas strictement convexe.
+
+    Nous supposons maintenant que la fonction convexe \( f\) n'est pas strictement convexe sur l'intervalle \( I\). Il existe \( x\neq y\in I\) et \( \lambda\in \mathopen] 0 , 1 \mathclose[\) tels que
+        \begin{equation}
+            f\big( \lambda x+(1-\lambda)y \big)=\lambda f(x)+(1-\lambda)f(y).
+        \end{equation}
+    Nous posons \( z=\lambda x+(1-\lambda)y\) et \( u\in\mathopen] x , z \mathclose[\) pour écrire des inégalités des pentes entre \( x<u<z<y\). Plus précisément si nous notons \( a\to b\) la pente de \( a\) à \( b\), c'est à dire \( a\to b=\frac{ f(b)-f(a) }{ b-a }\), alors les inégalités des pentes pour \( x<u<z\) puis \( u<z<y\) donnent
+        \begin{equation}        \label{EqooBMEFooMpoEzd}
+            x\to z\leq u\to z\leq z\to y.
+        \end{equation}
+        Voyons maintenant qu'en réalité \( z\to y=x\to z\). En effet en replaçant
+        \begin{equation}
+            f(y)=\frac{ f(z)-\lambda f(x) }{ 1-\lambda }
+        \end{equation}
+        et
+        \begin{equation}
+            y=\frac{ \lambda x }{ 1-\lambda }
+        \end{equation}
+        dans l'expression \( z\to y=\frac{ f(y)-f(z) }{ y-z }\) nous obtenons
+        \begin{equation}
+            z\to y=\frac{ f(y)-f(z) }{ y-z }=\frac{ f(z)-f(x) }{ z-x }=x\to z.
+        \end{equation}
+        Les inégalités \eqref{EqooBMEFooMpoEzd} sont donc des égalités :
+        \begin{equation}
+            \frac{ f(z)-f(x) }{ z-x }=\frac{ f(z)-f(u) }{ z-u }=\frac{ f(y)-f(z) }{ y-z }.
+        \end{equation}
+        Nous avons donc montré que le nombre \( a=\frac{ f(z)-f(u) }{ z-u }\) ne dépend pas de \( u\). Nous avons alors
+        \begin{equation}
+            f(z)-f(u)=a(z-u) 
+        \end{equation}
+        ou encore :
+        \begin{equation}
+            f(u)=f(z)-a(z-u),
+        \end{equation}
+    ce qui signifie que sur \( \mathopen] x , z \mathclose[\), la fonction \( f\) est affine.
+\end{proof}
+
+\begin{proposition} \label{PROPooQPOSooDZlUAJ}
+    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) est convexe si et seulement si son graphe est au dessus de chacune de ses tangentes.
+\end{proposition}
+
+\begin{proof}
+    \begin{subproof}
+        \item[Sens direct]
+            Soient \( x,y\in I\). Nous voulons :
+            \begin{equation}
+                f(y)\geq f(x)+f'(x)(y-x).
+            \end{equation}
+            Étant donné que nous aurons besoin, dans le quotient différentiel de quelque chose comme \( f(x+t)-f(x)\) nous écrivons la définition \eqref{EQooYNAPooFePQZy} de la convexité en inversant les rôles de \( x\) et \( y\) et en manipulant un peu :
+            \begin{subequations}
+                \begin{align}
+                    f\big( ty+(1-t)x \big)\leq tf(y)+(1-t)f(x)\\
+                    f\big( x+t(y-x) \big)\leq tf(y)+(1-t)f(x)\\
+                    f\big(  x+t(y-x)  \big)=f(x)\leq tf(y)-tf(x)
+                \end{align}
+            \end{subequations}
+            Nous divisons par \( t\) :
+            \begin{equation}
+                \frac{ f\big( x+t(y-x) \big)-f(x) }{ t }\leq f(y)-f(x).
+            \end{equation}
+            Le passage à la limite \( t\to 0\) donne
+            \begin{equation}
+                (y-x)f'(x)\leq f(y)-f(x),
+            \end{equation}
+            ce qu'il fallait.
+        \item[Sens inverse]
+            Pour tout \( x,y\in I\) nous supposons avoir
+            \begin{equation}        \label{EQooEXXIooHXJnER}
+                f(y)\geq f(x)+f'(x)(y-x).
+            \end{equation}
+            Si nous supposons \( x\neq y\) et si nous posons \( z=\lambda x+(1-\lambda)y\) nous voulons prouver que
+            \begin{equation}
+                f(z)\leq \lambda f(x)+(1-\lambda)f(y).
+            \end{equation}
+            Pour cela nous écrivons l'inégalité \eqref{EQooEXXIooHXJnER} avec les couples \( (x,z)\) et \( (y,z)\) :
+            \begin{subequations}
+                \begin{align}
+                    f(x)\geq f(z)+f'(z)'(x-z)\\
+                    f(y)\geq f(z)+f'(z)'(y-z)
+                \end{align}
+            \end{subequations}
+            En multipliant la première par \( \lambda\) et la seconde par \( (1-\lambda)\) et en sommant,
+            \begin{subequations}
+                \begin{align}
+                    \lambda f(x)+(1-\lambda)f(y)&\geq \lambda f(z)+\lambda f'(z)(x-z)+(1-\lambda)f(z)+(1-\lambda)f'(z)(y-z)\\
+                    &=f(z)+f'(z)\big( \lambda(x-z)+(1-\lambda)(y-z) \big)\\
+                    &=f(z).
+                \end{align}
+            \end{subequations}
+    \end{subproof}
+\end{proof}
+
+\begin{proposition}[\cite{MonCerveau}] \label{PropNIBooSbXIKO}
+    Soit \( f\colon \eR\to \eR \) une fonction convexe et \( a\in \eR\). Il existe une constante \( c_a\in \eR\) telle que pour tout \( x\) nous ayons
+    \begin{equation}    \label{EqSKIooSeAekM}
+        f(x)-f(a)\geq c_a(x-a).
+    \end{equation}
+    Autrement dit, le graphe de la fonction \( f\) est toujours au dessus de la droite d'équation
+    \begin{equation}
+        y=f(a)+c_a(x-a).
+    \end{equation}
+\end{proposition}
+
+\begin{proof}
+    Les dérivées à gauche et à droite de \( f\) données par le lemme \ref{LemKLTsHIQ} sont les candidats tout cuits pour être coefficient directeur de la droite que l'on cherche. Nous allons prouver qu'en posant
+    \begin{equation}
+        c_a=\inf_{t>a}\tau_a(t),
+    \end{equation}
+    la droite \( y=f(a)+c_a(x-a)\) répond à la question\footnote{En prenant l'autre, \( c_a'=\sup_{t<a}\tau_a(t)\), ça fonctionne aussi. En pensant à une fonction affine par morceaux, on remarque qu'en choisissant un nombre entre les deux, nous avons plus facilement une inégalité stricte dans \eqref{EqSKIooSeAekM}.}.
+
+    Nous devons prouver que le nombre \( \Delta_x=f(x)-\big( f(a)+c_a(x-a) \big)\) est positif pour tout \( x\).
+    \begin{subproof}
+
+    \item[Si \( x>a\)]
+        
+        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est positif :
+        \begin{subequations}
+            \begin{align}
+                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
+                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
+                &\geq 0
+            \end{align}
+        \end{subequations}
+        parce que \( t\to\tau_a(t)\) est croissante et que \( x>a\).
+
+    \item[Si \( x<a\)]
+        
+        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est négatif :
+        \begin{subequations}
+            \begin{align}
+                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
+                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
+                &\leq 0
+            \end{align}
+        \end{subequations}
+        parce que \( t\to\tau_a(t)\) est croissante et que \( x<a\).
+    \end{subproof}
+\end{proof}
+
+\begin{proposition}[\cite{MonCerveau}] \label{PropPEJCgCH}
+    Si \( g\) est une fonction convexe, il existe deux suites réelles \( (a_n)\) et \( (b_n)\) telles que
+    \begin{equation}
+        g(x)=\sup_{n\in \eN}(a_nx+b_n).
+    \end{equation}
+\end{proposition}
+\index{fonction!convexe}
+\index{densité!de \( \eQ\) dans \( \eR\)!utilisation}
+
+\begin{proof}
+    Pour \( u\in \eR\) nous considérons \( a(u)\) et \( b(u)\) tels que la droite \( y(x)=a(u)x+b(u)\) vérifie \( y(u)=g(u)\) et \( y(x)\leq g(x)\) pour tout \( x\). Cela est possible par la proposition \ref{PropNIBooSbXIKO}. Il s'agit d'une droite coupant le graphe de \( g\) en \( x=u\) et restant en dessous. Nous considérons alors \( (u_n)\) une suite quelconque dense dans \( \eR\) (disons les rationnels pour fixer les idées) et nous posons
+    \begin{subequations}
+        \begin{numcases}{}
+            a_n=a(u_n)\\
+            b_n=b(u_n).
+        \end{numcases}
+    \end{subequations}
+    Si \( q\in \eQ\) alors \( a_nx+b_n\leq g(x)\) pour tout \( n\) et \( g(q)\) est le supremum qui est atteint pour le \( n\) tel que \( u_n=q\). Si maintenant \( x\) n'est pas dans \( \eQ\) il faut travailler plus.
+
+    Nous prenons \( (\tilde q_n)\), une sous-suite de \( (q_n)\) convergeant vers \( x\) et \( N\) suffisamment grand pour que pour tout \( n\geq N\) on ait \( | \tilde q_n-x |\leq \epsilon\) et \( | g(\tilde q_n)-g(x) |\leq \epsilon\); cela est possible grâce à la continuité de \( g\) (lemme \ref{LemKLTsHIQ}). Ensuite les sous-suites \( (\tilde a_n)\) et \( (\tilde b_n)\) sont celles qui correspondent :
+    \begin{equation}
+        \tilde a_n\tilde q_n+\tilde b_n=g(\tilde q_n).
+    \end{equation}
+    Nous considérons la majoration
+    \begin{subequations}
+        \begin{align}
+            | \tilde a_nx+\tilde b_n-g(x) |&\leq| \tilde a_nx+\tilde b_n-(\tilde a_n\tilde q_n+\tilde b_n) |+\underbrace{| \tilde a_n\tilde q_n+\tilde b_n-g(\tilde q_n) |}_{=0}+\underbrace{| g(\tilde q_n)-g(x) |}_{\leq \epsilon}\\
+            &\leq | \tilde a_n | |x-\tilde q_n |+\epsilon\\
+            &=\epsilon\big( | \tilde a_n |+1 \big).
+        \end{align}
+    \end{subequations}
+    Il nous reste à montrer que \( | \tilde a_n |\) est borné par un nombre ne dépendant pas de \( n\) (pour les \( n>N\)).
+
+    Vu que la droite de coefficient directeur \( \tilde a_n\) et passant par le point \( \big( \tilde q_n,g(\tilde q_n) \big)\) reste en dessous du graphe de \( g\), nous avons pour tout \( n\) et tout \( y\in \eR\) l'inégalité
+    \begin{equation}
+        g(y)\geq \tilde a_n(y-\tilde q_n)+g(\tilde q_n)\in \tilde a_nB(y-x,\epsilon)+B\big( g(x),\epsilon \big).
+    \end{equation}
+    Si \( \tilde a_n\) n'est pas borné vers le haut, nous prenons \( y\) tel que \( B(y-x,\epsilon)\) soit minoré par un nombre \( k\) strictement positif et nous obtenons
+    \begin{equation}
+        g(y)\geq k\tilde a_n+l
+    \end{equation}
+    avec \( k\) et \( l\) indépendants de \( n\). Cela donne \( g(y)=\infty\). Si au contraire \( \tilde a_n\) n'est pas borné vers le bas, nous prenons $y$ tel que \( B(y-x,\epsilon)\) est majoré par un nombre \( k\) strictement négatif. Nous obtenons encore \( g(y)=\infty\).
+
+    Nous concluons que \( | \tilde a_n |\) est bornée.
+\end{proof}
+
+\begin{lemma}[\cite{KXjFWKA}]   \label{LemXOUooQsigHs}
+    L'application
+    \begin{equation}
+        \begin{aligned}
+            \phi\colon S^{++}(n,\eR)&\to \eR \\
+            A&\mapsto \det(A) 
+        \end{aligned}
+    \end{equation}
+    est \defe{log-convave}{concave!log-concave}\index{log-concave}, c'est à dire que l'application \( \ln\circ\phi\) est concave\footnote{La définition \ref{DEFooELGOooGiZQjt} du logarithme ne fonctionne que pour les réels strictement positifs. C'est le cas du déterminant d'une matrice réelle symétrique strictement définie positive.}. De façon équivalente, si \( A,B\in S^{++}\) et si \( \alpha+b=1\), alors
+    \begin{equation}    \label{EqSPKooHFZvmB}
+        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
+    \end{equation}
+\end{lemma}
+Ici \( S^{++}\) est l'ensemble des matrices symétriques strictement définies positives, définition \ref{DefAWAooCMPuVM}.
+
+\begin{proof}
+    Nous commençons par prouver que l'équation \eqref{EqSPKooHFZvmB} est équivalente à la log-concavité du déterminant. Pour cela il suffit de remarquer que les propriétés de croissance et d'additivité du logarithme donnent l'équivalence entre
+    \begin{equation}
+        \ln\Big( \det(\alpha A+\beta B) \Big)\geq \ln\Big( \det(\alpha A) \Big)+\ln\Big( \det(\beta B) \Big),
+    \end{equation}
+    et
+    \begin{equation}    \label{EqTJYooBWiRrn}
+        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
+    \end{equation}
+
+    Le théorème de pseudo-réduction simultanée, corollaire \ref{CorNHKnLVA}, appliqué aux matrices \( A\) et \( B\) nous donne une matrice inversible \( Q\) telle que
+    \begin{subequations}
+        \begin{numcases}{}
+            B=Q^tDQ\\
+            A=Q^tQ
+        \end{numcases}
+    \end{subequations}
+    avec 
+    \begin{equation}
+        D=\begin{pmatrix}
+            \lambda_1    &       &       \\
+                &   \ddots    &       \\
+                &       &   \lambda_n
+        \end{pmatrix},
+    \end{equation}
+    \( \lambda_i>0\). Nous avons alors
+    \begin{equation}
+        \det(A)^{\alpha}\det(B)^{\beta}=\det(Q)^{2\alpha}\det(Q)^{2\beta}\det(D)^{\beta}=\det(Q)^2\det(D)^{\beta}
+    \end{equation}
+    (parce que \( \alpha+\beta=1\)) et
+    \begin{equation}
+        \det(\alpha A+\beta B)=\det(\alpha Q^tQ+\beta Q^tDQ)=\det\big( Q^t(\alpha\mtu+\beta D)Q \big)=\det(Q)^2\det(\alpha\mtu+\beta D).
+    \end{equation}
+    L'inégalité \eqref{EqTJYooBWiRrn} qu'il nous faut prouver se réduit donc  à
+    \begin{equation}
+        \det(\alpha \mtu+\beta D)\geq \det(D)^{\beta}.
+    \end{equation}
+    Vue la forme de \( D\) nous avons
+    \begin{equation}
+        \det(\alpha\mtu+\beta D)=\prod_{i=1}^n(\alpha+\beta\lambda_i)
+    \end{equation}
+    et 
+    \begin{equation}
+        \det(D)^{\beta}=\big( \prod_{i=1}^{n}\lambda_i \big)^{\beta}.
+    \end{equation}
+    Il faut donc prouver que
+    \begin{equation}\label{EqGFLooOElciS}    
+        \prod_{i=1}^n(\alpha+\beta\lambda_i)\geq \big( \prod_{i=1}^n\lambda_i \big)^{\beta}.
+    \end{equation}
+    Cette dernière égalité de produit sera prouvée en passant au logarithme. Vu que le logarithme est concave par l'exemple \ref{ExPDRooZCtkOz}, nous avons pour chaque \( i\) que
+    \begin{equation}
+        \ln(\alpha+\beta\lambda_i)\geq \alpha\ln(1)+\beta\ln(\lambda_i)=\beta\ln(\lambda_i).
+    \end{equation}
+    En sommant cela sur \( i\) et en utilisant les propriétés de croissance et de multiplicativité du logarithme nous obtenons successivement
+    \begin{subequations}
+        \begin{align}
+            \sum_{i=1}^n\ln(\alpha+\beta\lambda_i)\geq \beta\sum_i\ln(\lambda_i)\\
+            \ln\big( \prod_i(\alpha+\beta\lambda_i) \big)\geq\ln\Big( \big( \prod_i\lambda_i \big)^{\beta} \Big)\\
+            \prod_i(\alpha+\beta\lambda_i)\geq\big( \prod_i\lambda_i \big)^{\beta},
+        \end{align}
+    \end{subequations}
+    ce qui est bien \eqref{EqGFLooOElciS}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{En dimension supérieure}
+%---------------------------------------------------------------------------------------------------------------------------
+
+\begin{definition}
+    Soit une partie convexe \( U\) de \( \eR^n\) et une fonction \( f\colon U\to \eR\). 
+    \begin{enumerate}
+        \item
+        La fonction \( f\) est \defe{convexe}{convexe!fonction sur \( \eR^n\)} si pour tout \( x,y\in U\) avec \( x\neq y\) et pour tout \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous avons
+            \begin{equation}
+                f\big( \theta x+(1-\theta)y \big)\leq \theta f(x)+(1-\theta)f(y).
+            \end{equation}
+        \item
+            Elle est \defe{strictement convexe}{strictement!convexe!sur \( \eR^n\)} si nous avons l'inégalité stricte.
+    \end{enumerate}
+\end{definition}
+
+\begin{proposition}[\cite{ooLJMHooMSBWki}]      \label{PROPooYNNHooSHLvHp}
+    Soit \( \Omega\) ouvert dans \( \eR^n\) et \( U\) convexe dans \( \Omega\), et une fonction différentiable \( f\colon U\to \eR\).
+    \begin{enumerate}
+        \item       \label{ITEMooRVIVooIayuPS}
+            La fonction \( f\) est convexe sur \( U\) si et seulement si pour tout \( x,y\in U\),
+            \begin{equation}
+                f(y)\geq f(x)+df_x(y-x).
+            \end{equation}
+        \item       \label{ITEMooCWEWooFtNnKl}
+            La fonction \( f\) est strictement convexe sur \( U\) si et seulement si pour tout \( x,y\in U\) avec \( x\neq y\),
+            \begin{equation}
+                f(y)>f(x)+df_x(y-x).
+            \end{equation}
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Nous avons quatre petites choses à démontrer.
+    \begin{subproof}
+    \item[\ref{ITEMooRVIVooIayuPS} sens direct]
+        Soit une fonction convexe \( f\). Nous avons :
+        \begin{equation}
+            f\big( (1-\theta)x+\theta y \big)\leq (1-\theta)f(x)+\theta f(y),
+        \end{equation}
+        donc
+        \begin{equation}
+            f\big( x+\theta(y-x) \big)-f(x)\leq \theta\big( f(y)-f(x) \big)
+        \end{equation}
+        Vu que \( \theta>0\) nous pouvons diviser par \( \theta\) sans changer le sens de l'inégalité :
+        \begin{equation}        \label{EQooAXXFooHWtiJh}
+            \frac{ f\big( x+\theta(y-x) \big)-f(x) }{ \theta }\leq f(y)-f(x).
+        \end{equation}
+        Nous prenons la limite \( \theta\to 0^+\). Cette limite est égale à a limite simple \( \theta\to 0\) et vaut (parce que \( f\) est différentiable) :
+        \begin{equation}
+            \frac{ \partial f }{ \partial (y-x) }(x)\leq f(y)-f(x),
+        \end{equation}
+        et aussi
+        \begin{equation}
+            df_x(y-x)\leq f(y)-f(x)
+        \end{equation}
+        par le lemme \ref{LemdfaSurLesPartielles}.
+    \item[\ref{ITEMooRVIVooIayuPS} sens inverse]
+        Pour tout \( a\neq b\) dans \( U\) nous avons
+        \begin{equation}        \label{EQooEALSooJOszWr}
+            f(b)\geq f(a)+df_a(b-a).
+        \end{equation}
+    Pour \( x\neq y\) dans \( U\) et pour \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous écrivons \eqref{EQooEALSooJOszWr} pour les couples \( \big( \theta x+(1-\theta)y,y \big)\) et \( \big( \theta x+(1-\theta)y,x \big)\). Ça donne :
+        \begin{equation}
+            f(y)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( \theta(y-x) \big),
+        \end{equation}
+        et
+        \begin{equation}
+            f(x)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( (1-\theta)(x-y) \big).
+        \end{equation}
+        La différentielle est linéaire; en multipliant la première par \( (1-\theta)\) et la seconde par \( \theta\) et en la somme, les termes en \( df\) se simplifient et nous trouvons
+        \begin{equation}
+            \theta f(x)+(1-\theta)f(y)\geq f\big( \theta x+(1-\theta)y \big).
+        \end{equation}
+    \item[\ref{ITEMooCWEWooFtNnKl} sens direct]
+        Nous avons encore l'équation \eqref{EQooAXXFooHWtiJh}, avec une inégalité stricte. Par contre, ça ne va pas être suffisant parce que le passage à la limite ne conserve pas les inégalités strictes. Nous devons donc être plus malins. 
+
+        Soient \( 0<\theta<\omega<1\). Nous avons \( (1-\theta)x+\theta y\in \mathopen[ x , (1-\omega)x+\omega y \mathclose]\), donc nous pouvons écrire \( (1-\theta)x+\theta y\) sous la forme \( (1-s)x+s\big( (1-\omega)x+\omega y \big)\). Il se fait que c'est bon pour \( s=\theta/\omega\) (et aussi que nous avons \( \theta/\omega<1\)). Donc nous avons
+        \begin{subequations}
+            \begin{align}
+            f\big( (1-\theta)x+\theta y \big)&=f\Big( (1-\frac{ \theta }{ \omega })x+\frac{ \theta }{ \omega }\big( (1-\omega)x+\omega y \big) \Big)\\
+            &<(1-\frac{ \theta }{ \omega })f(x)+\frac{ \theta }{ \omega }f\big( (1-\omega)x+\omega y \big).
+            \end{align}
+        \end{subequations}
+        Cela nous permet d'écrire
+        \begin{equation}
+            \frac{ f\big( (1-\theta)x+\theta y \big)-f(x) }{ \theta }<\frac{ f\big( (1-\omega)x+\omega y \big) }{ \omega }<f(y)-f(x).
+        \end{equation}
+        Le seconde inégalité est le pendant de \eqref{EQooAXXFooHWtiJh}. Maintenant en passant à la limite pour \( \theta\) nous conservons une inégalité stricte par rapport à \( f(y)-f(x)\) :
+        \begin{equation}
+            df_x(y-x)<f(y)-f(x).
+        \end{equation}
+    \end{subproof}
+\end{proof}
+
+% Il faut laisser les sauts de lignes suivants, pour rechercher efficacement les références vers le futur.
+Avant de lire la proposition suivante, il faut relire la proposition \ref{PROPooFWZYooUQwzjW} et ce qui s'y rapporte. 
+Lire aussi la remarque \ref{REMooVRPQooIybxmp} qui indique
+qu'il n'y a pas de réciproque dans l'énoncé \ref{ITEMooHAGQooYZyhQk}.     
+\begin{proposition}[\cite{ooLJMHooMSBWki}]
+    Soit une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) et un convexe \( U\subset \Omega\).
+    \begin{enumerate}
+        \item       \label{ITEMooZQCAooIFjHOn}
+            La fonction \( f\) est convexe sur \( U\) si et seulement si
+            \begin{equation}        \label{EQooIBDCooJYdiBb}
+                (d^2f)_x(y-x,y-x)\geq 0
+            \end{equation}
+            pour tout \( x,y\in U\).
+        \item       \label{ITEMooHAGQooYZyhQk}
+            Si pour tout \( x\neq y\) dans \( U\) nous avons
+            \begin{equation}
+                (d^2f)_x(y-x,y-x)>0
+            \end{equation}
+            alors la fonction \( f\) est strictement convexe sur \( U\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{remark}      \label{REMooYCRKooEQNIkC}
+    Notons que la condition \eqref{EQooIBDCooJYdiBb} n'est pas équivalente à demander \( (d^2f)_x(h,h)\geq 0\) pour tout \( h\). En effet nous ne demandons la positivité que dans les directions atteignables comme différence de deux éléments de \( U\). La partie \( U\) n'est pas spécialement ouverte; elle pourrait n'être qu'une droite dans \( \eR^3\). Dans ce cas, demander que \( f\) (qui est \( C^2\) sur l'ouvert \( \Omega\)) soit convexe sur \( U\) ne demande que la positivité de \( (d^2f)_x\) appliqué à des vecteurs situés sur la droite \( U\).
+\end{remark}
+
+\begin{proof}
+    Il y a trois parties à démontrer.
+    \begin{subproof}
+    \item[\ref{ITEMooZQCAooIFjHOn} sens direct]
+
+        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci au jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
+        \begin{equation}
+            f(x+th)=f(x)+tdf_x(h)+\frac{ t^2 }{2}(d^2_x)(h,h)+t^2\| h \|^2\alpha(th)
+        \end{equation}
+        avec \( \lim_{s\to 0}\alpha(s)=0\). Le fait que \( f\) soit convexe donne
+        \begin{equation}
+            0\leq f(x+th)-f(x)-tdf_x(h),
+        \end{equation}
+        et donc
+        \begin{equation}
+            0\leq \frac{ t^2 }{2}(d^2f)_x(h,h)+f^2\| h \|^2\alpha(th).
+        \end{equation}
+        En multipliant par \( 2\) et en divisant par \( t^2\),
+        \begin{equation}
+            0\leq (d^2f)_x(h,h)+2\| h \|^2\alpha(th).
+        \end{equation}
+        En prenant \( t\to 0\) nous avons bien  \( (d^2f)_x(y-x,y-x)\geq 0\).
+
+    \item[\ref{ITEMooZQCAooIFjHOn} sens inverse]
+
+        Soient \( x,y\in U\). Nous écrivons Taylor en version de la proposition \ref{PROPooWWMYooPOmSds} :
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\frac{ 1 }{2}(d^2f)_z(y-x,y-x)
+        \end{equation}
+    pour un certain \( z\in\mathopen] x , y \mathclose[\). En vertu de ce qui a été dit dans la remarque \ref{REMooYCRKooEQNIkC} nous ne pouvons pas évoquer l'hypothèse \eqref{EQooIBDCooJYdiBb} pour conclure que \( (d^2f)_z(y-x,y-x)\geq 0\). Il y a deux manières de nous sortir du problème :
+        \begin{itemize}
+            \item Trouver \( s\in U\) tel que \( y-x=s-z\).
+            \item Trouver un multiple de \( y-x\) qui soit de la forme \( y-x\).
+        \end{itemize}
+        La première approche ne fonctionne pas parce que \( s=y-x+z\) n'est pas garanti d'être dans \( U\); par exemple avec \( x=1\), \( z=2\), \( y=3\) et \( U=\mathopen[ 0 , 3 \mathclose]\). Dans ce cas \( s=4\notin U\).
+
+        Heureusement nous avons \( z=\theta x+(1-\theta)y\), donc \( z-x=(1-\theta)(y-x)\). Dans ce cas la bilinéarité de \( (d^2f)_z\) donne\footnote{Si vous avez bien suivi, la bilinéarité est contenue dans la proposition \ref{PROPooFWZYooUQwzjW}.}
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{\geq 0}.
+        \end{equation}
+        Nous en déduisons que \( f\) est convexe par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooRVIVooIayuPS}.
+    \item[\ref{ITEMooHAGQooYZyhQk}]
+
+        Le raisonnement que nous venons de faire pour le sens inverse de \ref{ITEMooZQCAooIFjHOn} tient encore, et nous avons
+        \begin{equation}
+            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{> 0}
+        \end{equation}
+        d'où nous déduisons la stricte convexité de \( f\) par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooCWEWooFtNnKl}.
+    \end{subproof}
+\end{proof}
+
+\begin{corollary}       \label{CORooMBQMooWBAIIH}
+    Avec la hessienne\ldots en cours d'écriture.
+\end{corollary}
+
+\begin{proof}
+    Cela va utiliser la proposition \ref{PropoExtreRn}.
+\end{proof}
+
+%--------------------------------------------------------------------------------------------------------------------------- 
+\subsection{Quelque inégalités}
+%---------------------------------------------------------------------------------------------------------------------------
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Inégalité de Jensen}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\index{inégalité!Jensen}
+\index{convexité!inégalité de Jensen}
+
+\begin{proposition}[Inégalité de Jensen]    \label{PropXIBooLxTkhU}
+    Soit \( f\colon \eR\to \eR\) une fonction convexe et des réels \( x_1\),\ldots,  \( x_n\). Soient des nombres positifs \( \lambda_1\),\ldots,  \( \lambda_n\) formant une combinaison convexe\footnote{Définition \ref{DefIMZooLFdIUB}.}. Alors
+    \begin{equation}
+        f\big( \sum_i\lambda_ix_i \big)\leq \sum_i\lambda_if(x_i).
+    \end{equation}
+\end{proposition}
+\index{inégalité!Jensen!pour une somme}
+
+\begin{proof}
+    Nous procédons par récurrence sur \( n\), en sachant que \( n=2\) est la définition de la convexité de \( f\). Vu que
+    \begin{equation}
+        \sum_{k=1}^n\lambda_kx_k=\lambda_nx_n+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n },
+    \end{equation}
+    nous avons
+    \begin{equation}
+        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)f\big( \sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n } \big).
+    \end{equation}
+    La chose à remarquer est que les nombres \( \frac{ \lambda_k }{ 1-\lambda_n }\) avec \( k\) allant de \( 1\) à \( n-1\) forment eux-mêmes une combinaison convexe. L'hypothèse de récurrence peut donc s'appliquer au second terme du membre de droite :
+    \begin{equation}
+        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_k }{ 1-\lambda_n }f(x_k)=\lambda_nf(x_n)+\sum_{k=1}^{n-1}\lambda_kf(x_k).
+    \end{equation}
+\end{proof}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Inégalité arithmético-géométrique}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+La proposition suivante dit que la moyenne arithmétique de nombres strictement positifs est supérieure ou égale à la moyenne géométrique.
+\begin{proposition}[Inégalité arithmético-géométrique\cite{CENooZKvihz}]    \label{PropWDPooBtHIAR}
+    Soient \( x_1\),\ldots, \( x_n\) des nombres strictement positifs. Nous posons
+    \begin{equation}
+        m_a=\frac{1}{ n }(x_1+\cdots +x_n)
+    \end{equation}
+    et
+    \begin{equation}
+        m_g=\sqrt[n]{x_1\ldots x_n}
+    \end{equation}
+    Alors \( m_g\leq m_a\) et \( m_g=m_a\) si et seulement si \( x_i=x_j\) pour tout \( i,j\).
+\end{proposition}
+\index{inégalité!arithmético-géométrique}
+
+\begin{proof}
+    Par hypothèse les nombres \( m_a\) et \( m_g\) sont tout deux strictement positifs, de telle sorte qu'il est équivalent de prouver \( \ln(m_g)\leq \ln(m_a)\) ou encore
+    \begin{equation}
+        \frac{1}{ n }\big( \ln(x_1)+\cdots +\ln(x_n) \big)\leq \ln\left( \frac{ x_1+\cdots +x_n }{ n } \right).
+    \end{equation}
+    Cela n'est rien d'autre que l'inégalité de Jensen de la proposition \ref{PropXIBooLxTkhU} appliquée à la fonction \( \ln\) et aux coefficients \( \lambda_i=\frac{1}{ n }\).
+\end{proof}
+
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+\subsubsection{Inégalité de Kantorovitch}
+%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
+
+\begin{proposition}[Inégalité de Kantorovitch\cite{EYGooOoQDnt}]    \label{PropMNUooFbYkug}
+    Soit \( A\) une matrice symétrique strictement définie positive dont les plus grandes et plus petites valeurs propres sont \( \lambda_{min}\) et \( \lambda_{max}\). Alors pour tout \( x\in \eR^n\) nous avons
+    \begin{equation}
+        \langle Ax, x\rangle \langle A^{-1}x, x\rangle \leq \frac{1}{ 4 }\left( \frac{ \lambda_{min} }{ \lambda_{max} }+\frac{ \lambda_{max} }{ \lambda_{min} } \right)^2\| x^4 \|.
+    \end{equation}
+\end{proposition}
+\index{inégalité!Kantorovitch}
+
+\begin{proof}
+    Sans perte de généralité nous pouvons supposer que \( \| x \|=1\). Nous diagonalisons\footnote{Théorème spectral \ref{ThoeTMXla}.} la matrice \( A\) par la matrice orthogonale  \( P\in\gO(n,\eR)\) : \( A=PDP^{-1}\) et \( A^{-1}=PD^{-1}P^{-1}\) où \( D\) est  une matrice diagonale formée des valeurs propres de \( A\).
+
+    Nous posons \( \alpha=\sqrt{\lambda_{min}\lambda_{max}}\) et nous regardons la matrice
+    \begin{equation}
+        \frac{1}{ \alpha }A+tA^{-1}
+    \end{equation}
+    dont les valeurs propres sont
+    \begin{equation}
+        \frac{ \lambda_i }{ \alpha }+\frac{ \alpha }{ \lambda_i }
+    \end{equation}
+    parce que les vecteurs propres de \( A\) et de \( A^{-1}\) sont les mêmes (ce sont les valeurs de la diagonale de \( D\)). Nous allons quelque peu étudier la fonction
+    \begin{equation}
+        \theta(x)=\frac{ x }{ \alpha }+\frac{ \alpha }{ x }.
+    \end{equation}
+    Elle est convexe en tant que somme de deux fonctions convexes. Elle a son minimum en \( x=\alpha\) et ce minimum vaut \( \theta(\alpha)=2\). De plus
+    \begin{equation}
+        \theta(\lambda_{max})=\theta(\lambda_{min})=\sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
+    \end{equation}
+    Une fonction convexe passant deux fois par la même valeur doit forcément être plus petite que cette valeur entre les deux\footnote{Je ne suis pas certain que cette phrase soit claire, non ?} : pour tout \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\),
+    \begin{equation}
+        \theta(x)\leq  \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
+    \end{equation}
+    
+    Nous sommes maintenant en mesure de nous lancer dans l'inégalité de Kantorovitch.
+    \begin{subequations}
+        \begin{align}
+            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }&\leq\frac{ 1 }{2}\left( \frac{ \langle Ax, x\rangle  }{ \alpha }+\alpha\langle A^{-1}x, x\rangle  \right)\label{subEqUKIooCWFSkwi}\\
+            &=\frac{ 1 }{2}\langle   \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x , x\rangle \\
+            &\leq\frac{ 1 }{2}\Big\| \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x \|\| x \| \label{subEqUKIooCWFSkwiii}\\
+            &\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \label{subEqUKIooCWFSkwiv}
+        \end{align}
+    \end{subequations}
+    Justifications :
+    \begin{itemize}
+        \item \ref{subEqUKIooCWFSkwi} par l'inégalité arithmético-géométrique, proposition \ref{PropWDPooBtHIAR}. Nous avons aussi inséré \( \alpha\frac{1}{ \alpha }\) dans le produit sous la racine.
+        \item \ref{subEqUKIooCWFSkwiii} par l'inégalité de Cauchy-Schwarz, théorème \ref{ThoAYfEHG}.
+        \item \ref{subEqUKIooCWFSkwiv} par la définition de la norme opérateur de la proposition \ref{DefNFYUooBZCPTr}
+    \end{itemize}
+    La norme opérateur est la plus grande des valeurs propres. Mais les valeurs propres de \( A/\alpha+\alpha A^{-1}\) sont de la forme \( \theta(\lambda_i)\), et tous les \( \lambda_i\) sont entre \( \lambda_{min} \) et \( \lambda_{max}\). Donc la plus grande valeur propre de \( A/\alpha+\alpha A^{-1}\) est \( \theta(x)\) pour un certain \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\). Par conséquent
+    \begin{equation}
+            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \leq \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
+    \end{equation}
+\end{proof}
diff --git a/tex/frido/77_series_fonctions.tex b/tex/frido/77_series_fonctions.tex
index 3295be401..b809d325d 100644
--- a/tex/frido/77_series_fonctions.tex
+++ b/tex/frido/77_series_fonctions.tex
@@ -1481,2215 +1481,3 @@ \subsection{Théorème de Cauchy-Arzella}
     \end{enumerate}
     Ces deux points sont illustrés dans les exemples \ref{EXooSHMMooHVfsMB} et \ref{EXooJNOMooYqUwTZ}.
 \end{remark}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-                    \section{Théorèmes d'inversion locale et de la fonction implicite}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Mise en situation}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Dans un certain nombre de situation, il n'est pas possible de trouver des solutions explicites aux équations qui apparaissent. Néanmoins, l'existence «théorique» d'une telle solution est souvent déjà suffisante. C'est l'objet du théorème de la fonction implicite.
-
-Prenons par exemple la fonction sur $\eR^2$ donnée par 
-\begin{equation}
-    F(x,y)=x^2+y^2-1.
-\end{equation}
-Nous pouvons bien entendu regarder l'ensemble des points donnés par $F(x,y)=0$. C'est le cercle dessiné à la figure \ref{LabelFigCercleImplicite}.
-\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite. Si on donne \( x\), nous ne pouvons pas savoir si nous parlons de \( P\) ou de \( P'\).}
-\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
-
-%\ref{LabelFigCercleImplicite}.
-%\newcommand{\CaptionFigCercleImplicite}{Un cercle pour montrer l'intérêt de la fonction implicite.}
-%\input{auto/pictures_tex/Fig_CercleImplicite.pstricks}
-
-Nous ne pouvons pas donner le cercle sous la forme $y=y(x)$ à cause du $\pm$ qui arrive quand on prend la racine carrée. Mais si on se donne le point $P$, nous pouvons dire que \emph{autour de $P$}, le cercle est la fonction
-\begin{equation}
-    y(x)=\sqrt{1-x^2}.
-\end{equation}
-Tandis que autour du point $P'$, le cercle est la fonction
-\begin{equation}
-    y(x)=-\sqrt{1-x^2}.
-\end{equation}
-Autour de ces deux point, donc, le cercle est donné par une fonction. Il n'est par contre pas possible de donner le cercle autour du point $Q$ sous la forme d'une fonction.
-
-Ce que nous voulons faire, en général, est de voir si l'ensemble des points tels que
-\begin{equation}
-    F(x_1,\ldots,x_n,y)=0
-\end{equation}
-peut être donné par une fonction $y=y(x_1,\ldots,x_n)$. En d'autre termes, est-ce qu'il existe une fonction $y(x_1,\ldots,x_n)$ telle que
-\begin{equation}
-    F\big( x_1,\ldots,x_n,y(x_1,\ldots,x_n)\big)=0.
-\end{equation}
-
-Plus généralement, soit une fonction
-\begin{equation}
-    \begin{aligned}
-        F\colon D\subset \eR^n\times \eR^m&\to \eR^m \\
-        (x,y)&\mapsto \big( F_1(x,y),\ldots, F_m(x,y) \big) 
-    \end{aligned}
-\end{equation}
-avec $x = (x_1,\ldots, x_n)$ et $y = (y_1,\ldots,y_m)$. Pour chaque $x$ fixé, on s'intéresse aux solutions du système de $m$ équations $F(x,y) = 0$ pour les inconnues $y$ ; en particulier, on voudrait pouvoir écrire $y = \varphi(x)$ vérifiant $F(x,\varphi(x)) = 0$.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème d'inversion locale}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma} \label{LemGZoqknC}
-    Soit \( E\) un espace de Banach (métrique complet) et \( \mO\) un ouvert de \( E\). Nous considérons une \( \lambda\)-contraction \( \varphi\colon \mO\to E\). Alors l'application
-    \begin{equation}
-    f\colon x\mapsto x+\varphi(x)
-    \end{equation}
-    est un homéomorphisme entre \( \mO\) et un ouvert de \( E\). De plus \( f^{-1}\) est Lipschitz de constante plus petite ou égale à \( (1-\lambda)^{-1}\).
-\end{lemma}
-Cette proposition utilise le théorème de point fixe de Picard \ref{ThoEPVkCL},
-et sera utilisée pour démontrer le théorème d'inversion locale \ref{ThoXWpzqCn}.
-% note que garder deux lignes ici est important pour vérifier les références vers le futur : la seconde ligne peut être ignorée, pas la seconde.
-
-\begin{proof}
-        Soient \( x_1,x_2\in\mO\). Nous posons \( y_1=f(x_1)\) et \( y_2=f(x_2)\). En vertu de l'inégalité de la proposition \ref{PropNmNNm} nous avons
-        \begin{subequations}    \label{subEqEBJsBfz}
-            \begin{align}
-            \big\| f(x_2)-f(x_1) \big\|&=\big\| x_2+\varphi(x_2)-x_1-\varphi(x_1) \big\|\\
-        &\geq \Big|        \| x_2-x_1 \|-\big\| \varphi(x_2)-\varphi(x_1) \big\|  \Big|\\
-    &\geq   (1-\lambda)\| x_2-x_1 \|.
-            \end{align}
-        \end{subequations}
-        À la dernière ligne les valeurs absolues sont enlevées parce que nous savons que ce qui est à l'intérieur est positif. Cela nous dit d'abord que \( f\) est injective parce que \( f(x_2)=f(x_1)\) implique \( x_2=x_1\). Donc \( f\) est inversible sur son image. Nous posons \( A=f(\mO)\) et nous devons prouver que que \( f^{-1}\colon A\to \mO\) est continue, Lipschitz de constante majorée par \( (1-\lambda)^{-1}\) et que \( A\) est ouvert.
-
-    Les inéquations \eqref{subEqEBJsBfz} nous disent que
-    \begin{equation}
-    \big\| f^{-1}(y_1)-f^{-1}(y_2) \big\|\leq \frac{ \| y_1-y_2 \| }{ 1-\lambda },
-    \end{equation}
-    c'est à dire que
-    \begin{equation}
-        f^{-1}\big( B(y,r) \big)\subset B\big( f^{-1}(y),\frac{ r }{ 1-\lambda } \big),
-    \end{equation}
-    ce qui signifie que \( f^{-1}\) est Lipschitz de constante souhaitée et donc continue.
-
-    Il reste à prouver que \( f(\mO)\) est ouvert. Pour cela nous prenons \( y_0=f(x_0)\) dans \( f(\mO)\) est nous prouvons qu'il existe \( \epsilon\) tel que \( B(y_0,\epsilon)\) soit dans \( f(\mO)\). Il faut donc que pour tout \( y\in B(y_0,\epsilon)\), l'équation \( f(x)=y\) ait une solution. Nous considérons l'application
-    \begin{equation}
-        L_y\colon x\mapsto y-\varphi(x).
-    \end{equation}
-    Ce que nous cherchons est un point fixe de \( L_y\) parce que si \( L_y(x)=x\) alors \( y=x+\varphi(x)=f(x)\). Vu que
-    \begin{equation}
-        \big\| L_y(x)-L_y(x') \big\|=\big\| \varphi(x)-\varphi(x') \big\|\leq\lambda\| x-x' \|,
-    \end{equation}
-    l'application \( L_y\) est une contraction de constante \( \lambda\). Par ailleurs \( x_0\) est un point fixe de \( L_{y_0}\), donc en vertu de la caractérisation \eqref{EqDZvtUbn} des fonctions Lipschitziennes, 
-    \begin{equation}
-        L_{y_0}\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B\big( L_{y_0}(x_0),\lambda\delta \big) }=\overline{ B(x_0,\lambda\delta) }.
-    \end{equation}
-    Vu que pour tout \( y\) et \( x\) nous avons \( L_y(x)=L_{y_0}(x)+y-y_0\),
-    \begin{equation}
-    L_y\big( \overline{ B(x_0,\delta) } \big)=L_{y_0}\big( \overline{ B(x_0,\delta) } \big)+(y-y_0)\subset \overline{ B(x_0,\lambda\delta) }+(y-y_0)\subset \overline{ B(x_0),\lambda\delta+\| y-y_0 \| }.
-    \end{equation}
-    Si \( \epsilon<(1-\lambda)\delta\) alors \( \lambda\delta+\| y-y_0 \|<\delta\). Un tel choix de \( \epsilon>0\) est possible parce que \( \lambda<1\). Pour une telle valeur de \( \epsilon\) nous avons
-    \begin{equation}
-        L_y\big( \overline{ B(x_0,\delta) } \big)\subset \overline{ B(x_0,\delta) }.
-    \end{equation}
-    Par conséquent \( L_y\) est une contraction sur l'espace métrique complet \( \overline{ B(x_0,\delta) }\), ce qui signifie que \( L_y\) y possède un point fixe par le théorème de Picard \ref{ThoEPVkCL}.
-\end{proof}
-
-Le théorème d'inversion locale s'énonce de la façon suivante dans \( \eR^n\) :
-\begin{theorem}[Inversion locale dans \( \eR^n\)]    \label{THOooQGGWooPBRNEX}      % Ne pas mettre de label ici parce qu'il faut référencer l'autre, celui dans Banach.
-    Soit \( f\in C^k(\eR^n,\eR^n)\) et \( x_0\in \eR^n\). Si \( df_{x_0}\) est inversible, alors il existe un voisinage ouvert \( U\) de \( x_0\) et \( V\) de \( f(x_0)\) tels que \( f\colon U\to V\) soit un \( C^k\)-difféomorphisme. (c'est à dire que \( f^{-1}\) est également de classe \( C^k\))
-\end{theorem}
-
-Nous allons le démontrer dans le cas un peu plus général (mais pas plus cher\footnote{Sauf la justification de la régularité de l'application \( A\mapsto A^{-1}\)}) des espaces de Banach en tant que conséquence du théorème de point fixe de Picard \ref{ThoEPVkCL}.
-
-\begin{theorem}[Inversion locale dans un espace de Banach\cite{OWTzoEK}] \label{ThoXWpzqCn}
-    Soit une fonction \( f\in C^p(E,F)\) avec \( p\geq 1\) entre deux espaces de Banach. Soit \( x_0\in E\) tel que \( df_{x_0}\) soit une bijection bicontinue\footnote{En dimension finie, une application linéaire est toujours continue et d'inverse continu.}. Alors il existe un voisinage ouvert \( V\) de \( x_0\) et \( W\) de \( f(x_0)\) tels que
-    \begin{enumerate}
-        \item
-        \( f\colon V\to W\) soit une bijection,
-    \item
-        \( f^{-1}\colon W\to V\) soit de classe \( C^p\).
-    \end{enumerate}
-\end{theorem}
-\index{application!différentiable}
-\index{théorème!inversion locale}
-
-\begin{proof}
-    Nous commençons par simplifier un peu le problème. Pour cela, nous considérons la translation \( T\colon x\mapsto x+x_0 \) et l'application linéaire
-    \begin{equation}
-        \begin{aligned}
-            L\colon \eR^n&\to \eR^n \\
-            x&\mapsto (df_{x_0})^{-1}x
-        \end{aligned}
-    \end{equation}
-    qui sont tout deux des difféomorphismes (\( L\) en est un par hypothèse d'inversibilité). Quitte à travailler avec la fonction \( k=L\circ f\circ T\), nous pouvons supposer que \( x_0=0\) et que \( df_{x_0}=\mtu\). Pour comprendre cela il faut utiliser deux fois la formule de différentielle de fonction composée de la proposition \ref{EqDiffCompose} :
-    \begin{equation}
-        dk_0(u)=dL_{(f\circ T)(0)}\Big( df_{T(0)}dT_0(u) \Big).
-    \end{equation}
-    Vu que \( L\) est linéaire, sa différentielle est elle-même, c'est à dire \( dL_{(f\circ T)(0)}=(df_{x_0})^{-1}\), et par ailleurs \( dT_0=\mtu\), donc
-    \begin{equation}
-        dk_0(u)=(df_{x_0})^{-1}\Big( df_{x_0}(u) \Big)=u,
-    \end{equation}
-    ce qui signifie bien que \( dk_0=\mtu\). Pour tout cela nous avons utilisé en plein le fait que \( df_{x_0}\) était inversible.
-
-Nous posons \( g=f-\mtu\), c'est à dire \( g(x)=f(x)-x\), qui a la propriété \( dg_0=0\). Étant donné que \( g\) est de classe \( C^1\), l'application\footnote{Ici \( \GL(F)\) est l'ensemble des applications linéaires, inversibles et continues de \( F\) dans lui-même. Ce ne sont pas spécialement des matrices parce que nous n'avons pas d'hypothèses sur la dimension de \( F\), finie ou non.}
-    \begin{equation}
-        \begin{aligned}
-            dg\colon E&\to \GL(F) \\
-            x&\mapsto dg_x 
-        \end{aligned}
-    \end{equation}
-    est continue. En conséquence de quoi nous avons un voisinage \( U'\) de \( 0 \) pour lequel
-    \begin{equation}    \label{EqSGTOfvx}
-        \sup_{x\in U'}\| dg_x \|<\frac{ 1 }{2}.
-    \end{equation}
-    Maintenant le théorème des accroissements finis \ref{ThoNAKKght} (\ref{val_medio_2} pour la dimension finie) nous indique que pour tout \( x,x'\in U'\) nous avons\footnote{Ici nous supposons avoir choisi \( U'\) convexe afin que tous les \( a\in \mathopen[ x , x' \mathclose]\) soient bien dans \( U'\) et donc soumis à l'inéquation \eqref{EqSGTOfvx}, ce qui est toujours possible, il suffit de prendre une boule.}
-    \begin{equation}
-        \| g(x')-g(x) \|\leq \sup_{a\in\mathopen[ x , x' \mathclose]}\| dg_a \| \cdot \| x-x' \|\leq \frac{ 1 }{2}\| x-x' \|,
-    \end{equation}
-    ce qui prouve que \( g\) est une contraction au moins sur l'ouvert \( U'\). Nous allons aussi donner une idée de la façon dont \( f\) fonctionne : si \( x_1,x_2\in U'\) alors
-    \begin{subequations}
-        \begin{align}
-            \| x_1-x_2 \|&=\| g(x_1)-f(x_1)-g(x_2)+f(x_2) \| \\
-            &\leq \| g(x_1)-g(x_2) \|+\| f(x_1)-f(x_2) \|\\
-            &\leq \frac{ 1 }{2}\| x_1-x_2 \|+\| f(x_1)-f(x_2) \|,
-        \end{align}
-    \end{subequations}
-    ce qui montre que
-    \begin{equation}
-        \| x_1-x_2 \|\leq 2\| f(x_1)-f(x_2) \|.
-    \end{equation}
-    Maintenant que nous savons que \( g\) est contractante de constante \( \frac{ 1 }{2}\) et que \( f=g+\mtu\) nous pouvons utiliser la proposition \ref{LemGZoqknC} pour conclure que \( f\) est un homéomorphisme sur un ouvert \( U\) (partie de \( U'\)) de \( E\) et \( f^{-1}\) a une constante de Lipschitz plus petite ou égale à \( (1-\frac{ 1 }{2})^{-1}=2\).
-
-    Nous allons maintenant prouver que \( f^{-1}\) est différentiable et que sa différentielle est donnée par \( (df^{-1})_{f(x)}=(df_x)^{-1}\).
-
-    Soient \( a,b\in U\) et \( u=b-a\). Étant donné que \( f\) est différentiable en \( a\), il existe une fonction \( \alpha\in o(\| u \|)\) telle que
-    \begin{equation}
-        f(b)-f(a)-df_a(u)=\alpha(u).
-    \end{equation}
-    En notant \( y_a=f(a)\) et \( y_b=f(b)\) et en appliquant \( (df_a)^{-1}\) à cette dernière équation,
-    \begin{equation}
-        (df_a)^{-1}(y_b-y_a)-u=(df_a)^{-1} \big( \alpha(u) \big).
-    \end{equation}
-    Vu que \( df_a\) est bornée (et son inverse aussi), le membre de droite est encore une fonction \( \beta\) ayant la propriété \( \lim_{u\to 0}\beta(u)/\| u \|=0\); en réordonnant les termes,
-    \begin{equation}
-        b-a=(df_a)^{-1}(y_b-y_a)+\beta(u)
-    \end{equation}
-    et donc
-    \begin{equation}
-        f^{-1}(y_b)-f^{-1}(y_a)-(df_a)^{-1}(y_b-y_a)=\beta(u),
-    \end{equation}
-    ce qui prouve que \( f^{-1}\) est différentiable et que \( (df^{-1})_{y_a}=(df_a)^{-1}\).
-
-    La différentielle \( df^{-1}\) est donc obtenue par la chaine
-    \begin{equation}
-    \xymatrix{%
-        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}     &   U'\ar[r]^-{df}&\GL(F)\ar[r]^-{\Inv}&\GL(F)
-       }
-    \end{equation}
-    où l'application \( \Inv\colon \GL(F)\to \GL(F)\) est l'application \( X\mapsto X^{-1}\) qui est de classe \(  C^{\infty}\) par le théorème \ref{ThoCINVBTJ}. D'autre part, par hypothèse \( df\) est une application de classe \( C^{k-1}\) et donc au minimum \( C^0\) parce que \( k\geq 1\). Enfin, l'application \( f^{-1}\colon f(U)\to U\) est continue (parce que la proposition \ref{LemGZoqknC} précise que \( f\) est un homéomorphisme). Donc toute la chaine est continue et \( df^{-1}\) est continue. Cela entraine immédiatement que \( f^{-1}\) est \( C^1\) et donc que toute la chaine est \( C^1\).
-
-    Par récurrence nous obtenons la chaine
-    \begin{equation}
-    \xymatrix{%
-        df^{-1}\colon f(U) \ar[r]^-{f^{-1}}_-{C^{k-1}}     &   U'\ar[r]^-{df}_-{C^{k-1}}&\GL(F)\ar[r]^-{\Inv}_-{ C^{\infty}}&\GL(F)
-       }
-    \end{equation}
-    qui prouve que \( df^{-1}\) est \( C^{k-1} \) et donc que \( f^{-1}\) est \( C^k\). La récurrence s'arrête ici parce que \( df\) n'est pas mieux que \( C^{k-1}\).
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Théorème de la fonction implicite}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Nous énonçons et le démontrons le théorème de la fonction implicite dans le cas d'espaces de Banach.
-\begin{theorem}[Théorème de la fonction implicite dans Banach\cite{SNPdukn}] \label{ThoAcaWho}
-    Soient \( E\), \( F\) et \( G\) des espaces de Banach et des ouverts \( U\subset E\), \( V\subset F\). Nous considérons une fonction \( f\colon U\times V\to G\) de classe \( C^r\) telle que\footnote{La notation \( d_y\) est la différentielle partielle de la définition \ref{VJM_CtSKT}.}
-    \begin{equation}
-        d_yf_{(x_0,y_0)}\colon F\to G
-    \end{equation}
-    soit un isomorphisme pour un certain \( (x_0,y_0)\in U\times V\).
-
-    Alors nous avons des voisinages \( U_0\) de \( x_0\) dans \( E\) et \( W_0\) de \( f(x_0,y_0)\) dans \( G\) et une fonction de classe \( C^r\) 
-    \begin{equation}
-        g\colon U_0\times W_0\to V
-    \end{equation}
-    telle que 
-    \begin{equation}
-        f\big( x,g(x,w) \big)=w
-    \end{equation}
-    pour tout \( (x,w)\in U_0\times W_0\).
-    
-    Cette fonction \( g\) est unique au sens suivant : il existe un voisinage \( V_0 \) de \( y_0\) tel que si \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) satisfont à \( f(x,y)=w\) alors \( y=g(x,w)\). Autrement dit, la fonction \( g\colon U_0\times W_0\to V_0\) est unique.
-\end{theorem}
-\index{théorème!fonction implicite dans Banach}
-
-\begin{proof}
-    Nous commençons par considérer la fonction
-    \begin{equation}
-        \begin{aligned}
-            \Phi\colon U\times V&\to E\times G \\
-            (x,y)&\mapsto \big( x,f(x,y) \big)
-        \end{aligned}
-    \end{equation}
-    et sa différentielle 
-    \begin{subequations}
-        \begin{align}
-            d\Phi_{(x_0,y_0)}(u,v)&=\Dsdd{ \big( x_0+tu,f(x_0+tu,y_0+tv) \big) }{t}{0}\\
-            &=\left( \Dsdd{ x_0+tu }{t}{0},\Dsdd{ f(x_0+tu,y_0+tv) }{t}{0} \right)\\
-            &=\left( u,df_{(x_0,y_0)}(u,v) \right).
-        \end{align}
-    \end{subequations}
-    Nous utilisons alors la proposition \ref{PropLDN_nHWDF} pour conclure que
-    \begin{equation}
-        d\Phi_{(x_0,y_0)}(u,v)=\big( u,(d_1f)_{(x_0,y_0)}(u)+(d_2f)_{(x_0,y_0)}(v) \big),
-    \end{equation}
-    mais comme par hypothèse \( (d_2f)_{(x_0,y_0)}\colon F\to G\) est un isomorphisme, l'application \( d\Phi_{(x_0,y_0)}\colon E\times F\to E\times G\) est également un isomorphisme. Par conséquent le théorème d'inversion locale \ref{ThoXWpzqCn} nous indique qu'il existe un voisinage \( \mO\) de \( (x_0,y_0)\) et \( \mP\) de \( \Phi(x_0,y_0)\) tels que \( \Phi\colon \mO\to \mP\) soit une bijection et \( \Phi^{-1}\colon \mP\to \mO\) soit de classe \( C^r\). Vu que \( \mP\) est un voisinage de
-    \begin{equation}
-        \Phi(x_0,y_0)=\big( x_0,f(x_0,y_0) \big),
-    \end{equation}
-    nous pouvons par \ref{PropDXR_KbaLC} le choisir un peu plus petit de telle sorte à avoir \( \mP=U_0\times W_0\) où \( U_0\) est un voisinage de \( x_0\) et \( W_0\) un voisinage de \( f(x_0,y_0)\). Dans ce cas nous devons obligatoirement aussi restreindre \( \mO\) à \( U_0\times V_0\) pour un certain voisinage \( V_0\) de \( y_0\). L'application \( \Phi^{-1}\) a obligatoirement la forme
-    \begin{equation}    \label{EqMHT_QrHRn}
-        \begin{aligned}
-            \Phi^{-1}\colon U_0\times W_0&\to U_0\times V_0 \\
-            (x,w)&\mapsto \big( x,g(x,w) \big) 
-        \end{aligned}
-    \end{equation}
-    pour une certaine fonction \( g\colon U_0\times W_0\to V\). Cette fonction \( g\) est la fonction cherchée parce qu'en appliquant \( \Phi\) à \eqref{EqMHT_QrHRn}, 
-    \begin{equation}
-        (x,w)=\Phi\big( x,g(x,w) \big)=\Big( x,f\big( x,g(x,w) \big) \Big),
-    \end{equation}
-    qui nous dit que pour tout \( x\in U_0\) et tout \( w\in W_0\) nous avons
-    \begin{equation}
-        f\big( x,g(x,w) \big)=w.
-    \end{equation}
-
-    Si vous avez bien suivi le sens de l'équation \eqref{EqMHT_QrHRn} alors vous avez compris l'unicité. Sinon, considérez \( (x,y)\in U_0\times V_0\) et \( w\in W_0\) tels que \( f(x,y)=w\). Alors \( \big( x,f(x,y) \big)=(x,w)\) et 
-    \begin{equation}
-        \Phi(x,y)=(x,w).
-    \end{equation}
-    Mais vu que \( \Phi\colon U_0\times V_0\to U_0\times W_0\) est une bijection, cette relation définit de façon univoque l'élément \( (x,y)\) de \( U_0\times V_0\), qui ne sera autre que \( g(x,w)\).
-\end{proof}
-
-Le théorème de la fonction implicite s'énonce de la façon suivante pour des espaces de dimension finie.
-% Attention : avant de citer ce théorème, voir s'il est suffisant. Ici \varphi a une variable; dans l'autre énoncé il en a deux.
-\begin{theorem}[Théorème de la fonction implicite en dimension finie]   \label{ThoRYN_jvZrZ}
-    Soit une fonction \( F\colon \eR^n\times \eR^m\to \eR^m\) de classe \( C^k\) et \( (\alpha,\beta)\in \eR^n\times \eR^m\) tels que
-    \begin{enumerate}
-        \item
-            \( F(\alpha,\beta)=0\),
-        \item
-            \( \frac{ \partial (F_1,\ldots, F_m) }{ \partial (y_1,\ldots, y_m) }\neq 0\), c'est à dire que \( (d_yF)_{(\alpha,\beta)} \) est inversible.
-    \end{enumerate}
-    Alors il existe un voisinage ouvert \( V\) de \( \alpha\) dans \( \eR^n\), un voisinage ouvert \( W\) de \( \beta\) dans \( \eR^m\) et une application \( \varphi\colon V\to W\) de classe \( C^k\)  telle que pour tout \( x\in V\) on ait
-    \begin{equation}
-        F\big( x,\varphi(x) \big)=0.
-    \end{equation}
-    De plus si \( (x,y)\in V\times W\) satisfait à \( F(x,y)=0\), alors \( y=\varphi(x)\).
-\end{theorem}
-\index{théorème!fonction implicite dans \( \eR^n\)}
-
-\begin{remark}\label{RemPYA_pkTEx}
-    Notons que cet énoncé est tourné un peu différemment en ce qui concerne le nombre de variables dont dépend la fonction implicite : comparez
-    \begin{subequations}
-        \begin{align}
-            f\big( x,g(x,w) \big)=w\\
-            F\big( x,\varphi(x) \big)=0.
-        \end{align}
-    \end{subequations}
-    Le deuxième est un cas particulier du premier en posant 
-    \begin{equation}
-        F(x,y)=f(x,y)-f(x_0,y_0)
-    \end{equation}
-    et donc en considérant \( w\) comme valant la constante \( f(x_0,y_0)\); dans ce cas la fonction \( g\) ne dépend plus que de la variable \( x\).
-
-\end{remark}
-
-\begin{example}
-    La remarque \ref{RemPYA_pkTEx} signifie entre autres que le théorème \ref{ThoAcaWho} est plus fort que \ref{ThoRYN_jvZrZ} parce que le premier permet de choisir la valeur d'arrivée. Parlons de l'exemple classique du cercle et de la fonction \( f(x,y)=x^2+y^2\). Nous savons que
-    \begin{equation}
-        f(\alpha,\beta)=1.
-    \end{equation}
-    Alors le théorème \ref{ThoAcaWho} nous donne une fonction \( g\) telle que
-    \begin{equation}
-        f(x,g(x,r))=r
-    \end{equation}
-    tant que \( x\) est proche de \( \alpha\), que \( r\) est proche de \( 1\) et que \( g\) donne des valeurs proches de \( \beta\).
-
-    L'énoncé \ref{ThoRYN_jvZrZ} nous oblige à travailler avec la fonction \( F(x,y)=x^2+y^2-1\), de telle sorte que
-    \begin{equation}
-        F(\alpha,\beta)=0,
-    \end{equation}
-    et que nous ayons une fonction \( \varphi\) telle que
-    \begin{equation}
-        F(x,\varphi(x))=0.
-    \end{equation}
-    La fonction \( \varphi\) ne permet donc que de trouver des points sur le cercle de rayon \( 1\).
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Exemple}
-%---------------------------------------------------------------------------------------------------------------------------
-    
-Le théorème de la fonction implicite a pour objet de donner l'existence de la fonction $\varphi$. Maintenant nous pouvons dire beaucoup de choses sur les dérivées de $\varphi$ en considérant la fonction
-\begin{equation}
-    x\mapsto F\big( x,\varphi(x) \big).
-\end{equation}
-Par définition de $\varphi$, cette fonction est toujours nulle. En particulier, nous pouvons dériver l'équation
-\begin{equation}
-    F\big( x,\varphi(x) \big)=0,
-\end{equation}
-et nous trouvons plein de choses.
-
-
-Prenons par exemple la fonction
-\begin{equation}
-    F\big( (x,y),z \big)=ze^z-x-y,
-\end{equation}
-et demandons nous ce que nous pouvons dire sur la fonction $z(x,y)$ telle que
-\begin{equation}
-    F\big( x,y,z(x,y) \big)=0,
-\end{equation}
-c'est à dire telle que
-\begin{equation}        \label{EqDefZImplExemple}
-    z(x,y) e^{z(x,y)}-x-y=0.
-\end{equation}
-pour tout $x$ et $y\in\eR$. Nous pouvons facilement trouver $z(0,0)$ parce que
-\begin{equation}
-    z(0,0) e^{z(0,0)}=0,
-\end{equation}
-donc $z(0,0)=0$.
-
-Nous pouvons dire des choses sur les dérivées de $z(x,y)$. Voyons par exemple $(\partial_xz)(x,y)$. Pour trouver cette dérivée, nous dérivons la relation \eqref{EqDefZImplExemple} par rapport à $x$. Ce que nous trouvons est
-\begin{equation}
-    (\partial_xz)e^z+ze^z(\partial_xz)-1=0.
-\end{equation}
-Cette équation peut être résolue par rapport à $\partial_xz$~:
-\begin{equation}
-    \frac{ \partial z }{ \partial x }(x,y)=\frac{1}{ e^z(1+z) }.
-\end{equation}
-Remarquez que cette équation ne donne pas tout à fait la dérivée de $z$ en fonction de $x$ et $y$, parce que $z$ apparaît dans l'expression, alors que $z$ est justement la fonction inconnue. En général, c'est la vie, nous ne pouvons pas faire mieux.
-
-Dans certains cas, on peut aller plus loin. Par exemple, nous pouvons calculer cette dérivée au point $(x,y)=(0,0)$ parce que $z(0,0)$ est connu :
-\begin{equation}
-    \frac{ \partial z }{ \partial x }(0,0)=1.
-\end{equation}
-Cela est pratique pour calculer, par exemple, le développement en Taylor de $z$ autour de $(0,0)$.
-
-\begin{example}
-    Est-ce que l'équation \( e^{y}+xy=0\) définit au moins localement une fonction \( y(x)\) ? Nous considérons la fonction
-    \begin{equation}
-        f(x,y)=\begin{pmatrix}
-            x    \\ 
-            e^{y}+xy    
-        \end{pmatrix}
-    \end{equation}
-    La différentielle de cette application est
-    \begin{equation}
-            df_{(0,0)}(u)=\frac{ d }{ dt }\Big[ f(tu_1,tu_2) \Big]_{t=0}
-            =\frac{ d }{ dt }\begin{pmatrix}
-                tu_1    \\ 
-                e^{tu_2}+t^2u_1u_2    
-            \end{pmatrix}_{t=0}
-            =\begin{pmatrix}
-                u_1    \\ 
-                u_2    
-            \end{pmatrix}.
-    \end{equation}
-    L'application \( f\) définit donc un difféomorphisme local autour des points \( (x_0,y_0)\) et \( f(x_0,y_0)\). Soit \( (u,0)\) un point dans le voisinage de \( f(x_0,y_0)\). Alors il existe un unique \( (x,y)\) tel que
-    \begin{equation}
-        f(x,y)=\begin{pmatrix}
-               x \\ 
-            e^y+xy    
-        \end{pmatrix}=
-        \begin{pmatrix}
-            u    \\ 
-                0
-        \end{pmatrix}.
-    \end{equation}
-    Nous avons automatiquement \( x=u\) et \( e^y+xy=0\). Notons toutefois que pour que ce procédé donne effectivement une fonction implicite \( y(x)\) nous devons avoir des points de la forme \( (u,0)\) dans le voisinage de \( f(x_0,y_0)\).
-\end{example}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Décomposition polaire (régularité)}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{normaltext}      \label{NomDJMUooTRUVkS}
-    Nous allons montrer que l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
-            A&\mapsto \sqrt{A} 
-        \end{aligned}
-    \end{equation}
-    est une difféomorphisme.
-
-    Cependant \( S^{++}(n,\eR)\) n'est pas un ouvert de \( \eM(n,\eR)\) et nous ne savons pas ce qu'est la différentielle d'une application non définie sur un ouvert. Nous allons donc en réalité montrer que l'application racine carré existe sur un voisinage de chacun des points de \( S^{++}(n,\eR)\). Et comme une union quelconque d'ouverts est un ouvert, la fonction \( f\) sera bien définie sur un ouvert de \( \eM(n,\eR)\).
-\end{normaltext}
-
-\begin{lemma}       \label{LemLBFOooDdNcgy}
-    L'application 
-    \begin{equation}
-        \begin{aligned}
-            f\colon S^{++}(n,\eR)&\to S^{++}(n,\eR) \\
-            A&\mapsto A^2 
-        \end{aligned}
-    \end{equation}
-    est un \(  C^{\infty}\)-difféomorphisme.
-\end{lemma}
-
-\begin{proof}
-    Prouvons d'abord que \( f\) prend ses valeurs dans \( S^{++}(n,\eR)\). Si \( A\in S^{++}(n,\eR)\) alors par la diagonalisation \ref{ThoeTMXla} elle s'écrit \( A=QDQ^{-1}\) où \( D\) est diagonale avec des nombres strictement positifs sur la diagonale. Avec cela, \( A^2=QD^2Q^{-1}\) où \( D^2\) contient encore des nombres strictement positifs sur la diagonale.
-
-    L'application \( f\) étant essentiellement des polynôme en les entrées de \( A\), elle est de classe \( C^{\infty}\).
-
-    Passons à l'étude de la différentielle. Comme mentionné en \ref{NomDJMUooTRUVkS} nous allons en réalité voir \( f\) sur un ouvert de \( \eM(n,\eR)\) autour de \( A\in S^{++}(n,\eR)\). Par conséquent si \( A\in S^{++}(n,\eR)\),
-    \begin{subequations}
-        \begin{align}
-            df\colon S^{++}(n,\eR)&\to \aL\big( \eM(n,\eR),\eM(n,\eR) \big)\\
-            df_A\colon \eM(n,\eR)&\to \eM(n,\eR).
-        \end{align}
-    \end{subequations}
-    Le calcul de \( df_A\) est facile. Soit \( u\in \eM(n,\eR)\) et faisons le calcul en utilisant la formule du lemme \eqref{LemdfaSurLesPartielles} :
-    \begin{subequations}
-        \begin{align}
-            df_A(u)&=\Dsdd{ f(A+tu) }{t}{0}\\
-            &=\Dsdd{ A^2+tAu+tuA+t^2u^2 }{t}{0}\\
-            &=Au+uA.
-        \end{align}
-    \end{subequations}
-    Nous allons utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} à la fonction \( f\). Dans la suite, \( A\) est une matrice de \( S^{++}(n,\eR)\).
-
-    \begin{subproof}
-        \item[\( df_A\) est injective]
-            Soit \( M\in \eM(n,\eR)\) dans le noyau de \( df_A\). En posant \( M'=A^{-1}MQ\) nous avons \( M=QM'Q^{-1}\) et on applique \( df_A\) à \( QM'Q^{-1}\) :
-            \begin{equation}
-                df_A(QM'Q^{-1})=Q\big( DM+MD \big)Q^{-1}.
-            \end{equation}
-            où \( D=\begin{pmatrix}
-                \lambda_1    &       &       \\
-                    &   \ddots    &       \\
-                    &       &   \lambda_n
-                \end{pmatrix}\) avec \( \lambda_i>0\). La matrice \( D\) est inversible. Nous avons \( M'=-DM'D^{-1}\), et en coordonnées,
-                \begin{subequations}
-                    \begin{align}
-                        M'_{ij}&=-\sum_{kl}D_{ikM'_{kl}}D^{-1}_{lj}\\
-                        &=-\sum_{kl}\lambda_i\delta_{ik}M'_{kl}\frac{1}{ \lambda_j }\delta_{lj}\\
-                        &=-\frac{ \lambda_i }{ \lambda_i }M'_{ij}.
-                    \end{align}
-                \end{subequations}
-                C'est à dire que \( M'_{ij}=-\frac{ \lambda_i }{ \lambda_j }M'_{ij}\) avec \( -\frac{ \lambda_i }{ \lambda_j }<0\). Cela implique \( M'=0\) et par conséquent \( M=0\).
-            \item[\( df_A\) est surjective]
-                Soit \( N\in \eM(n,\eR)\); nous cherchons \( M\in \eM(n,\eR)\) tel que \( df_A(M)=N\). Nous posons \( N'=Q^{-1} NQ\) et \( M=QM'Q^{-1}\), ce qui nous donne à résoudre \( df_D(M')=N'\). Passons en coordonnées :
-                \begin{subequations}
-                    \begin{align}
-                        (DM'+M'D)_{ij}&=\sum_k(\delta_{ik}\lambda_iM'_{kj}+M'_{ik}\delta_{kj}\lambda_j)\\&
-                        &=M'_{ij}(\lambda_i+\lambda_j)
-                    \end{align}
-                \end{subequations}
-                où \( \lambda_i+\lambda_j\neq 0\). Il suffit donc de prendre la matrice \( M'\) donnée par
-                \begin{equation}
-                    M'_{ij}=\frac{1}{ \lambda_i+\lambda_j }N'_{ij}
-                \end{equation}
-                pour que \( df_A(M')=N'\).
-    \end{subproof}
-    
-    Le théorème d'inversion locale donne un voisinage \( V\) de $A$ dans \( \eM(n,\eR)\) et un voisinage \( W\) de \( A^2\) dans \( \eM(n,\eR)\) tels que \( f\colon V\to W\) soit une bijection  et \( f^{-1}\colon W\to V\) soit de même régularité, en l'occurrence \( C^{\infty}\).
-\end{proof}
-
-\begin{remark}
-    Oui, il y a des matrices non symétriques qui ont une unique racine carré.
-\end{remark}
-
-La proposition suivante, qui dépend du le théorème d'inversion locale par le lemme \ref{LemLBFOooDdNcgy}, donne plus de régularité à la décomposition polaire donnée dans le théorème \ref{ThoLHebUAU}.
-\begin{proposition}[Décomposition polaire : cas réel (suite)]       \label{PropWCXAooDuFMjn}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \gO(n,\eR)\times S^{++}(n,\eR)&\to \GL(n,\eR) \\
-            (Q,S)&\mapsto SQ 
-        \end{aligned}
-    \end{equation}
-    est un difféomorphisme de classe \( C^{\infty}\).
-\end{proposition}
-
-\begin{proof}
-    Si \( M\) est donnée dans \( \GL(n,\eR)\) alors la décomposition polaire\footnote{Proposition \ref{ThoLHebUAU}.} \( M=QS\) est donnée par \( S=\sqrt{MM^t}\) et \( Q=MS^{-1}\). Autrement dit, si nous considérons la fonction de décomposition polaire
-    \begin{equation}
-        f\colon \gO(n,\eR)\times S^{++}(n,\eR)\to \GL(n,\eR)
-    \end{equation}
-    alors
-    \begin{equation}
-        f^{-1}(M)=\big(  M(\sqrt{MM^t})^{-1},\sqrt{MM^t}  \big).
-    \end{equation}
-    Nous avons vu dans le lemme \ref{LemLBFOooDdNcgy} que la racine carré était un \( C^{\infty}\)-difféomorphisme. Le reste n'étant que des produits de matrice, la régularité est de mise.
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Théorème de Von Neumann}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{lemma}[\cite{KXjFWKA}]
-    Soit \( G\), un sous groupe fermé de \( \GL(n,\eR)\) et 
-    \begin{equation}
-        \mL_G=\{ m\in \eM(n,\eR)\tq  e^{tm}\in G\,\forall t\in\eR \}.
-    \end{equation}
-    Alors \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\).
-\end{lemma}
-
-\begin{proof}
-    Si \( m\in\mL_G\), alors \( \lambda m\in\mL_G\) par construction. Le point délicat à prouver est le fait que si \( a,b\in \mL_G\), alors \( a+b\in\mL_G\). Soit \( a\in \eM(n,\eR)\); nous savons qu'il existe une fonction \( \alpha_a\colon \eR\to \eM\) telle que
-    \begin{equation}
-        e^{ta}=\mtu+ta+\alpha_a(t)
-    \end{equation}
-    et 
-    \begin{equation}
-        \lim_{t\to 0} \frac{ \alpha(t) }{ t }=0.
-    \end{equation}
-    Si \( a\) et \( b\) sont dans \( \mL_G\), alors \(  e^{ta} e^{tb}\in G\), mais il n'est pas vrai en général que cela soit égal à \(  e^{t(a+b)}\). Pour tout \( k\in \eN\) nous avons
-    \begin{equation}
-        e^{a/k} e^{b/k}=\left( \mtu+\frac{ a }{ k }+\alpha_a(\frac{1}{ k }) \right)\left( \mtu+\frac{ b }{ k }+\alpha_b(\frac{1}{ k }) \right)=\mtu+\frac{ a+b }{2}+\beta\left( \frac{1}{ k } \right)
-    \end{equation}
-   où \( \beta\colon \eR\to \eM\) est encore une fonction vérifiant \( \beta(t)/t\to 0\). Si \( k\) est assez grand, nous avons
-   \begin{equation}
-       \left\| \frac{ a+b }{ k }+\beta(\frac{1}{ k })  \right\|<1,
-   \end{equation}
-   et nous pouvons profiter du lemme \ref{LemQZIQxaB} pour écrire alors
-   \begin{equation}
-       \left(  e^{a/k} e^{b/k} \right)^k= e^{k\ln\big(\mtu+\frac{ a+b }{ k }+\beta(\frac{1}{ k })\big)}.
-   \end{equation}
-   Ce qui se trouve dans l'exponentielle est
-   \begin{equation}
-       k\left[ \frac{ a+b }{ k }+\alpha( \frac{1}{ k })+\sigma\left( \frac{ a+b }{ k }+\alpha(\frac{1}{ k }) \right) \right].
-   \end{equation}
-   Les diverses propriétés vues montrent que le tout tend vers \( a+b\) lorsque \( k\to \infty\). Par conséquent
-   \begin{equation}
-       \lim_{k\to \infty} \left(  e^{a/k} e^{b/k} \right)^k= e^{a+b}.
-   \end{equation}
-   Ce que nous avons prouvé est que pour tout \( t\), \(  e^{t(a+b)}\) est une limite d'éléments dans \( G\) et est donc dans \( G\) parce que ce dernier est fermé.
-\end{proof}
-
-Vu que \( \mL_G\) est un sous-espace vectoriel de \( \eM(n,\eR)\), nous pouvons considérer un supplémentaire \( M\).
-
-\begin{lemma}   \label{LemHOsbREC}
-    Il n'existe pas se suites \( (m_k)\) dans \( M\setminus\{ 0 \}\) convergeant vers zéro et telle que \(  e^{m_k}\in G\) pour tout \( k\).
-\end{lemma}
-
-\begin{proof}
-    Supposons que nous ayons \( m_k\to 0\) dans \( M\setminus\{ 0 \}\) avec \(  e^{m_k}\in G\). Nous considérons les éléments \( \epsilon_k=\frac{ m_k }{ \| m_k \| }\) qui sont sur la sphère unité de \(\GL(n,\eR)\). Quitte à prendre une sous-suite, nous pouvons supposer que cette suite converge, et vu que \( M\) est fermé, ce sera vers \( \epsilon\in M\) avec \( \| \epsilon \|=1\). Pour tout \( t\in \eR\) nous avons
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}.
-    \end{equation}
-    En vertu de la décomposition d'un réel en partie entière et décimale, pour tout \( k\) nous avons \( \lambda_k\in \eZ\) et \( | \mu_k |\leq \frac{ 1 }{2}\) tel que \( t/\| m_k \|=\lambda_k+\mu_k\). Avec ça,
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}\exp\Big( \frac{ t }{ m_k }m_k \Big)=\lim_{k\to \infty}  e^{\lambda_km_k} e^{\mu_km_k}.
-    \end{equation}
-    Pour tout \( k\) nous avons \(  e^{\lambda_km_k}\in G\). De plus \( | \mu_k |\) étant borné et \( m_k\) tendant vers zéro nous avons \(  e^{\mu_km_k}\to 1\). Au final
-    \begin{equation}
-        e^{t\epsilon}=\lim_{k\to \infty}  e^{t\epsilon_k}\in G
-    \end{equation}
-    Cela signifie que \( \epsilon\in\mL_G\), ce qui est impossible parce que nous avions déjà dit que \( \epsilon\in M\setminus\{ 0 \}\).
-\end{proof}
-
-\begin{lemma}   \label{LemGGTtxdF}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \mL_G\times M&\to \GL(n,\eR) \\
-            l,m&\mapsto  e^{l} e^{m} 
-        \end{aligned}
-    \end{equation}
-    est un difféomorphisme local entre un voisinage de \( (0,0)\) dans \( \eM(n,\eR)\) et un voisinage de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
-\end{lemma}
-Notons que nous ne disons rien de \(  e^{\eM(n,\eR)}\). Nous n'allons pas nous embarquer à discuter si ce serait tout \( \GL(n,\eR)\)\footnote{Vu les dimensions y'a tout de même peu de chance.} ou bien si ça contiendrait ne fut-ce que \( G\).
-
-\begin{proof}
-    Le fait que \( f\) prenne ses valeurs dans \( \GL(n,\eR)\) est simplement dû au fait que les exponentielles sont toujours inversibles. Nous considérons ensuite la différentielle : si \( u\in \mL_G\) et \( v\in M\) nous avons
-    \begin{equation}
-        df_{(0,0)}(u,v)=\Dsdd{ f\big( t(u,v) \big) }{t}{0}=\Dsdd{  e^{tu} e^{tv} }{t}{0}=u+v.
-    \end{equation}
-    L'application \( df_0\) est donc une bijection entre \( \mL_G\times M\) et \( \eM(n,\eR)\). Le théorème d'inversion locale \ref{ThoXWpzqCn} nous assure alors que \( f\) est une bijection entre un voisinage de \( (0,0)\) dans \( \mL_G\times M\) et son image. Mais vu que \( df_0\) est une bijection avec \( \eM(n,\eR)\), l'image en question contient un ouvert autour de \( \mtu\) dans \( \exp\big( \eM(n,\eR) \big)\).
-\end{proof}
-
-\begin{theorem}[Von Neumann\cite{KXjFWKA,ISpsBzT,Lie_groups}]       \label{ThoOBriEoe}
-    Tout sous-groupe fermé de \( \GL(n,\eR)\) est une sous-variété de \( \GL(n,\eR)\).
-\end{theorem}
-\index{théorème!Von Neumann}
-\index{exponentielle!de matrice!utilisation}
-
-\begin{proof}
-    Soit \( G\) un tel groupe; nous devons prouver que c'est localement difféomorphe à un ouvert de \( \eR^n\). Et si on est pervers, on ne va pas faire localement difféomorphe à un ouvert de \( \eR^n\), mais à un ouvert d'un espace vectoriel de dimension finie. Nous allons être pervers.
-
-    Étant donné que pour tout \( g\in G\), l'application 
-    \begin{equation}
-        \begin{aligned}
-            L_g\colon G&\to G \\
-            h&\mapsto gh 
-        \end{aligned}
-    \end{equation}
-    est de classe \(  C^{\infty}\) et d'inverse \(  C^{\infty}\), il suffit de prouver le résultat pour un voisinage de \( \mtu\).
-
-    Supposons d'abord que \( \mL_G=\{ 0 \}\). Alors \( 0\) est un point isolé de \( \ln(G)\); en effet si ce n'était pas le cas nous aurions un élément \( m_k\) de \( \ln(G)\) dans chaque boule \( B(0,r_k)\). Nous aurions alors \( m_k=\ln(a_k)\) avec \( a_k\in G\) et donc
-    \begin{equation}
-        e^{m_k}=a_k\in G.
-    \end{equation}
-    De plus \( m_k\) appartient forcément à \( M\) parce que \( \mL_G\) est réduit à zéro. Cela nous donnerait une suite \( m_k\to 0\) dans \( M\) dont l'exponentielle reste dans \( G\). Or cela est interdit par le lemme \ref{LemHOsbREC}. Donc \( 0\) est un point isolé de \( \ln(G)\). L'application \(\ln\) étant continue\footnote{Par le lemme \ref{LemQZIQxaB}.}, nous en déduisons que \( \mtu\) est isolé dans \( G\). Par le difféomorphisme \( L_g\), tous les points de \( G\) sont isolés; ce groupe est donc discret et par voie de conséquence une variété.
-
-    Nous supposons maintenant que \( \mL_G\neq\{ 0 \}\). Nous savons par la proposition \ref{PropXFfOiOb} que 
-    \begin{equation}
-        \exp\colon \eM(n,\eR)\to \eM(n,\eR)
-    \end{equation}
-    est une application \(  C^{\infty}\) vérifiant \( d\exp_0=\id\). Nous pouvons donc utiliser le théorème d'inversion locale \ref{ThoXWpzqCn} qui nous offre donc l'existence d'un voisinage \( U\) de \( 0\) dans \( \eM(n,\eR)\) tel que \( W=\exp(U)\) soit un ouvert de \( \GL(n,\eR)\) et que \( \exp\colon U\to W\) soit un difféomorphisme de classe \(  C^{\infty}\).
-
-    Montrons que quitte à restreindre \( U\) (et donc \( W\) qui reste par définition l'image de \( U\) par \( \exp\)), nous pouvons avoir \( \exp\big( U\cap\mL_G \big)=W\cap G\). D'abord \( \exp(\mL_G)\subset G\) par construction. Nous avons donc \( \exp\big( U\cap\mL_G \big)\subset W\cap G\). Pour trouver une restriction de \( U\) pour laquelle nous avons l'égalité, nous supposons que pour tout ouvert \( \mO\) dans \( U\), 
-    \begin{equation}
-        \exp\colon \mO\cap\mL_G\to \exp(\mO)\cap G
-    \end{equation}
-    ne soit pas surjective. Cela donnerait un élément de \( \mO\cap\complement\mL_G\) dont l'image par \( \exp\) n'est pas dans \( G\). Nous construisons ainsi une suite en considérant une boule \( B(0,\frac{1}{ k })\) inclue à \( U\) et \( x_k\in B(0,\frac{1}{ k })\cap\complement\mL_G\) vérifiant \(  e^{x_k}\in G\). Vu le choix des boules nous avons évidemment \( x_k\to 0\).
-
-    L'élément \(  e^{x_k}\) est dans \(  e^{\eM(n,\eR)}\) et le difféomorphisme du lemme \ref{LemGGTtxdF}\quext{Il me semble que l'utilisation de ce lemme manque à l'avant-dernière ligne de la preuve chez \cite{KXjFWKA}.} nous donne \( (l_k,m_k)\in \mL_G\times M\) tel que \(  e^{l_k} e^{m_k}= e^{x_k}\). À ce point nous considérons \( k\) suffisamment grand pour que \(  e^{x_k}\) soit dans la partie de l'image de \( f\) sur lequel nous avons le difféomorphisme. Plus prosaïquement, nous posons
-    \begin{equation}
-        (l_k,m_k)=f^{-1}( e^{x_k})
-    \end{equation}
-    et nous profitons de la continuité pour permuter la limite avec \( f^{-1}\) :
-    \begin{equation}
-        \lim_{k\to \infty} (l_k,m_k)=f^{-1}\big( \lim_{k\to \infty}  e^{x_k} \big)=f^{-1}(\mtu)=(0,0).
-    \end{equation}
-    En particulier \( m_k\to 0\) alors que \(  e^{m_k}= e^{x_k} e^{-l_k}\in G\). La suite \( m_k\) viole le lemme \ref{LemHOsbREC}. Nous pouvons donc restreindre \( U\) de telle façon à avoir
-    \begin{equation}
-        \exp\big( U\cap\mL_G \big)=W\cap G.
-    \end{equation}
-    Nous avons donc un ouvert de \( \mL_G\) (l'ouvert \( U\cap\mL_G\)) qui est difféomorphe avec l'ouvert \( W\cap G\) de \( G\). Donc \( G\) est une variété et accepte \( \mL_G\) comme carte locale.
-
-\end{proof}
-
-\begin{remark}
-    En termes savants, nous avons surtout montré que si \( G\) est un groupe de Lie d'algèbre de Lie \( \lG\), alors l'exponentielle donne un difféomorphisme local entre \( \lG\) et \( G\).
-\end{remark}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-\section{Recherche d'extrema}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema à une variable}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-Soit $f\colon A\subset \eR\to \eR$ et $a\in A$. Le point $a$ est un \defe{maximum local}{maximum!local} de $f$ s'il existe un voisinage $\mU$ de $a$ tel que $f(a)\geq f(x)$ pour tout $x\in\mU\cap A$. Le point $a$ est un \defe{maximum global}{maximum!global} si $f(a)\geq g(x)$ pour tout $x\in A$.
-\end{definition}
-
-La proposition basique à utiliser lors de la recherche d'extrema est la suivante :
-\begin{proposition}     \label{PROPooNVKXooXtKkuz}
-Soit $f\colon A\subset\eR\to \eR$ et $a\in\Int(A)$. Supposons que $f$ est dérivable en $a$. Si $a$ est un \href{http://fr.wikipedia.org/wiki/Extremum}{extremum} local, alors $f'(a)=0$.
-\end{proposition}
-
-La réciproque n'est pas vraie, comme le montre l'exemple de la fonction $x\mapsto x^3$ en $x=0$ : sa dérivée est nulle et pourtant $x=0$ n'est ni un maximum ni un minimum local. 
-
-Cette proposition ne sert donc qu'à sélectionner des \emph{candidats} extremum. Afin de savoir si ces candidats sont des extrema, il y a la proposition suivante.
-\begin{proposition}
-Soit $f\colon I\subset \eR\to \eR$, une fonction de classe $C^k$ au voisinage d'un point $a\in\Int I$. Supposons que
-\begin{equation}
-    f'(a)=f''(a)=\ldots=f^{(k-1)}(a)=0,
-\end{equation}
-et que
-\begin{equation}
-    f^{(k)}(a)\neq 0.
-\end{equation}
-Dans ce cas,
-\begin{enumerate}
-
-\item
-Si $k$ est pair, alors $a$ est un point d'extremum local de $f$, c'est un minimum si $f^{(k)}(a)>0$, et un maximum si $f^{(k)}(a)<0$,
-\item
-Si $k$ est impair, alors $a$ n'est pas un extremum local de $f$.
-
-\end{enumerate}
-\end{proposition}
-
-Note : jusqu'à présent nous n'avons rien dit des extrema \emph{globaux} de $f$. Il n'y a pas grand chose à en dire. Si un point d'extremum global est situé dans l'intérieur du domaine de $f$, alors il sera extremum local (a fortiori). Ou alors, le maximum global peut être sur le bord du domaine. C'est ce qui arrive à des fonctions strictement croissantes sur un domaine compact.
-
-Une seule certitude : si une fonction est continue sur un compact, elle possède une minimum et un maximum global par le théorème \ref{ThoMKKooAbHaro}.
-
-Soit une fonction $f\colon I\to \eR$, et soit $a\in I$. Si $f'(a)>0$, alors la tangente au graphe de $f$ au point $\big( a,f(a) \big)$ sera une droite croissante (coefficient directeur positif). Cela ne veut pas spécialement dire que la fonction elle-même sera croissante, mais en tout cas cela est un bon indice.
-
-\begin{example}
-	Si $f(x)=x^2$, il est connu que $f'(x)=2x$. Nous avons donc que $f'$ est positive si $x\geq 0$ et $f'>$ est négative si $x<0$. Cela correspond bien au fait que $x^2$ est décroissante sur $\mathopen] -\infty , 0 \mathclose[$ et croissante sur $\mathopen] 0 , \infty \mathclose[$.
-\end{example}
- 
-Sur la figure \ref{LabelFigWIRAooTCcpOV}, nous avons dessiné la fonction $f(x)=x\cos(x)$ et sa dérivée. Nous voyons que partout où la dérivée est négative, la fonction est décroissante tandis que, inversement, partout où la dérivée est positive, la fonction est croissante.
-\newcommand{\CaptionFigWIRAooTCcpOV}{La fonction $f(x)=x\cos(x)$ en bleu et sa dérivée en rouge.}
-\input{auto/pictures_tex/Fig_WIRAooTCcpOV.pstricks}
-
-Les extrema de la fonction $f$ sont donc placés là où $f'$ change de signe. En effet si $f'(x)<0$ pour $x<a$ et $f'(x)>0$ pour $x>a$, la fonction est décroissante jusqu'à $a$ et est ensuite croissante. Cela signifie que la fonction connait un creux en $a$. Le point $a$ est donc un minimum de la fonction.
-
-Attention cependant. Le fait que $f'(a)=0$ ne signifie pas automatiquement que $f$ a un maximum ou un minimum en $a$. Nous avons par exemple tracé sur la figure \ref{LabelFigVBOIooRHhKOH} les fonctions $x^3$ et sa dérivée. Il est à noter que, conformément à ce que l'on pense, certes la dérivée s'annule en $x=0$, mais elle ne change pas de signe.
-
-\newcommand{\CaptionFigVBOIooRHhKOH}{La dérivée de $x^3$ s'annule en $x=0$, mais ce n'est ni un minimum ni un maximum.}
-\input{auto/pictures_tex/Fig_VBOIooRHhKOH.pstricks}
- 
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Extrema libre}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}      \label{DEFooYJLZooLkEAYf}
-Un point $a$ à l'intérieur du domaine d'une fonction $f\colon A\subset\eR^n\to \eR$ est un \defe{point critique}{critique!point} de $f$ lorsque $df(a)=0$. 
-\end{definition}
-
-Ces points sont analogues aux points où la dérivée d'une fonction sur $\eR$ s'annule. Les points critiques de $f$ sont dons les candidats à être des points d'extremum.
-
-Dans le cas d'une fonction de deux variables,l la proposition \ref{PROPooFWZYooUQwzjW} nous permet de voir \( (d^2f)_a\) comme étant la matrice 
-\begin{equation}
-    d^2f(a)=\begin{pmatrix}
-    \frac{ d^2f  }{ dx^2 }(a)   &   \frac{ d^2f  }{ dx\,dy }(a) \\ 
-    \frac{ d^2f  }{ dy\,dx }(a)     &   \frac{ d^2f  }{ dy^2 }(a)
-\end{pmatrix}.
-\end{equation}
-Dans le cas d'une fonction $C^2$, cette matrice est symétrique.
-
-\begin{proposition}[\cite{ooZSEQooEGRdCK}] \label{PropUQRooPgJsuz}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et \( a\in \Omega\). Soit une fonction \( f\colon \Omega\to \eR\) différentiable en \( a\). Si \( a\) est un extremum local de \( f\), alors \( a\) est un point critique de \( f\).
-\end{proposition}
-
-\begin{proof}
-    Nous supposons que \( a\) est un maximum local (ce sera la même chose si \( a\) est un minimum). Soit \( r>0\) tel que \( f(x)\leq f(a)\) pour tout \( x\in B(a,r)\) (et tel que cette boule reste dans \( \Omega\)). Soit \( u\in \eR^n\) assez petit pour que \( a\pm u\in B(a,r)\) de sorte que la définition suivante ait un sens :
-    \begin{equation}
-        \begin{aligned}
-            g\colon \mathopen[ -1 , 1 \mathclose]&\to \eR \\
-            t&\mapsto f(a+tu) 
-        \end{aligned}
-    \end{equation}
-    Cette fonction est différentiable en \( t=0\) (composée de fonctions différentiables, proposition \ref{PROPooBWZFooTxKavX}) et a un maximum local en \( t=0\). Donc \( g'(0)=0\) par la proposition \ref{PROPooNVKXooXtKkuz}. Donc
-    \begin{equation}
-        0=\Dsdd{ f(a+tu) }{t}{0}=df_a(u).
-    \end{equation}
-\end{proof}
-
-\begin{proposition}[\cite{ooOQEZooBaRMjY,MonCerveau}]     \label{PropoExtreRn}
-    Soit un ouvert \( \Omega\) de \( \eR^n\) et une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) ainsi que \( a\in\Omega\).
-    \begin{enumerate}
-        \item   \label{ITEMooCVFVooWltGqI}
-            Si $a$ est un point critique\footnote{Définition \ref{DEFooYJLZooLkEAYf}.} de $f$, et si $d^2f_a$ est strictement définie positive\footnote{La fonction \( f\) est de classe \( C^2\), donc les dérivées croisées sont égales et \( d^2f\) est symétrique. La définition \ref{DefAWAooCMPuVM} s'applique donc.}, alors $a$ est un minimum local strict de $f$,
-        \item\label{ItemPropoExtreRn}
-            Si $a$ est un minimum local, alors $(d^2f)_a$ est semi définie positive.
-    \end{enumerate}
-\end{proposition}
-\index{extrema}
-
-\begin{proof}
-    Pour \ref{ItemPropoExtreRn}. Si \( a\) est un minimum local, nous savons déjà que \( df_a=0\) par la proposition \ref{PropUQRooPgJsuz}. Nous écrivons le développement de Taylor de \( f\) à l'ordre \( 2\) de la proposition \ref{PROPooTOXIooMMlghF} :
-    \begin{equation}
-        f(a+h)=f(a)+df_a(h)+\frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|).
-    \end{equation}
-    En prenant \( h\) assez petit pour que \( a+h\) ne sorte pas de la boule dans laquelle \( a\) est un minimum, nous avons \( f(a+h)-f(a)>0\). Donc
-    \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(h,h)+\| h \|^2\alpha(\| h \|)>0
-    \end{equation}
-    Nous divisons cela par \( \| h \|^2\) et notons \( e_h=h/\| h \|\) :
-    \begin{equation}
-        \frac{ 1 }{2}(d^2f)_a(e_h,e_h)+\alpha(\| h \|)>0.
-    \end{equation}
-    À la limite \( h\to 0\), le premier terme est constant tandis que le deuxième tend vers zéro. À la limite,
-    \begin{equation}
-        (d^2f)_a(e_h,e_h)\geq 0.
-    \end{equation}
-    La caractérisation du lemme \ref{LemWZFSooYvksjw}\ref{ITEMooMOZYooWcrewZ} nous dit alors que \( (d^2f)_a\) est semi-définie positive.
-\end{proof}
-
-La partie \ref{ItemPropoExtreRn} est tout à fait comparable au fait bien connu que, pour une fonction $f\colon \eR\to \eR$, si le point $a$ est minimum local, alors $f'(a)=0$ et $f''(a)>0$.
-
-Notons que le point \ref{ItemPropoExtreRn} ne parle pas de minimum strict, et donc pas de matrice \emph{strictement} définie positive.
-
-La méthode pour chercher les extrema de $f$ est donc de suivre le points suivants :
-\begin{enumerate}
-    \item
-        Trouver les candidats extrema en résolvant $\nabla f=(0,0)$,
-    \item
-        écrire $d^2f(a)$ pour chacun des candidats
-    \item
-        calculer les valeurs propres de $d^2f(a)$, déterminer si la matrice est définie positive ou négative,
-    \item
-        conclure.
-\end{enumerate}
-
-Une conséquence de la proposition \ref{PropcnJyXZ}\ref{ItemluuFPN}\footnote{La matrice $d^2f(a)$ est toujours symétrique quand $f$ est de classe $C^2$.} est que si \( \det M<0\), alors le point \( a\) n'est pas  un extrema dans le cas où $M=d^2f(a)$ par le point \ref{ItemPropoExtreRn} de la proposition \ref{PropoExtreRn}.
-
-\begin{example}
-    Soit la fonction \( f(x,y)=x^4+y^4-4xy\). C'est une fonction différentiable sans problèmes. D'abord sa différentielle est
-    \begin{equation}
-        df=\big(4x^3-4y;4y^3-4x),
-    \end{equation}
-    et la matrice des dérivées secondes est
-    \begin{equation}
-        M=d^2f(x,y)=\begin{pmatrix}
-            12x^2    &   -4    \\ 
-            -4    &   12y^2    
-        \end{pmatrix}.
-    \end{equation}
-    Nous avons \( fd=0\) pour les trois points \( (0,0)\), \( (1,1)\) et \( -1,-1\).
-
-    Pour le point \( (0,0)\) nous avons
-    \begin{equation}
-        M=\begin{pmatrix}
-            0    &   -4    \\ 
-            -4    &   0    
-        \end{pmatrix},
-    \end{equation}
-    dont les valeurs propres sont \( 4\) et \( -4\). Elle n'est donc semi-définie ou définie rien du tout. Donc \( (0,0)\) n'est pas un extremum local.
-
-    Au contraire pour les points \( (1,1)\) et \( (-1,-1)\) nous avons
-    \begin{equation}
-        M=\begin{pmatrix}
-            12    &   -4    \\ 
-            -4    &   12    
-        \end{pmatrix},
-    \end{equation}
-    dont les valeurs propres sont \( 16\) et \( 8\). La matrice \( d^2f\) y est donc définie positive. Ces deux points sont donc extrema locaux.
-\end{example}
-
-%---------------------------------------------------------------------------------------------------------------------------
-\subsection{Un peu de recettes de cuisine}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{enumerate}
-\item Rechercher les points critiques, càd les $(x,y)$ tels que
-\[\begin{cases} \frac{\partial f}{\partial x}(x,y) = 0 \\ \frac{\partial f}{\partial y}(x,y) = 0 \end{cases} \]
-En effet, si $(x_0,y_0)$ est un extrémum local de $f$, alors $\frac{\partial f}{\partial x}(x_0,y_0) = 0 = \frac{\partial f}{\partial y}(x_0,y_0)$.
-\item Déterminer la nature des points critiques: «test» des dérivées secondes:
-\[\text{On pose }H(x_0,y_0) = \frac{\partial^2 f}{\partial x^2}(x_0,y_0)\frac{\partial f^2}{\partial y^2}(x_0,y_0) - \left(\frac{\partial^2 f}{\partial x\partial y}(x_0,y_0)\right)^2\]
-\begin{enumerate}
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) > 0 \Longrightarrow (x_0,y_0)$ est un minimum local de $f$.
-\item Si $H(x_0,y_0) > 0$ et $\frac{\partial^2 f}{\partial x^2}(x_0,y_0) < 0 \Longrightarrow (x_0,y_0)$ est un maximum local de $f$.
-\item Si $H(x_0,y_0) < 0 \Longrightarrow f$ a un point de selle en $(x_0,y_0)$.
-\item Si $H(x_0,y_0) = 0 \Longrightarrow$ on ne peut rien conclure.
-\end{enumerate}
-\end{enumerate}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Extrema liés}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Soit $f$, une fonction sur $\eR^n$, et $M\subset \eR^n$ une variété de dimension $m$. Nous voulons savoir quelle sont les plus grandes et plus petites valeurs atteintes par $f$ sur $M$.
-
-Pour ce faire, nous avons un théorème qui permet de trouver des extrema \emph{locaux} de $f$ sur la variété. Pour rappel, $a\in M$ est une \defe{extrema local de $f$ relativement}{extrema!local!relatif} à l'ensemble $M$ s'il existe une boule $B(a,\epsilon)$ telle que $f(a)\leq f(x)$ pour tout $x\in B(a,\epsilon)\cap M$.
-
-\begin{theorem}[Extrema lié \cite{ytMOpe}] \label{ThoRGJosS}
-    Soit \( A\), un ouvert de \( \eR^n\) et
-    \begin{enumerate}
-        \item
-            une fonction (celle à minimiser) $f\in C^1(A,\eR)$,
-        \item 
-            des fonctions (les contraintes) $G_1,\ldots,G_r\in C^1(A,\eR)$,
-        \item
-            $M=\{ x\in A\tq G_i(x)=0\,\forall i\}$,
-        \item
-            un extrema local $a\in M$ de $f$ relativement à $M$.
-    \end{enumerate}
-    Supposons que les gradients $\nabla G_1(a)$, \ldots,$\nabla G_r(a)$ soient linéairement indépendants. Alors $a=(x_1,\ldots,x_n)$ est une solution de \( \nabla L(a)=0\) où
-    \begin{equation}
-        L(x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_r)=f(x_1,\ldots,x_n)+\sum_{i=1}^r\lambda_iG_i(x_1,\ldots,x_n).
-    \end{equation}
-    Autrement dit, si \( a\) est un extrema lié, alors \( \nabla f(a)\) est une combinaisons des \( \nabla G_i(a)\), ou encore il existe des \( \lambda_i\) tels que
-    \begin{equation}    \label{EqRDsSXyZ}
-        df(a)=\sum_i\lambda_idG_i(a).
-    \end{equation}
-\end{theorem}
-\index{théorème!inversion locale!utilisation}
-\index{extrema!lié}
-\index{théorème!extrema!lié}
-\index{application!différentiable!extrema lié}
-\index{variété}
-\index{rang!différentielle}
-\index{forme!linéaire!différentielle}
-La fonction $L$ est le \defe{lagrangien}{lagrangien} du problème et les variables \( \lambda_i\) sont les \defe{multiplicateurs de Lagrange}{multiplicateur!de Lagrange}\index{Lagrange!multiplicateur}.
-
-\begin{proof}
-    Si \( r=n\) alors les vecteurs linéairement indépendantes \( \nabla G_i(a) \) forment une base de \( \eR^n\) et donc évidemment les \( \lambda_i\) existent. Nous supposons donc maintenant que \( r<n\). Nous notons \( (z_i)_{i=1\ldots n}\) les coordonnées sur \( \eR^n\).
-    
-    La matrice
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{ \partial G_1 }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_1 }{ \partial z_n }(a)    \\
-            \vdots    &   \ddots    &   \vdots    \\
-            \frac{ \partial G_r }{ \partial z_1 }(a)    &   \cdots    &   \frac{ \partial G_r }{ \partial z_n }(a)
-        \end{pmatrix}
-    \end{equation}
-    est de rang \( r\) parce que les lignes sont par hypothèses linéairement indépendantes. Nous nommons \( (y_i)_{i=1,\ldots, r}\) un choix de \( r\) parmi les \( (z_i)\) tels que
-    \begin{equation}
-        \det\begin{pmatrix}
-            \frac{ \partial G_1 }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_1 }{ \partial y_r }    \\
-            \vdots    &   \ddots    &   \vdots    \\
-            \frac{ \partial G_r }{ \partial y_1 }    &   \ldots    &   \frac{ \partial G_r }{ \partial y_r }
-        \end{pmatrix}\neq 0.
-    \end{equation}
-    Nous identifions \( \eR^n\) à \( \eR^s\times \eR^r\) dans lequel \( \eR^r\) est la partie générée par les \( (y_i)_{i=1,\ldots, r}\). Nous nommons \( (x_j)_{j=1,\ldots, s}\) les coordonnées sur \( \eR^s\). Autrement dit, les coordonnées sur \( \eR^n\) sont \( x_1,\ldots, x_s,y_1,\ldots, y_r\). Dans ces coordonnées, nous nommons \( a=(\alpha,\beta)\) avec \( \alpha\in \eR^s\) et \( \beta\in \eR^r\).
-
-    Si nous notons \( G=(G_1,\ldots, G_r)\), le théorème de la fonction implicite (théorème \ref{ThoAcaWho})  nous dit qu'il existe un voisinage \( U'\) de \( \alpha\in \eR^n\), un voisinage \( V'\) de \( \beta\in \eR^r\) et une fonction \( \varphi\colon U'\to V'\) de classe \( C^1\) telle que si \( (x,y)\in U'\times V'\), alors
-    \begin{equation}
-        g(x,y)=0
-    \end{equation}
-    si et seulement si \( y=\varphi(x)\). Nous posons maintenant
-    \begin{subequations}
-        \begin{align}
-            \psi(x)&=(x,\varphi(x))\\
-            h(x)&=f\big( \psi(x) \big).
-        \end{align}
-    \end{subequations}
-    Nous avons \( \psi(\alpha)=a\) et \( \psi(x)\in M\) pour tout \( x\in U'\). La fonction \( h\) a donc un extrema local en \( \alpha\) et donc les dérivées partielles de \( h\) y sont nulles. Cela signifie que
-    \begin{equation}
-        0=\frac{ \partial h }{ \partial x_i }(\alpha)=\sum_{j=1}^n\frac{ \partial f }{ \partial x_j }\frac{ \partial x_j }{ \partial x_i }+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }\frac{ \partial \varphi_k }{ \partial x_i },
-    \end{equation}
-    c'est à dire
-    \begin{equation}
-        \frac{ \partial f }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial f }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
-    \end{equation}
-    pour tout \( i=1,\ldots, s\). D'autre part pour tout $k$, la fonction \( l_k(x)=G_k\big( x,\varphi(x) \big)\) est constante et vaut zéro; ses dérivées partielles sont donc nulles :
-    \begin{equation}
-        \frac{ \partial l }{ \partial x_i }(\alpha)=\frac{ \partial G_k }{ \partial x_i }(\alpha)+\sum_{k=1}^r\frac{ \partial G_k }{ \partial y_k }(a)\frac{ \partial \varphi_k }{ \partial x_i }(\alpha)=0
-    \end{equation}
-    pour tout \( i=1,\ldots, s\) et \( k=1,\ldots, r\).
-    
-    Les \( s\) premières colonnes de la matrice
-    \begin{equation}
-        \begin{pmatrix}
-            \frac{ \partial f }{ \partial x_1 }   &   \cdots    &   \frac{ \partial f }{ \partial x_s }    &   \frac{ \partial f }{ \partial y_1 }    &   \cdots    &   \frac{ \partial f }{ \partial y_r }\\  
-            \frac{ \partial G_1 }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial x_s }    &   \frac{ \partial G_1 }{ \partial y_1 }    &   \cdots    &   \frac{ \partial G_1 }{ \partial y_r }\\
-            \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots    &   \vdots\\
-            \frac{ \partial G_r }{ \partial x_1 }    &   \cdots    &   \frac{ \partial G_r }{ \partial x_s }    &   \frac{ \partial G_r }{ \partial y_1 }    &  \cdots   & \frac{ \partial G_r }{ \partial y_r }  
-        \end{pmatrix}
-    \end{equation}
-    s'expriment en terme des \( r\) dernières. La matrice est donc au maximum de rang \( r\). Notons que la première ligne est \( \nabla f\) et les \( r\) suivantes sont les \( \nabla G_i\). Vu que ces lignes sont des vecteurs liés, il existe \( \mu_0,\ldots, \mu_r\) tels que
-    \begin{equation}
-        \mu_0\nabla f+\sum_{i=1}^r\mu_i\nabla G_i=0.
-    \end{equation}
-    Par hypothèse les \( \nabla G_i\) sont linéairement indépendants, ce qui nous dit que \( \mu_0\neq 0\). Donc nous avons ce qu'il nous faut :
-    \begin{equation}
-        \nabla f(a)=\sum_i\frac{ \mu_i }{ \mu_0 } \nabla G_i(a).
-    \end{equation}
-
-    Notons qu'au vu de l'expression \eqref{EqRDsSXyZ}, le fait que les formes \( \{ dG_i(a) \}_{1\leq i\leq r}\) forment une partie libre dans \( (\eR^n)^*\) implique que les \( \lambda_i\) sont uniques.
-\end{proof}
-
-La proposition suivante est la même que \ref{ThoRGJosS}.
-\begin{proposition} \label{PropfPPUxh}
-    Soit \( U\), un ouvert de \( \eR^n\) et des fonctions de classe \( C^1\) \( f,g_1,\ldots, g_r\colon U\to \eR\). Nous considérons
-    \begin{equation}
-        \Gamma=\{ x\in U\tq g_1(x)=\ldots=g_r(x)=0 \}.
-    \end{equation}
-    Soit \( a\) un extrémum de \( f|_{\Gamma}\). Supposons que les formes \( dg_1,\ldots, dg_r\) soient linéairement indépendantes en \( a\). Alors il existe \( \lambda_1,\ldots, \lambda_r\) dans \( \eR\) tel que
-    \begin{equation}
-        df_a=\sum_{i=1}^r\lambda_i(dg_i)_a.
-    \end{equation}
-\end{proposition}
-
-En pratique les candidats extrema locaux sont tous les points où les gradients ne sont pas linéairement indépendants, plus tous les points donnés par l'équation $\nabla L=0$. Parmi ces candidats, il faut trouver lesquels sont maxima ou minima, locaux ou globaux.
-
-L'existence d'extrema locaux se prouve généralement en invoquant de la compacité, et en invoquant le lemme suivant qui permet de réduire le problème à un compact.
-
-\begin{lemma}       \label{LemmeMinSCimpliqueS}
-    Soit $S$, une partie de $\eR^n$ et $C$, un ouvert de $\eR^n$. Si $a\in\Int S$ est un minimum local relatif à $S\cap C$, alors il est un minimum local par rapport à $S$.
-\end{lemma}
-
-\begin{proof}
-    Nous avons que $\forall x\in B(a,\epsilon_1)\cap S\cap C$, $f(x)\geq f(x)$. Mais étant donné que $C$ est ouvert, et que $a\in C$, il existe un $\epsilon_2$ tel que $B(a,\epsilon_2)\subset C$. En prenant $\epsilon=\min\{ \epsilon_1,\epsilon_2 \}$, nous trouvons que $f(x)\geq f(a)$ pour tout $x\in B(a,\epsilon)\cap(S\cap C)=B(a,\epsilon)\cap S$.
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Fonctions convexes}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-\begin{definition}[\cite{JFihMcQ}]  \label{DefVQXRJQz}
-    Une fonction $f$ d’un intervalle $I$ de \( \eR\) vers \( \eR\) est dite \defe{convexe}{fonction!convexe}\index{convexité!fonction} lorsque, pour tous \( x_1\) et \( x_2\) de $I$ et tout $\lambda$ dans $[0, 1]$ nous avons
-    \begin{equation}        \label{EQooYNAPooFePQZy}
-        f\big(\lambda\, x_1+(1-\lambda)\, x_2\big) \leq \lambda\, f(x_1)+(1-\lambda)\, f(x_2)
-    \end{equation}
-    Si l'inégalité est stricte, alors nous disons que la fonction \( f\) est \defe{strictement convexe}{convexité!stricte}.
-
-    Une fonction est \defe{concave}{concave} si son opposée est convexe.
-\end{definition}
-
-
-\begin{normaltext}[\cite{GYfviRu}]
-    Les différents résultats pour les fonctions convexes s'adaptent généralement sans mal aux fonctions strictement convexes. Une nuance cependant : de même que les fonctions dérivables convexes sont celles qui ont une dérivée croissante, les fonctions dérivables strictement convexes sont celles qui ont une dérivée strictement croissante (proposition \ref{PropYKwTDPX}). En revanche, il ne faudrait pas croire que la dérivée seconde d'une fonction dérivable strictement convexe est nécessairement une fonction à valeurs strictement positives (voir théorème \ref{ThoGXjKeYb}) : la dérivée d'une fonction strictement croissante peut s'annuler occasionnellement, ou plus exactement peut s'annuler sur un ensemble de points d'intérieur vide. Penser à \( x\mapsto x^4\) pour un exemple de fonction strictement convexe dont la dérivée seconde s'annule.
-\end{normaltext}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Inégalité des pentes}
-%---------------------------------------------------------------------------------------------------------------------------
-
-Dans l'étude des fonctions convexes nous allons souvent utiliser la fonction \defe{taux d'accroissement}{taux d'accroissement} qui est, pour \( \alpha\) dans le domaine de convexité de \( f\) définie par
-\begin{equation}    \label{EqRYBazWd}
-    \begin{aligned}
-        \tau_{\alpha}\colon I\setminus\{ \alpha \}&\to \eR \\
-        x&\mapsto \frac{ f(x)-f(\alpha) }{ x-\alpha }. 
-    \end{aligned}
-\end{equation}
-
-\begin{proposition}[Inégalité des pentes\cite{OJIMBtv}] \label{PropMDMGjGO}
-    Soit \( f\) une fonction convexe sur un intervalle \( I\subset \eR\). Alors pour tout \( a<b<c\) dans \( I\) nous avons\footnote{Les inégalités sont strictes si la fonction \( f\) est strictement convexe.}
-    \begin{equation}
-        \frac{ f(b)-f(a)  }{ b-a }\leq\frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
-    \end{equation}
-    En d'autres termes,
-    \begin{equation}
-        \tau_a(b)\leq\tau_a(c)\leq \tau_b(c),
-    \end{equation}
-    c'est à dire que \( \tau\) est croissante en ses deux arguments.
-\end{proposition}
-\index{inégalité!des pentes}
-
-\begin{proof}
-    D'abord les inégalités \( a<b<c\) impliquent \( 0<b-a<c-a\) et donc
-    \begin{equation}
-        \lambda=\frac{ b-a }{ c-a }<1.
-    \end{equation}
-    L'astuce est de remarquer que \( (1-\lambda)a+\lambda c=b\). Donc \( \lambda\) a toutes les bonnes propriétés pour être utilisé dans la définition de la convexité :
-    \begin{equation}
-        f\big( (1-\lambda)a+\lambda c \big)\leq \lambda f(c)+(1-\lambda)f(a),
-    \end{equation}
-    c'est à dire
-    \begin{equation}
-        f(b)-f(a)\leq \lambda\big( f(c)-f(a) \big)
-    \end{equation}
-    ou encore, en remplaçant \( \lambda\) par sa valeur :
-    \begin{equation}
-        \frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(c)-f(a) }{ c-a }.
-    \end{equation}
-    Cela fait déjà une des inégalités à savoir.
-    
-    D'autre part en partant de \( -a<-b<-c\) nous posons
-    \begin{equation}
-        0<\lambda=\frac{ c-b }{ c-a }.
-    \end{equation}
-    Nous avons à nouveau \( b=(1-\lambda)c+\lambda a\) et nous pouvons obtenir la seconde inégalité
-    \begin{equation}
-        \frac{ f(c)-f(a) }{ c-a }\leq \frac{ f(c)-f(b) }{ c-b }.
-    \end{equation}
-\end{proof}
-
-Géométriquement, l'inégalité des pentes se comprend facilement : le coefficient angulaire de la corde du graphe augmente. Donc si \( x<y<z\), le coefficient moyen entre \( x\) et \( y\) est plus petit que celui entre \( x\) et \( z\) qui est plus petit que celui entre \( y\) et \( z\).
-
-Donc si le coefficient angulaire moyen entre \( a\) et \( b+u\) vaut celui entre \( a\) et \( b\), ce coefficient ne peut qu'être constant entra \( a\) et \( b\) : sinon il serait plus grand entre \( b\) et \( b+u\) et la moyenne sur \( a\to b+u\) serait plus grande que sa moyenne sur \( a\to b\). Mais avoir un coefficient angulaire constant signifie être une droite.
-
-En résumé, si une fonction est convexe et non strictement convexe, alors son graphe est une droite. C'est en gros cela que la proposition \ref{PROPooOCOEooEGybmS} clarifiera.
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Convexité et régularité}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{lemma}[\cite{GYfviRu}]   \label{LemKLTsHIQ}
-    Une fonction convexe sur un ouvert
-    \begin{enumerate}
-        \item
-            y admet des dérivées à gauche et à droite en chaque point,
-        \item
-            y est continue.
-    \end{enumerate}
-\end{lemma}
-
-\begin{proof}
-    Soit \( I=\mathopen] a , b \mathclose[\) un intervalle sur lequel \( f\) est convexe et \( \alpha\in I\). Nous allons prouver que \( f\) est continue en \( \alpha\). Nous considérons \( \tau_{\alpha}\) le taux d'accroissement définit par \eqref{EqRYBazWd}; c'est une fonction croissante comme précisé dans l'inégalité des trois pentes \ref{PropMDMGjGO} et de plus \( \tau_{\alpha}(x)\) est bornée supérieurement par \( \tau_{\alpha}(b)\) pour \( x<\alpha\) et inférieurement par \( \tau_{\alpha}(a)\) pour \( x>\alpha\). Les limites existent donc et sont finies par la proposition \ref{PropMTmBYeU}. Autrement dit les limites
-        \begin{subequations}
-            \begin{align}
-                \lim_{x\to \alpha+} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^+} \tau_{\alpha}(x)=\inf_{t>\alpha}\tau_{\alpha}(t)\\
-                \lim_{x\to \alpha^-} \frac{ f(x)-f(\alpha) }{ x-\alpha }&=\lim_{x\to \alpha^-} \tau_{\alpha}(x)=\sup_{t<\alpha}\tau_{\alpha}(t).
-            \end{align}
-        \end{subequations}
-        existent et sont finies, c'est à dire que la fonction \( f\) admet une dérivée à gauche et à droite.
-
-        Pour tout \( x\) nous avons les inégalités
-        \begin{equation}
-            \tau_{\alpha}(a)\leq \frac{ f(x)-f(\alpha) }{ x-\alpha }\leq \tau_{\alpha}(b).
-        \end{equation}
-        En posant \( k=\max\{ \tau_{\alpha}(a),\tau_{\alpha}(b) \}\) nous avons
-        \begin{equation}
-            \big| f(x)-f(\alpha) \big|\leq k| x-\alpha |.
-        \end{equation}
-        La fonction est donc Lipschitzienne et par conséquent continue par la proposition \ref{PropFZgFTEW}.
-\end{proof}
-
-\begin{remark}
-    Les dérivées à gauche et à droite ne sont a priori pas égales. Penser par exemple à une fonction affine par morceaux dont les pentes augmentent à chaque morceau.
-\end{remark}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Dérivées d'une fonction convexe}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{proposition}[\cite{RIKpeIH,ooGCESooQzZtVC,MonCerveau}] \label{PropYKwTDPX}
-    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) 
-    \begin{enumerate}
-        \item       \label{ITEMooUTSAooJvhZNm}
-            est convexe si et seulement si sa dérivée est croissante sur \( I\).
-        \item       \label{ITEMooLLSIooFwkxtV}
-            est strictement convexe si et seulement si sa dérivée est strictement croissante sur \( I\)
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-
-
-    Pour la preuve de \ref{ITEMooUTSAooJvhZNm} et \ref{ITEMooLLSIooFwkxtV}, nous allons démontrer les énoncés «non stricts»  et indiquer ce qu'il faut changer pour obtenir les énoncés «stricts».
-    \begin{subproof}
-    \item[Sens direct]
-    Nous supposons que \( f\) est convexe. Soient \( a<b\) dans \( I\) et \( x\in\mathopen] a , b \mathclose[\). D'après l'inégalité des pentes \ref{PropMDMGjGO},
-        \begin{equation}        \label{EqATDLooIcqdDI}
-            \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(a) }{ b-a }\leq \frac{ f(b)-f(x) }{ b-x }.
-        \end{equation}
-        En faisant la limite \( x\to a\) nous avons
-        \begin{equation}
-            f'(a)\leq \frac{ f(b)-f(a) }{ b-a }
-        \end{equation}
-        et la limite \( x\to b\) donne
-        \begin{equation}
-            \frac{ f(b)-f(a) }{ b-a }\leq f'(b).
-        \end{equation}
-        Ici les inégalités sont non a priori strictes, même si \( f\) est strictement convexe : même avec des inégalités strictes dans \eqref{EqATDLooIcqdDI}, le passage à la limite rend l'inégalité non stricte. Quoi qu'il en soit nous avons 
-        \begin{equation}        \label{EqQGVMooBpuvNr}
-            f'(a)\leq f'(b).
-        \end{equation}
-    \item[Sens direct : strict]
-         Nous savons déjà que \( f'\) est croissante. Si \eqref{EqQGVMooBpuvNr} était une égalité, alors \( f'\) serait constante sur \( \mathopen] a , b \mathclose[\) parce qu'en prenant \( c\) entre \( a\) et \( b\) nous aurions \( f'(a)\leq f'(c)\leq f'(b)\) avec \( f'(a)=f'(b)\). Donc \( f'(a)=f'(c)\). Avoir \( f'\) constante sur un intervalle est contraire à la stricte convexité.
-
-         \item[Sens réciproque]
-
-             Nous supposons que \( f'\) est croissante et nous considérons \( a<b\) dans \( I\) ainsi que \( \lambda\in \mathopen[ 0 , 1 \mathclose]\). Nous posons \( x=\lambda a+(1-\lambda)b\), et nous savons que \( a\leq x\leq b\). Le théorème des accroissements finis \ref{ThoAccFinis} donne \( c_1\in\mathopen] a , x \mathclose[\) et \( c_2\in \mathopen] x , b \mathclose[\) tels que
-                 \begin{equation}
-                     f'(c_1)=\frac{ f(x)-f(a) }{ x-a }
-                 \end{equation}
-                 et 
-                 \begin{equation}
-                     f'(c_2)=\frac{ f(b)-f(x) }{ b-x }.
-                 \end{equation}
-                 Et en plus \( c_1<c_2\). Vu que \( f'\) est croissante nous avons \( f'(c_1)\leq f'(c_2)\) et donc
-                 \begin{equation}       \label{EqSAOCooWAwClQ}
-                     \frac{ f(x)-f(a) }{ x-a }\leq\frac{ f(b)-f(x) }{ b-x }.
-                 \end{equation}
-                 En remplaçant \( x\) par sa valeur en termes de \( \lambda\), \( a\) et \( b\) nous avons \( x-a=(1-\lambda)(b-a)\) et \( b-x=\lambda(b-a)\), et l'inégalité \eqref{EqSAOCooWAwClQ} nous donne
-                 \begin{equation}
-                     f(x)\leq \lambda f(a)+(1-\lambda)f(b).
-                 \end{equation}
-             \item[Sens réciproque : strict]
-                 Si \( f'\) est strictement croissante, nous avons \( f'(c_2)<f'(c_2)\) et les inégalité suivantes sont strictes, ce qui donne
-                 \begin{equation}
-                     f(x)< \lambda f(a)+(1-\lambda)f(b).
-                 \end{equation}
-    \end{subproof}
-\end{proof}
-
-\begin{theorem}[\cite{RIKpeIH}] \label{ThoGXjKeYb}
-    Une fonction \( f\) de classe \( C^2\) est convexe si et seulement si \( f''\) est positive.
-\end{theorem}
-
-\begin{proof}
-    La fonction est \( C^2\), donc \( f''\) est positive si et seulement si \( f'\) est croissante (proposition \ref{PropGFkZMwD}) alors que la proposition \ref{PropYKwTDPX} nous jure que \( f\) sera convexe si et seulement si \( f'\) est croissante.
-\end{proof}
-
-\begin{remark}      \label{REMooVRPQooIybxmp}
-    Une fonction peut être strictement convexe sans que sa dérivée seconde ne soit toujours strictement positive. En exemple : \( x\mapsto x^4\) est strictement convexe alors que sa dérivée seconde s'annule en zéro.
-\end{remark}
-
-\begin{example} \label{ExPDRooZCtkOz}
-    Quelques exemples utilisant le théorème \ref{ThoGXjKeYb}
-    \begin{enumerate}
-        \item
-    La fonction \( x\mapsto x^2\) est convexe parce que sa dérivée seconde est la constante (positive) \( 2\).
-\item La fonction \( x\mapsto\frac{1}{ x }\) est convexe sur \( \eR^+\setminus\{ 0 \}\) (sa dérivée seconde est \( 2x^{-3}\)).
-\item
-    La fonction exponentielle est également convexe.
-\item
-    La fonction \( \ln\) est concave parce que la dérivée seconde de \( -\ln\) est \( \frac{1}{ x^2 }\) qui est strictement positif.
-    \end{enumerate}
-\end{example}
-
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Graphe d'une fonction convexe}
-%---------------------------------------------------------------------------------------------------------------------------
-
-L'idée principale du graphe d'une fonction convexe est qu'il est toujours au dessus du graphe de ses tangentes (lorsqu'elles existent). Lorsqu'elles n'existent pas, le lemme \ref{LemKLTsHIQ} donne des coefficients directeurs de droites qui vont rester en dessous du graphe de la fonction.
-
-\begin{proposition}[\cite{ooKCFNooVrqYhc}]      \label{PROPooOCOEooEGybmS}
-    Une fonction convexe est strictement convexe si et seulement s'il n'existe aucun intervalle de longueur non nulle sur lequel elle coïncide avec une fonction affine.
-\end{proposition}
-
-\begin{proof}
-    Si sur l'intervalle (non réduit à un point) \( \mathopen[ x , y \mathclose]\), la fonction convexe \( f\) coïncide avec une fonction affine, alors \( f(t)=at+b\) et pour \( \lambda\in\mathopen] 0 , 1 \mathclose[\) nous avons
-        \begin{equation}
-                f\big( \lambda x+(1-\lambda)y \big)=a\lambda x+a(1-\lambda)y+b=\lambda f(x)+(1-\lambda)f(y)
-        \end{equation}
-        où nous avons remplacé \( b\) par \( \lambda b+(1-\lambda)b\). Par conséquent la fonction n'est pas strictement convexe.
-
-    Nous supposons maintenant que la fonction convexe \( f\) n'est pas strictement convexe sur l'intervalle \( I\). Il existe \( x\neq y\in I\) et \( \lambda\in \mathopen] 0 , 1 \mathclose[\) tels que
-        \begin{equation}
-            f\big( \lambda x+(1-\lambda)y \big)=\lambda f(x)+(1-\lambda)f(y).
-        \end{equation}
-    Nous posons \( z=\lambda x+(1-\lambda)y\) et \( u\in\mathopen] x , z \mathclose[\) pour écrire des inégalités des pentes entre \( x<u<z<y\). Plus précisément si nous notons \( a\to b\) la pente de \( a\) à \( b\), c'est à dire \( a\to b=\frac{ f(b)-f(a) }{ b-a }\), alors les inégalités des pentes pour \( x<u<z\) puis \( u<z<y\) donnent
-        \begin{equation}        \label{EqooBMEFooMpoEzd}
-            x\to z\leq u\to z\leq z\to y.
-        \end{equation}
-        Voyons maintenant qu'en réalité \( z\to y=x\to z\). En effet en replaçant
-        \begin{equation}
-            f(y)=\frac{ f(z)-\lambda f(x) }{ 1-\lambda }
-        \end{equation}
-        et
-        \begin{equation}
-            y=\frac{ \lambda x }{ 1-\lambda }
-        \end{equation}
-        dans l'expression \( z\to y=\frac{ f(y)-f(z) }{ y-z }\) nous obtenons
-        \begin{equation}
-            z\to y=\frac{ f(y)-f(z) }{ y-z }=\frac{ f(z)-f(x) }{ z-x }=x\to z.
-        \end{equation}
-        Les inégalités \eqref{EqooBMEFooMpoEzd} sont donc des égalités :
-        \begin{equation}
-            \frac{ f(z)-f(x) }{ z-x }=\frac{ f(z)-f(u) }{ z-u }=\frac{ f(y)-f(z) }{ y-z }.
-        \end{equation}
-        Nous avons donc montré que le nombre \( a=\frac{ f(z)-f(u) }{ z-u }\) ne dépend pas de \( u\). Nous avons alors
-        \begin{equation}
-            f(z)-f(u)=a(z-u) 
-        \end{equation}
-        ou encore :
-        \begin{equation}
-            f(u)=f(z)-a(z-u),
-        \end{equation}
-    ce qui signifie que sur \( \mathopen] x , z \mathclose[\), la fonction \( f\) est affine.
-\end{proof}
-
-\begin{proposition} \label{PROPooQPOSooDZlUAJ}
-    Une fonction dérivable sur un intervalle \( I\) de \( \eR\) est convexe si et seulement si son graphe est au dessus de chacune de ses tangentes.
-\end{proposition}
-
-\begin{proof}
-    \begin{subproof}
-        \item[Sens direct]
-            Soient \( x,y\in I\). Nous voulons :
-            \begin{equation}
-                f(y)\geq f(x)+f'(x)(y-x).
-            \end{equation}
-            Étant donné que nous aurons besoin, dans le quotient différentiel de quelque chose comme \( f(x+t)-f(x)\) nous écrivons la définition \eqref{EQooYNAPooFePQZy} de la convexité en inversant les rôles de \( x\) et \( y\) et en manipulant un peu :
-            \begin{subequations}
-                \begin{align}
-                    f\big( ty+(1-t)x \big)\leq tf(y)+(1-t)f(x)\\
-                    f\big( x+t(y-x) \big)\leq tf(y)+(1-t)f(x)\\
-                    f\big(  x+t(y-x)  \big)=f(x)\leq tf(y)-tf(x)
-                \end{align}
-            \end{subequations}
-            Nous divisons par \( t\) :
-            \begin{equation}
-                \frac{ f\big( x+t(y-x) \big)-f(x) }{ t }\leq f(y)-f(x).
-            \end{equation}
-            Le passage à la limite \( t\to 0\) donne
-            \begin{equation}
-                (y-x)f'(x)\leq f(y)-f(x),
-            \end{equation}
-            ce qu'il fallait.
-        \item[Sens inverse]
-            Pour tout \( x,y\in I\) nous supposons avoir
-            \begin{equation}        \label{EQooEXXIooHXJnER}
-                f(y)\geq f(x)+f'(x)(y-x).
-            \end{equation}
-            Si nous supposons \( x\neq y\) et si nous posons \( z=\lambda x+(1-\lambda)y\) nous voulons prouver que
-            \begin{equation}
-                f(z)\leq \lambda f(x)+(1-\lambda)f(y).
-            \end{equation}
-            Pour cela nous écrivons l'inégalité \eqref{EQooEXXIooHXJnER} avec les couples \( (x,z)\) et \( (y,z)\) :
-            \begin{subequations}
-                \begin{align}
-                    f(x)\geq f(z)+f'(z)'(x-z)\\
-                    f(y)\geq f(z)+f'(z)'(y-z)
-                \end{align}
-            \end{subequations}
-            En multipliant la première par \( \lambda\) et la seconde par \( (1-\lambda)\) et en sommant,
-            \begin{subequations}
-                \begin{align}
-                    \lambda f(x)+(1-\lambda)f(y)&\geq \lambda f(z)+\lambda f'(z)(x-z)+(1-\lambda)f(z)+(1-\lambda)f'(z)(y-z)\\
-                    &=f(z)+f'(z)\big( \lambda(x-z)+(1-\lambda)(y-z) \big)\\
-                    &=f(z).
-                \end{align}
-            \end{subequations}
-    \end{subproof}
-\end{proof}
-
-\begin{proposition}[\cite{MonCerveau}] \label{PropNIBooSbXIKO}
-    Soit \( f\colon \eR\to \eR \) une fonction convexe et \( a\in \eR\). Il existe une constante \( c_a\in \eR\) telle que pour tout \( x\) nous ayons
-    \begin{equation}    \label{EqSKIooSeAekM}
-        f(x)-f(a)\geq c_a(x-a).
-    \end{equation}
-    Autrement dit, le graphe de la fonction \( f\) est toujours au dessus de la droite d'équation
-    \begin{equation}
-        y=f(a)+c_a(x-a).
-    \end{equation}
-\end{proposition}
-
-\begin{proof}
-    Les dérivées à gauche et à droite de \( f\) données par le lemme \ref{LemKLTsHIQ} sont les candidats tout cuits pour être coefficient directeur de la droite que l'on cherche. Nous allons prouver qu'en posant
-    \begin{equation}
-        c_a=\inf_{t>a}\tau_a(t),
-    \end{equation}
-    la droite \( y=f(a)+c_a(x-a)\) répond à la question\footnote{En prenant l'autre, \( c_a'=\sup_{t<a}\tau_a(t)\), ça fonctionne aussi. En pensant à une fonction affine par morceaux, on remarque qu'en choisissant un nombre entre les deux, nous avons plus facilement une inégalité stricte dans \eqref{EqSKIooSeAekM}.}.
-
-    Nous devons prouver que le nombre \( \Delta_x=f(x)-\big( f(a)+c_a(x-a) \big)\) est positif pour tout \( x\).
-    \begin{subproof}
-
-    \item[Si \( x>a\)]
-        
-        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est positif :
-        \begin{subequations}
-            \begin{align}
-                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
-                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
-                &\geq 0
-            \end{align}
-        \end{subequations}
-        parce que \( t\to\tau_a(t)\) est croissante et que \( x>a\).
-
-    \item[Si \( x<a\)]
-        
-        Nous divisons par \( x-a\) et nous devons prouver que \( \frac{ \Delta_x }{ x-a }\) est négatif :
-        \begin{subequations}
-            \begin{align}
-                \frac{ \Delta_x }{ x-a }&=\frac{ f(x)-f(a) }{ x-a }-c_a\\
-                &=\tau_a(x)-\inf_{t>a}\tau_a(t)\\
-                &\leq 0
-            \end{align}
-        \end{subequations}
-        parce que \( t\to\tau_a(t)\) est croissante et que \( x<a\).
-    \end{subproof}
-\end{proof}
-
-\begin{proposition}[\cite{MonCerveau}] \label{PropPEJCgCH}
-    Si \( g\) est une fonction convexe, il existe deux suites réelles \( (a_n)\) et \( (b_n)\) telles que
-    \begin{equation}
-        g(x)=\sup_{n\in \eN}(a_nx+b_n).
-    \end{equation}
-\end{proposition}
-\index{fonction!convexe}
-\index{densité!de \( \eQ\) dans \( \eR\)!utilisation}
-
-\begin{proof}
-    Pour \( u\in \eR\) nous considérons \( a(u)\) et \( b(u)\) tels que la droite \( y(x)=a(u)x+b(u)\) vérifie \( y(u)=g(u)\) et \( y(x)\leq g(x)\) pour tout \( x\). Cela est possible par la proposition \ref{PropNIBooSbXIKO}. Il s'agit d'une droite coupant le graphe de \( g\) en \( x=u\) et restant en dessous. Nous considérons alors \( (u_n)\) une suite quelconque dense dans \( \eR\) (disons les rationnels pour fixer les idées) et nous posons
-    \begin{subequations}
-        \begin{numcases}{}
-            a_n=a(u_n)\\
-            b_n=b(u_n).
-        \end{numcases}
-    \end{subequations}
-    Si \( q\in \eQ\) alors \( a_nx+b_n\leq g(x)\) pour tout \( n\) et \( g(q)\) est le supremum qui est atteint pour le \( n\) tel que \( u_n=q\). Si maintenant \( x\) n'est pas dans \( \eQ\) il faut travailler plus.
-
-    Nous prenons \( (\tilde q_n)\), une sous-suite de \( (q_n)\) convergeant vers \( x\) et \( N\) suffisamment grand pour que pour tout \( n\geq N\) on ait \( | \tilde q_n-x |\leq \epsilon\) et \( | g(\tilde q_n)-g(x) |\leq \epsilon\); cela est possible grâce à la continuité de \( g\) (lemme \ref{LemKLTsHIQ}). Ensuite les sous-suites \( (\tilde a_n)\) et \( (\tilde b_n)\) sont celles qui correspondent :
-    \begin{equation}
-        \tilde a_n\tilde q_n+\tilde b_n=g(\tilde q_n).
-    \end{equation}
-    Nous considérons la majoration
-    \begin{subequations}
-        \begin{align}
-            | \tilde a_nx+\tilde b_n-g(x) |&\leq| \tilde a_nx+\tilde b_n-(\tilde a_n\tilde q_n+\tilde b_n) |+\underbrace{| \tilde a_n\tilde q_n+\tilde b_n-g(\tilde q_n) |}_{=0}+\underbrace{| g(\tilde q_n)-g(x) |}_{\leq \epsilon}\\
-            &\leq | \tilde a_n | |x-\tilde q_n |+\epsilon\\
-            &=\epsilon\big( | \tilde a_n |+1 \big).
-        \end{align}
-    \end{subequations}
-    Il nous reste à montrer que \( | \tilde a_n |\) est borné par un nombre ne dépendant pas de \( n\) (pour les \( n>N\)).
-
-    Vu que la droite de coefficient directeur \( \tilde a_n\) et passant par le point \( \big( \tilde q_n,g(\tilde q_n) \big)\) reste en dessous du graphe de \( g\), nous avons pour tout \( n\) et tout \( y\in \eR\) l'inégalité
-    \begin{equation}
-        g(y)\geq \tilde a_n(y-\tilde q_n)+g(\tilde q_n)\in \tilde a_nB(y-x,\epsilon)+B\big( g(x),\epsilon \big).
-    \end{equation}
-    Si \( \tilde a_n\) n'est pas borné vers le haut, nous prenons \( y\) tel que \( B(y-x,\epsilon)\) soit minoré par un nombre \( k\) strictement positif et nous obtenons
-    \begin{equation}
-        g(y)\geq k\tilde a_n+l
-    \end{equation}
-    avec \( k\) et \( l\) indépendants de \( n\). Cela donne \( g(y)=\infty\). Si au contraire \( \tilde a_n\) n'est pas borné vers le bas, nous prenons $y$ tel que \( B(y-x,\epsilon)\) est majoré par un nombre \( k\) strictement négatif. Nous obtenons encore \( g(y)=\infty\).
-
-    Nous concluons que \( | \tilde a_n |\) est bornée.
-\end{proof}
-
-\begin{lemma}[\cite{KXjFWKA}]   \label{LemXOUooQsigHs}
-    L'application
-    \begin{equation}
-        \begin{aligned}
-            \phi\colon S^{++}(n,\eR)&\to \eR \\
-            A&\mapsto \det(A) 
-        \end{aligned}
-    \end{equation}
-    est \defe{log-convave}{concave!log-concave}\index{log-concave}, c'est à dire que l'application \( \ln\circ\phi\) est concave\footnote{La définition \ref{DEFooELGOooGiZQjt} du logarithme ne fonctionne que pour les réels strictement positifs. C'est le cas du déterminant d'une matrice réelle symétrique strictement définie positive.}. De façon équivalente, si \( A,B\in S^{++}\) et si \( \alpha+b=1\), alors
-    \begin{equation}    \label{EqSPKooHFZvmB}
-        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
-    \end{equation}
-\end{lemma}
-Ici \( S^{++}\) est l'ensemble des matrices symétriques strictement définies positives, définition \ref{DefAWAooCMPuVM}.
-
-\begin{proof}
-    Nous commençons par prouver que l'équation \eqref{EqSPKooHFZvmB} est équivalente à la log-concavité du déterminant. Pour cela il suffit de remarquer que les propriétés de croissance et d'additivité du logarithme donnent l'équivalence entre
-    \begin{equation}
-        \ln\Big( \det(\alpha A+\beta B) \Big)\geq \ln\Big( \det(\alpha A) \Big)+\ln\Big( \det(\beta B) \Big),
-    \end{equation}
-    et
-    \begin{equation}    \label{EqTJYooBWiRrn}
-        \det(\alpha A+\beta B)\geq \det(A)^{\alpha}\det(B)^{\beta}.
-    \end{equation}
-
-    Le théorème de pseudo-réduction simultanée, corollaire \ref{CorNHKnLVA}, appliqué aux matrices \( A\) et \( B\) nous donne une matrice inversible \( Q\) telle que
-    \begin{subequations}
-        \begin{numcases}{}
-            B=Q^tDQ\\
-            A=Q^tQ
-        \end{numcases}
-    \end{subequations}
-    avec 
-    \begin{equation}
-        D=\begin{pmatrix}
-            \lambda_1    &       &       \\
-                &   \ddots    &       \\
-                &       &   \lambda_n
-        \end{pmatrix},
-    \end{equation}
-    \( \lambda_i>0\). Nous avons alors
-    \begin{equation}
-        \det(A)^{\alpha}\det(B)^{\beta}=\det(Q)^{2\alpha}\det(Q)^{2\beta}\det(D)^{\beta}=\det(Q)^2\det(D)^{\beta}
-    \end{equation}
-    (parce que \( \alpha+\beta=1\)) et
-    \begin{equation}
-        \det(\alpha A+\beta B)=\det(\alpha Q^tQ+\beta Q^tDQ)=\det\big( Q^t(\alpha\mtu+\beta D)Q \big)=\det(Q)^2\det(\alpha\mtu+\beta D).
-    \end{equation}
-    L'inégalité \eqref{EqTJYooBWiRrn} qu'il nous faut prouver se réduit donc  à
-    \begin{equation}
-        \det(\alpha \mtu+\beta D)\geq \det(D)^{\beta}.
-    \end{equation}
-    Vue la forme de \( D\) nous avons
-    \begin{equation}
-        \det(\alpha\mtu+\beta D)=\prod_{i=1}^n(\alpha+\beta\lambda_i)
-    \end{equation}
-    et 
-    \begin{equation}
-        \det(D)^{\beta}=\big( \prod_{i=1}^{n}\lambda_i \big)^{\beta}.
-    \end{equation}
-    Il faut donc prouver que
-    \begin{equation}\label{EqGFLooOElciS}    
-        \prod_{i=1}^n(\alpha+\beta\lambda_i)\geq \big( \prod_{i=1}^n\lambda_i \big)^{\beta}.
-    \end{equation}
-    Cette dernière égalité de produit sera prouvée en passant au logarithme. Vu que le logarithme est concave par l'exemple \ref{ExPDRooZCtkOz}, nous avons pour chaque \( i\) que
-    \begin{equation}
-        \ln(\alpha+\beta\lambda_i)\geq \alpha\ln(1)+\beta\ln(\lambda_i)=\beta\ln(\lambda_i).
-    \end{equation}
-    En sommant cela sur \( i\) et en utilisant les propriétés de croissance et de multiplicativité du logarithme nous obtenons successivement
-    \begin{subequations}
-        \begin{align}
-            \sum_{i=1}^n\ln(\alpha+\beta\lambda_i)\geq \beta\sum_i\ln(\lambda_i)\\
-            \ln\big( \prod_i(\alpha+\beta\lambda_i) \big)\geq\ln\Big( \big( \prod_i\lambda_i \big)^{\beta} \Big)\\
-            \prod_i(\alpha+\beta\lambda_i)\geq\big( \prod_i\lambda_i \big)^{\beta},
-        \end{align}
-    \end{subequations}
-    ce qui est bien \eqref{EqGFLooOElciS}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{En dimension supérieure}
-%---------------------------------------------------------------------------------------------------------------------------
-
-\begin{definition}
-    Soit une partie convexe \( U\) de \( \eR^n\) et une fonction \( f\colon U\to \eR\). 
-    \begin{enumerate}
-        \item
-        La fonction \( f\) est \defe{convexe}{convexe!fonction sur \( \eR^n\)} si pour tout \( x,y\in U\) avec \( x\neq y\) et pour tout \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous avons
-            \begin{equation}
-                f\big( \theta x+(1-\theta)y \big)\leq \theta f(x)+(1-\theta)f(y).
-            \end{equation}
-        \item
-            Elle est \defe{strictement convexe}{strictement!convexe!sur \( \eR^n\)} si nous avons l'inégalité stricte.
-    \end{enumerate}
-\end{definition}
-
-\begin{proposition}[\cite{ooLJMHooMSBWki}]      \label{PROPooYNNHooSHLvHp}
-    Soit \( \Omega\) ouvert dans \( \eR^n\) et \( U\) convexe dans \( \Omega\), et une fonction différentiable \( f\colon U\to \eR\).
-    \begin{enumerate}
-        \item       \label{ITEMooRVIVooIayuPS}
-            La fonction \( f\) est convexe sur \( U\) si et seulement si pour tout \( x,y\in U\),
-            \begin{equation}
-                f(y)\geq f(x)+df_x(y-x).
-            \end{equation}
-        \item       \label{ITEMooCWEWooFtNnKl}
-            La fonction \( f\) est strictement convexe sur \( U\) si et seulement si pour tout \( x,y\in U\) avec \( x\neq y\),
-            \begin{equation}
-                f(y)>f(x)+df_x(y-x).
-            \end{equation}
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Nous avons quatre petites choses à démontrer.
-    \begin{subproof}
-    \item[\ref{ITEMooRVIVooIayuPS} sens direct]
-        Soit une fonction convexe \( f\). Nous avons :
-        \begin{equation}
-            f\big( (1-\theta)x+\theta y \big)\leq (1-\theta)f(x)+\theta f(y),
-        \end{equation}
-        donc
-        \begin{equation}
-            f\big( x+\theta(y-x) \big)-f(x)\leq \theta\big( f(y)-f(x) \big)
-        \end{equation}
-        Vu que \( \theta>0\) nous pouvons diviser par \( \theta\) sans changer le sens de l'inégalité :
-        \begin{equation}        \label{EQooAXXFooHWtiJh}
-            \frac{ f\big( x+\theta(y-x) \big)-f(x) }{ \theta }\leq f(y)-f(x).
-        \end{equation}
-        Nous prenons la limite \( \theta\to 0^+\). Cette limite est égale à a limite simple \( \theta\to 0\) et vaut (parce que \( f\) est différentiable) :
-        \begin{equation}
-            \frac{ \partial f }{ \partial (y-x) }(x)\leq f(y)-f(x),
-        \end{equation}
-        et aussi
-        \begin{equation}
-            df_x(y-x)\leq f(y)-f(x)
-        \end{equation}
-        par le lemme \ref{LemdfaSurLesPartielles}.
-    \item[\ref{ITEMooRVIVooIayuPS} sens inverse]
-        Pour tout \( a\neq b\) dans \( U\) nous avons
-        \begin{equation}        \label{EQooEALSooJOszWr}
-            f(b)\geq f(a)+df_a(b-a).
-        \end{equation}
-    Pour \( x\neq y\) dans \( U\) et pour \( \theta\in\mathopen] 0 , 1 \mathclose[\) nous écrivons \eqref{EQooEALSooJOszWr} pour les couples \( \big( \theta x+(1-\theta)y,y \big)\) et \( \big( \theta x+(1-\theta)y,x \big)\). Ça donne :
-        \begin{equation}
-            f(y)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( \theta(y-x) \big),
-        \end{equation}
-        et
-        \begin{equation}
-            f(x)\geq f\big( \theta x+(1-\theta)y \big)+df_{\theta x+(1-\theta)y}\big( (1-\theta)(x-y) \big).
-        \end{equation}
-        La différentielle est linéaire; en multipliant la première par \( (1-\theta)\) et la seconde par \( \theta\) et en la somme, les termes en \( df\) se simplifient et nous trouvons
-        \begin{equation}
-            \theta f(x)+(1-\theta)f(y)\geq f\big( \theta x+(1-\theta)y \big).
-        \end{equation}
-    \item[\ref{ITEMooCWEWooFtNnKl} sens direct]
-        Nous avons encore l'équation \eqref{EQooAXXFooHWtiJh}, avec une inégalité stricte. Par contre, ça ne va pas être suffisant parce que le passage à la limite ne conserve pas les inégalités strictes. Nous devons donc être plus malins. 
-
-        Soient \( 0<\theta<\omega<1\). Nous avons \( (1-\theta)x+\theta y\in \mathopen[ x , (1-\omega)x+\omega y \mathclose]\), donc nous pouvons écrire \( (1-\theta)x+\theta y\) sous la forme \( (1-s)x+s\big( (1-\omega)x+\omega y \big)\). Il se fait que c'est bon pour \( s=\theta/\omega\) (et aussi que nous avons \( \theta/\omega<1\)). Donc nous avons
-        \begin{subequations}
-            \begin{align}
-            f\big( (1-\theta)x+\theta y \big)&=f\Big( (1-\frac{ \theta }{ \omega })x+\frac{ \theta }{ \omega }\big( (1-\omega)x+\omega y \big) \Big)\\
-            &<(1-\frac{ \theta }{ \omega })f(x)+\frac{ \theta }{ \omega }f\big( (1-\omega)x+\omega y \big).
-            \end{align}
-        \end{subequations}
-        Cela nous permet d'écrire
-        \begin{equation}
-            \frac{ f\big( (1-\theta)x+\theta y \big)-f(x) }{ \theta }<\frac{ f\big( (1-\omega)x+\omega y \big) }{ \omega }<f(y)-f(x).
-        \end{equation}
-        Le seconde inégalité est le pendant de \eqref{EQooAXXFooHWtiJh}. Maintenant en passant à la limite pour \( \theta\) nous conservons une inégalité stricte par rapport à \( f(y)-f(x)\) :
-        \begin{equation}
-            df_x(y-x)<f(y)-f(x).
-        \end{equation}
-    \end{subproof}
-\end{proof}
-
-% Il faut laisser les sauts de lignes suivants, pour rechercher efficacement les références vers le futur.
-Avant de lire la proposition suivante, il faut relire la proposition \ref{PROPooFWZYooUQwzjW} et ce qui s'y rapporte. 
-Lire aussi la remarque \ref{REMooVRPQooIybxmp} qui indique
-qu'il n'y a pas de réciproque dans l'énoncé \ref{ITEMooHAGQooYZyhQk}.     
-\begin{proposition}[\cite{ooLJMHooMSBWki}]
-    Soit une fonction \( f\colon \Omega\to \eR\) de classe \( C^2\) et un convexe \( U\subset \Omega\).
-    \begin{enumerate}
-        \item       \label{ITEMooZQCAooIFjHOn}
-            La fonction \( f\) est convexe sur \( U\) si et seulement si
-            \begin{equation}        \label{EQooIBDCooJYdiBb}
-                (d^2f)_x(y-x,y-x)\geq 0
-            \end{equation}
-            pour tout \( x,y\in U\).
-        \item       \label{ITEMooHAGQooYZyhQk}
-            Si pour tout \( x\neq y\) dans \( U\) nous avons
-            \begin{equation}
-                (d^2f)_x(y-x,y-x)>0
-            \end{equation}
-            alors la fonction \( f\) est strictement convexe sur \( U\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{remark}      \label{REMooYCRKooEQNIkC}
-    Notons que la condition \eqref{EQooIBDCooJYdiBb} n'est pas équivalente à demander \( (d^2f)_x(h,h)\geq 0\) pour tout \( h\). En effet nous ne demandons la positivité que dans les directions atteignables comme différence de deux éléments de \( U\). La partie \( U\) n'est pas spécialement ouverte; elle pourrait n'être qu'une droite dans \( \eR^3\). Dans ce cas, demander que \( f\) (qui est \( C^2\) sur l'ouvert \( \Omega\)) soit convexe sur \( U\) ne demande que la positivité de \( (d^2f)_x\) appliqué à des vecteurs situés sur la droite \( U\).
-\end{remark}
-
-\begin{proof}
-    Il y a trois parties à démontrer.
-    \begin{subproof}
-    \item[\ref{ITEMooZQCAooIFjHOn} sens direct]
-
-        Soit une fonction convexe \( f\) sur \( U\). Soient aussi \( x,y\in U\) et \( h=y-x\). Nous utilisons ma version préférée de Taylor\footnote{Si vous présentez ceci au jury d'un concours, vous devriez être capable de raconter ce que signifie \( d^2f\), et pourquoi nous l'utilisons comme une \( 2\)-forme.} : celui de la proposition \ref{PROPooTOXIooMMlghF} :
-        \begin{equation}
-            f(x+th)=f(x)+tdf_x(h)+\frac{ t^2 }{2}(d^2_x)(h,h)+t^2\| h \|^2\alpha(th)
-        \end{equation}
-        avec \( \lim_{s\to 0}\alpha(s)=0\). Le fait que \( f\) soit convexe donne
-        \begin{equation}
-            0\leq f(x+th)-f(x)-tdf_x(h),
-        \end{equation}
-        et donc
-        \begin{equation}
-            0\leq \frac{ t^2 }{2}(d^2f)_x(h,h)+f^2\| h \|^2\alpha(th).
-        \end{equation}
-        En multipliant par \( 2\) et en divisant par \( t^2\),
-        \begin{equation}
-            0\leq (d^2f)_x(h,h)+2\| h \|^2\alpha(th).
-        \end{equation}
-        En prenant \( t\to 0\) nous avons bien  \( (d^2f)_x(y-x,y-x)\geq 0\).
-
-    \item[\ref{ITEMooZQCAooIFjHOn} sens inverse]
-
-        Soient \( x,y\in U\). Nous écrivons Taylor en version de la proposition \ref{PROPooWWMYooPOmSds} :
-        \begin{equation}
-            f(y)=f(x)+df_x(y-x)+\frac{ 1 }{2}(d^2f)_z(y-x,y-x)
-        \end{equation}
-    pour un certain \( z\in\mathopen] x , y \mathclose[\). En vertu de ce qui a été dit dans la remarque \ref{REMooYCRKooEQNIkC} nous ne pouvons pas évoquer l'hypothèse \eqref{EQooIBDCooJYdiBb} pour conclure que \( (d^2f)_z(y-x,y-x)\geq 0\). Il y a deux manières de nous sortir du problème :
-        \begin{itemize}
-            \item Trouver \( s\in U\) tel que \( y-x=s-z\).
-            \item Trouver un multiple de \( y-x\) qui soit de la forme \( y-x\).
-        \end{itemize}
-        La première approche ne fonctionne pas parce que \( s=y-x+z\) n'est pas garanti d'être dans \( U\); par exemple avec \( x=1\), \( z=2\), \( y=3\) et \( U=\mathopen[ 0 , 3 \mathclose]\). Dans ce cas \( s=4\notin U\).
-
-        Heureusement nous avons \( z=\theta x+(1-\theta)y\), donc \( z-x=(1-\theta)(y-x)\). Dans ce cas la bilinéarité de \( (d^2f)_z\) donne\footnote{Si vous avez bien suivi, la bilinéarité est contenue dans la proposition \ref{PROPooFWZYooUQwzjW}.}
-        \begin{equation}
-            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{\geq 0}.
-        \end{equation}
-        Nous en déduisons que \( f\) est convexe par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooRVIVooIayuPS}.
-    \item[\ref{ITEMooHAGQooYZyhQk}]
-
-        Le raisonnement que nous venons de faire pour le sens inverse de \ref{ITEMooZQCAooIFjHOn} tient encore, et nous avons
-        \begin{equation}
-            f(y)=f(x)+df_x(y-x)+\underbrace{\frac{ 1 }{2}\frac{1}{ (1-\theta)^2 }(d^2f)_z(z-x,z-x)}_{> 0}
-        \end{equation}
-        d'où nous déduisons la stricte convexité de \( f\) par la proposition \ref{PROPooYNNHooSHLvHp}\ref{ITEMooCWEWooFtNnKl}.
-    \end{subproof}
-\end{proof}
-
-\begin{corollary}       \label{CORooMBQMooWBAIIH}
-    Avec la hessienne\ldots en cours d'écriture.
-\end{corollary}
-
-\begin{proof}
-    Cela va utiliser la proposition \ref{PropoExtreRn}.
-\end{proof}
-
-%--------------------------------------------------------------------------------------------------------------------------- 
-\subsection{Quelque inégalités}
-%---------------------------------------------------------------------------------------------------------------------------
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Inégalité de Jensen}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\index{inégalité!Jensen}
-\index{convexité!inégalité de Jensen}
-
-\begin{proposition}[Inégalité de Jensen]    \label{PropXIBooLxTkhU}
-    Soit \( f\colon \eR\to \eR\) une fonction convexe et des réels \( x_1\),\ldots,  \( x_n\). Soient des nombres positifs \( \lambda_1\),\ldots,  \( \lambda_n\) formant une combinaison convexe\footnote{Définition \ref{DefIMZooLFdIUB}.}. Alors
-    \begin{equation}
-        f\big( \sum_i\lambda_ix_i \big)\leq \sum_i\lambda_if(x_i).
-    \end{equation}
-\end{proposition}
-\index{inégalité!Jensen!pour une somme}
-
-\begin{proof}
-    Nous procédons par récurrence sur \( n\), en sachant que \( n=2\) est la définition de la convexité de \( f\). Vu que
-    \begin{equation}
-        \sum_{k=1}^n\lambda_kx_k=\lambda_nx_n+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n },
-    \end{equation}
-    nous avons
-    \begin{equation}
-        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)f\big( \sum_{k=1}^{n-1}\frac{ \lambda_kx_k }{ 1-\lambda_n } \big).
-    \end{equation}
-    La chose à remarquer est que les nombres \( \frac{ \lambda_k }{ 1-\lambda_n }\) avec \( k\) allant de \( 1\) à \( n-1\) forment eux-mêmes une combinaison convexe. L'hypothèse de récurrence peut donc s'appliquer au second terme du membre de droite :
-    \begin{equation}
-        f\big( \sum_{k=1}^n\lambda_kx_k \big)\leq \lambda_nf(x_n)+(1-\lambda_n)\sum_{k=1}^{n-1}\frac{ \lambda_k }{ 1-\lambda_n }f(x_k)=\lambda_nf(x_n)+\sum_{k=1}^{n-1}\lambda_kf(x_k).
-    \end{equation}
-\end{proof}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Inégalité arithmético-géométrique}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-La proposition suivante dit que la moyenne arithmétique de nombres strictement positifs est supérieure ou égale à la moyenne géométrique.
-\begin{proposition}[Inégalité arithmético-géométrique\cite{CENooZKvihz}]    \label{PropWDPooBtHIAR}
-    Soient \( x_1\),\ldots, \( x_n\) des nombres strictement positifs. Nous posons
-    \begin{equation}
-        m_a=\frac{1}{ n }(x_1+\cdots +x_n)
-    \end{equation}
-    et
-    \begin{equation}
-        m_g=\sqrt[n]{x_1\ldots x_n}
-    \end{equation}
-    Alors \( m_g\leq m_a\) et \( m_g=m_a\) si et seulement si \( x_i=x_j\) pour tout \( i,j\).
-\end{proposition}
-\index{inégalité!arithmético-géométrique}
-
-\begin{proof}
-    Par hypothèse les nombres \( m_a\) et \( m_g\) sont tout deux strictement positifs, de telle sorte qu'il est équivalent de prouver \( \ln(m_g)\leq \ln(m_a)\) ou encore
-    \begin{equation}
-        \frac{1}{ n }\big( \ln(x_1)+\cdots +\ln(x_n) \big)\leq \ln\left( \frac{ x_1+\cdots +x_n }{ n } \right).
-    \end{equation}
-    Cela n'est rien d'autre que l'inégalité de Jensen de la proposition \ref{PropXIBooLxTkhU} appliquée à la fonction \( \ln\) et aux coefficients \( \lambda_i=\frac{1}{ n }\).
-\end{proof}
-
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-\subsubsection{Inégalité de Kantorovitch}
-%///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
-
-\begin{proposition}[Inégalité de Kantorovitch\cite{EYGooOoQDnt}]    \label{PropMNUooFbYkug}
-    Soit \( A\) une matrice symétrique strictement définie positive dont les plus grandes et plus petites valeurs propres sont \( \lambda_{min}\) et \( \lambda_{max}\). Alors pour tout \( x\in \eR^n\) nous avons
-    \begin{equation}
-        \langle Ax, x\rangle \langle A^{-1}x, x\rangle \leq \frac{1}{ 4 }\left( \frac{ \lambda_{min} }{ \lambda_{max} }+\frac{ \lambda_{max} }{ \lambda_{min} } \right)^2\| x^4 \|.
-    \end{equation}
-\end{proposition}
-\index{inégalité!Kantorovitch}
-
-\begin{proof}
-    Sans perte de généralité nous pouvons supposer que \( \| x \|=1\). Nous diagonalisons\footnote{Théorème spectral \ref{ThoeTMXla}.} la matrice \( A\) par la matrice orthogonale  \( P\in\gO(n,\eR)\) : \( A=PDP^{-1}\) et \( A^{-1}=PD^{-1}P^{-1}\) où \( D\) est  une matrice diagonale formée des valeurs propres de \( A\).
-
-    Nous posons \( \alpha=\sqrt{\lambda_{min}\lambda_{max}}\) et nous regardons la matrice
-    \begin{equation}
-        \frac{1}{ \alpha }A+tA^{-1}
-    \end{equation}
-    dont les valeurs propres sont
-    \begin{equation}
-        \frac{ \lambda_i }{ \alpha }+\frac{ \alpha }{ \lambda_i }
-    \end{equation}
-    parce que les vecteurs propres de \( A\) et de \( A^{-1}\) sont les mêmes (ce sont les valeurs de la diagonale de \( D\)). Nous allons quelque peu étudier la fonction
-    \begin{equation}
-        \theta(x)=\frac{ x }{ \alpha }+\frac{ \alpha }{ x }.
-    \end{equation}
-    Elle est convexe en tant que somme de deux fonctions convexes. Elle a son minimum en \( x=\alpha\) et ce minimum vaut \( \theta(\alpha)=2\). De plus
-    \begin{equation}
-        \theta(\lambda_{max})=\theta(\lambda_{min})=\sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
-    \end{equation}
-    Une fonction convexe passant deux fois par la même valeur doit forcément être plus petite que cette valeur entre les deux\footnote{Je ne suis pas certain que cette phrase soit claire, non ?} : pour tout \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\),
-    \begin{equation}
-        \theta(x)\leq  \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
-    \end{equation}
-    
-    Nous sommes maintenant en mesure de nous lancer dans l'inégalité de Kantorovitch.
-    \begin{subequations}
-        \begin{align}
-            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }&\leq\frac{ 1 }{2}\left( \frac{ \langle Ax, x\rangle  }{ \alpha }+\alpha\langle A^{-1}x, x\rangle  \right)\label{subEqUKIooCWFSkwi}\\
-            &=\frac{ 1 }{2}\langle   \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x , x\rangle \\
-            &\leq\frac{ 1 }{2}\Big\| \big( \frac{ A }{ \alpha }+\alpha A^{-1} \big)x \|\| x \| \label{subEqUKIooCWFSkwiii}\\
-            &\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \label{subEqUKIooCWFSkwiv}
-        \end{align}
-    \end{subequations}
-    Justifications :
-    \begin{itemize}
-        \item \ref{subEqUKIooCWFSkwi} par l'inégalité arithmético-géométrique, proposition \ref{PropWDPooBtHIAR}. Nous avons aussi inséré \( \alpha\frac{1}{ \alpha }\) dans le produit sous la racine.
-        \item \ref{subEqUKIooCWFSkwiii} par l'inégalité de Cauchy-Schwarz, théorème \ref{ThoAYfEHG}.
-        \item \ref{subEqUKIooCWFSkwiv} par la définition de la norme opérateur de la proposition \ref{DefNFYUooBZCPTr}
-    \end{itemize}
-    La norme opérateur est la plus grande des valeurs propres. Mais les valeurs propres de \( A/\alpha+\alpha A^{-1}\) sont de la forme \( \theta(\lambda_i)\), et tous les \( \lambda_i\) sont entre \( \lambda_{min} \) et \( \lambda_{max}\). Donc la plus grande valeur propre de \( A/\alpha+\alpha A^{-1}\) est \( \theta(x)\) pour un certain \( x\in\mathopen[ \lambda_{min} , \lambda_{max} \mathclose]\). Par conséquent
-    \begin{equation}
-            \sqrt{\langle Ax, x\rangle \langle A^{-1}x, x\rangle }\leq \frac{ 1 }{2}\| \frac{ A }{ \alpha }+\alpha A^{-1} \| \leq \sqrt{\frac{ \lambda_{min} }{ \lambda_{max} }}+\sqrt{\frac{ \lambda_{max} }{ \lambda_{min} }}.
-    \end{equation}
-\end{proof}
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Algorithme du gradient à pas optimal}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
-
-Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
-\begin{itemize}
-    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
-    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
-\end{itemize}
-
-\begin{definition}  \label{DefQXPooYSygGP}
-    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
-    \begin{equation}
-        \{ x\in P\tq f(x)\leq M \}
-    \end{equation}
-    est borné.
-\end{definition}
-En langage imagé la coercivité de \( f\) s'exprime par la limite
-\begin{equation}
-    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
-\end{equation}
-
-
-Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
-\begin{proposition}     \label{PROPooYRLDooTwzfWU}
-    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Alors :
-    \begin{enumerate}
-        \item
-            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
-        \item
-            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
-        \item
-            Ils sont égaux : \( \bar x=x^*\).
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-
-    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
-
-    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
-
-    \begin{subproof}
-    \item[\( f\) est strictement convexe]
-        
-        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
-    \end{equation}
-    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
-
-\item[\( f\) est coercive]
-    Montrons à présent que \( f\) est coercive. Nous avons :
-    \begin{subequations}
-        \begin{align}
-            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
-            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
-            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
-        \end{align}
-    \end{subequations}
-    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
-    \end{subproof}
-
-    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
-    \begin{equation}
-        \{ x\in \eR^n\tq f(x)\leq M \}
-    \end{equation}
-    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
-
-    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
-    \begin{equation}
-        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
-    \end{equation}
-    ce qui contredit la minimalité de \( f(\bar x_1)\).
-
-    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
-    \begin{equation}
-        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
-    \end{equation}
-    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
-\end{proof}
-
-\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
-    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
-    \begin{equation}
-        \begin{aligned}
-            f\colon \eR^n&\to \eR \\
-            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
-        \end{aligned}
-    \end{equation}
-    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
-    \begin{equation}
-        x_{k+1}=x_k+t_kd_k
-    \end{equation}
-    où
-    \begin{itemize}
-        \item 
-    \( d_k=-(\nabla f)(x_k)\) 
-\item
-    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
-    \end{itemize}
-
-    Alors pour tout \( k\geq 0\) nous avons
-    \begin{equation}
-        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
-    \end{equation}
-    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
-\end{proposition}
-
-\begin{proof}
-    Décomposition en plusieurs points.
-    \begin{subproof}
-    \item[Existence de \( \bar x\)]
-        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
-    \item[Si \( (\nabla f)(x_k)=0\)]
-    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
-    \begin{equation}
-        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
-    \end{equation}
-    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
-    \begin{equation}
-        (\nabla f)(a)=Aa+b.
-    \end{equation}
-    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
-
-    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
-        \item[\( t_k\) est bien défini]
-
-            Pour \( t\in \eR\) nous avons
-            \begin{equation}    \label{EqKEHooYaazQi}
-                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
-            \end{equation}
-            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
-            \begin{equation}    \label{EqVWJooWmDSER}
-                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
-            \end{equation}
-            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
-
-        \item[La valeur de \( d_{k+1}\)]
-
-            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
-            \begin{equation}
-                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
-            \end{equation}
-            parce que \( -Ax_k-b=d_k\).
-            
-            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
-            \begin{equation}
-                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
-            \end{equation}
-            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
-        
-        \item[Calcul de \( f(x_{k+1})\)] 
-
-            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
-            \begin{equation}
-                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
-            \end{equation}
-            
-        \item[Encore du calcul \ldots]
-
-            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
-            \begin{equation}
-                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
-            \end{equation}
-            Ayant cela en tête nous pouvons calculer :
-            \begin{subequations}
-                \begin{align}
-                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
-                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
-                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
-                    &=2\big( f(x_k)-\bar f \big)
-                \end{align}
-            \end{subequations}
-            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
-
-        \item[Erreur sur la valeur du minimum]
-
-            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
-            \begin{subequations}
-                \begin{align}
-                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
-                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
-                \end{align}
-            \end{subequations}
-            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
-            \begin{equation}
-                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
-            \end{equation}
-            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
-            \begin{equation}
-                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
-            \end{equation}
-            Par récurrence nous avons alors
-            \begin{equation}    \label{eqANKooNPfCFj}
-                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
-            \end{equation}
-            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
-
-        \item[Erreur sur la position du minimum]
-
-            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
-            \begin{subequations}
-                \begin{align}
-                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
-                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
-                \end{align}
-            \end{subequations}
-            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
-
-            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
-            \begin{equation}    \label{EqVMRooUMXjig}
-                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
-            \end{equation}
-            
-        \item[Conclusion]
-
-            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
-            \begin{equation}
-                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
-            \end{equation}
-            c'est à dire
-            \begin{equation}
-                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
-            \end{equation}
-    \end{subproof}
-\end{proof}
-
-Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
-
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
-\section{Ellipsoïde de John-Loewer}
-%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
-
-Soit \( q\) une forme quadratique sur \( \eR^n\) ainsi que \( \mB\) une base orthonormée de \( \eR^n\) dans laquelle la matrice de  \( q\) est diagonale. Dans cette base, la forme \( q\) est donnée par la proposition \ref{PropFWYooQXfcVY} :
-\begin{equation}
-    q(x)=\sum_i\lambda_ix_i
-\end{equation}
-où les \( \lambda_i\) sont les valeurs propres de \( q\).
-
-Plus généralement nous notons \( mat_{\mB}(q)\)\nomenclature[A]{\( mat_{\mB}(q)\)}{matrice de \( q\) dans la base \( \mB\)} la matrice de \( q\) dans la base \( \mB\) de \( \eR^n\).
-
-\begin{proposition} \label{PropOXWooYrDKpw}
-    Soit \( \mB\) une base orthonormée de \( \eR^n\) et l'application\footnote{L'ensemble \( Q(E)\) est l'ensemble des formes quadratiques sur \( E\).}
-    \begin{equation}
-        \begin{aligned}
-            D\colon Q(\eR^n)&\to \eR \\
-            q&\mapsto \det\big( mat_{\mB}(q) \big) .
-        \end{aligned}
-    \end{equation}
-    Alors :
-    \begin{enumerate}
-        \item
-            La valeur et \( D\) ne dépend pas du choix de la base orthonormée \( \mB\).
-        \item
-            La fonction \( D\) est donnée par la formule \( D(q)=\prod_i\lambda_i\) où les \( \lambda_i\) sont les valeurs propres de \( q\).
-        \item
-            La fonction \( D\) est continue.
-    \end{enumerate}
-\end{proposition}
-
-\begin{proof}
-    Soit \( q\) une forme quadratique sur \( \eR^n\). Nous considérons \( \mB\) une base de diagonalisation de \( q\) :
-    \begin{equation}
-        q(x)=\sum_i\lambda_ix_i
-    \end{equation}
-    où les \( x_i\) sont les composantes de \( x\) dans la base \( \mB\). Par définition, la matrice \( mat_{\mB}(q)\) est la matrice diagonale contenant les valeurs propres de \( q\).
-
-    Nous considérons aussi \( \mB_1\), une autre base orthonormées de \( \eR^n\). Nous notons \( S=mat_{\mB_1}(q)\); étant symétrique, cette matrice se diagonalise par une matrice orthogonale : il existe \( P\in\gO(n,\eR)\) telle que
-    \begin{equation}
-        S=P mat_{\mB}(q)P^t;
-    \end{equation}
-    donc \( \det(S)=\det(PP^t)\det\big( \diag(\lambda_1,\ldots, \lambda_n) \big)=\lambda_1\ldots\lambda_n\). Ceci prouve en même temps que \( D\) ne dépend pas du choix de la base et que sa valeur est le produit des valeurs propres.
-
-    Passons à la continuité. L'application déterminant \( \det\colon S_n(\eR^n)\to \eR\) est continue car polynôme en les composantes. D'autre par l'application \( mat_{\mB}\colon Q(\eR^n)\to S_n(\eR)\) est continue par la proposition \ref{PropFSXooRUMzdb}. L'application  \( D\) étant la composée de deux applications continues, elle est continue.
-\end{proof}
-
-\begin{proposition}[Ellipsoïde de John-Loewner\cite{KXjFWKA}]   \label{PropJYVooRMaPok}
-    Soit \( K\) compact dans \( \eR^n\) et d'intérieur non vide. Il existe une unique ellipsoïde\footnote{Définition \ref{DefOEPooqfXsE}.} (pleine) de volume minimal contenant \( K\).
-\end{proposition}
-\index{déterminant!utilisation}
-\index{extrema!volume d'un ellipsoïde}
-\index{convexité!utilisation}
-\index{compacité!utilisation}
-
-\begin{proof}
-    Nous subdivisons la preuve en plusieurs parties.
-    \begin{subproof}
-        \item[À propos de volume d'un ellipsoïde]
-
-            Soit \( \ellE\) un ellipsoïde. La proposition \ref{PropWDRooQdJiIr} et son corollaire \ref{CorKGJooOmcBzh} nous indiquent que 
-            \begin{equation}
-                \ellE=\{ x\in \eR^n\tq q(x)\leq 1 \}
-            \end{equation}
-            pour une certaine forme quadratique strictement définie positive \( q\). De plus il existe une base orthonormée \( \mB=\{ e_1,\ldots, e_n \}\) de \( \eR^n\) telle que 
-            \begin{equation}    \label{EqELBooQLPQUj}
-                q(x)=\sum_{i=1}^na_ix_i^2
-            \end{equation}
-            où \( x_i=\langle e_i, x\rangle \) et les \( a_i\) sont tous strictement positifs. Nous nommons \( \ellE_q\) l'éllipsoïde associée à la forme quadratique \( q\) et \( V_q\) son volume que nous allons maintenant calculer\footnote{Le volume ne change pas si nous écrivons l'inégalité stricte au lieu de large dans le domaine d'intégration; nous le faisons pour avoir un domaine ouvert.} :
-            \begin{equation}
-                V_q=\int_{\sum_ia_ix_i^2<1}dx
-            \end{equation}
-            Cette intégrale est écrite de façon plus simple en utilisant le \( C^1\)-difféomorphisme
-            \begin{equation}
-                \begin{aligned}
-                    \varphi\colon \ellE_q&\to B(0,1) \\
-                    x&\mapsto \Big( x_1\sqrt{a_1},\ldots, x_n\sqrt{a_n} \Big). 
-                \end{aligned}
-            \end{equation}
-            Le fait que \( \varphi\) prenne bien ses valeurs dans \( B(0,1)\) est un simple calcul : si \( x\in\ellE_q\), alors
-            \begin{equation}
-                \sum_i\varphi(x)_i^2=\sum_ia_ix_i^2<1.
-            \end{equation}
-            Cela nous permet d'utiliser le théorème de changement de variables \ref{THOooUMIWooZUtUSg} :
-            \begin{equation}
-                V_q=\int_{\sum_ia_ix_i^2<1}dx=\frac{1}{ \sqrt{a_1\ldots a_n} }\int_{B(0,1)}dx.
-            \end{equation}
-            %TODO : le volume de la sphère dans \eR^n. Mettre alors une référence ici.
-            La dernière intégrale est le volume de la sphère unité dans \( \eR^n\); elle n'a pas d'importance ici et nous la notons \( V_0\). La proposition \ref{PropOXWooYrDKpw} nous permet d'écrire \(V_q\) sous la forme
-            \begin{equation}
-                V_q=\frac{ V_0 }{ \sqrt{D(q)} }.
-            \end{equation}
-            
-        \item[Existence de l'ellipsoïde]
-
-            Nous voulons trouver un ellipsoïde contenant \( K\) de volume minimal, c'est à dire une forme quadratique \( q\in Q^{++}(\eR^n)\) telle que
-            \begin{itemize}
-                \item \( D(q)\) soit maximal
-                \item \( q(x)\leq 1\) pour tout \( x\in K\).
-            \end{itemize}
-            Nous considérons l'ensemble des candidats semi-définis positifs.
-            \begin{equation}
-                A=\{ q\in Q^+\tq q(x)\leq 1\forall x\in K \}.
-            \end{equation}
-            Nous allons montrer que \( A\) est convexe, compact et non vide dans \( Q(\eR^n)\); il aura ainsi un maximum de la fonction continue \( D\) définie sur \( Q(\eR^n)\). Nous montrerons ensuite que le maximum est dans \( Q^{++}\). L'unicité sera prouvée à part.
-
-            \begin{subproof}
-            \item[Non vide]
-                L'ensemble \( K\) est compact et donc borné par \( M>0\). La forme quadratique \( q\colon x\mapsto \| x \|^2/M^2\) est dans \( A\) parce que si \( x\in K\) alors 
-                \begin{equation}
-                    q(x)=\frac{ \| x \|^2 }{ M^2 }\leq 1.
-                \end{equation}
-            \item[Convexe]
-                Soient \( q,q'\in A\) et \( \lambda\in\mathopen[ 0 , 1 \mathclose]\). Nous avons encore \( \lambda q+(1-\lambda)q'\in Q^+\) parce que 
-                \begin{equation}
-                    \lambda q(x)+(1-\lambda)q'(x)\geq 0
-                \end{equation}
-                dès que \( q(x)\geq 0\) et \( q'(x)\geq 0\).
-            D'autre part si \( x\in K\) nous avons
-            \begin{equation}
-                \lambda q(x)+(1-\lambda)q'(x)\leq \lambda+(1-\lambda)=1.
-            \end{equation}
-            Donc \( \lambda q+(1-\lambda)q'\in A\).
-
-        \item[Fermé]
-
-            Pour rappel, la topologie de \( Q(\eR^n)\) est celle de la norme \eqref{EqZYBooZysmVh}. Nous considérons une suite \( (q_n)\) dans \( A\) convergeant vers \( q\in Q(\eR^n)\) et nous allons prouver que \( q\in A\), de sorte que la caractérisation séquentielle de la fermeture (proposition \ref{PropLFBXIjt}) conclue que \( A\) est fermé. En nommant \( e_x\) le vecteur unitaire dans la direction \( x\) nous avons
-            \begin{equation}
-                \big| q(x) \big|=\big| \| x \|^2q(e_x) \big|\leq \| x \|^2N(q),
-            \end{equation}
-            de sorte que notre histoire de suite convergente  donne pour tout \( x\) :
-            \begin{equation}
-                \big| q_n(x)-q(x) \big|\leq \| x \|^2N(q_n-q)\to 0.
-            \end{equation}
-            Vu que \( q_n(x)\geq 0\) pour tout \( n\), nous devons aussi avoir \( q(x)\geq 0\) et donc \( q\in Q^+\) (semi-définie positive). De la même manière si \( x\in K\) alors \( q_n(x)\leq 1\) pour tout \( n\) et donc \( q(x)\leq 1\). Par conséquent \( q\in A\) et \( A\) est fermé.
-
-        \item[Borné]
-
-            La partie \( K\) de \( \eR^n\) est borné et d'intérieur non vide, donc il existe \( a\in K\) et \( r>0\) tel que \( \overline{ B(a,r) }\subset K\). Si par ailleurs \( q\in A\) et \( x\in\overline{ B(0,r) }\) nous avons \( a+x\in K\) et donc \( q(a+x)\leq 1\). De plus \( q(-a)=q(a)\leq 1\), donc
-            \begin{equation}
-                \sqrt{q(x)}=\sqrt{q\big( x+a-a \big)}\leq \sqrt{q(x+a)}+\sqrt{q(-a)}\leq 2
-            \end{equation}
-            par l'inégalité de Minkowski \ref{PropACHooLtsMUL}. Cela prouve que si \( x\in\overline{ B(0,r) }\) alors \( q(x)\leq 4\). Si par contre \( x\in\overline{ B(0,1) }\) alors \( rx\in\overline{ B(0,r) } \) et 
-            \begin{equation}
-                0\leq q(x)=\frac{1}{ r^2 }q(rx)\leq \frac{ 4 }{ r^2 },
-            \end{equation}
-            ce qui prouve que \( N(q)\leq \frac{ 4 }{ r^2 }\) et que \( A\) est borné.
-
-
-            \end{subproof}
-
-            L'ensemble \( A\) est compact parce que fermé et borné, théorème de Borel-Lebesgue \ref{ThoXTEooxFmdI}. L'application continue \( D\colon Q(\eR^n)\to \eR\) de la proposition \ref{PropOXWooYrDKpw} admet donc un maximum sur le compact \( A\). Soit \( q_0\) ce maximum.
-
-            Nous montrons que \( q_0\in Q^{++}(\eR^d)\). Nous savons que l'application \( f\colon x\mapsto \frac{ \| x \|^2 }{ M^2 }\) est dans \( A\) et que \( D(f)>0\). Vu que \( q_0\) est maximale pour \( D\), nous avons
-            \begin{equation}
-                D(q_0)\geq D(f)>0.
-            \end{equation}
-            Donc \( q_0\in Q^{++}\).
-
-        \item[Unicité]
-
-            S'il existe une autre ellipsoïde de même volume que celle associée à la forme quadratique \( q_0\), nous avons une forme quadratique \( q\in Q^{++}\) telle que \( q(x)\leq 1\) pour tout \( x\in K\). C'est à dire que nous avons \( q_0,q\in A\) tels que \( D(q_0)=D(q)\).
-
-            Nous considérons la base canonique \( \mB_c\) de \( \eR^n\) et nous posons \( S=mat_{\mB_c}(q)\), \( S_0=mat_{\mB_c}(q_0)\). Étant donné que \( A\) est convexe, \( (q_0+q)/2\in A\) et nous allons prouver que cet élément de \( A\) contredit la maximalité de \( q_0\). En effet
-            \begin{equation}
-                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)
-            \end{equation}
-            Nous allons utiliser le lemme \ref{LemXOUooQsigHs} qui dit que le logarithme est log-concave sous la forme de l'équation \eqref{EqSPKooHFZvmB} avec \( \alpha=\beta=\frac{ 1 }{2}\) :
-            \begin{equation}    \label{eqBHJooYEUDPC}
-                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)>\sqrt{\det(S)}\sqrt{\det(S_0)}=\det(S_0)=D(q_0).
-            \end{equation}
-            Nous avons utilisé le fait que \( D(q_0)=D(q)\) qui signifie que \( \det(S_0)=\det(S)\). L'inéquation \eqref{eqBHJooYEUDPC} contredit la maximalité de \( D(q_0)\) et donne donc l'unicité.
-    \end{subproof}
-\end{proof}
-
diff --git a/tex/frido/79_inversion_locale.tex b/tex/frido/79_inversion_locale.tex
index 1616656eb..49f4c39f6 100644
--- a/tex/frido/79_inversion_locale.tex
+++ b/tex/frido/79_inversion_locale.tex
@@ -3,6 +3,425 @@
 %   Laurent Claessens
 % See the file fdl-1.3.txt for copying conditions.
 
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Algorithme du gradient à pas optimal}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Une idée pour trouver un minimum à une fonction est de prendre un point \( p\) au hasard, calculer le gradient \(\nabla f(p) \) et suivre la direction \(-\nabla f(p)\) tant que ça descend. Une fois qu'on est «dans le creux», recalculer le gradient et continuer ainsi.
+
+Nous allons détailler cet algorithme dans un cas très particulier d'une matrice \( A\) symétrique et strictement définie positive. 
+\begin{itemize}
+    \item Dans la proposition \ref{PROPooYRLDooTwzfWU} nous montrons que résoudre le système linéaire \( Ax=-b\) est équivalent à minimiser une certaine fonction.
+    \item La proposition \ref{PropSOOooGoMOxG} donnera une méthode itérative pour trouver ce minimum.
+\end{itemize}
+
+\begin{definition}  \label{DefQXPooYSygGP}
+    Si \( X\) est un espace vectoriel normé et \( f\colon X\to \eR\cup\{ \pm\infty \}\) nous disons que \( f\) est \defe{coercive}{coercive} sur le domaine non borné \( P\) de \( X\) si pour tout \( M\in \eR\), l'ensemble
+    \begin{equation}
+        \{ x\in P\tq f(x)\leq M \}
+    \end{equation}
+    est borné.
+\end{definition}
+En langage imagé la coercivité de \( f\) s'exprime par la limite
+\begin{equation}
+    \lim_{\substack{\| x \|\to \infty\\x\in P}}f(x)=+\infty.
+\end{equation}
+
+
+Nous rappelons que \( S^{++}(n,\eR)\) est l'ensemble des matrice symétriques strictement définies positives définies en \ref{NORMooAJLHooQhwpvr}.
+\begin{proposition}     \label{PROPooYRLDooTwzfWU}
+    Soit \( A\in S^{++}(n,\eR)\) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Alors :
+    \begin{enumerate}
+        \item
+            Il existe un unique \( \bar x\in \eR^n\) tel que \( A\bar x=-b\).
+        \item
+            Il existe un unique \( x^*\in \eR^n\) minimisant \( f\).
+        \item
+            Ils sont égaux : \( \bar x=x^*\).
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+
+    Une matrice symétrique strictement définie positive est inversible, entre autres parce qu'elle se diagonalise par des matrices orthogonales (qui sont inversibles) et que la matrice diagonalisée est de déterminant non nul : tous les éléments diagonaux sont strictement positifs. Voir le théorème spectral symétrique \ref{ThoeTMXla}.
+
+    D'où l'unicité du \( \bar x\) résolvant le système \( Ax=-b\) pour n'importe quel \( b\).
+
+    \begin{subproof}
+    \item[\( f\) est strictement convexe]
+        
+        Nous utilisons la proposition \ref{CORooMBQMooWBAIIH}. La fonction \( f\) s'écrit
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_kb_kx_k.
+    \end{equation}
+    Elle est de classe \( C^2\) sans problèmes, et il est vite vu que \( \frac{ \partial^2f }{ \partial x_i\partial x_j }=A_{ij}\), c'est à dire que \( A\) est la matrice hessienne de \( f\). Cette matrice étant définie positive par hypothèse, la fonction \( f\) est convexe.
+
+\item[\( f\) est coercive]
+    Montrons à présent que \( f\) est coercive. Nous avons :
+    \begin{subequations}
+        \begin{align}
+            | f(x) |&=\big| \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle  \big|\\
+            &\geq\frac{ 1 }{2}| \langle Ax, x\rangle  |-| \langle b, x\rangle  |\\
+            &\geq\frac{ 1 }{2}\lambda_{max}\| x \|^2-\| b \|\| x \|
+        \end{align}
+    \end{subequations}
+    Pour la dernière ligne nous avons nommé \( \lambda_{max}\) la plus grande valeur propre de \( A\) et utilisé Cauchy-Schwarz pour le second terme. Nous avons donc bien \( | f(x) |\to \infty\) lorsque \( \| x \|\to\infty\) et la fonction \( f\) est coercive.
+    \end{subproof}
+
+    Soit \( M\) une valeur atteinte par \( f\). L'ensemble
+    \begin{equation}
+        \{ x\in \eR^n\tq f(x)\leq M \}
+    \end{equation}
+    est fermé (parce que \( f\) est continue) et borné parce que \( f\) est coercive. Cela est donc compact\footnote{Théorème \ref{ThoXTEooxFmdI}} et \( f\) atteint un minimum qui sera forcément dedans. Cela est pour l'existence d'un minimum.
+
+    Pour l'unicité du minimum nous invoquons la convexité : si \( \bar x_1\) et \( \bar x_2\) sont deux points réalisant le minimum de \( f\), alors
+    \begin{equation}
+        f\left( \frac{ \bar x_1+\bar x_2 }{2} \right)<\frac{ 1 }{2}f(\bar x_1)+\frac{ 1 }{2}f(\bar x_2)=f(\bar x_1),
+    \end{equation}
+    ce qui contredit la minimalité de \( f(\bar x_1)\).
+
+    Nous devons maintenant prouver que \( \bar x\) vérifie l'équation \( A\bar x=-b\). Vu que \( \bar x\) est minimum local de \( f\) qui est une fonction de classe \( C^2\), le théorème des minima locaux \ref{PropUQRooPgJsuz} nous indique que \( \bar x\) est solution de \( \nabla f(x)=0\). Calculons un peu cela avec la formule
+    \begin{equation}
+        df_x(u)=\Dsdd{ f(x+tu) }{t}{0}=\frac{ 1 }{2}\big( \langle Ax, u\rangle +\langle Au, x\rangle  \big)+\langle b, u\rangle =\langle Ax, u\rangle +\langle b, u\rangle =\langle Ax+b, u\rangle .
+    \end{equation}
+    Donc demander \( df_x(u)=0\) pour tout \( u\) demande \( Ax+b=0\).
+\end{proof}
+
+\begin{proposition}[Gradient à pas optimal] \label{PropSOOooGoMOxG}
+    Soit \( A\in S^{++}(n,\eR)\) (\( A\) est une matrice symétrique strictement définie positive) et \( b\in \eR^n\). Nous considérons l'application
+    \begin{equation}
+        \begin{aligned}
+            f\colon \eR^n&\to \eR \\
+            x&\mapsto \frac{ 1 }{2}\langle Ax, x\rangle +\langle b, x\rangle . 
+        \end{aligned}
+    \end{equation}
+    Soit \( x_0\in \eR^n\). Nous définissons la suite \( (x_k)\) par
+    \begin{equation}
+        x_{k+1}=x_k+t_kd_k
+    \end{equation}
+    où
+    \begin{itemize}
+        \item 
+    \( d_k=-(\nabla f)(x_k)\) 
+\item
+    \( t_k\) est la valeur minimisant la fonction \( t\mapsto f(x_k+td_k)\) sur \( \eR\).
+    \end{itemize}
+
+    Alors pour tout \( k\geq 0\) nous avons
+    \begin{equation}
+        \| x_k-\bar x \|\leq K \left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^k
+    \end{equation}
+    où \( c_2(A)=\frac{ \lambda_{max} }{ \lambda_{min} }\) est le rapport ente la plus grande et la plus petite valeur propre\quext{Cela est certainement très lié au conditionnement de la matrice \( A\), voir la proposition \ref{PROPooNUAUooIbVgcN}.} de la matrice \( A\) et \( \bar x\) est l'unique élément de \( \eR^n\) à minimiser \( f\).
+\end{proposition}
+
+\begin{proof}
+    Décomposition en plusieurs points.
+    \begin{subproof}
+    \item[Existence de \( \bar x\)]
+        Le fait que \( \bar x\) existe et soit unique est la proposition \ref{PROPooYRLDooTwzfWU}.
+    \item[Si \( (\nabla f)(x_k)=0\)]
+    D'abord si \( \nabla f(x_k)=0\), c'est que \( x_{k+1}=x_k\) et l'algorithme est terminé : la suite est stationnaire. Pour dire que c'est gagné, nous devons prouver que \( x_k=\bar x\). Pour cela nous écrivons (à partir de maintenant «\( x_k\)» est la \( k\)\ieme composante de \( x\) qui est une variable, et non le \( x_k\) de la suite)
+    \begin{equation}
+        f(x)=\frac{ 1 }{2}\sum_{kl}A_{kl}x_lx_k+\sum_{k}b_kx_k
+    \end{equation}
+    et nous calculons \( \frac{ \partial f }{ \partial x_i }(a)\) en tenant compte du fait que \( \frac{ \partial x_k }{ \partial x_i }=\delta_{ki}\). Le résultat est que \( (\partial_if)(a)=(Ax+b)_i\) et donc que
+    \begin{equation}
+        (\nabla f)(a)=Aa+b.
+    \end{equation}
+    Vu que \( A\) est inversible (symétrique définie positive), il existe un unique \( a\in \eR^n\) qui vérifie cette relation. Par la proposition \ref{PROPooYRLDooTwzfWU}, cet élément est le minimum \( \bar x\).
+
+    Cela pour dire que si \( a\in \eR^n\) vérifie \( (\nabla f)(a)=0\) alors \( a=\bar x\). Nous supposons donc à partir de maintenant que \( \nabla f(x_k)\neq 0\) pour tout \( k\).
+        \item[\( t_k\) est bien défini]
+
+            Pour \( t\in \eR\) nous avons
+            \begin{equation}    \label{EqKEHooYaazQi}
+                f(x_k+td_k)=f(x_k)+\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle +t\langle \underbrace{Ax_k+b}_{=-d_k}, d_k\rangle=\frac{ 1 }{2}t^2\langle Ad_k, d_k\rangle -t_k\| d_k \|^2 +f(x_k).
+            \end{equation}
+            qui est un polynôme du second degré en \( t\). Le coefficient de \( t^2\) est \( \frac{ 1 }{2}\langle Ad_k, d_k\rangle >0\) parce que \( d_k\neq 0\) et \( A\) est strictement définie positive. Par conséquent la fonction \( t\mapsto f(x_k+td_k)\) admet bien un unique minimum. Nous pouvons même calculer \( t_k\) parce que l'on connaît pas cœur le sommet d'une parabole :
+            \begin{equation}    \label{EqVWJooWmDSER}
+                t_k=-\frac{ \langle Ax_k+b, d_k\rangle  }{ \langle Ad_k, d_k\rangle  }=\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }
+            \end{equation}
+            parce que \( d_k=-\nabla f(x_k)=-(Ax_k+b)\).
+
+        \item[La valeur de \( d_{k+1}\)]
+
+            Par définition, \( d_{k+1}=-\nabla f(x_{k+1})=-(Ax_{k+1}+b)\). Mais \( x_{k+1}=x_k+t_kd_k\), donc
+            \begin{equation}
+                d_{k+1}=-Ax_k-t_kAd_k-b=d_k-t_kAd_k
+            \end{equation}
+            parce que \( -Ax_k-b=d_k\).
+            
+            Par ailleurs, \( \langle d_{k+1}, d_k\rangle =0\) parce que
+            \begin{equation}
+                \langle d_{k+1}, d_k\rangle =\langle d_k, d_k\rangle -t_k\langle d_k, Ad_k\rangle =\| d_k \|^2-\frac{ \| d_k \|^2 }{ \langle Ad_k, d_k\rangle  }\langle d_k, Ad_k\rangle =0
+            \end{equation}
+            où nous avons utilisé la valeur \eqref{EqVWJooWmDSER} de \( t_k\).
+        
+        \item[Calcul de \( f(x_{k+1})\)] 
+
+            Nous repartons de \eqref{EqKEHooYaazQi} où nous substituons la valeur \eqref{EqVWJooWmDSER} de \( t_k\) :
+            \begin{equation}
+                f(x_{k+1})=f(x_k)+\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }.
+            \end{equation}
+            
+        \item[Encore du calcul \ldots]
+
+            Vu que le produit \( \langle Ad_k, d_k\rangle \) arrive tout le temps, nous allons étudier \( \langle A^{-1}d_k, d_k\rangle \). Le truc malin est d'essayer d'exprimer ça en termes de \( \bar x\) et \( \bar f=f(\bar x)\). Pour cela nous calculons \( f(\bar x)\) :
+            \begin{equation}
+                \bar f=f(\bar x)=f(-A^{-1} b)=-\frac{ 1 }{2}\langle b, A^{-1}b\rangle .
+            \end{equation}
+            Ayant cela en tête nous pouvons calculer :
+            \begin{subequations}
+                \begin{align}
+                    \langle A^{-1}d_k, d_k\rangle &=\langle A^{-1}(Ax_k+b), Ax_k+b\rangle \\
+                    &=\langle x_k, Ax_k\rangle +\langle A^{-1}b, Ax_k\rangle +\langle b, x_k\rangle+\underbrace{\langle A^{-1}b, b\rangle}_{-2\bar f} \\
+                    &=\langle x_k, Ax_k\rangle +2\langle x_k, b\rangle  -2\bar f \label{subeqVIIooVzZlRc}\\
+                    &=2\big( f(x_k)-\bar f \big)
+                \end{align}
+            \end{subequations}
+            où nous avons utilisé le fait que \( \langle x, Ay\rangle =\langle Ax, y\rangle \) parce que \( A\) est symétrique.
+
+        \item[Erreur sur la valeur du minimum]
+
+            Nous voulons à présent estimer la différence \( f(x_{k+1})-\bar f\). Pour cela nous mettons en facteur \( f(x_k)-\bar f\) dans \( f(x_{k+1}-\bar f)\); et d'ailleurs c'est pour cela que nous avons calculé \( \langle A^{-1}d_k, d_k\rangle \) : parce que ça fait intervenir \( f(x_k)-\bar f\).
+            \begin{subequations}
+                \begin{align}
+                    f(x_{k+1})-\bar f&=f(x_k)-\frac{ 1 }{2}\frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle  }-\bar f\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ 1 }{2}\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \big( f(x_k)-\bar f \big) } \right)\\
+                    &=\big( f(x_k)-\bar f \big)\left( 1-\frac{ \| d_k \|^{4} }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  } \right).\label{subeqGFDooRAwAJk}
+                \end{align}
+            \end{subequations}
+            Nous traitons le dénominateur à l'aide de l'inégalité de Kantorovitch \ref{PropMNUooFbYkug}. Nous avons
+            \begin{equation}
+                \frac{ \| d_k \|^4 }{ \langle Ad_k, d_k\rangle \langle A^{-1}d_k, d_k\rangle  }\geq \frac{ \| d_k \|^4 }{ \frac{1}{ 4 }\left( \sqrt{c_2(A)}+\frac{1}{ \sqrt{c_2(A)} } \right)^2\| d_k \|^4 }=\frac{ 4c_2(A) }{ (c_2(A)+1)^2 }.
+            \end{equation}
+            Mettre cela dans \eqref{subeqGFDooRAwAJk} est un calcul d'addition de fractions :
+            \begin{equation}
+                f(x_{k+1})-\bar f\leq \big( f(x_k)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^2.
+            \end{equation}
+            Par récurrence nous avons alors
+            \begin{equation}    \label{eqANKooNPfCFj}
+                f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}.
+            \end{equation}
+            Notons qu'il n'y a pas de valeurs absolues parce que \( \bar f\) étant le minimum de \( f\), les deux côtés de l'inégalité sont automatiquement positifs.
+
+        \item[Erreur sur la position du minimum]
+
+            Nous voulons à présent étudier la norme de \( x_k-\bar x\). Pour cela nous l'écrivons directement avec la définition de \( f\) en nous souvenant que \( b=-A\bar x\) :
+            \begin{subequations}
+                \begin{align}
+                    f(x_k)-\bar f&=\frac{ 1 }{2}\langle Ax_k, x_k\rangle +\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle +\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\langle Ax_k, x_k\rangle -\frac{ 1 }{2}\langle A\bar x, x_k\rangle-\frac{ 1 }{2}\langle A\bar x, x_k\rangle +\frac{ 1 }{2}\langle A\bar x, \bar x\rangle \\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), x_k\rangle +\langle A\bar x, \bar x-x_k\rangle  \Big)\\
+                    &=\frac{ 1 }{2}\Big( \langle A(x_k-\bar x), (x_k-\bar x)\rangle  \Big)
+                \end{align}
+            \end{subequations}
+            où à la dernière ligne nous avons fait \( \langle A\bar x, \bar x-x_k\rangle =\langle \bar x, A(\bar x-x_k)\rangle \) en vertu de la symétrie de \( A\).
+
+            Les produits de la forme \( \langle Ay, y\rangle \) sont majorés par \( \lambda_{min}\| y \|^2\) parce que \( \lambda_{min}\) est la plus grande valeur propre de \( A\). Dans notre cas,
+            \begin{equation}    \label{EqVMRooUMXjig}
+                f(x_k)-\bar f\geq \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2
+            \end{equation}
+            
+        \item[Conclusion]
+
+            En combinant les inéquations \eqref{EqVMRooUMXjig} et \eqref{eqANKooNPfCFj} nous trouvons
+            \begin{equation}
+                \frac{ 1 }{2}\lambda_{min}\| x_k-\bar x \|^2\leq f(x_k)-\bar f\leq \big( f(x_0)-\bar f \big)\left( \frac{ c_2(A)-1 }{ c_2(A)+1 } \right)^{2k}, 
+            \end{equation}
+            c'est à dire
+            \begin{equation}
+                \| x_k-\bar x \|\leq \sqrt{\frac{ 2\big( f(x_0)-\bar f \big) }{ \lambda_{min} +1}}^{2k}.
+            \end{equation}
+    \end{subproof}
+\end{proof}
+
+Notons que lorsque \( c_2(A)\) est proche de \( 1\) la méthode converge rapidement. Par contre si \( c_2(A)\) est proche de zéro, la méthode converge lentement.
+
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
+\section{Ellipsoïde de John-Loewer}
+%+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+
+Soit \( q\) une forme quadratique sur \( \eR^n\) ainsi que \( \mB\) une base orthonormée de \( \eR^n\) dans laquelle la matrice de  \( q\) est diagonale. Dans cette base, la forme \( q\) est donnée par la proposition \ref{PropFWYooQXfcVY} :
+\begin{equation}
+    q(x)=\sum_i\lambda_ix_i
+\end{equation}
+où les \( \lambda_i\) sont les valeurs propres de \( q\).
+
+Plus généralement nous notons \( mat_{\mB}(q)\)\nomenclature[A]{\( mat_{\mB}(q)\)}{matrice de \( q\) dans la base \( \mB\)} la matrice de \( q\) dans la base \( \mB\) de \( \eR^n\).
+
+\begin{proposition} \label{PropOXWooYrDKpw}
+    Soit \( \mB\) une base orthonormée de \( \eR^n\) et l'application\footnote{L'ensemble \( Q(E)\) est l'ensemble des formes quadratiques sur \( E\).}
+    \begin{equation}
+        \begin{aligned}
+            D\colon Q(\eR^n)&\to \eR \\
+            q&\mapsto \det\big( mat_{\mB}(q) \big) .
+        \end{aligned}
+    \end{equation}
+    Alors :
+    \begin{enumerate}
+        \item
+            La valeur et \( D\) ne dépend pas du choix de la base orthonormée \( \mB\).
+        \item
+            La fonction \( D\) est donnée par la formule \( D(q)=\prod_i\lambda_i\) où les \( \lambda_i\) sont les valeurs propres de \( q\).
+        \item
+            La fonction \( D\) est continue.
+    \end{enumerate}
+\end{proposition}
+
+\begin{proof}
+    Soit \( q\) une forme quadratique sur \( \eR^n\). Nous considérons \( \mB\) une base de diagonalisation de \( q\) :
+    \begin{equation}
+        q(x)=\sum_i\lambda_ix_i
+    \end{equation}
+    où les \( x_i\) sont les composantes de \( x\) dans la base \( \mB\). Par définition, la matrice \( mat_{\mB}(q)\) est la matrice diagonale contenant les valeurs propres de \( q\).
+
+    Nous considérons aussi \( \mB_1\), une autre base orthonormées de \( \eR^n\). Nous notons \( S=mat_{\mB_1}(q)\); étant symétrique, cette matrice se diagonalise par une matrice orthogonale : il existe \( P\in\gO(n,\eR)\) telle que
+    \begin{equation}
+        S=P mat_{\mB}(q)P^t;
+    \end{equation}
+    donc \( \det(S)=\det(PP^t)\det\big( \diag(\lambda_1,\ldots, \lambda_n) \big)=\lambda_1\ldots\lambda_n\). Ceci prouve en même temps que \( D\) ne dépend pas du choix de la base et que sa valeur est le produit des valeurs propres.
+
+    Passons à la continuité. L'application déterminant \( \det\colon S_n(\eR^n)\to \eR\) est continue car polynôme en les composantes. D'autre par l'application \( mat_{\mB}\colon Q(\eR^n)\to S_n(\eR)\) est continue par la proposition \ref{PropFSXooRUMzdb}. L'application  \( D\) étant la composée de deux applications continues, elle est continue.
+\end{proof}
+
+\begin{proposition}[Ellipsoïde de John-Loewner\cite{KXjFWKA}]   \label{PropJYVooRMaPok}
+    Soit \( K\) compact dans \( \eR^n\) et d'intérieur non vide. Il existe une unique ellipsoïde\footnote{Définition \ref{DefOEPooqfXsE}.} (pleine) de volume minimal contenant \( K\).
+\end{proposition}
+\index{déterminant!utilisation}
+\index{extrema!volume d'un ellipsoïde}
+\index{convexité!utilisation}
+\index{compacité!utilisation}
+
+\begin{proof}
+    Nous subdivisons la preuve en plusieurs parties.
+    \begin{subproof}
+        \item[À propos de volume d'un ellipsoïde]
+
+            Soit \( \ellE\) un ellipsoïde. La proposition \ref{PropWDRooQdJiIr} et son corollaire \ref{CorKGJooOmcBzh} nous indiquent que 
+            \begin{equation}
+                \ellE=\{ x\in \eR^n\tq q(x)\leq 1 \}
+            \end{equation}
+            pour une certaine forme quadratique strictement définie positive \( q\). De plus il existe une base orthonormée \( \mB=\{ e_1,\ldots, e_n \}\) de \( \eR^n\) telle que 
+            \begin{equation}    \label{EqELBooQLPQUj}
+                q(x)=\sum_{i=1}^na_ix_i^2
+            \end{equation}
+            où \( x_i=\langle e_i, x\rangle \) et les \( a_i\) sont tous strictement positifs. Nous nommons \( \ellE_q\) l'éllipsoïde associée à la forme quadratique \( q\) et \( V_q\) son volume que nous allons maintenant calculer\footnote{Le volume ne change pas si nous écrivons l'inégalité stricte au lieu de large dans le domaine d'intégration; nous le faisons pour avoir un domaine ouvert.} :
+            \begin{equation}
+                V_q=\int_{\sum_ia_ix_i^2<1}dx
+            \end{equation}
+            Cette intégrale est écrite de façon plus simple en utilisant le \( C^1\)-difféomorphisme
+            \begin{equation}
+                \begin{aligned}
+                    \varphi\colon \ellE_q&\to B(0,1) \\
+                    x&\mapsto \Big( x_1\sqrt{a_1},\ldots, x_n\sqrt{a_n} \Big). 
+                \end{aligned}
+            \end{equation}
+            Le fait que \( \varphi\) prenne bien ses valeurs dans \( B(0,1)\) est un simple calcul : si \( x\in\ellE_q\), alors
+            \begin{equation}
+                \sum_i\varphi(x)_i^2=\sum_ia_ix_i^2<1.
+            \end{equation}
+            Cela nous permet d'utiliser le théorème de changement de variables \ref{THOooUMIWooZUtUSg} :
+            \begin{equation}
+                V_q=\int_{\sum_ia_ix_i^2<1}dx=\frac{1}{ \sqrt{a_1\ldots a_n} }\int_{B(0,1)}dx.
+            \end{equation}
+            %TODO : le volume de la sphère dans \eR^n. Mettre alors une référence ici.
+            La dernière intégrale est le volume de la sphère unité dans \( \eR^n\); elle n'a pas d'importance ici et nous la notons \( V_0\). La proposition \ref{PropOXWooYrDKpw} nous permet d'écrire \(V_q\) sous la forme
+            \begin{equation}
+                V_q=\frac{ V_0 }{ \sqrt{D(q)} }.
+            \end{equation}
+            
+        \item[Existence de l'ellipsoïde]
+
+            Nous voulons trouver un ellipsoïde contenant \( K\) de volume minimal, c'est à dire une forme quadratique \( q\in Q^{++}(\eR^n)\) telle que
+            \begin{itemize}
+                \item \( D(q)\) soit maximal
+                \item \( q(x)\leq 1\) pour tout \( x\in K\).
+            \end{itemize}
+            Nous considérons l'ensemble des candidats semi-définis positifs.
+            \begin{equation}
+                A=\{ q\in Q^+\tq q(x)\leq 1\forall x\in K \}.
+            \end{equation}
+            Nous allons montrer que \( A\) est convexe, compact et non vide dans \( Q(\eR^n)\); il aura ainsi un maximum de la fonction continue \( D\) définie sur \( Q(\eR^n)\). Nous montrerons ensuite que le maximum est dans \( Q^{++}\). L'unicité sera prouvée à part.
+
+            \begin{subproof}
+            \item[Non vide]
+                L'ensemble \( K\) est compact et donc borné par \( M>0\). La forme quadratique \( q\colon x\mapsto \| x \|^2/M^2\) est dans \( A\) parce que si \( x\in K\) alors 
+                \begin{equation}
+                    q(x)=\frac{ \| x \|^2 }{ M^2 }\leq 1.
+                \end{equation}
+            \item[Convexe]
+                Soient \( q,q'\in A\) et \( \lambda\in\mathopen[ 0 , 1 \mathclose]\). Nous avons encore \( \lambda q+(1-\lambda)q'\in Q^+\) parce que 
+                \begin{equation}
+                    \lambda q(x)+(1-\lambda)q'(x)\geq 0
+                \end{equation}
+                dès que \( q(x)\geq 0\) et \( q'(x)\geq 0\).
+            D'autre part si \( x\in K\) nous avons
+            \begin{equation}
+                \lambda q(x)+(1-\lambda)q'(x)\leq \lambda+(1-\lambda)=1.
+            \end{equation}
+            Donc \( \lambda q+(1-\lambda)q'\in A\).
+
+        \item[Fermé]
+
+            Pour rappel, la topologie de \( Q(\eR^n)\) est celle de la norme \eqref{EqZYBooZysmVh}. Nous considérons une suite \( (q_n)\) dans \( A\) convergeant vers \( q\in Q(\eR^n)\) et nous allons prouver que \( q\in A\), de sorte que la caractérisation séquentielle de la fermeture (proposition \ref{PropLFBXIjt}) conclue que \( A\) est fermé. En nommant \( e_x\) le vecteur unitaire dans la direction \( x\) nous avons
+            \begin{equation}
+                \big| q(x) \big|=\big| \| x \|^2q(e_x) \big|\leq \| x \|^2N(q),
+            \end{equation}
+            de sorte que notre histoire de suite convergente  donne pour tout \( x\) :
+            \begin{equation}
+                \big| q_n(x)-q(x) \big|\leq \| x \|^2N(q_n-q)\to 0.
+            \end{equation}
+            Vu que \( q_n(x)\geq 0\) pour tout \( n\), nous devons aussi avoir \( q(x)\geq 0\) et donc \( q\in Q^+\) (semi-définie positive). De la même manière si \( x\in K\) alors \( q_n(x)\leq 1\) pour tout \( n\) et donc \( q(x)\leq 1\). Par conséquent \( q\in A\) et \( A\) est fermé.
+
+        \item[Borné]
+
+            La partie \( K\) de \( \eR^n\) est borné et d'intérieur non vide, donc il existe \( a\in K\) et \( r>0\) tel que \( \overline{ B(a,r) }\subset K\). Si par ailleurs \( q\in A\) et \( x\in\overline{ B(0,r) }\) nous avons \( a+x\in K\) et donc \( q(a+x)\leq 1\). De plus \( q(-a)=q(a)\leq 1\), donc
+            \begin{equation}
+                \sqrt{q(x)}=\sqrt{q\big( x+a-a \big)}\leq \sqrt{q(x+a)}+\sqrt{q(-a)}\leq 2
+            \end{equation}
+            par l'inégalité de Minkowski \ref{PropACHooLtsMUL}. Cela prouve que si \( x\in\overline{ B(0,r) }\) alors \( q(x)\leq 4\). Si par contre \( x\in\overline{ B(0,1) }\) alors \( rx\in\overline{ B(0,r) } \) et 
+            \begin{equation}
+                0\leq q(x)=\frac{1}{ r^2 }q(rx)\leq \frac{ 4 }{ r^2 },
+            \end{equation}
+            ce qui prouve que \( N(q)\leq \frac{ 4 }{ r^2 }\) et que \( A\) est borné.
+
+
+            \end{subproof}
+
+            L'ensemble \( A\) est compact parce que fermé et borné, théorème de Borel-Lebesgue \ref{ThoXTEooxFmdI}. L'application continue \( D\colon Q(\eR^n)\to \eR\) de la proposition \ref{PropOXWooYrDKpw} admet donc un maximum sur le compact \( A\). Soit \( q_0\) ce maximum.
+
+            Nous montrons que \( q_0\in Q^{++}(\eR^d)\). Nous savons que l'application \( f\colon x\mapsto \frac{ \| x \|^2 }{ M^2 }\) est dans \( A\) et que \( D(f)>0\). Vu que \( q_0\) est maximale pour \( D\), nous avons
+            \begin{equation}
+                D(q_0)\geq D(f)>0.
+            \end{equation}
+            Donc \( q_0\in Q^{++}\).
+
+        \item[Unicité]
+
+            S'il existe une autre ellipsoïde de même volume que celle associée à la forme quadratique \( q_0\), nous avons une forme quadratique \( q\in Q^{++}\) telle que \( q(x)\leq 1\) pour tout \( x\in K\). C'est à dire que nous avons \( q_0,q\in A\) tels que \( D(q_0)=D(q)\).
+
+            Nous considérons la base canonique \( \mB_c\) de \( \eR^n\) et nous posons \( S=mat_{\mB_c}(q)\), \( S_0=mat_{\mB_c}(q_0)\). Étant donné que \( A\) est convexe, \( (q_0+q)/2\in A\) et nous allons prouver que cet élément de \( A\) contredit la maximalité de \( q_0\). En effet
+            \begin{equation}
+                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)
+            \end{equation}
+            Nous allons utiliser le lemme \ref{LemXOUooQsigHs} qui dit que le logarithme est log-concave sous la forme de l'équation \eqref{EqSPKooHFZvmB} avec \( \alpha=\beta=\frac{ 1 }{2}\) :
+            \begin{equation}    \label{eqBHJooYEUDPC}
+                D\left( \frac{ q+q_0 }{ 2 }\right)=\det\left( \frac{ S+S_0 }{2} \right)>\sqrt{\det(S)}\sqrt{\det(S_0)}=\det(S_0)=D(q_0).
+            \end{equation}
+            Nous avons utilisé le fait que \( D(q_0)=D(q)\) qui signifie que \( \det(S_0)=\det(S)\). L'inéquation \eqref{eqBHJooYEUDPC} contredit la maximalité de \( D(q_0)\) et donne donc l'unicité.
+    \end{subproof}
+\end{proof}
+
+
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 
 \section{Formes quadratiques, signature, et lemme de Morse}
 %+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

From abb7d9763b79b60061202f69605e9b1bf04cbce6 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 27 Jun 2017 08:00:36 +0200
Subject: [PATCH 62/64] =?UTF-8?q?(organisation)=20Renomme=20les=20fichiers?=
 =?UTF-8?q?=20du=20chapitre=20"Int=C3=A9garion"=20pour=20plus=20de=20coh?=
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---
 mazhe.tex                                        | 16 ++++++++--------
 ..._series_fonctions.tex => 155_Integration.tex} |  0
 ..._series_fonctions.tex => 169_Integration.tex} |  0
 .../{2_calcul_integral.tex => 2_Integration.tex} |  0
 ...tegrales_multiples.tex => 38_Integration.tex} |  0
 ...s_series_fonctions.tex => 70_Integration.tex} |  0
 ...s_series_fonctions.tex => 72_Integration.tex} |  0
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 rename tex/frido/{169_suites_series_fonctions.tex => 169_Integration.tex} (100%)
 rename tex/frido/{2_calcul_integral.tex => 2_Integration.tex} (100%)
 rename tex/frido/{38_Chap_integrales_multiples.tex => 38_Integration.tex} (100%)
 rename tex/frido/{70_suites_series_fonctions.tex => 70_Integration.tex} (100%)
 rename tex/frido/{72_suites_series_fonctions.tex => 72_Integration.tex} (100%)
 rename tex/frido/{73_Chap_integrales_multiples.tex => 73_Integration.tex} (100%)

diff --git a/mazhe.tex b/mazhe.tex
index d0831f205..f3667e754 100644
--- a/mazhe.tex
+++ b/mazhe.tex
@@ -201,14 +201,14 @@ \chapter{Retour sur les groupes}
 \input{62_representations}
 
 \chapter{Intégration}
-\input{70_suites_series_fonctions}
-\input{71_suites_series_fonctions}
-\input{169_suites_series_fonctions}
-\input{72_suites_series_fonctions}
-\input{155_suites_series_fonctions}
-\input{73_Chap_integrales_multiples}
-\input{38_Chap_integrales_multiples}
-\input{2_calcul_integral}
+\input{70_Integration}
+\input{71_Integration}
+\input{169_Integration}
+\input{72_Integration}
+\input{155_Integration}
+\input{73_Integration}
+\input{38_Integration}
+\input{2_Integration}
 
 \chapter{Suites et séries de fonctions}
 \input{75_series_fonctions}
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From fb11fe81dd737325ce533769c4917b1fe2ccc5d3 Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 27 Jun 2017 18:22:10 +0200
Subject: [PATCH 63/64] typo

---
 tex/frido/174_series_fonctions.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/tex/frido/174_series_fonctions.tex b/tex/frido/174_series_fonctions.tex
index f610e0d13..ea95866ec 100644
--- a/tex/frido/174_series_fonctions.tex
+++ b/tex/frido/174_series_fonctions.tex
@@ -549,7 +549,7 @@ \section{Vitesses de $x^{\alpha}$, de l'exponentielle et du logarithme}
     \begin{equation}        \label{EqooilOz}
         \exp\left( \sum_{p\in P_x}\frac{1}{ p } \right)\geq\prod_{p\in P_x}\left( 1+\frac{1}{ p } \right)\geq \sum_{q\in S_x}\frac{1}{ q }.
     \end{equation}
-    La première inégalité est simplement le fait que \( 1+u\leq e^u\) si \( u\geq 0\) (directe de la définition \ref{ThoRWOZooYJOGgR}). Les inégalités suivantes proviennent du fait que le logarithme est une primite de la fonction inverse (proposition \ref{ExZLMooMzYqfK}) :
+    La première inégalité est simplement le fait que \( 1+u\leq e^u\) si \( u\geq 0\) (directe de la définition \ref{ThoRWOZooYJOGgR}). Les inégalités suivantes proviennent du fait que le logarithme est une primitive de la fonction inverse (proposition \ref{ExZLMooMzYqfK}) :
     \begin{equation}
         \ln(x)\leq \sum_{n\geq x}\int_{n}^{n+1}\frac{dt}{ t }\leq \sum_{n\geq x}\frac{1}{ n }.
     \end{equation}

From 5cfcf7b186369af06b5dec9cc7bfd53b5b50dfdd Mon Sep 17 00:00:00 2001
From: Laurent Claessens <laurent@claessens-donadello.eu>
Date: Tue, 27 Jun 2017 18:25:22 +0200
Subject: [PATCH 64/64] =?UTF-8?q?(organisation=20du=20d=C3=A9p=C3=B4t)=20A?=
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