figure caption updates

main.tex

@@ -284,15 +284,15 @@ \section{Example I and performance analysis}\label{sec:perf}
 
 \begin{figure}[ht]
 \includegraphics[width=1.0\columnwidth]{time_bfreqs.pdf}
-\caption{(a) Execution time of the dual fermion calculation for the Hubbard model in $2$ dimensions with ``atomic limit'' input at $U=20$, $\beta = 1$ as a function of the number of bosonic frequencies $N_{\Omega}$ at $N_{\omega} = 48,~N_k = 16$; (b) Systematic error $\delta_g$ of the dual fermion Green's function $\tilde G_{i\omega, k}$ at $i\omega = i\pi / \beta, k = (0,0)$ for a varied number of bosonic frequencies $N_\Omega$ plotted in a logarithmic scale. }
+\caption{(a) Execution time of the dual fermion calculation for the Hubbard model in $2$ dimensions with ``atomic limit'' input at $U=20$, $\beta = 1$ as a function of the number of bosonic frequencies $N_{\Omega}$ at $N_{\omega} = 48,~N_k = 16$; (b) Systematic error $\delta_g$ of the dual fermion Green's function $\tilde G_{i\omega, k}$ at $i\omega = i\pi / \beta, k = (0,0)$ as a function of bosonic frequencies $N_\Omega$, plotted on a logarithmic scale. }
 \label{fig:benchmark_b}
 \end{figure}
 
 Fig. \ref{fig:benchmark_b} shows the performance of the \texttt{opendf} code upon the change of the total number of bosonic frequencies $N_{\Omega}$ in the vertex $\gamma_{\Omega}$ for a fixed number of fermionic frequencies $N_{\omega}=48$ for a $16 \times 16$ k-space grid. The computational effort, indicated by the time to convergence in Fig.~\ref{fig:benchmark_b}(a), grows linearly in $N_{\Omega}$. The error $\delta_g$, as defined in Eqn.~\ref{eqn:deltag} and shown in frame (b), is of the order of a percent and decreases with a power law.
 
 \begin{figure}[ht]
 \includegraphics[width=1.0\columnwidth]{time_ffreqs.pdf}
-\caption{(a) Execution time of the dual fermion calculation for the Hubbard model in $2$ dimensions with ``atomic limit'' input at $U=20$, $\beta = 1$ as a function of the number of fermionic frequencies $N_{\omega}$ at $N_{\Omega} = 3$, $N_k = 8$; (b) Error $\delta_g$ of $G_{i\omega, k}$ at $i\omega = i\pi / \beta, k = (0,0)$ as a function of $1/N_{\omega}$ at the same parameters.}
+\caption{(a) Execution time of the dual fermion calculation for the Hubbard model in $2$ dimensions with ``atomic limit'' input at $U=20$, $\beta = 1$ as a function of the number of fermionic frequencies $N_{\omega}$ at $N_{\Omega} = 3$, $N_k = 8$; (b) Error $\delta_g$ of $G_{i\omega, k}$ at $i\omega = i\pi / \beta, k = (0,0)$ as a function of $1/N_{\omega}$, for the same parameters.}
 \label{fig:benchmark_f}
 \end{figure}