deletions | additions       diff --git a/Absolute Formulation.tex b/Absolute Formulation.tex  index 239b5cf..9a56331 100644  --- a/Absolute Formulation.tex  +++ b/Absolute Formulation.tex 
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\begin{equation}  GFE_a=\frac{d}{dt}\sum [\gamma_i(t)k_a(t)]  \end{equation}  Simplifyingfurther  we find\begin{equation}  GFE_a=\sum[\gamma_i\frac{d}{dt}[k_a]]+\sum[k_a(\frac{-S_i}{(\Sigma S_i)^2}\frac{d}{dt}[(\Sigma S_i)]+(\Sigma S_i)^{-1}\frac{d}{dt}[S_i])]  \end{equation}  Now substituting in the definition of the $\gamma_i$ and distributing the sum we find the following.  \begin{equation}  GFE_a=\sum[\gamma_i\frac{d}{dt}[k_a]]+\sum[-k_a\gamma_i \frac{d}{dt}[\ln (\Sigma S_i)]]+\sum[\frac{k_a}{\Sigma S_i}\frac{d}{dt}[S_i]]  \end{equation}  For simplicity lets call these three terms A, B, and C respectively. Term A is already simplified however lets analyze terms B and C individually.  \begin{equation}  B: \sum[-k_a\gamma\frac{d}{dt}[\ln \Sigma S_i(t)]]  \end{equation}  This sum can really be broken up the derivative term is already being summed therefore we can write the following.  \begin{equation}  -(\sum[k_a\gamma_i])(\frac{d}{dt}[\ln \Sigma S_i])=-(\sum[k_a\gamma_i])(\frac{1}{\Sigma S_i}\Sigma\frac{d}{dt}[ S_i])  \end{equation}  From here we remember that $\frac{d}{dt}[S_i]=k_aS_i$ and we find a nice simplification.  \begin{equation}  -(\sum[k_a\gamma_i])(\sum[k_a\gamma_i])=-(\sum k_a\gamma_i)^2=(\overline{k})^2  \end{equation}  With this form we can see the GFE forming, we now need to analyze term C.  \begin{equation}  C:\sum[\frac{k_a}{\Sigma S_i}\frac{d}{dt}[S_i]]=\sum[\frac{k_a S_i}{\Sigma S_i S_i}\frac{d}{dt}[S_i]]=\sum[k_a\gamma_i\frac{d}{dt}[\ln S_i]]=\sum[k_a^2\gamma_i]  \end{equation}  This last term can easily be identified as $=I_a=\overline{k_a^2}$ which completes our GFE for the absolute formulation.   \begin{equation}  GFE_a=\sum[\gamma_i\frac{d}{dt}[k_a]]-(\sum k_a\gamma_i)^2+\sum[k_a^2\gamma_i]=\sum[\gamma_i\frac{d}{dt}[k_a]]-(\overline{k})^2+\overline{k_a^2}  \end{equation}