deletions | additions       diff --git a/The Relative Formulation.tex b/The Relative Formulation.tex  index 24fc443..1e44c61 100644  --- a/The Relative Formulation.tex  +++ b/The Relative Formulation.tex 
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\section{Relative Formulation} formulation}  \subsection{Rate}  To use the framework we have previosly developed we need to generalize the theory to handle multiple S(t) functions all contributing to the model. To do this we need to make some useful definitions and see how the connect to both the formulation analyzing chemical kinetics by Ross, and to the work proposed by Fisher himself. First we can define three different forms of rates. The first we will call the relative rate, the all $\gamma$ formulation.  \begin{equation}  k_{rel}\equiv \frac{d}{dt}[\ln\gamma_i]=\frac{d}{dt}\left[\ln S_i-\ln\sum_i S_i\right] \end{equation}  \subsection{Fisher Information} information}  Using these definitions we can then formulate the associated Fisher Information for each of these rates. The relative Fisher Information $I_r$, and the absolute Fisher Information $I_a$.  \begin{equation}  I_r=\sum_i \gamma_i k_{rel}=\sum_i \gamma_i \frac{d}{dt}[\ln\gamma_i] \end{equation}  This is where the "all $\gamma$" formulation derives its name.  \subsection{GFE to FE}  \subsection{Fisher equations}  We can now define the assoicated Fisher Equation for the previously defined Fisher Information. 
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\end{equation}  For this derivation we will need some other definitions, being $k_r=\frac{d}{dr}[ln\gamma_i]$ and that $k_a=\frac{d}{dt}[\ln S(t)]=k_aS_i$ which we will use later. the derivation begins exactly as the absolute case.  \begin{equation}  GFE_r=\sum[\gamma_i\frac{d}{dt}[k_r]]+\sum[k_r\frac{d}{dt}[]\gamma_i] \end{equation}  As in the other derivation doing the expanded derivatives finds the following three terms which will be denoted as A, B, and C for simplification.  \begin{equation}  \sum[\gamma_i\frac{d}{dt}[k_r]]+\sum[\frac{-k_r S_i}{\Sigma S_i}\frac{d}{dt}[\ln \Sigma S_i]]+\sum[\frac{k_r}{\Sigma S_i}\frac{d}{dt}[S_i]] \end{equation}  We can now simplify the B and C terms to write the final form of the $GFE_r$.  \begin{equation}  B:\sum[\frac{-k_r S_i}{\Sigma S_i}\frac{d}{dt}[\ln \Sigma S_i]]=-(\Sigma k_r \gamma_i)(\frac{1}{\Sigma S_i}\Sigma \frac{d}{dt}[S_i])=-(\Sigma k_r \gamma_i)(\Sigma k_a \gamma_i) \end{equation}  Where the absolute rate coefficient appears through the previous definition of $\frac{d}{dt}[S_i]$. We can now look at the final term.  \begin{equation}  C:\sum[\frac{k_r}{\Sigma S_i}\frac{d}{dt}[S_i]]=\sum[k_r\gamma_i\frac{d}{dt}[\ln S_i]]=\sum[k_r\gamma_ik_aS_i] \end{equation}  Therefore we can write the simplified form of the $GFE_r$ as follows.   \begin{equation}  GFE_r=\sum[\gamma_i\frac{d}{dt}[k_r]]+-(\Sigma k_r \gamma_i)(\Sigma k_a \gamma_i)+\sum[k_r\gamma_ik_aS_i] \end{equation}  \subsection{Inequality}