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| -rw-r--r-- | frsb_kac_new.tex | 10 |
1 files changed, 10 insertions, 0 deletions
diff --git a/frsb_kac_new.tex b/frsb_kac_new.tex index e1ac91d..e7b65a9 100644 --- a/frsb_kac_new.tex +++ b/frsb_kac_new.tex @@ -425,6 +425,16 @@ We also have from the first (before the diagonal ansatz) $Df'(Q)=I-RQ^{-1}Rf'(Q) =(R_df''(1)+R_d^{-1}-\mu)I+(\hat\beta-D_d/R_d)f'(Q) \end{equation} +Always true: +\begin{equation} + 0=-\mu R+\hat\beta Rf'(Q)+R(R\odot f''(Q))+I-Df'(Q) +\end{equation} +Insert $D=D_dI+\delta D$, $R=R_dI+\delta R$, and $D_d=\hat\beta R_d$ +\begin{equation} + 0=-\hat\beta \mu R_d I +\hat\beta R_df'(Q)+R_d^2f''(1)I+I-\hat\beta R_df'(Q) + -\mu\delta R +\end{equation} + \begin{equation} 0=(\hat\beta Q+R) f'(Q)+(R\odot f''(Q)-\mu I)Q \end{equation} |