From 55dcf34e0283712a453ba0461c5b6ffd9364a7ea Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 20 Jun 2022 15:57:24 +0200
Subject: Some unpolished work notes.

---
 frsb_kac_new.tex | 10 ++++++++++
 1 file changed, 10 insertions(+)

(limited to 'frsb_kac_new.tex')

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}
-- 
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