From eaf4b97a1ae5feefd1b379b04dffc9287f082cc3 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 31 Jul 2024 22:58:39 +0200
Subject: Threw in some notes.

---
 topology.tex | 16 ++++++++++++++--
 1 file changed, 14 insertions(+), 2 deletions(-)

diff --git a/topology.tex b/topology.tex
index 8c21ee9..84b871d 100644
--- a/topology.tex
+++ b/topology.tex
@@ -117,12 +117,24 @@ $\mathbb M(1)=\frac1N\phi(1)\cdot\mathbf x_0$, the result is
   \overline{\chi}&=\int d\mathbb Q\,d\mathbb M\,d\sigma_0\,\exp\Bigg\{
     \frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
     -\frac M2\log\operatorname{sdet}f(\mathbb Q) \notag \\
-    &\qquad+N\int d1\,\left[
-      \mathbb M(1)+\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
+    &+N\int d1\,\left[
+      (1+\hat\beta\bar\theta_1\theta_1)\mathbb M(1)+\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big)
     \right]
   \Bigg\}
 \end{align}
 
+\begin{equation}
+  0=\int d2\,(\mathbb Q(1,2)-\mathbb M(1)\mathbb M(2)^T)(1+\bar\theta_2\theta_2\hat\beta)+\mathbb M(1)
+\end{equation}
+
+\[
+  D=\beta R
+  \quad
+  \beta=-\frac{m+\sum_aR_{1a}}{\sum_aC_{1a}}
+  \quad
+  \hat m=0
+\]
+
 \begin{equation}
   \begin{aligned}
     &\frac1N\log\overline\chi
-- 
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