From 1ff97060e45ba44d503a95995986b682a605b38d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 28 Jun 2024 17:21:48 +0200
Subject: Fixed some minus signs due to an old convention.

---
 marginal.tex | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/marginal.tex b/marginal.tex
index bb4a96d..b3896db 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -893,7 +893,7 @@ the contributions from the marginal pieces of the calculation, and is given by
     &\mathcal U_\mathrm{SSG}(\hat\lambda,Q,X,\hat X\mid\beta,\lambda^*,\mu,C)
     =\hat\lambda\lambda^*
     +\lim_{n\to0}\lim_{m_1\cdots m_n\to0}\frac1n\Bigg\{
-    \frac12\log\det Q+
+    \frac12\log\det Q-
       \sum_{a=1}^n\bigg(
         \sum_{\alpha=1}^{m_a}\beta\mu Q_{aa}^{\alpha\alpha}
         +\hat\lambda\mu Q_{aa}^{11}
@@ -1124,7 +1124,7 @@ the complexity is likewise given by
 \begin{align}
     &\mathcal U_\mathrm{MSG}(\hat q,\hat\lambda,Q^{11},Q^{22},Q^{12}\mid\beta,\lambda^*,\omega_1,\omega_2) \notag \\
     &\quad=\lim_{m\to0}\bigg\{\sum_{\alpha=1}^m\left[\hat q^\alpha(Q^{11,\alpha\alpha}+Q^{22,\alpha\alpha}-1)-\beta(\omega_1Q^{11,\alpha\alpha}+\omega_2Q^{22,\alpha\alpha}-2\epsilon Q^{12,\alpha\alpha})\right]
-    +\hat\lambda(\omega_1Q^{11,11}+\omega_2Q^{22,11}-2\epsilon Q^{12,11}) \notag \\
+    -\hat\lambda(\omega_1Q^{11,11}+\omega_2Q^{22,11}-2\epsilon Q^{12,11}) \notag \\
     &\qquad\qquad+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{\alpha\gamma}^m(Q^{ii,\alpha\gamma})^2+2\beta\hat\lambda\sum_\alpha^m(Q^{ii,1\alpha})^2+\hat\lambda^2(Q^{ii,11})^2\right]
     +\frac12\log\det\begin{bmatrix}
       Q^{11}&Q^{12}\\
@@ -1162,12 +1162,12 @@ limit of $m\to0$ is taken, we find
           +2(q^{ii}_0)^2
           -2(\tilde q^{ii}_0)^2
         \right)
-        -2\beta\hat\lambda\left(
+        +2\beta\hat\lambda\left(
           (\tilde q^{ii}_d)^2-(\tilde q^{ii}_0))^2
         \right)
         +\hat\lambda^2(\tilde q^{ii}_d)^2
       \right]
-      +\hat\lambda\tilde q^{ii}_d\omega_i
+      -\hat\lambda\tilde q^{ii}_d\omega_i
       -\beta(\tilde q^{ii}_d-q^{ii}_d)\omega_i
       \right\} \notag \\
       &+\frac12\log\bigg[
@@ -1191,9 +1191,9 @@ limit of $m\to0$ is taken, we find
       \bigg]
       \notag \\
       &-\log\left[(q^{11}_d-q^{11}_0)(q^{22}_d-q^{22}_0)-(q^{12}_d-q^{12}_0)^2\right]
-      -2\epsilon\big[\hat\lambda\tilde q^{12}_d
-        -\beta(\tilde q^{12}_d-q^{12}_d)\big]
-      +\hat q(q^{11}_d+q^{22}_d-1)+\hat{\tilde q}(\tilde q^{11}_d+\tilde q^{22}_d-1)
+      +2\epsilon\big[\hat\lambda\tilde q^{12}_d
+        +\beta(\tilde q^{12}_d-q^{12}_d)\big]
+      -\hat q(q^{11}_d+q^{22}_d-1)+\hat{\tilde q}(\tilde q^{11}_d+\tilde q^{22}_d-1)
       \label{eq:multispherical.ansatz}
   \end{align}
 \end{widetext}
-- 
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