From 68a93429a68ecd107b9982467b770185c844d214 Mon Sep 17 00:00:00 2001
From: Yingjie Wang <phywyj@gmail.com>
Date: Fri, 10 Apr 2026 01:18:27 -0400
Subject: [PATCH] update: auto commit

---
 report.tex | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/report.tex b/report.tex
index f4defc5..87cf30a 100644
--- a/report.tex
+++ b/report.tex
@@ -702,12 +702,14 @@ $}
 
   \begin{frame}{Searching for a first order Z4c}{old workflow: by hand}
     We introduce the following auxiliary variables:
-    \begin{equation}
-      \dlapse{_i} \definedby \Partial{_i} \lapse \qc
-      \dshift{^i_j} \definedby \Partial{_j} \shift{^i} \qc
-      \dchi*{_i} \definedby \spaceD*{_i} \ln(\chi) \qc
-      \dsmetric*{_i_j_k} \definedby \Partial{_i} \smetric*{_j_k},
-    \end{equation}
+    \[
+    \scalebox{0.8}{$\displaystyle
+        \dlapse{_i} \definedby \Partial{_i} \lapse \qc
+          \dshift{^i_j} \definedby \Partial{_j} \shift{^i} \qc
+          \dchi*{_i} \definedby \spaceD*{_i} \ln(\chi) \qc
+          \dsmetric*{_i_j_k} \definedby \Partial{_i} \smetric*{_j_k},
+    $}
+    \]
     and replace all the spatical derivatives of the original variables with the new auxiliary variables, we got this
     \[
     \scalebox{0.6}{$\displaystyle