From d7d9f7181cab211528c1e1b19297e0d294662205 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Fri, 21 Mar 2025 12:32:21 +0000
Subject: [PATCH] Update on Overleaf.
---
isospectral.tex | 2 +-
main.tex | 5 +++--
2 files changed, 4 insertions(+), 3 deletions(-)
diff --git a/isospectral.tex b/isospectral.tex
index ffb2fae..7cab952 100644
--- a/isospectral.tex
+++ b/isospectral.tex
@@ -4,7 +4,7 @@
\begin{definition}[Laplace-Beltrami operator]
a generalization of the Laplace operator to more general spaces
\begin{align*}
- \Delta f \coloneq \divergence \nabla f
+ \Delta f \coloneq \divergence \nabla f.
\end{align*}
\end{definition}
\item The spectrum of this operator is a subset of $[0, \infty)$. We call it the \emph{spectrum} of the manifold.
diff --git a/main.tex b/main.tex
index 3c2d366..49cb633 100644
--- a/main.tex
+++ b/main.tex
@@ -4,6 +4,7 @@
\usecolortheme{lily}
\setbeamertemplate{navigation symbols}{}
+
% boadilla seems to be the only one with enough space for stuff.compile it now it is very toxic but works
\usepackage{graphicx} % Required for inserting images
@@ -120,7 +121,7 @@ code-for-last-col = \color{blue}
\pause
\item For the majority of groups $\Gamma$, the anisotropies $\frac{\delta T}{T}$ do not coincide with observations.
\pause
- \item The only groups for which it does are $\Gamma = O^*$ and $\Gamma = I^*$ — the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120 respectively.
+ \item The only groups for which they do are $\Gamma = O^*$ and $\Gamma = I^*$ — the \textcolor{red}{binary octahedral} and \textcolor{red}{binary icosahedral} groups of order 48 and 120 respectively.
\pause
\end{itemize}
\begin{figure}[H]
@@ -158,7 +159,7 @@ code-for-last-col = \color{blue}
\begin{frame}{Statistical Isotropy and Hypothesis}
\small
\textbf{Theoretical Expectation:}\\
- From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2$.\\
+ From the perspective of an Earth-based observer, we can view the CMB as a function defined on the celestial sphere $\mathbb{S}^2.$\\
\begin{itemize}
\pause
\item The CMB temperature fluctuations can be expanded as: