From 82d3eee42e23733a5e2f213c35dff7900d28f2fe Mon Sep 17 00:00:00 2001
From: Bela Gabriel Schneider <b.g.schneider@student.rug.nl>
Date: Thu, 20 Mar 2025 13:28:02 +0000
Subject: [PATCH 1/7] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index f2bae47..5777250 100644
--- a/main.tex
+++ b/main.tex
@@ -154,7 +154,7 @@ code-for-last-col = \color{blue}
\begin{itemize}
\item Multi-connected space: it has non-contractable loops
- \item Inhomogeneous space: it does not look the same from every point on the
+ \item Inhomogeneous space: it does not look the same from every point in space
\end{itemize}
\end{frame}
From b078bcbeaf6380fb8ea59384ccf13fc37b05d00c Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Thu, 20 Mar 2025 13:29:21 +0000
Subject: [PATCH 2/7] Update on Overleaf.
From fafc08cde805156d03d811fc57f3066ff6911009 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Thu, 20 Mar 2025 13:30:10 +0000
Subject: [PATCH 3/7] Update on Overleaf.
---
main.tex | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/main.tex b/main.tex
index 5777250..408a36d 100644
--- a/main.tex
+++ b/main.tex
@@ -77,14 +77,14 @@ code-for-last-col = \color{blue}
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{mug-neighbourhoods.png}
- \caption{The prototypical example of a manifold -- a mug. Image source: [13]}
+ \caption{The prototypical example of a manifold a mug. Image source: [13]}
\label{fig:mug-neighbourhoods}
\end{figure}
\begin{figure}[H]
\centering
%$ \captionsetup{width=.75\linewidth}
- \includegraphics[width=0.27\linewidth]{Contractible loops.png}
+ \includegraphics[width=0.2\linewidth]{Contractible loops.png}
\caption{Diagram showing two double tori with (non)-contractible paths. Image source[7]}
\label{fig:CoLoop}
\end{figure}
From 1e9fd6a7e4dc8f1a05f78340074238f025b54f4f Mon Sep 17 00:00:00 2001
From: Bela Gabriel Schneider <b.g.schneider@student.rug.nl>
Date: Thu, 20 Mar 2025 13:31:27 +0000
Subject: [PATCH 4/7] Update on Overleaf.
---
main.tex | 9 +++++----
1 file changed, 5 insertions(+), 4 deletions(-)
diff --git a/main.tex b/main.tex
index 408a36d..cbe80ef 100644
--- a/main.tex
+++ b/main.tex
@@ -98,12 +98,12 @@ code-for-last-col = \color{blue}
\begin{frame}[fragile]{CMB Anisotropy of Homogeneous Spherical Spaces}
\begin{itemize}
- \item Manifolds $M := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
+ \item Manifolds $\textcolor{blue}{M} := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
\item Helmholtz equation on $M$ given by
- $$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
+ $$(\Delta + E_\textcolor{red}{\beta}^M)\psi_\textcolor{red}{\beta}^{M, i} = 0$$
- \item In fact $E_\beta^m = \beta^2-1$ for $\beta \in \mathbb{N}$ we call $\beta$ a wave number
+ \item In fact $E_\textcolor{red}{\beta}^m = \textcolor{red}{\beta}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$ we call $\textcolor{red}{\beta}$ a wave number
\item The set of all possible wave numbers [for which there exists a non-zero solution] depends on $H$
@@ -154,7 +154,8 @@ code-for-last-col = \color{blue}
\begin{itemize}
\item Multi-connected space: it has non-contractable loops
- \item Inhomogeneous space: it does not look the same from every point in space
+ \item Inhomogeneous space: it does not look identical from every point in space
+ \item
\end{itemize}
\end{frame}
From ef49338c137ab197ba70d158af000ce47bff0c54 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Thu, 20 Mar 2025 13:31:32 +0000
Subject: [PATCH 5/7] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index cbe80ef..ede4a46 100644
--- a/main.tex
+++ b/main.tex
@@ -101,7 +101,7 @@ code-for-last-col = \color{blue}
\item Manifolds $\textcolor{blue}{M} := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
\item Helmholtz equation on $M$ given by
- $$(\Delta + E_\textcolor{red}{\beta}^M)\psi_\textcolor{red}{\beta}^{M, i} = 0$$
+ $$(\Delta + E_\textcolor{red}{\beta}^M\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{M, i} = 0$$
\item In fact $E_\textcolor{red}{\beta}^m = \textcolor{red}{\beta}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$ we call $\textcolor{red}{\beta}$ a wave number
From 2c0b4a32b925bffd3590ded6baade9d40718e882 Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Thu, 20 Mar 2025 13:31:33 +0000
Subject: [PATCH 6/7] Update on Overleaf.
---
main.tex | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index ede4a46..ec90553 100644
--- a/main.tex
+++ b/main.tex
@@ -101,7 +101,7 @@ code-for-last-col = \color{blue}
\item Manifolds $\textcolor{blue}{M} := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
\item Helmholtz equation on $M$ given by
- $$(\Delta + E_\textcolor{red}{\beta}^M\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{M, i} = 0$$
+ $$(\Delta + E_\textcolor{red}{\beta}^\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{M, i} = 0$$
\item In fact $E_\textcolor{red}{\beta}^m = \textcolor{red}{\beta}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$ we call $\textcolor{red}{\beta}$ a wave number
From 7d3b7dff1601ff87a2fd06a786ed92a3ceb42ada Mon Sep 17 00:00:00 2001
From: "juso.koc" <juso.koc@gmail.com>
Date: Thu, 20 Mar 2025 13:31:42 +0000
Subject: [PATCH 7/7] Update on Overleaf.
---
main.tex | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/main.tex b/main.tex
index ec90553..64ba803 100644
--- a/main.tex
+++ b/main.tex
@@ -100,8 +100,8 @@ code-for-last-col = \color{blue}
\begin{itemize}
\item Manifolds $\textcolor{blue}{M} := \mathbb{S}^3 /_ \sim$ where $\sim $ identifies the orbits of finite $H \leq SO(4)$ finite
- \item Helmholtz equation on $M$ given by
- $$(\Delta + E_\textcolor{red}{\beta}^\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{M, i} = 0$$
+ \item Helmholtz equation on $\textcolor{blue}{M}$ given by
+ $$(\Delta + E_\textcolor{red}{\beta}^\textcolor{blue}{M})\psi_\textcolor{red}{\beta}^{\textcolor{blue}{M}, i} = 0$$
\item In fact $E_\textcolor{red}{\beta}^m = \textcolor{red}{\beta}^2-1$ for $\textcolor{red}{\beta} \in \mathbb{N}$ we call $\textcolor{red}{\beta}$ a wave number