ddd/bachelor-prep-presentation
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Merge branch 'master' of https://git.overleaf.com/67dbe68df0414d63b309f98a

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prescientmoon 2025-03-20 14:31:43 +01:00
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main.tex
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@ -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}
@ -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
$$(\Delta + E_\beta^M)\psi_\beta^{M, i} = 0$$
\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 $E_\beta^m = \beta^2-1$ 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 on the
\item Inhomogeneous space: it does not look identical from every point in space
\item
\end{itemize}
\end{frame}