Update slides

1 files changed, 171 insertions, 74 deletions

diff --git a/presentation/KugelBSc.tex b/presentation/KugelBSc.tex
index 7128908..ac56d51 100644
--- a/presentation/KugelBSc.tex
+++ b/presentation/KugelBSc.tex
@@ -1,4 +1,4 @@
-\documentclass[xetex, onlymath]{beamer}
+\documentclass[xetex, onlymath, aspectratio=169]{beamer}
 \usefonttheme{serif}
 \usetheme{hsr}
 
@@ -46,6 +46,8 @@
 \usepackage{tikz}
 \usetikzlibrary{calc}
 \usepackage{xcolor}
+\usepackage{pgfplots}
+\pgfplotsset{compat=1.9}
 
 % source code
 \usepackage{listings}
@@ -117,29 +119,46 @@
 \section{Fourier on \(\mathbb{R}^2\)}
 
 \begin{frame}{Nice Periodic Functions}
-	\begin{definition}
-		A function
-		\[
-			f : \mathbb{R}^2 \to \mathbb{C}
-		\]
-		is a ``nice periodic function'' when it is
-		\begin{itemize}
-			\item smooth,
-			\item differentiable,
-			\item \textcolor{gray}{(abs.)} integrable,
-			\item periodic on \([0, 1] \times [0, 1]\), i.e.
-				\[
-					f(\mu, \nu) = f(\mu + 1, \nu) = f(\mu, \nu + 1).
-				\]
-		\end{itemize}
-	\end{definition}
+	\begin{columns}
+		\begin{column}{.7\linewidth}
+    	\begin{definition}
+    		A function
+    		\[
+    			f : \mathbb{R}^2 \to \mathbb{C}
+    		\]
+    		is a ``nice periodic function'' when it is
+    		\begin{itemize}
+    			\item smooth,
+    			\item differentiable,
+    			\item \textcolor{gray}{(abs.)} integrable,
+    			\item periodic on \([0, 1] \times [0, 1]\), i.e.
+    				\[
+    					f(\xi, \eta) = f(\xi + 1, \eta) = f(\xi, \eta + 1).
+    				\]
+    		\end{itemize}
+    	\end{definition}
+		\end{column}
+		\begin{column}{.3\linewidth}
+			\begin{center}
+				\begin{tikzpicture}[
+					axis/.style = {
+						very thick, -latex, draw = black
+					},
+				]
+					\draw[lightgray] (0, 0) grid (3, 3);
+					\draw[axis] (0, 0) -- (3.3, 0) node[right] {\(\xi\)};
+					\draw[axis] (0, 0) -- (0, 3.3) node[above] {\(\eta\)};
+				\end{tikzpicture}
+			\end{center}
+		\end{column}
+	\end{columns}		
 \end{frame}
 
 \begin{frame}{Function Space}
 	\begin{block}{Basis Functions}
   	The space of nice periodic functions is spanned by the (also nice) functions
   	\[
-  		B_{m, n}(\mu, \nu) = e^{i2\pi m\mu} e^{i2\pi n\nu}.
+  		B_{m, n}(\xi, \eta) = e^{i2\pi m\xi} e^{i2\pi n\eta}.
   	\]
 	\end{block}
 \end{frame}
@@ -150,14 +169,14 @@
 
 \begin{frame}{Inner Product}
 	\begin{definition}<1->
-		Let \(f(\mu, \nu)\) and \(g(\mu, \nu)\) be nice periodic functions. Their inner product is
+		Let \(f(\xi, \eta)\) and \(g(\xi, \eta)\) be nice periodic functions. Their inner product is
 		\[
-			\langle f, g \rangle = \iint_{[0, 1]^2} f g^* \, d\mu d\nu.
+			\langle f, g \rangle = \iint_{[0, 1]^2} f(\xi, \eta) \overline{g}(\xi, \eta) \, d\xi d\eta.
 		\]
 	\end{definition}
 
 	\begin{definition}<2->
-		For a nice periodic function \(f(\mu, \nu)\): the numbers
+		For a nice periodic function \(f(\xi, \eta)\): the numbers
 		\[
 			c_{m, n} = \langle f, B_{m, n} \rangle
 		\]
@@ -169,8 +188,8 @@
 	\begin{theorem}
 		For nice periodic functions:
   	\[
-  		f(\mu, \nu) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}}
-				c_{m, n} B_{m, n} (\mu, \nu)
+  		f(\xi, \eta) = \sum_{m \in \mathbb{Z}} \sum_{n \in \mathbb{Z}}
+				c_{m, n} B_{m, n} (\xi, \eta)
   	\]
 		where
 		\[
@@ -183,7 +202,7 @@
 	
 	\centering
 
-	{\huge\bfseries\itshape Why \(B_{m, n} = e^{i2\pi m\mu} e^{i2\pi n\nu}\)?}
+	{\huge\bfseries\itshape Why \(B_{m, n} = e^{i2\pi m\xi} e^{i2\pi n\eta}\)?}
 	\vspace{3em}
 	
 	{\huge\bfseries\itshape Because
@@ -195,19 +214,19 @@
 \begin{frame}{The Problem}
 	\begin{block}{Fourier's Problem}<1->
 		\[
-			\nabla^2 f(\mu, \nu)
-			  = \frac{\partial^2 f}{\partial \mu^2} + \frac{\partial^2 f}{\partial \nu^2}
-			  = \lambda f(\mu, \nu)
+			\nabla^2 f(\xi, \eta)
+			  = \frac{\partial^2 f}{\partial \xi^2} + \frac{\partial^2 f}{\partial \eta^2}
+			  = \lambda f(\xi, \eta)
 		\]
 	\end{block}
 	\begin{alertblock}{Solution}<2->
 		Separation ansatz:
 		\[
-			f(\mu, \nu) = M(\mu) N(\nu)
+			f(\xi, \eta) = M(\xi) N(\eta)
 		\]
 		Resulting ODEs:
 		\begin{align*}
-			\frac{d^2 M}{d \mu^2} &= \kappa M(\mu), & \frac{d^2 N}{d \nu^2} &= (\lambda - \kappa) N(\nu)
+			\frac{d^2 M}{d \xi^2} &= \kappa M(\xi), & \frac{d^2 N}{d \eta^2} &= (\lambda - \kappa) N(\eta)
 		\end{align*}
 	\end{alertblock}
 \end{frame}
@@ -217,7 +236,7 @@
 \begin{frame}{Spherical Coordinates}
 	\begin{columns}
 		\begin{column}{.6\linewidth}
-			\includegraphics[width=\linewidth]{figures/spherical-coordinates}
+			\includegraphics[height=.9\paperheight]{figures/spherical-coordinates}
 		\end{column}
 		\begin{column}{.4\linewidth}
 			\noindent
@@ -241,14 +260,14 @@
 	\uncover<1->{
   	Cartesian Laplacian
   	\[
-  		\nabla^2 \equiv \frac{\partial^2}{\partial \mu^2} + \frac{\partial^2}{\partial \nu^2}
+  		\nabla^2 := \frac{\partial^2}{\partial \xi^2} + \frac{\partial^2}{\partial \eta^2}
   	\]
 	}
 	
 	\uncover<2->{
   	Spherical Laplacian
   	\[
-  		\nabla^2 \equiv
+  		\nabla^2 :=
   			\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)
     			+ \frac{1}{r^2} \onslide<3-> \underbrace{ \onslide<2-> \left[
   						\frac{1}{\sin\vartheta} \frac{\partial}{\partial \vartheta}
@@ -262,33 +281,56 @@
 	\uncover<4->{
   	Surface Spherical Laplacian
   	\[
-  		\nabla^2_s \equiv r^2 \nabla^2
+  		\nabla^2_s := r^2 \nabla^2
   			- \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)
   	\]
 	}
 \end{frame}
 
-\bgroup
-\setbeamercolor{background canvas}{bg=black}
-\setbeamertemplate{navigation symbols}{}
-\begin{frame}{Geometrical Intuition}
+\begin{frame}[fragile]{Geometrical Intuition}
+	\only<1>{
+		\begin{center}
+  		\begin{tikzpicture}
+				\begin{axis}[
+					clip = false,
+					width = .8\linewidth, height = .8\paperheight,
+					xtick = \empty, ytick = \empty,
+					colormap name = viridis,
+					axis lines = middle,
+					axis line style = {ultra thick, -latex}
+				]
+        	\addplot+[
+						smooth, mark=none, line width = 3pt, mesh,
+						point meta=explicit,
+					] file {figures/laplacian-1d.dat};
+        \end{axis}
+  		\end{tikzpicture}
+		\end{center}
+	}
+	\only<2>{
+		\includegraphics[width=\linewidth]{figures/laplacian-3d}
+	}
+	\only<3>{
+		\begin{center}
+			\includegraphics[height=.7\paperheight]{figures/laplacian-sphere}
+		\end{center}
+	}
 \end{frame}
-\egroup
 
 \begin{frame}{Where is \(\nabla^2_s\) useful?}
-  	To do brain scans, apparently \cite{carvalhaes_surface_2015}
+	To do brain scans, apparently \cite{carvalhaes_surface_2015}
+		\begin{center}
 		\only<1>{
-			\begin{figure}
-				\includegraphics[width=.8\linewidth, clip=100 0 0 0]{figures/eeg-photo}
-				\caption{Electroencephalogram (EEG). Image from Wikimedia \cite{baburov__2009}.}
-			\end{figure}
+			\includegraphics[width=.8\linewidth, clip, trim=0 20 0 20]{figures/eeg-photo}
+			\nocite{baburov__2009}
+			% \caption{Electroencephalogram (EEG). Image from Wikimedia \cite{baburov__2009}.}
 		}
 		\only<2>{
-    	\begin{figure} \centering
-    		\includegraphics[width=\linewidth]{figures/surface-laplacian-eeg}
-    		\caption{Surface Laplacian in EEG. Taken from \cite{ries_role_2013}.}
-    	\end{figure}
+    	\includegraphics[width=\linewidth]{figures/surface-laplacian-eeg}
+			\nocite{ries_role_2013}
+    	% \caption{Surface Laplacian in EEG. Taken from \cite{ries_role_2013}.}
 		}
+	\end{center}
 \end{frame}
 
 \begin{frame}{Brain Scans}
@@ -296,25 +338,38 @@
 		\begin{column}{.6\linewidth}
     	Electrodynamics
     	\begin{align*}
-	    		\nabla^2 \phi
-    			&= \bm{\nabla \cdot} \bm{\nabla} \phi \qquad
-    				\color{lightgray} \left( \phi = \int_\mathsf{A}^\mathsf{B} \vec{E} \bm{\cdot} d\vec{s} \right) \\
-    			&= \bm{\nabla \cdot} \vec{E} \\
-    		  &\color{lightgray}= \int_{\Omega} (\bm{\nabla \cdot} \vec{E}) \bm{\cdot} d\vec{s}
-    		   = \oint_{\partial \Omega} \vec{E} \bm{\cdot} d\vec{s} \\
-    			&= \frac{\rho}{\varepsilon}
+	    	\onslide<1->{
+					\nabla^2 \phi &= \bm{\nabla \cdot} \bm{\nabla} \phi \qquad
+						\color{lightgray} \left(
+						\phi = \int_\mathsf{A}^\mathsf{B} \vec{E} \bm{\cdot} d\vec{l}
+					\right) \\
+				}
+				\onslide<2->{
+					&= \bm{\nabla \cdot} \vec{E} \\
+				}
+				\onslide<3->{
+		  		&\color{lightgray}= \int_{\Omega} (\bm{\nabla \cdot} \vec{E}) \bm{\cdot} d\vec{s}
+		   		= \oint_{\partial \Omega} \vec{E} \bm{\cdot} d\vec{s} \\
+				}
+				\onslide<4->{
+					&= \frac{\rho}{\varepsilon}
+				}
     	\end{align*}
-    	So over the scalp
-    	\[
-    		\nabla^2_s \phi 
-    			= \frac{\rho_s}{\varepsilon}
-    			= \text{Current flow in the brain}
-    	\]
+			\uncover<5->{
+      	So over the scalp
+      	\[
+      		\nabla^2_s \phi
+      			= \frac{\rho_s}{\varepsilon}
+      			= \text{Current flow in the brain}
+      	\]
+			}
 		\end{column}
 		\begin{column}{.4\linewidth}
-			\centering
-			\includegraphics[width=\linewidth]{figures/flux}
-			\nocite{maschen_divergence_2013}
+			\uncover<2->{
+  			\centering
+  			\includegraphics[width=\linewidth]{figures/flux}
+  			\nocite{maschen_divergence_2013}
+			}
 		\end{column}
 	\end{columns}
 \end{frame}
@@ -344,7 +399,7 @@
 		\[
 			\frac{d^2\Phi}{d\varphi^2} = \kappa \Phi(\varphi)
 				\implies \Phi(\varphi) = e^{im\varphi},
-				\quad m \in \mathbb{Z}
+				\quad \textcolor{gray}{m \in \mathbb{Z}}
 		\]
 	\end{alertblock}
 \end{frame}
@@ -388,7 +443,7 @@
 
 \begin{frame}
 	\centering
-	\includegraphics[width=\linewidth]{figures/legendre-polynomials}
+	\includegraphics[height=\paperheight]{figures/legendre-polynomials}
 \end{frame}
 
 \begin{frame}{Associated Legendre Polynomials}
@@ -407,7 +462,7 @@
 
 \begin{frame}
 	\centering
-	\includegraphics[width=\linewidth]{figures/associated-legendre-polynomials}
+	\includegraphics[height=\paperheight]{figures/associated-legendre-polynomials}
 \end{frame}
 
 
@@ -427,6 +482,11 @@
 	\end{alertblock}
 \end{frame}
 
+\begin{frame}{What do they look like?}
+	\Large \bfseries 
+	Python Magic
+\end{frame}
+
 \bgroup
 \setbeamercolor{background canvas}{bg=black}
 \setbeamertemplate{navigation symbols}{}
@@ -434,6 +494,16 @@
 \end{frame}
 \egroup
 
+\begin{frame}{Research Question}
+	\begin{block}{Recurrence Relation(s)?}
+		\begin{align*}
+			\tilde{Y}_{m+1, n} &\stackrel{?}{=} f(\tilde{Y}_{m, n}, \tilde{Y}_{m-1, n}, \tilde{Y}_{m, n-1}, \ldots) \\
+			\tilde{Y}_{m, n+1} &\stackrel{?}{=} f(\tilde{Y}_{m, n}, \tilde{Y}_{m-1, n}, \tilde{Y}_{m, n-1}, \ldots) \\
+			\tilde{Y}_{m+1, n+1} &\stackrel{?}{=} f(\tilde{Y}_{m, n}, \tilde{Y}_{m-1, n}, \tilde{Y}_{m, n-1}, \ldots)
+		\end{align*}
+	\end{block}
+\end{frame}
+
 \section{Fourier on \(S^2\)}
 
 \begin{frame}{Basis functions?}
@@ -443,10 +513,10 @@
 		The inner product of nice functions \(f(\varphi, \vartheta)\) and \(g(\varphi, \vartheta)\) from \(S^2\) to \(\mathbb{C}\) is
 		\[
 			\langle f, g \rangle
-				= \iint_{S^2} f g^* \, d\Omega
+				= \iint_{S^2} f(\varphi, \vartheta) \overline{g}(\varphi, \vartheta) \, d\Omega
 				\uncover<3->{
   				= \int\limits_0^{2\pi} \int\limits_0^{\pi}
-  					f(\varphi, \vartheta) g^*(\varphi, \vartheta)
+  					f(\varphi, \vartheta) \overline{g}(\varphi, \vartheta)
   					\sin\vartheta \, d\vartheta d\varphi
 				}
 		\]
@@ -531,7 +601,7 @@
 			\end{alertblock}
 		\end{column}
 		\begin{column}{.5\linewidth}<2->
-				\begin{block}{Angular M. and KE}
+				\begin{block}{Angular Momentum and KE}
 				\[
 					\vec{L} = \vec{r}\bm{\times}{\vec{p}},
 					\quad
@@ -549,6 +619,33 @@
 	\end{columns}
 \end{frame}
 
+\iffalse
+\begin{frame}{Intuition for the Operators}
+	\begin{columns}
+		\begin{column}{.5\linewidth}
+			\begin{block}{Plane wave}<1->
+				\[
+					\Psi(\vec{x}, t) = \exp i(\vec{k} \bm{\cdot} \vec{x} + \omega t )
+				\]
+			\end{block}
+			\begin{block}{Spatial derivative}<2->
+				\begin{align*}
+					\uncover<2->{
+  					\bm{\nabla} \Psi(\vec{x}, t) &= 
+  						\bm{\nabla} \exp i(\vec{k} \bm{\cdot} \vec{x} + \omega t ) \\
+					}
+					\uncover<3->{
+						&= ik \exp i(\vec{k} \bm{\cdot} \vec{x} + \omega t ) \\
+						&= ik \Psi(\vec{x}, t)
+					}
+				\end{align*}
+			\end{block}
+		\end{column}
+		\begin{column}{.5\linewidth}
+		\end{column}
+	\end{columns}
+\end{frame}
+\fi
 
 \bgroup
 \setbeamercolor{background canvas}{bg=black}
@@ -627,7 +724,7 @@
 					+ \frac{1}{r^2} \frac{\partial}{\partial r} \left(
 							r^2 \frac{\partial}{\partial r}
 						\right)}_\text{Kinetic Energy}
-					+ U(\vec{r})
+					+\, U(\vec{r})
 				\Bigg] \Psi(\vec{r}) = E \Psi(\vec{r})
 			}
 			% Introduce E_kr
@@ -650,18 +747,18 @@
 	\begin{columns}
 		\begin{column}{.6\linewidth}
 			\Large 
-			\uncover<11->{
+			\uncover<6>{
 				\Large
 				\textit{But why?} \\[2em]
 			}
 			
-			\uncover<12->{
+			\uncover<6>{
 				\bfseries
 				Hydrogen atom has radial symmetry!
 			}
 		\end{column}
 		\begin{column}{.35\linewidth}
-    	\uncover<11->{
+    	\uncover<6>{
     		\includegraphics[width=\linewidth]{figures/hydrogen}
 				\nocite{depiep_electron_2013}
     	}
@@ -675,7 +772,7 @@
 % \section{Other applications}
 
 \begin{frame}{Bibliography}
-	\renewcommand*{\bibfont}{\tiny}
+	\renewcommand*{\bibfont}{\scriptsize}
 	\printbibliography
 \end{frame}