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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} |