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authorNao Pross <np@0hm.ch>2022-05-27 16:29:48 +0200
committerNao Pross <np@0hm.ch>2022-05-27 16:29:48 +0200
commita5e0c8fb18eeabe0cfe00b8ce35d46a7edc6b29b (patch)
tree4b7cea2f9f339e8a30806bfe854175f691447029 /presentation
parentFix presentation Makefile (diff)
downloadFourierOnS2-a5e0c8fb18eeabe0cfe00b8ce35d46a7edc6b29b.tar.gz
FourierOnS2-a5e0c8fb18eeabe0cfe00b8ce35d46a7edc6b29b.zip
Update slides
Diffstat (limited to 'presentation')
-rw-r--r--presentation/KugelBSc.pdfbin4880758 -> 5336061 bytes
-rw-r--r--presentation/KugelBSc.tex245
-rw-r--r--presentation/Makefile2
-rw-r--r--presentation/figures/laplacian-1d.dat500
-rw-r--r--presentation/figures/laplacian-1d.py32
-rw-r--r--presentation/figures/laplacian-3d.jpgbin0 -> 265680 bytes
-rw-r--r--presentation/figures/laplacian-sphere.jpgbin0 -> 162808 bytes
7 files changed, 704 insertions, 75 deletions
diff --git a/presentation/KugelBSc.pdf b/presentation/KugelBSc.pdf
index 255b64b..5367852 100644
--- a/presentation/KugelBSc.pdf
+++ b/presentation/KugelBSc.pdf
Binary files differ
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}
diff --git a/presentation/Makefile b/presentation/Makefile
index 7bba7be..8b6b4fa 100644
--- a/presentation/Makefile
+++ b/presentation/Makefile
@@ -10,7 +10,7 @@ BIBFILE := $(DOCNAME).bib
all: $(DOCNAME).pdf
clean:
- @rm -v $(DOCNAME)-blx.bib $(DOCNAME).aux $(DOCNAME).bbl $(DOCNAME).blg $(DOCNAME).log $(DOCNAME).nav $(DOCNAME).out $(DOCNAME).run.xml $(DOCNAME).snm $(DOCNAME).toc
+ @rm -fv $(DOCNAME)-blx.bib $(DOCNAME).aux $(DOCNAME).bbl $(DOCNAME).blg $(DOCNAME).log $(DOCNAME).nav $(DOCNAME).out $(DOCNAME).run.xml $(DOCNAME).snm $(DOCNAME).toc
$(DOCNAME).pdf : $(SOURCES) $(BIBFILE)
$(TEX) $(TEXARGS) $<
diff --git a/presentation/figures/laplacian-1d.dat b/presentation/figures/laplacian-1d.dat
new file mode 100644
index 0000000..6622398
--- /dev/null
+++ b/presentation/figures/laplacian-1d.dat
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diff --git a/presentation/figures/laplacian-1d.py b/presentation/figures/laplacian-1d.py
new file mode 100644
index 0000000..65d29d0
--- /dev/null
+++ b/presentation/figures/laplacian-1d.py
@@ -0,0 +1,32 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+@np.vectorize
+def fn(x):
+ return (x ** 2) * 2 / 100 + (1 + x / 4) + np.sin(x)
+
+@np.vectorize
+def ddfn(x):
+ return 2 * 5 / 100 - np.sin(x)
+
+x = np.linspace(0, 10, 500)
+y = fn(x)
+ddy = ddfn(x)
+
+cmap = ddy - np.min(ddy)
+cmap = cmap * 1000 / np.max(cmap)
+
+plt.plot(x, y)
+plt.plot(x, ddy)
+# plt.plot(x, cmap)
+
+plt.show()
+
+fname = "laplacian-1d.dat"
+np.savetxt(fname, np.array([x, y, cmap]).T, delimiter=" ")
+
+# with open(fname, "w") as f:
+# # f.write("x y cmap\n")
+# for xv, yv, cv in zip(x, y, cmap):
+# f.write(f"{xv} {yv} {cv}\n")
diff --git a/presentation/figures/laplacian-3d.jpg b/presentation/figures/laplacian-3d.jpg
new file mode 100644
index 0000000..8ab1f4f
--- /dev/null
+++ b/presentation/figures/laplacian-3d.jpg
Binary files differ
diff --git a/presentation/figures/laplacian-sphere.jpg b/presentation/figures/laplacian-sphere.jpg
new file mode 100644
index 0000000..bcffd17
--- /dev/null
+++ b/presentation/figures/laplacian-sphere.jpg
Binary files differ