Merge remote-tracking branch 'upstream/master'

29 files changed, 1794 insertions, 4 deletions

diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
index d46dc7f..6ac5665 100644
--- a/vorlesungen/slides/8/Makefile.inc
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -28,5 +28,25 @@ chapter8 =								\
 	../slides/8/tokyo/bahn0.tex					\
 	../slides/8/tokyo/bahn1.tex					\
 	../slides/8/tokyo/bahn2.tex					\
+	../slides/8/chrind.tex						\
+	../slides/8/chrindprop.tex					\
+	../slides/8/chroma1.tex						\
+	../slides/8/amax.tex						\
+	../slides/8/subgraph.tex					\
+	../slides/8/chrwilf.tex						\
+	../slides/8/weitere.tex						\
+	../slides/8/wavelets/funktionen.tex				\
+	../slides/8/wavelets/laplacebasis.tex				\
+	../slides/8/wavelets/vektoren.tex				\
+	../slides/8/wavelets/fourier.tex				\
+	../slides/8/wavelets/lokalisierungsvergleich.tex		\
+	../slides/8/wavelets/frequenzlokalisierung.tex			\
+	../slides/8/wavelets/dilatation.tex				\
+	../slides/8/wavelets/matrixdilatation.tex			\
+	../slides/8/wavelets/gundh.tex					\
+	../slides/8/wavelets/dilbei.tex					\
+	../slides/8/wavelets/frame.tex					\
+	../slides/8/wavelets/framekonstanten.tex			\
+	../slides/8/wavelets/beispiel.tex				\
 	../slides/8/chapter.tex
 
diff --git a/vorlesungen/slides/8/amax.tex b/vorlesungen/slides/8/amax.tex
new file mode 100644
index 0000000..951400a
--- /dev/null
+++ b/vorlesungen/slides/8/amax.tex
@@ -0,0 +1,86 @@
+%
+% amax.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\alpha_{\text{max}}$ und $d$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Definition}
+$\alpha_{\text{max}}$ ist der grösste Eigenwert der Adjazenzmatrix
+\end{block}
+\uncover<2->{
+\begin{block}{Fakten}
+\begin{itemize}
+\item<3->
+Der Eigenwert $\alpha_{\text{max}}$ ist einfach
+\item<4->
+Es gibt einen positiven Eigenvektor $f$ zum Eigenwert $\alpha_{\text{max}}$
+\item<5->
+$f$ maximiert
+\[
+\frac{\langle Af,f\rangle}{\langle f,f\rangle}
+=
+\alpha_{\text{max}}
+\]
+\end{itemize}
+Herkunft: Perron-Frobenius-Theorie positiver Matrizen (nächste Woche)
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<6->{%
+\begin{block}{Mittlerer Grad}
+\[
+\overline{d}
+=
+\frac1{n} \sum_{v} \operatorname{deg}(v)
+\le
+\alpha_{\text{max}}
+\le
+d
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<7->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item Konstante Funktion $1$ anstelle von $f$:
+\[
+\frac{\langle A1,1\rangle}{\langle 1,1\rangle}
+\uncover<8->{=
+\frac{\sum_v \operatorname{deg}(v)}{n}}
+\uncover<9->{=
+\overline{d}}
+\uncover<10->{\le
+\alpha_{\text{max}}}
+\]
+\item<11-> Komponenten von $Af$ summieren:
+\begin{align*}
+\uncover<12->{
+\alpha_{\text{max}}
+f(v) &= (Af)(v)}\uncover<13->{ = \sum_{u\sim v} f(u)}
+\\
+\uncover<14->{\alpha_{\text{max}}
+\sum_{v}f(v)
+&=
+\sum_v
+\operatorname{deg}(v) f(v)}
+\\
+&\uncover<15->{\le
+d\sum_v f(v)}
+\;
+\uncover<16->{\Rightarrow
+\;
+\alpha_{\text{max}} \le d}
+\end{align*}
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex
index 6a0b13f..69b7231 100644
--- a/vorlesungen/slides/8/chapter.tex
+++ b/vorlesungen/slides/8/chapter.tex
@@ -30,3 +30,24 @@
 \folie{8/tokyo/bahn1.tex}
 \folie{8/tokyo/bahn2.tex}
 
+\folie{8/chrind.tex}
+\folie{8/chrindprop.tex}
+\folie{8/chroma1.tex}
+\folie{8/amax.tex}
+\folie{8/subgraph.tex}
+\folie{8/chrwilf.tex}
+\folie{8/weitere.tex}
+
+\folie{8/wavelets/funktionen.tex}
+\folie{8/wavelets/laplacebasis.tex}
+\folie{8/wavelets/fourier.tex}
+\folie{8/wavelets/lokalisierungsvergleich.tex}
+\folie{8/wavelets/frequenzlokalisierung.tex}
+\folie{8/wavelets/dilatation.tex}
+\folie{8/wavelets/matrixdilatation.tex}
+\folie{8/wavelets/gundh.tex}
+\folie{8/wavelets/frame.tex}
+\folie{8/wavelets/dilbei.tex}
+\folie{8/wavelets/framekonstanten.tex}
+\folie{8/wavelets/beispiel.tex}
+
diff --git a/vorlesungen/slides/8/chrind.tex b/vorlesungen/slides/8/chrind.tex
new file mode 100644
index 0000000..bd406ab
--- /dev/null
+++ b/vorlesungen/slides/8/chrind.tex
@@ -0,0 +1,231 @@
+%
+% chrind.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Chromatische Zahl und Unabhängigkeitszahl}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Chromatische Zahl}
+$\operatorname{chr}(G)=\mathstrut$
+minimale Anzahl Farben, die zum Einfärben eines Graphen $G$ nötig sind derart,
+dass benachbarte Knoten verschiedene Farben haben.
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\Ra{2}
+\def\Ri{1}
+\def\e{1.0}
+\def\r{0.2}
+
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\only<1>{
+        \fill[color=white] (PA) circle[radius=\r];
+        \fill[color=white] (PB) circle[radius=\r];
+        \fill[color=white] (PC) circle[radius=\r];
+        \fill[color=white] (PD) circle[radius=\r];
+        \fill[color=white] (PE) circle[radius=\r];
+        \fill[color=white] (QA) circle[radius=\r];
+        \fill[color=white] (QB) circle[radius=\r];
+        \fill[color=white] (QC) circle[radius=\r];
+        \fill[color=white] (QD) circle[radius=\r];
+        \fill[color=white] (QE) circle[radius=\r];
+}
+
+\only<2->{
+        \fill[color=blau] (PA) circle[radius=\r];
+        \fill[color=rot] (PB) circle[radius=\r];
+        \fill[color=rot] (PC) circle[radius=\r];
+        \fill[color=gruen] (PD) circle[radius=\r];
+        \fill[color=gruen] (PE) circle[radius=\r];
+
+        \fill[color=rot] (QA) circle[radius=\r];
+        \fill[color=blau] (QB) circle[radius=\r];
+        \fill[color=gruen] (QC) circle[radius=\r];
+        \fill[color=rot] (QD) circle[radius=\r];
+        \fill[color=blau] (QE) circle[radius=\r];
+}
+
+\draw (PA) circle[radius=\r];
+\draw (PB) circle[radius=\r];
+\draw (PC) circle[radius=\r];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QA) circle[radius=\r];
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QD) circle[radius=\r];
+\draw (QE) circle[radius=\r];
+
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{chr} G = 3$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Unabhängigkeitszahl}
+$\operatorname{ind}(G)=\mathstrut$
+maximale Anzahl nicht benachbarter Knoten
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\Ra{2}
+\def\Ri{1}
+\def\e{1.0}
+\def\r{0.2}
+
+\definecolor{rot}{rgb}{0.8,0,0.8}
+\definecolor{gruen}{rgb}{0.2,0.6,0.2}
+\definecolor{blau}{rgb}{1,0.6,0.2}
+\definecolor{gelb}{rgb}{0,0,1}
+
+\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)});
+\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)});
+\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)});
+\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)});
+\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)});
+
+\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)});
+\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)});
+\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)});
+\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)});
+\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)});
+
+\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle;
+\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle;
+\draw (PA)--(QA);
+\draw (PB)--(QB);
+\draw (PC)--(QC);
+\draw (PD)--(QD);
+\draw (PE)--(QE);
+
+\foreach \n in {1,...,7}{
+	\only<\n>{\node[color=white] at (1,2.9) {$\n$};}
+}
+
+\fill[color=white] (PA) circle[radius=\r];
+\fill[color=white] (PB) circle[radius=\r];
+\fill[color=white] (PC) circle[radius=\r];
+\fill[color=white] (PD) circle[radius=\r];
+\fill[color=white] (PE) circle[radius=\r];
+\fill[color=white] (QA) circle[radius=\r];
+\fill[color=white] (QB) circle[radius=\r];
+\fill[color=white] (QC) circle[radius=\r];
+\fill[color=white] (QD) circle[radius=\r];
+\fill[color=white] (QE) circle[radius=\r];
+
+\only<4->{
+        \fill[color=rot] (QA) circle[radius={1.5*\r}];
+        \fill[color=rot!40] (QB) circle[radius=\r];
+        \fill[color=rot!40] (QE) circle[radius=\r];
+        \fill[color=rot!40] (PA) circle[radius=\r];
+}
+
+\only<5->{
+        \fill[color=blau] (PB) circle[radius={1.5*\r}];
+        \fill[color=blau!40] (PD) circle[radius=\r];
+        \fill[color=blau!40] (PE) circle[radius=\r];
+        \fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r];
+}
+
+\only<6->{
+        \fill[color=gruen] (PC) circle[radius={1.5*\r}];
+        \fill[color=gruen!40] (QC) circle[radius=\r];
+        \fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r];
+        \fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r];
+}
+
+\only<7->{
+        \fill[color=gelb] (QD) circle[radius={1.5*\r}];
+        \fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r];
+        \fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r];
+        \fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r];
+}
+
+\only<-3|handout:0>{
+	\draw (QA) circle[radius=\r];
+}
+\only<4->{
+	\draw (QA) circle[radius={1.5*\r}];
+}
+
+\only<-4|handout:0>{
+	\draw (PB) circle[radius=\r];
+}
+\only<5->{
+	\draw (PB) circle[radius={1.5*\r}];
+}
+
+\only<-5|handout:0>{
+	\draw (PC) circle[radius=\r];
+}
+\only<6->{
+	\draw (PC) circle[radius={1.5*\r}];
+}
+
+\only<-6|handout:0>{
+	\draw (QD) circle[radius=\r];
+}
+\only<7->{
+	\draw (QD) circle[radius={1.5*\r}];
+}
+
+\draw (PA) circle[radius=\r];
+\draw (PD) circle[radius=\r];
+\draw (PE) circle[radius=\r];
+
+\draw (QB) circle[radius=\r];
+\draw (QC) circle[radius=\r];
+\draw (QE) circle[radius=\r];
+
+\only<4|handout:0>{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 1$};
+}
+\only<5|handout:0>{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 2$};
+}
+\only<6|handout:0>{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 3$};
+}
+\only<7->{
+\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 4$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chrindprop.tex b/vorlesungen/slides/8/chrindprop.tex
new file mode 100644
index 0000000..094588c
--- /dev/null
+++ b/vorlesungen/slides/8/chrindprop.tex
@@ -0,0 +1,62 @@
+%
+% chrindprop.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zusammenhang zwischen $\operatorname{chr}G$ und $\operatorname{ind}G$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Proposition}
+Ist $G$ ein Graph mit $n$ Knoten, dann gilt
+\[
+\operatorname{chr}G
+\cdot
+\operatorname{ind}G
+\ge n
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Beispiel}
+Peterson-Graph $K$ hat $n=10$ Knoten:
+\[
+\operatorname{chr}(K)
+\cdot
+\operatorname{ind}(K)
+=
+3\cdot 4
+\ge 
+10
+=
+n
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<3->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<4-> eine minimale Färbung hat $\operatorname{chr}(G)$ Farben
+\item<5-> Sie teilt die Knoten in $\operatorname{chr}(G)$
+gleichfarbige Mengen auf
+\item<6-> Jede einfarbige Menge von Knoten ist unabhängig, d.~h.~sie 
+besteht aus Knoten, die nicht miteinander verbunden sind.
+\item<7-> Jede einfarbige Menge enthält höchstens $\operatorname{ind}(G)$
+\item<8-> Die Gesamtzahl der Knoten ist
+\[
+n\uncover<9->{=\sum_{\text{Farbe}}\underbrace{|V_{\text{Farbe}}|}_{\le \operatorname{ind}(G)}}
+\uncover<10->{\le
+\operatorname{chr}(G)
+\cdot
+\operatorname{ind}(G)}
+\]
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chroma1.tex b/vorlesungen/slides/8/chroma1.tex
new file mode 100644
index 0000000..6a55704
--- /dev/null
+++ b/vorlesungen/slides/8/chroma1.tex
@@ -0,0 +1,56 @@
+%
+% chroma1.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Schranke für $\operatorname{chr}(G)$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Proposition}
+Ist $G$ ein Graph mit maximalem Grad $d$, dann gilt
+\[
+\operatorname{chr}(G) \le d + 1
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Beispiel}
+\begin{itemize}
+\item<3->
+Peterson-Graph $G$: maximaler Grad ist $d=3$, aber
+\[
+\operatorname{chr}(G)
+=
+3
+< d+1=4
+\]
+\item<4->
+Voller Graph $V$: maximaler Grad ist $d=n-1$,
+\[
+\operatorname{chr}(V) = n = d+1
+\]
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\uncover<4->{%
+\begin{proof}[Beweis]
+Mit vollständiger Induktion, d.~h.~Annahme: Graphen mit $<n$ Knoten und
+maximalem Grad $d$ lassen sich mit höchstens $d+1$ Farben färben.
+\begin{itemize}
+\item<5-> $X$ ein Graph mit $n$ Knoten
+\item<6-> entferne den Knoten $v\in X$, $X'=X\setminus\{v\}$
+\item<7-> $X'$ lässt sich mit höchstens $d+1$ Farben einfärben
+\item<8-> $v$ hat höchstens $d$ Nachbarn, die höchsten $d$ verschiedene
+Farben haben
+\item<9-> Es bleibt eine Farbe für $v$
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/chrwilf.tex b/vorlesungen/slides/8/chrwilf.tex
new file mode 100644
index 0000000..7edb10e
--- /dev/null
+++ b/vorlesungen/slides/8/chrwilf.tex
@@ -0,0 +1,115 @@
+%
+% chrwilf.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\def\kante#1#2{
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (#1) -- (#2);
+}
+\def\knoten#1#2{
+	\uncover<8->{
+		\fill[color=#2!30] (#1) circle[radius=0.2];
+		\draw[color=#2] (#1) circle[radius=0.2];
+	}
+	\only<-7>{
+		\draw (#1) circle[radius=0.2];
+	}
+}
+\def\R{1.5}
+\definecolor{rot}{rgb}{1,0,0}
+\definecolor{gruen}{rgb}{0,0.6,0}
+\definecolor{blau}{rgb}{0,0,1}
+\begin{frame}[t]
+\frametitle{Schranke für die chromatische Zahl}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Wilf)}
+$\uncover<2->{\operatorname{chr}(X) \le 1+}\alpha_{\text{max}} \le\uncover<2->{ 1 + }d$
+\end{block}
+\uncover<3->{%
+\begin{block}{Beispiel}
+\begin{align*}
+\uncover<4->{d&= 4}
+&&\uncover<5->{\Rightarrow& \operatorname{chr}(G) &\le 5}\\
+\uncover<6->{\alpha_{\text{max}} &= 
+2.9565}
+&&\uncover<7->{\Rightarrow& \operatorname{chr}(G) &\le 3}\\
+\uncover<4->{\overline{d} &= \frac{24}{9}=\rlap{$2.6666$}}
+\end{align*}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\coordinate (A) at (0:\R);
+\coordinate (B) at (40:\R);
+\coordinate (C) at (80:\R);
+\coordinate (D) at (120:\R);
+\coordinate (E) at (160:\R);
+\coordinate (F) at (200:\R);
+\coordinate (G) at (240:\R);
+\coordinate (H) at (280:\R);
+\coordinate (I) at (320:\R);
+
+\knoten{A}{rot}
+\knoten{B}{blau}
+\knoten{C}{gruen}
+\knoten{D}{blau}
+\knoten{E}{rot}
+\knoten{F}{blau}
+\knoten{G}{rot}
+\knoten{H}{gruen}
+\knoten{I}{blau}
+
+\kante{A}{B}
+\kante{B}{C}
+\kante{C}{D}
+\kante{D}{E}
+\kante{E}{F}
+\kante{F}{G}
+\kante{G}{H}
+\kante{H}{I}
+\kante{I}{A}
+
+\kante{A}{C}
+\kante{A}{D}
+\kante{D}{G}
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<9->{%
+\begin{proof}[Beweis]
+Induktion nach der Grösse $n$ des Graphen.
+\begin{itemize}
+\item<10->
+Entferne $v\in X$ mit minimalem Grad: $X'=X\setminus \{v\}$
+\item<11->
+Induktionsannahme:
+\[
+\operatorname{chr}(X')
+\le
+1+
+\alpha_{\text{max}}' 
+\]
+\item<12->
+$X'$ kann mit höhcstens $1+\alpha_{\text{max}}'\le 1+\alpha_{\text{max}}$
+Farben eingefärbt werden.
+\item<13->
+Wegen
+\(
+\deg(v) \le \overline{d} \le \alpha_{\text{max}}
+\)
+hat $v$ höchstens $\alpha_{\text{max}}$ Nachbarn, um $v$ zu färben,
+braucht man also höchstens $1+\alpha_{\text{max}}$ Farben.
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex
index 952c85b..10f88cd 100644
--- a/vorlesungen/slides/8/inzidenz.tex
+++ b/vorlesungen/slides/8/inzidenz.tex
@@ -5,6 +5,8 @@
 %
 \bgroup
 \definecolor{darkgreen}{rgb}{0,0.6,0}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
 \begin{frame}[t]
 \frametitle{Inzidenz- und Adjazenzmatrix}
 \vspace{-20pt}
@@ -67,7 +69,7 @@
 \vspace{-10pt}
 \uncover<5->{%
 \begin{block}{Definition}
-\vspace{-15pt}
+%\vspace{-15pt}
 \begin{align*}
 B(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante $j$ endet in Knoten $i$}\\
 A(G)_{ij}&=1&&\Leftrightarrow&&\text{Kante zwischen Knoten $i$ und $j$}
diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex
index 5f2f51a..43e5330 100644
--- a/vorlesungen/slides/8/inzidenzd.tex
+++ b/vorlesungen/slides/8/inzidenzd.tex
@@ -5,6 +5,8 @@
 %
 \bgroup
 \definecolor{darkgreen}{rgb}{0,0.6,0}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
 \begin{frame}[t]
 \frametitle{Inzidenz- und Adjazenz-Matrix}
 \vspace{-20pt}
@@ -67,7 +69,7 @@
 \vspace{-15pt}
 \uncover<5->{%
 \begin{block}{Definition}
-\vspace{-20pt}
+%\vspace{-20pt}
 \begin{align*}
 B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\
 B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\
diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex
index 1d8b725..93333bc 100644
--- a/vorlesungen/slides/8/produkt.tex
+++ b/vorlesungen/slides/8/produkt.tex
@@ -56,7 +56,7 @@
 \end{center}
 \vspace{-15pt}
 \begin{block}{Berechne}
-\vspace{-20pt}
+%\vspace{-20pt}
 \begin{align*}
 \uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t
 \end{align*}
diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex
index 425fe1c..62180d9 100644
--- a/vorlesungen/slides/8/spanningtree.tex
+++ b/vorlesungen/slides/8/spanningtree.tex
@@ -3,6 +3,7 @@
 %
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
+\bgroup
 \begin{frame}
 \frametitle{Spannbäume}
 
@@ -121,7 +122,7 @@ Wieviele Spannbäume gibt es?
 \begin{column}{0.56\hsize}
 \uncover<5->{%
 \begin{block}{Laplace-Matrix}
-\vspace{-15pt}
+%\vspace{-15pt}
 \[
 L=
 \tiny
@@ -162,3 +163,4 @@ L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{arr
 \end{columns}
 
 \end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/subgraph.tex b/vorlesungen/slides/8/subgraph.tex
new file mode 100644
index 0000000..f3005f9
--- /dev/null
+++ b/vorlesungen/slides/8/subgraph.tex
@@ -0,0 +1,60 @@
+%
+% subgraph.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\alpha_{\text{max}}$ eines Untergraphen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz}
+$X'$ ein echter Untergraph von $X$ mit Adjazenzmatrix $A'$ und grösstem
+Eigenwert $\alpha_{\text{max}}'$
+\[
+\alpha_{\text{max}}' \le \alpha_{\text{max}}
+\]
+\end{block}
+\uncover<2->{$V'$ die Knoten von $X'$}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{proof}[Beweis]
+\begin{itemize}
+\item<4->
+$f'$ der positive Eigenvektor von $A'$
+\item<5->
+Definiere 
+\[
+g(v)
+=
+\begin{cases}
+f'(v) &\qquad v\in V'\\
+0     &\qquad \text{sonst}
+\end{cases}
+\]
+\item<6-> Skalarprodukte:
+\begin{align*}
+\uncover<7->{\langle f',f'\rangle &= \langle g,g\rangle}
+\\
+\uncover<8->{\langle A'f',f'\rangle &\le \langle Ag,g\rangle}
+\end{align*}
+\item<9-> Vergleich
+\[
+\alpha_{\text{max}}'
+=
+\frac{\langle A'f',f'\rangle}{\langle f',f'\rangle}
+\uncover<10->{\le 
+\frac{\langle Ag,g\rangle}{\langle g,g\rangle}}
+\uncover<11->{\le
+\alpha_{\text{max}}}
+\]
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/Makefile b/vorlesungen/slides/8/wavelets/Makefile
new file mode 100644
index 0000000..3b4a5ce
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/Makefile
@@ -0,0 +1,8 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+vektoren.tex:	ev.m
+	octave ev.m
diff --git a/vorlesungen/slides/8/wavelets/beispiel.tex b/vorlesungen/slides/8/wavelets/beispiel.tex
new file mode 100644
index 0000000..dcc33d4
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/beispiel.tex
@@ -0,0 +1,44 @@
+%
+% beispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{
+\node at (0,0) [rotate=-90]
+{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wavelets auf einer Kugel}
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\only<1>{ \bild{0.6}{wavelets-phi-sphere-334.pdf} }
+
+\only<2>{ \bild{0.6}{wavelets-psi-5-sphere-334.pdf} }
+\only<3>{ \bild{0.6}{wavelets-psi-4-sphere-334.pdf} }
+\only<4>{ \bild{0.6}{wavelets-psi-3-sphere-334.pdf} }
+\only<5>{ \bild{0.6}{wavelets-psi-2-sphere-334.pdf} }
+\only<6>{ \bild{0.6}{wavelets-psi-1-sphere-334.pdf} }
+
+\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; }
+\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; }
+\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; }
+\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; }
+\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; }
+\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; }
+
+\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; }
+\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; }
+\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; }
+\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; }
+\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; }
+\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; }
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/dilatation.tex b/vorlesungen/slides/8/wavelets/dilatation.tex
new file mode 100644
index 0000000..881f760
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/dilatation.tex
@@ -0,0 +1,62 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Dilatation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Dilatation in $\mathbb{R}$}
+$f\colon \mathbb{R}\to\mathbb{R}$
+Definition im Ortsraum:
+\[
+(D_af)(x)
+=
+\frac{1}{\sqrt{|a|}}
+f\biggl(\frac{x}{a}\biggr)
+\]
+\uncover<2->{%
+Dilatation im Frequenzraum:
+\[
+\widehat{D_af}(\omega)
+=
+D_{1/a}\hat{f}(\omega)
+\]}
+\uncover<3->{%
+Spektrum wird mit $1/a$ skaliert!}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{``Dilatation'' auf einem Graphen}
+\begin{itemize}
+\item<5-> Dilatation auf dem Graphen gibt es nicht
+\item<6-> Dilatation im Spektrum $\{\lambda_1,\dots,\lambda_n\}$ gibt es nicht
+\item<7-> ``Spektrale Dilatation'' verwenden
+\begin{enumerate}
+\item<8-> Start: $e_k$
+\item<9-> Fourier-Transformation: $\chi^te_k$
+\item<10-> Spektrum skalieren: mit
+$D_{1/a}g$ filtern
+\item<11-> Rücktransformation
+\[
+D_{g,a}e_k
+=
+\chi
+\uncover<12->{\operatorname{diag}(\tilde{D}_{1/a}g(\lambda_*))
+\chi^t e_k}
+\]
+\end{enumerate}
+\end{itemize}
+
+
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/dilbei.tex b/vorlesungen/slides/8/wavelets/dilbei.tex
new file mode 100644
index 0000000..fc66a0a
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/dilbei.tex
@@ -0,0 +1,46 @@
+%
+% beispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{
+\node at (0,0) [rotate=-90]
+{\includegraphics[width=#1\textwidth]{../../../SeminarWavelets/buch/papers/sgwt/images/#2}};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wavelets einer Strecke}
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\only<1>{ \bild{0.6}{wavelets-psi-line-5-10.pdf} }
+\only<2>{ \bild{0.6}{wavelets-psi-line-4-10.pdf} }
+\only<3>{ \bild{0.6}{wavelets-psi-line-3-10.pdf} }
+\only<4>{ \bild{0.6}{wavelets-psi-line-2-10.pdf} }
+\only<5>{ \bild{0.6}{wavelets-psi-line-1-10.pdf} }
+
+\only<6>{ \bild{0.6}{wavelets-phi-line-10.pdf} }
+
+\only<1>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_1$}; }
+\only<2>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_2$}; }
+\only<3>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_3$}; }
+\only<4>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_4$}; }
+\only<5>{ \node at (-7.6,2.8) [right] {Bandpass mit $g_5$}; }
+\only<6>{ \node at (-7.6,2.8) [right] {Tiefpass mit $h$}; }
+
+
+\only<1>{ \node at (-7.6,2) [right] {$D_{g,1/a_1}\chi_*$}; }
+\only<2>{ \node at (-7.6,2) [right] {$D_{g,1/a_2}\chi_*$}; }
+\only<3>{ \node at (-7.6,2) [right] {$D_{g,1/a_3}\chi_*$}; }
+\only<4>{ \node at (-7.6,2) [right] {$D_{g,1/a_4}\chi_*$}; }
+\only<5>{ \node at (-7.6,2) [right] {$D_{g,1/a_5}\chi_*$}; }
+
+\only<6>{ \node at (-7.6,2) [right] {$D_{h}\chi_*$}; }
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/ev.m b/vorlesungen/slides/8/wavelets/ev.m
new file mode 100644
index 0000000..7f4dd55
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/ev.m
@@ -0,0 +1,97 @@
+#
+# ev.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+L = [
+    2, -1,  0, -1,  0;
+   -1,  4, -1, -1, -1;
+    0, -1,  2,  0, -1;
+   -1, -1,  0,  3, -1;
+    0, -1, -1, -1,  3
+];
+
+[v, lambda] = eig(L);
+
+function knoten(fn, wert, punkt)
+	if (wert > 0) 
+		farbe = sprintf("red!%02d", round(100 * wert));
+	else
+		farbe = sprintf("blue!%02d", round(-100 * wert));
+	end
+	fprintf(fn, "\t\\fill[color=%s] %s circle[radius=0.25];\n",
+		farbe, punkt);
+	fprintf(fn, "\t\\draw %s circle[radius=0.25];\n", punkt);
+endfunction
+
+function vektor(fn, v, name, lambda)
+	fprintf(fn, "\\def\\%s{\n", name);
+	fprintf(fn, "\t\\coordinate (A) at ({0*\\a},0);\n");
+	fprintf(fn, "\t\\coordinate (B) at ({1*\\a},0);\n");
+	fprintf(fn, "\t\\coordinate (C) at ({2*\\a},0);\n");
+	fprintf(fn, "\t\\coordinate (D) at ({0.5*\\a},{-\\b});\n");
+	fprintf(fn, "\t\\coordinate (E) at ({1.5*\\a},{-\\b});\n");
+	fprintf(fn, "\t\\draw (A) -- (B);\n");
+	fprintf(fn, "\t\\draw (A) -- (D);\n");
+	fprintf(fn, "\t\\draw (B) -- (C);\n");
+	fprintf(fn, "\t\\draw (B) -- (D);\n");
+	fprintf(fn, "\t\\draw (B) -- (E);\n");
+	fprintf(fn, "\t\\draw (C) -- (E);\n");
+	fprintf(fn, "\t\\draw (D) -- (E);\n");
+	fprintf(fn, "\t\\node at (-2.8,{-0.5*\\b}) [right] {$\\lambda=%.4f$};\n",
+		round(1000 * abs(lambda)) / 10000);
+	w = v / max(abs(v));
+	knoten(fn, w(1,1), "(A)");
+	knoten(fn, w(2,1), "(B)");
+	knoten(fn, w(3,1), "(C)");
+	knoten(fn, w(4,1), "(D)");
+	knoten(fn, w(5,1), "(E)");
+	fprintf(fn, "}\n");
+endfunction
+
+function punkt(fn, x, wert)
+	fprintf(fn, "({%.4f*\\c},{%.4f*\\d})", x, wert);
+endfunction
+
+function funktion(fn, v, name, lambda)
+	fprintf(fn, "\\def\\%s{\n", name);
+	fprintf(fn, "\t\\draw[color=red,line width=1.4pt]\n\t\t");
+	punkt(fn, -2, v(1,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, -1, v(4,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, 0, v(2,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, 1, v(5,1));
+	fprintf(fn, " --\n\t\t");
+	punkt(fn, 2, v(3,1));
+	fprintf(fn, ";\n");
+	fprintf(fn, "\t\\draw[->] ({-2.1*\\c},0) -- ({2.1*\\c},0);\n");
+	fprintf(fn, "\t\\draw[->] (0,{-1.1*\\d}) -- (0,{1.1*\\d});\n");
+	for x = (-2:2)
+		fprintf(fn, "\t\\fill ({%d*\\c},0) circle[radius=0.05];\n", x);
+	endfor
+	fprintf(fn, "}\n");
+endfunction
+
+fn = fopen("vektoren.tex", "w");
+
+vektor(fn, v(:,1), "vnull", lambda(1,1));
+funktion(fn, v(:,1), "fnull", lambda(1,1));
+
+vektor(fn, v(:,2), "vone", lambda(2,2));
+funktion(fn, v(:,2), "fone", lambda(2,2));
+
+vektor(fn, v(:,3), "vtwo", lambda(3,3));
+funktion(fn, v(:,3), "ftwo", lambda(3,3));
+
+vektor(fn, v(:,4), "vthree", lambda(4,4));
+funktion(fn, v(:,4), "fthree", lambda(4,4));
+
+vektor(fn, v(:,5), "vfour", lambda(5,5));
+funktion(fn, v(:,5), "ffour", lambda(5,5));
+
+fclose(fn);
+
+
diff --git a/vorlesungen/slides/8/wavelets/fourier.tex b/vorlesungen/slides/8/wavelets/fourier.tex
new file mode 100644
index 0000000..3195ec8
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/fourier.tex
@@ -0,0 +1,86 @@
+%
+% fourier.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Fourier-Transformation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Gegeben: Funktion $f$ auf dem Graphen
+\\
+\uncover<2->{%
+Gesucht: Koeffizienten $\hat{f}$ der Darstellung in der Laplace-Basis}
+\end{block}
+\uncover<3->{%
+\begin{block}{Definition $\chi$-Matrix}
+Eigenwerte $0=\lambda_1<\lambda_2\le \dots \le \lambda_n$ von $L$
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}
+\node at (-1.9,0) [left] {$\chi=\mathstrut$};
+\node at (0,0) {$\left(\raisebox{0pt}[1.7cm][1.7cm]{\hspace{3.5cm}}\right)$};
+
+\fill[color=blue!20] (-1.7,-1.7) rectangle (-1.1,1.7);
+\draw[color=blue] (-1.7,-1.7) rectangle (-1.1,1.7);
+\node at (-1.4,0) [rotate=90] {$v_1=\mathstrut$EV zum EW $\lambda_1$\strut};
+
+\fill[color=blue!20] (-1.0,-1.7) rectangle (-0.4,1.7);
+\draw[color=blue] (-1.0,-1.7) rectangle (-0.4,1.7);
+\node at (-0.7,0) [rotate=90] {$v_2=\mathstrut$EV zum EW $\lambda_2$\strut};
+
+\fill[color=blue!20] (1.1,-1.7) rectangle (1.7,1.7);
+\draw[color=blue] (1.1,-1.7) rectangle (1.7,1.7);
+\node at (1.4,0) [rotate=90] {$v_n=\mathstrut$EV zum EW $\lambda_n$\strut};
+
+\node at (0.4,0) {$\dots$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Transformation}
+$L$ symmetrisch
+\\
+\uncover<5->{$\Rightarrow$ 
+Die Eigenvektoren von $L$ können orthonormiert gewählt werden}
+\\
+\uncover<6->{$\Rightarrow$ 
+Koeffizienten können durch Skalarprodukte ermittelt werden:}
+\uncover<7->{%
+\[
+\hat{f}(k)
+=
+\hat{f}(\lambda_k)
+\uncover<8->{=
+\langle v_k, f\rangle
+\quad\Rightarrow\quad
+\hat{f}}
+\uncover<9->{=
+\chi^tf}
+\]}
+\uncover<10->{%
+$\chi$ ist die {\em Fourier-Transformation}}
+\end{block}}
+\uncover<11->{%
+\begin{block}{Rücktransformation}
+Eigenvektoren orthonormiert
+\\
+\uncover<12->{$\Rightarrow$
+$\chi$ orthogonal}
+\uncover<13->{
+\[
+\chi\chi^t = I
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/frame.tex b/vorlesungen/slides/8/wavelets/frame.tex
new file mode 100644
index 0000000..4d0c7d1
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/frame.tex
@@ -0,0 +1,66 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Graph Wavelet Frame}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Frame-Vektoren}
+Zu Dilatationsfaktoren $A=\{a_i\,|\,i=1,\dots,N\}$
+konstruiere das Frame
+\begin{align*}
+F=
+\{&D_he_1,\dots,D_he_n,\\
+  &Dg_1e_1,\dots,Dg_1e_n,\\
+  &Dg_2e_1,\dots,Dg_2e_n,\\
+  &\dots\\
+  &Dg_Ne_1,\dots,Dg_Ne_n\}
+\end{align*}
+\uncover<2->{Notation:
+\begin{align*}
+v_{0,k}
+&=
+D_he_k
+\\
+v_{i,k}
+&=
+Dg_ie_k
+\end{align*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Frameoperator}
+\begin{align*}
+\mathcal{T}\colon \mathbb{R}^n\to\mathbb{R}^{nN}
+:
+v
+&\mapsto
+\begin{pmatrix}
+\uncover<4->{\langle D_he_1,v\rangle}\\
+\uncover<4->{\vdots}\\
+\uncover<4->{\langle D_he_n,v\rangle}\\
+\hline
+\uncover<5->{\langle D_{g_1}e_1,v\rangle}\\
+\uncover<5->{\vdots}\\
+\uncover<5->{\langle D_{g_1}e_n,v\rangle}\\
+\hline
+\uncover<6->{\vdots}\\
+\uncover<6->{\vdots}\\
+\hline
+\uncover<7->{\langle D_{g_N}e_1,v\rangle}\\
+\uncover<7->{\vdots}\\
+\uncover<7->{\langle D_{g_N}e_n,v\rangle}
+\end{pmatrix}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/framekonstanten.tex b/vorlesungen/slides/8/wavelets/framekonstanten.tex
new file mode 100644
index 0000000..a436536
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/framekonstanten.tex
@@ -0,0 +1,71 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+%\setlength{\abovedisplayskip}{5pt}
+%\setlength{\belowdisplayskip}{5pt}
+\frametitle{Framekonstanten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Menge $\mathcal{F}$ von Vektoren heisst ein Frame,
+falls es Konstanten $A$ und $B$ gibt derart, dass
+\[
+A\|v\|^2
+\le
+\|\mathcal{T}v\|^2
+\sum_{b\in\mathcal{F}} |\langle b,v\rangle|^2
+\le
+B\|v\|^2
+\]
+\uncover<2->{$A>0$ garantiert Invertierbarkeit}
+\end{block}
+\uncover<3->{%
+\begin{block}{$\|\mathcal{T}v\|$ für Graph-Wavelets}
+\begin{align*}
+\|\mathcal{T}v\|^2
+&=
+\sum_k |\langle D_he_k,v\rangle|^2
++
+\sum_{i,k} |\langle D_{g_i}e_k, v\rangle|^2
+\\
+&\uncover<4->{=
+\sum_k |h(\lambda_k) \hat{v}(k)|^2
++
+\sum_{k,i} |g_i(\lambda_k) \hat{v}(k)|^2}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{$A$ und $B$}
+Frame-Norm-Funktion
+\begin{align*}
+f(\lambda)
+&=
+h(\lambda)
++
+\sum_i g_i(\lambda)
+\\
+&\uncover<6->{=
+h(\lambda)
++
+\sum_i g(a_i\lambda)}
+\end{align*}
+\uncover<7->{Abschätzung für Frame-Konstanten
+\begin{align*}
+A&\uncover<8->{=
+\min_{i} f(\lambda_i)}
+\\
+B&\uncover<9->{=
+\max_{i} f(\lambda_i)}
+\end{align*}}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex
new file mode 100644
index 0000000..c78e6dd
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/frequenzlokalisierung.tex
@@ -0,0 +1,78 @@
+%
+% frequenzlokalisierung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\def\kurve#1#2{
+        \draw[color=#2,line width=1.4pt]
+                plot[domain=0:6.3,samples=400]
+                        ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1});
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lokalisierung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Bandpass}
+Gegeben durch $g(\lambda)\ge 0$:
+\begin{align*}
+g(0)                             &= 0\\
+\lim_{\lambda\to\infty}g(\lambda)&= 0
+\end{align*}
+\vspace{-10pt}
+\begin{enumerate}
+\item<3-> Fourier-transformieren
+\item<4-> Amplituden mit $g(\lambda)$ multiplizieren
+\item<5-> Rücktransformieren
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Tiefpass}
+Gegeben durch $h(\lambda)\ge0$:
+\begin{align*}
+h(0)                             &= 1\\
+\lim_{\lambda\to\infty}h(\lambda)&= 0
+\end{align*}
+\vspace{-10pt}
+\begin{enumerate}
+\item<8-> Fourier-Transformation
+\item<9-> Amplituden mit $h(\lambda)$ multiplizieren
+\item<10-> Rücktransformation
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.8]
+
+\uncover<2->{
+\begin{scope}[xshift=-4.5cm]
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+\kurve{3}{red}
+\draw[->] (0,-0.1) -- (0,3.3);
+\end{scope}
+}
+
+\uncover<7->{
+\begin{scope}[xshift=4.5cm]
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+\draw[color=darkgreen,line width=1.4pt]
+        plot[domain=0:6.3,samples=100]
+                ({\x},{3*exp(-(\x/0.5)*(\x/0.5)});
+
+\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}];
+\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/funktionen.tex b/vorlesungen/slides/8/wavelets/funktionen.tex
new file mode 100644
index 0000000..2e3ae9b
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/funktionen.tex
@@ -0,0 +1,78 @@
+%
+% funktionen.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\knoten#1#2{
+	\draw #1 circle[radius=0.25];
+	\node at #1 {$#2$};
+}
+\def\kante#1#2{
+	\draw[shorten >= 0.25cm,shorten <= 0.25cm] #1 -- #2;
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Funktionen auf einem Graphen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Ein Graph $G=(V,E)$, eine Funktion auf dem Graphen ist
+\[
+f\colon V \to \mathbb{R} : v\mapsto f(v)
+\]
+Knoten: $V=\{1,\dots,n\}$
+\\
+\uncover<2->{%
+Vektorschreibweise
+\[
+f = \begin{pmatrix}
+f(1)\\f(2)\\\vdots\\f(n)
+\end{pmatrix}
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Matrizen}
+Adjazenz-, Grad- und Laplace-Matrix operieren auf Funktionen auf Graphen:
+\[
+L
+=
+\begin{pmatrix*}[r]
+ 2&-1& 0&-1& 0\\
+-1& 4&-1&-1&-1\\
+ 0&-1& 2& 0&-1\\
+-1&-1& 0& 3&-1\\
+ 0&-1&-1&-1& 3\\
+\end{pmatrix*}
+\]
+\end{block}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{2}
+\coordinate (A) at (0,0);
+\coordinate (B) at (\a,0);
+\coordinate (C) at ({2*\a},0);
+\coordinate (D) at ({0.5*\a},{-0.5*sqrt(3)*\a});
+\coordinate (E) at ({1.5*\a},{-0.5*sqrt(3)*\a});
+\knoten{(A)}{1}
+\knoten{(B)}{2}
+\knoten{(C)}{3}
+\knoten{(D)}{4}
+\knoten{(E)}{5}
+\kante{(A)}{(B)}
+\kante{(B)}{(C)}
+\kante{(A)}{(D)}
+\kante{(B)}{(D)}
+\kante{(B)}{(E)}
+\kante{(C)}{(E)}
+\kante{(D)}{(E)}
+\end{tikzpicture}
+\end{center}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/gundh.tex b/vorlesungen/slides/8/wavelets/gundh.tex
new file mode 100644
index 0000000..2d6c677
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/gundh.tex
@@ -0,0 +1,85 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\kurve#1#2{
+        \draw[color=#2,line width=1.4pt]
+                plot[domain=0:6.3,samples=400]
+                        ({\x},{7*\x*exp(-(\x/#1)*(\x/#1))/#1});
+}
+
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wavelets}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Mutterwavelets + Dilatation}
+Eine Menge von Dilatationsfaktoren
+\[
+A= \{a_1,a_2,\dots,a_N\}
+\]
+wählen\uncover<2->{, und mit Funktionen
+\[
+{\color{blue}g_i} = \tilde{D}_{1/a_i}{\color{red}g}
+\]
+die Standardbasisvektoren filtern}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{
+\begin{block}{Vaterwavelets}
+Tiefpass mit Funktion ${\color{darkgreen}h(\lambda)}$,
+Standardbasisvektoren mit ${\color{darkgreen}h}$ filtern:
+\[
+D_{\color{darkgreen}h}e_k
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\kurve{1}{red}
+\uncover<4->{
+\foreach \k in {0,...,4}{
+        \pgfmathparse{0.30*exp(ln(2)*\k)}
+        \xdef\l{\pgfmathresult}
+        \kurve{\l}{blue}
+}
+}
+
+\node[color=red] at ({0.7*1},3) [above] {$g(\lambda)$};
+\uncover<4->{
+\node[color=blue] at ({0.7*0.3*16},3) [above] {$g_i(\lambda)$};
+}
+
+\draw[->] (0,-0.1) -- (0,3.3);
+\end{scope}
+
+\begin{scope}[xshift=7cm]
+
+\uncover<6->{
+\draw[->] (-0.1,0) -- (6.6,0) coordinate[label={$\lambda$}];
+
+\draw[color=darkgreen,line width=1.4pt]
+        plot[domain=0:6.3,samples=100]
+                ({\x},{3*exp(-(\x/0.5)*(\x/0.5)});
+
+\draw[->] (0,-0.1) -- (0,3.3) coordinate[label={right:$\color{darkgreen}h(\lambda)$}];
+}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/laplacebasis.tex b/vorlesungen/slides/8/wavelets/laplacebasis.tex
new file mode 100644
index 0000000..ced4c09
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/laplacebasis.tex
@@ -0,0 +1,62 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\a{2}
+\def\b{0.8}
+\def\c{1}
+\def\d{0.6}
+\input{../slides/8/wavelets/vektoren.tex}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Laplace-Basis}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\begin{scope}[yshift=-0.4cm,xshift=-5.5cm]
+\fnull
+\end{scope}
+
+\begin{scope}[yshift=-1.8cm,xshift=-5.5cm]
+\fone
+\end{scope}
+
+\begin{scope}[yshift=-3.2cm,xshift=-5.5cm]
+\ftwo
+\end{scope}
+
+\begin{scope}[yshift=-4.6cm,xshift=-5.5cm]
+\fthree
+\end{scope}
+
+\begin{scope}[yshift=-6.0cm,xshift=-5.5cm]
+\ffour
+\end{scope}
+
+\begin{scope}[yshift=0cm]
+\vnull
+\end{scope}
+
+\begin{scope}[yshift=-1.4cm]
+\vone
+\end{scope}
+
+\begin{scope}[yshift=-2.8cm]
+\vtwo
+\end{scope}
+
+\begin{scope}[yshift=-4.2cm]
+\vthree
+\end{scope}
+
+\begin{scope}[yshift=-5.6cm]
+\vfour
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex
new file mode 100644
index 0000000..d6575d0
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/lokalisierungsvergleich.tex
@@ -0,0 +1,46 @@
+%
+% lokalisierungsvergleich.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lokalisierung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ortsraum}
+Ortsraum$\mathstrut=V$
+\begin{itemize}
+\item<3-> Standardbasis
+\item<5-> lokalisiert in den Knoten
+\item<7-> die meisten $\hat{f}(k)$ gross
+\item<9-> vollständig delokalisiert im Frequenzraum
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Frequenzraum}
+\uncover<2->{Frequenzraum $\mathstrut=\{\lambda_1,\lambda_2,\dots,\lambda_n\}$}
+\begin{itemize}
+\item<4-> Laplace-Basis
+\item<6-> lokalisiert in den Eigenwerten
+\item<8-> die meisten Komponenten gross
+\item<10-> vollständig delokalisiert im Ortsraum
+\end{itemize}
+\end{block}
+\end{column}
+\end{columns}
+\uncover<11->{%
+\begin{block}{Plan}
+Gesucht sind Funktionen auf dem Graphen derart, die
+\begin{enumerate}
+\item<12-> in der Nähe einzelner Knoten konzentriert/lokalisiert sind und
+\item<13-> deren Fourier-Transformation in der Nähe einzelner Eigenwerte 
+konzentriert/lokalisiert ist
+\end{enumerate}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/matrixdilatation.tex b/vorlesungen/slides/8/wavelets/matrixdilatation.tex
new file mode 100644
index 0000000..3536736
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/matrixdilatation.tex
@@ -0,0 +1,39 @@
+%
+% matrixdilatation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Dilatation in Matrixform}
+Dilatationsfaktor $a$, skaliertes Wavelet beim Knoten $k$ mit Spektrum
+$\tilde{D}_{1/a}g$
+\begin{align*}
+D_{g,a}e_k
+&=
+\chi
+\begin{pmatrix}
+g(a\lambda_1)&      0      & \dots  &      0      \\
+      0      &g(a\lambda_2)& \dots  &      0      \\
+   \vdots    &    \vdots   & \ddots &    \vdots   \\
+      0      &      0      & \dots  &g(a\lambda_n)
+\end{pmatrix}
+\chi^t
+e_k
+\intertext{\uncover<2->{``verschmierter'' Standardbasisvektor am Knoten $k$}}
+\uncover<2->{D_he_k
+&=
+\chi
+\begin{pmatrix}
+h(\lambda_1)&     0      & \dots  &     0      \\
+     0      &h(\lambda_2)& \dots  &     0      \\
+  \vdots    &   \vdots   & \ddots &   \vdots   \\
+     0      &     0      & \dots  &h(\lambda_n)
+\end{pmatrix}
+\chi^t
+e_k}
+\end{align*}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wavelets/vektoren.tex b/vorlesungen/slides/8/wavelets/vektoren.tex
new file mode 100644
index 0000000..2315d53
--- /dev/null
+++ b/vorlesungen/slides/8/wavelets/vektoren.tex
@@ -0,0 +1,200 @@
+\def\vnull{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.0000$};
+	\fill[color=red!100] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=red!100] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!100] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=red!100] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!100] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\fnull{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.4472*\d}) --
+		({-1.0000*\c},{0.4472*\d}) --
+		({0.0000*\c},{0.4472*\d}) --
+		({1.0000*\c},{0.4472*\d}) --
+		({2.0000*\c},{0.4472*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vone{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.1586$};
+	\fill[color=blue!100] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=blue!00] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!100] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=blue!41] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!41] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\fone{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{-0.6533*\d}) --
+		({-1.0000*\c},{-0.2706*\d}) --
+		({0.0000*\c},{-0.0000*\d}) --
+		({1.0000*\c},{0.2706*\d}) --
+		({2.0000*\c},{0.6533*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vtwo{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.3000$};
+	\fill[color=red!100] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=blue!00] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!100] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=blue!100] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=blue!100] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\ftwo{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.5000*\d}) --
+		({-1.0000*\c},{-0.5000*\d}) --
+		({0.0000*\c},{-0.0000*\d}) --
+		({1.0000*\c},{-0.5000*\d}) --
+		({2.0000*\c},{0.5000*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vthree{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.4414$};
+	\fill[color=red!41] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=red!00] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=blue!41] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=blue!100] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!100] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\fthree{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.2706*\d}) --
+		({-1.0000*\c},{-0.6533*\d}) --
+		({0.0000*\c},{0.0000*\d}) --
+		({1.0000*\c},{0.6533*\d}) --
+		({2.0000*\c},{-0.2706*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
+\def\vfour{
+	\coordinate (A) at ({0*\a},0);
+	\coordinate (B) at ({1*\a},0);
+	\coordinate (C) at ({2*\a},0);
+	\coordinate (D) at ({0.5*\a},{-\b});
+	\coordinate (E) at ({1.5*\a},{-\b});
+	\draw (A) -- (B);
+	\draw (A) -- (D);
+	\draw (B) -- (C);
+	\draw (B) -- (D);
+	\draw (B) -- (E);
+	\draw (C) -- (E);
+	\draw (D) -- (E);
+	\node at (-2.8,{-0.5*\b}) [right] {$\lambda=0.5000$};
+	\fill[color=red!25] (A) circle[radius=0.25];
+	\draw (A) circle[radius=0.25];
+	\fill[color=blue!100] (B) circle[radius=0.25];
+	\draw (B) circle[radius=0.25];
+	\fill[color=red!25] (C) circle[radius=0.25];
+	\draw (C) circle[radius=0.25];
+	\fill[color=red!25] (D) circle[radius=0.25];
+	\draw (D) circle[radius=0.25];
+	\fill[color=red!25] (E) circle[radius=0.25];
+	\draw (E) circle[radius=0.25];
+}
+\def\ffour{
+	\draw[color=red,line width=1.4pt]
+		({-2.0000*\c},{0.2236*\d}) --
+		({-1.0000*\c},{0.2236*\d}) --
+		({0.0000*\c},{-0.8944*\d}) --
+		({1.0000*\c},{0.2236*\d}) --
+		({2.0000*\c},{0.2236*\d});
+	\draw[->] ({-2.1*\c},0) -- ({2.1*\c},0);
+	\draw[->] (0,{-1.1*\d}) -- (0,{1.1*\d});
+	\fill ({-2*\c},0) circle[radius=0.05];
+	\fill ({-1*\c},0) circle[radius=0.05];
+	\fill ({0*\c},0) circle[radius=0.05];
+	\fill ({1*\c},0) circle[radius=0.05];
+	\fill ({2*\c},0) circle[radius=0.05];
+}
diff --git a/vorlesungen/slides/8/weitere.tex b/vorlesungen/slides/8/weitere.tex
new file mode 100644
index 0000000..46a3da0
--- /dev/null
+++ b/vorlesungen/slides/8/weitere.tex
@@ -0,0 +1,43 @@
+%
+% weitere.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Weitere Resultate der spektralen Graphentheorie}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Satz (Hoffmann)}
+\[
+\operatorname{chr} X \ge 1 + \frac{\alpha_{\text{max}}}{-\alpha_{\text{min}}}
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Satz (Hoffmann)}
+\[
+\operatorname{ind} X \le n \biggl(1-\frac{d_{\text{min}}}{\lambda_{\text{max}}}\biggr)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Korollar}
+Für einen regulären Graphen mit $n$ Knoten gilt
+\begin{align*}
+\operatorname{ind} X
+&\le
+\frac{n}{\displaystyle 1-\frac{d}{\alpha_{\text{min}}}}
+\\
+\operatorname{chr} X
+&\ge
+1-\frac{d}{\alpha_{\text{min}}}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/wilf.m b/vorlesungen/slides/8/wilf.m
new file mode 100644
index 0000000..49dc161
--- /dev/null
+++ b/vorlesungen/slides/8/wilf.m
@@ -0,0 +1,22 @@
+#
+# wilf.m -- chromatische Zahl für einen Graphen
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+N = 9;
+A = zeros(N,N);
+
+for i = (1:N)
+	j = 1 + rem(i, N)
+	A(i,j) = 1;
+endfor
+for i = (1:3:N-3)
+	j = 1 + rem(i + 2, N)
+	A(i,j) = 1;
+endfor
+
+A(1,3) = 1;
+
+A = A + A'
+
+eig(A)