Merge pull request #1 from AndreasFMueller/master

update

33 files changed, 2991 insertions, 0 deletions

diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
new file mode 100644
index 0000000..d46dc7f
--- /dev/null
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -0,0 +1,32 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter8 =								\
+	../slides/8/dgraph.tex						\
+	../slides/8/graph.tex						\
+	../slides/8/grad.tex						\
+	../slides/8/inzidenz.tex					\
+	../slides/8/inzidenzd.tex					\
+	../slides/8/diffusion.tex					\
+	../slides/8/laplace.tex						\
+	../slides/8/produkt.tex						\
+	../slides/8/fourier.tex						\
+	../slides/8/spanningtree.tex					\
+	../slides/8/pfade/adjazenz.tex					\
+	../slides/8/pfade/langepfade.tex				\
+	../slides/8/pfade/beispiel.tex					\
+	../slides/8/pfade/gf.tex					\
+	../slides/8/floyd-warshall/problem.tex				\
+	../slides/8/floyd-warshall/rekursion.tex			\
+	../slides/8/floyd-warshall/iteration.tex			\
+	../slides/8/floyd-warshall/wegiteration.tex			\
+	../slides/8/floyd-warshall/wege.tex				\
+	../slides/8/tokyo/google.tex					\
+	../slides/8/tokyo/bahn0.tex					\
+	../slides/8/tokyo/bahn1.tex					\
+	../slides/8/tokyo/bahn2.tex					\
+	../slides/8/chapter.tex
+
diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex
new file mode 100644
index 0000000..6a0b13f
--- /dev/null
+++ b/vorlesungen/slides/8/chapter.tex
@@ -0,0 +1,32 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{8/graph.tex}
+\folie{8/dgraph.tex}
+\folie{8/grad.tex}
+\folie{8/inzidenz.tex}
+\folie{8/inzidenzd.tex}
+\folie{8/diffusion.tex}
+\folie{8/laplace.tex}
+\folie{8/produkt.tex}
+\folie{8/fourier.tex}
+\folie{8/spanningtree.tex}
+
+\folie{8/pfade/adjazenz.tex}
+\folie{8/pfade/langepfade.tex}
+\folie{8/pfade/beispiel.tex}
+\folie{8/pfade/gf.tex}
+
+\folie{8/floyd-warshall/problem.tex}
+\folie{8/floyd-warshall/rekursion.tex}
+\folie{8/floyd-warshall/iteration.tex}
+\folie{8/floyd-warshall/wegiteration.tex}
+\folie{8/floyd-warshall/wege.tex}
+
+\folie{8/tokyo/google.tex}
+\folie{8/tokyo/bahn0.tex}
+\folie{8/tokyo/bahn1.tex}
+\folie{8/tokyo/bahn2.tex}
+
diff --git a/vorlesungen/slides/8/dgraph.tex b/vorlesungen/slides/8/dgraph.tex
new file mode 100644
index 0000000..6b5864a
--- /dev/null
+++ b/vorlesungen/slides/8/dgraph.tex
@@ -0,0 +1,100 @@
+%
+% dgraph.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}
+\frametitle{Gerichteter Graph}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2.4}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\uncover<3->{
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+	\draw[color=white,line width=5pt] (B) -- (D);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+}
+
+\uncover<2->{
+	\draw (A) circle[radius=0.2];
+	\draw (B) circle[radius=0.2];
+	\draw (C) circle[radius=0.2];
+	\draw (D) circle[radius=0.2];
+	\draw (E) circle[radius=0.2];
+
+	\node at (A) {$1$};
+	\node at (B) {$2$};
+	\node at (C) {$3$};
+	\node at (D) {$4$};
+	\node at (E) {$5$};
+}
+\node at (0,0) {$G$};
+
+\uncover<3->{
+	\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+	\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+	\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+	\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+	\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+	\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+	\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+}
+
+\uncover<7->{
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=red]
+                (E) to[out=-18,in=-126,distance=2cm] (E);
+}
+
+\uncover<9->{
+	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm,color=darkgreen]
+		(D) to[out=120,in=-120] (C);
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Definition}
+Ein gerichteter Graph $G=(V,E)$ ist
+\begin{enumerate}
+\item<2-> Eine Menge $V$ von Knoten (Vertizes)
+$V=\{v_1,v_2,\dots\}$
+\item<3->
+Eine Menge $E$ von gerichteten Kanten
+(Edges)
+\[
+E\subset \{ (v_1,v_2)\;|\; v_i\in V\}
+\]
+\end{enumerate}
+\end{block}
+\vspace{-30pt}
+\uncover<6->{%
+\begin{block}{Achtung}
+\begin{itemize}
+\item<6-> Kanten sind {\em geordnete} Paare
+\uncover<7->{$\Rightarrow$ {\color{red}Schleifen} sind möglich}
+\item<8-> Kanten sind immer ``Einbahnstrassen''
+\item<9-> {\color{darkgreen}Gegenrichtung explizit angeben}
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/diffusion.tex b/vorlesungen/slides/8/diffusion.tex
new file mode 100644
index 0000000..0d07a27
--- /dev/null
+++ b/vorlesungen/slides/8/diffusion.tex
@@ -0,0 +1,89 @@
+%
+% diffusion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Diffusion}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D);
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A);
+
+\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm]
+	(A) -- (E);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(A) -- (C);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(B) -- (D);
+
+\fill[color=red] (A) circle[radius=0.3];
+\fill[color=red!50] (B) circle[radius=0.3];
+\fill[color=white] (C) circle[radius=0.3];
+\fill[color=blue!50] (D) circle[radius=0.3];
+\fill[color=blue] (E) circle[radius=0.3];
+
+\draw (A) circle[radius=0.3];
+\draw (B) circle[radius=0.3];
+\draw (C) circle[radius=0.3];
+\draw (D) circle[radius=0.3];
+\draw (E) circle[radius=0.3];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\begin{block}{Knotenfunktion}
+$f\colon V\to \mathbb{R}$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Fluss}
+Je grösser die Differenz zu den Nachbarn, desto grösser der Fluss in 
+den Knoten:
+\begin{align*}
+\frac{df(v)}{dt}
+&=
+\kappa \sum_{\text{$v'$ Nachbar von $v$}} (f(v')-f(v))
+\end{align*}
+``Wärmeleitungsgleichung''
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/burgerking.png b/vorlesungen/slides/8/floyd-warshall/burgerking.png
new file mode 100644
index 0000000..cf4211d
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/burgerking.png
diff --git a/vorlesungen/slides/8/floyd-warshall/fw.tex b/vorlesungen/slides/8/floyd-warshall/fw.tex
new file mode 100644
index 0000000..99929fb
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/fw.tex
@@ -0,0 +1,680 @@
+%
+% fw.tex -- Durchführung des Floyd-Warshall Algorithmus
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+
+\definecolor{wegelb}{rgb}{1,0.6,0}
+\definecolor{weghell}{rgb}{1,0.9,0.6}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\begin{columns}[t]
+\begin{column}{0.44\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+
+\def\r{2.2}
+\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)});
+\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)});
+\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)});
+\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)});
+\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)});
+
+\def\knoten#1#2#3{
+	\fill[color=#3] #1 circle[radius=0.3];
+	\draw[line width=1pt] #1 circle[radius=0.3];
+	\node at #1 {$#2$};
+}
+
+\def\kante#1#2#3{
+	\draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm] #1 -- #2;
+	\fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22];
+	\node at ($0.5*#1+0.5*#2$) {$#3$};
+}
+
+% Wege über 1
+% 3--1--5
+\only<4>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E);
+}
+
+% Wege über 2
+% 5--2--3
+\only<6>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(C);
+}
+% 5--2--4
+\only<7>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(D);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(D);
+}
+
+% Wege über 3
+% 2--3--1
+\only<9>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A);
+}
+% 2--3--5
+\only<10>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E);
+}
+% 4--3--1
+\only<11>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A);
+}
+% 4--3--5
+\only<12>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E);
+}
+% 5--3--1
+\only<13>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(C)--(A);
+}
+% 5--3--5
+\only<14>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(B)--(C)--(A)--(E);
+}
+
+% Wege über 4
+% 2--4--1
+\only<16>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C)--(A);
+}
+% 2--4--3
+\only<17>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C);
+}
+% 2--4--5
+\only<18>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(D)--(C)--(A)--(E);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C)--(A)--(E);
+}
+% 5--4--1
+\only<19>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(B)--(C)--(A);
+}
+% 5--4--3
+\only<20>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(B)--(C);
+}
+% 5--4--5
+\only<21>{
+	\draw[->,line width=7pt,color=red!50] 
+		(E)--(D)--(C)--(A)--(E);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(E)--(B)--(C)--(A)--(E);
+}
+% 1--5--1
+\only<23>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(B)--(C)--(A);
+}
+% 1--5--2
+\only<24>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(B);
+}
+% 1--5--3
+\only<25>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(B)--(C);
+}
+% 1--5--4
+\only<26>{
+	\draw[->,line width=7pt,color=red!50] 
+		(A)--(E)--(D);
+}
+% 2--5--1
+\only<27>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C)--(A);
+}
+% 2--5--2
+\only<28>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(B);
+}
+% 2--5--3
+\only<29>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(C);
+}
+% 2--5--4
+\only<30>{
+	\draw[->,line width=7pt,color=red!50] 
+		(B)--(C)--(A)--(E)--(D);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(B)--(D);
+}
+% 3--5--1
+\only<31>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(C)--(A);
+}
+% 3--5--2
+\only<32>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(B);
+}
+% 3--5--3
+\only<33>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(B)--(C);
+}
+% 3--5--4
+\only<34>{
+	\draw[->,line width=7pt,color=red!50] 
+		(C)--(A)--(E)--(D);
+}
+% 4--5--1
+\only<35>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(D)--(C)--(A);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(D)--(C)--(A);
+}
+% 4--5--2
+\only<36>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(B);
+}
+% 4--5--3
+\only<37>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(D)--(C);
+	\draw[->,line width=7pt,color=blue!50,dashed] 
+		(D)--(C);
+}
+% 4--5--4
+\only<38>{
+	\draw[->,line width=7pt,color=red!50] 
+		(D)--(C)--(A)--(E)--(D);
+}
+
+
+\uncover<40>{
+	\draw[->,color=red!50,line width=7pt]
+		(B)--(C)--(A)--(E)--(D);
+}
+
+\kante{(A)}{(E)}{1}
+\kante{(B)}{(C)}{2}
+\kante{(B)}{(D)}{13}
+\kante{(C)}{(A)}{3}
+\kante{(D)}{(C)}{6}
+\kante{(E)}{(B)}{5}
+\kante{(E)}{(D)}{6}
+
+\only<1>{
+	\knoten{(A)}{}{white}
+	\knoten{(B)}{}{white}
+	\knoten{(C)}{}{white}
+	\knoten{(D)}{}{white}
+	\knoten{(E)}{}{white}
+}
+
+\only<2->{
+	\knoten{(A)}{1}{white}
+	\knoten{(B)}{2}{white}
+	\knoten{(C)}{3}{white}
+	\knoten{(D)}{4}{white}
+	\knoten{(E)}{5}{white}
+}
+
+\only<4>{
+	\knoten{(A)}{1}{darkgreen!50}
+}
+\only<6-7>{
+	\knoten{(B)}{2}{darkgreen!50}
+}
+\only<9-14>{
+	\knoten{(C)}{3}{darkgreen!50}
+}
+\only<16-21>{
+	\knoten{(D)}{4}{darkgreen!50}
+}
+\only<23-38>{
+	\knoten{(E)}{5}{darkgreen!50}
+}
+
+\end{tikzpicture}
+\end{center}
+\begin{block}{Aufgabe}
+Finde den kürzesten Weg von 2 nach 4
+\end{block}
+\end{column}
+\begin{column}{0.5\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex,scale=0.8]
+
+\def\punkt#1#2{
+	({#2-0.5},{0.5-(#1)})
+}
+\def\punktoff#1#2{
+	({#2-0.7},{0.7-(#1)})
+}
+\def\feld#1#2#3{
+	\ifthenelse{\boolean{wegweiser}}{
+		\fill[color=white]
+			({#2-1},{1-#1}) rectangle ({#2-0.45},{0.45-#1});
+		\node at \punktoff{#1}{#2} {$#3$};
+	}{
+		\fill[color=white]
+			({#2-1},{1-#1}) rectangle ({#2},{-#1});
+		\node at \punkt{#1}{#2} {$#3$};
+	}
+}
+\def\verbindung#1#2#3{
+	\draw[->,line width=5pt,color=red!20,shorten >= 0.2cm,shorten <= 0.2cm]
+		\punkt{#1}{#2}--\punkt{#2}{#3};
+	\node at (5,-5.5) [left] {$#1\rightsquigarrow #2\rightsquigarrow #3$\strut};
+}
+\def\Infty{{}}
+\def\wegweiser#1#2#3{
+	\ifthenelse{\boolean{wegweiser}}{
+		\ifnum #2 = #3
+		\fill[color=weghell]
+			({#2-0.45},{0.45-#1}) rectangle ({#2-0.05},{0.05-#1});
+		\else
+		\fill[color=wegelb]
+			({#2-0.45},{0.45-#1}) rectangle ({#2-0.05},{0.05-#1});
+		\fi
+		\node at ({#2-0.25},{0.25-#1}) {\tiny #3};
+	}{}
+}
+
+% direkte Wege
+\uncover<3->{
+	\feld{1}{1}{\Infty}
+	\feld{1}{2}{\Infty}
+	\feld{1}{3}{\Infty}
+	\feld{1}{4}{\Infty}
+	\feld{1}{5}{1}
+	\wegweiser{1}{5}{5}
+
+	\feld{2}{1}{\Infty}
+	\feld{2}{2}{\Infty}
+	\feld{2}{3}{2}
+	\wegweiser{2}{3}{3}
+	\feld{2}{4}{13}
+	\wegweiser{2}{4}{4}
+	\feld{2}{5}{\Infty}
+
+	\feld{3}{1}{3}
+	\wegweiser{3}{1}{1}
+	\feld{3}{2}{\Infty}
+	\feld{3}{3}{\Infty}
+	\feld{3}{4}{\Infty}
+	\feld{3}{5}{\Infty}
+
+	\feld{4}{1}{\Infty}
+	\feld{4}{2}{\Infty}
+	\feld{4}{3}{6}
+	\wegweiser{4}{3}{3}
+	\feld{4}{4}{\Infty}
+	\feld{4}{5}{\Infty}
+
+	\feld{5}{1}{\Infty}
+	\feld{5}{2}{5}
+	\wegweiser{5}{2}{2}
+	\feld{5}{3}{\Infty}
+	\feld{5}{4}{6}
+	\wegweiser{5}{4}{4}
+	\feld{5}{5}{\Infty}
+}
+
+\uncover<3-3>{
+	\node at (-0.8,-5.5) [right] {direkte Verbindungen};
+}
+
+\uncover<4-4>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $1$:\strut};
+}
+
+% Wege über 1
+% 3-1-5
+\uncover<4>{
+	\verbindung{3}{1}{5}
+	\feld{3}{5}{\color{red}4}
+	\wegweiser{3}{5}{1}
+}
+\uncover<5->{
+	\feld{3}{5}{4}
+	\wegweiser{3}{5}{1}
+}
+
+\uncover<6-7>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $2$:\strut};
+}
+
+% Wege über 2
+% 5-2-3
+\uncover<6>{
+	\verbindung{5}{2}{3}
+	\feld{5}{3}{\color{red}7}
+	\wegweiser{5}{3}{2}
+}
+\uncover<7->{
+	\feld{5}{3}{7}
+	\wegweiser{5}{3}{2}
+}
+% 5-2-4
+\uncover<7>{
+	\verbindung{5}{2}{4}
+	\feld{5}{4}{\color{blue}6}
+}
+
+\uncover<9-14>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $3$:\strut};
+}
+
+% Wege über 3
+% 2-3-1
+\uncover<9>{
+	\verbindung{2}{3}{1}
+	\feld{2}{1}{\color{red}5}
+	\wegweiser{2}{1}{3}
+}
+\uncover<10->{
+	\feld{2}{1}{5}
+	\wegweiser{2}{1}{3}
+}
+% 2-3-5
+\uncover<10>{
+	\verbindung{2}{3}{5}
+	\feld{2}{5}{\color{red}6}
+	\wegweiser{2}{5}{3}
+}
+\uncover<11->{
+	\feld{2}{5}{6}
+	\wegweiser{2}{5}{3}
+}
+% 4-3-1
+\uncover<11>{
+	\verbindung{4}{3}{1}
+	\feld{4}{1}{\color{red}9}
+	\wegweiser{4}{1}{3}
+}
+\uncover<12->{
+	\feld{4}{1}{9}
+	\wegweiser{4}{1}{3}
+}
+% 4-3-5
+\uncover<12>{
+	\verbindung{4}{3}{5}
+	\feld{4}{5}{\color{red}10}
+	\wegweiser{4}{5}{3}
+}
+\uncover<13->{
+	\feld{4}{5}{10}
+	\wegweiser{4}{5}{3}
+}
+% 5-3-1
+\uncover<13>{
+	\verbindung{5}{3}{1}
+	\feld{5}{1}{\color{red}10}
+	\wegweiser{5}{1}{2}
+}
+\uncover<14->{
+	\feld{5}{1}{10}
+	\wegweiser{5}{1}{2}
+}
+% 5-3-5
+\uncover<14>{
+	\verbindung{5}{3}{5}
+	\feld{5}{5}{\color{red}11}
+	\wegweiser{5}{5}{2}
+}
+\uncover<15->{
+	\feld{5}{5}{11}
+	\wegweiser{5}{5}{2}
+}
+
+\uncover<16-21>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $4$:\strut};
+}
+
+% Wege über 4
+% 2-4-1
+\uncover<16>{
+	\verbindung{2}{4}{1}
+	\feld{2}{1}{\color{blue}5}
+}
+% 2-4-3
+\uncover<17>{
+	\verbindung{2}{4}{3}
+	\feld{2}{3}{\color{blue}2}
+}
+% 2-4-5
+\uncover<18>{
+	\verbindung{2}{4}{5}
+	\feld{2}{5}{\color{blue}6}
+}
+% 5-4-1
+\uncover<19>{
+	\verbindung{5}{4}{1}
+	\feld{5}{1}{\color{blue}10}
+}
+% 5-4-3
+\uncover<20>{
+	\verbindung{5}{4}{3}
+	\feld{5}{3}{\color{blue}7}
+}
+% 5-4-5
+\uncover<21>{
+	\verbindung{5}{4}{5}
+	\feld{5}{5}{\color{blue}11}
+}
+
+% Wege über 5
+\uncover<23-38>{
+	\node[color=darkgreen] at (-0.8,-5.5) [right] {Wege über $5$:\strut};
+}
+
+% Wege über 5
+% 1-5-1
+\uncover<23>{
+	\verbindung{1}{5}{1}
+	\feld{1}{1}{\color{red}11}
+	\wegweiser{1}{1}{5}
+}
+\uncover<24->{
+	\feld{1}{1}{11}
+	\wegweiser{1}{1}{5}
+}
+% 1-5-2
+\uncover<24>{
+	\verbindung{1}{5}{2}
+	\feld{1}{2}{\color{red}6}
+	\wegweiser{1}{2}{5}
+}
+\uncover<25->{
+	\feld{1}{2}{6}
+	\wegweiser{1}{2}{5}
+}
+% 1-5-3
+\uncover<25>{
+	\verbindung{1}{5}{3}
+	\feld{1}{3}{\color{red}8}
+	\wegweiser{1}{3}{5}
+}
+\uncover<26->{
+	\feld{1}{3}{8}
+	\wegweiser{1}{3}{5}
+}
+% 1-5-4
+\uncover<26>{
+	\verbindung{1}{5}{4}
+	\feld{1}{4}{\color{red}7}
+	\wegweiser{1}{4}{5}
+}
+\uncover<27->{
+	\feld{1}{4}{7}
+	\wegweiser{1}{4}{5}
+}
+% 2-5-1
+\uncover<27>{
+	\verbindung{2}{5}{1}
+	\feld{2}{1}{\color{blue}5}
+}
+% 2-5-2
+\uncover<28>{
+	\verbindung{2}{5}{2}
+	\feld{2}{2}{\color{red}11}
+	\wegweiser{2}{2}{3}
+}
+\uncover<29->{
+	\feld{2}{2}{11}
+	\wegweiser{2}{2}{3}
+}
+% 2-5-3
+\uncover<29>{
+	\verbindung{2}{5}{3}
+	\feld{2}{3}{\color{blue}2}
+}
+% 2-5-4
+\uncover<30>{
+	\verbindung{2}{5}{4}
+	\feld{2}{4}{\color{red}12}
+	\wegweiser{2}{4}{3}
+}
+\uncover<31->{
+	\feld{2}{4}{12}
+	\wegweiser{2}{4}{3}
+}
+% 3-5-1
+\uncover<31>{
+	\verbindung{3}{5}{1}
+	\feld{3}{1}{\color{blue}3}
+}
+% 3-5-2
+\uncover<32>{
+	\verbindung{3}{5}{2}
+	\feld{3}{2}{\color{red}9}
+	\wegweiser{3}{2}{1}
+}
+\uncover<33->{
+	\feld{3}{2}{9}
+	\wegweiser{3}{2}{1}
+}
+% 3-5-3
+\uncover<33>{
+	\verbindung{3}{5}{3}
+	\feld{3}{3}{\color{red}11}
+	\wegweiser{3}{3}{1}
+}
+\uncover<34->{
+	\feld{3}{3}{11}
+	\wegweiser{3}{3}{1}
+}
+% 3-5-4
+\uncover<34>{
+	\verbindung{3}{5}{4}
+	\feld{3}{4}{\color{red}10}
+	\wegweiser{3}{4}{1}
+}
+\uncover<35->{
+	\feld{3}{4}{10}
+	\wegweiser{3}{4}{1}
+}
+% 4-5-1
+\uncover<35>{
+	\verbindung{4}{5}{1}
+	\feld{4}{1}{\color{blue}9}
+}
+% 4-5-2
+\uncover<36>{
+	\verbindung{4}{5}{2}
+	\feld{4}{2}{\color{red}15}
+	\wegweiser{4}{2}{3}
+}
+\uncover<37->{
+	\feld{4}{2}{15}
+	\wegweiser{4}{2}{3}
+}
+% 4-5-3
+\uncover<37>{
+	\verbindung{4}{5}{3}
+	\feld{4}{3}{\color{blue}6}
+}
+% 4-5-4
+\uncover<38>{
+	\verbindung{4}{5}{4}
+	\feld{4}{4}{\color{red}16}
+	\wegweiser{4}{4}{3}
+}
+\uncover<39->{
+	\feld{4}{4}{16}
+	\wegweiser{4}{4}{3}
+}
+
+
+\uncover<3->{
+
+	\foreach \x in {0,...,5}{
+		\draw[line width=0.7pt] (\x,0.8)--(\x,-5);
+	}
+	\foreach \y in {0,...,-5}{
+		\draw[line width=0.7pt] (-0.8,\y)--(5,\y);
+	}
+	\draw[line width=1.4pt] (0,0)--(5,0)--(5,-5)--(0,-5)--cycle;
+
+	\node at (0.5,0.5) {$1$};
+	\node at (1.5,0.5) {$2$};
+	\node at (2.5,0.5) {$3$};
+	\node at (3.5,0.5) {$4$};
+	\node at (4.5,0.5) {$5$};
+
+	\node at (-0.5,-0.5) {$1$};
+	\node at (-0.5,-1.5) {$2$};
+	\node at (-0.5,-2.5) {$3$};
+	\node at (-0.5,-3.5) {$4$};
+	\node at (-0.5,-4.5) {$5$};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\uncover<40>{
+	\vspace{-22pt}
+	\begin{block}{Lösung}
+	Der kürzeste Weg von 2 nach 4 ist 2---3---1---5---4
+	\end{block}
+}
+
+\end{column}
+\end{columns}
+
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/iteration.tex b/vorlesungen/slides/8/floyd-warshall/iteration.tex
new file mode 100644
index 0000000..d7e782d
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/iteration.tex
@@ -0,0 +1,14 @@
+%
+% iteration.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\newboolean{wegweiser}
+\begin{frame}[fragile]
+\frametitle{Floyd-Warshall: Iteration}
+\setboolean{wegweiser}{false}
+\input{../slides/8/floyd-warshall/fw.tex}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/macdonalds.png b/vorlesungen/slides/8/floyd-warshall/macdonalds.png
new file mode 100644
index 0000000..c442dfb
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/macdonalds.png
diff --git a/vorlesungen/slides/8/floyd-warshall/problem.tex b/vorlesungen/slides/8/floyd-warshall/problem.tex
new file mode 100644
index 0000000..93f8229
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/problem.tex
@@ -0,0 +1,146 @@
+%
+% graph.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[fragile]
+\frametitle{Problem: Kürzeste Wege}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\blob#1#2#3{
+	\fill[color=#3] #1 circle[radius=0.2];
+	\draw[line width=0.7pt] #1 circle[radius=0.2];
+	\node at #1 {{\tiny #2}};
+}
+
+\def\kante#1#2{
+	\draw[line width=0.7pt,shorten >= 0.2,shorten >= 0.2] #1 -- #2 ;
+}
+
+\def\a{72}
+\def\r{1.0}
+\def\R{2.0}
+\def\RR{3.5}
+
+\coordinate (A1) at ({\r*cos(0*\a)},{\r*sin(0*\a)});
+\coordinate (B1) at ({\r*cos(1*\a)},{\r*sin(1*\a)});
+\coordinate (C1) at ({\r*cos(2*\a)},{\r*sin(2*\a)});
+\coordinate (D1) at ({\r*cos(3*\a)},{\r*sin(3*\a)});
+\coordinate (E1) at ({\r*cos(4*\a)},{\r*sin(4*\a)});
+
+\coordinate (F1) at ({\R*cos(0*\a)},{\R*sin(0*\a)});
+\coordinate (G1) at ({\R*cos(1*\a)},{\R*sin(1*\a)});
+\coordinate (H1) at ({\R*cos(2*\a)},{\R*sin(2*\a)});
+\coordinate (I1) at ({\R*cos(3*\a)},{\R*sin(3*\a)});
+\coordinate (J1) at ({\R*cos(4*\a)},{\R*sin(4*\a)});
+
+\coordinate (K1) at ({\RR*cos(0.5*\a)},{\RR*sin(0.5*\a)});
+\coordinate (L1) at ({\RR*cos(1.5*\a)},{\RR*sin(1.5*\a)});
+\coordinate (M1) at ({\RR*cos(2.5*\a)},{\RR*sin(2.5*\a)});
+\coordinate (N1) at ({\RR*cos(3.5*\a)},{\RR*sin(3.5*\a)});
+\coordinate (O1) at ({\RR*cos(4.5*\a)},{\RR*sin(4.5*\a)});
+
+\kante{(A1)}{(C1)}
+\kante{(C1)}{(E1)}
+\kante{(E1)}{(B1)}
+\kante{(B1)}{(D1)}
+\kante{(D1)}{(A1)}
+
+\kante{(F1)}{(G1)}
+\kante{(G1)}{(H1)}
+\kante{(H1)}{(I1)}
+\kante{(I1)}{(J1)}
+\kante{(J1)}{(F1)}
+
+\kante{(A1)}{(F1)}
+\kante{(B1)}{(G1)}
+\kante{(C1)}{(H1)}
+\kante{(D1)}{(I1)}
+\kante{(E1)}{(J1)}
+
+\kante{(K1)}{(L1)}
+\kante{(L1)}{(M1)}
+\kante{(M1)}{(N1)}
+\kante{(N1)}{(O1)}
+\kante{(O1)}{(K1)}
+
+\kante{(F1)}{(K1)}
+\kante{(G1)}{(L1)}
+\kante{(H1)}{(M1)}
+\kante{(I1)}{(N1)}
+\kante{(J1)}{(O1)}
+
+\kante{(F1)}{(O1)}
+\kante{(G1)}{(K1)}
+\kante{(H1)}{(L1)}
+\kante{(I1)}{(M1)}
+\kante{(J1)}{(N1)}
+
+\uncover<2>{
+	\draw[line width=2pt,color=red] (M1)--(H1)--(G1)--(B1);
+}
+
+\uncover<3>{
+	\draw[line width=2pt,color=red] (M1)--(L1)--(G1)--(B1);
+}
+
+\uncover<4>{
+	\draw[line width=2pt,color=red] (M1)--(I1)--(D1)--(B1);
+}
+
+\uncover<5>{
+	\draw[line width=2pt,color=red] (M1)--(I1)--(D1)--(A1)--(F1);
+}
+
+\uncover<6->{
+	\draw[line width=2pt,color=red] (M1)--(I1)--(J1)--(F1);
+}
+
+\uncover<2-4>{
+	\blob{(B1)}{1}{red!20}
+	\blob{(M1)}{12}{red!20}
+}
+\uncover<5-8>{
+	\blob{(M1)}{1}{red!20}
+	\blob{(F1)}{12}{red!20}
+}
+
+\blob{(A1)}{0}{white}
+\uncover<1>{
+	\blob{(B1)}{1}{white}
+}
+\uncover<5-8>{
+	\blob{(B1)}{12}{white}
+}
+\blob{(C1)}{2}{white}
+\blob{(D1)}{3}{white}
+\blob{(E1)}{4}{white}
+\uncover<1-4>{
+	\blob{(F1)}{5}{white}
+}
+\blob{(G1)}{6}{white}
+\blob{(H1)}{7}{white}
+\blob{(I1)}{8}{white}
+\blob{(J1)}{9}{white}
+\blob{(K1)}{10}{white}
+\blob{(L1)}{11}{white}
+\uncover<1>{
+	\blob{(M1)}{12}{white}
+}
+\blob{(N1)}{13}{white}
+\blob{(O1)}{14}{white}
+
+\node at (6,0) {\begin{minipage}{5cm}
+\begin{itemize}
+\item<3-> Nicht eindeutig
+\item<5-> geradeste Wege sind nicht unbedingt die kürzesten
+\item<7-> Gewichtung der Kanten
+($\text{Schnellstrassen}\ne\text{Feldwege}$)
+\item<8-> Orientierung der Kanten?
+\end{itemize}
+\end{minipage}};
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
diff --git a/vorlesungen/slides/8/floyd-warshall/rekursion.tex b/vorlesungen/slides/8/floyd-warshall/rekursion.tex
new file mode 100644
index 0000000..c664e41
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/rekursion.tex
@@ -0,0 +1,108 @@
+%
+% rekursion.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[fragile]
+\frametitle{Rekursion}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\blob#1#2#3{
+        \fill[color=#3] #1 circle[radius=0.3];
+        \draw[line width=0.7pt] #1 circle[radius=0.3];
+        \node at #1 {{#2}};
+}
+
+\def\kante#1#2{
+        \draw[line width=0.7pt,shorten >= 0.3,shorten >= 0.3] #1 -- #2 ;
+}
+
+\coordinate (A) at (0,0);
+\coordinate (B1) at (2,2);
+\coordinate (B2) at (2,1);
+\coordinate (B3) at (2,0);
+\coordinate (B4) at (2,-1);
+\coordinate (B5) at (2,-2);
+
+\draw[line width=1.9pt,color=gray] (A)--(B1);
+\draw[line width=1.9pt,color=gray] (A)--(B2);
+\draw[line width=1.9pt,color=gray] (A)--(B3);
+\draw[line width=1.9pt,color=gray] (A)--(B4);
+\draw[line width=1.9pt,color=gray] (A)--(B5);
+
+\coordinate (Z) at (10,0);
+
+\begin{scope}
+\clip (2,-2.3) rectangle (10,2.3);
+\foreach \y in{-10,...,10}{
+	\draw[line width=1.9pt,color=gray]
+		(2,\y)--(10,{\y-8});
+	\draw[line width=1.9pt,color=gray]
+		(2,\y)--(10,{\y+8});
+}
+\end{scope}
+
+\uncover<2>{
+\draw[line width=4pt,color=red] (A)--(B1)--(5,-1)--(8,2)--(Z);
+}
+
+\uncover<3>{
+\draw[line width=4pt,color=red] (A)--(B2)--(3,0)--(4,1)--(5,0)--(6,1)--(8.5,-1.5)--(Z);
+}
+
+\uncover<4>{
+\draw[line width=4pt,color=red] (A)--(B3)--(2.5,0.5)--(3.5,-0.5)--(5,1.0)--(7,-1)--(9,1)--(Z);
+}
+
+\uncover<5>{
+\draw[line width=4pt,color=red] (A)--(B4)--(3,0)--(4,1)--(5,0)--(6,1)--(7,0)
+	--(6.0,-1.0)--(7,-2)--(7.5,-1.5)--(7,-1)--(7.5,-0.5)
+	--(8.5,-1.5)--(Z);
+}
+
+\uncover<6->{
+	\draw[line width=4pt,color=red] (A)--(B5)--(6,2);
+}
+\uncover<7->{
+	\draw[line width=4pt,color=red] (6,2)--(7,1)--(5,-1);
+}
+\uncover<8->{
+	\draw[line width=4pt,color=red] (5,-1)--(6,-2)--(8,0)--(9,-1);
+}
+\uncover<9->{
+	\draw[line width=4pt,color=red] (9,-1)--(Z);
+}
+
+\blob{(A)}{$A$}{red!20}
+\blob{(B1)}{$B_1$}{white}
+\blob{(B2)}{$B_2$}{white}
+\blob{(B3)}{$B_3$}{white}
+\blob{(B4)}{$B_4$}{white}
+\blob{(B5)}{$B_5$}{white}
+
+\blob{(Z)}{$Z$}{red!20}
+
+\uncover<6->{
+	\node at (6,2) {\includegraphics[width=1.5cm]{../slides/8/floyd-warshall/macdonalds.png}};
+}
+
+\uncover<7->{
+	\node at (5,-1) {\includegraphics[width=1.5cm]{../slides/8/floyd-warshall/starbucks.png}};
+}
+
+\uncover<8->{
+	\node at (9,-1) {\includegraphics[width=2cm]{../slides/8/floyd-warshall/burgerking.png}};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\begin{block}{Abstieg}
+Für den kürzesten Weg von $A$ nach $Z$ suche denjenigen Nachbarn $B_i$
+von $A$, der den kürzesten Weg von $B_i$ nach $Z$ hat.
+\uncover<7->{$\Rightarrow$ wir brauchen {\color{red}alle} kürzesten Wege!}
+\end{block}
+
+\end{frame}
diff --git a/vorlesungen/slides/8/floyd-warshall/starbucks.png b/vorlesungen/slides/8/floyd-warshall/starbucks.png
new file mode 100644
index 0000000..a28dbf7
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/starbucks.png
diff --git a/vorlesungen/slides/8/floyd-warshall/wege.tex b/vorlesungen/slides/8/floyd-warshall/wege.tex
new file mode 100644
index 0000000..7ff62a1
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/wege.tex
@@ -0,0 +1,26 @@
+%
+% wege.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{Wege statt Weglänge?}
+\begin{columns}[t]
+\begin{column}{0.48\hsize}
+\begin{block}{Wege speichern?}
+\uncover<3->{Es reicht, einen Wegweiser zum nächsten Knoten zu speichern}
+\end{block}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\hsize}
+\uncover<2->{%
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/floyd-warshall/wegweiser.jpg}
+\end{center}}%
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/floyd-warshall/wegiteration.tex b/vorlesungen/slides/8/floyd-warshall/wegiteration.tex
new file mode 100644
index 0000000..84ec679
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/wegiteration.tex
@@ -0,0 +1,13 @@
+%
+% wegiteration.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\newboolean{wegweiser}
+\begin{frame}[fragile]
+\frametitle{Floyd-Warshall: Wegweiser}
+\setboolean{wegweiser}{true}
+\input{../slides/8/floyd-warshall/fw.tex}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg
new file mode 100644
index 0000000..33aebe3
--- /dev/null
+++ b/vorlesungen/slides/8/floyd-warshall/wegweiser.jpg
diff --git a/vorlesungen/slides/8/fourier.tex b/vorlesungen/slides/8/fourier.tex
new file mode 100644
index 0000000..86d8086
--- /dev/null
+++ b/vorlesungen/slides/8/fourier.tex
@@ -0,0 +1,83 @@
+%
+% fourier.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\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}{Algebra}
+Die Laplace-Matrix eines Graphen ist symmetrisch
+\uncover<2->{%
+
+$\Rightarrow$
+Es gibt eine Basis aus Eigenvektoren $g_i\in\mathbb{R}^n$ von $L(G)$:
+\begin{align*}
+L(G)g_i&=\lambda_i g_i
+\end{align*}}
+\end{block}
+\uncover<12->{%
+\vspace{-20pt}
+\begin{block}{Fourier-Transformation}
+Jedes $f\in\mathbb{R}^n$ kann durch die $g_i$ ausgedrückt werden
+\begin{align*}
+\uncover<13->{
+f&= a_1 g_1 + \dots + a_n g_n
+}
+\\
+\uncover<14->{
+&= \hat{f}_1 g_1 + \dots + \hat{f}_ng_n = \sum_{k=1}^n \hat{f}_kg_k
+}
+\end{align*}
+\uncover<15->{%
+Zerlegung nach Zeitkonstante $\lambda_i$
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Anwendung}
+Wärmeleitungsgleichung
+\begin{align*}
+\uncover<4->{
+\frac{d}{dt}f &= L(G) f
+}
+\intertext{\uncover<5->{{\usebeamercolor[fg]{title}Ansatz:}}}
+\uncover<6->{
+f&=a_1g_1T_1(t)+\dots + a_ng_nT_n(t)
+}
+\\
+\uncover<7->{
+\frac{d}{dt}f
+&=
+a_1g_1\dot{T}_1(t) + \dots + a_1g_1 \dot{T}_n(t)
+}
+\\
+\uncover<8->{
+&=
+a_1Lg_1 + \dots + a_nLg_n
+}
+\\
+\uncover<9->{
+&=
+a_1\lambda_1 g_1 + \dots + a_n\lambda_n g_n
+}
+\\
+\uncover<10->{
+\dot{T}_i(t) &= \lambda_i T_i(t)
+}
+\uncover<11->{
+\quad
+\Rightarrow
+\quad
+T_i(t) = e^{\lambda_it} \uncover<-9>{T_i(0)}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/grad.tex b/vorlesungen/slides/8/grad.tex
new file mode 100644
index 0000000..a232828
--- /dev/null
+++ b/vorlesungen/slides/8/grad.tex
@@ -0,0 +1,84 @@
+%
+% grad.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Grad}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+%\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+%\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+%\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+%\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+%\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+%\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+%\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+
+\end{tikzpicture}
+\end{center}
+\begin{block}{Definition}
+Der Grad
+$\deg v$
+eines Knotens $v\in V$ ist die Anzahl der Kanten mit Ende in $v$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Gradmatrix}
+Diagonalmatrix mit $d_{ii}=\deg v_i$
+\[
+D(G)
+=
+\begin{pmatrix}
+3&0&0&0&0\\
+0&3&0&0&0\\
+0&0&3&0&0\\
+0&0&0&2&0\\
+0&0&0&0&1
+\end{pmatrix}
+\]
+\end{block}
+\begin{block}{Satz}
+Die Summe der Grade ist gerade:
+\[
+\sum_{i=1}^n\deg v_i = \operatorname{Spur} D(G) \equiv 0 \mod 2
+\]
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/graph.tex b/vorlesungen/slides/8/graph.tex
new file mode 100644
index 0000000..32150af
--- /dev/null
+++ b/vorlesungen/slides/8/graph.tex
@@ -0,0 +1,117 @@
+%
+% graph.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Graph}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2.4}
+
+\begin{scope}
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\uncover<3->{
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+	\draw[color=white,line width=5pt] (B) -- (D);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+}
+
+\uncover<2->{
+	\draw (A) circle[radius=0.2];
+	\draw (B) circle[radius=0.2];
+	\draw (C) circle[radius=0.2];
+	\draw (D) circle[radius=0.2];
+	\draw (E) circle[radius=0.2];
+
+	\node at (A) {$1$};
+	\node at (B) {$2$};
+	\node at (C) {$3$};
+	\node at (D) {$4$};
+	\node at (E) {$5$};
+}
+\node at (0,0) {$G$};
+
+\uncover<3->{
+	\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$)
+		[above right] {$\scriptstyle 1$};
+	\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$)
+		[above left] {$\scriptstyle 2$};
+	\node at ($0.5*(C)+0.5*(D)+(0.05,0)$)
+		[left] {$\scriptstyle 3$};
+	\node at ($0.5*(D)+0.5*(E)$)
+		[below] {$\scriptstyle 4$};
+	\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$)
+		[below right] {$\scriptstyle 5$};
+	\node at ($0.6*(A)+0.4*(C)$)
+		[above] {$\scriptstyle 6$};
+	\node at ($0.4*(B)+0.6*(D)$)
+		[left] {$\scriptstyle 7$};
+}
+
+\uncover<8->{
+	\draw[shorten >= 0.2cm,shorten <= 0.2cm]
+		(E) to[out=-18,in=-126,distance=2cm] (E);
+
+	\draw[color=red,line width=4pt] ($(E)+(-0.5,-0.5)+(0,-0.5)$)
+		-- ($(E)+(0.5,0.5)+(0,-0.5)$);
+	\draw[color=red,line width=4pt] ($(E)+(-0.5,0.5)+(0,-0.5)$)
+		-- ($(E)+(0.5,-0.5)+(0,-0.5)$);
+}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Ein Graph $G=(V,E)$ ist
+\begin{enumerate}
+\item<2->
+Eine Menge $V$ von Knoten (Vertizes):
+$V=\{v_1,v_2,\dots\}$
+\item<3->
+Eine Menge $E$ von Kanten (Edges):
+\[
+E\subset
+\left\{ e = \{v_1,v_2\}\;\left|\; \begin{minipage}{1.3cm}\raggedright
+$v_i\in V$\\
+$v_1\ne v_2$
+\end{minipage}
+\right.
+\right\}
+\]
+\end{enumerate}
+\end{block}
+\vspace{-20pt}
+\uncover<5->{%
+\begin{block}{Achtung:}
+\begin{itemize}
+\item<6-> Kanten sind Mengen
+\uncover<7->{$\Rightarrow$ zwei verschiedene Knoten}
+\uncover<8->{$\Rightarrow$ Keine Schleifen}
+\item<9-> Kanten sind ungerichtet, keine ``Einbahnstrassen''
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex
new file mode 100644
index 0000000..952c85b
--- /dev/null
+++ b/vorlesungen/slides/8/inzidenz.tex
@@ -0,0 +1,150 @@
+%
+% inzidenz.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Adjazenzmatrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+{\color<2->{darkgreen}
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+}
+
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+%\draw[shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\only<-2>{
+\fill[color=white] (B) circle[radius=0.2];
+}
+\only<3->{
+\fill[color=red!20] (B) circle[radius=0.2];
+}
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 4$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 5$};
+{\color<2->{darkgreen}
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 6$};
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Definition}
+\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 $j$}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dy{0.48}
+\def\dx{0.54}
+
+
+\begin{scope}
+\uncover<3->{
+\fill[color=red!20] (1.8,1.8) rectangle (4.75,2.15);
+}
+\uncover<2->{
+\fill[color=darkgreen!40,opacity=0.5] (4.46,0.36) rectangle (4.79,2.65);
+}
+\foreach \y in {1,...,5}{
+	\node[color=gray] at (5.3,{2.45-(\y-1)*\dy}) {\tiny $\y$};
+}
+\foreach \y in {1,...,6}{
+	\node[color=gray] at ({1.92+(\y-1)*\dx},2.90) {\tiny $\y$};
+}
+\draw[color=gray] (1.8,2.75) -- (4.7,2.75);
+\draw[color=gray] (5.2,2.55) -- (5.2,0.45);
+\node[color=gray] at ({1.92+2.5*\dx},3.1) {\tiny Kanten};
+\node[color=gray] at (5.3,{2.45-2*\dy}) [above,rotate=-90] {\tiny Knoten};
+\end{scope}
+
+\uncover<4->{
+\begin{scope}
+\uncover<3->{
+\fill[color=red!20] (1.8,-1.16) rectangle (4.25,-0.77);
+\fill[color=red!20] (2.3,-2.6) rectangle (2.63,-0.29);
+}
+\foreach \y in {1,...,5}{
+	\node[color=gray] at (4.7,{-0.5-(\y-1)*\dy}) {\tiny $\y$};
+	\node[color=gray] at ({1.92+(\y-1)*\dx},-0.1) {\tiny $\y$};
+}
+\draw[color=gray] (1.8,-0.22) -- (4.2,-0.22);
+\draw[color=gray] (4.6,-0.4) -- (4.6,-2.55);
+\node[color=gray] at ({1.92+2*\dx},0.1) {\tiny Knoten};
+\node[color=gray] at (4.7,{-0.5-2*\dy}) [above,rotate=-90] {\tiny Knoten};
+\end{scope}
+}
+
+\node (0,0) [right] {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix}
+1&0&0&1&1&0\\
+1&1&0&0&0&1\\
+0&1&1&0&1&0\\
+0&0&1&0&0&1\\
+0&0&0&1&0&0
+\end{pmatrix}
+\\[12pt]
+\uncover<4->{
+A(G)
+&=
+\begin{pmatrix}
+0&1&1&0&1\\
+1&0&1&1&0\\
+1&1&0&1&0\\
+0&1&1&0&0\\
+1&0&0&0&0
+\end{pmatrix}
+\end{aligned}}$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/inzidenzd.tex b/vorlesungen/slides/8/inzidenzd.tex
new file mode 100644
index 0000000..5f2f51a
--- /dev/null
+++ b/vorlesungen/slides/8/inzidenzd.tex
@@ -0,0 +1,164 @@
+%
+% inzidenzd.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Adjazenz-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+{\color<2->{darkgreen}
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+}
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\only<-2>{
+\fill[color=white] (B) circle[radius=0.2];
+}
+\only<3->{
+\fill[color=red!20] (B) circle[radius=0.2];
+}
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+{\color<2->{darkgreen}
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-15pt}
+\uncover<5->{%
+\begin{block}{Definition}
+\vspace{-20pt}
+\begin{align*}
+B(G)_{ij}&=-1&&\Leftrightarrow&&\text{Kante $j$ von $i$}\\
+B(G)_{kj}&=+1&&\Leftrightarrow&&\text{Kante $j$ nach $k$}\\
+A(G)_{ij}&=\phantom{-}1&&\Leftrightarrow&&\text{Kante von $i$ nach $j$}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dx{0.84}
+\def\dy{0.48}
+
+\begin{scope}[xshift=4cm,yshift=3cm]
+\uncover<3->{
+\fill[color=red!20] 
+({-0.67-(7-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2})
+rectangle
+({-0.67-(7-7)*\dx+0.2},{-0.38-(2-1)*\dy+0.16});
+}
+\uncover<2->{
+\fill[color=darkgreen!40,opacity=0.5] 
+({-0.67-(7-7)*\dx-0.4},{-0.38-(5-1)*\dy-0.2})
+rectangle
+({-0.67-(7-7)*\dx+0.2},{-0.38-(1-1)*\dy+0.16});
+}
+%\draw (0,0) circle[radius=0.05];
+\foreach \x in {1,...,7}{
+	\node[color=gray] at ({-0.67-(7-\x)*\dx},0.0) {\tiny $\x$};
+}
+\draw[color=gray] ({-0.72-6*\dx},-0.1) -- (-0.6,-0.1);
+\foreach \y in {1,...,5}{
+	\node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$};
+}
+\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4);
+\node[color=gray] at ({-0.67-(7-4)*\dx},0.04) [above] {\tiny Kanten};
+\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy})
+	[above,rotate=-90] {\tiny Knoten};
+\end{scope}
+
+\uncover<4->{
+\begin{scope}[xshift=2.32cm,yshift=-0.24cm]
+%\draw (0,0) circle[radius=0.05];
+\fill[color=red!20] 
+({-0.67-(5-1)*\dx-0.4},{-0.38-(2-1)*\dy-0.2})
+rectangle
+({-0.67-(5-5)*\dx+0.2},{-0.38-(2-1)*\dy+0.16});
+\fill[color=red!20] 
+({-0.67-(5-2)*\dx-0.4},{-0.38-(5-1)*\dy-0.2})
+rectangle
+({-0.67-(5-2)*\dx+0.2},{-0.38-(1-1)*\dy+0.16});
+\foreach \x in {1,...,5}{
+	\node[color=gray] at ({-0.67-(5-\x)*\dx},0.0) {\tiny $\x$};
+}
+\draw[color=gray] ({-0.72-4*\dx},-0.1) -- (-0.6,-0.1);
+\foreach \y in {1,...,5}{
+	\node[color=gray] at ({0},{-0.38-(\y-1)*\dy}) {\tiny $\y$};
+}
+\draw[color=gray] (-0.1,-0.28) -- (-0.1,-2.4);
+\node[color=gray] at ({-0.67-(5-3)*\dx},0.04) [above] {\tiny Knoten};
+\node[color=gray] at ({0.00},{-0.38-(3-1)*\dy})
+	[above,rotate=-90] {\tiny Knoten};
+\end{scope}
+}
+
+\node at (0,0) {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix*}[r]
+-1& 0& 0& 0&+1&-1& 0\\
++1&-1& 0& 0& 0& 0&-1\\
+ 0&+1&-1& 0& 0&+1& 0\\
+ 0& 0&+1&-1& 0& 0&+1\\
+ 0& 0& 0&+1&-1& 0& 0
+\end{pmatrix*}
+\\[20pt]
+\uncover<4->{
+A(G)
+&=
+\begin{pmatrix*}[r]
+ 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\
+\phantom{-}1& 0&\phantom{-}1&\phantom{-}1& 0\\
+\phantom{-}1&\phantom{-}1& 0&\phantom{-}1& 0\\
+ 0&\phantom{-}1&\phantom{-}1& 0&\phantom{-}1\\
+\phantom{-}1& 0& 0&\phantom{-}1& 0
+\end{pmatrix*}}
+\end{aligned}$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/8/laplace.tex b/vorlesungen/slides/8/laplace.tex
new file mode 100644
index 0000000..a1c364d
--- /dev/null
+++ b/vorlesungen/slides/8/laplace.tex
@@ -0,0 +1,213 @@
+%
+% laplace.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Laplace-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (D);
+
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (B);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (C) -- (D);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (D) -- (E);
+\draw[shorten >= 0.3cm,shorten <= 0.3cm] (E) -- (A);
+
+\uncover<2-4>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+}
+
+\uncover<3-7>{
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(A) -- (C);
+}
+
+\uncover<4-13>{
+\draw[->,color=darkgreen,line width=8pt,shorten <= 0.25cm,shorten >= 0cm]
+	(A) -- (E);
+}
+
+\uncover<5->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(A) -- (B);
+}
+
+\uncover<6-8>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+}
+
+\uncover<7-10>{
+\draw[->,color=darkgreen,line width=4pt,shorten <= 0.25cm,shorten >= 0.15cm]
+	(B) -- (D);
+}
+
+\uncover<8->{
+\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm]
+	(A) -- (C);
+}
+
+\uncover<9->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(B) -- (C);
+}
+
+\uncover<10-11>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+}
+
+\uncover<11->{
+\draw[<->,color=darkgreen,line width=4pt,shorten <= 0.15cm,shorten >= 0.15cm]
+	(B) -- (D);
+}
+
+\uncover<12->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(C) -- (D);
+}
+
+\uncover<13-14>{
+\draw[->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+}
+
+\uncover<14->{
+\draw[<->,color=darkgreen,line width=8pt,shorten <= 0cm,shorten >= 0cm]
+	(A) -- (E);
+}
+
+\uncover<15->{
+\draw[<->,color=darkgreen,line width=2pt,shorten <= 0.25cm,shorten >= 0.25cm]
+	(D) -- (E);
+}
+
+\fill[color=red] (A) circle[radius=0.3];
+\fill[color=red!50] (B) circle[radius=0.3];
+\fill[color=white] (C) circle[radius=0.3];
+\fill[color=blue!50] (D) circle[radius=0.3];
+\fill[color=blue] (E) circle[radius=0.3];
+
+\draw (A) circle[radius=0.3];
+\draw (B) circle[radius=0.3];
+\draw (C) circle[radius=0.3];
+\draw (D) circle[radius=0.3];
+\draw (E) circle[radius=0.3];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+
+\end{tikzpicture}
+\end{center}
+\uncover<16->{%
+\begin{block}{Definition}
+Laplace-Matrix
+\[
+L(G) = D(G) - A(G)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{align*}
+f
+&=
+\begin{pmatrix}
+f(1)\\
+f(2)\\
+f(3)\\
+f(4)\\
+f(5)
+\end{pmatrix}
+\\
+\frac{df}{dt}
+&=
+-\kappa
+\begin{pmatrix*}[r]
+\only<1>{\phantom{-0}}
+	\only<2>{\phantom{-}1}
+	\only<3>{\phantom{-}2}
+	\only<4->{\phantom{-}3}
+	&\only<1>{\phantom{-0}}\only<2->{{-1}}%
+	&\only<-2>{\phantom{-0}}\only<3->{{-1}}%
+	&\uncover<16->{ 0}
+	&\only<-3>{\phantom{-0}}\only<4->{{-1}}\\
+\only<-4>{\phantom{-0}}\only<5->{{-1}}
+	&\only<-4>{\phantom{-0}}
+	\only<5>{\phantom{-}1}
+	\only<6>{\phantom{-}2}
+	\only<7->{\phantom{-}3}
+	&\only<-5>{\phantom{-0}}\only<6->{{-1}}
+	&\only<-6>{\phantom{-0}}\only<7->{{-1}}
+	&\uncover<16->{ 0}\\
+\only<-7>{\phantom{-0}}\only<8->{{-1}}
+	&\only<-8>{\phantom{-0}}\only<9->{{-1}}
+	&\only<-7>{\phantom{-0}}
+	\only<8>{\phantom{-}1}
+	\only<9>{\phantom{-}2}
+	\only<10->{\phantom{-}3}
+	&\only<-9>{\phantom{-0}}\only<10->{{-1}}
+	&\uncover<16->{ 0}\\
+\uncover<16->{ 0}
+	&\only<-10>{\phantom{-0}}\only<11->{{-1}}
+	&\only<-11>{\phantom{-0}}\only<12->{{-1}}
+	&\only<-10>{\phantom{-0}}
+	\only<11>{\phantom{-}1}
+	\only<12>{\phantom{-}2}
+	\only<13->{\phantom{-}3}
+	&\only<-12>{\phantom{-0}}\only<13->{{-1}}\\
+\only<-13>{\phantom{-0}}\only<14->{{-1}}
+	&\uncover<16->{ 0}
+	&\uncover<16->{ 0}
+	&\only<-14>{\phantom{-0}}\only<15->{{-1}}
+	&\only<-13>{\phantom{-0}}
+	\only<14>{\phantom{-}1}
+	\only<15->{\phantom{-}2}
+\end{pmatrix*}
+\begin{pmatrix}
+f(1)\\
+f(2)\\
+f(3)\\
+f(4)\\
+f(5)
+\end{pmatrix}
+\\
+\uncover<17->{
+&=
+-\kappa L f}
+\end{align*}
+\vspace{-20pt}
+\uncover<18->{%
+\begin{block}{Rekonstruktion}
+Der Graph lässt sich aus $L$ rekonstruieren
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+
+\egroup
+
+
diff --git a/vorlesungen/slides/8/pfade/adjazenz.tex b/vorlesungen/slides/8/pfade/adjazenz.tex
new file mode 100644
index 0000000..f923262
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/adjazenz.tex
@@ -0,0 +1,97 @@
+%
+% adjazenz.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.5,0,0}
+\begin{frame}[fragile]
+\newboolean{pfeilspitzen}
+\setboolean{pfeilspitzen}{true}
+\frametitle{Adjazenz-Matrix}
+
+\begin{columns}[t]
+\begin{column}{0.48\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\r{2.2}
+\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)});
+\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)});
+\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)});
+\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)});
+\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)});
+
+\def\knoten#1#2{
+        \fill[color=white] #1 circle[radius=0.3];
+        \draw[line width=1pt] #1 circle[radius=0.3];
+        \node at #1 {$#2$};
+}
+
+\def\kante#1#2#3{
+        \ifthenelse{\boolean{pfeilspitzen}}{
+                \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }{
+                \draw[line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }
+%        \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22];
+%        \node at ($0.5*#1+0.5*#2$) {$#3$};
+}
+
+\kante{(A)}{(E)}{1}
+\kante{(B)}{(C)}{2}
+\kante{(B)}{(D)}{13}
+\kante{(C)}{(A)}{3}
+\kante{(D)}{(C)}{6}
+\kante{(E)}{(B)}{5}
+\kante{(E)}{(D)}{6}
+
+\knoten{(A)}{1}
+\knoten{(B)}{2}
+\knoten{(C)}{3}
+\knoten{(D)}{4}
+\knoten{(E)}{5}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\hsize}
+\[
+a_{{\color{darkred}i}{\color{blue}j}}
+=
+\begin{cases}
+1&\quad\text{\# Kanten von ${\color{blue}j}$ nach ${\color{darkred}i}$}\\
+0&\quad\text{sonst}
+\end{cases}
+\]
+\begin{center}
+\begin{tikzpicture}[>=latex]
+\node at (0,0) {$\displaystyle
+A=
+\begin{pmatrix}
+0&0&1&0&0\\
+0&0&0&0&1\\
+0&1&0&1&0\\
+0&1&0&0&1\\
+1&0&0&0&0
+\end{pmatrix}
+$};
+\def\s{0.54}
+\foreach \x in {1,...,5}{
+	\node[color=blue] at ({-0.71+(\x-1)*\s},1.4) {\tiny $\x$};
+}
+\node[color=blue] at ({-0.71+2*\s},1.7) {von};
+\def\r{0.48}
+\foreach \y in {1,...,5}{
+	\node[color=darkred] at ({-0.71+5*\s},{0.02+(3-\y)*\r}) {\tiny $\y$};
+}
+\node[color=darkred] at ({-0.4+5*\s},{0.02}) [rotate=90] {nach};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/pfade/beispiel.tex b/vorlesungen/slides/8/pfade/beispiel.tex
new file mode 100644
index 0000000..43685f3
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/beispiel.tex
@@ -0,0 +1,404 @@
+%
+% beispiel.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\newboolean{pfeilspitzen}
+
+\def\knoten#1#2{
+        \fill[color=white] #1 circle[radius=0.3];
+        \draw[line width=1pt] #1 circle[radius=0.3];
+        \node at #1 {$#2$};
+}
+
+\def\kante#1#2#3{
+        \ifthenelse{\boolean{pfeilspitzen}}{
+                \draw[->,line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }{
+                \draw[line width=1pt,shorten >= 0.3cm,shorten <= 0.3cm]
+                        #1 -- #2;
+        }
+%        \fill[color=white,opacity=0.7] ($0.5*#1+0.5*#2$) circle[radius=0.22];
+%        \node at ($0.5*#1+0.5*#2$) {$#3$};
+}
+
+\begin{frame}
+\setboolean{pfeilspitzen}{true}
+\frametitle{Beispiel}
+\begin{columns}[t]
+\begin{column}{0.37\hsize}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\def\r{2.2}
+\coordinate (A) at ({\r*cos(-54+0*72)},{\r*sin(-54+0*72)});
+\coordinate (C) at ({\r*cos(-54+1*72)},{\r*sin(-54+1*72)});
+\coordinate (D) at ({\r*cos(-54+2*72)},{\r*sin(-54+2*72)});
+\coordinate (B) at ({\r*cos(-54+3*72)},{\r*sin(-54+3*72)});
+\coordinate (E) at ({\r*cos(-54+4*72)},{\r*sin(-54+4*72)});
+
+\kante{(A)}{(E)}{1}
+\kante{(B)}{(C)}{2}
+\kante{(B)}{(D)}{13}
+\kante{(C)}{(A)}{3}
+\kante{(D)}{(C)}{6}
+\kante{(E)}{(B)}{5}
+\kante{(E)}{(D)}{6}
+
+\knoten{(A)}{1}
+\knoten{(B)}{2}
+\knoten{(C)}{3}
+\knoten{(D)}{4}
+\knoten{(E)}{5}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.59\hsize}
+
+\only<1>{
+\begin{block}{Pfade der Länge 1}
+\[
+A=
+\begin{pmatrix}
+0&0&1&0&0\\
+0&0&0&0&1\\
+0&1&0&1&0\\
+0&1&0&0&1\\
+1&0&0&0&0
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<2>{
+\begin{block}{Pfade der Länge 2}
+\[
+A^2=\begin{pmatrix}
+   0 & 1 & 0 & 1 & 0 \\
+   1 & 0 & 0 & 0 & 0 \\
+   0 & 1 & 0 & 0 & 2 \\
+   1 & 0 & 0 & 0 & 1 \\
+   0 & 0 & 1 & 0 & 0
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<3>{
+\begin{block}{Pfade der Länge 3}
+\[
+A^3=\begin{pmatrix}
+   0 & 1 & 0 & 0 & 2 \\
+   0 & 0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 \\
+   2 & 0 & 0 & 0 & 1 \\
+   1 & 0 & 1 & 0 & 0 \\
+   0 & 1 & 0 & 1 & 0
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<4>{
+\begin{block}{Pfade der Länge 4}
+\[
+A^4=\begin{pmatrix}
+   2%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 & 0 & 1 \\
+   0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 1 & 0 \\
+   1 & 0 & 2%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 \\
+   0 & 1 & 1 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 \\
+   0 & 1 & 0 & 0 & 2%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<5>{
+\begin{block}{Pfade der Länge 5}
+\[
+A^5=\begin{pmatrix}
+   1 & 0 & 2 & 0 & 0 \\
+   0 & 1 & 0 & 0 & 2 \\
+   0 & 2 & 1 & 2 & 0 \\
+   0 & 2 & 0 & 1 & 2 \\
+   2 & 0 & 0 & 0 & 1
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<6>{
+\begin{block}{Pfade der Länge 6}
+\[
+A^6=\begin{pmatrix}
+   0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 2 & 1 & 2 & 0 \\
+   2 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 & 1 \\
+   0 & 3 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 1 & 4 \\
+   2 & 1 & 0 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 3 \\
+   1 & 0 & 2 & 0 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<7>{
+\begin{block}{Pfade der Länge 7}
+\[
+A^7=\begin{pmatrix}
+   0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 3 & 0 & 1 & 4 \\
+   1 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 2 & 0 & 0 \\
+   4 & 1 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 4 \\
+   3 & 0 & 2 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 1 \\
+   0 & 2 & 1 & 2 & 0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<8>{
+\begin{block}{Pfade der Länge 8}
+\[
+A^8=\begin{pmatrix}
+   4 & 1 & 0 & 0 & 4 \\
+   0 & 2 & 1 & 2 & 0 \\
+   4 & 0 & 4 & 0 & 1 \\
+   1 & 2 & 3 & 2 & 0 \\
+   0 & 3 & 0 & 1 & 4
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<9>{
+\begin{block}{Pfade der Länge 9}
+\[
+A^9=\begin{pmatrix}
+   4 & 0 & 4 & 0 & 1 \\
+   0 & 3 & 0 & 1 & 4 \\
+   1 & 4 & 4 & 4 & 0 \\
+   0 & 5 & 1 & 3 & 4 \\
+   4 & 1 & 0 & 0 & 4
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<10>{
+\begin{block}{Pfade der Länge 10}
+\[
+A^{10}=\begin{pmatrix}
+   1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 4 & 4 & 4 & 0 \\
+   4 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 0 & 0 & 4 \\
+   0 & 8 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 4 & 8 \\
+   4 & 4 & 0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+& 8 \\
+   4 & 0 & 4 & 0 & 1%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<11>{
+\begin{block}{Pfade der Länge 15}
+\[
+A^{15}=\begin{pmatrix}
+    1 & 20 &  6 & 12 & 16 \\
+   12 &  1 &  8 &  0%
+\begin{picture}(0,0)
+\color{red}\put(-3,4){\circle{12}}
+\end{picture}%
+&  6 \\
+   16 & 18 &  1 &  6 & 32 \\
+   20 &  6 &  8 &  1 & 18 \\
+    6 &  8 & 12 &  8 &  1
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<12>{
+\begin{block}{Pfade der Länge 20}
+\[
+A^{20}=\begin{pmatrix}
+   33 & 56 &  8 & 24 & 80 \\
+   24 & 17 & 32 & 16 &  8 \\
+   80 & 32 & 33 &  8 & 80 \\
+   56 & 24 & 48 & 17 & 32 \\
+    8 & 48 & 24 & 32 & 33
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<13>{
+\begin{block}{Pfade der Länge 25}
+\[
+A^{25}=\begin{pmatrix}
+   193 & 120 &  74 &  40 & 240 \\
+    40 & 113 &  80 &  80 &  74 \\
+   240 & 114 & 193 &  74 & 160 \\
+   120 & 154 & 160 & 113 & 114 \\
+    74 & 160 &  40 &  80 & 193
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<14>{
+\begin{block}{Pfade der Länge 30}
+\[
+A^{30}=\begin{pmatrix}
+   673 & 348 & 460 & 188 & 560 \\
+   188 & 433 & 160 & 240 & 460 \\
+   560 & 648 & 673 & 460 & 536 \\
+   348 & 700 & 400 & 433 & 648 \\
+   460 & 400 & 188 & 160 & 673
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\only<15>{
+\begin{block}{Pfade der Länge 35}
+\[
+A^{35}=\begin{pmatrix}
+   1793%
+\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3)
+& 1644 & 1806 & 1108 & 1632 \\
+   1108 & 1233 &  536 &  560 & 1806 \\
+   1632 & 2914 & 1793%
+\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3)
+& 1806 & 2752 \\
+   1644 & 2366 & 1096 & 1233 & 2914 \\
+   1806 & 1096 & 1108 &  536 & 1793%
+\color{red}\drawline(-23,-3)(-23,10)(2,10)(2,-3)(-23,-3)
+\end{pmatrix}
+\]
+\end{block}
+}
+
+\end{column}
+\end{columns}
+\vbox to2cm{
+\vfill
+\only<3>{
+	\begin{block}{Kürzester Verbindung von 3 nach 2}
+	Der Weg 3---1---6---2 ist die kürzeste Verbindung von 3 nach 2
+	\end{block}
+}
+\only<4>{
+	\begin{block}{Kürzeste Zyklen}
+	Jeder Knoten liegt auf einem Zyklus der Länge 4,
+	dies sind die kürzesten Zyklen.
+	1, 3 und 5 liegen auf beiden Zyklen, 2 und 4 nur auf einem.
+	\end{block}
+}
+\only<6>{
+	\begin{block}{Zyklen der Länge 6}
+	{\em Keine} Zyklen der Länge 6
+	\end{block}
+}
+\only<7>{
+	\begin{block}{Zyklen der Länge 7}
+	{\em Keine} Zyklen der Länge 7
+	\end{block}
+}
+\only<10>{
+	\begin{block}{Zyklen der Länge 10}
+	Genau ein Zyklus der Länge 10
+	\end{block}
+}
+\only<11>{
+	\begin{block}{Verbindung von 4 nach 2}
+	{\em Keine} Verbindung der Länge 15 von 4 nach 2
+	\end{block}
+}
+\only<15>{
+	\begin{block}{Zyklen der Länge 35}
+	Es gibt 1793 Zyklen, die 1 enthalten, und sie enthalten alle auch 3 und 5
+	\end{block}
+}
+}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/pfade/gf.tex b/vorlesungen/slides/8/pfade/gf.tex
new file mode 100644
index 0000000..e89a1fb
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/gf.tex
@@ -0,0 +1,54 @@
+%
+% gf.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\definecolor{darkred}{rgb}{0.8,0,0}
+\frametitle{Erzeugende Funktion}
+Alle Weglängen zusammen:
+\[
+\uncover<7->{f({\color{darkred}t})=}
+\uncover<4->{E+}
+A
+\uncover<3->{{\color{darkred}t}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{2!}}
+A^2
+\uncover<3->{{\color{darkred}t^2}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{3!}}
+A^3
+\uncover<3->{{\color{darkred}t^3}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{4!}}
+A^4
+\uncover<3->{{\color{darkred}t^4}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{5!}}
+A^5
+\uncover<3->{{\color{darkred}t^5}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{6!}}
+A^6
+\uncover<3->{{\color{darkred}t^6}}
+\uncover<2->{+}
+\uncover<5->{\frac{1}{7!}}
+A^7
+\uncover<3->{{\color{darkred}t^7}}
+\dots
+\uncover<6->{= e^{A{\color{darkred}t}}}
+\]
+\uncover<4->{%
+heisst {\em\usebeamercolor[fg]{title} \only<5->{exponentiell} erzeugende Funktion}
+der Wege-Anzahlen}
+
+\begin{itemize}
+\item<8->
+Begriff der Entropie auf einem Graphen
+\item<9->
+Wahrscheinlichkeit, dass ein Zufallsspaziergänger auf einem Graphen an
+einem bestimmten Knoten vorbeikommt
+\end{itemize}
+
+\end{frame}
diff --git a/vorlesungen/slides/8/pfade/langepfade.tex b/vorlesungen/slides/8/pfade/langepfade.tex
new file mode 100644
index 0000000..8c0dd0d
--- /dev/null
+++ b/vorlesungen/slides/8/pfade/langepfade.tex
@@ -0,0 +1,59 @@
+%
+% langepfade.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.5,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}
+\frametitle{Wieviele Pfade der Länge $k$?}
+\begin{definition}
+Anzahl Pfade der Länge $k$ zwischen zwei Knoten
+\[
+a_{{\color{darkred}i}{\color{blue}j}}^{(k)} 
+=
+\#\{\text{Pfade der Länge $k$ von $\color{blue}j$ nach $\color{darkred}i$}\},
+\qquad
+A^{(k)}
+=
+\left(
+a_{{\color{darkred}i}{\color{blue}j}}^{(k)}
+\right)
+\]
+\end{definition}
+\uncover<2->{
+{\usebeamercolor[fg]{title}Spezialfall:} $A^{(1)}=A$.
+}
+
+\uncover<3->{
+\begin{block}{Rekursionsformel}
+\vspace{-25pt}
+\begin{align*}
+a_{{\color{darkred}i}{\color{blue}{\color{blue}j}}}^{(k)}
+&\uncover<4->{=
+\sum_{{\color{darkgreen}l}=1}^n
+\#\{\text{Pfade der Länge $1$ von $\color{darkgreen}l$ nach $\color{darkred}i$}\}}
+\\[-11pt]
+&\uncover<4->{\qquad\qquad\times
+\#\{\text{Pfade der Länge $k-1$ von $\color{blue}j$ nach $\color{darkgreen}l$}\}}
+\\
+&\uncover<5->{=
+\sum_{{\color{darkgreen}l}=1}^n
+a_{{\color{darkred}i}{\color{darkgreen}l}}^{(1)}
+\cdot
+a_{{\color{darkgreen}l}{\color{blue}j}}^{(k-1)}}
+\\
+\uncover<6->{
+\Rightarrow\qquad
+A^{(k)}}
+&\uncover<6->{=
+A\;A^{(k-1)}}
+\uncover<7->{
+\qquad\Rightarrow\qquad
+A^{(k)} = A^k}
+\end{align*}
+\end{block} }
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/produkt.tex b/vorlesungen/slides/8/produkt.tex
new file mode 100644
index 0000000..1d8b725
--- /dev/null
+++ b/vorlesungen/slides/8/produkt.tex
@@ -0,0 +1,100 @@
+%
+% produkt.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Inzidenz- und Laplace-Matrix}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.2}
+
+\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
+\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
+\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
+\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
+\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
+\draw[color=white,line width=5pt] (B) -- (D);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);
+
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
+\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
+
+\draw (A) circle[radius=0.2];
+\draw (B) circle[radius=0.2];
+\draw (C) circle[radius=0.2];
+\draw (D) circle[radius=0.2];
+\draw (E) circle[radius=0.2];
+
+\node at (A) {$1$};
+\node at (B) {$2$};
+\node at (C) {$3$};
+\node at (D) {$4$};
+\node at (E) {$5$};
+\node at (0,0) {$G$};
+
+\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
+\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
+\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
+\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
+\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
+\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
+\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};
+
+\end{tikzpicture}
+\end{center}
+\vspace{-15pt}
+\begin{block}{Berechne}
+\vspace{-20pt}
+\begin{align*}
+\uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\dx{0.84}
+\def\dy{0.48}
+
+\node at (0,0) {$\displaystyle
+\begin{aligned}
+B(G)
+&=
+\begin{pmatrix*}[r]
+-1& 0& 0& 0&+1&-1& 0\\
++1&-1& 0& 0& 0& 0&-1\\
+ 0&+1&-1& 0& 0&+1& 0\\
+ 0& 0&+1&-1& 0& 0&+1\\
+ 0& 0& 0&+1&-1& 0& 0
+\end{pmatrix*}
+\\[20pt]
+\uncover<2->{
+L(G)
+&=
+\begin{pmatrix*}[r]
+ 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{-1}\\
+\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}\\
+\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{ 0}\\
+\uncover<3->{ 0}&\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}\\
+\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{ 0}&\uncover<3->{-1}& 2
+\end{pmatrix*}}
+\end{aligned}$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex
new file mode 100644
index 0000000..425fe1c
--- /dev/null
+++ b/vorlesungen/slides/8/spanningtree.tex
@@ -0,0 +1,164 @@
+%
+% spanningtree.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{Spannbäume}
+
+\vspace{-16pt}
+
+\begin{columns}[t]
+
+\begin{column}{0.40\hsize}
+\begin{block}{Netzwerk}
+Alle Knoten erreichen, Schleifen vermeiden $\Rightarrow$ Spannbaum
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,scale=0.18]
+
+\coordinate (A) at ( 1.2927,-15.0076);
+\coordinate (B) at ( 5.0261,- 7.7143);
+\coordinate (C) at ( 4.9260,-13.0335);
+\coordinate (D) at (12.2094,-22.9960);
+\coordinate (F) at (17.8334,-13.4687);
+\coordinate (G) at ( 6.4208,-10.2438);
+\coordinate (H) at (17.2367,- 3.1047);
+\coordinate (K) at (24.3760,- 3.0293);
+\coordinate (L) at (23.2834,- 1.3563);
+\coordinate (M) at (28.7093,- 4.0627);
+
+\fill (A) circle[radius=0.5];
+\fill (B) circle[radius=0.5];
+\fill (C) circle[radius=0.5];
+\fill (D) circle[radius=0.5];
+\fill (F) circle[radius=0.5];
+\fill (G) circle[radius=0.5];
+\fill (H) circle[radius=0.5];
+\fill (K) circle[radius=0.5];
+\fill (L) circle[radius=0.5];
+\fill (M) circle[radius=0.5];
+
+%\uncover<1-4>{
+%\node at (A) [above] {$A$};
+%\node at (B) [above] {$B$};
+%\node at (C) [below] {$C$};
+%\node at (D) [below] {$D$};
+%\node at (F) [below right] {$F$};
+%\node at (G) [above] {$G$};
+%\node at (H) [above] {$H$};
+%\node at (K) [above right] {$K$};
+%\node at (L) [above] {$L$};
+%\node at (M) [above] {$M$};
+%}
+
+\uncover<5->{
+\node at (A) [above] {$1$};
+\node at (B) [above] {$2$};
+\node at (C) [below] {$3$};
+\node at (D) [below] {$4$};
+\node at (F) [below right] {$5$};
+\node at (G) [above] {$6$};
+\node at (H) [above] {$7$};
+\node at (K) [above right] {$8$};
+\node at (L) [above] {$9$};
+\node at (M) [above] {$10$};
+}
+
+\draw (L)--(H);
+\draw (L)--(K);
+\draw (L)--(M);
+
+\draw (H)--(B);
+\draw (H)--(G);
+\draw (H)--(F);
+\draw (H)--(K);
+
+\draw (K)--(F);
+\draw (K)--(M);
+
+\draw (M)--(F);
+\draw (M)--(D);
+
+\draw (B)--(A);
+\draw (B)--(C);
+\draw (B)--(G);
+
+\draw (G)--(C);
+\draw (G)--(F);
+
+\draw (F)--(D);
+
+\draw (C)--(F);
+\draw (C)--(A);
+\draw (C)--(D);
+
+\draw (A)--(D);
+
+\uncover<2>{
+\draw[line width=2pt,join=round]
+	(A)--(D)--(C)--(F)--(G)--(B)--(H)--(K)--(L)--(M);
+}
+
+\uncover<3>{
+\draw[line width=2pt,join=round]
+	(M)--(D)--(A)--(C)--(G)--(B)--(H)--(L)--(K)--(F);
+}
+
+\uncover<4->{
+\draw[line width=2pt] (M)--(K)--(L)--(H)--(F)--(D);
+\draw[line width=2pt] (F)--(G)--(C)--(A);
+\draw[line width=2pt] (G)--(B);
+}
+
+\end{tikzpicture}
+\end{center}
+\vspace{-10pt}
+Wieviele Spannbäume gibt es?
+\end{block}
+\end{column}
+
+\begin{column}{0.56\hsize}
+\uncover<5->{%
+\begin{block}{Laplace-Matrix}
+\vspace{-15pt}
+\[
+L=
+\tiny
+\begin{pmatrix}
+ 3&-1&-1&-1& 0& 0& 0& 0& 0& 0\\
+-1& 4&-1& 0& 0&-1&-1& 0& 0& 0\\
+-1&-1& 5&-1&-1&-1& 0& 0& 0& 0\\
+-1& 0&-1& 4&-1& 0& 0& 0& 0&-1\\
+ 0& 0&-1&-1& 6&-1&-1&-1& 0&-1\\
+ 0&-1&-1& 0&-1& 4&-1& 0& 0& 0\\
+ 0&-1& 0& 0&-1&-1& 5&-1&-1& 0\\
+ 0& 0& 0& 0&-1& 0&-1& 4&-1&-1\\
+ 0& 0& 0& 0& 0& 0&-1&-1& 3&-1\\
+ 0& 0& 0&-1&-1& 0& 0&-1&-1& 4\\
+\end{pmatrix}
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<6->{%
+\begin{block}{Satz von Kirchhoff}
+Die Anzahl der Spannbäume eines Netzwerkes ist ein Kofaktor
+des Laplaceoperators
+\vspace{-5pt}
+\[
+\det L_{ij} = 
+\left|
+L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{array}\right.
+\right|
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<7->{%
+{\usebeamercolor[fg]{title}Beispiel:} 41524
+}
+
+\end{column}
+
+\end{columns}
+
+\end{frame}
diff --git a/vorlesungen/slides/8/tokyo/bahn0.tex b/vorlesungen/slides/8/tokyo/bahn0.tex
new file mode 100644
index 0000000..9c39712
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/bahn0.tex
@@ -0,0 +1,11 @@
+%
+% bahn.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/tokyo/tokyosubway.pdf}
+\end{center}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/bahn1.tex b/vorlesungen/slides/8/tokyo/bahn1.tex
new file mode 100644
index 0000000..6ac3344
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/bahn1.tex
@@ -0,0 +1,28 @@
+%
+% bahn.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Tokyo Bahn-Netz}
+\begin{center}
+\begin{tabular}{rl}
+882&Bahnstationen\\
+108&Bahnlinien\\
+29&Bahngesellschaften\\
+20\,000\,000&Passagiere täglich\\
+7\,000\,000&Passagiere alleine in Shinjuku\\
+\end{tabular}
+\end{center}
+\uncover<2->{
+\begin{block}{Dirichlet-Zerlegung und Bahnhöfe}
+\begin{center}
+\uncover<3->{Passagiere wählen den nächsten Bahnhöfe}\\
+\uncover<4->{$\Downarrow$}\\
+\uncover<5->{Bahnhöfe definieren eine Dirichletzerlegung der Stadt}
+\end{center}
+\end{block}
+}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/bahn2.tex b/vorlesungen/slides/8/tokyo/bahn2.tex
new file mode 100644
index 0000000..4adc1bf
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/bahn2.tex
@@ -0,0 +1,12 @@
+%
+% bahn.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/tokyo/shinjuku-subway-map.jpg}
+\end{center}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/google.tex b/vorlesungen/slides/8/tokyo/google.tex
new file mode 100644
index 0000000..d37c98d
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/google.tex
@@ -0,0 +1,11 @@
+%
+% google.tex
+%
+% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\begin{center}
+\includegraphics[width=\hsize]{../slides/8/tokyo/transportnetworkgraph.png}
+\end{center}
+\end{frame}
+
diff --git a/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg
new file mode 100644
index 0000000..1c513da
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/shinjuku-subway-map.jpg
diff --git a/vorlesungen/slides/8/tokyo/tokyosubway.pdf b/vorlesungen/slides/8/tokyo/tokyosubway.pdf
new file mode 100644
index 0000000..6b84a8d
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/tokyosubway.pdf
diff --git a/vorlesungen/slides/8/tokyo/transportnetworkgraph.png b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png
new file mode 100644
index 0000000..4a11183
--- /dev/null
+++ b/vorlesungen/slides/8/tokyo/transportnetworkgraph.png