add new slides

13 files changed, 249 insertions, 494 deletions

diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
index 233835a..d46dc7f 100644
--- a/vorlesungen/slides/8/Makefile.inc
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -28,10 +28,5 @@ chapter8 =								\
 	../slides/8/tokyo/bahn0.tex					\
 	../slides/8/tokyo/bahn1.tex					\
 	../slides/8/tokyo/bahn2.tex					\
-	../slides/8/markov/google.tex					\
-	../slides/8/markov/markov.tex					\
-	../slides/8/markov/irreduzibel.tex				\
-	../slides/8/markov/stationaer.tex				\
-	../slides/8/markov/pf.tex					\
 	../slides/8/chapter.tex
 
diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex
index ac06775..6a0b13f 100644
--- a/vorlesungen/slides/8/chapter.tex
+++ b/vorlesungen/slides/8/chapter.tex
@@ -30,10 +30,3 @@
 \folie{8/tokyo/bahn1.tex}
 \folie{8/tokyo/bahn2.tex}
 
-\folie{8/markov/google.tex}
-\folie{8/markov/markov.tex}
-\folie{8/markov/stationaer.tex}
-\folie{8/markov/irreduzibel.tex}
-\folie{8/markov/pf.tex}
-
-
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/inzidenz.tex b/vorlesungen/slides/8/inzidenz.tex
index 87578df..952c85b 100644
--- a/vorlesungen/slides/8/inzidenz.tex
+++ b/vorlesungen/slides/8/inzidenz.tex
@@ -126,8 +126,8 @@ B(G)
 1&0&0&1&1&0\\
 1&1&0&0&0&1\\
 0&1&1&0&1&0\\
-0&0&1&0&0&0\\
-0&0&0&1&0&1
+0&0&1&0&0&1\\
+0&0&0&1&0&0
 \end{pmatrix}
 \\[12pt]
 \uncover<4->{
diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/8/markov/google.tex
deleted file mode 100644
index d1ec31d..0000000
--- a/vorlesungen/slides/8/markov/google.tex
+++ /dev/null
@@ -1,123 +0,0 @@
-%
-% google.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Google-Matrix}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-\def\r{2.4}
-\coordinate (A) at (0,0);
-\coordinate (B) at (0:\r);
-\coordinate (C) at (60:\r);
-\coordinate (D) at (120:\r);
-\coordinate (E) at (180:\r);
-
-\foreach \a in {2,...,5}{
-	\fill[color=white] ({60*(\a-2)}:\r) circle[radius=0.2];
-	\draw ({60*(\a-2)}:\r) circle[radius=0.2];
-	\node at ({60*(\a-2)}:\r) {$\a$};
-}
-\fill[color=white] (A) circle[radius=0.2];
-\draw (A) circle[radius=0.2];
-\node at (A) {$1$};
-
-{\color<6>{red}
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
-}
-
-{\color<7>{red}
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) to[out=-150,in=-30] (E);
-}
-
-{\color<8>{red}
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-90,in=30] (A);
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-30,in=90] (B);
-}
-
-{\color<9>{red}
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (C);
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (A);
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
-}
-
-{\color<10>{red}
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);
-	\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) to[out=90,in=-150] (D);
-}
-
-\end{tikzpicture}
-\end{center}
-\vspace{-10pt}
-\renewcommand{\arraystretch}{1.1}
-\uncover<5->{
-\begin{align*}
-H&=\begin{pmatrix}
-\uncover<6->{0          }
-	&\uncover<7->{0          }
-	&\uncover<8->{{\color<8>{red}\frac{1}{2}}}
-	&\uncover<9->{{\color<9>{red}\frac{1}{3}}}
-	&\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\
-\uncover<6->{{\color<6>{red}\frac{1}{2}}}
-	&\uncover<7->{0          }
-	&\uncover<8->{{\color<8>{red}\frac{1}{2}}}
-	&\uncover<9->{0          }
-	&\uncover<10->{0          }\\
-\uncover<6->{{\color<6>{red}\frac{1}{2}}}
-	&\uncover<7->{{\color<7>{red}\frac{1}{2}}}
-	&\uncover<8->{0          }
-	&\uncover<9->{{\color<9>{red}\frac{1}{3}}}
-	&\uncover<10->{0          }\\
-\uncover<6->{0          }
-	&\uncover<7->{0          }
-	&\uncover<8->{0          }
-	&\uncover<9->{0          }
-	&\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\
-\uncover<6->{0          }
-	&\uncover<7->{{\color<7>{red}\frac{1}{2}}}
-	&\uncover<8->{0          }
-	&\uncover<9->{{\color<9>{red}\frac{1}{3}}}
-	&\uncover<10->{0          }
-\end{pmatrix}
-\\
-\uncover<11->{
-h_{ij}
-&=
-\frac{1}{\text{Anzahl Links ausgehend von $j$}}
-}
-\end{align*}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\begin{block}{Aufgabe}
-Bestimme die Wahrscheinlichkeit $p(i)$, mit der sich ein Surfer
-auf der Website $i$ befindet
-\end{block}
-\uncover<2->{
-\begin{block}{Navigation}
-$p(i) = P(i,\text{vor Navigation})$,
-\uncover<3->{$p'(i)=P(i,\text{nach Navigation})$}
-\uncover<4->{
-\[
-p'(i) = \sum_{j=1}^n h_{ij} p(j)
-\]}
-\end{block}}
-\vspace{-15pt}
-\begin{block}{Freier Wille}
-\vspace{-12pt}
-\[
-G = \alpha H + (1-\alpha)\frac{UU^t}{n}
-\]
-Google-Matrix
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
diff --git a/vorlesungen/slides/8/markov/irreduzibel.tex b/vorlesungen/slides/8/markov/irreduzibel.tex
deleted file mode 100644
index 87e90e4..0000000
--- a/vorlesungen/slides/8/markov/irreduzibel.tex
+++ /dev/null
@@ -1,136 +0,0 @@
-%
-% irreduzibel.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Irreduzible Markovkette}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\r{2}
-\coordinate (A) at ({\r*cos(0*60)},{\r*sin(0*60)});
-\coordinate (B) at ({\r*cos(1*60)},{\r*sin(1*60)});
-\coordinate (C) at ({\r*cos(2*60)},{\r*sin(2*60)});
-\coordinate (D) at ({\r*cos(3*60)},{\r*sin(3*60)});
-\coordinate (E) at ({\r*cos(4*60)},{\r*sin(4*60)});
-\coordinate (F) at ({\r*cos(5*60)},{\r*sin(5*60)});
-
-\uncover<-2>{
-\draw (A) -- (B);
-\draw (A) -- (C);
-\draw (A) -- (D);
-\draw (A) -- (E);
-\draw (A) -- (F);
-
-\draw (B) -- (A);
-\draw (B) -- (C);
-\draw (B) -- (D);
-\draw (B) -- (E);
-\draw (B) -- (F);
-
-\draw (C) -- (A);
-\draw (C) -- (B);
-\draw (C) -- (D);
-\draw (C) -- (E);
-\draw (C) -- (F);
-
-\draw (D) -- (A);
-\draw (D) -- (B);
-\draw (D) -- (C);
-\draw (D) -- (E);
-\draw (D) -- (F);
-
-\draw (E) -- (A);
-\draw (E) -- (B);
-\draw (E) -- (C);
-\draw (E) -- (D);
-\draw (E) -- (F);
-
-\draw (F) -- (A);
-\draw (F) -- (B);
-\draw (F) -- (C);
-\draw (F) -- (D);
-\draw (F) -- (E);
-}
-
-\uncover<3->{
-
-\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt] (A) to[out=90,in=-30] (B);
-\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt] (A) -- (C);
-\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) -- (A);
-\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt] (B) to[out=150,in=30] (C);
-\draw[->,color=black!20,shorten >= 0.15cm,line width=3pt] (B) to[out=-90,in=-150,distance=1cm] (B);
-\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) to[out=-60,in=180] (A);
-\draw[->,color=black!50,shorten >= 0.15cm,line width=3pt] (C) -- (B);
-
-\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
-	(D) to[out=-90,in=150] (E);
-\draw[->,color=black!30,shorten >= 0.15cm,line width=3pt]
-	(E) -- (D);
-\draw[->,color=black!70,shorten >= 0.15cm,line width=3pt]
-	(E) to[out=-30,in=-150] (F);
-\draw[->,color=black!40,shorten >= 0.15cm,line width=3pt]
-	(F) -- (E);
-\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
-	(F) to[out=120,in=0] (D);
-\draw[->,color=black!60,shorten >= 0.15cm,line width=3pt]
-	(D) -- (F);
-}
-
-\fill[color=white] (A) circle[radius=0.2];
-\fill[color=white] (B) circle[radius=0.2];
-\fill[color=white] (C) circle[radius=0.2];
-\fill[color=white] (D) circle[radius=0.2];
-\fill[color=white] (E) circle[radius=0.2];
-\fill[color=white] (F) 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];
-\draw (F) 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 (F) {$6$};
-
-\end{tikzpicture}
-\end{center}
-\uncover<2->{%
-\begin{block}{Irreduzibel}
-Graph zusammenhängend $\Rightarrow$
-Keine Zerlegung in Teilgraphen möglich
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<3->{%
-\begin{block}{Reduzibel}
-Die Zustandsmenge zerfällt in zwei disjunkte Teilmengen $V=V_1\cup V_2$
-und es gibt keine Übergängen zwischen den Mengen:
-\uncover<4->{%
-\begin{align*}
-P
-&=
-\begin{pmatrix*}[l]
-0  &0.2&0.5&   &   &   \\
-0.3&0.2&0.5&   &   &   \\
-0.7&0.6&0  &   &   &   \\
-   &   &   &0  &0.3&0.4\\
-   &   &   &0.4&0  &0.6\\
-   &   &   &0.6&0.7&0  
-\end{pmatrix*}
-\end{align*}}%
-\uncover<5->{%
-$P$ zerfällt in zwei Blöcke die unabhängig voneinander analysiert werden können
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
diff --git a/vorlesungen/slides/8/markov/markov.tex b/vorlesungen/slides/8/markov/markov.tex
deleted file mode 100644
index e92ff0f..0000000
--- a/vorlesungen/slides/8/markov/markov.tex
+++ /dev/null
@@ -1,111 +0,0 @@
-%
-% markov.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\begin{frame}[t]
-\frametitle{Markovketten}
-\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 >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!40]
-	(A) -- (C);
-\draw[color=white,line width=8pt] (B) -- (D);
-\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!80]
-	(B) -- (D);
-
-\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!60]
-	(A) -- (B);
-\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black!20]
-	(B) -- (C);
-\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black]
-	(C) -- (D);
-\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black]
-	(D) -- (E);
-\draw[->,shorten >= 0.1cm,shorten <= 0.1cm,line width=4pt,color=black]
-	(E) -- (A);
-
-\fill[color=white] (A) circle[radius=0.2];
-\fill[color=white] (B) circle[radius=0.2];
-\fill[color=white] (C) circle[radius=0.2];
-\fill[color=white] (D) circle[radius=0.2];
-\fill[color=white] (E) 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 ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 0.6$};
-\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 0.2$};
-\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 1$};
-\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 1$};
-\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 1$};
-\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 0.4$};
-\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 0.8$};
-
-\end{tikzpicture}
-\end{center}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Verteilung}
-\begin{itemize}
-\item<8->
-Welche stationäre Verteilung auf den Knoten stellt sich ein?
-\item<9->
-$P(i)=?$
-\end{itemize}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<2->{%
-\begin{block}{\strut\mbox{Übergang\only<3->{s-/Wahrscheinlichkeit}smatrix}}
-$P_{ij} = P(i | j)$, Wahrscheinlichkeit, in den Zustand $i$ überzugehen,
-\begin{align*}
-P
-&=
-\begin{pmatrix}
-   &   & & &1\phantom{.0}\\
-0.6&   & & & \\
-0.4&0.2& & & \\
-   &0.8&1\phantom{.0}& & \\
-   &   & &1\phantom{.0}& 
-\end{pmatrix}
-\end{align*}
-\end{block}}
-\vspace{-10pt}
-\uncover<4->{%
-\begin{block}{Eigenschaften}
-\begin{itemize}
-\item<5-> $P_{ij}\ge 0\;\forall i,j$
-\item<6-> Spaltensumme:
-\(
-\displaystyle
-\sum_{i=1}^n P_{ij} = 1\;\forall j
-\)
-\end{itemize}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
diff --git a/vorlesungen/slides/8/markov/pf.tex b/vorlesungen/slides/8/markov/pf.tex
deleted file mode 100644
index da2ef2b..0000000
--- a/vorlesungen/slides/8/markov/pf.tex
+++ /dev/null
@@ -1,53 +0,0 @@
-%
-% pf.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Perron-Frobenius-Theorie}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Positive Matrizen und Vektoren}
-$P\in M_{m\times n}(\mathbb{R})$ 
-\begin{itemize}
-\item<2->
-$P$ heisst positiv, $P>0$, wenn $p_{ij}>0\;\forall i,j$
-\item<3->
-$P\ge 0$, wenn $p_{ij}\ge 0\;\forall i,j$
-\end{itemize}
-\end{block}
-\uncover<4->{%
-\begin{block}{Beispiele}
-\begin{itemize}
-\item<5->
-Adjazenzmatrix $A(G)$
-\item<6->
-Gradmatrix $D(G)$
-\item<7->
-Wahrscheinlichkeitsmatrizen
-\end{itemize}
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<8->{%
-\begin{block}{Satz}
-Es gibt einen positiven Eigenvektor $p$ von $P$ zum Eigenwert $1$
-\end{block}}
-\uncover<9->{%
-\begin{block}{Satz}
-$P$ irreduzible Matrix, $P\ge 0$, hat einen Eigenvektor $p$, $p\ge 0$,
-zum Eigenwert $1$
-\end{block}}
-\uncover<10->{%
-\begin{block}{Potenzmethode}
-Falls $P\ge 0$ einen eindeutigen Eigenvektor $p$ hat\uncover<11->{,
-dann konveriert die rekursiv definierte Folge
-\[
-p_{n+1}=\frac{Pp_n}{\|Pp_n\|}, p_0 \ge 0, p_0\ne 0
-\]}%
-\uncover<12->{$\displaystyle\lim_{n\to\infty} p_n = p$}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
diff --git a/vorlesungen/slides/8/markov/stationaer.tex b/vorlesungen/slides/8/markov/stationaer.tex
deleted file mode 100644
index 92fab16..0000000
--- a/vorlesungen/slides/8/markov/stationaer.tex
+++ /dev/null
@@ -1,57 +0,0 @@
-%
-% stationaer.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Stationäre Verteilung}
-%\vspace{-15pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Zeitentwicklung}
-\begin{itemize}
-\item<2->
-$P$ eine Wahrscheinlichkeitsmatrix
-\item<3->
-$p_0\in\mathbb{R}^n$ Verteilung zur Zeit $t=0$ bekannt
-\item<4->
-$p_k\in\mathbb{R}^n$ Verteilung zur Zeit $t=k$
-\end{itemize}
-\uncover<5->{%
-Entwicklungsgesetz
-\begin{align*}
-P(i,t=k)
-&=
-\sum_{j=1}^n P_{ij} P(j,t=k-1)
-\\
-\uncover<6->{
-p_k &= Pp_{k-1}
-}
-\end{align*}}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Stationär}
-Bedingung: $p_{k\mathstrut} = p_{k-1}$
-\uncover<8->{
-\begin{align*}
-\Rightarrow
-Pp &= p
-\end{align*}}\uncover<9->{%
-Eigenvektor zum Eigenwert $1$}
-\end{block}}
-\uncover<10->{%
-\begin{block}{Fragen}
-\begin{enumerate}
-\item<11->
-Gibt es eine stationäre Verteilung?
-\item<12->
-Gibt es einen Eigenvektor mit Einträgen $\ge 0$?
-\item<13->
-Gibt es mehr als eine Verteilung?
-\end{enumerate}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
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/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