9 files changed, 260 insertions, 132 deletions
diff --git a/vorlesungen/slides/2/funktionenalgebra.tex b/vorlesungen/slides/2/funktionenalgebra.tex
index e3339c3..9116be4 100644
--- a/vorlesungen/slides/2/funktionenalgebra.tex
+++ b/vorlesungen/slides/2/funktionenalgebra.tex
@@ -31,12 +31,12 @@ f(x)\cdot g(x)
 \begin{align*}
 \|f\cdot g\|_\infty
 &=
-\sup_{x\in[0,1]} f(x)g(x)
+\sup_{x\in[0,1]} |f(x)g(x)|
 \\
 \uncover<4->{
 &\le
-\sup_{x\in[0,1]}f(x)
-\sup_{y\in[0,1]}g(y)
+\sup_{x\in[0,1]}|f(x)|
+\sup_{y\in[0,1]}|g(y)|
 }
 \\
 \uncover<5->{
diff --git a/vorlesungen/slides/2/skalarprodukt.tex b/vorlesungen/slides/2/skalarprodukt.tex
index 2a9784f..99d8a73 100644
--- a/vorlesungen/slides/2/skalarprodukt.tex
+++ b/vorlesungen/slides/2/skalarprodukt.tex
@@ -13,7 +13,7 @@
 \begin{block}{Positiv definite, symmetrische Bilinearform}
 $\langle \;\,,\;\rangle\colon V\times V\to \mathbb{R}$
 \begin{itemize}
-\item
+\item<2->
 Bilinear:
 \begin{align*}
 \langle \alpha u+\beta v,w\rangle
@@ -28,18 +28,19 @@ Bilinear:
 +
 \beta\langle u,w\rangle
 \end{align*}
-\item
+\item<3->
 Symmetrisch: $\langle u,v\rangle = \langle v,u\rangle$
-\item
+\item<4->
 $\langle x,x\rangle >0 \quad\forall x\ne 0$
 \end{itemize}
 \end{block}
 \end{column}
 \begin{column}{0.48\textwidth}
+\uncover<5->{%
 \begin{block}{Positive definite, hermitesche Sesquilinearform}
 $\langle \;\,,\;\rangle\colon V\times V\to \mathbb{C}$
 \begin{itemize}
-\item
+\item<6->
 Sesquilinear:
 \begin{align*}
 \langle \alpha u+\beta v,w\rangle
@@ -54,40 +55,42 @@ Sesquilinear:
 +
 \beta\langle u,w\rangle
 \end{align*}
-\item
+\item<7->
 Hermitesch: $\langle u,v\rangle = \overline{\langle v,u\rangle}$
-\item
+\item<8->
 $\langle x,x\rangle >0 \quad\forall x\ne 0$
 \end{itemize}
-\end{block}
+\end{block}}
 \end{column}
 \end{columns}
 \begin{columns}[t,onlytextwidth]
 \begin{column}{0.28\textwidth}
+\uncover<9->{%
 \begin{block}{$2$-Norm}
 $\|v\|_2^2 = \langle v,v\rangle$
 \\
 $\|v\|_2 = \sqrt{\langle v,v\rangle}$
-\end{block}
+\end{block}}
 \end{column}
 \begin{column}{0.78\textwidth}
+\uncover<10->{%
 \begin{itemize}
-\item $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$
-\item $\| \lambda v \|_2
+\item<11-> $\|v\|_2 = \sqrt{\langle v,v\rangle} > 0\quad\forall v\ne 0$
+\item<12-> $\| \lambda v \|_2
 =
 \sqrt{\langle \lambda v,\lambda v\rangle\mathstrut}
 =
 \sqrt{\overline{\lambda}\lambda\langle v,v\rangle}
 =
 |\lambda|\cdot \|v\|_2$
-\item
+\item<13->
 \raisebox{-8pt}{
 $\begin{aligned}
 \|u+v\|_2^2 &= \|u\|_2^2 + 2{\color{red}\operatorname{Re}\langle u,v\rangle} + \|v\|_2^2
 \\
 (\|u\|_2+\|v\|_2)^2 &= \|u\|_2^2 + 2{\color{red}\|u\|_2\|v\|_2} + \|v\|_2^2
 \end{aligned}$}
-\end{itemize}
+\end{itemize}}
 \end{column}
 \end{columns}
 \end{frame}
diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex
index 36e3be2..f0c4401 100644
--- a/vorlesungen/slides/5/unitaer.tex
+++ b/vorlesungen/slides/5/unitaer.tex
@@ -20,11 +20,11 @@ $U$ unitär lässt das Skalarprodukt invariant
 \uncover<2->{%
 $\lambda$ ein Eigenwert mit Eigenvektor $v$:
 \begin{align*}
-\langle u,v\rangle
+\langle v,v\rangle
 &=
 \langle Uu,Uv\rangle
-\uncover<3->{= \langle \lambda u,\lambda v\rangle}
-\uncover<4->{= |\lambda|^2 \langle u,v\rangle}
+\uncover<3->{= \langle \lambda v,\lambda v\rangle}
+\uncover<4->{= |\lambda|^2 \langle v,v\rangle}
 \\
 \uncover<5->{\Rightarrow\;|\lambda|&=1}
 \end{align*}}
diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc
new file mode 100644
index 0000000..e8c7502
--- /dev/null
+++ b/vorlesungen/slides/8/Makefile.inc
@@ -0,0 +1,11 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter8 =								\
+	../slides/8/dgraph.tex						\
+	../slides/8/graph.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..761ea63
--- /dev/null
+++ b/vorlesungen/slides/8/chapter.tex
@@ -0,0 +1,7 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{8/graph.tex}
+\folie{8/dgraph.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/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/Makefile.inc b/vorlesungen/slides/Makefile.inc
index 20929e4..fe22282 100644
--- a/vorlesungen/slides/Makefile.inc
+++ b/vorlesungen/slides/Makefile.inc
@@ -9,7 +9,8 @@ include ../slides/2/Makefile.inc
 include ../slides/3/Makefile.inc
 include ../slides/4/Makefile.inc
 include ../slides/5/Makefile.inc
+include ../slides/8/Makefile.inc
 
 slides = 								\
 	$(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4)	\
-	$(chapter5)
+	$(chapter5) $(chapter8)
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 5f08a8f..30aef37 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -3,117 +3,6 @@
 %
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-
-%\folie{3/motivation.tex}
-%\folie{3/inverse.tex}
-%\folie{3/polynome.tex}
-%\folie{3/division.tex}
-%\folie{3/division2.tex}
-%\folie{3/ringstruktur.tex}
-%\folie{3/teilbarkeit.tex}
-%\folie{3/faktorisierung.tex}
-%\folie{3/faktorzerlegung.tex}
-%\folie{3/einsetzen.tex}
-%\folie{3/maximalergrad.tex}
-%\folie{3/minimalbeispiel.tex}
-%\folie{3/minimalpolynom.tex}
-%\folie{3/drehmatrix.tex}
-%\folie{3/drehfaktorisierung.tex}
-%\folie{3/fibonacci.tex}
-%\folie{3/operatoren.tex}
-%\folie{3/adjunktion.tex}
-%\folie{3/adjalgebra.tex}
-
-%\folie{4/ggt.tex}
-%\folie{4/euklidmatrix.tex}
-%\folie{4/euklidbeispiel.tex}
-%\folie{4/euklidtabelle.tex}
-%\folie{4/fp.tex}
-%\folie{4/division.tex}
-%\folie{4/gauss.tex}
-% \folie{4/dh.tex}
-% XXX \folie{4/frobenius.tex}
-
-%\folie{4/divisionpoly.tex}
-%\folie{4/euklidpoly.tex}
-%\folie{4/polynomefp.tex}
-%\folie{4/alpha.tex}
-
-% XXX \folie{4/f2.tex}
-%\folie{4/schieberegister.tex}
-
-% XXX Idee der elliptischen Kurve
-% XXX \folie{4/ecidee.tex}
-                                                              
-
-\section{Eigenwertproblem}
-% XXX Motivation: beliebige Funktionen f(A) berechnen
-%\folie{5/motivation.tex}
-%\folie{5/charpoly.tex}
-
-\section{Invariante Unterräume}
-%\folie{5/kernbild.tex}
-%\folie{5/ketten.tex}
-%\folie{5/dimension.tex}
-%\folie{5/folgerungen.tex}
-%\folie{5/injektiv.tex}
-%\folie{5/nilpotent.tex}
-%\folie{5/eigenraeume.tex}
-%\folie{5/zerlegung.tex}
-%\folie{5/normalnilp.tex}
-%\folie{5/bloecke.tex}
-
-% Jordan Normalform
-\section{Jordan-Normalform}
-%\folie{5/jordanblock.tex}
-%\folie{5/jordan.tex}
-% XXX Diagonalform
-% XXX \folie{5/diagonalform.tex}
-%\folie{5/reellenormalform.tex}
-% XXX \folie{5/hessenberg.tex}
-
-\section{Satz von Cayley-Hamilton}
-%\folie{5/cayleyhamilton.tex}
-
-\section{Matrixnormen}
-%\folie{2/norm.tex}
-%\folie{2/skalarprodukt.tex}
-%\folie{2/cauchyschwarz.tex}
-%\folie{2/funktionenraum.tex}
-%\folie{2/polarformel.tex}
-%\folie{2/operatornorm.tex}
-%\folie{2/funktionenalgebra.tex}
-%\folie{2/linearformnormen.tex}
-%\folie{2/frobeniusnorm.tex}
-%\folie{2/frobeniusanwendung.tex}
-
-\section{Approximation mit Polynomen}
-% XXX Stone-Weierstrass
-% XXX \folie{5/stoneweierstrass.tex}
-% XXX Spektrum einer Matrix
-%\folie{5/spektrum.tex}
-\folie{5/normal.tex}
-\folie{5/unitaer.tex}
-% XXX Approximation einer Funktion auf dem Spektrum
-% XXX \folie{5/spektrumapproximation.tex}
-% XXX Approximation einer Matrix in der erzeugten Algebra
-% XXX \folie{5/matrixapproximation.tex}
-% XXX Gelfand-Transformation
-% XXX \folie{5/gelfandtransformation.tex}
-
-\section{Potenzreihen}
-% Konvergenzradius
-%\folie{5/konvergenzradius.tex}
-%\folie{5/krbeispiele.tex}
-%\folie{5/spektralgelfand.tex}
-%\folie{5/Aiteration.tex}
-%\folie{5/satzvongelfand.tex}
-%\folie{5/logarithmusreihe.tex}
-
-\section{Differentialgleichungen}
-%\folie{5/potenzreihenmethode.tex}
-%\folie{5/exponentialfunktion.tex}
-% XXX Sinus und Cosinus, Eulerscher Satz
-% XXX \folie{5/sinuscosinus.tex}
-%\folie{5/hyperbolisch.tex}
+\folie{8/dgraph.tex}
+\folie{8/graph.tex}