add new slides

42 files changed, 1041 insertions, 132 deletions

diff --git a/vorlesungen/slides/0/Makefile.inc b/vorlesungen/slides/0/Makefile.inc
index ab91126..a6bb320 100644
--- a/vorlesungen/slides/0/Makefile.inc
+++ b/vorlesungen/slides/0/Makefile.inc
@@ -10,6 +10,8 @@ chapter0 =								\
 	../slides/0/resourcen.tex					\
 	../slides/0/latextipps.tex					\
 	../slides/0/nextsteps.tex					\
-	../slides/0/themen.tex						\
+	../slides/0/themen1.tex						\
+	../slides/0/themen2.tex						\
+	../slides/0/themen3.tex						\
 	../slides/0/chapter.tex
 
diff --git a/vorlesungen/slides/0/chapter.tex b/vorlesungen/slides/0/chapter.tex
index a860830..6e09557 100644
--- a/vorlesungen/slides/0/chapter.tex
+++ b/vorlesungen/slides/0/chapter.tex
@@ -7,5 +7,7 @@
 \folie{0/was.tex}
 \folie{0/resourcen.tex}
 \folie{0/latextipps.tex}
-\folie{0/themen.tex}
+\folie{0/themen1.tex}
+\folie{0/themen2.tex}
+\folie{0/themen3.tex}
 \folie{0/nextsteps.tex}
diff --git a/vorlesungen/slides/0/themen.tex b/vorlesungen/slides/0/themen.tex
deleted file mode 100644
index c50f32d..0000000
--- a/vorlesungen/slides/0/themen.tex
+++ /dev/null
@@ -1,68 +0,0 @@
-%
-% themen.tex
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\begin{frame}
-\frametitle{Seminararbeitsthemen I}
-\begin{enumerate}[<+->]
-\item
-Verkehrsnetze und Verkehrsfluss
-\item
-Mittelwert von Matrizen
-\item
-Pascal-Matrizen
-\item 
-Stirling-Matrizen
-\item
-Vandermonde-Matrix
-\item
-Probabilistische Matrix-Produkt-Kontrolle
-\item
-Der Satz von Furrer-Hungerbühler-Jantschgi
-\item
-Clifford-Algebren
-\end{enumerate}
-\end{frame}
-
-\begin{frame}
-\frametitle{Seminararbeitsthemen II}
-\begin{enumerate}[<+->]
-\setcounter{enumi}{7}
-\item
-Schnelle Matrixmultiplikation
-\item
-Parkettierungen mit Dominosteinen zählen
-\item
-Punktgruppen und Kristallographie
-\item
-Symmetriegruppen und Machine Learning
-\item
-Floyd-Warshall-Algorithmus
-\item
-Laser
-\item
-Munkres-Algorithmus
-\end{enumerate}
-\end{frame}
-
-\begin{frame}
-\frametitle{Seminararbeitsthemen III}
-\begin{enumerate}[<+->]
-\setcounter{enumi}{14}
-\item
-Iwasawa-Zerlegung
-\item 
-Reed-Solomon Code
-\item
-Pauli- und Dirac-Matrizen
-\item
-Klassifikation der Lie-Gruppen
-\item
-Iterierte Funktionsschemata
-\item
-QR-Codes
-\item
-McEliece-Kryptosystem
-\end{enumerate}
-\end{frame}
diff --git a/vorlesungen/slides/0/themen1.tex b/vorlesungen/slides/0/themen1.tex
new file mode 100644
index 0000000..756e037
--- /dev/null
+++ b/vorlesungen/slides/0/themen1.tex
@@ -0,0 +1,27 @@
+%
+% themen1.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}
+\frametitle{Seminararbeitsthemen I}
+\begin{enumerate}[<+->]
+\item
+Verkehrsnetze und Verkehrsfluss
+\item
+Mittelwert von Matrizen
+\item
+Pascal-Matrizen
+\item 
+Stirling-Matrizen
+\item
+Vandermonde-Matrix
+\item
+Probabilistische Matrix-Produkt-Kontrolle
+\item
+Der Satz von Furrer-Hungerbühler-Jantschgi
+\item
+Clifford-Algebren
+\end{enumerate}
+\end{frame}
+
diff --git a/vorlesungen/slides/0/themen2.tex b/vorlesungen/slides/0/themen2.tex
new file mode 100644
index 0000000..1fbdab3
--- /dev/null
+++ b/vorlesungen/slides/0/themen2.tex
@@ -0,0 +1,27 @@
+%
+% themen2.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Seminararbeitsthemen II}
+\begin{enumerate}[<+->]
+\setcounter{enumi}{7}
+\item
+Schnelle Matrixmultiplikation
+\item
+Parkettierungen mit Dominosteinen zählen
+\item
+Punktgruppen und Kristallographie
+\item
+Symmetriegruppen und Machine Learning
+\item
+Floyd-Warshall-Algorithmus
+\item
+Laser
+\item
+Munkres-Algorithmus
+\end{enumerate}
+\end{frame}
+
diff --git a/vorlesungen/slides/0/themen3.tex b/vorlesungen/slides/0/themen3.tex
new file mode 100644
index 0000000..5a6bdf5
--- /dev/null
+++ b/vorlesungen/slides/0/themen3.tex
@@ -0,0 +1,26 @@
+%
+% themen3.tex
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\begin{frame}
+\frametitle{Seminararbeitsthemen III}
+\begin{enumerate}[<+->]
+\setcounter{enumi}{14}
+\item
+Iwasawa-Zerlegung
+\item 
+Reed-Solomon Code
+\item
+Pauli- und Dirac-Matrizen
+\item
+Klassifikation der Lie-Gruppen
+\item
+Iterierte Funktionsschemata
+\item
+QR-Codes
+\item
+McEliece-Kryptosystem
+\end{enumerate}
+\end{frame}
diff --git a/vorlesungen/slides/2/Makefile.inc b/vorlesungen/slides/2/Makefile.inc
index 34dadf6..c857fec 100644
--- a/vorlesungen/slides/2/Makefile.inc
+++ b/vorlesungen/slides/2/Makefile.inc
@@ -15,5 +15,7 @@ chapter2 =								\
 	../slides/2/funktionenalgebra.tex				\
 	../slides/2/frobeniusnorm.tex					\
 	../slides/2/frobeniusanwendung.tex				\
+	../slides/2/quotient.tex					\
+	../slides/2/quotientv.tex					\
 	../slides/2/chapter.tex
 
diff --git a/vorlesungen/slides/2/chapter.tex b/vorlesungen/slides/2/chapter.tex
index 38eb21f..49e656a 100644
--- a/vorlesungen/slides/2/chapter.tex
+++ b/vorlesungen/slides/2/chapter.tex
@@ -13,3 +13,5 @@
 \folie{2/funktionenalgebra.tex}
 \folie{2/frobeniusnorm.tex}
 \folie{2/frobeniusanwendung.tex}
+\folie{2/quotient.tex}
+\folie{2/quotientv.tex}
diff --git a/vorlesungen/slides/2/images/Makefile b/vorlesungen/slides/2/images/Makefile
new file mode 100644
index 0000000..8bce5c9
--- /dev/null
+++ b/vorlesungen/slides/2/images/Makefile
@@ -0,0 +1,32 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	quotient1.jpg quotient2.jpg quotient1.pdf quotient2.pdf
+
+quotient1.png:	quotient1.pov quotient.inc
+	povray +A0.1 +W1920 +H1080 -Oquotient1.png quotient1.pov
+
+quotient1.jpg:	quotient1.png Makefile
+	convert -extract 1360x1040+330+20 quotient1.png \
+		-density 300 -units PixelsPerInch quotient1.jpg
+
+quotient2.png:	quotient2.pov quotient.inc
+	povray +A0.1 +W1920 +H1080 -Oquotient2.png quotient2.pov
+
+quotient2.jpg:	quotient2.png Makefile
+	convert -extract 1360x1040+330+20 quotient2.png \
+		-density 300 -units PixelsPerInch quotient2.jpg
+
+quotient:	quotient.ini quotient.inc quotient.pov
+	rm -rf quotient
+	mkdir quotient
+	povray +A0.1 -Oquotient/0.png -W1920 -H1080 quotient.ini
+
+quotient1.pdf:	quotient1.tex quotient1.jpg
+	pdflatex quotient1.tex
+
+quotient2.pdf:	quotient2.tex quotient2.jpg
+	pdflatex quotient2.tex
+
diff --git a/vorlesungen/slides/2/images/quotient.inc b/vorlesungen/slides/2/images/quotient.inc
new file mode 100644
index 0000000..3fa49d1
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient.inc
@@ -0,0 +1,186 @@
+//
+// quotient.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.035;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+	location <8, 15, -50>
+	look_at <0.4, 0.2, 0.4>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<-4, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.1 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+#macro kasten()
+	box { <-0.5,-0.5,-0.5>, <1.5,1,1.5> }
+#end
+
+
+arrow(<-0.6,0,0>, <1.6,0,0>, at, White)
+arrow(<0,0,-0.6>, <0,0,1.6>, at, White)
+arrow(<0,-0.6,0>, <0,1.2,0>, at, White)
+
+#declare U = <-1,3,-0.5>;
+#declare V1 = <1,0.2,0>;
+#declare V2 = <0,0.2,1>;
+
+#macro gerade(richtung, farbe)
+	intersection {
+		kasten()
+		cylinder { -U + richtung, U + richtung, at }
+		pigment {
+			color farbe
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#declare A = <0.8, -0.2, 0>;
+#declare B = <0.2, 0.8, 0>;
+
+#macro ebene(vektor1, vektor2)
+#declare n = vcross(vektor1,vektor2);
+
+
+intersection {
+	kasten()
+	plane { n, 0.005 }
+	plane { -n, 0.005 }
+	pigment {
+		color rgbf<0.8,0.8,1,0.7>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+intersection {
+	kasten()
+	union {
+		#declare Xstep = 0.45;
+		#declare X = -5 * Xstep;
+		#while (X < 5.5 * Xstep)
+			cylinder { X*vektor1 - 5*vektor2, X*vektor1 + 5*vektor2, at/2 }
+			#declare X = X + Xstep;
+		#end
+		#declare Ystep = 0.45;
+		#declare Y = -5 * Ystep;
+		#while (Y < 5.5 * Ystep)
+			cylinder { -5*vektor1 + Y*vektor2, 5*vektor1 + Y*vektor2, at/2 }
+			#declare Y = Y + Ystep;
+		#end
+	}
+	pigment {
+		color rgb<0.9,0.9,1>
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+
+
+gerade(O, Red)
+
+#declare gruen = rgb<0.2,0.4,0.2>;
+#declare blau = rgb<0,0.4,0.8>;
+#declare rot = rgb<1,0.4,0.0>;
+
+#macro repraesentanten(vektor1, vektor2) 
+
+#declare d1 = A.x*vektor1 + A.y*vektor2;
+#declare d2 = B.x*vektor1 + B.y*vektor2;
+
+arrow(0, d1 + d2, at, rot)
+gerade(d1 + d2, rot)
+
+gerade(d1, blau)
+arrow(O, d1, at, blau)
+cylinder { d1, d1 + d2, 0.6 * at
+	pigment {
+		color gruen
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+gerade(d2, gruen)
+arrow(O, d2, at, gruen)
+cylinder { d2, d1 + d2, 0.6 * at
+	pigment {
+		color blau
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+
+#end
+
+#macro vektorraum(s)
+#declare b1 = V1 + s * 0.03 * U;
+#declare b2 = V2 + s * 0.03 * U;
+
+ebene(b1, b2)
+repraesentanten(b1, b2)
+#end
+
diff --git a/vorlesungen/slides/2/images/quotient.ini b/vorlesungen/slides/2/images/quotient.ini
new file mode 100644
index 0000000..f62b21a
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient.ini
@@ -0,0 +1,7 @@
+Input_File_Name="quotient.pov"
+Initial_Frame=0
+Final_Frame=100
+Initial_Clock=-1
+Final_Clock=1
+Cyclic_Animation=off
+Pause_when_Done=off
diff --git a/vorlesungen/slides/2/images/quotient1.jpg b/vorlesungen/slides/2/images/quotient1.jpg
new file mode 100644
index 0000000..aeb713e
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.jpg
diff --git a/vorlesungen/slides/2/images/quotient1.pov b/vorlesungen/slides/2/images/quotient1.pov
new file mode 100644
index 0000000..60bab7f
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.pov
@@ -0,0 +1,8 @@
+//
+// quotient1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "quotient.inc"
+
+vektorraum(-1)
diff --git a/vorlesungen/slides/2/images/quotient1.tex b/vorlesungen/slides/2/images/quotient1.tex
new file mode 100644
index 0000000..30d82d2
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient1.tex
@@ -0,0 +1,29 @@
+%
+% quotient1.tex -- Vektorraumquotient
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.7,0,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\node at (0,0) {\includegraphics[width=8cm]{quotient1.jpg}};
+
+\node[color=blue] at (0.7,-1.3) {$v$};
+\node[color=darkgreen] at (-1.0,0.1) {$w$};
+\node[color=orange] at (2.5,0.1) {$v+w$};
+\node[color=darkred] at (-2.1,-0.9) {$0$};
+\node[color=darkred] at (-3.1,2.4) {$U$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/vorlesungen/slides/2/images/quotient2.jpg b/vorlesungen/slides/2/images/quotient2.jpg
new file mode 100644
index 0000000..345cf22
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.jpg
diff --git a/vorlesungen/slides/2/images/quotient2.pov b/vorlesungen/slides/2/images/quotient2.pov
new file mode 100644
index 0000000..771425d
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.pov
@@ -0,0 +1,8 @@
+//
+// quotient2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "quotient.inc"
+
+vektorraum(1)
diff --git a/vorlesungen/slides/2/images/quotient2.tex b/vorlesungen/slides/2/images/quotient2.tex
new file mode 100644
index 0000000..607fd03
--- /dev/null
+++ b/vorlesungen/slides/2/images/quotient2.tex
@@ -0,0 +1,29 @@
+%
+% quotient2.tex -- Vektorraumquotient
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\definecolor{darkred}{rgb}{0.7,0,0}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\node at (0,0) {\includegraphics[width=8cm]{quotient2.jpg}};
+
+\node[color=blue] at (0.57,-0.94) {$v$};
+\node[color=darkgreen] at (-1.15,0.65) {$w$};
+\node[color=orange] at (2.15,1) {$v+w$};
+\node[color=darkred] at (-2.1,-0.9) {$0$};
+\node[color=darkred] at (-3.1,2.4) {$U$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/vorlesungen/slides/2/quotient.tex b/vorlesungen/slides/2/quotient.tex
new file mode 100644
index 0000000..24b0523
--- /dev/null
+++ b/vorlesungen/slides/2/quotient.tex
@@ -0,0 +1,110 @@
+%
+% quotient.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.7,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\s{0.3}
+\def\punkt#1#2{({#1-3*#2},{8*#2})}
+\def\gerade#1{
+\draw[darkgreen,line width=1.4pt]
+	\punkt{#1}{1}
+	--
+	\punkt{#1}{-1};
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quotientenraum}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Einen Unterraum ``ignorieren''}
+{\usebeamercolor[fg]{title}Gegeben:} $U\subset V$ ein Unterraum
+\\
+{\usebeamercolor[fg]{title}Gesucht:} Eine Projektion auf einen Vektorraum,
+in dem die Richtungen in $U$ zu $0$ gemacht werden
+\end{block}
+\uncover<2->{%
+\begin{block}{Projektion}
+In $V$ Klassen bilden:
+\[
+\pi
+\colon
+v\mapsto
+\llbracket v\rrbracket
+=
+v+U
+\]
+\end{block}}
+\vspace{-12pt}
+\uncover<3->{%
+\begin{block}{Quotientenraum}
+\vspace{-12pt}
+\begin{align*}
+V/U
+&=
+\{ v+U\;|\; v\in V \}
+\\
+\uncover<4->{\pi(\lambda v)&=\lambda v+U= \lambda \pi(v)}
+\\
+\uncover<5->{\pi(v+w)
+&=
+v+w+U}
+\ifthenelse{\boolean{presentation}}{
+\only<6>{=
+v+U+w+U}}{}
+\uncover<7->{=
+\pi(v) + \pi(w)}
+\phantom{blubb}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (U) at (-3,8);
+\def\t{0.03}
+\begin{scope}
+\clip (-2,-2) rectangle (4,4.8);
+\draw[color=darkred,line width=2pt] (-3,8) -- (1.5,-4);
+\node[color=darkred] at (-1.45,4.6) {$U$};
+\node[color=darkred] at (-0.05,-0.05) [above left] {$0$};
+
+\gerade{2.5}
+
+\ifthenelse{\boolean{presentation}}{
+	\foreach \n in {8,...,25}{
+		\pgfmathparse{(\n-12)*0.04}
+		\xdef\s{\pgfmathresult}
+		\only<\n>{
+			\draw[color=blue,line width=1.2pt]
+				\punkt{-5}{-2*\s} -- \punkt{5}{2*\s};
+			\draw[->,color=blue,line width=2pt]
+				(0,0) -- \punkt{2.5}{\s};
+			\node[color=blue] at \punkt{2.5}{\s}
+				[above right] {$v'$};
+		}
+	}
+}{
+	\xdef\s{0.35}
+	\draw[color=blue,line width=1.2pt]
+		\punkt{-5}{-2*\s} -- \punkt{5}{2*\s};
+	\draw[->,color=blue,line width=2pt] (0,0) -- \punkt{2.5}{\s};
+	\node[color=blue] at \punkt{2.5}{\s} [above right] {$v'$};
+}
+
+\draw[->,color=darkgreen,line width=1.4pt] (0,0) -- \punkt{2.5}{0.1};
+
+\node[color=darkgreen] at \punkt{2.5}{0.1} [above right] {$v$};
+
+\end{scope}
+\draw[->] (-2,0) -- (4,0) coordinate[label={$x$}];
+\draw[->] (0,-2) -- (0,5) coordinate[label={right:$x$}];
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/2/quotientv.tex b/vorlesungen/slides/2/quotientv.tex
new file mode 100644
index 0000000..dc01f21
--- /dev/null
+++ b/vorlesungen/slides/2/quotientv.tex
@@ -0,0 +1,62 @@
+%
+% quotientv.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkred}{rgb}{0.7,0,0}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Quotient}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.33\textwidth}
+\begin{block}{Repräsentanten}
+Jeder Unterraum $W\subset V$ mit 
+$W\cap U = \{0\}$
+kann als Menge von Repräsentanten
+für
+\begin{align*}
+V/U
+&=
+\{v+U\;|\;v \in V\}
+\\
+&\simeq W
+\end{align*}
+dienen.
+\end{block}
+\uncover<3->{%
+\begin{block}{Orthogonalraum}
+Mit Skalarprodukt ist
+$W=U^\perp$ eine bevorzugte Wahl
+\end{block}}
+\end{column}
+\begin{column}{0.66\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\only<1>{
+	\node at (0,0)
+	{\includegraphics[width=8.5cm]{../slides/2/images/quotient1.jpg}};
+	\node[color=darkgreen] at (-0.5,0.3) {$v$};
+	\node[color=blue] at (0.7,-1.4) {$w$};
+	\node[color=orange] at (2.7,0.1) {$v+w$};
+	\fill[color=white,opacity=0.5] (3.7,1.0) circle[radius=0.25];
+	\node at (3.7,1.0) {$W$};
+}
+\only<2->{
+	\node at (0,0)
+	{\includegraphics[width=8.5cm]{../slides/2/images/quotient2.jpg}};
+	\node[color=darkgreen] at (-0.75,0.95) {$v$};
+	\node[color=blue] at (0.6,-1.05) {$w$};
+	\node[color=orange] at (2.36,1.05) {$v+w$};
+	\fill[color=white,opacity=0.5] (3.7,2.9) circle[radius=0.25];
+	\node at (3.7,2.9) {$W$};
+}
+\node[color=darkred] at (-3.3,2.6) {$U$};
+\node[color=darkred] at (-2.25,-1.0) {$0$};
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc
index 7f52cb1..f2edc80 100644
--- a/vorlesungen/slides/3/Makefile.inc
+++ b/vorlesungen/slides/3/Makefile.inc
@@ -12,6 +12,8 @@ chapter3 =								\
 	../slides/3/division2.tex					\
 	../slides/3/ringstruktur.tex					\
 	../slides/3/teilbarkeit.tex					\
+	../slides/3/ideal.tex						\
+	../slides/3/quotientenring.tex					\
 	../slides/3/faktorisierung.tex					\
 	../slides/3/faktorzerlegung.tex					\
 	../slides/3/einsetzen.tex					\
diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex
index 0f049e7..deec12e 100644
--- a/vorlesungen/slides/3/chapter.tex
+++ b/vorlesungen/slides/3/chapter.tex
@@ -10,6 +10,8 @@
 \folie{3/division2.tex}
 \folie{3/ringstruktur.tex}
 \folie{3/teilbarkeit.tex}
+\folie{3/ideal.tex}
+\folie{3/quotientenring.tex}
 \folie{3/faktorisierung.tex}
 \folie{3/faktorzerlegung.tex}
 \folie{3/einsetzen.tex}
diff --git a/vorlesungen/slides/3/ideal.tex b/vorlesungen/slides/3/ideal.tex
new file mode 100644
index 0000000..f7f432e
--- /dev/null
+++ b/vorlesungen/slides/3/ideal.tex
@@ -0,0 +1,63 @@
+%
+% ideal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ideal}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Voraussetzungen}
+$R$ ein Ring, $r\in R$
+\end{block}
+\uncover<2->{%
+\begin{block}{Vielfache\uncover<4->{ = Hauptideal}}
+Die Menge aller Elemente, die durch $r$ teilbar sind\uncover<3->{:
+\[
+(r)=rR
+\]}
+\uncover<4->{heisst {\em Hauptideal}}
+\end{block}}
+\uncover<5->{%
+\begin{block}{Ideal}
+$I\subset R$ mit
+\(RI\subset I\), \(I+I\subset I\)
+\end{block}}
+\uncover<6->{%
+\begin{block}{Hauptidealring}
+Jedes Ideal von $R$ ist ein Hauptideal
+\\
+\uncover<7->{{\usebeamercolor[fg]{title}Beispiele:}
+$\mathbb{Z}$,
+$\Bbbk[X]$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Grösster gemeinsamer Teiler}
+$a,b\in R$
+\begin{align*}
+\uncover<9->{(a) + (b)
+&= aR + bR}
+\intertext{\uncover<10->{ist eine Ideal }\uncover<11->{$\Rightarrow$ ein Hauptideal}}
+&\uncover<12->{= cR}\uncover<13->{ = \operatorname{ggT}(a,b)R}
+\end{align*}
+\uncover<14->{Existenz des $\operatorname{ggT}(a,b)$ ist eine
+gemeinsame Eigenschaft}
+\end{block}}
+\uncover<15->{%
+\begin{block}{Allgemein}
+\begin{itemize}
+\item<16->
+Alle euklidischen Ringe sind Hauptidealringe
+\item<17->
+Alle solchen Ringe verwenden den gleichen Algorithmus
+für $\operatorname{ggT}(a,b)$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/3/quotientenring.tex b/vorlesungen/slides/3/quotientenring.tex
new file mode 100644
index 0000000..4aa9e43
--- /dev/null
+++ b/vorlesungen/slides/3/quotientenring.tex
@@ -0,0 +1,59 @@
+%
+% Quotientenring.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quotientenring}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Quotientenring}
+$I\subset R$ ein Ideal
+\\
+\uncover<2->{
+$R/I$ hat eine Ringstruktur:
+\begin{align*}
+\uncover<3->{\pi(s)&=s+I}
+\\
+\uncover<4->{\pi(s)\pi(r)&= (s+I)(r+I)}\\
+            &\uncover<5->{= sr +\underbrace{sI + rI}_{\subset RI\subset I} + II = sr+I}
+\\
+\uncover<6->{\pi(s)+\pi(r)&= (s+I)+(r+I)}\\
+	     &\uncover<7->{=s+r+I=\pi(s+r)}
+\end{align*}}
+\end{block}
+\vspace{-15pt}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Beispiele}
+\begin{itemize}
+\item
+$\mathbb{Z}/(n)=\mathbb{Z}/n\mathbb{Z}$,
+$\mathbb{F}_p=\mathbb{Z}/(p)=\mathbb{Z}/p\mathbb{Z}$
+\item<8->
+$p\in\Bbbk[X]$
+$\Rightarrow$
+$\Bbbk[X]/(p)$ ist ein Ring
+\end{itemize}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Algebraideal}
+$I\subset A$
+\begin{itemize}
+\item<10->
+$I$ ein Unterraum von $A$ als Vektorraum
+\item<11->
+$I$ ein Ideal von $A$ als Ring
+\end{itemize}
+\end{block}}
+\uncover<12->{%
+\begin{block}{Quotientenalgebra}
+$A/I$ ist eine Algebra
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
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/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
diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc
new file mode 100644
index 0000000..fa6c29b
--- /dev/null
+++ b/vorlesungen/slides/9/Makefile.inc
@@ -0,0 +1,14 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter9 =								\
+	../slides/9/google.tex						\
+	../slides/9/markov.tex						\
+	../slides/9/irreduzibel.tex					\
+	../slides/9/stationaer.tex					\
+	../slides/9/pf.tex						\
+	../slides/9/chapter.tex
+
diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex
new file mode 100644
index 0000000..9e26587
--- /dev/null
+++ b/vorlesungen/slides/9/chapter.tex
@@ -0,0 +1,14 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+
+
+\folie{9/google.tex}
+\folie{9/markov.tex}
+\folie{9/stationaer.tex}
+\folie{9/irreduzibel.tex}
+\folie{9/pf.tex}
+
+
diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/9/google.tex
index d1ec31d..d1ec31d 100644
--- a/vorlesungen/slides/8/markov/google.tex
+++ b/vorlesungen/slides/9/google.tex
diff --git a/vorlesungen/slides/8/markov/irreduzibel.tex b/vorlesungen/slides/9/irreduzibel.tex
index 87e90e4..87e90e4 100644
--- a/vorlesungen/slides/8/markov/irreduzibel.tex
+++ b/vorlesungen/slides/9/irreduzibel.tex
diff --git a/vorlesungen/slides/8/markov/markov.tex b/vorlesungen/slides/9/markov.tex
index e92ff0f..e92ff0f 100644
--- a/vorlesungen/slides/8/markov/markov.tex
+++ b/vorlesungen/slides/9/markov.tex
diff --git a/vorlesungen/slides/8/markov/pf.tex b/vorlesungen/slides/9/pf.tex
index da2ef2b..da2ef2b 100644
--- a/vorlesungen/slides/8/markov/pf.tex
+++ b/vorlesungen/slides/9/pf.tex
diff --git a/vorlesungen/slides/8/markov/stationaer.tex b/vorlesungen/slides/9/stationaer.tex
index 92fab16..92fab16 100644
--- a/vorlesungen/slides/8/markov/stationaer.tex
+++ b/vorlesungen/slides/9/stationaer.tex
diff --git a/vorlesungen/slides/Makefile b/vorlesungen/slides/Makefile
index dd8b06e..d3c7a17 100644
--- a/vorlesungen/slides/Makefile
+++ b/vorlesungen/slides/Makefile
@@ -11,9 +11,18 @@ include Makefile.inc
 
 files = common.tex $(slides)
 
+catalog:	slides-catalog.pdf
+presentation:	slides-presentation.pdf
+handout:	slides-handout.pdf
+
 slides-handout.pdf:	slides-handout.tex slides.tex $(files)
 	pdflatex slides-handout.tex
 
+slides-catalog.pdf:	slides-handout.pdf
+	pdfjam --outfile slides-catalog.pdf \
+		--paper a4paper --nup 2x5 \
+		slides-handout.pdf
+
 slides-presentation.pdf:	slides-presentation.tex slides.tex $(files)
 	pdflatex slides-presentation.tex
 
diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc
index fe22282..4bf9348 100644
--- a/vorlesungen/slides/Makefile.inc
+++ b/vorlesungen/slides/Makefile.inc
@@ -10,7 +10,8 @@ include ../slides/3/Makefile.inc
 include ../slides/4/Makefile.inc
 include ../slides/5/Makefile.inc
 include ../slides/8/Makefile.inc
+include ../slides/9/Makefile.inc
 
 slides = 								\
 	$(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4)	\
-	$(chapter5) $(chapter8)
+	$(chapter5) $(chapter8) $(chapter9)
diff --git a/vorlesungen/slides/slides.tex b/vorlesungen/slides/slides.tex
index 0dc6c6e..b606375 100644
--- a/vorlesungen/slides/slides.tex
+++ b/vorlesungen/slides/slides.tex
@@ -3,19 +3,77 @@
 %
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\title[Intro]{Intro}
-\section{Intro}
+\def\titel{
+% title slide for this chapter
 \begin{frame}
 \titlepage
 \end{frame}
+\ifthenelse{\boolean{presentation}}{}{
+% add an empty slide for alignment in the slide catalog
+\begin{frame}
+\end{frame}
+}
+}
+
+\title[MathSem]{Mathematisches Seminar: Matrizen}
+\section{Intro}
+\titel
+\author[]{}
 \subtitle{}
 \input{0/chapter.tex}
 
 \title[Grundlagen]{Grundlagen}
 \section{Grundlagen}
-\begin{frame}
-\titlepage
-\end{frame}
-\subtitle{}
+\titel
 \input{1/chapter.tex}
 
+\title[Vektoren/Matrizen]{Vektoren und Matrizen}
+\section{Vektoren und Matrizen}
+\titel
+\input{2/chapter.tex}
+
+\title[Polynome]{Polynome}
+\section{Polynome}
+\titel
+\input{3/chapter.tex}
+
+\title[Endliche Körper]{Endliche Körper}
+\section{Endliche Körper}
+\titel
+\input{4/chapter.tex}
+
+\title[EW/EV]{Eigenwerte und Eigenvektoren}
+\section{Eigenwerte und Eigenvektoren}
+\titel
+\input{5/chapter.tex}
+
+%\title[Permutationen]{Permutationen}
+%\section{Permutationen}
+%\titel
+%\input{6/chapter.tex}
+
+%\title[Matrizengruppen]{Matrizengruppen}
+%\section{Matrizengruppen}
+%\titel
+%\input{7/chapter.tex}
+
+\title[Graphen]{Graphen}
+\section{Graphen}
+\titel
+\input{8/chapter.tex}
+
+\title[Wahrscheinlichkeit]{Wahrscheinlichkeitsmatrizen}
+\section{Wahrscheinlichkeitsmatrizen}
+\titel
+\input{9/chapter.tex}
+
+%\title[Krypto]{Anwendungen in Kryptographie und Codierungstheorie}
+%\section{Krypto}
+%\titel
+%\input{a/chapter.tex}
+
+%\title[Homologie]{Homologie}
+%\section{Homologie}
+%\titel
+%\input{b/chapter.tex}
+
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 411414e..2423a3f 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -3,44 +3,5 @@
 %
 % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-%\folie{8/dgraph.tex}
-%\folie{8/graph.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}
-
-% XXX \folie{8/schwingungen.tex}
-% XXX \folie{8/eigenvektorbasis.tex}
-% XXX \folie{8/spektralegraphentheorie.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/wege.tex}
-%\folie{8/floyd-warshall/wegiteration.tex}
-
-% Markov-Ketten
-% - Google-Matrix
-% - Stationär
-
-%\folie{8/tokyo/google.tex}
-%\folie{8/tokyo/bahn0.tex}
-%\folie{8/tokyo/bahn1.tex}
-%\folie{8/tokyo/bahn2.tex}
-
-
-%\folie{8/markov/google.tex}
-%\folie{8/markov/markov.tex}
-%\folie{8/markov/irreduzibel.tex}
-%\folie{8/markov/stationaer.tex}
-%\folie{8/markov/pf.tex}
+\folie{3/ideal.tex}
+\folie{3/quotientenring.tex}